<table><caption>**Table 7**
Values of $v \sin i$ We Derived for the Stars in Our Late-K Stellar Sample</caption><thead><tr><th>Star</th><th>FWHM<br>(Å ± 3σ)<br>HARPS</th><th>FWHM<br>(Å ± 3σ)<br>SOPHIE</th><th>V<sub>rad</sub><br>km s<sup>-1</sup></th><th>v sin i<br>km s<sup>-1</sup> ± 3σ<br>HARPS</th><th>v sin i<br>km s<sup>-1</sup> ± 3σ<br>SOPHIE</th><th>v sin i<br>km s<sup>-1</sup><br>Others</th><th>P/sin i<br>(days ± 3σ)</th></tr></thead><tbody><tr><td>GJ 1056</td><td>0.1327 0.0028</td><td>...</td><td>+5.05</td><td>1.82-0.38 + 0.32</td><td>...</td><td>...</td><td>19.1 + 7.2-4.3</td></tr><tr><td>GJ 1066</td><td>0.1334 0.0028</td><td>...</td><td>+26.08</td><td>1.91-0.37 + 0.29</td><td>...</td><td>...</td><td>17.7 + 6.0-3.5</td></tr><tr><td>GJ 1067</td><td>...</td><td>0.1672 0.0034</td><td>-10.10</td><td>...</td><td>2.08-0.62 + 0.45</td><td>...</td><td>16.2 + 9.3-4.3</td></tr><tr><td>GJ 1177A</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>6.1</td><td>4.78 + 1.5-1.1</td></tr><tr><td>GJ 1213</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>0.0</td><td>&gt;31.6</td></tr><tr><td>GJ 1267</td><td>0.1366 0.0028</td><td>...</td><td>-5.18</td><td>2.24-0.28 + 0.27</td><td>...</td><td>...</td><td>13.7 + 3.4-2.6</td></tr><tr><td>GJ 1279</td><td>0.1347 0.0028</td><td>...</td><td>+4.81</td><td>2.06-0.35 + 0.27</td><td>...</td><td>1.0</td><td>17.1 + 5.2-3.3</td></tr><tr><td>GJ 3072</td><td>0.1277 0.0028</td><td>...</td><td>+9.22</td><td>1.10-0.53 + 0.42</td><td>...</td><td>...</td><td>26.3 + 31-9.6</td></tr><tr><td>GJ 3411</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>6.0</td><td>5.60 + 2.0-1.4</td></tr><tr><td>GJ 3494</td><td>0.1387 0.0028</td><td>...</td><td>+7.35</td><td>2.44-0.27 + 0.27</td><td>...</td><td>...</td><td>13.7 + 3.4-2.7</td></tr><tr><td>GJ 3551</td><td>0.1285 0.0028</td><td>...</td><td>+58.98</td><td>1.22-0.48 + 0.41</td><td>...</td><td>2.85</td><td>29.3 + 23-9.4</td></tr><tr><td>GJ 3996</td><td>0.1334 0.0028</td><td>...</td><td>+5.70</td><td>1.91-0.37 + 0.29</td><td>...</td><td>...</td><td>17.8 + 7.3-4.5</td></tr><tr><td>GJ 4140</td><td>0.1279 0.0028</td><td>...</td><td>+7.29</td><td>1.13-0.52 + 0.42</td><td>...</td><td>...</td><td>23.6 + 25-8.2</td></tr><tr><td>GJ 9073B</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>2.70</td><td>11.8 + 6.6-4.2</td></tr><tr><td>GJ 9250</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>0.0</td><td>&gt;33.5</td></tr><tr><td>GJ 9299</td><td>0.1324 0.0028</td><td>...</td><td>+27.90</td><td>1.78-0.39 + 0.33</td><td>...</td><td>...</td><td>18.9 + 7.8-4.6</td></tr><tr><td>GJ 9667</td><td>...</td><td>...</td><td>-12.01</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>GJ 9714</td><td>0.1394 0.0028</td><td>...</td><td>-14.75</td><td>2.51-0.27 + 0.27</td><td>...</td><td>...</td><td>13.1 + 3.0-2.4</td></tr><tr><td>GJ 9827</td><td>0.1300 0.0028</td><td>...</td><td>+31.99</td><td>1.45-0.43 + 0.38</td><td>...</td><td>...</td><td>21.7 + 13-6.8</td></tr><tr><td>Gl 14</td><td>...</td><td>0.1766 0.0034</td><td>+2.67</td><td>...</td><td>3.25-0.37 + 0.34</td><td>2.6</td><td>10.2 + 2.4-1.8</td></tr><tr><td>Gl 40A</td><td>0.1316 0.0028</td><td>...</td><td>+14.73</td><td>1.67-0.40 + 0.36</td><td>...</td><td>...</td><td>18.3 + 7.8-4.5</td></tr><tr><td>Gl 45</td><td>0.1295 0.0028</td><td>...</td><td>+6.68</td><td>1.37-0.44 + 0.39</td><td>...</td><td>...</td><td>21.9 + 13-6.3</td></tr><tr><td>Gl 50</td><td>0.1285 0.0028</td><td>...</td><td>-0.59</td><td>1.22-0.48 + 0.41</td><td>...</td><td>...</td><td>26.0 + 27-11</td></tr><tr><td>Gl 52</td><td>...</td><td>0.1687 0.0034</td><td>-40.77</td><td>...</td><td>2.28-0.52 + 0.45</td><td>2.0</td><td>13.5 + 5.7-3.3</td></tr><tr><td>Gl 57</td><td>0.12361 0.0028</td><td>...</td><td>+26.67</td><td>0</td><td>...</td><td>...</td><td>&gt;38</td></tr><tr><td>Gl 105.5</td><td>...</td><td>...</td><td>+73.02</td><td>...</td><td>...</td><td>2.72</td><td>12.9 + 5.3-3.4</td></tr><tr><td>Gl 112.1</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>5.48</td><td>6.75 + 2.2-1.5</td></tr><tr><td>Gl 116</td><td>...</td><td>0.1583 0.0034</td><td>-49.71</td><td>...</td><td>&lt;1</td><td>2.4</td><td>9.68 + 4.4-2.6</td></tr><tr><td>Gl 123</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>0.90</td><td>36.6 + 21-11</td></tr><tr><td>Gl 142</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>2.70</td><td>13.1 + 5.1-3.3</td></tr><tr><td>Gl 143.1</td><td>0.1304 0.0028</td><td>...</td><td>+13.64</td><td>1.51-0.43 + 0.37</td><td>...</td><td>...</td><td>21.2 + 12-6.3</td></tr><tr><td>Gl 146</td><td>0.1306 0.0028</td><td>...</td><td>+21.63</td><td>1.54-0.42 + 0.37</td><td>...</td><td>...</td><td>21.1 + 11-5.6</td></tr><tr><td>Gl 153A</td><td>...</td><td>0.1721 0.0034</td><td>+2.54</td><td>...</td><td>2.73-0.45 + 0.41</td><td>2.8</td><td>10.6 + 3.8-2.6</td></tr><tr><td>Gl 156</td><td>0.1275 0.0028</td><td>...</td><td>+62.31</td><td>1.07-0.55 + 0.43</td><td>...</td><td>...</td><td>30.5 + 37-11</td></tr><tr><td>Gl 162.2</td><td>0.1472 0.0028</td><td>...</td><td>-4.22</td><td>3.21-0.22 + 0.22</td><td>...</td><td>...</td><td>10.5 + 1.8-1.6</td></tr><tr><td>Gl 182</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>1.86<sup>a</sup></td></tr><tr><td>Gl 186A</td><td>0.133 0.0020</td><td>...</td><td>+113.5</td><td>1.86-0.37 + 0.31</td><td>...</td><td>...</td><td>16.7 + 6.9-4.2</td></tr><tr><td>Gl 186B</td><td>0.137 0.0040</td><td>...</td><td>+113.5</td><td>2.28-0.27 + 0.27</td><td>...</td><td>...</td><td>13.6 + 3.8-3.0</td></tr><tr><td>Gl 191</td><td>0.1309 0.0028</td><td>...</td><td>+244.89</td><td>1.58-0.42 + 0.37</td><td>...</td><td>...</td><td>7.9 + 3.7-2.0</td></tr><tr><td>Gl 208</td><td>0.1355 0.0028</td><td>0.1756 0.0034</td><td>+21.83</td><td>2.14-0.32 + 0.27</td><td>3.15-0.40 + 0.34</td><td>...</td><td>8.91 + 2.2-1.6</td></tr><tr><td>Gl 221</td><td>0.1252 0.0028</td><td>...</td><td>+23.18</td><td>(0.63)</td><td>...</td><td>...</td><td>(49.1)</td></tr><tr><td>Gl 226.2</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>0.53</td><td>57.6 + 74-24</td></tr><tr><td>Gl 249</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>10.46</td><td>3.27 + 0.7-0.5</td></tr><tr><td>Gl 256</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>7.97</td><td>4.46 + 1.1-0.9</td></tr><tr><td>Gl 296</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>3.8</td><td>8.55 + 4.2-3.0</td></tr><tr><td>Gl 322</td><td>...</td><td>0.1643 0.0034</td><td>-22.02</td><td>...</td><td>(1.58)</td><td>...</td><td>(18.6)</td></tr><tr><td>Gl 389.1</td><td>0.1354 0.0028</td><td>...</td><td>-3.47</td><td>2.13-0.32 + 0.27</td><td>...</td><td>...</td><td>15.8 + 5.3-3.6</td></tr><tr><td>Gl 397</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>2.34</td><td>15.5 + 7.3-4.3</td></tr><tr><td>Gl 412.3A</td><td>0.142 0.001</td><td>...</td><td>+18.17</td><td>2.76-0.27 + 0.26</td><td>...</td><td>...</td><td>11.8 + 3.3-2.7</td></tr><tr><td>Gl 412.3B</td><td>0.0901 0.002</td><td>...</td><td>+18.17</td><td>&lt;1</td><td>...</td><td>...</td><td>...</td></tr><tr><td>Gl 414A</td><td>...</td><td>0.1643 0.0034</td><td>-16.12</td><td>...</td><td>(1.58)</td><td>2.7,3.0</td><td>(20.8)</td></tr><tr><td>Gl 416</td><td>...</td><td>0.1697 0.0034</td><td>-11.70</td><td>...</td><td>2.41-0.47 + 0.45</td><td>2.85</td><td>16.3 + 5.9-3.9</td></tr><tr><td>Gl 421B</td><td>0.1281 0.0028</td><td>...</td><td>+8.10</td><td>1.16-0.51 + 0.42</td><td>...</td><td>1.90,2.00</td><td>20.0 + 37-14</td></tr><tr><td>Gl 425B</td><td>0.1491 0.0028</td><td>...</td><td>-1.28</td><td>3.35-0.21 + 0.21</td><td>...</td><td>4.20</td><td>12.3 + 3.7-3.3</td></tr><tr><td>Gl 455.1</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>3.68</td><td>7.76 + 2.4-1.8</td></tr><tr><td>Gl 466</td><td>...</td><td>0.1677 0.0034</td><td>...</td><td>...</td><td>2.15-0.57 + 0.45</td><td>...</td><td>17.7 + 10-5.5</td></tr><tr><td>Gl 496.1</td><td>0.1289 0.0028</td><td>...</td><td>+39.29</td><td>1.28-0.46 + 0.40</td><td>...</td><td>...</td><td>27.5 + 20-8.7</td></tr><tr><td>Gl 509.1</td><td>...</td><td>0.1771 0.0034</td><td>-6.26</td><td>...</td><td>3.30-0.35 + 0.34</td><td>...</td><td>10.7 + 2.6-2.1</td></tr><tr><td>Gl 517</td><td>0.2702 0.0028</td><td>...</td><td>-22.50</td><td>9.77 ± 0.13</td><td>...</td><td>8.3,10.8</td><td>3.05 ± 0.28</td></tr><tr><td>Gl 522</td><td>0.1307 0.0028</td><td>...</td><td>+46.29</td><td>1.55-0.42 + 0.37</td><td>...</td><td>...</td><td>21.6 + 11-6.0</td></tr><tr><td>Gl 524.1</td><td>0.1279 0.0028</td><td>...</td><td>+6.16</td><td>1.13-0.52 + 0.42</td><td>...</td><td>...</td><td>22.0 + 24-7.9</td></tr><tr><td>Gl 542.2</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>3.21</td><td>12.2 + 4.3-3.0</td></tr></tbody></table>
<table><caption>**Table 7**
(Continued)</caption><thead><tr><th>Star</th><th>FWHM<br>(Å ± 3σ)</th><th>FWHM<br>(Å ± 3σ)</th><th>V<sub>rad</sub><br>(km s<sup>-1</sup>)</th><th>v sin i<br>(km s<sup>-1</sup> ± 3σ)</th><th>v sin i<br>(km s<sup>-1</sup> ± 3σ)</th><th>v sin i<br>(km s<sup>-1</sup>)</th><th>P/sin i<br>(days ± 3σ)</th></tr></thead><tbody><tr><td>Gl 546</td><td>...</td><td>0.1692 0.0034</td><td>-36.85</td><td>...</td><td>2.35-0.49 + 0.45</td><td>3.0</td><td>13.8 + 5.4-3.4</td></tr><tr><td>Gl 558</td><td>...</td><td>0.1643 0.0034</td><td>-55.14</td><td>...</td><td>(1.58)</td><td>...</td><td>(19.5)</td></tr><tr><td>Gl 562</td><td>...</td><td>0.1657 0.0034</td><td>+42.68</td><td>...</td><td>1.84-1.09 + 0.50</td><td>1.90</td><td>14.3 + 28-5.2</td></tr><tr><td>Gl 570.2</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>2.63</td><td>13.5 + 8.5-5.3</td></tr><tr><td>Gl 571</td><td>0.1307 0.0028</td><td>...</td><td>+3.86</td><td>1.55-0.42 + 0.37</td><td>...</td><td>...</td><td>18.9 + 9.9-5.4</td></tr><tr><td>Gl 571.1</td><td>0.1271 0.0028</td><td>...</td><td>+15.86</td><td>1.01-0.61 + 0.43</td><td>...</td><td>...</td><td>35.6 + 66-14</td></tr><tr><td>Gl 576</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>6.43</td><td>4.06 + 1.3-1.0</td></tr><tr><td>Gl 583</td><td>0.1334 0.0028</td><td>...</td><td>-17.83</td><td>1.91-0.37 + 0.29</td><td>...</td><td>...</td><td>15.9 + 5.8-3.5</td></tr><tr><td>Gl 585.1</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>0.0</td><td>>31.1</td></tr><tr><td>Gl 619</td><td>...</td><td>0.1672 0.0034</td><td>+7.19</td><td>...</td><td>2.08-0.62 + 0.45</td><td>2.7</td><td>15.5 + 8.5-3.9</td></tr><tr><td>Gl 626</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>6.4</td><td>5.31 + 1.6-1.1</td></tr><tr><td>Gl 629.1</td><td>0.1342 0.0028</td><td>...</td><td>-76.44</td><td>2.01-0.37 + 0.27</td><td>...</td><td>2.7</td><td>17.1 + 7.3-4.5</td></tr><tr><td>Gl 659B</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>3.53</td><td>8.47 + 3.3-2.4</td></tr><tr><td>Gl 673</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>3.67</td><td>9.02 + 2.6-1.8</td></tr><tr><td>Gl 707</td><td>0.1377 0.0028</td><td>...</td><td>-2.26</td><td>2.35-0.27 + 0.27</td><td>...</td><td>...</td><td>14.0 + 3.1-2.4</td></tr><tr><td>Gl 708</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>5.3</td><td>6.68 + 2.5-1.7</td></tr><tr><td>Gl 710</td><td>0.1348 0.0028</td><td>...</td><td>-13.24</td><td>2.07-0.34 + 0.27</td><td>...</td><td>...</td><td>14.9 + 4.8-3.1</td></tr><tr><td>Gl 726</td><td>0.1314 0.0028</td><td>...</td><td>+16.82</td><td>1.64-0.41 + 0.37</td><td>...</td><td>...</td><td>18.0 + 7.9-4.5</td></tr><tr><td>Gl 728</td><td>...</td><td>0.1611 0.0034</td><td>-21.78</td><td>...</td><td>0</td><td>...</td><td>>35</td></tr><tr><td>Gl 747.2</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>14.10</td><td>3.18 + 0.6-0.5</td></tr><tr><td>Gl 747.3A</td><td>0.1398 0.0027</td><td>...</td><td>-15.81</td><td>2.55 ± 0.27</td><td>...</td><td>...</td><td>13.9 + 3.2-2.6</td></tr><tr><td>Gl 747.3B</td><td>0.1263 0.0023</td><td>...</td><td>+11.44</td><td>(0.86)</td><td>...</td><td>...</td><td>(41)</td></tr><tr><td>Gl 757</td><td>0.1262 0.0028</td><td>...</td><td>+53.68</td><td>(0.84)</td><td>...</td><td>...</td><td>(33)</td></tr><tr><td>Gl 763</td><td>...</td><td>...</td><td>-58.54</td><td>...</td><td>...</td><td>2.1</td><td>13.5</td></tr><tr><td>Gl 773</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>6.5</td><td>4.89 + 1.4-1.0</td></tr><tr><td>Gl 782</td><td>0.1341 0.0028</td><td>...</td><td>+23.12</td><td>2.00-0.37 + 0.27</td><td>...</td><td>...</td><td>15.1 + 5.1-3.0</td></tr><tr><td>Gl 786.1</td><td>...</td><td>0.1638 0.0034</td><td>+13.56</td><td>...</td><td>(1.46)</td><td>...</td><td>(22.0)</td></tr><tr><td>Gl 795A</td><td>...</td><td>...</td><td>-43.51</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>Gl 795B</td><td>...</td><td>...</td><td>-32.16</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>Gl 798</td><td>0.1289 0.0028</td><td>...</td><td>-45.71</td><td>1.28-0.46 + 0.40</td><td>...</td><td>...</td><td>21.7 + 15-6.6</td></tr><tr><td>Gl 801</td><td>0.1332 0.0028</td><td>...</td><td>-72.45</td><td>1.88-0.37 + 0.30</td><td>...</td><td>...</td><td>17.4 + 7.0-4.3</td></tr><tr><td>Gl 818</td><td>0.1346 0.0028</td><td>...</td><td>-65.45</td><td>2.05-0.35 + 0.27</td><td>...</td><td>2.00</td><td>18.7 + 5.7-3.5</td></tr><tr><td>Gl 820B</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>48.0</td></tr><tr><td>Gl 826.1</td><td>0.1362 0.0028</td><td>...</td><td>+19.01</td><td>2.21-0.29 + 0.27</td><td>...</td><td>...</td><td>14.2 + 3.6-2.6</td></tr><tr><td>Gl 828.1</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>1.40</td><td>21.1 + 21-8.4</td></tr><tr><td>Gl 830</td><td>0.1322 0.0028</td><td>...</td><td>-83.43</td><td>1.75-0.39 + 0.34</td><td>...</td><td>3.68</td><td>20.6 + 8.3-4.9</td></tr><tr><td>Gl 836.9</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>6.65</td><td>6.67 + 3.2-2.4</td></tr><tr><td>Gl 847A</td><td>0.1259 0.0028</td><td>...</td><td>-30.55</td><td>(0.78)</td><td>...</td><td>...</td><td>(34)</td></tr><tr><td>Gl 857.1</td><td>...</td><td>0.2072 0.0034</td><td>-6.85</td><td>...</td><td>6.01-0.23 + 0.23</td><td>6.2</td><td>7.66 + 1.0-0.96</td></tr><tr><td>Gl 884</td><td>0.1309 0.0028</td><td>...</td><td>+17.16</td><td>1.58-0.42 + 0.37</td><td>...</td><td>3.00,3.21</td><td>19.5 + 9.2-5.1</td></tr><tr><td>Gl 885A</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>6.27</td><td>3.98 + 2.4-1.7</td></tr><tr><td>Gl 889A</td><td>0.1348 0.0028</td><td>...</td><td>+16.35</td><td>2.07-0.35 + 0.27</td><td>...</td><td>...</td><td>15.2 + 5.0-3.2</td></tr><tr><td>Gl 891</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>2.5</td><td>11.3 + 5.7-3.5</td></tr><tr><td>Gl 894</td><td>0.1402 0.0028</td><td>...</td><td>+0.09</td><td>2.59-0.27 + 0.27</td><td>...</td><td>...</td><td>13.2 + 3.6-2.9</td></tr><tr><td>Gl 895.5</td><td>0.1391 0.0028</td><td>...</td><td>+2.18</td><td>2.48-0.27 + 0.27</td><td>...</td><td>...</td><td>11.9 + 3.1-2.5</td></tr><tr><td>Gl 898</td><td>0.1488 0.0028</td><td>...</td><td>+0.25</td><td>3.33 ± 0.21</td><td>...</td><td>3.37</td><td>10.6 + 1.8-1.6</td></tr><tr><td>Gl 900</td><td>...</td><td>...</td><td>-10.41</td><td>...</td><td>...</td><td>13.0</td><td>2.54 + 0.5-0.4</td></tr><tr><td>Gl 906</td><td>...</td><td>0.1618 0.0034</td><td>-20.49</td><td>...</td><td>(0.33)</td><td>...</td><td>>86</td></tr><tr><td>Gl 907.1</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>3.4</td><td>14.0 + 5.6-3.9</td></tr><tr><td>HIP 3121</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>17.0</td><td>2.21 + 0.5-0.4</td></tr><tr><td>HIP 6290</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>3.21</td><td>6.73 + 2.5-1.7</td></tr><tr><td>HIP 11437</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>5.0</td><td>8.98 + 3.0-2.3</td></tr><tr><td>HIP 14593A</td><td>0.1298 0.0028</td><td>...</td><td>+17.47</td><td>1.42-0.43 + 0.38</td><td>...</td><td>...</td><td>26.6 + 18-9.0</td></tr><tr><td>HIP 19410A</td><td>0.146 0.0030</td><td>...</td><td>+14.36</td><td>3.11-0.24 + 0.21</td><td>...</td><td>...</td><td>8.92 + 2.5-2.1</td></tr><tr><td>HIP 19410B</td><td>0.0936 0.0031</td><td>...</td><td>+9.32</td><td>&lt;1</td><td>...</td><td>...</td><td>...</td></tr><tr><td>HIP 19410C</td><td>0.156 0.0031</td><td>...</td><td>+0.22</td><td>3.88-0.21 + 0.20</td><td>...</td><td>...</td><td>5.06 + 1.3-1.1</td></tr><tr><td>HIP 21865</td><td>0.1337 0.0028</td><td>...</td><td>+53.29</td><td>1.95-0.37 + 0.28</td><td>...</td><td>...</td><td>17.3 + 6.4-3.9</td></tr><tr><td>HIP 40170</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>0.53</td><td>53.9 + 71-23</td></tr><tr><td>HIP 42108A</td><td>0.135 0.0029</td><td>...</td><td>+27.10</td><td>2.09-0.34 + 0.27</td><td>...</td><td>...</td><td>13.5 + 4.9-3.2</td></tr><tr><td>HIP 42108B</td><td>0.134 0.0029</td><td>...</td><td>+38.56</td><td>1.99-0.37 + 0.27</td><td>...</td><td>...</td><td>14.1 + 5.7-3.4</td></tr><tr><td>HIP 42910</td><td>0.1361 0.0028</td><td>...</td><td>+0.23</td><td>2.19-0.30 + 0.27</td><td>...</td><td>...</td><td>19.0 + 6.3-4.6</td></tr><tr><td>HIP 50773</td><td>...</td><td>0.1628 0.0034</td><td>-16.67</td><td>...</td><td>(1.07)</td><td>...</td><td>(28.4)</td></tr><tr><td>HIP 51073</td><td>0.1325 0.0028</td><td>...</td><td>+57.04</td><td>1.79-0.38 + 0.33</td><td>...</td><td>...</td><td>18.5 + 9.3-5.7</td></tr><tr><td>HIP 51263A</td><td>0.1299 0.0028</td><td>...</td><td>+83.87</td><td>1.44-0.38 + 0.33</td><td>...</td><td>...</td><td>23.6 + 14-7.4</td></tr></tbody></table>
<table><caption>**Table 7**
(Continued)</caption><thead><tr><th>Star</th><th>FWHM<br>(Å ± 3σ)</th><th>FWHM<br>(Å ± 3σ)</th><th>V<sub>rad</sub><br>(km s<sup>-1</sup>)</th><th>v sin i<br>(km s<sup>-1</sup> ± 3σ)</th><th>v sin i<br>(km s<sup>-1</sup> ± 3σ)</th><th>v sin i<br>(km s<sup>-1</sup>)</th><th>P/sin i<br>(days ± 3σ)</th></tr></thead><tbody><tr><td>HIP 51263B</td><td>0.1165 ± 0.0028</td><td>...</td><td>+46.32</td><td>&lt;1</td><td>...</td><td>...</td><td>&gt;70</td></tr><tr><td>HIP 53175</td><td>...</td><td>0.1702 ± 0.0034</td><td>+14.72</td><td>...</td><td>2.48-0.45 + 0.45</td><td>...</td><td>15.0 + 6.4-4.4</td></tr><tr><td>HIP 56838</td><td>0.1319 ± 0.0028</td><td>...</td><td>+39.26</td><td>1.71-0.40 + 0.35</td><td>...</td><td>...</td><td>21.4 + 9.6-5.6</td></tr><tr><td>HIP 58945</td><td>...</td><td>0.1682 ± 0.0034</td><td>-7.02</td><td>...</td><td>2.21-0.54 + 0.45</td><td>...</td><td>14.3 + 7.8-4.4</td></tr><tr><td>HIP 59247</td><td>...</td><td>0.1702 ± 0.0034</td><td>+4.45</td><td>...</td><td>2.48-0.45 + 0.45</td><td>...</td><td>13.8 + 6.1-4.2</td></tr><tr><td>HIP 60438</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>17.9</td><td>2.45 + 0.6-0.6</td></tr><tr><td>HIP 60501</td><td>...</td><td>0.1672 ± 0.0034</td><td>+6.13</td><td>...</td><td>2.08-0.62 + 0.45</td><td>...</td><td>25.0 + 18-8.6</td></tr><tr><td>HIP 72044</td><td>...</td><td>0.1643 ± 0.0034</td><td>-30.11</td><td>...</td><td>(1.58)</td><td>...</td><td>(19.1)</td></tr><tr><td>HIP 76550</td><td>0.1271 ± 0.0028</td><td>...</td><td>-64.50</td><td>1.01-0.61 + 0.43</td><td>...</td><td>...</td><td>22.9 + 49-10</td></tr><tr><td>HIP 78395</td><td>0.1299 ± 0.0028</td><td>...</td><td>+25.52</td><td>1.44-0.38 + 0.33</td><td>...</td><td>...</td><td>21.6 + 13-6.9</td></tr><tr><td>HIP 80083</td><td>0.1304 ± 0.0028</td><td>...</td><td>-4.81</td><td>1.51-0.43 + 0.37</td><td>...</td><td>...</td><td>18.1 + 14-7.3</td></tr><tr><td>HIP 95223</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>1.90</td><td>18.0 + 11-5.8</td></tr><tr><td>HIP 103150</td><td>0.1230 ± 0.0028</td><td>...</td><td>+15.77</td><td>&lt;1</td><td>...</td><td>...</td><td>&gt;30</td></tr><tr><td>HIP 110245</td><td>0.1475 ± 0.0028</td><td>...</td><td>-8.87</td><td>3.23 ± 0.21</td><td>...</td><td>...</td><td>12.3 + 3.0-2.6</td></tr><tr><td>HIP 110714</td><td>0.1294 ± 0.0028</td><td>...</td><td>+6.62</td><td>1.36-0.45 + 0.39</td><td>...</td><td>...</td><td>24.6 + 17-8.3</td></tr><tr><td>HIP 113597</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>6.27</td><td>7.46 + 2.7-1.9</td></tr><tr><td>HIP 116011</td><td>...</td><td>0.1638 ± 0.0034</td><td>+8.78</td><td>...</td><td>(1.46)</td><td>...</td><td>(19.4)</td></tr><tr><td>LHS 3060</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>4.9</td><td>6.20 + 1.4-1.1</td></tr><tr><td>STKM 1-607</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>40.0</td><td>...</td></tr></tbody></table>

**Notes.** Also listed: FWHM of the cross-correlation profile and *P/sin i*.

ª Rutten (1987).

<figure><img src="image_10.png" /><figcaption>**Figure 6.** A portion of the wavelength range used for our cross-correlation analysis of the late-K dwarf Gl 798: 5730 Å–5850 Å. This range yields very clean cross-correlation profiles for late-K dwarfs. For Gl 798, the S/N is ≈380: the variations in the spectrum near the continuum level are actually numerous blended weak absorption lines, each of which contributes to the cross-correlation function for this star.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>**Figure 7.** Examples of cross-correlation functions obtained from our analysis of HARPS spectra of five late-K dwarfs in our sample: Gl 57 (0 km s⁻¹), GJ 9299 (1.78 km s⁻¹), GJ 9714 (2.51 km s⁻¹), Gl 425B (3.35 km s⁻¹), and Gl 517 (9.77 km s⁻¹) (from narrowest to broadest). The FWHM of the peak for any particular star (in units of Å) is used to determine the value of v sin i for that star.</figcaption></figure>

256) with the center to limb darkening after Claret (2000). We computed synthetic profiles for the following range of v sin i values (in km s⁻¹): 0.05, 0.25, 0.50, 0.75, 1.00, 1.50, 2.00, 3.00, 4.0, 5.0, 7.0, and 10.0.

We convolve our cross-correlation functions of our template stars (Gl 57 for HARPS and Gl 728 for SOPHIE) with our rotational profiles. The FWHM of Gl 57 is 0.1236 Å (6.40 km s⁻¹), and for Gl 728 it is 0.1611 Å (8.35 km s⁻¹). Some of the cross-correlation profiles of Gl 57 convolved with our theoretical rotational profiles are shown in Figure 8 in order of increasing FWHM: 0, 1.0, 2.0, 3.0, 4.0, 5.0, 7.0, and 10.0 km s⁻¹. One can see in this figure that the cross-correlation profile broadened at 1.0 km s⁻¹ is clearly distinguishable from the unbroadened cross-correlation profile of Gl 57 (solid line).

We measured the FWHM of the rotational profiles and show these measurements as a function of the theoretical v sin i in Figure 9 for HARPS and SOPHIE. We use this diagram to derive v sin i values from the direct FWHM measures. In general, the correct v sin i values are smaller than the corresponding FWHM values, assuming a Gaussian profile.

In Table 7, we list the values we have derived for v sin i for our late-K dwarfs. Rotation was detected in most of our sample (92 stars), with v sin i typically in the range 1.0–3.5 km s⁻¹.

With HARPS, we also observed a few stars of particular interest: five spectroscopic binaries (Gl 186AB, Gl 412.3AB, Gl 747.3AB, HIP 42108AB, HIP 51263AB) and one

spectroscopic triple system (HIP 19410ABC). For two of these stars, the components are well separated (Gl 747.3AB and HIP 51263AB). We show their cross-correlation profiles in Figure 10. One can see in both cases that the secondary peak is much weaker than the primary peak. This indicates that the secondary has a later spectral type than the primary. For these two spectroscopic binaries, we proceeded as in the case of single stars: we measured the FWHM directly from the correlation profiles. For the other stars, the two or three components are blended. In these cases we applied multi-Gaussian fits to the cross-correlation profiles. We show these profiles in Figure 11 together with the fits and the profiles of each component.

In the case of Gl 186AB, the secondary is significantly weaker than the primary. It is therefore most likely not a late-K dwarf. The same applies to the secondary and tertiary components of HIP 19410. In contrast, the primary and secondary components of HIP 42108AB have almost the same intensity. In this case, we assume that both components are late-K dwarfs. Note the relatively good fits that we obtain with multi-Gaussians. We give the FWHM of the cross-correlation peaks for all these stars in Table 7 together with the uncertainties on the fits.

We would like to emphasize that a few of the active stars in our dK6 stellar sample are candidate young stars that may not yet have contracted to the main sequence (MS). A reliable way to identify the (few) pre-main-sequence (PMS) stars in our
<figure><img src="image_12.png" /><figcaption>**Figure 8.** The cross-correlation profile of Gl 57 convolved with our theoretical rotational profiles, in order of increasing FWHM: 0, 1.0, 2.0, 3.0, 4.0, 5.0, 7.0, and 10.0 km s⁻¹. One can see in this figure that the cross-correlation profile broadened at 1.0 km s⁻¹ is clearly distinguishable from the unbroadened cross-correlation profile of Gl 57 (solid line).</figcaption></figure>

samples is the stellar radius: the fact is that PMS stars that have not yet contracted to the MS have abnormally large radii. We identified three such stars in our dK6 sample, GJ 1177A, GJ 182, and GJ 425B, as possible PMS stars (see Table 1). Of these, Gl 425B is a rather low-activity star. As such, the abnormally large radius for this star is probably due to binarity. Nevertheless, we shall find (in Paper II) that these stars do correlate well with the MS stars in the RAC. In our M2 sample (see Table 3), we have also identified two PMS stars: GJ 1264 and GJ 803. But again, as we shall see in Paper II, these PMS stars do correlate very well with the other MS stars in the context of the RAC. In our M3 sample, GJ 277A is a possible PMS star (Table 4). In our M4 sample, GJ 2069A, GJ 3322, GJ 4185A, GJ 4338B, GJ 669A, and GJ 812A are possible PMS stars according to their radii (Table 5). We emphasize that all these stars do not rotate especially fast. There are many MS stars that rotate faster. These PMS stars are supposed to actually spin up as they contract to the MS, and young MS stars are supposed to be the fastest rotators (e.g., Barnes 2003). That is what we actually observe in our samples of stars: young MS stars are the fastest rotators (e.g., among our M4 sample, GJ 3631, GJ 3789, GJ 4020B, GJ 4338B, GJ 431, GJ 630.1, GJ 791.2A) with $v \sin i$ in excess of 15 km s⁻¹ and up to 56 km s⁻¹.

All our RACs will turn out (in Paper II) to demonstrate that PMS stars do not depart significantly from the overall RACs of the MS stars with the same spectral subtype. Numerous previous studies also found similar results, such as Mamajek & Hillebrand (2008), Browning et al. (2010), Christian et al. (2011), and West et al. (2015). This finding is rather intriguing since PMS stars may have internal structures that are different

from MS stars; some PMS stars may even be fully convective. More investigation is required to confirm this result, but so far, in our samples, we found no evidence that PMS stars obey RACs that are significantly different from those of MS stars (see Paper II).

## 5. SOURCES OF UNCERTAINTY IN THE MEASURES

For dM2 stars (Paper XIV), we showed that, by using four different spectrographs and the results from other authors, it was possible to attain a detection limit of 1 km s⁻¹ in $v \sin i$, with a precision of about 0.3 km s⁻¹. This empirical analysis includes all sources of uncertainties on the measures.

In the present study, as was the case for our dM2 stars, our cross-correlation profiles have a very high S/N (several hundreds; see Figure 7). This is because many spectral lines enter into our calculation of the cross-correlations. It is therefore in theory possible to measure subtle differences between different cross-correlation profiles of only a few percent. However, as in the case of dM2 stars (mentioned above), as well as in the case of dM3 stars (see HM), the greatest uncertainty arises from the background of the cross-correlation functions. There are secondary peaks in the cross-correlation function backgrounds, and this represents a problem in defining the zero level. In fact, these secondary features repeat in all of our cross-correlations because all of our stars have quite similar spectral types. For dK6 stars, we found that the noise in the background is a bit greater than that of dM2 and especially dM3 stars. This is due to the nature of the spectral range we use for the cross-correlations. In dM3 stars, there are hundreds of narrow spectral lines present in the
<figure><img src="image_13.png" /><figcaption>**Figure 9.** The true $v \sin i$ (in km s$^{-1}$) as a function of the measured FWHM (in Å) of the cross-correlation peak. These curves are used to compute a star's $v \sin i$ from HARPS (continuous line) and SOPHIE (dashed line) measurements of FWHM. The typical uncertainty on these curves is ±0.005 Å on the $x$ axis.</figcaption></figure>

spectrum, and these lines are evenly distributed across the selected wavelength range. However, for dK6 stars, the spectral lines are stronger, and there are fewer of them within our selected wavelength range. This results in a larger background noise when we calculate the cross-correlation functions.

In order to assess the noise in the background from one star to another, we averaged the backgrounds, after normalization of the cross-correlation peaks, of four stars: Gl 45, Gl 57, Gl 162.2, and Gl 425B. We then subtracted this average background from the raw cross-correlation profiles. The resulting profiles and remaining background fluctuations are shown in Figure 12. One can notice some differences in the broadenings of the cross-correlation functions for Gl 57, Gl 45, Gl 162.2, Gl 425B, but also some variations in the cross-correlation backgrounds. This gives us a direct estimate of the typical background “noise.” This noise (i.e., the secondary peaks) has an amplitude of about $\pm 3\sigma = 3\%$ of the main peak (Figure 12). The noise level in the dK6 analysis implies an uncertainty of $3\sigma = 0.0028$ Å in the FWHM, i.e., $3\sigma = 0.40 \text{ km s}^{-1}$ on $v \sin i$ for HARPS spectra for $v \sin i = 2 \text{ km s}^{-1}$. For SOPHIE spectra we have an uncertainty of about $3\sigma = 0.0034$ Å. However, these uncertainties on the FWHM yield a variable uncertainty on $v \sin i$ depending on where the data fall relative to the curves in Figure 9. Indeed, because the curves in Figure 9 are significantly steeper at small FWHM, the uncertainty in $v \sin i$ increases the smaller the value of $v \sin i$. The curves in Figure 9 show that $v \sin i$ is not

detected within $\pm 3\sigma$ below $0.88 \text{ km s}^{-1}$ for HARPS and $1.67 \text{ km s}^{-1}$ for SOPHIE.

In order to further assess the precision of our measurements from our cross-correlations, we plot in Figure 13 the values of $v \sin i$ obtained from the PCA-based inversion of the stellar parameters (Section 3.1) versus the values of $v \sin i$ measured from our cross-correlations: the agreement between the results from these two independent methods is reasonable if one considers the relatively high uncertainty from the PCA-based inversion method (1.5 km s$^{-1}$). We find that the mean of the differences between the two sets of measures is only 0.633 km s$^{-1}$. The standard deviation of the differences in this set of measurements compared to the one-to-one relationship (see Figure 13) is $\sigma = 0.421 \text{ km s}^{-1}$. This means that the typical $3\sigma$ uncertainty for these measures is $1.263 \text{ km s}^{-1}$. This is better than the expected uncertainty of the PCA-based inversion method (1.5 km s$^{-1}$; see Section 3.1).

We searched the literature for other $v \sin i$ or rotational period $P$ measurements for our K6, M2, M3, and M4 targets (we used the tutorial developed by Paletou & Zolotukhin 2014).<fnref n="8" /> We found several measurements of $v \sin i$ (Vogt et al. 1983; Marcy & Chen 1992; Tokovinin 1992; Favata et al. 1995; Delfosse et al. 1998; Mohanty & Basri 2003; Nordstrom et al. 2004; Glebocki & Gnacinski 2005; Torres et al. 2006; Jenkins et al. 2009; Schroder et al. 2009; Martínez-Arnáiz et al. 2010) and $P$ (Pizzolato et al. 2003; Kiraga &

<footnote n="8">⁸ http://www.astropy.org/</footnote>
<figure><img src="image_14.png" /><figcaption>**Figure 10.** Cross-correlation functions for three spectroscopic binaries in our late-K dwarf sample: Gl 186AB, Gl 747.3AB, and HIP 51263AB. For two of these binaries, the two components are well separated, and we do not need to use multi-Gaussian fits in our analysis. Moreover, in these two cases, the secondary is an M dwarf of later spectral type, and as a result it is considerably fainter than the primary. In the case of Gl 186AB, we analyzed this system using a multi-Gaussian fit to the cross-correlation function. We illustrate the best fit (superposed), as well as the profiles of the individual components.</figcaption></figure>

Stepień 2007; Irwin et al. 2011). In cases where a rotation period *P* was reported in these papers, we derived the *v sin i* from *P* assuming sin *i* = 1.

We show in Figure 14 our results for *v sin i* versus the
results from other spectrographs (e.g., SOPHIE as a function of
HARPS (Paper XIV, HM), 19 measurements in total) and the
measurements from other authors (73 measurements). The
filled circles denote results obtained for dK6 stars in this paper.
The overall agreement between our measures and those of other
authors is good. The agreement goes down to *v sin i* as low as
0.5 km s⁻¹. For results below 1 km s⁻¹ we have 13 data points.
For these points, the mean difference between our results and
others is found to be 0.37 km s⁻¹. The standard deviation of
the differences in this set of measurements compared to the
one-to-one relationship (see Figure 14) is σ = 0.27 km s⁻¹.

This means that the typical 3σ uncertainty for this subset of
measures is only 0.81 km s⁻¹.

For the slow rotators, that is, for dK6, dM2, dM3, and dM4
stars (with no evidence for emission in Hα), the mean of the
differences between our *v sin i* results and those reported by
other authors is only 0.50 km s⁻¹. The standard deviation of
the differences in this set of measurements compared to the
one-to-one relationship is σ = 0.32 km s⁻¹. This means that
the typical 3σ uncertainty for the slow rotators is only 0.96
km s⁻¹. These figures are lower than the ones we derived from
the uncertainties arising from features in the background of the
cross-correlations (see above and HM). This indicates that the
uncertainty on our *v sin i* measures when using the cross-
correlation technique depends essentially on the S/N in the
cross-correlation peaks and that the effects of the uncertainties
<figure><img src="image_15.png" /><figcaption>**Figure 11.** Cross-correlation functions for two spectroscopic binaries and one triple system in our late-K dwarf sample: Gl 412.3AB, HIP 19410ABC, and HIP 42108AB. We analyzed these systems using multi-Gaussian fits to the cross-correlation functions. In each figure we illustrate the best fit (superposed), as well as the profiles of the individual components.</figcaption></figure>

on the background continuum have been overestimated. Therefore, the uncertainties on $v \sin i$ and $P/\sin i$ derived in Table 7 are only safe upper limits. These results confirm that both HARPS and SOPHIE are very stable instruments over the long term. Therefore, our spectra are well suited for reliable measurements of small rotational broadenings in K and M dwarfs.

For the fast rotators, i.e., for dK6e, dM2e, dM3e, and dM4e stars (with clear evidence for emission in $H_α$), the mean of the differences between our $v \sin i$ results and those reported by others is $0.99 \text{ km s}^{-1}$. The standard deviation of the differences in this set of measurements compared to the one-to-one relationship is $\sigma = 0.60 \text{ km s}^{-1}$. This means that the typical $3\sigma$ uncertainty for the fast rotators is about $1.80 \text{ km s}^{-1}$. This uncertainty is about two times larger than for the slow rotators.

This is probably due to the presence of large spots on the surfaces of these stars.

The reason we can attain precisions in $v \sin i$ as good as those mentioned above is that we have paid attention to several factors in our analysis. (1) The choice of the wavelength interval is crucial in obtaining cross-correlation profiles where the S/N is high and the background noise is as low as possible. (2) The method of subtracting the background noise is essential in order to recover with the highest precision the true cross-correlation profiles. (3) The spectra that we selected for analysis are mostly high S/N spectra and were all reduced with an identical automated technique. (4) We use only spectrographs that are very stable in radial velocity and that use image scramblers to avoid seeing effects on the FWHM of the spectral lines. By combining these factors, we have found that it is
<figure><img src="image_16.png" /><figcaption>**Figure 12.** The cross-correlation profiles and the variations in the background continuum for four late-K dwarfs: Gl 57 (solid line), Gl 45 (dashed line), Gl 162.2 (dotted–dashed line), and Gl 425B (dotted line). This diagram illustrates the variations in the background continuum from one star to another and yields an estimate of the error on the FWHM measurements of the cross-correlation peaks.</figcaption></figure>

possible to measure subtle differences between different cross-correlation profiles at the level of as little as a few percent.

## 6. THE DISTRIBUTIONS OF ROTATIONAL PERIODS AMONG M AND K DWARFS

We derived the projected rotation periods *P*/sin*i* for all our dK6 targets according to the *v sin i* results in Table 7 and the radii given in Table 1. Values of *P*/sin*i* are given in Table 7 for our dK6 target stars, along with their uncertainties.

We show in Figure 15 the histogram of *P*/sin*i* values for dK6 stars, together with the analogous results for dM2 stars (upper panel). Also shown are the histograms of *P*/sin*i* values for dM3 and dM4 stars (lower panel). We see that the peak of the distribution for dK6 stars lies at a mean projected period in the range 13–22 days. We also see that the longest *P*/sin*i* that we have detected for dK6 stars is 30–35 days (see Figure 15): these longest periods overlap with the range of upper limits (~33–66 days) that were estimated in Section 1 above, given the limitations (0.5–1 km s⁻¹) on measuring *v sin i*.

The histogram for dK6 stars in Figure 15 has a shape different from that of dM2 stars, in spite of their relatively close spectral types. In dM2 stars, in addition to a main peak at about 11 days, there may be a second (smaller) peak at shorter periods (4 days): if these two peaks are really distinct, we could refer to them as belonging to slow and fast rotators, respectively, with more slow rotators than fast. Among the dK6 sample, there is

also an analog of a short-period peak at about 3–7 days. In dM2 stars, the distribution among the slow rotators is compact, extending to periods no larger than 16–20 days, whereas in dK6 stars, the distribution is more spread out, extending to 25 days. The distribution for dM4 stars is not bimodal as in dM2 and dK6 stars. For dM4 stars, there is a hint that there are more fast rotators than slow rotators. For dM3 stars, our data set suggests that the distribution is also bimodal, with two distinct groups of slow and fast rotators.

The principal conclusion that we draw from the two panels in Figure 15 is that there appears to be a global shift of the distributions from dK6 stars to dM4 stars. Specifically, mid-M dwarfs rotate in general faster than early-M dwarfs, a fact already well established among M dwarfs. The differences between the four histograms in Figure 15 indicate the value of selecting stars in narrowly restricted spectral subtypes: this selection suggests that different subtypes among the M dwarfs may exhibit distinctive rotational properties. Finally, we note that the mean period of the dK6 star histogram (17 days) is intermediate between the mean period of the dM2 star histogram (11 days) and that of dK4 stars (≈35 days, Paper XVI). This is an indication of the overall trend toward shorter rotation periods as we go from dK4 to dM4 (see HM).

Formally, we have calculated a mean value and a median value for each of the four histograms shown in Figure 15. For dK6 stars, we find a mean of 15.8 days and a median of
<figure><img src="image_17.png" /><figcaption>**Figure 13.** Comparison between our cross-correlation results for $v \sin i$ (x axis) and the results from our PCA-based inversion of the stellar parameters (Section 3.1) (y axis). The solid line indicates the one-to-one relation. Differences between the two independent methods do not exceed 0.63 km s$^{-1}$ on average.</figcaption></figure>

16.0 days. For dM2 stars, the mean and median are 10.8 and 11.5 days, respectively. For dM3 stars, the mean and median are 18.0 and 20.5 days, respectively. For dM4 stars, the mean and median are 4.9 and 5.0 days, respectively. Thus, whether we consider the mean period or the median period, we can state the following: as we go from dK6 to dM2 to dM4, there is a trend toward shorter periods. However, there is an exception to this trend at spectral subtype dM3: there, the mean/median period does not fit the trend, but is definitely longer than the dK6–dM2–dM4 trend would predict. In fact, the dM3 mean/median period is actually longer than the results for dK6. Thus, as already suggested by the smaller data set used by HM, the dM3 stars in our sample have markedly slower rotations than the trend between dK6 and dM4.

In HM, histograms were derived for $P/\sin i$ values among dM2 stars (85 targets), dM3 stars (80 targets), and dM4 stars (21 targets) (see Figure 9 in HM). In the present paper, we have obtained a fourth histogram, for dK6 stars. Even a casual inspection shows that there are differences between the histograms (see Figure 15). The question is: can a random distribution of inclination angles produce the four distributions, assuming the actual period distribution is the same for all three samples?

To address this, we have applied the Kolmogorov–Smirnov (K-S) two-sided test to each histogram in order to test the null hypothesis that the stars in any one sample are drawn from the

same population as the stars in another sample. For any pair of samples, we construct the cumulative distributions $F(1)$ and $F(2)$ for periods ranging from 44 to 1 day. Extracting numerical values for the K-S statistic $D(1, 2)$, we find that, when we compare a given distribution to the distribution of any other spectral subtype, the null hypothesis (namely, that the distributions are drawn from the same population) is rejected at a confidence level better than 0.001%. This suggests that the $P/\sin i$ distribution (i.e., the rotational braking mechanism) in dM3 stars is controlled by different factors than in dM2 or dM4 stars. The K-S analysis also indicates that the dK6 stars have a distribution that differs significantly from those of the other dM spectral subtypes.

But in this regard, an anonymous referee has pointed out that histograms can suffer from slight distortions as a result of choices related to binning. In view of this, the referee suggests that, when comparing the rotational period distributions in Figure 15, it may be more useful to compare smoothed continuous distributions rather than raw histograms. Different possibilities are available for smoothing, such as kernel density estimates, splines, and bezier. Here, the software package (gnuplot) that is available to one of the authors (DJM) offers bezier smoothing. Using this, the smoothed rotation distributions were used to construct cumulative distributions for all four spectral subtypes. For each pair of spectral subtypes, the K-S statistic $D(1, 2)$ was extracted. The maximum value of $D$
<figure><img src="image_18.png" /><figcaption>**Figure 14.** Comparison between our results for $v \sin i$ (x axis) and results from other authors (y axis). We show the K6 dwarf measurements of other sources as a function of the present measurements as filled circles. Included in the plot are data for all of our samples of M dwarfs (including subtypes K6, M2, M3, and M4). The solid line indicates the one-to-one relation. The differences between our results and the others do not exceed 0.50 km s$^{-1}$ on average for the low-activity stars.</figcaption></figure>

(1,2), normalized by a factor that depends on the sizes of the two distributions being compared, leads to a coefficient $c(\alpha)$ that was found to have the following numerical values for various sample pairs: 5.067 (K6 versus M4), 4.623 (M3 versus M4), 3.602 (M2 versus M4), 2.901 (M2 versus M3), 2.201 (K6 versus M2), and 1.382 (K6 versus M3). Statistically speaking, values of $c(\alpha)$ in excess of 1.95 correspond to a probability $\alpha$ of less than 0.001 that a pair of distributions has been drawn from the same parent population. The above numerical results indicate that the rotational distributions of K6 and M3 stars (for which $c(\alpha) = 1.382$) might originate from the same population. But in the case of all the other pairs, (K6,M2), (K6,M4), (M2, M3), (M2,M4), and (M3,M4), the bezier-smoothed distributions have a chance of less than 0.1% of being drawn from the same population. This suggests that the $P/\sin i$ distribution (which is determined by the rotational braking mechanism) in dM3 stars is controlled by factors different from those that operate in dM2 or dM4 stars, but that dK6 stars might have a braking mechanism similar to that in dM3 stars.

7. CONCLUSION

In this paper, we have derived numerical values of various parameters of K and M dwarfs in preparation for a follow-up study of RACs (in a future Paper II). For M dwarfs, values of $T_{\text{eff}}$ have been determined from the $(R-I)_C$ color using the

empirical calibration of Mann et al. (2015). For late-K dwarfs, values of $T_{\text{eff}}$ have been determined from the $(R-I)_C$ color using the calibration of Kenyon & Hartmann (1995). These determinations have led to values of $T_{\text{eff}}$ for a total of 612 stars with spectral subtypes dK4, dK6, dM2, dM3, and dM4. In an independent approach, we have also determined several stellar parameters ($T_{\text{eff}}$, log(g), and [M/H]) using the PCA-based inversion technique of Paletou et al. (2015a) for a sample of 105 late-K dwarfs. We also compile all effective temperatures from the literature for this sample of late-K dwarfs. Together with our determinations of $T_{\text{eff}}$ from the $(R-I)_C$ color, this allows us to obtain reliable values of $T_{\text{eff}}$ for our sample of late-K dwarfs. With the values of $T_{\text{eff}}$ available, we then derived the stellar radii for our sample of K–M dwarfs, based on parallax measurements from Hipparcos. We find that our radii agree well with the interferometric radii reported by Boyajian et al. (2012): in the $T_{\text{eff}}$–radius plane, we find that our estimates of radii are within 0.19%, 6.3%, 4.3%, and 8.2% of the Boyajian results for our late-K, dM2, dM3, and dM4 stellar samples, respectively.

We also propose empirical relationships between radius and metallicity for dK6, dM3, and dM4 stars, similar to that reported for dM2 stars by Houdebine (2011b). This allows us to derive metallicity for all our targets, and these will be used in Paper II to correct the RACs for metallicity effects.
<figure><img src="image_19.png" /><figcaption>**Figure 15.** Histograms of stars as a function of *P/sini* for dK6 stars, dM2 stars, dM3 stars, and dM4 stars. The peak of the distribution shifts from (about) 17 days for dK6 stars to 10 days for dM2 stars, and to 3 days for dM4 stars.</figcaption></figure>

We present a cross-correlation analysis of HARPS and SOPHIE spectra of 105 late-K dwarfs, which has allowed us to detect $v \sin i$ in 92 of these stars. In combination with our previous $v \sin i$ measurements in M and K dwarfs (Papers VII, XIV, XVI, XVII, HM) and from the compilations of $v \sin i$ measurements from the literature for dK6, dM3, and dM4 stars, we now derive $P/\sin i$ measures for a sample of 418 K and M dwarfs. This allows us to investigate the distributions of $P/\sin i$ for each spectral subtype separately. We find that the $P/\sin i$ distributions are statistically distinct from one spectral subtype to another at a 99.9% confidence level. Therefore, if one wants to investigate certain properties of K and M dwarfs that are related to rotation (such as RACs), it is important to consider the different spectral subtypes separately.

The parameters that we have derived in this paper for stars belonging to various spectral subtypes from dK4 to dM4 will be used as input in Paper II to construct separate RACs for each spectral subtype.

This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. DJM is supported in part by the NASA Space Grant program. This research was based on data obtained from the ESO Science Archive Facility and the Observatoire de Haute Provence SOPHIE database. This research made use of Astropy<fnref n="9" />, a community-developed core Python package for astronomy (Astropy Collaboration, 2013). This research has made use of the VizieR catalog access tool, CDS, Strasbourg, France. The original description of the VizieR service was published in A&AS 143, 23.

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