TITLE: How to show that planetary orbits are conic sections
QUESTION [6 upvotes]: It's all very fun to use differential equations to show that the path of a particle follows a parabola under constant acceleration, a circle when the angular momentum is just enough to allow for zero radial acceleration and an ellipse when it is in a general orbit. (still haven't done the hyperbola).
But a question that has always bugged me is why do these objects all follow conic sections under gravity? Intuitively, it seems like the inverse square law in polar coordinates is responsible. But an object accelerating under constant force still takes on a parabola. Is this related to the fact that angular momentum is constant under gravity due to zero torque? (Kepler's second law)

REPLY [2 votes]: The planetary orbits are conics section in the approximation of a central attractive gravitational force from a fixed Sun. 
The fact that in this case the orbits are conic sections depends not only by the fact that the force depends only on $r$ as $F=kr^n$ (central force ) but also from the fact that $ n=-2$ for the gravity. 
The nature of the conic depends from the total specific energy $E_T$: we have an ellipse if the $E_T<0$ ( the system is closed), a parabola if  $E_T=0$ (an open critical system) or an hyperbola if $E_T>0$ (open system).
There are plenty of resources in Internet about this topic (as  suggested by Mark McClure or as here).
The case of a falling body in a constant force field is different from these situations because here we does not have a central force, and the fact that the solution is a parabola has the simple motivation that the equations of motion are second ordere differential equations and a function that has a constant second derivative is a second degree polynimial.