Statistical mechanics of collisionless orbits. IV. Distribution of angular momentum

0.69 MB·

by  Liliya L. R. Williams

The Astrophysical Journal,accepted (2014)
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
STATISTICAL MECHANICS OF COLLISIONLESS ORBITS. IV. DISTRIBUTION OF ANGULAR MOMENTUM
Liliya L. R. Williams1, Jens Hjorth2, and Rados law Wojtak2
Received 2013 August 12; accepted 2014 January 8
ABSTRACT
It has been shown in previous work that DARKexp, which is a theoretically derived, maximum
entropy, one shape parameter model for isotropic collisionless systems, provides very good fits to
4
simulated and observed dark-matter halos. Specifically, it fits the energy distribution, N(E), and
1
the density profiles, including the central cusp. Here, we extend DARKexp N(E) to include the
0
distribution in angular momentum, L2, for spherically symmetric systems. First, we argue, based on
2
theoretical,semi-analytical,andsimulationresults,thatwhiledark-matterhalosarerelaxedinenergy,
n they are not nearly as relaxed in angular momentum, which precludes using maximum entropy to
a uniquely derive N(E,L2). Instead, we require that when integrating N(E,L2) over squared angular
J
momentaoneretrievestheDARKexpN(E). StartingwithageneralexpressionforN(E,L2)weshow
0
how the distribution of particles in L2 is related to the shape of the velocity distribution function,
2
VDF, and velocity anisotropy profile, β(r). We then demonstrate that astrophysicallyrealistic halos,

] as judged by the VDF shape and β(r), must have linear or convex distributions in L2, for each
O separate energy bin. The distribution in energy of the most bound particles must be nearly flat, and
become more tilted in favorofradialorbits for less bound particles. These results areconsistentwith
C
numericalsimulationsandrepresentanimportantsteptowardsderiving the full distributionfunction
.
h for spherically symmetric dark-matter halos.
p Subject headings: dark matter — galaxies: halos
-
o
tr 1. INTRODUCTION 2010, hereafter Paper I, for a discussion). Several
s other works have appeared since (Stiavelli & Bertin
a The full dynamical description of relaxed dark-matter
1987;Hjorth & Madsen1991;Spergel & Hernquist1992;
[ halos is of fundamental importance for our basic under-
Chavanis1998;Levin et al.2008;Lapi & Cavaliere2009;
standing as well as for more practical applications in
1 Pontzen & Governato2013). Some ofthese relyonarbi-
galaxyformationandevolution,and cosmology. N-body
v traryassumptions,whileothersrequiremanyparameters
simulations have converged on the properties of dark-
5 to fit halos adequately.
matter halos (Navarro et al. 2004; Stadel et al. 2009;
8 Our statistical mechanics approach (Paper I) differs
Navarro et al. 2010), though some uncertainty may still
0 from the previous ones in that we work in a different
remain arising from finite resolution effects.
5
state space, which we argue is more appropriate for col-
. A lot ofwork has been devotedto attempts to explain
1 the density and velocity structure of relaxed halos by lisionless systems, and uses an accurate description of
0 low occupation numbers, which appears to be impor-
examining the dynamical processes at work, like mass
4 tant for self-gravitatingsystems (see also Madsen 1996).
accretion rate, conservation of radial action, radial or-
1 In Paper I we derived the differential mass, or energy
bit instability, etc. (e.g., Le Delliou & Henriksen 2003;
: distribution, dM/dE = N(E) for collisionless material
v Williams et al. 2004; Lu et al. 2006; Salvador-Sol´e et al.
under the assumption that the final steady-state con-
i 2007; Ascasibar et al. 2007; Dalal et al. 2010). Though
X figuration represents the most likely state, and there-
suchphenomenologicalargumentsarevaluableandbring
fore can be obtained as a maximum entropy state. Our
r insight to the problem, they do not arise directly from
a model is called DARKexp, and its energy distribution is
fundamental physics. Because the properties of virial-
ized dark matter halos appear universal, and are only N(ǫ)=exp(φ0−ǫ)−1, where ǫ=β˜E is the dimension-
weakly dependent on initial conditions, like cosmologi- lessenergy,β˜is the inversethermodynamictemperature
cal model, local density, etc., it is reasonable to assume (β˜ < 0), and φ = β˜Φ is the dimensionless central po-
0 0
that the structure of halos is governed by physics more tential.
fundamental than that described by phenomenology. Unlike other theoretically motivated density profiles
Motivated by the possibility of a first principles solu- (King 1966; Lynden-Bell 1967; Madsen 1996), DARK-
tion,severalgroupshaveattemptedastatisticalmechan- exp predicts central density cusps. The asymptotic,
ics approach. To our knowledge, the first attempts were small r density slope for all central potentials, φ , is
0
made by Ogorodnikov (1957) and Lynden-Bell (1967), dlnρ/dlnr =−1,butforradiiaccessibletoN-bodysim-
with somewhat limited success (see Hjorth & Williams ulations,thecentralslopevariesdependingonφ0: φ0<∼4
1School of Physics and Astronomy, University of Min- shyasvteeminsnheravseloipnenserbseltowpeeesnsh−a1lloawnedr−th2a.n−1,whileφ0>∼5
nesota, 116 Church Street SE, Minneapolis, MN 55455, USA;
DARKexpappears to be a verygooddescriptor of the
[email protected]
2DarkCosmologyCentre, NielsBohrInstitute, Universityof energydistribution anddensity profile of dynamicalsys-
Copenhagen, Juliane Maries Vej 30, DK–2100 Copenhagen Ø, tems: galaxyandgalaxyclustersizedark-matterhalosin
Denmark;[email protected],[email protected]
2 Williams et al.
simulations (Williams et al. 2010, hereafter Paper III), having any mixing in L2.
observed galaxy clusters (Beraldo e Silva et al. 2013), Numerically simulated dark-matter halos in equilib-
and even many globular clusters (Williams et al. 2012). rium are relaxed in energy as evidenced by their being
Beraldo e Silva et al. (2013) compared a range of theo- well fit with DARKexp (Paper III). Figure 1 contains
retical and phenomenological models to relaxed galaxy four individual halos and their DARKexp fits that make
clusters whose profiles were estimated using strong and up the average shown in that paper. The fit at very
weak lensing. DARKexp did better than other theoreti- large negative energies (i.e. for very bound particles)
cal models, and performed as well as the best empirical andintermediateenergiesisverygood;atsmallnegative
fitting functions. energies (right side of each panel) most of the particles
The DARKexp model derivedin Paper I has one limi- are quite far from the halo center, and so may not be in
tation: itdescribesthedistributionofparticlesinenergy equilibrium.
only, i.e., N(E), and implicitly assumes that the distri- Are the halos equally well relaxed in L2? Appar-
bution in angular momentum corresponds to that of a ently not. Wojtak et al. (2013) show that the princi-
system with an isotropic velocity ellipsoid. While veloc- pal axes of cosmologically simulated halos are aligned
ityanisotropy(hereafter,anisotropy)affectstheshapeof with the local velocity ellipsoids, and the alignment is
N(E) only weakly, one still would like to know the full strongestintheinnermostshells. Moreover,theprincipal
dynamical description of a system, including the distri- axes are aligned with the large scale structure filaments
bution of particles in angular momentum, L. Systems (Libeskind et al. 2013) which implies that the angular
considered in this work have no net rotation, so the an- distribution of angular momentum, L, retains the mem-
gularmomentumvectorisreducedtoitsmodulus. Dark- ory of the formation process, down to the inner most
matterhalosinsimulations,andprobablyintheUniverse regions. Ifmixinginanglewerecomplete,asrequiredby
arenotisotropic. Thereforeoneneedsto extendDARK- fullrelaxation,thesealignmentsshouldhavebeenerased.
exp N(E) to include L2. Why are halos not well mixed in L2? We speculate
ThispaperisdevotedtoestimatingN(E,L2)fromba- that this is because in a collapsing system, radial forces,
sicargumentsandsimplemodels. InSection2wediscuss and hence changes in radial forces, tend to be larger
the degree of mixing in energy and angular momentum and longer lasting than tangential ones. The former
and argue that a maximum entropy approach may not are largely responsible for mixing in E, while the later
be applicable to the problem in hand. Instead we pro- are exclusively responsible for mixing in L2. In other
poseanintegralconstraintonN(E,L2). InSection3we words, radial fluctuations in the gravitational potential
generate and characterize halos obeying this constraint aremoredominantthantangentialones,wherethelatter
and compare to simulated halos. Section 4 provides a are brought about by ellipticity, substructure and merg-
summary and an outlook. ing.
2. GENERAL CONSIDERATIONS FOR N(E,L2) 2.2. Failure of maximum entropy arguments for the
angular momenta
2.1. The lack of mixing of angular momenta
As a consequence, this observation implies that the-
Relaxation into equilibrium, or at least into a long-
oretical maximum entropy approaches that assume full
livedsteadystatecanbedrivenbyanydynamicalprocess
mixing in angle andrequire L2 to be partof the entropy
that mixes, or redistributes, particle energies and angu-
(e.g., Pontzen & Governato 2013), likely cannot capture
lar momenta, thereby causing them to be uncorrelated
the properties of simulated dark-matter halos. To check
withthe correspondinginitialvalues. Dark-matterhalos
thatfreeredistributionormixingofL2isnottakingplace
insimulationsarenearlyrelaxedsystems,thoughthede-
in collapsing systems, we considered the final state of a
gree of relaxationis still not clear. In a collisionless sys-
system assuming that it does. In other words, we apply
tem, mixing in energy is achieved because the particles
a maximum entropy argument to L2 as well as to E. To
exchange energy with the global time-varying potential
do this we extend our derivationpresented in Paper I to
(Lynden-Bell 1967), while mixing in angularmomentum
include angular momentum. As in that paper, a maxi-
isaccomplishedthroughtorqueswhicharepresentinany
mum entropy procedure is applied, where in addition to
system that deviates from spherical symmetry.
the total energy and total mass, a quantity relating to
We now argue that dark-matter halos are not as well
the total angular momentum is also held fixed. Apart
relaxedinangularmomentumastheyareinenergy. This
from the fact that there are now three, rather than two,
difference in the degree of relaxation in the two param-
Lagrange multipliers, the maximization of entropy pro-
eters, E and L2, is possible because relaxation in en-
cedureisthesameastheoneusedtoderivetheisotropic
ergy and in angular momentum can proceed relatively
DARKexp. The final derived distribution has the form
independently of each other, at least in some systems.
One example is the Extended Secondary Infall Model N(E,L2)∝exp(β˜Φ −β˜E−γLξ)−1, (1)
0
(ESIM), described in Williams & Hjorth (2010, here-
after Paper II) and some earlier works (Ryden & Gunn which is analogous to Equation (30) of Hjorth (1994).
1987;Williams et al.2004). ESIMsystemsrelaxthrough (Note that β˜in the aboveis not to be confused with the
spherically symmetric collapse in which particles/orbits velocity anisotropy. We denote the velocity anisotropy
keep their initial L2 throughout the evolution. Even by β or β(r).)
though L2 of individual particles do not change at all, In general, there is no analytical way to derive the
the corresponding E does change significantly, and the densityprofilesfromEquation(1). Wethereforeusedan
final virialized halos are well fit with DARKexp. There- iterative procedure, similar to the one used in Paper II,
foremixingandrelaxationinE canbeachieved,without to obtain the corresponding density profiles. We exper-
DARKexp: Distribution of angular momentum 3
imented with several combinations of parameters γ and the density profile, with little dependence on the veloc-
ξ (see Appendix A for details). Some parameter combi- ity anisotropy. The velocity anisotropy is defined in the
nations did not produce viable halos, i.e., did not con- usualway,β(r)=1−[σ (r)/σ (r)]2,whereσ andσ are
θ r θ r
verge. Those that did converge had velocity anisotropy velocity dispersions in one of the tangential directions,
profilesthatwereisotropicatsmallradiiandbecametan- and radial direction, respectively. Anisotropy is deter-
gentially anisotropic at large radii. Because no dynami- mined by the distribution of particles/orbitsin L2. This
callyproducedcollisionlesssystemsisknowntohavesuch meansthatagivenρ(r), coupledwithdifferentformsfor
anisotropy, and because it seems unlikely that any dy- β(r), will result in nearly the same N(E). The insensi-
namicalprocesswouldleadtosuchasystem,weconclude tivityofN(E)to velocityanisotropywaspointedoutby
that systems described by Equation (1), with γ 6= 0, do Binney & Tremaine (2008) (their Section4.4 andFigure
not exist in simulations, or the real Universe. 4.15b)using isotropicandfully radiallyanisotropicJaffe
Onecouldarguethatthechoiceofapowerlawformfor models.
L in Equation (1) is limiting, and there could be other This lack of sensitivity of N(E) to the veloc-
functional forms that would produce realistic systems. ity anisotropy ensures that DARKexp density pro-
While possible, we speculate that the general behaviour file, derived for isotropic orbits, should also describe
of solutions will be similar to that displayed by Equa- anisotropiccosmologicalN-body simulatedhalos,aswas
tion (1), regardless of the what function of L is used. shown explicitly in Paper III. The recent finding by
A more thorough investigation is needed to explore this Beraldo e Silva et al.(2013)thatDARKexpdensitypro-
question. files provide very good fits to observed galaxy clusters,
Though Equation (1) cannot be solved analytically to whosegalaxies(Biviano & Poggianti2009;Biviano et al.
produce ρ(r) in general, in Appendix B we consider a 2013)anddarkmatter (Host et al.2009)appearto have
special case that can be solved analytically. In this case radialvelocityanisotropyatlargeradii,isalsoconsistent
the density profile is withtheenergydistributionbeinglargelyindependentof
anisotropy.
ρ(r)= 1 − β˜ 2/(ξ−2) 2+ξ r−4(ξ−1)/(ξ−2), (2) In Section 3 we will use Equation (3) to arrive at the
4π(cid:16) γξ(cid:17) (cid:16)2−ξ(cid:17) full distribution of particles in E and L2, or N(E,L2).
where β˜ < 0, γ < 0, and −2 < ξ < 0. Equation (2)
is a power law, ranging in slope between ρ ∝ r−2 and 3. DISTRIBUTION OF L2 IN HALOS
ρ ∝ r−3. These are grossly inconsistent with the pro-
We generateas widea rangeofN(E,L2)distributions
files obtained in simulations, whose slopes are not con-
aspossible,allsatisfyingEquation(3). N(E)isidentical
stant, and steepen with radius from around −1 to −3
forallsystems,andthedensityprofilesaresimilarforall
well within the virial radius (e.g., Navarro et al. 1997).
halos(DARKexpφ =4). Themajordifferencebetween
Because Equation (1) does not seem to produce sys- 0
systems is in the velocity dependent properties, which
tems with realistic density profiles and constant or in-
are related to the L2 distribution. We then ask how the
creasing velocity anisotropy profiles, we conclude that
distributionof particles in angularmomentumrelates to
the maximum entropy argument cannot be used to de-
the astrophysically relevant halo quantities, namely the
rive the energy and angular momentum distribution of
velocity distribution function, VDF, and the anisotropy
dynamicallyevolvedcollisionlesssystems. Thecombined
profile, β(r). Here we will mostly deal with the radial
evidence of the arguments presented above leads us to
VDF, which is a histogram of radial speeds of particles
conclude that dynamically evolved collisionless systems
(with respect to the halo center) in a specified radial
are not as well relaxed in angular momentum as they
range within the halo.
are in energy; in other words, the particle angular mo-
menta L, and hence their moduli L are not as freely InordertofindarelationbetweenN(E,L2),theVDF
and β(r), we need to quantify these properties in a suc-
redistributed during evolution as their energies.
cinct way, such that, for example, the entire β(r) profile
2.3. Proposed integral form for N(E,L2) isrepresentedbyasinglevalue. WedothisinSection3.2.
Eventhoughallourhalosarephysicallypossible,notall
Our hypothesis that mixing in E is achievedrelatively
are astrophysically realistic. For example, an astrophys-
quicklyandefficiently,whilemixinginL2isnot,suggests
ically realistic halo cannot have large (spherically aver-
that the general form for N(E,L2) should be
aged) β(r) at small radii. We then isolate N(E,L2) dis-
L2max(E) tributionsthatgiverisetoastrophysicallyrealisticVDFs
N (E)= N(E,L2)dL2. (3) and anisotropy profiles.
DARKexp Z
0
Here N(E,L2) is non-separable, and upon integration
3.1. Generating the L2 distributions
over all angular momenta gives DARKexp N(E). This
meansthatsystemsthathaveDARKexpN(E)canhave The general method for generating N(E,L2) uses
arangeofL2distributions. ThedifferenttypesofL2dis- Equation (3). We start with the DARKexp form for
tributionscan,forexample,comeaboutasaconsequence N(E), and at each energy distribute particles in L2 ac-
of different formationscenarios,suchas cosmologicalvs. cording to some prescription. (Same method as used in
isolated collapse. Paper III.) We work with dimensionless energy units, ǫ,
Equation(3)isalsoconsistentwithanunrelatedprop- defined in Paper I, and ℓ2 = L2/L2 , where L (ǫ) is
circ circ
ertyofdifferentialenergydistributionsingeneral. Differ- theangularmomentumforacircularorbitatthatenergy.
ential energy distributions, N(E), depend primarily on Weuseatotalofeightdifferentprescriptions. Hereisone
4 Williams et al.
example: (inset) VDF. We denote the radial VDF distribution by
N (u ), where u =v /σ , and σ is the radial velocity
N(ǫ,ℓ2)=NDARKexp(ǫ)h1− c+a 1(cid:16)φ0φ−0 ǫ(cid:17)bi−1 (4) dniousrprmerasrliiozendattot1h,atr0r∞adNiruusr.(ruTr)hdeuarre=ra1u.ndFeorreaallchthVeDhFaloiss
φ −ǫ b c we considered tanRgential VDFs peak at ut = 0. This is
× 1−a 0 ℓ2 . not so for radial VDF, which sometimes show deficits of
h (cid:16) φ0 (cid:17) (cid:16) (cid:17) i orbits at small radial speeds. In this case the third ra-
The first square bracket contains the normalization fac- dialbinfromthecenter(bluehistogram)showsacentral
tor, which depends on energy, while the second square ‘crater’. The lower right panel shows the density pro-
bracket contains the dependence on L and energy. This file, ρ(r) aslog(ρr2) (thick solidline), ofDARKexpwith
makesN(E,L2)non-separable(asistrueforallourpre- φ0 = 4, and the anisotropy profile, β(r) (dashed line).
scriptions). Another example is The thin solid line is the NFW profile shown here for
comparison. Thehorizontalaxisisinunitsofr−2,thera-
N(ǫ,ℓ2)=N (ǫ) b−1c1/b φ0−ǫ −a/b dius where the logarithmicdensity slope, dlnρ(r)/dlnr,
DARKexp h (cid:16) φ0 (cid:17) is equal to −2.
× tcirc e−tt1/b−1dt −1e−t, (5) 3.2. Characterizing the L2 distributions
Z0 i ToseehowthedistributioninL2isrelatedtotheVDF
t=c−1 φ0−ǫ aℓ2b, shape and β(r), we need to characterize all three quan-
(cid:16) φ0 (cid:17) tities with simple parameters.
We start with the L2 distribution. For any
where the expression in the square brackets is the nor- given energy ǫ, the distribution of particles in ℓ2 ≡
malizationfactor,andisproportionaltothelowerincom- (L/Lcirc)2 is denoted by NL2(ℓ2), which is normalized,
plete gamma function, with t being t when L=L ,
orℓ=1. Notethatconstantscair,cb,andcinEquationsc(ir4c) 01NL2(ℓ2)dℓ2 = 1, where, by definition, ℓ2 runs from 0
and(5) arenotthe same. DifferentrealizationsofEqua- tRo1. WedefinethecurvatureoftheNL2(ℓ2)distribution
tions(4)and(5)usearangeofvaluesfortheseconstants. for a given ǫ as
We used density profiles for DARKexp φ = 4; other
0 1
rveasluuletss.of Eφa0chanLd2otdhisetrritbyupteiosnofrepsuroltfisleisngaivedisffimerielnart K˜(ǫ)= R0[NL2(ℓ2)−NL2(ℓ2m)]dℓ2 −1, K= K˜(ǫ)
anisotropy profile. Figure 2 shows the anisotropy pro- [NL2(0)−NL2(ℓ2m)]ℓ2m/2 D Eallǫ
filesofallthehalosusedinthispaper. Theyspanawide (6)
rangeofpossibilities. Thesubsetoftheseprofilesthatare where ℓm is the largest ℓ where NL2(ℓ2) is non-zero. In
similar to those in cosmological N-body ΛCDM simula- the upper left panel of Figure 4, the distributions corre-
tions areshownas blackcurvesin Figure3. The simula- spondingtothemostboundparticles(blackandredhis-
tionsarerepresentedherebytworecentworks. Theblue tograms)haveℓ =1,whilethedistributioncorrespond-
m
curveis the averageprofile fromFig. 3bof Ludlow et al. ing to the least bound particles (blue histogram) has
(2011). The greencurves are the averageand upper and ℓ2 = 0.76, and near circular orbits are completely ab-
lower limits of relaxed systems taken from Lemze et al. m
sent. Inwords,Equation(6)isanexpressionforthecur-
(2012) (blue curves in their Fig. 13). Because their ra-
dius is in units of the virial radius while our systems do vatureofthelineconnectingthehighestpointofNL2(ℓ2),
not have a defined virial radius, we scaled their horizon- i.e.,atℓ=0andthelowestpoint,NL2(ℓ2 =jm2). K(ǫ)is
0forstraight-linedistributions,negativeforconvex(sag-
taasltahxeisLtuodhloawveettheals.a(m20e1v1e)lodcaittya.anTishoitsrompayyvanloutebaetrt−h2e ging)NL2(ℓ2)distributions,andpositiveforconcave(up-
optimal scaling, but whatever scaling one adopts, it is ward‘bulging’)NL2(ℓ2)distributions. Thecurvaturesof
clear that different simulations do not completely agree all six histograms in this figure are positive, K >0.
with each other. This is also clear from the examina- We define two more quantities for NL2(ℓ2). The aver-
tion of the seven β(r) profiles presented in Fig. 11b of age vertical extent of the NL2(ℓ2) histograms,
Navarro et al. (2010).
In Figure 4 we show an example of one of our halos. ∆¯ = NL2(ℓ=0)−NL2(ℓ=1) , (7)
The upper left panel shows one possible prescription for D Eallǫ
the distribution in (L/Lcirc)2 of particles binned by en- andthe vertical rms dispersion betweenthe NL2(ℓ2)his-
ergy into six energy ǫ bins. The most bound particles tograms, which we call ∆ . It quantifies how spread
rms
(black and red histograms) show nearly uniform distri- out the histograms at various energies are. If the distri-
bution in L2. A perfectly uniform distribution would butions in NL2(ℓ2) coincide at all energies, then ∆rms =
resultinanisotropic system(Williams et al.2010). The 0. For the halo shown in Figure 4, ∆¯ = 1.26 and
leastbound particles(magentaandblue histograms)are ∆ =0.26.
rms
biased toward radial orbits, and the near circular orbits Next, we quantify the shape of the VDF. Because our
arecompletely absentattheseenergies. Theupper right ultimategoalistoseparateplausiblehalosfromimplausi-
panelhasthe full N(E,L2)distribution,shownwithlin- ble ones we are especially interestedin the crater,which
ear and log vertical axis. The magenta line represents we consider unphysical. (VDFs of N-body simulations
circular orbits. The linear plot shows that for the least are either flat-topped or peaked at small speeds, see
bound particles, at ǫ>∼2, near circular orbits are absent. Kuhlen et al. 2010; Hansen & Sparre 2012). Let urm be
Thelowerleftpanelhasradial(mainplot)andtangential the radialspeedwhere N (u ) is the largest. For VDFs
ur r
DARKexp: Distribution of angular momentum 5
thatpeakatur =0,urm =0,butforthosewithacrater, For less negative energies the NL2 distributions favor
u >0. The fractional area of the radial VDF crater is low angular momentum orbits, but in such a way that
rm
urm NL2 is convex. The larger the ∆¯ the larger the velocity
anisotropy at large r. These anisotropy profiles are not
Υ= Z [Nur(ur=urm)−Nur(ur)]dur. (8) dissimilarto thoseseenincosmologicalsimulations,rep-
resentedhere by two recent works,Ludlow et al. (2011);
0
Lemze et al. (2012); see Section 3.1 for details.
In the case of Figure 4 the blue VDF in the lower left
Unfortunately, Figure 6 does not provide a com-
panel has Υ=0.058.
pletely clean separation because the shapes of VDF and
Finally, we quantify the shape of the anisotropy pro-
anisotropy profiles depend on the detailed properties of
file, β(r). We consider β(r) to be realistic if it is either
the L2 distribution,whichcannotbe fully capturedwith
increasing or staying constant with radius, and nearly
simple global parameters. In Figure 8 we show three
zero(i.e., isotropic)atverysmallradii. (We removeand
more systems inside the diagonal rectangle of Figure 6.
donotconsidersystemswithanisotropyprofilesthatare
The top set of panels shows a system that is correctly
monotonicallyfallingatallradii.) Withthatinmind,we
define β0 to be the value at log(r/r−2)=−1.5 and βmin cealivmei,naantedditbyhaosuVr DcrFitecrriaataerssu.nTrehaelimstiidc:dlietsseNtLo2fipsacnoenls-
tobetheminimumvalueattainedintheradialrangebe-
isasystemforwhichourselectioncriteriafailbyasmall
tween log(r/r−2) = −1.5 and 0. βmin is not always the amount: the system’s B = 0.106, which is just outside
sameasβ becausesomeanisotropyprofileshaveminima
0 ourlimitof0.1. Itisrepresentedbyamagentasquarein
between log(r/r−2) = −1.5 and 0, and then increase at theupperportionofthediagonalrectangle. Thebottom
larger r. We quantify the β profile with B = β +β .
0 min set of panels contains a system where our criteria fail,
Thisdefinitionissomewhatarbitrary,andothervariants
but in the opposite sense: they eliminate a realistic sys-
of B can be adopted. The anisotropy profile shown in
the lower right panel of Figure 4 has B =0.02. tem because it has a slightly concave NL2 shape, K >0
(one of the blue dots with a red cross through it).
The bottom panels of Figures 7 and 8 were generated
3.3. Relating the L2 distributions to the halo velocity
usingtheEquation(4)prescriptionfordistributingorbits
properties
in L2, while the top panels of Figure 7 and the middle
Having characterized NL2, the VDF, and β(r) with panels of Figure 8 were generated using Equation (5).
simple parameters, we can now ask how the distribu- Equation(5)(withconstantc>∼0.2)isprobablythebest
tionofangularmomentum,NL2,isreflectedinthehalos’ among the ones we tried in terms of generating systems
VDF and β(r). In Figure 5 we plot ∆¯ vs. ∆rms, both of thattendtoliemostlyinthediagonalrectangleofrealis-
whicharedeterminedfromNL2. Redtrianglepointsrep- tic halos. Note that its dependence on L2 is of exponen-
resent halos with VDF craters, Υ>0. Magenta squares tial form e−t(L2), similar to the exponential dependence
representhaloswithB >0.1. Bothofthesetypesofsys-
on energy in the DARKexp N(E).
tems are unrealistic. It is immediately obvious from the
ThesystemsinFigure7canbecomparedqualitatively
plot that these two types of systems tend to inhabit dif-
with typical halos from N-body simulations, shown in
ferent parts of the plot, roughly separated by a straight
Figure 9. This is based on a sample of 36 cluster-size
diagonalline. Mostofthesystemsclosetothelineareas-
relaxed halos extracted from a z = 0 snapshot of an
trophysically realistic (blue filled dots). However, there
N-body simulation of a standard ΛCDM cosmological
are some systems with VDF craters that are mixed in
model(fordetailsofthesimulationandthehalocatalog,
(red triangles among blue dots).
see Wojtak et al. 2008). This halo sample has already
Figure6showsonlyaportionoftheFigure5diagram;
been used for calculating the six-dimensional distribu-
it cuts out systems with large ∆¯ and ∆ , which have
rms tion function as a function of energy and angular mo-
large anisotropies, close to 1, at large radii. In Figure 6
mentum, and testing its phenomenological model with
we show that VDF crater systems (Υ > 0) can be iden-
radially changing anisotropy (Wojtak et al. 2008). Each
tified andhence eliminatedby using another propertyof
halo contains from 5×105 to 5×106 particles inside its
the NL2(ℓ2) distributions, namely the curvature K, de- virial sphere defined in terms of the mean overdensity,
fined by Equation (6). Points marked with red crosses
hρi/ρ ≈ 100, where ρ is the present critical den-
have K > 0. There is a very close correspondence be- crit crit
sity. Theenergydistributionofthesehaloswasalsoused
tween red triangles and red crosses. In other words,
in Paper III.
NL2(ℓ2) distributions that are concave (K > 0) almost We conclude that realistic systems are characterized
alwayshaveVDFcraters(Υ>0),andsystemsthathave
VDF craters are almost always concave. by two main properties of their NL2: (1) NL2 shapes
are straight or convex (K ≤ 0), and (2) the dispersion
Figure6showsthatconvexK <0systemsrestrictedto
lieintheboxaroundthediagonallinehaverealisticVDF in NL2 for different energies (∆rms) is proportional to
andβ(r)profiles. Thisselectionuses∆¯,∆ andK,i.e., the averageverticalextent of the NL2 distributions (∆¯),
rms i.e., the systems lie inside the diagonal rectangular box
it is based solely on the shape of NL2(ℓ2). In Figure 6 of Figure 6.
wehavethereforeaccomplishedourgoalofisolatingNL2
shapes that gives rise to realistic systems.
4. SUMMARY AND CONCLUSIONS
Figure 7 shows three examples of realistic halos (blue
dots inside the diagonal rectangle in Figure 6). Note WhileDARKexpN(E)wasarrivedatthroughstatisti-
thatallthree are isotropicatsmallradii,whichis a con- cal mechanics maximum entropy analysis, the extension
sequence of NL2 being nearly flat at very large negative toN(E,L2)consideredinthis paperisnot. Instead,our
energies (black and red histograms in the left panels). starting premise was Equation (3), which is the integral
6 Williams et al.
equation for the isotropic DARKexp, and is based on cles. WegivetwoexamplesofaprescriptionforN(E,L2)
the assumption that obtaining the distribution of parti- that generates realistic systems: Equations (4) and (5).
cles in E is different from that in L2. The two different The approximate constant ranges for Equation (4) are:
treatments—maximumentropyforE anda phenomeno- 1<a<2, 0.5<b<1.5, and 0.5<c<1.5, and for Equa-
logical approach for L2—are appropriate because, as we tion (5): 1<a<3, 0<b<1.5, and c>0.2.
argueinSection2,self-gravitatingcollapsingsystemsare WhatcangiverisetosuchN([L/L ]2)distributions?
circ
free to redistribute their particles in E, but not in L2. We argued above that because the particles are not well
Because the redistribution of angular momentum is not mixed in L2, the distribution in L2 probably depends
unrestricted, notall portions of the L2 space are equally on the details of the initial conditions and dynamics
accessible, making maximum entropy arguments inap- of halo collapse, like radial orbit, and other instabili-
propriate. ties, and is possibly somewhat different for isolated vs.
By investingating the properties of a large num- cosmological collapses. The advantage of our approach
ber of halos all consistent with Equation (3) we con- is that because we considered all astrophysically real-
clude that astrophysically realistic halos must have istic systems, it encompasses both of these, as well as
N([L/Lcirc]2) distributions (for each energy separately) other possible cases. Our next step is to use the form of
that are linear or somewhat convex in (L/L )2. Also, N([L/L ]2) obtained here to generate the distribution
circ circ
the N([L/L ]2) distribution for most bound particles function f(E,L2), which can be compared to f(E,L2)
circ
(largestnegativeenergies)mustbeuniform,andbecome measured from N-body simulations (e.g., Wojtak et al.
more tilted in favor of radial orbits for less bound parti- 2008).
APPENDIX
APPENDIX A
Suppose some L-distribution is conserved as a part of a statistical mechanical approach, which also conserves total
massandenergy. The distributioninenergyandangularmomentumthatcorrespondstothe mostlikelystate isgiven
by
N(E,L2)∝exp(β˜Φ −β˜E−γLξ)−1. (A1)
0
This is analogous to Equation (30) of Hjorth (1994). We show below that the density profiles that correspond to
Equation (A1) are not the same as for DARKexp, and that the velocity anisotropy profiles become more tangential
with increasing radius. Thus these systems are very different from those seen in numerical simulations.
We note that not all (γ, ξ) parameter combinations produce density profiles. Systems with γ > 0 or ξ < 0 did not
converge, and neither did systems with γ ≪ −10 or ξ ≫ 1. In Figure 10 we show two systems that did converge.
Their parameters are γ =−1, ξ =0.5, φ =6 (top panels), and γ =−5, ξ =0.2, φ =6. Other systems have similar
0 0
general characteristics.
APPENDIX B
Here we consider a special sub-class of systems described by Equation (1), which produce analytical solutions.
TheN(E,L2)distributioncanberepresentedintheE vs.L2 plane. ForeveryE thereisamaximumL=L =L
circ c
that corresponds to the circular orbit of that energy. The set of these L forms an ’upper’ envelope in the E vs. L2
c
plane. In a general case the density of particles along that envelope will vary as a function of E. Let us suppose that
there is a sub-class of systems where the density is constant. Then Equation (A1) reduces to N(E ,L2) =const. on
c c
that envelope. Equivalently, β˜Φ −β˜E −γLξ =const. Differentiating, we get
0 c c
dE γ
c
=− =const, (B1)
dLξ β˜
c
because γ andβ˜areconstants fora givensystem. The lefthand side ofEquation(B1)canbe expresseddifferently, as
it refers to circular orbits. Circular speed v is given by v2 =rdΦ/dr, and the corresponding energy is E = 1v2+Φ.
c c c 2 c
Using these we get E = 1rΦ′+Φ, and L2 =r3Φ′, where primes denote differentiation with respect to radius. Next,
c 2 c
we can obtain expressions for dE /dr and dLξ/dr, and hence
c c
dE 1 1−(ξ/2)
c = r3Φ′ . (B2)
dLξ ξr2(cid:16) (cid:17)
c
Combining Equations (B1) and (B2) we get
β˜ 2/(ξ−2)
Φ′(r)= − r(2−3ξ)/(ξ−2). (B3)
(cid:16) γξ(cid:17)
Integrating, we have
β˜ 2/(ξ−2) (2−ξ)
Φ(r)= − r2ξ/(2−ξ)+C, (B4)
(cid:16) γξ(cid:17) 2ξ
DARKexp: Distribution of angular momentum 7
and applying the Poisson equation finally gives Equation (2):
1 β˜ 2/(ξ−2) 2+ξ
ρ(r)= − r−4(ξ−1)/(ξ−2). (B5)
4π(cid:16) γξ(cid:17) (cid:16)2−ξ(cid:17)
Not all combinations of parameters β˜, γ, ξ are allowed. The factor [−β˜/(γξ)]2/(ξ−2) in the above equations tells us
that β˜/(γξ) has to be negative. There are four ways of realizing this, and we discuss these cases separately. In Ia and
Ib, the inverse temperature is negative, β˜< 0, as in DARKexp, but in IIa and IIb, it is positive. Note that all four
casesguaranteethatdE /dL >0,i.e. thatasthe energyofacircularorbitisincreasing,its angularmomentummust
c c
increase as well. On the other hand, Equation (B2) can have either sign.
Case Ia: β˜<0, γ >0 and ξ >0.
In Equation (A1) the term γLξ is independent of energy, so for β˜E →β˜Φ , N(E,L2) can get as small as −1, i.e. it
0
can become negative. This is not allowed, and so this combination of parameters is ruled out.
Case Ib: β˜<0, γ <0 and ξ <0.
N(E,L2) cannot become negative, so in general, such solutions are allowed. Looking at the factor (2+ξ)/(2−ξ) in
Equation(B5) we see that only −2<ξ <0 are allowed. The factor (2−ξ)/(2ξ) in Equation(B4) guaranteesthat the
potential is negative, so constant C can be set to zero. The two limiting solutions are: as ξ →0, ρ(r) →r−2, and as
ξ →−2, ρ(r)→r−3.
Case IIa: β˜>0, γ <0 and ξ >0.
N(E,L2) in Equation (A1) can become negative, especially when E →0, and −γLξ cannot compensate for negative
β˜Φ . Therefore this combination of parameters is ruled out.
0
Case IIb: β˜>0, γ >0 and ξ <0.
N(E,L2) is always negative, so this combination of parameters is not allowed.
We conclude that if Lξ is treated the same way as total mass and total energy, then maximizing entropy subject
to condition of Equation (B1) gives potential and density profiles that are power laws in radius, with the values of
constants β˜, γ, ξ given by Case Ib, and the allowed density profile slopes spanning the range from −2 to −3. Just
likeinAppendix A,weconcludethatthe solutionsobtainedhereareverydifferentfromDARKexp,andverydifferent
from systems obtained in numerical simulations.
It is interesting to note that Equation (A1) does not reduce to DARKexp for isotropic systems. When ξ → 0,
Equation(A1) becomes N(E,L2)∝exp(β˜Φ −β˜E−γ)−1. Because γ cannot be 0 (see EquationB1), the N(E,L2)
0
distribution does not reduce to DARKexp for any allowed parameter values.
The Dark CosmologyCentre (DARK) is funded by the Danish National ResearchFoundation. L.L.R.W. wouldlike
to thank DARK for their hospitality. The authors are grateful to Stefan Gottl¨ober, who kindly allowed one of his
CLUES simulations,
(http://www.clues-project.org/simulations.html)to be used in this paper. The simulation has been performed
at the Leibniz Rechenzentrum (LRZ) Munich.
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DARKexp: Distribution of angular momentum 9
Figure 2. Anisotropyprofilesforallthehalosusedinthispaper. Thereddashedcurvecorrespondsto 1[−dlnρ(r)/dlnr]ofDARKexp
2
φ0 = 4, and represents the upper limit on the anisotropy derived by An&Evans (2006); Ciotti&Morganti (2010). We disallowed
monotonicallydecreasingvaluesofβ(r). Smallfluctuationsinβ(r)areduetonumericalnoise. Wehighlightincolorβ(r)profilesfromtwo
prescriptions: Equation4and5areshownasmagentaandbluecurves,respectively.
10 Williams et al.
Figure 3. Similar to Fig. 2. Here we compare our anisotropy profiles (black curves) with those from numerical simulations. The blue
curve isthe average profile fromFig. 3b of Ludlowetal. (2011). The green curves arethe average and upper and lower limitsof relaxed
systemstakenfromLemzeetal.(2012)(bluecurvesintheirFig.13). Becausetheirradiusisinunitsofthevirialradiuswhileoursystems
donot haveadefined virialradius,wescaled their horizontal axis tohave the samevelocity anisotropyvalue atr−2 as the Ludlowetal.
(2011)data. Asubsetofourmodelsthatfitcomfortablywithinthegreencurveboundsareshowninblack.