Newton's laws of motion
Laws in physics about force and motion
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Newton's laws of motion are three physical laws that describe the relationship between the motion of an object and the forces acting on it. These laws, which provide the basis for Newtonian mechanics, can be paraphrased as follows:
The three laws of motion were first stated by Isaac Newton in his "Philosophiæ Naturalis Principia Mathematica" ("Mathematical Principles of Natural Philosophy"), originally published in 1687. Newton used them to investigate and explain the motion of many physical objects and systems. In the time since Newton, new insights, especially around the concept of energy, built the field of classical mechanics on his foundations. Limitations to Newton's laws have also been discovered; new theories are necessary when objects move at very high speeds (special relativity), are very massive (general relativity), or are very small (quantum mechanics).
Prerequisites.
Newton's laws are often stated in terms of "point" or "particle" masses, that is, bodies whose volume is negligible. This is a reasonable approximation for real bodies when the motion of internal parts can be neglected, and when the separation between bodies is much larger than the size of each. For instance, the Earth and the Sun can both be approximated as pointlike when considering the orbit of the former around the latter, but the Earth is not pointlike when considering activities on its surface.
The mathematical description of motion, or kinematics, is based on the idea of specifying positions using numerical coordinates. Movement is represented by these numbers changing over time: a body's trajectory is represented by a function that assigns to each value of a time variable the values of all the position coordinates. The simplest case is one-dimensional, that is, when a body is constrained to move only along a straight line. Its position can then be given by a single number, indicating where it is relative to some chosen reference point. For example, a body might be free to slide along a track that runs left to right, and so its location can be specified by its distance from a convenient zero point, or origin, with negative numbers indicating positions to the left and positive numbers indicating positions to the right. If the body's location as a function of time is formula_1, then its average velocity over the time interval from formula_2 to formula_3 is formula_4Here, the Greek letter formula_5 (delta) is used, per tradition, to mean "change in". A positive average velocity means that the position coordinate formula_6 increases over the interval in question, a negative average velocity indicates a net decrease over that interval, and an average velocity of zero means that the body ends the time interval in the same place as it began. Calculus gives the means to define an "instantaneous" velocity, a measure of a body's speed and direction of movement at a single moment of time, rather than over an interval. One notation for the instantaneous velocity is to replace formula_5 with the symbol formula_8, for example,formula_9This denotes that the instantaneous velocity is the derivative of the position with respect to time. It can roughly be thought of as the ratio between an infinitesimally small change in position formula_10 to the infinitesimally small time interval formula_11 over which it occurs. More carefully, the velocity and all other derivatives can be defined using the concept of a limit. A function formula_12 has a limit of formula_13 at a given input value formula_2 if the difference between formula_15 and formula_13 can be made arbitrarily small by choosing an input sufficiently close to formula_2. One writes, formula_18Instantaneous velocity can be defined as the limit of the average velocity as the time interval shrinks to zero:formula_19 "Acceleration" is to velocity as velocity is to position: it is the derivative of the velocity with respect to time. Acceleration can likewise be defined as a limit:formula_20Consequently, the acceleration is the "second derivative" of position, often written formula_21.
Position, when thought of as a displacement from an origin point, is a vector: a quantity with both magnitude and direction.1 Velocity and acceleration are vector quantities as well. The mathematical tools of vector algebra provide the means to describe motion in two, three or more dimensions. Vectors are often denoted with an arrow, as in formula_22, or in bold typeface, such as formula_23. Often, vectors are represented visually as arrows, with the direction of the vector being the direction of the arrow, and the magnitude of the vector indicated by the length of the arrow. Numerically, a vector can be represented as a list; for example, a body's velocity vector might be formula_24, indicating that it is moving at 3 metres per second along the horizontal axis and 4 metres per second along the vertical axis. The same motion described in a different coordinate system will be represented by different numbers, and vector algebra can be used to translate between these alternatives.4
The study of mechanics is complicated by the fact that household words like "energy" are used with a technical meaning. Moreover, words which are synonymous in everyday speech are not so in physics: "force" is not the same as "power" or "pressure", for example, and "mass" has a different meaning than "weight". The physics concept of "force" makes quantitative the everyday idea of a push or a pull. Forces in Newtonian mechanics are often due to strings and ropes, friction, muscle effort, gravity, and so forth. Like displacement, velocity, and acceleration, force is a vector quantity.
Laws.
First law.
Translated from Latin, Newton's first law reads,
"Every object perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon."
Newton's first law expresses the principle of inertia: the natural behavior of a body is to move in a straight line at constant speed. A body's motion preserves the status quo, but external forces can perturb this.
The modern understanding of Newton's first law is that no inertial observer is privileged over any other. The concept of an inertial observer makes quantitative the everyday idea of feeling no effects of motion. For example, a person standing on the ground watching a train go past is an inertial observer. If the observer on the ground sees the train moving smoothly in a straight line at a constant speed, then a passenger sitting on the train will also be an inertial observer: the train passenger "feels" no motion. The principle expressed by Newton's first law is that there is no way to say which inertial observer is "really" moving and which is "really" standing still. One observer's state of rest is another observer's state of uniform motion in a straight line, and no experiment can deem either point of view to be correct or incorrect. There is no absolute standard of rest. Newton himself believed that absolute space and time existed, but that the only measures of space or time accessible to experiment are relative.
"The change of motion of an object is proportional to the force impressed; and is made in the direction of the straight line in which the force is impressed."114
Second law.
By "motion", Newton meant the quantity now called momentum, which depends upon the amount of matter contained in a body, the speed at which that body is moving, and the direction in which it is moving. In modern notation, the momentum of a body is the product of its mass and its velocity:
formula_25
where all three quantities can change over time.
In common cases the mass formula_26 does not change with time and the derivative acts only upon the velocity. Then force equals the product of the mass and the time derivative of the velocity, which is the acceleration:
formula_27
As the acceleration is the second derivative of position with respect to time, this can also be written
formula_28
Newton's second law, in modern form, states that the time derivative of the momentum is the force:4.1
formula_29
When applied to systems of variable mass, the equation above is only valid only for a fixed set of particles. Applying the derivative as in
formula_30
can lead to incorrect results. For example, the momentum of a water jet system must include the momentum of the ejected water:
formula_31
 The forces acting on a body add as vectors, and so the total force on a body depends upon both the magnitudes and the directions of the individual forces. When the net force on a body is equal to zero, then by Newton's second law, the body does not accelerate, and it is said to be in mechanical equilibrium. A state of mechanical equilibrium is "stable" if, when the position of the body is changed slightly, the body remains near that equilibrium. Otherwise, the equilibrium is "unstable."
A common visual representation of forces acting in concert is the free body diagram, which schematically portrays a body of interest and the forces applied to it by outside influences. For example, a free body diagram of a block sitting upon an inclined plane can illustrate the combination of gravitational force, "normal" force, friction, and string tension.
Newton's second law is sometimes presented as a "definition" of force, i.e., a force is that which exists when an inertial observer sees a body accelerating. This is sometimes regarded as a potential tautology — acceleration implies force, force implies acceleration. However, Newton's second law not only merely defines the force by the acceleration: forces exist as separate from the acceleration produced by the force in a particular system. The same force that is identified as producing acceleration to an object can then be applied to any other object, and the resulting accelerations (coming from that same force) will always be inversely proportional to the mass of the object. What Newton's Second Law states is that all the effect of a force onto a system can be reduced to two pieces of information: the magnitude of the force, and it's direction, and then goes on to specify what the effect is.
Beyond that, an equation detailing the force might also be specified, like Newton's law of universal gravitation. By inserting such an expression for formula_32 into Newton's second law, an equation with predictive power can be written. Newton's second law has also been regarded as setting out a research program for physics, establishing that important goals of the subject are to identify the forces present in nature and to catalogue the constituents of matter.13412-2
However, forces can often be measured directly with no acceleration being involved, such as through weighing scales. By postulating a physical object that can be directly measured independently from acceleration, Newton made a objective physical statement with the second law alone, the predictions of which can be verified even if no force law is given.
"To every action, there is always opposed an equal reaction; or, the mutual actions of two bodies upon each other are always equal, and directed to contrary parts."116
Third law.
Overly brief paraphrases of the third law, like "action equals reaction" might have caused confusion among generations of students: the "action" and "reaction" apply to different bodies. For example, consider a book at rest on a table. The Earth's gravity pulls down upon the book. The "reaction" to that "action" is "not" the support force from the table holding up the book, but the gravitational pull of the book acting on the Earth.
Newton's third law relates to a more fundamental principle, the conservation of momentum. The latter remains true even in cases where Newton's statement does not, for instance when force fields as well as material bodies carry momentum, and when momentum is defined properly, in quantum mechanics as well. In Newtonian mechanics, if two bodies have momenta formula_33 and formula_34 respectively, then the total momentum of the pair is formula_35, and the rate of change of formula_36 is formula_37 By Newton's second law, the first term is the total force upon the first body, and the second term is the total force upon the second body. If the two bodies are isolated from outside influences, the only force upon the first body can be that from the second, and vice versa. By Newton's third law, these forces have equal magnitude but opposite direction, so they cancel when added, and formula_36 is constant. Alternatively, if formula_36 is known to be constant, it follows that the forces have equal magnitude and opposite direction.
Candidates for additional laws.
Various sources have proposed elevating other ideas used in classical mechanics to the status of Newton's laws. For example, in Newtonian mechanics, the total mass of a body made by bringing together two smaller bodies is the sum of their individual masses. Frank Wilczek has suggested calling attention to this assumption by designating it "Newton's Zeroth Law". Another candidate for a "zeroth law" is the fact that at any instant, a body reacts to the forces applied to it at that instant. Likewise, the idea that forces add like vectors (or in other words obey the superposition principle), and the idea that forces change the energy of a body, have both been described as a "fourth law".
Moreover, some texts organize the basic ideas of Newtonian mechanics into different postulates, other than the three laws as commonly phrased, with the goal of being more clear about what is empirically observed and what is true by definition.9
Examples.
The study of the behavior of massive bodies using Newton's laws is known as Newtonian mechanics. Some example problems in Newtonian mechanics are particularly noteworthy for conceptual or historical reasons.
Uniformly accelerated motion.
If a body falls from rest near the surface of the Earth, then in the absence of air resistance, it will accelerate at a constant rate. This is known as free fall. The speed attained during free fall is proportional to the elapsed time, and the distance traveled is proportional to the square of the elapsed time. Importantly, the acceleration is the same for all bodies, independently of their mass. This follows from combining Newton's second law of motion with his law of universal gravitation. The latter states that the magnitude of the gravitational force from the Earth upon the body is
formula_40
where formula_26 is the mass of the falling body, formula_42 is the mass of the Earth, formula_43 is Newton's constant, and formula_44 is the distance from the center of the Earth to the body's location, which is very nearly the radius of the Earth. Setting this equal to formula_45, the body's mass formula_26 cancels from both sides of the equation, leaving an acceleration that depends upon formula_43, formula_42, and formula_44, and formula_44 can be taken to be constant. This particular value of acceleration is typically denoted formula_51:
formula_52
If the body is not released from rest but instead launched upwards and/or horizontally with nonzero velocity, then free fall becomes projectile motion. When air resistance can be neglected, projectiles follow parabola-shaped trajectories, because gravity affects the body's vertical motion and not its horizontal. At the peak of the projectile's trajectory, its vertical velocity is zero, but its acceleration is formula_51 downwards, as it is at all times. Setting the wrong vector equal to zero is a common confusion among physics students.
Uniform circular motion.
When a body is in uniform circular motion, the force on it changes the direction of its motion but not its speed. For a body moving in a circle of radius formula_44 at a constant speed formula_55, its acceleration has a magnitudeformula_56and is directed toward the center of the circle. The force required to sustain this acceleration, called the centripetal force, is therefore also directed toward the center of the circle and has magnitude formula_57. Many orbits, such as that of the Moon around the Earth, can be approximated by uniform circular motion. In such cases, the centripetal force is gravity, and by Newton's law of universal gravitation has magnitude formula_58, where formula_42 is the mass of the larger body being orbited. Therefore, the mass of a body can be calculated from observations of another body orbiting around it.130
Newton's cannonball is a thought experiment that interpolates between projectile motion and uniform circular motion. A cannonball that is lobbed weakly off the edge of a tall cliff will hit the ground in the same amount of time as if it were dropped from rest, because the force of gravity only affects the cannonball's momentum in the downward direction, and its effect is not diminished by horizontal movement. If the cannonball is launched with a greater initial horizontal velocity, then it will travel farther before it hits the ground, but it will still hit the ground in the same amount of time. However, if the cannonball is launched with an even larger initial velocity, then the curvature of the Earth becomes significant: the ground itself will curve away from the falling cannonball. A very fast cannonball will fall away from the inertial straight-line trajectory at the same rate that the Earth curves away beneath it; in other words, it will be in orbit (imagining that it is not slowed by air resistance or obstacles).
Harmonic motion.
Consider a body of mass formula_26 able to move along the formula_61 axis, and suppose an equilibrium point exists at the position formula_62. That is, at formula_62, the net force upon the body is the zero vector, and by Newton's second law, the body will not accelerate. If the force upon the body is proportional to the displacement from the equilibrium point, and directed to the equilibrium point, then the body will perform simple harmonic motion. Writing the force as formula_64, Newton's second law becomes
formula_65
This differential equation has the solution
formula_66
where the frequency formula_67 is equal to formula_68, and the constants formula_69 and formula_70 can be calculated knowing, for example, the position and velocity the body has at a given time, like formula_71.
One reason that the harmonic oscillator is a conceptually important example is that it is good approximation for many systems near a stable mechanical equilibrium. For example, a pendulum has a stable equilibrium in the vertical position: if motionless there, it will remain there, and if pushed slightly, it will swing back and forth. Neglecting air resistance and friction in the pivot, the force upon the pendulum is gravity, and Newton's second law becomes formula_72where formula_13 is the length of the pendulum and formula_74 is its angle from the vertical. When the angle formula_74 is small, the sine of formula_74 is nearly equal to formula_74 (see small-angle approximation), and so this expression simplifies to the equation for a simple harmonic oscillator with frequency formula_78.
A harmonic oscillator can be "damped," often by friction or viscous drag, in which case energy bleeds out of the oscillator and the amplitude of the oscillations decreases over time. Also, a harmonic oscillator can be "driven" by an applied force, which can lead to the phenomenon of resonance.
Objects with variable mass.
Newtonian physics treats matter as being neither created nor destroyed, though it may be rearranged. It can be the case that an object of interest gains or loses mass because matter is added to or removed from it. In such a situation, Newton's laws can be applied to the individual pieces of matter, keeping track of which pieces belong to the object of interest over time. For instance, if a rocket of mass formula_79, moving at velocity formula_80, ejects matter at a velocity formula_81 relative to the rocket, then
formula_82
where formula_32 is the net external force (e.g., a planet's gravitational pull).
Fan and sail.
The fan and sail example is a situation studied in discussions of Newton's third law. In the situation, a fan is attached to a cart or a sailboat and blows on its sail. From the third law, one would reason that the force of the air pushing in one direction would cancel out the force done by the fan on the sail, leaving the entire apparatus stationary. However, because the system is not entirely enclosed, there are conditions in which the vessel will move; for example, if the sail is built in a manner that redirects the majority of the airflow back towards the fan, the net force will result in the vessel moving forward.
Work and energy.
The concept of energy was developed after Newton's time, but it has become an inseparable part of what is considered "Newtonian" physics. Energy can broadly be classified into kinetic, due to a body's motion, and potential, due to a body's position relative to others. Thermal energy, the energy carried by heat flow, is a type of kinetic energy not associated with the macroscopic motion of objects but instead with the movements of the atoms and molecules of which they are made. According to the work-energy theorem, when a force acts upon a body while that body moves along the line of the force, the force does "work" upon the body, and the amount of work done is equal to the change in the body's kinetic energy. In many cases of interest, the net work done by a force when a body moves in a closed loop — starting at a point, moving along some trajectory, and returning to the initial point — is zero. If this is the case, then the force can be written in terms of the gradient of a function called a scalar potential:303
formula_84
This is true for many forces including that of gravity, but not for friction; indeed, almost any problem in a mechanics textbook that does not involve friction can be expressed in this way.19 The fact that the force can be written in this way can be understood from the conservation of energy. Without friction to dissipate a body's energy into heat, the body's energy will trade between potential and (non-thermal) kinetic forms while the total amount remains constant. Any gain of kinetic energy, which occurs when the net force on the body accelerates it to a higher speed, must be accompanied by a loss of potential energy. So, the net force upon the body is determined by the manner in which the potential energy decreases.
Rigid-body motion and rotation.
A rigid body is an object whose size is too large to neglect and which maintains the same shape over time. In Newtonian mechanics, the motion of a rigid body is often understood by separating it into movement of the body's center of mass and movement around the center of mass.
Center of mass.
Significant aspects of the motion of an extended body can be understood by imagining the mass of that body concentrated to a single point, known as the center of mass. The location of a body's center of mass depends upon how that body's material is distributed. For a collection of pointlike objects with masses formula_85 at positions formula_86, the center of mass is located at formula_87 where formula_42 is the total mass of the collection. In the absence of a net external force, the center of mass moves at a constant speed in a straight line. This applies, for example, to a collision between two bodies. If the total external force is not zero, then the center of mass changes velocity as though it were a point body of mass formula_42. This follows from the fact that the internal forces within the collection, the forces that the objects exert upon each other, occur in balanced pairs by Newton's third law. In a system of two bodies with one much more massive than the other, the center of mass will approximately coincide with the location of the more massive body.
Rotational analogues of Newton's laws.
When Newton's laws are applied to rotating extended bodies, they lead to new quantities that are analogous to those invoked in the original laws. The analogue of mass is the moment of inertia, the counterpart of momentum is angular momentum, and the counterpart of force is torque.
Angular momentum is calculated with respect to a reference point. If the displacement vector from a reference point to a body is formula_90 and the body has momentum formula_36, then the body's angular momentum with respect to that point is, using the vector cross product, formula_92 Taking the time derivative of the angular momentum gives formula_93 The first term vanishes because formula_94 and formula_95 point in the same direction. The remaining term is the torque, formula_96 When the torque is zero, the angular momentum is constant, just as when the force is zero, the momentum is constant. The torque can vanish even when the force is non-zero, if the body is located at the reference point (formula_97) or if the force formula_32 and the displacement vector formula_90 are directed along the same line.
The angular momentum of a collection of point masses, and thus of an extended body, is found by adding the contributions from each of the points. This provides a means to characterize a body's rotation about an axis, by adding up the angular momenta of its individual pieces. The result depends on the chosen axis, the shape of the body, and the rate of rotation.28
Multi-body gravitational system.
Newton's law of universal gravitation states that any body attracts any other body along the straight line connecting them. The size of the attracting force is proportional to the product of their masses, and inversely proportional to the square of the distance between them. Finding the shape of the orbits that an inverse-square force law will produce is known as the Kepler problem. The Kepler problem can be solved in multiple ways, including by demonstrating that the Laplace–Runge–Lenz vector is constant, or by applying a duality transformation to a 2-dimensional harmonic oscillator. However it is solved, the result is that orbits will be conic sections, that is, ellipses (including circles), parabolas, or hyperbolas. The eccentricity of the orbit, and thus the type of conic section, is determined by the energy and the angular momentum of the orbiting body. Planets do not have sufficient energy to escape the Sun, and so their orbits are ellipses, to a good approximation; because the planets pull on one another, actual orbits are not exactly conic sections.
If a third mass is added, the Kepler problem becomes the three-body problem, which in general has no exact solution in closed form. That is, there is no way to start from the differential equations implied by Newton's laws and, after a finite sequence of standard mathematical operations, obtain equations that express the three bodies' motions over time. Numerical methods can be applied to obtain useful, albeit approximate, results for the three-body problem. The positions and velocities of the bodies can be stored in variables within a computer's memory; Newton's laws are used to calculate how the velocities will change over a short interval of time, and knowing the velocities, the changes of position over that time interval can be computed. This process is looped to calculate, approximately, the bodies' trajectories. Generally speaking, the shorter the time interval, the more accurate the approximation.
Chaos and unpredictability.
Nonlinear dynamics.
Newton's laws of motion allow the possibility of chaos. That is, qualitatively speaking, physical systems obeying Newton's laws can exhibit sensitive dependence upon their initial conditions: a slight change of the position or velocity of one part of a system can lead to the whole system behaving in a radically different way within a short time. Noteworthy examples include the three-body problem, the double pendulum, dynamical billiards, and the Fermi–Pasta–Ulam–Tsingou problem.
Newton's laws can be applied to fluids by considering a fluid as composed of infinitesimal pieces, each exerting forces upon neighboring pieces. The Euler momentum equation is an expression of Newton's second law adapted to fluid dynamics. A fluid is described by a velocity field, i.e., a function formula_100 that assigns a velocity vector to each point in space and time. A small object being carried along by the fluid flow can change velocity for two reasons: first, because the velocity field at its position is changing over time, and second, because it moves to a new location where the velocity field has a different value. Consequently, when Newton's second law is applied to an infinitesimal portion of fluid, the acceleration formula_101 has two terms, a combination known as a "total" or "material" derivative. The mass of an infinitesimal portion depends upon the fluid density, and there is a net force upon it if the fluid pressure varies from one side of it to another. Accordingly, formula_102 becomes
formula_103
where formula_104 is the density, formula_105 is the pressure, and formula_106 stands for an external influence like a gravitational pull. Incorporating the effect of viscosity turns the Euler equation into a Navier–Stokes equation:
formula_107
where formula_108 is the kinematic viscosity.
Singularities.
It is mathematically possible for a collection of point masses, moving in accord with Newton's laws, to launch some of themselves away so forcefully that they fly off to infinity in a finite time. This unphysical behavior, known as a "noncollision singularity", depends upon the masses being pointlike and able to approach one another arbitrarily closely, as well as the lack of a relativistic speed limit in Newtonian physics.
It is not yet known whether or not the Euler and Navier–Stokes equations exhibit the analogous behavior of initially smooth solutions "blowing up" in finite time. The question of existence and smoothness of Navier–Stokes solutions is one of the Millennium Prize Problems.
Relation to other formulations of classical physics.
Classical mechanics can be mathematically formulated in multiple different ways, other than the "Newtonian" description (which itself, of course, incorporates contributions from others both before and after Newton). The physical content of these different formulations is the same as the Newtonian, but they provide different insights and facilitate different types of calculations. For example, Lagrangian mechanics helps make apparent the connection between symmetries and conservation laws, and it is useful when calculating the motion of constrained bodies, like a mass restricted to move along a curving track or on the surface of a sphere. Hamiltonian mechanics is convenient for statistical physics, leads to further insight about symmetry, and can be developed into sophisticated techniques for perturbation theory. Due to the breadth of these topics, the discussion here will be confined to concise treatments of how they reformulate Newton's laws of motion.
Lagrangian.
Lagrangian mechanics differs from the Newtonian formulation by considering entire trajectories at once rather than predicting a body's motion at a single instant.109 It is traditional in Lagrangian mechanics to denote position with formula_109 and velocity with formula_110. The simplest example is a massive point particle, the Lagrangian for which can be written as the difference between its kinetic and potential energies:
formula_111
where the kinetic energy is
formula_112
and the potential energy is some function of the position, formula_113. The physical path that the particle will take between an initial point formula_114 and a final point formula_115 is the path for which the integral of the Lagrangian is "stationary". That is, the physical path has the property that small perturbations of it will, to a first approximation, not change the integral of the Lagrangian. Calculus of variations provides the mathematical tools for finding this path.485 Applying the calculus of variations to the task of finding the path yields the Euler–Lagrange equation for the particle,
formula_116
Evaluating the partial derivatives of the Lagrangian gives
formula_117
which is a restatement of Newton's second law. The left-hand side is the time derivative of the momentum, and the right-hand side is the force, represented in terms of the potential energy.737
Landau and Lifshitz argue that the Lagrangian formulation makes the conceptual content of classical mechanics more clear than starting with Newton's laws. Lagrangian mechanics provides a convenient framework in which to prove Noether's theorem, which relates symmetries and conservation laws. The conservation of momentum can be derived by applying Noether's theorem to a Lagrangian for a multi-particle system, and so, Newton's third law is a theorem rather than an assumption.124
Hamiltonian.
In Hamiltonian mechanics, the dynamics of a system are represented by a function called the Hamiltonian, which in many cases of interest is equal to the total energy of the system.742 The Hamiltonian is a function of the positions and the momenta of all the bodies making up the system, and it may also depend explicitly upon time. The time derivatives of the position and momentum variables are given by partial derivatives of the Hamiltonian, via Hamilton's equations.203 The simplest example is a point mass formula_26 constrained to move in a straight line, under the effect of a potential. Writing formula_109 for the position coordinate and formula_120 for the body's momentum, the Hamiltonian is
formula_121
In this example, Hamilton's equations are
formula_122
and
formula_123
Evaluating these partial derivatives, the former equation becomes
formula_124
which reproduces the familiar statement that a body's momentum is the product of its mass and velocity. The time derivative of the momentum is
formula_125
which, upon identifying the negative derivative of the potential with the force, is just Newton's second law once again.742
As in the Lagrangian formulation, in Hamiltonian mechanics the conservation of momentum can be derived using Noether's theorem, making Newton's third law an idea that is deduced rather than assumed.251
Among the proposals to reform the standard introductory-physics curriculum is one that teaches the concept of energy before that of force, essentially "introductory Hamiltonian mechanics".
Hamilton–Jacobi.
The Hamilton–Jacobi equation provides yet another formulation of classical mechanics, one which makes it mathematically analogous to wave optics.284 This formulation also uses Hamiltonian functions, but in a different way than the formulation described above. The paths taken by bodies or collections of bodies are deduced from a function formula_126 of positions formula_127 and time formula_128. The Hamiltonian is incorporated into the Hamilton–Jacobi equation, a differential equation for formula_129. Bodies move over time in such a way that their trajectories are perpendicular to the surfaces of constant formula_129, analogously to how a light ray propagates in the direction perpendicular to its wavefront. This is simplest to express for the case of a single point mass, in which formula_129 is a function formula_132, and the point mass moves in the direction along which formula_129 changes most steeply. In other words, the momentum of the point mass is the gradient of formula_129:
formula_135
The Hamilton–Jacobi equation for a point mass is
formula_136
The relation to Newton's laws can be seen by considering a point mass moving in a time-independent potential formula_137, in which case the Hamilton–Jacobi equation becomes
formula_138
Taking the gradient of both sides, this becomes
formula_139
Interchanging the order of the partial derivatives on the left-hand side, and using the power and chain rules on the first term on the right-hand side,
formula_140
Gathering together the terms that depend upon the gradient of formula_129,
formula_142
This is another re-expression of Newton's second law. The expression in brackets is a "total" or "material" derivative as mentioned above, in which the first term indicates how the function being differentiated changes over time at a fixed location, and the second term captures how a moving particle will see different values of that function as it travels from place to place:
formula_143
Relation to other physical theories.
Thermodynamics and statistical physics.
In statistical physics, the kinetic theory of gases applies Newton's laws of motion to large numbers (typically on the order of the Avogadro number) of particles. Kinetic theory can explain, for example, the pressure that a gas exerts upon the container holding it as the aggregate of many impacts of atoms, each imparting a tiny amount of momentum.62
The Langevin equation is a special case of Newton's second law, adapted for the case of describing a small object bombarded stochastically by even smaller ones.235 It can be writtenformula_144where formula_145 is a drag coefficient and formula_146 is a force that varies randomly from instant to instant, representing the net effect of collisions with the surrounding particles. This is used to model Brownian motion.
Electromagnetism.
Newton's three laws can be applied to phenomena involving electricity and magnetism, though subtleties and caveats exist.
Coulomb's law for the electric force between two stationary, electrically charged bodies has much the same mathematical form as Newton's law of universal gravitation: the force is proportional to the product of the charges, inversely proportional to the square of the distance between them, and directed along the straight line between them. The Coulomb force that a charge formula_147 exerts upon a charge formula_148 is equal in magnitude to the force that formula_148 exerts upon formula_147, and it points in the exact opposite direction. Coulomb's law is thus consistent with Newton's third law.
Electromagnetism treats forces as produced by "fields" acting upon charges. The Lorentz force law provides an expression for the force upon a charged body that can be plugged into Newton's second law in order to calculate its acceleration.85 According to the Lorentz force law, a charged body in an electric field experiences a force in the direction of that field, a force proportional to its charge formula_109 and to the strength of the electric field. In addition, a "moving" charged body in a magnetic field experiences a force that is also proportional to its charge, in a direction perpendicular to both the field and the body's direction of motion. Using the vector cross product,formula_152
If the electric field vanishes (formula_153), then the force will be perpendicular to the charge's motion, just as in the case of uniform circular motion studied above, and the charge will circle (or more generally move in a helix) around the magnetic field lines at the cyclotron frequency formula_154.222 Mass spectrometry works by applying electric and/or magnetic fields to moving charges and measuring the resulting acceleration, which by the Lorentz force law yields the mass-to-charge ratio.
Collections of charged bodies do not always obey Newton's third law: there can be a change of one body's momentum without a compensatory change in the momentum of another. The discrepancy is accounted for by momentum carried by the electromagnetic field itself. The momentum per unit volume of the electromagnetic field is proportional to the Poynting vector.
There is subtle conceptual conflict between electromagnetism and Newton's first law: Maxwell's theory of electromagnetism predicts that electromagnetic waves will travel through empty space at a constant, definite speed. Thus, some inertial observers seemingly have a privileged status over the others, namely those who measure the speed of light and find it to be the value predicted by the Maxwell equations. In other words, light provides an absolute standard for speed, yet the principle of inertia holds that there should be no such standard. This tension is resolved in the theory of special relativity, which revises the notions of "space" and "time" in such a way that all inertial observers will agree upon the speed of light in vacuum.
Special relativity.
In special relativity, the rule that Wilczek called "Newton's Zeroth Law" breaks down: the mass of a composite object is not merely the sum of the masses of the individual pieces.33 Newton's first law, inertial motion, remains true. A form of Newton's second law, that force is the rate of change of momentum, also holds, as does the conservation of momentum. However, the definition of momentum is modified. Among the consequences of this is the fact that the more quickly a body moves, the harder it is to accelerate, and so, no matter how much force is applied, a body cannot be accelerated to the speed of light. Depending on the problem at hand, momentum in special relativity can be represented as a three-dimensional vector, formula_155, where formula_26 is the body's rest mass and formula_145 is the Lorentz factor, which depends upon the body's speed. Alternatively, momentum and force can be represented as four-vectors.107
Newton's third law must be modified in special relativity. The third law refers to the forces between two bodies at the same moment in time, and a key feature of special relativity is that simultaneity is relative. Events that happen at the same time relative to one observer can happen at different times relative to another. So, in a given observer's frame of reference, action and reaction may not be exactly opposite, and the total momentum of interacting bodies may not be conserved. The conservation of momentum is restored by including the momentum stored in the field that describes the bodies' interaction.
Newtonian mechanics is a good approximation to special relativity when the speeds involved are small compared to that of light.131
General relativity.
General relativity is a theory of gravity that advances beyond that of Newton. In general relativity, the gravitational force of Newtonian mechanics is reimagined as curvature of spacetime. A curved path like an orbit, attributed to a gravitational force in Newtonian mechanics, is not the result of a force deflecting a body from an ideal straight-line path, but rather the body's attempt to fall freely through a background that is itself curved by the presence of other masses. A remark by John Archibald Wheeler that has become proverbial among physicists summarizes the theory: "Spacetime tells matter how to move; matter tells spacetime how to curve." Wheeler himself thought of this reciprocal relationship as a modern, generalized form of Newton's third law. The relation between matter distribution and spacetime curvature is given by the Einstein field equations, which require tensor calculus to express.43
The Newtonian theory of gravity is a good approximation to the predictions of general relativity when gravitational effects are weak and objects are moving slowly compared to the speed of light.327
Quantum mechanics.
Quantum mechanics is a theory of physics originally developed in order to understand microscopic phenomena: behavior at the scale of molecules, atoms or subatomic particles. Generally and loosely speaking, the smaller a system is, the more an adequate mathematical model will require understanding quantum effects. The conceptual underpinning of quantum physics is very different from that of classical physics. Instead of thinking about quantities like position, momentum, and energy as properties that an object "has", one considers what result might "appear" when a measurement of a chosen type is performed. Quantum mechanics allows the physicist to calculate the probability that a chosen measurement will elicit a particular result. The expectation value for a measurement is the average of the possible results it might yield, weighted by their probabilities of occurrence.
The Ehrenfest theorem provides a connection between quantum expectation values and Newton's second law, a connection that is necessarily inexact, as quantum physics is fundamentally different from classical. In quantum physics, position and momentum are represented by mathematical entities known as Hermitian operators, and the Born rule is used to calculate the expectation values of a position measurement or a momentum measurement. These expectation values will generally change over time; that is, depending on the time at which (for example) a position measurement is performed, the probabilities for its different possible outcomes will vary. The Ehrenfest theorem says, roughly speaking, that the equations describing how these expectation values change over time have a form reminiscent of Newton's second law. However, the more pronounced quantum effects are in a given situation, the more difficult it is to derive meaningful conclusions from this resemblance.
History.
The concepts invoked in Newton's laws of motion — mass, velocity, momentum, force — have predecessors in earlier work, and the content of Newtonian physics was further developed after Newton's time. Newton combined knowledge of celestial motions with the study of events on Earth and showed that one theory of mechanics could encompass both.
As noted by scholar I. Bernard Cohen, Newton's work was more than a mere synthesis of previous results, as he selected certain ideas and further transformed them, with each in a new form that was useful to him, while at the same time proving false of certain basic or fundamental principles of scientists such as Galileo Galilei, Johannes Kepler, René Descartes, and Nicolaus Copernicus. He approached natural philosophy with mathematics in a completely novel way, in that instead of a preconceived natural philosophy, his style was to begin with a mathematical construct, and build on from there, comparing it to the real world to show that his system accurately accounted for it.
Antiquity and medieval background.
Aristotle and "violent" motion.
The subject of physics is often traced back to Aristotle, but the history of the concepts involved is obscured by multiple factors. An exact correspondence between Aristotelian and modern concepts is not simple to establish: Aristotle did not clearly distinguish what we would call speed and force, used the same term for density and viscosity, and conceived of motion as always through a medium, rather than through space. In addition, some concepts often termed "Aristotelian" might better be attributed to his followers and commentators upon him. These commentators found that Aristotelian physics had difficulty explaining projectile motion. Aristotle divided motion into two types: "natural" and "violent". The "natural" motion of terrestrial solid matter was to fall downwards, whereas a "violent" motion could push a body sideways. Moreover, in Aristotelian physics, a "violent" motion requires an immediate cause; separated from the cause of its "violent" motion, a body would revert to its "natural" behavior. Yet, a javelin continues moving after it leaves the thrower's hand. Aristotle concluded that the air around the javelin must be imparted with the ability to move the javelin forward.
Philoponus and impetus.
John Philoponus, a Byzantine Greek thinker active during the sixth century, found this absurd: the same medium, air, was somehow responsible both for sustaining motion and for impeding it. If Aristotle's idea were true, Philoponus said, armies would launch weapons by blowing upon them with bellows. Philoponus argued that setting a body into motion imparted a quality, impetus, that would be contained within the body itself. As long as its impetus was sustained, the body would continue to move. In the following centuries, versions of impetus theory were advanced by individuals including Nur ad-Din al-Bitruji, Avicenna, Abu'l-Barakāt al-Baghdādī, John Buridan, and Albert of Saxony. In retrospect, the idea of impetus can be seen as a forerunner of the modern concept of momentum. The intuition that objects move according to some kind of impetus persists in many students of introductory physics.
Inertia and the first law.
The French philosopher René Descartes introduced the concept of inertia by way of his "laws of nature" in "The World" ("Traité du monde et de la lumière") written 1629–33. However, "The World" purported a heliocentric worldview, and in 1633 this view had given rise a great conflict between Galileo Galilei and the Roman Catholic Inquisition. Descartes knew about this controversy and did not wish to get involved. "The World" was not published until 1664, ten years after his death.
The modern concept of inertia is credited to Galileo. Based on his experiments, Galileo concluded that the "natural" behavior of a moving body was to keep moving, until something else interfered with it. In "Two New Sciences" (1638) Galileo wrote:<templatestyles src="Template:Blockquote/styles.css" />Imagine any particle projected along a horizontal plane without friction; then we know, from what has been more fully explained in the preceding pages, that this particle will move along this same plane with a motion which is uniform and perpetual, provided the plane has no limits.Galileo recognized that in projectile motion, the Earth's gravity affects vertical but not horizontal motion. However, Galileo's idea of inertia was not exactly the one that would be codified into Newton's first law. Galileo thought that a body moving a long distance inertially would follow the curve of the Earth. This idea was corrected by Isaac Beeckman, Descartes, and Pierre Gassendi, who recognized that inertial motion should be motion in a straight line. Descartes published his laws of nature (laws of motion) with this correction in "Principles of Philosophy" ("Principia Philosophiae") in 1644, with the heliocentric part toned down.
<templatestyles src="Template:Blockquote/styles.css" />First Law of Nature: Each thing when left to itself continues in the same state; so any moving body goes on moving until something stops it.<templatestyles src="Template:Blockquote/styles.css" />Second Law of Nature: Each moving thing if left to itself moves in a straight line; so any body moving in a circle always tends to move away from the centre of the circle.
According to American philosopher Richard J. Blackwell, Dutch scientist Christiaan Huygens had worked out his own, concise version of the law in 1656. It was not published until 1703, eight years after his death, in the opening paragraph of "De Motu Corporum ex Percussione".
<templatestyles src="Template:Blockquote/styles.css" />Hypothesis I: Any body already in motion will continue to move perpetually with the same speed and in a straight line unless it is impeded.
According to Huygens, this law was already known by Galileo and Descartes among others.
Force and the second law.
Christiaan Huygens, in his "Horologium Oscillatorium" (1673), put forth the hypothesis that "By the action of gravity, whatever its sources, it happens that bodies are moved by a motion composed both of a uniform motion in one direction or another and of a motion downward due to gravity." Newton's second law generalized this hypothesis from gravity to all forces.
One important characteristic of Newtonian physics is that forces can act at a distance without requiring physical contact. For example, the Sun and the Earth pull on each other gravitationally, despite being separated by millions of kilometres. This contrasts with the idea, championed by Descartes among others, that the Sun's gravity held planets in orbit by swirling them in a vortex of transparent matter, "aether". Newton considered aetherial explanations of force but ultimately rejected them. The study of magnetism by William Gilbert and others created a precedent for thinking of "immaterial" forces, and unable to find a quantitatively satisfactory explanation of his law of gravity in terms of an aetherial model, Newton eventually declared, "I feign no hypotheses": whether or not a model like Descartes's vortices could be found to underlie the "Principia"'s theories of motion and gravity, the first grounds for judging them must be the successful predictions they made. And indeed, since Newton's time every attempt at such a model has failed.
Momentum conservation and the third law.
Johannes Kepler suggested that gravitational attractions were reciprocal — that, for example, the Moon pulls on the Earth while the Earth pulls on the Moon — but he did not argue that such pairs are equal and opposite. In his "Principles of Philosophy" (1644), Descartes introduced the idea that during a collision between bodies, a "quantity of motion" remains unchanged. Descartes defined this quantity somewhat imprecisely by adding up the products of the speed and "size" of each body, where "size" for him incorporated both volume and surface area. Moreover, Descartes thought of the universe as a plenum, that is, filled with matter, so all motion required a body to displace a medium as it moved.
During the 1650s, Huygens studied collisions between hard spheres and deduced a principle that is now identified as the conservation of momentum. Christopher Wren would later deduce the same rules for elastic collisions that Huygens had, and John Wallis would apply momentum conservation to study inelastic collisions. Newton cited the work of Huygens, Wren, and Wallis to support the validity of his third law.
Newton arrived at his set of three laws incrementally. In a 1684 manuscript written to Huygens, he listed four laws: the principle of inertia, the change of motion by force, a statement about relative motion that would today be called Galilean invariance, and the rule that interactions between bodies do not change the motion of their center of mass. In a later manuscript, Newton added a law of action and reaction, while saying that this law and the law regarding the center of mass implied one another. Newton probably settled on the presentation in the "Principia," with three primary laws and then other statements reduced to corollaries, during 1685.
After the "Principia".
Newton expressed his second law by saying that the force on a body is proportional to its change of motion, or momentum. By the time he wrote the "Principia," he had already developed calculus (which he called "the science of fluxions"), but in the "Principia" he made no explicit use of it, perhaps because he believed geometrical arguments in the tradition of Euclid to be more rigorous.15 Consequently, the "Principia" does not express acceleration as the second derivative of position, and so it does not give the second law as formula_158. This form of the second law was written (for the special case of constant force) at least as early as 1716, by Jakob Hermann; Leonhard Euler would employ it as a basic premise in the 1740s. Euler pioneered the study of rigid bodies and established the basic theory of fluid dynamics. Pierre-Simon Laplace's five-volume "Traité de mécanique céleste" (1798–1825) forsook geometry and developed mechanics purely through algebraic expressions, while resolving questions that the "Principia" had left open, like a full theory of the tides.
The concept of energy became a key part of Newtonian mechanics in the post-Newton period. Huygens' solution of the collision of hard spheres showed that in that case, not only is momentum conserved, but kinetic energy is as well (or, rather, a quantity that in retrospect we can identify as one-half the total kinetic energy). The question of what is conserved during all other processes, like inelastic collisions and motion slowed by friction, was not resolved until the 19th century. Debates on this topic overlapped with philosophical disputes between the metaphysical views of Newton and Leibniz, and variants of the term "force" were sometimes used to denote what we would call types of energy. For example, in 1742, Émilie du Châtelet wrote, "Dead force consists of a simple tendency to motion: such is that of a spring ready to relax; living force is that which a body has when it is in actual motion." In modern terminology, "dead force" and "living force" correspond to potential energy and kinetic energy respectively. Conservation of energy was not established as a universal principle until it was understood that the energy of mechanical work can be dissipated into heat. With the concept of energy given a solid grounding, Newton's laws could then be derived within formulations of classical mechanics that put energy first, as in the Lagrangian and Hamiltonian formulations described above.
Modern presentations of Newton's laws use the mathematics of vectors, a topic that was not developed until the late 19th and early 20th centuries. Vector algebra, pioneered by Josiah Willard Gibbs and Oliver Heaviside, stemmed from and largely supplanted the earlier system of quaternions invented by William Rowan Hamilton.
Notes.
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References.
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Force
Influence that can change motion of an object
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A force is an influence that can cause an object to change its velocity unless counterbalanced by other forces. The concept of force makes the everyday notion of pushing or pulling mathematically precise. Because the magnitude and direction of a force are both important, force is a vector quantity. The SI unit of force is the newton (N), and force is often represented by the symbol F.
Force plays an important role in classical mechanics. The concept of force is central to all three of Newton's laws of motion. Types of forces often encountered in classical mechanics include elastic, frictional, contact or "normal" forces, and gravitational. The rotational version of force is torque, which produces changes in the rotational speed of an object. In an extended body, each part applies forces on the adjacent parts; the distribution of such forces through the body is the internal mechanical stress. In the case of multiple forces, if the net force on an extended body is zero the body is in equilibrium.
In modern physics, which includes relativity and quantum mechanics, the laws governing motion are revised to rely on fundamental interactions as the ultimate origin of force. However, the understanding of force provided by classical mechanics is useful for practical purposes.
Development of the concept.
Philosophers in antiquity used the concept of force in the study of stationary and moving objects and simple machines, but thinkers such as Aristotle and Archimedes retained fundamental errors in understanding force. In part, this was due to an incomplete understanding of the sometimes non-obvious force of friction and a consequently inadequate view of the nature of natural motion. A fundamental error was the belief that a force is required to maintain motion, even at a constant velocity. Most of the previous misunderstandings about motion and force were eventually corrected by Galileo Galilei and Sir Isaac Newton. With his mathematical insight, Newton formulated laws of motion that were not improved for over two hundred years.
By the early 20th century, Einstein developed a theory of relativity that correctly predicted the action of forces on objects with increasing momenta near the speed of light and also provided insight into the forces produced by gravitation and inertia. With modern insights into quantum mechanics and technology that can accelerate particles close to the speed of light, particle physics has devised a Standard Model to describe forces between particles smaller than atoms. The Standard Model predicts that exchanged particles called gauge bosons are the fundamental means by which forces are emitted and absorbed. Only four main interactions are known: in order of decreasing strength, they are: strong, electromagnetic, weak, and gravitational. High-energy particle physics observations made during the 1970s and 1980s confirmed that the weak and electromagnetic forces are expressions of a more fundamental electroweak interaction.
Pre-Newtonian concepts.
Since antiquity the concept of force has been recognized as integral to the functioning of each of the simple machines. The mechanical advantage given by a simple machine allowed for less force to be used in exchange for that force acting over a greater distance for the same amount of work. Analysis of the characteristics of forces ultimately culminated in the work of Archimedes who was especially famous for formulating a treatment of buoyant forces inherent in fluids.
Aristotle provided a philosophical discussion of the concept of a force as an integral part of Aristotelian cosmology. In Aristotle's view, the terrestrial sphere contained four elements that come to rest at different "natural places" therein. Aristotle believed that motionless objects on Earth, those composed mostly of the elements earth and water, were in their natural place when on the ground, and that they stay that way if left alone. He distinguished between the innate tendency of objects to find their "natural place" (e.g., for heavy bodies to fall), which led to "natural motion", and unnatural or forced motion, which required continued application of a force. This theory, based on the everyday experience of how objects move, such as the constant application of a force needed to keep a cart moving, had conceptual trouble accounting for the behavior of projectiles, such as the flight of arrows. An archer causes the arrow to move at the start of the flight, and it then sails through the air even though no discernible efficient cause acts upon it. Aristotle was aware of this problem and proposed that the air displaced through the projectile's path carries the projectile to its target. This explanation requires a continuous medium such as air to sustain the motion.
Though Aristotelian physics was criticized as early as the 6th century, its shortcomings would not be corrected until the 17th century work of Galileo Galilei, who was influenced by the late medieval idea that objects in forced motion carried an innate force of impetus. Galileo constructed an experiment in which stones and cannonballs were both rolled down an incline to disprove the Aristotelian theory of motion. He showed that the bodies were accelerated by gravity to an extent that was independent of their mass and argued that objects retain their velocity unless acted on by a force, for example friction. Galileo's idea that force is needed to change motion rather than to sustain it, further improved upon by Isaac Beeckman, René Descartes, and Pierre Gassendi, became a key principle of Newtonian physics.
In the early 17th century, before Newton's "Principia", the term "force" () was applied to many physical and non-physical phenomena, e.g., for an acceleration of a point. The product of a point mass and the square of its velocity was named (live force) by Leibniz. The modern concept of force corresponds to Newton's (accelerating force).
Newtonian mechanics.
Sir Isaac Newton described the motion of all objects using the concepts of inertia and force. In 1687, Newton published his magnum opus, "Philosophiæ Naturalis Principia Mathematica". In this work Newton set out three laws of motion that have dominated the way forces are described in physics to this day. The precise ways in which Newton's laws are expressed have evolved in step with new mathematical approaches.
First law.
Newton's first law of motion states that the natural behavior of an object at rest is to continue being at rest, and the natural behavior of an object moving at constant speed in a straight line is to continue moving at that constant speed along that straight line. The latter follows from the former because of the principle that the laws of physics are the same for all inertial observers, i.e., all observers who do not feel themselves to be in motion. An observer moving in tandem with an object will see it as being at rest. So, its natural behavior will be to remain at rest with respect to that observer, which means that an observer who sees it moving at constant speed in a straight line will see it continuing to do so.
Second law.
According to the first law, motion at constant speed in a straight line does not need a cause. It is "change" in motion that requires a cause, and Newton's second law gives the quantitative relationship between force and change of motion.
Newton's second law states that the net force acting upon an object is equal to the rate at which its momentum changes with time. If the mass of the object is constant, this law implies that the acceleration of an object is directly proportional to the net force acting on the object, is in the direction of the net force, and is inversely proportional to the mass of the object.
A modern statement of Newton's second law is a vector equation:
formula_1
where formula_2 is the momentum of the system, and formula_3 is the net (vector sum) force.399 If a body is in equilibrium, there is zero "net" force by definition (balanced forces may be present nevertheless). In contrast, the second law states that if there is an "unbalanced" force acting on an object it will result in the object's momentum changing over time.
In common engineering applications the mass in a system remains constant allowing as simple algebraic form for the second law. By the definition of momentum,
formula_4
where "m" is the mass and formula_5 is the velocity. If Newton's second law is applied to a system of constant mass, "m" may be moved outside the derivative operator. The equation then becomes
formula_6
By substituting the definition of acceleration, the algebraic version of Newton's second law is derived:
formula_7
Third law.
Whenever one body exerts a force on another, the latter simultaneously exerts an equal and opposite force on the first. In vector form, if formula_8 is the force of body 1 on body 2 and formula_9 that of body 2 on body 1, then
formula_10
This law is sometimes referred to as the "action-reaction law", with formula_11 called the "action" and formula_12 the "reaction".
Newton's Third Law is a result of applying symmetry to situations where forces can be attributed to the presence of different objects. The third law means that all forces are "interactions" between different bodies. and thus that there is no such thing as a unidirectional force or a force that acts on only one body.
In a system composed of object 1 and object 2, the net force on the system due to their mutual interactions is zero:
formula_13
More generally, in a closed system of particles, all internal forces are balanced. The particles may accelerate with respect to each other but the center of mass of the system will not accelerate. If an external force acts on the system, it will make the center of mass accelerate in proportion to the magnitude of the external force divided by the mass of the system.
Combining Newton's Second and Third Laws, it is possible to show that the linear momentum of a system is conserved in any closed system. In a system of two particles, if formula_14 is the momentum of object 1 and formula_15 the momentum of object 2, then
formula_16
Using similar arguments, this can be generalized to a system with an arbitrary number of particles. In general, as long as all forces are due to the interaction of objects with mass, it is possible to define a system such that net momentum is never lost nor gained.ch.12
Defining "force".
Some textbooks use Newton's second law as a "definition" of force. However, for the equation formula_17 for a constant mass formula_18 to then have any predictive content, it must be combined with further information. Moreover, inferring that a force is present because a body is accelerating is only valid in an inertial frame of reference. The question of which aspects of Newton's laws to take as definitions and which to regard as holding physical content has been answered in various ways, which ultimately do not affect how the theory is used in practice. Notable physicists, philosophers and mathematicians who have sought a more explicit definition of the concept of force include Ernst Mach and Walter Noll.
Combining forces.
Forces act in a particular direction and have sizes dependent upon how strong the push or pull is. Because of these characteristics, forces are classified as "vector quantities". This means that forces follow a different set of mathematical rules than physical quantities that do not have direction (denoted scalar quantities). For example, when determining what happens when two forces act on the same object, it is necessary to know both the magnitude and the direction of both forces to calculate the result. If both of these pieces of information are not known for each force, the situation is ambiguous.
Historically, forces were first quantitatively investigated in conditions of static equilibrium where several forces canceled each other out. Such experiments demonstrate the crucial properties that forces are additive vector quantities: they have magnitude and direction. When two forces act on a point particle, the resulting force, the "resultant" (also called the "net force"), can be determined by following the parallelogram rule of vector addition: the addition of two vectors represented by sides of a parallelogram, gives an equivalent resultant vector that is equal in magnitude and direction to the transversal of the parallelogram. The magnitude of the resultant varies from the difference of the magnitudes of the two forces to their sum, depending on the angle between their lines of action.ch.12
Free-body diagrams can be used as a convenient way to keep track of forces acting on a system. Ideally, these diagrams are drawn with the angles and relative magnitudes of the force vectors preserved so that graphical vector addition can be done to determine the net force.
As well as being added, forces can also be resolved into independent components at right angles to each other. A horizontal force pointing northeast can therefore be split into two forces, one pointing north, and one pointing east. Summing these component forces using vector addition yields the original force. Resolving force vectors into components of a set of basis vectors is often a more mathematically clean way to describe forces than using magnitudes and directions. This is because, for orthogonal components, the components of the vector sum are uniquely determined by the scalar addition of the components of the individual vectors. Orthogonal components are independent of each other because forces acting at ninety degrees to each other have no effect on the magnitude or direction of the other. Choosing a set of orthogonal basis vectors is often done by considering what set of basis vectors will make the mathematics most convenient. Choosing a basis vector that is in the same direction as one of the forces is desirable, since that force would then have only one non-zero component. Orthogonal force vectors can be three-dimensional with the third component being at right angles to the other two.ch.12
Equilibrium.
When all the forces that act upon an object are balanced, then the object is said to be in a state of equilibrium. Hence, equilibrium occurs when the resultant force acting on a point particle is zero (that is, the vector sum of all forces is zero). When dealing with an extended body, it is also necessary that the net torque be zero. A body is in "static equilibrium" with respect to a frame of reference if it at rest and not accelerating, whereas a body in "dynamic equilibrium" is moving at a constant speed in a straight line, i.e., moving but not accelerating. What one observer sees as static equilibrium, another can see as dynamic equilibrium and vice versa.
Static.
Static equilibrium was understood well before the invention of classical mechanics. Objects that are not accelerating have zero net force acting on them.
The simplest case of static equilibrium occurs when two forces are equal in magnitude but opposite in direction. For example, an object on a level surface is pulled (attracted) downward toward the center of the Earth by the force of gravity. At the same time, a force is applied by the surface that resists the downward force with equal upward force (called a normal force). The situation produces zero net force and hence no acceleration.
Pushing against an object that rests on a frictional surface can result in a situation where the object does not move because the applied force is opposed by static friction, generated between the object and the table surface. For a situation with no movement, the static friction force "exactly" balances the applied force resulting in no acceleration. The static friction increases or decreases in response to the applied force up to an upper limit determined by the characteristics of the contact between the surface and the object.
A static equilibrium between two forces is the most usual way of measuring forces, using simple devices such as weighing scales and spring balances. For example, an object suspended on a vertical spring scale experiences the force of gravity acting on the object balanced by a force applied by the "spring reaction force", which equals the object's weight. Using such tools, some quantitative force laws were discovered: that the force of gravity is proportional to volume for objects of constant density (widely exploited for millennia to define standard weights); Archimedes' principle for buoyancy; Archimedes' analysis of the lever; Boyle's law for gas pressure; and Hooke's law for springs. These were all formulated and experimentally verified before Isaac Newton expounded his three laws of motion.ch.12
Dynamic.
Dynamic equilibrium was first described by Galileo who noticed that certain assumptions of Aristotelian physics were contradicted by observations and logic. Galileo realized that simple velocity addition demands that the concept of an "absolute rest frame" did not exist. Galileo concluded that motion in a constant velocity was completely equivalent to rest. This was contrary to Aristotle's notion of a "natural state" of rest that objects with mass naturally approached. Simple experiments showed that Galileo's understanding of the equivalence of constant velocity and rest were correct. For example, if a mariner dropped a cannonball from the crow's nest of a ship moving at a constant velocity, Aristotelian physics would have the cannonball fall straight down while the ship moved beneath it. Thus, in an Aristotelian universe, the falling cannonball would land behind the foot of the mast of a moving ship. When this experiment is actually conducted, the cannonball always falls at the foot of the mast, as if the cannonball knows to travel with the ship despite being separated from it. Since there is no forward horizontal force being applied on the cannonball as it falls, the only conclusion left is that the cannonball continues to move with the same velocity as the boat as it falls. Thus, no force is required to keep the cannonball moving at the constant forward velocity.
Moreover, any object traveling at a constant velocity must be subject to zero net force (resultant force). This is the definition of dynamic equilibrium: when all the forces on an object balance but it still moves at a constant velocity. A simple case of dynamic equilibrium occurs in constant velocity motion across a surface with kinetic friction. In such a situation, a force is applied in the direction of motion while the kinetic friction force exactly opposes the applied force. This results in zero net force, but since the object started with a non-zero velocity, it continues to move with a non-zero velocity. Aristotle misinterpreted this motion as being caused by the applied force. When kinetic friction is taken into consideration it is clear that there is no net force causing constant velocity motion.ch.12
Examples of forces in classical mechanics.
Some forces are consequences of the fundamental ones. In such situations, idealized models can be used to gain physical insight. For example, each solid object is considered a rigid body.
Gravitational force or Gravity.
What we now call gravity was not identified as a universal force until the work of Isaac Newton. Before Newton, the tendency for objects to fall towards the Earth was not understood to be related to the motions of celestial objects. Galileo was instrumental in describing the characteristics of falling objects by determining that the acceleration of every object in free-fall was constant and independent of the mass of the object. Today, this acceleration due to gravity towards the surface of the Earth is usually designated as formula_19 and has a magnitude of about 9.81 meters per second squared (this measurement is taken from sea level and may vary depending on location), and points toward the center of the Earth. This observation means that the force of gravity on an object at the Earth's surface is directly proportional to the object's mass. Thus an object that has a mass of formula_18 will experience a force:
formula_21
For an object in free-fall, this force is unopposed and the net force on the object is its weight. For objects not in free-fall, the force of gravity is opposed by the reaction forces applied by their supports. For example, a person standing on the ground experiences zero net force, since a normal force (a reaction force) is exerted by the ground upward on the person that counterbalances his weight that is directed downward.ch.12
Newton's contribution to gravitational theory was to unify the motions of heavenly bodies, which Aristotle had assumed were in a natural state of constant motion, with falling motion observed on the Earth. He proposed a law of gravity that could account for the celestial motions that had been described earlier using Kepler's laws of planetary motion.
Newton came to realize that the effects of gravity might be observed in different ways at larger distances. In particular, Newton determined that the acceleration of the Moon around the Earth could be ascribed to the same force of gravity if the acceleration due to gravity decreased as an inverse square law. Further, Newton realized that the acceleration of a body due to gravity is proportional to the mass of the other attracting body. Combining these ideas gives a formula that relates the mass (formula_22) and the radius (formula_23) of the Earth to the gravitational acceleration:
formula_24
where formula_25 is the distance between the two objects' centers of mass and formula_26 is the unit vector pointed in the direction away from the center of the first object toward the center of the second object.
This formula was powerful enough to stand as the basis for all subsequent descriptions of motion within the Solar System until the 20th century. During that time, sophisticated methods of perturbation analysis were invented to calculate the deviations of orbits due to the influence of multiple bodies on a planet, moon, comet, or asteroid. The formalism was exact enough to allow mathematicians to predict the existence of the planet Neptune before it was observed.
Electromagnetic.
The electrostatic force was first described in 1784 by Coulomb as a force that existed intrinsically between two charges. The properties of the electrostatic force were that it varied as an inverse square law directed in the radial direction, was both attractive and repulsive (there was intrinsic polarity), was independent of the mass of the charged objects, and followed the superposition principle. Coulomb's law unifies all these observations into one succinct statement.
Subsequent mathematicians and physicists found the construct of the "electric field" to be useful for determining the electrostatic force on an electric charge at any point in space. The electric field was based on using a hypothetical "test charge" anywhere in space and then using Coulomb's Law to determine the electrostatic force. Thus the electric field anywhere in space is defined as
formula_27
where formula_28 is the magnitude of the hypothetical test charge. Similarly, the idea of the "magnetic field" was introduced to express how magnets can influence one another at a distance. The Lorentz force law gives the force upon a body with charge formula_28 due to electric and magnetic fields:
formula_30
where formula_3 is the electromagnetic force, formula_32 is the electric field at the body's location, formula_33 is the magnetic field, and formula_5 is the velocity of the particle. The magnetic contribution to the Lorentz force is the cross product of the velocity vector with the magnetic field.
The origin of electric and magnetic fields would not be fully explained until 1864 when James Clerk Maxwell unified a number of earlier theories into a set of 20 scalar equations, which were later reformulated into 4 vector equations by Oliver Heaviside and Josiah Willard Gibbs. These "Maxwell's equations" fully described the sources of the fields as being stationary and moving charges, and the interactions of the fields themselves. This led Maxwell to discover that electric and magnetic fields could be "self-generating" through a wave that traveled at a speed that he calculated to be the speed of light. This insight united the nascent fields of electromagnetic theory with optics and led directly to a complete description of the electromagnetic spectrum.
Normal.
When objects are in contact, the force directly between them is called the normal force, the component of the total force in the system exerted normal to the interface between the objects. The normal force is closely related to Newton's third law. The normal force, for example, is responsible for the structural integrity of tables and floors as well as being the force that responds whenever an external force pushes on a solid object. An example of the normal force in action is the impact force on an object crashing into an immobile surface.ch.12
Friction.
Friction is a force that opposes relative motion of two bodies. At the macroscopic scale, the frictional force is directly related to the normal force at the point of contact. There are two broad classifications of frictional forces: static friction and kinetic friction.
The static friction force (formula_35) will exactly oppose forces applied to an object parallel to a surface up to the limit specified by the coefficient of static friction (formula_36) multiplied by the normal force (formula_37). In other words, the magnitude of the static friction force satisfies the inequality:
formula_38
The kinetic friction force (formula_39) is typically independent of both the forces applied and the movement of the object. Thus, the magnitude of the force equals:
formula_40
where formula_41 is the coefficient of kinetic friction. The coefficient of kinetic friction is normally less than the coefficient of static friction.
Tension.
Tension forces can be modeled using ideal strings that are massless, frictionless, unbreakable, and do not stretch. They can be combined with ideal pulleys, which allow ideal strings to switch physical direction. Ideal strings transmit tension forces instantaneously in action–reaction pairs so that if two objects are connected by an ideal string, any force directed along the string by the first object is accompanied by a force directed along the string in the opposite direction by the second object. By connecting the same string multiple times to the same object through the use of a configuration that uses movable pulleys, the tension force on a load can be multiplied. For every string that acts on a load, another factor of the tension force in the string acts on the load. Such machines allow a mechanical advantage for a corresponding increase in the length of displaced string needed to move the load. These tandem effects result ultimately in the conservation of mechanical energy since the work done on the load is the same no matter how complicated the machine.ch.12
Spring.
A simple elastic force acts to return a spring to its natural length. An ideal spring is taken to be massless, frictionless, unbreakable, and infinitely stretchable. Such springs exert forces that push when contracted, or pull when extended, in proportion to the displacement of the spring from its equilibrium position. This linear relationship was described by Robert Hooke in 1676, for whom Hooke's law is named. If formula_42 is the displacement, the force exerted by an ideal spring equals:
formula_43
where formula_44 is the spring constant (or force constant), which is particular to the spring. The minus sign accounts for the tendency of the force to act in opposition to the applied load.ch.12
Centripetal.
For an object in uniform circular motion, the net force acting on the object equals:
formula_45
where formula_18 is the mass of the object, formula_47 is the velocity of the object and formula_25 is the distance to the center of the circular path and formula_49 is the unit vector pointing in the radial direction outwards from the center. This means that the net force felt by the object is always directed toward the center of the curving path. Such forces act perpendicular to the velocity vector associated with the motion of an object, and therefore do not change the speed of the object (magnitude of the velocity), but only the direction of the velocity vector. More generally, the net force that accelerates an object can be resolved into a component that is perpendicular to the path, and one that is tangential to the path. This yields both the tangential force, which accelerates the object by either slowing it down or speeding it up, and the radial (centripetal) force, which changes its direction.ch.12
Continuum mechanics.
Newton's laws and Newtonian mechanics in general were first developed to describe how forces affect idealized point particles rather than three-dimensional objects. In real life, matter has extended structure and forces that act on one part of an object might affect other parts of an object. For situations where lattice holding together the atoms in an object is able to flow, contract, expand, or otherwise change shape, the theories of continuum mechanics describe the way forces affect the material. For example, in extended fluids, differences in pressure result in forces being directed along the pressure gradients as follows:
formula_50
where formula_51 is the volume of the object in the fluid and formula_52 is the scalar function that describes the pressure at all locations in space. Pressure gradients and differentials result in the buoyant force for fluids suspended in gravitational fields, winds in atmospheric science, and the lift associated with aerodynamics and flight.ch.12
A specific instance of such a force that is associated with dynamic pressure is fluid resistance: a body force that resists the motion of an object through a fluid due to viscosity. For so-called "Stokes' drag" the force is approximately proportional to the velocity, but opposite in direction:
formula_53
where:
More formally, forces in continuum mechanics are fully described by a stress tensor with terms that are roughly defined as
formula_56
where formula_57 is the relevant cross-sectional area for the volume for which the stress tensor is being calculated. This formalism includes pressure terms associated with forces that act normal to the cross-sectional area (the matrix diagonals of the tensor) as well as shear terms associated with forces that act parallel to the cross-sectional area (the off-diagonal elements). The stress tensor accounts for forces that cause all strains (deformations) including also tensile stresses and compressions.
Fictitious.
There are forces that are frame dependent, meaning that they appear due to the adoption of non-Newtonian (that is, non-inertial) reference frames. Such forces include the centrifugal force and the Coriolis force. These forces are considered fictitious because they do not exist in frames of reference that are not accelerating.ch.12 Because these forces are not genuine they are also referred to as "pseudo forces".
In general relativity, gravity becomes a fictitious force that arises in situations where spacetime deviates from a flat geometry.
Concepts derived from force.
Rotation and torque.
Forces that cause extended objects to rotate are associated with torques. Mathematically, the torque of a force formula_3 is defined relative to an arbitrary reference point as the cross product:
formula_59
where formula_60 is the position vector of the force application point relative to the reference point.
Torque is the rotation equivalent of force in the same way that angle is the rotational equivalent for position, angular velocity for velocity, and angular momentum for momentum. As a consequence of Newton's first law of motion, there exists rotational inertia that ensures that all bodies maintain their angular momentum unless acted upon by an unbalanced torque. Likewise, Newton's second law of motion can be used to derive an analogous equation for the instantaneous angular acceleration of the rigid body:
formula_61
where
This provides a definition for the moment of inertia, which is the rotational equivalent for mass. In more advanced treatments of mechanics, where the rotation over a time interval is described, the moment of inertia must be substituted by the tensor that, when properly analyzed, fully determines the characteristics of rotations including precession and nutation.
Equivalently, the differential form of Newton's Second Law provides an alternative definition of torque:
formula_64
where formula_65 is the angular momentum of the particle.
Newton's Third Law of Motion requires that all objects exerting torques themselves experience equal and opposite torques, and therefore also directly implies the conservation of angular momentum for closed systems that experience rotations and revolutions through the action of internal torques.
Yank.
The yank is defined as the rate of change of force
formula_66
The term is used in biomechanical analysis, athletic assessment and robotic control. The second ("tug"), third ("snatch"), fourth ("shake"), and higher derivatives are rarely used.
Kinematic integrals.
Forces can be used to define a number of physical concepts by integrating with respect to kinematic variables. For example, integrating with respect to time gives the definition of impulse:
formula_67
which by Newton's Second Law must be equivalent to the change in momentum (yielding the Impulse momentum theorem).
Similarly, integrating with respect to position gives a definition for the work done by a force:
formula_68
which is equivalent to changes in kinetic energy (yielding the work energy theorem).
Power "P" is the rate of change d"W"/d"t" of the work "W", as the trajectory is extended by a position change formula_69 in a time interval d"t":
formula_70
so
formula_71
with formula_72 the velocity.
Potential energy.
Instead of a force, often the mathematically related concept of a potential energy field is used. For instance, the gravitational force acting upon an object can be seen as the action of the gravitational field that is present at the object's location. Restating mathematically the definition of energy (via the definition of work), a potential scalar field formula_73 is defined as that field whose gradient is equal and opposite to the force produced at every point:
formula_74
Forces can be classified as conservative or nonconservative. Conservative forces are equivalent to the gradient of a potential while nonconservative forces are not.ch.12
Conservation.
A conservative force that acts on a closed system has an associated mechanical work that allows energy to convert only between kinetic or potential forms. This means that for a closed system, the net mechanical energy is conserved whenever a conservative force acts on the system. The force, therefore, is related directly to the difference in potential energy between two different locations in space, and can be considered to be an artifact of the potential field in the same way that the direction and amount of a flow of water can be considered to be an artifact of the contour map of the elevation of an area.ch.12
Conservative forces include gravity, the electromagnetic force, and the spring force. Each of these forces has models that are dependent on a position often given as a radial vector formula_60 emanating from spherically symmetric potentials. Examples of this follow:
For gravity:
formula_76
where formula_77 is the gravitational constant, and formula_78 is the mass of object "n".
For electrostatic forces:
formula_79
where formula_80 is electric permittivity of free space, and formula_81 is the electric charge of object "n".
For spring forces:
formula_82
where formula_44 is the spring constant.ch.12
For certain physical scenarios, it is impossible to model forces as being due to a simple gradient of potentials. This is often due a macroscopic statistical average of microstates. For example, static friction is caused by the gradients of numerous electrostatic potentials between the atoms, but manifests as a force model that is independent of any macroscale position vector. Nonconservative forces other than friction include other contact forces, tension, compression, and drag. For any sufficiently detailed description, all these forces are the results of conservative ones since each of these macroscopic forces are the net results of the gradients of microscopic potentials.ch.12
The connection between macroscopic nonconservative forces and microscopic conservative forces is described by detailed treatment with statistical mechanics. In macroscopic closed systems, nonconservative forces act to change the internal energies of the system, and are often associated with the transfer of heat. According to the Second law of thermodynamics, nonconservative forces necessarily result in energy transformations within closed systems from ordered to more random conditions as entropy increases.ch.12
Units.
The SI unit of force is the newton (symbol N), which is the force required to accelerate a one kilogram mass at a rate of one meter per second squared, or kg·m·s−2.The corresponding CGS unit is the dyne, the force required to accelerate a one gram mass by one centimeter per second squared, or g·cm·s−2. A newton is thus equal to 100,000 dynes.
The gravitational foot-pound-second English unit of force is the pound-force (lbf), defined as the force exerted by gravity on a pound-mass in the standard gravitational field of 9.80665 m·s−2. The pound-force provides an alternative unit of mass: one slug is the mass that will accelerate by one foot per second squared when acted on by one pound-force. An alternative unit of force in a different foot–pound–second system, the absolute fps system, is the poundal, defined as the force required to accelerate a one-pound mass at a rate of one foot per second squared.
The pound-force has a metric counterpart, less commonly used than the newton: the kilogram-force (kgf) (sometimes kilopond), is the force exerted by standard gravity on one kilogram of mass. The kilogram-force leads to an alternate, but rarely used unit of mass: the metric slug (sometimes mug or hyl) is that mass that accelerates at 1 m·s−2 when subjected to a force of 1 kgf. The kilogram-force is not a part of the modern SI system, and is generally deprecated, sometimes used for expressing aircraft weight, jet thrust, bicycle spoke tension, torque wrench settings and engine output torque.
 See also "Ton-force".
Revisions of the force concept.
At the beginning of the 20th century, new physical ideas emerged to explain experimental results in astronomical and submicroscopic realms. As discussed below, relativity alters the definition of momentum and quantum mechanics reuses the concept of "force" in microscopic contexts where Newton's laws do not apply directly.
Special theory of relativity.
In the special theory of relativity, mass and energy are equivalent (as can be seen by calculating the work required to accelerate an object). When an object's velocity increases, so does its energy and hence its mass equivalent (inertia). It thus requires more force to accelerate it the same amount than it did at a lower velocity. Newton's Second Law,
formula_84
remains valid because it is a mathematical definition. But for momentum to be conserved at relativistic relative velocity, formula_47, momentum must be redefined as:
formula_86
where formula_87 is the rest mass and formula_88 the speed of light.
The expression relating force and acceleration for a particle with constant non-zero rest mass formula_18 moving in the formula_90 direction at velocity formula_47 is:
formula_92
where
formula_93
is called the Lorentz factor. The Lorentz factor increases steeply as the relative velocity approaches the speed of light. Consequently, the greater and greater force must be applied to produce the same acceleration at extreme velocity. The relative velocity cannot reach formula_88.§15–8
If formula_47 is very small compared to formula_88, then formula_97 is very close to 1 and
formula_98
is a close approximation. Even for use in relativity, one can restore the form of
formula_99
through the use of four-vectors. This relation is correct in relativity when formula_100 is the four-force, formula_18 is the invariant mass, and formula_102 is the four-acceleration.
The "general" theory of relativity incorporates a more radical departure from the Newtonian way of thinking about force, specifically gravitational force. This reimagining of the nature of gravity is described more fully below.
Quantum mechanics.
Quantum mechanics is a theory of physics originally developed in order to understand microscopic phenomena: behavior at the scale of molecules, atoms or subatomic particles. Generally and loosely speaking, the smaller a system is, the more an adequate mathematical model will require understanding quantum effects. The conceptual underpinning of quantum physics is different from that of classical physics. Instead of thinking about quantities like position, momentum, and energy as properties that an object "has", one considers what result might "appear" when a measurement of a chosen type is performed. Quantum mechanics allows the physicist to calculate the probability that a chosen measurement will elicit a particular result. The expectation value for a measurement is the average of the possible results it might yield, weighted by their probabilities of occurrence.
In quantum mechanics, interactions are typically described in terms of energy rather than force. The Ehrenfest theorem provides a connection between quantum expectation values and the classical concept of force, a connection that is necessarily inexact, as quantum physics is fundamentally different from classical. In quantum physics, the Born rule is used to calculate the expectation values of a position measurement or a momentum measurement. These expectation values will generally change over time; that is, depending on the time at which (for example) a position measurement is performed, the probabilities for its different possible outcomes will vary. The Ehrenfest theorem says, roughly speaking, that the equations describing how these expectation values change over time have a form reminiscent of Newton's second law, with a force defined as the negative derivative of the potential energy. However, the more pronounced quantum effects are in a given situation, the more difficult it is to derive meaningful conclusions from this resemblance.
Quantum mechanics also introduces two new constraints that interact with forces at the submicroscopic scale and which are especially important for atoms. Despite the strong attraction of the nucleus, the uncertainty principle limits the minimum extent of an electron probability distribution and the Pauli exclusion principle prevents electrons from sharing the same probability distribution. This gives rise to an emergent pressure known as degeneracy pressure. The dynamic equilibrium between the degeneracy pressure and the attractive electromagnetic force give atoms, molecules, liquids, and solids stability.
Quantum field theory.
In modern particle physics, forces and the acceleration of particles are explained as a mathematical by-product of exchange of momentum-carrying gauge bosons. With the development of quantum field theory and general relativity, it was realized that force is a redundant concept arising from conservation of momentum (4-momentum in relativity and momentum of virtual particles in quantum electrodynamics). The conservation of momentum can be directly derived from the homogeneity or symmetry of space and so is usually considered more fundamental than the concept of a force. Thus the currently known fundamental forces are considered more accurately to be "fundamental interactions".
While sophisticated mathematical descriptions are needed to predict, in full detail, the result of such interactions, there is a conceptually simple way to describe them through the use of Feynman diagrams. In a Feynman diagram, each matter particle is represented as a straight line (see world line) traveling through time, which normally increases up or to the right in the diagram. Matter and anti-matter particles are identical except for their direction of propagation through the Feynman diagram. World lines of particles intersect at interaction vertices, and the Feynman diagram represents any force arising from an interaction as occurring at the vertex with an associated instantaneous change in the direction of the particle world lines. Gauge bosons are emitted away from the vertex as wavy lines and, in the case of virtual particle exchange, are absorbed at an adjacent vertex. The utility of Feynman diagrams is that other types of physical phenomena that are part of the general picture of fundamental interactions but are conceptually separate from forces can also be described using the same rules. For example, a Feynman diagram can describe in succinct detail how a neutron decays into an electron, proton, and antineutrino, an interaction mediated by the same gauge boson that is responsible for the weak nuclear force.
Fundamental interactions.
All of the known forces of the universe are classified into four fundamental interactions. The strong and the weak forces act only at very short distances, and are responsible for the interactions between subatomic particles, including nucleons and compound nuclei. The electromagnetic force acts between electric charges, and the gravitational force acts between masses. All other forces in nature derive from these four fundamental interactions operating within quantum mechanics, including the constraints introduced by the Schrödinger equation and the Pauli exclusion principle. For example, friction is a manifestation of the electromagnetic force acting between atoms of two surfaces. The forces in springs, modeled by Hooke's law, are also the result of electromagnetic forces. Centrifugal forces are acceleration forces that arise simply from the acceleration of rotating frames of reference.
The fundamental theories for forces developed from the unification of different ideas. For example, Newton's universal theory of gravitation showed that the force responsible for objects falling near the surface of the Earth is also the force responsible for the falling of celestial bodies about the Earth (the Moon) and around the Sun (the planets). Michael Faraday and James Clerk Maxwell demonstrated that electric and magnetic forces were unified through a theory of electromagnetism. In the 20th century, the development of quantum mechanics led to a modern understanding that the first three fundamental forces (all except gravity) are manifestations of matter (fermions) interacting by exchanging virtual particles called gauge bosons. This Standard Model of particle physics assumes a similarity between the forces and led scientists to predict the unification of the weak and electromagnetic forces in electroweak theory, which was subsequently confirmed by observation.
Gravitational.
Newton's law of gravitation is an example of "action at a distance": one body, like the Sun, exerts an influence upon any other body, like the Earth, no matter how far apart they are. Moreover, this action at a distance is "instantaneous." According to Newton's theory, the one body shifting position changes the gravitational pulls felt by all other bodies, all at the same instant of time. Albert Einstein recognized that this was inconsistent with special relativity and its prediction that influences cannot travel faster than the speed of light. So, he sought a new theory of gravitation that would be relativistically consistent. Mercury's orbit did not match that predicted by Newton's law of gravitation. Some astrophysicists predicted the existence of an undiscovered planet (Vulcan) that could explain the discrepancies. When Einstein formulated his theory of general relativity (GR) he focused on Mercury's problematic orbit and found that his theory added a correction, which could account for the discrepancy. This was the first time that Newton's theory of gravity had been shown to be inexact.
Since then, general relativity has been acknowledged as the theory that best explains gravity. In GR, gravitation is not viewed as a force, but rather, objects moving freely in gravitational fields travel under their own inertia in straight lines through curved spacetime – defined as the shortest spacetime path between two spacetime events. From the perspective of the object, all motion occurs as if there were no gravitation whatsoever. It is only when observing the motion in a global sense that the curvature of spacetime can be observed and the force is inferred from the object's curved path. Thus, the straight line path in spacetime is seen as a curved line in space, and it is called the "ballistic trajectory" of the object. For example, a basketball thrown from the ground moves in a parabola, as it is in a uniform gravitational field. Its spacetime trajectory is almost a straight line, slightly curved (with the radius of curvature of the order of few light-years). The time derivative of the changing momentum of the object is what we label as "gravitational force".
Electromagnetic.
Maxwell's equations and the set of techniques built around them adequately describe a wide range of physics involving force in electricity and magnetism. This classical theory already includes relativity effects. Understanding quantized electromagnetic interactions between elementary particles requires quantum electrodynamics (or QED). In QED, photons are fundamental exchange particles, describing all interactions relating to electromagnetism including the electromagnetic force.
Strong nuclear.
There are two "nuclear forces", which today are usually described as interactions that take place in quantum theories of particle physics. The strong nuclear force is the force responsible for the structural integrity of atomic nuclei, and gains its name from its ability to overpower the electromagnetic repulsion between protons.
The strong force is today understood to represent the interactions between quarks and gluons as detailed by the theory of quantum chromodynamics (QCD). The strong force is the fundamental force mediated by gluons, acting upon quarks, antiquarks, and the gluons themselves. The strong force only acts "directly" upon elementary particles. A residual is observed between hadrons (notably, the nucleons in atomic nuclei), known as the nuclear force. Here the strong force acts indirectly, transmitted as gluons that form part of the virtual pi and rho mesons, the classical transmitters of the nuclear force. The failure of many searches for free quarks has shown that the elementary particles affected are not directly observable. This phenomenon is called color confinement.232
Weak nuclear.
Unique among the fundamental interactions, the weak nuclear force creates no bound states. The weak force is due to the exchange of the heavy W and Z bosons. Since the weak force is mediated by two types of bosons, it can be divided into two types of interaction or "vertices" — charged current, involving the electrically charged W+ and W− bosons, and neutral current, involving electrically neutral Z0 bosons. The most familiar effect of weak interaction is beta decay (of neutrons in atomic nuclei) and the associated radioactivity. This is a type of charged-current interaction. The word "weak" derives from the fact that the field strength is some 1013 times less than that of the strong force. Still, it is stronger than gravity over short distances. A consistent electroweak theory has also been developed, which shows that electromagnetic forces and the weak force are indistinguishable at a temperatures in excess of approximately . Such temperatures occurred in the plasma collisions in the early moments of the Big Bang.
References.
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Mercury (planet)
First planet from the Sun
Mercury is the first planet from the Sun and the smallest in the Solar System. In English, it is named after the ancient Roman god (Mercury), god of commerce and communication, and the messenger of the gods. Mercury is classified as a terrestrial planet, with roughly the same surface gravity as Mars. The surface of Mercury is heavily cratered, as a result of countless impact events that have accumulated over billions of years. Its largest crater, Caloris Planitia, has a diameter of , which is about one-third the diameter of the planet (). Similarly to the Earth's Moon, Mercury's surface displays an expansive rupes system generated from thrust faults and bright ray systems formed by impact event remnants.
Mercury's sidereal year (88.0 Earth days) and sidereal day (58.65 Earth days) are in a 3:2 ratio. This relationship is called spin–orbit resonance, and "sidereal" here means "relative to the stars". Consequently, one solar day (sunrise to sunrise) on Mercury lasts for around 176 Earth days: twice the planet's sidereal year. This means that one side of Mercury will remain in sunlight for one Mercurian year of 88 Earth days; while during the next orbit, that side will be in darkness all the time until the next sunrise after another 88 Earth days.
Combined with its high orbital eccentricity, the planet's surface has widely varying sunlight intensity and temperature, with the equatorial regions ranging from at night to during sunlight. Due to the very small axial tilt, the planet's poles are permanently shadowed. This strongly suggests that water ice could be present in the craters. Above the planet's surface is an extremely tenuous exosphere and a faint magnetic field that is strong enough to deflect solar winds. Mercury has no natural satellites.
As of the early 2020s, many broad details of Mercury's geological history are still under investigation or pending data from space probes. Like other planets in the Solar System, Mercury was formed approximately 4.5 billion years ago. Its mantle is highly homogeneous, which suggests that Mercury had a magma ocean early in its history, like the Moon. According to current models, Mercury may have a solid silicate crust and mantle overlying a solid outer core, a deeper liquid core layer, and a solid inner core. There are many competing hypotheses about Mercury's origins and development, some of which incorporate collision with planetesimals and rock vaporization.
Nomenclature.
Historically, humans knew Mercury by different names depending on whether it was an evening star or a morning star. By about 350 BC, the ancient Greeks had realized the two stars were one. They knew the planet as , meaning "twinkling", and , for its fleeting motion, a name that is retained in modern [[Greek language|Greek]] ( ). The Romans named the planet after the swift-footed Roman messenger god, [[Mercury (mythology)|Mercury]] (Latin ), whom they equated with the Greek Hermes, because it moves across the sky faster than any other planet, though some associated the planet with [[Apollo]] instead, as detailed by [[Pliny the Elder]]. The [[astronomical symbol]] for Mercury is a stylized version of Hermes' [[caduceus]]; a [[Christian cross]] was added in the 16th century:[[File:Mercury symbol (fixed width).svg|16px|☿]].
Physical characteristics.
[[File:Terrestrial planet size comparisons-2.jpg|thumb|upright=1.4|Mercury to scale among the [[Inner Solar System]] [[planetary-mass object]]s beside the Sun, arranged by the order of their orbits outward from the Sun (from left: Mercury, [[Venus]],
[[Earth]], the [[Moon]], [[Mars]] and [[Ceres (dwarf planet)|Ceres]])]]
Mercury is one of four [[terrestrial planet]]s in the [[Solar System]], which means it is a rocky body like Earth. It is the smallest planet in the Solar System, with an [[equator]]ial [[radius]] of . Mercury is also [[list of Solar System objects by radius|smaller]]—albeit more massive—than the largest [[natural satellite]]s in the Solar System, [[Ganymede (moon)|Ganymede]] and [[Titan (moon)|Titan]]. Mercury consists of approximately 70% metallic and 30% [[silicate]] material.
Internal structure.
[[File:Mercury with magnetic field.svg|left|thumb|upright=1.6|Mercury's internal structure and magnetic field]]
Mercury appears to have a solid silicate [[Crust (geology)|crust]] and mantle overlying a solid, metallic outer core layer, a deeper liquid core layer, and a solid inner core. The composition of the iron-rich core remains uncertain, but it likely contains nickel, silicon and perhaps sulfur and carbon, plus trace amounts of other elements. The planet's density is the second highest in the Solar System at 5.427 g/cm3, only slightly less than Earth's density of 5.515 g/cm3. If the effect of [[gravitational compression]] were to be factored out from both planets, the materials of which Mercury is made would be denser than those of Earth, with an uncompressed density of 5.3 g/cm3 versus Earth's 4.4 g/cm3. Mercury's density can be used to infer details of its inner structure. Although Earth's high density results appreciably from gravitational compression, particularly at the [[planetary core|core]], Mercury is much smaller and its inner regions are not as compressed. Therefore, for it to have such a high density, its core must be large and rich in iron.
The radius of Mercury's core is estimated to be , based on interior models constrained to be consistent with a [[moment of inertia factor]] of . Hence, Mercury's core occupies about 57% of its volume; for Earth this proportion is 17%. Research published in 2007 suggests that Mercury has a molten core. The mantle-crust layer is in total thick. Projections differ as to the size of the crust specifically; data from the "Mariner 10" and "MESSENGER" probes suggests a thickness of , whereas an [[Airy isostacy]] model suggests a thickness of . One distinctive feature of Mercury's surface is the presence of numerous narrow ridges, extending up to several hundred kilometers in length. It is thought that these were formed as Mercury's core and mantle cooled and contracted at a time when the crust had already solidified.
Mercury's core has a higher iron content than that of any other planet in the Solar System, and several theories have been proposed to explain this. The most widely accepted theory is that Mercury originally had a metal–silicate ratio similar to common [[chondrite]] meteorites, thought to be typical of the Solar System's rocky matter, and a mass approximately 2.25 times its current mass. Early in the Solar System's history, Mercury may have been struck by a [[planetesimal]] of approximately <templatestyles src="Fraction/styles.css" />1⁄6 Mercury's mass and several thousand kilometers across. The impact would have stripped away much of the original crust and mantle, leaving the core behind as a relatively major component. A similar process, known as the [[giant impact hypothesis]], has been proposed to explain the formation of Earth's Moon.
Alternatively, Mercury may have formed from the [[solar nebula]] before the Sun's energy output had stabilized. It would initially have had twice its present mass, but as the [[protostar|protosun]] contracted, temperatures near Mercury could have been between 2,500 and 3,500 K and possibly even as high as 10,000 K. Much of Mercury's surface rock could have been vaporized at such temperatures, forming an atmosphere of "rock vapor" that could have been carried away by the [[solar wind]]. A third hypothesis proposes that the solar nebula caused [[drag (physics)|drag]] on the particles from which Mercury was [[Accretion (astrophysics)|accreting]], which meant that lighter particles were lost from the accreting material and not gathered by Mercury.
Each hypothesis predicts a different surface composition, and two space missions have been tasked with making observations of this composition. The first "[[MESSENGER]]", which ended in 2015, found higher-than-expected potassium and sulfur levels on the surface, suggesting that the giant impact hypothesis and vaporization of the crust and mantle did not occur because said potassium and sulfur would have been driven off by the extreme heat of these events. "[[BepiColombo]]", which will arrive at Mercury in 2025, will make observations to test these hypotheses. The findings so far would seem to favor the third hypothesis; however, further analysis of the data is needed.
Surface geology.
Mercury's surface is similar in appearance to that of the Moon, showing extensive [[Lunar mare|mare]]-like plains and heavy cratering, indicating that it has been geologically inactive for billions of years. It is more [[heterogeneous]] than the surface of [[Mars]] or the Moon, both of which contain significant stretches of similar geology, such as [[Lunar mare|maria]] and plateaus. [[Albedo]] features are areas of markedly different reflectivity, which include impact craters, the resulting ejecta, and [[ray system]]s. Larger albedo features correspond to higher reflectivity plains. Mercury has "[[wrinkle-ridge]]s" (dorsa), Moon-like [[highland]]s, mountains (montes), plains (planitiae), escarpments (rupes), and valleys ([[Vallis (planetary geology)|valles]]).
[[File:Unmasking the Secrets of Mercury.jpg|thumb|left|[[MESSENGER#Scientific instruments|MASCS]] spectrum scan of Mercury's surface by "MESSENGER"]]
The planet's mantle is chemically heterogeneous, suggesting the planet went through a [[magma ocean]] phase early in its history. Crystallization of minerals and convective overturn resulted in a layered, chemically heterogeneous crust with large-scale variations in chemical composition observed on the surface. The crust is low in iron but high in sulfur, resulting from the stronger early [[Chemical reduction|chemically reducing]] conditions than is found on other terrestrial planets. The surface is dominated by iron-poor [[pyroxene]] and [[olivine]], as represented by [[enstatite]] and [[forsterite]], respectively, along with sodium-rich [[plagioclase]] and minerals of mixed magnesium, calcium, and iron-sulfide. The less reflective regions of the crust are high in carbon, most likely in the form of graphite.
Names for features on Mercury come from a variety of sources and are set according to the [[IAU]] [[planetary nomenclature]] system. Names coming from people are limited to the deceased. Craters are named for artists, musicians, painters, and authors who have made outstanding or fundamental contributions to their field. Ridges, or dorsa, are named for scientists who have contributed to the study of Mercury. Depressions or [[fossa (geology)|fossae]] are named for works of architecture. Montes are named for the word "hot" in a variety of languages. [[Plain]]s or planitiae are named for [[Mercury (god)|Mercury]] in various languages. [[Escarpment]]s or [[rupēs]] are named for ships of scientific expeditions. Valleys or valles are named for abandoned cities, towns, or settlements of antiquity.
Impact basins and craters.
[[File:PIA19421-Mercury-Craters-MunchSanderPoe-20150416.jpg|thumb|left|Enhanced-color image of craters [[Munch (crater)|Munch]] (left), [[Sander (crater)|Sander]] (center), and [[Poe (crater)|Poe]] (right) amid volcanic plains (orange) near [[Caloris Basin]]]]
Mercury was heavily bombarded by comets and [[asteroid]]s during and shortly following its formation 4.6 billion years ago, as well as during a possibly separate subsequent episode called the [[Late Heavy Bombardment]] that ended 3.8 billion years ago. Mercury received impacts over its entire surface during this period of intense crater formation, facilitated by the lack of any [[atmosphere]] to slow impactors down. During this time Mercury was [[volcano|volcanically]] active; basins were filled by [[magma]], producing smooth plains similar to the maria found on the Moon. One of the most unusual craters is [[Apollodorus (crater)|Apollodorus]], or "the Spider", which hosts a series of radiating troughs extending outwards from its impact site.
[[Craters on Mercury]] range in diameter from small bowl-shaped cavities to [[multi-ringed impact basin]]s hundreds of kilometers across. They appear in all states of degradation, from relatively fresh rayed craters to highly degraded crater remnants. Mercurian craters differ subtly from lunar craters in that the area blanketed by their ejecta is much smaller, a consequence of Mercury's stronger surface gravity. According to [[International Astronomical Union]] rules, each new crater must be named after an artist who was famous for more than fifty years, and dead for more than three years, before the date the crater is named.
The largest known crater is [[Caloris Planitia]], or Caloris Basin, with a diameter of . The impact that created the Caloris Basin was so powerful that it caused [[lava]] eruptions and left a concentric mountainous ring ~ tall surrounding the [[impact crater]]. The floor of the Caloris Basin is filled by a geologically distinct flat plain, broken up by ridges and fractures in a roughly polygonal pattern. It is not clear whether they were volcanic lava flows induced by the impact or a large sheet of impact melt.
At the [[antipodes|antipode]] of the Caloris Basin is a large region of unusual, hilly terrain known as the "Weird Terrain". One hypothesis for its origin is that shock waves generated during the Caloris impact traveled around Mercury, converging at the basin's antipode (180 degrees away). The resulting high stresses fractured the surface. Alternatively, it has been suggested that this terrain formed as a result of the convergence of ejecta at this basin's antipode.
[[File:EW1027346412Gnomap.png|thumb|Tolstoj basin is along the bottom of this image of Mercury's limb]]
Overall, 46 impact basins have been identified. A notable basin is the -wide, multi-ring [[Tolstoj Basin]] that has an ejecta blanket extending up to from its rim and a floor that has been filled by smooth plains materials. [[Beethoven Basin]] has a similar-sized ejecta blanket and a -diameter rim. Like the Moon, the surface of Mercury has likely incurred the effects of [[space weathering]] processes, including solar wind and [[micrometeorite]] impacts.
Plains.
There are two geologically distinct plains regions on Mercury. Gently rolling, hilly [[Inter-crater plains on Mercury|plains in the regions between craters]] are Mercury's oldest visible surfaces, predating the heavily cratered terrain. These inter-crater plains appear to have obliterated many earlier craters, and show a general paucity of smaller craters below about in diameter.
Smooth plains are widespread flat areas that fill depressions of various sizes and bear a strong resemblance to lunar maria. Unlike lunar maria, the smooth plains of Mercury have the same albedo as the older inter-crater plains. Despite a lack of unequivocally volcanic characteristics, the localization and rounded, lobate shape of these plains strongly support volcanic origins. All the smooth plains of Mercury formed significantly later than the Caloris basin, as evidenced by appreciably smaller crater densities than on the Caloris ejecta blanket.
Compressional features.
An unusual feature of Mercury's surface is the numerous compression folds, or [[rupes]], that crisscross the plains. These exist on the Moon, but are much more prominent on Mercury. As Mercury's interior cooled, it contracted and its surface began to deform, creating [[wrinkle ridge]]s and [[lobate scarp]]s associated with [[thrust fault]]s. The scarps can reach lengths of and heights of . These compressional features can be seen on top of other features, such as craters and smooth plains, indicating they are more recent. Mapping of the features has suggested a total shrinkage of Mercury's radius in the range of ~. Most activity along the major thrust systems probably ended about 3.6–3.7 billion years ago. Small-scale thrust fault scarps have been found, tens of meters in height and with lengths in the range of a few kilometers, that appear to be less than 50 million years old, indicating that compression of the interior and consequent surface geological activity continue to the present.
Volcanism.
[[File:Picasso crater.png|thumb|[[Picasso (crater)|Picasso crater]]—the large arc-shaped pit located on the eastern side of its floor is postulated to have formed when subsurface magma subsided or drained, causing the surface to collapse into the resulting void.]]
There is evidence for [[pyroclastic flow]]s on Mercury from low-profile [[shield volcano]]es. Fifty-one pyroclastic deposits have been identified, where 90% of them are found within impact craters. A study of the degradation state of the impact craters that host pyroclastic deposits suggests that pyroclastic activity occurred on Mercury over a prolonged interval.
A "rimless depression" inside the southwest rim of the Caloris Basin consists of at least nine overlapping volcanic vents, each individually up to in diameter. It is thus a "[[compound volcano]]". The vent floors are at least below their brinks and they bear a closer resemblance to volcanic craters sculpted by explosive eruptions or modified by collapse into void spaces created by magma withdrawal back down into a conduit. Scientists could not quantify the age of the volcanic complex system but reported that it could be on the order of a billion years.
Surface conditions and exosphere.
[[File:North pole of Mercury -- NASA.jpg|thumb|Composite of the north pole of Mercury, where NASA confirmed the discovery of a large volume of water ice, in permanently dark craters that are found there.]]
The surface temperature of Mercury ranges from . It never rises above 180 K at the poles, due to the absence of an atmosphere and a steep temperature gradient between the equator and the poles. At [[perihelion]], the equatorial [[subsolar point]] is located at latitude 0°W or 180°W, and it climbs to a temperature of about . During [[aphelion]], this occurs at 90° or 270°W and reaches only . On the dark side of the planet, temperatures average . The intensity of [[sunlight]] on Mercury's surface ranges between 4.59 and 10.61 times the [[solar constant]] (1,370 W·m−2).
Although daylight temperatures at the surface of Mercury are generally extremely high, observations strongly suggest that ice (frozen water) exists on Mercury. The floors of deep craters at the poles are never exposed to direct sunlight, and temperatures there remain below 102 K, far lower than the global average. This creates a [[Cold trap (astronomy)|cold trap]] where ice can accumulate. Water ice strongly reflects [[radar]], and observations by the 70-meter [[Goldstone Solar System Radar]] and the [[Very Large Array|VLA]] in the early 1990s revealed that there are patches of high radar [[Reflection (physics)|reflection]] near the poles. Although ice was not the only possible cause of these reflective regions, astronomers thought it to be the most likely explanation. The presence of [[ice|water ice]] was confirmed using "MESSENGER" images of craters at the north pole.
The icy crater regions are estimated to contain about 1014–1015 kg of ice, and may be covered by a layer of [[regolith]] that inhibits [[Sublimation (phase transition)|sublimation]]. By comparison, the [[Antarctica|Antarctic]] ice sheet on Earth has a mass of about 4×1018 kg, and Mars's south polar cap contains about 1016 kg of water. The origin of the ice on Mercury is not yet known, but the two most likely sources are from [[outgassing]] of water from the planet's interior and deposition by impacts of comets.
Mercury is too small and hot for its [[gravity]] to retain any significant [[atmosphere]] over long periods of time; it does have a tenuous surface-bounded [[exosphere]] at a surface pressure of less than approximately 0.5 nPa (0.005 picobars). It includes [[hydrogen]], [[helium]], [[oxygen]], [[sodium]], [[calcium]], [[potassium]], [[magnesium]], [[silicon]], and [[hydroxide]], among others. This exosphere is not stable—atoms are continuously lost and replenished from a variety of sources. [[Hydrogen atom]]s and [[helium atom]]s probably come from the solar wind, [[diffusion|diffusing]] into Mercury's [[magnetosphere]] before later escaping back into space. The [[radioactive decay]] of elements within Mercury's crust is another source of helium, as well as sodium and potassium. Water vapor is present, released by a combination of processes such as comets striking its surface, [[sputtering]] creating water out of hydrogen from the solar wind and oxygen from rock, and sublimation from reservoirs of water ice in the permanently shadowed polar craters. The detection of high amounts of water-related ions like O+, OH−, and [[hydronium|H3O+]] was a surprise. Because of the quantities of these ions that were detected in Mercury's space environment, scientists surmise that these molecules were blasted from the surface or exosphere by the solar wind.
Sodium, potassium, and calcium were discovered in the atmosphere during the 1980s–1990s, and are thought to result primarily from the vaporization of surface rock struck by micrometeorite impacts including presently from [[Comet Encke]]. In 2008, magnesium was discovered by "MESSENGER". Studies indicate that, at times, sodium emissions are localized at points that correspond to the planet's magnetic poles. This would indicate an interaction between the magnetosphere and the planet's surface.
According to NASA, Mercury is not a suitable planet for Earth-like life. It has a [[exosphere|surface boundary exosphere]] instead of a layered atmosphere, extreme temperatures, and high solar radiation. It is unlikely that any living beings can withstand those conditions. Some parts of the subsurface of Mercury may have been [[Planetary habitability|habitable]], and perhaps [[life form]]s, albeit likely primitive [[microorganism]]s, may have existed on the planet.
Magnetic field and magnetosphere.
[[File:Mercury Magnetic Field NASA.jpg|thumb|Graph showing relative strength of Mercury's magnetic field]]
Despite its small size and slow 59-day-long rotation, Mercury has a significant, and apparently global, [[magnetic field]]. According to measurements taken by "Mariner 10", it is about 1.1% the strength of [[Earth's magnetic field|Earth's]]. The magnetic-field strength at Mercury's equator is about 300 [[Tesla (unit)|nT]]. Like that of Earth, Mercury's magnetic field is [[dipolar]] and nearly aligned with the planet's spin axis (10° dipolar tilt, compared to 11° for Earth). Measurements from both the "Mariner 10" and "MESSENGER" space probes have indicated that the strength and shape of the magnetic field are stable.
It is likely that this magnetic field is generated by a [[Dynamo theory|dynamo]] effect, in a manner similar to the magnetic field of Earth. This dynamo effect would result from the circulation of the planet's iron-rich liquid core. Particularly strong [[tidal heating]] effects caused by the planet's high orbital eccentricity would serve to keep part of the core in the liquid state necessary for this dynamo effect.
Mercury's magnetic field is strong enough to deflect the solar wind around the planet, creating a magnetosphere. The planet's magnetosphere, though small enough to fit within Earth, is strong enough to trap solar wind [[plasma (physics)|plasma]]. This contributes to the space weathering of the planet's surface. Observations taken by the "Mariner 10" spacecraft detected this low energy plasma in the magnetosphere of the planet's nightside. Bursts of energetic particles in the planet's magnetotail indicate a dynamic quality to the planet's magnetosphere.
During its second flyby of the planet on October 6, 2008, "MESSENGER" discovered that Mercury's magnetic field can be extremely "leaky". The spacecraft encountered magnetic "tornadoes"—twisted bundles of magnetic fields connecting the planetary magnetic field to interplanetary space—that were up to 800 km wide or a third of the radius of the planet. These twisted magnetic flux tubes, technically known as [[flux transfer event]]s, form open windows in the planet's magnetic shield through which the solar wind may enter and directly impact Mercury's surface via [[magnetic reconnection]]. This also occurs in Earth's magnetic field. The "MESSENGER" observations showed the reconnection rate was ten times higher at Mercury, but its proximity to the Sun only accounts for about a third of the reconnection rate observed by "MESSENGER".
Orbit, rotation, and longitude.
Mercury has the most [[Orbital eccentricity|eccentric]] orbit of all the planets in the Solar System; its eccentricity is 0.21 with its distance from the Sun ranging from . It takes 87.969 Earth days to complete an orbit. The diagram illustrates the effects of the eccentricity, showing Mercury's orbit overlaid with a circular orbit having the same [[semi-major axis]]. Mercury's higher velocity when it is near perihelion is clear from the greater distance it covers in each 5-day interval. In the diagram, the varying distance of Mercury to the Sun is represented by the size of the planet, which is inversely proportional to Mercury's distance from the Sun.
This varying distance to the Sun leads to Mercury's surface being flexed by [[tidal bulge]]s raised by the [[Sun]] that are about 17 times stronger than the Moon's on Earth. Combined with a 3:2 [[#Spin-orbit resonance|spin–orbit resonance]] of the planet's rotation around its axis, it also results in complex variations of the surface temperature. The resonance makes a single [[solar day]] (the length between two [[meridian (astronomy)|meridian]] transits of the Sun) on Mercury last exactly two Mercury years, or about 176 Earth days.
Mercury's orbit is inclined by 7 degrees to the plane of Earth's orbit (the [[ecliptic]]), the largest of all eight known solar planets. As a result, [[transits of Mercury]] across the face of the Sun can only occur when the planet is crossing the plane of the ecliptic at the time it lies between Earth and the Sun, which is in May or November. This occurs about every seven years on average.
Mercury's [[axial tilt]] is almost zero, with the best measured value as low as 0.027 degrees. This is significantly smaller than that of [[Jupiter]], which has the second smallest axial tilt of all planets at 3.1 degrees. This means that to an observer at Mercury's poles, the center of the Sun never rises more than 2.1 [[arcminutes]] above the horizon. By comparison, the [[angular size]] of the Sun as seen from Mercury ranges from <templatestyles src="Fraction/styles.css" />1+1⁄4 to 2 degrees across.
At certain points on Mercury's surface, an observer would be able to see the Sun peek up a little more than two-thirds of the way over the horizon, then reverse and set before rising again, all within the same [[Extraterrestrial skies#Mercury|Mercurian day]]. This is because approximately four Earth days before perihelion, Mercury's angular [[orbital speed|orbital velocity]] equals its angular [[rotational velocity]] so that the Sun's [[improper motion|apparent motion]] ceases; closer to perihelion, Mercury's angular orbital velocity then exceeds the angular rotational velocity. Thus, to a hypothetical observer on Mercury, the Sun appears to move in a [[apparent retrograde motion|retrograde]] direction. Four Earth days after perihelion, the Sun's normal apparent motion resumes. A similar effect would have occurred if Mercury had been in synchronous rotation: the alternating gain and loss of rotation over a revolution would have caused a libration of 23.65° in longitude.
For the same reason, there are two points on Mercury's equator, 180 degrees apart in [[longitude]], at either of which, around perihelion in alternate Mercurian years (once a Mercurian day), the Sun passes overhead, then reverses its apparent motion and passes overhead again, then reverses a second time and passes overhead a third time, taking a total of about 16 Earth-days for this entire process. In the other alternate Mercurian years, the same thing happens at the other of these two points. The amplitude of the retrograde motion is small, so the overall effect is that, for two or three weeks, the Sun is almost stationary overhead, and is at its most brilliant because Mercury is at perihelion, its closest to the Sun. This prolonged exposure to the Sun at its brightest makes these two points the hottest places on Mercury. Maximum temperature occurs when the Sun is at an angle of about 25 degrees past noon due to [[Diurnal temperature variation#Temperature lag|diurnal temperature lag]], at 0.4 Mercury days and 0.8 Mercury years past sunrise. Conversely, there are two other points on the equator, 90 degrees of longitude apart from the first ones, where the Sun passes overhead only when the planet is at aphelion in alternate years, when the apparent motion of the Sun in Mercury's sky is relatively rapid. These points, which are the ones on the equator where the apparent retrograde motion of the Sun happens when it is crossing the horizon as described in the preceding paragraph, receive much less solar heat than the first ones described above.
Mercury attains an inferior conjunction (nearest approach to Earth) every 116 Earth days on average, but this interval can range from 105 days to 129 days due to the planet's eccentric orbit. Mercury can come as near as to Earth, and that is slowly declining: The next approach to within is in 2679, and to within in 4487, but it will not be closer to Earth than until 28,622. Its period of retrograde motion as seen from Earth can vary from 8 to 15 days on either side of an inferior conjunction. This large range arises from the planet's high orbital eccentricity. Essentially, because Mercury is closest to the Sun, when taking an average over time, Mercury is most often the closest planet to the Earth, and—in that measure—it is the closest planet to each of the other planets in the Solar System.
Longitude convention.
The longitude convention for Mercury puts the zero of longitude at one of the two hottest points on the surface, as described above. However, when this area was first visited, by "Mariner 10", this zero meridian was in darkness, so it was impossible to select a feature on the surface to define the exact position of the meridian. Therefore, a small crater further west was chosen, called [[Hun Kal]], which provides the exact reference point for measuring longitude. The center of Hun Kal defines the 20° west meridian. A 1970 International Astronomical Union resolution suggests that longitudes be measured positively in the westerly direction on Mercury. The two hottest places on the equator are therefore at longitudes 0° W and 180° W, and the coolest points on the equator are at longitudes 90° W and 270° W. However, the "MESSENGER" project uses an east-positive convention.
Spin-orbit resonance.
[[File:Mercury's orbital resonance.svg|thumb|After one orbit, Mercury has rotated 1.5 times, so after two complete orbits the same hemisphere is again illuminated.]]
For many years it was thought that Mercury was synchronously [[tidally locked]] with the Sun, [[rotating]] once for each orbit and always keeping the same face directed towards the Sun, in the same way that the same side of the Moon always faces Earth. Radar observations in 1965 proved that the planet has a 3:2 spin-orbit resonance, rotating three times for every two revolutions around the Sun. The eccentricity of Mercury's orbit makes this resonance stable—at perihelion, when the solar tide is strongest, the Sun is nearly stationary in Mercury's sky.
The 3:2 resonant tidal locking is stabilized by the variance of the tidal force along Mercury's eccentric orbit, acting on a permanent dipole component of Mercury's mass distribution. In a circular orbit there is no such variance, so the only resonance stabilized in such an orbit is at 1:1 (e.g., Earth–Moon), when the tidal force, stretching a body along the "center-body" line, exerts a torque that aligns the body's axis of least inertia (the "longest" axis, and the axis of the aforementioned dipole) to always point at the center. However, with noticeable eccentricity, like that of Mercury's orbit, the tidal force has a maximum at perihelion and therefore stabilizes resonances, like 3:2, ensuring that the planet points its axis of least inertia roughly at the Sun when passing through perihelion.
The original reason astronomers thought it was synchronously locked was that, whenever Mercury was best placed for observation, it was always nearly at the same point in its 3:2 resonance, hence showing the same face. This is because, coincidentally, Mercury's rotation period is almost exactly half of its synodic period with respect to Earth. Due to Mercury's 3:2 spin-orbit resonance, a solar day lasts about 176 Earth days. A [[sidereal day]] (the period of rotation) lasts about 58.7 Earth days.
Simulations indicate that the orbital eccentricity of Mercury varies [[chaos theory|chaotically]] from nearly zero (circular) to more than 0.45 over millions of years due to [[Perturbation (astronomy)|perturbations]] from the other planets. This was thought to explain Mercury's 3:2 spin-orbit resonance (rather than the more usual 1:1), because this state is more likely to arise during a period of high eccentricity. However, accurate modeling based on a realistic model of tidal response has demonstrated that Mercury was captured into the 3:2 spin-orbit state at a very early stage of its history, within 20 (more likely, 10) million years after its formation.
Numerical simulations show that a future [[Secular resonance|secular]] [[Orbital resonance|orbital resonant]] interaction with the perihelion of Jupiter may cause the eccentricity of Mercury's orbit to increase to the point where there is a 1% chance that the orbit will be destabilized in the next five billion years. If this happens, Mercury may fall into the Sun, collide with Venus, be ejected from the Solar System, or even disrupt the rest of the inner Solar System.
Advance of perihelion.
[[File:Drehung der Apsidenlinie light.svg|right|thumb|[[Apsidal precession]] of Mercury's orbit]]
In 1859, the French mathematician and astronomer [[Urbain Le Verrier]] reported that the slow [[precession]] of Mercury's orbit around the Sun could not be completely explained by [[Newtonian mechanics]] and perturbations by the known planets. He suggested, among possible explanations, that another planet (or perhaps instead a series of smaller "corpuscules") might exist in an orbit even closer to the Sun than that of Mercury, to account for this perturbation. Other explanations considered included a slight oblateness of the Sun. The success of the search for [[Neptune]] based on its perturbations of the orbit of [[Uranus]] led astronomers to place faith in this possible explanation, and the hypothetical planet was named [[Vulcan (hypothetical planet)|Vulcan]], but no such planet was ever found.
The observed [[perihelion precession]] of Mercury is 5,600 [[arcseconds]] (1.5556°) per century relative to Earth, or per century relative to the inertial [[International Celestial Reference Frame|ICRF]]. Newtonian mechanics, taking into account all the effects from the other planets and including 0.0254 arcseconds per century due to the oblateness of the Sun, predicts a precession of 5,557 arcseconds (1.5436°) per century relative to Earth, or per century relative to ICRF. In the early 20th century, [[Albert Einstein]]'s [[general theory of relativity]] provided the explanation for the observed precession, by formalizing gravitation as being mediated by the curvature of spacetime. The effect is small: just per century (or 0.43 arcsecond per year, or 0.1035 arcsecond per orbital period) for Mercury; it therefore requires a little over 12.5 million orbits, or 3 million years, for a full excess turn. Similar, but much smaller, effects exist for other Solar System bodies: 8.6247 arcseconds per century for Venus, 3.8387 for Earth, 1.351 for Mars, and 10.05 for [[1566 Icarus]].
Observation.
[[File:Mercury.jpg|thumb|Image mosaic by "Mariner 10", 1974]]
Mercury's [[apparent magnitude]] is calculated to vary between −2.48 (brighter than [[Sirius]]) around [[superior conjunction]] and +7.25 (below the limit of naked-eye visibility) around [[inferior conjunction]]. The mean apparent magnitude is 0.23 while the standard deviation of 1.78 is the largest of any planet. The mean apparent magnitude at superior conjunction is −1.89 while that at inferior conjunction is +5.93. Observation of Mercury is complicated by its proximity to the Sun, as it is lost in the Sun's glare for much of the time. Mercury can be observed for only a brief period during either morning or evening twilight.
Ground-based telescope observations of Mercury reveal only an illuminated partial disk with limited detail. The [[Hubble Space Telescope]] cannot observe Mercury at all, due to safety procedures that prevent its pointing too close to the Sun. Because the shift of 0.15 revolutions of Earth in a Mercurian year makes up a seven-Mercurian-year cycle (0.15 × 7 ≈ 1.0), in the seventh Mercurian year, Mercury follows almost exactly (earlier by 7 days) the sequence of phenomena it showed seven Mercurian years before.
Like the Moon and Venus, Mercury exhibits [[Lunar phase|phases]] as seen from Earth. It is "new" at [[inferior conjunction]] and "full" at superior conjunction. The planet is rendered invisible from Earth on both of these occasions because of its being obscured by the Sun, except at its new phase during a transit. Mercury is technically brightest as seen from Earth when it is at a full phase. Although Mercury is farthest from Earth when it is full, the greater illuminated area that is visible and the [[opposition effect|opposition brightness surge]] more than compensates for the distance. The opposite is true for Venus, which appears brightest when it is a [[crescent]], because it is much closer to Earth than when [[gibbous]].
[[File:PIA19247-Mercury-NPolarRegion-Messenger20150316.jpg|thumb|left|False-color map showing the maximum temperatures of the north polar region]]
[[File:Mercury Venus Moon over San Jose 08 Jan 2024.png|thumb|right|Mercury (lower left) as seen from [[San Jose, California]] with Venus and the Moon.]]
Mercury is best observed at the first and last quarter, although they are phases of lesser brightness. The first and last quarter phases occur at greatest [[elongation (astronomy)|elongation]] east and west of the Sun, respectively. At both of these times, Mercury's separation from the Sun ranges anywhere from 17.9° at perihelion to 27.8° at aphelion. At greatest "western" elongation, Mercury rises at its earliest before sunrise, and at greatest "eastern" elongation, it sets at its latest after sunset.
[[File:PIA19422-Mercury-CarnegieRupes-MDIS-MLA-20150416.jpg|thumb|right|False-color image of [[Carnegie Rupes]], a tectonic landform—high terrain (red); low (blue).]]
Mercury is more often and easily visible from the [[Southern Hemisphere]] than from the [[Northern Hemisphere|Northern]]. This is because Mercury's maximum western elongation occurs only during early autumn in the Southern Hemisphere, whereas its greatest eastern elongation happens only during late winter in the Southern Hemisphere. In both of these cases, the angle at which the planet's orbit intersects the horizon is maximized, allowing it to rise several hours before sunrise in the former instance and not set until several hours after sundown in the latter from southern mid-latitudes, such as Argentina and South Africa.
An alternate method for viewing Mercury involves observing the planet with a [[telescope]] during daylight hours when conditions are clear, ideally when it is at its greatest elongation. This allows the planet to be found easily, even when using telescopes with apertures. However, great care must be taken to obstruct the Sun from sight because of the extreme risk for eye damage. This method bypasses the limitation of twilight observing when the ecliptic is located at a low elevation (e.g. on autumn evenings). The planet is higher in the sky and less atmospheric effects affect the view of the planet. Mercury can be viewed as close as 4° to the Sun near superior conjunction when it is almost at its brightest.
Mercury can, like several other planets and the brightest stars, be seen during a total [[solar eclipse]].
Observation history.
Ancient astronomers.
[[File:Mercury-bonatti.png|thumb|Mercury, from "Liber astronomiae", 1550]]
The earliest known recorded observations of Mercury are from the [[MUL.APIN]] tablets. These observations were most likely made by an [[Assyria]]n astronomer around the 14th century BC. The [[cuneiform]] name used to designate Mercury on the MUL.APIN tablets is transcribed as UDU.IDIM.GU\U4.UD ("the jumping planet"). [[Babylonian astronomy|Babylonian records]] of Mercury date back to the 1st millennium BC. The [[Babylonia]]ns called the planet [[Nabu]] after the messenger to the gods in [[Babylonian mythology|their mythology]].
The [[Greeks in Egypt|Greco]]-[[Ancient Egypt|Egyptian]] astronomer [[Ptolemy]] wrote about the possibility of planetary transits across the face of the Sun in his work "Planetary Hypotheses". He suggested that no transits had been observed either because planets such as Mercury were too small to see, or because transits were too infrequent.
[[File:Shatir500.jpg|thumb|left|[[Ibn al-Shatir]]'s model for the appearances of Mercury, showing the multiplication of [[epicycles]] using the [[Tusi couple]], thus eliminating the Ptolemaic eccentrics and [[equant]].]]
In [[ancient China]], Mercury was known as "the Hour Star" ("Chen-xing" ). It was associated with the direction north and the phase of water in the [[Five Phases]] system of metaphysics. Modern [[Chinese culture|Chinese]], [[Korean culture|Korean]], [[Japanese culture|Japanese]] and [[Vietnamese culture|Vietnamese]] cultures refer to the planet literally as the "water star" (), based on the [[Five elements (Chinese philosophy)|Five elements]]. [[Hindu mythology]] used the name [[Budha]] for Mercury, and this god was thought to preside over Wednesday. The god [[Odin]] (or Woden) of [[Germanic paganism]] was associated with the planet Mercury and Wednesday. The [[Maya civilization|Maya]] may have represented Mercury as an owl (or possibly four owls; two for the morning aspect and two for the evening) that served as a messenger to the [[underworld]]. Mercury was sometimes known as [[Stilbon (mythology)|Stilbon]] ([[Greek language|Greek]]: Στίλβων) meaning 'the shining, glittering'.
In [[medieval Islamic astronomy]], the [[Al-Andalus|Andalusian]] astronomer [[Abū Ishāq Ibrāhīm al-Zarqālī]] in the 11th century described the [[deferent]] of Mercury's [[geocentric orbit]] as being oval, like an egg or a [[Pine nut|pignon]], although this insight did not influence his astronomical theory or his astronomical calculations. In the 12th century, [[Ibn Bajjah]] observed "two planets as black spots on the face of the Sun", which was later suggested as the transit of Mercury and/or Venus by the [[Maragheh observatory|Maragha]] astronomer [[Qotb al-Din Shirazi]] in the 13th century. Most such medieval reports of transits were later taken as observations of [[sunspot]]s.
In India, the [[Kerala school of astronomy and mathematics|Kerala school]] astronomer [[Nilakantha Somayaji]] in the 15th century developed a partially heliocentric planetary model in which Mercury orbits the Sun, which in turn orbits Earth, similar to the [[Tychonic system]] later proposed by [[Tycho Brahe]] in the late 16th century.
Ground-based telescopic research.
The first telescopic observations of Mercury were made by [[Thomas Harriot]] and [[Galileo]] from 1610. In 1612, [[Simon Marius]] observed the brightness of Mercury varied with the planet's orbital position and concluded it had phases "in the same way as Venus and the Moon". In 1631, [[Pierre Gassendi]] made the first telescopic observations of the transit of a planet across the Sun when he saw a transit of Mercury predicted by [[Johannes Kepler]]. In 1639, [[Giovanni Zupi]] used a telescope to discover that the planet had orbital phases similar to Venus and the Moon. The observation demonstrated conclusively that Mercury orbited the Sun.
A rare event in astronomy is the passage of one planet in front of another ([[occultation]]), as seen from Earth. Mercury and Venus occult each other every few centuries, and the event of May 28, 1737, is the only one historically observed, having been seen by [[John Bevis]] at the [[Royal Greenwich Observatory]]. The next occultation of Mercury by Venus will be on December 3, 2133.
The difficulties inherent in observing Mercury meant that it was far less studied than the other planets. In 1800, [[Johann Schröter]] made observations of surface features, claiming to have observed mountains. [[Friedrich Bessel]] used Schröter's drawings to erroneously estimate the rotation period as 24 hours and an axial tilt of 70°. In the 1880s, [[Giovanni Schiaparelli]] mapped the planet more accurately, and suggested that Mercury's rotational period was 88 days, the same as its orbital period due to tidal locking. This phenomenon is known as [[synchronous rotation]]. The effort to map the surface of Mercury was continued by [[Eugenios Antoniadi]], who published a book in 1934 that included both maps and his own observations. Many of the planet's surface features, particularly the [[List of albedo features on Mercury|albedo features]], take their names from Antoniadi's map.
In June 1962, Soviet scientists at the [[Institute of Radio-engineering and Electronics]] of the [[USSR Academy of Sciences]], led by [[Vladimir Kotelnikov]], became the first to bounce a radar signal off Mercury and receive it, starting radar observations of the planet. Three years later, radar observations by Americans [[Gordon H. Pettengill]] and Rolf B. Dyce, using the [[Arecibo radio telescope]] in [[Puerto Rico]], showed conclusively that the planet's rotational period was about 59 days. The theory that Mercury's rotation was synchronous had become widely held, and it was a surprise to astronomers when these radio observations were announced. If Mercury were tidally locked, its dark face would be extremely cold, but measurements of radio emission revealed that it was much hotter than expected. Astronomers were reluctant to drop the synchronous rotation theory and proposed alternative mechanisms such as powerful heat-distributing winds to explain the observations.
In 1965, Italian astronomer [[Giuseppe Colombo]] noted that the rotation value was about two-thirds of Mercury's orbital period, and proposed that the planet's orbital and rotational periods were locked into a 3:2 rather than a 1:1 resonance. Data from "Mariner 10" subsequently confirmed this view. This means that Schiaparelli's and Antoniadi's maps were not "wrong". Instead, the astronomers saw the same features during every "second" orbit and recorded them, but disregarded those seen in the meantime, when Mercury's other face was toward the Sun, because the orbital geometry meant that these observations were made under poor viewing conditions.
Ground-based optical observations did not shed much further light on Mercury, but radio astronomers using interferometry at microwave wavelengths, a technique that enables removal of the solar radiation, were able to discern physical and chemical characteristics of the subsurface layers to a depth of several meters. Not until the first space probe flew past Mercury did many of its most fundamental morphological properties become known. Moreover, technological advances have led to improved ground-based observations. In 2000, high-resolution [[lucky imaging]] observations were conducted by the [[Mount Wilson Observatory]] Hale telescope. They provided the first views that resolved surface features on the parts of Mercury that were not imaged in the "Mariner 10" mission. Most of the planet has been mapped by the Arecibo radar telescope, with resolution, including polar deposits in shadowed craters of what may be water ice.
Research with space probes.
[[File:MESSENGER Assembly.jpg|thumb|left|"MESSENGER" being prepared for launch]]
[[File:PIA18389-MarsCuriosityRover-MercuryTransitsSun-20140603.gif|thumb|Mercury transiting the [[Sun]] as viewed by the Mars rover [[Curiosity (rover)|"Curiosity"]] (June 3, 2014).]]
Reaching Mercury from Earth poses significant technical challenges, because it orbits so much closer to the Sun than Earth. A Mercury-bound spacecraft launched from Earth must travel over into the Sun's gravitational [[potential well]]. Mercury has an [[orbital speed]] of , whereas Earth's orbital speed is . Therefore, the spacecraft must make a larger change in [[velocity]] ([[delta-v]]) to get to Mercury and then enter orbit, as compared to the delta-v required for, say, [[Exploration of Mars|Mars planetary missions]].
The [[potential energy]] liberated by moving down the Sun's potential well becomes [[kinetic energy]], requiring a delta-v change to do anything other than pass by Mercury. Some portion of this [[delta-v budget]] can be provided from a [[gravity assist]] during one or more fly-bys of Venus. To land safely or enter a stable orbit the spacecraft would rely entirely on rocket motors. [[Aerobraking]] is ruled out because Mercury has a negligible atmosphere. A trip to Mercury requires more rocket fuel than that required to [[escape velocity|escape]] the Solar System completely. As a result, only three space probes have visited it so far. A proposed alternative approach would use a [[solar sail]] to attain a Mercury-synchronous orbit around the Sun.
"Mariner 10".
[[File:Mariner 10.jpg|thumb|"Mariner 10", the first probe to visit Mercury]]
The first spacecraft to visit Mercury was NASA's "Mariner 10" (1974–1975). The spacecraft used the gravity of Venus to adjust its orbital velocity so that it could approach Mercury, making it both the first spacecraft to use this gravitational "slingshot" effect and the first NASA mission to visit multiple planets. "Mariner 10" provided the first close-up images of Mercury's surface, which immediately showed its heavily cratered nature, and revealed many other types of geological features, such as the giant scarps that were later ascribed to the effect of the planet shrinking slightly as its iron core cools. Unfortunately, the same face of the planet was lit at each of "Mariner 10"'s close approaches. This made close observation of both sides of the planet impossible, and resulted in the mapping of less than 45% of the planet's surface.
The spacecraft made three close approaches to Mercury, the closest of which took it to within of the surface. At the first close approach, instruments detected a magnetic field, to the great surprise of planetary geologists—Mercury's rotation was expected to be much too slow to generate a significant [[dynamo]] effect. The second close approach was primarily used for imaging, but at the third approach, extensive magnetic data were obtained. The data revealed that the planet's magnetic field is much like Earth's, which deflects the solar wind around the planet. For many years after the "Mariner 10" encounters, the origin of Mercury's magnetic field remained the subject of several competing theories.
On March 24, 1975, just eight days after its final close approach, "Mariner 10" ran out of fuel. Because its orbit could no longer be accurately controlled, mission controllers instructed the probe to shut down. "Mariner 10" is thought to be still orbiting the Sun, passing close to Mercury every few months.
"MESSENGER".
[[File:Details of MESSENGER's Impact Location.jpg|thumb|Estimated details of the impact of "MESSENGER" on April 30, 2015]]
A second NASA mission to Mercury, named "MESSENGER" (MErcury Surface, Space ENvironment, GEochemistry, and Ranging), was launched on August 3, 2004. It made a fly-by of Earth in August 2005, and of Venus in October 2006 and June 2007 to place it onto the correct trajectory to reach an orbit around Mercury. A first fly-by of Mercury occurred on January 14, 2008, a second on October 6, 2008, and a third on September 29, 2009. Most of the hemisphere not imaged by "Mariner 10" was mapped during these fly-bys. The probe successfully entered an elliptical orbit around the planet on March 18, 2011. The first orbital image of Mercury was obtained on March 29, 2011. The probe finished a one-year mapping mission, and then entered a one-year extended mission into 2013. In addition to continued observations and mapping of Mercury, "MESSENGER" observed the 2012 [[solar maximum]].
[[File:MercuryTopo.png|thumb|right|Topography of Mercury based on MDIS (Mercury Dual Imaging System) data]] The mission was designed to clear up six key issues: Mercury's high density, its geological history, the nature of its magnetic field, the structure of its core, whether it has ice at its poles, and where its tenuous atmosphere comes from. To this end, the probe carried imaging devices that gathered much-higher-resolution images of much more of Mercury than "Mariner 10", assorted [[spectrometer]]s to determine the abundances of elements in the crust, and [[magnetometer]]s and devices to measure velocities of charged particles. Measurements of changes in the probe's orbital velocity were expected to be used to infer details of the planet's interior structure. "MESSENGER"'s final maneuver was on April 24, 2015, and it crashed into Mercury's surface on April 30, 2015. The spacecraft's impact with Mercury occurred at 3:26:01 p.m. [[Eastern Time Zone|EDT]] on April 30, 2015, leaving a crater estimated to be in diameter.
"BepiColombo".
The [[European Space Agency]] and the [[Japanese Space Agency]] developed and launched a joint mission called "BepiColombo", which will orbit Mercury with two probes: one to map the planet and the other to study its magnetosphere. Launched on October 20, 2018, "BepiColombo" is expected to reach Mercury in 2025. It will release a magnetometer probe into an elliptical orbit, then chemical rockets will fire to deposit the mapper probe into a circular orbit. Both probes will operate for one terrestrial year. The mapper probe carries an array of spectrometers similar to those on "MESSENGER", and will study the planet at many different wavelengths including [[infrared]], [[ultraviolet]], [[X-ray]] and [[gamma ray]]. "BepiColombo" conducted the first of its six planned Mercury flybys on October 1, 2021, and the sixth was completed on January 9, 2025. The spacecraft will enter the planet's orbit in 2026.
"Perseverance" rover.
On March 5, 2024, NASA released images of transits of the moon [[Deimos (moon)|Deimos]], the moon [[Phobos (moon)|Phobos]] and the planet Mercury as viewed by the "Perseverance" rover on the planet Mars.
See also.
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Notes.
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References.
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External links.
[[Category:Mercury (planet)| ]]
[[Category:Planets of the Solar System]]
[[Category:Terrestrial planets]]
[[Category:Astronomical objects known since antiquity]]
[[Category:Solar System]]