<table><caption>**Table 7**
The Values of *v sin i*, *P/sin i* and [M/H] Compiled from Paper I for Our Targets in Our List of dM4 Stars</caption><thead><tr><th>Star</th><th>v sin i<br>(km s<sup>-1</sup>)</th><th>P/sin i<br>(days)</th><th>[M/H]<br>(dex)</th><th>[M/H]<sup>a</sup><br>(dex)</th><th>Ca II EW<br>(Å)</th><th>F<sub>HK</sub><br>(10<sup>4</sup> erg s<sup>-1</sup> cm<sup>-2</sup>)</th><th>F<sub>HK</sub><sup>b</sup><br>(10<sup>4</sup> erg s<sup>-1</sup> cm<sup>-2</sup>)</th></tr></thead><tbody><tr><td>GJ 1001A</td><td>1.15</td><td>12.81–5.66+18.02</td><td>...</td><td>+0.047</td><td>-0.235</td><td>0.377</td><td>0.338</td></tr><tr><td>GJ 1005AB</td><td>3.00</td><td>3.41–0.96+1.44</td><td>-0.47</td><td>-0.210</td><td>...</td><td>...</td><td>...</td></tr><tr><td>GJ 1034</td><td>4.50</td><td>2.05–0.61+0.79</td><td>-0.38</td><td>-0.275</td><td>...</td><td>...</td><td>...</td></tr><tr><td>GJ 1065</td><td>4.00</td><td>2.70–0.65+0.88</td><td>0.20</td><td>-0.172</td><td>-0.25</td><td>0.431</td><td>0.640</td></tr><tr><td>GJ 1105</td><td>2.00</td><td>6.93–2.28+4.24</td><td>0.24</td><td>+0.022</td><td>-0.62</td><td>1.068</td><td>0.615</td></tr><tr><td>GJ 1129</td><td>3.00</td><td>4.62–1.94+2.91</td><td>0.05</td><td>+0.022</td><td>-0.328</td><td>0.463</td><td>0.440</td></tr><tr><td>GJ 1134</td><td>4.10</td><td>3.11–0.74+1.00</td><td>-0.16</td><td>-0.036</td><td>...</td><td>...</td><td>...</td></tr><tr><td>GJ 1207</td><td>10.70</td><td>1.18–0.21+0.25</td><td>-0.15</td><td>-0.042</td><td>-7.098</td><td>11.10</td><td>12.23</td></tr><tr><td>GJ 1254</td><td>4.00</td><td>5.39–1.41+1.90</td><td>0.22</td><td>+0.122</td><td>...</td><td>...</td><td>...</td></tr><tr><td>GJ 1289</td><td>2.60</td><td>4.34–1.27+2.04</td><td>0.12</td><td>-0.137</td><td>-1.680</td><td>2.256</td><td>3.093</td></tr><tr><td>GJ 2036B</td><td>19.8</td><td>0.429–0.07+0.08</td><td>...</td><td>-0.327</td><td>-14.49</td><td>19.66</td><td>19.66</td></tr><tr><td>GJ 2069A</td><td>6.43</td><td>3.20–0.98+1.34</td><td>0.27</td><td>+0.102</td><td>-16.91</td><td>24.53</td><td>13.17</td></tr><tr><td>GJ 3092B</td><td>3.80</td><td>4.87–1.24+1.70</td><td>...</td><td>+0.064</td><td>...</td><td>...</td><td>...</td></tr><tr><td>GJ 3149B</td><td>4.00</td><td>3.02–1.26+1.71</td><td>0.09</td><td>-0.078</td><td>-5.647</td><td>8.193</td><td>6.660</td></tr><tr><td>GJ 3283B</td><td>6.00</td><td>1.94–0.67+0.93</td><td>...</td><td>-0.111</td><td>-6.372</td><td>8.101</td><td>10.46</td></tr><tr><td>GJ 3322</td><td>7.68</td><td>9.20–4.57+5.94</td><td>...</td><td>...</td><td>-10.58</td><td>16.30</td><td>16.30</td></tr><tr><td>GJ 3398</td><td>4.00</td><td>3.11–0.81+1.10</td><td>...</td><td>-0.054</td><td>...</td><td>...</td><td>...</td></tr><tr><td>GJ 3631AB</td><td>19.10</td><td>0.49–0.16+0.18</td><td>-0.64</td><td>-0.268</td><td>-19.29</td><td>26.51</td><td>26.51</td></tr><tr><td>GJ 3707</td><td>2.0</td><td>7.36–3.53+6.55</td><td>0.35</td><td>+0.047</td><td>-0.341</td><td>0.531</td><td>0.477</td></tr><tr><td>GJ 3789</td><td>56.50</td><td>0.227–0.07+0.07</td><td>0.07</td><td>-0.033</td><td>-19.31</td><td>28.02</td><td>23.85</td></tr><tr><td>GJ 3801</td><td>4.00</td><td>3.67–0.88+1.20</td><td>0.14</td><td>+0.047</td><td>...</td><td>...</td><td>...</td></tr><tr><td>GJ 3804</td><td>2.50</td><td>6.64–2.04+3.32</td><td>0.09</td><td>+0.055</td><td>-0.299</td><td>0.582</td><td>0.513</td></tr><tr><td>GJ 3873</td><td>4.00</td><td>6.90–2.89+3.91</td><td>0.11</td><td>+0.245</td><td>-0.465</td><td>0.693</td><td>0.538</td></tr><tr><td>GJ 3900</td><td>3.00</td><td>5.15–2.95+4.43</td><td>0.46</td><td>+0.050</td><td>-0.453</td><td>0.698</td><td>0.622</td></tr><tr><td>GJ 3907</td><td>2.50</td><td>5.39–1.91+3.11</td><td>0.06</td><td>+0.006</td><td>...</td><td>...</td><td>...</td></tr><tr><td>GJ 4020B</td><td>16.16</td><td>1.10–0.24+0.27</td><td>...</td><td>+0.060</td><td>-12.94</td><td>17.38</td><td>15.14</td></tr><tr><td>GJ 4030</td><td>4.50</td><td>3.90–1.14+1.49</td><td>-0.04</td><td>+0.059</td><td>...</td><td>...</td><td>...</td></tr><tr><td>GJ 4049B</td><td>2.50</td><td>3.81–1.16+1.89</td><td>-0.13</td><td>-0.255</td><td>...</td><td>...</td><td>...</td></tr><tr><td>GJ 4063AB</td><td>2.17</td><td>4.99–2.93+5.16</td><td>-0.54</td><td>-0.169</td><td>-0.533</td><td>0.918</td><td>3.183</td></tr><tr><td>GJ 4108</td><td>4.50</td><td>2.04–0.60+0.78</td><td>...</td><td>-0.278</td><td>...</td><td>...</td><td>...</td></tr><tr><td>GJ 4248</td><td>1.58</td><td>7.75–2.71+6.04</td><td>...</td><td>-0.066</td><td>...</td><td>...</td><td>...</td></tr><tr><td>GJ 4333</td><td>2.50</td><td>8.64–2.51+4.09</td><td>0.25</td><td>+0.123</td><td>-0.574</td><td>0.871</td><td>0.656</td></tr><tr><td>GJ 4338B</td><td>14.50</td><td>1.53–0.29+0.33</td><td>...</td><td>+0.134</td><td>-10.95</td><td>15.05</td><td>11.05</td></tr><tr><td>GJ 4378A</td><td>4.00</td><td>4.88–1.69+2.29</td><td>...</td><td>+0.081</td><td>-2.839</td><td>4.501</td><td>3.735</td></tr><tr><td>Gl 15B</td><td>1.90</td><td>5.01–1.57+3.01</td><td>-0.95</td><td>-0.255</td><td>-0.921</td><td>1.390</td><td>2.500</td></tr><tr><td>Gl 46</td><td>1.29</td><td>16.32–6.60+18.1</td><td>0.15</td><td>+0.112</td><td>-0.295</td><td>0.554</td><td>0.392</td></tr><tr><td>Gl 54.1</td><td>2.50</td><td>3.00–0.89+1.45</td><td>-0.33</td><td>-0.402</td><td>-6.318</td><td>7.039</td><td>15.05</td></tr><tr><td>Gl 84.1B</td><td>2.00</td><td>12.45–4.27+7.94</td><td>-0.18</td><td>+0.190</td><td>-0.625</td><td>1.217</td><td>0.786</td></tr><tr><td>Gl 105B</td><td>2.45</td><td>6.61–1.95+3.21</td><td>-0.09</td><td>+0.053</td><td>-0.330</td><td>0.465</td><td>0.572</td></tr><tr><td>Gl 166C</td><td>2.90</td><td>4.47–1.17+1.78</td><td>-0.10</td><td>-0.024</td><td>-4.921</td><td>6.607</td><td>6.982</td></tr><tr><td>GL 169.1A</td><td>1.94</td><td>7.80–2.44+4.62</td><td>0.29</td><td>+0.049</td><td>...</td><td>...</td><td>...</td></tr><tr><td>GL 179</td><td>2.50</td><td>7.53–2.35+3.84</td><td>0.22</td><td>+0.065</td><td>-0.695</td><td>1.372</td><td>0.827</td></tr><tr><td>GL 203</td><td>4.00</td><td>2.67–0.68+0.92</td><td>-0.23</td><td>-0.179</td><td>-0.121</td><td>0.239</td><td>0.406</td></tr><tr><td>Gl 206A</td><td>6.37</td><td>1.95–0.62+0.84</td><td>0.19</td><td>-0.054</td><td>-8.154</td><td>15.21</td><td>17.22</td></tr><tr><td>Gl 206B</td><td>6.11</td><td>2.04–0.65+0.90</td><td>0.19</td><td>-0.054</td><td>-4.922</td><td>9.179</td><td>10.39</td></tr><tr><td>GL 213</td><td>2.70</td><td>4.76–1.29+2.03</td><td>-0.16</td><td>-0.030</td><td>-0.311</td><td>0.439</td><td>0.635</td></tr><tr><td>GL 232</td><td>3.10</td><td>3.05–0.80+1.18</td><td>-0.25</td><td>-0.259</td><td>-0.36</td><td>0.484</td><td>0.879</td></tr><tr><td>GL 234A</td><td>4.73</td><td>2.72–0.57+0.74</td><td>...</td><td>-0.030</td><td>-4.830</td><td>5.490</td><td>5.883</td></tr><tr><td>GL 268A</td><td>10.65</td><td>1.57–0.32+0.38</td><td>0.16</td><td>+0.056</td><td>-5.95</td><td>6.736</td><td>5.921</td></tr><tr><td>GL 268B</td><td>10.65</td><td>1.57–0.32+0.38</td><td>0.16</td><td>+0.056</td><td>-3.37</td><td>3.815</td><td>3.353</td></tr><tr><td>GL 273</td><td>2.36</td><td>6.99–1.91+3.22</td><td>-0.08</td><td>+0.055</td><td>-0.432</td><td>0.750</td><td>0.902</td></tr><tr><td>Gl 277B</td><td>6.40</td><td>3.39–0.87+1.19</td><td>0.10</td><td>+0.125</td><td>-6.040</td><td>9.867</td><td>7.399</td></tr><tr><td>GL 285</td><td>6.16</td><td>2.95–0.71+0.98</td><td>0.24</td><td>+0.062</td><td>-16.04</td><td>18.45</td><td>10.62</td></tr><tr><td>GL 299</td><td>2.96</td><td>3.30–0.90+1.35</td><td>-0.43</td><td>-0.239</td><td>-0.375</td><td>0.416</td><td>1.120</td></tr><tr><td>GL 300</td><td>3.00</td><td>5.16–1.34+2.01</td><td>0.20</td><td>+0.050</td><td>-0.654</td><td>0.781</td><td>0.696</td></tr><tr><td>GL 317</td><td>2.50</td><td>9.37–2.56+4.18</td><td>-0.10</td><td>+0.160</td><td>-0.641</td><td>1.248</td><td>0.863</td></tr><tr><td>GL 319B</td><td>3.70</td><td>7.17–1.82+2.53</td><td>0.04</td><td>+0.223</td><td>-0.236</td><td>0.422</td><td>0.385</td></tr><tr><td>GL 324B</td><td>2.36</td><td>6.11–1.90+3.20</td><td>0.42</td><td>+0.039</td><td>-0.73</td><td>0.961</td><td>0.878</td></tr><tr><td>GL 375A</td><td>10.0</td><td>2.35–0.48+0.58</td><td>...</td><td>+0.161</td><td>-8.709</td><td>16.71</td><td>11.53</td></tr><tr><td>GL 375B</td><td>10.0</td><td>2.35–0.48+0.58</td><td>...</td><td>+0.161</td><td>-7.965</td><td>15.29</td><td>10.55</td></tr><tr><td>GL 402</td><td>2.40</td><td>5.74–1.65+2.75</td><td>0.12</td><td>+0.019</td><td>-0.690</td><td>1.035</td><td>0.991</td></tr><tr><td>Gl 431</td><td>20.36</td><td>0.960–0.15+0.17</td><td>...</td><td>+0.081</td><td>-8.790</td><td>17.47</td><td>14.50</td></tr><tr><td>Gl 445</td><td>2.25</td><td>6.48–1.90+3.28</td><td>-0.30</td><td>+0.044</td><td>-0.183</td><td>0.332</td><td>0.662</td></tr></tbody></table>
<table><caption>**Table 7**
(Continued)</caption><thead><tr><th>Star</th><th>v sin i<br>(km s<sup>-1</sup>)</th><th>P/ sin i<br>(days)</th><th>[M/H]<br>(dex)</th><th>[M/H]<sup>a</sup><br>(dex)</th><th>Ca II EW<br>(Å)</th><th>F<sub>HK</sub><br>(10<sup>4</sup> erg s<sup>-1</sup> cm<sup>-2</sup>)</th><th>F<sub>HK</sub><sup>b</sup><br>(10<sup>4</sup> erg s<sup>-1</sup> cm<sup>-2</sup>)</th></tr></thead><tbody><tr><td>Gl 447</td><td>2.20</td><td>4.51–1.33+2.32</td><td>-0.04</td><td>-0.230</td><td>-0.444</td><td>0.535</td><td>0.909</td></tr><tr><td>GL 458BC</td><td>5.40</td><td>1.74–0.45+0.65</td><td>-0.53</td><td>-0.262</td><td>...</td><td>...</td><td>...</td></tr><tr><td>Gl 469</td><td>1.58</td><td>12.81–4.94+11.0</td><td>0.14</td><td>+0.095</td><td>-0.47</td><td>0.854</td><td>0.619</td></tr><tr><td>GL 486</td><td>2.25</td><td>7.99–2.37+4.10</td><td>-0.05</td><td>+0.061</td><td>-0.242</td><td>0.413</td><td>0.463</td></tr><tr><td>Gl 487A</td><td>1.65</td><td>8.74–3.05+6.54</td><td>-0.04</td><td>+0.039</td><td>-1.125</td><td>2.335</td><td>2.134</td></tr><tr><td>Gl 487B</td><td>1.22</td><td>11.87–4.79+14.05</td><td>-0.04</td><td>+0.039</td><td>-1.125</td><td>2.335</td><td>2.134</td></tr><tr><td>Gl 487C</td><td>2.60</td><td>5.55–1.55+2.47</td><td>-0.04</td><td>+0.039</td><td>-1.125</td><td>2.335</td><td>2.134</td></tr><tr><td>GL 490B</td><td>8.60</td><td>2.27–0.56+0.71</td><td>0.07</td><td>+0.081</td><td>-7.35</td><td>10.66</td><td>9.073</td></tr><tr><td>GL 512B</td><td>4.00</td><td>2.63–0.94+1.27</td><td>0.12</td><td>-0.190</td><td>-0.470</td><td>0.602</td><td>0.932</td></tr><tr><td>GL 520C</td><td>8.20</td><td>1.50–0.35+0.45</td><td>-0.14</td><td>-0.063</td><td>-7.65</td><td>11.44</td><td>15.79</td></tr><tr><td>GL 544B</td><td>4.66</td><td>2.44–0.80+1.03</td><td>-0.13</td><td>-0.129</td><td>...</td><td>...</td><td>...</td></tr><tr><td>GL 545</td><td>4.00</td><td>3.07–1.08+1.46</td><td>-0.12</td><td>-0.063</td><td>...</td><td>...</td><td>...</td></tr><tr><td>Gl 553.1</td><td>2.50</td><td>6.68–2.04+3.33</td><td>0.26</td><td>+0.056</td><td>-0.186</td><td>0.371</td><td>0.326</td></tr><tr><td>GL 555</td><td>2.60</td><td>5.74–1.58+2.53</td><td>0.20</td><td>+0.048</td><td>-0.458</td><td>0.644</td><td>0.577</td></tr><tr><td>GL 568AB</td><td>4.00</td><td>4.78–1.18+1.60</td><td>0.45</td><td>+0.073</td><td>...</td><td>...</td><td>...</td></tr><tr><td>GL 592</td><td>3.00</td><td>5.36–1.53+2.29</td><td>0.14</td><td>+0.053</td><td>-0.212</td><td>0.336</td><td>0.297</td></tr><tr><td>GL 609</td><td>3.00</td><td>4.57–1.25+1.88</td><td>-0.06</td><td>+0.017</td><td>-0.401</td><td>0.582</td><td>0.668</td></tr><tr><td>GL 630.1AB</td><td>27.50</td><td>0.484–0.09+0.10</td><td>-0.34</td><td>-0.003</td><td>-8.51</td><td>10.09</td><td>22.07</td></tr><tr><td>GL 643AB</td><td>2.73</td><td>4.45–1.26+1.97</td><td>-0.21</td><td>-0.074</td><td>-0.197</td><td>0.326</td><td>0.529</td></tr><tr><td>GL 644B</td><td>2.73</td><td>4.10–1.18+1.84</td><td>-0.21</td><td>-0.144</td><td>-0.316</td><td>0.555</td><td>0.900</td></tr><tr><td>Gl 669A</td><td>6.30</td><td>3.84–1.01+1.39</td><td>0.08</td><td>+0.176</td><td>-5.277</td><td>9.231</td><td>6.155</td></tr><tr><td>GL 682</td><td>3.00</td><td>4.80–1.22+1.83</td><td>0.11</td><td>+0.039</td><td>-0.476</td><td>0.701</td><td>0.641</td></tr><tr><td>GL 695B</td><td>7.33</td><td>3.03–0.72+0.95</td><td>...</td><td>+0.136</td><td>...</td><td>...</td><td>...</td></tr><tr><td>Gl 699</td><td>2.02</td><td>4.99–1.49+2.75</td><td>-0.40</td><td>-0.220</td><td>-0.363</td><td>0.532</td><td>0.883</td></tr><tr><td>GL 720B</td><td>2.30</td><td>7.02–2.06+3.52</td><td>-0.16</td><td>+0.053</td><td>-0.29</td><td>0.478</td><td>0.691</td></tr><tr><td>GL 725B</td><td>5.09</td><td>4.33–1.42+2.12</td><td>-0.57</td><td>-0.155</td><td>-0.298</td><td>0.561</td><td>2.084</td></tr><tr><td>GL 729</td><td>4.72</td><td>2.11–0.41+0.53</td><td>-0.14</td><td>-0.227</td><td>-5.570</td><td>9.099</td><td>12.56</td></tr><tr><td>GL 732A</td><td>...</td><td>...</td><td>-0.01</td><td>+0.056</td><td>-3.462</td><td>5.489</td><td>5.617</td></tr><tr><td>GL 781.1B</td><td>9.20</td><td>2.17–0.56+0.70</td><td>-0.07</td><td>+0.090</td><td>-4.466</td><td>6.700</td><td>7.872</td></tr><tr><td>GL 783.2B</td><td>1.76</td><td>8.14–3.48+7.09</td><td>-0.15</td><td>+0.036</td><td>...</td><td>...</td><td>...</td></tr><tr><td>GL 791.2A</td><td>31.90</td><td>0.351–0.05+0.05</td><td>0.06</td><td>-0.144</td><td>-7.775</td><td>9.500</td><td>13.23</td></tr><tr><td>GL 812A</td><td>10.00</td><td>2.95–0.71+0.87</td><td>0.12</td><td>+0.284</td><td>-2.814</td><td>4.922</td><td>3.734</td></tr><tr><td>GL 860B</td><td>4.05</td><td>3.25–0.71+0.96</td><td>-0.13</td><td>-0.012</td><td>-2.055</td><td>4.084</td><td>4.198</td></tr><tr><td>Gl 865A</td><td>7.09</td><td>1.90–0.53+0.71</td><td>-0.10</td><td>+0.006</td><td>-8.70</td><td>15.22</td><td>15.01</td></tr><tr><td>Gl 865B</td><td>6.20</td><td>2.17–0.64+0.88</td><td>-0.10</td><td>+0.006</td><td>-4.88</td><td>8.536</td><td>8.419</td></tr><tr><td>Gl 867B</td><td>7.01</td><td>2.48–0.54+0.72</td><td>0.15</td><td>+0.059</td><td>-7.722</td><td>14.40</td><td>10.19</td></tr><tr><td>GL 873A</td><td>5.99</td><td>3.02–0.70+0.98</td><td>-0.05</td><td>+0.062</td><td>-9.347</td><td>16.73</td><td>14.50</td></tr><tr><td>GL 876A</td><td>2.57</td><td>7.09–1.95+3.13</td><td>...</td><td>+0.062</td><td>-0.473</td><td>0.779</td><td>0.675</td></tr><tr><td>GL 896A</td><td>15.81</td><td>1.14–0.18+0.21</td><td>0.10</td><td>+0.062</td><td>-9.470</td><td>13.02</td><td>11.29</td></tr><tr><td>GL 896B</td><td>15.81</td><td>1.14–0.18+0.21</td><td>0.10</td><td>+0.062</td><td>-9.156</td><td>12.59</td><td>10.92</td></tr><tr><td>LHS 2795</td><td>6.40</td><td>(2.38)</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>LHS 3279</td><td>30.0</td><td>(0.508)</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>G 97-52B</td><td>4.00</td><td>(3.81)</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr></tbody></table>

**Notes.** We also give the Ca II EW (from the compilation of Table 6), surface fluxes $F_{HK}$, and the surface fluxes corrected for the metallicity effects.

a Metallicity from the radius–metallicity relation.

b Surface flux corrected for metallicity effects.

(This table is available in machine-readable form.)

in our combined dM4+dM4e sample:

$$F_{\text{Ca II}} = 8.38 \pm 2.05 \times 10^5 \times (P/\sin i)^{-2.564 \pm 0.19}. \quad (24)$$

The gradient in Equation (24) (−2.564 ± 0.19) is significantly steeper than the gradient that we determined for our dK6 (−1.047 ± 0.042: Table 1) and dM2 stars (−1.575 ± 0.058: Table 1), and is also steeper than the gradient for our dM3 stars (−2.020 ± 0.11: Table 1). This confirms that the slope of the linear LSFs increases in absolute value when the spectral type increases from dK4 to dM4. This increase is particularly marked at the TTCC (M3) and beyond.

For $P/\sin i \ge 1.8$ days we obtain the following homogeneous linear LSF for the stars in our combined dM4+dM4e

sample:

$$F_{\text{Ca II}} = 7.94 \pm 2.05 \times 10^5 \times (P/\sin i)^{-2.526 \pm 0.13}. \quad (25)$$

The gradient in Equation (25) (−2.526 ± 0.18) is very close to that determined for the heteroscedastic linear fit (−2.564 ± 0.19: Table 1). As a whole, we find comparable results from the heteroscedastic and homoscedastic linear fits, suggesting that the errors on the measures do not play a large role in the determination of the linear fits (Table 1).

In Figure 12 we observe a flattening (“saturation”) for the fastest rotating dM4e stars for $P/\sin i < 1.6$ days. This is a range in periods where saturation is expected to occur (see Section 1.4). Although our data set is not yet complete enough
<figure><img src="image_15.png" /><figcaption>Figure 13. Values of the slopes of the heteroscedastic linear LSFs (solid lines) and the homoscedastic linear LSFs (dashed lines) to the RACs as a function of the infra-red color (R-I)C. We indicate the fits to the combined samples of high +low-activity stars (dK+dKe, dM+dMe) in each spectral sub-type (labelled "Low+High Activity"). The lower curves are the fits to low-activity stars only (dK, dM, labelled "Low Activity"). We also indicate the fits to the high-activity star sub-samples in the unsaturated regime (dKe, dMe, labelled "High Activity"). The slopes for the combined samples increase significantly from dK4 to dM4, with only minor differences between heteroscedastic and homoscedastic fits. For the low-activity stars, as a whole, the slopes for the heteroscedastic and homoscedastic models agree well and the results indicate that the slopes remain almost constant within the error bars from dK4 to dM4. The magnitudes of the slopes for the (unsaturated) high-activity stars lie significantly above those for the low-activity stars at spectral types dK4, dK6, and dM2. At the TTCC (M3) and beyond (M4), the slopes for the (unsaturated) high-activity stars significantly diminish.</figcaption></figure>

to be conclusive of a saturation phenomenon, we assume here that, for P/sin i shorter than ~2 days, saturation occurs. In this period range, we observe 12 fast rotators with a mean surface flux of ~1.54 x 10⁵ erg s⁻¹ cm⁻².

We also performed separate linear LSFs to the sub-samples of only the low-activity dM4 and the sub-set of high-activity dM4e stars (all of which we assume are in the unsaturated regime), respectively (see Table 1). The heteroscedastic linear LSF to the dM4 low-activity stars yields:

$$F_{Ca\ II} = 3.08 \pm 1.25 \times 10^4 \times (P/\sin i)^{-0.825 \pm 0.35}. \quad (26)$$

Therefore, the gradient of the RAC for the low-activity stars is again much shallower than that of the combined sample of low and high-activity stars (unsaturated), and by more than 3σ. Therefore, the linear LSF to the full sample is somehow inadequate. We note also that the slopes of the heteroscedastic linear LSF to the sub-samples of only the low-activity stars remain approximately constant (within measurement errors) from dK4 to dM4. This point will be further developed in Section 3.7.

The homoscedastic linear LSF to the dM4 low-activity stars yields:

$$F_{Ca\ II} = 3.80 \pm 1.89 \times 10^4 \times (P/\sin i)^{-0.91 \pm 0.39}. \quad (27)$$

Again, the gradient of the RAC for the low-activity stars is shallower than that of the combined sample of low and high-activity stars, and by more than 3σ. We find that for the

low-activity dM4 stars, both the homoscedastic and heteroscedastic models give similar results for the mean value of the slope of the RAC. And this mean value (0.825–0.91) is remarkably similar (within 1σ) to the mean values for low-activity stars in the other four spectral sub-types: 0.624, 0.637, 0.891, 0.93. However, there is one aspect in which the dM4 slow rotators differ from the other four sub-types: the dM4 stars have significantly larger values of σ associated with the slope of the RAC in both the homo- and heteroscedastic LSFs. We will return to this topic in Section 3.7.2.

We find that the χ² is much poorer for the heteroscedastic model (1.966) than that for the homoscedastic LSF (0.057). Nevertheless, both fits are statistically highly significant (better than 96%).

The heteroscedastic linear LSF to the 23 unsaturated dM4e high-activity stars yields:

$$F_{Ca\ II} = 2.27 \pm 0.58 \times 10^5 \times (P/\sin i)^{-0.951 \pm 0.31}. \quad (28)$$

Therefore, the gradient of the RAC for the high-activity stars is also much shallower than that of the combined sample of low and high-activity stars, and by more than 3σ. Therefore, the linear LSF to the full sample seems rather inadequate. This point is straightforward if one looks at the linear heteroscedastic LSFs in Figure 12. It is obvious that the LSFs to the low-activity stars, and to the high-activity stars cannot be reproduced by a simple linear or quadratic function. We note that the dM4e data confirm the decrease in the slope of the LSF fits to unsaturated stars as we pass through the TTCC. In addition, we also performed a linear LSF to the full sample of dM4e stars (unsaturated+saturated) in order to compare with the previous fit. We find a slope of −0.47 for the full sub-sample of dM4e stars (Table 1). This result again suggests that the slopes for the sub-samples of low-activity stars and of high-activity stars are of the same order after the TTCC. The parameter R/δ is 18.90 for the high-activity star sub-sample and 19.13 for the low-activity star sub-sample. These figures are still very high, although that of the low-activity sub-sample has decreased compares to other spectral types. Therefore, the parameter δ = (error)/√n is still much smaller than the period domains of the RACs in both cases. As a consequence, these LSFs should be established with a good confidence level, which is in agreement with the high statistical significances we obtain for these fits (Table 1).

We find that the linear fits (to the unsaturated dM4+dM4e samples) are incapable of reproducing the slopes among the low-activity stars and the (unsaturated) high-activity stars. We face the same problem as we did in dealing with the case of dM3 stars: the problem is that there exist high activity, relatively slow rotators and low-activity, relatively fast rotators. These two types of stars overlap in their range of values of P/sin i.

The dM3 and especially the dM4 data sets lead us to question the validity of fitting both the low-activity and high-activity (unsaturated) sub-samples with a single RAC. Instead, it seems to us that it might be better to consider the possibility that two distinct RACs are present in the data. To the extent that the RACs are determined with dynamo operation, this leads us to wonder: is there perhaps a distinct dynamo process operating in low-activity dM4 stars from the dynamo process that is at work in (unsaturated) high-activity dM4e stars? The slopes of the linear LSF to the low-activity stars for our five stellar samples are rather homogeneous, even for the dM4 stars,
although the latter stars are almost certainly fully convective. This result suggests that there may exist a single dynamo regime for the slow rotators from dK4 to dM4 within errors (see Section 3.7.2). This important result should be tested with larger samples of low-activity stars.

There exist more and more unsaturated high-activity- relatively slow rotators when we move to the spectral sub- types dM3 and especially dM4. These relatively slow rotators overlap in P/sin i with the low-activity stars. This poses an important problem: we cannot fit both the low-activity and (unsaturated) high-activity samples satisfactorily with a single RAC. We suggest that the dynamo mechanisms behave differently among the high-activity stars when crossing the TTCC. At the TTCC (dM3) and beyond, the data suggest that there exist two different dynamo regimes for low and high- activity stars (see Section 3.7). These differences in these dynamo regimes could be related to differential rotation: in the young high-activity stars differential rotation is expected to be larger and it should vanish with age, i.e., in the low-activity stars. Large differential rotation would boost the magnetic field generation in these fully convective stars, such that we could have high activity, relatively slow rotators with large differ- ential rotation.

Now that we have assembled enough data to construct RACs at five different spectral sub-types for late K and M dwarfs, it is worthwhile to examine the systematics of RACs as a function of spectral sub-type.

3.7. *The Slopes of the RACs: Variation with Spectral Sub-type*

Now that we have values of the RAC slopes for five different spectral sub-types, we provide an overview by plotting the slopes as a function of the infra-red color index $(R-I)_C$ in Figure 13. In this figure, results are plotted separately for three distinct samples of stars: the low-activity stars only (lower curves), the (unsaturated) high-activity stars only (labelled “High Activity”), and the combined samples low+high-activity stars (labelled “Low+High Activity”). In this figure, the heteroscedastic models are plotted as solid lines, and the homoscedastic models are plotted as the dashed lines.

3.7.1. *Linear LSFs: Combined Samples of Low Plus High Activity Stars*

In Figure 13, the upper curves refer to the combined samples: the plotted points indicate the mean values of the RAC gradients that we have obtained for each of our five sub- samples by means of linear heteroscedastic (solid line) and homoscedastic (dashed line) LSFs. Attached to each mean value of the gradient is plotted the $1\sigma$ uncertainty in the mean value of the gradient.

We see that the value of the mean gradient becomes monotonically steeper as $(R-I)_C$ increases. The heteroscedastic (solid line) and homoscedastic (dashed line) models give similar results. There are some significant differences (above the $3\sigma$ level) between the two models for the dK6 sample, but this does not affect the overall trend observed for the gradient.

The fact that the magnitude of the gradient increases monotonically with increasing spectral type implies that the magnitude of the effects of rotation on the mechanisms of the dynamo increases systematically as we examine our combined samples at later spectral types. If this result is connected with the physics of the dynamo, and if it is physically permissible to

consider low-activity stars in the same context as (unsaturated) high-activity stars, then the results in Figure 13 may provide an important new constraint on the dynamo mechanism(s). The magnitude of the gradient increases monotonically as we go from the spectral sub-type dK4 to the spectral sub-type dM4. The results in Figure 13 suggest that, if the TTCC occurs at dM3, the effects of rotation on chromospheric emission are larger in fully convective stars than in early M type and late K dwarfs where a radiative core persists. If this result is taken at face value, it does not appear to be consistent with the suggestions of Durney et al. (1993), mentioned in the first paragraph of Introduction.

Previous studies have also reported a variation in the RAC gradient of linear fits (e.g., Stepień 1989, 1993, 1994) as a function of spectral type: the RAC was found to be steepest (i.e., most negative) at dF6 ($B-V = 0.45$) and shallowest close to dK6 ($B-V = 1.41$; KH). However, the most complete study (Stepień 1989) did not include any stars that are cooler than dK6. Thus, the study we report here is complementary to Stepień's work, extending that work toward cooler stars as far as dM4 (($B-V = 1.60$)). However, our results indicate that the trend noted by Stepień between dF6 and dK6 (namely, the RAC becomes shallower in cool stars) does not continue at later spectral types. On the contrary, we find that the trend in the gradient reverses: according to our samples, the RAC for the combined samples becomes steeper as we go toward dM4.

It is natural to wonder if the pronounced steepening of the RAC gradient is a signature of changes in the dynamo mechanism when crossing the TTCC. As noted in the Introduction, it appears that the TTCC lies between subtypes dM2 and dM4. In this context, we consider it significant that, among the stars in our samples, the RAC gradient for the combined samples steepens rapidly between spectral subtypes dM2 and dM4. Moreover, HM report a notable lengthening in the mean rotation periods of inactive and active stars also at spectral subtype M3. Also at M3, coronal loops in active stars become noticeably longer (see HM for a discussion of loop length data reported by Mullan et al. 2006).

Our data lead us to believe that something interesting takes place in the spectral subtype range dM2–dM4 as regards the rotational and activity parameters. Is the interesting behavior between dM2 and dM4 perhaps associated with changes in the dynamo regime when crossing the TTCC? Possibly, although we have not yet been successful in identifying a physical explanation that accounts for the results reported above. However, whatever the physical mechanism that is responsible for the increased steepnesses in Figure 13, the rise in the steepness of the gradient for the combined sample from sub- type K4 to sub-type M2 indicates that the changes occurring in the RAC slopes (and therefore perhaps also in the dynamo mechanism(s)) may be progressive and may begin even earlier than sub-type dM2.

3.7.2. *Linear LSFs: Low Activity Stars Only*

Now let us consider only the low-activity stars, in order to position ourselves as much as possible in the unsaturated regime of the CaII fluxes, and therefore (hopefully) also in the unsaturated regime of the dynamo. The continuous line and dashed line in Figure 13 (lower curves) are a plot of the slopes of the heteroscedastic and homoscedastic linear LSFs respectively to the RACs for the low-activity stars sub-samples only. The parameter $R/\delta$ lies between 19.13 and 39.68 for these low-activity
star sub-samples. Therefore, the parameter $\delta = \frac{\langle\text{error}\rangle}{\sqrt{n}}$, which is a normalized estimate of the mean error on the measurements, is much smaller than the period domains of the RACs in all cases. Hence, these LSFs should be established to a relatively high confidence level, which agrees with the high statistical significances we obtain for these fits (Table 1).

In this regard, we note that the process of applying a LSF to a data set leads not only to a mean value for the slope: the LSF also yields a standard deviation $\sigma$ for the slope. We consider it worthwhile to note that the values of $\sigma$ contain information about the robustness of the slope.

Figure 13 shows that the slopes for the slow rotators (dK, dM only) do not behave the same as for the combined samples of stars (dK+dKe or dM+dMe). The gradient $a$ for the low-activity stars remains almost constant (within the error bars) between dK4 and dM4 at a level of $a = -(0.8–0.9)$. More specifically, if we consider not merely the absolute value of $a$, but also the statistical significance of its value compared to its own $\sigma$, we find the following pattern: in dK4 stars, the magnitude of $a$ is 5.6 times its own $\sigma$, in dK6 stars, $a$ is 6.1 times its $\sigma$, in dM2 stars, $a$ is 7.4 times its $\sigma$, and in dM3 stars, $a$ is 4.3 times its $\sigma$. All four of these cases have slopes that are statistically highly significant. The chromospheric emission in these cases does indeed depend sensitively on rotation. But in M4 stars, $a$ has a value that is only 2.3 times its own $\sigma$ for the homoscedastic model and 2.4 times its own $\sigma$ for the heteroscedastic model. In this case, at the $3\sigma$ level of significance, the mean slope that we have derived for the RAC of our sample of low-activity dM4 stars is formally consistent with zero. In a situation where the slope of the RAC is zero (or formally consistent with zero), the meaning is that the chromospheric emission in low-activity dM4 stars does not depend on rotation at all (or is consistent with zero sensitivity to rotation).

Although our LSFs of the slow rotators seem well established because of the large number of measures we have in our sub-samples, we still consider these results as preliminary and they should be confirmed with measures from a higher resolution spectrograph (such as ESO-ESPRESSO), because our errors on the individual measures are still large compared to the period domains of the RACs.

### 3.7.3. Linear LSFs: High Activity Stars Only

We also show the slopes of the heteroscedastic and homoscedastic LSFs to the sub-samples of only the (unsaturated) high-activity stars in Figure 13. The parameter $R/\delta$ lies between 17.20 to 23.45 for these high-activity star sub-samples (the case of dK4e stars is not considered because we have too few measures). Therefore, the parameter $\delta = \frac{\langle\text{error}\rangle}{\sqrt{n}}$, which is the normalized estimate of the mean error on the measurements, is much smaller than the period domains of the RACs in all cases. Hence, these LSFs should be established with a rather good confidence level, which agrees with the high statistical significances we obtain for these correlations (≥90%, Table 1). However, we consider these results as still preliminary because the domains of the RACs are relatively small compared to the typical uncertainties on individual measures. We note that these slopes behave quite differently from the slopes of the low-activity stars and the slopes of the combined samples of low+high-activity stars. For the high-activity stars, we observe that the slopes are clearly steeper than those for the low-activity

<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /><figcaption>**Figure 14.** Upper panel: RACs for the logarithm of the Ca II resonance line surface fluxes and $\log(P/\sin i)$ for combined samples of low+high-activity stars in each sub-type dK4, dK6, dM2, dM3, and dM4. Note the major differences that exist between different spectral sub-types as regards the surface fluxes of the Ca II lines and the rotation periods. Lower panel: RACs for the same five spectral sub-types, but for the sub-samples of low-activity stars only (see the text).</figcaption></figure>

stars at spectral types dK6 and dM2. At these spectral types, they are also steeper than the slopes of the combined samples of low+high-activity stars. These higher slopes suggest the existence of two different dynamo regimes for the low and the high-activity sub-samples for these stars.

However, things change at the TTCC. At the TTCC (dM3), the slopes for the high-activity stars are found to diminish dramatically, by a factor of about 2. Although there remains a large uncertainty on this slope for dM3e stars, the fall of the slopes at the TTCC is confirmed by the measure of the slope for dM4e stars with a much higher confidence level. Beyond the
TTCC (dM4), the slope continues to diminish. In fact, the slope at dM4 reaches a value that is very similar to the slope we have obtained for the low-activity star sub-samples (although we emphasize that the overall shapes of the RACs are very different: see Figures 8 and 12). This dramatic decrease in steepness among (unsaturated) high-activity stars at M3 and M4 suggests that the dynamo mechanisms operating in dK4, dK6, and dM2 high-activity stars (perhaps an α-Ω dynamo?) may be different from those operating in the fully convective dM3 and dM4 high-activity stars (perhaps an α² dynamo?). In this regard, we detect no major changes in the slopes at the TTCC for the low-activity star sub-samples apart from a decreased statistical significance of the RAC slope. This point is further discussed in the next sub-section.

We emphasize that, although our LSFs to the high-activity stars seem relatively well established, in our sub-samples we have only few measures (from 8 to 23). Therefore, these results should be confirmed with larger stellar samples and are still preliminary.

### 3.7.4. Approaching the TTCC?

These results lead us to suggest a five-part conclusion.

(i) The gradients of the RACs that we have obtained in our combined samples of low+high-activity stars unambiguously increase with increasing spectral type with a high level of confidence. But it also appears that the linear LSFs only provide a poor description of the full RACs.

(ii) There appears to be a significant dichotomy between the low-activity and the (unsaturated) high-activity sub-samples. This leads us to consider that there may exist two distinct dynamo regimes in dKe, dMe and dK, dM stars.

(iii) The gradients of the combined samples suggest that there are changes operating in the dynamo mechanisms before, at, and after the TTCC. In addition, the gradients to the high-activity sub-samples point to important changes in the dynamo mechanisms occurring at the TTCC and beyond.

(iv) We can confidently assert that in low-activity dK4, dK6, dM2, and dM3 stars, there is a robust (>4σ) increase in Ca II flux as the period decreases. That is, the dynamo in unsaturated dK4–dM3 stars is clearly sensitive to rotation.

(v) But in low-activity M4 stars, the sensitivity of the dynamo to rotation, although still perhaps present, is not as robust, and may even be zero (at the 3σ level of significance). Thus, even in the low-activity stars in our samples, a rotational dynamo is certainly contributing significantly to chromospheric heating in dK4–dM3 stars. But in dM4 low-activity stars, the signs of rotational control over chromospheric emission are less significant (in a statistical sense).

These two last points, if they can be confirmed, might be relevant in the context of Durney et al. (1993), who suggested that rotation would be a controlling influence in the case of interface dynamos, but should not be as effective as a controlling influence in distributed dynamos. On the one hand, the robust sensitivity to rotation that we have found in dK4–dM3 low-activity stars could be evidence for interface dynamos in those stars. On the other hand, in dM4 low-activity stars, the weaker evidence for statistically significant rotational

sensitivity might presage the lessening of the effects of an interface dynamo. In this context, M3 might be considered to be the latest spectral sub-type to have definitive evidence for the presence of a radiative core (so that an interface dynamo is even possible). It is worth recalling that HM also concluded that at M3, something unusual happens to the rotational braking. Could the HM result, in combination with our results for the RAC slope, be a sign that the TTCC occurs between M3 and M4? This will be tested in a future study using an even finer grained spectral type sampling at about the TTCC using new data.

We suggest that in Figure 13, the striking difference between the combined sample (upwardly rising lines) and the low-activity stars only (lower lines) may be associated with the fact that the combined sample might contain two distinct regimes of dynamo operation (high and low), whereas the low-activity sample contains only a single regime (low). Suppose that the separation of dM4e stars from dM4 stars in Figure 12 is associated with two distinct dynamo regimes: if so, the attempt to fit a LSF to the combined sample of dM4+dM4e stars may involve an attempt to “force” two different types of dynamos into a single mold. Such an attempt would not be as physically meaningful as attempting to fit a truly homogeneous population (e.g., the low-activity stars alone). Analogous arguments can be made for dM3 stars (Figure 8) and dM2 stars (Figure 5). According to this argument, it may be more profitable to focus separately on the different curves of the low-activity stars and the high-activity stars respectively in Figure 13 in order to gain insight into dynamo theory.

We conclude that for the low-activity stars only, the gradient is definitely shallower than that for the samples of the low plus high-activity stars. But we also conclude that the gradient for the low-activity stars only does not vary significantly with spectral type. We also note that the differences between the slopes for the low-activity stars and for the low+high-activity stars increase drastically with increasing spectral type. This result emphasizes that the efficiency of the dynamo mechanisms increases drastically at short rotation periods, and also increases with decreasing stellar mass. We also conclude that there are no significant changes occurring in the efficiency of the dynamo mechanisms for low-activity stars as a function of stellar mass when moving from dK4 to dM4.

### 3.7.5. Different Dynamo Regimes Among Low Activity and High Activity Stars at the TTCC and Beyond?

In the plots that we have compiled of each RAC for our five spectral sub-types (Figures 2, 3, 5, 8, 12), we overplot the separate heteroscedastic linear LSF to the low-activity stars and to the high-activity stars for direct comparison with the heteroscedastic linear LSF to the low+high-activity samples. As we saw in the previous sections, the overplots show clearly that the linear LSF to the low+high-activity stars fails completely to reproduce the slopes of the linear LSF that we have derived both from the slow rotators and the fast (unsaturated) rotators. We show that the low+high-activity samples cannot be described by a single RAC.

In view of this, it seems to us that, to the extent that the RAC is determined by a dynamo mechanism, we need to invoke two different dynamo regimes. The first (we suggest) is at work among dK and dM stars (i.e., low-activity stars). This first regime seems rather constant from dK4 to dM4 and is therefore independent (to a first approximation) of the spectral type.
A second dynamo regime (we suggest) is at work among dKe and dMe stars. This second regime exhibits steeper RAC slopes for the spectral types dK4, dK6, and dM2, and shallower slopes for the dM3 and dM4 spectral types. The data suggest that the second regime deviates significantly from the first regime at the TTCC and beyond. The main problem is that, among the stars in our samples at spectral sub-types dM3 and dM4, there are interlopers, namely, a certain number of high-activity, relatively slow rotators and a certain number of low-activity relatively fast rotators that overlap in $P/\sin i$. These interlopers make it impossible to reproduce the entire data sets with a single RAC. Our results suggest that, whether we are considering low-activity stars alone, or the combined samples of low+high activity, the properties of the RACs undergo changes of some kind, especially at the TTCC and beyond. This conclusion is especially true for dM4 stars.

The findings we present here can be considered as providing evidence in favor of the following hypothesis: the dynamo mechanism in fully convective stars is different from the dynamo mechanisms in stars where a tachocline exists. We propose that differential rotation (probably important in the young dMe stars and less important in the older dM stars) plays an important role in defining the RACs among dM and dMe stars at the TTCC and beyond.

### 3.8. The RACs for Five Different Spectral Sub-types

We bring together the heteroscedastic linear fits to the low +high-activity RACs (in terms of surface fluxes) for our five spectral sub-types in Figure 14 (upper panel). In this section, we discuss two key aspects of the figure, and how they vary as a function of spectral type: (i) the absolute magnitudes of the Ca II flux and (ii) the slopes of the RAC in the unsaturated regime.

There are several striking trends in this figure. First, with increasing spectral type, the curves tend to shift downward and to the right, i.e., the overall level of chromospheric flux decreases, and the periods tend to be shorter. Important differences between the RACs of dK4, dM2, and dM4 stars have already been reported by Houdebine (2012b). Also, variations in the RAC properties at different spectral types was previously reported by Stepień (1993, 1994). He found that the $log(R'_{HK})-P_{rot}$ relationships globally shifted toward shorter rotation periods as (B-V) increases from 0.52 (dF7.5) to 1.15 (i.e., spectral type dK5, according to KH). On the other hand, in an earlier study, the opposite tendency emerged: Stepień (1989) found that the $log(\Delta F_{Ca II})-P_{rot}$ relationships were globally shifted toward longer rotation periods for (B-V) from 0.45 (dF6) to 0.66 (dG5) and that they stabilize for (B-V) from 0.75 (dG8.5) to 1.40 (dK6). Also, Patten & Simon (1996) proposed that the projected rotation period becomes longer as one move from spectral types G1 to M2. In view of this complicated behavior, it seems that in the $F_{Ca II}-P_{rot}$ RACs, first the rotation period becomes longer between dF6 and dG7, then remains unchanged between dG7 and dK6, and then the period starts to become shorter between dK6 and dM4; the latter behavior persists into later spectral types.

In Figure 14, in the context of the combined samples (upper panel), one can see that the steepness of the RACs (at least in the unsaturated regime) increases noticeably as we go toward later spectral type (as already shown in Figure 13). We have not detected any saturation behavior for stars in our dK4, dK6, or dM2 stars, whereas we may observe a saturation in dM3 and

dM4 stars for the most rapid rotators. Might this saturated-unsaturated behavior also point toward a different dynamo mechanism in dM3 and dM4 stars from the dynamo mechanism in dK4, dK6, and dM2 stars? We also observe that the largest Ca II surface fluxes are found in the stars with the earliest spectral types (dK4, dK6, dM2). For dM4 stars, there is a pronounced decrease in the absolute magnitudes of the surface fluxes for the fastest (unsaturated) rotators compared to dM2 stars. But even among relatively slow rotators (with periods longer than 10 days), where any effects of saturation are expected to be minimal, the Ca II surface fluxes are observed to decline by more than 100 between dK4 and dM4.

Surveying the results, we see that dK4 and dK6 stars have similar RACs, with a slightly larger gradient for the dK6 stars. Then the RAC falls to lower surface fluxes for dM2 stars and the RAC steepens. The RAC for dM3 stars is distinctly different in shape compared to that of the dM2 stars: dM3e stars exhibit saturation whereas the dM2e stars do not. Also, dM3 stars overlap in rotation periods with dM2 stars. This is due to the fact that we found (in HM) that dM3 stars possess abnormally long rotation periods for both high-activity and low-activity stars compared to the adjoining dM2 and dM4 stars. In HM, it was suggested that the anomaly of relatively long rotational periods at type M3 might be related to an empirical report (Mullan et al. 2006) that flaring loop lengths undergo an increase to larger values at dM3. In dM4 stars, the (unsaturated) RAC shape parallels that of dM3.

The trends in surface fluxes of Ca II in Figure 14 suggest that, whatever is controlling chromospheric emission in our sample stars, the mechanism is less effective (by factors of 10 or more), and leads to different RAC shapes, in dM3 and dM4 stars than in dM2, dK6, and dK4 stars.

As regards the latter stars, it might at first sight seem that we should be confident that interface dynamos are at work (e.g., Mullan et al. 2015). (Unfortunately, we cannot be absolutely confident that *only* an ID is at work in the dM2, dK6, and dK4 stars: the results of Brown et al. (2010) indicate that a DD can also operate in the convection zone of partially convective stars. If both ID and DD are in fact simultaneously operative in a certain star, current dynamo models give no quantitative results for the relative strengths of magnetic fields that would be generated by the different dynamos. Thus, although we use the words “fairly confident” in the present context, the question of the relative importance of ID and DD in a partially convective star remains ambiguous.

Could it be that the different RAC shapes in dM3 and dM4 stars might be due to the absence of an interface dynamo? If so, TTCC may be occurring between dM2 and dM3 sub-types. We also note that the changes in the RACs are progressive from dK4 to dM4, and that the fall in the fluxes begins at about dM2 and continues from dM3 to dM4. However, whatever the correct physical interpretation in terms of different dynamos eventually turns out to be, our data show that the efficiency of the dynamo mechanisms falls progressively from dK6 to dM4, and that this fall is particularly pronounced after the TTCC at spectral sub-type M4.

#### 3.8.1. Low Activity Stars Only

In the lower panel of Figure 14, we show results of the heteroscedastic models for low-activity stars only. The most striking difference between the lower and upper panels in
<table><caption>**Table 8**
Values of log(L<sub>HK</sub>), log(L<sub>bol</sub>), R'<sub>HK</sub>, and R<sub>0</sub> for Our Samples of dK4, dK6, dM2, dM3, and dM4 Stars</caption><thead><tr><th>Star Name</th><th>log(L<sub>HK</sub>) (erg s<sup>-1</sup>)</th><th>log(L<sub>HK</sub>)<sup>a</sup> (erg s<sup>-1</sup>)</th><th>log(L<sub>bol</sub>) (erg s<sup>-1</sup>)</th><th>R'<sub>HK</sub></th><th>R'<sub>HK</sub><sup>a</sup></th><th>log(L<sub>X</sub>) (erg s<sup>-1</sup>)</th><th>log(L<sub>X</sub>/L<sub>bol</sub>)</th><th>R<sub>0</sub></th></tr></thead><tbody><tr><td></td><td></td><td></td><td>K4</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>GJ 4061</td><td>28.062</td><td>...</td><td>32.840</td><td>-4.778</td><td>...</td><td>...</td><td>...</td><td>0.1801</td></tr><tr><td>GJ 4322B</td><td>27.739</td><td>...</td><td>32.993</td><td>-5.254</td><td>...</td><td>28.164</td><td>-4.829</td><td>0.7267</td></tr><tr><td>Gl 106</td><td>28.338</td><td>...</td><td>32.999</td><td>-4.661</td><td>...</td><td>28.384</td><td>-4.615</td><td>0.4069</td></tr><tr><td>Gl 131</td><td>28.322</td><td>...</td><td>32.661</td><td>-4.338</td><td>...</td><td>...</td><td>...</td><td>0.6554</td></tr><tr><td>Gl 160.2</td><td>27.948</td><td>...</td><td>32.638</td><td>-4.690</td><td>...</td><td>27.342</td><td>-5.296</td><td>0.5172</td></tr><tr><td>Gl 413</td><td>27.659</td><td>...</td><td>32.705</td><td>-5.046</td><td>...</td><td>...</td><td>...</td><td>1.0236</td></tr><tr><td>Gl 416</td><td>28.354</td><td>...</td><td>32.891</td><td>-4.537</td><td>...</td><td>...</td><td>...</td><td>0.7809</td></tr><tr><td>Gl 517</td><td>28.876</td><td>...</td><td>32.589</td><td>-3.713</td><td>...</td><td>29.554</td><td>-3.035</td><td>0.0252</td></tr><tr><td>Gl 570A</td><td>27.940</td><td>...</td><td>32.971</td><td>-5.031</td><td>...</td><td>27.544</td><td>-5.427</td><td>0.7265</td></tr><tr><td>Gl 664</td><td>28.201</td><td>...</td><td>32.777</td><td>-4.576</td><td>...</td><td>27.748</td><td>-5.029</td><td>0.2794</td></tr><tr><td>Gl 698A</td><td>28.398</td><td>...</td><td>32.819</td><td>-4.421</td><td>...</td><td>28.041</td><td>-4.778</td><td>...</td></tr><tr><td>Gl 707</td><td>28.354</td><td>...</td><td>32.665</td><td>-4.312</td><td>...</td><td>...</td><td>...</td><td>0.3142</td></tr><tr><td>Gl 719AB</td><td>29.338</td><td>...</td><td>32.792</td><td>-3.454</td><td>...</td><td>29.808</td><td>-2.984</td><td>0.0833</td></tr><tr><td>Gl 727</td><td>28.441</td><td>...</td><td>33.031</td><td>-4.590</td><td>...</td><td>28.320</td><td>-4.711</td><td>0.4797</td></tr><tr><td>Gl 775</td><td>28.362</td><td>...</td><td>32.996</td><td>-4.634</td><td>...</td><td>27.813</td><td>-5.183</td><td>0.6563</td></tr><tr><td>Gl 818</td><td>28.280</td><td>...</td><td>32.829</td><td>-4.549</td><td>...</td><td>...</td><td>...</td><td>0.6411</td></tr><tr><td>Gl 820A</td><td>28.135</td><td>...</td><td>32.780</td><td>-4.645</td><td>...</td><td>27.447</td><td>-5.333</td><td>0.6712</td></tr><tr><td>Gl 879</td><td>28.391</td><td>...</td><td>32.910</td><td>-4.519</td><td>...</td><td>28.326</td><td>-4.584</td><td>0.2816</td></tr><tr><td>Gl 898</td><td>28.502</td><td>...</td><td>32.642</td><td>-4.140</td><td>...</td><td>28.107</td><td>-4.535</td><td>0.1517</td></tr><tr><td>MCC 266</td><td>28.310</td><td>...</td><td>32.963</td><td>-4.653</td><td>...</td><td>...</td><td>...</td><td>0.6229</td></tr><tr><td>MCC 522</td><td>28.966</td><td>...</td><td>32.960</td><td>-3.994</td><td>...</td><td>...</td><td>...</td><td>0.1131</td></tr></tbody></table>

**Note.**

<sup>a</sup> Corrected from metallicity effects.

(This table is available in its entirety in machine-readable form.)

Figure 14 is that the low-activity stars do not exhibit any evidence for saturated behavior. Moreover, a striking characteristic of the lower panel is that the five RACs are almost all parallel to one another (as already noted in Figure 13). These features support our claim that low-activity stars may provide us with a possible “window” into a more homogeneous sample in terms of the dynamo mechanism: it seems likely that only unsaturated dynamo operation is at work in the lower panel of Figure 14.

We note that surface fluxes for unsaturated dM4 stars are again found to be 10–100 times smaller than those in unsaturated dK4 and dK6 stars. And we also observe a similar global pattern in both panels (upper and lower): there is a trend toward shorter periods with increasing spectral type.

It is clear that the overall level of Ca II flux is diminishing as we go from K4 to M4: the *b* coefficient of the linear LSF to the RAC of dK4 stars is larger than the linear LSF to the RAC of dM4 stars by a factor of 25. This suggests that the dM4 stars at any given period are generating mechanical energy flux some 25 times less effectively than dK4 stars with the same rotational period. Even in the much narrower gap between dM3 and dM4, the *b* value for the linear LSF to the RAC has decreased by a factor of almost 10 at any particular rotational period. These results indicate that dM4 stars really are suffering from a weakening of the ability to generate mechanical energy (whether in magnetic form or in acoustic form or in a combination of both forms) compared to dM3 and earlier sub-types.

We conclude that the overall level of dynamo efficiency (as regards chromospheric heating) diminishes markedly as the stellar mass decreases. However, we also find that, as regards the low-activity stars (i.e., unsaturated dynamos), there is an almost universal dependency of the dynamo efficiency on the

rotation period: i.e., the level of activity varies approximately as ~P/ sin i<sup>-0.80</sup>. This universal feature is missing when we attempt to analyze samples that combine stars of both low-activity and high activity. For the combined samples, our results indicate that the dynamo efficiency decreases differently and drastically with decreasing rotation period as we move from dK4 dwarfs to M4 dwarfs.

### 3.9. $R'_{HK}$ as a Function of the Rossby Number

In the spirit of searching for RACs among a variety of parameters (in case the correlations are obscured more in certain cases), we now set aside the rotation period that we have used so far, and explore how activity depends on Rossby number. Dynamo theory suggests that the dynamo efficiency, and therefore the magnetic field strength and its direct diagnostic in the chromosphere, the Ca II lines, should scale with the dynamo number (Montesinos et al. 2001) $N_D$ given by:

$$N_D \sim \frac{\Delta\Omega r_{cz} L}{R_0^2 \nu\Omega d^2} \qquad (29)$$

where $R_0 = P/\tau_c$ is the Rossby number, $\tau_c$ the convective overturning time, $\Omega$ is a characteristic rotation rate in the lower convection zone, $r_{cz}$ the radius at the base of the convection zone, *L* the characteristic length-scale for the differential rotation in the overshoot region just below the convection zone, $\nu$ the ratio of the diffusivities in the two layers directly below and above the overshoot region and the quantity *d* is given by $d \sim \sqrt{\eta\tau_c}$ where $\eta$ is the turbulent diffusivity in the layer just above the overshoot region. Making reasonable assumptions for late-type stars, the dynamo number simplifies to
<figure><img src="image_18.png" /><figcaption>**Figure 15.** Upper panel: $R'_{HK}$ as a function of the Rossby number, $R_0$. Solid line: the mean empirical relationship for dK4 stars. The data for dK4 stars is a good representation of the F, G, early K type star correlation. For dK6 stars, the mean values of $R'_{HK}$ (long-dashed line) lie on average a factor of 3 below the dK4 stars. For dM2 stars, the mean values of $R'_{HK}$ (dotted curve) lie a factor of about 10 below the dK4 stars. Lower panel: dM3 stars (dotted-dashed curve) and dM4 stars (double-dotted-dashed curve) lie a factor of about 20 and 90 below the dK4 stars, respectively. Our data indicate a gradual decrease in the fluxes in M dwarfs that continues into the fully convective M3 and M4 dwarfs, in agreement with our findings on the RACs based on the Ca II stellar surface fluxes (Section 3.6). This suggests a progressive change in the dynamo mechanisms from a shell dynamo to a distributive dynamo.</figcaption></figure>
<figure><img src="image_19.png" /><figcaption>Figure 16. $L_X/L_{bol}$ as a function of the Rossby number, $R_0$, for our five different spectral types. We overplot the running means for dK4, dK6, dM2, dM3, and dM4 stars. Note the important differences in the behavior of $L_X/L_{bol}$ (this figure) compared to that of $R'_{HK}$ (see Figure 15).</figcaption></figure>

(Noyes et al. 1984, hereafter N84):

$$N_D \sim (\Omega\tau_c)^2 \sim R_0^{-2}. \qquad (30)$$

Therefore, a good evaluation of the efficiency of dynamo mechanisms can be done through plotting the chromospheric indices $R'_{HK} = L_{HK}/L_{bol}$ (where $L_{HK}$ is the luminosity in the H and K lines of Ca II) as a function of the Rossby number $R_0 = P/\tau_c$ (e.g., N84, Stepien 1989, 1994; Hempelmann et al. 1995; Patten & Simon 1996; Montesinos et al. 2001).

Montesinos et al. (2001) argue that the dimensionless factor,

$$\frac{\Delta\Omega r_{cz} L}{\nu\Omega d^2} \qquad (31)$$

plays a role in the scatter of the $R'_{HK}$ versus $R_0$ diagrams. This dimensionless factor depends notably on the internal differential rotation, which is mostly confined to the overshoot region just below the convection zone as well as on the turbulent magnetic diffusivities, neither of which are currently reliably known. In view of this lack of information, we shall for simplicity here assume this dimensionless factor to be constant for our samples of late-type dwarfs.

In order to evaluate Rossby numbers in the stars of our sub-samples, we have adopted values of $\tau_c$ from the results of Spada et al. (2013) who gives the relationship between stellar mass $M_*$ and $\tau_c$. We derived a relationship between the stellar radius, $R_*$, and $\tau_c$ using the mass-radius relationships of Spada

et al. (2013) for [Fe/H] = 0. The results of Spada et al. (2013) on $\tau_c$ are in reasonably good agreement with other calculations (e.g., Kim & Demarque 1996). Recent values of $\tau_c$ agree to some extent with the values of Noyes et al. (1984) for solar masses down to $M_* \sim 0.8 M_\odot$ but differ substantially for lower mass stars. We compare the values of $\tau_c$ from Spada et al. (2013) and Noyes et al. (1984) for our five spectral sub-types in Paper I. The large increase in the numerical values of $\tau_c$, which we take from Spada et al. (2013), at low masses has important consequences for the $R'_{HK}/R_0$ relationships in M dwarfs as we shall see below.

We use the $\tau_c/R_*$ tabulation to derive $R_0$ for each star in our five samples of stars. We also computed the luminosity, $L_{HK}$, in the Ca II lines according to the continuum surface fluxes derived above from the models of de Laverny et al. (2012). We list the Ca II luminosities $L_{HK}$ and $R_0$ for our five samples of stars in Table 8. We also computed the bolometric luminosities $L_{bol}$ and the activity index $R'_{HK} = L_{HK}/L_{bol}$ for all our targets. We list the results in Table 8. We further computed $L_{HK}$ and $R'_{HK}$ corrected for metallicity effects for dM2, dM3, and dM4 stars.

Mullan & MacDonald (2001) suggest that the ratio of the X-ray luminosity created by large-scale fields $L_X(L)$ to that of the X-ray luminosity created by turbulent fields $L_X(t)$ may attain factors of 5–10. Although the case of chromospheric lines does not necessarily track the coronal X-ray emission, one might also expect a decrease in chromospheric emission when
the dynamo mechanisms change from a shell dynamo to a distributive dynamo. Note, however that in the case of the chromosphere some modeling calculations (e.g., Ulmschneider & Musielak 2003; Fawzy et al. 2002; Ulmschneider et al. 2005a, 2005b) lead one to expect a basal flux in the chromospheric lines. Such basal flux has not been observed in our data sets (e.g., this study, Papers XV and XVIII). However, some previous authors claim to have detected such fluxes for various spectral types (e.g., Wilson 1968; Schrijver 1987; Rutten et al. 1991; Strassmeier et al. 1994; Fawzy et al. 2002). Note however that M dwarfs were usually excluded from those studies. On the other hand, there do exist certain data sets for line fluxes of Mg II and Lyα in inactive M dwarfs that have been found to be consistent with ab initio models of acoustically heated chromospheres (Mullan & Cheng 1993): thus, there seems to be little reason to exclude the concept of “basal” fluxes from a discussion of dM chromospheres. Here we investigate the $R'_{HK} - R_0$ relationships from dK4 stars to the fully convective dM3 and dM4 stars and search for a signature of the change in the dynamo mechanisms at the TTCC.

In Figure 15, we plot $log(R'_{HK})$ as a function of $log(R_0)$ for our five different spectral types. We found, using the values of $\tau_c$ from N84, that the mean of the measurements for dK4 stars (the dK4 data smoothed with a Gaussian of FWHM = 0.1) agree well with the correlation for F, G, and K type stars found by N84. Thus, our RAC data for dK4 stars are consistent with the N84 study and represents well a low-mass extension of the correlation for F, G, and early K type stars.

In Figure 15 (upper panel), we can see that for the dK6 stars (filled triangles), the mean values (the data smoothed with a Gaussian of FWHM = 0.1, dashed line) lies slightly below the correlation for dK4 stars, typically by a factor from 1.5 to 5.0, i.e., a factor of 3 on average. The dK6 curve crosses the dK4 curve at $log(R_0) \sim -0.2$. There is a large scatter in the data for dK4, dK6, and dM2 stars in spite of the fact that we corrected the data for the effects of metallicity in dM2 stars (otherwise, the scatter is even larger): this is explained by the variations in radii of our stars at a given spectral sub-type, and the large variations of $\tau_c$ with radius. This scatter increases for the spectral types dM3 and dM4 Figure 15(b)) because, for these stars, there is a large increase in $\tau_c$ with decreasing stellar radius and within a small range in radius (see Spada et al. 2013).

For our dM2 stars, the situation is more striking: the data (dotted line in Figure 15(a)) now lie much lower than the dK4 curve. There is a clear and large difference between our dK4 data and our dM2 data. Over the common range in $R_0$, the curve for dM2 stars is roughly parallel to that of the dK4 stars, but is shifted below by a factor from 6 to 16. Our data for the dM2 stars cover a larger range in $R_0$ than in previous studies, notably for small values of $R_0$ (i.e., stars with short rotational periods). One can see that the correlation suddenly rises for $log R_0$ less than -1.3. At small values of $R_0$, $log(R'_{HK})$ reaches values of about -4.0 or even larger.

Moving later in spectral sub-type, we show in Figure 15 (lower panel) the analogous data for our dM3 and dM4 stars. We note that, in this diagram, the mean data for these stars (dash-dotted line) generally lie below the correlation of dK4 stars (solid line), and also significantly below our dM2 data (dotted line). The large variations in $\tau_c$ between M2, M3, and M4 stars imply that the curves for M3 and M4 stars are

systematically shifted toward lower $R_0$. But we emphasize that also, globally, the values of $R'_{HK}$ decrease from M2 to M4 (Figure 15, lower panel). The mean curve for our dM3 stars lies significantly below the mean curve for dM2 stars. This difference is even more pronounced for the sample of dM4 stars. At $log(R_0) = -1.2$, we find that the dM3 curve lies a factor of 20 below that of the dK4 stars, and that the dM4 curve lies a factor of about 100 below the dK4 curve (Figure 15, lower panel). The curves for dM2, dM3, and dM4 stars are more or less parallel (except at very low values of $R_0$ for dM3 and dM4 stars, where there are only a few data points). This highlights the consistency of the decreases in the $R'_{HK}-R_0$ curves when going from mid-K type stars to the fully convective dM4 stars.

The decrease from dM2 to dM3 stars is typically a factor of 2 (at $log(R_0) = -1.0$, whereas the decrease from dM3 to dM4 stars is typically a factor of 4 (taken at $log(R_0) = -1.5$).

Christian et al. (2011) have reported a saturation (or even “super-saturation”) phenomenon in the chromospheric emission for fast rotators in young clusters at about $log(R'_{HK}) = -4.08$ but with a significant scatter (±0.5 in the log-transformed data). In our stellar samples, we do not have enough very fast rotators to confirm the existence of super-saturation. However, we note that $R'_{HK}$ measures generally lie below -3.8 for all our samples of late-K and M dwarfs. We also find that this high level of activity is attained for different values of $R_0$ that depend on the spectral type. For dK4 stars, this high level of activity is attained at about $log(R_0) \sim -1.0$, whereas for dM4 stars, it is attained at about $log(R_0) \sim -2.7$.

Our data sets suggest that M dwarfs (especially M3 and M4) generally do not follow the same relationship as F, G, and K dwarfs. This had not been noted previously because the M dwarf data sample was too sparse. If this difference is due to a difference in dynamo mechanisms, then the change in dynamo mechanisms seems to be progressive, as is illustrated by our dK6 and dM2 data sets. Therefore, we cannot conclude from the results in Figure 15 that there is any abrupt change at the TTCC (if this occurs no later than M4): instead, there seems to be a gradual decrease in the efficiency of the dynamo mechanisms that may start as early as spectral type dK6. This gradual change was suggested by Mullan & MacDonald (2001). The efficiency of an interface (shell) dynamo may fall off between dK4 and dM2 as the radiative core occupies an increasingly small proportion of the stellar radius (Mullan et al. 2015), and the present data suggest that this decrease continues at least as far as M4. The latter point is strengthened by our results in Paper XV, which showed a progressive change from high fluxes (large radii M2 dwarfs, partially convective) to very low fluxes (M2 subdwarfs, fully convective). The $R'_{HK}-R_0$ diagram may be one of the most sensitive tests for detecting changes occurring in the dynamo mechanisms. However, the results from these RAC relationships have to be compared with the other magnetic activity indicators we study here. Note that with the present data there is evidence in the $R'_{HK}-R_0$ diagrams that gradual changes in the dynamo efficiency are occurring all the way to M4. Other magnetic activity diagnostics, including the mean fluxes (Section 3.8), also point to a continued change from K4 to M4.

The results in Figure 15 strongly suggest that, as far as chromospheric heating is concerned, there may be a gradual decrease in dynamo efficiency in M dwarfs compared to F, G,
<figure><img src="image_20.png" /><figcaption>**Figure 17.** $L_X$ vs. $L_{HK}$ for the stars in our dK6, dM3, and dM4 samples for which ROSAT X-ray data are available. As regards the empirical correlations (solid lines), the slopes are very similar: 1.54, 1.74, and 1.37 for dK6, dM3, and dM4 stars, respectively. Note the differences in the scales on both axes for the different panels: for a given value of chromospheric emission, the dM4 stars have a much larger coronal emission than the dK6 stars.</figcaption></figure>

and K dwarfs. It is generally believed that an α-Ω type of dynamo dominates in F, G, and K dwarfs. In early M dwarfs, an interface dynamo is probably still operating (Mullan et al. 2015), although eventually, in fully convective M dwarfs (beyond the TTCC), the dynamo mechanism is expected to behave differently. The most striking conclusion of the present section is that, as regards chromospheric heating, there are almost two orders of magnitude decrease in going from dK4 to dM4. This further strengthens our previous findings (see Figure 14) that, in the context of chromospheric heating, the mechanical fluxes decrease by a factor of order 100 as we go from dK4 to dM4.

### 3.10. The Coronal RAC: $L_X/L_{bol}$ as a Function of the Rossby Number

Now we turn to the corona, where deposition of mechanical heating manifests itself in the form of X-ray luminosity $L_X$. Mullan & MacDonald (2001) found that, in a sample of some 40 ROSAT measures of dMe stars by Fleming et al. (1993), $L_X/L_{bol}$ was found to remain essentially invariant over the spectral range from early M to at least M7, at least for the most active stars (see also Browning et al. 2010). This result, which implies that coronal heating efficiency in the most active stars remains unchanged at the TTCC, now appears worth re-examining in the context of our findings (see Figure 15) of changes in
<figure><img src="image_21.png" /><figcaption>**Figure 18.** $L_X$ as a function of $L_{HK}$ for all five of our spectral sub-types. The gradients of the empirical correlations are similar for the five spectral types. But the more pronounced aspect of this figure is that, at a given chromospheric flux, the coronal flux increases toward later spectral types. For instance, for a chromospheric Ca II luminosity of $10^{27.5}$ erg s$^{-1}$, the X-ray luminosity increases by a factor of 110 from spectral type dK4 to dM4.</figcaption></figure>

chromospheric heating before and near the TTCC. Our sample is now several times larger than that of Fleming et al. (1993). We therefore investigate here the behavior of $L_X/L_{bol}$ as a function of $R_0$ for our five different spectral sub-types, to see if we can identify any signature of the TTCC in the coronal data.

In order to illustrate the sensitivity of coronal heating to rotation, in Figure 16 we plot $L_X/L_{bol}$ as a function of $R_0$ for our five different spectral sub-types. We also plot in this diagram the running mean curves for our dK4, dK6, dM2, dM3, and dM4 stars. The differences between the behaviors of $R'_{HK}$ (Figure 15) and $L_X/L_{bol}$ (Figure 16) are significant. For dK4, dK6, and dM2 stars, $L_X/L_{bol}$ rises slowly as $R_0$ decreases for the slow rotators ($\log(R_0) = 0$ to $-1.0$), and suddenly rises abruptly for the rapid rotators ($\log(R_0) < -1.0$). For the fast rotators ($\log(R_0) < -1.5$), $L_X/L_{bol}$ seems to saturate at a level of about $10^{-3.0}-10^{-2.5}$. Patten & Simon (1996) also observed a similar behavior among their samples of stars in young clusters (IC 2391, α Persei, Pleiades, Hyades, and main-sequence stars in the field). They also observed a saturation at $L_X/L_{bol} \sim 10^{-3.0}$ for $\log(R_0) < -0.6$. On the other hand, they did not observe a plateau-like of behavior from $-3.0 < \log(R_0) < -1.3$ such as our dM2, dM3, and dM4 star data display in Figure 16. Instead they observe a strong rise in $L_X/L_{bol}$ from $10^{-6.5}$ to $10^{-3.0}$, more like the behavior of our data from $-1.3 < \log(R_0) < 0.3$ in Figure 16. Overall, our impression is that all stars in our five sub-samples follow a more or less similar correlation and show evidence for saturation for $\log(R_0) < -1.3$.

We should emphasize that our $L_X$ data (taken from Hunsch et al. 1999) are severely biased toward the most active stars because of the limited sensitivity of ROSAT. Therefore, our samples of measures for the least active stars ($\log R_0 > -1.3$) are inevitably biased in a negative sense, and may not be dependable when we calculate the means of $L_X/L_{bol}$ as a function of $R_0$.

### 3.11. Comparison and Contrast between Chromospheric and Coronal Heating Rates

In order to facilitate comparison of Figure 15 (lower panel: chromosphere) and Figure 16 (corona), we use similar notation for the lines that illustrate average values of the quantities at different spectral sub-types. Thus, in Figure 15 (lower), the dM3/dM4 stars (dotted line) lie lowest in the figure, whereas in Figure 16, the same stars lie at about the same levels as other spectral sub-types in the figure. For purposes of the present discussion, let us suppose that dM3 and dM4 stars may be labelled as at, or later than, the TTCC; we refer to these stars by the shorthand notation TTCC+. Stars at spectral types dM2 and earlier are referred to as TTCC-. In terms of this notation, our results indicate that, in terms of the chromosphere, TTCC+ stars are definitely weaker emitters than TTCC- stars (see Figure 14 lower panel). On the other hand, in terms of the corona, TTCC+ stars have similar emissions to those from TTCC- stars.

Thus, when we compare stars on different sides of the TTCC, the trend in coronal activity does not behave in the same way as the trend in chromospheric activity.

In order to explore this difference between activity at the chromospheric and coronal levels of M dwarf atmospheres, we turn now to investigate how $L_X$ varies as a function of $L_{HK}$, and also as a function of spectral type.

#### 3.11.1. $L_X$ as a Function of $L_{HK}$

In a previous study, Schrijver et al. (1992) reported on the inter-relation between $L_X$ and $L_{HK}$ with nearly simultaneous observations of a sample of 26 F5–K3 main-sequence stars. Schrijver et al. (1992) obtained a good correlation over three orders of magnitude in $L_{HK}$ and over four orders of magnitude in $L_X$. They found that $L_X$ increased faster than $L_{HK}$: $L_X \propto L_{HK}^{1.50 \pm 0.20}$. In the present study, we compiled $L_{HK}$ and $L_X$ for our five different spectral subtypes. We find that the LSF to our samples give:

$$L_X = 4.57 \ 10^{-16} L_{HK}^{1.54 \pm 0.29} \text{ erg s}^{-1} \quad (32)$$

for our dK4 stars,

$$L_X = 7.94 \ 10^{-17} L_{HK}^{1.57 \pm 0.21} \text{ erg s}^{-1} \quad (33)$$

for our dK6 stars,

$$L_X = 1.00 \ 10^{-15} L_{HK}^{1.57 \pm 0.17} \text{ erg s}^{-1} \quad (34)$$

for our dM2 stars,

$$L_X = 7.94 \ 10^{-20} L_{HK}^{1.74 \pm 0.11} \text{ erg s}^{-1} \quad (35)$$

for our dM3 stars, and

$$L_X = 2.51 \ 10^{-9} L_{HK}^{1.37 \pm 0.12} \text{ erg s}^{-1} \quad (36)$$

for our dM4 stars.

The correlations are in all cases highly significant (see Table 1). The only exception occurs in the dK6 stars: this is because in this sample we have only few high-activity stars. Since the slope for dK6 stars is the same as for dK4 and dM2 stars, we believe the dK6 correlation is relevant for comparison to other sample correlations. Perhaps the smaller range of $L_X$ values among dK6 stars also contributes to low significance. In Figure 17, we show the $L_X$ versus $L_{HK}$ correlations we found for our dK6, dM3, and dM4 stellar samples.
Including late-K and M dwarfs of all five sub-types, we find an average slope of 1.56 in our $log(L_X)-log(L_{HK})$ relation-ships. This is essentially identical to the value reported by Schrijver et al. (1992) for F5-K3 stars. Therefore, the same slope seems to prevail for all main-sequence stars from F5 to M4. This result emphasizes that, in low mass stars, it is a general result that the coronal emission grows faster than the chromospheric emission, from low-activity stars to high-activity stars. It is interesting to note that, when investigating IUE spectra of F to K type stars, Ayres et al. (1981) found also a power law slope of about 1.5 between the $10^5$ K line fluxes and the Mg II line fluxes. However, in their correlation between the soft X-rays and Mg II fluxes, they found a larger slope.

However, what is more striking than the similarity in gradients found for these correlations, is the multiplicative coefficients in the correlations. We find that this factor changes from $4.47 \times 10^{-16}$ for dK4 stars to $2.51 \times 10^{-9}$ for dM4 stars. This large difference indicates that the empirical correlations differ greatly from one spectral sub-type to another even if the gradients are comparable. To illustrate this, we plot the LSF of $L_X$ versus $L_{HK}$ in Figure 18 for our five spectral sub-types. In all cases, it is apparent that the correlations have slopes that do not differ greatly. But the absolute values of fluxes from the chromospheres differ greatly. E.g., dK4 and dK6 stars have $L_{HK}$ values mainly in the range $10^{27.5-29.5}$ erg s$^{-1}$, whereas the chromospheres of dM4 stars emit mainly in a much lower range, $10^{25-27.5}$ erg s$^{-1}$. On the other hand, emissions from the *coronae* of dK4 and dM4 stars overlap in the range $10^{27.5-28}$ erg s$^{-1}$.

As a result, at any given chromospheric emission level, the coronal emission strongly increases as we consider later spectral sub-types. E.g., for a chromospheric Ca II luminosity of $10^{27.5}$ erg s$^{-1}$, the X-ray luminosity increases by a factor of at least 110 between dK4 and dM4. Therefore, in the case of M dwarfs, the situation is quite different from that of earlier F, G, and K spectral types (e.g., Schrijver et al. 1992): stars from F to G to K all follow a common correlation between $L_X$ and $L_{HK}$. In a similar vein, the large differences we observe among M dwarfs explains the large scatter observed in the $L_X-L_{HK}$ correlation reported by Panagi & Mathioudakis (1993) for K and M dwarfs.

It is natural to inquire at this point, what makes M dwarfs so different from F-G-K dwarfs as regards the ratio of coronal to chromospheric emission? One possibility has been suggested by Mullan (1984). Coronal heating depends ultimately on tapping into the reservoir of mechanical energy associated with convective motions. In the context of electrodynamic coupling, the efficiency with which energy contained in that reservoir can be conveyed to the corona depends on two timescales, $\tau_c$ for convection, and $\tau_A$ for the coronal loops in which the energy is to be deposited. In the Sun, these timescales differ by factors of order 100. This large difference explains why the coronal energy flux from the Sun amounts to only 1% or less of the mechanical energy flux in convection. But Mullan (1984) predicted that, as one goes down the main sequence, there could come a point where the two timescales $\tau_c$ and $\tau_A$ could be comparable. This would occur by reduction in $\tau_c$ due to smaller convection cells, and increases in $\tau_A$ due to longer coronal loops. By making some assumptions, the prediction was made that $\tau_c \approx \tau_A$ would occur among stars with $T_{eff} \approx 3400$ K (see Mullan 1984: Section VI). What spectral type does this correspond to? According to Rajpurohit et al. (2013), the corresponding I-J color is 1.2-1.3, and the spectral type is M3.

Moreover, $I-J = 1.2$ corresponds to V-I color = 2.2, which is also the color at which coronal loops become longer (Mullan et al. 2006).

At the spectral type where $\tau_c \approx \tau_A$, the coronal loops would reach a resonance in their efficiency to tap into the convective reservoir. As a result, the efficiency of coronal heating in dM3 stars would be larger than in the Sun by 80-170. This range overlaps with the excess by factors of order 110 reported in Figure 18 above. Because a resonant process is at work in this model, the increase in coronal heating efficiency is expected to build up in spectral types as these approach closer to the resonance, i.e., approaching M3. Thus, the full increase by factors of 80-170 should be realized at M3, but smaller enhancements are expected at (say) M2, K6, and K4.

It is important to note that the electrodynamic resonance works only for the corona: there is no analogous process occurring in the chromosphere. Heating of the chromosphere, whether in an F star or an M star, continues to rely on localized dissipation of acoustic waves (for the “basal” component; Mullan & Cheng 1993) and on localized dissipation of currents, via the conductivity tensor, in a partially ionized medium (for the “magnetic” contribution; e.g., Kazeminezhad & Goodman 2006).

Even if the resonant model (Mullan 1984) turns out to be incorrect, at least the empirical results explain the $L_X/L_{bol}-R_0$ relationships reported above. We can now interpret the $L_X/L_{bol}-R_0$ relationships (in Figure 16) vis-a-vis the $R'_{HK}-R_0$ relationships (in Figure 15). Indeed, in late-K and M dwarfs, the 110-fold increase in X-ray coronal emission from spectral sub-types K4 and M4 compensates for the 90-fold decrease in dynamo/chromospheric efficiency from the same spectral types. As a result, in Figure 16, the M dwarfs have about the same $L_X/L_{bol}$ values as the dK4 and dK6 stars, whereas in Figure 15, we found a drop-off for dM2 stars of about a factor of 10 as regards the chromospheric heating. Note, however, that in Figure 16, there is some convergence in $L_X/L_{bol}$ for $R_0 < -1.3$ for all spectral types.

This discussion may help us to understand why Mullan & MacDonald (2001) failed to identify any signature of the TTCC in $L_X/L_{bol}$ data: their failure may result from one aspect of coronal properties in M dwarfs. As a result of these properties, and their variation with spectral type in the vicinity of an electrodynamic resonance that (perhaps coincidentally) overlaps with the TTCC, the coronal ratio $L_X/L_{bol}$ is not really suited for diagnosing the changing properties of the dynamo mechanisms before and at the TTCC. In view of the data presented in the present paper, we are now disposed to believe that the chromospheric emission (e.g., the Ca II luminosity) is better suited to respond to a signature of the TTCC.

### 3.12. The RACs and the Mean Rotation Periods

Can the properties of the RACs discussed above explain the mean $P/\sin i$ values obtained by HM for stars ranging in spectral sub-type from dK4 to dM4 stars? In particular, can they help in understanding the HM results of unexpectedly long rotation periods for dM3 stars? To answer this, we note that the mean $P/\sin i$ values of the slow rotators tend to decrease toward later spectral types. As a matter of fact, this trend is reproduced in the lower end of the RACs (Figure 14). Therefore, the decrease in the mean $P/\sin i$ at later spectral types could be due to the fact that the dynamo mechanisms
become inefficient at a rotation period that decreases toward the later spectral type (Figure 14). However, this is not completely the case for the dM3 RAC, which extends to somewhat longer periods than the dM2 RAC. Globally, the dM3 RAC is shifted toward longer periods with respect to the dM2 RAC, such that the mean *P*/sin*i* for dM3 stars is longer than for dM2 stars (as mentioned by HM). For dM4 stars, the whole of the RAC is shifted toward shorter rotation periods compared to dM3 stars. This explains why the mean *P*/sin*i* for dM4 stars is clearly smaller than that for dM3 stars. In fact, the dynamo mechanism in dM4 stars apparently becomes so inefficient at long periods that our sample contains not a single dM4 star with *P*/sin*i* longer than 10 days (see Figure 14). This can be contrasted with the much longer *P*/sin*i* values for dM3 stars: as large as 30 days (see Figure 8). Our discovery in the present paper that the level of chromospheric emission in dM4 stars is definitely smaller than in dM3 stars (suggesting that dynamo action in dM4 stars is less effective than in dM3 stars) could explain why the mean *P*/sin*i* is much shorter at dM4 than at dM3: dM4 stars are not as good at generating magnetic fields and, as a result, they do not have access to as good a “magnetic brake” as dM3 stars. The dM4 stars just keep on spinning fast, whereas the dM3 stars are braked. This might supplement (or replace) the hypothesis of long loop lengths (cf. HM) as the reason for slow rotation at dM3.

Also among dM2 stars, there are no stars rotating as slowly as the slowest dM3 stars (see Figure 5). So dM2 stars do not have access to as good a brake as dM3 stars. And yet, their dynamo effectiveness seems to be as high as, or higher than, the dM3 stars (see Figure 14 above). In this case, the onset of increased loop lengths at dM3 (cf. HM) would help to explain why dM3 stars have access to a better magnetic brake than dM2 stars.

For the fast rotators, we found (in HM) that the trend for the mean *P*/sin*i* is to become, in general, shorter as the spectral type becomes later, except at sub-type dM3 where the mean period is found to be significantly larger. The minima of the RACs show the opposite trend, i.e., the minimum *P*/sin*i* decreases with increasing spectral type. However, only very few active stars are at the extremes of the RACs: most active stars lie at longer periods. In the case of dM3 and dM4 stars, most active stars lie in the upper part of the rising slope of the RAC (see Figures 8 and 12). We also refer to Figures 3 and 5 for dK6 and dM2 stars, respectively. Therefore, the trend observed in the active stars can be interpreted by the RACs only if one considers their locations on these curves. Nevertheless, we note again that the upper end of the RAC for dM3 stars (in its rising part) is shifted again toward periods that are clearly longer than those for dM2 and dM4 stars. This again explains at least partly the abnormally large mean *P*/sin*i* of the active dM3 stars (see HM).

In summary, we find that the properties of the RACs can explain at least partly the observed trends in the mean *P*/sin*i* and also the abnormally long *P*/sin*i* values reported for dM3 stars by HM. However, this does not exclude another explanation, possibly involving a change in loop lengths (HM), which might also contribute to the unusual rise in the mean *P*/sin*i* at dM3.

**4. Comparison with Other Studies of Rotation and Activity**

West & Basri (2009) investigated the rotation and activity in $H_α$ in a sample of 14 late-type M dwarfs (M6–M7). They found

that many of these objects are rotating relatively fast (>3.5 km s⁻¹) but also have $H_α$ in absorption. These rotational velocities imply rather short rotation periods for these small objects. This confirms our observed trend (see HM), that in general among late type dwarfs, rotation periods diminish with increasing spectral type. West & Basri (2009) also derived an empirical relationship between $L_{H_α}/L_{bol}$ and $v \sin i$ from a compilation of data: they found that $L_{H_α}/L_{bol}$ increases typically by two orders of magnitude for a change in $v \sin i$ of only 5 km s⁻¹. This implies a steep gradient in the RAC. Although our spectral range does not go to spectral subtypes that are as late as those of West and Basri, our results are not inconsistent with theirs in the sense that the steepest slope we have found for the RAC occurs among the latest spectral type in our sample (dM4, see Table 1), which is closest to the spectral types of the West-Basri sample.

Browning et al. (2010) analyzed the rotational broadening and activity in the Ca II lines for a sample of 123 M dwarfs. Unfortunately, because of limitations on their spectral resolution, they could measure $v \sin i$ for only seven stars, namely, those in which the projected rotational speeds were >2.5 km s⁻¹. They found that the rotation was detected mostly in stars later than M3 rather than in the range M0–M2.5. This is also consistent with our findings. They also found, in agreement with our results, that there is a “gap” in the measures of $L_{Ca II}/L_{bol}$ between the active stellar group and the low-activity stellar group (see Figure 11 above). They found a rough relationship between $L_{Ca II}/L_{bol}$ and $v \sin i$. However, most of their measures lie in the saturated regime, whereas in the present work, we have chosen to perform our RAC analyses on stars that lie in the unsaturated regime (so that we may examine stars that probably have dynamos also in the unsaturated regime).

Wright et al. (2011) reported on $L_X/L_{bol}$ as a function of $R_0$ for a sample of (824) stars, which is significantly larger than we have analyzed here: the Wright et al. sample is three times larger than the number we used for chromospheric RACs, and about 10 times larger than the number of stars we used for the coronal RACs (Figure 16). However, the sample of Wright et al. extends over a much broader range of spectral types than we have studied here, from spectral type F to M5. They found a correlation between $L_X/L_{bol}$ and $R_0$ although there is a significant scatter among the data. Their results reveal a saturated regime and an unsaturated regime, with a break at $\log(R_0) \approx -0.1$. The saturation value of $L_X/L_{bol}$ was found to be at a log value of ~-3. In the non-saturated regime the power law fit between $\log(L_X/L_{bol})$ and $\log(P)$ was found to have a slope of -2.18 ± 0.16. If the unsaturated (coronal) stars in their sample are analogous to our low-activity (chromospheric) stars, then their coronal slope (-2.2) is much steeper than what we found for the chromospheric slope (-0.8) (Table 1). The increase in steepness of the coronal RAC relative to the chromospheric slope (by an amount of about 1.4 in the slope) is reminiscent of our results in Section 3.11, Equations (32)–(36) where the slopes of $L_X$ are steeper than the slope of $L_{HK}$ by values, which (within errors) also overlap with 1.4 (except for dM2 stars).

Rebassa-Mansergas et al. (2013) studied a sample of white dwarf/M dwarf binaries from the Sloan Digital Sky Survey (SDSS DR7). They found indications that magnetic braking is less efficient beyond the fully convective boundary. This again agrees with our results that M4 dwarfs rotate faster than early M type dwarfs. They also studied the $L_{H_α}/L_{bol} - v \sin i$ relationship
and found mostly that stars with $v \sin i = 5$ km s$^{-1}$ are all in the saturated regime. They also found a rapid increase in $L_{H\alpha}/L_{bol}$ (by 2 orders of magnitude) as $v \sin i$ increases from 2 km s$^{-1}$ to $v \sin i = 5$ km s$^{-1}$. This agrees with the similar diagram of West & Basri (2009), and again suggests a steep gradient in the RAC between low-activity and high-activity late-type M dwarfs (as we have found for our samples: see Table 1).

Robertson et al. (2013) investigated the magnetic activity level in $H\alpha$ for a sample of 93 K5–M5 dwarfs. They found, in agreement with our results on the RACs, that early type M dwarfs (M0–M2) tend to have higher levels of activity than later type dwarfs and that, in general, $\log(L_{H\alpha}/L_{bol})$ continuously decreases from −3.6 to −3.95 as the spectral subtype increases from M0 to M5. This again agrees with our global finding that the surface fluxes in the Ca II lines decrease from K4 to M4.

Recently, an important paper has been published by West et al. (2015: W15) reporting on an analysis of chromospheric RACs in a sample of 238 M dwarfs, using $H\alpha$ emission as a measure of chromospheric “activity.” The sample stars had originally been selected as targets for planet searches, but W15 used the photometry to search for rotation periods *P* for the stars themselves. Photometric periods are not subject to the $\sin i$ uncertainty that affects the $P/\sin i$ we have obtained in the present study. In constructing their RAC, W15 identified 164 stars for which values of *P* and $H\alpha$ data were available. The *P* values ranged from 0.3 to 100 days. The advantage of using photometry (over spectroscopy) to obtain *P* values is clear: there are no observational limits imposed by attempts to extract $v \sin i$ values from spectroscopic data. As a result, photometry allows W15 to determine rotational periods for slow rotators that are beyond the capabilities that we have used here. As a result, the W15 upper limit on *P* is several times longer than we have been able to report on in the present paper. In order to study variations of RAC with spectral type, W15 divided their stars into two groups: M1–M4 (64 stars, mainly M3 and M4), and M5–M8 (100 stars, mainly M5 and M6). Their plots of RAC (in their Figures 7 and 8) show that the M1–M4 stars have a negative slope: a LSF yields a slope of $-0.19 \pm 0.036$. For the M5–M8 stars, the LSF gives a formal slope of essentially zero ($-0.016 \pm 0.050$). These results suggest that since a finite (though small in absolute value) negative slope exists for the RAC in M1–M4 stars, rotation might well play a role in the chromospheric emission in M1–M4 stars. On the other hand, since the RAC slope was found to be essentially zero for M5–M8 stars, rotation may play little or no role in determining chromospheric emission in M5–M8 stars. At first sight, this result might be considered as evidence for the suggestion of Durney et al. (1993) that there might be a change in dynamo mode between M1–M4 and M5–M8. But we would like to suggest another possible interpretation, as follows.

In view of our own results for the RAC’s in M4, M3, and M2 stars, we find it a matter of interest that the slope of the chromospheric RAC determined by W15 for M1–M4 stars ($-0.19 \pm 0.036$) is much shallower than the values we have found. For our combined samples of dM+dMe stars, we have found slopes of $-1.5$ to $-2.5$ (see Table 1). And even when we confine attention to the slow rotators, we still find slopes for M2, M3, and M4 dwarfs ($-0.89$ to $-0.93$), which are clearly significantly steeper than W15 report for their M1–M4 sample. Our results indicate that chromospheric emission in M2–M4 stars in our samples exhibit much stronger sensitivity to rotation

than W15 have reported for either of their samples. Is it possible that this dichotomy arises from selection effects? We note that the data of W15 include mainly “active” stars with $H\alpha$ in emission, i.e., fast rotators, whereas our samples (especially the slow rotator samples) are biased toward slow rotators ($H\alpha$ not in emission). Now, fast rotators are more likely to be in the saturated regime of the dynamo (see Section 1.1): in that regime, the RAC flattens out, and takes on a slope that is close to zero. In fact, in a discussion of the systematic differences between their M1–M4 sample and their M5–M8 sample, W15 point out that “At similar rotation periods, a much larger fraction of the late-type M dwarfs [M5–M8] are active” than is the case for the M1–M4 stars. Visual inspection of Figure 6 in W15 suggests that some 65% of their M1–M4 stars are labelled “active” while almost 90% of their M5–M8 stars are “active.” This is consistent with the W15 statement that a “much larger fraction” of the M5–M8 stars are in the saturated regime. This leads us to wonder if the presence of saturated behavior is contributing to the (essentially) zero slope reported by W15 for M5–M8 stars. And even among the M1–M4 stars of W15, where the fraction of active stars is admittedly smaller than in the M5–M8 sample, that fraction is by no means small: some 65% of the M1–M4 stars are “active,” and therefore may, lie in the saturated regime. Suppose that 65% of the M1–M4 stars in the W15 sample lie in the saturated regime (with a slope of zero), while the remaining 35% lie in the unsaturated regime, where the slope is finite (and negative), with a value of $-a$. Then the complete W15 sample of M1–M4 stars would have an RAC with an average slope $a$ ($W15) = (0.65 \times 0) + (0.35 \times (-a))$. For slow rotators with spectral sub-types extending from dK4 to dM4, we have found that $-a$ can take on values in the range from $-0.6$ to $-0.9$. This leads us to predict that $a(W15)$ could range from $-0.21$ to $-0.32$. In fact, the empirical slope reported by W15 ($-0.19 \pm 0.036$) contains, within a $3\sigma$ range, values of $a$ (W15) extending from $-0.08$ to $-0.30$. This overlaps extensively with our predicted range of slopes.

Thus, it is possible that (i) the existence of a small but finite negative slope obtained by W15 for M1–M4 stars is due to the presence of a minority (35%) of stars in the unsaturated regime and (ii) the change in slope in going from M1–M4 to M5–M8 is due to the larger number of stars in the saturated regime among the M5–M8 sample. In order to test these possibilities, and in order to get a more definitive test of a change in dynamo mode at the TTCC, we suggest that it would be best to concentrate on stars in the unsaturated regime. So far, our own studies have not yet reached into M5–M8 stars: it will be a matter of great interest to determine if the W15 findings of significant change in RAC slope between M4 and M5 can be replicated using unsaturated stars.

## 5. Conclusion

In this section, we first (Section 5.1) summarize the approach we have adopted in order to study dynamos in low-mass stars. Then we go on (Sections 5.2–5.5) to describe how we have quantitatively characterized the observational data in terms of chromospheric observations. A discussion of coronal data follows (Section 5.6). In Section 5.7, we present the major conclusions of our study in terms of three hypotheses about dynamo action in low-mass dwarf stars. Last, we discuss evidence that has a bearing on crossing the TTCC (Section 5.8).
## 5.1. Dynamos: Saturated and Unsaturated

In order to account for the presence of active chromospheres and coronae in low-mass stars, dynamos are believed to be operative. Dynamos may rely on rotation plus turbulence, or on turbulence alone, to generate magnetic fields that then, as a result of “magnetic activity” (of some kind) lead to heating of the chromospheric and coronal gas. It is possible that different types of dynamos are at work in stars of different masses. In this paper, we bring together observational evidence to see if it is possible to identify signatures of different types of dynamos. Our study is based on the assumption that different types of dynamos are expected to lead to different answers to the following question: how does the amount of “activity” behave as a function of “rotation”? In order to address this, we need to compile data that provide us with quantitative measures of “activity” and “rotation.”

An important aspect of the present paper is that we measure rotational speeds in stars that are rotating more slowly than has been reported in previous studies. Why are we interested in slow rotators? The reason has to do with the physics of dynamos: theoretical work suggests that rotational dynamo operation can saturate when the rotational speed reaches a certain limit. It seems to us that it would be difficult to extract information from a saturated dynamo, where the activity level no longer depends on how fast the star is rotating. For this reason, in this paper, we prefer to deal with dynamos in an unsaturated condition, where an increase in rotation leads to a clear and measurable increase in activity. This is what has driven us in the present paper to undertake a concerted effort toward identifying stars with the slowest possible rotations; to do this, we rely on spectroscopic data that were obtained with the highest possible resolution.

## 5.2. Data being Used in this Paper

Data have been compiled on chromospheric emission and (projected) rotational periods for a sample of 418 stars (Paper I) ranging in spectral sub-type from dK4 to dM4 (42 dK4, 118 dK6, 94 dM2, 81 dM3, 83 dM4).

The analysis that we apply to our data has the goal to derive a RAC for stars in each spectral sub-type, and then see if there are any systematic variations in the RACs as the spectral sub-type approaches the limit where main-sequence stars make a TTCC. The location of the TTCC is a matter of some dispute, possibly as early as dM2, possibly as late as dM4. Our choice of spectral sub-types is meant to overlap with TTCC.

## 5.3. Constructing RACs at Various Spectral Sub-types: Chromospheric Data

As a quantitative measure of “activity,” we use the mean surface flux $F_{Ca II}$ of emission in the Ca II H and K lines. As a quantitative measure of “rotation,” we combine the projected rotational speeds $v \sin i$ with stellar radii to obtain a “projected rotation period” $P/\sin i$. We construct an RAC for stars in each spectral sub-type by plotting (in log-log format) the surface flux versus $P/\sin i$. The RACs that we have obtained for stars in each of our five spectral sub-types can be found in Figures 2, 3, 5, 8, and 12.

## 5.4. Chromospheric Data: the Slopes of the RACs

A general feature of many RACs is that as rotational periods become shorter, the Ca II surface flux becomes larger, up to a point. Beyond that point, shorter periods do not lead to any increase in Ca II flux: at these shortest periods, the RAC is probably “saturated.” A key aspect of the present paper is that we discuss only the unsaturated part of the RAC. In this case, we obtain least squares fits of the RAC to a function of the form $F_{Ca II} = b(P/\sin i)^a$. In a log-log plot, $a$ is the slope of the RAC, and is a negative number. The coefficient $b$ is a measure of the amplitude of chromospheric heating.

Numerical values of the slope $a$, which we have obtained from least squares fitting to our data for the various spectral sub-types, are shown in Figure 13. The different curves show the results we have obtained when we group our target stars in three different ways. (i) The curve that rises monotonically from lower left to upper right refers to the combined sample of all “unsaturated” stars, both those with low activity (dK, dM) and those with high activity (dKe, dMe). (ii) The curve that runs (almost) horizontally near the lower boundary of the figure refers to stars that are confined to the low-activity sub-samples (dK, dM). (iii) The “jagged” curve extending from middle left to lower right refers to stars that are confined to the high-activity sub-samples (dKe, dMe). The results in Figure 13 will guide our discussion of dynamos in the final sub-section.

## 5.5. Chromospheric Data: the Amplitude of Chromospheric Heating

As a second step in obtaining RACs, we consider, not the slopes, but the overall level of chromospheric heating in terms of the ratio of $L_{HK}$ to $L_{bol}$. Based on this ratio, we have obtained RACs based on a more physically relevant parameter (the Rossby number $R_o$). These RACs (see Figure 15) show that, at the longest rotation periods ($\log(R_o) = -1.2$) in our data set, chromospheric heating in dM3 stars is less effective by a factor of about 20 than in dK4 stars. Our results suggest that the chromospheric heating efficiency in dM4 stars is less effective by a factor of about 100 than in F-G-K type stars. We also observe a progressive decline in the location of the RACs as we go from dK4 to dM4: dK6 stars lie slightly below (a factor of 3) the RAC of F, G, and K type stars, while dM2 stars lie a factor of 10 below the RAC of F, G, and K type stars. The data point to a conclusion that we consider reliable: the efficiency of chromospheric heating decreases progressively between dK4 and dM4. The amplitude of the overall decrease is 20–90. This also implies that the overall efficiency of the dynamo mechanisms also decreases by the same factors when moving from K4 to M4 dwarfs.

## 5.6. Coronal Data

Turning now to the RAC associated with coronal emission, we find a very different behavior from the chromospheric data (Figure 16). The coronal emission $L_X/L_{bol}$ does not decrease significantly as we go from dK4 to dM4. The lack of decrease is especially marked when we compare the coronal emission to the chromospheric emission (see Figure 18): for a given value of $L_{HK}$ (say $10^{27.5}$ erg s⁻¹), the value of $L_X$ in dM4 stars is larger by a factor of order 100 compared to $L_X$ in dK4/dK6 stars. Thus, while the chromospheric heating efficiency is decreasing as we go from dK4 to dM4 (by a factor of
up to 100), the coronal heating efficiency is simultaneously increasing (by a factor of 100 or so).

In terms of a dynamo interpretation, this raises the question: which part of a stellar atmosphere should we study in order to obtain more reliable information about dynamo efficiency in M dwarfs, the chromosphere or the corona? The behaviors are so different that it is not clear that the same information about the dynamo will emerge from both data sets. We have argued (Section 3.11) that it may be preferable to concentrate on the chromosphere.

In view of this, we now present some conclusions that, in our opinion, help to bring order to the chromospheric data that have been analyzed in the present paper.

## 5.7. Hypotheses about Two Distinct Dynamos in Low-mass Stars

Inspection of Figure 13 leads us to offer the following hypotheses.

(i) In the case of low-activity stars, we see that the RAC slope *a* is essentially unchanged as we go from dK4 to dM4. To the extent that a particular RAC slope is associated with a particular dynamo mechanism, our results indicate that low-activity stars have essentially the same dynamo at work in stars that range in spectral sub-type from dK4 to dM4. Such a dynamo must have something to do with a physical property that is present in all low mass stars from dK4 to dM4, i.e., on both sides of the TTCC. What property is common to all such stars? The answer is: all of them have a deep convective envelope in which turbulence provides a ready supply of energy to drive a distributed dynamo (DD), i.e., a dynamo that could be described by either an $\alpha^2$ model or an $\alpha^2\Omega$ model. This leads us to Hypothesis (A): low-activity stars from dK4 to dM4 may be dominated by a turbulent (DD) dynamo.

(ii) In the case of the RAC for high-activity stars, the aspect of Figure 13 that is most likely to catch the eye is probably the “jagged” behavior at early spectral types. But we would like to draw attention first to a different aspect of the RACs: namely, the values of the slopes *a* at the latest spectral types (dM3e and dM4e). In these cases, our results indicate that the RAC slopes for high-activity stars are essentially the same as for the low-activity stars. Once again relying on a putative association between an RAC slope and a dynamo mechanism, these results suggest that, at spectral types dM3 and dM4, the dynamo mechanism in low- and high-activity stars are actually the same. It has already been suggested (see (i)) that a DD is at work in all low-activity stars. This leads us to Hypothesis (B): in the high-activity stars at the latest sub-types in our samples (dM3e and dM4e), a DD is at work. This is not a surprising conclusion: dM3e and dM4e stars are completely convective (CC), so they also have access to DD operation in the form of either an $\alpha^2$ dynamo model or an $\alpha^2-\Omega$ dynamo model.

(iii) Moving now to high-activity stars at earlier spectral types (dK4e, dK6e, and dM2e), we see in Figure 13 that the RAC slopes are very different from the slopes of the low-activity stars. This suggests that high-activity stars in the range dK4e–dM2e have access to a different kind of dynamo from the type (DD) that may dominate in low-activity stars. What might give rise to a non-DD type of dynamo in stars with spectral types in the range dK4–dM2? The answer is surely related to the fact that such stars have interfaces between the outer convective envelope and an inner radiative core. This leads to Hypothesis (C): in high-activity stars at the earliest

spectral types in our samples, an interface dynamo (ID) is at work. It seems probable that such a dynamo could be described by an $\alpha-\Omega$ dynamo model. To be sure, dK4e–dM2e stars also have deep convective envelopes: therefore a DD is probably also at work. But the clear difference in RAC slopes between low-activity stars in the range dK4–dM2 and high-activity stars in the range dK4e–dM2e indicates that the DD (with its shallow RAC slope) is not playing a dominant role in dK4e–dM2e stars.

(iv) If our hypotheses have any validity, we can conclude that the primary quantitative difference between ID and DD is this: the RAC slopes are steeper (by up to 1.5 units) among the ID stars than the DD stars. Thus, the ID stars are much more sensitive to rotation than the DD stars. Specifically, if we compare two stars that differ in rotation by a factor of (say) 10, two stars where ID operates will differ in activity level by a factor that is 30 times larger than the difference in activity level of two stars where DD operates.

(v) The conclusion in item (iv) is reminiscent of a suggestion that was made by Durney et al. (1993): the RAC in a star with ID should depend sensitively on period, but the RAC in a star with DD should not. The results in Figure 13 are at least partially consistent with Durney et al.: ID stars (as we identify them) are definitely more sensitive to rotation than DD stars (as we identify them). Admittedly, we have not found that the DD has zero slope for its RAC: but we have found that the slope is at least smaller than in the ID stars.

The fact that the DD stars have an RAC with a non-zero slope suggests that rotation $\Omega$ does have some effect on the activity level in these stars. In terms of the two options that we have proposed in Hypotheses A and B above, it might be preferable to conclude that the dominant dynamo model in DD stars may be the $\alpha^2-\Omega$ model.

Finally, we make a point about the overall methodology that has been ultimately responsible for this paper. The opening sentence in item (i) above, which provides a start to our three hypotheses, would not have been possible if we did not have access to a large sample of low-activity stars. Such stars are slow rotators. Therefore, if the lead author of this study (E.R. H.) had not paid attention to extracting the lowest possible values of $v/\sin i$ (of order 1 km s$^{-1}$) in as many stars as possible, our sample of low-activity stars might have been so small as to prevent us from drawing statistically significant conclusions.

## 5.8. Any Evidence for Crossing the TTCC?

The theoretical concept that main-sequence stars undergo a TTCC at a particular mass (in the vicinity of spectral types M2–M4) has been in the literature for 50 years or more. Nevertheless, the search for an empirical signature that might support this concept has yielded no definitive evidence. For example, Mullan & MacDonald (2001) sought such evidence in X-ray data, but were unable to identify any signature of the TTCC. Also, in our own coronal data (Figure 16), we see no definitive sign of a transition. Neither could we find evidence for the TTCC in the RACs of chromospheric data when we combined low- and high-activity stars (Figure 13), monotonically rising line from lower left to upper right. Nor could we find any evidence of TTCC when we confined our attention to the chromospheric RACs of low-activity stars (lowest lying line in Figure 13.
However, in one particular sample, our study has led to what we believe is a potentially valuable signature. Specifically, in our sample of high-activity stars between dK4e and dM4e, the slopes of the chromospheric RAC (Figure 13, “jagged” line) consists of two distinct regimes. In one regime, the slopes overlap with the (shallow) slopes of low-activity stars. In the other regime, the slopes are found to be much steeper. We interpret the transition between the two regimes as evidence for a transition between dynamo modes. The transition occurs between dM2e and dM3e. We suggest that this cross-over from steep to shallow RAC slopes may provide empirical evidence that the TTCC has been crossed.

We have shown how empirical information about rotation and chromospheric emission may contribute to understanding some aspects of dynamo mechanisms in cool dwarfs. Data for M dwarfs with spectral types later than M4, and also for L and T dwarfs, would be of great interest to complete our view on the dynamo mechanisms in low-mass stars. For example, is there any evidence that the efficiency of coronal heating actually decreases after we pass though the electrodynamic resonance that is expected to occur around dM3-dM4? It would also be of interest to investigate the variations of rotational and chromospheric properties by using finer grained samples of stars near the TTCC, i.e., using samples of stars all of which are confined to spectral types of dM2.5, dM3.5, and dM4.5 stars. This could contribute to better understanding of the behavior of possible changes in the dynamo mechanism(s) at the TTCC.

This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. D.J.M. is supported in part by the NASA Space Grant program. This study 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. We also used the tutorial developed by Paletou & Zolotukhin (2014).<fnref n="10" /> This research was achieved using the POLLUX database (http://pollux.graal.univ-montp2.fr) operated at LUPM (Université Montpellier-CNRS, France) with the support of the PNPS and INSU. 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.

# Appendix
## Quadratic Heteroscedastic Regression Models

This appendix is devoted to classical mathematical results on stochastic linear regression models. Our purpose is to show how to better take into account the measurement errors of $P/\sin i$ in order to explain the variability of Ca II surfaces fluxes. In the previous literature, only linear or quadratic homoscedastic LSF were investigated in order to explain this variability. Our strategy is slightly different as we propose to make use of linear or quadratic heteroscedastic LSF in order to better explain this variability.

More precisely, consider one of our five groups of dK4, dK6, dM2, dM3, or dM4 stars, for example dM2. Let $n > 3$ be the total number of the dM2 stars under study. For each dM2 star,

denote by $x_k$ the $\log(P/\sin i)$ associated measure and by $\sigma_k > 0$ the measurement error of $\log(P/\sin i)$. Moreover, let $\dot{Y}_k$ be the $\log(Ca II Flux)$ measure of this dM2 star. Then, we shall deal with the linear or quadratic heteroscedastic regression models, respectively given, for $k = 1, \dots, n$, by

$$Y_k = a + bx_k + \sigma_k \varepsilon_k, \quad (37)$$

and

$$Y_k = a + bx_k + cx_k^2 + \sigma_k \varepsilon_k, \quad (38)$$

where $a, b$, and $c$ are unknown parameters and the random noise ($\varepsilon_k$) is a standard Gaussian white noise with mean zero and unknown variance $\tau^2$. The important point is the crucial role played by the error term $\sigma_k$. The variance of the additive noise clearly depends on the explanatory variable $x_k$ as it is given by $\sigma_k^2 \tau^2$ where $\sigma_k$ is the measurement error associated with $x_k$. We shall only focus our attention on the quadratic heteroscedastic regression model given by (38) inasmuch as the linear regression model (37) is a particular case of (38).

The quadratic heteroscedastic regression model (38) can be rewritten into the matrix form

$$Y = X\theta + \Gamma^{1/2}\varepsilon, \quad (39)$$

where the vector of observations $Y$, the vector of unknown parameters $\theta$, and the vector containing the random noises $\varepsilon$, are respectively given by

$$Y = \begin{pmatrix} Y_1 \\ \vdots \\ Y_n \end{pmatrix}, \quad \theta = \begin{pmatrix} a \\ b \\ c \end{pmatrix}, \quad \varepsilon = \begin{pmatrix} \varepsilon_1 \\ \vdots \\ \varepsilon_n \end{pmatrix}.$$

In addition, the heteroscedastic matrix $\Gamma = \text{diag}(\sigma_1^2, \sigma_2^2, \dots, \sigma_n^2)$ and the design matrix $X$ is given by

$$X = \begin{pmatrix} 1 & x_1 & x_1^2 \\ 1 & x_2 & x_2^2 \\ \vdots & \vdots & \vdots \\ 1 & x_n & x_n^2 \end{pmatrix}.$$

On the one hand, the least squares estimator (LSE) of $\theta$ is the value $\hat{\theta}$ that minimizes the strictly convex function $\tilde{\Delta}(\theta) = \|Y - X\theta\|^2$. Straightforward calculation leads to

$$\tilde{\theta} = (X'X)^{-1}X'Y.$$

However, this estimator does not taken into account the heteroscedasticity of model (39). On the other hand, the weighted least squares estimator (WLSE) of $\theta$ is the value $\hat{\theta}$, which minimizes the strictly convex function

$$\hat{\Delta}(\theta) = (Y - X\theta)'\Gamma^{-1}(Y - X\theta).$$

It is not hard to see that

$$\hat{\theta} = (X'\Gamma^{-1}X)^{-1}X'\Gamma^{-1}Y. \quad (40)$$

We immediately deduce from (39) and (40) that

$$\hat{\theta} = \theta + (X'\Gamma^{-1}X)^{-1}X'\Gamma^{-1/2}\varepsilon.$$

Consequently, as $\varepsilon$ is a $n$-dimensional Gaussian vector $N(0, \tau^2 I)$ where $I$ stands for the identity matrix of order $n$, we

<footnote n="9">⁹ http://www.astropy.org/ and http://astroquery.readthedocs.org/en/latest/.</footnote>

<footnote n="10">¹⁰ http://www.astropy.org/</footnote>
obtain that $\hat{\theta}$ is a three-dimensional Gaussian vector,

$$ \hat{\theta} = \begin{pmatrix} \hat{a} \\ \hat{b} \\ \hat{c} \end{pmatrix} \sim \mathcal{N}(\theta, \tau^2 (X'\Gamma^{-1}X)^{-1}). \quad (41) $$

Hereafter, the linear least squares fit (LSF) is the line

$$ y = \hat{a} + \hat{b}x, \quad (42) $$

while the quadratic LSF is the curve

$$ y = \hat{a} + \hat{b}x + \hat{c}x^2. \quad (43) $$

Furthermore, denote by $H$ the hat matrix

$$ H = X(X'\Gamma^{-1}X)^{-1}X'\Gamma^{-1} \quad \text{and} \quad L = I - H. $$

It follows from (39) to (40) that $\hat{Y} = HY = X\hat{\theta}$, which implies that $HY$ has an $\mathcal{N}(X\theta, \tau^2H\Gamma)$ distribution and $LY = Y - \hat{Y}$ has an $\mathcal{N}(0, \tau^2L\Gamma)$ distribution. Hence, as $\text{rank}(L) = n - 3$, we obtain that the sum of squared errors (SSE) has a chi-squared distribution,

$$ \|\Gamma^{-1/2}(Y - \hat{Y})\|^2 \sim \tau^2 \chi^2(n - 3). \quad (44) $$

The SSE is a way to evaluate the discrepancy between the data $Y$ and its estimate $\hat{Y}$ and a small value of the SSE indicates a tight fit of our model to the data. Let us recall the celebrated decomposition of total sum of squares, which is in fact a direct application of Pythagoras's theorem

$$ \|\Gamma^{-1/2}Y\|^2 = \|\Gamma^{-1/2}\hat{Y}\|^2 + \|\Gamma^{-1/2}(Y - \hat{Y})\|^2. $$

Finally, a natural estimator of the variance $\tau^2$ is

$$ \hat{\tau}^2 = \frac{\|\Gamma^{-1/2}(Y - \hat{Y})\|^2}{n - 3}. \quad (45) $$

It is not hard to see that the random vector $\hat{\theta}$ and $\hat{\tau}^2$ are independent. Consequently, we deduce from (41) that for any real number $x$,

$$ (\hat{a} - a) + (\hat{b} - b)x + (\hat{c} - c)x^2 \sim \mathcal{N}(0, \tau^2\xi(x)) $$

where

$$ \xi(x) = \begin{pmatrix} 1 \\ x \\ x^2 \end{pmatrix}^t (X'\Gamma^{-1}X)^{-1} \begin{pmatrix} 1 \\ x \\ x^2 \end{pmatrix}. $$

Dividing on both sides by $\hat{\tau}$, the ratio has a Student distribution,

$$ \frac{(\hat{a} - a) + (\hat{b} - b)x + (\hat{c} - c)x^2}{\hat{\tau}\sqrt{\xi(x)}} \sim t(n - 3). \quad (46) $$

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