THE VELOCITY ANISOTROPY OF DISTANT MILKY WAY HALO STARS FROM HUBBLE SPACE TELESCOPE PROPER MOTIONS

During the course of the Sohn et al. (2012a) M31 PM study, PM catalogs for individual stars in the three HST fields were created. As the M31 stars were used to align first- and second-epoch data in the Sohn et al. (2012a) study, these PMs are measured relative to the M31 stars. To obtain absolute PMs, we added the mean absolute PMs of the M31 stars for each field. For this we used the values reported by Sohn et al. (2012a), as determined with respect to stationary background galaxies. The PM errors were calculated based on the positional repeatability seen in the multiple first-epoch ACS/WFC measurements. We adopt similar errors for WFC3/UVIS as for ACS/WFC, which may be a slight overestimate since the WFC3/UVIS pixel scale is smaller.

In the following section we select potential foreground Milky Way halo stars in the three HST/ACS fields. The proper motions of these stars are then extracted by cross-matching with the Sohn et al. (2012a) catalog of individual star PM measurements. In the magnitude range under consideration (21.5 ≲ m_F814W ≲ 25.5; see below), the average uncertainty in the proper motion measurements is σ_μ ∼ 0.05 mas yr⁻¹. The corresponding velocity error depends on the distance, but for example is only 5 km s⁻¹ at D = 20 kpc. This is considerably more accurate than previous proper motion measurements of individual stars in the halo (with σ_μ ∼ 1–4 mas yr⁻¹; see, e.g., Munn et al. 2004; Casetti-Dinescu et al. 2006; Bramich et al. 2008).

We select candidate foreground Milky Way halo stars using the color–magnitude diagrams (CMDs) of the HST/ACS M31 fields. The main-sequence turn-off (MSTO) for Milky Way halo stars is located at brighter (apparent) magnitudes than the M31 MSTO, but at similar colors blueward of m_F606W − m_F814W ∼ −0.3. In Figure 1, our color–magnitude selection box is shown on the CMDs of the three M31 fields. We target halo stars in a sparsely populated region of the CMD, where we expect the least contamination from Milky Way disk stars and M31 stars. We select 13, 9, and 3 stars from the Spheroid, Disk and Stream fields respectively, giving a total of 23 candidate halo stars from the three M31 fields. The extracted PMs and HST/ACS STMAG magnitudes are given in Table 1.

In Figure 2 we compare our CMD halo star selection with the Besançon Galaxy model (Robin et al. 2003). We select stars from this model in a 1 deg² field-of-view (FOV) centered on M31. We convert Johnson–Cousins UBVRI photometry to HST/ACS STMAG using the relations given by Sirianni et al. (2005). The stars inside our color–magnitude selection box are predominantly halo MS stars, and a small fraction (7%) are disk stars. Most of these disk stars are intrinsically faint white dwarfs, which have similar (blue) colors to MSTO stars. Note that if we scale this 1 deg² FOV to the HST/ACS FOV then the Besançon model predicts N ∼ 5 halo stars per HST pointing. Our sample of N = 23 stars from three pointings is slightly higher than this prediction; this is due to contamination by M31 stars in our sample, which we now discuss.

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Figure 3 shows the Galactic proper motion components of the Besançon Galaxy model stars which lie within our color–magnitude selection box. The gray crosses are halo stars and the magenta circles are disk stars. The disk stars have a much higher spread in PM than the halo stars. This is because the stars belonging to the Milky Way disk are much closer than the halo stars, so they tend to have much larger proper motions. The black contour indicates the PM region containing 99% of the halo stars. Within this region there is a very small amount of disk contamination (∼1%), and the low PM disk stars have a negligible effect on our analysis described in Section 3 (see discussion in Section 4.2.3). The blue squares (and red triangles) show the PMs of the 23 stars selected from the HST/ACS images. Encouragingly, the distribution of PMs closely resembles the halo population of the Besançon model.

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The dashed yellow circle indicates the region in PM space consistent with the net PM of M31 ((μ_{l, M31}, μ_{b, M31}) = (0.04, −0.03) ± 0.01 mas yr⁻¹; Sohn et al. 2012a ⁴). Note that this region in PM space has a radius of ∼0.5 mas yr⁻¹. There are N = 9 stars in our sample that lie within this region, which are highlighted by red triangles. Given the uncertainties in our proper motion measurements and the internal velocity dispersion in the M31 fields (V_int ∼ 125 km s⁻¹; see van der Marel et al. 2012b), we find that these nine stars are typically within 3–4 σ of the net motion of M31. In the following analysis we excise these stars from our sample as they are, presumably, horizontal branch, asymptotic giant branch, and/or variable M31 stars. In fact, four of these stars have been identified as variable stars in M31 from independent studies (Brown et al. 2004; Jeffery et al. 2011).

Of the remaining N = 14 stars, one lies outside the 99% contour of the Besançon model, this star is highlighted with a magenta circle. Given the prediction of 7% disk contamination from the Besançon model, it is likely that we may have ∼1 disk star in our sample of 14 stars, and the location of this star in PM space makes it the most probable candidate. In our analysis below, we only consider stars within the 99% contour defined by the Besançon model, and so we do not include this star. However, we note that these two biases in PM space (due to M31 stars and disk stars) are fully accounted for in our analysis. Furthermore, we verify that a slightly different choice of contour, which includes this possible disk star, does not change the final velocity ellipsoid results by more than ∼15 km s⁻¹, which is less than our statistical uncertainties (∼30 km s⁻¹). In the following section we outline how we model the tangential velocity components of the remaining N = 13 halo stars.

The tangential motion of the halo stars depends on their PM and distance: v_t = 4.74047 μD. Here, the PM is in mas yr⁻¹ and the heliocentric distance is in kpc.

The large spread in absolute magnitude of stars near the MSTO means that we cannot determine an accurate distance to individual halo stars. Therefore, we cannot measure the velocity ellipsoid directly, despite the fact that we have very accurate PMs. However, we can determine a probability distribution function (pdf) of halo star distances, based on a suitably chosen model for the star formation history and 3D density distribution of the halo (both of which have been well constrained by existing studies). This allows a statistical determination of the velocity ellipsoid, without knowing the actual 3D velocities of individual stars.

We calibrate the pdf of halo star absolute magnitudes via a two step process: First, we use suitably chosen isochrones to describe the distribution of absolute magnitudes as a function of color. Second, we translate this distribution into a continuous, analytic pdf that can be used in a maximum likelihood routine (see Section 3.2).

3.1.1. Weighted Isochrones

We use the VandenBerg et al. (2006) isochrones, calibrated for HST/ACS STMAG photometry (Brown et al. 2005), to model the absolute magnitudes of our selected halo stars. In Figure 4 we show isochrones for a range of metallicities (−2.3 ⩽ [Fe/H] ⩽ −1.0) and ages (10 ⩽ T/Gyr ⩽ 14) applicable for a halo population. In the color range of our sample, m_F606W − m_F814W < −0.3, the halo stars are close to the MSTO, but they can also lie at fainter magnitudes on the MS, or brighter magnitudes on the red giant branch (RGB). Thus, the absolute magnitudes can range from 3 ≲ M_F814W ≲ 7.5, and our sample of halo stars potentially spans a distance range, 10 ≲ D/kpc ≲ 100.

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We model the absolute magnitudes of the halo stars as a function of color, and apply three weighting factors to the isochrones.

• 1.  
  Initial-mass-function (IMF) weight: We assume a Salpeter IMF and weight the contribution from different mass stars accordingly. Low mass MS stars thus have higher weight than higher mass RGB stars.
• 2.  
  Metallicity weight: We assume the halo population has a Gaussian distribution of metallicities with mean [Fe/H] = −1.9 and dispersion σ = 0.5 (see, e.g., Xue et al. 2008). We assume the metallicity distribution is constant with radius; this is in good agreement with recent studies which find no significant metallicity gradient in the stellar halo (Z. Ma et al., in preparation).
• 3.  
  Age weight: We weight the different age isochrones by assuming a Gaussian distribution of ages with mean 〈T〉 = 12 Gyr and dispersion σ = 2 Gyr (see, e.g., Kalirai 2012).

In Figure 5 we show the absolute magnitude distribution in six color bins, weighted by IMF, metallicity and age. The dashed blue line shows a double-Gaussian fit to each color bin. At redder colors the distribution of absolute magnitudes separates into the MS and RGB populations, while at bluer colors the populations merge into the MSTO.

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3.1.2. Analytic Absolute Magnitude Calibration

In Figure 6 we show the parameters from the double-Gaussian fits (amplitude, mean and sigma) for both the RGB and MS populations as a function of color. The dashed red lines are polynomial (second or third order) fits to the relations. We use these polynomial relations to define a continuous, double-Gaussian pdf for absolute magnitude as a function of color:

$$\begin{eqnarray} && G(M_{{\rm F814W}} | m_{{\rm F606W}} - m_{{\rm F814W}})= G_1(A_1,M_1,\sigma _1,M_{{\rm F814W}}) \nonumber \\ &&\quad + G_2(A_2, M_2, \sigma _2,M_{{\rm F814W}}), \end{eqnarray} \tag{ 1 }$$

where G(A, M, σ, x) = A exp[ − (x − M)²/(2σ²)], and A, M, and σ (amplitude, mean, and sigma) are polynomial functions of m_F606W − m_F814W color.

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The red squares in Figure 5. show the absolute magnitude distributions reproduced by our analytic model. These are in good agreement with the (non-analytic) distributions derived from the weighted isochrones.

In the following section, our absolute magnitude pdf is used in a maximum likelihood routine to determine the tangential components of the halo velocity ellipsoid.

In this section, we outline the maximum likelihood method used to derive the tangential velocity ellipsoid components of the Milky Way halo. We assume that the velocity distribution in the plane of the sky follows a Gaussian distribution, with constant (σ_l, σ_b, v_{l, 0}, v_{b, 0}) over the radial range spanned by our data:

$$\begin{equation} F_v(v_l, v_b) \propto \mathrm{exp}\left[-\frac{(v_l-v_{l,0})^2}{2 \sigma ^2_{l}}\right] \mathrm{exp}\left[-\frac{(v_b-v_{b,0})^2}{2 \sigma ^2_{b}}\right]. \end{equation} \tag{ 2 }$$

In the halo, where D ≫ R₀ ≃ 8.5 kpc, the Galactic coordinates, l, b are good approximations to the spherical coordinates, ϕ, θ respectively. For each star, there are six observables: the angular position on the sky (l, b), the PM (μ_l, μ_b), and the two photometric magnitudes (m_F814W, m_F606W). Observed heliocentric velocities are converted to Galactocentric ones by assuming a circular speed of 240 km s⁻¹ (e.g., Reid et al. 2009; McMillan 2011; Schönrich 2012) at the position of the Sun (R₀ = 8.5 kpc) with a solar peculiar motion (U, V, W) = (11.1, 12.24, 7.25) km s⁻¹ (Schönrich et al. 2010). Here, U is directed toward the Galactic center, V is positive in the direction of Galactic rotation, and W is positive towards the North Galactic Pole. In the direction of M31, the velocity of the sun projects to: (v_l, v_b) = (− 139.5, 83.7) km s⁻¹.

The halo pdf at fixed m_F606W − m_F814W color, in increments of absolute magnitude, apparent magnitude, Galactic PM, and solid angle, F(y), where y is defined as y = y(M_F814W, m_F814W, μ_l, μ_b, Ω), is given by

$$\begin{equation} F \, \, \Delta \textbf {y} \propto F_v \, \rho \, D^5 \, G \, \, \Delta \textbf {y}. \end{equation} \tag{ 3 }$$

Here, D = D(M_F814W, m_F814W) is the heliocentric distance, F_v = F_v(D, μ_l, μ_b) is the velocity distribution function given in Equation (2), ρ = ρ(D, l, b) is the density distribution of halo stars, G = G(M_F814W|m_F606W − m_F814W) is the absolute magnitude pdf defined in Equation (1) and $\Delta \textbf {y}= \Delta M_{{\rm F814W}} \Delta m_{{\rm F814W}} \Delta \mu _l \Delta \mu _b \Delta \Omega$ is the volume element. We assume the halo stars follow the broken power-law density profile derived by Deason et al. (2011b), but we comment on the effect of the density profile parameterization on our results in Section 4.2.2. We marginalize over the absolute magnitude coordinate, $\bar{F} = \int F \, \mathrm{d} M_{{\rm F814W}}$, and define the likelihood function:

$$\begin{equation} L=\prod \bar{F}(\sigma _l, \sigma _b, v_{l,0}, v_{b,0}, \textbf {x}). \end{equation} \tag{ 4 }$$

Here, x denotes the observables (m_F814W, m_F606W − m_F814W, μ_l, μ_b, l, b), and σ_l, σ_b, v_{l, 0}, v_{b, 0} are free parameters. We use a brute force grid method to find the maximum likelihood parameters. The net motion in Galactic latitude, v_{b, 0}, is set to zero, while we allow for net motion in Galactic longitude, v_{l, 0} ∼ −v_{ϕ, 0}, which approximates the net rotational velocity of the halo.

The kinematical biases that we have introduced (see Figure 3) are accounted for through a renormalization of the likelihood function over the region in observable space that passes our cuts. In Equation (4), $\bar{F}(p,\textbf {x})$ is the normalized probability that a star exists at observables x, given model parameters p = (σ_l, σ_b, v_{l, 0}, v_{b, 0}). We define F_sel(x) as our selection function, which states the probability that a star with observables x is in our sample. Thus, F_sel(x) is 1 or 0, depending on our adopted cuts. The normalized probability that a star is found in our sample, at observables x, given model parameters p is given by $\bar{F}(p,\textbf {x}) F_{\rm sel}(\textbf {x}) / N(p)$. Here, $N(p)= \int _{\textbf {x}_c} \bar{F}(p, \textbf {x}) F_{\rm sel}(\textbf {x})$ is the normalization factor, and x_c denotes the region in x-space that passes our selection cuts. We calculate these integrals numerically. This procedure reduces the likelihood to

$$\begin{equation} L=\prod \frac{\bar{F}(p,\textbf {x})}{N(p)} \end{equation} \tag{ 5 }$$

since, by definition, F_sel(x_i) = 1 for any star i = 1, ..., N that makes it into our sample.

In the following section, we discuss the results of applying this maximum likelihood algorithm to our sample of halo stars.

In Figure 7 we show the results of our maximum likelihood analysis (see also Table 2). The resulting velocity moments⁵ in the plane of the sky are: $\sigma _b =83^{+24}_{-16}$ km s⁻¹, $\sigma _l=94^{+28}_{-18}$ km s⁻¹, and $v_{l,0}=-75^{+28}_{-29}$ km s⁻¹. We calculate the mean and root-mean-square spread of the halo star distances using the pdf of our favored parameters, e.g., $\langle D \rangle = 1/N \, \sum ^N_i \int F_{ml} D \mathrm{d} D$, where F_ml denotes the pdf assuming our maximum likelihood parameters. We find 〈D〉 = 19 ± 1 kpc and σ_D = 6 ± 2 kpc. In Galactocentric radii, this corresponds to r ∼ 24 kpc.

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Table 2. Summary of Our Main Results

Velocity Ellipsoid (km s−1)
Galactic	$\langle v^2_{\rm los} \rangle ^{1/2}=105^{+5}_{-5}$	$\langle v^2_{b} \rangle ^{1/2}=83^{+24}_{-16}$	$\langle v^2_{l} \rangle ^{1/2}=123^{+29}_{-23}$	$\langle v_{l} \rangle =-75^{+28}_{-29}$
Spherical polars	$\langle v^2_{r} \rangle ^{1/2}=107^{+6}_{-5}$	$\langle v^2_{\theta } \rangle ^{1/2}=86^{+22}_{-16}$	$\langle v^2_{\phi } \rangle ^{1/2}=121^{+28}_{-21}$	$\langle v_{\phi } \rangle =73^{+28}_{-27}$
Velocity Anisotropy
	$\beta =0.0^{+0.2}_{-0.4}$	$\sqrt{\frac{ \langle v^2_t \rangle }{ \langle v^2_r \rangle }} = 1.0^{+0.2}_{-0.1}$	$\sqrt{\frac{\langle v^2_\phi \rangle }{ \langle v^2_\theta \rangle }}= 1.4^{+0.4}_{-0.3}$
Position
	l = 121°	b = −21°	〈D〉 = 19 ± 1 ± 6 kpc	〈r〉 = 24 ± 1 ± 6 kpc

Notes. We give the velocity ellipsoid in Galactic and spherical coordinate systems and the resulting velocity anisotropy. Note that, the LOS velocity dispersion has been estimated by previous studies in the literature (see text for more details). We also give the approximate location of our three HST fields in the plane of the sky, as well as the average heliocentric and Galactocentric distances for our sample. For the latter quantities we list two uncertainties, the first being the error in the mean, and the second being the root-mean-square spread of the sample.

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Our maximum likelihood velocity ellipsoid quantities are converted into a spherical polar coordinate system using a Monte Carlo method. We generate stars which follow the stellar halo density profile (Deason et al. 2011b), and assign v_l and v_b velocity distributions drawn from our likelihood constraints on σ_l, σ_b, and v_{l, 0}. In the approximate distance range of our sample, 10 ≲ D/kpc ≲ 30, the LOS velocity distribution of stellar halo stars is approximately Gaussian with σ_los = 105 ± 5 km s⁻¹ (see Sirko et al. 2004; Xue et al. 2008; Brown et al. 2010); we apply this LOS velocity distribution to our Monte Carlo samples. The v_r, v_θ, v_ϕ velocity components are then calculated from the generated star positions and v_los, v_b, v_l velocities. In Table 2 we show the resulting velocity ellipsoids in spherical polar coordinates. As expected, the v_θ, v_ϕ velocity distributions closely follow the v_b, v_l coordinates.

To calculate the velocity anisotropy we consider the ratio of tangential and radial pressures, i.e., $\langle V^2_t \rangle / \langle V^2_r \rangle$, where $\langle V^2_t \rangle = \sigma ^2_\phi +\langle v_\phi \rangle ^2 + \sigma ^2_\theta$. The resulting velocity anisotropy suggests isotropic velocity pressure, with $\langle V^2_t \rangle / \langle V^2_r \rangle =1.0^{+0.2}_{-0.1}$ or $\beta =0.0^{+0.2}_{-0.4}$; this is in contrast to the highly radial values found in the solar neighborhood: β ∼ 0.7 or (σ_r, σ_ϕ, σ_θ) ∼ (140, 80, 80) km s⁻¹ (Smith et al. 2009; Bond et al. 2010; Carollo et al. 2010). However, several studies estimating velocity anisotropy from LOS velocity alone, in the distance range, 10 ≲ D/kpc ≲ 30, also find a more tangential velocity ellipsoid than the solar neighborhood (Sirko et al. 2004; Deason et al. 2011a; Kafle et al. 2012). We note that a significant contribution to the tangential pressure comes from angular momentum, as v_ϕ ∼ 70 km s⁻¹. In the following section, we discuss some potential systematics which could bias our results. Encouragingly, we find that systematics play a minor role in the total error budget.

4.2.1. Stellar Isochrones

4.2.2. Density Profile

4.2.3. Disk Contamination

4.2.4. Solar Motion

In Figure 8 we show velocity anisotropy as a function of radius, based on various estimates from the literature. When our results are compared to solar neighborhood measures of β at smaller radii, and LOS velocity estimates at larger radii, we begin to see that the velocity anisotropy may have a "dip" at r ∼ 20 kpc, rather than a constant or continuous decline. Furthermore, the location of this possible dip is coincident with a break in the stellar halo density: Deason et al. (2011b; see also Sesar et al. 2011; Watkins et al. 2009) find that the stellar halo density profile declines more rapidly beyond a break radius of r ∼ 16–26 kpc. The origin of this break radius is still uncertain: Deason et al. (2013) recently suggested that the break radius in the Milky Way may be due to a shell-type structure built up from the aggregation of accreted stars at apocenter. However, Beers et al. (2012) claim that the break radius signifies a transition between two halo populations of different origins (see discussion below). In the former scenario, the kinematic signature of a shell—a predominance of tangential motion at the turn-around radius, plus a cold radial velocity dispersion—is intriguingly similar to the apparent "dip" in velocity anisotropy that we observe. At present, the evidence for a rise again in velocity anisotropy beyond r ∼ 30 kpc relies solely on LOS velocities, where the derived anisotropy is intimately linked to the assumptions of the halo potential. Therefore, it is important to independently determine the tangential motion of halo stars beyond the break radius in order to confirm/falsify the apparent dip in anisotropy.

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It is possible that some of our halo stars belong to the "TriAnd" overdensity. This structure subtends an area of at least 50° × 60° in the constellations of Triangulum and Andromeda, and is located at a similar distance to our halo sample (16 ≲ D/kpc ≲ 25; Rocha-Pinto et al. 2004; Majewski et al. 2004; Martin et al. 2007). The nature of TriAnd has been debated in the literature; Rocha-Pinto et al. (2004; see also Majewski et al. 2004) propose either a bound core of a very dark-matter-dominated dwarf or a portion of a tidal stream, while Johnston et al. (2012) suggest that TriAnd is an apogalactian piece of a disrupted dwarf galaxy. The conclusions we draw here are most consistent with this last interpretation.

In recent years, an additional formation mechanism for halo stars has been put forward. Studies based on hydrodynamical cosmological simulations suggests that halo stars can form in situ from gas in the parent galaxy in addition to formation in external dwarf galaxies and then subsequent accretion (Zolotov et al. 2009; Font et al. 2011). McCarthy et al. (2012) showed that these in situ stars can have significant prograde rotation and therefore increased tangential pressure support from angular momentum. Therefore, this alternative formation mechanism for halo stars could also explain the more tangentially biased halo star orbits. However, this does not explain why the orbits of halo stars in the solar neighborhood—presumably where in situ stars are more dominant over accreted stars—have such strongly radial orbits.

Finally, our sample of halo stars is small (N = 13) and covers a narrow FOV, so we cannot reject the possibility that our results are biased by local substructure. However, we do not find any obvious clustering of our sample in PM space, so it seems unlikely that the majority of our halo stars belong to a common stream. Furthermore, the agreement of our results with studies using LOS velocities in a similar radial regime but different directions on the sky suggests that the isotropic velocity anisotropy derived here is a true global property.

We find a significant signal of prograde rotation in our halo sample, with 〈v_ϕ〉 ∼ 70 km s⁻¹. As mentioned above, this signal could be due to the presence of a shell-type structure in this radial regime, or the halo could have a net rotation—perhaps due to the influence of stars born in situ. Previous work on the rotation of halo populations have found contrasting results. Frenk & White (1980) and Zinn (1985) found evidence of net prograde rotation in the Milky Way globular cluster population (with 〈v_ϕ〉 ≃ 50–60 km s⁻¹). More recently, Deason et al. (2011a) find that BHB stars with [Fe/H] > −2 have a net streaming motion of 〈v_ϕ〉 ∼ 50 km s⁻¹ in a similar radial range to this study. However, Carollo et al. (2010) claim that their sample of outer (D > 15 kpc) halo stars have a net retrograde rotation with v_ϕ ∼ −80 km s⁻¹ (but see Schönrich et al. 2011).

It is clear that a confused picture surrounds the rotational properties of the halo. This is something we hope to address in the future with additional HST fields.

The above discussion illustrates the importance of measuring the radial velocity anisotropy profile in order to understand the formation mechanism of the stellar halo. We hope to extend this present study to multiple HST fields with long-baseline multi-epoch photometry. This will enable us to move beyond small number statistics, cover a wider area on the sky, and probe farther out into the stellar halo. Studies using LOS velocity measurements rely on assumptions of the halo potential and are limited to within r ≲ 30–40 kpc: using PMs, we can more directly measure the tangential motion of halo stars. Furthermore, the depth of the HST fields allows us to study MS stars; the dominant population of the stellar halo. Current spectroscopic studies and the upcoming Gaia mission (with magnitude limit V < 20) are limited to intrinsically bright halo stars, such as BHB or RGB stars, which may not be unbiased tracers.

Finally, an independent measure of velocity anisotropy is vital in order to derive the mass profile of our Galaxy. Recently, Deason et al. (2012b) found that the radial velocity dispersion of halo stars declines rapidly at large radii. At present, the degeneracy between tracer density, anisotropy and halo mass cannot be disentangled. However, a measure of the tangential motion of these distant halo stars will allow us to address whether or not this cold radial velocity dispersion is due to a shift in pressure from radial to tangential components. Our method, using multi-epoch HST images, may be the only way to measure velocity anisotropy at these large distances. The upcoming Gaia mission will measure proper motions for an unprecedented number of halo stars with V < 20, and will likely revolutionize our understanding of the inner stellar halo. However, even with bright halo tracers (e.g., BHB or Carbon stars), the PM accuracy of Gaia, σ_μ ∼ 0.3 mas yr⁻¹ (σ_V ∼ 140 km s⁻¹ at D ∼ 100 kpc), will be unable to accurately constrain the tangential motion of very distant halo stars.

The PM accuracies achievable with just a few orbits of HST time (see Table 1) are well below the halo velocity dispersion, even at D = 100 kpc. Therefore, PM accuracy is no impediment at all for studies of this kind. However, the small FOV of HST means that there are only a handful of halo stars per field. Therefore, to find the rarer stars that are at larger distances than the mean for our sample (〈r〉 ∼ 24 kpc), many fields need to be observed.