text: Algorithm 1 Preprocessing of the EMAGS IV. ALGORITHM OVERVIEW Input: EMAGS The present algorithm automatically generates object la- Output: Border mask and list of initialization points bels in the EMAGS to enable their use as ground truth Spatial and temporal smoothing or comparison data. Every object is traced through the for each time step do considered sequence using the best ﬁtting of length and Extract edges width, which is obtained over multiple time steps. The result Generate possible object border lines is an automatically labeled EMAGS, where ideally every Calculate center points occurring object has its correct dimension and position in end for every time step, even if the true dimensions are only observed return border mask and initialization points in few time steps. As the algorithm consists of multiple complex steps, this section gives an overview over the whole procedure, while the individual parts are explained separately in Sec. IV. The pseudocode in Algorithm 2 introduces the Algorithm 2 Overview Input: EMAGS Output: labeled EMAGS preprocess EMAGS to calculate initialization points and border mask while get initialization point do Object initialization: connected component, polygon, velocity proﬁle Start temporal search for forward step, backward step do while in sequence and object plausible do Object silhouette prediction Get connected component search starting points Extract connected component: ﬁrst blob Blob reduction via outlier removal Calculate velocity proﬁle Object plausibility check Fig. 3. One time step of the preprocessed data is shown, referring to one Construct blob polygon and get reference point slice from the EMAGS. The occupancy information is used as background, Update object width and length estimation where black means occupied and white means not occupied. The border mask is shown in blue lines and the calculated initialization points are Construct object polygon plotted in green. A zoomed excerpt is integrated in the bottom left, including end while two possible objects. One is marked with a single initialization point, the Start backward step with best object estimates from other with multiple. A third occupied area is seen, but it does not move over time, so it is not marked as possible object. forward step end for Delete initialization points covered by extracted object Object and trajectory consistency validation of these points as initialization points for the extraction Orientation correction for standing objects algorithm explained in the following sections. One slice, i.e. (optional) Temporal trajectory smoothing time step, of the preprocessing result is shown in Fig. 3 Write object to result including a zoomed view showing initialization points in end while detail. The border mask is plotted in blue, where each marked return labeled EMAGS point is part of the border of a possible object. The green points are the initialization points marking an inner point of a possible object. It is possible for an object to have idea of the main processing steps. Fig. 4 shows the main steps multiple or no initialization points in a speciﬁc time step, in detail in four rows of example pictures. The ﬁrst two rows as the preprocessing is a coarse ﬁrst evaluation. However, illustrate the forward pass, while backward processing is every object that has a clear appearance at least once in the depicted in the two bottom rows. The ﬁrst row shows in green sequence will be marked with an initialization point in that the predicted visible silhouette of the last object extraction time step. This data is the output of preprocessing and will be drawn over a grayscale DOGMa, where dark pixels refer to used in the main algorithm to extract actual objects with their high P . A red cross illustrates cells within the predicted O correct shapes. At this point, there is no temporal connection silhouette that ﬁt best to the expected object velocity, P , O established between the initialization points, as it is not clear and blob center. These cells are used to start the connected if every initialization point marks an actual object. component (blob) extraction. Blue pixels refer to the current ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Algorithm

text: Preprocessing

text: of

text: the

text: EMAGS

text: IV.

text: ALGORITHM

text: OVERVIEW

text: Input:

text: EMAGS

text: The

text: present

text: algorithm

text: automatically

text: generates

text: object

text: la-

text: Output:

text: Border

text: mask

text: and

text: list

text: of

text: initialization

text: points

text: bels

text: in

text: the

text: EMAGS

text: to

text: enable

text: their

text: use

text: as

text: ground

text: truth

text: Spatial

text: and

text: temporal

text: smoothing

text: or

text: comparison

text: data.

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text: object

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text: edges

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text: Sec.

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text: − see (3.6). By k we denote the j-th column of the matrix [x (f )]N 1 . Applying Hadamard’s (cid:18) j (cid:19) j (cid:96) j,(cid:96)=0 inequality and (cid:89)then arithmetic-g(cid:0)eometric me(cid:1)an inequality we obtain − N 1 K 2 ≤ 2 σj( )X (cid:18)d(cid:89)et[xj(f(cid:96))](cid:19)(cid:18) (cid:89) (cid:19)(cid:18) (cid:89) (cid:89) (cid:19) j=0 − − − ≤ h1 1 (cid:107) (cid:107) N 1 (cid:107) (cid:107) L hmi 1 (cid:107) (cid:107) (cid:18) (cid:88)kj 2e (cid:19) (cid:18) kj 2e (cid:88) (cid:19)kj 2e j=0 j=hmL i=2 j=hmi−1 − − − (3.9) ≤ 1 h1 1 (cid:107) (cid:107) h1 1 N 1 (cid:107) (cid:107) N hmL h (cid:89) (cid:18) kj 2e N −(cid:88)h (cid:19) kj 2e 1 j=0 mL j=hmL × L 1 hmi−1 (cid:107) (cid:107) hmi−hmi−1 − k 2 . i=2 hmi hmi−1 j=hmi−1 j e From (3.(cid:88)5), (3.7), and(cid:88)(3.8(cid:88)) we have (cid:88) − − − − h1 1 (cid:107) (cid:107) h1 1 N 1 | | ≤ N 1 (cid:107) (cid:107) ≤ K ≤ k 2 = x (f ) 2 f 2 NA2d ( )2 NA2(C + ε)2 j e j (cid:96) (cid:96) e 0 X 0 j=0 j=0 (cid:96)=0 (cid:96)=0 K ≤ (since d ( )(cid:88) C , by our a(cid:88)ssump(cid:88)tion), and for i = 2, . . . , L, 0 X 0 − − − hmi 1 (cid:107) (cid:107) hmi 1 N 1 | | k 2 = x (f ) 2 j e (cid:88) j (cid:96) j=hmi−1 j=hmi−1 (cid:96)=0 − N 1 ≤ ≤ − − dist(f ; Q(V − ))2(cid:107)·(cid:107) N(C + ε)2A22 2s(mi 2) . (cid:96) mi 1 e 0 (cid:96)=0 Similarly, we h(cid:88)ave (cid:88) − − N 1 N 1 (cid:107) (cid:107) ≤ ≤ − − k 2 dist(f ; Q(V ))2(cid:107)·(cid:107) N(C + ε)2A22 2s(n 1) . j e (cid:96) n 0 e j=hmL (cid:96)=0 (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) Insert(cid:89)ing this into (3.9) we ﬁnd − − − − N 1 σ (K) 2 ≤ NA2(C0(cid:18)+ ε)2 h1 N(C0 + ε)−2A22(cid:19) 2s(n 1) N hmL j X (cid:89) h N h j=0 1 mL − − − × L N(C0(cid:18)+ ε)−2A22 2s(mi 2) hmi hmi−1 (cid:19) (3.10) (cid:89) i=2 hmi hmi−1 − − − L − − − = MA2N(C0 + ε)2N 2 2s(n 1)(N hmL) 2 2s(mi 2)(hmi hmi−1) , i=2 (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) where we put (cid:89) − − N h1 N N hmL L N hmi hmi−1 M = − − . h1 N hmL i=2 hmi hmi−1 6 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: 4 One can also notice an increasing of the energy when → → 1 n n, i.e., n 0, a regime in which the incom- 1,0 2,0 pressibility condition is no longer fulﬁlled. In this case the system can not be described as a magnetic vortex, but as a single (or a few-particle) state bound in the core of the quantum vortex. xx Single impurity trapped in the core of a vortex 0.5 In the limit of extreme diluteness of the second com- ponent of the mixture, the system must be described as a single particle in an eﬀective potential given by the inter- action with the density of the majority component which hosts the defect. Therefore from Eq. (8) one can write 0 a Schr¨odinger equation for the wave function Ψ of the -10 -5 0 5 10 2 ξ impurity in the form: x / s (cid:126) FIG. 1: The dot-dashed red (dot-dot-dashed blue) line re- − 2 ∇ | | 2Ψ + g Ψ 2Ψ = µ Ψ . (12) spresents the density (normalized to the total background 2m 2 12 1 2 2 2 density) along the x-axis of component 1 (2) in a magnetic √ vortex. The solid (dashed) black line shows the total den- For the order parameter Ψ (r, θ) = n f (r) exp(iθ) 1 1,0 1 sity along the x-axis of the magnetic vortex (ground state). we assume the known solution for a single component a = a = 54.54 a and a = 50.78 a . The total number 11 22 B 12 B vortex with healing length equal to ξ , since in this limit of particles is N = 105 and the mass is that of 23Na. The d Ψ is not aﬀected by the interaction with the impurity. insets pictorically displays the case where the density in the 1 In this conﬁguration, the impurity sees the vortex core second component is small enough to avoid fulﬁlling the in- compressibility condition. In this case, one obtains a bound as a trapping potential as shown in the inset of Fig. 1. state in the core of the vortex. When we add more atoms of the minority compo- nent the width of Ψ is enlarged due to the repulsive 2 (cid:40) intraspecies interaction and the width of the vortex is en- (cid:113) √ larged. An important question is what will be the fate of ≤ the ﬁlling of the vortex core when the numbers of atoms f1(r) = 12 ξr1s − rξs22 iiff rr > √22 ξξss . (10) oWstfaettehweeilvlmoslihvnoeowsriitnnytuomcotemhreipcoamnllaeygnnitnebtSiececcovtmoioretnsexltadhriagstecrutshaseneddlobclaaerlfigosereerd.. We also derive a simple model to estimate the threshold for the vortex proﬁle which captures the main physics between the two regimes in terms of the atoms of the of a single-component vortex line, [? ] with a rescaled minority component. Indeed, as long as the number of healing length ξs. By using the Ansatz (10) the magnetic atoms in component 2 is small, they can be hosted in the vortex energy can (cid:90)be written as vortical region, while after a certain critical number they will diﬀuse outside the vortex core, constituting, at large (cid:126) ∇ distances, a uniform gas with density n . (cid:18)(cid:18)2n n((cid:19)f )(cid:18)2 (cid:19) 2,0 E =E + 1,0 − 1 dr = E + IV. MAGNETIC VORTICES IN A 2D MV π(cid:126)1V2n (cid:16) 2m− nn(cid:17)(cid:19)n1,0f122n − n 1V − HARMONIC TRAP 1,0 1 ln − 1,0 m n 2n 2n The calculation of magnetic vortices in the presence 1,0 1,0 of harmonic trapping is motivated by several reasons. n − n ln 1 1,0 , (11) Experimentally, harmonically trapped gases are in fact n 2n 1,0 well suited to produce vortical conﬁgurations and, conse- quently, their study represents a topic of primary interest. (cid:126) where E1V = π 2n1,0/m ln (1.46 R/ξs) is the energy Moreover, the presence of harmonic trapping is particu- of a single component vortex [27] with a rescaled heal- larly useful to investigate the buoyancy eﬀect in the case ing length ξs. From Eq. (11), one can cheack that for of unequal intra-species interactions, as we will discuss in δg small enough the magnetic vortex has a smaller en- Appendix . In the following, we will consider Bose gases ergy→cost than the single component one. In the limit hosted by an axially symmetric harmonic trap n n, the correction vanishes and the magnetic vor- 2,0 1 tex has a lower energy for any value δg < g. We will later V (r, z) = m(ω2r2 + ω2z2) , ext 2 z show that this conclusion is veriﬁed numerically also in (cid:29) presence of a trap and without imposing explicitly the with r2 = x2 + y2. 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text: a

text: magnetic

text: vortex,

text: but

text: as

text: a

text: single

text: (or

text: a

text: few-particle)

text: state

text: bound

text: in

text: the

text: core

text: of

text: the

text: quantum

text: vortex.

text: xx

text: Single

label: section

text: impurity

text: trapped

text: in

text: the

text: core

text: of

text: a

text: vortex

text: In

text: the

text: limit

text: of

text: extreme

text: diluteness

text: of

text: the

text: second

text: com-

text: ponent

text: of

text: the

text: mixture,

text: the

text: system

text: must

text: be

text: described

text: as

text: a

text: single

text: particle

text: in

text: an

text: eﬀective

text: potential

text: given

text: by

text: the

text: inter-

text: action

text: with

text: the

text: density

text: of

text: the

text: majority

text: component

text: which

text: hosts

text: the

text: defect.

text: Therefore

text: from

text: Eq.

text: one

text: can

text: write

text: a

text: Schr¨odinger

text: equation

text: for

text: the

text: wave

text: function

text: Ψ

text: of

text: the

text: ξ

text: impurity

text: in

text: the

text: form:

text: x

text: s

text: FIG.

text: The

text: dot-dashed

text: red

text: (dot-dot-dashed

text: blue)

text: line

text: re-

text: 2Ψ

text: g

text: Ψ

text: 2Ψ

text: µ

text: Ψ

text: spresents

text: the

text: density

text: (normalized

text: to

text: the

text: total

text: background

text: density)

text: along

text: the

text: x-axis

text: of

text: component

text: in

text: a

text: magnetic

text: vortex.

text: The

text: solid

text: (dashed)

text: black

text: line

text: shows

text: the

text: total

text: den-

text: For

text: the

text: order

text: parameter

text: Ψ

text: (r,

text: θ)

text: n

text: f

text: (r)

text: exp(iθ)

text: sity

text: along

text: the

text: x-axis

text: of

text: the

text: magnetic

text: vortex

text: (ground

text: state).

text: we

text: assume

text: the

text: known

text: solution

text: for

text: a

text: single

text: component

text: a

text: a

text: a

text: and

text: a

text: a

text: The

text: total

text: number

text: B

text: B

text: vortex

text: with

text: healing

text: length

text: equal

text: to

text: ξ

text: since

text: in

text: this

text: limit

text: of

text: particles

text: is

text: N

text: and

text: the

text: mass

text: is

text: that

text: of

text: The

text: d

text: Ψ

text: is

text: not

text: aﬀected

text: by

text: the

text: interaction

text: with

text: the

text: impurity.

text: insets

text: pictorically

text: displays

text: the

text: case

text: where

text: the

text: density

text: in

text: the

text: In

text: this

text: conﬁguration,

text: the

text: impurity

text: sees

text: the

text: vortex

text: core

text: second

text: component

text: is

text: small

text: enough

text: to

text: avoid

text: fulﬁlling

text: the

text: in-

text: compressibility

text: condition.

text: In

text: this

text: case,

text: one

text: obtains

text: a

text: bound

text: as

text: a

text: trapping

text: potential

text: as

text: shown

text: in

text: the

text: inset

text: of

text: Fig.

text: state

text: in

text: the

text: core

text: of

text: the

text: vortex.

text: When

text: we

text: add

text: more

text: atoms

text: of

text: the

text: minority

text: compo-

text: nent

text: the

text: width

text: of

text: Ψ

text: is

text: enlarged

text: due

text: to

text: the

text: repulsive

text: intraspecies

text: interaction

text: and

text: the

text: width

text: of

text: the

text: vortex

text: is

text: en-

text: larged.

text: An

text: important

text: question

text: is

text: what

text: will

text: be

text: the

text: fate

text: of

text: the

text: ﬁlling

text: of

text: the

text: vortex

text: core

text: when

text: the

text: numbers

text: of

text: atoms

text: iiff

text: rr

text: ξξss

text: We

text: also

text: derive

text: a

text: simple

text: model

text: to

text: estimate

text: the

text: threshold

text: for

text: the

text: vortex

text: proﬁle

text: which

text: captures

text: the

text: main

text: physics

text: between

text: the

text: two

text: regimes

text: in

text: terms

text: of

text: the

text: atoms

text: of

text: the

text: of

text: a

text: single-component

text: vortex

text: line,

text: with

text: a

text: rescaled

text: minority

text: component.

text: Indeed,

text: as

text: long

text: as

text: the

text: number

text: of

text: healing

text: length

text: ξs.

text: By

text: using

text: the

text: Ansatz

text: the

text: magnetic

text: atoms

text: in

text: component

text: is

text: small,

text: they

text: can

text: be

text: hosted

text: in

text: the

text: vortex

text: energy

text: can

text: written

text: as

text: vortical

text: region,

text: while

text: after

text: a

text: certain

text: critical

text: number

text: they

text: will

text: diﬀuse

text: outside

text: the

text: vortex

text: core,

text: constituting,

text: at

text: large

text: distances,

text: a

text: uniform

text: gas

text: with

text: density

text: n

text: E

text: =E

text: dr

text: E

text: IV.

text: MAGNETIC

text: VORTICES

text: IN

text: A

text: MV

text: n

text: HARMONIC

text: TRAP

text: ln

text: m

text: n

text: The

text: calculation

text: of

text: magnetic

text: vortices

text: in

text: the

text: presence

text: of

text: harmonic

text: trapping

text: is

text: motivated

text: by

text: several

text: reasons.

text: n

text: n

text: ln

text: Experimentally,

text: harmonically

text: trapped

text: gases

text: are

text: in

text: fact

text: n

text: well

text: suited

text: to

text: produce

text: vortical

text: conﬁgurations

text: and,

text: conse-

text: quently,

text: their

text: study

text: represents

text: a

text: topic

text: of

text: primary

text: interest.

text: where

text: π

text: ln

text: R/ξs)

text: is

text: the

text: energy

text: Moreover,

text: the

text: presence

text: of

text: harmonic

text: trapping

text: is

text: particu-

text: of

text: a

text: single

text: component

text: vortex

text: with

text: a

text: rescaled

text: heal-

text: larly

text: useful

text: to

text: investigate

text: the

text: buoyancy

text: eﬀect

text: in

text: the

text: case

text: ing

text: length

text: ξs.

text: From

text: Eq.

text: one

text: can

text: cheack

text: that

text: for

text: of

text: unequal

text: intra-species

text: interactions,

text: as

text: we

text: will

text: discuss

text: in

text: δg

text: small

text: enough

text: the

text: magnetic

text: vortex

text: has

text: a

text: smaller

text: en-

text: Appendix

text: In

text: the

text: following,

text: we

text: will

text: consider

text: Bose

text: gases

text: ergy→cost

text: than

text: the

text: single

text: component

text: one.

text: In

text: the

text: limit

text: hosted

text: by

text: an

text: axially

text: symmetric

text: harmonic

text: trap

text: n

text: n,

text: the

text: correction

text: vanishes

text: and

text: the

text: magnetic

text: vor-

text: tex

text: has

text: a

text: lower

text: energy

text: for

text: any

text: value

text: δg

text: g.

text: We

text: will

text: later

text: V

text: (r,

text: z)

text: ext

text: z

text: show

text: that

text: this

text: conclusion

text: is

text: veriﬁed

text: numerically

text: also

text: in

text: presence

text: of

text: a

text: trap

text: and

text: without

text: imposing

text: explicitly

text: the

text: with

text: We

text: also

text: assume

text: ω

text: ω,

text: in

text: such

text: a

text: z

text: incompressibility

text: condition.

text: way

text: that

text: the

text: z

text: degree

text: of

text: freedom

text: is

text: frozen

text: in

text: the

text: ground

text: n

text: )x(

text: n

text: ))xx((

text: nn

text: ##LTFigure##

text: ##LTLine##

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text: mercury is forced into the samples using increasing external pressure. The volume of the injected mercury as the function of the pressure is the so-called saturation curve. The modeling of mercury porosimetry was ﬁrst studied by Chatzis and Androutsopoulos [16–18]. An external pressure driven access-limited invasion percolation model called porcolation was introduced in [19]. The porcolation method can be used for real networks provided that the statistical information required for the network generation is available. According to recent studies [20], the modeling of granular porous media can be eﬀectively and accurately done based on processing 2D/3D images. The main objective of the current work is to study saturation properties of both regular and irregular networks. We carried out porcolation simulations on square/cubic networks, networks based on exotic graphs like the Sierpin´ski triangle and carpet, and also on localized and completely random networks. Saturation curves were determined with OpenPNM. This paper is structured as follows: in Section the theoretical background of diﬀerent percolation models is presented. Section 3 presents saturation curves of porcolation simulations on square and cubic networks. In Section saturation curves for Sierpin´ski triangle and Sierpin´ski carpet networks are shown. In Section the locality properties of irregular 3D networks were investigated with a newly developed graph generation model and saturation curves were also obtained for networks with diﬀerent pore degree distribution. Section concludes the paper. 2 Percolation models Percolation theory was introduced in 1957 by Broadbent and Hammersley [21]. They investigated how the random properties of a medium inﬂuence the percolation of a ﬂuid through it. In the following four percolation models are presented. 2.1 Ordinary Percolation There are two fundamentally diﬀerent types of the ordinary percolation model: bond-percolation and site-percolation [22] (Figure 1). (a) (b) Figure 1: (a) Site-percolation, (b) Bond-percolation on square lattice. In site-percolation (Figure 1(a)) each lattice site is occupied with some probability P. Occupied sites having one common side are called neighbors [23], while a group of neighboring occupied sites is called a cluster. Clusters have a crucial role in percolation theory, since the existence of spanning cluster (a cluster that connects opposite boundaries) means that the invading ﬂuid (the ﬂuid that enters the medium under pressure) can percolate through the medium. If the occupation probability P is small, there is only a slight chance of having a spanning cluster. On the other hand, if P is nearly 1, there will almost certainly be a spanning cluster. The critical value of occupation probability (P ), at which an inﬁnite cluster appears crit in an inﬁnite lattice is 0.593 [24] and 0.312 [25] for two dimensional square and three dimensional cubic networks, respectively. In bond-percolation (Figure 1(b)) it is not the lattice sites, but the connecting bonds that are occupied with probability P. Site- and bond-percolation yields diﬀerent critical probabilities for the same lattice, but the values can be calculated from each other [26,27]. 2 ##LTFigure## ##LTFigure##

text: mercury

text: is

text: forced

text: into

text: the

text: samples

text: using

text: increasing

text: external

text: pressure.

text: The

text: volume

text: of

text: the

text: injected

text: mercury

text: as

text: the

text: function

text: of

text: the

text: pressure

text: is

text: the

text: so-called

text: saturation

text: curve.

text: The

text: modeling

text: of

text: mercury

text: porosimetry

text: was

text: ﬁrst

text: studied

text: by

text: Chatzis

text: and

text: Androutsopoulos

text: An

text: external

text: pressure

text: driven

text: access-limited

text: invasion

text: percolation

text: model

text: called

text: porcolation

text: was

text: introduced

text: in

text: The

text: porcolation

text: method

text: can

text: be

text: used

text: for

text: real

text: networks

text: provided

text: that

text: the

text: statistical

text: information

text: required

text: for

text: the

text: network

text: generation

text: is

text: available.

text: According

text: to

text: recent

text: studies

text: the

text: modeling

text: of

text: granular

text: porous

text: media

text: can

text: be

text: eﬀectively

text: and

text: accurately

text: done

text: based

text: on

text: processing

text: images.

text: The

text: main

text: objective

text: of

text: the

text: current

text: work

text: is

text: to

text: study

text: saturation

text: properties

text: of

text: both

text: regular

text: and

text: irregular

text: networks.

text: We

text: carried

text: out

text: porcolation

text: simulations

text: on

text: square/cubic

text: networks,

text: networks

text: based

text: on

text: exotic

text: graphs

text: like

text: the

text: Sierpin´ski

text: triangle

text: and

text: carpet,

text: and

text: also

text: on

text: localized

text: and

text: completely

text: random

text: networks.

text: Saturation

text: curves

text: were

text: determined

text: with

text: OpenPNM.

text: This

text: paper

text: is

text: structured

text: as

text: follows:

text: in

text: Section

text: the

text: theoretical

text: background

text: of

text: diﬀerent

text: percolation

text: models

text: is

text: presented.

text: Section

text: presents

text: saturation

text: curves

text: of

text: porcolation

text: simulations

text: on

text: square

text: and

text: cubic

text: networks.

text: In

text: Section

text: saturation

text: curves

text: for

text: Sierpin´ski

text: triangle

text: and

text: Sierpin´ski

text: carpet

text: networks

text: are

text: shown.

text: In

text: Section

text: the

text: locality

text: properties

text: of

text: irregular

text: networks

text: were

text: investigated

text: with

text: a

text: newly

text: developed

text: graph

text: generation

text: model

text: and

text: saturation

text: curves

text: were

text: also

text: obtained

text: for

text: networks

text: with

text: diﬀerent

text: pore

text: degree

text: distribution.

text: Section

text: concludes

text: the

text: paper.

text: Percolation

text: models

text: Percolation

text: theory

text: was

text: introduced

text: in

text: by

text: Broadbent

text: and

text: Hammersley

text: They

text: investigated

text: how

text: the

text: random

text: properties

text: of

text: a

text: medium

text: inﬂuence

text: the

text: percolation

text: of

text: a

text: ﬂuid

text: through

text: it.

text: In

text: the

text: following

text: four

text: percolation

text: models

text: are

text: presented.

text: Ordinary

text: Percolation

text: There

text: are

text: two

text: fundamentally

text: diﬀerent

text: types

text: of

text: the

text: ordinary

text: percolation

text: model:

text: bond-percolation

text: and

text: site-percolation

text: (Figure

text: (a)

text: (b)

text: Figure

text: (a)

text: Site-percolation,

text: (b)

text: Bond-percolation

text: on

text: square

text: lattice.

text: In

text: site-percolation

text: (Figure

text: each

text: lattice

text: site

text: is

text: occupied

text: with

text: some

text: probability

text: P.

text: Occupied

text: sites

text: having

text: one

text: common

text: side

text: are

text: called

text: neighbors

text: while

text: a

text: group

text: of

text: neighboring

text: occupied

text: sites

text: is

text: called

text: a

text: cluster.

text: Clusters

text: have

text: a

text: crucial

text: role

text: in

text: percolation

text: theory,

text: since

text: the

text: existence

text: of

text: spanning

text: cluster

text: (a

text: cluster

text: that

text: connects

text: opposite

text: boundaries)

text: means

text: that

text: the

text: invading

text: ﬂuid

text: (the

text: ﬂuid

text: that

text: enters

text: the

text: medium

text: under

text: pressure)

text: can

text: percolate

text: through

text: the

text: medium.

text: If

text: the

text: occupation

text: probability

text: P

text: is

text: small,

text: there

text: is

text: only

text: a

text: slight

text: chance

text: of

text: having

text: a

text: spanning

text: cluster.

text: On

text: the

text: other

text: hand,

text: if

text: P

text: is

text: nearly

text: there

text: will

text: almost

text: certainly

text: be

text: a

text: spanning

text: cluster.

text: The

text: critical

text: value

text: of

text: occupation

text: probability

text: (P

text: at

text: which

text: an

text: inﬁnite

text: cluster

text: appears

text: crit

text: in

text: an

text: inﬁnite

text: lattice

text: is

text: and

text: for

text: two

text: dimensional

text: square

text: and

text: three

text: dimensional

text: cubic

text: networks,

text: respectively.

text: In

text: bond-percolation

text: (Figure

text: it

text: is

text: not

text: the

text: lattice

text: sites,

text: but

text: the

text: connecting

text: bonds

text: that

text: are

text: occupied

text: with

text: probability

text: P.

text: Site-

text: and

text: bond-percolation

text: yields

text: diﬀerent

text: critical

text: probabilities

text: for

text: the

text: same

text: lattice,

text: but

text: the

text: values

text: can

text: be

text: calculated

text: from

text: each

text: other

text: ##LTFigure##

text: ##LTFigure##


text: d 0 0.25 d 1 d 2 d 3 0.20 0.15 0.10 0.05 0.00 2 3 4 5 6 7 8 9 10 11 Pore degree Figure 14: The discrete pore degree distributions 0.6 d 0 d 1 0.5 d 2 d 3 0.4 0.3 0.2 0.1 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Pressure p Figure 15: Saturation for diﬀerent pore degree distributions 11 FDP S noitarutaS ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: d

text: d

text: d

text: d

text: Pore

text: degree

text: Figure

text: The

label: caption

text: discrete

text: pore

text: degree

text: distributions

text: d

text: d

text: d

text: d

text: Pressure

text: p

text: Figure

text: Saturation

text: for

text: diﬀerent

text: pore

text: degree

text: distributions

text: FDP

text: S

text: noitarutaS

text: ##LTFigure##

text: ##LTFigure##

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text: 11 Janka, H.-T. 2012, Ann. Rev. Nucl. Part. Sc., 62, 407 Reaclib. 2013, Database of Joint Institute of Nuclear K¨appeler, F., Gallino, R., Bisterzo, S., & Aoki, W. 2011, Rev. Astrophysics, http://www.jinaweb.org Mod. Phys., 83, 157 Smartt, S. J., Chen, T.-W., Jerkstrand, A., Coughlin, M., Kasen, D., Metzger, B., Barnes, J., Quataert, E., & Kankare, E., Sim, S. A., Fraser, M., Inserra, C., Maguire, K., Ramirez-Ruiz, E. 2017, Nature, 551, 80 Chambers, K. C., Huber, M. E., Kru¨hler, T., Leloudas, G., Korobkin, O., Rosswog, S., Arcones, A., & Winteler, C. 2012, Magee, M., Shingles, L. J., Smith, K. W., Young, D. R., Tonry, MNRAS, 426, 1940 J., Kotak, R., Gal-Yam, A., Lyman, J. D., Homan, D. S., Lodders, K. 2003, ApJ, 591, 1220 Agliozzo, C., Anderson, J. P., Angus, C. R., Ashall, C., Lugaro, M., Davis, A. M., Gallino, R., Pellin, M. J., Straniero, Barbarino, C., Bauer, F. E., Berton, M., Botticella, M. T., O., & K¨appeler, F. 2003, ApJ, 593, 486 Bulla, M., Bulger, J., Cannizzaro, G., Cano, Z., Cartier, R., Meyer, B. S. 1994, ARA&A, 32, 153 Cikota, A., Clark, P., De Cia, A., Della Valle, M., Denneau, L., Meyer, B. S. & Brown, J. S. 1997, ApJS, 112, 199 Dennefeld, M., Dessart, L., Dimitriadis, G., Elias-Rosa, N., Meyer, B. S., Clayton, D. D., & The, L.-S. 2000, ApJ, 540, L49 Firth, R. E., Flewelling, H., Flo¨rs, A., Franckowiak, A., Nishimura, N., Takiwaki, T., & Thielemann, F.-K. 2015, ApJ, Frohmaier, C., Galbany, L., Gonza´lez-Gait´an, S., Greiner, J., 810, 109 Gromadzki, M., Guelbenzu, A. N., Guti´errez, C. P., NuDat2. 2013, National Nuclear Data Center, information Hamanowicz, A., Hanlon, L., Harmanen, J., Heintz, K. E., extracted from the NuDat 2 database, Heinze, A., Hernandez, M.-S., Hodgkin, S. T., Hook, I. M., http://www.nndc.bnl.gov/nudat2/ Izzo, L., James, P. A., Jonker, P. G., Kerzendorf, W. E., Klose, Otsuki, K., Tagoshi, H., Kajino, T., & Wanajo, S. 2000, ApJ, S., Kostrzewa-Rutkowska, Z., Kowalski, M., Kromer, M., 533, 424 Kuncarayakti, H., Lawrence, A., Lowe, T. B., Magnier, E. A., Pellin, M. J., Davis, A. M., Lewis, R. S., Amari, S., & Clayton, Manulis, I., Martin-Carrillo, A., Mattila, S., McBrien, O., R. N. 1999, in Lunar & Planet. Sci. Conf., Vol. 30, 1969 Mu¨ller, A., Nordin, J., O’Neill, D., Onori, F., Palmerio, J. T., Pellin, M. J., Savina, M. R., Calaway, W. F., Tripa, C. E., Pastorello, A., Patat, F., Pignata, G., Podsiadlowski, P., Pumo, Barzyk, J. G., Davis, A. M., Gyngard, F., Amari, S., Zinner, M. L., Prentice, S. J., Rau, A., Razza, A., Rest, A., Reynolds, E., Lewis, R. S., & Clayton, R. N. 2006, in Lunar & Planet. T., Roy, R., Ruiter, A. J., Rybicki, K. A., Salmon, L., Schady, Sci. Conf., Vol. 37, 2041 P., Schultz, A. S. B., Schweyer, T., Seitenzahl, I. R., Smith, M., Pignatari, M., Gallino, R., Heil, M., Wiescher, M., Ka¨ppeler, F., Sollerman, J., Stalder, B., Stubbs, C. W., Sullivan, M., Szegedi, Herwig, F., & Bisterzo, S. 2010, ApJ, 710, 1557 H., Taddia, F., Taubenberger, S., Terreran, G., van Soelen, B., Pignatari, M., G¨obel, K., Reifarth, R., & Travaglio, C. 2016, Vos, J., Wainscoat, R. J., Walton, N. A., Waters, C., Weiland, International Journal of Modern Physics E, 25, 1630003 H., Willman, M., Wiseman, P., Wright, D. E., Wyrzykowski, Pruet, J., Hoﬀman, R. D., Woosley, S. E., Janka, H.-T., & Buras, L(cid:32) ., & Yaron, O. 2017, Nature, 551, 75 R. 2006, ApJ, 644, 1028 Thompson, T. A., Burrows, A., & Meyer, B. S. 2001, ApJ, 562, Qian, Y.-Z. 2014, Journal of Physics G Nuclear Physics, 41, 887 044002 Travaglio, C., Gallino, R., Rauscher, T., Dauphas, N., R¨opke, Qian, Y.-Z. & Wasserburg, G. J. 2007, Phys. Rep., 442, 237 F. K., & Hillebrandt, W. 2014, ApJ, 795, 141 Qian, Y.-Z. & Woosley, S. E. 1996, ApJ, 471, 331 Wanajo, S. 2006, ApJ, 647, 1323 Raiteri, C. M., Gallino, R., Busso, M., Neuberger, D., & Wanajo, S., Janka, H.-T., & Kubono, S. 2011, ApJ, 729, 46 Kaeppeler, F. 1993, ApJ, 419, 207 Witti, J., Janka, H.-T., & Takahashi, K. 1994, A&A, 286, 841 Rauscher, T. & Thielemann, F.-K. 2000, At. Data Nucl. Data Woosley, S. E. & Hoﬀman, R. D. 1992, ApJ, 395, 202 Tables, 75, 1

text: Janka,

label: reference

text: H.-T.

text: Ann.

text: Rev.

text: Nucl.

text: Part.

text: Sc.,

text: Reaclib.

text: Database

text: of

text: Joint

text: Institute

text: of

text: Nuclear

text: K¨appeler,

text: F.,

text: Gallino,

text: R.,

text: Bisterzo,

text: S.,

text: Aoki,

text: W.

text: Rev.

text: Astrophysics,

text: Mod.

text: Phys.,

text: Smartt,

text: S.

text: J.,

text: Chen,

text: T.-W.,

text: Jerkstrand,

text: A.,

text: Coughlin,

text: M.,

text: Kasen,

text: D.,

text: Metzger,

text: B.,

text: Barnes,

text: J.,

text: Quataert,

text: E.,

text: Kankare,

text: E.,

text: Sim,

text: S.

text: A.,

text: Fraser,

text: M.,

text: Inserra,

text: C.,

text: Maguire,

text: K.,

text: Ramirez-Ruiz,

text: E.

text: Nature,

text: Chambers,

text: K.

text: C.,

text: Huber,

text: M.

text: E.,

text: Kru¨hler,

text: T.,

text: Leloudas,

text: G.,

text: Korobkin,

text: O.,

text: Rosswog,

text: S.,

text: Arcones,

text: A.,

text: Winteler,

text: C.

text: Magee,

text: M.,

text: Shingles,

text: L.

text: J.,

text: Smith,

text: K.

text: W.,

text: Young,

text: D.

text: R.,

text: Tonry,

text: MNRAS,

text: J.,

text: Kotak,

text: R.,

text: Gal-Yam,

text: A.,

text: Lyman,

text: J.

text: D.,

text: Homan,

text: D.

text: S.,

text: Lodders,

text: K.

text: ApJ,

text: Agliozzo,

text: C.,

text: Anderson,

text: J.

text: P.,

text: Angus,

text: C.

text: R.,

text: Ashall,

text: C.,

text: Lugaro,

text: M.,

text: Davis,

text: A.

text: M.,

text: Gallino,

text: R.,

text: Pellin,

text: M.

text: J.,

text: Straniero,

text: Barbarino,

text: C.,

text: Bauer,

text: F.

text: E.,

text: Berton,

text: M.,

text: Botticella,

text: M.

text: T.,

text: O.,

text: K¨appeler,

text: F.

text: ApJ,

text: Bulla,

text: M.,

text: Bulger,

text: J.,

text: Cannizzaro,

text: G.,

text: Cano,

text: Z.,

text: Cartier,

text: R.,

text: Meyer,

text: B.

text: S.

text: ARA&A,

text: Cikota,

text: A.,

text: Clark,

text: P.,

text: De

text: Cia,

text: A.,

text: Della

text: Valle,

text: M.,

text: Denneau,

text: L.,

text: Meyer,

text: B.

text: S.

text: Brown,

text: J.

text: S.

text: ApJS,

text: Dennefeld,

text: M.,

text: Dessart,

text: L.,

text: Dimitriadis,

text: G.,

text: Elias-Rosa,

text: N.,

text: Meyer,

text: B.

text: S.,

text: Clayton,

text: D.

text: D.,

text: The,

text: L.-S.

text: ApJ,

text: Firth,

text: R.

text: E.,

text: Flewelling,

text: H.,

text: Flo¨rs,

text: A.,

text: Franckowiak,

text: A.,

text: Nishimura,

text: N.,

text: Takiwaki,

text: T.,

text: Thielemann,

text: F.-K.

text: ApJ,

text: Frohmaier,

text: C.,

text: Galbany,

text: L.,

text: Gonza´lez-Gait´an,

text: S.,

text: Greiner,

text: J.,

text: Gromadzki,

text: M.,

text: Guelbenzu,

text: A.

text: N.,

text: Guti´errez,

text: C.

text: P.,

text: National

text: Nuclear

text: Data

text: Center,

text: information

text: Hamanowicz,

text: A.,

text: Hanlon,

text: L.,

text: Harmanen,

text: J.,

text: Heintz,

text: K.

text: E.,

text: extracted

text: from

text: the

text: NuDat

text: database,

text: Heinze,

text: A.,

text: Hernandez,

text: M.-S.,

text: Hodgkin,

text: S.

text: T.,

text: Hook,

text: I.

text: M.,

text: Izzo,

text: L.,

text: James,

text: P.

text: A.,

text: Jonker,

text: P.

text: G.,

text: Kerzendorf,

text: W.

text: E.,

text: Klose,

text: Otsuki,

text: K.,

text: Tagoshi,

text: H.,

text: Kajino,

text: T.,

text: Wanajo,

text: S.

text: ApJ,

text: S.,

text: Kostrzewa-Rutkowska,

text: Z.,

text: Kowalski,

text: M.,

text: Kromer,

text: M.,

text: Kuncarayakti,

text: H.,

text: Lawrence,

text: A.,

text: Lowe,

text: T.

text: B.,

text: Magnier,

text: E.

text: A.,

text: Pellin,

text: M.

text: J.,

text: Davis,

text: A.

text: M.,

text: Lewis,

text: R.

text: S.,

text: Amari,

text: S.,

text: Clayton,

text: Manulis,

text: I.,

text: Martin-Carrillo,

text: A.,

text: Mattila,

text: S.,

text: McBrien,

text: O.,

text: R.

text: N.

text: in

text: Lunar

text: Planet.

text: Sci.

text: Conf.,

text: Vol.

text: Mu¨ller,

text: A.,

text: Nordin,

text: J.,

text: O’Neill,

text: D.,

text: Onori,

text: F.,

text: Palmerio,

text: J.

text: T.,

text: Pellin,

text: M.

text: J.,

text: Savina,

text: M.

text: R.,

text: Calaway,

text: W.

text: F.,

text: Tripa,

text: C.

text: E.,

text: Pastorello,

text: A.,

text: Patat,

text: F.,

text: Pignata,

text: G.,

text: Podsiadlowski,

text: P.,

text: Pumo,

text: Barzyk,

text: J.

text: G.,

text: Davis,

text: A.

text: M.,

text: Gyngard,

text: F.,

text: Amari,

text: S.,

text: Zinner,

text: M.

text: L.,

text: Prentice,

text: S.

text: J.,

text: Rau,

text: A.,

text: Razza,

text: A.,

text: Rest,

text: A.,

text: Reynolds,

text: E.,

text: Lewis,

text: R.

text: S.,

text: Clayton,

text: R.

text: N.

text: in

text: Lunar

text: Planet.

text: T.,

text: Roy,

text: R.,

text: Ruiter,

text: A.

text: J.,

text: Rybicki,

text: K.

text: A.,

text: Salmon,

text: L.,

text: Schady,

text: Sci.

text: Conf.,

text: Vol.

text: P.,

text: Schultz,

text: A.

text: S.

text: B.,

text: Schweyer,

text: T.,

text: Seitenzahl,

text: I.

text: R.,

text: Smith,

text: M.,

text: Pignatari,

text: M.,

text: Gallino,

text: R.,

text: Heil,

text: M.,

text: Wiescher,

text: M.,

text: Ka¨ppeler,

text: F.,

text: Sollerman,

text: J.,

text: Stalder,

text: B.,

text: Stubbs,

text: C.

text: W.,

text: Sullivan,

text: M.,

text: Szegedi,

text: Herwig,

text: F.,

text: Bisterzo,

text: S.

text: ApJ,

text: H.,

text: Taddia,

text: F.,

text: Taubenberger,

text: S.,

text: Terreran,

text: G.,

text: van

text: Soelen,

text: B.,

text: Pignatari,

text: M.,

text: G¨obel,

text: K.,

text: Reifarth,

text: R.,

text: Travaglio,

text: C.

text: Vos,

text: J.,

text: Wainscoat,

text: R.

text: J.,

text: Walton,

text: N.

text: A.,

text: Waters,

text: C.,

text: Weiland,

text: International

text: Journal

text: of

text: Modern

text: Physics

text: E,

text: H.,

text: Willman,

text: M.,

text: Wiseman,

text: P.,

text: Wright,

text: D.

text: E.,

text: Wyrzykowski,

text: Pruet,

text: J.,

text: Hoﬀman,

text: R.

text: D.,

text: Woosley,

text: S.

text: E.,

text: Janka,

text: H.-T.,

text: Buras,

text: Yaron,

text: O.

text: Nature,

text: R.

text: ApJ,

text: Thompson,

text: T.

text: A.,

text: Burrows,

text: A.,

text: Meyer,

text: B.

text: S.

text: ApJ,

text: Qian,

text: Y.-Z.

text: Journal

text: of

text: Physics

text: G

text: Nuclear

text: Physics,

text: Travaglio,

text: C.,

text: Gallino,

text: R.,

text: Rauscher,

text: T.,

text: Dauphas,

text: N.,

text: R¨opke,

text: Qian,

text: Y.-Z.

text: Wasserburg,

text: G.

text: J.

text: Phys.

text: Rep.,

text: F.

text: K.,

text: Hillebrandt,

text: W.

text: ApJ,

text: Qian,

text: Y.-Z.

text: Woosley,

text: S.

text: E.

text: ApJ,

text: Wanajo,

text: S.

text: ApJ,

text: Raiteri,

text: C.

text: M.,

text: Gallino,

text: R.,

text: Busso,

text: M.,

text: Neuberger,

text: D.,

text: Wanajo,

text: S.,

text: Janka,

text: H.-T.,

text: Kubono,

text: S.

text: ApJ,

text: Kaeppeler,

text: F.

text: ApJ,

text: Witti,

text: J.,

text: Janka,

text: H.-T.,

text: Takahashi,

text: K.

text: A&A,

text: Rauscher,

text: T.

text: Thielemann,

text: F.-K.

text: At.

text: Data

text: Nucl.

text: Data

text: Woosley,

text: S.

text: E.

text: Hoﬀman,

text: R.

text: D.

text: ApJ,

text: Tables,


text: 3 9 Y (0) (Arcones & Bliss 2014). The evolution of the ra- e 8 2s dius r(t) used to calculate that of the νe and ν¯e ﬂuxes is 5s shown in Fig. 1b. 7 8s In Figs. 1c and 1d, we illustrate the nucleosynthesis 6 for our baseline models assuming Y (0) = 0.45 (neutron- e 5 rich) and 0.60 (proton-rich), respectively. It can be seen 4 that the variations of T(t) and S among the three tra- 0 3 jectories have a minor impact on the ﬁnal abundances for neutron-rich conditions when the same Y (0) is used. 2 e However, the situation is markedly diﬀerent for proton- 1 rich conditions, where increasingly heavier nuclei are pro- 0 duced for the wind mass elements ejected at later times. 10-2 10-1 100 This is because the expansion aﬀects the νp-process in t (s) such conditions through the determination of both the number ratio of protons to seed nuclei and the ν¯ ﬂux e at the onset of the process. Speciﬁcally, the expansion ∼ 109 time scale for T 6 to 3 GK sets the number ratio of protons to seed nuclei at the onset of the νp-process. In addition, this time scale also determines the radius of the 108 mass element, and hence the ν¯ ﬂux it receives, at this e onset. Finally, the combination of this ν¯ ﬂux and the ∼ e expansion time scale for T 2 to 1 GK determines the 2s 107 extent of neutron production by ν¯ absorption on pro- 5s e tons during the νp-process. Clearly, a suﬃcient neutron 8s abundance is required to overcome the bottlenecks on 106 the path to heavier nuclei via (n, p) reactions. The slow- 10-2 10-1 100 ing down of the mass element near the wind termination t (s) helps to fulﬁll this requirement. The trajectory ejected 10-2 at t = 2 s has the fastest initial expansion and the mass 2s Mo Ru pb 10-3 element is already at a large radius at the onset of the 5s νp-process (see Fig. 1b). Therefore, for this trajectory 10-4 8s ν¯ absorption on protons is strongly reduced during the e 10-5 νp-process even with the slowing down of the expansion near the wind termination. In contrast, the overall ex- 10-6 pansion for the trajectory ejected at t = 8 s is slower pb 10-7 so that the νp-process becomes much more eﬃcient and reaches signiﬁcantly heavier nuclei. 10-8 In the following discussion, we will focus on the trajec- 10-9 tory ejected at tpb = 8 s. Pertinent results for the other 5 10 15 20 25 30 35 40 45 50 55 two trajectories are also presented. As described above, Atomic number Z our study is parametric in that we adopt ﬁxed evolution 10-2 of temperature T(t) and radius r(t) for a wind trajectory 2s Mo Ru while varying S and Y (0) to obtain a range of ρ(t) and 10-3 e 5s Y (t), respectively. We note that only accurate modeling e 10-4 8s of a speciﬁc astrophysical environment can yield a self- consistent set of T(t), ρ(t), and Y (t). In the case of the 10-5 e neutrino-driven wind, this would require accurate neu- 10-6 trino transport in the proto-neutron star and accurate simulation of the CCSN explosion (Arcones & Thiele- 10-7 mann 2013; Arcones & Bliss 2014). While our parametric 10-8 approach can only serve as approximation to the rigor- ous astrophysical models, it captures the salient features 10-9 5 10 15 20 25 30 35 40 45 50 55 of such models that are important to heavy-element nu- Atomic number Z cleosynthesis. Further, it is eﬃcient for surveying a wide range of possibilities. The results from our parametric Fig. 1.— Baseline models for nucleosynthesis. (a) Time evolution − models should provide good guidance in ﬁnding the con- of T/GK (solid curves) and log(ρ/g cm 3) (dashed curves) for tra- jectories of wind mass elements ejected at t = 2 (black curves), ditions for producing Mo and Ru isotopes of interest. pb 5 (green curves), and 8 s (red curves), respectively. (b) Time evo- 3. RESULTS ON MO AND RU ISOTOPES lution of the radius r for each trajectory. The region of T = 2 to We are interested in the conditions under which var- 1 GK important to the νp-process is indicated as that between the ious Mo and Ru isotopes can be made in environments solid circle and triangle. (c) The ﬁnal elemental (number) abun- dances for each trajectory assuming Ye(0) = 0.45 (neutron-rich). similar to the neutrino-driven wind. In conventional (d) Same as (c), but for Ye(0) = 0.60 (proton-rich). terms, the seven stable isotopes of Mo (Ru) fall into four categories: (1) 92,94Mo (96,98Ru) as the p-only nuclei, spectra so that they are consistent with the speciﬁed )mc( r Y ecnadnubA Y ecnadnubA )3-mc g( 0ρ 01gol ,)KG( T ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## 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text: Fig.

text: Baseline

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text: nucleosynthesis.

text: (a)

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text: in

text: ﬁnding

text: the

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text: of

text: T/GK

text: (solid

text: curves)

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text: (dashed

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text: jectories

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text: Mo

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text: (green

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text: (b)

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text: the

text: ious

text: Mo

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text: Ru

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text: be

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text: solid

text: circle

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text: triangle.

text: (c)

text: The

text: ﬁnal

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text: (neutron-rich).

text: similar

text: to

text: the

text: neutrino-driven

text: wind.

text: In

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text: (d)

text: Same

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text: (c),

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text: for

text: (proton-rich).

text: terms,

text: the

text: seven

text: stable

text: isotopes

text: of

text: Mo

text: (Ru)

text: fall

text: into

text: four

text: categories:

text: as

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text: T

text: ##LTFigure##

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