<figure><img src="image_1.png" /></figure>

# Erratum: “Galactic Forces Rule the Dynamics of Milky Way Dwarf Galaxies”
*(2018, ApJ, 860, 76)*

Francois Hammer<fnref n="1" />, Yanbin Yang<fnref n="1" />, Frederic Arenou<fnref n="1" />, Carine Babusiaux<fnref n="1" />,<fnref n="2" />, Jianling Wang<fnref n="1" />,<fnref n="3" />, Mathieu Puech<fnref n="1" />, and Hector Flores<fnref n="1" />

Received 2018 September 10; published 2018 October 24

## 1. Erratum

This work has shown that under the assumption of the impulse approximation, the dynamics and velocity dispersions of the dSph progenitors can be largely accounted for by the gravitational forces of the Milky Way (MW). This implies that the dynamical crossing time in these objects should be calculated using the stellar mass contribution only, rather than using the observed $\sigma_{los}$ as incorrectly done in the published article. The corresponding calculations confirm the validity of the impulse approximation and then the conclusions of the paper.

Here we recalculate the characteristic times in a manner more consistent with former studies made in the context of the impulse approximation (see, e.g., Aguilar & White 1985). In fact the astrophysical context is rather similar to that considered for globular clusters that are progressively destroyed by the MW tidal shocks (Aguilar 1993) within the framework of the impulse approximation. As for globular clusters, in dSphs the impulse approximation would require that their stars feel the fluctuations of the MW gravitational potential over timescales that are smaller when compared to the intrinsic dynamical timescale, particularly in the outer parts. The former timescale, or encountering time, can be approximated by the ratio between the distance to the MW and the initial velocity of the encountering ($V_{inf}$), before the dSph has been captured by the MW. The latter, dubbed as the standard crossing time following Aguilar & White (1985), can be approximated by the ratio of $r_{half}$ to the velocity dispersion attributable to the sole stellar mass, which is $\sigma_{dSph,stars} = (M_{stellar}/(2\mu r_{half}))^{1/2}$ with $\mu = 930 M_{\odot} pc^{-1} km^{-2} s^{-2}$ (Wolf et al. 2010). To calculate these values, we approximate the initial velocities by that of the LMC ($V_{inf} \sim 200 \text{ kms}^{-1}$ after tracing back it at 500 kpc, see Hammer et al. 2015) and we use a $M_{stellar}/L_{V}$ ratio of 2.5 for dSphs without young or intermediate age stars, and 1.5 for Carina, Fornax, and Leo I (see, e.g., Lelli et al. 2017 and references therein). It has been demonstrated by Aguilar & White (1985) that the impulse approximation holds even when the encountering to the crossing time ratio reaches values up to 5 (see also Binney & Tremaine 1987).

All dSphs are within a regime for which the impulse approximation holds, since $t_{enc}/t_{cross}$ ranges from 0.25 to 5, but for the most distant ones, Leo I and Leo II, which show a ratio larger than 10. For example, if the impulse approximation applies to globular clusters (Aguilar 1993), it should also apply to Ultra Faint Dwarfs since they share similar stellar masses, while their radii (and crossing times) are ~20 larger on average. We provide a revised version of our Table 1, for which only the tenth and eleventh columns have been modified in accordance to the text of this Note. There is another change that should be noted that includes the second sentence of the second paragraph of Section 5. It should read, “However, during a first passage, one can consider that the Galactic acceleration affects instantaneously dSph kinematics of stars (Aguilar & White 1985) since the crossing time ($t_{cross} \sim r_{half}/\sigma_{dSph,stars}$) is larger or comparable to the encounter time ($t_{enc} \sim D_{MW}/V_{inf}$) for dSphs lying within the VPOS.”

<footnote n="1">¹GEPI, Observatoire de Paris, Université PSL, CNRS, Place Jules Janssen F-92195, Meudon, France; francois.hammer@obspm.fr</footnote>

<footnote n="2">²Université de Grenoble-Alpes, CNRS, IPAG, F-38000 Grenoble, France</footnote>

<footnote n="3">³NAOC, Chinese Academy of Sciences, A20 Datun Road, 100012 Beijing, People's Republic of China</footnote>
<table><caption>**Table 1**

Essential Data for 21 dSphs Lying in the VPOS, Completed by That for Sagittarius and Crater2 (See the Text)</caption><thead><tr><th>Name<br>(1)</th><th>D<sub>MW</sub><br>(kpc)<br>(2)</th><th>r<sub>half</sub><br>(kpc)<br>(3)</th><th>θ<br>(degr.)<br>(4)</th><th>σ<sub>los</sub><br>(km s<sup>-1</sup>)<br>(5)</th><th>M<sub>stellar</sub><br>(M<sub>☉</sub>)<br>(6)</th><th>M<sub>tot</sub>/M<sub>stellar</sub><br>(log)<br>(7)</th><th>g<sub>MW</sub>/g<sub>dSphstar</sub><br>(log)<br>(8)</th><th>v<sub>gsr</sub><br>(km s<sup>-1</sup>)<br>(9)</th><th>t<sub>cross</sub><br>10<sup>8</sup>yr<br>(10)</th><th>t<sub>enc</sub>/t<sub>cross</sub><br>(11)</th><th>References<br>(12)</th></tr></thead><tbody><tr><td>Segue</td><td>28</td><td>0.029 ± 0.007</td><td>18.3 ± 8</td><td>4.3 ± 1.2</td><td>330 ± 210</td><td>3.275<sup>+0.483</sup><sub>-0.6835</sub></td><td>3.06 ± 0.3473</td><td>117.4</td><td>2.207</td><td>0.621</td><td>(1), (2)</td></tr><tr><td>UrsaMaj.II</td><td>38</td><td>0.14 ± 0.025</td><td>-5.9 ± 4</td><td>6.7 ± 1.4</td><td>4000 ± 1900</td><td>3.261<sup>+0.3692</sup><sub>-0.4697</sub></td><td>3.17 ± 0.2584</td><td>-35.8</td><td>5.22</td><td>0.385</td><td>(3), (4)</td></tr><tr><td>BootesII</td><td>40</td><td>0.051 ± 0.017</td><td>9.99 ± 55</td><td>10.5 ± 7.4</td><td>1000 ± 800</td><td>3.814<sup>+0.7367</sup><sub>-6.861</sub></td><td>2.865 ± 0.4528</td><td>-125.7</td><td>2.27</td><td>0.841</td><td>(5), (6)</td></tr><tr><td>SegueII</td><td>41</td><td>0.034 ± 0.005</td><td>23.0 ± 17</td><td>3.4 ± 1.8</td><td>850 ± 170</td><td>2.729<sup>+0.4027</sup><sub>-5.445</sub></td><td>2.57 ± 0.1546</td><td>48.93</td><td>2.66</td><td>0.771</td><td>(7)</td></tr><tr><td>Willman1</td><td>43</td><td>0.025 ± 0.006</td><td>47.41 ± 5</td><td>4.3 ± 1.8</td><td>1000 ± 700</td><td>2.729<sup>+0.5764</sup><sub>-1.194</sub></td><td>2.204 ± 0.369</td><td>31.65</td><td>1.29</td><td>1.63</td><td>(8), (9)</td></tr><tr><td>ComaBer.</td><td>45</td><td>0.077 ± 0.01</td><td>56.6 ± 10</td><td>4.6 ± 0.8</td><td>3700 ± 1700</td><td>2.708<sup>+0.3369</sup><sub>-0.4013</sub></td><td>2.587 ± 0.2295</td><td>76.47</td><td>2.57</td><td>0.819</td><td>(1), (4)</td></tr><tr><td>Bootes</td><td>64</td><td>0.242 ± 0.021</td><td>62.97 ± 6</td><td>6.5 ± 2.0</td><td>30,000 ± 6000</td><td>2.597<sup>+0.2969</sup><sub>-0.5089</sub></td><td>2.462 ± 0.1151</td><td>96.46</td><td>5.80</td><td>0.54</td><td>(10), (9)</td></tr><tr><td>Draco</td><td>76</td><td>0.196 ± 0.012</td><td>2.49 ± 2</td><td>9.1 ± 1.2</td><td>0.27 ± 0.04 10<sup>6</sup></td><td>1.843<sup>+0.1685</sup><sub>-0.2015</sub></td><td>1.219 ± 0.0836</td><td>-110.5</td><td>2.04</td><td>1.9</td><td>(11), (12)</td></tr><tr><td>UrsaMinor</td><td>78</td><td>0.28 ± 0.015</td><td>7.84 ± 5</td><td>9.5 ± 1.2</td><td>0.20 ± 0.09 10<sup>6</sup></td><td>2.166<sup>+0.2954</sup><sub>-0.3254</sub></td><td>1.643 ± 0.2011</td><td>-96.11</td><td>3.21</td><td>1.19</td><td>(11), (13)</td></tr><tr><td>Sculptor</td><td>86</td><td>0.26 ± 0.039</td><td>-64.5 ± 1</td><td>9.2 ± 1.1</td><td>1.4 ± 0.6 10<sup>6</sup></td><td>1.261<sup>+0.2943</sup><sub>-0.3304</sub></td><td>0.6725 ± 0.227</td><td>86.56</td><td>1.08</td><td>3.86</td><td>(14), (15)</td></tr><tr><td>Sextans(I)</td><td>89</td><td>0.682 ± 0.117</td><td>5.95 ± 5</td><td>7.9 ± 1.3</td><td>0.41 ± 0.19 10<sup>6</sup></td><td>2.081<sup>+0.3386</sup><sub>-0.4029</sub></td><td>2.022 ± 0.2507</td><td>78.58</td><td>5.29</td><td>0.824</td><td>(16), (17)</td></tr><tr><td>UrsaMajor</td><td>102</td><td>0.318 ± 0.045</td><td>49.87 ± 3</td><td>11.9 ± 3.5</td><td>14,000 ± 4000</td><td>3.572<sup>+0.3297</sup><sub>-0.5276</sub></td><td>2.739 ± 0.1749</td><td>-10.24</td><td>7.92</td><td>0.631</td><td>(18), (4)</td></tr><tr><td>Carina</td><td>107</td><td>0.241 ± 0.023</td><td>-8.85 ± 5</td><td>6.6 ± 1.2</td><td>0.24 ± 0.10 10<sup>6</sup></td><td>1.705<sup>+0.3198</sup><sub>-0.3859</sub></td><td>1.233 ± 0.1993</td><td>22.24</td><td>2.27</td><td>2.262</td><td>(19), (17)</td></tr><tr><td>Hercules</td><td>126</td><td>0.33 ± 0.063</td><td>12.47 ± 4</td><td>3.7 ± 0.9</td><td>36,000 ± 11,000</td><td>2.163<sup>+0.3154</sup><sub>-0.454</sub></td><td>2.224 ± 0.2126</td><td>129.7</td><td>4.15</td><td>1.523</td><td>(1), (17)</td></tr><tr><td>Fornax</td><td>149</td><td>0.668 ± 0.034</td><td>59.06 ± 1</td><td>11.7 ± 0.9</td><td>14.0 ± 4.0 10<sup>6</sup></td><td>0.879<sup>+0.1894</sup><sub>-0.201</sub></td><td>0.1357 ± 0.132</td><td>-20.71</td><td>1.69</td><td>4.081</td><td>(21), (17)</td></tr><tr><td>LeoIV</td><td>155</td><td>0.116 ± 0.03</td><td>88.2 ± 9</td><td>3.3 ± 1.7</td><td>8700 ± 4600</td><td>2.226<sup>+0.5442</sup><sub>-6.095</sub></td><td>1.795 ± 0.3216</td><td>14.34</td><td>4.85</td><td>1.564</td><td>(1), (4)</td></tr><tr><td>CanesV.II</td><td>161</td><td>0.074 ± 0.012</td><td>13.09 ± 9</td><td>4.6 ± 1.0</td><td>7900 ± 3600</td><td>2.362<sup>+0.5442</sup><sub>-6.095</sub></td><td>1.421 ± 0.2432</td><td>-103.3</td><td>1.71</td><td>4.614</td><td>(22), (8)</td></tr><tr><td>LeoV</td><td>179</td><td>0.042 ± 0.005</td><td>60.2 ± 13</td><td>2.4 ± 1.9</td><td>4500 ± 2600</td><td>1.795<sup>+0.6641</sup><sub>-5.398</sub></td><td>1.102 ± 0.2717</td><td>59.5</td><td>1.45</td><td>5.864</td><td>(23), (23)</td></tr><tr><td>CanesV.</td><td>218</td><td>0.564 ± 0.036</td><td>16.73 ± 4</td><td>7.6 ± 0.4</td><td>0.23 ± 0.03 10<sup>6</sup></td><td>2.216<sup>+0.1071</sup><sub>-0.1138</sub></td><td>1.515 ± 0.0793</td><td>68.41</td><td>4.86</td><td>2.123</td><td>(24), (8)</td></tr><tr><td>LeoII</td><td>236</td><td>0.151 ± 0.017</td><td>60.4 ± 10</td><td>6.6 ± 0.7</td><td>0.59 ± 0.18 10<sup>6</sup></td><td>1.112<sup>+0.2261</sup><sub>-0.2519</sub></td><td>-0.0935 ± 0.165</td><td>22.97</td><td>1.07</td><td>10.39</td><td>(25), (12)</td></tr><tr><td>LeoI</td><td>258</td><td>0.246 ± 0.019</td><td>16.45 ± 3</td><td>9.2 ± 1.4</td><td>3.4 ± 1.1 10<sup>6</sup></td><td>0.851<sup>+0.2593</sup><sub>-0.3043</sub></td><td>0.4922 ± 0.156</td><td>178.9</td><td>0.707</td><td>18.91</td><td>(25), (26)</td></tr><tr><td>Sagittarius</td><td>18</td><td>1.55 ± 0.05</td><td>-11.4 ± 2</td><td>11.4 ± 0.7</td><td>17.0 ± 3.0 10<sup>6</sup></td><td>1.138<sup>+0.1286</sup><sub>-0.1357</sub></td><td>2.052 ± 0.0817</td><td>160.8</td><td>2.68</td><td>0.346</td><td>(27), (28)</td></tr><tr><td>Crater2</td><td>117</td><td>1.066 ± 0.084</td><td>...</td><td>2.7 ± 0.3</td><td>0.17 ± 0.03 10<sup>6</sup></td><td>1.735<sup>+0.1705</sup><sub>-0.1954</sub></td><td>2.624 ± 0.1043</td><td>-71.79</td><td>132</td><td>0.041</td><td>(29), (29)</td></tr></tbody></table>

Note. Column 1: name; Column 2: distance to the MW; Column 3: 2D-projected half-light radius; Column 4: measured line-of-sight velocity dispersion from Walker et al. (2009a); Column 5: dSph major-axis PA angle relatively to the MW disk plane; Column 6: total stellar mass evaluated with $M_☉/L_☉ = 1$ in V-band; Column 7: total-to-stellar mass estimated within $r_{half}$ and that is equal to $(\sigma_{los}/\sigma_{dSph,stars})^2$; Column 8: acceleration ratio as defined by Equation (1) of Hammer et al. (2018); Column 9: galactic rest-frame velocities of dSphs; Column 10: crossing time defined as $r_{half}/\sigma_{star}$; Column 11: ratio of the encounter time ($t_{enc} = D_{MW}/V_{inf}$) to the crossing time; Column 12: references for discovery then for distance estimate (see notes below). (1) Belokurov et al. (2007), (2) Simon et al. (2011), (3) Zucker et al. (2006a), (4) Simon & Geha (2007), (5) Walsh et al. (2007), (6) Koch et al. (2009), (7) Belokurov et al. (2009), (8) Willman et al. (2005a), (9) Martin et al. (2007), (10) Belokurov et al. (2006), (11) Wilson (1955), (12) Walker et al. (2007), (13) Walker et al. (2009a), (14) Shapley (1938a), (15) Pietrzyński et al. (2008), (16) Irwin et al. (1990), (17) Walker et al. (2009b), (18) Willman et al. (2005b), (19) Cannon et al. (1977), (20) Adén et al. (2009), (21) Shapley (1938b), (22) Sakamoto & Hasegawa (2006), (23) Belokurov et al. (2008), (24) Zucker et al. (2006b), (25) Harrington & Wilson (1950), (26) Harrington & Wilson (1950), (27) Ibata et al. (1994), (28) Monaco et al. (2004), (29) Caldwell et al. (2017).
ORCID iDs

François Hammer https://orcid.org/0000-0002-2165-5044

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