Dissipative dark matter and theAndromeda plane of satellites

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Citation Randall, Lisa, and Jakub Scholtz. 2015. “Dissipative Dark Matterand the Andromeda Plane of Satellites.” Journal of Cosmologyand Astroparticle Physics 2015 (09) (September 24): 057–057.doi:10.1088/1475-7516/2015/09/057.

Published Version doi:10.1088/1475-7516/2015/09/057

Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:28265549

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Dissipative Dark Matter and the Andromeda Plane of Satellites

Lisa Randall and Jakub ScholtzDepartment of Physics, Harvard University, Cambridge, MA 02138, USA

December 8, 2014

Abstract

We show that dissipative dark matter can potentially explain the large observed mass to light ratioof the dwarf satellite galaxies that have been observed in the recently identified planar structure aroundAndromeda, which are thought to result from tidal forces during a galaxy merger. Whereas dwarf galaxiescreated from ordinary disks would be dark matter poor, dark matter inside the galactic plane not onlyprovides a source of dark matter, but one that is more readily bound due to the dark matter’s lowervelocity. This initial N-body study shows that with a thin disk of dark matter inside the baryonic disk,mass-to-light ratios as high as O(30) can be generated when tidal forces pull out patches of sizes similar tothe scales of Toomre instabilities of the dark disk. A full simulation will be needed to confirm this result.

1 Introduction

Recent observations of dwarf satellites in the An-dromeda galaxy [1] show them to have a distributionwith about half (15 out of 27) in a single plane about14 kpc thick and 400 kpc in radius. Furthermore,most of these (13 out of 15) are rotating in the samedirection. This surprising result contradicts the ex-pectation from usual cold dark matter (CDM) mod-els, which tend to predict a more isotropic distribu-tion. Similar claims have been made about the MilkyWay [2], which also exhibits more planar dwarf galaxydistribution than expected. Furthermore, other evenmore recent results show such a planar distributionof satellite galaxies to be surprisingly common, withmany galaxies located in a plane [3], although [4] cau-tions that this claim may be too sensitive to the se-lection criteria for the galaxies investigated.

Of course, this surprising result can potentially tellus about non-standard phenomena only if standardexplanations do not successfully address the data.Galactic dynamics can certainly modify spatial distri-butions from naive expectations. Indeed, tidal forcesarising from merging galaxies might well account forthe spatial distribution [5, 6]. However, even if thedynamics of tidal stripping explains the spatial struc-ture, a puzzle still remains. Studies [7, 8] show ev-idence for large mass to light (M/L) ratios in thesesatellites; we have summarized the results of thesestudies in table 1 and figure 1. Although these M/L

values come with large uncertainties, figure 1 showsthere is a consistent trend indicating many of thesedwarf galaxies are dominated by dark matter. Also,at this point, there is no obvious difference betweenthe in-the-plane and the out-of-the-plane satellites interms of their M/L values. Both in-the-plane andout-of-the-plane populations display a range of M/Lvalues – and our model will result in a similar rangeof values for the in-the-plane satellites.

Such large M/L values cannot be accounted for byjust baryonic matter and are therefore inconsistent iftidal forces are responsible for creation of these dwarfsatellites. According to conventional scenarios, notenough ordinary cold dark matter would be trappedin these structures to agree with measured mass tolight ratios since tidal stripping does not incorporateCDM. Unless an explanation for the sizable amountof nonluminous matter contained in these dwarfs, theresults will remain perplexing.

In fact, there is no known universally accepted res-olution to this puzzle. Yang et al [9] propose forthe Milky Way Galaxy that, over time, satellites ab-sorb dark matter and lose baryons, thereby raisingthe mass to light ratio. However, this resolution isspecific to the Milky Way satellites and it is not clearhow applicable it is to other systems. Another pro-posed explanation is that there is simply a good dealof nonluminous ordinary matter. But no evidence tosupport this hypothesis is observed. An alternativeexplanation was put forth by Foot et al. [10], who

1

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412.

1839

v1 [

astr

o-ph

.GA

] 4

Dec

201

4

dSph M/L(1) M/L(2) M/L(3)And I 21± 5 – 18± 7And III 14± 12 – 45± 14And IX 88+167

−88 – 404± 150And XI 78+183

−78 216+115−87 215± 162

And XII 77+176−77 0+194 0+194

And XIV 63± 55 – 71± 46And XVI 39+79

−39 – 4.2+6.4−4.2

And XVII – – 12+16−12

And XXV – 10.3+7.0−6.7 10± 8

And XXVI – 325+243−225 318± 179

Cass II – 308+269−219 318± 257

NGC 147 3± 0.5 – –NGC 185 4± 0.5 – –

Table 1: Collected values of M/L for co-rotating in-the-plane dwarf satellites of the Andromeda Galaxy.The M/L(1) values come from [7], we have computedthe M/L error bars from their data. The M/L(2)values and error bars are from Table 4 of [8]. Wehave also used the same method as [7] to computethe M/L(3) values and error bars based on data fromTable 5 of [8].

104 105 106 107100

101

102

103

LHLSUNL

ML

HMS

UN

L SU

NL

Out of the plane

In the plane, co-rot

In the plane, counter

Figure 1: Comparison between out-of-the-plane(green), co-rotating in-the-plane (blue) and counter-rotating in-the-plane (red) dwarf satellites of the An-dromeda Galaxy. Neither population appears dis-tinct in terms of their M/L values. The M/L val-ues and error bars have been obtained by applyingthe method from [7] to data from Table 5 of [8] andtherefore agree with column M/L(3) of table 1 in thiswork.

suggest that a disk of mirror dark matter might playa role, though they have no explicit mechanism ordynamics for how this came about.

Given our limited understanding of dark matter,results such as these are worthy of attention. If darkmatter continues to elude detection, it could be thatexploring matter spatial distributions throughout theuniverse will give us best clues as to whether darkmatter has interactions other than gravitational.

In this paper, we propose an admittedly specula-tive idea based on the Partially Interacting Dark Mat-ter (PIDM) scenario proposed by Fan et al. [11]. InPIDM scenarios most dark matter is non-interactingor extremely weakly interacting and resides in asmooth spherical halo, but a small fraction of thedark matter has self-interactions. Double Disk DarkMatter (DDDM) includes dissipative self-interactionsin the dark sector. If the dark matter particle is heav-ier than the proton the DDDM forms a thin disk ofdark matter embedded in the plane of the galacticdisk. We propose that tidal forces acting on such adisk, with a thicker disk of baryonic matter and athinner disk containing a small fraction of dark mat-ter, could account for observations of planes of TidalDwarf Galaxies with large mass-luminosity ratio. Al-though the dissipative dark matter constitutes onlya fraction of the mass of the initial disk, a thin diskis indicative of a dark matter mass heavier than thatof the proton as well as a velocity lower than that forbaryons. Because of the lower initial velocity and thelack of interaction with the more energetic baryons,dark matter particles are more likely to be trappedinto bound structures when tidal forces pull them out,and hence large mass to light ratios are expected.

1.1 PIDM

Reference [11] proposes that the dark matter is com-posed of at least two different sectors. The majorityof dark matter (DM) has all the assumed propertiesof CDM – it does not interact very much with ei-ther the Standard Model or with itself. However, asmall portion of the dark matter is self-interactingand dissipative. A particularly simple model con-sists of two fermions X and C charged under a U(1)′

that only mediates forces among dark sector particles.The assumed mass of X is bigger than proton mass,mX > 1 GeV, while C is light: mC . me. The U(1)′

vector (the dark photon) is massless and therefore of-fers the possibility of carrying away arbitrarily smallamounts of energy and so both X and C are able todissipate energy into the dark radiation. Whereas the

2

fermion X constitutes majority of the energy densityof the PIDM, fermion C acts a coolant because itsThompson scattering cross-section with dark photonsis much larger:

σT =8π

3

α2D

m2C

(1)

We will work with a scenario in which a galaxy con-tains equal number of oppositely charged X and Cparticles so that at sufficiently low temperatures Xand C should combine into a hydrogen-like boundstate with binding energy:

B =α2DmC

2(2)

When the temperature of the dark sector drops belowT ∼ B/20 we expect the majority of the gas to recom-bine [11]. Indeed, we assume that the gas of unboundXs and Cs continues cooling via bremsstrahlung (andinitially via Compton scattering on dark radiation)until it reaches this temperature, below which thecooling essentially stops.

The recombination temperature, T , can be compa-rable or even lower than the temperatures availableto the baryonic sector. If the X particle is heavierthan the proton, this would imply a thinner disk forthe interacting component of dark matter. In fact,for a common binding energy, the thickness of thedisk scales inversely with the mass of X and so aPIDM disk that is more narrow than the visible galac-tic plane should form. This disk is very likely alignedwith the baryonic disk due to their mutual gravita-tional attraction as well as common tidal torques act-ing on both disks. However, because of the X particlemass, we expect the dark matter disk to be narrowerthan the disk of ordinary matter. Moreover, what-ever pressure support the baryonic disk enjoys, thePIDM disk may lack. This means that dynamics forthe dark disk can be complicated. Depending on thelocal surface densities and T , the PIDM disk can beunstable and form clumps of characteristic size as de-scribed in section 2.3.

2 Methodology

2.1 Basic Dynamics

Our analysis uses as a starting point the work of Ham-mer et al., who propose that Andromeda is the re-sult of at least one merger of two large objects whosemasses differ by a factor of about three. If this is thecase, tidal forces will likely affect the overall structure

of the galaxy, ejecting some material from its progeni-tors that can remain bound into the overall structure.Using numerical simulations, the authors of [6] haveshown they can closely model the size and spatial dis-tribution of the observed tidal streams, which theythen assume can fragment into the observed dwarfgalaxies. With appropriate simulation parameters,their explanation seems to match the satellites’ pla-nar distribution well. However, they cannot explainthe large amount of nonluminous matter these satel-lites contain.

We apply the Hammer mechanism to our scenariowith progenitor galaxies that contain a thicker disk ofbaryonic matter and a thinner disk of PIDM embed-ded inside. As in the Hammer scenario, tidal forcespull out large amounts of matter from the disk, in ourcase both baryonic and dark. This is typically not thecase in the ordinary CDM scenario, because the totalamount of CDM in the galactic disk is insignificant.In our scenario too, initially, the amount of baryonicmatter far exceeds the amount of dark matter that ispulled out of the disk, with baryons dominating byroughly a factor of five or six.

However, a significant feature of PIDM is that thedispersion velocity of the dark matter can be muchlower than that of baryons. Therefore, when tidalforces pull both baryons and dark matter from thegalactic plane, dark matter particles are much morelikely to remain bound into an emergent structurewhile baryons are more likely to evaporate. Due tothe fact that the process of evaporation of unboundbaryons and PIDM is out of equilibrium, highly timedependent, and strongly influenced by energy trans-fer between the particles, we currently cannot analyt-ically predict the amount of baryons and PIDM thatstay bound. We instead perform numerical simula-tions to evaluate the fraction of baryons that mightremain.

Our goal in this paper is to show that given tidaldisruption, a large fraction of dark matter will remainbound in any dwarf satellite galaxies that might form.Therefore, in our simulations, we assume tidal forcespull out a patch of material from the original galaxy,taking the tidal disruption as a given and focusingsolely on the patch and how it evolves with no othermatter around it. The size of the patch can be deter-mined by the scales on which the tidal forces operatebut also by other scales such as the size of gravita-tionally unstable objects in the dark disk. The patchcannot be much smaller than these unstable regions,because once these instabilities form they are much

3

harder to split. On the other hand, if this patchbecomes much bigger than the size of gravitationalinstabilities we can separate the behavior into evolu-tion of each gravitational instability and further in-teraction of these instabilities. We will describe thesegravitational instabilities, clumps, in section 2.3. Ide-ally a full simulation would model the gravitationalinstabilities and the tidal disruption and determinethe characteristic scales for us. Our intention in thispaper is only to show that PIDM leads to a muchlarger fraction of dark matter than the usual CDMscenario.

A typical initial state has significantly morebaryons. About 10−20% of these would be bound ini-tially because their initial velocity is smaller than theescape velocity from the entire cloud. However, sincebaryons have much higher spatial overlap with them-selves than with the PIDM, a lot of these baryonsgain enough energy to become free. The PIDM par-ticles on the other hand, start off with lower velocityand are furthermore less likely to interact with thehigher velocity baryons and hence do not evaporateas much. Based on these simulations, we find this tobe the case, with the final dark matter mass oftenexceeding that in the baryons by at least a factor of10.

2.2 Progenitor Galaxies

Since the smaller progenitor galaxy (SPG) is morelikely to get tidally disrupted, we assume that thepatch that eventually forms the tidal dwarfs comesfrom the SPG. In order to characterize this patch weneed to know the SPG’s surface density, its local den-sity at the disrupted region and hence its scale radius,the local orbital velocity that tells us both the stabi-lizing angular momentum and which also contributesto the total kinetic energy in the patch, as well as thedispersion velocities for both the PIDM and for thebaryons. The dispersion velocity of the PIDM followsfrom the model, as shown in section 1.1. The remain-ing parameters can be estimated by rescaling the val-ues from the Milky Way galaxy to the mass of thesmaller of the progenitors estimated by [5]. Hammeret al. argue that Andromeda is a result of a mergerof two galaxies with mass ratio of about three, whichmakes the SPG’s mass about MAndromeda/4. We usethe Tully-Fisher relationship L ∝ v4 and assume thatthe mass to light ratio is similar to rescale the MilkyWay properties in order to determine the SPG prop-erties similarly to what was done in [12].

By rescaling the mass and velocity according to the

Tully-Fisher relationship, we find the best estimatefor radial disk scale to be half that of the Milky Way,or rd = 1.5kpc. Using the surface density at theSun’s location to normalize and applying the abovescale length yields at the center of the SPG Σ0 =860M/pc2 (which corresponds to 60M/pc2 in thesolar neighborhood). The rotation curve flattens outat R = 2kpc and reaches the asymptotic velocity v0 ∼165km/s.

Since the mass of Andromeda is uncertain, we alsoredo our analysis assuming Andromeda is larger by afactor of 1.5. In that case the mass of the SPG is thatof Milky Way rescaled by a factor of 1.5/4 = 0.375.This means the SPG radial scale is rd = 1.84kpc andthe orbital velocity in the flat rotation curve goes tov0 ∼ 183km/s.

2.3 Clumps

As we argued in section 2.1, each patch consists ofone or more gravitational instabilities, clumps. Bothbaryons and PIDM are subject to gravitational in-stabilities that may cause clumping in the originalsmooth distribution of matter within the disks. Ina uniform three dimensional distribution the crite-rion for existence of gravitational instabilities is theJeans mass. The Toomre criterion is the analog ofthe Jeans mass in a two dimensional system with an-gular momentum, such as disks. The Toomre crite-rion is a good indicator of the onset of instabilitiesas it describes the time evolution of an amplitude ofa small density perturbation with a particular wave-length. When ω(k), the oscillation frequency of theamplitude of a density perturbation with wavelengthk, gains a non-zero imaginary component, these den-sity perturbations grow (and quickly violate the lin-ear assumption used to derive this expression). Theexpression for ω(k) is simple:

κ2 − 2πGΣ(R)|k|+ σ2k2 = ω(k)2 < 0, (3)

where κ =√

2Ω is the epicyclic frequency of the disk(√

2 corresponds to a flat rotation curve), σ is the ve-locity dispersion of the matter in the disk and Σ(R)is the local surface density of the disk. Notice thatlarge κ2 stabilizes the disk and so does large σ2. Inthe first case, the disk is angular momentum sup-ported and in the latter it is pressure supported. Onthe other hand, the term proportional to GΣ lowersω2 and induces instabilities, just as we would expect.If the disk is unstable (ω2(k) < 0 for some k) there

4

0 20 40 60 80 100-4000

-3000

-2000

-1000

0

1000

2000

k@kpc-1D

Ω2 @Gy

r-2 D

k+k- k*

Figure 2: Illustration of the Toomre Criterion withparameters that correspond to local properties ofSPG at R = 4kpc. The red marked range of ks corre-sponds to density perturbations that grow with time.The largest and smallest k modes that exhibit grow-ing behavior are marked k− and k+, while the modethat initially grows the fastest is marked k∗.

are three wavenumbers of interest:

k+ =πGΣ

σ2+

√(πGΣ

σ2

)2

− κ2

σ2(4)

k− =πGΣ

σ2−

√(πGΣ

σ2

)2

− κ2

σ2(5)

k∗ =πGΣ

σ2(6)

Whereas k+ and k− correspond to the smallest andlargest unstable regions (for which ω(k±) = 0), re-spectively, k∗ is the wavenumber of the least sta-ble perturbation (ω2(k∗) = min(ω2) < 0). Figure 2shows the different k-numbers of interest.

Notice that 2/k∗ is equal to the characteristic scaleheight of the disk hD (under the assumption σ =σr = σz). This means that the most unstable regionis essentially the same as the region for which thethree-dimensional Jeans instability applies. However,we will focus here not on the most unstable but onthe largest unstable region of the dark disk.

In our model the velocity dispersion is purely ther-mal and the PIDM cools down to a temperature oforder T ∼ B/20 = α2

DmC/40, (B is the binding en-ergy of the dark hydrogen). As a result the velocitydispersion obeys a simple relation:

σ =αDc√

40

√mC

mX(7)

The velocity dispersion σ determines the scale height

of the disk hD:

hD =σ2

πGΣdm=

α2Dc

2

40πGΣdm

mC

mX, (8)

and therefore we can, and will, treat hD as a freeparameter in our study.

In our N-body simulations we set the size of thepatch pulled out of the SPG to be of the same orderas the size of the largest unstable region in the darkdisk. This is a self-consistent approach because all theparticles from a particular clump that might end upin the same gravitationally bound object are includedin the simulation. However, since we do not know theexact size of the patch that gets pulled out, we treatx as a free parameter and show results for a range ofx’s.

2.4 Simulation

How much of the matter that gets lifted out of thedisk remains bound in the tidal stream and how muchof the matter flows out? The answer depends onthe kinetic energy of the material that is pulled out.Because baryons under our dark disk assumptionhave bigger velocity than the dark matter particles,baryons evaporate out of the region at a higher ratethan the dark matter. Even many of those baryonson the low velocity tail that initially had low enoughvelocity to be bound will be removed by their in-teractions with other energetic baryons, which willkick them out of the bound system. So even thoughbaryons dominate the initial tidally produced clump,dark matter can dominate in the end. However, it issubtle to calculate the values in this complicated dy-namical system where energy is exchanged betweenbaryons and dark matter so we choose to do a crudedynamical simulation to evaluate what will occur.

In order to determine how many particles of eachtype stay bound to the patch that has been pulled outby tidal forces we perform a series of N-body simu-lations. We replace the original smooth density dis-tribution of matter by N particles that are modeledby Plummer spheres (defined in appendix A) withsmoothing length ε of the same order as the initialinterparticle spacing. We numerically integrate theequations of motion of these particles with a leapfrogalgorithm, as defined in appendix B. We chose theleapfrog algorithm for its relative simplicity and highdegree of energy conservation, which is important be-cause our analysis depends on determining the fi-nal distributions of binding energies of these parti-

5

cles. The first bN/6c particles are assigned param-eters to represent PIDM and the last d5N/6e repre-sent baryons. Our simulation takes the mass of eachparticle to be the same m = Mtot/N . At the end ofthe simulation we evaluate the binding energy of eachparticle:

EiB = Ei

k + Eip =

1

2mv2i −

∑i 6=j

Gm2

|ri − rj |(9)

If the binding energy EB is negative, we consider theparticle bound and assume it will become part of atidal dwarf galaxy. Conversely, if the binding energyis positive, the particle escapes and does not belong tothe bound structure. We do not observe a significantnumber of binaries1 and therefore asymptotically thisbinding energy represents true binding energy withrespect to the remnant body. In order to test this,we also compute the potential energy with respect tothe largest group of particles found by a Friends ofFriends algorithm [13]. We call these mutually dis-joint groups cores. The binding energy with respectto this core is:

EiB(core) =

1

2m(vi − vcore)2 −

∑j∈core

Gm2

|ri − rj |, (10)

where vcore is the velocity of center of mass of thecore. We confirm these two different definitions ofbinding energy give us similar numbers of bound andfree particles at the end of our simulation. However,in some scenarios the final state consists of more thanone self-bound object. In this case we compute thebinding energies with respect to two largest cores andcompare the results. In order to reduce the depen-dence of our numerical integration on the representa-tion of the initial state we repeat each simulation tentimes with exactly the same parameters but differentrandom number seeds and average over the quantitiesextracted from these simulations.

2.5 Initial State

The spatial distributions of each species are uniformin the xy−plane inside a square of size x, which wetake to be a range of values of order x = 2π/k−with densities that scale as an exponential in the z-

1We believe this is due to relatively large ε.

direction with scale heights hD and hB .

ρb =Σb(R)

2hBexp

(− |z|hB

)(11)

ρdm =Σdm(R)

2hDexp

(− |z|hD

)(12)

The velocity distributions contain two contributions.Aside from the Boltzmann distributed velocities,there is a velocity field corresponding to the bulk ro-tation of the particles with a flat rotation curve withvelocity v0 with the center of the square displaced byR from the SPG’s center.vxvy

vz

Bulk

=v0√

(x+R)2 + y2

−yx+R

0

(13)

The second contribution to the velocity distributionsis the Boltzmann distribution for each componentwith dispersion velocities σdm and σb.

2.6 Simulation Consistency Checks

In order to determine if our simulation parameters,such as ∆t and ε, were suitably chosen, we keep trackof three observables:

δ ≡ min(∆r/ε) (14)

η ≡ max

(∆~v.∆~r

∆~r.∆~r∆t

)(15)

e ≡ Ef

Ei− 1 (16)

The first observable δ shows the degree to whichsmoothing plays a role in the simulation. When δ issmall there are particles in the simulation that movevery close to each other. Although this is not a pri-ori wrong, we would like to know if this behavior issystematic or not. The second variable η representsthe maximum relative distance by which two parti-cles move towards each other per time step duringthe course of the simulation. If for any two particlesη > 1 the simulation makes too large time steps andmay miss some of the detailed dynamics. Finally thethird variable e represents the fractional change inthe total energy of the system. Since our results arebased on energies of particles in the simulation werequire that e be as small as possible. Figure 3 showsthe distribution of values of these variables for acrossall 600 individual simulations in the Benchmark set.

6

-3.5 -3.0 -2.5 -2.0 -1.5

0

50

100

150

200

log10HeL

freq

uen

cy

Energy Conservation HExtended SimulationL

-2.0 -1.5 -1.0 -0.5 0.0 0.5

0

20

40

60

80

100

120

log10HΗL

freq

uen

cy

Time Step Checks HExtended SimulationL

-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0

0

20

40

60

80

100

120

log10H∆L

freq

uen

cy

Degree of Softening HExtended SimulationL

Figure 3: These histograms show the distributions oflogarithms of the variables we use to check internalconsistency of our simulations.

You can see that in the rare cases the total energyfluctuates by up to 1%, however, this is the case foronly a few simulations and this is the most extremedeviation during each simulation. Similarly, only afew simulations contain time step during which a par-ticle moved far compared to its distance to any otherparticles, as can be seen from the distribution of the ηvariable. Finally, we see that particles do get withinthe softening distance ε because the minimum valueof δ reached down to 10−2 in some simulations.

2.7 Dependence on number of parti-cles in the simulation

Naturally, we would like our results to be indepen-dent of the number of particles we use to simulatethe continuous distributions of baryons and matter.We verify this by running a series of simulations withidentical parameters but different total numbers ofparticles N ∈ 50, 100, 200, 400, 800, 1600. We pro-cess each simulation and plot the resulting fractionsof bound dark matter and baryons in figure 4. Al-though the fraction of bound particles of each kindshows a mild dependence on the total number of par-ticles used in the simulations, these variations arewell within the error bands of each other as soon asN ≥ 100.

3 Simulations

3.1 Set 1: The Benchmark Set

We are interested in determining how the mass of thefinal state and the dark matter baryon ratio dependon x, the size of the initial patch, and hD, the thick-ness of the dark matter disk. Therefore, in this set ofsimulations we have fixed all other parameters:

R = 4kpc

ε = 0.02kpc

N = 1024

Σdm = 107M/kpc2

Σbar = 5× 107M/kpc2

hB = 0.3kpc

v0 = 180kpc/Gyr

and scanned through the sets

hD/kpc ∈ 0.01, 0.02, 0.03, 0.04, 0.05, 0.06x/kpc ∈ 0.3, 0.4, 0.5, 0.6, 0.8, 1.0

7

õõóóííáá

çç

ôô

0.00 0.05 0.10 0.15 0.20

0.00

0.01

0.02

0.03

0.04

fDM

f BA

R

õ N=1600

ó N=800

í N=400

á N=200

ç N=100

ô N=50

õõóóííááçç

ôô

0.00 0.05 0.10 0.15 0.20

0.00

0.05

0.10

0.15

0.20

fDM

f BA

R

õ N=1600

ó N=800

í N=400

á N=200

ç N=100

ô N=50

Figure 4: Convergence test: Although the fractionof bound particles of each kind shows a mild depen-dence on the total number of particles used in thesimulations, these variations are well within the errorbands we derive for these quantities. The results areshown for two different scenarios.

We have chosen the lower limit on the patch sizex = 0.3kpc because this is the scale of the Toomreinstability for hD = 0.01kpc. Moreover, the scale-height of baryons is also hB = 0.3kpc and so itseems unphysical to pull out any smaller patches.We stopped the simulation after ∆t = 2.5Gyr anddetermined the number of bound DM particles andbound baryon particles by the method described insection 2.4. The results show that indeed the boundstates contain a high fraction of dark matter – in gen-eral dominating over the baryonic component. Wefind that both final dark matter to baryon ratio andfinal mass of the resulting clumps depend on the sizeof the initial patch for reasons we now explain. Wehave ran the same analysis after ∆t = 0.5Gyr andthe results were very similar. This makes us believewe are probing an asymptotic final state and that thefinal clumps we observe have virialized.

First, we study the dependence of the final clumpmass on the size of the initial patch. Naively, we ex-pect that since the DM component is very cold, mostof it stays bound, while the baryons are very hot andinteracting and therefore many of them evaporate.Therefore, we predict that the mass of the final clumpis close to the initial mass of the DM component andso follows the relation m = Σdmx

2. Indeed, figure 5

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1.000.500.30 0.70

1 ´ 104

5 ´ 104

1 ´ 105

5 ´ 105

1 ´ 106

5 ´ 106

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SU

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Set 1: Total Bound Mass vs patch size

æ hD=10pc

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ì hD=30pc

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Figure 5: Dependence of the final mass on the size ofthe initial patch.

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SS1: Total Bound Mass vs disk thickness

æ x=0.3kpc

à x=0.4kpc

ì x=0.5kpc

ò x=0.6kpc

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ç x=1.0kpc

Figure 6: Dependence of the final mass on the thick-ness of the dark disk.

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æ hD=10pc

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Figure 7: Dependence of the final DM to baryon ratioon the size of the initial patch.

confirms this. This relation breaks down for thickerdisks. In those cases the Jeans mass of the clumps be-comes larger than their actual mass and the clumpsare not stable, losing a significant portion of theirdark matter component to evaporation.

The final ratio of dark matter to baryons shows

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oSet 1: DMB ratio vs disk thickness

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Figure 8: Dependence of the final DM to baryon ratioon the thickness of the dark disk.

strong dependence on the size of the initial patch.As the patch size x increases the amount of angularmomentum in this patch increases as x4. As a re-sult the dark matter distribution inside a large patchhas a lot of angular momentum and forms a bar. Thisbar throws away the components with the largest spe-cific angular momentum leaving behind a low angularmomentum core. During the formation of the bar,before its re-collapse, the dark matter particles aremuch closer in velocities to the baryons. Since thereare many more baryons in the phase space occupiedby the dark matter we expect lower dark matter tobaryon ratio of the final bound state. This can beseen in figures 7 and 8.

Our simulations show that PIDM can produceclumps of matter and hence possibly dwarf satel-lite galaxies with a highly increased dark matter tobaryon ratio from tidal interactions between merginggalaxies. Because this ratio is highly dependent onthe scale of the objects that are pulled out by thetidal forces, a dedicated full simulation of this eventis essential to determining the dark matter to baryonratio of the final bound object.

3.2 Set 2: Increased Mass Set

We have repeated our analysis with the same pa-rameters but with increased mass of the AndromedaGalaxy by a factor of 1.5. The SPG is then the sameas the Milky Way scaled down by a factor of 0.375.This changes the characteristic length scales and ro-tational velocities as we have indicated in section 2.2.The patches we investigated were still pulled out fromR = 4.0kpc from the center of the smaller progenitorgalaxy in order to be able to compare to scenariosfrom the Benchmark Set. We find that the change in

the dark matter baryon ratios were not statisticallysignificant. However, we observe that the total boundmass increased by a factor of 2−3 compared to the theBenchmark Set. When we rescaled Andromeda massby a factor of 1.5, the radial scale factor of the disk in-creased by a factor of

√1.5. Therefore, at R = 4.0kpc

the density increased by a factor of ∼ 1.65. Thismeans that the increase in mass was not linear, andthe essential non-linearity of this gravitational systembecame apparent.

3.3 Set 3: Different Location in theDisk

We performed another study in which we changed thelocation of the original patch while keeping all otherparameters fixed. We chose a patch one scale radiusfurther away from the center of the SPG – at R =5.5kpc, as opposed to the original R = 4.0kpc used inthe Benchmark Set. We did not explore any patchescloser to the center of the SPG, because these patchesare much less likely to get pulled out (objects closerto the center are more strongly bound). Similarly,we did not chose patches any further from the centerof the SPG because the final masses of the resultingbound objects become too small. However, note thatif more than one patch get pulled out of the SPGand subsequently merge they can still form massiveobjects.

We find that the final dark matter to baryon ratiosare not statistically different between the R = 4.0kpcand 5.5kpc scenarios. While we expect the finalbound masses to be at least e−1 lower, they in factturn out to be lower by a factor of ∼ 5, most likelydue to non-linearity of the system just as it was thecase in set 2.

3.4 Set 4: Fixed Mass Set

Our previous simulations have shown a strong de-pendence on the size of the initial patch. However, insimulations with fixed number of particles and surfacedensity, increasing the patch size means that eachparticle has to have a larger mass. In order to checkthat this scaling is not an artifact of our choices ofparticle mass we set up another set of simulations. Inthis set we changed the number of particles in pro-portion to the total simulated mass with every othervariable fixed as in the previous case. Table 2 showsthe values of N we used in order to fix the mass ofeach particle to 104M.

9

x[kpc] N(Extended Run) N(Fixed Mass Run)0.3 1024 5400.4 1024 9600.5 1024 15000.6 1024 21600.8 1024 3840

Table 2: Values of N for different patch sizes for boththe Extended Run and the Fixed Mass Run.

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æ hD=10pc

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Figure 9: Dependence of the final mass on the sizeof the initial patch. The results from the BenchmarkSet are dotted.

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Figure 10: Dependence of the final dark matter masson the thickness of the dark disk. The results fromthe Benchmark Set are dotted.

The results are very similar to the Benchmark Setand we do not see any systematic deviations, as canbe seen from figures 9, 10, 11 and 12.

4 Conclusion

In this paper we have investigated a possible expla-nation for the planar distribution of satellite dwarf

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Figure 11: Dependence of the final DM to baryonratio on the size of the initial patch. The resultsfrom the Benchmark Set are dotted. We have addeda small offset to the x values for different series inorder to allow the reader to resolve all the error bars.

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Figure 12: Dependence of the final DM to baryonratio on the thickness of the dark disk. The resultsfrom the Benchmark Set are dotted. We have addeda small offset to the hD values for different series inorder to allow the reader to resolve all the error bars.

galaxies with high dark matter content. We find thatpresence of a thin dark disk formed from cooled dis-sipative dark matter during galactic collisions can re-sult in formation of DDDM rich tidal dwarf galaxiesafter galactic mergers. In our simulations, the finaldark matter to baryon ratio could reach as high as∼ 30. Such numbers are in agreement with some ofthe observed M/L values for the tidal dwarf in theplane around Andromeda galaxy. However, this ra-tio depends on the size of the region that gets ejectedduring the merger of the two galaxies. Our study doesnot determine the scale of these patches and furtherfull simulations of galactic mergers containing darkdisks have to be performed in order to answer thisquestion.

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The experimental values of M/L show a significantrange and our model could account for this by identi-fying different final values of M/L with different sizedpatches or patches that were pulled out from differ-ent positions in the disk of one of the proto-galaxies.A full N-body simulation of the merger would allowus to predict the distribution of M/L values for thesetidal galaxies and put additional constraints on ourmodel. It will be interesting when measurements andN-body simulations advance to the point we will beable to constrain the parameters of these models.

We expect that out-of-the-plane satellites havelarge M/L values. At this point, it seems coinci-dental that they are not statistically different fromin-the-plane satellites in terms of their M/L.

There are several potentially testable consequencesof this model as well as exciting questions in need ofanswer:

We predict that these tidal dwarf galaxies are dom-inated by their DDDM component. As a result theirdensity profiles may deviate from the CDM and bary-onic density profiles we would expect to find in thesetidal dwarf galaxies. We are currently investigatingwhat shape these profiles take in order to allow futurecomparison with experimental data and distinguishPIDM dominated galaxies from ones with high usualCDM content.

Another observational strategy could be to searchfor radiation from these objects. We do not expectany annihilation signals from these DDDM rich tidaldwarf galaxies, because the DDDM was treated asasymmetric. However, if there is non-zero kineticmixing between the ordinary and dark photons, itmay be possible to detect radiation from the excita-tions of the bound states in DDDM and a detailedstudy is necessary to investigate the potential con-straints on these models.

It will also be of interest to study further the metal-licity of the dwarf galaxies. In present day galax-ies we observe a correlation between metallicity ofgas clouds and their temperature (higher metallicityallows faster cooling to lower temperatures). Sincewe also observe a correlation between velocity dis-tributions (temperature) and baryon abundance inthe tidal dwarf our scenario predicts tidal dwarfswith higher metallicity should contain larger baryoniccomponent as can be observed in the dwarf galaxies.However, a more quantitative statement can be onlyobtained after detailed full simulation which tracksmetallicity of the baryonic gas.

In conclusion, our work illustrates the possibility

of achieving high mass to luminosity ratios in tidaldwarf galaxies. This mechanism is unique to thePIDM scenario, which has just the right propertiesto allow for low velocity dark matter that can boundinto satellites. A dedicated full N-body simulationof galactic mergers with embedded DDDM disks canhelp confirm these predictions.

5 Acknowledgements

We are grateful to Matthew Walker for alerting usto the issue of the planarity of dwarf galaxies in An-dromeda and other galaxies and for many helpful dis-cussions on the subject. We would also like to thankScott Tremaine, Lars Hernquist, James Guillochon,Shy Genel and especially Matt Reece for useful dis-cussions and comments on our manuscript. The workof LR was supported in part by NSF grants PHY-0855591 and PHY-1216270. JS would like to thankthe Center For Fundamental Laws of Nature Initia-tive.

A Plummer Sphere

In order to model an extended distribution of masswe use the Plummer Sphere[14]. The mass densitydistribution for this model is:

ρ(r, ε) =3M

4π

ε2

(r2 + ε2)5/2

(17)

This gives simple expressions for the potential andforce between the two particles:

Φ(r) = − GM2

(r2 + ε2)1/2

(18)

~F (r) = − GM2~r

(r2 + ε2)3/2

(19)

(20)

Notice that the potential and force reduce to theirNewtonian expressions in the limit ε → 0. However,the choice ε 6= 0 regulates the singularity at r = 0.

B Leapfrog Algorithm

In order to numerically integrate the equations ofmotion we have to choose a particular integrationmethod. We chose a relatively simple Leapfrog algo-rithm. This algorithm works with positions at inte-gers steps and velocities at half-integers steps, which

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is where its name comes from. Schematically it canbe written down as:

xi = xi−1 + vi−1/2dt

ai = F (xi)/m

vi+1/2 = vi−1/2 + aidt

xi+1 = xi + vi+1/2dt

...

where F (x) is the net force acting on the particle. Inour case it is the sum of the gravitational forces fromall other particles in the simulation.

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