LOCAL GROUP DWARF SPHEROIDALS: CORRELATED DEVIATIONS FROM THE BARYONIC TULLY–FISHER RELATION

At first glance, the dwarfs follow the same trend as the spirals, just with larger scatter. Our first concern is therefore whether the data are accurate enough to falsify the null hypothesis that all of the dwarf spheroidals are consistent with the BTFR. The classical dwarfs, for which the data are best, adhere fairly well to the BTFR. Indeed, Carina, Fornax, Leo I, Leo II, Sculptor, and Sextans follow the BTFR as well as rotating disks do.

The M31 dwarfs also lie near to the BTFR, albeit with larger scatter. Three (And I, X, and XIV) are consistent with the BTFR while the other three from the Kalirai et al. (2010) sample (And II, III, and VII) sit somewhat above it. This might happen if Equation (1) understates the circular velocity. The kinematic data we adopt for the M31 dwarfs are very recent. If we look back to a time when the data for the classical dwarfs were in a similarly early state, their scatter goes up (Gerhard & Spergel 1992; Milgrom 1995). We therefore consider the M31 satellites to be broadly consistent with the BTFR, given the uncertainties. And II is the most deviant case, being about 2σ away. This is the situation for the most recent measurement, σ_los = 7.3 ±  0.8 km s⁻¹ (Kalirai et al. 2010). This case illustrates the modest yet important way in which the data can evolve. If we adopt σ_los = 9.3 ± 2.7 km s⁻¹ as measured by Côté et al. (1999), no discrepancy with the BTFR is inferred.

The majority of the MW ultrafaint dwarfs, together with Draco and Ursa Minor, lie systematically below the BTFR. Not all of the recently discovered dwarfs deviate. Canes Venatici I, Leo T, and Leo IV are all consistent with the BTFR within the errors. However, Bootes I, Canes Venatici II, Coma Berenices, Segue 1, Ursa Major I, Ursa Major II, and Willman 1 all deviate substantially and significantly (>2σ) from the BTFR. This corresponds to deviation by at least an order of magnitude (F⁻¹_b>10). If a population of as-yet undiscovered ultrafaint dSphs exist with large dispersions and extremely low surface brightnesses (Bovill & Ricotti 2009; Bullock et al. 2010), we expect them to deviate still farther from the BTFR.

Hercules is an interesting case. It either deviates or it does not, depending on which published velocity dispersion we adopt. For consistency with Wolf et al. (2010), we use here the velocity dispersion σ_los = 5.1 ± 0.9 km s⁻¹ found by Simon & Geha (2007). For this σ_los, Hercules deviates substantially from the BTFR, being off by a factor F_b⁻¹ ≈ 25 in mass, with a 68% confidence interval 7 < F⁻¹_b < 45. Put this way, it would seem safe to say that it deviates by an order of magnitude. However, if instead we adopt the more recent measurement⁴ of σ_los = 3.7 ± 0.9 km s⁻¹ reported by Adén et al. (2009), we find F⁻¹_b ≈ 7 with a 68% confidence interval 0.2 < F⁻¹_b < 13. The central value again deviates by a large factor, and the uncertainties in the two measurements overlap. However, the ±1σ boundaries now encompass the BTFR, so we would no longer consider this to be a deviant case.

The case of Hercules illustrates an important point. A small and formally consistent difference (1.4 ± 1.3 km s⁻¹ in this case) can have an impact on how we perceive the results under consideration here. It is therefore important to take the formally significant deviations with a grain of salt. This is a rapidly evolving field; even with heroic efforts the velocity dispersion data for the ultrafaint dwarfs are not of the same quality as those for the classical dwarfs, nor can they be given the small number of stars in these systems.

Given the cautionary tale of Hercules, one possible interpretation is that there are no real deviations from the BTFR. This hypothesis predicts that the scatter will decrease as the data improve, with the more deviant cases migrating toward the BTFR. There is room to improve the data by weeding out nonmember stars (Serra et al. 2009) and binaries (Minor et al. 2010), both of which inflate σ_los. Measuring proper motions to constrain anisotropies will give a more accurate assessment of V_c (e.g., Strigari et al. 2007). Of course, we must also consider the degree to which it is appropriate to compare σ_los measured at small radii in dwarfs to V_c measured at large radii in disk galaxies.

There are substantial difficulties with attributing all deviations from the BTFR to errors. One is their sheer magnitude: the discrepancies F_b are an order of magnitude in some cases, and two orders of magnitude for some of the ultrafaint dwarfs. In the most extreme case, Segue 1, F⁻¹_b ≈ 230. To place this object on the BTFR would require that its baryonic mass increase by this factor. Alternatively, since the BTFR is steep, a reduction in the observed σ = 4.3 ± 1.1 km s⁻¹ (Geha et al. 2009) to σ_los = 1.1 km s⁻¹ would also suffice. While a 3σ change may be conceivable in one object, it would have to happen in numerous cases. Moreover, the residuals are asymmetric: many more systems have F_b < 1 than F_b>1. This asymmetry is larger than indicated by the formal errors, though we caution that the errors are asymmetric in the same sense. As always, a systematic error could be responsible, such as the inflation of σ_los by binaries. Any such systematic would have to similarly affect many different objects observed by a number of independent groups while not affecting the objects that do adhere to the BTFR.

Another problem with attributing deviations from the BTFR entirely to uncertainties is that the residuals correlate with the physical properties of the dwarfs (Figures 2 and 3). It seems unlikely that the observed correlations are mere flukes. If not, they will persist as data continue to accumulate.

If the BTFR is a property shared by isolated galaxies when initially formed, we might seek to reconcile the Local Group satellites with it by extracting baryons before they form stars. Mechanisms to do this come in a variety of flavors. In this section, we consider astrophysical effects that systematically suppress the cooling or retention of gas in dwarfs.

3.2.1. Cosmic Reionization

The reionization of the universe at z>6 is one mechanism by which the formation of stars in low-mass halos might be suppressed. Reionization heats the gas, inhibiting further star formation in small halos (V_c ≲ 20 km s⁻¹ at the time of reinoization: see, e.g., Bullock et al. 2000; Kravtsov 2010). Halos that remain this small may never experience further star formation, persisting as "fossil" dwarfs containing only ancient stars (Ricotti & Gnedin 2005; Bovill & Ricotti 2009).

The results of simulations (Crain et al. 2007; Hoeft & Gottloeber 2010) are compared to the data in Figure 4. Here, we plot the detected baryon fraction, which is the fraction of baryons f_d = M_b/(f_bM₅₀₀) that are detected in any given halo relative to the cosmic baryon fraction f_b (McGaugh et al. 2010). Note that f_d differs from the BTFR residual F_b.

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Reionization models like those of Crain et al. (2007) and Hoeft & Gottloeber (2010) result in a sharp truncation of the cold baryon content of halos at a particular mass scale. Above this mass scale, reionization does nothing to prevent the baryons from cooling and forming stars. In contrast, the data deviate smoothly and gradually from the universal baryon fraction, not abruptly as suggested by models consisting of reionization only. Thus, while the model of Crain et al. (2007) appears to work well for V_c < 20 km s⁻¹, it does not explain dwarf irregulars with V_c ≳ 30 km s⁻¹. At this slightly larger scale, reionization is not expected to impede the condensation of cool gas and subsequent star formation, yet the mass of baryons detected in the form of stars and cold gas is an order of magnitude shy of the cosmic baryon fraction.

We might suppose that there is some fundamental difference between dSph and dIrr galaxies that separately determines their location in Figure 4. However, the continuity of the BTFR to most of the classical dwarf spheroidals suggests a common origin. It appears that physics beyond reionization is needed to explain the observed trend in detected baryon fraction f_d with halo mass.

Reionization might act in addition to whatever physics drives the main trend in Figure 4 by causing a sharp low-velocity⁵ cutoff. This might explain the residual correlation with luminosity exhibited by the ultrafaint dwarfs (at least those with L_V < 10⁵ L_☉ in Figure 2). However, it does nothing to explain the trend of the residuals with ellipticity (Figure 3).

3.2.2. Stellar Feedback

3.2.3. Ram Pressure Stripping

Tides are a mechanism that acts on the stars directly. The correlation of BTFR residuals with ellipticity and tidal susceptibility as well as with luminosity would occur naturally if some of a dwarf's original stars have been stripped away by tidal perturbations as it orbits the MW. Under this hypothesis, the deviant dwarfs originally adhered to the BTFR, but have had much of their original luminosity removed. Much of the stellar halo of the MW might have been built up this way through hierarchical merging (Johnston 1998; Bullock & Johnston 2005).

Analyses of the dynamics of the dwarf spheroidals typically assume spherical symmetry. The large ellipticities of many of the dwarfs call this assumption into question. It is manifestly untrue in Canes Venatici II, Ursa Minor, Ursa Major I and II, and Hercules, all of which have ≳ 0.5. In this context, it is important to determine whether some dwarfs show dynamical evidence of disruption (e.g., Muñoz et al. 2008, 2010).

The fact that the degree of deviation from the BTFR correlates with ellipticity (Figure 3) may be a clue that tidal disruption is playing a role. A further clue is provided by Figure 5, which shows that the ellipticity of the dwarfs also correlates with distance from their host. While we do not expect all dwarf spheroidals to be perfectly round, neither do we expect their intrinsic shape to depend on distance unless some environmental factor is playing a role (see also Bellazzini et al. 1996).

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If ellipticity is an indicator of tidal disruption then the correlation of deviations with luminosity follows naturally (cf. Piatek & Pryor 1995; Muñoz et al. 2008). The more disrupted a galaxy, the more elliptical it becomes, and the more stars it has lost. The more stars a dwarf has lost, the further it deviates from the BTFR. An obvious observational test of this hypothesis would be to search for streams associated with the deviant dwarfs (e.g., Muñoz et al. 2005; Fellhauer et al. 2007; Sales et al. 2008; Newberg et al. 2010). The age and metallicity of stars in these streams should be consistent with having been drawn from the distribution present in the parent body.

Sagittarius is one example where this has clearly happened. It is now out of equilibrium, so we would not expect it to reside on the BTFR. If we adopt a luminosity of ∼2 × 10⁷ L_☉ (Mateo et al. 1996), the BTFR predicts σ_los ∼ 16 km s⁻¹, somewhat larger than the observed ∼11 km s⁻¹ (Ibata et al. 1997; Bellazzini et al. 2008). Given the uncertainties, it is not obvious that this constitutes a significant discrepancy. However, if the larger total luminosity estimate (∼10⁸ L_☉) of Niederste-Ostholt et al. (2010) is correct, then Sagittarius should initially have had σ_los ∼ 24 km s⁻¹. We need to understand how luminosity and velocity dispersion evolve as disruption takes place.

A satellite in orbit around a much larger parent galaxy is obviously subject to tidal disruption (Johnston 1998), but it is less obvious how σ_los will evolve as mass is stripped (see, e.g., Klimentowski et al. 2009; Łokas et al. 2010). Naively, we would expect the initial stripping to primarily affect the outer dark halo first. Stars would not be stripped in significant numbers until much of the dark halo was already gone (Peñarrubia et al. 2008).

We can be more quantitative about the propensity of a dwarf to be stripped by returning to the subject of tidal susceptibility (Equation (3)). It is conventional to define the tidal radius (Keenan 1981a, 1981b) as

$$\begin{equation} r_{t,D} = D \left(\frac{m}{3M}\right)^{1/3}. \end{equation} \tag{ 4 }$$

This is approximately the radius where a star contained in a satellite of mass m is subject to being lost to the host mass M, though the details depend on the orbit. Equation (4) implicitly assumes circular orbits; eccentric orbits are most susceptible to tidal stripping near pericenter where $D \rightarrow \mathit {a}(1-\mathit {e})$ (where $\mathit {a}$ is the semi-major axis and $\mathit {e}$ is the eccentricity of the orbit). Equation (4) provides an estimate for when stripping may occur, bearing in mind that the current galactocentric distances of the dwarfs could be larger than their pericenter distances.

Figure 6 shows the sizes of the dwarfs in terms of tidal radii. For a star orbiting at the deprojected 3D half-light radius, we set m = M_1/2 = 3σ²_losr_1/2/G (Wolf et al. 2010). The distance D is the galactocentric distance of each dwarf from its host, and the mass of the host is the mass contained within D: M = V²D/G. For the MW, we use V_MW(D) from McGaugh (2008) while for M31 we assume that a constant velocity suffices at the location of the M31 dwarfs and adopt V_M31 = 250 km s⁻¹ (Chemin et al. 2009).

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The tidal radii of the dwarfs are typically an order of magnitude or more larger than their half-light radii. By this criterion, the stars of all the dwarfs are safely cocooned deep inside their dark matter halos and are in no danger of being stripped (Peñarrubia et al. 2009). The vast majority of the dark matter must be tidally stripped before the stars become exposed to stripping (Peñarrubia et al. 2008). It therefore appears that these dwarfs should not currently be losing stars. This is somewhat surprising, given the large deficit F_b of some of these dwarfs, and the correlation with tidal susceptibility seen in Figure 3.

Photometric tidal radii are less certain than the deprojected 3D half-light radius plotted in Figure 6, but correlate very well with it. On average, r_t,phot = (3.1 ± 0.8)r_1/2 for the dwarfs with photometrically well measured tidal radii. We mark this location in Figure 6. The mean size of the dwarfs is only r_1/2 = 0.06r_t,D. To bring the photometric tidal radii into agreement with the dynamical tidal radii computed with Equation (4) would require an adjustment to the mass of nearly 2 orders of magnitude: m ≈ 0.01 M_1/2. This seems like a lot to ask, but there are several possibilities.

First, M_1/2 may be overestimated because the disrupting systems are not in equilibrium. The true mass might be less, allowing tidal escape. In this case, the velocity dispersion is artificially inflated by escaping stars. Though qualitatively attractive, this effect is probably not large enough. From numerical simulations, Łokas et al. (2010) estimate that the inflation of the velocity dispersion due to stripping should result in an overestimate in mass of at most 60%. Nevertheless, one can imagine that the amount of dark matter is less than it might seem.

In this context, it is worth noting that at least some dark matter is required. Equation (4) differs from Equation (3) only by a factor of 3^1/3, yet the result in the left panel of Figure 6 differs from that in the lower right panel of Figure 3 because there we set m = M_b. Dwarfs lacking in dark matter are quite susceptible to tidal influences (Kroupa 1997). Indeed, using the baryonic mass M_b in place of the dynamical mass M_1/2 in Equation (4) results in tidal radii that are much smaller than the observed half-light radii. If this were the case, the dwarfs should have disintegrated long ago.

A second possibility is that M_1/2 is correct and the observed dwarfs are not yet the parents of any streams. The streams observed in the halo would have been produced by the dissolution of other dwarfs. These objects dissolved in the past and are now completely gone. While this may have happened, it does nothing to explain the deviations F_b, nor their correlation with ellipticity.

A third possibility is that the orbits of the dwarfs are highly radial, a situation that is widely expected in ΛCDM (e.g., Bullock & Johnston 2005; Klimentowski et al. 2009; Peñarrubia et al. 2009; Łokas et al. 2010; Sales et al. 2010). If we replace D with $\mathit {a}(1-\mathit {e})$, r_t,D shrinks as $\mathit {e} \rightarrow 1$. To explain Figure 6, we need the deviant dwarfs to be on orbits with $\mathit {e} \gtrsim 0.9$.

Most of the damage would occur during pericenter passage, when the least bound stars are lost. The typical dwarf is not currently close to its pericenter, but may bear the scars of its close passages in the form of reduced luminosity and deviation from the BTFR. In this case, we imagine that the streams in the halo have been built up by stars lost during successive pericenter passages.

This last possibility provides an appealing explanation for the correlation of F_b with ellipticity as well as luminosity. However, we caution that the velocity dispersion is subject to evolution during this process as well as the luminosity and shape of a dwarf. In the simulations of Peñarrubia et al. (2008), the velocity dispersion decreases as luminosity is lost, in a proportion that almost maintains the slope of the BTFR. Such an evolution will not reproduce the trend observed here, which requires a more rapid loss of luminosity with respect to velocity dispersion. This motivates further simulation work that probes a wider range of initial conditions and halo models.

In the context of our empirically motivated stellar stripping hypothesis, we expect three broad classes of objects. Dwarf satellites that have not yet ventured too close to their massive host will reside on the BTFR. The classical dwarfs (excluding Ursa Minor and Draco) and the dwarfs of M31⁶ fall in this category. Dwarfs that deviate from the BTFR do so because they have lost some stars during pericenter passages. Greater deviation is a sign of greater damage, from either closer pericenters or multiple pericenter passages. Ursa Minor, Draco, and most of the MW ultrafaint dwarfs fall in this category. Finally, some of the observed streams may have come from parent bodies that have been totally disrupted by this process.

The stellar stripping hypothesis makes several predictions. The dwarfs that deviate from the BTFR should be the parents of stellar streams. These stars should have properties consistent with the parent body: their ages and abundance patters should be consistent with the same distributions. The streams should be in orbits identifiable with their parents. It appears likely that stars would largely be liberated during pericenter passage, so there is in principle a further prediction about the phase at which stars were injected into the stream. Pericenter passage is a brief portion of the orbit, so we would not expect any of the observed dwarfs to currently be in the process of losing stars. At their present galactocentric distances, all of the dwarfs should be safely bound in what remains of their dark matter cocoons. Observation of streams being liberated at present would require some further interpretation (Section 3.4).

In order to liberate stars as envisioned under the stellar stripping hypothesis, the disrupting satellites need to be on highly elliptical orbits. Though we do not yet know the orbits of all the dwarfs, there are some constraints. Carina has $\mathrm{\it e} \approx 0.67$ (Piatek et al. 2003) and Ursa Minor has $\mathrm{\it e} \approx 0.39$ (Piatek et al. 2005). Carina adheres to the BTFR but Ursa Minor does not, so their observed orbits are the opposite of what we would expect in this hypothesis. While disfavored, a pericenter passage that is sufficiently small (∼20 kpc) to liberate some stars from Ursa Minor is not yet excluded by the orbital data (Lux et al. 2010). Even if this were the case it would remain curious that Ursa Minor deviates from the BTFR and Carina does not. Clearly, better orbital constraints are desirable, as are searches for liberated stars.

An alternative to the ΛCDM paradigm is the modified Newtonian dynamics (MOND) hypothesized by Milgrom (1983). In MOND, there is no dark matter; rather, the apparent need for dark matter stems from a change to the force law that occurs at an acceleration scale a₀ ≲ 1 Å s⁻². Above this scale, all is normal: the gravitational acceleration can be calculated from the observed distribution of baryonic matter and Newton's Law of Gravity: g = g_N. Below this scale (a ≪ a₀), the modification $g \rightarrow \sqrt{a_0 g_N}$ applies. This idea has its problems (Clowe et al. 2004; Angus et al. 2008), but has also had more success than seems to be widely appreciated (Sanders & McGaugh 2002; Bekenstein 2006).

Dwarf spheroidals provide a strong test of MOND because their low surface densities place them deep in the modified regime. In this regime, the BTFR is an absolute consequence of the force law for isolated objects: $g = \sqrt{a_0 g_N}$ leads to a₀GM_b = V⁴_c for point masses. The gas-dominated dwarf irregular galaxies studied by Stark et al. (2009) and Trachternach et al. (2009) fall along the BTFR, consistent with MOND. Indeed, this is confirmation of a prediction that has zero free parameters: the gas mass dominates M_b and these galaxies fall along the MOND prediction, which is indistinguishable from the best-fit BTFR.

The deviation of the Local Group dwarf satellites from the BTFR would appear to falsify MOND (Gerhard & Spergel 1992; Milgrom 1995; Sánchez-Salcedo & Hernandez 2007; Angus 2008; Serra et al. 2009; Hernandez et al. 2010). However, the Local Group dwarf satellites are not isolated. In MOND, the criterion for isolation is whether the internal acceleration of an object, its self-gravity g_int, exceeds that imposed by external systems, g_ext. If g_int < g_ext, the external field modifies what would otherwise happen to the same object in isolation. This external field effect is a unique feature of MOND that has no analog in the conventional context.

Among the dwarfs considered here, Leo T is the only example that is isolated enough to be in the pure MOND regime such that g_ext < g_int < a₀. As such, it should obey the BTFR, and within the errors, it does. Sagittarius is an example that is clearly not isolated, with g_int < g_ext < a₀. This places it in the quasi-Newtonian regime (Milgrom 1986) where the mass estimator is different (Milgrom 1995) and the BTFR is no longer absolute. The quasi-Newtonian mass estimator basically just corrects the normal Newtonian mass estimator by a factor G → G_eff = Ga₀/g_ext (Milgrom 1986). Using a₀ = 1.2 Å s⁻² (Begeman et al. 1991) and g_ext = V²_MW/D with V_MW = 210 km s⁻¹ (Xue et al. 2008; McGaugh 2008) at galactocentric D = 19 kpc (Bellazzini et al. 2008), the mass-to-light ratio of Sagittarius is ϒ^V_* ≈ 1.2 M_☉/L_☉ for the total luminosity estimated by Ibata et al. (1997). This is surprisingly reasonable, but would be reduced if we adopted the larger luminosity estimate of Niederste-Ostholt et al. (2010). Perhaps this reasonable seeming mass-to-light ratio is a fluke since all the same assumptions apply to the analysis here as in the Newtonian case: stability and sphericity. Neither condition is satisfied in Sagittarius, which is highly elliptical and far from equilibrium.

For the classical dwarfs excluding Draco and Ursa Minor, the inferred mass-to-light ratios are reasonable for stellar populations (Angus 2008; Serra et al. 2009). Hernandez et al. (2010) even find that lower mass-to-light ratios tend to occur in dwarfs with more recent star-forming events, as expected from a stellar populations perspective. This is not trivial, as the mass-to-light ratio computed in MOND is extremely sensitive to the accuracy of the data simply because the mass depends on a high power of the velocity dispersion. Even small errors in identifying members can have a noticeable effect (Serra et al. 2009). So far, however, Draco and Ursa Minor persist⁷ in being problem cases with uncomfortably high (∼10 M_☉/L_☉) mass-to-light ratios (Gerhard & Spergel 1992; Milgrom 1995).

For the ultrafaint dwarfs, the mass-to-light ratios are not at all reasonable (Sánchez-Salcedo & Hernandez 2007). Though the BTFR is no longer absolute when the external field dominates, it should remain a reasonable proxy for MOND (Milgrom 1986). In this approximation, the inverse of the deviations F⁻¹_b are proxy estimates for the mass-to-light ratios in MOND⁸ as they measure how remote each dwarf is from the mean ϒ^V_* = 1.3 M_☉/L_☉ of Mateo et al. (1998) and Martin et al. (2008a). From this perspective, ϒ^V_* ≈ F_b⁻¹>10 is quite unreasonable. The cases with F_b⁻¹ ≈ 100 are simply absurd.

The ultrafaint dwarfs may therefore be fatal to MOND, provided that both their data and the assumptions underlying the analysis are valid (Section 3.1). The assumption of sphericity is obviously not satisfied in many of the most deviant cases, though it seems unlikely that geometric corrections alone could explain the large deviations seen in these objects. The assumption that the dwarfs are in dynamical equilibrium is perhaps more consequential. If they are, they should not exhibit discrepancies of 1 or 2 orders of magnitude. If they are not, then MOND may become similar to the stellar stripping hypothesis (Section 3.3), though not identical. Tidal forces are stronger in MOND than in conventional gravity (Brada & Milgrom 2000b) as the long range boost to the effective force becomes more important than the extra dark mass (Angus & McGaugh 2008). This opens the possibility that stripping could be occurring at present.

We can estimate the tidal radii of the dwarfs in MOND just as we did conventionally in Section 3.3. Zhao & Tian (2006) find that the expression for the tidal radius in MOND is nearly identical to that in the conventional case (Equation (4)), differing only by a factor of (2/3)^1/3:

$$\begin{equation} r_{t,M} = D \left(\frac{m}{2M}\right)^{1/3}. \end{equation} \tag{ 5 }$$

What is really different here are the masses m and M of the satellite and host. Here, we simply set m = M_b, the baryonic mass of each dwarf estimated from its luminosity as before. For the host mass M, we take M_MW = 6 × 10¹⁰ M_☉ for the MW (Flynn et al. 2006; McGaugh 2008), and M_M31 = 2M_MW for M31 (Chemin et al. 2009). Note that these are the baryonic masses of the two host galaxies, as there is no dynamical dark matter in MOND.

Contrary to the case for dark matter, the residuals from the BTFR correlate strongly with the size of the dwarfs as measured by MONDian tidal radii (Figure 6). The closer the half-light radius is to the tidal radius, the further the dwarf deviates from the BTFR. Intriguingly, the size scale at which the deviation occurs corresponds well to the photometrically measured tidal radius: $r_{t,M} \approx r_{t,\rm phot}$. Galaxies that adhere to the BTFR are safely within their tidal radii such that $r_{t,\rm phot} < r_{t,M}$ while the deviant cases typically have $r_{t,\rm phot} \gtrsim r_{t,M}$. Dwarfs with a larger fraction of their stars exceeding the tidal radius appear to have lost more light. This implies that tidal disruption may be the cause of the deviance of Draco, Ursa Minor, and the MW ultrafaint dwarfs. The MONDian interpretation of these systems would thus appear to be that they are in the process of being tidally distorted and ultimately shredded in the field of the MW. The small F_b are not large MONDian M_*/L_V because the assumption of stable equilibrium does not hold. Rather, a small F_b is an indicator of the degree of tidal disruption.

The deviations F_b in Figure 6 appear to follow a straight line that continues from well below the BTFR to a bit above it. Satellites should not have F_b>1 as a result of tides in MOND. Rather, once objects become isolated enough, they should adhere to the BTFR with no net residuals. So this plot should go up to unity, then remain there. Presumably this is what we would see if we could include the isolated disk galaxies⁹ from Figure 1. The perception of a linear rise to F_b>1 is due to three satellites of M31: And II, III, and VII. For these three objects, the implied MOND mass-to-light ratios are uncomfortably low: ϒ_* ≈ 0.2 M_☉/L_☉ would be required to bring them onto the BTFR. Given the uncertainties in both the analysis and the data (Section 3.1), we hesitate to put too much weight on these few cases.

The correlation seen in the MONDian portion of Figure 6 is quite strong. With the exception of the most extreme outlier, it is consistent with little intrinsic scatter. How great the intrinsic scatter is depends sensitively on what we take for the uncertainty in ϒ_*. We would not expect a perfect correlation, as there should be some variation due to the orbits of the dwarfs. The expression for the tidal radius implicitly assumes circular orbits; more generally we should replace $D \rightarrow \mathit {a}(1-\mathit {e})$. Dwarfs on radial orbits should be more susceptible to disruption, while retrograde orbits may be somewhat more stable. While it is probably an overinterpretation of the data, we should at least point out that this effect might be perceptible. Two or three of the ultrafaint dwarfs (Canes Venatici II, Willman 1, and perhaps also the most discrepant case, Segue 1) sit below the main correlation. One might infer from this that they are on more eccentric orbits than the other dwarfs. Sextans, on the other hand, remains consistent with the BTFR in spite of having a half-light radius approaching its tidal radius. This situation might persist longer if it happens to be on a retrograde orbit. Another possibility is that Sextans is currently making its first close approach, so it has not yet suffered disruption. Indeed, Brada & Milgrom (2000a) show that dwarfs on mildly elliptical orbits may not disrupt at all if they are only exposed to pronounced tidal effects for a brief period near the pericenters of their orbits.

The numerical simulations of Brada & Milgrom (2000a) show that disrupting satellites form stellar streams (see their Figure 5). Thus, the MOND hypothesis for explaining deviations from the BTFR becomes very similar to the stellar stripping hypothesis (Section 3.3). An important difference is that we need not invoke highly eccentric orbits for the dwarfs. Since $r_{t,M} \approx r_{t,\rm phot}$, stellar stripping may be ongoing at present.

The severity of the effect of the external field can be quantified (Brada & Milgrom 2000a) by

$$\begin{equation} \gamma = \left(\frac{D}{r}\right)^{3/2} \left(\frac{m}{M}\right)^{1/2}. \end{equation} \tag{ 6 }$$

This measures how adiabatic the effects of the external field are. In effect, γ is the number of orbits a star should make within a dwarf for every orbit the dwarf makes about its host. For specificity, we compute γ for a star at the deprojected 3D half-light radius r_1/2 with m = M_b/2. Equation (6) applies when the field of the host dominates: m/r² < M/D². Based on their baryonic masses, all dwarfs are in this regime¹⁰ except Leo T. For this galaxy, γ = (D/r)(m/M)^1/4 (Brada & Milgrom 2000a).

In order for a dwarf to remain in equilibrium, there must be enough internal orbits for the dwarf to adjust to variations in the external potential. This raises the question of how many orbits suffice to maintain an adiabatic variation. Certainly, there should be at least two internal orbits per significant change in the potential. The potential of the host is due entirely to the baryonic mass as there is no dark matter halo in MOND, so a dwarf on a polar circular orbit sees the host vary from face-on to edge-on in one quarter of an orbit. This is a significant variation in the potential that will only be magnified for non-circular orbits. Combining these criteria, we suggest that a minimum value for the adiabatic condition to hold is γ>8. This crude argument appears to be consistent with the numerical simulations of Brada & Milgrom (2000a, see their Figure 4): satellites that remain distant enough from their host to always have γ>8 experience only small perturbations. Satellites reaching γ ≲ 8 near the pericenters of their orbits puff up in size and decline in velocity dispersion, but recover as they recede. Satellites that enter the regime γ < 2 experience unchecked growth in size, subjecting them to tidal disruption and the formation of tidal streams. The velocity dispersions of these systems initially decline but then grow. This is qualitatively the right vector for an evolutionary track that would explain the deviations from the BTFR, but considerably more work would be required to predict such a track or even to check how sensitive the evolution is to parameters such as orbital eccentricity.

Figure 7 shows the deviations from the BTFR and the ellipticity of the dwarfs plotted as a function of γ computed at the half-light radius. Correlations with both are apparent. The deviation sets in around γ ≈ 10, consistent with the above reasoning. The most distorted dwarfs are those with the least time to react to changes in the external field. Dwarfs with large γ have many orbits to adjust to changes in the potential; these objects are mostly round. While it may be possible to avoid distorting the shapes of dwarfs via tides when they are protected by a cocoon of dark matter (Muñoz et al. 2008), tidal distortion seems inevitable in MOND (Brada & Milgrom 2000a).

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We note that the plot of the deviations in Figure 7 is very similar to Figure 6. One interesting difference is the case of Leo T, which is in the pure MOND regime. Consequently, it executes fewer internal orbits per external orbit than it would if it were in the quasi-Newtonian regime like the rest of the dwarfs. This moves it closer to the clump of dwarfs that adhere to the BTFR in Figure 7 than in Figure 6. We also note that Leo T is the only dwarf considered here that retains gas. Tides can strip gas, with the consequences discussed in Section 3.2.3. It may therefore be significant that all of the remaining dwarfs are dominated by the external field of the host. The absence of gas in dwarfs within ∼250 kpc might be a signature of the transition to domination by the external field in MOND.

The structure of Figures 6 and Figure 7 should be similar in MOND, but this behavior is not guaranteed. It only follows if the MOND formula for γ (Equation (6)) provides the correct timescale. This would not follow had we used a Newtonian estimator for the timescale, either with dark matter or without. With dark matter, orbits within the dwarfs are more rapid thanks to the extra mass, and they always have time to equilibrate (γ ≫ 10). Without dark matter in the dwarfs themselves, the dark matter halos of the hosts dominate and the dwarfs should dissolve rapidly (γ ≪ 1). Only when we use the baryonic mass for both dwarf and host in Equation (6) does the onset of the deviation from the BTFR correspond to when internal and external orbital timescales are comparable. This is one sense in which the external field effect is unique to MOND and appears to be consistent with the data for the MW dwarf spheroidals.

A separate question is whether the number of dwarfs currently near total dissolution is reasonable. By reasonable, we mean a dissolution timescale that is shorter than a Hubble time so that there is time for deviation from the BTFR to have occurred, but not so short that we must be catching them at a special time. Unfortunately, it is not clear what the dissolution timescale is (see Sanchez-Salcedo & Lora 2010). The trends in Figure 5 are suggestive of progressive mass loss, but we do not know what the mass loss rate is, so cannot quantify $t_{\rm dissolve} = |M_b/\dot{M}_b|$. As a proxy, we can estimate the crossing time $t_{\rm cross} = 2r_{t,\rm phot}/\sigma _{\rm los}$. Presumably it takes a number of crossing times for a dwarf to dissolve, though it is unclear how many. Most dwarfs have 10⁸ < t_cross < 10⁹ years. The MW satellite with the longest crossing time is Sextans (∼1 Gyr), which is a fair fraction of its orbital period (∼3 Gyr if a = D). This object provides an interesting constraint, as it has had time to complete four circular orbits and a dozen crossing times in a Hubble time. This seems like plenty of time for a low γ system so close to its tidal radius to have suffered some damage, and yet it remains consistent with the BTFR. If instead it is in a somewhat eccentric orbit that only now places it in the regime of tidal susceptibility, its crossing time is long enough that perhaps we do not (yet) see deviation in this case simply because the system has not had time to react. On the opposite extreme, the smallest of the ultrafaint dwarfs, Segue 1 and Willman 1, have the shortest crossing times: t_cross ≈ 5 × 10⁷ years. This is uncomfortably short. Both are far from the BTFR, so perhaps have been caught just before total dissolution. Nevertheless, these cases would appear to pose a dissolution timescale problem. It is hard to say how severe this problem is without knowledge of the initial population. The mass function is not known in MOND, nor is it even clear whether the dwarfs are primordial or tidal in origin (Gentile et al. 2007).

The properties of the deviant dwarfs provide a strong test of the MOND hypothesis. If they are stable, undisturbed systems, then their mass-to-light ratios are unacceptably large and MOND fails. If instead they are being tidally stripped, this situation is naturally understood in the context of MOND for which the tidal radii are comparable to the luminous size. This differs from the stellar stripping hypothesis in the dark matter context in that tidal disruption should be going on now for the deviant dwarfs, and not just when the dwarfs are near the pericenters of their orbits. In addition, the fact that the tidal radii of the dwarfs seem to be adequately estimated by assuming a ≈ D implies that their orbits are not highly eccentric. The distribution of orbital eccentricities of the dwarfs may therefore provide another test to distinguish MOND from ΛCDM.