THE MASSES OF LOCAL GROUP DWARF SPHEROIDAL GALAXIES: THE DEATH OF THE UNIVERSAL MASS PROFILE

We focus our analysis here on the inclusion of the masses of the M31 dSphs into the universal density profile of Walker et al. (2009). In their work, the authors were spurred on by the earlier results of Strigari et al. (2008) that showed that all of the MW dSphs for which kinematic data were available were consistent with having the same mass contained within a radius of 300 pc (roughly 1 × 10⁷ M_☉) despite spanning six decades in luminosity. Strigari et al. (2008) used this result to argue that it was possible that all dSphs inhabited a universal dark matter halo, where the density as a function of radius was identical, irrespective of the number of stars the halo hosted. However, Wolf et al. (2010) demonstrated that extrapolations to both larger and smaller radii than the true half-light radius are extremely uncertain in cases where the velocity anisotropy is unknown, and this is true for all Local Group dSphs. For objects with r_half ≪ 300 pc, one has to extrapolate to radii inhabited by no tracers, where tidal stripping may have removed the outer dark matter envelope (Peñarrubia et al. 2008b). That means that for some galaxies, extrapolating out to 300 pc could overestimate the enclosed mass by several orders of magnitude. In the interest of trying to measure a more meaningful mass for these objects to determine whether dSphs truly resided within a universal halo, Walker et al. (2009) measured the velocity dispersion, and hence mass, within the half-light radius of the MW dSphs. Then, by treating each velocity dispersion measurement from MW dSphs as a measurement of the velocity dispersion at a given radius (the half-light radius of the dSph in question) within a single dark matter halo, they could map out the velocity dispersion profile for this singular halo. In particular, they tested the cosmologically motivated Navarro–Frenk–White (NFW; Navarro et al. 1997) density profile:

$$\begin{equation} r_{\rm half}\sigma _v^2=\frac{2\eta R_SV_{\mathrm{max}}^2}{5}\times \left[\frac{\mathrm{ln}(1+r_{\mathrm{half}}/R_S)-\frac{r_{\mathrm{half}}/R_S}{1+r_{\mathrm{half}}/R_S}}{\rm {ln}(1+\eta)-\frac{\eta }{1+\eta }}\right], \end{equation} \tag{ 1 }$$

where V_max is the maximum circular velocity of the halo, R_S is the scale radius of the halo, and η = 2.16. They also used a cored density profile where:

$$\begin{eqnarray} r_{\mathrm{half}}\sigma _v^2&=&\frac{2\eta R_SV_{\mathrm{max}}^2}{5(\mathrm{ln}[1+\eta ])+\frac{2}{1+\eta }-\frac{1}{2(1+\eta)^2}-\frac{3}{2}} \nonumber \\ &&\times \left[\mathrm{ln}(1+r_{\rm half}/R_S)+\frac{2}{1+r_{\rm half}/R_S}\right.\nonumber \\ &&\left. -\frac{1}{2(1+r_{\rm half}/R_S)^2}-\frac{3}{2}\right], \end{eqnarray} \tag{ 2 }$$

with η = 4.42, α = 1, and γ = 0. The results of this study showed that the MW dSphs were consistent with having formed with a universal mass profile, although the authors noted that there was significant scatter about this relation, a factor of two greater than expected from the observational uncertainties alone. Later that same year, a revised study of the mass of the Hercules dSph (Adén et al. 2009), which provided a better treatment of the contaminating foreground population, determined a much lower value for the velocity dispersion of this object (3.72 ± 0.91 km s⁻¹ versus 5.1 ± 0.9 km s⁻¹ from Simon & Geha 2007). With their revised value, they showed that the mass of Hercules was not consistent with the universal mass profile. As Hercules is likely significantly affected by tides (Adén et al. 2009; Martin & Jin 2010), this is perhaps not unexpected. Similarly, an analysis of the Böotes I dSph by Koposov et al. (2011), who used multi-epoch observations taken with the Very Large Telescope and implemented an enhanced data reduction approach to measure extremely precise radial velocities, measured a velocity dispersion of $\sigma _v=4.6^{+0.8}_{-0.6}{\rm \,km\,s^{-1}}$, significantly lower than that of ∼6.5 km s⁻¹ reported in previous studies. This also renders the Böotes I dSph inconsistent with the universal mass profile.

In Walker et al. (2009), velocity dispersions for only two M31 dSphs (And II and IX) were available. We therefore fit NFW and cored density profiles (Equations (1) and (2)) to the velocity dispersions of the entire dSph population with L > 2 × 10⁴ L_☉ (ensuring we probe the same luminosity regime in both the MW and M31), to see how well these populations can be fit with a single density profile. Both profiles have two free parameters of interest to fit, the circular velocity of the halo, V_max and the scale radius R_S. To constrain these values, we use a maximum likelihood fitting routine to determine the most probable values for these parameters by maximizing the likelihood function, $\mathcal {L}$, defined as:

$$\begin{eqnarray} \mathcal {L}(\lbrace r_{h,i},\sigma _{v,i},\delta _{\sigma _v,i}\rbrace |V_{\rm max},R_S)&=&\prod _{i=0}^{N} \frac{1}{\sqrt{2\pi \delta _{\sigma _v,i}^2}} \nonumber \\ &&\times{\rm exp}\left[-\frac{(\sigma _{\mathrm{profile}}-\sigma _{v,i})^2}{2\delta _{\sigma _v,i}^2}\right] \nonumber\\ \end{eqnarray} \tag{ 3 }$$

where σ_profile is the velocity dispersion as predicted by Equations (1) and (2) for a dSph with half-light radius r_{h, i}; σ_{v, i} is the measured velocity dispersion of the i^th dSph; and $\delta _{\sigma _v,i}$ is the uncertainty on the measured dispersion. We include only the uncertainty in σ_v in our method, neglecting that of the half-light radius, as the velocity dispersion parameter has a greater impact on the mass profile, as it is proportional to the square of σ_v, depending only linearly on r_half.

We show the results of this fit in Figure 1. The red triangles represent MW dSphs brighter than L = 2 × 10⁴ L_☉, while open triangles represent those fainter than this cut. The blue circles are the M31 dSphs. The magenta dot-dashed line shows our best-fit NFW profile to the whole population (with V_max = 14.7 ± 0.5 km s⁻¹ and R_S = 876 ± 284 pc), whilst the cyan dashed line is the best-fit core profile (with V_max = 14.0 ± 0.4 km s⁻¹ and R_S = 242 ± 124 pc) where the best-fit V_max (R_S) parameter is determined by marginalizing the 2D maximum likelihood contours over R_S (V_max). In all cases, the quoted uncertainties are derived by assuming that the likelihood functions have a Gaussian-like distribution, allowing us to project the marginalized maximum likelihood contours to the value at which $2\,\mathrm{ln}\,(\mathcal {L})$ has decreased by the square of the confidence interval of interest, which in this case is the 1σ (i.e., 68%) confidence interval. Clearly, neither of these mass profiles is a good fit for many of the Local Group dSphs. This is statistically demonstrated by the reduced χ² values for these fits (χ² = 4.1 for both profiles). Of the 39 objects, 24 are outliers at >1σ, with ∼1/5 of the population being outliers at the 3σ level.

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From the above analysis, it is clear that the scatter about the best fitting profiles is significant and well beyond what we can hope to explain with measurement uncertainties. But do we really expect that all low luminosity galaxies should reside in dark matter halos with identical density profiles? The dark matter subhalos produced in, for example, the Aquarius simulations (Springel et al. 2008) demonstrate a range of possible values of V_max and R_s for these objects. Further, work by e.g., Zolotov et al. (2012), Brooks & Zolotov (2012) and Vera-Ciro et al. (2013) demonstrate that the infall time, host mass, and presence of baryons can all affect the dark matter structures of subhalos. As such, scatter in density profiles is completely expected, and differences between the satellite populations of the Milky Way and Andromeda might also be seen that could tell us about the evolutionary histories of the two systems.

To investigate this, we introduce a mass-scatter term, $\sigma _{V_{\rm max}}$, into our maximum likelihood fitting algorithm (Equation (3)) replacing $\delta _{\sigma _v,i}$ with δ_{tot, i}, which is the combination of the measured uncertainty in the velocity dispersion measurements and the mass-scatter term, such that $\delta _{\mathrm{tot},i}=\sqrt{\vphantom{A^A}\smash{{{\delta _{\sigma _v,i}^2+\sigma _{V_{\rm max}}^2}}}}$.

If the mass profiles of the Andromeda and Milky Way dSphs are truly similar within their inner regions, our algorithm should find best-fit values of V_max, R_S, and $\sigma _{V_{\rm max}}$ that are broadly consistent when fitting the two populations separately and as a whole. In Figure 2, we show the likelihood contours for these parameters. In the top three panels, we overlay the 1σ and 2σ confidence interval contours (again, defined as the region of parameter space where $2\,\mathrm{ln}\,(\mathcal {L})$ decreases by the square of the confidence interval in question) for the NFW fits to the Milky Way (red dashed), Andromeda (blue dot-dashed), and the full sample (black solid) for our three free parameters (marginalized over the 3rd parameter not displayed in each 2D subplot), with solid points representing the best-fit values in each case. In the second row of subplots, we show the one-dimensional marginalized relative likelihood functions for V_max, R_S, and $\sigma _{V_{\rm max}}$ for each of these fits. The lower six panels show the same, but for the cored fits. In the NFW case, while the best-fit values for R_S in each case seem dramatically different for the MW and M31 at first glance with $R_S=1034^{+1508}_{-524}{\rm \,pc}$ and $R_S=322^{+247}_{-143}{\rm \,pc}$  respectively, their uncertainties are such that they agree within 1σ. The amount of scatter in mass at a given radius is also very similar, with $\sigma _{V_{\rm max}}=2.9^{+0.7}_{-0.5}{\rm \,km\,s^{-1}}$ and $\sigma _{V_{\rm max}}=3.9^{+0.7}_{-0.6}{\rm \,km\,s^{-1}}$ for the MW and M31, respectively. The preferred values for V_max ($V_{\rm max}=18.4^{+2.9}_{-3.1}{\rm \,km\,s^{-1}}$ and $V_{\rm max}=12.8^{+1.4}_{-1.2}{\rm \,km\,s^{-1}}$  respectively), however, are marginally less consistent, with M31 preferring a lower value of V_max (and hence, lower masses) than the MW system. For the core profiles, we get a similar result, with the values for R_S ($R_S=253^{+143}_{-99}{\rm \,pc}$ and $R_S=142^{+72}_{-53}{\rm \,pc}$), and $\sigma _{V_{\rm max}}$ ($\sigma _{V_{\rm max}}=2.9^{+0.8}_{-0.6}{\rm \,km\,s^{-1}}$and $\sigma _{V_{\rm max}}=3.8^{+0.7}_{-0.6}{\rm \,km\,s^{-1}}$) for the MW and M31 agreeing within 1σ, and marginally inconsistent values of V_max ($V_{\rm max}=16.2^{+2.8}_{-2.1}{\rm \,km\,s^{-1}}$ and $V_{\rm max}=12.8^{+1.3}_{-1.1}{\rm \,km\,s^{-1}}$).

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In the top two panels of Figure 3, we overplot the best-fit relations from this analysis in the r_half–σ_v plane. In the left panel, we show the best-fit NFW profile for the MW (red line) and M31 (blue line) dSphs, with the best-fit core profiles in the right panel. The shaded regions represent the scatter we derived convolved with the 1σ uncertainties for V_max and $\sigma _{V_{\rm max}}$. Both the MW NFW and Core profiles provide an excellent fit to all the dSphs barring the Hercules dSph. However, as it is likely to be highly tidally disturbed (Adén et al. 2009; Martin & Jin 2010), this is not too surprising. For the M31 fits, we see three systems that are outliers at the ∼2–3σ level: And VI, VII, and XXV. And XXV has previously been identified as an unusually low-mass system in C13, so this inconsistency is perhaps not unexpected. However, And VI and VII are thought to represent fairly typical satellites, with velocity dispersions similar to their MW counterpart Fornax, which has a comparable half-light radius to these two objects.

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The differences in the preferred V_max has a striking visual effect on the resulting best-fit profiles for the MW and M31. While in both the NFW and Core case, the relations for MW and M31 populations track each other well at the smaller radius (lower mass) end (albeit with greater scatter in M31), at larger radii, there appears to be a divergence between the two systems. Both the NFW and Core profiles for M31 begin to turnover at ∼600 pc, while the MW profile continues to rise (turning over at ∼1200 pc in the cored case). This turnover radius is interesting, as there are only three MW dSphs with half-light radii ≳ 600 pc, one of which is the tidally disrupting Sagittarius (Sgr) and thus is excluded from our fits, the other two being Fornax and Sextans. In M31, there are seven galaxies (And I, II, VII, XIX, XXI, XXIII, and XXV), three of which (And XIX, XXI, and XXV) are curiously very low mass for their size. In C13, they were measured to be 3σ outliers to the best-fit mass profiles of Walker et al. (2009) and, as can be seen in Figure 3, they are significant outliers to the best-fit MW relation (2, 2.5 and 3σ, respectively). In addition, despite having very similar half-light radii, Sgr (r_half = 1550 ± 50 pc) and And XIX ($r_{\rm half}=2072^{+1092}_{-422}$ pc) have very different velocity dispersions (σ_v = 11.7 ± 0.7 km s⁻¹ and $\sigma _v=4.7^{+1.6}_{-1.4}{\rm \,km\,s^{-1}}$).

To see if it is these outliers driving the differences between the MW and M31 relations, we repeat the same fits performed above, but without all systems that were found to lie at or greater than 3σ from the best-fit profiles of Walker et al. (2009). This included the three M31 outliers, and additionally the MW dSphs, Hercules, and CVn I, which are also outliers to the Walker et al. (2009) relation. In Figure 4, we show the same contours as in Figure 2 for NFW and Core profile fits, only this time with these outliers omitted. The exclusion of Hercules and CVn I from the MW fits has only a slight effect on these profiles, but removing the low mass M31 outliers is more substantial. The agreement between R_s, V_max and $\sigma _{V_{\rm max}}$ is significantly better (as shown in Table 1). In the lower two panels of Figure 3, we show these best-fit profiles in the r_half–σ_v plane. Barring the five excluded dSphs and the M33 satellite, And XXII (Chapman et al. 2013), all the Local Group objects have velocity dispersions that are well described by the M31 and MW relations. The best-fit profiles to the whole Local Group system are shown in Figure 5, and have preferred values for the NFW (core) parameters of $R_S=664^{+412}_{-232}{\rm \,pc}$ ($R_S=225^{+70}_{-55}{\rm \,pc}$), $V_{\rm max}=16.2^{+2.6}_{-1.7}{\rm \,km\,s^{-1}}$ ($V_{\rm max}=15.6^{+1.5}_{-1.3}{\rm \,km\,s^{-1}}$), and $\sigma _{V_{\rm max}}=2.9^{+0.5}_{-0.4}{\rm \,km\,s^{-1}}$ ($\sigma _{V_{\rm max}}=2.8^{+0.5}_{-0.4}{\rm \,km\,s^{-1}}$). Therefore, whilst dSph galaxies do not live within dark matter halos with identical density profiles, the vast majority do inhabit statistically similar halos with a well-defined mass range at any given radius.

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Table 1. Best-fit Parameters from Mass Profile Fits to MW and M31 dSph Data Using NFW and Cored Profiles

Model	Full			M31			MW
	Vmax	RS	$\sigma _{V_{\rm max}}$	Vmax	RS	$\sigma _{V_{\rm max}}$	Vmax	RS	$\sigma _{V_{\rm max}}$
	( km s−1)	(pc)	( km s−1)	( km s−1)	(pc)	( km s−1)	( km s−1)	(pc)	( km s−1)
NFW	$13.6^{+1.3}_{-1.0}$	$408^{+221}_{-143}$	3.6 ± 0.5	$12.8^{+1.4}_{-1.2}$	$322^{+247}_{-143}$	$3.9^{+0.7}_{-0.6}$	$18.4^{+2.9}_{-3.1}$	$1034^{+1508}_{-524}$	$2.9^{+0.7}_{-0.5}$
NFW (minus outliers)	$16.2^{+2.6}_{-1.7}$	$664^{+412}_{-232}$	$2.9^{+0.5}_{-0.4}$	$16.7^{+3.5}_{-2.4}$	$790^{+828}_{-349}$	$3.2^{+0.7}_{-0.6}$	$18.7^{+4.9}_{-4.1}$	$708^{+1816}_{-391}$	$2.4^{+0.7}_{-0.5}$
Core	$13.5^{+1.1}_{-0.9}$	$165^{+58}_{-47}$	$3.5^{+0.5}_{-0.4}$	$12.8^{+1.3}_{-1.1}$	$142^{+72}_{-53}$	$3.8^{+0.7}_{-0.6}$	$16.2^{+2.8}_{-2.1}$	$253^{+143}_{-99}$	$2.9^{+0.8}_{-0.6}$
Core (minus outliers)	$15.6^{+1.5}_{-1.3}$	$225^{+70}_{-55}$	$2.8^{+0.5}_{-0.4}$	$15.7^{+2.1}_{-1.7}$	$257^{+108}_{-88}$	$3.2^{+0.7}_{-0.6}$	$15.9^{+3.1}_{-2.1}$	$208^{+119}_{-82}$	$2.5^{+0.8}_{-0.6}$

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In Figure 5, we show the mass within the half-light radii of the dSphs (calculated using the Walker et al. 2009 mass estimator, where $M_{\rm half}=580\; r_{\rm half}\; \sigma _v^2$, tabulated in Table 2) with the best-fit NFW and Core relations when excluding the outliers. At all radii, the total scatter is less than half a magnitude in mass. For example, at r_half = 10 pc, 100 pc, and 1000 pc, the average masses from the cored profile are M_half = 2.3 × 10⁴ M_☉, 1.1 × 10⁶ M_☉, and 5.4 × 10⁷ M_☉, and the scatter (i.e., half the distance outlined by the shaded band) is M_scatter = 1.2 × 10⁴ M_☉, 0.6 × 10⁶ M_☉, and 2.5 × 10⁷ M_☉, which is ∼50% of the average mass in each case. The numbers for the best fit NFW profile are almost identical.

Table 2. The Masses, Mass-to-Light Ratios, and V_max Values for Local Group dSphs as Derived in This Work

Name	Mhalf	[M/L]half	Vc, 1/2
	(× 107 M☉)	( M☉/ L☉)	( km s−1)
And I	5.05 ± 1.35	22.4 ± 8.5	16.1 ± 4.4
And II	3.31 ± 0.7	27.6 ± 10.8	12.3 ± 2.6
And III	1.95 ± 0.46	39.0 ± 12.9	14.7 ± 3.7
And V	2.30 ± 0.40	78.0 ± 19.5	16.6 ± 3.3
And VI	4.67 ± 0.9	27.5$^{+7.61}_{-6.85}$	19.5$^{+4.2}_{-3.9}$
And VII	7.61 ± 0.9	9.0 ± 1.6	20.5 ± 2.7
And IX	2.25 ± 0.7	302 ± 132	17.2 ± 6.0
And X	0.46 ± 0.2	61.2$^{+52}_{-49}$	10.1$^{+5.1}_{-4.3}$
And XI	0.41$^{+0.3}_{-0.2}$	165.5$^{+ 196}_{- 142}$	12.0$^{+11.0}_{-8.1}$
And XIII	0.26$^{+0.22}_{-0.16}$	126.6$^{+ 153}_{- 108}$	9.2$^{+10.1}_{-6.4}$
And XIV	0.42 ± 0.2	41.6 ± 28.2	8.4 ± 5.2
And XV	0.22 ± 0.11	9.0$^{+7.1}_{-7.0}$	6.3$^{+3.4}_{-3.3}$
And XVI	0.24$^{+0.08}_{-0.06}$	11.6$^{+3.9}_{-2.9}$	8.8$^{+3.2}_{-2.7}$
And XVII	0.13$^{+0.33}_{-0.19}$	12.82$^{+ 44.79}_{- 26.38}$	4.5$^{+11.2}_{-4.5}$
And XVIII	1.5 ± 0.5	44.8 ± 27.1	15.3 ± 6.1
And XIX	2.7$^{+1.9}_{-1.2}$	118.0$^{+ 124.2}_{- 85.2}$	7.4$^{+6.6}_{-3.8}$
And XX	0.30$^{+0.28}_{-0.17}$	213.0$^{+ 282.2}_{- 171.0}$	11.2$^{+12.0}_{-7.0}$
And XXI	1.2$^{+0.5}_{-0.4}$	29.8$^{+18.7}_{- 16.5}$	7.1$^{+3.1}_{-2.7}$
And XXII	0.10$^{+0.11}_{-0.08}$	69.7$^{+ 102.2}_{- 82.4}$	4.4$^{+4.7}_{-3.9}$
And XXIII	3.4$^{+0.8}_{-0.7}$	68.4$^{+ 46.4}_{- 46.1}$	11.2$^{+2.8}_{-2.6}$
And XXV	0.33$^{+0.19}_{-0.18}$	10.2$^{+ 10.0}_{-9.5}$	4.7$^{+2.9}_{-2.6}$
And XXVI	0.83$^{+0.72}_{-0.46}$	279.8$^{+ 383}_{- 277.6}$	13.7$^{+ 15.7}_{-9.6}$
And XXVIII	0.53$^{+0.36}_{-0.27}$	50.5$^{+ 51.0}_{- 39.1}$	10.4$^{+7.7}_{-5.8}$
And XXIX	0.68$^{+0.23}_{-0.22}$	67.8$^{+ 38.6}_{- 37.5}$	9.0$^{+3.4}_{-3.2}$
And XXX	2.1$^{+2.0}_{-1.2}$	300.0$^{+ 433.1}_{- 302.3}$	18.6$^{+17.7}_{-11.4}$
Scl	1.3 ± 0.3	18.2 ± 9.8	14.5 ± 3.9
LeoT	0.58 ± 0.22	196.9 ± 120.0	11.8 ± 5.1
UMaI	2.6$^{+1.2}_{-1.1}$	3731$^{+ 2577}_{- 2524}$	18.8$^{+8.9}_{-8.5}$
LeoIV	0.13 ± 0.10	299.1 ± 54.2	5.2 ± 4.0
Com	0.09 ± 0.03	510.8 ± 309.0	7.3 ± 2.2
CVnII	0.09 ± 0.03	229.9$^{+ 158}_{- 152}$	7.3$^{+3.0}_{-2.6}$
LeoI	1.2 ± 0.3	7.1 ± 3.3	14.5 ± 3.5
LeoII	0.38 ± 0.07	12.9 ± 5.2	10.4 ± 2.2
Car	0.6 ± 0.2	50.7 ± 28.9	10.4 ± 3.0
UMi	1.5 ± 0.3	146.6 ± 76.5	15.0 ± 2.9
Dra	0.94 ± 0.18	69.7 ± 22.0	14.4 ± 3.0
For	5.3 ± 0.6	7.6 ± 2.5	18.5 ± 2.4
Sex	2.5 ± 0.71	120.4 ± 74.4	12.5 ± 4.2
Boo	0.30$^{+0.08}_{-0.06}$	198.0$^{+ 83.4}_{- 69.1}$	7.3$^{+2.0}_{-1.6}$
CVnI	1.9 ± 0.2	164.3 ± 31.2	12.0 ± 1.4
Herc	0.26$^{+0.11}_{-0.10}$	145.6$^{+ 95.8}_{- 89.7}$	5.8$^{+2.8}_{-2.4}$
LeoV	0.04 ± 0.05	197.5 ± 339.1	3.8 ± 4.4
Wil1	0.03 ± 0.02	536.2 ± 613.8	6.8 ± 4.6
UmaII	0.26 ± 0.10	1319.1 ± 961	9.0 ± 3.9
Seg	0.02 ± 0.01	1374.7$^{+ 1453.47}_{- 1234.16}$	5.8$^{+3.9}_{-2.8}$
Seg2	0.02 ± 0.02	536.4 ± 588.7	4.1$^{+0.9}_{-4.1}$

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Our decision to exclude And XIX, XXI, and XXV was based on their designation as significant (>3σ) low-mass outliers in C13 and to the derived MW profile. There are several other potentially low-mass systems that were identified in C13 (namely And XXII) and T12 (And XIV, And XV, and XXII also), which we did not exclude, simply because their associated uncertainties place them much closer to the regime of expected mass from the MW system. If they too were shown to be truly low mass with subsequent observations, this would imply that the M31 dwarf spheroidal systems do have greater scatter toward lower masses in their mass profiles compared with the MW.

Briefly, we compare the best-fit values of V_max and the scatter in this term with recent cosmological and semi-analytical models to deduce whether the values we statistically obtain for the Local Group dSphs compare favorably with our theoretical understanding of galaxy formation and evolution.

If we naively compare to dark matter only simulations, such as the subhalos in the Aquarius simulations (Springel et al. 2008) of six MW-mass dark matter halos, we find that our measured values of V_max are lower than would be expected. The same discrepancy was pointed out by Boylan-Kolchin et al. (2012). While the MW dSphs have 12 ≲ V_max ≲ 25 km s⁻¹, they found at least 10 subhalos in each Aquarius host with V_max > 25 km s⁻¹. This discrepancy is referred to as the "TBTF" problem and would seem to persist when including M31 dSphs.

If we instead compare with models where baryons are taken into account, do we still see such inconsistencies? In Rashkov et al. (2012), dark matter subhalos from the high-resolution Via Lactea II simulations are populated with baryons at the time of infall into their host halo by dynamically tagging dark matter particles as stars. These systems are then traced until z = 0, where their final properties are compared to observations. These simulations are able to reproduce many observed properties of MW dSphs (such as velocity dispersions, sizes, metalicities, number count, etc.), and the present day values of V_max for the 10 most luminous subhalos are more compatible with observations, having 10 < V_max < 40 km s⁻¹ (∼50% of which are less than 20 km s⁻¹). Our average V_max plus scatter term gives a statistical range for the V_max of the Local Group dSphs of ∼12–22 km s⁻¹. So while the bulk of their sample is consistent, there remain too many high-mass dSph satellites to be fully consistent. We can also compare our calculated values of R_S with those of the Rashkov et al. (2012) simulations. The range of R_max (which is the radius at which the circular velocity of the halo is at a maximum, i.e., equal to V_max) for their 10 most luminous subhalos ranges from ∼790–5400 pc (assuming an NFW profile). According to Peñarrubia et al. (2008a), R_max ∼ 2R_S, so this corresponds to scale radii for the Rashkov et al. (2012) halos of 395 pc ≲ R_s ≲ 2700 pc which is consistent with the scale radius of $R_S=664^{+412}_{-232}{\rm \,pc}$ that we find for our combined NFW profile (with outliers excluded), suggesting that these subhalos are similarly dense to the Local Group dSphs.

Bovill & Ricotti (2011a) model satellites within the Local Volume from reionization until today, tracing the merger histories and tidal interactions of these objects as they merge to form more massive galaxies. As with the Rashkov et al. (2012) study, they are able to reproduce many of the observed properties of MW and M31 dSphs. For satellites with similar luminosities to those we fit in this work (i.e., L ≳ 10⁴ L_☉), they measure 10 ≲ V_max ≲ 30 km s⁻¹ which is, again, largely consistent with the range of V_max we find. In this instance, the Bovill & Ricotti (2011a) model produces more bright, massive satellites than we see in the Local Group. They discuss this in Bovill & Ricotti (2011b) as the "missing bright satellite" problem. However, for the systems with comparable luminosities, there is significant overlap in their masses.

From these comparisons, we are content that the best-fit profiles to the MW, M31, and total Local Group dSph we have derived are not hugely at odds with predictions from simulations. Some tension remains at the higher end of the subhalo mass range, as the simulations we compare with identify at least a few subhalos with greater V_max than are compatible with observations.

The notion that the lower central masses observed in the outlying dSphs is the result of some physical process that has moved them away from a more typical mass is appealing, and the possibility has been briefly discussed in other works, particularly Collins et al. (2011) and C13. There, the authors point out that dSphs that are more extended at a given luminosity, such as the extreme outliers, And XIX, XXI, and XXV, also tend to have lower central masses. One of the MW low mass outliers, the MW Herc object, is also already thought to be a tidally disrupting system, transitioning into a stellar stream (Martin & Jin 2010).

Outside of the Local Group, a number of tidally disrupting dwarf galaxy systems have recently been observed whose half-light radii are also much more extended than would be expected from the r_half–L relation (Brasseur et al. 2011), resulting in very low surface brightnesses. These include NCG 4449B (M_V = −13.4 and r_half = 2.7 kpc, Rich et al. 2012; Martínez-Delgado et al. 2012), and the Hydra dwarf galaxy, HCC-087 (M_V = −11.6 and r_half = 3.1 kpc; Koch et al. 2012). Their closest analog within the Local Group is And XIX (M_V = −9.3, $r_{\rm half}=2.1^{+1.0}_{-0.4}{\rm \,kpc}$; N. F. Martin et al., in preparation) making it an outlier to the Brasseur et al. (2011) relation, as well as falling below the mass expectation for a galaxy of its radial scale. Taking these examples into account, perhaps these properties (low surface brightness and/or low mass) are indicators of a system undergoing significant tidal interaction with its host.

Mass loss of subhalos from interactions with their hosts has also been studied in numerical models, both in a dark matter only context (e.g., Tormen et al. 1998; Klypin et al. 1999; Ghigna et al. 2000; Hayashi et al. 2003; Zentner & Bullock 2003; Kravtsov et al. 2004b; Kazantzidis et al. 2004) and with the inclusion of baryons (e.g., Peñarrubia et al. 2010; D'Onghia et al. 2010; Zolotov et al. 2012; Brooks & Zolotov 2012). In all cases, as these systems orbit their host, their dark matter is stripped, lowering their densities and masses at all radii. After the dark matter is removed, the stars reach a dynamic equilibrium with their lower density potential, causing a drop in the central mass. In simulations where baryonic physics are included, the mass losses from subhalos as a result of tidal interactions with a host are more pronounced than for the dark matter only case. Further, the size of the mass reduction increases with earlier infall times and more radial orbits. In Zolotov et al. (2012), they demonstrated that a subhalo accreted at z > 6 Gyr in a smoothed particle hydrodynamic (SPH) simulation would experience a greater reduction in its mass than is seen with a dark matter only set up. Similarly, the mass of subhalos on radial orbits in the SPH simulation also experience a more significant drop in mass than their dark matter only counterparts. In all cases, the presence of a massive baryonic disk in the host galaxy (such as those hosted by the Galaxy and M31) reduces the masses of the satellite population at a much greater rate than in the dark matter only case.

One could therefore argue that the outliers seen in this study, such as Hercules, And XIX, XXI, and XXV, may have fallen in to their host galaxies earlier, and onto more radial orbits where they interact more significantly with their host, leading to a more pronounced mass loss. It is difficult to properly model the orbital properties of these objects, but recent work by Watkins et al. (2013) modeled the orbital properties of M31 dSphs by combining the timing argument with phase-space distribution functions. This work found no evidence to suggest that the M31 outliers are on very radial orbits, nor do they seem to have experienced particularly close passages with M31 itself, perhaps ruling out this option.

A prime example of a tidally disrupting dSph within the MW is the Sgr dSph. This object is currently undergoing violent tidal disruption, yet it has a velocity dispersion that is entirely consistent with the best-fit NFW and Cored mass profiles to both the MW alone and to the full Local Group, perhaps arguing against the mechanism we have outlined above. However, Sgr is currently near the pericenter of its orbit, only ∼20 kpc from the Galactic center (Law & Majewski 2010). The outliers we refer to are located further out (D_host > 70 kpc for all outliers; Martin & Jin 2010; Koposov et al. 2011; Conn et al. 2013), and so we do not expect them to be currently experiencing significant tidal distortions, rather that their past interactions with their host have removed more mass from their centers than their more "typical" counterparts.

In summary, numerical models have demonstrated that tidal mechanisms are able to lower the masses of dSphs, and could explain the lower than expected masses of the Local Group outliers, Herc, And XIV, XV, XVI, XIX, XXI, and XXV if they have experienced more significant past interactions with their host.

For many years, kinematic studies of low surface brightness galaxies have shown that the mass profiles of these objects are less centrally dense than expected. They are more compatible with flatter, cored halo functions, rather than the cuspier NFW profiles seen in simulations (e.g., Flores & Primack 1994; de Blok & Bosma 2002; de Blok et al. 2003; de Blok 2005). Many have argued that this is a result of bursty, energetic star formation and supernova (SN) within these galaxies. These processes drive mass out from the center of the halo, flattening the high density cusp into a lower density core, leading to a lower central mass than predicted by pure dark matter simulations (e.g., Navarro et al. 1996; Dekel & Woo 2003; Read & Gilmore 2005; Mashchenko et al. 2006; Pontzen & Governato 2012; Governato et al. 2012; Macciò et al. 2012). Could the lower than expected central masses of the Local Group dSphs also be caused by feedback?

Zolotov et al. (2012) and Brooks & Zolotov (2012) compared a dark matter only simulation with an SPH simulation of a MW type galaxy in a cosmological context to see whether the inclusion of baryons and feedback in the latter can produce satellite galaxies with lower central masses and densities. For galaxies with a stellar mass M_* > 10⁷ M_☉ (M_V ≲ −12) at the time of infall, feedback can reduce the central mass of dSph galaxies. Below this mass, the galaxies have an insufficient total mass to retain enough gas beyond reionization to continue with the significant, bursty star formation processes required to remove mass from their centers. The Local Group outliers discussed above have M_V ≳ −10, so unless they have been significantly tidally stripped by their hosts after falling in, i.e., experienced total (dark matter plus baryonic) mass losses of greater than ∼90% (Peñarrubia et al. 2008b), feedback cannot explain their current masses. This is supported by the findings of Garrison-Kimmel et al. (2013), where they model the dynamical effect of SN feedback on the mass distribution of dark matter halos. To match the current observed central masses in MW dSph galaxies, one would need to deposit 100% of the energy resulting from 40,000 SN directly to the dark matter halos, which is greater than the expected total number of SN to have ever occurred in the majority of these systems. The work of Peñarrubia et al. (2012) also support the findings of these works, namely, that fainter dSphs should not be able to significantly lower their central masses via feedback, and if they were able, they would serve to exacerbate the missing satellite problem.

In Di Cintio et al. (2014), they also study the effect of feedback in subhalos on their mass profiles, using a suite of galaxies from the MaGICC project (Stinson et al. 2010, 2013). Their findings show that the mass of stars formed per halo mass is the most important factor for the shaping of the central mass profiles of galaxies. Objects with stellar-to-halo mass ratios of M_*/M_halo ≲ 0.01 are not able to alter their dark matter distributions, but as this fraction increases, so does the ability to flatten their central mass profiles. They find that this process is maximally efficient in galaxies with M_* ≈ 10^8.5 M_☉, and below this, the central dark matter slope increases once more. As such, their findings are similar to those of Zolotov et al. (2012). This seeming consensus on the amount of baryons required to efficiently reshape the dark matter mass profile of a dwarf galaxies via feedback means that, in principle, based on their current luminosities, only four of the dSphs discussed in this work (And II, And VII, Fornax, and Leo I; McConnachie 2012) have enough explosive energy at their disposal to reduce their central densities with feedback alone. For the remaining MW and M31 objects, another mechanism, such as tides, would need to be invoked to explain their low masses.

From a plethora of works (e.g., Moore et al. 1999; Ghigna et al. 2000; Kravtsov et al. 2004a; Zentner et al. 2005; van den Bosch et al. 2005; Zheng et al. 2005; Giocoli et al. 2008; Springel et al. 2008), we know that the number of subhalos within a host halo, scales with the mass of the host halo itself. Therefore, when comparing the mass of halos we observe within the MW or Andromeda with those found in simulations, it is important that we select a simulated galaxy of the same mass. Unfortunately, in the case of both M31 and the MW, the total masses of these systems are actually quite uncertain, ranging from ∼0.7 to 2.7 × 10¹² M_☉ for the Milky Way (e.g., Wilkinson & Evans 1999; Xue et al. 2008; Li & White 2008; Watkins et al. 2010; Piffl et al. 2013) and ∼0.8 to 2.2 × 10¹² M_☉ (e.g., Evans & Wilkinson 2000; Evans et al. 2000; Li & White 2008; Guo et al. 2010; Watkins et al. 2010), making this difficult. From the point of view of abundance matching, a galaxy as luminous as the MW should be hosted by a halo with an even higher mass than these estimates (∼3 × 10¹² M_☉; Behroozi et al. 2013), which could imply that our Galaxy is a significant outlier when compared with the bulk of the galaxies within the Universe.

Vera-Ciro et al. (2013) investigated the effect of varying the mass of the host halo on both the number and dynamics of subhalos using the Aquarius simulations (Springel et al. 2008). They found that they were able to match both these quantities when using a simulated halo whose mass was consistent with the lower bound of observational constraints for the MW, 8 × 10¹¹ M_☉. This immediately eliminates the TBTF problem, as the most massive simulated subhalos have V_max ≲ 25 km s⁻¹. Di Cintio et al. (2012) also find that they can match the number and dynamics of MW satellites using halos from the CLUES simulations with masses of 5–7 × 10¹¹ M_☉ (Gottloeber et al. 2010; Di Cintio et al. 2012). Thus, if the virial mass of the MW is at the lower end of current observational estimates, the masses we measure for its subhalos would be much more inline with predictions from numerical simulations. However, it is worth noting that such a low mass for the MW would lessen the probability of hosting massive satellites like the LMC and SMC, and may just replace the missing massive satellite problem with a found massive satellite problem (e.g., Boylan-Kolchin et al. 2011a; Busha et al. 2011). It is also in conflict with a recent estimate of the mass of the MW from Boylan-Kolchin et al. (2013) who use the Aquarius simulations to demonstrate that the 3D space motion of the Leo I dSph puts a lower limit of ∼1 × 10¹² M_☉ on the mass of the MW at a confidence level of 95%. Vera-Ciro et al. (2013) also find that the Andromeda satellites can be best matched if the host mass is 1.77 × 10¹² M_☉, implying that the mass of M31 is roughly twice that of the MW. This value is compatible with the best estimate for the Andromeda mass when using the full dwarf galaxy satellite population as a tracer (L. Watkins 2013, private communication). If Andromeda is more massive than the MW, it could explain the fact that M31 has more (in number terms) massive non-dSph satellites (NGC 147, NGC 185, NGC 205, M32, and M33) than the MW (LMC and SMC). It may also explain the low-mass outliers, particularly the statistically significant And XIX, XXI, and XXV. In particular, the very low value of V_{c, 1/2} we derive for And XXV strongly indicates that it is currently residing in a dark matter subhalo with a maximum circular velocity below the mass limit expected for luminous galaxy formation, suggesting it has experienced some physical process that has lowered its mass significantly over the course of its evolution. A more massive host would imply that the dSphs (which are more susceptible to tidal disruption than the more massive satellites) within this system have experienced greater tidal forces over the course of their evolution, which could lower their masses below the SF threshold of 10–15 km s⁻¹ (Peñarrubia et al. 2008b; Koposov et al. 2009).

Precisely pinning down the correct viral masses of the MW and M31 is clearly an important step toward better understanding the masses of the dwarf galaxies we observe within the Local Group in a cosmological context. Without precise mass estimates, it is difficult for us to quantify discrepancies with theoretical expectations, like those of the TBTF problem, and might help us to explain the differences between the masses of dSphs we see around M31 and the MW.