New tools for probing the phase space structure of dark matter halos
aa r X i v : . [ a s t r o - ph . C O ] F e b New Tools for Probing the Phase Space Structureof Dark Matter Halos
Monica Valluri ∗ , Victor P. Debattista † , Thomas Quinn ∗∗ and Ben Moore ‡ ∗ University of Michigan † University of Central Lancashire ∗∗ University of Washington ‡ University of Zürich
Abstract.
We summarize recent developments in the use of spectral methods for analyzing largenumbers of orbits in N -body simulations to obtain insights into the global phase space structure ofdark matter halos. The fundamental frequencies of oscillation of orbits can be used to understandthe physical mechanism by which the shapes of dark matter halos evolve in response to the growthof central baryonic components. Halos change shape primarily because individual orbits changetheir shapes adiabatically in response to the growth of a baryonic component, with those at smallradii become preferentially rounder. Chaotic scattering of orbits occurs only when the central pointmass is very compact and is equally effective for centrophobic long-axis tube orbits as it is forcentrophilic box orbits. Keywords:
Methods: N -body simulations – galaxies: evolution – galaxies: formation – galaxies:dark matter – galaxies: kinematics and dynamics – cosmology: dark matter PACS:
INTRODUCTION
While collisionless N -body simulations produce dark matter halos that are triaxial orprolate, simulations which include the effects of baryons result in more spherical oraxisymmetric halos[1, 2, 3]. It has been suggested that the change in shape could bethe consequence of chaotic scattering of the box orbits that form the "back bone" oftriaxial galaxies [4]. Alternatively, halo shapes might change due the adiabatic responseof orbits to the change in the central potential. In a recent paper Valluri et al. [5] showedthat by applying a spectral method to analyze large numbers of randomly selected orbitsin N -body simulations it was possible to clearly distinguish between these two options.In this paper we present a brief summary of some of their main results. FREQUENCY ANALYSIS OF N -BODY ORBITS Prolate and triaxial dark matter halos were formed from the merger of spherical NFWhalos [6] and baryonic components (representing a disk, an elliptical galaxy or a massivecompact nucleus) were grown adiabatically with time. The evolution of the halo wasfollowed using
PKDGRAV an efficient, multi-stepping, parallel tree code [7]. After thebaryonic component was grown to full strength it was artifically "evaporated" to allowsus to test for the importance of chaos in the evolution of the system [3, hereafter D08
New Tools for Probing the Phase Space Structure of Dark Matter Halos October 30, 2018 1
IGURE 1.
For ∼ w a ) versus after ( w b ) the growth of a baryoniccomponent are shown. Left: effect of an extended baryonic component (disk with 3 kpc scale length).
Right: effect of a hard spherical point masses with 0.1 kpc softening. and references therein]. In each system several thousand orbits were selected and theirtrajectories evolved in a frozen potential in each phase of the evolution of each halo.The initial triaxial/prolate phase is referred to as phase a , once the baryonic componentis grown to full strength the halo is in phase b , and after the baryonic component had"evaporated" and the system returned to equilibrium the halo is in phase c . Each orbitwas analyzed using a code that decompose its phase space trajectory to obtain its threefundamental frequencies of oscillation [8, 9]. The fundamental frequencies were thenused to obtain a complete picture of the properties of individual orbits: to distinguishbetween regular and chaotic orbits, to classify regular orbits into major orbit families, toquantify the average shape of an orbit and relate its shape to the shape of the halo, andto identify the major resonant families of the halo which determine its structure [5].
RESULTS
The largest of the three fundamental frequencies in each of the three phases of theevolution of our models (referred to as w a , w b and w c respectively), can be compared todistinguish between adiabatic and chaotic evolution. In the case of a primarily adiabaticresponse, particles deep in the potential (large w a ) are expected to experience a greaterincrease in frequency than particles further from the center. In Figure 1 we comparethe frequencies of orbits in an initially triaxial halo ( w a ) with their frequency in thepresence of a baryonic component ( w b ). For the extended disk ( Left ) (as well as otherextended baryonic components), w b increases monotonically with w a with fairly smallscatter, indicating that the orbits in this potential responded adiabatically (the dashed lineshows the 1:1 correlation between the two frequencies). However, a hard central pointmass ( Right ) results in significant scattering in w b with small values of w a (i.e. weaklybound orbits) having some of the largest values of w b . w b sometimes decreased insteadof increasing providing further evidence for chaotic scattering in this case.The orbital frequencies were used to distinguish between regular and chaotic orbitsand to classify the regular orbits into major families. We studied both triaxial and prolatehalos and our orbital analysis showed that the initial triaxial halo was composed of 84- New Tools for Probing the Phase Space Structure of Dark Matter Halos October 30, 2018 2
IGURE 2.
The change in frequency of an orbit w ac versus pericentric radii r peri for triaxial halo ( Left )and prolate halo (
Right ).
86% box orbits, 11-12% long-axis tube orbits, 2% short axis tubes, and 1-2% chaoticorbits. In contrast the prolate halo had 15% box orbits, 78% long-axis tubes, 7% short-axis tubes and no chaotic orbits. To test the hypothesis that it is the box orbits that aremost significantly scattered by a central point mass we define the fractional change
D w ac in the frequency of an orbit from its value in the initial halo ( w a ) to its value after thebaryonic component was evaporated ( w c ) to measure the amount of chaotic scattering.Figure 2 shows that orbits with smaller pericenter radii r peri experience a significantlylarger change in frequency D w ac than orbits at large pericenter radii. This is true for boththe triaxial halo ( Left ) which is dominated by centrophilic box orbits and the prolatehalo (
Right ) which is dominated by long-axis tubes. Thus chaotic scattering is equallystrong for the centrophobic long-axis tube orbits and centrophilic box orbits contrary tothe prevailing view.In a triaxial potential in which the major, intermediate and short axes are along theCartesian directions x , y , z respectively, the oscillation frequencies satisfy the condition | w x | < | w y | < | w z | for any orbit with the same over-all shape as the density distribution.For such an orbit we can define a shape parameter c s ≡ | w y / w z | − | w x / w z | which ispositive for orbits with elongation along the major axis of the figure, with larger valuesof c s implying a greater degree of elongation. Since orbits closer to the central potentialare more significantly affected by the baryonic component, we expect them to becomerounder ( c s →
0) than orbits further out. Figure 3 shows how the orbital shape parameter c s for various orbital types varies as a function of the pericenter radius r peri for a triaxialhalo ( Left ) and after a disk was grown in this halo with symmetry axis parallel to theshort axis (
Right ). In each plot the curves show the mean shape distribution of orbits ofa given orbital type at each radius, with orbital types indicated in the line-legends. In theinitial triaxial halo boxes, long-axis tubes and chaotic orbits are significantly elongated( c s ∼ .
35) both at small and large radii. After the growth of the short-axis disk theorbits at small radii become axisymmetric ( c s ∼
0) while orbits at large radii (especiallyboxes) remain quite elongated. This change in orbital shape at small radii was seen inall halos regardless of the radial scale length of the baryonic component.
New Tools for Probing the Phase Space Structure of Dark Matter Halos October 30, 2018 3
IGURE 3.
The mean orbital shape parameter c s for different orbital types (indicated by line legends)vs. pericentric radius r peri . IMPLICATIONS
The analysis of fundamental frequencies of orbits in N -body halos is a powerful tech-nique that allows us to identify the primary physical processes that cause halo shapes tochange in response to the growth of a baryonic component. We confirmed the conclusionreached by D08 that chaos is not an important driver of shape evolution but found thatsignificant chaotic scattering does occur when the baryonic component is in the form ofa hard central point mass (of scale length ∼ . ACKNOWLEDGMENTS
MV would like to thank the organizers of the conference at UCLAN and the Universityof Malta for organizing an excellent meeting.
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