Space Motions of the Dwarf Spheroidal Galaxies Draco and Sculptor Based on HST Proper Motions with a ∼10 yr Time Baseline

We downloaded both the regular flat-fielded _flt.fits and corrected _flc.fits images from the Mikulski Archive for Space Telescopes (MAST). The latter images are pre-processed for the imperfect CTE using the pixel-based correction algorithms of Anderson & Bedin (2010). The PM measurements were performed using both sets of images since we were uncertain how the current version of the CTE correction routine we used performs on images obtained using the POST-FLASH option. We found that the _flc.fits images taken with the POST-FLASH option were somewhat overcorrected for the imperfect CTE, which causes systematics in our PM measurements. Therefore, the final results were all derived from the _flt.fits images. We carefully examined the individual _flt.fits images for both epochs and found that the level of CTE loss for the images taken in 2012–2013 is comparable to that of the 2004–2006 data thanks to the 2012–2013 POST-FLASH observations. In the end, this has worked to our advantage for PM measurements since the impact of CTE loss on astrometry was found to almost cancel out when computing the difference in positions of objects between the two epochs. As we discuss below, we also used local corrections for mitigating the residual systematics.

As the first step, we processed the_flt.fits images using the img2xym_WFC.09x10 program from Anderson & King (2006) to obtain a position and a flux for each star in each exposure. We applied corrections to the positions using the known ACS/WFC geometric distortions. We then created a high-resolution stacked image for each field using the first- (for the Draco dSph) and second-epoch (for the Sculptor dSph) images. Stars and galaxies were then identified from the stacked images. Photometric measures from the img2xym_WFC.09x10 program were used to create a CMD for each field, and this was used to identify member stars of our target dSphs. The subsequent analysis steps are different depending on which type of background objects are used as stationary references sources. We discuss further steps for each case below.

For each star and background galaxy, a template was constructed from the high-resolution stacked image. This template was used to measure a position for each object in each individual exposure in each epoch. Templates were fitted directly to the images of the epoch from which they were created (first epoch for Draco, and second epoch for Sculptor). For fitting templates to the images of the other epoch, we included 7 × 7 pixel convolution kernels to allow for PSF differences between epochs. These kernels were derived using bright and isolated Draco/Sculptor stars distributed throughout the fields.

The template-based positions of stars for multiple exposures were averaged and used to define first- (for the Draco dSph) or second-epoch (for the Sculptor dSph) reference frames. We used the positions of the stars in each of the second- (Draco) or first-epoch (Sculptor) exposures to transform the template-measured positions of the galaxies into the reference frames. We then took the difference between the first- and second-epoch positions of galaxies to obtain the relative displacement of the galaxies with respect to the dSph stars. To remove any remaining systematic PM residuals associated with the detector position and brightness of sources (e.g., due to imperfect CTE corrections), we derived and applied a local correction for each background galaxy using stars of similar brightness that lie in the vicinity. Finally, we multiply the relative displacements of the galaxies by −1 to obtain the mean absolute displacement of the dSph stars, since in reality the galaxies/QSOs are stationary and the stars are moving. Multiplying the resulting displacements by the pixel scale of our reference images ($50\,\mathrm{mas}\,{\mathrm{pix}}^{-1}$) and dividing by the time baseline turns our results into actual PMs.

For the DRACO-F1 field, we have data obtained in 2004, 2006, and 2013 as described in Section 2.1. For the final PM measurement, we used the 2004 and 2013 data as our first and second epoch, respectively. The 2006 data were used to provide an extra check (see Section 2.3.1), but they were not included in our final results.

Because the DRACO-F3 field was observed with F555W, which has only about half the bandwidth of F606W, the background galaxies are significantly fainter than those detected in the F606W images. After attempting to construct and fit templates to the background galaxies in this field, we concluded that the overall quality of positional measurements was too poor to include in our results. For this reason, the PM results for the DRACO-F3 field are only reported using QSOs as stationary reference sources (see Section 2.2.3).

All three of our Draco fields include QSOs in them, and we use these objects to provide an independent measurement of the Draco PM. For the DRACO-F1 field, Pryor et al. (2015) used two QSOs, one detected in the top ACS/WFC chip (WFC1), and the other detected in the bottom chip (WFC2). Both QSOs were easily identified in our images thanks to Figure 2 of Pryor et al. (2015). However, due to the increase in individual exposure times for our GO-12966 data, we found that the QSO located in WFC1 is slightly saturated in the images taken in 2012, making its positional measurement unreliable. We therefore decided to only use the QSO detected in WFC2. This QSO, and the QSOs in the other two Draco fields were detected in the 2012 data with counts well below the saturation limits.

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For measuring PMs using QSOs as reference sources, we used the positions of QSOs and stars measured based on the library PSFs by the img2xym_WFC.09x10, instead of using the template-based positions described in Section 2.2.2. This is because the PSFs of QSOs are virtually indistinguishable from the PSFs of stars, and because the library-based positions are more accurate than the template-based positions. We start by only selecting member stars of Draco, based on their CMD properties, that are detected on the same ACS/WFC image quadrant as the QSO. The positions of these stars in each individual image are corrected for the known geometric distortions and subsequently averaged separately for the first- and second-epoch data. The positions of stars in the first-epoch data are used to define a reference frame. We then used the positions of stars in the second epoch to transform the position of the QSO into the reference frame. As a result, the PM of Draco stars can be inferred by taking the difference between the first-epoch reference QSO position and the second-epoch transformed QSO position, multiplying the results by −1, converting pixels to mas, and dividing by the time baseline.

The QSOs we used for measuring PMs are typically brighter than most of the Draco member stars we used for setting up the reference frame. For example, in the same quadrant as the QSO located in the DRACO-F1 field, there are only 18 out of 132 Draco stars that are brighter than the QSO. For the other two fields, the situation is worse: only 2 out of 68 and 1 out of 42 Draco stars are brighter than the QSOs in fields DRACO-F2 and DRACO-F3, respectively. This can potentially cause CTE-related systematics since the CTE degradation is known to be a strong function of the brightness of a source, and we are using stars of different brightnesses than the QSOs to define the reference frame. To correct for this effect, the procedure described above was iterated using Draco stars in different brightness ranges to define the reference frame. In our first iteration, the measurement was carried out using stars brighter than an instrumental magnitude of ${m}_{\mathrm{instr}}=-9.00$.¹⁰ In subsequent iterations, we decreased this limit in steps of 0.5 mag until the faint limit was ${m}_{\mathrm{instr}}=-11.50$. For each step, we compute the PMs of the QSOs and the median magnitude of stars used in the transformation process. This provides a relation between the brightness of stars and the measured PMs. We fit a line to this relation and computed the PM for the case of stars having the same brightness as the QSOs. These relations are monotonic, implying that we are indeed correcting for the residual CTE effect. The final PMs of Draco stars with respect to the stationary QSOs were then derived using the same procedure as outlined in Section 2.2.2. The final PM uncertainties were obtained by taking the quadrature sum of the uncertainties in the average positions at the two epochs and the uncertainties from fitting the lines to the PM versus brightness relation.

As with the case of using background galaxies as stationary references, we used the DRACO-F1 field's 2006 data only as an extra check, and our final PM for this field was obtained using 2004 versus 2013 data.

Our PM results for the Draco dSph are presented in Table 2, and the corresponding PM diagram is shown in Figure 2(a). PM measurements using different background sources are plotted with different symbols, and results from each field are plotted in different colors. The PM results for the DRACO-F1 field in Table 2 were derived using a data set with a time baseline of 9 yr (2004 vs. 2013). However, since we have images acquired in 2006 for this field, we used them as an external check by measuring PMs of Draco stars using 2006 data as the first epoch and 2014 data as the second epoch. We followed the same procedure outlined in Section 2.2 to obtain PMs using both QSO and background galaxies as stationary references. The resulting 7 yr baseline PMs are consistent within $1.5\sigma$ compared to the 9 yr baseline PMs listed in the first two lines of Table 2, with slightly larger uncertainties as expected from the shorter time baseline.¹¹ This provides an additional check on our PM results for the DRACO-F1 field.

Table 2.  Final PM Results for the Draco dSph

	${\mu }_{W}$ a	${\mu }_{N}$ b	${\sigma }_{{\mu }_{W}}$	${\sigma }_{{\mu }_{N}}$
Field	($\,\mathrm{mas}\,{\mathrm{yr}}^{-1}$)		($\,\mathrm{mas}\,{\mathrm{yr}}^{-1}$)
F1 (galaxies)	−0.0168	−0.1958	0.0290	0.0294
F1 (QSOs)	−0.0463	−0.2025	0.0188	0.0164
F2 (galaxies)	−0.0526	−0.1812	0.0264	0.0265
F2 (QSOs)	−0.0825	−0.1478	0.0174	0.0179
F3 (QSOs)	−0.0512	−0.1386	0.0263	0.0348
Weighted average	−0.0562	−0.1765	0.0099	0.0100

Notes.

^aPM in direction of west. Note that ${\mu }_{W}=-{\mu }_{\alpha }\ast \cos \delta$. ^bPM in direction of north. Note that ${\mu }_{N}={\mu }_{\delta }$.

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The uncertainties in the measurements are dominated by the random errors in the reference frame set by background galaxies or QSOs. These random errors are independent from each other. We therefore calculate the average PM of Draco by taking the error-weighted mean of the five measurements provided in Table 2, which yields

$$\begin{eqnarray}&&({\mu }_{W},{\mu }_{N})\\ &&\quad =\,(-0.0562\pm 0.0099,-0.1765\pm 0.0100)\,\mathrm{mas}\,{\mathrm{yr}}^{-1}.\end{eqnarray} \tag{ 1 }$$

The final average of the five data points and associated uncertainties in each coordinate are plotted as a black cross in Figure 2(a).

Overall, we find that measurements using different objects as stationary references agree well with each other. This provides confidence in our Draco PM results, and more generally in the PM measurement technique using background galaxies as stationary objects. Measurements for different fields also agree to within the error bars. To test the statistical agreement among the individual measurements listed in Table 2, we calculate the quantity

$$\begin{eqnarray}&&{\chi }^{2}=\displaystyle \sum _{i}\left[{\left(\displaystyle \frac{{\mu }_{W,i}-{\overline{\mu }}_{W}}{{\rm{\Delta }}{\mu }_{W,i}}\right)}^{2}+{\left(\displaystyle \frac{{\mu }_{N,i}-{\overline{\mu }}_{N}}{{\rm{\Delta }}{\mu }_{N,i}}\right)}^{2}\right].\end{eqnarray} \tag{ 2 }$$

This quantity is expected to follow a probability distribution with an expectation value of the number of degrees of freedom (${N}_{\mathrm{DF}}$) with a dispersion of $\sqrt{2{N}_{\mathrm{DF}}}$. Since we have five independent measurements each in two directions on the sky, the ${\chi }^{2}$ is then expected to have a value of 8 ± 4. From Table 2 and Equation (2), we find ${\chi }^{2}=11.1$. Therefore, we find that our measurements in Table 2 are consistent within our quoted uncertainties.

Our final 1d PM uncertainty for Draco is $10\,\mu \mathrm{as}\,{\mathrm{yr}}^{-1}$ in each direction. This is smaller than any other measurement uncertainties we have achieved using our PM measurement techniques and therefore may appear to be beyond HST's astrometric capabilities. However, this small uncertainty is mainly due to the time baselines being longer than our previous studies. For example, in our M31 study (Sohn et al. 2012), we achieved a 1d PM uncertainty of $\sim 12\,\mu \mathrm{as}\,{\mathrm{yr}}^{-1}$ for time baselines of 5–7 yr averaging results from three separate fields. Our Draco data also consist of measurements from three separate fields, but the time baselines are about 2.5 yr longer than in the M31 work. Simply scaling uncertainties of the M31 work by this time baseline ratio gives an estimated uncertainty of $8.5\,\mu \mathrm{as}\,{\mathrm{yr}}^{-1}$, which is consistent with our measured uncertainty for Draco. In our Leo I study (Sohn et al. 2013), we achieved a 1d PM uncertainty of $29\,\mu \mathrm{as}\,{\mathrm{yr}}^{-1}$ using two epochs of ACS/WFC data separated by 5 yr for a single field. Scaling by the time baseline ratio and dividing by $\sqrt{3}$ to account for the difference in the number of fields used for the measurement gives $10\,\mu \mathrm{as}\,{\mathrm{yr}}^{-1}$, which again is consistent with our PM uncertainty for Draco. We conclude that our PM measurement uncertainties for Draco are in line with expectations from our previous studies.

In both of our previous studies mentioned above, we carried out detailed analyses to argue that there were no systematic errors in excess of the random errors. Since our smaller random errors here are merely due to a longer time baseline (which reduces random and systematic errors equally) and the higher number of fields, the same conclusions should hold true. Nevertheless, the difference between QSO and background galaxy results for DRACO-F1 and DRACO-F2 may indicate a small systematic effect. This is likely a problem with the QSO measurements since they (1) sample only one region on the detector, (2) do not average over multiple background sources, and (3) require a magnitude correction as demonstrated in Section 2.2.3. However, systematics are a problem only if they are correlated between different measurements, but we find no evidence for any such effects. Therefore, upon averaging, these systematics should decrease as $\sqrt{N}$ as we have assumed in the averaging of our results.

As shown in Figure 1, our field locations for measuring the absolute PM are offset from the center of Draco by angular distances of 12'–30'. If the internal motions of Draco stars in tangential directions are significantly large, our PM measurement may not represent the center of mass (COM) motion of Draco. This is particularly true if the tangential motions show a systematic pattern (e.g., clockwise or counterclockwise rotation). To check this, we have subtracted the average PM of Draco from the PMs of each of our target fields and plotted the residual 2d motions on the sky in the left panel of Figure 4. We do not detect any rotational sign from the residual motions.

We also carried out additional checks as follows. Whereas so far there are no internal PM measurements of Draco, the line of sight (LOS) velocity shows a slight rotation sign at the level of $6\,\mathrm{km}\,{{\rm{s}}}^{-1}$ at a radius of 30' along Draco's major axis (Kleyna et al. 2001). However, given that this is smaller than the central LOS velocity dispersion of Draco ($\sigma =9.1\pm 1.2\,\mathrm{km}\,{{\rm{s}}}^{-1};$ Wilkinson et al. 2004), it has been claimed as nonsignificant (Kleyna et al. 2002). At the distance of Draco, $6\,\mathrm{km}\,{{\rm{s}}}^{-1}$ is equivalent to $0.017\,\mathrm{mas}\,{\mathrm{yr}}^{-1}$, which is about twice the size of our final random error in Table 2. If Draco is rotating at this speed on the sky, our residual motions above would have shown systematic rotational signs, but we do not detect such signs. We note that Draco appears quite elongated on the sky ($e=0.30;$ Odenkirchen et al. 2001), suggesting that it is seen at high inclination. This implies that the rotation in the plane of the sky should be less than that along the LOS. Finally, even if the rotational motion was systematically affecting our PM results for each field, the final average should represent the systemic tangential motions of Draco given that our three target fields sample stars on both sides of the dwarf near the major axis at similar angular distances. For the reasons stated above, we adopt our PM results in Equation (1) as our final measurement for Draco.

In Figure 2(b), we compare our new PM results with the two recent PM measurements using data obtained with HST (Pryor et al. 2015) and the Subaru Telescope (Casetti-Dinescu & Girard 2016). As mentioned in Section 2.1, the measurement by Pryor et al. (2015) was obtained using 2 yr time baseline data for the DRACO-F1 field. While the two HST results are consistent within 1σ in ${\mu }_{N}$, they are discrepant at the $\sim 2\sigma$ level in ${\mu }_{W}$, despite using the same field (albeit with a shorter time baseline), the same type of objects (QSOs and background galaxies), and similar techniques as used in this study. The source of this discrepancy is unclear, but it is reasonable to assume that PM results with longer time baselines (in this case, our results) are less subject to systematics. The comparison with the Subaru results shows even larger discrepancies. Given that HST is less prone to systematics related to atmospheric effects and instrumental change, we believe that our results are more reliable.

Our PM results for the Sculptor dSph are presented in Table 3, and the corresponding PM diagram is shown in Figure 3(a). Since we only used background galaxies as stationary references for this galaxy, each field has a single measurement. The error-weighted mean of the two fields in Table 3 gives

$$\begin{eqnarray}&&({\mu }_{W},\,{\mu }_{N})\\ &&\quad =\,(-0.0296\pm 0.0209,\,-0.1358\pm 0.0214)\,\mathrm{mas}\,{\mathrm{yr}}^{-1}.\end{eqnarray} \tag{ 3 }$$

As evident in Figure 3(a), the independent measurements from our two observed fields are consistent with each other within $1\sigma$. Indeed, we find ${\chi }^{2}=0.3$, which is in line with the expected value of 2 ± 2.

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Table 3.  Final PM Results for the Sculptor dSph

	${\mu }_{W}$	${\mu }_{N}$	${\sigma }_{{\mu }_{W}}$	${\sigma }_{{\mu }_{N}}$
Field	($\,\mathrm{mas}\,{\mathrm{yr}}^{-1}$)		($\,\mathrm{mas}\,{\mathrm{yr}}^{-1}$)
F1 (galaxies)	−0.0368	−0.1222	0.0367	0.0368
F2 (galaxies)	−0.0262	−0.1428	0.0254	0.0263
Weighted average	−0.0296	−0.1358	0.0209	0.0214

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For Sculptor, Battaglia et al. (2008) find a radial velocity gradient of ${7.6}_{-2.2}^{+3.0}\,\,\mathrm{km}\,{{\rm{s}}}^{-1}$ deg^–1 along its projected major axis, probably due to intrinsic rotation. Our target fields are located near the minor axis at 7–9 arcmin from the center of Sculptor. The residual 2d motions of our target fields after subtracting the average PM of Sculptor are shown as color arrows in the right panel of Figure 4. We note that the residual motions are too small to show compared to the average PM of Sculptor, demonstrating that the internal motions among the fields are negligible. Indeed, our 1d PM uncertainty at the distance of Sculptor is $8.6\,\mathrm{km}\,{{\rm{s}}}^{-1}$, so even if we assume that Sculptor has tangential motions at the same level of the radial velocity gradient, our PM uncertainties are comparable to this. Therefore, no correction for the COM motion of Sculptor is required, and we adopt Equation (3) as our final PM measurement for Sculptor.

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We compare our PM results with the HST measurement by Piatek et al. (2006). In their study, Piatek et al. (2006) used QSOs in two different fields to measure the absolute PM of Sculptor. The two measurements agree with each other within $1\sigma$, with our 1d PM uncertainty being ∼6 times smaller than that of Piatek et al. (2006). While both measurements employed the astrometric powers of HST, Piatek et al. (2006) used STIS data with time baselines of 2–3 yr, while we used ACS/WFC data separated by 11 yr. Field locations are significantly different, and so these two measurements can be considered as completely independent. The agreement between the two PM measurements, despite using different types of background sources in different fields observed with different detectors, highlights the success in using HST instruments as tools for measuring absolute PMs of dwarf galaxies in the MW halo.

For Draco, the Galactocentric $(X,\,Y,\,Z)$ position is

$$\begin{eqnarray}&&{{\boldsymbol{r}}}_{\mathrm{Dra}}=(-4.3,\,62.3,\,43.3)\,\mathrm{kpc}.\end{eqnarray} \tag{ 4 }$$

To calculate the 3D space velocity of Draco, we adopt a heliocentric LOS velocity of ${v}_{\mathrm{LOS}}=-292.8\pm 0.4\,\mathrm{km}\,{{\rm{s}}}^{-1}$, estimated by applying the chemodynamical model of Walker et al. (2015a) to the spectroscopic data set of Walker et al. (2015b). Combining this with our PM results in Section 2.3.1, the Galactocentric velocity $({V}_{X},\,{V}_{Y},\,{V}_{Z})$ of Draco becomes

$$\begin{eqnarray}&&{{\boldsymbol{v}}}_{\mathrm{Dra}}=(61.0,\,16.3,\,-173.0)\pm (6.4,\,5.8,\,3.2)\,\mathrm{km}\,{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 5 }$$

The uncertainties listed here and hereafter were obtained from a Monte Carlo (MC) scheme by propagating all observed uncertainties (distance, velocity, and their correlations), including those for the Sun. The corresponding Galactocentric radial and tangential velocities are then

$$\begin{eqnarray}&&{({V}_{\mathrm{rad}},{V}_{\tan })}_{\mathrm{Dra}}=(-88.6,\,161.4)\pm (4.4,\,5.6)\,\mathrm{km}\,{{\rm{s}}}^{-1},\end{eqnarray} \tag{ 6 }$$

and the observed total velocity of Draco with respect to the MW is

$$\begin{eqnarray}&&{V}_{\mathrm{tot},\mathrm{Dra}}\equiv | {{\boldsymbol{v}}}_{\mathrm{Dra}}| =184.1\pm 4.3\,\mathrm{km}\,{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 7 }$$

For Sculptor, the Galactocentric position is

$$\begin{eqnarray}&&{{\boldsymbol{r}}}_{\mathrm{Scl}}=(-5.2,-9.8,-85.4)\,\mathrm{kpc}.\end{eqnarray} \tag{ 8 }$$

We adopt a heliocentric LOS velocity of ${v}_{\mathrm{LOS}}=111.5\,\pm 0.3\,\mathrm{km}\,{{\rm{s}}}^{-1}$, obtained by applying the model of Walker et al. (2015a) to the spectroscopic data of Walker et al. (2009), and combining this with our PM results for Sculptor, we obtain a Galactocentric velocity of

$$\begin{eqnarray}&&{{\boldsymbol{v}}}_{\mathrm{Scl}}=(36.0,\,186.3,\,-96.7)\pm (8.8,\,10.9,\,1.3)\,\mathrm{km}\,{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 9 }$$

The Galactocentric radial and tangential velocities are

$$\begin{eqnarray}&&{({V}_{\mathrm{rad}},{V}_{\tan })}_{\mathrm{Scl}}=(72.6,\,200.2)\pm (1.3,\,10.8)\,\mathrm{km}\,{{\rm{s}}}^{-1},\end{eqnarray} \tag{ 10 }$$

and the total velocity of Sculptor with respect to the MW is

$$\begin{eqnarray}&&{V}_{\mathrm{tot},\mathrm{Scl}}\equiv | {{\boldsymbol{v}}}_{\mathrm{Scl}}| =213.0\pm 9.9\,\mathrm{km}\,{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 11 }$$

The escape velocity of a tracer object provides first-order insights into the enclosed mass at its distance. The escape velocity ${v}_{\mathrm{esc}}$ for a point mass ${M}_{\mathrm{MW}}$ is defined as

$$\begin{eqnarray}&&{v}_{\mathrm{esc}}=\sqrt{2{{GM}}_{\mathrm{MW}}/r},\end{eqnarray} \tag{ 12 }$$

where r is the Galactocentric distance to the tracer object. According to cosmological simulations, it is unlikely to find an unbound satellite at the present epoch near an MW-size galaxy (Boylan-Kolchin et al. 2013; but see Section 4.2 of this paper). Therefore, by forcing Draco and Sculptor to be bound to the MW, we can use the equation above to calculate the lower limit on the enclosed MW mass. Using the total velocities from Equations (7) and (11), we arrive at lower limits of the enclosed MW mass $0.3\times {10}^{12}\,{M}_{\odot }$ and $0.5\times {10}^{12}\,{M}_{\odot }$ at distances of ${R}_{\mathrm{GC}}=76$ and 86 kpc, respectively.

Using the older PM measurement by Pryor et al. (2015) and Piatek et al. (2006), the total velocities of Draco and Sculptor become ${V}_{\mathrm{tot},\mathrm{Dra}}=225.9\,\mathrm{km}\,{{\rm{s}}}^{-1}$ and ${V}_{\mathrm{tot},\mathrm{Scl}}=248.1\,\mathrm{km}\,{{\rm{s}}}^{-1}$, respectively. These imply lower limits of enclosed MW masses of $0.9\times {10}^{12}\,{M}_{\odot }$ and $1.2\times {10}^{12}\,{M}_{\odot }$ at R_GC = 76 and 86 kpc, respectively. In conclusion, our new PM measurements allow significantly lower MW masses based on the escape velocities.