Indirect Dark Matter Detection for Flattened Dwarf Galaxies Jason L. Sanders,1, ∗ N. Wyn Evans,1, † Alex GeringerSameth,2, ‡ and Walter Dehnen3, § 1
arXiv:1604.05493v4 [astroph.GA] 16 Sep 2016
Institute of Astronomy, Madingley Rd, Cambridge, CB3 0HA 2 McWilliams Center for Cosmology, Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213 3 Department of Physics & Astronomy, University of Leicester, University Road, Leicester, LE1 7RH (Dated: September 19, 2016) Gammaray experiments seeking to detect evidence of dark matter annihilation in dwarf spheroidal galaxies require knowledge of the distribution of dark matter within these systems. We analyze the effects of flattening on the annihilation (J) and decay (D) factors of dwarf spheroidal galaxies with both analytic and numerical methods. Flattening has two consequences: first, there is a geometric effect as the squeezing (or stretching) of the dark matter distribution enhances (or diminishes) the Jfactor; second, the line of sight velocity dispersion of stars must hold up the flattened baryonic component in the flattened dark matter halo. We provide analytic formulae and a simple numerical approach to estimate the correction to the J and Dfactors required over simple spherical modeling. The formulae are validated with a series of equilibrium models of flattened stellar distributions embedded in flattened darkmatter distributions. We compute corrections to the J and Dfactors for the Milky Way dwarf spheroidal galaxies under the assumption that they are all prolate or all oblate and find that the hierarchy of Jfactors for the dwarf spheroidals is slightly altered (typical correction factors for an ellipticity of 0.4 are 0.75 for the oblate case and 1.6 for the prolate case). We demonstrate that spherical estimates of the Dfactors are very insensitive to the flattening and introduce uncertainties significantly less than the uncertainties in the Dfactors from the other obper cent servables for all the dwarf spheroidals (for example, +10 −3 per cent for a typical ellipticity of 0.4). We conclude by investigating the spread in correction factors produced by triaxial figures and provide uncertainties in the Jfactors for the dwarf spheroidals using different physicallymotivated assumptions for their intrinsic shape and axis alignments. We find that the uncertainty in the Jfactors due to triaxiality increases with the observed ellipticity and, in general, introduces uncertainties of a factor of 2 in the Jfactors. We discuss our results in light of the reported gammaray signal from the highlyflattened ultrafaint Reticulum II. Tables of the J and Dfactors for the Milky Way dwarf spheroidal galaxies are provided (assuming an oblate or prolate structure) along with a table of the uncertainty on these factors arising from the unknown triaxiality. PACS numbers: 95.35.+d, 95.55.Ka, 12.60.i, 98.52.Wz
I.
INTRODUCTION
In recent years, gammaray observations of Milky Way dwarf spheroidal galaxies (dSphs) have led to great strides in sensitivity to dark matter annihilation. Here the goal is to probe particles which interact with the Standard Model with the wellmotivated weakscale annihilation cross section hσvi ' 3 × 10−26 cm3 s−1 . Particles having this cross section will exist today with an abundance equal to that observed for dark matter ΩDM , making this socalled relic cross section a natural target for experimental searches for annihilation. Combined analyses of dSphs using data from Fermi Large Area Telescope (LAT) first ruled out the relic cross section for dark matter particle masses of a few tens of GeV [1, 2] and followup analyses incorporating more dSphs and increased observation time continue to improve sensitivity [e.g. 3–5]. For higher dark matter masses (M & TeV), the three ma
∗ Electronic
address: address: ‡ Electronic address: § Electronic address: † Electronic
[email protected] [email protected] [email protected] [email protected]
jor Cherenkov telescope collaborations continue to invest significant time on pointed observations of Milky Way dSphs. The resulting upper limits are two to three orders of magnitude from the relic cross section [6–10], but the situation bodes well for the future CTA project [e.g. 11]. An exciting development in this field is the recent and ongoing discovery of large numbers of new Milky Way satellites made possible by widearea photometric surveys [e.g. 12–15]. Since 2015 the number of known Milky Way satellites has approximately doubled thanks to Southern hemisphere data from the Dark Energy Survey and PanSTARRS. These new dSphs have the potential to significantly build on current efforts to uncover evidence of dark matter annihilation [e.g. 16–20]. Intriguingly, the first of these new dwarf spheroidal galaxies discovered, Reticulum II, shows indications of a gammaray signal exceeding background in the FermiLAT data [17]. Two methods of modeling the gammaray background yield falsealarm probabilities of p = 0.0001 and 0.01 for detecting such a signal. Subsequent analysis [19] confirmed the results of [17] and argued that the Reticulum II signal was consistent with the gammaray excess reported from the Galactic Center and claimed
2 as dark matter. With a reprocessing of the raw Fermi data [18], the FermiLAT Collaboration found an increased probability for a background fluctuation explaining the Reticulum II signal (p = 0.05) and concluded the signal is insignificant. Making sense of the results of [17] and [18] is complicated by the fact that the two datasets are only partially independent, sharing approximately half the detector events. A separate analysis is needed to compute joint probabilities of background fluctuation in the partially correlated datasets. In this work we follow a different path towards assessing dark matter interpretations of gammaray signals. Rather than analyzing the gammaray data, we consider the determination of the dark matter content of the Milky Way’s dSphs, a necessary ingredient for performing optimized combined searches using dSphs. A critical test of any alleged dark matter signal from dSphs is that the amplitude of the gammaray signal must scale amongst the dSphs according to their Jfactors (see, e.g., [21, 22]). The Jfactor is the square of the dark matter density integrated along the line of sight and over the solid angle of the observation, Z Z J= ρDM 2 (`, Ω)d`dΩ. (1) While annihilating dark matter models are theoretically better motivated, there are models in which dark matter decays [23]. In these models, the relevant astrophysical factor is the Dfactor, which is the dark matter density integrated along the line of sight and over the solid angle of the observation. Robust determinations of the relative Jfactors is of prime importance. For instance, the FermiLAT Collaboration [18], under the assumption that each of eight considered dSphs was equally likely to produce a signal, further diluted the significance of the Reticulum II gammaray excess to p = 1 − (1 − 0.05)8 = 0.33, concluding that it is insignificant. However, there are reasons to doubt the usefulness of this argument as Reticulum II is closer and very highly flattened, both of which can enhance the amplitude of an annihilation signal compared to other dSphs. Therefore, we require accurate relative estimates of the J and Dfactors, but unfortunately the data on the most tempting dSph candidates are often of limited quality. Motivated by this [24, hereafter Paper I] provided simple formulae for the J and Dfactors for a spherical NFW profile and infinite spherical cusps. The formulae relied on the empirical law that the mass within the halflight radius is well constrained as [25, 26] Mh = M (Rh ) ≈
5 hσ 2 iRh , 2G los
(2)
where Rh is the (projected) halflight radius of the stars 2 and hσlos i is the luminosity weighted squared lineofsight velocity dispersion. However, an entirely characteristic feature of dSphs is in the name – spheroidal! They are flattened (with a typical ellipticity between 0.3 and 0.5), and some of the
ultrafaints are very highly flattened with ellipticities exceeding 0.5, such as Hercules [27], Ursa Major I [28], Ursa Major II [29], and indeed Reticulum II [30]. Therefore, the underlying physical model of a spherical dark halo containing a round distribution of stars may fail to capture important aspects of the physics. Here we extend the scope of spherical analyses, to account for the effects of flattening in both the stellar and dark matter profiles. Bonnivard et al. [31] provided a systematic investigation of Jfactors of flattened figures. Here, two mildly triaxial numerical models of dSphs (created for The Gaia Challenge) were viewed along each of the short, medium and long axes. This investigation revealed that the projection effects can have a significant impact on the velocity dispersion, and concluded that the Jfactors constructed by Jeans analyses can vary from the true values by ∼ 2.5. Recently, [32] computed Jfactor estimates for the dSphs using axisymmetric Jeans modelling. These authors attributed the differences between their measured Jfactors and those from spherical analyses primarily to other modelling assumptions. It is natural to expect that the dissipationless dark matter distribution is rounder – or at least no more flattened – than the dissipative baryonic component. So, large classical dSphs which appear roundish on the sky (such as Leo I and II) may have almost spherical dark matter halos. However, the dark halos of the ultrafaints are expected to be more highly flattened than those of the classical dSphs, as it is known that baryonic feedback effects drive the dark matter distribution towards sphericity [33, 34]. The ultrafaints have such a puny baryonic content that pure dissipationless simulations [35, 36], which find strongly triaxial and nearly prolate dark halos, may be a much better guide to the true shape. For instance, recent simulations have found that the baryonic distribution is just ∼ 10 per cent flatter than the darkmatter distribution for darkmatter halos of 1010 M [37]. Throughout this paper, we work under the assumption that the dark matter distribution is flattened in the same way as the stellar distribution. The effects of flattening can be understood qualitatively for a few simple configurations. The simplest is the faceon case when the darkmatter and stellar distributions are flattened along the lineofsight. Observationally, the isophotes still appear circular and the measured halflight radius remains the same, but we have increased (decreased) the density of dark matter in the oblate (prolate) case. Naturally, this effect – which we refer to as the geometric factor – gives rise to a larger (smaller) Jfactor than a spherical analysis would infer. But, we must also consider the effect of flattening on the lineofsight velocity dispersion, which we call the kinematic factor. For the oblate case, the stellar distribution is more compressed, so the lineofsight dispersion is now smaller than the spherically averaged dispersion. Less contained mass is inferred and so the spherical Jfactor underestimates the total Jfactor. Therefore, for faceon viewing of an oblate figure, both the geometric and kine
3
II.
MADETOMEASURE FLATTENED EQUILIBRIA
We begin our analysis of the Jfactors of flattened dSphs with numerical models constructed by the madetomeasure (M2M) methods [38] as implemented by Dehnen [39]. The models are twocomponent: dark and stellar. Each component has a target density of the form ρ(m) ∝ p−1 q −1
m −γ rs
1+
m α (γ−β)/α rs
sech
m , (3) rt
x/arcmin
x/arcmin
Oblate
Prolate
10 2 10 1 10 0 10 1 10 2 10 1
10 0
10 1
R/arcmin
10 2 10 5 0
vlos /km s
30 20 10 0 10 20 30 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00
y/arcmin
30 20 10 0 10 20 30 10 3
p(vlos )/(km s −1 ) −1
y/arcmin
30 20 10 0 10 20 30 30 20 10 0 10 20 30
M ¯ /arcmin −2
matic effects cause the Jfactor inferred from a spherical analysis to be an underestimate of the true value. For the prolate case, the velocity dispersion is larger than the sphericallyaveraged dispersion and so more mass is inferred and the spherical Jfactor is an overestimate. When the dSph is viewed edgeon such that it appears flattened in the sky, the combined result of the kinematic and geometric effects is less clear. For oblate figures, the density is increased over the spherical case, whilst the halflight radius remains the same. These geometric effects cause the Jfactor assuming sphericity to be an underestimate. However, the kinematic factor works the other way, as the measured velocity dispersion is greater than the spherical average. We will see that the combination of these two competing effects leads to a small decrease in the true Jfactor over that inferred from a spherical analysis. For the prolate case, we have the converse situation with the geometric factor leading to an overestimate whilst the kinematic factor leads to an underestimate. However, now the stretching of the stellar profile in the sky causes the halflight radius to increase. We will see that the net result is an increase in the true Jfactor over the spherical Jfactor. This qualitative explanation is tested in Section II where we construct equilibrium models of the Reticulum II galaxy via the madetomeasure method. We explore a range of different flattenings and provide simple fits for the correction factors. In Section III, we use these fits to derive Jfactors for the known dwarf spheroidals under the assumption that they are either prolate or oblate. In Section IV, we build intuition for our numerical results by considering two families of axisymmetric equilibria for which analytic progress is possible and present a more rapid general approach for estimating the correction factors using the virial theorem. Section V extends these findings to the triaxial case and demonstrates how the correction factors vary for a triaxial figure as a function of the viewing angle. In Section VI, we discuss the constraints and evidence on the intrinsic shapes and alignments of the Milky Way dSphs and give estimates of the uncertainties in the Jfactors of the dSphs due to unknown triaxiality. In Section VII we summarize our findings and discuss possible implications for the claimed signal from Reticulum II in light of our work.
5 10 15 −1
FIG. 1: Reticulum II M2M equilibria of a flattened Plummer distribution of stars in a flattened NFW dark halo. The top left panel shows the logarithm of the projected mass distribution of an oblate model viewed edgeon with axis ratio 0.4. The contours are logarithmically spaced. The top right panel shows the logarithm of the projected mass distribution of a prolate model viewed edgeon with axis ratio 0.4. Note the ‘X’shape in the prolate case. The bottom left panel shows the surface density profiles in elliptical bins with Plummer profile fits (oblate in blue, prolate in dashed green). The bottom right panel shows the lineofsight velocity distributions (oblate in blue, prolate in dashed green).
where m2 = x2 + (y/p)2 + (z/q)2 . This is the familiar double powerlaw with scale radius rs , with an exponential taper at the tidal radius rt . For r rs , the density falls like r−γ , whilst for r rs , it falls like r−β . The case α = 1, β = 3, γ = 1 is the NFW dark halo. Plummer models are often used to describe the light profiles of dSphs [see e.g., 40, 41]. They correspond to the parameters α = 2, β = 5, γ = 1 and rt = ∞. We begin by constructing two flattened spheroidal (p = 1) models of the Reticulum II dSph. For both models, the dark halo is a NFW model (α = 1, β = 3, γ = 1, rs = 1, rt = 10). The stars follow a Plummer profile (α = 2, β = 5, γ = 0, rs = 0.5, rt = 9). The chosen ratio of the dark matter scale radius to the stellar scale radius lies within the measured range for the Local Group dSphs (∼ 1.25 to ∼ 30) [42]. The two models differ in their shape. The first model is oblate in both the stars and the dark matter with an axis ratio of q = 0.4 (chosen to match the observed axis ratio of Reticulum II of 0.39 [30]). The second model is prolate with an axis ratio of q = 2.5. When viewed along the xaxis both models appear flattened with axis ratio 0.4. In addition, we construct a third spherical model as a reference. This has the same parameters, but without the flattening in either
4 the dark matter or the stars. The dark NFW halos source the potential (computed using a biorthonormal basis expansion [39]) in which the weights of the Plummer models are adjusted until the target densities are reached. No other constraints on the distribution functions are used. We use a 107 particle realization of the flattened NFW distribution to compute the potential. The constraints on the Plummer model are generated with 100 realizations of 106 particles and 106 particles are used in the M2M simulation. To check convergence, the models were run turning off the weight adjustment in the M2M code. Both flattened models exhibit a slow drift in the density constraint suggesting they are not perfect equilibrium models. However, this is almost certainly true for the actual dSphs which reside in the tidal field of the Milky Way. Reticulum II has a halflight major axis length of 5.63 arcmin, is at a distance of ∼ 30 kpc [30] and has a lineofsight velocity dispersion of 3.22 km s−1 [43]. To match the final models to the observed constraints on Reticulum II, we compute the projected halflight major axis length (fitted with a Plummer model) and the projected lineofsight velocity dispersion. We then compute the scale factors R and V that scale the radial distributions and the velocity distributions to the observations. The corresponding total mass of the dark matter profile (set to unity in the simulation) is then scaled by a factor M = RV 2 . For the spherical model we match the halflight major axis length to an ‘ellipticitycorrected’ radius given by the geometric mean of the halflight major and minor axis lengths. This is related√to the observed halflight major axis length Rh as Rh 1 − where is the ellipticity. In Figs. 1, we show the final projected distributions of the two flattened models. Note that for the prolate case, the models do not completely reproduce the target density profile as there is a clear ‘X’ shape in the (x, y) plane. Additionally, we show the surface density of the two models (using a masstolight ratio of 500, [43]) and the lineofsight velocity distributions. The prolate velocity distribution is slightly peakier than the oblate case but such a small difference would not be detectable observationally. To explore the effects of adjusting the stellar and darkmatter profiles, we also build two further models, one with a central cusp in the stellar profile (γ = 1) and one with a cored darkmatter profile with parameters α = 1, β = 4 and γ = 0.
A.
source is given by1 Z +∞ Z Dθ Z 2π 1 dz dR R dφ ρ2DM , J(θ) = 2 D −∞ 0 0
(4)
where D is the distance to the source (30 kpc for Reticulum II) and θ is the beam angle. Similarly, the Dfactor is given by Z +∞ Z Dθ Z 2π 1 D(θ) = 2 dz dR R dφ ρDM . (5) D −∞ 0 0 In Table I we report the J and Dfactors at θ = 0.5◦ (the typical observational resolution). We also show the J and Dfactors for the spherical model computed from the formulae of Paper I. We see that these formula underestimate the Jfactor by a factor 1.2 and the Dfactor by a factor 1.05. We also record the correction factor between the prolate / oblate models and the spherical models using the notation FJ = log10 (J/Jsph ), FD = log10 (D/Dsph ).
(6)
The oblate model with NFW dark matter and Plummer light has a Jfactor that is diminished by a factor 1.4 over the spherical model and a Dfactor that is diminished by a factor 1.3. On the other hand, the prolate model has an enhancement in the Jfactor by a factor 3.4 and a small decrease in the Dfactor of 10 per cent. The nearprolate model with a cuspy stellar profile produces a very similar Jfactor to the Plummer prolate model, but here the Dfactor is enhanced over the spherical model by 20 per cent. Finally, in a similar fashion to the prolate NFW profile, the prolate cored dark matter profile also produces an enhancement in the Jfactor of a factor 3 and a small diminution in the Dfactor of order 10 per cent. B.
A Range of Flattenings
We have established that a prolate model of Reticulum II viewed edgeon produces a significant enhancement in the Jfactor over its spherical counterpart, whilst an oblate model has a slight diminution. However, the observed dSphs span a whole range of ellipticities, so we now go on to explore models with a variety of flattenings. We construct 3 oblate M2M models with the same parameters as the spherical reference model in Table I but with flattenings q = 0.5, 0.6, 0.7, and similarly 3 prolate M2M models with flattenings q = 1.423, 1.667, 2. Again the M2M models are normalized to match the lineofsight velocity dispersion and halflight majoraxis length of Reticulum II.
J and Dfactors 1
For our five models of Reticulum II, we proceed to calculate the J and Dfactors. The Jfactor for a distant
When computing these integrals numerically, we have found it useful to perform the coordinate transformation tan χ = z/rs where rs is the scale radius of the density profile.
5 −5
TABLE I: J and Dfactors for a beam angle of 0.5◦ for a series of Reticulum II models. The Jfactors are in units of GeV2 cm and the Dfactors are in units of GeV cm−2 . Each model was normalized such that the lineofsight velocity dispersion and halflight major axis length matched that of Reticulum II. For the spherical model, an ‘ellipticitycorrected’ halflight radius of √ Rh 1 − (where is the ellipticity) was used to scale the models. Note the correction factors are with respect to the spherical NFW, spherical Plummer model in the first row not with respect to the corresponding spherical model. Model Spherical NFW Spherical Plummer
Paper I J Paper I D log10 (J(0.5◦ )) log10 (D(0.5◦ ))
FJ
FD
18.56
17.56
18.64
17.58
0.00 0.00
Oblate NFW, p = 1, q = 0.4 Oblate Plummer, p = 1, q = 0.4


18.45
17.62
−0.19 0.05
Prolate NFW, p = 1, q = 2.5 Prolate Plummer, p = 1, q = 2.5


19.05
17.67
0.40 0.09
Nearprolate NFW, p = 0.5, q = 0.4 Nearprolate cuspy Plummer α? = 2, β? = 5, γ? = 1, p = 0.4, q = 0.38


19.01
17.78
0.37 0.20
Prolate cored DM αDM = 1, βDM = 4, γDM = 0, p = 1, q = 2.5 Prolate Plummer, p = 1, q = 2.5


18.90
17.66
0.26 0.08
The J and Dfactors for our series of models are plotted in Fig. 2. All models are viewed such that they appear maximally flattened (along the shortaxis for the prolate cases and along the longaxis for the oblate cases). We also show the J and Dfactors computed using the simple formulae (equations 15 and 19) from Paper I. We see that this formula disagrees with the spherical case by ∼ 0.2 due to the use of the empirical relation for the halflight mass. As shown in Paper I, for most dSphs this is less than the uncertainty in the Jfactor due to uncertainties in the lineofsight velocity dispersion and halflight radius. The prolate models produce a sequence of more enhanced Jfactor at all angles as we increase the flattening q. The oblate models produce a similar sequence of decreasing J as we decrease the flattening q. These trends are reproduced in the Dfactor. Note the asymmetry with q in both J and D: the equivalent flattening for an prolate model produces a larger difference from the spherical model than the corresponding oblate model. With this sequence of models, we also investigate how the Jfactor for an apparently round dSph changes as the dSph is flattened along the line of sight. In Fig. 3, we show the range of J and Dfactors for the set of flattened models viewed faceon such that the isophotes appear round and all models have the same halflight radius. We see that the range of possible Jfactors with flattening along the line of sight varies by a factor of 10. The oblate models all have a similar decrease in the Jfactor. The Dfactor is unaffected by flattening along the line of sight. For the series of flattened M2M models, we compute the correction factors FJ and FD by comparing each model with the spherical model with the same lineofsight velocity dispersion and the ‘ellipticitycorrected’
TABLE II: Slopes η of the base10 logarithms of the correction factors with respect to log10 q fitted to the madetomeasure models of Section II. The prolate and oblate cases are treated separately. The multiplicative factor by which a J or Dfactor from a spherical analysis must be corrected is given by q η . Note the spherical models to which we√compare use an ‘ellipticitycorrected’ halflight radius of Rh 1 − where = 1 − q is the ellipticity in the oblate case and = 1 − 1/q in the prolate case. View η Oblate (q < 1) η Prolate (q > 1) FJ Edgeon 0.534 0.899 Faceon 1.647 1.181 FD Edgeon 0.056 0.177 Faceon 0.335 0.089
halflight radius. The trends of FJ and FD with respect to q are very smooth so we opt to fit the corrections from the models with a simple functional form
Ffit = η log10 (q)
(7)
where we fit q < 1 and q > 1 separately. The values of η chosen are given in Table II. Although our fit is an extrapolation for q < 0.4 and q > 2.5, we will see that it agrees well with the more involved models of Section IV. In reality, the correction factors are a function of the beam angle. We have found that the correction factors are very insensitive to the beam angle so this formula is appropriate for all dSphs irrespective of their size compared to the resolution of the instrument.
6
q
q 19.5
log 10 (J(θ)/ GeV 2 cm −5 )
19.2 19.0 18.8 18.6 18.4 18.2 18.0 17.8 18.0
Edgeon
19.0 18.5 18.0 17.5
Faceon
18.0
17.5 17.0 16.5 16.0 15.5 15.0
17.5 17.0 16.5 16.0 15.5 15.0
0.0
0.1
0.2
0.3
0.4
0.5
θ/ deg FIG. 2: J and Dfactors as a function of beam angle for a range of flattened models viewed edgeon with identical lineofsight velocity dispersions and halflight majoraxis lengths. The models are coloured by the flattening in the density of both the stars and dark matter, q. The spherical model is shown with the shortdashed line, whilst the analytic formula for the NFW model (equations (15) and (19) from Paper I) is shown with the longdashed line.
III.
0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50
log 10 (D(θ)/ GeV cm −2 )
log 10 (D(θ)/ GeV cm −2 )
log 10 (J(θ)/ GeV 2 cm −5 )
0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50
J AND DFACTORS FOR THE MILKY WAY DSPHS
We now apply the corrections to the J and Dfactors of the observed dSphs. They are listed in Table III along with their measured ellipticities = 1 − b/a where b/a is the observed axis ratio. We take the majority of the ellipticities and ±1σ errorbars from the review of [44]. The ellipticities of the new dSphs discovered in the Dark Energy Survey are taken from [30], the ellipticity of Pisces II is taken from [45] and that of Hydra II from [46]. For both Leo T and Horologium I, only upperbounds on the ellipticity are available. For each dSph, we compute the correction factor assuming the dSph is either oblate or prolate and observed edgeon. We draw samples from the error distributions of the ellipticities and compute the median and ±1σ values of the correction factors for both the J and Dfactors using equation (7). The baseline spherical model to which we √are comparing uses an effective halflight radius of Rh 1 − . We combine these estimates with the spheri
0.0
0.1
0.2
0.3
0.4
0.5
θ/ deg FIG. 3: J and Dfactors as a function of beam angle for a range of flattened models viewed faceon with identical lineofsight velocity dispersions and halflight majoraxis lengths. The models are coloured by the flattening in the density of both the stars and dark matter, q. The spherical model is shown with the shortdashed line and the model using the formulae from Paper I is shown with the longdashed line.
cal estimates computed in Paper I (adding the errors in quadrature). The results of this procedure are reported in Tables III and IV. We show this data in Fig. 4. We plot the distribution of J and Dfactors for the dSphs assuming they are spherical, oblate or prolate. Ursa Major I has the largest ellipticity and hence the largest prolate correction factor (a factor of ∼ 4). Reticulum II, Ursa Major II and Hercules all have ellipticities ∼ 0.6 and so the prolate correction factors are approximately ∼ 2.2 − 2.7. For ellipticities less than ∼ 0.4, the correction factors are less than the errors on the spherical Jfactors. For every dSph the correction to the Dfactors is smaller than the errors in the spherical Dfactor. Hence, we conclude that flattening has a negligible effect on the Dfactor estimates. If the entire population of dSphs is prolate then only Tucana II and Willman 1 have potentially higher Jfactors than Reticulum II, with both Ursa Major II and Segue 1 having a very similar Jfactor to Reticulum II. We remark that Tucana II is consistent with having circular isophotes [30], whilst the assumption of dynamical equilibrium for Willman 1 is dubious [47]. Similarly, Ursa
7 Major II appears to be in process of severe tidal disruption [29]. Finally, the Jfactor of Segue 1 has been shown to be extremely sensitive to the presence of foreground contaminants [e.g., 48, 49]. These final three dSphs have been marked in red in Figure 4 to indicate their dubious Jfactors. Therefore, it is possible that the Reticulum II gammaray signal may be due to annihilation if the dwarf has a prolate shape. We can robustly conclude from Figure 4 that if Reticulum II has a prolate shape then an observed annihilation signal from only Reticulum II is not in tension with the lack of signals from all the other dSphs irrespective of their shapes. If, however, Reticulum II is oblate and has an observed annihilation signal we begin to have some tension if there is a lack of signal from the other dSphs. The majority of this tension arises from those problematic dSphs already mentioned. However, if both Ursa Minor and Tucana II have prolate shapes it becomes unlikely that they both have smaller J factors than an oblate Reticulum II.
1. For a given viewing angle (ϑ, ϕ), we find the measured ellipticity and orientation of the observed minor axis (using, for instance, equations (A1,A2,A6) of [50]) and compute the elliptical halflight radius Rh0 . This gives us a length scaling R = Rh /Rh0 , and encodes the geometric factor described in the introduction. 2. In principle, to solve for the kinematics of the stars in the dark matter potential, we could use the axisymmetric Jeans equations. However, there are degeneracies in the solution space and only a few algorithms exist for solution [51, 52]. As we need only match an integrated quantity, we use the virial 2 theorem to compute hσlos i as 02 hσlos i=
SEMIANALYTIC MODELS
Numerical M2M models provide a robust method for determining the corrections required when modeling flattened systems as spherical. However, they are computationally expensive to construct so cannot be employed in a Markov Chain Monte Carlo analysis that requires many models. We have provided a simple fitting formula for our model setup, but there will be some variation in the correction factors depending on, for instance, the light profile, the density profile of the dark matter, and the ratio of the scale lengths of the light to the dark matter. We now proceed to understand and reproduce the results of the M2M models using simpler methods. In subsection A, we describe a general virial method to compute Jfactors for flattened halo models. This numerical algorithm can be applied to any dark matter density, but in the two following subsections, we provide analytic shortcuts to evaluate the Jfactors for two specific families – the infinite flattened cusps and the flat rotation curve halos. Readers primarily interested in the results, rather than the details of the methods, should skip to subsection D, where we compare our models to the M2M results. Figures 6 and 7 provide summary plots, which show the range of correction factors as a function of the flattening of the stellar density.
A.
The Virial Method
We can construct approximate equilibrium models much more cheaply than with the full M2M apparatus by using the virial theorem. The two constraints provided by the data are the integrated lineofsight velocity 2 dispersion hσlos i and the halflight majoraxis length Rh . We describe a method to match these observations given density models for the light (ρ? ) and dark matter (ρDM ).
(8)
where Z
IV.
Wlos 2 Wlos hσtot i = , W M
Wlos =
d3 x ρ? Rij xj
∂ΦDM Rki . ∂xk
(9)
ΦDM is the dark matter potential (generically computed using a multipole expansion [53]), M is the total dark matter mass and Rij is the projection matrix along the lineofsight from coordinates aligned with the principal axes of the dSph. We have used the Einstein summation convention. For triaxial symmetry, the crossterms in the integral vanish so we need only project the velocity dispersions along the principal axes. This gives us a ve2 02 locity scaling V = hσlos i/hσlos i, and encodes the kinematic factor described in the introduction. 3. We compute a mass scaling M = V 2 R. The initial model is scaled by M and R and the J and Dfactors are computed. These can be compared to the spherical model with the same lineofsight velocity dispersion and halflight radius. This algorithm is completely general. For some special choices of stellar and dark matter density, the integration in the virial theorem can be performed analytically. We now give two examples – infinite flattened cusps and flat rotation curve halos – for which the virial integrals can be done. This means that the behavior of the Jfactor at fixed observables (line of sight velocity dispersion and halflight radii) can be mapped out analytically as a function of flattening or concentration.
B.
Flattened Cusps
Let us take the dark matter halo as an axisymmetric cusp stratified on similar concentric spheroids with an axis ratio q. If the cusps have the same mass Mh within the spheroidal halflight radius mh , then the mass
Name Hydra II Leo T Leo II Segue 2 Leo I Horologium I Fornax Draco Sculptor Carina Sextans Coma Berenices Bo¨ otes I Canes Venatici I Tucana II Pisces II Grus I Willman 1 Segue 1 Leo IV Leo V Canes Venatici II Ursa Minor Reticulum II Ursa Major II Hercules Ursa Major I
log10 (Jsph (0.5◦ )) +0.20 0.01−0.01 16.56+0.87 −1.85 < 0.10 17.32+0.38 −0.37 0.13+0.05 17.44+0.25 −0.05 −0.25 0.15+0.10 17.11+0.85 −0.10 −1.76 0.21+0.03 17.80+0.28 −0.03 −0.28 < 0.28 18.64+0.95 −0.39 0.30+0.01 18.15+0.16 −0.01 −0.16 0.31+0.02 18.86+0.24 −0.02 −0.24 +0.03 0.32−0.03 18.65+0.29 −0.29 0.33+0.05 17.99+0.34 −0.05 −0.34 0.35+0.05 17.87+0.29 −0.05 −0.29 0.38+0.14 18.67+0.33 −0.14 −0.32 0.39+0.06 16.65+0.64 −0.06 −0.38 0.39+0.03 17.27+0.11 −0.03 −0.11 0.39+0.10 19.05+0.87 −0.20 −0.58 0.40+0.10 17.90+1.14 −0.10 −0.80 0.41+0.20 17.96+0.90 −0.28 −1.93 0.47+0.08 19.29+0.91 −0.08 −0.62 0.48+0.13 19.41+0.39 −0.13 −0.40 +0.11 0.49−0.11 16.64+0.90 −0.90 16.94+1.05 0.50+0.15 −0.72 −0.15 0.52+0.11 17.65+0.40 −0.11 −0.40 0.56+0.05 19.15+0.25 −0.05 −0.24 0.59+0.02 18.71+0.84 −0.03 −0.32 0.63+0.05 19.38+0.39 −0.05 −0.39 0.68+0.08 16.83+0.45 −0.08 −0.45 0.80+0.04 18.48+0.25 −0.04 −0.25 Oblate FJ −0.002+0.002 −0.052 −0.012+0.008 −0.009 −0.032+0.011 −0.017 −0.037+0.019 −0.042 −0.055+0.008 −0.010 −0.034+0.024 −0.028 −0.083+0.003 −0.003 −0.086+0.006 −0.007 −0.090+0.010 −0.011 −0.093+0.016 −0.020 −0.100+0.016 −0.020 −0.108+0.039 −0.071 −0.114+0.020 −0.026 −0.115+0.011 −0.012 −0.114+0.053 −0.049 −0.119+0.032 −0.048 −0.117+0.064 −0.117 −0.147+0.030 −0.040 −0.151+0.045 −0.076 −0.156+0.041 −0.064 −0.159+0.053 −0.096 −0.171+0.042 −0.065 −0.191+0.024 −0.029 −0.207+0.016 −0.012 −0.231+0.028 −0.035 −0.266+0.050 −0.069 −0.373+0.041 −0.053
Prolate FJ log10 (Jobl (0.5◦ )) log10 (Jpro (0.5◦ )) 0.004+0.087 16.56+0.87 16.56+0.87 −0.003 −1.85 −1.85 +0.38 0.020+0.015 17.31 17.34+0.38 −0.014 −0.37 −0.37 0.054+0.029 17.41+0.25 17.49+0.25 −0.018 −0.25 −0.25 0.063+0.070 17.07+0.85 17.17+0.85 −0.032 −1.76 −1.76 0.092+0.017 17.75+0.28 17.89+0.28 −0.014 −0.28 −0.28 0.058+0.047 18.61+0.95 18.70+0.95 −0.040 −0.39 −0.39 0.139+0.006 18.07+0.16 18.29+0.16 −0.005 −0.16 −0.16 0.145+0.012 18.77+0.24 19.00+0.24 −0.011 −0.24 −0.24 +0.019 +0.29 0.151−0.016 18.56−0.29 18.80+0.29 −0.29 0.156+0.034 17.90+0.34 18.15+0.34 −0.026 −0.34 −0.34 0.169+0.034 17.77+0.29 18.04+0.29 −0.028 −0.29 −0.29 0.182+0.120 18.56+0.33 18.85+0.35 −0.066 −0.33 −0.33 0.192+0.043 16.54+0.64 16.84+0.64 −0.033 −0.38 −0.38 0.194+0.020 17.15+0.11 17.46+0.11 −0.018 −0.11 −0.11 0.193+0.082 18.94+0.87 19.24+0.87 −0.090 −0.58 −0.59 0.200+0.081 17.78+1.14 18.10+1.14 −0.053 −0.80 −0.80 0.198+0.196 17.84+0.90 18.16+0.92 −0.107 −1.93 −1.93 0.247+0.067 19.14+0.91 19.54+0.91 −0.051 −0.62 −0.62 0.255+0.128 19.26+0.39 19.66+0.41 −0.076 −0.41 −0.41 +0.108 +0.90 0.262−0.068 16.48−0.90 16.90+0.91 −0.90 0.268+0.161 16.78+1.05 17.21+1.06 −0.089 −0.73 −0.73 0.288+0.109 17.48+0.40 17.94+0.41 −0.071 −0.41 −0.41 0.321+0.049 18.96+0.25 19.47+0.25 −0.040 −0.24 −0.24 0.348+0.020 18.50+0.84 19.06+0.84 −0.027 −0.32 −0.32 0.389+0.060 19.15+0.39 19.77+0.39 −0.047 −0.39 −0.39 0.447+0.116 16.56+0.45 17.28+0.46 −0.084 −0.46 −0.46 0.629+0.089 18.11+0.25 19.11+0.27 −0.070 −0.26 −0.26
TABLE III: Annihilation correction factors for dwarf spheroidals due to their observed ellipticity (note Leo T and Horologium I only have upperbounds on the −5 ellipticity). We report the spherical Jfactor for a beam angle of 0.5◦ in units of GeV2 cm along with the corrections FJ assuming the galaxy is observed exactly edgeon and is either oblate or prolate. We report the resultant Jfactors for these cases as Jobl and Jpro in units of GeV2 cm−5 . The dSphs are ordered by their ellipticity.
8
Name Hydra II Leo T Leo II Segue 2 Leo I Horologium I Fornax Draco Sculptor Carina Sextans Coma Berenices Bo¨ otes I Canes Venatici I Tucana II Pisces II Grus I Willman 1 Segue 1 Leo IV Leo V Canes Venatici II Ursa Minor Reticulum II Ursa Major II Hercules Ursa Major I
log10 (Dsph (0.5◦ )) +0.20 0.01−0.01 16.89+0.44 −0.92 < 0.10 17.35+0.37 −0.37 0.13+0.05 17.62+0.25 −0.05 −0.25 0.15+0.10 17.08+0.86 −0.10 −1.75 0.21+0.03 17.89+0.28 −0.03 −0.28 < 0.28 17.78+0.47 −0.20 0.30+0.01 18.26+0.17 −0.01 −0.17 0.31+0.02 18.39+0.25 −0.02 −0.25 +0.03 +0.29 0.32−0.03 18.33−0.29 0.33+0.05 17.98+0.34 −0.05 −0.34 +0.05 0.35−0.05 18.07+0.29 −0.29 18.06+0.32 0.38+0.14 −0.32 −0.14 0.39+0.06 17.28+0.64 −0.06 −0.38 17.78+0.11 0.39+0.03 −0.11 −0.03 0.39+0.10 18.45+0.88 −0.20 −0.58 0.40+0.10 17.41+0.57 −0.10 −0.40 0.41+0.20 17.59+0.46 −0.28 −0.96 0.47+0.08 18.03+0.91 −0.08 −0.62 0.48+0.13 18.17+0.39 −0.39 −0.13 +0.11 +0.90 0.49−0.11 17.22−0.90 0.50+0.15 17.23+1.05 −0.15 −0.70 +0.11 0.52−0.11 17.37+0.40 −0.40 0.56+0.05 18.45+0.24 −0.05 −0.24 0.59+0.02 17.93+0.85 −0.03 −0.32 18.48+0.39 0.63+0.05 −0.05 −0.39 0.68+0.08 17.38+0.45 −0.08 −0.45 0.80+0.04 18.15+0.25 −0.04 −0.25 Oblate FD −0.000+0.000 −0.005 −0.001+0.001 −0.001 −0.003+0.001 −0.002 −0.004+0.002 −0.004 −0.006+0.001 −0.001 −0.004+0.002 −0.003 −0.009+0.000 −0.000 −0.009+0.001 −0.001 −0.009+0.001 −0.001 −0.010+0.002 −0.002 −0.011+0.002 −0.002 −0.011+0.004 −0.007 −0.012+0.002 −0.003 −0.012+0.001 −0.001 −0.012+0.006 −0.005 −0.012+0.003 −0.005 −0.012+0.007 −0.012 −0.015+0.003 −0.004 −0.016+0.005 −0.008 −0.016+0.004 −0.007 −0.017+0.006 −0.010 −0.018+0.004 −0.007 −0.020+0.002 −0.003 −0.022+0.002 −0.001 −0.024+0.003 −0.004 −0.028+0.005 −0.007 −0.039+0.004 −0.006
Prolate FD log10 (Dobl (0.5◦ )) log10 (Dpro (0.5◦ )) 0.001+0.017 16.89+0.44 16.89+0.44 −0.001 −0.92 −0.92 +0.37 0.004+0.003 17.35 17.35+0.37 −0.003 −0.37 −0.37 0.011+0.006 17.62+0.25 17.63+0.25 −0.004 −0.25 −0.25 0.012+0.014 17.08+0.86 17.09+0.86 −0.006 −1.75 −1.75 0.018+0.003 17.88+0.28 17.91+0.28 −0.003 −0.28 −0.28 0.011+0.009 17.78+0.47 17.79+0.47 −0.008 −0.20 −0.20 0.027+0.001 18.25+0.17 18.29+0.17 −0.001 −0.17 −0.17 0.029+0.002 18.38+0.25 18.42+0.25 −0.002 −0.25 −0.25 +0.004 +0.29 0.030−0.003 18.32−0.29 18.36+0.29 −0.29 0.031+0.007 17.97+0.34 18.01+0.34 −0.005 −0.34 −0.34 0.033+0.007 18.06+0.29 18.10+0.29 −0.005 −0.29 −0.29 0.036+0.024 18.05+0.32 18.10+0.32 −0.013 −0.32 −0.32 0.038+0.008 17.27+0.64 17.32+0.64 −0.007 −0.38 −0.38 0.038+0.004 17.77+0.11 17.82+0.11 −0.004 −0.11 −0.11 0.038+0.016 18.44+0.88 18.49+0.88 −0.018 −0.58 −0.58 0.039+0.016 17.40+0.57 17.45+0.57 −0.010 −0.40 −0.40 0.039+0.039 17.58+0.46 17.63+0.46 −0.021 −0.96 −0.96 0.049+0.013 18.01+0.91 18.08+0.91 −0.010 −0.62 −0.62 0.050+0.025 18.15+0.39 18.22+0.39 −0.015 −0.39 −0.39 +0.021 +0.90 0.052−0.013 17.20−0.90 17.27+0.90 −0.90 0.053+0.032 17.21+1.05 17.28+1.05 −0.018 −0.70 −0.70 0.057+0.022 17.35+0.40 17.43+0.40 −0.014 −0.40 −0.40 0.063+0.010 18.43+0.24 18.51+0.24 −0.008 −0.24 −0.24 0.069+0.004 17.91+0.85 18.00+0.85 −0.005 −0.32 −0.32 0.077+0.012 18.46+0.39 18.56+0.39 −0.009 −0.39 −0.39 0.088+0.023 17.35+0.45 17.47+0.45 −0.017 −0.45 −0.45 0.124+0.018 18.11+0.25 18.27+0.25 −0.014 −0.25 −0.25
TABLE IV: As Table III, but for the decay correction factors. The Dfactors are quoted in units of GeV cm−2 .
9
10
log 10 (J(0. 5 ◦ )/ GeV 2 cm −5 )
22
Spherical
21
Oblate
Prolate
20
Reticulum II +1σ prolate
19
Reticulum II +1σ oblate
18 17 16 15 14
4 2
1 1
7 6
2 3
8 9
6 10 11 5 13 15 14 12 9 16 17 18 22 21 25 20 19 27 26 23 24 5 11 12 7 8 15 14 13 10 17 19 16 22 24 25 20 18 23 26 21 27
Segue 1 Ursa Major II Willman 1 Ursa Minor Tucana II Draco Reticulum II Coma Berenices Sculptor Horologium I Ursa Major I Fornax Carina Grus I Pisces II Sextans Leo I Canes Venatici II Leo II Leo T Canes Venatici I Segue 2 Leo V Hercules Boötes I Leo IV Hydra II
13
3 4
20
log 10 (D(0. 5 ◦ )/ GeV cm −2 )
Spherical
Oblate
Prolate
19 18 17 16
Segue 1 Ursa Major II Willman 1 Ursa Minor Tucana II Draco Reticulum II Coma Berenices Sculptor Horologium I Ursa Major I Fornax Carina Grus I Pisces II Sextans Leo I Canes Venatici II Leo II Leo T Canes Venatici I Segue 2 Leo V Hercules Boötes I Leo IV Hydra II
15
FIG. 4: J (top) and Dfactors (bottom) integrated over a beam angle of 0.5◦ for 27 dSphs. The diamonds with red errorbars are computed assuming a spherical model and are taken from Paper I. The circles with blue errorbars show the spherical Jfactors adjusted by the oblate correction factors marginalized over the uncertainty in the ellipticity (assuming the galaxy is observed edgeon) and the squares with black errorbars show the spherical Jfactors adjusted by the prolate correction factors marginalized over the uncertainty in the ellipticity (assuming the galaxy is observed edgeon). The dSphs are ordered by their median spherical Jfactors. The top set of red numbers gives the ordering of the upperlimits on the spherical Jfactors, and the bottom set of black numbers gives the ordering of the upperlimits on the prolate Jfactors. The gray dashed lines show the 1σ upperlimit for the Reticulum II assuming it is prolate or oblate. The three dSphs with red names have unknown additional systematic uncertainties due to the presence of contaminants or the questionable assumption of dynamical equilibrium.
11 enclosed is M (m) = Mh
m mh
3−γDM for m ≤ rt
(10)
and M = Mh (rt /mh )3−γDM otherwise. m2 = x2 + y 2 + z 2 q −2 = R2 +z 2 q −2 and rt is a hard truncation ellipsoidal radius. The dark matter density is ρDM (m) =
Mh 3 − γDM DM mγDM 4πqm3−γ h
for m ≤ rt ,
(11)
and zero otherwise. Note the factor of q in the denominator which comes from the Jacobian. It means that the oblate models (q < 1) in the sequence have an increased density as compared to their spherical progenitor, whilst the prolate models (q > 1) have a decreased density. The spherical member of the family obeys the empirical law (2). As the mass Mh is preserved along the sequence, we can still use Eq. (2) for the flattened cusps provided we correct the observables – the line of sight velocity dispersion and the projected halflight radius – to the spherical parent. For comparison purposes, it is useful to define the Jfactor and Dfactor of the infinite spherical cusp (rt → ∞, equations (8) and (11) in Paper I) as Jsph =
2 1 hσlos iRh 2 Dθ 3−2γDM P (γDM ), (12) D2 Rh3 G Rh
Dsph
1 hσ 2 iRh Dθ 3−γDM = 2 los Q(γDM ), D G Rh
(13)
(14)
If an oblate model is viewed along the short axis, or a prolate model is viewed along the long axis, then it appears round. The line of sight coincides with the symmetry or z axis. The geometric corrections are then straightforward to evaluate as Jgeo,face =
1 , q
Dgeo,face = 1.
Jgeo,edge =
q 2−γDM 2πq 2
2π
Z
dθ(cos2 θ + q −2 sin2 θ)1/2−γDM ,
0
(16) and Dgeo,edge =
q 1−γDM /2 2πq
Z
(15)
This case is very simple because both the field of view and the surface density contours are circular. Note that both factors are independent of the slope of the density profile γDM . If an infinite (rt → ∞) oblate or prolate model is viewed edgeon, it appears flattened with axis ratio q. The line of sight then coincides with, say, the y direction. Observationally, the effective radius of a flattened
2π
dθ(cos2 θ+q −2 sin2 θ)1/2−γDM /2 .
0
(17) The factors can only be reduced to a single quadrature due to the mismatch between the circular beam aperture and the elliptical isophotes. For oblate (prolate) models, the geometric correction leads to an increase (decrease) in the Jfactor as compared to a spherical model with the same Mh if γDM ≤ 2. If the dark matter halo is truncated at a finite ellipsoidal radius rt < Dθ, the beam encloses all the dark matter and the edgeon geometric factors reduce to Jgeo,edge = q 1−γDM ,
where both P (γDM ) and Q(γDM ) are constants given in Paper I. In these expressions, the halflight radius Rh is the ‘ellipticitycorrected’ halflight radius that includes a √ factor of 1 − . The J and D factors for our axisymmetric models can now be written in the form J = Jsph Jgeo Jkin , D = Dsph Dgeo Dkin .
model is always measured along the projected major axis. For an oblate model, the measured effective radius is Rh whereas for the prolate model, the effective radius of its spherical progenitor is actually Rh /q. Additionally, comparison with the ‘ellipticitycorrected’ √ spherical model gives rise to an additional factor of 1 − in the √ effective radius which equals q for the oblate case and p 1/q for the prolate case. The geometric corrections (i.e. the ratio of the J and Dfactors to those for a spherical model with the same Mh ) are now
Dgeo,edge = q 1−γDM /2 .
(18)
These equations are preferable as for γDM < 3 they correspond to finite mass models and for γDM < 3/2 they produce finite Jfactors. We have found that they give much better representations of the correction factors for more general models. We have computed the ratio of the J and Dfactors to those of the spherical model with the same Mh . As Mh is estimated from the lineofsight velocity dispersion, we must now compute the ratio of the true Mh to that computed using only the lineofsight velocity dispersion. This ratio is the kinematic correction, which we compute using the tensor virial theorem [e.g., 54, 55]. The effect of flattening on the kinematics of the stars is given by R 3 2 2 d x ρ? (σRR + σφφ ) TRR WRR R = = . (19) 3 2 Tzz Wzz d x ρ? σzz where T and W are the kinetic energy and potential energy tensors [53, 55, 56]. The stellar density in dSphs is well approximated by a Plummer or King profile. Such laws do not lead to tractable integrals in the virial theorem (19). Instead, we approximate the stellar density as a powerlaw stratified on similar concentric spheroids with m2? = R2 + z 2 q?−2 and so q? is the stellar flattening. This means we can take advantage of equations (1924) in [55], which give the virial ratios for stellar populations whose density is a pure scalefree powerlaw declining like
12
1/2 1/2 1 1 + 1 + 8q 2 2
(20)
This formula is given in refs [57, 58]. As is well known, the equipotentials are always rounder than the density contours, so that qφ ≈ 1 even if the dark halo is quite flattened. Then for γ? = 3, we have from [55] √ Q? Q − Q? ArcsinhQ √ , 2[ Q? ArcsinhQ − Q]
2 hσxx i = 2 i √ hσzz Q Q − Q? ArcsinQ ?√ , 2[ Q? ArcsinQ − Q]
Tensor virial
®
0.4
®
0.2 0.0
0.2
γ =2
0.4
γ =4
qφ =
γ =3 0.6 2 / σ2 ) log 10 ( σxx zz
distance−γ? . Note that as the all the considered models have infinite mass we must work with the ratios of the velocity dispersions. If we assume the equipotentials are spheroidally stratified, the correction is a function of Q? = qφ2 /q?2 , where qφ is the flattening of the dark halo equipotentials, which is related to the flattening q in the dark halo density via
0.6 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
q
Q? > 1 (21) Q? < 1,
FIG. 5: Kinematic ratio for oblate and prolate figures. Each line shows the prediction from a stellar axisymmetric cusp with density slope γ? flattened with axis ratio q embedded in a halo also with flattening q. The black points show the numerical results from the M2M models.
where (√ Q(Q? ) =
√
Q? − 1 1 − Q?
Q? > 1 Q? < 1
(22)
Our formulae are appropriate if rt → ∞ or the stellar profile is truncated at a smaller radius than the darkmatter profile. This virial ratio is unity when Q? = 1. This follows because if the stellar density is constant on the equipotentials, then the velocity dispersion is isotropic. It is greater than unity when Q? > 1 (that is, when the model is oblate), as the globally averaged velocity dispersion component along the long or x axis must be larger than that along the short or z axis. It is less than unity when Q? < 1, as the roles of the x and z axes are now reversed for the prolate figure. Formulae for other values of γ? , or stellar density falloff, are given in Appendix A. We plot the logarithm of the virial ratio in Fig. 5 along with the ratio calculated from the M2M models. The line for γ? = 3 agrees well with the M2M data. The green dashed line shows the virial ratio computed using the virial method. For a stellar density stratified on the same concentric ellipsoids as the darkmatter density, the virial ratio is simply a function of the shape of the ellipsoids and is independent of the radial density profile so the plotted line has a very simple functional form [53, 59, 60]. It can be shown that along the sequence of models the total luminosityaveraged square velocity dispersion σtot is constant. Therefore, for a given model the kinematic factor is the ratio of the total velocity dispersion to the lineofsight velocity dispersion. When viewed down the xaxis, the kinematic correction factor is Jkin,edge =
hσ 2 i 2 2 2 hσzz i 2 tot = + 2 2 hσlos i 3 3hσxx i
(23)
This is the ratio of the squared velocity dispersion along the line of sight to the average value. This is smaller (larger) than unity for oblate (prolate) models. When viewed down the zaxis, the kinematic correction factor is Jkin,face =
1 3
+
2 2 2hσxx i 2 i 3hσzz
(24)
This is smaller (larger) than unity for prolate (oblate) models. Note that, as the Dfactors are proportional to √ 2 , the Dfactor kinematic factor Dkin = Jkin . σtot C.
Flat Rotation Curve Models
A simple but widelyused model of a dark halo has potentialdensity pair [53, 57] ρDM (R, z) =
2 −2 2 2 2 v02 (2qφ + 1)Rd + R + z (2 − qφ ) , 4πGqφ2 (Rd 2 + R2 + z 2 qφ2 )2
ΦDM (R, z) =
v02 ln(Rd 2 + R2 + z 2 qφ−2 ) 2
(25)
Here, v0 is a velocity scale that is the asymptotic value of the flat rotation curve, whilst Rd is the dark matter lengthscale while qφ is the axis ratio of the equipotentials. The dark √ matter density is everywhere positive provided qφ > 1/ 2, so the model can be oblate, spherical or prolate. Unless qφ = 1, the flattening of the dark matter density changes with radius such that the oblate models become more oblate in the outskirts whilst the prolate models become more prolate. At large radii qφ is related to the isodensity flattening q via equation (20).
13 The dark halo is cusped if Rd = 0, but the cusp is isothermal and so much more severe than in the NFW model. The Jfactor for the model viewed along the zaxis or symmetry axis is J=
h v04 3(1 − y) − 4qφ2 (y 3 − 1) 3 96Rd D2 G2 qφ
∆21 = qφ2 − q?2 ,
i + qφ4 (8 − 3y − 2y 3 − 3y 5 ) , (26) p with y = Rd / Rd2 + D2 θ2 . At large angles, y → 0 and so the asymptotic value is J→
v04 [3 + 4qφ2 + 8qφ4 ]. 96Rd D2 G2 qφ3
v02 Rd qφ (Dθ/Rd )2 p . GD2 1 + (Dθ/Rd )2
(28)
Note that the total mass of the model is not finite so the Dfactor does not tend to a finite value as θ → ∞. Viewed edgeon, two of the integrations for the Jfactor are analytic, leaving a final integral over the spherical aperture to be performed numerically J(θ → ∞) = Z ×
v04 1536πRd D2 G2 qφ8
2π
dφ 0
G1 (qφ ) + G2 (qφ ) cos(2φ) + G3 (qφ ) cos(4φ) , (cos2 φ + qφ−2 sin2 φ)3 (29)
G1 (qφ ) = 120 + 280Q2 + 221Q4 + 64Q6 + 9Q8 , G2 (qφ ) = 4Q2 (14 + 3Q2 (3 + Q2 )2 ), 4
2
(30)
4
G3 (qφ ) = Q (7 + 8Q + 3Q ), and Q2 = qφ2 − 1. The Dfactor can also be expressed as a single quadrature but the expression is too bulky to present here. Into this dark halo, we embed a population of stars to model the dSph, namely ρ(R, z) =
ρ0 Rc β? . (Rc 2 + R2 + z 2 q?−2 )β? /2
Here, ρ0 is a normalization constant, while q is the axis ratio of the spheroidal isodensity contours. If β? = 5, this is the familiar Plummer model. If the scalelength of the stars Rc is equal to the scalelength of the dark matter Rd , and the flattening of the stellar density q? is equal to the flattening of the dark matter equipotentials qφ , then the phase space distribution function is an isothermal [57]. We derive more general formulae below, but note that this simple limit enables an easy check of the correctness of our results.
∆22 = Rd 2 − Rc 2 ,
D2 = qφ2 Rd 2 − q?2 Rc 2 . (31)
Then the velocity dispersions are 2 hσRR i
v 2 qφ Rc 2 h = 04 4 2 2∆41 Rd 3 DArccosh ∆1 ∆2 D − ∆1 D
(27)
Similarly, the Dfactor for a model viewed along the zaxis is given by D=
As both the density and the potential are simple, we can calculate the velocity dispersions seen on viewing the stellar distribution along the short or long axis. We give the results for β? = 5 here, and delegate other formulae to the Appendix B. We begin by defining
2
(2qφ2 Rd 2
+
q φ Rd q? Rc
q?2 (Rc 2 −3Rd 2 ))Arccosh
qφ q?
qφ q?
i − qφ ∆21 ∆22 D2 , v02 q?2 Rc 2 h 2 2 2 2 q ∆ ∆ D − qφ ∆1 D4 Arccosh ∆41 ∆22 D2 ? 1 2 qφ Rd i . + qφ ∆41 DRd 3 Arccosh q? Rc
2 hσzz i =
(32)
The formulae hold generally on using the identity (for S < 1) ArccoshS ≡ −i arccos S These formulae give the line of sight velocity dispersion of an axisymmetric Plummer model viewed along the short and long axes in a dark halo of arbitrary flattening and lengthscale. If Rc = Rd , then h v02 qφ 2 5q?4 qφ − 7q?2 qφ3 + 2qφ5 hσRR i = 2 4(qφ − q?2 )3 i qφ 4 2 2 1/2 , + 3q? (qφ − q? ) Arccosh q h v02 q?2 2 hσzz i = q 4 + qφ2 q?2 − 2q?4 2(qφ2 − q?2 )3 φ i qφ . (33) − 3q?2 qφ (qφ2 − q?2 )1/2 Arccosh q If additionally q? = qφ , then 2 hσRR i=
2v02 , 5
2 hσzz i=
v02 . 5
(34)
With the line of sight velocity dispersion in hand, we can simply rescale the model so that the Jfactors are computed for models with the same observables (line of sight velocity dispersion and halflight radius) as the flattenings and the ratio of dark to luminous scalelength Rc /Rd varies. D.
Comparisons
The models in Sections IV B and IV C are complementary. The infinite cusps allow us to vary the central slope of the dark matter. The cored models allow
14 !
!
q η fit
q η fit
Cusp, γ = 3 Core, β = 5, Rd /Rc = 20
Virial Method
Cusp, γ = 3 Core, β = 5, Rd /Rc = 20
Virial Method
1.0 Viewed faceon
Viewed faceon 0.4 0.2
0.0
Rd
FD
FJ
0.5
/R
c=
0.5
20
0
1.0 0.0
0.5
1.0
1.5
c =2
γ =2
γ = 4
2.5
Rd /
Rc
0.2
Rd /R
2.0
0.0
3.0
3.5
=2
4.0
0.0
0.5
1.0
1.5
q η fit
γDM = 0.
5
0.5
.5
γ DM = 0
FD
Rd /Rc = 200 FJ
3.0
3.5
4.0
1.0 Viewed edgeon
0.5
γDM = 1. 5
2 Rc =
R d/
0.5
2.5
Cusp, γDM = 1, γ = 3 Core, β = 5, Rd /Rc = 20
Virial Method
1.0 Viewed edgeon
0.0
2.0
q Cusp, γDM = 1, γ = 3 Core, β = 5, Rd /Rc = 20
Virial Method
00
Rd /Rc = 2
0.4
q q η fit
γ =2 γ =4
0.0
γDM = 1. 5 R /R = 200 d c Rd /Rc = 2
0.5
1.0
1.0 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
q FIG. 6: Jfactor correction factors for oblate and prolate figures viewed faceon (top) and edgeon (bottom). The black points show the numerical results from the M2M models of stellar Plummer models flattened with axis ratio q embedded in NFW darkmatter halos also of axis ratio q. The dashed green line shows the results of the virial method of Section IV A. The blue band shows a range of axisymmetric cusp models from Section IV B. The central line corresponds to a model with γDM = 1, γ? = 3. In the top panel, we have varied the slope of the light profile (note the faceon correction factor is independent of γDM in this case). In the bottom panel, we have varied the slope of the dark matter. In both panels, the red band shows a series of cored flat rotation curve models from Section IV C. The central line has outer stellar density profile of β? = 5 and ratio of darkmatter to stellar scale radii of Rd /Rc = 20. The band corresponds to varying this scale radii ratio by a factor of ten.
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
q FIG. 7: Dfactor correction factors for oblate and prolate figures viewed faceon (top) and edgeon (bottom). See the caption of Figure 6 for details on each line.
us to vary the ratio of the luminous to the dark matter lengthscale. Taken together, a gamut of possibilities of dark halo cusps, density profiles and lengthscales can be swept out. The base10 logarithms of the correction factors (6) are plotted as a function of dark halo flattening q in Fig. 6. The flattening of the stellar population q? is the same as that of the dark halo q. For all plots we use the observed parameters of Reticulum II with a beam angle of 0.5◦ . The correction factors are a function of the beam angle but we have found that this dependence is very weak and that the correction factors are essentially independent of the size of the dSph with respect to the beam size. Therefore, the reported correction factors are appropriate for all beam angles. We show the correction factors for models viewed face
15 on (top panels) and edgeon (bottom panels). Any correction factor between these extremes should be possible as the inclination angle can be varied between these extremes. The correction factors computed from the M2M Plummer models in NFW halos with axis ratio q are shown as black points. They are in good agreement with the results of the virial method of Section IV A applied to the selfsame model, which are shown as a green curve. Note that the exception to this is the q = 2.5 prolate model. As noted in Section II, the MadetoMeasure method in this case produces an equilibrium figure that significantly deviates from a spheroidally stratified Plummer model. We also show as blue bands the range of results for the axisymmetric cusps of Section IV B, in which both the slopes of the dark matter and stellar cusps are allowed to vary. Finally, the red band shows a series of cored flat rotation curve models from Section IV C. The central line has outer stellar density slope of β? = 5 and ratio of darkmatter to stellar scale radii of Rd /Rc = 20. The band corresponds to varying this scale radii ratio by a factor of ten. As we move along the curves, the models have the same lineofsight velocity dispersion and the same halflight radius. As the models make different assumptions as to the dark matter density and potential, we do not expect these curves to match up exactly with the M2M models, but it is encouraging that they all show similar trends. When an oblate model is viewed along the short axis or faceon, it appears circular, but there is always a boost to the Jfactor. For flattenings of q = q? = 0.5, this can be a factor of 3 boost over the spherical Jfactor. When the model is viewed edgeon, or along the long axis, then it appears flattened with isophotes of ellipticity 1 − q? . However, the geometric and kinematic corrections work in different directions, the former to boost, the latter to reduce, the Jfactor. The net effect is less significant than in the faceon case and is ∼ 0.1−0.2 dex for 0.4 < q < 0.7. When a prolate model is viewed along the long or zaxis, it appears round. Here, the geometric and the kinematic corrections both diminish the Jfactor. Although the model looks round on the sky, its Jfactor can be substantially less than computed by a spherical analysis. For example, if the true flattening is q = q? = 2.5, then the Jfactor is decreased by a factor . 0.3. When the model is viewed edgeon, the isophotes have ellipticity 1 − q?−1 , and the kinematic and geometric factors act in opposite directions with the net result being a small boost. If the flattening is q = q? = 2.5 the Jfactor is increased by ∼ 0.3 dex over the spherical estimate. The blue bands give an indication of how the correction factors vary as the darkmatter density slope is adjusted. We find steeper cusps give smaller corrections for models viewed edgeon, but has no effect on the corrections for models viewed faceon as the geometric factor is independent of γDM . For the faceon case we see that making the slope of the stellar density profile steeper produces larger corrections to the Jfactor. From the red bands we observe that making the dark
matter halo more extended (increasing Rd /Rc ) produces larger corrections for the faceon case but smaller corrections for the edgeon case. The width of the red bands when the lengthscales Rc /Rd is varied is at most 0.5 dex, even at the most extreme flattenings. Most of the Milky Way dSphs are rounder than q? = 0.5. In this regime, the red band is thinner, and gives rise to an uncertainty of at most ∼ 0.25 dex. This suggests that varying the concentration of the dark matter halo will not have a significant effect on the flattened Jfactors. This is corroborated by experiments with the Plummer profile embedded in the NFW profile. Additionally, making the stellar density in these models fall off more rapidly increases the magnitude of the Jfactor correction factors when viewing faceon but decreases the magnitude of the correction when viewing edgeon. The equivalent results for the Dfactors are shown in Fig. 7. The correction factors for the M2M models models are all . 0.2 dex, and suggest that for most applications, the spherical approximation suffices for the Dfactors. Note that in the faceon case, the cored models give a similar approximation of the correction factors as the cuspy models whilst for the edgeon case the cored models more faithfully represent the true correction factors than the cuspy models. Increasing the outer stellar density slope for the flat rotation curve models increases the magnitude of Dfactor correction factors when viewing faceon but has little effect for the edgeon case. Finally, we note that for an (α, β, γ) stellar model of equation (3) embedded in another (α, β, γ) darkmatter model the Jfactor correction factors are very insensitive to the choice of the density slopes of the stellar and darkmatter distributions and the ratio of the stellar to darkmatter scalelengths. The same is broadly true for the Dfactor correction factors except that the edgeon correction factor has a weak dependence with the outer slope of the dark matter profile. This is slightly at odds with the flat rotation curve model results but this may be due to the flat rotation curve models having a density flattening that varies with radius whilst the (α, β, γ) models have a constant density flattening.
V.
THE EFFECTS OF TRIAXIALITY
Generically, we might expect the light and darkmatter distributions in dwarf spheroidals to be triaxial [61]. Triaxiality can introduce additional flattening (stretching) along the lineofsight and so naturally increases (decreases) the Jfactor and gives rise to larger (smaller) correction factors. Here we extend the formulae given in the previous section to account for intrinsic triaxial shapes. We begin by focusing on the infinite cusp models where some analytic progress can be made before moving on to consider more general density profiles. We extend the models of equation (11) and introduce an intermediatetomajor axis ratio p in addition to the minortomajor axis ratio q. Here we restrict q < p < 1
16 such that a prolate model has p = q = 6 1. It is conventional to use a triaxiality parameter T to describe the figures defined by T =
1 − p2 . 1 − q2
(35)
Note that figures with T = 0 are oblate whilst those with T = 1 are prolate. The density for the triaxial cusp models is
ρDM (m) =
3 − γDM Mh 3−γDM mγDM 4πpqmh
for m ≤ rt , (36)
and zero otherwise. m2 = x2 + y 2 p−2 + z 2 q −2 and rt is a truncation ellipsoidal radius. When an infinite (rt → ∞) model is viewed along the z axis, the observed flattening is p and the geometric factor is a combination of equation (15) and (16) such that Z p2−γDM 2π Jgeo,z = dθ(cos2 θ + p−2 sin2 θ)1/2−γDM . 2πp2 q 0 (37) In this case, the observed majoraxis length corresponds to the intrinsic model scale radius. For γDM ≤ 2, the integral is a monotonic function of q that is greater than unity for q < 1 and less than unity for q > 1. If viewed along the yaxis the observed flattening is q and the Jfactor is given by
If the infinite model is observed along a line of sight oriented with spherical polar angles (ϕ, ϑ) with respect to the intrinsic Cartesian coordinates of the model, the geometric factor must be computed with the full threedimensional integrals as Z Dα Z 2π Z ∞ 1 0 dR RρDM 2 (x), dθ dz Jgeo = Jsph D2 −∞ 0 0 (41) ˆ the ˆ + R sin θϑˆ + z 0 rˆ with (ˆ ˆ ϑ) where x = R cos θϕ r , ϕ, set of spherical polar unit vectors. Making the model finite with rt < Dθ produces J/Jsph = 1/(pq). However, for this general viewing angle calculation of the observed scale radius seems intractable. The kinematic fac2 2 tors Jkin = (hσtot i/hσlos i)2 can be derived for these more general viewing angles as h1 i2 1 + f1 + f2 Jkin = , 2 2 2 3 cos2 ϑ + f1 sin ϑ cos2 ϕ + f2 sin ϑ sin ϕ (42) where Rπ R 2π 2 dθ 0 dφ F (θ, φ) sin3 θ cos2 φ i hσxx 0 f1 = 2 = R π > f2 R 2π hσzz i dθ 0 dφ F (θ, φ) sin θ cos2 θ 0 Rπ R 2π 2 hσyy i dθ 0 dφ F (θ, φ) sin3 θ sin2 φ 0 f2 = 2 = R π > 1, R 2π hσzz i dθ 0 dφ F (θ, φ) sin θ cos2 θ 0 (43) and
2−γDM
2π
F (θ, φ) = (sin2 θ cos2 φ+P?2 sin2 θ sin2 φ+Q2? cos2 θ)−γDM /2 . (44) 0 (38) Here P? = pφ /p and Q? = qφ /q with pφ and qφ being and again the observed majoraxis length coincides with the axis ratios of the dark matter potential. We see that the intrinsic model scale radius. When viewed along the when viewing down the major axis (ϑ = π/2, ϕ = 0) majoraxis, the observed flattening is q/p and the obthe kinematic correction factor is less than unity whilst served majoraxis length coincides with the intermediviewing down the minor axis (ϑ = 0, ϕ = 0) produces ate axis so the resultant measured scalelength must be a kinematic correction factor greater than unity. Genscaled by a factor 1/p. This gives rise to a geometric erally, we find that Jkin,x < Jkin,y < Jkin,z such that factor of the total correction factors for γDM < 3/2 obey the hiZ erarchy FJ,x < FJ,y < FJ,z . We have found that the (q/p)2−γDM 2π 2 2 −2 1/2−γDM effects of triaxiality seem to be in accordance with our Jgeo,x = dθ(cos θ+(q/p) sin θ) . 2π(q/p)2 p 0 expectation from the axisymmetric case. When there is (39) additional flattening along the lineofsight the geometric As with the axisymmetric case, the infinite cusps have and kinematic correction factors combine to increase the limited use and it is more practical to use models with correction factor, whilst additional stretching decreases finite truncation ellipsoidal radii rt < Dθ. In this case, the correction factor. the geometric factors are given by We now compute general triaxial correction factors using the method of Section IV A. We show an example of 1−γDM Jgeo,x = (qp) , the Jfactor correction factors for the Reticulum II model 1−γDM (40) Jgeo,y = q /p, presented in Section II but with stellar minortomajor axis ratio q = 0.4 and stellar intermediatetomajor axis Jgeo,z = p1−γDM /q. ratio p = 0.73. We assume the darkmatter distribution For the models that produce a finite Jfactor (γDM < is flattened in the same way as the stellar distribution. 3/2), we find Jgeo,x < Jgeo,y < Jgeo,z . For the This model has triaxiality parameter T = 0.55, (which astrophysicallymotivated case of γDM = 1 the geometwas deemed the bestfit to the Local Group dSphs by ric factors are simply Jgeo,x = 1, Jgeo,y = 1/p and [61]). The base10 logarithm of the correction factor for Jgeo,z = 1/q. all viewing angles is given in Fig. 8. We see that, in Jgeo,y =
q 2πq 2 p
Z
dθ(cos2 θ + q −2 sin2 θ)1/2−γDM ,
17 VI. INTRINSIC SHAPES AND AXIS ALIGNMENTS OF DWARF SPHEROIDAL GALAXIES
FIG. 8: Jfactor correction factors for a triaxial model of Reticulum II: each point on the sphere is colored by the correction factor when viewing the model along the radial vector that passes through that point. The black contours show the observed ellipticity when viewed from that direction. The small ellipsoid shows an isodensity contour for the considered model which has axis ratios p = 0.73 and q = 0.4 in both the stellar and darkmatter distributions. The largest correction factor is achieved when viewing the model down the short axis (z) whilst the smallest correction factor is achieved when viewing the model down the long axis (x). When viewing down the intermediate axis (y) the observed ellipticity matches that of Reticulum II.
agreement with the simple predictions from the infinite cusp models, the largest correction factor occurs when the model is viewed down the short axis (the z axis) whilst the smallest is when viewing down the long axis (the x axis). The black contours on the sphere show lines of constant observed ellipticity. We see that for this figure an observed ellipticity of 0.3 gives rise to a variation in the correction factor of 0.6 dex. In conclusion, additional flattening along the lineofsight can lead to an increase in the correction factors. For a general triaxial figure the largest correction factor is obtained when viewing the model down the short axis whilst the smallest correction factor is yielded when viewing the model down the long axis.
We have presented corrections to the J and Dfactors based on the assumption that the dSphs are prolate or oblate figures with axes aligned with the lineofsight. Such configurations are quite unlikely as we anticipate that generically the dSph principal axes are misaligned with the lineofsight. In this section we will discuss what is known regarding the intrinsic shapes of the dSphs and how this translates into observed properties via their axes alignment with respect to the lineofsight. For a given individual galaxy, we have a couple of probes of its intrinsic shape [62–64]. The first of these is the presence of isophotal twisting, that is the change in the orientation of the major axis of the isodensity contours with onsky distance from the galaxy center. Isophotal twisting is a clear signature of a triaxial figure with varying axis ratios with radius, although isophotal twisting may also be caused by tidal effects [65]. Another indicator of triaxiality is evidence of kinematic misalignment between the axis of rotation and the minor axis of the projected density. For entire populations of galaxies, progress can be made by analyzing statistics of the population [e.g. 50]. Recently, [61] demonstrated that under the assumption that the intrinsic axes of the dSphs are randomly oriented, the Local Group dSph population is best reproduced by triaxial models with mean triaxiality T¯ = 0.55+0.21 −0.22 and a mean intrinsic ellipticity (E = 1 − (c/a)) ¯ = 0.51+0.07 . The assumption of random orientation of E −0.06 is perhaps to be questioned, particularly for the Milky Way dSphs. Darkmatter only simulations [66, 67] have demonstrated that the major axes of subhalos tend to be aligned with the radial direction to the center of their host halo and this picture has been corroborated when baryons have been included [68]. The main exception to this is near the subhalo’s pericentric passage where the major axis is briefly aligned perpendicular to the radial direction. Most of the dSphs are distant enough for the radial direction and our lineofsight to approximately coincide, which suggests that for many dSphs the observed flattening corresponds to the intermediatetominor axis ratio and that there is significant stretching of the dSphs along the lineofsight. As demonstrated in this paper, this gives rise to overestimates of the Jfactors from spherical analyses for the prolate faceon models and for the triaxial model viewed down the major axis. Based on this discussion, we now compute the expected J correction factors with their associated uncertainties under a number of assumptions regarding the intrinsic shape and alignment of the dSphs. We use the emcee package from [69] to draw 500 samples of (T, E, ϑ, ϕ) i.e. the triaxiality, the intrinsic ellipticity and the two viewing angles. Our likelihood is the distribution of the observed ellipticity for each dSph given by, for instance, equation
18
1. Uniform (U): T ∼ U(0, 1), E ∼ U(0, 0.95), cos ϑ ∼ U(0, 1), ϕ ∼ U(0, π/2), 2. Viewing down the majoraxis (R): T ∼ U(0, 1), E ∼ U(0, 0.95), ϑ ∼ N (π/2, 0.1 rad), ϕ ∼ N (0, 0.1 rad), 3. S´ anchezJanssen et al. [61] priors (T): T ∼ N (0.55, 0.04), E ∼ N (0.51, 0.12), cos ϑ ∼ U(0, 1), ϕ ∼ U(0, π/2), where U(a, b) is a uniform distribution from a to b and N (µ, σ) is a normal distribution with mean µ and standard deviation σ. For each sample we compute the base10 logarithm of the correction factor FJ to construct a distribution of correction factors. In Figure 9, we show the full 1D distributions of the correction factors for Reticulum II. All three prior assumptions produce a correction factor distribution that peaks near zero. The broadest distribution corresponds to the case where uniform priors have been adopted in all parameters. In this case, the largest correction factors correspond to models with high intrinsic ellipticity E viewed down the short axis. These models have triaxiality T close to unity so are near prolate models. The smallest correction factors correspond to models with low intrinsic ellipticity viewed down the longaxis. For the prior assumption that we are viewing along the major axis, we find the median correction factor peaks at ∼ −0.2dex. For this prior assumption, there is an approximate onetoone relationship between T and E as well as T and FJ . Models with smaller T correspond to smaller E and hence smaller amplitude correction factors as these models are approximately edgeon oblate, whilst larger T and larger E produce larger amplitude negative correction factors as these models are nearer prolate stretched along the lineofsight. For the prior assumption that the models have some fixed triaxiality and intrinsic ellipticity, the largest correction factors correspond to viewing angles nearer the short axis and the smallest correction factors correspond to viewing angles closer to the long axis. The medians and ±1σ errorbars of the correction factors for all the dSphs computed for the three prior assumptions are given in Table V. This information is also displayed in Figure 10. If we assume the dSphs are preferentially viewed along the major axis, the median correction factor is less than unity and is weakly decreasing with increasing ellipticity. As the dSphs become more flattened on the sky, they are forced to become more extended along the lineofsight and so the Jfactor decreases. The medians of the correction factors for dSphs with small ellipticity under the uniform prior assumption is around 0.05 dex and the upper errorbars are in general
6
Uniform (U) Major axis (R) SJ 2016 (T)
5 4
dN/dF J
(A1,A2) of [50]. For those dSphs with upperbounds on their ellipticity we use a normal distribution with mean zero and standard deviation of half the upperbound. We consider three different prior distributions on the parameters (T, E, ϑ, ϕ):
3 2 1 0 0.4
0.2
0.0
0.2
0.4
0.6
0.8
1.0
FJ
FIG. 9: Distribution of the base10 logarithm of the Jfactor correction factors for triaxial models of Reticulum II under three different assumptions on the prior distributions of the intrinsic triaxiality, ellipticity and viewing angles as described in the text.
larger than the lower errorbars but the spread encompasses zero. The median shift is due to the asymmetry in the correction factors between oblate and prolate models seen in Fig. 6. This suggests that all Jfactors are underestimated by ∼ 10 percent but naturally this conclusion is very sensitive to the exact prior assumptions. For small ellipticity, the spread in the correction factors for the uniform and fixed shape priors are approximately equal with the spread weakly increasing with increasing ellipticity for the uniform case. For the fixed shape prior the spread decreases at large ellipticity as models viewed down the minor axis become inconsistent with the observed ellipticity. We have fitted a simple relation to the uncertainty in the correction factors from the uniform priors ∆FJ (the average of the ±1σ uncertainties) as a function of ellipticity as √ ∆FJ ≈ 0.4 , (45) which gives a fractional uncertainty in the Jfactor of √ ∆J ≈ 100.4 − 1. J
(46)
This expression gives 50 per cent uncertainty for ≈ 0.2, a factor 1.8 uncertainty for ≈ 0.4 and a factor 2.3 uncertainty for ≈ 0.6. For small , ∆J/J ≈ 0.9. We conclude for a typical dSph ellipticity of 0.4 there is approximately a factor of two uncertainty in the Jfactors due to the unknown triaxiality and alignment of the dSph.
VII.
DISCUSSION AND CONCLUSIONS
Flattening is a crucial attribute of a dwarf spheroidal galaxy. Both the dark halo and the stellar distribution
19 TABLE V: Median and ±1σ J correction factors for the known dSphs for three different assumptions about the intrinsic shapes and alignments. The first correction factor (marked ‘U’) uses flat uniform priors on the triaxiality, minortomajor axis ratio and viewing angles. The second (marked ‘R’) uses a normal prior on the viewing angles centered on (θ = π/2, φ = 0) with width 5◦ . The third (marked ‘T’) uses a normal prior on the triaxiality and minortomajor axis ratios with means (0.55, 0.49) and widths (0.04, 0.12) (based on the fits to the Local Group dSphs of S´ anchezJanssen et al. [61]. Name Hydra II Leo T Leo II Segue 2 Leo I Horologium I Fornax Draco Sculptor Carina Sextans Coma Berenices Bo¨ otes I Canes Venatici I Tucana II Pisces II Grus I Willman 1 Segue 1 Leo IV Leo V Canes Venatici II Ursa Minor Reticulum II Ursa Major II Hercules Ursa Major I
Ellipticity 0.01+0.20 −0.01 < 0.10 0.13+0.05 −0.05 0.15+0.10 −0.10 0.21+0.03 −0.03 < 0.28 0.30+0.01 −0.01 0.31+0.02 −0.02 0.32+0.03 −0.03 0.33+0.05 −0.05 0.35+0.05 −0.05 0.38+0.14 −0.14 0.39+0.06 −0.06 0.39+0.03 −0.03 0.39+0.10 −0.20 0.40+0.10 −0.10 0.41+0.20 −0.28 0.47+0.08 −0.08 0.48+0.13 −0.13 0.49+0.11 −0.11 0.50+0.15 −0.15 0.52+0.11 −0.11 0.56+0.05 −0.05 +0.02 0.59−0.03 0.63+0.05 −0.05 0.68+0.08 −0.08 0.80+0.04 −0.04
FJ,U −0.01+0.10 −0.02 −0.00+0.07 −0.02 0.02+0.18 −0.07 0.01+0.25 −0.07 0.03+0.28 −0.11 0.00+0.13 −0.04 0.04+0.38 −0.13 0.04+0.28 −0.13 0.04+0.36 −0.15 0.06+0.31 −0.15 0.06+0.27 −0.16 0.04+0.32 −0.15 0.04+0.42 −0.15 0.07+0.27 −0.17 0.03+0.33 −0.13 0.07+0.36 −0.18 0.01+0.29 −0.12 0.08+0.33 −0.21 0.05+0.35 −0.18 0.07+0.32 −0.20 0.05+0.33 −0.18 0.09+0.33 −0.22 0.03+0.41 −0.18 +0.29 0.08−0.23 0.06+0.32 −0.22 0.08+0.35 −0.23 0.02+0.39 −0.19
can be flattened. The ultrafaint dSphs have many fewer baryons than the classical dwarfs so it is anticipated that feedback effects have a weaker effect on the shape of the dark matter distribution in the ultrafaints. Therefore, for the ultrafaints, a flattened stellar distribution probably corresponds to a flattened dark matter distribution. Of these ultrafaints, Reticulum II is an interesting object as it is particularly nearby and also one of the most highly flattened of all the ultrafaints, at least as judged by the stellar light. On these grounds, we might well expect that flattening may provide an explanation as to why a gammaray signal may have been seen towards Reticulum II as opposed to other ultrafaints. We have explored the impact of flattening on the J and Dfactors, which control the expected dark matter annihilation and decay signals from the dSphs. The effects of flattening on these factors can be decomposed into two separate corrections: the geometric and the kinematic factors. The first of these corresponds to the increase (decrease) in darkmatter density produced by squeezing (stretching) the models. The latter corresponds to how the observed velocity dispersion relates to the total velocity dispersion or the enclosed dark matter mass. When viewing oblate (prolate) models faceon, these two fac
FJ,R −0.05+0.04 −0.12 −0.06+0.04 −0.13 −0.11+0.06 −0.14 −0.11+0.07 −0.13 −0.16+0.07 −0.15 −0.08+0.06 −0.13 −0.19+0.07 −0.15 −0.19+0.06 −0.13 −0.21+0.08 −0.13 −0.20+0.08 −0.13 −0.21+0.08 −0.12 −0.21+0.08 −0.13 −0.22+0.07 −0.12 −0.22+0.07 −0.13 −0.20+0.09 −0.14 −0.20+0.07 −0.12 −0.21+0.11 −0.12 −0.24+0.07 −0.12 −0.23+0.06 −0.12 −0.24+0.07 −0.11 −0.24+0.08 −0.11 −0.23+0.06 −0.11 −0.25+0.06 −0.12 +0.07 −0.26−0.10 −0.26+0.06 −0.09 −0.24+0.06 −0.09 −0.22+0.08 −0.08
FJ,T 0.07+0.19 −0.17 0.03+0.08 −0.06 0.09+0.15 −0.15 0.09+0.21 −0.17 0.13+0.20 −0.23 0.09+0.19 −0.15 0.06+0.33 −0.21 0.05+0.30 −0.17 0.04+0.28 −0.19 0.03+0.27 −0.18 0.00+0.24 −0.16 0.03+0.28 −0.16 0.00+0.24 −0.18 −0.01+0.17 −0.14 0.01+0.22 −0.14 0.00+0.21 −0.16 0.01+0.24 −0.14 −0.01+0.16 −0.13 −0.01+0.19 −0.16 −0.02+0.15 −0.15 −0.01+0.20 −0.15 −0.02+0.15 −0.13 −0.03+0.08 −0.11 +0.08 −0.04−0.12 −0.04+0.07 −0.13 −0.05+0.07 −0.10 −0.06+0.06 −0.10
tors act together to increase (decrease) the Jfactor over a spherical analysis, whereas, when viewing these models edgeon, the two factors compete and result in a decrease (increase) in the Jfactor over a spherical analysis. We have used MadetoMeasure techniques [38, 39] to build numerical equilibrium models of Reticulum II. These reproduce the flattened shape, the majoraxis length and the line of sight velocity dispersion of Reticulum II. For the models with a prolate dark matter halo with ellipticity ∼ 0.6 viewed edgeon, flattening could cause an additional amplification of ∼ 2 − 2.5 for Reticulum II over that expected for spherical dark halos. This factor could be still larger if the stellar profile falls off more slowly than a Plummer law (which could increase the kinematic factor). It could also be larger if the dark halo of Reticulum II is triaxial (as anticipated from darkmatteronly simulations) and hence more flattened along the lineofsight. However, this scenario is disfavored by darkmatter simulations with and without baryons that produce subhalos which preferentially point towards the center of their host halo and so we might anticipate dSphs to be elongated along the lineofsight. We corroborated the results of the MadetoMeasure simulations with a simpler virial method that allows for
20
0.8
Uniform (U)
Major axis (R)
SJ 2016 (T)
0.6
FJ
0.4 0.2 0.0 0.2
Hydra II Leo T Leo II Segue 2 Leo I Horologium I Fornax Draco Sculptor Carina Sextans Coma Berenices Tucana II Canes Venatici I Boötes I Pisces II Grus I Willman 1 Segue 1 Leo IV Leo V Canes Venatici II Ursa Minor Reticulum II Ursa Major II Hercules Ursa Major I
0.4
FIG. 10: Medians and ±1σ errorbars for the base10 logarithms of the Jfactor correction factors for all the dSphs ranked by their ellipticity. The three different errorbars correspond to three different prior assumptions regarding the intrinsic shapes and alignments of the dSphs as described in the text.
more rapid calculation of the correction factors for general geometries. A simple fitting relation has been provided for rapid estimation of the correction factors for the oblate and prolate cases. Additionally, we have inspected two cases where some analytic progress can be made in the computation of the Jfactors. This has allowed us to characterize how the correction factors change as a function of the stellar and darkmatter distributions. We found that the correction factors for the MadetoMeasure models agree well with the trends seen in the analytic models. We used our models to estimate the Jfactors for the dSphs under the assumption that the figures are aligned with the lineofsight and are either oblate or prolate. The ranking of the Jfactors of the dSphs is slightly altered when accounting for flattening under the assumption that all the dSphs are either prolate or oblate. Typical correction factors for a dSph with ellipticity 0.4 are 0.75 in the oblate case and 1.6 in the prolate case. We also demonstrated that the corrections to the Dfactors are much smaller than the scatter in the spherical Dfactor from the other observables. For instance, a dSph with ellipticity 0.4 has a Dfactor correction factor of 0.97 in the oblate case and 1.1 in the prolate case. Therefore, we concluded that flattening is unimportant for Dfactor computation. We concluded our discussion of the effects of flatten
ing by computing correction factors for triaxial figures. The findings from the axisymmetric cases were found to simply extend when considering triaxiality. The largest Jfactor correction factor corresponds to viewing the figure along the minor axis, whilst the smallest corresponds to viewing the figure along the major axis. We found that for a Reticulum IIlike model the Jfactor correction factor varies by a factor of ∼ 6 − 10 as one changes the viewing angle. For a fixed observed ellipticity, the correction factor can vary by a factor of ∼ 4. We demonstrated that for the known dSphs the uncertainty in the correction factors due to unknown triaxiality increases with the observed ellipticity of the dSph and is typically a factor of two for ∼ 0.4. If all dSphs have their major axes aligned with the lineofsight (as suggested by some simulations), the correction factors decrease as a function of observed ellipticity and are typically a factor 1/2 for 0.4 . . 0.6. Deviations from sphericity in both the light profile and the dark matter are important. This suggests fundamental limitations to the spherical Jeans modeling which is common in the field (although see [32] for Jfactors computed using axisymmetric Jeans modeling). In particular, increasingly sophisticated statistical techniques [70, 71] will fail to include an inherent uncertainty if the assumption of a spherical stellar density profiles in a spherical dark halo breaks down. The uncertainties,
21 which are different for different dSphs, must be accounted for joint analyses of multiple dSphs. Spherical Jeans modeling is probably most useful for large classical dwarf spheroidal galaxies that look nearly round (such as Leo I or Fornax). It ignores important uncertainties for the ultrafaints, which is unfortunate as these are the most promising targets of all for indirect dark matter detection. We hope that the work presented here – a systematic foray into the domain of flattening – is the beginning of a systematic exploration of more general flattened and triaxial dark halo shapes. Finally, this study was partly inspired by the gammaray detection [17] toward the very flattened ultrafaint, Reticulum II. Our work demonstrates that Reticulum II could have a Jfactor that is higher than spherical analyses suggest if it is a prolate figure. However, the correction for the prolate shape does not make Reticulum II stand out as the dSph with the highest Jfactor nor does a lack of signal from the other dSphs create any tension, irrespective of the shapes of the other dSphs. If, however, Reticulum II is an oblate figure the Jfactor is lower than that found through spherical analyses and lack of signal from the other dSphs may give rise to some tension if
the other dSphs (such as Ursa Minor) are prolate. More generically we have demonstrated that unknown triaxiality produces uncertainty in the Jfactor for Reticulum II of a factor of ∼ 2. In general, we have shown that the effect of flattening on expected dark matter annihilation fluxes cannot be ignored. Indeed, flattening can shift expected signals by amounts larger than error bars due to current velocity dispersion measurements. These currently unknown shifts change the ranking of dSph targets for gammaray experiments. However, if the orientations of the Milky Way dSphs can be determined, the results presented here can help pin down relative Jfactors and allow tests of dark matter explanations of gammaray detections.
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Acknowledgments
JLS acknowledges financial support from the Science and Technology Facilities Council (STFC) of the United Kingdom. Figure 1 was produced using the Pynbody package [72].
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Appendix A: Virial ratios for cusped models
In Section IV B we presented the virial ratio 2 2 hσxx i/hσzz i for a flattened cusped model with density slope γ? = 3. Here we provide formulae for other values
23 so that when q? = qφ , we obtain
of γ? . For γ? = 2, the virial ratio is 2 Q? T (Q) − Q hσxx i = , 2 i hσzz 2[Q − T (Q)]
(A1)
where T = Arctanh for Q? < 1 and T = Arctan for Q? > 1, and Q is defined in equation (22). For γ? = 4, the virial ratio is 2 hσxx i Q? [(Q2 − 1)T (Q) + Q] = 2 hσzz i 2[Q? T (Q) − Q]
(A2)
where we have corrected a typographical error in eq. (21) of Agnello and Evans [55]. For γ? = 5, the virial ratio is simply 2 hσxx i = Q? . 2 hσzz i
(A3)
A plot of the ratios as a function of flattening are given as Figure 1 in [55]
2 hσRR i=
v02 , 2
In Section IV C, we presented formulae for the velocity dispersions of a cored stellar profile with outer density slope β? = 5 embedded in a cored dark matter density profile. Here we provide equivalent formulae for the cases β? = 4 and β? = 6. Let us recall the definitions ∆21 = qφ2 − q?2 ,
∆22 = Rd 2 − Rc 2 ,
(B1)
and let us introduce the function Rd ∆1 Rc ∆1 i 1 h Arctan − Arctan . F = ∆1 ∆2 q? ∆2 qφ ∆ 2 (B2) Notice that if Rc > Rd or q? > q, this function remains welldefined on using the identity (for S < 0) √ √ 1 1 √ Arctan S ≡ √ Arctanh −S. −S S For the case β? = 4, v02 qφ Rc h 2 2 (qφ Rd + q?2 (Rc 2 −2Rd 2 ))F (B3) ∆21 ∆22 i − qφ Rc + q? Rd , i v 2 q 2 Rc h q? 2 qφ F − . (B4) hσzz i = 0 ?2 ∆1 q? Rc + qφ Rd
2 hσRR i =
If Rc = Rd , then the velocity dispersions are a lot simpler. A careful Taylor expansion gives 2 hσRR i = 2 hσzz i =
2qφ (2q? + qφ )v02 , 3(q? + qφ )2 q?2 v02 . (q? + qφ )2
(B5)
v02 . 4
(B6)
Finally, we give the results for β? = 6, v02 qφ Rc 2 h Rd (Rc 2 + 2Rd 2 )∆21 − q? qφ Rc ∆22 ∆21 ∆42 q? Rc + qφ Rd i − Rc (3qφ2 Rd 2 + q?2 (Rc 2 − 4Rd 2 ))F ,
2 hσRR i =
(B7)
2
2 hσzz i =
q? qφ Rc Rd + qφ2 Rd − q?2 ∆22 i v02 q?2 Rc 2 h −qφ Rc F + . 2 2 ∆ 1 ∆2 (q? Rc + qφ Rd )2
If Rc = Rd , then 2 hσRR i = 2v02 2 i = v02 hσzz
Appendix B: Velocity dispersions for flat rotation curve models
2 hσzz i=
qφ (8q?2 + 9q? qφ + 3qφ2 ) , 15(q + qφ )3
q?2 (3q? + qφ ) . 3(q? + qφ )3
(B8)
and if additionally q? = qφ , we recover 2 hσRR i=
v02 , 3
2 hσzz i=
v02 . 6
(B9)