ANDROMEDA DWARFS IN LIGHT OF MOND. II. TESTING PRIOR PREDICTIONS

The giant and satellite galaxies of the Local Group offer various ways of testing the predictions of ΛCDM and modified Newtonian dynamics (MOND). The number statistics of satellites around the Milky Way and Andromeda is one test of the ΛCDM paradigm, leading to the well-known missing satellite problem (e.g., Moore et al. 1999). More recently, the phase-space distributions of the satellites around their host galaxy have been shown to be an additional challenge for ΛCDM (Kroupa 2012). Rather than resembling the orbit distribution of subhalos in structure formation simulations, many of the satellites are organized into rather thin, rotating disks, as shown for the Milky Way by Pawlowski et al. (2012) and for M31 by Ibata et al. (2013). Whether the dSph satellites of the Local Group are primordial dwarfs residing in subhalos or tidally formed remnants is another possible test to distinguish ΛCDM and MOND (e.g., Gentile et al. 2007). For example, Zhao et al. (2013) have made a detailed account of the timing argument in MOND, showing that a near collision between the Milky Way and Andromeda ∼10 Gyr ago might lead to the formation of the thick disk and the observed properties of the disks of satellites. On the other hand, we have perhaps been wrong to assume that dSph galaxies are all primordial residents of Galactic subhalos, as some might arrive through filamentary accretion (Stewart et al. 2013; Wang et al. 2013). This might explain the observed phase space distribution in ΛCDM while also easing the challenge of suppressing dwarf formation in subhalos without squelching it entirely (Somerville 2002). Shaya & Tully (2013) find that the observed planarity arises fairly naturally in reconstructions of the orbits of Local Group galaxies that fit the available observations, though it is not obvious that the smooth density component that they invoke is consistent with the expectations of structure formation simulations.

The amplitude of the mass discrepancy in each galaxy provides another test. In a recent analysis (McGaugh & Milgrom 2013, hereafter Paper I), we considered the internal dynamics of the dwarf spheroidal (dSph) satellites of Andromeda. We employed MOND (Milgrom 1983) to calculate the velocity dispersion of stars in each of the dSph satellites of Andromeda, many of which have only recently been discovered.

MOND posits that the mass discrepancies observed in galactic systems are not the result of dark matter, but rather of modified dynamics at low accelerations a ≲ a₀ ≈ 10⁻¹⁰ m s⁻². In MOND, the dynamics in the limit a  ≪  a₀ are required to be invariant under spacetime scaling, (t, r) → λ(t, r) (Milgrom 2009). The MOND calculations of the expected velocity dispersions in Paper I are full-fledged predictions, as they use inflexible procedures, with no leeway for reckoning with known values of the dispersions.

When we wrote Paper I, velocity dispersion data were available for 17 of the dSph satellites of M31. The predicted velocity dispersions were consistent with the majority of the data within its stated uncertainty. In a number of cases (most notably And II, but also And I, III, VII, and XIV), the predictions were in good agreement with multiple independent observations. In some cases (most notably And V and And IX), the prediction agreed better with one observation than another. These cases fall in the category of inflexible expectations: the known velocity dispersions play no role in the prediction.

Paper I also predicted the velocity dispersions for another 10 dwarfs that did not yet have velocity dispersion measurements. Since that time, data for these dwarfs have appeared in the literature, providing the opportunity to check our predictions. We compare our predictions to the new observations in Section 2. In Section 3 we predict the velocity dispersions of several more new dwarfs. In Section 4 we discuss the results and consider whether they can be understood in the context of dark matter. We summarize our conclusions in Section 5.

Velocity dispersion data for the 10 dSph galaxies predicted a priori in Paper I have been obtained by Tollerud et al. (2013) and Collins et al. (2013). Tollerud et al. (2013) measure the velocity dispersions of And XXVIII and XXIX based on observations of 18 and 24 stars, respectively. Collins et al. (2013) provide comprehensive measurements of very nearly all the known dwarfs of Andromeda. These monumental observational efforts provide the basis for testing our predictions, which are based on the photometric data compiled by McConnachie (2012).

The predictions from Table 2 of Paper I are reproduced in Table 1 together with the new observations. In addition to the 10 a priori cases, we include in Table 1 three more dwarfs (And XVIII, XXI, and XXII) that fall in the numerical sequence and have new or improved data. We have not adjusted the predictions in any way. These are compared to the data in Figure 1.

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Table 1. Predicted and Observed Velocity Dispersions

Dwarf	LV	r1/2	σiso	σefe	σobs, 1	σobs, 2	Ref.
	(105 L☉)	(pc)	(km s−1)		(km s−1) (#)
And XVII	2.6	381	$4.5^{+0.9}_{-0.7}\pm 0.4$	$2.5^{+1.0}_{-0.7}\pm 0.5$	$2.9^{+2.2}_{-1.9}$ (8)	...	1, 2
And XVIIIa	6.3	417	$5.6_{-0.9}^{+1.1}\pm 2.6$	...	⩽2.7 (4)	9.7 ± 2.3 (22)	1, 2, 3
And XIX	4.1	2244	$5.0^{+1.0}_{-0.8}\pm 0.7$	$2.6^{+1.1}_{-0.8}\pm 0.8$	$4.7^{+1.6}_{-1.4}$ (26)	...	1, 2
And XX	0.28	165	$2.6^{+0.5}_{-0.4}\pm 0.5$	$2.1^{+0.9}_{-0.6}\pm 0.9$	$7.1^{+3.9}_{-2.5}$ (4)	...	1, 2
And XXIa	4.6	1023	$5.2_{-0.8}^{+1.0}\pm 0.8$	$3.7^{+1.5}_{-1.1}\pm 1.3$	$4.5^{+1.2}_{-1.0}$ (32)	7.2 ± 5.5 (6)	1, 2, 3
And XXIIa	0.35	340	$2.7_{-0.4}^{+0.5}\pm 0.5$	$2.3^{+1.0}_{-0.7}\pm 1.0$	$2.8^{+1.9}_{-1.4}$ (10)	$3.5_{-2.5}^{+4.2}$ (7)	1, 2, 3
And XXIII	10.	1372	$6.4^{+1.2}_{-1.0}\pm 0.7$	$4.4^{+1.8}_{-1.3}\pm 1.0$	7.1 ± 1.0 (42)	...	1, 2
And XXIV	0.94	489	$3.5^{+0.7}_{-0.6}\pm 0.4$	$2.8^{+1.2}_{-0.8}\pm 0.7$	⩽7.3 (12)	...	1, 2
And XXV	6.5	945	$5.7^{+1.1}_{-0.9}\pm 0.7$	$3.5^{+1.5}_{-1.0}\pm 0.8$	$3.0^{+1.2}_{1.1}$ (26)	...	1, 2
And XXVI	0.59	296	$3.1^{+0.6}_{-0.5}\pm 0.4$	$2.0^{+0.8}_{-0.6}\pm 0.5$	$8.6^{+2.8}_{-2.2}$ (6)	...	1, 2
And XXVIIb	1.2	579	$3.7^{+0.7}_{-0.6}\pm 0.4$	$1.8^{+0.7}_{-0.5}\pm 0.4$	14.8 (8)	...	1, 2
And XXVIII	2.1	284	$4.3^{+0.8}_{-0.7}\pm 0.9$	...	$6.6^{+2.9}_{-2.1}$ (17)	4.9 ± 1.6 (18)	1, 2, 4
And XXIX	1.8	481	$4.1^{+0.8}_{-0.7}\pm 0.3$	$3.8^{+1.6}_{-1.1}\pm 0.6$	...	5.7 ± 1.2 (24)	1, 4

Notes. The MOND velocity dispersions predicted a priori in Paper I are reproduced here for comparison to the data. The first set of uncertainties corresponds to the range of assumed $\Upsilon _*= 2^{+2}_{-1}\;{M}_{\odot }/{L}_{\odot }$. This is a convenient depiction but not is an uncertainty in the usual sense, as the mean and range of ϒ_* is unknown. The second set is the conventional uncertainty propagating the observational errors around an assumed ϒ_* = 2 M_☉/L_☉. ^aInitial velocity dispersion measurements for And XVIII, XXI, and XXII by Tollerud et al. (2012) predate Paper I, though only that of And XVIII involved more than seven stars. The predicted velocity dispersions for all the remaining dwarfs in this table are completely a priori. ^bCollins et al. (2013) find that And XXVII is not in equilibrium, so the measured velocity dispersion does not provide a test of the equilibrium prediction. References. Luminosities and half-light radii are from (1) McConnachie (2012). We assume that the three-dimensional r_1/2 = 1.33R_e (Wolf et al. 2010). Measured velocity dispersions are adopted from (2) Collins et al. (2013, always given as σ_{obs, 1}), (3) Tollerud et al. (2012), and (4) Tollerud et al. (2013). The number of stars contributing to the measurement is given in parentheses.

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We calculated the characteristic velocity dispersion of each dwarf galaxy for three assumed values of their stellar mass-to-light ratio: ϒ_* = 1, 2, and 4 M_☉/L_☉ in the V band. We report this as a plus or minus range in Table 1. This is a convenient depiction, but is not an uncertainty in the usual sense. It simply illustrates the effect of a factor of two variation around a nominal mass-to-light ratio of ϒ_* = 2 M_☉/L_☉, which is a plausible estimate given our present state of knowledge about the stellar populations of Local Group dwarfs (e.g., Martin et al. 2008).

We propagate the observational uncertainties in the photometric properties of the dwarfs into the predicted velocity dispersions. This observational uncertainty is reported in Table 1 separately from the theoretical uncertainty in mass-to-light ratio. One should of course bear in mind that astronomical uncertainties are rarely well described by conventional random error distributions, though this is implicit in error propagation. We are, however, more concerned with the potential for systematic uncertainties not described by the error bars. For example, the luminosity of And II trebled from early to later analyses (see Paper I).

2.1. Isolated and EFE Regimes

2.2. Individual Cases

2.3. Systematics Effects

2.4. Comparison of Pairs of Similar Dwarfs in the Isolated and EFE Regimes

This field is growing at a remarkable rate. Not only have velocity dispersion measurements appeared for the dwarfs discussed above within weeks of the predictions of Paper I, but new dwarfs continue to be discovered. In this section we predict the velocity dispersions of three recently identified dwarfs: And XXX (also known as Cass II; Collins et al. 2013), And XXXI, and And XXXII (also known as Lac I and Cass III, respectively; Martin et al. 2013). We follow exactly the same procedure as described in Paper I. Predictions for these objects are reported in Table 2.

Table 2. Predicted Velocity Dispersions

Dwarf	LV	r1/2	σiso	σefe	gin
	(105 L☉)	(pc)	(km s−1)		(a0)
And XXX	1.4	356	$3.8^{+0.7}_{-0.6}\pm 0.7$	$3.5^{+1.5}_{-1.0}\pm 1.3$	0.013
And XXXI	41.	1216	$9.0^{+1.7}_{-1.4}\pm 1.5$	...	0.024
And XXXII	71.	1941	$10.3^{+1.9}_{-1.6}\pm 1.7$	$10.3^{+4.3}_{-3.0}\pm 3.5$	0.020

Notes. Predictions for And XXX/Cass II are based on the data of Collins et al. (2013) and those for And XXXI/Lac I and And XXXII/Cass III are based on the data of Martin et al. (2013). The first uncertainty is that from a factor of two in the mass-to-light ratio and the second is that from observational errors, as per Table 1. The characteristic acceleration at the half-light radius is given in the last column.

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Collins et al. (2013) provide an initial velocity dispersion estimate of $11.8^{+7.7}_{-4.7}\;\mathrm{km}\,\mathrm{s}^{-1}$ for And XXX. This is rather larger than our predicted value of ≈4 km s⁻¹. However, the significance of this difference is <2σ. Indeed, the measured velocity dispersion is non-zero at only the 2.5σ level. The measurement is based on only eight stars, and the analysis is beset by a considerable galactic foreground in both position and velocity space. This complicates identification of true members, heightening the concern over interlopers contaminating the result. A further complication in this case may be the proximity of And XXX to NGC 147 and NGC 185.

And XXXI and And XXXII should provide good tests. Both are relatively bright, quite large (r_1/2 > 1 kpc), and reasonably remote from Andromeda itself. The predicted velocity dispersion of both is ≈10 km s⁻¹. And XXXI is expected to be in the isolated regime, for which our prediction is more precise. Due to its large size (r_1/2 ≈ 2 kpc), And XXXII straddles the boundary between isolated and EFE regimes in spite of its large distance from M31. Nevertheless, there is not much flexibility in our prediction for either dwarf (Table 2).

By and large, MOND performs quite well in predicting the velocity dispersions of the dSph satellites of Andromeda. This is achieved with reasonable stellar mass-to-light ratios, eliminating the mass discrepancies in these systems. In contrast, the Newtonian dynamical mass-to-light ratios exceed 10 and frequently approach ∼50. Such large mass discrepancies were not anticipated by conventional theory. Only MOND predicted this from the outset: the low surface brightness of dSph systems imply very low accelerations, necessarily pointing to large mass discrepancies.

This paper has focussed on a priori predictions. The other cases discussed in Paper I already had some measurements of the velocity dispersion, but these in no way inform the MOND predictions. In only a few cases are the data somewhat in tension with the predictions. Given the nature of astronomical data and the potential for systematic errors, it would be surprising if there were not a few such cases.

A pertinent question is how this compares to the dark matter paradigm. What does ΛCDM predict? Why is MOND-like phenomenology (McGaugh 2004) ever observed in ΛCDM? Is the predictive success of MOND a problem for the dark matter picture (e.g., Sanders 2009)?

The ΛCDM paradigm does not make clear and inescapable predictions as MOND does for individual dSph galaxies. The velocity dispersions is determined almost wholly by the dark matter, so it is not obvious that the luminosity should have much predictive power. For example, the work of Guo et al. (2010) implies σ∝L^0.1

Most baryons in these tiny dwarfs must either be missing (i.e., blown out by feedback) or themselves be dark. This is not a subtle effect: >99% of the primordially available baryons fail to form stars (McGaugh et al. 2010). A 1% variation in the number of baryons expelled is the difference between a dwarf that is twice as bright as typical and one that has no stars at all. It is unlikely that the scatter resulting from any expulsion mechanism can be so small (e.g., Benítez-Llambay et al. 2013), so it is remarkable that a procedure based on the luminous mass like the one we have employed should provide detailed, accurate predictions.

There have of course been efforts to predict the properties of dwarf satellites in the context of ΛCDM. Peñarrubia et al. (2008a) find that halo mass is not simply related to luminosity, as we just argued ourselves. They also predict that larger dwarfs will have larger velocity dispersions at a given luminosity. This follows essentially from the stars probing a larger portion of the rising rotation curve of the dark matter halo. As discussed in Section 2.2, there are now examples of large dwarfs with small velocity dispersions, in contradiction to this expectation. This is an important instance in which the predictions of ΛCDM and MOND are clearly distinct. It is the prediction of MOND that is realized in the data.

We have used new data to test our published predictions (Paper I) for the velocity dispersions of the dSph satellites of Andromeda. The a priori predictions fare well. MOND is not merely consistent with the data, it has the ability to predict it in advance. This is particularly notable in the case of several large, diffuse dwarfs for which MOND correctly anticipated small velocity dispersions of only a few km s⁻¹ when large dispersions (≳ 10 km s⁻¹) had been anticipated.

We note that a further test is provided by the dichotomy in MOND between isolated objects and those dominated by the EFE. In the isolated regime, the predicted velocity dispersion depends only on the luminous mass. In the regime where the external field of the host dominates, the velocity dispersion also depends on the size of the dwarf and the strength of the external field. Consequently, photometrically identical dwarfs are predicted to have different velocity dispersions if one is in the isolated regime and the other is influenced by the external field imposed by the host galaxy. A comparison of similar pairs of dwarfs shows tentative evidence for this dichotomy, which is not expected in ΛCDM.

The work of S.M. is supported in part by NSF grant AST AST0908370 and NASA ADAP grant NNX11AF89G.