P et $\dot{P}$ pour un échantillon de modes p sur la séquence $M = 0.4758 + 0.0022 M_{\odot}$

<table><thead><tr><th rowspan="2">âge<br>(Ma)</th><th rowspan="2">Y<sub>c</sub></th><th colspan="2"><i>p</i><sub>0,0</sub></th><th colspan="2"><i>p</i><sub>0,2</sub></th><th colspan="2"><i>p</i><sub>0,4</sub></th><th colspan="2"><i>p</i><sub>0,6</sub></th></tr><tr><th><i>P</i><br>(s)</th><th><i>P&#775;</i><br>(s/Ma)</th><th><i>P</i><br>(s)</th><th><i>P&#775;</i><br>(s/Ma)</th><th><i>P</i><br>(s)</th><th><i>P&#775;</i><br>(s/Ma)</th><th><i>P</i><br>(s)</th><th><i>P&#775;</i><br>(s/Ma)</th></tr></thead><tbody><tr><td>4.49</td><td>0.893</td><td>313.57</td><td>+2.19×10<sup>0</sup></td><td>186.28</td><td>+1.30×10<sup>0</sup></td><td>144.31</td><td>+7.09×10<sup>-1</sup></td><td>114.50</td><td>+7.80×10<sup>-1</sup></td></tr><tr><td>19.95</td><td>0.741</td><td>346.55</td><td>+2.03×10<sup>0</sup></td><td>205.85</td><td>+1.21×10<sup>0</sup></td><td>154.78</td><td>+6.14×10<sup>-1</sup></td><td>126.21</td><td>+7.12×10<sup>-1</sup></td></tr><tr><td>24.07</td><td>0.709</td><td>354.76</td><td>+1.97×10<sup>0</sup></td><td>210.75</td><td>+1.17×10<sup>0</sup></td><td>157.27</td><td>+6.06×10<sup>-1</sup></td><td>129.09</td><td>+6.91×10<sup>-1</sup></td></tr><tr><td>37.88</td><td>0.603</td><td>381.62</td><td>+1.94×10<sup>0</sup></td><td>226.78</td><td>+1.16×10<sup>0</sup></td><td>166.28</td><td>+6.90×10<sup>-1</sup></td><td>138.50</td><td>+6.69×10<sup>-1</sup></td></tr><tr><td>55.12</td><td>0.469</td><td>416.06</td><td>+2.10×10<sup>0</sup></td><td>247.58</td><td>+1.28×10<sup>0</sup></td><td>179.22</td><td>+8.37×10<sup>-1</sup></td><td>150.20</td><td>+7.16×10<sup>-1</sup></td></tr><tr><td>67.76</td><td>0.360</td><td>443.55</td><td>+2.20×10<sup>0</sup></td><td>264.21</td><td>+1.33×10<sup>0</sup></td><td>190.31</td><td>+8.78×10<sup>-1</sup></td><td>159.27</td><td>+6.74×10<sup>-1</sup></td></tr><tr><td>77.17</td><td>0.274</td><td>463.28</td><td>+1.89×10<sup>0</sup></td><td>276.10</td><td>+1.16×10<sup>0</sup></td><td>198.18</td><td>+7.83×10<sup>-1</sup></td><td>164.90</td><td>+5.14×10<sup>-1</sup></td></tr><tr><td>84.13</td><td>0.206</td><td>474.91</td><td>+1.45×10<sup>0</sup></td><td>283.25</td><td>+8.86×10<sup>-1</sup></td><td>203.09</td><td>+5.95×10<sup>-1</sup></td><td>167.58</td><td>+1.96×10<sup>-1</sup></td></tr><tr><td>89.27</td><td>0.154</td><td>479.29</td><td>-1.67×10<sup>-1</sup></td><td>285.91</td><td>-1.05×10<sup>-1</sup></td><td>204.75</td><td>-1.19×10<sup>-1</sup></td><td>167.05</td><td>-5.21×10<sup>-1</sup></td></tr><tr><td>93.17</td><td>0.117</td><td>478.06</td><td>+4.80×10<sup>-1</sup></td><td>285.21</td><td>+3.36×10<sup>-1</sup></td><td>204.20</td><td>+2.81×10<sup>-1</sup></td><td>165.75</td><td>+4.47×10<sup>-1</sup></td></tr><tr><td>94.45</td><td>0.107</td><td>477.54</td><td>-2.75×10<sup>0</sup></td><td>284.94</td><td>-1.67×10<sup>0</sup></td><td>204.07</td><td>-1.14×10<sup>0</sup></td><td>166.16</td><td>-3.62×10<sup>-1</sup></td></tr><tr><td>96.22</td><td>0.130</td><td>473.50</td><td>+3.92×10<sup>0</sup></td><td>282.43</td><td>+2.36×10<sup>0</sup></td><td>202.29</td><td>+1.62×10<sup>0</sup></td><td>165.20</td><td>+8.31×10<sup>-1</sup></td></tr><tr><td>97.60</td><td>0.109</td><td>482.91</td><td>+3.34×10<sup>0</sup></td><td>288.17</td><td>+2.06×10<sup>0</sup></td><td>206.27</td><td>+1.43×10<sup>0</sup></td><td>167.34</td><td>+6.91×10<sup>-1</sup></td></tr><tr><td>99.44</td><td>0.086</td><td>474.93</td><td>-6.54×10<sup>0</sup></td><td>283.37</td><td>-3.95×10<sup>0</sup></td><td>202.93</td><td>-2.77×10<sup>0</sup></td><td>164.95</td><td>-2.04×10<sup>0</sup></td></tr><tr><td>101.10</td><td>0.070</td><td>466.75</td><td>-4.67×10<sup>0</sup></td><td>278.43</td><td>-2.82×10<sup>0</sup></td><td>199.43</td><td>-1.99×10<sup>0</sup></td><td>162.00</td><td>-1.69×10<sup>0</sup></td></tr><tr><td>102.92</td><td>0.054</td><td>456.58</td><td>-6.24×10<sup>0</sup></td><td>272.30</td><td>-3.75×10<sup>0</sup></td><td>195.13</td><td>-2.63×10<sup>0</sup></td><td>158.57</td><td>-2.16×10<sup>0</sup></td></tr><tr><td>104.62</td><td>0.041</td><td>444.91</td><td>-7.76×10<sup>0</sup></td><td>265.28</td><td>-4.69×10<sup>0</sup></td><td>190.18</td><td>-3.33×10<sup>0</sup></td><td>154.37</td><td>-2.77×10<sup>0</sup></td></tr><tr><td>106.06</td><td>0.031</td><td>432.21</td><td>-9.95×10<sup>0</sup></td><td>257.60</td><td>-6.01×10<sup>0</sup></td><td>184.74</td><td>-4.21×10<sup>0</sup></td><td>150.19</td><td>-2.93×10<sup>0</sup></td></tr><tr><td>107.23</td><td>0.023</td><td>418.90</td><td>-1.33×10<sup>1</sup></td><td>249.59</td><td>-7.96×10<sup>0</sup></td><td>179.13</td><td>-5.61×10<sup>0</sup></td><td>146.19</td><td>-4.47×10<sup>0</sup></td></tr><tr><td>108.16</td><td>0.017</td><td>405.41</td><td>-1.52×10<sup>1</sup></td><td>241.48</td><td>-9.15×10<sup>0</sup></td><td>173.42</td><td>-6.37×10<sup>0</sup></td><td>141.63</td><td>-4.48×10<sup>0</sup></td></tr><tr><td>110.22</td><td>0.002</td><td>330.19</td><td>-7.91×10<sup>1</sup></td><td>196.39</td><td>-4.72×10<sup>1</sup></td><td>141.59</td><td>-3.38×10<sup>1</sup></td><td>116.85</td><td>-2.99×10<sup>1</sup></td></tr><tr><td>110.52</td><td>0.000</td><td>300.31</td><td>-1.31×10<sup>2</sup></td><td>178.60</td><td>-7.81×10<sup>1</sup></td><td>128.83</td><td>-5.60×10<sup>1</sup></td><td>105.61</td><td>-4.84×10<sup>1</sup></td></tr><tr><td>110.66</td><td>0.000</td><td>280.07</td><td>-1.51×10<sup>2</sup></td><td>166.55</td><td>-9.00×10<sup>1</sup></td><td>120.19</td><td>-6.45×10<sup>1</sup></td><td>98.21</td><td>-5.51×10<sup>1</sup></td></tr></tbody></table>

<table><thead><tr><th rowspan="2">âge<br>(Ma)</th><th rowspan="2">Y<sub>c</sub></th><th colspan="2"><i>p</i><sub>0,8</sub></th><th colspan="2"><i>p</i><sub>1,4</sub></th><th colspan="2"><i>p</i><sub>2,4</sub></th><th colspan="2"><i>p</i><sub>3,4</sub></th></tr><tr><th><i>P</i><br>(s)</th><th><i>P&#775;</i><br>(s/Ma)</th><th><i>P</i><br>(s)</th><th><i>P&#775;</i><br>(s/Ma)</th><th><i>P</i><br>(s)</th><th><i>P&#775;</i><br>(s/Ma)</th><th><i>P</i><br>(s)</th><th><i>P&#775;</i><br>(s/Ma)</th></tr></thead><tbody><tr><td>4.49</td><td>0.893</td><td>93.06</td><td>+5.94×10<sup>-1</sup></td><td>154.76</td><td>+1.08×10<sup>0</sup></td><td>149.94</td><td>+8.98×10<sup>-1</sup></td><td>138.55</td><td>+7.86×10<sup>-1</sup></td></tr><tr><td>19.95</td><td>0.741</td><td>102.00</td><td>+5.48×10<sup>-1</sup></td><td>171.00</td><td>+1.00×10<sup>0</sup></td><td>162.95</td><td>+7.27×10<sup>-1</sup></td><td>150.46</td><td>+7.37×10<sup>-1</sup></td></tr><tr><td>24.07</td><td>0.709</td><td>104.23</td><td>+5.35×10<sup>-1</sup></td><td>175.07</td><td>+9.74×10<sup>-1</sup></td><td>165.79</td><td>+6.60×10<sup>-1</sup></td><td>153.45</td><td>+7.19×10<sup>-1</sup></td></tr><tr><td>37.88</td><td>0.603</td><td>111.60</td><td>+5.36×10<sup>-1</sup></td><td>188.34</td><td>+9.62×10<sup>-1</sup></td><td>174.04</td><td>+5.47×10<sup>-1</sup></td><td>163.30</td><td>+7.18×10<sup>-1</sup></td></tr><tr><td>55.12</td><td>0.469</td><td>121.24</td><td>+5.98×10<sup>-1</sup></td><td>205.58</td><td>+1.06×10<sup>0</sup></td><td>183.41</td><td>+5.88×10<sup>-1</sup></td><td>176.38</td><td>+8.19×10<sup>-1</sup></td></tr><tr><td>67.76</td><td>0.360</td><td>129.03</td><td>+6.14×10<sup>-1</sup></td><td>219.32</td><td>+1.09×10<sup>0</sup></td><td>191.43</td><td>+6.61×10<sup>-1</sup></td><td>186.87</td><td>+7.93×10<sup>-1</sup></td></tr><tr><td>77.17</td><td>0.274</td><td>134.49</td><td>+5.32×10<sup>-1</sup></td><td>229.12</td><td>+9.55×10<sup>-1</sup></td><td>197.61</td><td>+6.43×10<sup>-1</sup></td><td>193.98</td><td>+7.44×10<sup>-1</sup></td></tr><tr><td>84.13</td><td>0.206</td><td>137.75</td><td>+3.86×10<sup>-1</sup></td><td>235.04</td><td>+7.34×10<sup>-1</sup></td><td>201.63</td><td>+4.67×10<sup>-1</sup></td><td>197.75</td><td>+1.28×10<sup>-1</sup></td></tr><tr><td>89.27</td><td>0.154</td><td>138.73</td><td>-1.28×10<sup>-1</sup></td><td>237.24</td><td>-8.61×10<sup>-2</sup></td><td>202.61</td><td>-2.46×10<sup>-1</sup></td><td>193.97</td><td>-2.00×10<sup>0</sup></td></tr><tr><td>93.17</td><td>0.117</td><td>138.28</td><td>+2.42×10<sup>-1</sup></td><td>236.68</td><td>+2.84×10<sup>-1</sup></td><td>201.74</td><td>+2.74×10<sup>-1</sup></td><td>189.43</td><td>+1.77×10<sup>0</sup></td></tr><tr><td>94.45</td><td>0.107</td><td>138.28</td><td>-7.07×10<sup>-1</sup></td><td>236.46</td><td>-1.38×10<sup>0</sup></td><td>201.71</td><td>-9.34×10<sup>-1</sup></td><td>192.40</td><td>+1.78×10<sup>0</sup></td></tr><tr><td>96.22</td><td>0.130</td><td>137.11</td><td>+1.06×10<sup>0</sup></td><td>234.38</td><td>+1.94×10<sup>0</sup></td><td>200.24</td><td>+1.48×10<sup>0</sup></td><td>193.30</td><td>+2.16×10<sup>-1</sup></td></tr><tr><td>97.60</td><td>0.109</td><td>139.75</td><td>+9.57×10<sup>-1</sup></td><td>239.11</td><td>+1.71×10<sup>0</sup></td><td>203.77</td><td>+1.21×10<sup>0</sup></td><td>193.77</td><td>+3.65×10<sup>-2</sup></td></tr><tr><td>99.44</td><td>0.086</td><td>137.53</td><td>-1.86×10<sup>0</sup></td><td>235.17</td><td>-3.25×10<sup>0</sup></td><td>200.50</td><td>-2.69×10<sup>0</sup></td><td>191.90</td><td>-2.15×10<sup>0</sup></td></tr><tr><td>101.10</td><td>0.070</td><td>135.15</td><td>-1.36×10<sup>0</sup></td><td>231.10</td><td>-2.32×10<sup>0</sup></td><td>196.99</td><td>-2.01×10<sup>0</sup></td><td>187.52</td><td>-2.51×10<sup>0</sup></td></tr><tr><td>102.92</td><td>0.054</td><td>132.22</td><td>-1.80×10<sup>0</sup></td><td>226.06</td><td>-3.08×10<sup>0</sup></td><td>192.71</td><td>-2.63×10<sup>0</sup></td><td>182.90</td><td>-3.41×10<sup>0</sup></td></tr><tr><td>104.62</td><td>0.041</td><td>128.82</td><td>-2.26×10<sup>0</sup></td><td>220.29</td><td>-3.86×10<sup>0</sup></td><td>187.72</td><td>-3.34×10<sup>0</sup></td><td>175.75</td><td>-4.22×10<sup>0</sup></td></tr><tr><td>106.06</td><td>0.031</td><td>125.18</td><td>-2.79×10<sup>0</sup></td><td>213.95</td><td>-4.96×10<sup>0</sup></td><td>182.37</td><td>-4.04×10<sup>0</sup></td><td>170.93</td><td>-2.26×10<sup>0</sup></td></tr><tr><td>107.23</td><td>0.023</td><td>121.44</td><td>-3.80×10<sup>0</sup></td><td>207.34</td><td>-6.56×10<sup>0</sup></td><td>176.96</td><td>-5.58×10<sup>0</sup></td><td>167.34</td><td>-5.75×10<sup>0</sup></td></tr><tr><td>108.16</td><td>0.017</td><td>117.56</td><td>-4.29×10<sup>0</sup></td><td>200.65</td><td>-7.55×10<sup>0</sup></td><td>171.32</td><td>-5.96×10<sup>0</sup></td><td>161.44</td><td>-3.94×10<sup>0</sup></td></tr><tr><td>110.22</td><td>0.002</td><td>96.12</td><td>-2.30×10<sup>1</sup></td><td>163.32</td><td>-3.91×10<sup>1</sup></td><td>140.14</td><td>-3.50×10<sup>1</sup></td><td>133.53</td><td>-4.47×10<sup>1</sup></td></tr><tr><td>110.52</td><td>0.000</td><td>87.46</td><td>-3.80×10<sup>1</sup></td><td>148.56</td><td>-6.49×10<sup>1</sup></td><td>126.94</td><td>-5.76×10<sup>1</sup></td><td>117.51</td><td>-6.30×10<sup>1</sup></td></tr><tr><td>110.66</td><td>0.000</td><td>81.61</td><td>-4.37×10<sup>1</sup></td><td>138.54</td><td>-7.48×10<sup>1</sup></td><td>118.09</td><td>-6.61×10<sup>1</sup></td><td>108.24</td><td>-6.79×10<sup>1</sup></td></tr></tbody></table>
P et $\dot{P}$ pour un échantillon de modes p sur la séquence $M = 0.4758 + 0.0032 M_{\odot}$

<table><thead><tr><th rowspan="2">âge (Ma)</th><th rowspan="2">Y<sub>c</sub></th><th colspan="2"><i>p</i><sub>0,0</sub></th><th colspan="2"><i>p</i><sub>0,2</sub></th><th colspan="2"><i>p</i><sub>0,4</sub></th><th colspan="2"><i>p</i><sub>0,6</sub></th></tr><tr><th><i>P</i> (s)</th><th><i>P&#775;</i> (s/Ma)</th><th><i>P</i> (s)</th><th><i>P&#775;</i> (s/Ma)</th><th><i>P</i> (s)</th><th><i>P&#775;</i> (s/Ma)</th><th><i>P</i> (s)</th><th><i>P&#775;</i> (s/Ma)</th></tr></thead><tbody><tr><td>1.79</td><td>0.927</td><td>348.97</td><td>+2.70×10<sup>0</sup></td><td>208.10</td><td>+1.61×10<sup>0</sup></td><td>151.61</td><td>+1.02×10<sup>0</sup></td><td>127.31</td><td>+9.25×10<sup>-1</sup></td></tr><tr><td>12.06</td><td>0.809</td><td>376.28</td><td>+2.57×10<sup>0</sup></td><td>224.48</td><td>+1.56×10<sup>0</sup></td><td>162.20</td><td>+1.05×10<sup>0</sup></td><td>136.58</td><td>+8.56×10<sup>-1</sup></td></tr><tr><td>19.53</td><td>0.742</td><td>395.04</td><td>+2.48×10<sup>0</sup></td><td>235.89</td><td>+1.50×10<sup>0</sup></td><td>170.06</td><td>+1.02×10<sup>0</sup></td><td>142.56</td><td>+7.42×10<sup>-1</sup></td></tr><tr><td>23.43</td><td>0.709</td><td>404.64</td><td>+2.44×10<sup>0</sup></td><td>241.67</td><td>+1.47×10<sup>0</sup></td><td>173.98</td><td>+9.94×10<sup>-1</sup></td><td>145.37</td><td>+7.07×10<sup>-1</sup></td></tr><tr><td>39.83</td><td>0.585</td><td>443.38</td><td>+2.35×10<sup>0</sup></td><td>265.06</td><td>+1.42×10<sup>0</sup></td><td>190.17</td><td>+9.89×10<sup>-1</sup></td><td>156.37</td><td>+6.03×10<sup>-1</sup></td></tr><tr><td>56.38</td><td>0.456</td><td>483.80</td><td>+2.59×10<sup>0</sup></td><td>289.57</td><td>+1.58×10<sup>0</sup></td><td>207.27</td><td>+1.11×10<sup>0</sup></td><td>166.01</td><td>+6.44×10<sup>-1</sup></td></tr><tr><td>68.54</td><td>0.348</td><td>516.94</td><td>+2.85×10<sup>0</sup></td><td>309.81</td><td>+1.74×10<sup>0</sup></td><td>221.54</td><td>+1.22×10<sup>0</sup></td><td>174.72</td><td>+7.55×10<sup>-1</sup></td></tr><tr><td>77.44</td><td>0.258</td><td>541.73</td><td>+2.59×10<sup>0</sup></td><td>324.96</td><td>+1.59×10<sup>0</sup></td><td>232.17</td><td>+1.12×10<sup>0</sup></td><td>181.44</td><td>+7.50×10<sup>-1</sup></td></tr><tr><td>83.92</td><td>0.189</td><td>555.78</td><td>+1.58×10<sup>0</sup></td><td>333.62</td><td>+9.79×10<sup>-1</sup></td><td>238.32</td><td>+6.98×10<sup>-1</sup></td><td>185.69</td><td>+4.73×10<sup>-1</sup></td></tr><tr><td>88.76</td><td>0.138</td><td>559.75</td><td>-1.26×10<sup>-1</sup></td><td>336.13</td><td>-5.47×10<sup>-2</sup></td><td>240.13</td><td>-2.96×10<sup>-2</sup></td><td>186.72</td><td>-1.02×10<sup>-1</sup></td></tr><tr><td>92.27</td><td>0.102</td><td>556.04</td><td>-2.10×10<sup>0</sup></td><td>333.91</td><td>-1.29×10<sup>0</sup></td><td>238.57</td><td>-9.21×10<sup>-1</sup></td><td>185.26</td><td>-7.76×10<sup>-1</sup></td></tr><tr><td>95.10</td><td>0.076</td><td>546.81</td><td>-4.64×10<sup>0</sup></td><td>328.28</td><td>-2.79×10<sup>0</sup></td><td>234.59</td><td>-1.94×10<sup>0</sup></td><td>182.11</td><td>-1.46×10<sup>0</sup></td></tr><tr><td>97.37</td><td>0.056</td><td>533.49</td><td>-7.03×10<sup>0</sup></td><td>320.27</td><td>-4.28×10<sup>0</sup></td><td>229.04</td><td>-3.01×10<sup>0</sup></td><td>177.96</td><td>-2.28×10<sup>0</sup></td></tr><tr><td>99.17</td><td>0.041</td><td>518.20</td><td>-1.05×10<sup>1</sup></td><td>310.92</td><td>-6.42×10<sup>0</sup></td><td>222.41</td><td>-4.54×10<sup>0</sup></td><td>172.86</td><td>-3.53×10<sup>0</sup></td></tr><tr><td>100.58</td><td>0.029</td><td>500.73</td><td>-1.41×10<sup>1</sup></td><td>300.32</td><td>-8.52×10<sup>0</sup></td><td>214.93</td><td>-6.00×10<sup>0</sup></td><td>167.10</td><td>-4.56×10<sup>0</sup></td></tr><tr><td>101.68</td><td>0.021</td><td>483.67</td><td>-2.33×10<sup>1</sup></td><td>289.99</td><td>-1.42×10<sup>1</sup></td><td>207.66</td><td>-1.01×10<sup>1</sup></td><td>161.55</td><td>-7.70×10<sup>0</sup></td></tr><tr><td>103.04</td><td>0.011</td><td>451.93</td><td>-6.50×10<sup>1</sup></td><td>270.67</td><td>-3.92×10<sup>1</sup></td><td>193.95</td><td>-2.77×10<sup>1</sup></td><td>151.08</td><td>-2.13×10<sup>1</sup></td></tr><tr><td>104.51</td><td>0.000</td><td>356.44</td><td>-1.37×10<sup>2</sup></td><td>213.10</td><td>-8.12×10<sup>1</sup></td><td>153.19</td><td>-5.71×10<sup>1</sup></td><td>119.70</td><td>-4.57×10<sup>1</sup></td></tr><tr><td>104.62</td><td>0.000</td><td>341.39</td><td>-2.10×10<sup>2</sup></td><td>204.17</td><td>-1.26×10<sup>2</sup></td><td>146.91</td><td>-8.98×10<sup>1</sup></td><td>114.67</td><td>-6.90×10<sup>1</sup></td></tr><tr><td>104.82</td><td>0.000</td><td>299.42</td><td>-6.68×10<sup>0</sup></td><td>178.98</td><td>-4.02×10<sup>0</sup></td><td>128.95</td><td>-2.89×10<sup>0</sup></td><td>100.88</td><td>-2.32×10<sup>0</sup></td></tr></tbody></table>

<table><thead><tr><th rowspan="2">âge (Ma)</th><th rowspan="2">Y<sub>c</sub></th><th colspan="2"><i>p</i><sub>0,8</sub></th><th colspan="2"><i>p</i><sub>1,4</sub></th><th colspan="2"><i>p</i><sub>2,4</sub></th><th colspan="2"><i>p</i><sub>3,4</sub></th></tr><tr><th><i>P</i> (s)</th><th><i>P&#775;</i> (s/Ma)</th><th><i>P</i> (s)</th><th><i>P&#775;</i> (s/Ma)</th><th><i>P</i> (s)</th><th><i>P&#775;</i> (s/Ma)</th><th><i>P</i> (s)</th><th><i>P&#775;</i> (s/Ma)</th></tr></thead><tbody><tr><td>1.79</td><td>0.927</td><td>102.27</td><td>+7.64×10<sup>-1</sup></td><td>172.79</td><td>+1.33×10<sup>0</sup></td><td>159.15</td><td>+7.42×10<sup>-1</sup></td><td>148.61</td><td>+1.02×10<sup>0</sup></td></tr><tr><td>12.06</td><td>0.809</td><td>110.05</td><td>+7.45×10<sup>-1</sup></td><td>186.36</td><td>+1.30×10<sup>0</sup></td><td>166.85</td><td>+7.66×10<sup>-1</sup></td><td>159.13</td><td>+1.03×10<sup>0</sup></td></tr><tr><td>19.53</td><td>0.742</td><td>115.49</td><td>+7.07×10<sup>-1</sup></td><td>195.84</td><td>+1.24×10<sup>0</sup></td><td>172.67</td><td>+7.86×10<sup>-1</sup></td><td>166.77</td><td>+9.90×10<sup>-1</sup></td></tr><tr><td>23.43</td><td>0.709</td><td>118.21</td><td>+6.90×10<sup>-1</sup></td><td>200.63</td><td>+1.21×10<sup>0</sup></td><td>175.75</td><td>+7.95×10<sup>-1</sup></td><td>170.56</td><td>+9.59×10<sup>-1</sup></td></tr><tr><td>39.83</td><td>0.585</td><td>129.27</td><td>+6.66×10<sup>-1</sup></td><td>219.95</td><td>+1.17×10<sup>0</sup></td><td>189.41</td><td>+8.74×10<sup>-1</sup></td><td>185.78</td><td>+8.78×10<sup>-1</sup></td></tr><tr><td>56.38</td><td>0.456</td><td>140.63</td><td>+7.29×10<sup>-1</sup></td><td>240.16</td><td>+1.30×10<sup>0</sup></td><td>205.02</td><td>+1.04×10<sup>0</sup></td><td>201.36</td><td>+1.14×10<sup>0</sup></td></tr><tr><td>68.54</td><td>0.348</td><td>149.81</td><td>+7.58×10<sup>-1</sup></td><td>256.84</td><td>+1.43×10<sup>0</sup></td><td>218.38</td><td>+1.14×10<sup>0</sup></td><td>212.12</td><td>+1.37×10<sup>1</sup></td></tr><tr><td>77.44</td><td>0.258</td><td>156.26</td><td>+6.71×10<sup>-1</sup></td><td>269.26</td><td>+1.30×10<sup>0</sup></td><td>228.16</td><td>+1.01×10<sup>0</sup></td><td>209.86</td><td>-1.25×10<sup>-1</sup></td></tr><tr><td>83.92</td><td>0.189</td><td>159.85</td><td>+3.77×10<sup>-1</sup></td><td>276.38</td><td>+8.04×10<sup>-1</sup></td><td>233.71</td><td>+6.46×10<sup>-1</sup></td><td>209.95</td><td>-1.82×10<sup>-1</sup></td></tr><tr><td>88.76</td><td>0.138</td><td>160.34</td><td>-2.75×10<sup>-1</sup></td><td>278.45</td><td>-4.21×10<sup>-2</sup></td><td>234.72</td><td>-4.66×10<sup>-1</sup></td><td>207.68</td><td>-6.43×10<sup>-1</sup></td></tr><tr><td>92.27</td><td>0.102</td><td>158.41</td><td>-7.56×10<sup>-1</sup></td><td>276.63</td><td>-1.07×10<sup>0</sup></td><td>231.23</td><td>-1.40×10<sup>0</sup></td><td>204.95</td><td>-1.00×10<sup>0</sup></td></tr><tr><td>95.10</td><td>0.076</td><td>155.68</td><td>-1.27×10<sup>0</sup></td><td>272.00</td><td>-2.28×10<sup>0</sup></td><td>226.51</td><td>-1.99×10<sup>0</sup></td><td>201.29</td><td>-1.61×10<sup>0</sup></td></tr><tr><td>97.37</td><td>0.056</td><td>152.10</td><td>-1.86×10<sup>0</sup></td><td>265.47</td><td>-3.51×10<sup>0</sup></td><td>221.34</td><td>-2.55×10<sup>0</sup></td><td>196.76</td><td>-2.49×10<sup>0</sup></td></tr><tr><td>99.17</td><td>0.041</td><td>147.94</td><td>-3.02×10<sup>0</sup></td><td>257.78</td><td>-5.28×10<sup>0</sup></td><td>215.23</td><td>-4.90×10<sup>0</sup></td><td>191.12</td><td>-3.93×10<sup>0</sup></td></tr><tr><td>100.58</td><td>0.029</td><td>142.87</td><td>-3.98×10<sup>0</sup></td><td>249.07</td><td>-6.99×10<sup>0</sup></td><td>207.07</td><td>-5.87×10<sup>0</sup></td><td>184.70</td><td>-5.04×10<sup>0</sup></td></tr><tr><td>101.68</td><td>0.021</td><td>138.05</td><td>-6.25×10<sup>0</sup></td><td>240.58</td><td>-1.17×10<sup>1</sup></td><td>199.62</td><td>-8.78×10<sup>0</sup></td><td>178.60</td><td>-8.48×10<sup>0</sup></td></tr><tr><td>103.04</td><td>0.011</td><td>129.55</td><td>-1.82×10<sup>1</sup></td><td>224.65</td><td>-3.24×10<sup>1</sup></td><td>187.68</td><td>-2.75×10<sup>1</sup></td><td>167.07</td><td>-2.37×10<sup>1</sup></td></tr><tr><td>104.51</td><td>0.000</td><td>102.79</td><td>-4.36×10<sup>1</sup></td><td>177.07</td><td>-6.73×10<sup>1</sup></td><td>147.23</td><td>-9.19×10<sup>1</sup></td><td>132.19</td><td>-5.11×10<sup>1</sup></td></tr><tr><td>104.62</td><td>0.000</td><td>97.99</td><td>-5.80×10<sup>1</sup></td><td>169.67</td><td>-1.05×10<sup>2</sup></td><td>137.13</td><td>-8.38×10<sup>1</sup></td><td>126.57</td><td>-7.64×10<sup>1</sup></td></tr><tr><td>104.82</td><td>0.000</td><td>86.39</td><td>-2.15×10<sup>0</sup></td><td>148.75</td><td>-4.19×10<sup>0</sup></td><td>120.36</td><td>-1.53×10<sup>1</sup></td><td>111.30</td><td>-3.06×10<sup>0</sup></td></tr></tbody></table>
P et $\dot{P}$ pour un échantillon de modes p sur la séquence $M = 0.4758 + 0.0042 M_{\odot}$

<table><thead><tr><th rowspan="2">âge<br>(Ma)</th><th rowspan="2">Y<sub>c</sub></th><th colspan="2"><i>p</i><sub>0,0</sub></th><th colspan="2"><i>p</i><sub>0,2</sub></th><th colspan="2"><i>p</i><sub>0,4</sub></th><th colspan="2"><i>p</i><sub>0,6</sub></th></tr><tr><th><i>P</i><br>(s)</th><th><i>P&#775;</i><br>(s/Ma)</th><th><i>P</i><br>(s)</th><th><i>P&#775;</i><br>(s/Ma)</th><th><i>P</i><br>(s)</th><th><i>P&#775;</i><br>(s/Ma)</th><th><i>P</i><br>(s)</th><th><i>P&#775;</i><br>(s/Ma)</th></tr></thead><tbody><tr><td>1.79</td><td>0.927</td><td>386.86</td><td>+3.10×10<sup>0</sup></td><td>231.59</td><td>+1.85×10<sup>0</sup></td><td>166.62</td><td>+1.27×10<sup>0</sup></td><td>138.31</td><td>+7.94×10<sup>-1</sup></td></tr><tr><td>12.06</td><td>0.809</td><td>418.38</td><td>+3.00×10<sup>0</sup></td><td>250.53</td><td>+1.83×10<sup>0</sup></td><td>179.77</td><td>+1.30×10<sup>0</sup></td><td>146.47</td><td>+7.95×10<sup>-1</sup></td></tr><tr><td>19.66</td><td>0.741</td><td>440.57</td><td>+2.83×10<sup>0</sup></td><td>264.13</td><td>+1.73×10<sup>0</sup></td><td>189.52</td><td>+1.23×10<sup>0</sup></td><td>152.36</td><td>+7.38×10<sup>-1</sup></td></tr><tr><td>23.84</td><td>0.707</td><td>452.26</td><td>+2.78×10<sup>0</sup></td><td>271.23</td><td>+1.68×10<sup>0</sup></td><td>194.53</td><td>+1.18×10<sup>0</sup></td><td>155.39</td><td>+7.25×10<sup>-1</sup></td></tr><tr><td>39.40</td><td>0.588</td><td>495.38</td><td>+2.78×10<sup>0</sup></td><td>297.37</td><td>+1.69×10<sup>0</sup></td><td>212.94</td><td>+1.19×10<sup>0</sup></td><td>167.40</td><td>+7.96×10<sup>-1</sup></td></tr><tr><td>55.93</td><td>0.456</td><td>543.54</td><td>+3.16×10<sup>0</sup></td><td>326.68</td><td>+1.93×10<sup>0</sup></td><td>233.56</td><td>+1.36×10<sup>0</sup></td><td>181.55</td><td>+9.76×10<sup>-1</sup></td></tr><tr><td>68.06</td><td>0.348</td><td>583.24</td><td>+3.24×10<sup>0</sup></td><td>351.03</td><td>+1.99×10<sup>0</sup></td><td>250.81</td><td>+1.41×10<sup>0</sup></td><td>194.26</td><td>+1.06×10<sup>0</sup></td></tr><tr><td>77.10</td><td>0.266</td><td>611.27</td><td>+2.97×10<sup>0</sup></td><td>368.30</td><td>+1.83×10<sup>0</sup></td><td>263.05</td><td>+1.29×10<sup>0</sup></td><td>203.43</td><td>+9.58×10<sup>-1</sup></td></tr><tr><td>83.80</td><td>0.197</td><td>628.81</td><td>+2.01×10<sup>0</sup></td><td>379.09</td><td>+1.24×10<sup>0</sup></td><td>270.66</td><td>+8.82×10<sup>-1</sup></td><td>209.08</td><td>+6.63×10<sup>-1</sup></td></tr><tr><td>88.45</td><td>0.151</td><td>635.07</td><td>+6.24×10<sup>-1</sup></td><td>383.03</td><td>+4.14×10<sup>-1</sup></td><td>273.50</td><td>+3.11×10<sup>-1</sup></td><td>211.23</td><td>+2.33×10<sup>-1</sup></td></tr><tr><td>91.08</td><td>0.124</td><td>635.24</td><td>-6.05×10<sup>-1</sup></td><td>383.20</td><td>-3.70×10<sup>-1</sup></td><td>273.65</td><td>-2.62×10<sup>-1</sup></td><td>211.31</td><td>-2.24×10<sup>-1</sup></td></tr><tr><td>93.82</td><td>0.099</td><td>631.36</td><td>-2.22×10<sup>0</sup></td><td>380.79</td><td>-1.37×10<sup>0</sup></td><td>271.92</td><td>-9.73×10<sup>-1</sup></td><td>209.91</td><td>-7.67×10<sup>-1</sup></td></tr><tr><td>96.05</td><td>0.080</td><td>624.68</td><td>-3.92×10<sup>0</sup></td><td>376.69</td><td>-2.40×10<sup>0</sup></td><td>269.03</td><td>-1.68×10<sup>0</sup></td><td>207.68</td><td>-1.29×10<sup>0</sup></td></tr><tr><td>98.37</td><td>0.061</td><td>612.73</td><td>-6.49×10<sup>0</sup></td><td>369.41</td><td>-3.95×10<sup>0</sup></td><td>263.93</td><td>-2.75×10<sup>0</sup></td><td>203.78</td><td>-2.10×10<sup>0</sup></td></tr><tr><td>100.23</td><td>0.046</td><td>597.76</td><td>-1.00×10<sup>1</sup></td><td>360.31</td><td>-6.08×10<sup>0</sup></td><td>257.58</td><td>-4.26×10<sup>0</sup></td><td>198.94</td><td>-3.24×10<sup>0</sup></td></tr><tr><td>101.76</td><td>0.034</td><td>579.58</td><td>-1.52×10<sup>1</sup></td><td>349.20</td><td>-9.36×10<sup>0</sup></td><td>249.76</td><td>-6.65×10<sup>0</sup></td><td>192.96</td><td>-5.12×10<sup>0</sup></td></tr><tr><td>102.99</td><td>0.025</td><td>560.90</td><td>-1.92×10<sup>1</sup></td><td>337.69</td><td>-1.17×10<sup>1</sup></td><td>241.58</td><td>-8.24×10<sup>0</sup></td><td>186.66</td><td>-6.31×10<sup>0</sup></td></tr><tr><td>103.97</td><td>0.018</td><td>542.10</td><td>-3.80×10<sup>1</sup></td><td>326.23</td><td>-2.31×10<sup>1</sup></td><td>233.51</td><td>-1.63×10<sup>1</sup></td><td>180.48</td><td>-1.25×10<sup>1</sup></td></tr><tr><td>105.76</td><td>0.005</td><td>474.16</td><td>-9.88×10<sup>1</sup></td><td>284.90</td><td>-5.98×10<sup>1</sup></td><td>204.32</td><td>-4.22×10<sup>1</sup></td><td>158.19</td><td>-3.26×10<sup>1</sup></td></tr><tr><td>106.48</td><td>0.000</td><td>403.00</td><td>-1.85×10<sup>2</sup></td><td>241.88</td><td>-1.11×10<sup>2</sup></td><td>173.90</td><td>-7.88×10<sup>1</sup></td><td>134.74</td><td>-6.09×10<sup>1</sup></td></tr><tr><td>106.58</td><td>0.000</td><td>384.48</td><td>-2.30×10<sup>2</sup></td><td>230.74</td><td>-1.38×10<sup>2</sup></td><td>166.02</td><td>-9.83×10<sup>1</sup></td><td>128.65</td><td>-7.55×10<sup>1</sup></td></tr><tr><td>106.77</td><td>0.000</td><td>340.87</td><td>-1.62×10<sup>2</sup></td><td>204.47</td><td>-9.79×10<sup>1</sup></td><td>147.34</td><td>-7.02×10<sup>1</sup></td><td>114.31</td><td>-5.33×10<sup>1</sup></td></tr></tbody></table>

<table><thead><tr><th rowspan="2">âge<br>(Ma)</th><th rowspan="2">Y<sub>c</sub></th><th colspan="2"><i>p</i><sub>0,8</sub></th><th colspan="2"><i>p</i><sub>1,4</sub></th><th colspan="2"><i>p</i><sub>2,4</sub></th><th colspan="2"><i>p</i><sub>3,4</sub></th></tr><tr><th><i>P</i><br>(s)</th><th><i>P&#775;</i><br>(s/Ma)</th><th><i>P</i><br>(s)</th><th><i>P&#775;</i><br>(s/Ma)</th><th><i>P</i><br>(s)</th><th><i>P&#775;</i><br>(s/Ma)</th><th><i>P</i><br>(s)</th><th><i>P&#775;</i><br>(s/Ma)</th></tr></thead><tbody><tr><td>1.79</td><td>0.927</td><td>113.44</td><td>+8.84×10<sup>-1</sup></td><td>192.24</td><td>+1.53×10<sup>0</sup></td><td>167.31</td><td>+1.06×10<sup>0</sup></td><td>162.61</td><td>+1.17×10<sup>0</sup></td></tr><tr><td>12.06</td><td>0.809</td><td>122.44</td><td>+8.62×10<sup>-1</sup></td><td>207.88</td><td>+1.52×10<sup>0</sup></td><td>178.63</td><td>+1.18×10<sup>0</sup></td><td>174.84</td><td>+1.23×10<sup>0</sup></td></tr><tr><td>19.66</td><td>0.741</td><td>128.80</td><td>+8.02×10<sup>-1</sup></td><td>219.17</td><td>+1.43×10<sup>0</sup></td><td>187.75</td><td>+1.16×10<sup>0</sup></td><td>184.11</td><td>+1.16×10<sup>0</sup></td></tr><tr><td>23.84</td><td>0.707</td><td>132.08</td><td>+7.79×10<sup>-1</sup></td><td>225.03</td><td>+1.39×10<sup>0</sup></td><td>192.44</td><td>+1.10×10<sup>0</sup></td><td>188.76</td><td>+1.07×10<sup>0</sup></td></tr><tr><td>39.40</td><td>0.588</td><td>144.12</td><td>+7.55×10<sup>-1</sup></td><td>246.55</td><td>+1.39×10<sup>0</sup></td><td>209.76</td><td>+1.14×10<sup>0</sup></td><td>205.45</td><td>+1.19×10<sup>0</sup></td></tr><tr><td>55.93</td><td>0.456</td><td>156.23</td><td>+7.21×10<sup>-1</sup></td><td>270.59</td><td>+1.58×10<sup>0</sup></td><td>229.22</td><td>+1.22×10<sup>0</sup></td><td>221.52</td><td>+3.07×10<sup>-1</sup></td></tr><tr><td>68.06</td><td>0.348</td><td>164.82</td><td>+6.85×10<sup>-1</sup></td><td>290.55</td><td>+1.63×10<sup>0</sup></td><td>245.23</td><td>+1.49×10<sup>0</sup></td><td>220.84</td><td>+3.88×10<sup>-2</sup></td></tr><tr><td>77.10</td><td>0.266</td><td>170.46</td><td>+5.27×10<sup>-1</sup></td><td>304.67</td><td>+1.49×10<sup>0</sup></td><td>255.45</td><td>+2.21×10<sup>-1</sup></td><td>224.31</td><td>+6.62×10<sup>-1</sup></td></tr><tr><td>83.80</td><td>0.197</td><td>173.56</td><td>+4.31×10<sup>-1</sup></td><td>313.43</td><td>+1.01×10<sup>0</sup></td><td>254.61</td><td>+2.61×10<sup>-1</sup></td><td>229.02</td><td>+6.31×10<sup>-1</sup></td></tr><tr><td>88.45</td><td>0.151</td><td>174.95</td><td>+2.98×10<sup>-2</sup></td><td>316.64</td><td>+3.33×10<sup>-1</sup></td><td>253.45</td><td>-2.05×10<sup>0</sup></td><td>231.04</td><td>+1.65×10<sup>-1</sup></td></tr><tr><td>91.08</td><td>0.124</td><td>174.50</td><td>-3.41×10<sup>-1</sup></td><td>316.75</td><td>-3.19×10<sup>-1</sup></td><td>246.26</td><td>-2.35×10<sup>0</sup></td><td>230.84</td><td>-3.60×10<sup>-1</sup></td></tr><tr><td>93.82</td><td>0.099</td><td>173.11</td><td>-6.84×10<sup>-1</sup></td><td>314.73</td><td>-1.12×10<sup>0</sup></td><td>242.16</td><td>-1.28×10<sup>0</sup></td><td>229.04</td><td>-9.12×10<sup>-1</sup></td></tr><tr><td>96.05</td><td>0.080</td><td>171.18</td><td>-1.08×10<sup>0</sup></td><td>311.37</td><td>-1.98×10<sup>0</sup></td><td>238.76</td><td>-1.66×10<sup>0</sup></td><td>226.49</td><td>-1.43×10<sup>0</sup></td></tr><tr><td>98.37</td><td>0.061</td><td>167.96</td><td>-1.77×10<sup>0</sup></td><td>305.42</td><td>-3.13×10<sup>0</sup></td><td>233.82</td><td>-3.05×10<sup>0</sup></td><td>222.25</td><td>-2.26×10<sup>0</sup></td></tr><tr><td>100.23</td><td>0.046</td><td>163.89</td><td>-2.67×10<sup>0</sup></td><td>298.00</td><td>-5.28×10<sup>0</sup></td><td>226.84</td><td>-4.07×10<sup>0</sup></td><td>217.07</td><td>-3.47×10<sup>0</sup></td></tr><tr><td>101.76</td><td>0.034</td><td>159.15</td><td>-3.95×10<sup>0</sup></td><td>288.93</td><td>-7.66×10<sup>0</sup></td><td>221.87</td><td>-2.58×10<sup>0</sup></td><td>210.64</td><td>-5.55×10<sup>0</sup></td></tr><tr><td>102.99</td><td>0.025</td><td>154.30</td><td>-5.26×10<sup>0</sup></td><td>279.52</td><td>-9.57×10<sup>0</sup></td><td>218.70</td><td>-8.39×10<sup>0</sup></td><td>203.81</td><td>-6.78×10<sup>0</sup></td></tr><tr><td>103.97</td><td>0.018</td><td>149.15</td><td>-9.82×10<sup>0</sup></td><td>270.14</td><td>-1.89×10<sup>1</sup></td><td>210.48</td><td>-1.22×10<sup>1</sup></td><td>197.17</td><td>-1.34×10<sup>1</sup></td></tr><tr><td>105.76</td><td>0.005</td><td>131.58</td><td>-2.81×10<sup>1</sup></td><td>236.24</td><td>-4.94×10<sup>1</sup></td><td>188.69</td><td>-4.98×10<sup>1</sup></td><td>173.24</td><td>-3.60×10<sup>1</sup></td></tr><tr><td>106.48</td><td>0.000</td><td>111.34</td><td>-5.37×10<sup>1</sup></td><td>200.68</td><td>+5.91×10<sup>1</sup></td><td>152.81</td><td>-8.45×10<sup>1</sup></td><td>147.30</td><td>-6.87×10<sup>1</sup></td></tr><tr><td>106.58</td><td>0.000</td><td>105.97</td><td>-6.05×10<sup>1</sup></td><td>206.60</td><td>-1.95×10<sup>2</sup></td><td>144.36</td><td>-8.37×10<sup>1</sup></td><td>140.43</td><td>-8.16×10<sup>1</sup></td></tr><tr><td>106.77</td><td>0.000</td><td>94.48</td><td>-3.70×10<sup>1</sup></td><td>169.60</td><td>-8.08×10<sup>1</sup></td><td>128.46</td><td>-5.55×10<sup>1</sup></td><td>124.93</td><td>-5.64×10<sup>1</sup></td></tr></tbody></table>
P et $\dot{P}$ pour un échantillon de modes g sur la séquence $M = 0.4690 + 0.0001 M_{\odot}$

<table><thead><tr><th rowspan="2">âge (Ma)</th><th rowspan="2">Y<sub>c</sub></th><th colspan="2">g<sub>3,1</sub></th><th colspan="2">g<sub>3,2</sub></th><th colspan="2">g<sub>3,3</sub></th><th colspan="2">g<sub>3,4</sub></th></tr><tr><th>P (s)</th><th>dot (s/Ma)</th><th>P (s)</th><th>dot (s/Ma)</th><th>P (s)</th><th>dot (s/Ma)</th><th>P (s)</th><th>dot (s/Ma)</th></tr></thead><tbody><tr><td>4.49</td><td>0.896</td><td>237.49</td><td>+1.96×10<sup>-1</sup></td><td>326.36</td><td>-8.26×10<sup>-1</sup></td><td>415.76</td><td>-1.68×10<sup>0</sup></td><td>501.63</td><td>-1.34×10<sup>0</sup></td></tr><tr><td>23.34</td><td>0.726</td><td>238.83</td><td>-1.79×10<sup>-1</sup></td><td>310.87</td><td>-8.14×10<sup>-1</sup></td><td>391.96</td><td>-4.32×10<sup>-1</sup></td><td>485.79</td><td>+1.64×10<sup>-1</sup></td></tr><tr><td>39.31</td><td>0.611</td><td>232.88</td><td>-4.43×10<sup>-1</sup></td><td>300.74</td><td>-2.86×10<sup>-1</sup></td><td>396.59</td><td>+6.79×10<sup>-1</sup></td><td>501.02</td><td>+1.26×10<sup>0</sup></td></tr><tr><td>58.04</td><td>0.476</td><td>225.24</td><td>-4.01×10<sup>-1</sup></td><td>302.20</td><td>+1.98×10<sup>-1</sup></td><td>409.35</td><td>+6.21×10<sup>-1</sup></td><td>520.96</td><td>+8.93×10<sup>-1</sup></td></tr><tr><td>71.77</td><td>0.366</td><td>217.52</td><td>-8.72×10<sup>-1</sup></td><td>303.03</td><td>-1.24×10<sup>-1</sup></td><td>415.74</td><td>+2.45×10<sup>-1</sup></td><td>530.96</td><td>+5.08×10<sup>-1</sup></td></tr><tr><td>81.82</td><td>0.274</td><td>210.76</td><td>+5.82×10<sup>-2</sup></td><td>302.42</td><td>+2.34×10<sup>-1</sup></td><td>417.57</td><td>+2.64×10<sup>-1</sup></td><td>534.01</td><td>+9.70×10<sup>-2</sup></td></tr><tr><td>89.23</td><td>0.204</td><td>210.38</td><td>-1.20×10<sup>0</sup></td><td>302.73</td><td>-7.29×10<sup>-1</sup></td><td>417.40</td><td>-7.63×10<sup>-1</sup></td><td>531.78</td><td>-9.41×10<sup>-1</sup></td></tr><tr><td>94.80</td><td>0.149</td><td>199.24</td><td>-1.95×10<sup>0</sup></td><td>295.15</td><td>-1.61×10<sup>0</sup></td><td>409.26</td><td>-1.96×10<sup>0</sup></td><td>521.25</td><td>-3.12×10<sup>0</sup></td></tr><tr><td>98.98</td><td>0.108</td><td>192.39</td><td>-1.69×10<sup>0</sup></td><td>288.23</td><td>-1.85×10<sup>0</sup></td><td>400.14</td><td>-2.39×10<sup>0</sup></td><td>506.39</td><td>-3.29×10<sup>0</sup></td></tr><tr><td>102.26</td><td>0.082</td><td>187.37</td><td>-8.83×10<sup>-1</sup></td><td>281.94</td><td>-1.78×10<sup>0</sup></td><td>391.55</td><td>-2.91×10<sup>0</sup></td><td>494.27</td><td>-5.27×10<sup>0</sup></td></tr><tr><td>104.84</td><td>0.059</td><td>185.13</td><td>-1.74×10<sup>0</sup></td><td>276.61</td><td>-2.92×10<sup>0</sup></td><td>382.42</td><td>-4.51×10<sup>0</sup></td><td>477.56</td><td>-6.73×10<sup>0</sup></td></tr><tr><td>106.88</td><td>0.044</td><td>179.62</td><td>-3.24×10<sup>0</sup></td><td>269.18</td><td>-3.97×10<sup>0</sup></td><td>371.91</td><td>-5.38×10<sup>0</sup></td><td>463.15</td><td>-8.29×10<sup>0</sup></td></tr><tr><td>108.49</td><td>0.032</td><td>174.10</td><td>-3.72×10<sup>0</sup></td><td>262.18</td><td>-5.15×10<sup>0</sup></td><td>362.14</td><td>-7.58×10<sup>0</sup></td><td>446.41</td><td>-1.29×10<sup>1</sup></td></tr><tr><td>109.75</td><td>0.023</td><td>169.09</td><td>-4.21×10<sup>0</sup></td><td>255.22</td><td>-5.32×10<sup>0</sup></td><td>351.99</td><td>-7.17×10<sup>0</sup></td><td>430.80</td><td>-8.97×10<sup>0</sup></td></tr><tr><td>111.04</td><td>0.014</td><td>160.51</td><td>-1.13×10<sup>1</sup></td><td>244.12</td><td>-1.60×10<sup>1</sup></td><td>336.87</td><td>-2.31×10<sup>1</sup></td><td>411.95</td><td>-3.23×10<sup>1</sup></td></tr><tr><td>112.97</td><td>0.000</td><td>125.31</td><td>-2.17×10<sup>1</sup></td><td>191.24</td><td>-3.31×10<sup>1</sup></td><td>257.27</td><td>-5.03×10<sup>1</sup></td><td>292.18</td><td>-7.69×10<sup>1</sup></td></tr></tbody></table>

<table><thead><tr><th rowspan="2">âge (Ma)</th><th rowspan="2">Y<sub>c</sub></th><th colspan="2">g<sub>3,5</sub></th><th colspan="2">g<sub>3,6</sub></th><th colspan="2">g<sub>1,4</sub></th><th colspan="2">g<sub>2,4</sub></th></tr><tr><th>P (s)</th><th>dot (s/Ma)</th><th>P (s)</th><th>dot (s/Ma)</th><th>P (s)</th><th>dot (s/Ma)</th><th>P (s)</th><th>dot (s/Ma)</th></tr></thead><tbody><tr><td>4.49</td><td>0.896</td><td>583.24</td><td>-5.85×10<sup>-1</sup></td><td>664.23</td><td>+1.90×10<sup>-1</sup></td><td>1137.51</td><td>-1.07×10<sup>0</sup></td><td>680.10</td><td>-1.24×10<sup>0</sup></td></tr><tr><td>23.34</td><td>0.726</td><td>580.43</td><td>+7.23×10<sup>-1</sup></td><td>674.38</td><td>+1.24×10<sup>0</sup></td><td>1132.19</td><td>+1.29×10<sup>0</sup></td><td>667.56</td><td>+4.91×10<sup>-1</sup></td></tr><tr><td>39.31</td><td>0.611</td><td>602.67</td><td>+1.62×10<sup>0</sup></td><td>702.65</td><td>+1.95×10<sup>0</sup></td><td>1172.53</td><td>+3.00×10<sup>0</sup></td><td>689.90</td><td>+1.75×10<sup>0</sup></td></tr><tr><td>58.04</td><td>0.476</td><td>628.18</td><td>+1.08×10<sup>0</sup></td><td>733.56</td><td>+1.24×10<sup>0</sup></td><td>1220.63</td><td>+2.02×10<sup>0</sup></td><td>717.81</td><td>+1.23×10<sup>0</sup></td></tr><tr><td>71.77</td><td>0.366</td><td>640.78</td><td>+8.03×10<sup>-1</sup></td><td>747.71</td><td>+9.91×10<sup>-1</sup></td><td>1243.13</td><td>+1.35×10<sup>0</sup></td><td>731.31</td><td>+6.83×10<sup>-1</sup></td></tr><tr><td>81.82</td><td>0.274</td><td>643.25</td><td>-8.24×10<sup>-1</sup></td><td>739.30</td><td>-4.59×10<sup>0</sup></td><td>1247.94</td><td>-1.11×10<sup>0</sup></td><td>734.87</td><td>-4.56×10<sup>-2</sup></td></tr><tr><td>89.23</td><td>0.204</td><td>629.30</td><td>-2.83×10<sup>0</sup></td><td>690.61</td><td>-6.35×10<sup>0</sup></td><td>1232.48</td><td>-2.33×10<sup>0</sup></td><td>730.32</td><td>-1.41×10<sup>0</sup></td></tr><tr><td>94.80</td><td>0.149</td><td>602.78</td><td>-7.99×10<sup>0</sup></td><td>661.08</td><td>-5.02×10<sup>0</sup></td><td>1207.94</td><td>-8.91×10<sup>0</sup></td><td>714.72</td><td>-4.83×10<sup>0</sup></td></tr><tr><td>98.98</td><td>0.108</td><td>566.42</td><td>-6.73×10<sup>0</sup></td><td>640.12</td><td>-4.65×10<sup>0</sup></td><td>1163.54</td><td>-8.84×10<sup>0</sup></td><td>691.82</td><td>-4.79×10<sup>0</sup></td></tr><tr><td>102.26</td><td>0.082</td><td>549.77</td><td>-5.43×10<sup>0</sup></td><td>626.60</td><td>-3.65×10<sup>0</sup></td><td>1134.37</td><td>-1.32×10<sup>1</sup></td><td>674.86</td><td>-7.53×10<sup>0</sup></td></tr><tr><td>104.84</td><td>0.059</td><td>533.23</td><td>-6.53×10<sup>0</sup></td><td>615.36</td><td>-6.11×10<sup>0</sup></td><td>1092.64</td><td>-1.56×10<sup>1</sup></td><td>651.03</td><td>-9.24×10<sup>0</sup></td></tr><tr><td>106.88</td><td>0.044</td><td>517.52</td><td>-1.06×10<sup>1</sup></td><td>599.26</td><td>-9.24×10<sup>0</sup></td><td>1059.67</td><td>-2.07×10<sup>1</sup></td><td>631.35</td><td>-1.18×10<sup>1</sup></td></tr><tr><td>108.49</td><td>0.032</td><td>496.90</td><td>-1.35×10<sup>1</sup></td><td>582.58</td><td>-1.19×10<sup>1</sup></td><td>1017.25</td><td>-3.09×10<sup>1</sup></td><td>607.33</td><td>-1.81×10<sup>1</sup></td></tr><tr><td>109.75</td><td>0.023</td><td>481.43</td><td>-1.06×10<sup>1</sup></td><td>567.01</td><td>-1.17×10<sup>1</sup></td><td>981.28</td><td>-2.12×10<sup>1</sup></td><td>585.94</td><td>-1.23×10<sup>1</sup></td></tr><tr><td>111.04</td><td>0.014</td><td>459.24</td><td>-3.27×10<sup>1</sup></td><td>541.78</td><td>-3.69×10<sup>1</sup></td><td>934.43</td><td>-7.93×10<sup>1</sup></td><td>559.25</td><td>-4.58×10<sup>1</sup></td></tr><tr><td>112.97</td><td>0.000</td><td>349.47</td><td>-6.90×10<sup>1</sup></td><td>418.81</td><td>-7.71×10<sup>1</sup></td><td>645.13</td><td>-1.85×10<sup>2</sup></td><td>390.44</td><td>-1.08×10<sup>2</sup></td></tr></tbody></table>
P et $\dot{P}$ pour un échantillon de modes g sur la séquence $M = 0.4758 + 0.0002 M_{\odot}$

<table><thead><tr><th rowspan="2">âge<br>(Ma)</th><th rowspan="2">Y<sub>c</sub></th><th colspan="2">g<sub>3,1</sub></th><th colspan="2">g<sub>3,2</sub></th><th colspan="2">g<sub>3,3</sub></th><th colspan="2">g<sub>3,4</sub></th></tr><tr><th>P<br>(s)</th><th>˙P<br>(s/Ma)</th><th>P<br>(s)</th><th>˙P<br>(s/Ma)</th><th>P<br>(s)</th><th>˙P<br>(s/Ma)</th><th>P<br>(s)</th><th>˙P<br>(s/Ma)</th></tr></thead><tbody><tr><td>1.79</td><td>0.928</td><td>240.56</td><td>+2.05×10<sup>-1</sup></td><td>332.20</td><td>-7.69×10<sup>-1</sup></td><td>425.16</td><td>-2.74×10<sup>0</sup></td><td>495.76</td><td>-1.83×10<sup>0</sup></td></tr><tr><td>12.06</td><td>0.811</td><td>242.42</td><td>+1.33×10<sup>-1</sup></td><td>324.54</td><td>-6.98×10<sup>-1</sup></td><td>402.56</td><td>-1.12×10<sup>0</sup></td><td>483.39</td><td>+3.92×10<sup>-2</sup></td></tr><tr><td>18.98</td><td>0.750</td><td>242.82</td><td>-4.20×10<sup>-2</sup></td><td>319.32</td><td>-9.17×10<sup>-1</sup></td><td>399.51</td><td>-1.81×10<sup>-1</sup></td><td>488.89</td><td>+1.05×10<sup>0</sup></td></tr><tr><td>21.73</td><td>0.727</td><td>242.40</td><td>-3.26×10<sup>-1</sup></td><td>316.34</td><td>-1.31×10<sup>0</sup></td><td>398.84</td><td>-3.11×10<sup>-1</sup></td><td>491.51</td><td>+8.69×10<sup>-1</sup></td></tr><tr><td>28.83</td><td>0.673</td><td>237.92</td><td>-6.39×10<sup>-1</sup></td><td>305.57</td><td>-1.18×10<sup>0</sup></td><td>397.08</td><td>+5.32×10<sup>-2</sup></td><td>497.34</td><td>+9.28×10<sup>-1</sup></td></tr><tr><td>46.87</td><td>0.537</td><td>232.21</td><td>-4.08×10<sup>-1</sup></td><td>306.07</td><td>+2.41×10<sup>-1</sup></td><td>410.49</td><td>+7.56×10<sup>-1</sup></td><td>518.98</td><td>+1.07×10<sup>0</sup></td></tr><tr><td>61.90</td><td>0.418</td><td>224.09</td><td>-3.83×10<sup>-1</sup></td><td>306.59</td><td>+2.74×10<sup>-1</sup></td><td>418.65</td><td>+6.69×10<sup>-1</sup></td><td>532.40</td><td>+9.67×10<sup>-1</sup></td></tr><tr><td>73.01</td><td>0.317</td><td>221.76</td><td>-1.92×10<sup>-1</sup></td><td>309.37</td><td>-3.05×10<sup>-1</sup></td><td>424.83</td><td>-1.33×10<sup>-2</sup></td><td>540.97</td><td>+1.37×10<sup>-1</sup></td></tr><tr><td>81.14</td><td>0.236</td><td>221.16</td><td>+2.63×10<sup>-1</sup></td><td>304.02</td><td>-5.70×10<sup>-1</sup></td><td>421.70</td><td>-3.77×10<sup>-1</sup></td><td>538.62</td><td>-3.56×10<sup>-1</sup></td></tr><tr><td>87.19</td><td>0.175</td><td>223.37</td><td>+1.57×10<sup>-1</sup></td><td>300.96</td><td>-7.70×10<sup>-1</sup></td><td>418.70</td><td>-1.02×10<sup>0</sup></td><td>533.82</td><td>-1.89×10<sup>0</sup></td></tr><tr><td>91.79</td><td>0.129</td><td>222.60</td><td>-4.20×10<sup>-1</sup></td><td>295.73</td><td>-1.42×10<sup>0</sup></td><td>411.20</td><td>-2.13×10<sup>0</sup></td><td>521.42</td><td>-2.99×10<sup>0</sup></td></tr><tr><td>95.14</td><td>0.099</td><td>220.37</td><td>-1.02×10<sup>0</sup></td><td>290.65</td><td>-1.58×10<sup>0</sup></td><td>403.28</td><td>-2.54×10<sup>0</sup></td><td>509.63</td><td>-4.74×10<sup>0</sup></td></tr><tr><td>97.94</td><td>0.073</td><td>216.28</td><td>-1.94×10<sup>0</sup></td><td>285.40</td><td>-2.44×10<sup>0</sup></td><td>394.76</td><td>-3.96×10<sup>0</sup></td><td>493.02</td><td>-6.72×10<sup>0</sup></td></tr><tr><td>100.12</td><td>0.054</td><td>211.15</td><td>-2.78×10<sup>0</sup></td><td>278.37</td><td>-4.14×10<sup>0</sup></td><td>384.42</td><td>-5.26×10<sup>0</sup></td><td>477.67</td><td>-7.38×10<sup>0</sup></td></tr><tr><td>101.71</td><td>0.041</td><td>206.07</td><td>-3.73×10<sup>0</sup></td><td>270.90</td><td>-5.05×10<sup>0</sup></td><td>374.95</td><td>-7.15×10<sup>0</sup></td><td>464.70</td><td>-9.44×10<sup>0</sup></td></tr><tr><td>103.17</td><td>0.030</td><td>199.90</td><td>-5.07×10<sup>0</sup></td><td>263.39</td><td>-5.25×10<sup>0</sup></td><td>364.33</td><td>-8.03×10<sup>0</sup></td><td>448.51</td><td>-1.18×10<sup>1</sup></td></tr><tr><td>104.28</td><td>0.022</td><td>194.28</td><td>-1.14×10<sup>1</sup></td><td>257.56</td><td>-1.26×10<sup>1</sup></td><td>355.42</td><td>-1.84×10<sup>1</sup></td><td>435.44</td><td>-2.54×10<sup>1</sup></td></tr><tr><td>106.49</td><td>0.005</td><td>169.15</td><td>-3.61×10<sup>1</sup></td><td>229.80</td><td>-5.69×10<sup>1</sup></td><td>314.73</td><td>-9.11×10<sup>1</sup></td><td>379.31</td><td>-1.37×10<sup>2</sup></td></tr><tr><td>107.23</td><td>0.000</td><td>142.43</td><td>-6.72×10<sup>1</sup></td><td>187.68</td><td>-1.20×10<sup>2</sup></td><td>247.34</td><td>-3.05×10<sup>2</sup></td><td>278.22</td><td>-2.64×10<sup>2</sup></td></tr></tbody></table>

<table><thead><tr><th rowspan="2">âge<br>(Ma)</th><th rowspan="2">Y<sub>c</sub></th><th colspan="2">g<sub>3,5</sub></th><th colspan="2">g<sub>3,6</sub></th><th colspan="2">g<sub>1,4</sub></th><th colspan="2">g<sub>2,4</sub></th></tr><tr><th>P<br>(s)</th><th>˙P<br>(s/Ma)</th><th>P<br>(s)</th><th>˙P<br>(s/Ma)</th><th>P<br>(s)</th><th>˙P<br>(s/Ma)</th><th>P<br>(s)</th><th>˙P<br>(s/Ma)</th></tr></thead><tbody><tr><td>1.79</td><td>0.928</td><td>535.15</td><td>-1.17×10<sup>0</sup></td><td>620.73</td><td>-4.31×10<sup>0</sup></td><td>1162.35</td><td>-6.38×10<sup>0</sup></td><td>686.22</td><td>-3.51×10<sup>0</sup></td></tr><tr><td>12.06</td><td>0.811</td><td>529.36</td><td>+6.41×10<sup>-1</sup></td><td>590.81</td><td>-1.17×10<sup>-1</sup></td><td>1119.83</td><td>+3.42×10<sup>-1</sup></td><td>662.37</td><td>+4.43×10<sup>-2</sup></td></tr><tr><td>18.98</td><td>0.750</td><td>539.15</td><td>+1.75×10<sup>0</sup></td><td>601.28</td><td>+1.95×10<sup>0</sup></td><td>1139.67</td><td>+3.39×10<sup>0</sup></td><td>672.10</td><td>+1.74×10<sup>0</sup></td></tr><tr><td>21.73</td><td>0.727</td><td>544.02</td><td>+1.86×10<sup>0</sup></td><td>606.13</td><td>+1.70×10<sup>0</sup></td><td>1147.70</td><td>+2.58×10<sup>0</sup></td><td>676.27</td><td>+1.35×10<sup>0</sup></td></tr><tr><td>28.83</td><td>0.673</td><td>558.73</td><td>+2.08×10<sup>0</sup></td><td>618.80</td><td>+1.83×10<sup>0</sup></td><td>1164.61</td><td>+2.52×10<sup>0</sup></td><td>685.17</td><td>+1.37×10<sup>0</sup></td></tr><tr><td>46.87</td><td>0.537</td><td>589.18</td><td>+1.43×10<sup>0</sup></td><td>649.50</td><td>+1.43×10<sup>0</sup></td><td>1217.30</td><td>+2.52×10<sup>0</sup></td><td>715.57</td><td>+1.48×10<sup>0</sup></td></tr><tr><td>61.90</td><td>0.418</td><td>610.41</td><td>+1.49×10<sup>0</sup></td><td>668.64</td><td>+1.29×10<sup>0</sup></td><td>1248.31</td><td>+2.16×10<sup>0</sup></td><td>733.89</td><td>+1.31×10<sup>0</sup></td></tr><tr><td>73.01</td><td>0.317</td><td>624.52</td><td>+7.17×10<sup>-1</sup></td><td>679.84</td><td>+3.11×10<sup>-1</sup></td><td>1266.22</td><td>+2.09×10<sup>-2</sup></td><td>745.14</td><td>+9.52×10<sup>-2</sup></td></tr><tr><td>81.14</td><td>0.236</td><td>625.99</td><td>-2.41×10<sup>-1</sup></td><td>670.66</td><td>-3.19×10<sup>0</sup></td><td>1257.32</td><td>-1.40×10<sup>0</sup></td><td>740.89</td><td>-6.33×10<sup>-1</sup></td></tr><tr><td>87.19</td><td>0.175</td><td>613.93</td><td>-5.27×10<sup>0</sup></td><td>645.68</td><td>-4.01×10<sup>0</sup></td><td>1239.91</td><td>-6.20×10<sup>0</sup></td><td>732.79</td><td>-3.04×10<sup>0</sup></td></tr><tr><td>91.79</td><td>0.129</td><td>584.94</td><td>-4.72×10<sup>0</sup></td><td>630.81</td><td>-2.57×10<sup>0</sup></td><td>1203.01</td><td>-7.74×10<sup>0</sup></td><td>713.68</td><td>-4.32×10<sup>0</sup></td></tr><tr><td>95.14</td><td>0.099</td><td>570.47</td><td>-6.57×10<sup>0</sup></td><td>621.49</td><td>-3.60×10<sup>0</sup></td><td>1173.18</td><td>-1.30×10<sup>1</sup></td><td>696.78</td><td>-7.07×10<sup>0</sup></td></tr><tr><td>97.94</td><td>0.073</td><td>547.52</td><td>-7.34×10<sup>0</sup></td><td>608.41</td><td>-5.63×10<sup>0</sup></td><td>1128.10</td><td>-1.65×10<sup>1</sup></td><td>672.14</td><td>-9.48×10<sup>0</sup></td></tr><tr><td>100.12</td><td>0.054</td><td>532.44</td><td>-8.68×10<sup>0</sup></td><td>594.06</td><td>-7.82×10<sup>0</sup></td><td>1092.71</td><td>-1.75×10<sup>1</sup></td><td>651.07</td><td>-1.03×10<sup>1</sup></td></tr><tr><td>101.71</td><td>0.041</td><td>514.47</td><td>-1.37×10<sup>1</sup></td><td>578.87</td><td>-1.19×10<sup>1</sup></td><td>1059.99</td><td>-2.46×10<sup>1</sup></td><td>632.56</td><td>-1.35×10<sup>1</sup></td></tr><tr><td>103.17</td><td>0.030</td><td>496.86</td><td>-1.06×10<sup>1</sup></td><td>562.39</td><td>-1.23×10<sup>1</sup></td><td>1021.33</td><td>-2.61×10<sup>1</sup></td><td>609.99</td><td>-1.58×10<sup>1</sup></td></tr><tr><td>104.28</td><td>0.022</td><td>485.14</td><td>-2.52×10<sup>1</sup></td><td>548.76</td><td>-2.86×10<sup>1</sup></td><td>992.40</td><td>-6.10×10<sup>1</sup></td><td>592.41</td><td>-3.54×10<sup>1</sup></td></tr><tr><td>106.49</td><td>0.005</td><td>429.47</td><td>-1.21×10<sup>2</sup></td><td>485.62</td><td>-1.32×10<sup>2</sup></td><td>857.59</td><td>-3.29×10<sup>2</sup></td><td>514.09</td><td>-1.93×10<sup>2</sup></td></tr><tr><td>107.23</td><td>0.000</td><td>340.12</td><td>-4.99×10<sup>2</sup></td><td>387.69</td><td>-5.38×10<sup>2</sup></td><td>614.25</td><td>-8.17×10<sup>2</sup></td><td>371.33</td><td>-3.91×10<sup>2</sup></td></tr></tbody></table>
P et $\dot{P}$ pour un échantillon de modes g sur la séquence $M = 0.4690 + 0.0007 M_{\odot}$

<table><thead><tr><th rowspan="2">âge<br>(Ma)</th><th rowspan="2">Y<sub>c</sub></th><th colspan="2"><i>g</i><sub>3,1</sub></th><th colspan="2"><i>g</i><sub>3,2</sub></th><th colspan="2"><i>g</i><sub>3,3</sub></th><th colspan="2"><i>g</i><sub>3,4</sub></th></tr><tr><th><i>P</i><br>(s)</th><th><i>P&#x0307;</i><br>(s/Ma)</th><th><i>P</i><br>(s)</th><th><i>P&#x0307;</i><br>(s/Ma)</th><th><i>P</i><br>(s)</th><th><i>P&#x0307;</i><br>(s/Ma)</th><th><i>P</i><br>(s)</th><th><i>P&#x0307;</i><br>(s/Ma)</th></tr></thead><tbody><tr><td>1.01</td><td>0.939</td><td>237.61</td><td>-1.86&times;10<sup>-3</sup></td><td>327.81</td><td>-3.80&times;10<sup>-1</sup></td><td>408.73</td><td>+1.52&times;10<sup>-1</sup></td><td>444.27</td><td>-3.15&times;10<sup>-2</sup></td></tr><tr><td>1.11</td><td>0.938</td><td>237.27</td><td>-1.57&times;10<sup>-3</sup></td><td>326.96</td><td>-3.80&times;10<sup>-1</sup></td><td>408.04</td><td>+1.50&times;10<sup>-1</sup></td><td>443.36</td><td>-3.15&times;10<sup>-2</sup></td></tr><tr><td>2.22</td><td>0.924</td><td>237.62</td><td>+3.97&times;10<sup>-2</sup></td><td>327.33</td><td>-4.23&times;10<sup>-1</sup></td><td>408.81</td><td>-9.14&times;10<sup>-2</sup></td><td>444.23</td><td>-3.65&times;10<sup>-2</sup></td></tr><tr><td>14.18</td><td>0.797</td><td>239.80</td><td>+5.61&times;10<sup>-2</sup></td><td>318.66</td><td>-9.01&times;10<sup>-1</sup></td><td>393.46</td><td>-1.26&times;10<sup>0</sup></td><td>446.16</td><td>+6.05&times;10<sup>-1</sup></td></tr><tr><td>33.89</td><td>0.647</td><td>237.47</td><td>+1.82&times;10<sup>-1</sup></td><td>303.47</td><td>-4.28&times;10<sup>-1</sup></td><td>394.80</td><td>+6.92&times;10<sup>-1</sup></td><td>471.07</td><td>+1.44&times;10<sup>0</sup></td></tr><tr><td>53.93</td><td>0.501</td><td>251.64</td><td>+9.72&times;10<sup>-1</sup></td><td>302.56</td><td>+1.54&times;10<sup>-1</sup></td><td>407.58</td><td>+6.14&times;10<sup>-1</sup></td><td>495.80</td><td>+1.12&times;10<sup>0</sup></td></tr><tr><td>68.48</td><td>0.386</td><td>266.01</td><td>+1.02&times;10<sup>0</sup></td><td>304.32</td><td>+4.09&times;10<sup>-2</sup></td><td>415.72</td><td>+4.44&times;10<sup>-1</sup></td><td>511.54</td><td>+1.01&times;10<sup>0</sup></td></tr><tr><td>79.11</td><td>0.290</td><td>276.68</td><td>+9.42&times;10<sup>-1</sup></td><td>304.32</td><td>+1.08&times;10<sup>-2</sup></td><td>418.76</td><td>+9.80&times;10<sup>-2</sup></td><td>520.28</td><td>+5.11&times;10<sup>-1</sup></td></tr><tr><td>86.89</td><td>0.212</td><td>282.50</td><td>+4.41&times;10<sup>-1</sup></td><td>302.97</td><td>-5.86&times;10<sup>-1</sup></td><td>416.98</td><td>-7.22&times;10<sup>-1</sup></td><td>520.98</td><td>-4.35&times;10<sup>-1</sup></td></tr><tr><td>92.67</td><td>0.155</td><td>283.32</td><td>-1.42&times;10<sup>-1</sup></td><td>298.51</td><td>-6.68&times;10<sup>-1</sup></td><td>410.89</td><td>-1.23&times;10<sup>0</sup></td><td>515.04</td><td>-1.75&times;10<sup>0</sup></td></tr><tr><td>97.02</td><td>0.113</td><td>280.97</td><td>-1.14&times;10<sup>0</sup></td><td>294.70</td><td>-1.59&times;10<sup>0</sup></td><td>403.81</td><td>-2.42&times;10<sup>0</sup></td><td>504.50</td><td>-3.14&times;10<sup>0</sup></td></tr><tr><td>100.26</td><td>0.082</td><td>275.46</td><td>-2.24&times;10<sup>0</sup></td><td>287.46</td><td>-2.63&times;10<sup>0</sup></td><td>393.58</td><td>-3.75&times;10<sup>0</sup></td><td>491.55</td><td>-5.17&times;10<sup>0</sup></td></tr><tr><td>102.86</td><td>0.059</td><td>268.66</td><td>-2.96&times;10<sup>0</sup></td><td>280.17</td><td>-3.05&times;10<sup>0</sup></td><td>382.95</td><td>-4.40&times;10<sup>0</sup></td><td>475.18</td><td>-7.37&times;10<sup>0</sup></td></tr><tr><td>104.83</td><td>0.043</td><td>261.77</td><td>-4.35&times;10<sup>0</sup></td><td>272.97</td><td>-4.64&times;10<sup>0</sup></td><td>372.65</td><td>-6.68&times;10<sup>0</sup></td><td>458.85</td><td>-9.33&times;10<sup>0</sup></td></tr><tr><td>106.42</td><td>0.031</td><td>253.85</td><td>-5.30&times;10<sup>0</sup></td><td>264.98</td><td>-4.69&times;10<sup>0</sup></td><td>361.83</td><td>-5.38&times;10<sup>0</sup></td><td>445.12</td><td>-6.30&times;10<sup>0</sup></td></tr><tr><td>107.67</td><td>0.023</td><td>246.22</td><td>-7.74&times;10<sup>0</sup></td><td>257.94</td><td>-7.15&times;10<sup>0</sup></td><td>352.30</td><td>-1.01&times;10<sup>1</sup></td><td>431.84</td><td>-1.49&times;10<sup>1</sup></td></tr><tr><td>108.71</td><td>0.016</td><td>238.18</td><td>-2.42&times;10<sup>1</sup></td><td>250.50</td><td>-2.48&times;10<sup>1</sup></td><td>341.78</td><td>-3.55&times;10<sup>1</sup></td><td>416.35</td><td>-5.09&times;10<sup>1</sup></td></tr><tr><td>110.80</td><td>0.001</td><td>187.61</td><td>-7.72&times;10<sup>1</sup></td><td>198.65</td><td>-1.02&times;10<sup>2</sup></td><td>267.49</td><td>-2.16&times;10<sup>2</sup></td><td>309.90</td><td>-2.99&times;10<sup>2</sup></td></tr><tr><td>110.95</td><td>0.000</td><td>176.04</td><td>-9.73&times;10<sup>1</sup></td><td>183.31</td><td>-1.07&times;10<sup>2</sup></td><td>235.04</td><td>-2.63&times;10<sup>2</sup></td><td>265.09</td><td>-2.17&times;10<sup>2</sup></td></tr><tr><td>111.16</td><td>0.000</td><td>155.60</td><td>-7.41&times;10<sup>1</sup></td><td>160.90</td><td>-5.27&times;10<sup>1</sup></td><td>179.78</td><td>-5.64&times;10<sup>1</sup></td><td>219.49</td><td>-1.03&times;10<sup>2</sup></td></tr><tr><td>111.54</td><td>0.000</td><td>127.43</td><td>+2.20&times;10<sup>-1</sup></td><td>140.88</td><td>+9.34&times;10<sup>-1</sup></td><td>158.33</td><td>-1.78&times;10<sup>0</sup></td><td>180.51</td><td>-3.84&times;10<sup>0</sup></td></tr></tbody></table>

<table><thead><tr><th rowspan="2">âge<br>(Ma)</th><th rowspan="2">Y<sub>c</sub></th><th colspan="2"><i>g</i><sub>3,5</sub></th><th colspan="2"><i>g</i><sub>3,6</sub></th><th colspan="2"><i>g</i><sub>1,4</sub></th><th colspan="2"><i>g</i><sub>2,4</sub></th></tr><tr><th><i>P</i><br>(s)</th><th><i>P&#x0307;</i><br>(s/Ma)</th><th><i>P</i><br>(s)</th><th><i>P&#x0307;</i><br>(s/Ma)</th><th><i>P</i><br>(s)</th><th><i>P&#x0307;</i><br>(s/Ma)</th><th><i>P</i><br>(s)</th><th><i>P&#x0307;</i><br>(s/Ma)</th></tr></thead><tbody><tr><td>1.01</td><td>0.939</td><td>529.36</td><td>-2.12&times;10<sup>0</sup></td><td>626.55</td><td>-3.70&times;10<sup>0</sup></td><td>1036.61</td><td>+4.34&times;10<sup>-1</sup></td><td>610.21</td><td>+1.74&times;10<sup>-1</sup></td></tr><tr><td>1.11</td><td>0.938</td><td>527.44</td><td>-2.12&times;10<sup>0</sup></td><td>624.18</td><td>-3.70&times;10<sup>0</sup></td><td>1035.30</td><td>+4.34&times;10<sup>-1</sup></td><td>609.24</td><td>+1.74&times;10<sup>-1</sup></td></tr><tr><td>2.22</td><td>0.924</td><td>526.73</td><td>-2.29&times;10<sup>0</sup></td><td>622.09</td><td>-3.67&times;10<sup>0</sup></td><td>1037.16</td><td>+4.94&times;10<sup>-1</sup></td><td>610.43</td><td>+1.94&times;10<sup>-1</sup></td></tr><tr><td>14.18</td><td>0.797</td><td>496.56</td><td>-1.35&times;10<sup>0</sup></td><td>590.68</td><td>-8.15&times;10<sup>-1</sup></td><td>1051.95</td><td>+2.13&times;10<sup>0</sup></td><td>617.16</td><td>+1.10&times;10<sup>0</sup></td></tr><tr><td>33.89</td><td>0.647</td><td>519.01</td><td>+1.86&times;10<sup>0</sup></td><td>622.39</td><td>+2.21&times;10<sup>0</sup></td><td>1112.77</td><td>+3.26&times;10<sup>0</sup></td><td>652.07</td><td>+1.93&times;10<sup>0</sup></td></tr><tr><td>53.93</td><td>0.501</td><td>543.56</td><td>+1.00&times;10<sup>0</sup></td><td>651.27</td><td>+1.14&times;10<sup>0</sup></td><td>1168.99</td><td>+2.52&times;10<sup>0</sup></td><td>685.43</td><td>+1.51&times;10<sup>0</sup></td></tr><tr><td>68.48</td><td>0.386</td><td>558.75</td><td>+9.85&times;10<sup>-1</sup></td><td>667.19</td><td>+8.72&times;10<sup>-1</sup></td><td>1203.28</td><td>+2.13&times;10<sup>0</sup></td><td>706.20</td><td>+1.31&times;10<sup>0</sup></td></tr><tr><td>79.11</td><td>0.290</td><td>566.79</td><td>+4.17&times;10<sup>-1</sup></td><td>670.72</td><td>-4.60&times;10<sup>-1</sup></td><td>1220.09</td><td>+7.02&times;10<sup>-1</sup></td><td>717.03</td><td>+5.55&times;10<sup>-1</sup></td></tr><tr><td>86.89</td><td>0.212</td><td>566.76</td><td>-5.19&times;10<sup>-1</sup></td><td>653.12</td><td>-5.10&times;10<sup>0</sup></td><td>1216.66</td><td>-1.84&times;10<sup>0</sup></td><td>716.54</td><td>-8.30&times;10<sup>-1</sup></td></tr><tr><td>92.67</td><td>0.155</td><td>559.99</td><td>-2.02&times;10<sup>0</sup></td><td>616.25</td><td>-6.05&times;10<sup>0</sup></td><td>1196.59</td><td>-5.50&times;10<sup>0</sup></td><td>706.73</td><td>-2.75&times;10<sup>0</sup></td></tr><tr><td>97.02</td><td>0.113</td><td>547.63</td><td>-3.68&times;10<sup>0</sup></td><td>591.16</td><td>-6.64&times;10<sup>0</sup></td><td>1165.25</td><td>-8.78&times;10<sup>0</sup></td><td>690.62</td><td>-4.68&times;10<sup>0</sup></td></tr><tr><td>100.26</td><td>0.082</td><td>532.49</td><td>-6.00&times;10<sup>0</sup></td><td>566.98</td><td>-7.58&times;10<sup>0</sup></td><td>1130.33</td><td>-1.37&times;10<sup>1</sup></td><td>671.53</td><td>-7.56&times;10<sup>0</sup></td></tr><tr><td>102.86</td><td>0.059</td><td>514.00</td><td>-7.98&times;10<sup>0</sup></td><td>547.99</td><td>-7.24&times;10<sup>0</sup></td><td>1087.43</td><td>-1.90&times;10<sup>1</sup></td><td>647.73</td><td>-1.06&times;10<sup>1</sup></td></tr><tr><td>104.83</td><td>0.043</td><td>497.34</td><td>-8.92&times;10<sup>0</sup></td><td>532.27</td><td>-9.37&times;10<sup>0</sup></td><td>1047.34</td><td>-2.15&times;10<sup>1</sup></td><td>624.66</td><td>-1.28&times;10<sup>1</sup></td></tr><tr><td>106.42</td><td>0.031</td><td>483.14</td><td>-8.58&times;10<sup>0</sup></td><td>517.28</td><td>-7.91&times;10<sup>0</sup></td><td>1017.08</td><td>-1.35&times;10<sup>1</sup></td><td>606.23</td><td>-8.21&times;10<sup>0</sup></td></tr><tr><td>107.67</td><td>0.023</td><td>469.22</td><td>-1.46&times;10<sup>1</sup></td><td>503.44</td><td>-1.44&times;10<sup>1</sup></td><td>986.39</td><td>-3.55&times;10<sup>1</sup></td><td>588.05</td><td>-2.07&times;10<sup>1</sup></td></tr><tr><td>108.71</td><td>0.016</td><td>454.07</td><td>-4.76&times;10<sup>1</sup></td><td>488.43</td><td>-5.01&times;10<sup>1</sup></td><td>949.52</td><td>-1.25&times;10<sup>2</sup></td><td>566.52</td><td>-7.21&times;10<sup>1</sup></td></tr><tr><td>110.80</td><td>0.001</td><td>354.53</td><td>-2.32&times;10<sup>2</sup></td><td>383.63</td><td>-2.70&times;10<sup>2</sup></td><td>687.66</td><td>-7.47&times;10<sup>2</sup></td><td>415.80</td><td>-4.29&times;10<sup>2</sup></td></tr><tr><td>110.95</td><td>0.000</td><td>319.75</td><td>-3.70&times;10<sup>2</sup></td><td>343.10</td><td>-2.82&times;10<sup>2</sup></td><td>575.58</td><td>-6.06&times;10<sup>2</sup></td><td>351.42</td><td>-3.12&times;10<sup>2</sup></td></tr><tr><td>111.16</td><td>0.000</td><td>242.07</td><td>-6.57&times;10<sup>1</sup></td><td>283.79</td><td>-1.31&times;10<sup>2</sup></td><td>448.39</td><td>-2.16&times;10<sup>2</sup></td><td>286.00</td><td>-1.42&times;10<sup>2</sup></td></tr><tr><td>111.54</td><td>0.000</td><td>217.10</td><td>-6.91&times;10<sup>0</sup></td><td>234.09</td><td>-1.09&times;10<sup>1</sup></td><td>366.21</td><td>-1.65&times;10<sup>1</sup></td><td>231.90</td><td>-9.36&times;10<sup>0</sup></td></tr></tbody></table>
P et $\dot{P}$ pour un échantillon de modes g sur la séquence $M = 0.4758 + 0.0012 M_{\odot}$

<table><thead><tr><th rowspan="2">âge<br>(Ma)</th><th rowspan="2">Y<sub>c</sub></th><th colspan="2">g<sub>3,1</sub></th><th colspan="2">g<sub>3,2</sub></th><th colspan="2">g<sub>3,3</sub></th><th colspan="2">g<sub>3,4</sub></th></tr><tr><th>P<br>(s)</th><th>Π<br>(s/Ma)</th><th>P<br>(s)</th><th>Π<br>(s/Ma)</th><th>P<br>(s)</th><th>Π<br>(s/Ma)</th><th>P<br>(s)</th><th>Π<br>(s/Ma)</th></tr></thead><tbody><tr><td>1.79</td><td>0.928</td><td>248.84</td><td>+1.26×10<sup>0</sup></td><td>333.92</td><td>+2.03×10<sup>-1</sup></td><td>387.32</td><td>+1.44×10<sup>0</sup></td><td>443.61</td><td>-2.07×10<sup>0</sup></td></tr><tr><td>12.06</td><td>0.810</td><td>261.85</td><td>+1.28×10<sup>0</sup></td><td>329.09</td><td>-1.82×10<sup>0</sup></td><td>395.80</td><td>-4.05×10<sup>-1</sup></td><td>425.93</td><td>-1.02×10<sup>0</sup></td></tr><tr><td>18.02</td><td>0.758</td><td>269.73</td><td>+1.39×10<sup>0</sup></td><td>319.79</td><td>+1.31×10<sup>-1</sup></td><td>394.13</td><td>+1.04×10<sup>0</sup></td><td>425.81</td><td>+1.37×10<sup>0</sup></td></tr><tr><td>20.52</td><td>0.737</td><td>273.17</td><td>+1.31×10<sup>0</sup></td><td>320.67</td><td>-5.21×10<sup>-1</sup></td><td>397.36</td><td>+7.25×10<sup>-1</sup></td><td>429.92</td><td>+1.44×10<sup>0</sup></td></tr><tr><td>25.27</td><td>0.699</td><td>278.77</td><td>+1.16×10<sup>0</sup></td><td>309.21</td><td>-2.47×10<sup>0</sup></td><td>394.41</td><td>-7.01×10<sup>-1</sup></td><td>433.54</td><td>+6.77×10<sup>-1</sup></td></tr><tr><td>37.76</td><td>0.605</td><td>293.61</td><td>+1.04×10<sup>0</sup></td><td>313.00</td><td>+1.17×10<sup>0</sup></td><td>407.50</td><td>+1.43×10<sup>0</sup></td><td>451.45</td><td>+1.60×10<sup>0</sup></td></tr><tr><td>55.13</td><td>0.470</td><td>304.15</td><td>+2.05×10<sup>-1</sup></td><td>322.59</td><td>+7.74×10<sup>-1</sup></td><td>418.17</td><td>+3.61×10<sup>-1</sup></td><td>473.71</td><td>+1.24×10<sup>0</sup></td></tr><tr><td>67.83</td><td>0.361</td><td>306.18</td><td>+3.39×10<sup>-1</sup></td><td>338.26</td><td>+1.38×10<sup>0</sup></td><td>425.33</td><td>+7.44×10<sup>-1</sup></td><td>490.76</td><td>+1.37×10<sup>0</sup></td></tr><tr><td>77.34</td><td>0.277</td><td>309.20</td><td>-2.72×10<sup>-2</sup></td><td>349.92</td><td>+1.04×10<sup>0</sup></td><td>430.55</td><td>+4.20×10<sup>-2</sup></td><td>502.03</td><td>+8.54×10<sup>-1</sup></td></tr><tr><td>84.38</td><td>0.209</td><td>307.30</td><td>-2.61×10<sup>-1</sup></td><td>355.99</td><td>+6.73×10<sup>-1</sup></td><td>428.57</td><td>-3.22×10<sup>-1</sup></td><td>506.08</td><td>+3.69×10<sup>-1</sup></td></tr><tr><td>88.05</td><td>0.172</td><td>306.35</td><td>-3.90×10<sup>-1</sup></td><td>357.81</td><td>+2.46×10<sup>-1</sup></td><td>426.78</td><td>-9.41×10<sup>-1</sup></td><td>506.31</td><td>-4.78×10<sup>-1</sup></td></tr><tr><td>92.52</td><td>0.128</td><td>302.02</td><td>-1.80×10<sup>0</sup></td><td>356.81</td><td>-7.68×10<sup>-1</sup></td><td>420.52</td><td>-1.41×10<sup>0</sup></td><td>501.89</td><td>-1.06×10<sup>0</sup></td></tr><tr><td>94.63</td><td>0.112</td><td>297.71</td><td>-2.01×10<sup>0</sup></td><td>354.89</td><td>-8.97×10<sup>-1</sup></td><td>415.98</td><td>-3.83×10<sup>0</sup></td><td>498.35</td><td>-3.07×10<sup>0</sup></td></tr><tr><td>95.42</td><td>0.104</td><td>297.36</td><td>+2.57×10<sup>0</sup></td><td>354.20</td><td>-8.95×10<sup>-1</sup></td><td>414.45</td><td>+2.82×10<sup>0</sup></td><td>496.24</td><td>-1.07×10<sup>0</sup></td></tr><tr><td>97.80</td><td>0.154</td><td>315.40</td><td>+4.14×10<sup>0</sup></td><td>354.43</td><td>+2.26×10<sup>0</sup></td><td>442.86</td><td>+7.18×10<sup>0</sup></td><td>505.67</td><td>+5.55×10<sup>0</sup></td></tr><tr><td>100.24</td><td>0.116</td><td>307.92</td><td>-5.28×10<sup>0</sup></td><td>361.35</td><td>+5.58×10<sup>-1</sup></td><td>432.14</td><td>-8.45×10<sup>0</sup></td><td>507.77</td><td>-6.59×10<sup>0</sup></td></tr><tr><td>102.90</td><td>0.083</td><td>298.68</td><td>-2.89×10<sup>0</sup></td><td>352.97</td><td>-4.98×10<sup>0</sup></td><td>417.75</td><td>-3.10×10<sup>0</sup></td><td>482.50</td><td>-4.19×10<sup>0</sup></td></tr><tr><td>105.04</td><td>0.063</td><td>292.49</td><td>-3.93×10<sup>0</sup></td><td>344.69</td><td>-3.49×10<sup>0</sup></td><td>407.10</td><td>-1.17×10<sup>1</sup></td><td>467.22</td><td>-3.06×10<sup>1</sup></td></tr><tr><td>105.78</td><td>0.056</td><td>289.58</td><td>-7.72×10<sup>0</sup></td><td>342.10</td><td>-1.04×10<sup>1</sup></td><td>398.46</td><td>-1.46×10<sup>1</sup></td><td>444.55</td><td>-1.62×10<sup>1</sup></td></tr><tr><td>110.20</td><td>0.014</td><td>255.46</td><td>-3.17×10<sup>1</sup></td><td>296.07</td><td>-3.86×10<sup>1</sup></td><td>333.79</td><td>-5.31×10<sup>1</sup></td><td>372.86</td><td>-4.54×10<sup>1</sup></td></tr><tr><td>112.16</td><td>0.000</td><td>193.24</td><td>-1.66×10<sup>2</sup></td><td>220.48</td><td>-2.11×10<sup>2</sup></td><td>229.67</td><td>-1.07×10<sup>2</sup></td><td>283.89</td><td>-1.75×10<sup>2</sup></td></tr><tr><td>112.26</td><td>0.000</td><td>176.62</td><td>-2.06×10<sup>1</sup></td><td>199.34</td><td>-5.81×10<sup>1</sup></td><td>218.98</td><td>-5.84×10<sup>1</sup></td><td>266.40</td><td>-9.37×10<sup>1</sup></td></tr><tr><td>112.74</td><td>0.000</td><td>166.72</td><td>-6.82×10<sup>0</sup></td><td>171.44</td><td>-4.40×10<sup>0</sup></td><td>190.94</td><td>-5.93×10<sup>0</sup></td><td>221.42</td><td>-1.20×10<sup>1</sup></td></tr></tbody></table>

<table><thead><tr><th rowspan="2">âge<br>(Ma)</th><th rowspan="2">Y<sub>c</sub></th><th colspan="2">g<sub>3,5</sub></th><th colspan="2">g<sub>3,6</sub></th><th colspan="2">g<sub>1,4</sub></th><th colspan="2">g<sub>2,4</sub></th></tr><tr><th>P<br>(s)</th><th>Π<br>(s/Ma)</th><th>P<br>(s)</th><th>Π<br>(s/Ma)</th><th>P<br>(s)</th><th>Π<br>(s/Ma)</th><th>P<br>(s)</th><th>Π<br>(s/Ma)</th></tr></thead><tbody><tr><td>1.79</td><td>0.928</td><td>544.39</td><td>-4.15×10<sup>0</sup></td><td>649.19</td><td>-4.97×10<sup>0</sup></td><td>994.66</td><td>-3.77×10<sup>0</sup></td><td>597.67</td><td>-2.81×10<sup>0</sup></td></tr><tr><td>12.06</td><td>0.810</td><td>510.26</td><td>-1.68×10<sup>0</sup></td><td>612.61</td><td>-7.47×10<sup>-1</sup></td><td>968.84</td><td>+4.03×10<sup>-3</sup></td><td>576.07</td><td>-6.97×10<sup>-1</sup></td></tr><tr><td>18.02</td><td>0.758</td><td>510.12</td><td>+1.86×10<sup>0</sup></td><td>619.70</td><td>+2.61×10<sup>0</sup></td><td>980.84</td><td>+3.84×10<sup>0</sup></td><td>579.92</td><td>+2.11×10<sup>0</sup></td></tr><tr><td>20.52</td><td>0.737</td><td>515.20</td><td>+1.49×10<sup>0</sup></td><td>626.08</td><td>+2.04×10<sup>0</sup></td><td>990.52</td><td>+3.26×10<sup>0</sup></td><td>585.57</td><td>+1.90×10<sup>0</sup></td></tr><tr><td>25.27</td><td>0.699</td><td>516.41</td><td>+1.40×10<sup>-1</sup></td><td>631.20</td><td>+9.75×10<sup>-1</sup></td><td>999.85</td><td>+1.79×10<sup>0</sup></td><td>590.36</td><td>+8.98×10<sup>-1</sup></td></tr><tr><td>37.76</td><td>0.605</td><td>536.11</td><td>+1.80×10<sup>0</sup></td><td>656.49</td><td>+2.13×10<sup>0</sup></td><td>1038.74</td><td>+3.29×10<sup>0</sup></td><td>613.54</td><td>+2.02×10<sup>0</sup></td></tr><tr><td>55.13</td><td>0.470</td><td>552.84</td><td>+6.53×10<sup>-1</sup></td><td>678.58</td><td>+7.18×10<sup>-1</sup></td><td>1082.35</td><td>+2.32×10<sup>0</sup></td><td>640.30</td><td>+1.44×10<sup>0</sup></td></tr><tr><td>67.83</td><td>0.361</td><td>563.44</td><td>+1.04×10<sup>0</sup></td><td>690.92</td><td>+1.66×10<sup>0</sup></td><td>1114.10</td><td>+2.56×10<sup>0</sup></td><td>660.19</td><td>+1.61×10<sup>0</sup></td></tr><tr><td>77.34</td><td>0.277</td><td>569.68</td><td>-2.73×10<sup>-1</sup></td><td>694.12</td><td>-3.15×10<sup>0</sup></td><td>1133.61</td><td>+1.08×10<sup>0</sup></td><td>672.83</td><td>+8.01×10<sup>-1</sup></td></tr><tr><td>84.38</td><td>0.209</td><td>563.99</td><td>-7.87×10<sup>-1</sup></td><td>655.98</td><td>-5.76×10<sup>0</sup></td><td>1136.69</td><td>+2.24×10<sup>-1</sup></td><td>675.83</td><td>+2.51×10<sup>-1</sup></td></tr><tr><td>88.05</td><td>0.172</td><td>558.31</td><td>-3.37×10<sup>0</sup></td><td>633.86</td><td>-7.38×10<sup>0</sup></td><td>1134.79</td><td>-2.11×10<sup>0</sup></td><td>675.32</td><td>-9.30×10<sup>-1</sup></td></tr><tr><td>92.52</td><td>0.128</td><td>543.86</td><td>+5.51×10<sup>-1</sup></td><td>606.44</td><td>-6.62×10<sup>-1</sup></td><td>1121.44</td><td>-1.76×10<sup>0</sup></td><td>668.29</td><td>-1.41×10<sup>0</sup></td></tr><tr><td>94.63</td><td>0.112</td><td>538.22</td><td>-1.26×10<sup>1</sup></td><td>595.82</td><td>-1.84×10<sup>1</sup></td><td>1113.42</td><td>-9.46×10<sup>0</sup></td><td>663.38</td><td>-4.62×10<sup>0</sup></td></tr><tr><td>95.42</td><td>0.104</td><td>529.79</td><td>-4.74×10<sup>-1</sup></td><td>585.39</td><td>+5.43×10<sup>0</sup></td><td>1107.03</td><td>-1.91×10<sup>0</sup></td><td>660.35</td><td>-8.94×10<sup>-1</sup></td></tr><tr><td>97.80</td><td>0.154</td><td>583.71</td><td>+1.35×10<sup>1</sup></td><td>682.69</td><td>+1.50×10<sup>1</sup></td><td>1141.42</td><td>+1.40×10<sup>1</sup></td><td>677.46</td><td>+8.00×10<sup>0</sup></td></tr><tr><td>100.24</td><td>0.116</td><td>550.77</td><td>-2.49×10<sup>1</sup></td><td>603.82</td><td>-3.76×10<sup>1</sup></td><td>1124.59</td><td>-3.08×10<sup>1</sup></td><td>674.57</td><td>-1.29×10<sup>1</sup></td></tr><tr><td>102.90</td><td>0.083</td><td>501.99</td><td>-9.93×10<sup>0</sup></td><td>569.13</td><td>-3.88×10<sup>0</sup></td><td>1026.94</td><td>-1.90×10<sup>1</sup></td><td>631.36</td><td>-7.28×10<sup>0</sup></td></tr><tr><td>105.04</td><td>0.063</td><td>489.96</td><td>-7.89×10<sup>0</sup></td><td>557.60</td><td>-9.65×10<sup>0</sup></td><td>1009.42</td><td>-4.08×10<sup>1</sup></td><td>615.64</td><td>-3.31×10<sup>1</sup></td></tr><tr><td>105.78</td><td>0.056</td><td>484.12</td><td>-1.40×10<sup>1</sup></td><td>550.46</td><td>-1.50×10<sup>1</sup></td><td>979.26</td><td>-3.11×10<sup>1</sup></td><td>591.14</td><td>-2.02×10<sup>1</sup></td></tr><tr><td>110.20</td><td>0.014</td><td>422.36</td><td>-4.85×10<sup>1</sup></td><td>484.37</td><td>-5.78×10<sup>1</sup></td><td>841.81</td><td>-1.01×10<sup>2</sup></td><td>502.03</td><td>-6.06×10<sup>1</sup></td></tr><tr><td>112.16</td><td>0.000</td><td>327.34</td><td>-2.02×10<sup>2</sup></td><td>371.10</td><td>-5.72×10<sup>2</sup></td><td>644.17</td><td>-1.11×10<sup>3</sup></td><td>383.26</td><td>-2.77×10<sup>2</sup></td></tr><tr><td>112.26</td><td>0.000</td><td>307.16</td><td>-1.56×10<sup>2</sup></td><td>313.86</td><td>-1.07×10<sup>2</sup></td><td>532.93</td><td>-2.54×10<sup>2</sup></td><td>355.52</td><td>-2.14×10<sup>2</sup></td></tr><tr><td>112.74</td><td>0.000</td><td>232.50</td><td>-8.88×10<sup>0</sup></td><td>262.54</td><td>-1.13×10<sup>1</sup></td><td>411.08</td><td>-2.44×10<sup>1</sup></td><td>252.80</td><td>-8.46×10<sup>0</sup></td></tr></tbody></table>
P et $\dot{P}$ pour un échantillon de modes g sur la séquence $M = 0.4758 + 0.0022 M_{\odot}$

<table><thead><tr><th rowspan="2">âge<br>(Ma)</th><th rowspan="2">Y<sub>c</sub></th><th colspan="2">g<sub>3,1</sub></th><th colspan="2">g<sub>3,2</sub></th><th colspan="2">g<sub>3,3</sub></th><th colspan="2">g<sub>3,4</sub></th></tr><tr><th>P<br>(s)</th><th>Π<br>(s/Ma)</th><th>P<br>(s)</th><th>Π<br>(s/Ma)</th><th>P<br>(s)</th><th>Π<br>(s/Ma)</th><th>P<br>(s)</th><th>Π<br>(s/Ma)</th></tr></thead><tbody><tr><td>4.49</td><td>0.893</td><td>286.30</td><td>+1.97×10<sup>0</sup></td><td>338.63</td><td>-1.03×10<sup>0</sup></td><td>387.88</td><td>+1.23×10<sup>0</sup></td><td>444.04</td><td>-1.98×10<sup>0</sup></td></tr><tr><td>19.95</td><td>0.741</td><td>308.43</td><td>+3.56×10<sup>-1</sup></td><td>325.59</td><td>-4.67×10<sup>-1</sup></td><td>401.41</td><td>+1.68×10<sup>-1</sup></td><td>427.16</td><td>+6.92×10<sup>-1</sup></td></tr><tr><td>24.07</td><td>0.709</td><td>308.72</td><td>-7.16×10<sup>-2</sup></td><td>324.78</td><td>+1.89×10<sup>-1</sup></td><td>401.79</td><td>+2.27×10<sup>-1</sup></td><td>432.06</td><td>+1.47×10<sup>0</sup></td></tr><tr><td>37.88</td><td>0.603</td><td>308.03</td><td>-1.62×10<sup>-2</sup></td><td>341.74</td><td>+1.61×10<sup>0</sup></td><td>413.61</td><td>+9.73×10<sup>-1</sup></td><td>454.36</td><td>+1.69×10<sup>0</sup></td></tr><tr><td>55.12</td><td>0.469</td><td>311.03</td><td>+4.98×10<sup>-1</sup></td><td>366.80</td><td>+1.49×10<sup>0</sup></td><td>426.80</td><td>+8.44×10<sup>-1</sup></td><td>484.56</td><td>+1.86×10<sup>0</sup></td></tr><tr><td>67.76</td><td>0.360</td><td>319.32</td><td>+6.38×10<sup>-1</sup></td><td>386.36</td><td>+1.51×10<sup>0</sup></td><td>437.84</td><td>+6.63×10<sup>-1</sup></td><td>508.53</td><td>+1.85×10<sup>0</sup></td></tr><tr><td>77.17</td><td>0.274</td><td>325.09</td><td>+7.07×10<sup>-1</sup></td><td>399.30</td><td>+1.21×10<sup>0</sup></td><td>441.58</td><td>+2.01×10<sup>-1</sup></td><td>524.76</td><td>+1.56×10<sup>0</sup></td></tr><tr><td>84.13</td><td>0.206</td><td>330.07</td><td>+5.73×10<sup>-1</sup></td><td>406.02</td><td>+6.28×10<sup>-1</sup></td><td>440.39</td><td>-8.23×10<sup>-1</sup></td><td>532.98</td><td>+5.64×10<sup>-1</sup></td></tr><tr><td>89.27</td><td>0.154</td><td>331.06</td><td>-3.52×10<sup>-1</sup></td><td>406.69</td><td>-5.49×10<sup>-1</sup></td><td>434.58</td><td>-1.06×10<sup>0</sup></td><td>532.92</td><td>-5.31×10<sup>-1</sup></td></tr><tr><td>93.17</td><td>0.117</td><td>329.41</td><td>-2.41×10<sup>-2</sup></td><td>403.71</td><td>-5.39×10<sup>-1</sup></td><td>428.71</td><td>-2.75×10<sup>0</sup></td><td>527.15</td><td>-3.02×10<sup>0</sup></td></tr><tr><td>94.45</td><td>0.107</td><td>329.10</td><td>-9.72×10<sup>-1</sup></td><td>403.07</td><td>-6.49×10<sup>-1</sup></td><td>427.91</td><td>+4.28×10<sup>0</sup></td><td>524.15</td><td>-2.86×10<sup>-1</sup></td></tr><tr><td>96.22</td><td>0.130</td><td>328.30</td><td>+2.29×10<sup>0</sup></td><td>403.72</td><td>+2.85×10<sup>0</sup></td><td>440.63</td><td>+2.38×10<sup>0</sup></td><td>529.53</td><td>+4.94×10<sup>0</sup></td></tr><tr><td>97.60</td><td>0.109</td><td>332.68</td><td>+1.05×10<sup>0</sup></td><td>407.98</td><td>+4.86×10<sup>-1</sup></td><td>435.62</td><td>-5.86×10<sup>0</sup></td><td>532.13</td><td>-4.61×10<sup>0</sup></td></tr><tr><td>99.44</td><td>0.086</td><td>327.18</td><td>-4.24×10<sup>0</sup></td><td>401.00</td><td>-5.10×10<sup>0</sup></td><td>426.74</td><td>-4.80×10<sup>0</sup></td><td>508.74</td><td>-1.57×10<sup>1</sup></td></tr><tr><td>101.10</td><td>0.070</td><td>321.38</td><td>-3.39×10<sup>0</sup></td><td>393.67</td><td>-4.58×10<sup>0</sup></td><td>417.02</td><td>-7.06×10<sup>0</sup></td><td>483.23</td><td>-1.67×10<sup>1</sup></td></tr><tr><td>102.92</td><td>0.054</td><td>314.35</td><td>-4.22×10<sup>0</sup></td><td>384.29</td><td>-5.18×10<sup>0</sup></td><td>404.23</td><td>-5.46×10<sup>0</sup></td><td>456.34</td><td>-7.67×10<sup>0</sup></td></tr><tr><td>104.62</td><td>0.041</td><td>306.41</td><td>-5.34×10<sup>0</sup></td><td>375.11</td><td>-6.33×10<sup>0</sup></td><td>395.90</td><td>-6.67×10<sup>0</sup></td><td>450.24</td><td>-6.90×10<sup>0</sup></td></tr><tr><td>106.06</td><td>0.031</td><td>297.77</td><td>-6.56×10<sup>0</sup></td><td>364.69</td><td>-7.68×10<sup>0</sup></td><td>383.95</td><td>-8.06×10<sup>0</sup></td><td>436.23</td><td>-6.31×10<sup>0</sup></td></tr><tr><td>107.23</td><td>0.023</td><td>288.99</td><td>-9.01×10<sup>0</sup></td><td>354.50</td><td>-1.08×10<sup>1</sup></td><td>374.15</td><td>-1.07×10<sup>1</sup></td><td>431.59</td><td>-7.13×10<sup>0</sup></td></tr><tr><td>108.16</td><td>0.017</td><td>280.20</td><td>-8.70×10<sup>0</sup></td><td>344.62</td><td>-7.80×10<sup>0</sup></td><td>365.86</td><td>-1.75×10<sup>0</sup></td><td>425.98</td><td>+3.05×10<sup>0</sup></td></tr><tr><td>110.22</td><td>0.002</td><td>230.02</td><td>-6.00×10<sup>1</sup></td><td>284.22</td><td>-8.44×10<sup>1</sup></td><td>302.19</td><td>-1.17×10<sup>2</sup></td><td>346.19</td><td>-1.60×10<sup>2</sup></td></tr><tr><td>110.52</td><td>0.000</td><td>207.24</td><td>-9.98×10<sup>1</sup></td><td>242.36</td><td>-2.37×10<sup>2</sup></td><td>260.92</td><td>-1.58×10<sup>2</sup></td><td>286.02</td><td>-2.52×10<sup>2</sup></td></tr><tr><td>110.66</td><td>0.000</td><td>191.88</td><td>-1.15×10<sup>2</sup></td><td>203.12</td><td>-3.02×10<sup>2</sup></td><td>237.89</td><td>-1.68×10<sup>2</sup></td><td>247.95</td><td>-2.82×10<sup>2</sup></td></tr></tbody></table>

<table><thead><tr><th rowspan="2">âge<br>(Ma)</th><th rowspan="2">Y<sub>c</sub></th><th colspan="2">g<sub>3,5</sub></th><th colspan="2">g<sub>3,6</sub></th><th colspan="2">g<sub>1,4</sub></th><th colspan="2">g<sub>2,4</sub></th></tr><tr><th>P<br>(s)</th><th>Π<br>(s/Ma)</th><th>P<br>(s)</th><th>Π<br>(s/Ma)</th><th>P<br>(s)</th><th>Π<br>(s/Ma)</th><th>P<br>(s)</th><th>Π<br>(s/Ma)</th></tr></thead><tbody><tr><td>4.49</td><td>0.893</td><td>543.99</td><td>-2.47×10<sup>0</sup></td><td>646.79</td><td>-1.56×10<sup>0</sup></td><td>982.02</td><td>-2.85×10<sup>0</sup></td><td>594.15</td><td>-2.40×10<sup>0</sup></td></tr><tr><td>19.95</td><td>0.741</td><td>519.34</td><td>+1.54×10<sup>-1</sup></td><td>635.63</td><td>+9.60×10<sup>-1</sup></td><td>956.42</td><td>+7.35×10<sup>-1</sup></td><td>570.36</td><td>+1.89×10<sup>-1</sup></td></tr><tr><td>24.07</td><td>0.709</td><td>522.17</td><td>+1.05×10<sup>0</sup></td><td>641.60</td><td>+1.76×10<sup>0</sup></td><td>962.45</td><td>+1.97×10<sup>0</sup></td><td>573.33</td><td>+1.09×10<sup>0</sup></td></tr><tr><td>37.88</td><td>0.603</td><td>541.05</td><td>+1.33×10<sup>0</sup></td><td>667.74</td><td>+1.76×10<sup>0</sup></td><td>996.59</td><td>+2.55×10<sup>0</sup></td><td>594.20</td><td>+1.64×10<sup>0</sup></td></tr><tr><td>55.12</td><td>0.469</td><td>559.74</td><td>+1.10×10<sup>0</sup></td><td>691.90</td><td>+1.21×10<sup>0</sup></td><td>1037.75</td><td>+2.50×10<sup>0</sup></td><td>621.59</td><td>+1.69×10<sup>0</sup></td></tr><tr><td>67.76</td><td>0.360</td><td>573.13</td><td>+7.80×10<sup>-1</sup></td><td>706.59</td><td>+1.10×10<sup>0</sup></td><td>1067.95</td><td>+1.87×10<sup>0</sup></td><td>642.83</td><td>+1.46×10<sup>0</sup></td></tr><tr><td>77.17</td><td>0.274</td><td>577.17</td><td>+1.15×10<sup>-1</sup></td><td>704.85</td><td>-2.69×10<sup>0</sup></td><td>1081.32</td><td>+1.19×10<sup>0</sup></td><td>654.45</td><td>+1.10×10<sup>0</sup></td></tr><tr><td>84.13</td><td>0.206</td><td>571.73</td><td>-2.37×10<sup>0</sup></td><td>663.76</td><td>-9.56×10<sup>0</sup></td><td>1084.93</td><td>-8.34×10<sup>-1</sup></td><td>659.61</td><td>+6.30×10<sup>-2</sup></td></tr><tr><td>89.27</td><td>0.154</td><td>557.85</td><td>-1.68×10<sup>0</sup></td><td>614.66</td><td>-6.50×10<sup>0</sup></td><td>1076.19</td><td>-1.93×10<sup>0</sup></td><td>656.67</td><td>-1.10×10<sup>0</sup></td></tr><tr><td>93.17</td><td>0.117</td><td>546.71</td><td>-7.00×10<sup>0</sup></td><td>597.21</td><td>-5.03×10<sup>0</sup></td><td>1063.54</td><td>-6.17×10<sup>0</sup></td><td>650.19</td><td>-2.46×10<sup>0</sup></td></tr><tr><td>94.45</td><td>0.107</td><td>543.24</td><td>+8.22×10<sup>0</sup></td><td>599.41</td><td>+1.63×10<sup>1</sup></td><td>1059.28</td><td>+3.84×10<sup>0</sup></td><td>648.13</td><td>+4.36×10<sup>-1</sup></td></tr><tr><td>96.22</td><td>0.130</td><td>568.04</td><td>+1.31×10<sup>0</sup></td><td>635.87</td><td>-1.57×10<sup>0</sup></td><td>1075.73</td><td>+4.30×10<sup>0</sup></td><td>653.19</td><td>+3.24×10<sup>0</sup></td></tr><tr><td>97.60</td><td>0.109</td><td>550.61</td><td>-1.62×10<sup>1</sup></td><td>605.96</td><td>-2.39×10<sup>1</sup></td><td>1068.49</td><td>-1.23×10<sup>1</sup></td><td>653.98</td><td>-3.46×10<sup>0</sup></td></tr><tr><td>99.44</td><td>0.086</td><td>532.06</td><td>-6.91×10<sup>0</sup></td><td>584.60</td><td>-7.80×10<sup>0</sup></td><td>1039.32</td><td>-1.70×10<sup>1</sup></td><td>639.58</td><td>-9.36×10<sup>0</sup></td></tr><tr><td>101.10</td><td>0.070</td><td>520.95</td><td>-6.44×10<sup>0</sup></td><td>568.53</td><td>-9.20×10<sup>0</sup></td><td>1011.00</td><td>-1.84×10<sup>1</sup></td><td>623.17</td><td>-1.21×10<sup>1</sup></td></tr><tr><td>102.92</td><td>0.054</td><td>509.20</td><td>-6.79×10<sup>0</sup></td><td>554.70</td><td>-7.52×10<sup>0</sup></td><td>980.69</td><td>-1.02×10<sup>1</sup></td><td>601.25</td><td>-7.60×10<sup>0</sup></td></tr><tr><td>104.62</td><td>0.041</td><td>496.60</td><td>-8.06×10<sup>0</sup></td><td>540.46</td><td>-8.76×10<sup>0</sup></td><td>967.63</td><td>-1.32×10<sup>1</sup></td><td>592.40</td><td>-8.65×10<sup>0</sup></td></tr><tr><td>106.06</td><td>0.031</td><td>483.18</td><td>-1.13×10<sup>1</sup></td><td>529.46</td><td>-5.48×10<sup>0</sup></td><td>941.32</td><td>-1.71×10<sup>1</sup></td><td>575.34</td><td>-1.04×10<sup>1</sup></td></tr><tr><td>107.23</td><td>0.023</td><td>469.77</td><td>-9.71×10<sup>0</sup></td><td>519.82</td><td>-1.63×10<sup>1</sup></td><td>921.60</td><td>-2.24×10<sup>1</sup></td><td>563.44</td><td>-1.48×10<sup>1</sup></td></tr><tr><td>108.16</td><td>0.017</td><td>456.16</td><td>-2.74×10<sup>1</sup></td><td>508.33</td><td>+6.32×10<sup>0</sup></td><td>901.79</td><td>-1.01×10<sup>1</sup></td><td>550.69</td><td>-4.10×10<sup>0</sup></td></tr><tr><td>110.22</td><td>0.002</td><td>374.86</td><td>-1.77×10<sup>1</sup></td><td>423.51</td><td>-2.10×10<sup>2</sup></td><td>741.64</td><td>-2.60×10<sup>2</sup></td><td>452.16</td><td>-1.75×10<sup>2</sup></td></tr><tr><td>110.52</td><td>0.000</td><td>335.95</td><td>-3.62×10<sup>2</sup></td><td>350.24</td><td>-2.70×10<sup>2</sup></td><td>611.64</td><td>-7.36×10<sup>2</sup></td><td>375.04</td><td>-3.90×10<sup>2</sup></td></tr><tr><td>110.66</td><td>0.000</td><td>269.93</td><td>-5.26×10<sup>2</sup></td><td>311.61</td><td>-2.79×10<sup>2</sup></td><td>489.80</td><td>-9.37×10<sup>2</sup></td><td>312.51</td><td>-4.75×10<sup>2</sup></td></tr></tbody></table>
P et $\dot{P}$ pour un échantillon de modes g sur la séquence $M = 0.4758 + 0.0032 M_{\odot}$

<table><thead><tr><th rowspan="2">âge<br>(Ma)</th><th rowspan="2">Y<sub>c</sub></th><th colspan="2"><i>g</i><sub>3,1</sub></th><th colspan="2"><i>g</i><sub>3,2</sub></th><th colspan="2"><i>g</i><sub>3,3</sub></th><th colspan="2"><i>g</i><sub>3,4</sub></th></tr><tr><th><i>P</i><br>(s)</th><th><i>P&#x0307;</i><br>(s/Ma)</th><th><i>P</i><br>(s)</th><th><i>P&#x0307;</i><br>(s/Ma)</th><th><i>P</i><br>(s)</th><th><i>P&#x0307;</i><br>(s/Ma)</th><th><i>P</i><br>(s)</th><th><i>P&#x0307;</i><br>(s/Ma)</th></tr></thead><tbody><tr><td>1.79</td><td>0.927</td><td>303.99</td><td>+1.96×10<sup>0</sup></td><td>343.98</td><td>-9.51×10<sup>-1</sup></td><td>395.69</td><td>+2.06×10<sup>0</sup></td><td>451.36</td><td>-3.37×10<sup>0</sup></td></tr><tr><td>12.06</td><td>0.809</td><td>320.40</td><td>+8.81×10<sup>-1</sup></td><td>337.47</td><td>+3.46×10<sup>-4</sup></td><td>411.56</td><td>+5.22×10<sup>-1</sup></td><td>429.01</td><td>+2.07×10<sup>-1</sup></td></tr><tr><td>19.53</td><td>0.742</td><td>319.48</td><td>-1.35×10<sup>0</sup></td><td>342.15</td><td>+1.19×10<sup>0</sup></td><td>409.86</td><td>-6.07×10<sup>-1</sup></td><td>442.16</td><td>+2.27×10<sup>0</sup></td></tr><tr><td>23.43</td><td>0.709</td><td>313.05</td><td>-1.49×10<sup>0</sup></td><td>347.54</td><td>+1.50×10<sup>0</sup></td><td>408.10</td><td>-1.37×10<sup>-1</sup></td><td>450.34</td><td>+2.04×10<sup>0</sup></td></tr><tr><td>39.83</td><td>0.585</td><td>315.30</td><td>+8.14×10<sup>-1</sup></td><td>374.13</td><td>+1.64×10<sup>0</sup></td><td>423.45</td><td>+1.08×10<sup>0</sup></td><td>486.11</td><td>+2.22×10<sup>0</sup></td></tr><tr><td>56.38</td><td>0.456</td><td>329.54</td><td>+1.23×10<sup>0</sup></td><td>401.31</td><td>+1.74×10<sup>0</sup></td><td>434.71</td><td>+6.78×10<sup>-1</sup></td><td>523.21</td><td>+2.36×10<sup>0</sup></td></tr><tr><td>68.54</td><td>0.348</td><td>348.29</td><td>+1.65×10<sup>0</sup></td><td>421.98</td><td>+1.49×10<sup>0</sup></td><td>444.65</td><td>+8.06×10<sup>-1</sup></td><td>552.33</td><td>+2.30×10<sup>0</sup></td></tr><tr><td>77.44</td><td>0.258</td><td>362.12</td><td>+1.43×10<sup>0</sup></td><td>432.12</td><td>+7.27×10<sup>-1</sup></td><td>452.01</td><td>+9.86×10<sup>-1</sup></td><td>567.52</td><td>+6.93×10<sup>-1</sup></td></tr><tr><td>83.92</td><td>0.189</td><td>369.76</td><td>+7.85×10<sup>-1</sup></td><td>432.99</td><td>-7.05×10<sup>-1</sup></td><td>458.05</td><td>+6.11×10<sup>-1</sup></td><td>564.85</td><td>-1.57×10<sup>0</sup></td></tr><tr><td>88.76</td><td>0.138</td><td>371.14</td><td>-3.15×10<sup>-1</sup></td><td>425.97</td><td>-2.13×10<sup>0</sup></td><td>458.40</td><td>-5.64×10<sup>-1</sup></td><td>550.34</td><td>-4.94×10<sup>0</sup></td></tr><tr><td>92.27</td><td>0.102</td><td>367.93</td><td>-1.61×10<sup>0</sup></td><td>417.11</td><td>-2.86×10<sup>0</sup></td><td>454.09</td><td>-2.03×10<sup>0</sup></td><td>530.61</td><td>-5.21×10<sup>0</sup></td></tr><tr><td>95.10</td><td>0.076</td><td>361.37</td><td>-3.11×10<sup>0</sup></td><td>407.80</td><td>-3.92×10<sup>0</sup></td><td>445.89</td><td>-3.88×10<sup>0</sup></td><td>515.54</td><td>-6.75×10<sup>0</sup></td></tr><tr><td>97.37</td><td>0.056</td><td>352.63</td><td>-4.61×10<sup>0</sup></td><td>397.72</td><td>-4.81×10<sup>0</sup></td><td>435.22</td><td>-5.46×10<sup>0</sup></td><td>495.67</td><td>-1.05×10<sup>1</sup></td></tr><tr><td>99.17</td><td>0.041</td><td>342.51</td><td>-7.00×10<sup>0</sup></td><td>386.85</td><td>-8.32×10<sup>0</sup></td><td>422.83</td><td>-9.18×10<sup>0</sup></td><td>475.96</td><td>-1.05×10<sup>1</sup></td></tr><tr><td>100.58</td><td>0.029</td><td>331.17</td><td>-8.77×10<sup>0</sup></td><td>375.33</td><td>-5.27×10<sup>0</sup></td><td>409.20</td><td>-8.17×10<sup>0</sup></td><td>460.17</td><td>-1.36×10<sup>1</sup></td></tr><tr><td>101.68</td><td>0.021</td><td>320.25</td><td>-1.52×10<sup>1</sup></td><td>365.22</td><td>-1.53×10<sup>1</sup></td><td>396.16</td><td>-1.84×10<sup>1</sup></td><td>443.29</td><td>-2.15×10<sup>1</sup></td></tr><tr><td>103.04</td><td>0.011</td><td>299.55</td><td>-4.32×10<sup>1</sup></td><td>344.43</td><td>-5.73×10<sup>1</sup></td><td>371.18</td><td>-5.56×10<sup>1</sup></td><td>414.11</td><td>-7.78×10<sup>1</sup></td></tr><tr><td>104.51</td><td>0.000</td><td>235.99</td><td>-1.22×10<sup>2</sup></td><td>260.14</td><td>-2.87×10<sup>2</sup></td><td>289.41</td><td>-2.53×10<sup>2</sup></td><td>299.75</td><td>-1.64×10<sup>2</sup></td></tr><tr><td>104.62</td><td>0.000</td><td>222.59</td><td>-2.43×10<sup>2</sup></td><td>228.54</td><td>-1.51×10<sup>2</sup></td><td>261.52</td><td>-2.21×10<sup>2</sup></td><td>281.74</td><td>-2.20×10<sup>2</sup></td></tr><tr><td>104.82</td><td>0.000</td><td>174.03</td><td>-1.76×10<sup>1</sup></td><td>198.42</td><td>-4.83×10<sup>1</sup></td><td>217.34</td><td>-4.34×10<sup>1</sup></td><td>237.79</td><td>-5.43×10<sup>1</sup></td></tr></tbody></table>

<table><thead><tr><th rowspan="2">âge<br>(Ma)</th><th rowspan="2">Y<sub>c</sub></th><th colspan="2"><i>g</i><sub>3,5</sub></th><th colspan="2"><i>g</i><sub>3,6</sub></th><th colspan="2"><i>g</i><sub>1,4</sub></th><th colspan="2"><i>g</i><sub>2,4</sub></th></tr><tr><th><i>P</i><br>(s)</th><th><i>P&#x0307;</i><br>(s/Ma)</th><th><i>P</i><br>(s)</th><th><i>P&#x0307;</i><br>(s/Ma)</th><th><i>P</i><br>(s)</th><th><i>P&#x0307;</i><br>(s/Ma)</th><th><i>P</i><br>(s)</th><th><i>P&#x0307;</i><br>(s/Ma)</th></tr></thead><tbody><tr><td>1.79</td><td>0.927</td><td>560.28</td><td>-5.45×10<sup>0</sup></td><td>671.69</td><td>-5.65×10<sup>0</sup></td><td>1003.38</td><td>-8.18×10<sup>0</sup></td><td>605.88</td><td>-5.04×10<sup>0</sup></td></tr><tr><td>12.06</td><td>0.809</td><td>519.15</td><td>-1.12×10<sup>0</sup></td><td>631.52</td><td>-4.39×10<sup>-1</sup></td><td>942.41</td><td>-1.44×10<sup>0</sup></td><td>566.95</td><td>-1.29×10<sup>0</sup></td></tr><tr><td>19.53</td><td>0.742</td><td>523.58</td><td>+8.64×10<sup>-1</sup></td><td>643.30</td><td>+1.82×10<sup>0</sup></td><td>950.47</td><td>+1.22×10<sup>0</sup></td><td>568.64</td><td>+5.36×10<sup>-1</sup></td></tr><tr><td>23.43</td><td>0.709</td><td>525.88</td><td>+6.56×10<sup>-1</sup></td><td>648.76</td><td>+1.36×10<sup>0</sup></td><td>953.24</td><td>+8.62×10<sup>-1</sup></td><td>570.13</td><td>+5.48×10<sup>-1</sup></td></tr><tr><td>39.83</td><td>0.585</td><td>550.59</td><td>+1.52×10<sup>0</sup></td><td>682.03</td><td>+1.95×10<sup>0</sup></td><td>995.86</td><td>+2.67×10<sup>0</sup></td><td>597.20</td><td>+1.84×10<sup>0</sup></td></tr><tr><td>56.38</td><td>0.456</td><td>567.72</td><td>+9.14×10<sup>-1</sup></td><td>704.27</td><td>+9.57×10<sup>-1</sup></td><td>1026.98</td><td>+1.92×10<sup>0</sup></td><td>622.62</td><td>+1.67×10<sup>0</sup></td></tr><tr><td>68.54</td><td>0.348</td><td>579.21</td><td>+8.33×10<sup>-1</sup></td><td>714.79</td><td>+8.01×10<sup>-1</sup></td><td>1051.27</td><td>+1.53×10<sup>0</sup></td><td>645.32</td><td>+1.83×10<sup>0</sup></td></tr><tr><td>77.44</td><td>0.258</td><td>586.58</td><td>+1.00×10<sup>0</sup></td><td>705.86</td><td>-4.59×10<sup>0</sup></td><td>1059.99</td><td>+6.34×10<sup>-1</sup></td><td>659.83</td><td>+1.44×10<sup>0</sup></td></tr><tr><td>83.92</td><td>0.189</td><td>594.05</td><td>+1.19×10<sup>0</sup></td><td>664.12</td><td>-6.25×10<sup>0</sup></td><td>1059.66</td><td>-1.32×10<sup>0</sup></td><td>666.51</td><td>+3.59×10<sup>-1</sup></td></tr><tr><td>88.76</td><td>0.138</td><td>594.52</td><td>-2.03×10<sup>0</sup></td><td>630.99</td><td>-9.72×10<sup>0</sup></td><td>1047.46</td><td>-3.55×10<sup>0</sup></td><td>664.27</td><td>-1.41×10<sup>0</sup></td></tr><tr><td>92.27</td><td>0.102</td><td>582.64</td><td>-3.80×10<sup>0</sup></td><td>596.03</td><td>-6.71×10<sup>0</sup></td><td>1030.73</td><td>-6.58×10<sup>0</sup></td><td>655.99</td><td>-3.48×10<sup>0</sup></td></tr><tr><td>95.10</td><td>0.076</td><td>569.81</td><td>-6.41×10<sup>0</sup></td><td>584.08</td><td>-4.68×10<sup>0</sup></td><td>1007.82</td><td>-9.17×10<sup>0</sup></td><td>642.91</td><td>-5.86×10<sup>0</sup></td></tr><tr><td>97.37</td><td>0.056</td><td>552.69</td><td>-7.38×10<sup>0</sup></td><td>569.36</td><td>-7.36×10<sup>0</sup></td><td>984.01</td><td>-1.26×10<sup>1</sup></td><td>627.03</td><td>-8.21×10<sup>0</sup></td></tr><tr><td>99.17</td><td>0.041</td><td>536.58</td><td>-1.33×10<sup>1</sup></td><td>553.26</td><td>-1.22×10<sup>1</sup></td><td>955.06</td><td>-2.08×10<sup>1</sup></td><td>608.64</td><td>-1.32×10<sup>1</sup></td></tr><tr><td>100.58</td><td>0.029</td><td>518.38</td><td>-6.67×10<sup>0</sup></td><td>534.98</td><td>-1.08×10<sup>1</sup></td><td>927.50</td><td>-1.25×10<sup>1</sup></td><td>589.57</td><td>-1.11×10<sup>1</sup></td></tr><tr><td>101.68</td><td>0.021</td><td>500.97</td><td>-2.14×10<sup>1</sup></td><td>517.60</td><td>-2.34×10<sup>1</sup></td><td>901.53</td><td>-4.07×10<sup>1</sup></td><td>571.23</td><td>-2.63×10<sup>1</sup></td></tr><tr><td>103.04</td><td>0.011</td><td>471.82</td><td>-7.24×10<sup>1</sup></td><td>485.83</td><td>-7.09×10<sup>1</sup></td><td>846.23</td><td>-1.47×10<sup>2</sup></td><td>535.50</td><td>-9.74×10<sup>1</sup></td></tr><tr><td>104.51</td><td>0.000</td><td>365.35</td><td>-5.63×10<sup>2</sup></td><td>381.58</td><td>-3.49×10<sup>2</sup></td><td>630.06</td><td>-7.66×10<sup>2</sup></td><td>392.26</td><td>-4.22×10<sup>2</sup></td></tr><tr><td>104.62</td><td>0.000</td><td>303.37</td><td>-2.69×10<sup>2</sup></td><td>343.15</td><td>-3.05×10<sup>2</sup></td><td>545.80</td><td>-5.77×10<sup>2</sup></td><td>345.87</td><td>-3.20×10<sup>2</sup></td></tr><tr><td>104.82</td><td>0.000</td><td>249.60</td><td>-3.40×10<sup>1</sup></td><td>282.16</td><td>-7.41×10<sup>1</sup></td><td>430.35</td><td>-1.50×10<sup>2</sup></td><td>281.97</td><td>-2.26×10<sup>1</sup></td></tr></tbody></table>
P et $\dot{P}$ pour un échantillon de modes g sur la séquence $M = 0.4758 + 0.0042 M_{\odot}$

<table><thead><tr><th rowspan="2">âge<br>(Ma)</th><th rowspan="2">Y<sub>c</sub></th><th colspan="2">g<sub>3,1</sub></th><th colspan="2">g<sub>3,2</sub></th><th colspan="2">g<sub>3,3</sub></th><th colspan="2">g<sub>3,4</sub></th></tr><tr><th>P<br>(s)</th><th>Π<br>(s/Ma)</th><th>P<br>(s)</th><th>Π<br>(s/Ma)</th><th>P<br>(s)</th><th>Π<br>(s/Ma)</th><th>P<br>(s)</th><th>Π<br>(s/Ma)</th></tr></thead><tbody><tr><td>1.79</td><td>0.927</td><td>322.77</td><td>+1.35×10<sup>0</sup></td><td>348.55</td><td>-3.27×10<sup>-1</sup></td><td>418.03</td><td>+3.57×10<sup>-1</sup></td><td>455.42</td><td>-1.51×10<sup>0</sup></td></tr><tr><td>12.06</td><td>0.809</td><td>331.57</td><td>-1.32×10<sup>-1</sup></td><td>350.47</td><td>+1.22×10<sup>0</sup></td><td>419.59</td><td>-2.58×10<sup>-1</sup></td><td>450.27</td><td>+1.51×10<sup>0</sup></td></tr><tr><td>19.66</td><td>0.741</td><td>325.33</td><td>-1.10×10<sup>0</sup></td><td>364.52</td><td>+1.97×10<sup>0</sup></td><td>416.62</td><td>-2.03×10<sup>-1</sup></td><td>470.97</td><td>+2.91×10<sup>0</sup></td></tr><tr><td>23.84</td><td>0.707</td><td>321.60</td><td>-5.46×10<sup>-1</sup></td><td>372.22</td><td>+1.80×10<sup>0</sup></td><td>417.00</td><td>+3.98×10<sup>-1</sup></td><td>481.87</td><td>+2.48×10<sup>0</sup></td></tr><tr><td>39.40</td><td>0.588</td><td>332.45</td><td>+1.37×10<sup>0</sup></td><td>401.72</td><td>+1.92×10<sup>0</sup></td><td>432.04</td><td>+9.06×10<sup>-1</sup></td><td>522.26</td><td>+2.70×10<sup>0</sup></td></tr><tr><td>55.93</td><td>0.456</td><td>358.14</td><td>+1.80×10<sup>0</sup></td><td>429.47</td><td>+1.25×10<sup>0</sup></td><td>445.60</td><td>+1.38×10<sup>0</sup></td><td>562.78</td><td>+1.84×10<sup>0</sup></td></tr><tr><td>68.06</td><td>0.348</td><td>381.22</td><td>+1.89×10<sup>0</sup></td><td>440.70</td><td>+7.20×10<sup>-1</sup></td><td>469.17</td><td>+2.19×10<sup>0</sup></td><td>578.59</td><td>+9.31×10<sup>-1</sup></td></tr><tr><td>77.10</td><td>0.266</td><td>397.33</td><td>+1.65×10<sup>0</sup></td><td>444.62</td><td>-7.68×10<sup>-2</sup></td><td>488.18</td><td>+1.99×10<sup>0</sup></td><td>582.65</td><td>-3.62×10<sup>-1</sup></td></tr><tr><td>83.80</td><td>0.197</td><td>406.75</td><td>+1.04×10<sup>0</sup></td><td>441.89</td><td>-5.25×10<sup>-1</sup></td><td>499.73</td><td>+1.29×10<sup>0</sup></td><td>574.85</td><td>-1.92×10<sup>0</sup></td></tr><tr><td>88.45</td><td>0.151</td><td>409.57</td><td>+8.17×10<sup>-2</sup></td><td>438.01</td><td>-1.52×10<sup>0</sup></td><td>503.40</td><td>+2.17×10<sup>-1</sup></td><td>560.34</td><td>-5.02×10<sup>0</sup></td></tr><tr><td>91.08</td><td>0.124</td><td>408.79</td><td>-7.06×10<sup>-1</sup></td><td>432.91</td><td>-2.18×10<sup>0</sup></td><td>502.81</td><td>-7.40×10<sup>-1</sup></td><td>546.48</td><td>-4.31×10<sup>0</sup></td></tr><tr><td>93.82</td><td>0.099</td><td>405.55</td><td>-1.67×10<sup>0</sup></td><td>426.44</td><td>-2.66×10<sup>0</sup></td><td>499.02</td><td>-2.06×10<sup>0</sup></td><td>534.21</td><td>-7.09×10<sup>0</sup></td></tr><tr><td>96.05</td><td>0.080</td><td>400.79</td><td>-2.63×10<sup>0</sup></td><td>419.82</td><td>-3.25×10<sup>0</sup></td><td>493.11</td><td>-3.26×10<sup>0</sup></td><td>515.69</td><td>-6.63×10<sup>0</sup></td></tr><tr><td>98.37</td><td>0.061</td><td>392.95</td><td>-4.30×10<sup>0</sup></td><td>411.48</td><td>-4.02×10<sup>0</sup></td><td>483.45</td><td>-5.30×10<sup>0</sup></td><td>503.86</td><td>-6.80×10<sup>0</sup></td></tr><tr><td>100.23</td><td>0.046</td><td>383.45</td><td>-5.84×10<sup>0</sup></td><td>402.66</td><td>-5.76×10<sup>0</sup></td><td>471.32</td><td>-7.85×10<sup>0</sup></td><td>486.47</td><td>-1.05×10<sup>1</sup></td></tr><tr><td>101.76</td><td>0.034</td><td>372.15</td><td>-9.43×10<sup>0</sup></td><td>392.42</td><td>-8.65×10<sup>0</sup></td><td>457.18</td><td>-1.15×10<sup>1</sup></td><td>472.37</td><td>-6.81×10<sup>0</sup></td></tr><tr><td>102.99</td><td>0.025</td><td>360.55</td><td>-1.19×10<sup>1</sup></td><td>381.77</td><td>-9.86×10<sup>0</sup></td><td>442.99</td><td>-1.48×10<sup>1</sup></td><td>464.00</td><td>-1.63×10<sup>1</sup></td></tr><tr><td>103.97</td><td>0.018</td><td>348.85</td><td>-2.39×10<sup>1</sup></td><td>372.11</td><td>-2.18×10<sup>1</sup></td><td>428.47</td><td>-2.91×10<sup>1</sup></td><td>448.04</td><td>-2.85×10<sup>1</sup></td></tr><tr><td>105.76</td><td>0.005</td><td>306.12</td><td>-7.18×10<sup>1</sup></td><td>333.10</td><td>-9.90×10<sup>1</sup></td><td>376.39</td><td>-1.16×10<sup>2</sup></td><td>397.02</td><td>-1.06×10<sup>2</sup></td></tr><tr><td>106.48</td><td>0.000</td><td>254.42</td><td>-3.65×10<sup>2</sup></td><td>261.86</td><td>-1.52×10<sup>2</sup></td><td>292.79</td><td>-3.17×10<sup>2</sup></td><td>320.71</td><td>-3.21×10<sup>2</sup></td></tr><tr><td>106.58</td><td>0.000</td><td>217.91</td><td>-1.66×10<sup>2</sup></td><td>246.68</td><td>-1.58×10<sup>2</sup></td><td>261.13</td><td>-2.07×10<sup>2</sup></td><td>288.57</td><td>-2.39×10<sup>2</sup></td></tr><tr><td>106.77</td><td>0.000</td><td>186.40</td><td>-7.38×10<sup>1</sup></td><td>216.72</td><td>-1.26×10<sup>2</sup></td><td>221.84</td><td>-1.02×10<sup>2</sup></td><td>243.22</td><td>-7.43×10<sup>1</sup></td></tr></tbody></table>

<table><thead><tr><th rowspan="2">âge<br>(Ma)</th><th rowspan="2">Y<sub>c</sub></th><th colspan="2">g<sub>3,5</sub></th><th colspan="2">g<sub>3,6</sub></th><th colspan="2">g<sub>1,4</sub></th><th colspan="2">g<sub>2,4</sub></th></tr><tr><th>P<br>(s)</th><th>Π<br>(s/Ma)</th><th>P<br>(s)</th><th>Π<br>(s/Ma)</th><th>P<br>(s)</th><th>Π<br>(s/Ma)</th><th>P<br>(s)</th><th>Π<br>(s/Ma)</th></tr></thead><tbody><tr><td>1.79</td><td>0.927</td><td>566.19</td><td>-5.82×10<sup>0</sup></td><td>678.38</td><td>-5.79×10<sup>0</sup></td><td>1010.81</td><td>-9.05×10<sup>0</sup></td><td>610.69</td><td>-5.54×10<sup>0</sup></td></tr><tr><td>12.06</td><td>0.809</td><td>522.76</td><td>-1.05×10<sup>0</sup></td><td>637.62</td><td>-3.30×10<sup>-1</sup></td><td>943.70</td><td>-1.51×10<sup>0</sup></td><td>568.50</td><td>-1.25×10<sup>0</sup></td></tr><tr><td>19.66</td><td>0.741</td><td>529.96</td><td>+1.42×10<sup>0</sup></td><td>652.10</td><td>+2.34×10<sup>0</sup></td><td>955.70</td><td>+2.18×10<sup>0</sup></td><td>573.13</td><td>+1.17×10<sup>0</sup></td></tr><tr><td>23.84</td><td>0.707</td><td>534.73</td><td>+1.15×10<sup>0</sup></td><td>660.06</td><td>+1.80×10<sup>0</sup></td><td>962.72</td><td>+1.71×10<sup>0</sup></td><td>577.34</td><td>+1.09×10<sup>0</sup></td></tr><tr><td>39.40</td><td>0.588</td><td>559.40</td><td>+1.33×10<sup>0</sup></td><td>692.91</td><td>+1.92×10<sup>0</sup></td><td>1002.93</td><td>+2.25×10<sup>0</sup></td><td>604.28</td><td>+1.80×10<sup>0</sup></td></tr><tr><td>55.93</td><td>0.456</td><td>578.59</td><td>+1.84×10<sup>0</sup></td><td>713.48</td><td>+6.22×10<sup>-1</sup></td><td>1026.10</td><td>+1.34×10<sup>0</sup></td><td>632.59</td><td>+2.05×10<sup>0</sup></td></tr><tr><td>68.06</td><td>0.348</td><td>608.85</td><td>+2.68×10<sup>0</sup></td><td>723.16</td><td>+1.56×10<sup>0</sup></td><td>1046.47</td><td>+1.82×10<sup>0</sup></td><td>661.59</td><td>+2.50×10<sup>0</sup></td></tr><tr><td>77.10</td><td>0.266</td><td>633.59</td><td>+3.08×10<sup>0</sup></td><td>717.84</td><td>-5.91×10<sup>0</sup></td><td>1058.20</td><td>+3.07×10<sup>-1</sup></td><td>682.52</td><td>+2.05×10<sup>0</sup></td></tr><tr><td>83.80</td><td>0.197</td><td>647.37</td><td>-4.37×10<sup>-1</sup></td><td>665.08</td><td>-5.52×10<sup>0</sup></td><td>1054.46</td><td>-1.21×10<sup>0</sup></td><td>693.54</td><td>+1.10×10<sup>0</sup></td></tr><tr><td>88.45</td><td>0.151</td><td>629.76</td><td>-7.73×10<sup>0</sup></td><td>654.53</td><td>-4.01×10<sup>-1</sup></td><td>1046.33</td><td>-2.53×10<sup>0</sup></td><td>695.65</td><td>-3.62×10<sup>-1</sup></td></tr><tr><td>91.08</td><td>0.124</td><td>608.44</td><td>-6.30×10<sup>0</sup></td><td>653.21</td><td>-9.80×10<sup>-1</sup></td><td>1037.28</td><td>-4.67×10<sup>0</sup></td><td>693.15</td><td>-1.57×10<sup>0</sup></td></tr><tr><td>93.82</td><td>0.099</td><td>595.54</td><td>-5.67×10<sup>0</sup></td><td>648.23</td><td>-2.60×10<sup>0</sup></td><td>1020.71</td><td>-7.18×10<sup>0</sup></td><td>686.29</td><td>-3.70×10<sup>0</sup></td></tr><tr><td>96.05</td><td>0.080</td><td>580.87</td><td>-5.52×10<sup>0</sup></td><td>640.87</td><td>-4.08×10<sup>0</sup></td><td>1003.55</td><td>-8.00×10<sup>0</sup></td><td>676.23</td><td>-4.94×10<sup>0</sup></td></tr><tr><td>98.37</td><td>0.061</td><td>570.27</td><td>-5.83×10<sup>0</sup></td><td>628.62</td><td>-6.77×10<sup>0</sup></td><td>983.88</td><td>-9.32×10<sup>0</sup></td><td>662.60</td><td>-7.69×10<sup>0</sup></td></tr><tr><td>100.23</td><td>0.046</td><td>555.93</td><td>-8.71×10<sup>0</sup></td><td>613.46</td><td>-9.50×10<sup>0</sup></td><td>962.83</td><td>-1.43×10<sup>1</sup></td><td>644.78</td><td>-1.12×10<sup>1</sup></td></tr><tr><td>101.76</td><td>0.034</td><td>542.78</td><td>-6.73×10<sup>0</sup></td><td>595.28</td><td>-1.51×10<sup>1</sup></td><td>936.57</td><td>-2.32×10<sup>1</sup></td><td>626.04</td><td>-1.41×10<sup>1</sup></td></tr><tr><td>102.99</td><td>0.025</td><td>534.50</td><td>-1.64×10<sup>1</sup></td><td>576.72</td><td>-1.91×10<sup>1</sup></td><td>907.98</td><td>-2.45×10<sup>1</sup></td><td>608.69</td><td>-2.00×10<sup>1</sup></td></tr><tr><td>103.97</td><td>0.018</td><td>518.43</td><td>-3.02×10<sup>1</sup></td><td>557.96</td><td>-3.77×10<sup>1</sup></td><td>883.94</td><td>-5.39×10<sup>1</sup></td><td>589.10</td><td>-3.89×10<sup>1</sup></td></tr><tr><td>105.76</td><td>0.005</td><td>464.33</td><td>-1.34×10<sup>2</sup></td><td>490.46</td><td>-1.25×10<sup>2</sup></td><td>787.48</td><td>-2.35×10<sup>2</sup></td><td>519.47</td><td>-1.75×10<sup>2</sup></td></tr><tr><td>106.48</td><td>0.000</td><td>367.63</td><td>-6.21×10<sup>2</sup></td><td>400.13</td><td>-5.73×10<sup>2</sup></td><td>618.09</td><td>-3.34×10<sup>2</sup></td><td>393.70</td><td>-4.67×10<sup>2</sup></td></tr><tr><td>106.58</td><td>0.000</td><td>305.53</td><td>-1.78×10<sup>2</sup></td><td>342.79</td><td>-3.17×10<sup>2</sup></td><td>584.70</td><td>-8.18×10<sup>2</sup></td><td>347.00</td><td>-2.32×10<sup>2</sup></td></tr><tr><td>106.77</td><td>0.000</td><td>271.71</td><td>-1.26×10<sup>2</sup></td><td>282.58</td><td>-1.37×10<sup>2</sup></td><td>429.36</td><td>-2.24×10<sup>2</sup></td><td>302.94</td><td>-1.20×10<sup>2</sup></td></tr></tbody></table>
Annexe D
Collection d'Articles

Dans cette annexe, nous mettons à la disposition du lecteur les articles publiés ou acceptés dans les révues scientifiques avec comité d'arbitrage et qui sont en rapport avec les développements présentés dans cette thèse. Ces publications sont présentées dans l'ordre suivant (voir la liste des références pour obtenir les détails sur chaque article):

- Page 397: Charpinet et al., 1996 (ApJL)

- Page 401: Charpinet et al., 1997a (ApJL)

- Page 405: Charpinet et al., 1997b (ApJL)

- Page 409: Billères et al., 1997 (ApJL)

- Page 413: Billères et al., 1998b (ApJL)

- Page 417: Koen et al., 1999 (MNRAS)

Plusieurs articles sont également publiés dans les revues sans comité d'arbitrage (principalement dans les compte-rendus de conférence). Ceux-ci sont mentionnés dans la liste suivante mais ne sont pas reproduits dans cette annexe. Ils sont accessibles à partir des informations données dans la liste des références.

- Workshop Européen sur les Naines Blanches (Blanes, Espagne 1996): Charpinet et al., 1997c

- Conférence sur les "Faint Blue Stars" (Schenectady, Etats-Unis 1996): Charpinet et al., 1998

- Workshop Européen sur les Naines Blanches (Tromsø, Norvège 1998): Charpinet et al., 1999

- Conférence sur les "Faint Blue Stars" (Schenectady, Etats-Unis 1996): Billères et al., 1998a

- IAU Symposium (Kyoto, Japon 1997): Fontaine et al., 1998
<figure><img src="image_121.png" /></figure>
THE ASTROPHYSICAL JOURNAL, 471: L103-L106, 1996 November 10

© 1996, The American Astronomical Society. All rights reserved. Printed in U.S.A.

# THE POTENTIAL OF ASTEROSEISM>OLOGY FOR HOT, SUBDWARF B STARS:
A NEW CLASS OF PULSATING STARS?

S. CHARPINET, G. FONTAINE, AND P. BRASSARD

Département de Physique, Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montréal, Québec, Canada, H3C 3J7;
charpinet@astro.umontreal.ca, fontaine@astro.umontreal.ca, brassard@astro.umontreal.ca

AND

B. DORMAN<fnref n="1" />

Laboratory for Astronomy and Solar Physics, NASA/GSFC, Greenbelt, MD 20771; ben.dorman@gsfc.nasa.gov

Received 1996 May 20; accepted 1996 August 23

## ABSTRACT

We present key sample results of a systematic survey of the pulsation properties of models of hot B subdwarfs. We use equilibrium structures taken from detailed evolutionary sequences of solar metallicity ($Z = 0.02$) supplemented by grids of static envelope models of various metallicities ($Z = 0.02, 0.04, 0.06, 0.08$, and $0.10$). We consider all pulsation modes with $l = 0, 1, 2$, and $3$ in the 80–1500 s period window, the interval currently most suitable for fast photometric detection techniques. We establish that significant driving is often present in hot B subdwarfs and is due to an opacity bump associated with heavy-element ionization. We find that models with $Z \ge 0.04$ show low radial order unstable modes; both radial and nonradial ($p, f$, and $g$) pulsations are excited. The unstable models have $T_{eff} \lesssim 30,000$ K and $log g \lesssim 5.7$, depending somewhat on the metallicity. We emphasize that metal enrichment need only occur locally in the driving region. On this basis, combined with the accepted view that local enrichments and depletions of metals are commonplace in the envelopes of hot B subdwarfs, we predict that some of these stars should show luminosity variations resulting from pulsational instabilities.

*Subject headings:* stars: interiors — stars: oscillations — subdwarfs

## 1. INTRODUCTION

In the last 20 years, considerable progress has been made in our understanding of the physical properties and evolutionary status of hot, hydrogen-rich, subdwarf B stars (see, e.g., Greenstein & Sargent 1974; Heber 1987; Saffer et al. 1994). Following the scenario originally proposed by Heber and collaborators (e.g., Heber et al. 1984), it is currently believed that subdwarf B stars are ~0.5 $M_\odot$ objects with hydrogen envelope masses that are small enough that, after core helium exhaustion on the extended horizontal branch (EHB), they never evolve toward the asymptotic giant branch (AGB). They remain at high effective temperatures ($T_{eff} \gtrsim 20,000$ K) throughout their core-burning evolution. In the post-HB, He-shell-burning stage of evolution the models are referred to as "AGB-manqué" objects (see Greggio & Renzini 1990) and are associated with the field subdwarf O stars (Dorman, O'Connell, & Rood 1995 and references therein). A number of post-EHB evolutionary studies have appeared in the literature in the last several years (see Dorman 1995 for a review).

While asteroseismology is proving to be an extremely powerful tool in studying other types of stars, its potential has not yet been studied for subdwarf B stars, partly because of the lack (until recently) of sufficiently realistic equilibrium structures. Furthermore, and to our knowledge, luminosity variations caused by pulsational instabilities have not been reported for these stars. However, pulsation theorists with a keen eye may have noticed for some time that the potential for driving pulsation modes appears to exist in the envelopes of subdwarf B stars. Indeed, early models of such envelopes (Wesemael et

al. 1982; Groth, Kodritzki, & Heber 1985) as well as their more sophisticated modern counterparts are all characterized by the presence of a He II–He III convection zone. By analogy with other types of pulsating stars (whose instabilities are always driven by one form or another of an opacity mechanism), one might expect that the opacity bump associated with this partial ionization zone could also excite pulsation modes in hot B subdwarfs.

Motivated in part by this observation, we undertook a systematic exploration of the potential of asteroseismology for subdwarf B stars. The purposes of the present Letter are (1) to report on the salient results of our investigation, (2) to call attention to the very real possibility that some subdwarf B stars may undergo stellar oscillations, and (3) to encourage searches for luminosity variations in such stars.

## 2. COMPUTATIONS AND RESULTS

The first batch of equilibrium models investigated in this study consists of full stellar models taken from five distinct evolutionary sequences. These were chosen in order to map a significant fraction of the region of the $T_{eff}-log g$ plane actually occupied by the known subdwarf B stars (see Saffer et al. 1994 and references therein). In this particular region, we retained about 25 models per sequence. The evolutionary models were computed with the methods described in Dorman (1992a, 1992b) and Dorman, Rood, & O'Connell (1993). The new models use the OPAL opacities described by Rogers & Iglesias (1992) computed in 1993 December, which adopted the element mix referred to as "Grevesse & Noels 1993." Where necessary (during He flashes), we used new low-temperature opacities by D. R. Alexander (1995, private

<footnote n="1">¹ NAS/NRC Resident Research Associate.</footnote>
<table><caption>TABLE 1
PULSATION PERIODS FOR A TYPICAL SUBDWARF B STAR MODEL</caption><thead><tr><th>k</th><th>l = 0</th><th>l = 1(p)</th><th>l = 1(g)</th><th>l = 2(f, p)</th><th>l = 2(g)</th><th>l = 3(f, p)</th><th>l = 3(g)</th></tr></thead><tbody><tr><td>0</td><td>384.13</td><td>...</td><td>...</td><td>277.45</td><td>...</td><td>270.11</td><td>...</td></tr><tr><td>1</td><td>281.70</td><td>379.97</td><td>399.07</td><td>257.90</td><td>370.99</td><td>222.67</td><td>306.83</td></tr><tr><td>2</td><td>226.74</td><td>279.72</td><td>669.19</td><td>223.56</td><td>407.72</td><td>205.82</td><td>355.99</td></tr><tr><td>3</td><td>191.25</td><td>225.28</td><td>965.35</td><td>192.39</td><td>576.19</td><td>187.56</td><td>428.29</td></tr><tr><td>4</td><td>173.54</td><td>188.71</td><td>1136.8</td><td>179.41</td><td>675.85</td><td>169.34</td><td>506.07</td></tr><tr><td>5</td><td>157.05</td><td>163.60</td><td>1311.82</td><td>157.95</td><td>774.48</td><td>153.11</td><td>563.83</td></tr><tr><td>6</td><td>138.66</td><td>147.90</td><td>...</td><td>139.38</td><td>897.80</td><td>135.79</td><td>655.69</td></tr><tr><td>7</td><td>114.61</td><td>134.07</td><td>...</td><td>126.07</td><td>1004.21</td><td>121.56</td><td>723.91</td></tr><tr><td>8</td><td>105.62</td><td>120.33</td><td>...</td><td>115.90</td><td>1182.10</td><td>110.75</td><td>846.59</td></tr><tr><td>9</td><td>96.96</td><td>108.84</td><td>...</td><td>106.04</td><td>1361.08</td><td>102.54</td><td>975.36</td></tr><tr><td>10</td><td>89.50</td><td>99.99</td><td>...</td><td>97.15</td><td>1429.47</td><td>95.03</td><td>1030.05</td></tr><tr><td>11</td><td>83.38</td><td>93.28</td><td>...</td><td>89.77</td><td>...</td><td>88.03</td><td>1130.15</td></tr><tr><td>12</td><td>...</td><td>87.18</td><td>...</td><td>83.87</td><td>...</td><td>81.88</td><td>1259.89</td></tr><tr><td>13</td><td>...</td><td>81.28</td><td>...</td><td>...</td><td>...</td><td>...</td><td>1337.26</td></tr><tr><td>14</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>1425.84</td></tr></tbody></table>

communication, described in Alexander & Ferguson 1994), which were computed with the same element mix. These smoothly match the OPAL opacity set within the hydrogen ionization zone. The other difference in the physics was the use of the Itoh et al. conductive opacities (Itoh et al. 1983, 1993; Itoh, Hayashi, & Kohyama 1993, 1994; Itoh & Kohyama 1994).

Each sequence describes the evolution of an AGB-manqué star with a core mass of 0.4758 $M_\odot$. The sequences differ in that different envelope masses are considered: $M_{\text{env}} = 0.0002$, 0.0012, 0.0022, 0.0032, and 0.0042 $M_\odot$. The composition of the envelopes is assumed to be solar ($X = 0.70388$, $Z = 0.01718$). The luminosity of each model is provided by He burning confined to a very small core. Residual H burning at the base of the H-rich envelope contributes negligibly to the luminosity until the AGB-manqué phase of the evolution.

We first carried out a detailed adiabatic survey of the evolutionary models with the help of the Galerkin finite-element code of Brassard et al. (1992). Specifically, for each equilibrium model considered, we computed pulsation periods in the adiabatic approximation for all modes with $l = 0, 1, 2$, and 3 in the 80–1500 s period window. This interval corresponds to the range of periods most easily detectable with present-day fast photometric techniques. Moreover, it is well known that surface cancellation effects render detection of luminosity variations excessively difficult for $l \ge 4$ in compact stars (Dziembowski 1977; Robinson, Kepler, & Nather 1982; Brassard 1987; Brassard, Fontaine, & Wesemael 1995), so such modes are not of direct observational interest. We will report elsewhere the detailed results of this extensive survey, including discussions of the effects of changing model parameters, the period evolution, the rates of period changes, and the effects of mode trapping and mode confinement caused by the composition interfaces in our doubly stratified equilibrium models. For the needs of the present Letter, in order to give an idea of the modes to be expected in the 80–1500 s period window, we simply report on some sample results.

Table 1 lists the pulsation periods for a typical model of a hot B subdwarf. This model belongs to the sequence with $M_{\text{core}} = 0.4758 M_\odot$ and $M_{\text{env}} = 0.0012 M_\odot$. It has an age of $\sim 8.4 \times 10^7$ yr (time elapsed since the zero-age HB phase), a surface gravity log $g = 5.46$, and an effective temperature $T_{\text{eff}} = 27,500$ K. The table gives the pulsation period (expressed in seconds) as a function of the radial order $k$. We distinguish between the radial modes ($l = 0$) and the $p$ and $g$ branches for

nonradial modes with $l = 1, 2$, and 3. Our results indicate that subdwarf B stars have rich period spectra, easily accessible with present-day observational techniques (white-light fast photometry, for example). Provided that we can demonstrate that some of these modes can be excited, this should motivate observational searches for pulsational instabilities in these stars.

Therefore, in the second step of our investigation, we carried out a stability analysis of our equilibrium models with the help of the finite-element nonadiabatic pulsation code briefly described in Fontaine et al. (1994) and Brassard, Fontaine, & Bergeron (1996). In order to understand the results and be able to identify the driving regions (if any), we considered the variations with depth of the so-called work integral. Figure 1 illustrates a typical case and refers to the equilibrium model whose periods are provided in Table 1. The solid curve corresponds to the integrand of the work integral for the $l = 2, f$-mode with a period of 277.45 s. (We emphasize that, while specific to this particular mode, the results shown here are also typical of all other modes of interest.) The values of $dW/dr$ are positive in driving regions and negative in damping regions. We find that driving is negligible in the He II–He III convection zone, contrary to our initial expectation. In retrospect, it appears that the convection zone is located too high in the envelope, in a region that contains very little mass; therefore, it carries practically no weight in terms of driving (or damping). Likewise, the H-burning shell is so weak in our models that no significant contribution to driving from the $\epsilon$ mechanism is observed. The important result here is that the main driving region in hot B subdwarfs is associated with an opacity bump due to heavy-element ionization. This is shown by the dashed curve which illustrates the run of the Rosseland opacity in the unperturbed model. As expected, the opacity shows a maximum in the convection zone due to He II–He III partial ionization, but it also shows a secondary maximum around $\log q \approx -9.2$, obviously associated with the driving region and which tends to disappear when the heavy-element content (assumed to be $Z \sim 0.02$ in this model) is decreased. The realization that driving is related to metallicity provides an important clue to the possible existence of real subdwarf B pulsators. This is similar in nature to the so-called Z-bump mechanism uncovered by others in different types of pulsating stars (e.g., Cox et al. 1992; Moskalik & Dziembowski 1992; Dziembowski & Pamiatnykh 1993; Gautschy & Saio 1993).
<figure><img src="image_122.png" /><figcaption>FIG. 1.—Run of the integrand of the work integral for the *f*-mode with *l* = 2 (*solid curve*) and of the Rosseland opacity (*dashed curve*) vs. fractional mass depth in a typical subdwarf B model. The locations of the He-burning core, the residual H-burning shell, and the He II–He III convection zone are indicated. The driving region (positive values of *dW/dr*) is clearly associated with an opacity bump, itself caused by heavy-element ionization.</figcaption></figure>

The specific mode considered in Figure 1 turns out to be
stable, the damping region below the driving region contrib-
uting somewhat more to the overall work integral. This result
is quite typical of all of the evolutionary models computed with
Z ~ 0.02. Indeed, in our detailed nonadiabatic survey of those
equilibrium models, we found no unstable mode. However, in
many cases, we found that the overall damping rates were very
small, suggesting to us that such modes could perhaps still be
excited in real stars (whose detailed structure might, of course,
be different from that of the particular models we considered).
In view of the correlation we uncovered between driving and
metallicity, and the fact that local variations of metallicity are
expected in subdwarf B stars (see below), it seemed natural to
investigate the effects of changing the metallicity.

In the final step of our analysis, we constructed a second
batch of equilibrium models taking into account variations of
Z in the H-rich envelope. Contrary to our previous models,
these are static (i.e., nonevolving) structures. The models were
built with a version of the stellar code described by Brassard &
Fontaine (1994) and Brassard et al. (1996), suitably modified
to produce envelope structures extending as deep as log
q = -0.05. The constitutive physics used is nearly identical to
that used previously in the construction of our full evolution-
ary models, apart from the use of slightly newer OPAL
opacities computed in 1995. Tests indicate that our ignoring of
the small He-burning core increases somewhat the values of
the periods of the *q*-modes (by 10%–20%, typically) but does
not affect those of the *p*-modes which, contrary to the former,
are formed essentially in the envelope region. We will provide
more details on these models elsewhere.

We constructed five different model grids, one each for a
fixed metal content Z = 0.02, 0.04, 0.06, 0.08, and 0.10 in the

H-rich envelope (X = 0.70). Each grid consists of 72 unper-
turbed models covering a rectangular region in the T<sub>en</sub>–log g
plane overlapping with the region where hot B subdwarfs are
actually found (4.34 ≤ log T<sub>eff</sub> ≤ 4.62 in steps of 0.04, and
4.8 ≤ log g ≤ 6.4 in steps of 0.2). In these exploratory calcu-
lations, we fixed the total mass of each model to 0.48 M<sub>☉</sub> and
the H-rich envelope mass to log q(H) = –4. Tests indicate
that stability does not depend on the value of log q(H).

In agreement with our expectations concerning the effects of
metallicity, we found unstable pulsation modes for models with
Z ≥ 0.04. For these models, there is a blue-edge temperature
located somewhere between log T<sub>eff</sub> = 4.46 and 4.50, which
shows a weak sensitivity to metallicity. We tentatively interpret
this as the consequence of the fact that metallicity primarily
affects the size of the opacity bump (and, hence, the magni-
tude of the driving), rather than its location in a star of a given
effective temperature. The blue edge itself must be due to the
outward displacement of the driving region to the outermost
layers (which carry little weight in the overall stability of a
mode) as the effective temperature increases. Note that our
actual value of the blue-edge temperature (T<sub>eff</sub> ~ 30,000 K)
may also depend on the assumption of uniform metallicity as
well as on the assumed mass of the models (currently 0.48
M<sub>☉</sub>), but this remains to be investigated. We find also that
stability does depend on the assumed surface gravity. Unstable
models have log g ≲ 5.7, with a weak dependence on metal-
licity, but this also remains to be investigated in more detail.
As expected, the driving is more efficient in the high-metallic-
ity models; a wider band of modes are excited in such models.
In all unstable models, the excited modes have low values of
the radial order k, and radial as well as nonradial pulsations
are predicted. The most strongly driven modes have typical
e-folding times of ~8 × 10<sup>-2</sup> yr, much shorter than the typical
evolutionary time of a hot B subdwarf (~10<sup>8</sup> yr). These linear
instabilities would normally develop into observable ampli-
tudes.

3. DISCUSSION AND CONCLUSION

The results presented in this paper pave the way to the
exciting possibility of being able to use asteroseismology to
probe the internal constitution of subdwarf B stars. A sub-
group of these stars may indeed constitute a new class of
pulsating stars. Of course, this possibility hinges here on the
question of metal enrichment in the driving region. We
emphasize the fact that this enrichment need only occur in the
driving region *itself* and not necessarily everywhere in the
envelope (as we assumed in the relatively crude envelope
models used in the present study). That heavy elements can
show local enrichments in certain regions of a hot B subdwarf
envelope, and local depletions in others, is not only plausible
but is also *expected*. Indeed, subdwarf B stars all show peculiar
surface abundances whose study constitutes a most active and
interesting subfield of stellar physics in its own right. It is
currently believed that the abundance anomalies observed in
the atmospheres of these stars result from the competition
between gravitational settling, radiative levitation, and weak
stellar winds (Michaud et al. 1985, 1989; Heber 1987;
Bergeron et al. 1988; Chayer et al. 1996). In the most recent
investigation, Chayer et al. (1996) have carried out new
computations of radiative levitation on metals in models of hot
B subdwarfs using the same tools considered previously in the
white dwarf context by Chayer et al. (1995) and Chayer,
L106

CHARPINET ET AL.

Fontaine, & Wesemael (1995). Of high relevance here, they found that metals, in particular iron (a major contributor to opacity), do levitate in the envelopes of these stars. Local overabundances of more than an order of magnitude can be built through this process. In the case of iron, the maximum overabundance occurs near $log q \approx -9, -10$. This result greatly enhances the possibility that driving occurs as a result of a local enhancement of the metal content.² Of course, a detailed study of the driving process in hot B subdwarfs will ultimately require the use of more detailed envelope structures taking into account diffusion processes. *In the meantime, we feel confident enough to risk the prediction that some

² A similar phenomenon may be responsible for mode excitation in rapidly oscillating Ap stars.

subdwarf B stars should show luminosity variations resulting from pulsational instabilities. We believe that the material presented in this paper warrants further theoretical investigations of the pulsation properties of these stars and, above all, searches for luminosity variations. We have undertaken both.³

This work was supported in part by the NSREC of Canada and by the fund FCAR (Québec). B. D. acknowledges partial support from NASA RTOP 188-41-51-03.

³ After this paper was completed and submitted, we learned of the exciting discovery of pulsating sdB stars at the South African Astronomical Observatory. This was reported by O'Donoghue at the 10th European Workshop on White Dwarfs held in Blanes (Spain) on 1996 June 17–21. Thus, there is indeed a new class of pulsating stars out there!

REFERENCES

Alexander, D. A., & Ferguson, J. W. 1994, ApJ, 437, 879

Bergeron, P., Wesemael, F., Michaud, G., & Fontaine, G. 1988, ApJ, 332, 964

Brassard, P. 1987, M.Sc. thesis, Univ. Montréal

Brassard, P., & Fontaine, G. 1994, in IAU Colloq. 147, The Equation of State in Astrophysics, ed. G. Chabrier & E. Schatzman (Cambridge: Cambridge Univ. Press), 560

Brassard, P., Fontaine, G., & Bergeron, P. 1996, in preparation

Brassard, P., Fontaine, G., & Wesemael, F. 1995, ApJS, 96, 545

Brassard, P., Pelletier, C., Fontaine, G., & Wesemael, F. 1992, ApJS, 80, 725

Chayer, P., et al. 1996, in preparation

Chayer, P., Fontaine, G., & Wesemael, F. 1995, ApJS, 99, 189

Chayer, P., Vennes, S., Pradhan, A. K., Thejll, P., Beauchamp, A., Fontaine, G., & Wesemael, F. 1995, ApJ, 454, 429

Cox, A. N., Morgan, S. M., Rogers, F. J., & Iglesias, C. 1992, ApJ, 393, 272

Dorman, B. 1992a, ApJS, 80, 701

———. 1992b, ApJS, 81, 221

———. 1995, in Proc. 32d Liège Astrophys. Colloq., Stellar Evolution: What Should Be Done, ed. A. Noels, D. Fraipont-Caro, N. Grevesse, & P. Demarque (Liège: Institut d'Astrophysique), 291

Dorman, B., O'Connell, R. W., & Rood, R. T. 1995, ApJ, 442, 105

Dorman, B., Rood, R. T., & O'Connell, R. W. 1993, ApJ, 415, 596

Dziembowski, W. 1977, Acta Astron., 27, 203

Dziembowski, W., & Pamiatnykh, A. A. 1992, MNRAS, 262, 204

Fontaine, G., Brassard, P., Wesemael, F., & Tassoul, M. 1994, ApJ, 428, L61

Gautschi, A., & Saio, H. 1993, MNRAS, 262, 213

Greenstein, J. L., & Sargent, A. I. 1974, ApJS, 28, 157

Greggio, L., & Renzini, A. 1990, ApJ, 364, 35

Groth, H. G., Kudritzki, R. P., & Heber, U. 1985, A&A, 152, 107

Heber, U. 1987, in IAU Colloq. 95, The Second Conference on Faint Blue Stars, ed. A. G. D. Philip, J. Liebert, & D. S. Hayes (Schenectady: Davis), 79

Heber, U., Hunger, K., Jonas, G., & Kudritzki, R. P. 1984, A&A, 130, 119

Itoh, N., Hayashi, H., & Kohyama, Y. 1993, ApJ, 418, 405

———, 1994, ApJ, 436, 418

Itoh, N., & Kohyama, Y. 1994, ApJ, 420, 943

Itoh, N., Kohyama, Y., Matsumoto, N., & Seki, M. 1993, ApJ, 404, 418

Itoh, N., Mitake, S., Iyetomi, H., & Ichimaru, S. 1983, 273, 774

Michaud, G., Bergeron, P., Heber, U., & Wesemael, F. 1989, ApJ, 338, 417

Michaud, G., Bergeron, P., Wesemael, F., & Fontaine, G. 1985, ApJ, 299, 741

Moskalik, P., & Dziembowski, W. 1992, A&A, 256, L5

Robinson, E. L., Kepler, S. O., & Nather, R. N. 1982, ApJ, 259, 219

Rogers, F. J., & Iglesias, C. A. 1992, ApJ, 401, 361

Saffer, R. A., Bergeron, P., Koester, D., & Liebert, J. 1994, ApJ, 432, 351

Wesemael, F., Winget, D. E., Cabot, W., Van Horn, H. M., & Fontaine, G. 1982, ApJ, 254, 221
A DRIVING MECHANISM FOR THE NEWLY DISCOVERED CLASS OF PULSATING SUBDWARF B STARS

S. CHARPINET, G. FONTAINE, AND P. BRASSARD

Département de Physique, Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montréal, Québec, Canada, H3C 3J7;
charpinet@astro.umontreal.ca, fontaine@astro.umontreal.ca, brassard@astro.umontreal.ca

P. CHAYER

Center for EUV Astrophysics, 2150 Kittredge Street, University of California at Berkeley, Berkeley, CA 94720-5030; chayer@cea.berkeley.edu

F. J. ROGERS AND C. A. IGLESIAS

Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, CA 94550; opal@coral.llnl.gov

AND

B. DORMAN<fnref n="1" />

Laboratory for Astronomy and Solar Physics, NASA/GSFC, Greenbelt, MD 20771; Ben.Dorman@gsfc.nasa.gov

Received 1997 January 24; accepted 1997 April 22

ABSTRACT

We present new calculations that strongly reinforce the idea—originally proposed by Charpinet et al.—that pulsation modes are driven through an opacity bump due to a local enhancement of the iron abundance in the envelopes of sdB stars. Our improved models incorporate nonuniform iron abundance distributions obtained through the condition of diffusive equilibrium between gravitational settling and radiative levitation. They also include special Rosseland opacity tables that take into account the large variations of the iron abundance about the cosmic value that are predicted by equilibrium radiative levitation theory. For representative models with $M = 0.48 M_\odot$ and log $g = 5.8$, we find strong instabilities for low-order radial and nonradial ($p$ and $f$) pulsation modes in the range 36,500 K $\gtrsim T_{\text{eff}} \gtrsim 29,000$ K. The four pulsating sdB stars currently known all have effective temperatures in that range. In addition, one of our models with $T_{\text{eff}} = 34,000$ K has a band of unstable modes with periods in the range 116–195 s, in excellent agreement with those of the known pulsators. We therefore claim that our proposed iron bump mechanism provides a natural explanation for the instabilities found in the newly discovered class of pulsating sdB stars.

Subject heading: stars: interiors — stars: oscillations — subdwarfs

In a recent effort (Charpinet et al. 1996, hereafter Paper I), some of us presented the first results of a systematic survey of the asteroseismological potential of a neglected class of stars, the hot B subdwarfs (sdB). These evolved, compact stars are found in the range 40,000 K $\gtrsim T_{\text{eff}} \gtrsim 24,000$ K, and have typical values $M \sim 0.5 M_\odot$ and log $g \sim 5.8$ (see Saffer et al. 1994 and references therein). They are identified with low-mass core helium burning models with outer H-rich envelopes that are too thin to sustain H-shell burning on the extended horizontal branch. Consequently, after core helium exhaustion, they do not evolve toward the asymptotic giant branch. They turn left in the HR diagram, contribute for a while to the subdwarf O population, and, ultimately, become white dwarfs (see, e.g., Dorman, Rood, & O'Connell 1993).

The investigations of Paper I led to the discovery of an efficient driving mechanism due to an opacity bump associated with heavy-element (essentially iron) ionization in some models of sdB stars. This is similar in nature to the so-called Z-bump mechanism uncovered by others in different types of pulsating stars (e.g., Cox et al. 1992; Moskalik & Dziembowski 1992; Dziembowski & Pamiatnykh 1992; Gautschy & Saio 1993). It was found that both radial and nonradial ($p, f$, and $g$) low-order modes could be excited in these models. On this basis, Paper I made the prediction that a subclass of sdB stars should show luminosity variations resulting from pulsational instabilities.

After Paper I was completed and submitted, we learned of the independent observational effort carried out at the South African Astronomical Observatory, which led to the exciting discoveries of the first real pulsating sdB stars (O'Donoghue et al. 1997a; see also Kilkenny et al. 1997b). The properties of the first four members of this new class of pulsating stars (dubbed EC 14026 objects, after the prototype) have been described in a recent series of papers (Kilkenny et al. 1997a; Koen et al. 1997; Stobie et al. 1997; O'Donoghue et al. 1997b). Of particular interest, these stars appear to crowd in a specific region of the $T_{\text{eff}}$-log $g$ plane ($T_{\text{eff}} \sim 35,000$ K, log $g \sim 5.9$) (O'Donoghue et al. 1997b). They are also multiperiodic with periods in the range 121–159 s. While we like to think that the driving mechanism uncovered in Paper I is responsible for the luminosity variations in the real pulsators, it remains to be seen that this is the case. We recall that, in the exploratory spirit of Paper I, the effects of metallicity on driving were investigated through relatively crude models with assumed uniform metal abundances. Unstable modes were found for enhanced metallicity ($Z \gtrsim 0.04$), low gravity (log $g \lesssim 5.7$), and relatively low temperature ($T_{\text{eff}} \lesssim 30,000$ K) models. However, it was pointed out there that enhanced metallicity need only occur locally in the driving region and, furthermore, that the derived theoretical boundaries of instability in the $T_{\text{eff}}$-log $g$ plane may depend on the assumption of uniform enhanced metallicity used in these relatively crude first-generation envelope models. In view of this, and considering the challenge posed above by the observational situation, we have computed more sophisticated and realistic envelope models with nonuni-

<footnote n="1">¹ Also, Department of Astronomy, University of Virginia, P.O. Box 3818, Charlottesville, VA 22903-0818.</footnote>
form metal abundances. The main purpose of this Letter is to report on the salient features of a nonadiabatic study of these second-generation models.

Our new envelope models are based on the assumption that an equilibrium state has been reached between gravitational settling and radiative levitation operating on a given metal in a sdB star. In the absence of other competing mechanisms (e.g., stellar winds, accretion, turbulence), it can be shown (see Fontaine & Chayer 1997) that this equilibrium state is reached rapidly, over timescales much smaller than a typical evolutionary timescale (~$10^8$ yr in the present case; Dorman et al. 1993). This approach leads to predicted surface abundances that are anomalous, as is invariably found in studies of sdB spectra (see, e.g., the review in Heber 1991). Of direct interest in the present context, it also leads to a nonuniform distribution of a given heavy element as a function of depth, with local enhancements and depletions, one which depends intimately on the atomic structure of that element. From an asteroseis- mological point of view, the variable abundance of an opaque metal (such as iron, for example) as a function of depth can potentially play a decisive role in the driving process through its influence on the Rosseland opacity distribution.

In the present effort, we idealized the situation by consid-
ering an outer envelope consisting of only iron in a back-
ground of hydrogen. This is justified on the grounds that iron
alone is responsible for the destabilization mechanism found
in Paper I. Also, sdB stars typically show large helium defi-
ciencies in their atmospheres. Thus, in a first step, we com-
puted the distribution of iron as a function of depth in a
hydrogen envelope through equilibrium radiative levitation
theory. We used the same tools developed by Chayer, Fon-
taine, & Wesemael (1995a) in a white dwarf context, and
incorporated some of the improvements discussed by Chayer
et al. (1995b).

The solid curves in Figure 1 give the distribution of iron in
the outer envelope of representative models of sdB stars with
*M* = 0.48 *M*☉, log *g* = 5.8, and various effective temperatures.
The integrated distribution over depth corresponds to the total
reservoir of iron that is maintained against settling by radiative
levitation in a given model. We note that the radiative support
increases monotonically with increasing effective temperature.
We note also that overabundances of iron are created over a
substantial fraction of the outer envelope. These overabun-
dances can become quite significant, particularly for the hotter
models. We point out that there is no problem in building up
such large overabundances; the total amount of iron in the
levitating reservoir is still a minute fraction of the total iron
content in the star.

In a second step, we computed special Rosseland opacity
tables to be able to include the very large variations of the iron
abundance about the normal value (both underabundances
and overabundances) that we obtained from our radiative
levitation calculations. For that task, we ran the OPAL codes
(Rogers & Iglesias 1992; Iglesias & Rogers 1996) for a series
of assumed nonstandard iron abundances that bracketed the
cosmic value. Not surprisingly, the effects of varying the iron
abundance on the Rosseland opacity can be quite drastic.
More details on these calculations will be reported elsewhere.

The dotted curves in Figure 1 show the Rosseland opacity distribution (interpolated on our special grid of OPAL data) computed on the basis of the variable abundance of iron. The opacity profile shows two bumps, both of which migrate toward the surface as the effective temperature increases. The

<figure><img src="image_123.png" /></figure>

FIG. 1.—Equilibrium abundance of iron (solid curve) as a function of the fractional mass depth log *q* [= log (1 − *M*(r)/*M*☉)] for a series of representative models of sdB stars with *M* = 0.48 *M*☉, log *g* = 5.8, and log *T*<sub>eff</sub> from 4.34 to 4.62 in steps of 0.04. In each panel the tip of the solid curve on the right hand side corresponds to the location of the Rosseland photosphere. The dashed horizontal line gives the normal value of the *Fe/H* number ratio. Also shown is the profile of the Rosseland opacity (dotted curve); its logarithmic value can be read on the right axis.

bump nearer to the stellar surface is essentially due to partial
ionization of hydrogen, while the second one, located deeper,
is due to the ionization of an electronic shell of iron. It is the
shape and the location of that second bump that play an
essential role in our driving mechanism.

The new opacity tables as well as the equilibrium distribu-
tions of iron were incorporated in a suitably modified version
of the stellar code described by Brassard & Fontaine (1994)
and Brassard, Fontaine, & Bergeron (1997). This code pro-
duces equilibrium static structures extending as deep as log
$q = -0.05$, which are well suited for pulsational analysis (see
Paper I). As noted above, we considered representative sdB
models with $M = 0.48 M_{\odot}$, $\log g = 5.8$, and covering a range of
effective temperature that overlaps completely with the range
in which the real sdB stars are found. As in Paper I, we
assumed, for each equilibrium structure, a He-rich core
($-0.05 \ge \log q > -4.00$) surrounded by a H-rich envelope
($\log q \le -4.00$). Figure 1 has presented that part (the outer
envelope) of some of these static structures that is most
relevant to the driving process.

Finally, as in Paper I, we carried out a stability analysis for
all modes with *l* = 0, 1, 2, and 3 in the 80–1500 s period
window for each equilibrium structure that we considered.
This was done with the help of the finite-element nonadiabatic
pulsation code briefly described in Fontaine et al. (1994) and
in Brassard et al. (1997). We found strongly excited low-order
radial and nonradial (*p* and *f*) pulsation modes in models in
the range 36,500 K $\gtrsim$ *T*<sub>eff</sub> $\gtrsim$ 29,000 K. The driving process is
the same as that uncovered in Paper I; it is a κ-mechanism
resulting from partial ionization of iron (the iron opacity bump
seen in Fig. 1). Contrary to our first generation models with
assumed uniform enhanced metallicity, however, the shape
<table><caption>TABLE 1
PULSATION PERIODS AND E-FOLDING TIMES FOR REPRESENTATIVE sdB MODEL</caption><thead><tr><th rowspan="2">k</th><th colspan="2">l = 0</th><th colspan="2">l = 1</th><th colspan="2">l = 2</th><th colspan="2">l = 3</th></tr><tr><th>P (s)</th><th>τ<sub>e</sub> (yr)</th><th>P (s)</th><th>τ<sub>e</sub> (yr)</th><th>P (s)</th><th>τ<sub>e</sub> (yr)</th><th>P (s)</th><th>τ<sub>e</sub> (yr)</th></tr></thead><tbody><tr><td>6.........</td><td>...</td><td>...</td><td>85.41</td><td>stable</td><td>83.27</td><td>stable</td><td>80.89</td><td>stable</td></tr><tr><td>5.........</td><td>86.91</td><td>stable</td><td>94.74</td><td>stable</td><td>93.27</td><td>stable</td><td>91.73</td><td>stable</td></tr><tr><td>4.........</td><td>96.05</td><td>stable</td><td>109.82</td><td>stable</td><td>108.19</td><td>stable</td><td>105.27</td><td>stable</td></tr><tr><td>3.........</td><td>110.74</td><td>stable</td><td>128.86</td><td>4.09 × 10<sup>-2</sup></td><td>122.51</td><td>4.20 × 10<sup>-2</sup></td><td>116.31</td><td>3.95 × 10<sup>-2</sup></td></tr><tr><td>2.........</td><td>132.72</td><td>3.49 × 10<sup>-2</sup></td><td>142.85</td><td>4.05 × 10<sup>-2</sup></td><td>140.85</td><td>3.17 × 10<sup>-2</sup></td><td>138.90</td><td>2.80 × 10<sup>-2</sup></td></tr><tr><td>1.........</td><td>144.76</td><td>5.73 × 10<sup>-2</sup></td><td>176.22</td><td>1.02 × 10<sup>0</sup></td><td>174.43</td><td>8.75 × 10<sup>-1</sup></td><td>165.27</td><td>7.82 × 10<sup>-1</sup></td></tr><tr><td>0.........</td><td>176.77</td><td>1.08 × 10<sup>0</sup></td><td>...</td><td>...</td><td>195.44</td><td>4.43 × 10<sup>2</sup></td><td>178.26</td><td>2.17 × 10<sup>0</sup></td></tr><tr><td>1.........</td><td>...</td><td>...</td><td>764.96</td><td>stable</td><td>466.40</td><td>stable</td><td>352.88</td><td>stable</td></tr><tr><td>2.........</td><td>...</td><td>...</td><td>1221.35</td><td>stable</td><td>722.75</td><td>stable</td><td>526.00</td><td>stable</td></tr><tr><td>3.........</td><td>...</td><td>...</td><td>1383.72</td><td>stable</td><td>804.62</td><td>stable</td><td>575.28</td><td>stable</td></tr><tr><td>4.........</td><td>...</td><td>...</td><td>...</td><td>...</td><td>1016.00</td><td>stable</td><td>732.24</td><td>stable</td></tr><tr><td>5.........</td><td>...</td><td>...</td><td>...</td><td>...</td><td>1295.40</td><td>stable</td><td>927.76</td><td>stable</td></tr><tr><td>6.........</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>1131.12</td><td>stable</td></tr><tr><td>7.........</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>1209.81</td><td>stable</td></tr><tr><td>8.........</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>1352.66</td><td>stable</td></tr></tbody></table>

and location of that bump are determined here in a more natural and realistic way, through the condition of diffusive equilibrium between gravitational settling and radiative levitation. Hence, through our more realistic approach, we now find that the instabilities are confined to a finite interval of effective temperature, an instability strip. Furthermore, the four known pulsators have effective temperatures (O'Donoghue et al. 1997b) that all fall within that range.

There is a natural explanation for the instability strip. Indeed, there is a lower limit (red edge) to the temperature that allows driving at which there is not enough iron supported by radiative levitation (see Fig. 1). Below $T_{eff} \sim 29,000$ K, the iron opacity bump is relatively broad and does not contribute enough to driving. With increasing $T_{eff}$, the iron opacity bump gets sharper and migrates toward the surface. The first effect favors driving by increasing the opacity derivatives, and the second effect allows the full “sampling” of the driving region. Ultimately, however, at high enough effective temperatures (blue edge), the opacity bump has migrated too high in the envelope in a region that contains very little mass and, therefore, carries practically no weight in terms of driving (or damping).

Table 1 gives typical results for one of our unstable models ($T_{eff} = 34,000$ K, $M = 0.48 M_{\odot}$, log g = 5.8). The periods and e-folding times (for the unstable modes) are reported. We find a band of excited periods comprising the lowest order radial and nonradial p modes. The e-folding times for the unstable modes are all much shorter than the typical evolutionary timescale of a sdB star ($\sim 10^8$ yr). Given enough energy, these modes would thus normally develop observable amplitudes. In contrast, all of the g modes investigated are stable.<fnref n="2" /> We note, in this context, that the g-mode periods are overestimated here by 10%–20% because of our use of static structures that neglect the small He-burning core (see Paper I). Because the radial and p modes are essentially envelope modes, this neglect does not practically affect their periods.

In Figure 2, we compare the period spectra observed in the four known EC 14026 stars (Kilkenny et al. 1997a; Koen et al. 1997; Stobie et al. 1997; O'Donoghue et al. 1997b) with the

theoretical period spectrum for our representative model. The latter spectrum is restricted to the 80–280 s interval, and includes all modes with $l = 0, 1, 2$, and 3. Those are radial and p (and f) modes (see Table 1). The band of excited theoretical periods covers the range 116–195 s. Given that the effective temperatures (found in the range 33,500–35,700 K) and, particularly, the surface gravities (found in the range 5.5–6.1) of the real pulsators are not, of course, exactly equal to the

<figure><img src="image_124.png" /><figcaption>FIG. 2.—Comparison of the period spectra observed in the four known EC 14026 stars with the theoretical period spectrum of a representative model with $M = 0.48 M_{\odot}$, log g = 5.8, and $T_{eff} = 34,000$ K. The periods for all modes with $l = 0, 1, 2$, and 3 and in the interval 80–280 s are shown. These correspond to the lowest order radial and nonradial p (and f) modes. The g modes have periods larger than 280 s (see Table 1). The unstable modes are represented by continuous segments, while the stable modes are represented by dotted line segments.</figcaption></figure>

<footnote n="2">² Our calculations do suggest some marginal g-mode instabilities in models cooler than 29,000 K (the red edge of the radial and p-mode instability strip discussed here), but this must be investigated further.</footnote>
L126

CHARPINET ET AL.

values of our representative model ($T_{eff}$ = 34,000 K, log g = 5.8), we find it gratifying that the observed periods all fall into that range. Figure 2 also shows that *p* modes are necessarily present in the real pulsators, since the density of radial modes is clearly insufficient.

The results presented in this paper strongly reinforce the idea that pulsation modes are driven through an opacity bump mechanism due to a local enhancement of the iron abundance in the envelopes of sdB stars (see Paper I). Our second-generation models did away with the crude assumption of uniform enhanced metallicity and incorporated nonuniform iron abundance distributions obtained through the condition of diffusive equilibrium between gravitational settling and radiative levitation. This important improvement leads to unstable models that now have effective temperatures comparable with the temperatures of the EC 14026 stars. Furthermore, the unstable modes in a model with atmospheric parameters representative of the real pulsators have periods comparable to those observed in these pulsators. Low-order radial and nonradial (*p* and *f*) modes are involved. On the basis of these agreements, we reiterate the suggestion that an iron bump mechanism may also be responsible for mode excitation in rapidly oscillating Ap stars.

There are many tasks to be tackled in the future. Firstly, we need to investigate in more detail how the results are affected by changing model parameters. In particular, we need to map the instability domain in the $T_{eff}$-log g plane. The effects of

metals other than iron are also of interest. In the longer run, we need to develop a consistent picture of sdB asteroseismology as seen from a spectral evolution point of view. For instance, Fontaine & Chayer (1997) have demonstrated quantitatively that the general helium deficiencies observed in sdB stars can be explained in terms of weak stellar winds. While the mass loss rates postulated remain too small for changing the equilibrium structure of a star and affect significantly the dynamics of the pulsations, they can still potentially alter the reservoir of levitating iron that has been invoked here. It remains to be seen how mass loss can affect the iron distribution in sdB stars, but we can envision, at the outset, the possibility that two stars with similar values of $T_{eff}$ and log g may suffer from mass loss at different rates. In that case, one star (the one with enough iron in its reservoir) could pulsate, while the other (with a depleted reservoir) could be stable.

It is a real pleasure for us to acknowledge stimulating discussions on EC 14026 stars with Darragh O'Donoghue, Dave Kilkenny, and Bob Stobie. This work was supported in part by the NSREC of Canada and by the fund FCAR (Québec). P. C. acknowledges partial support from NASA contracts NCC5-138 and NAG5-2620. The work of F. J. R. and C. A. I. was supported under the auspices of the Department of Energy by Lawrence Livermore National Laboratory. B. D. acknowledges support from NASA grants NAG5-700 and NAGW-4106.

REFERENCES

Brassard, P., & Fontaine, G. 1994, in IAU Colloq. 147, The Equation of State in Astrophysics, ed. G. Chabrier & E. Schatzman (Cambridge: Cambridge Univ. Press), 560

Brassard, P., Fontaine, G., & Bergeron, P. 1997, in preparation

Charpinet, S., Fontaine, G., Brassard, P., & Dorman, B. 1996, ApJ, 471, L103 (Paper I)

Chayer, P., Fontaine, G., & Wesemael, F. 1995a, ApJS, 99, 189

Chayer, P., Vennes, S., Pradhan, A. K., Thejl, P., Beauchamp, A., Fontaine, G., & Wesemael, F. 1995b, ApJ, 454, 429

Cox, A. N., Morgan, S. M., Rogers, F. J., & Iglesias, C. A. 1992, ApJ, 393, 272

Dorman, B., Rood, R. T., & O'Connell, R. W. 1993, ApJ, 415, 596

Dziembowski, W., & Pamiatykh, A. A. 1992, MNRAS, 262, 204

Fontaine, G., Brassard, P., Wesemael, F., & Tassoul, M. 1994, ApJ, 428, L61

Fontaine, G., & Chayer, P. 1997, in Proc. 3d Conf. on Faint Blue Stars, ed. A. G. D. Philip, J. Liebert, & R. A. Saffer (Schenectady: Davis), in press

Gautschy, A., & Saio, H. 1993, MNRAS, 262, 213

Heber, U. 1991, in IAU Symp. 145, Evolution of Stars: The Photospheric Abundance Connection, ed. G. Michaud & A. Tutukov (Dordrecht: Kluwer), 363

Iglesias, C. A., & Rogers, F. J. 1996, ApJ, 464, 943

Kilkenny, D., Koen, C., O'Donoghue, D., & Stobie, R. S. 1997a, MNRAS, 285, 640

Kilkenny, D., O'Donoghue, D., Koen, C., Stobie, R. S., Lynas-Gray, A. E., & Kawaler, S. D. 1997b, in Proc. 3d Conf. on Faint Blue Stars, ed. A. G. D. Philip, J. Liebert, & R. A. Saffer (Schenectady: Davis), in press

Koen, C., Kilkenny, D., O'Donoghue, D., Van Wyk, F., & Stobie, R. S. 1997, MNRAS, 285, 645

Moskalik, P. & Dziembowski, W. 1992, A&A, 256, L5

O'Donoghue, D., Koen, C., Kilkenny, D., Stobie, R. S., Lynas-Gray, A. E., & Kawaler, S. D. 1997a, in Proc. 10th Eur. Workshop on White Dwarfs, ed. J. Isern, M. Hernanz, & E. Garcia-Berro (Dordrecht: Kluwer), in press

O'Donoghue, D., Lynas-Gray, A. E., Kilkenny, D., Stobie, R. S., & Koen, C. 1997b, MNRAS, 285, 657

Rogers, F. J., & Iglesias, C. A. 1992, ApJ, 401, 361

Saffer, R. A., Bergeron, P., Koester, D., & Liebert, J. 1994, ApJ, 432, 351

Stobie, R. S., Kawaler, S. D., Kilkenny, D., O'Donoghue, D., & Koen, C. 1997, MNRAS, 285, 651
THE ASTROPHYSICAL JOURNAL, 489:L149–L152, 1997 November 10

© 1997. The American Astronomical Society. All rights reserved. Printed in U.S.A.

GRAVITY-MODE INSTABILITIES IN MODELS OF POST-EXTREME HORIZONTAL BRANCH STARS: ANOTHER CLASS OF PULSATING STARS?

S. CHARPINET, G. FONTAINE, AND P. BRASSARD

Département de Physique, Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montréal, Québec H3C 3J7, Canada;

charpinet@astro.umontreal.ca, fontaine@astro.umontreal.ca, brassard@astro.umontreal.ca

AND

BEN DORMAN<fnref n="1" />

Laboratory for Astronomy and Solar Physics, NASA/GSFC, Greenbelt, MD 20771; Ben.Dorman@gsfc.nasa.gov

Received 1997 July 30; accepted 1997 September 11; published 1997 October 13

ABSTRACT

We present new results of a stability analysis of realistic models of post-extreme horizontal branch stars. We find that *g*-mode instabilities develop in some of these models as a result of a potent *ε*-mechanism associated with the presence of an active H-burning shell. The *ε*-process drives low-order and low-degree *g* modes with typical periods in the range 40–125 s. The unstable models populate a broad instability strip covering the interval 76,000 K $\ge T_{\text{eff}} \ge$ 44,000 K and are identified with low-mass DAO white dwarfs. They descend from stars that reach the zero-age “extended” horizontal branch with H-rich envelope masses $M_{\text{env}} \ge 0.0010 M_{\odot}$. Our computations indicate that some DAO stars should show luminosity variations resulting from pulsational instabilities. We suggest looking for brightness variations in six particularly promising candidates.

*Subject headings:* stars: interiors — stars: oscillations — subdwarfs — white dwarfs

1. ASTROPHYSICAL CONTEXT

We have recently embarked in a systematic investigation of the asteroseismological potential of stellar models on the extreme horizontal branch (EHB) and beyond. This was made possible thanks to significant progress in our ability to compute increasingly sophisticated and realistic models for this relatively neglected phase of stellar evolution (see, e.g., Dorman 1995 for a review). The models of interest are low-mass objects ($M \le 0.5 M_{\odot}$) with outer H-rich envelopes too thin to reach the AGB after core helium exhaustion. Such models cannot sustain significant H-shell burning during core helium burning evolution. The core He burning phase, of typical length $\sim 10^8$ yr, is identified with subdwarf B (sdB) stars. The stars have atmospheric parameters found in the ranges 40,000 K $\ge T_{\text{eff}} \ge 24,000$ K and $6.2 \ge \log g \ge 5.1$ (see Saffer et al. 1994, and references therein).

A stability analysis of stellar models in the sdB phase of evolution has led us to the discovery of an efficient driving mechanism due to an opacity bump associated with iron ionization (Charpinet et al. 1996). It was found that both radial and nonradial (*p*, *f*, and *g*) low-order modes could be excited in some of these models. On this basis, we made the prediction that a subclass of sdB stars should show luminosity variations resulting from pulsational instabilities. The independent and exciting discoveries of the first real pulsating sdB stars in South Africa (Kilkenny et al. 1997; Koen et al. 1997; Stobie et al. 1997; O'Donoghue et al. 1997) gave us confidence in the basic validity of our models. They also led to further developments on the observational front (Billères et al. 1997) and to refinements of the physical description of our iron bump mechanism (Charpinet et al. 1997).

While the observational and theoretical foundations for the pulsating sdB stars now appear to be well established, the asteroseismological potential of their descendants, the post-EHB stars, has not been investigated and remains of great interest.

During the post-core-exhaustion phase the stars contract and the H-burning shell finally “turns on.” With insufficient hydrogen energy to force a red giant star envelope, the stars live in the so-called “AGB-Manqué” phase (Greggio & Renzini 1990) for a period similar to the equivalent Early (i.e., pre-thermal pulsing) AGB evolution phase, 2–4 $\times 10^7$ yr. These post-EHB, He shell burning models are associated with the field subdwarf O stars (Dorman, O’Connell, & Rood 1995, and references therein). A majority of these stars cluster around $T_{\text{eff}} \sim 45,000$ K and $\log g \sim 5.5$ (Dreizler 1993). Ultimately, the models join the white dwarf cooling tracks near $T_{\text{eff}} \sim 80,000$ K and are identified, in the early white dwarf phase, with the low-gravity DAO white dwarfs (Bergeron et al. 1994).

From an asteroseismological standpoint, the ignition of hydrogen at the base of the H-rich outer layer in these hot, post-EHB models is particularly interesting. Up to 50% of the luminosity of the star may be provided by hydrogen burning in a thin shell in these evolved models. The location of the H-burning shell strongly suggests that a potent $\epsilon$-mechanism might drive pulsation modes there (see below). In contrast, the He-burning shell appears to be located too deep at the outset to play a key role in destabilizing modes. Moreover, we do not expect the iron bump mechanism uncovered in sdB models to be relevant to post-EHB models as they are too hot (see Charpinet et al. 1997).

In this Letter, we report on the salient features of a stability analysis we carried out for realistic models of post-EHB stars, and on our discovery of pulsational instabilities in some of them.

2. STABILITY ANALYSIS OF POST-EHB MODELS

The equilibrium models employed in this investigation are full stellar models taken from seven distinct evolutionary sequences. The sequences span the evolution from the zero-age EHB (ZAEHB) to the cool white dwarf phase. The evolutionary models were computed with the methods described in Dorman (1992a, 1992b) and Dorman, Rood, & O’Connell (1993) but included improvements in the constitutive physics. The new

<footnote n="1">¹ Also, Department of Astronomy, University of Virginia, P.O. Box 3818, Charlottesville, VA 22903-0818.</footnote>
<figure><img src="image_125.png" /></figure>

FIG. 1.—Typical evolutionary tracks for EHB and post-EHB models in the $T_{\text{eff}}$–log g diagram. Three tracks are illustrated, each corresponding to the same initial core mass, 0.4758 $M_{\odot}$, but with ZAEHB H-rich envelope masses of 0.0042, 0.0012, and 0.0002 $M_{\odot}$. The individual models along a track are represented by small crosses and are joined together by dotted straight line segments. They cover the evolution from the ZAEHB to the white dwarf phase. Superimposed, in the upper right corner, are the positions of 213 real sdB stars (small dots). Likewise, the positions of the known DAO (filled circles) and hot DA (open circles) white dwarfs are illustrated in the lower part of the diagram. The thick segments along two of the tracks in the DAO region correspond to the positions of unstable models driven by the $\epsilon$ mechanism in the H-burning shell.

models use the OPAL opacities described by Rogers & Iglesias (1992) computed in 1993 December, which adopted the element mix referred to as “Grevesse & Noels 1993.” Where necessary (during He flashes), we used new low-temperature opacities by D. R. Alexander (1995, private communication, described in Alexander & Ferguson 1994), which were computed with the same element mix. These smoothly match the OPAL opacity set within the hydrogen ionization zone. The other difference in the input physics was the use of the Itoh et al. (1983; Itoh, Hayashi, & Koyama 1993a, 1994a; Itoh et al. 1993b; Itoh & Kohyama 1994b) conductive opacities.

Five of the sequences (the same as those considered by Charpinet et al. 1996) correspond to the evolution of an AGB-Manqué star with a core mass of 0.4758 $M_{\odot}$. The sequences differ in that different initial envelope masses on the ZAEHB are considered: $M_{\text{env}} = 0.0002, 0.0012, 0.0022, 0.0032$, and $0.0042 M_{\odot}$. Two additional sequences with core masses of $0.4690 M_{\odot}$ and ZAEHB envelope masses of 0.0001 and 0.0007 $M_{\odot}$ have been added in the meantime to provide a better mapping of the sdB region in the $T_{\text{eff}}$–log g diagram. In all cases, the composition of the envelopes was assumed to be solar ($X = 0.70388, Z = 0.01718$), derived from calibrating a solar model sequence to $\log L/L_{\odot} = 0, T_{\text{eff}} = 5770$ K at age 4.6 Gyr.

Figure 1 illustrates the evolutionary tracks for three of our sequences in the $T_{\text{eff}}$–log g plane. For comparison purposes, the positions of 213 known sdB stars (according to R. A. Saffer

1997, private communication) are shown in the upper right region of the diagram, while the positions of the DAO and hot DA white dwarfs analyzed by Bergeron et al. (1994) are shown in the lower part of the figure. Of prime interest in the present context is the conclusion of Bergeron et al. (1994) that six of the DAO stars in their sample (those distributed about our evolutionary tracks) are post-EHB objects. In contrast, the majority of the hot white dwarfs they discuss have higher gravities (see Fig. 1), and must be considered as post-AGB stars (see Bergeron et al. 1994).

In the present effort, we have carried out a stability analysis for all of our post-EHB models, i.e., those beyond the sdB phase itself. Those models are characterized by double He and H shell burning. We considered all modes with $l = 0, 1, 2$, and 3 in the 5–500 s period window. This was done with the finite-element nonadiabatic pulsation code briefly described in Fontaine et al. (1994) and in Brassard, Fontaine, & Bergeron (1997). We found that, in four of our sequences, the $\epsilon$ mechanism produced by the H-burning shell is sufficiently potent to drive low-order g-modes in models located at the beginning of the white dwarf branch of the evolutionary tracks. The unstable models define a broad strip covering the range 76,000 K $\ge T_{\text{eff}} \ge 44,000$ K and are identified with the DAO phase of post-EHB evolution. The thick line segments along two evolutionary tracks in Figure 1 show a mapping of the instability strip. The sequences with the smaller H-rich envelope masses on the ZAEHB (i.e., those with $M_{\text{env}} = 0.0001, 0.0002,$
No. 2, 1997

ANOTHER CLASS OF PULSATING STARS?

L151

<table><caption>TABLE 1
PULSATION PERIODS AND e-FOLDING TIMES FOR A REPRESENTATIVE DAO MODEL</caption><thead><tr><th rowspan="2">k</th><th colspan="2">l = 1</th><th colspan="2">l = 2</th><th colspan="2">l = 3</th></tr><tr><th>P (s)</th><th>τ<sub>e</sub> (yr)</th><th>P (s)</th><th>τ<sub>e</sub> (yr)</th><th>P (s)</th><th>τ<sub>e</sub> (yr)</th></tr></thead><tbody><tr><td>3 (p) ......</td><td>10.99</td><td>Stable</td><td>10.48</td><td>Stable</td><td>9.78</td><td>Stable</td></tr><tr><td>2 (p) ......</td><td>13.63</td><td>Stable</td><td>11.90</td><td>Stable</td><td>11.24</td><td>Stable</td></tr><tr><td>1 (p) ......</td><td>15.61</td><td>Stable</td><td>15.19</td><td>Stable</td><td>14.75</td><td>Stable</td></tr><tr><td>0 (f) ......</td><td>...</td><td>...</td><td>20.70</td><td>Stable</td><td>17.18</td><td>Stable</td></tr><tr><td>1 (g) ......</td><td>82.37</td><td>6.99 × 10<sup>6</sup></td><td>54.11</td><td>9.98 × 10<sup>5</sup></td><td>42.41</td><td>2.42 × 10<sup>5</sup></td></tr><tr><td>2 (g) ......</td><td>109.78</td><td>9.01 × 10<sup>4</sup></td><td>69.33</td><td>1.34 × 10<sup>4</sup></td><td>50.25</td><td>1.14 × 10<sup>4</sup></td></tr><tr><td>3 (g) ......</td><td>122.73</td><td>1.14 × 10<sup>4</sup></td><td>74.59</td><td>4.70 × 10<sup>4</sup></td><td>59.59</td><td>Stable</td></tr><tr><td>4 (g) ......</td><td>158.70</td><td>Stable</td><td>101.64</td><td>Stable</td><td>76.55</td><td>Stable</td></tr><tr><td>5 (g) ......</td><td>187.84</td><td>Stable</td><td>114.16</td><td>Stable</td><td>87.22</td><td>Stable</td></tr></tbody></table>

and 0.0007 $M_\odot$) have less active H-burning shells throughout the evolution. While the $\epsilon$ mechanism due to such shells still produces significant local driving, particularly within the instability strip, it is not strong enough to overcome radiative damping processes in these models.

A typical unstable model is model 32, belonging to the sequence with $M_{\text{core}} = 0.4758 M_\odot$ and $M_{\text{env}} = 0.0012 M_\odot$; its evolutionary track is illustrated in Figure 1. It has an age of $1.39 \times 10^8$ yr (time elapsed since the ZAEHB), a surface gravity $\log g = 7.24$, a luminosity $L = 6.31 L_\odot$, and an effective temperature $T_{\text{eff}} = 55,560$ K. Table 1 gives the periods and the e-folding times (for the unstable modes) of the lowest order pulsation modes for this model. The radial modes ($l = 0$) are all stable and are not listed in the table. In this, and in all other models showing instabilities, only the g-modes are excited; the p-modes are never driven. The table indicates that the lowest order g-modes ($k = 1, 2$, and/or 3) are excited and span a period

range 42–123 s. The e-folding times must be compared with the time it takes for a post-EHB star to cross the instability strip. Since this time is approximately equal to $4.0 \times 10^6$ yr, substantially longer than most of the e-folding times, it would appear that the excited pulsations have plenty of time to develop observable amplitudes.

Figure 2 illustrates some structural properties for our representative model. The upper panel indicates, among other things, the location of the He and H burning shells as well as the magnitude of the nuclear energy generation rate there (dotted curve). On the scale used here, the contribution of the He-burning shell is insignificant. The lower panel shows the compositional stratification as well as the luminosity profile. The latter demonstrates that about 40% of the total luminosity of the model is due to H-shell burning at the base of the H-rich envelope. At the epoch of the model, some 68% of the hydrogen in the original ZAEHB envelope of 0.0012 $M_\odot$ has been consumed. Explicit tests carried out by switching off $\epsilon_{\text{H}}$ and/or $\epsilon_{\text{He}}$ in the pulsation calculations demonstrate, as implied by the upper panel of Figure 2, that all the driving is caused by the $\epsilon$ mechanism generated by the H-burning shell. The latter is essentially driven by the CN cycle, the p-p luminosity being negligible in this and the other unstable models. It is worth pointing out here that the metallicity would have to be lowered by several orders of magnitude for the p-p contribution to become dominant. For instance, models with $Z = 10^{-4}$ are already CN driven.

### 3. DISCUSSION

The g-mode instabilities uncovered here are very similar in nature to the pulsational instabilities discussed by Kawaler (1988; see also Sienkiewicz 1980) in the context of the much

<figure><img src="image_126.png" /><figcaption>FIG. 2.—(Upper panel) Dotted curve shows the magnitude of the nuclear energy generation rate in, and the location of, the He and H burning shell. The other curves refer to the amplitude of the relative Lagrangian perturbation of the temperature for three g-mode overtones with $l = 1$. Only the mode with $k = 2$ is excited. (Lower panel) Solid curve gives the luminosity profile as a function of fractional mass depth ($\log q = \log [1 - M(r)/M_\odot]$) in our representative model. The other curves show the composition stratification.</figcaption></figure>
more luminous and hotter central stars of planetary nebulae. Standard models of these post-AGB stars also show a double-shell burning structure. As in Kawaler (1988), the $ϵ$ process provides here both the driving force and a filter mechanism (selecting only a few modes as unstable), but, unlike the case for those very luminous models, only the H-burning shell contributes to the process in the present situation.

The g-modes are favored for instability because, in the course of post-EHB evolution, their region of formation migrates from the core in the sdB phase (Charpinet et al. 1996, 1997) to the outer envelope in the cool pulsating white dwarfs of the ZZ Ceti type (see, e.g., Brassard et al. 1992). In the post-EHB DAO phase, g-modes are mostly sensitive to the physical conditions in the deep envelope, where the H-burning shell is located. In contrast, the He-burning shell is located in the core (see Fig. 2, upper panel), is less active, and thus plays no significant role.

The g-modes that are driven are those with oscillation amplitudes that are large in the H shell burning region. In practice, this means that the largest maximum in the temperature perturbation of a mode (usually the maximum located nearest to the surface) should nearly coincide with the H-burning shell. As the radial index *k* of a mode increases, this maximum in the temperature perturbation moves outward, ultimately reaching a location beyond the shell. Those modes can no longer be efficiently driven. This then provides a filter, a cutoff in *k*, beyond which the g-modes are stable.

In the upper panel of Figure 2, we have plotted the absolute value of the Lagrangian temperature perturbation of three modes as a function of depth in our representative model. Note that this function is normalized for each mode, so the only information of interest here is related to the relative strengths and locations of maxima for a given mode. The three modes illustrated belong to the sequence with *l* = 1. The mode with *k* = 2 has a period of 109.78 s and is unstable (see Table 1). The behavior of its temperature perturbation (solid curve) illustrates what was written in the previous paragraph, namely that its largest maximum nearly overlaps with the H-burning shell. This maximizes the efficiency of $ϵ$ driving for that mode.

The modes with $k \ge 4$ (and $l = 1$) are all stable in our representative model. The dot-dashed curve in the upper panel of Figure 2 shows that the largest maximum in the temperature perturbation of the *k* = 5 mode (a stable mode with a period of 187.84 s; see Table 1) is located well above the shell burning

region, near log $q \approx -5.8$, where radiative damping is important. In this specific case, the temperature perturbation also shows a node just below the H-burning shell, so the conditions are particularly unfavorable for efficient $ϵ$ driving for this mode. Finally, we illustrate also the behavior of the normalized temperature perturbation for a mode with $k = 8$ (not listed in Table 1, but stable and with a period of 272.70 s). Here a secondary maximum is located right in the middle of the H-burning shell, but the ensuing driving is overcome by damping in the regions above where the relative amplitude of the temperature perturbation becomes much larger.

### 4. CONCLUSION

The results presented in this paper strongly suggest that non-radial pulsations may be present in post-EHB evolution, and thus the methods of asteroseismology may shed light on the structure of stars in this phase. We have found that g-mode instabilities develop in realistic models of post-EHB stars. The unstable models are the descendants of stars with relatively massive ($M_{\text{env}} \ge 0.0010 M_{\odot}$) H-rich envelopes on the ZAEHB, and pulsate because of the presence of an active H-burning shell at the base of the envelope. They define a broad instability strip covering the range 76,000 K $\ge T_{\text{eff}} \ge 44,000$ K and are identified with low-mass DAO white dwarfs. Low-order and low-degree g-modes with typical periods around ~80 s are predicted to be unstable in that instability strip.

Specifically, we suggest looking for luminosity variations in those DAO objects already identified by Bergeron et al. (1994) as post-EHB stars. Those are HZ 34, GD 651, Ton 353, PG 0834+501, Feige 55, and PG 0134+181. Of course, there is no a priori guarantee that those stars have ZAHB progenitors with envelopes in the correct mass range, since the region of temperature/gravity space occupied by the models with instabilities must be quite thin. Also, weak mass loss (often invoked to explain the abundance anomalies of sdB stars) may succeed in thinning that envelope to the point, perhaps, of reducing significantly the $ϵ$ driving. In any case, the proposed observations would provide interesting constraints on the amount of hydrogen left over in these DAO stars.

This work was supported in part by the NSREC of Canada and by the fund FCAR (Québec). B. D. acknowledges support from NASA grants NAG5-700 and NAGW-4106.

### REFERENCES

* Alexander, D. A., & Ferguson, J. W. 1994, ApJ, 437, 879

* Bergeron, P., Wesemael, F., Beauchamp, A., Wood, M. A., Lamontagne, R., Fontaine, G., & Liebert, J. 1994, ApJ, 432, 305

* Billères, M., Fontaine, G., Brassard, P., Charpinet, S., Liebert, J., Saffer, R. A., & Vauclair, G. 1997, ApJ, 487, L81

* Brassard, P., Fontaine, G., & Bergeron, P. 1997, in preparation

* Brassard, P., Fontaine, G., Wesemael, F., & Tassoul, M. 1992, ApJS, 81, 747

* Charpinet, S., Fontaine, G., Brassard, P., & Dorman, B. 1996, ApJ, 471, L103

* Charpinet, S., Fontaine, G., Brassard, P., Chayer, P., Rogers, F. J., Iglesias, C. A., & Dorman, B. 1997, ApJ, 483, L123

* Dorman, B. 1992a, ApJS, 80, 701

* —, 1992b, ApJS, 81, 221

* —, 1995, in Proc. 32d Liège Astrophysical Colloq., Stellar Evolution: What Should Be Done, ed. A. Noels, D. Fraipont-Caro, N. Grevesse, & P. Demarque (Liège: Institut d'Astrophysique), 291

* Dorman, B., O'Connell, R. W., & Rood, R. T. 1995, ApJ, 442, 105

* Dorman, B., Rood, R. T., & O'Connell, R. W. 1993, ApJ, 415, 596

* Dreizler, S. 1993, in White Dwarfs: Advances in Observation and Theory, ed. M. A. Barstow (NATO ASI Ser. 403), 287

* Fontaine, G., Brassard, P., Wesemael, F., & Tassoul, M. 1994, ApJ, 428, L61

* Greggio, L., & Renzini, A. 1990, ApJ, 364, 35

* Itoh, N., Hayashi, H., & Kohyama, Y. 1993a, ApJ, 418, 405

* —, 1994a, ApJ, 436, 418

* Itoh, N., & Kohyama, Y. 1994b, ApJ, 420, 943

* Itoh, N., Kohyama, Y. Matsumoto, N., & Seki, M. 1993b, ApJ, 404, 418

* Itoh, N., Mitake, S., Iyetomi, H., & Ichimaru, S. 1983, 273, 774

* Kawaler, S. D. 1988, ApJ, 334, 220

* Kilkenny, D., Koen, C., O'Donoghue, D., & Stobie, R. S. 1997b, MNRAS, 285, 640

* Koen, C., Kilkenny, D., O'Donoghue, D., Van Wyk, F., & Stobie, R. S. 1997, MNRAS, 285, 645

* O'Donoghue, D., Lynas-Gray, A. E., Kilkenny, D., Stobie, R. S., & Koen, C. 1997, MNRAS, 285, 657

* Rogers, F. J., & Iglesias, C. A. 1992, ApJ, 401, 361

* Saffer, R. A., Bergeron, P., Koester, D., & Liebert, J. 1994, ApJ, 432, 351

* Sienkiewicz, R. 1980, A&A, 85, 295

* Stobie, R. S., Kawaler, S. D., Kilkenny, D., O'Donoghue, D., & Koen, C. 1997, MNRAS, 285, 651
THE ASTROPHYSICAL JOURNAL, 487: L81–L84, 1997 September 20

© 1997, The American Astronomical Society. All rights reserved. Printed in U.S.A.

DISCOVERY OF *p*-MODE INSTABILITIES IN THE HOT SUBDWARF B STAR PG 1047+003

M. BILLÈRES,<fnref n="1" /> G. FONTAINE, P. BRASSARD, AND S. CHARPINET

Département de Physique, Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montréal, Québec, H3C 3J7 Canada;
billères@astro.umontreal.ca, fontaine@astro.umontreal.ca, brassard@astro.umontreal.ca,
charpinet@astro.umontreal.ca

J. LIEBERT

Steward Observatory, University of Arizona, Tucson, AZ 85721; liebert@as.arizona.edu

R. A. SAFFER

Department of Astronomy and Astrophysics, Villanova University, Villanova, PA 19085; saffer@ast.vill.edu

AND

G. VAUCLAIR

Observatoire Midi-Pyrénées, 14 Avenue E. Belin, F-31400, Toulouse, France; gerardv@obs-mip.fr

Received 1997 June 16; accepted 1997 July 11

ABSTRACT

We report the discovery of multiperiodic luminosity variations in the hot B subdwarf PG 1047+003. At least five periodicities are seen in the light curve, from 104.2 s to 161.9 s, but others are also present at lower amplitudes in that interval. The largest oscillation has an amplitude of ≈9.2 millimag in white light and a period of 142.2 s. With atmospheric parameters $T_{\text{eff}} \sim 34,370$ K and $\log g \sim 5.7$ for PG 1047+003, these variations are identified with low-order radial and nonradial (*p* and *f*) pulsation modes. The similarity of periods and derived stellar parameters indicate that PG 1047+003 is a genuine member of the EC 14026 class, the latest and newest family of pulsators in the field of asteroseismology. However, it shows no evidence of a binary companion, implying that the mechanism for driving pulsations is internal to the star. We also report on the current status of our ongoing survey to search for additional sdB pulsators.

*Subject headings:* stars: interiors — stars: oscillations — subdwarfs

1. INTRODUCTION

Charpinet et al. (1996) recently presented the first results of a systematic survey of the asteroseismological properties of a neglected class of stars, the hot B subdwarfs (sdB). These evolved, compact stars are found in the range 40,000 K ≳ $T_{\text{eff}}$ ≳ 24,000 K and have typical values $M \sim 0.5 M_{\odot}$ and $\log g \sim 5.8$ (see Saffer et al. 1994, and references therein). They are identified with low-mass core helium burning models with outer H-rich envelopes that are too thin to sustain H-shell burning on the extended horizontal branch (see, e.g., Dorman, Rood, & O’Connell 1993).

Charpinet et al. (1996) found that low-order radial and nonradial (*p*, *f*, and *g*) modes were excited in some of their models because of a κ mechanism associated to heavy-element (essentially iron) ionization. On this basis, they made the prediction that a subclass of sdB stars should be pulsating. Motivated by this prediction, we initiated, at the end of 1996 May, a search for luminosity variations in this class of objects. Our goals are (1) to search for sdB pulsators in the northern hemisphere, (2) to investigate the extent of this phenomenon in the $T_{\text{eff}}$–log g plane, and (3) to map the empirical instability strip in that diagram. Our sample of target stars is based on the estimates of atmospheric parameters taken from the lists of sdB stars provided by Allard (1986; see also Wesemael et al. 1997), Moehler, Heber, & de Boer (1990), Saffer et al. (1994), and unpublished results by one of us (R. A. S.).

Unknown to us at the time we initiated our search, and quite independently of the theoretical prediction of Charpinet et al.

(1996), a remarkable observational effort, carried out at the South African Astronomical Observatory, led to the exciting discoveries of the first real pulsating sdB stars. This was initially reported in 1996 June by O’Donoghue et al. (1997a; see also Kilkenny et al. 1997a). The properties of the first four members of this new class of pulsating stars (named EC 14026 objects, after the prototype) have been described in a recent series of papers (Kilkenny et al. 1997b; Koen et al. 1997; Stobie et al. 1997; O’Donoghue et al. 1997b). The real pulsators are all multiperiodic with periods in the interval 121–159 s, and they are found in the ranges 35,700 K ≳ $T_{\text{eff}}$ ≳ 33,500 K and 6.10 ≳ log g ≳ 5.54. They are part of binary systems containing dimmer, cooler F or G-type companions.

The location of the EC 14026 stars in the $T_{\text{eff}}$–log g plane is in apparent conflict with Charpinet et al. (1996), who suggested that pulsating sdB stars should have $T_{\text{eff}} \lesssim 30,000$ K and $\log g \lesssim 5.7$. However, these authors also warned that these theoretical boundaries could be dependent on the assumption of *uniform* metallicity used in their so-called first generation models. Indeed, motivated by the discovery of the first EC 14026 stars, Charpinet et al. (1997) considered more sophisticated second generation models in which the crude assumption of uniform metallicity has been replaced by the more realistic condition of diffusive equilibrium between gravitational settling and radiative levitation on the metal–iron–responsible for the driving process. In that second study, Charpinet et al. (1997) have uncovered a theoretical instability strip, 36,500 K ≳ $T_{\text{eff}}$ ≳ 29,000 K (for log g = 5.8 models), in which the four currently known EC 14026 stars fall, and obtained an excellent agreement between the expected and

<footnote n="1">¹ Also Observatoire Midi-Pyrénées, 14 Avenue E. Belin, F-31400, Toulouse, France.</footnote>
the observed periods. Low-order radial and low-order and low-degree nonradial (*p* and *f*) modes are involved.

In this Letter, we report on the discovery of multiperiodic luminosity variations in the hot B subdwarf PG 1047+003, a star with characteristics very similar to those of the known EC 14026 stars. In addition, we briefly discuss the current status of our observational search for these pulsators. A previous report has been presented by Billères et al. (1997).

## 2. OBSERVATIONS

PG 1047+003 is a relatively bright sdB star with $B_{pg}$ = 12.98 (Green, Schmidt, & Liebert 1986) and $y = 13.48$ (Wesemael et al. 1992). Its Strömgren color indices ($b-y = -0.133$, $u-b = -0.245$) lead to an index $Q' = -0.038$, from which we can estimate an effective temperature $T_{eff} \approx 33,500$ K (see Wesemael et al. 1992). On the basis of optical spectroscopy and *IUE* spectrophotometry, Allard (1986) has given two possible solutions for the atmospheric parameters, depending on *IUE* calibration: $T_{eff} \approx 32,000$ K and $log\,g \approx 5.4$ or $T_{eff} \approx 39,500$ K and $log\,g \approx 6.4$. A recent reanalysis of part of these data (the optical spectroscopy only) using new model atmospheres leads to $T_{eff} \approx 32,470$ K and $log\,g \approx 5.2$ (Wesemael et al. 1997). The uncertainties on the atmospheric parameters of PG 1047+003 are thus fairly large, and better observations would be highly desirable. For our present needs, however, it is sufficient to give the same weight to these various estimates, and we thus find $T_{eff} \approx 34,370$ K and $log\,g \approx 5.7$. At this effective temperature, PG 1047+003 also falls, along with the known EC 14026 stars, within the theoretical instability strip uncovered by Charpinet et al. (1997).

We observed PG 1047+003 for 16,090 s at the Mount Bigelow 1.6 m telescope on 1997 March 18. As usual in our survey, white-light photometry was used to maximize the photon count rate. The photometric observations were gathered with LAPOUNE, the portable Montréal three-channel photometer, which uses Hamamatsu R647-04 photomultipliers. Simultaneous observations of the target star, a sky patch, and a (constant) comparison star were gathered. The sky contribution was subtracted point by point from the light curve, and the extinction was removed using cubic spline functions which fit through the light curve with nodes separated by 1000 s. The net average count rate for the target star was ~22,000 counts s⁻¹. The light curve gathered for the comparison star during our PG 1047+003 run (which was used to monitor the quality of the night) clearly reveals that we have observed under excellent photometric conditions. Throughout the run, a constant sampling time of 10 s was used. More details on the instrument, observing technique, and data reduction procedure will be presented elsewhere.

Figure 1 shows a part of the sky-subtracted, extinction-corrected light curve which we have obtained for PG 1047+003. It is expressed in terms of residual amplitude in mill magnitudes. The light curve reveals, clearly and unambiguously, the presence of complex, multiperiodic brightness modulations. The variations appear to be dominated by a pseudo-period of 140–145 s, but it is clear that several modes are involved. For instance, peak-to-peak variations can sometimes reach close to 50 millimag, but, on other occasions (e.g., the segment between 3100 and 3600 s), destructive interference between modes does occur.

A standard Fourier analysis of the light curve reveals more quantitatively its complex structure. This is illustrated in

<figure><img src="image_127.png" /><figcaption>FIG. 1.—First part of the light curve of PG 1047+003, observed on 1997 March 18 with the Mount Bigelow 1.6 m telescope, expressed in residual amplitude in millimag. The light curve is continuous, starting at the upper left, and each point represents a sampling time of 10 s. The segment shown here corresponds to the first 10,000 s of a total light curve 16,090 s long.</figcaption></figure>

Figure 2, which shows the low-frequency part of the Fourier (amplitude) spectrum of the light curve. The spectrum in the region from 20 mHz out to the Nyquist frequency (50 mHz) is entirely consistent with noise and is not illustrated. The noise level in that spectral region is less than ~0.6 millimag.

Figure 2 shows that there are many significant components in the Fourier spectrum of the light curve of PG 1047+003. In particular, our light curve is dominated by two periodicities, one at 7.035 mHz (142.2 s) with an amplitude $\Delta m \approx 9.2$ millimag, and the other at 6.885 mHz (145.2 s) with an amplitude $\Delta m \approx 6.4$ millimag. Other obvious peaks in the Fourier spectrum are found at 6.178, 8.927, and 9.600 mHz, corresponding to periods 161.9, 112.0, and 104.2 s, and amplitudes $\Delta m \approx 3.6, 1.7$, and 1.7 millimag. In addition, there are

<figure><img src="image_128.png" /><figcaption>FIG. 2.—Low-frequency part of the Fourier (amplitude) spectrum of the light curve of PG 1047+003, observed on 1997 March 18 with the Mount Bigelow 1.6 m telescope. The amplitude axis is expressed in terms of the variations (in millimag) about the mean intensity of the star. Five periodicities are obvious, but others are present as well.</figcaption></figure>
No. 1, 1997

P-MODE INSTABILITIES IN PG 1047+003

L83

other periodicities, particularly near the main complex of
peaks, but higher temporal resolution is clearly needed here to
identify them without ambiguity.

In order to interpret our observations, we have computed
oscillation periods of models of sdB stars with atmospheric
parameters around those adopted above for PG 1047+003. To
carry out that task, we used the same tools as in Charpinet et
al. (1996, 1997). Details on the model building and pulsation
codes will be reported elsewhere. A comparison indicates that
the observed periods (104–162 s) correspond to low-order,
low-degree radial and nonradial (p and f) pulsation modes, as
in the known EC 14026 stars (Stobie et al. 1997; Kilkenny et al.
1997a).

3. DISCUSSION AND CONCLUSION

The observations presented in this paper demonstrate that PG 1047+003 is a multiperiodic pulsator with atmospheric parameters, periods, and amplitudes very similar to those of the EC 14026 stars. Unlike the latter objects, however, the optical colors and the optical spectrum of PG 1047+003 show no indication for the presence of a companion. A direct imaging search in the infrared has also failed to detect such a companion (Thejll, Thiessen, & Jiminez 1994). In contrast, the known EC 14026 pulsators are all part of binary systems containing main sequence late-type companions (O'Donoghue et al. 1997a; Kilkenny et al. 1997a), and the question arises as to the role of the companions on the mechanism that excites pulsation modes in these stars. We suggest that these compan- ions play no significant role. Indeed, it has been known for some time (Allard et al. 1987; and see also Allard et al. 1994) that the majority of sdB stars are part of such binary systems, so it may not be a surprise that the pulsators discovered in South Africa are binaries. Also, Saffer (1991)has demon- strated that the majority of the sdB binaries cannot be close binaries (i.e, with orbital periods of a few days and less, corresponding to separations of the order of 1 AU), so the possibility of direct interaction appears remote. In addition, from a theoretical point of view, the driving mechanism proposed by Charpinet et al. (1996, 1997) is independent of influences external to the sdB star itself. We thus suggest that the binary nature is not an essential ingredient to the EC 14026 phenomenon and conclude that PG 1047+003 is a genuine addition to the small, but rapidly growing family of EC 14026 stars.

Finally, we report on our ongoing search for more sdB
pulsators. We have, so far, gathered observations at the CFHT
in 1996 June, at the Mont Mégantic 1.6 m telescope in 1996
August, at the MDM 1.3 m telescope in 1996 November, and
at the Mount Bigelow 1.6 m telescope in 1997 March. A total
of 32 sdB stars have been found “constant,” in the sense that
we rule out white-light brightness variations with amplitudes
larger than 1 millimag (or less, depending on the telescope and
brightness of the target) in the 20–3600 s period window for
these objects. Their distribution in the T<sub>eff</sub>–log g plane is shown
in Figure 3, where we have also indicated the position of PG
1047+003 as well as those of the previously known EC 14026

<figure><img src="image_129.png" /><figcaption>FIG. 3.—The current distribution in the T<sub>eff</sub>-log g diagram of the sdB stars observed in our ongoing survey. The small filled circles represent the 32 stars found nonvariable. The location of PG 1047+003 as well as those of the four previously known EC 14026 stars are indicated by a dot surrounded by a small circle. A number identifies each of the pulsators.</figcaption></figure>

stars according to O'Donoghue et al. (1997a). With due
caution associated with the uncertainties in the individual
determinations of the atmospheric parameters of sdB stars,
coupled with the fact that estimates of these parameters suffer
from systematic differences from one study to another (see
Wesemael et al. 1997), there is already a hint that both
pulsators and nonpulsators may coexist in the same part of the
T<sub>eff</sub>-log g diagram. If confirmed, this coexistence may not be so
surprising. Indeed, it is well known that sdB stars are all
chemically peculiar. Their abundance anomalies are widely
believed to be caused by diffusion processes coupled to weak
stellar winds or, possibly, accretion in the rare close binaries
(see, e.g., Fontaine & Chayer 1997, and references therein).
Following Charpinet et al. (1997), it is then possible to
envision the possibility that two stars with similar values of T<sub>eff</sub>
and log g may suffer from mass loss at different rates. In that
case, the star undergoing a relatively weak wind may retain
enough iron in its levitating reservoir to excite pulsation modes
(in the scenario proposed by Charpinet et al. 1996, 1997).
while the star suffering from a more severe mass loss rate may
have lost most of its metal and, therefore, be stable.

It is a real pleasure for us to acknowledge stimulating
discussions on EC 14026 stars with Darragh O'Donoghue,
Dave Kilkenny, and Bob Stobie. We also wish to thank the
directors, staff, and members of Time Allocation Committees
of the CFHT, Michigan-Dartmouth-MIT, Mont Mégantic, and
Steward Observatories. This work was supported in part by the
NSREC of Canada and by the fund FCAR (Québec).

REFERENCES

Allard, F. 1986, M.Sc. thesis, Université de Montréal

Allard, F., Wesemael, F., Fontaine, G., Bergeron, P., & Lamontagne, R. 1994,
AJ, 107, 1565

Allard, F., Wesemael, F., Lamontagne, R., Bergeron, P., & Fontaine, G. 1987,
in IAU Colloq. 65, The Second Conference on Faint Blue Stars, ed. A. G. D.
Philip, D. S. Hayes, & J. Liebert (Schenectady: Davis), 611

Billères, M., Fontaine, G., Charpinet, S., Brassard, P., & Urbain, F. 1997, in
Proc. 3d Conf. Faint Blue Stars, ed. A. G. D. Philip, J. Liebert, & R. A. Saffer
(Schenectady: Davis), in press

Charpinet, S., Fontaine, G., Brassard, P., Chayer, P., Rogers, F. J., Iglesias,
C. A., & Dorman, B. 1997, ApJ, in press

Charpinet, S., Fontaine, G., Brassard, P., & Dorman, B. 1996, ApJ, 471, L103
L84

BILLÈRES ET AL.

Dorman, B., Rood, R.T., & O'Connell, R.W. 1993, ApJ, 415, 596

Fontaine, G., & Chayer, P. 1997, in Proc. 3d Conf. Faint Blue Stars, ed. A. G. D. Philip, J. Liebert, & R. A. Saffer (Schenectady: Davis), in press

Green, R. F., Schmidt, M., & Liebert, J. 1986, ApJS, 61, 305

Kilkenny, D., O'Donoghue, D., Koen, C., Stobie, R. S., Lynas-Gray, A. E., & Kawaler, S. D. 1997a, in Proc. 3d Con. Faint Blue Stars, ed. A. G. D. Philip, J. Liebert, & R. A. Saffer (Schenectady: Davis), in press

Kilkenny, D., Koen, C., O'Donoghue, D., & Stobie, R. S. 1997b, MNRAS, 285, 640

Koen, C., Kilkenny, D., O'Donoghue, D., Van Wyk, F., & Stobie, R. S. 1997, MNRAS, 285, 645

Moehler, S., Heber, U., & de Boer, K. S. 1990, A&A, 239, 265

O'Donoghue, D., Koen, C., Kilkenny, D., Stobie, R. S., Lynas-Gray, A. E., & Kawaler, S. D. 1997a, in Proc. 10th European Workshop on White Dwarfs, ed. J. Isern, M. Hernanz, & E. Garcia-Berro (Dordrecht: Kluwer), in press

O'Donoghue, D., Lynas-Gray, A. E., Kilkenny, D., Stobie, R. S., & Koen, C. 1997b, MNRAS, 285, 657

Saffer, R. A. 1991, Ph.D. thesis, Univ. of Arizona

Saffer, R. A., Bergeron, P., Koester, D., & Liebert, J. 1994, ApJ, 432, 351

Stobie, R. S., Kawaler, S. D., Kilkenny, D., O'Donoghue, D., & Koen, C. 1997, MNRAS, 285, 651

Thejll, P., Theissen, A., & Jiminez, R. 1994, A&A, 292, 457

Wesemael, F., Allard, F., Lamontagne, R., Fontaine, G., & Bergeron, P. 1997, in Proc. 3d Conf. Faint Blue Stars, ed. A. G. D. Philip, J. Liebert, & R. A. Saffer (Schenectady: Davis), in press

Wesemael, F., Fontaine, G., Bergeron, P., Lamontagne, R., & Green, R. F. 1992, AJ, 104, 203
THE ASTROPHYSICAL JOURNAL, 494:L75–L78, 1998 February 10

© 1998. The American Astronomical Society. All rights reserved. Printed in U.S.A.

DETECTION OF *p*-MODE PULSATIONS IN THE HOT SUBDWARF B STAR KPD 2109+4401

M. BILLÈRES,<fnref n="1" /> G. FONTAINE, P. BRASSARD, AND S. CHARPINET

Département de Physique, Université de Montréal, C. P. 6128, Succ. Centre-Ville, Montréal, Québec H3C 3J7, Canada;
billeres@astro.umontreal.ca, fontaine@astro.umontreal.ca, brassard@astro.umontreal.ca, charpinet@astro.umontreal.ca

JAMES LIEBERT

Steward Observatory, University of Arizona, Tucson, AZ 85721; liebert@as.arizona.edu

R. A. SAFFER

Department of Astronomy and Astrophysics, Villanova University, Villanova, PA 19085; saffer@ast.vill.edu

P. BERGERON

Lockheed Martin Canada, 6111 Royalmount Avenue, Montréal, Québec H4P 1K6, Canada; pierre.bergeron@lmco.com

AND

G. VAUCLAIR

Observatoire Midi-Pyrénées, 14 Avenue E. Belin, F-31400, Toulouse, France; gerardv@obs-mip.fr

Received 1997 October 29; accepted 1997 December 10; published 1998 January 30

ABSTRACT

We report the discovery of multiperiodic luminosity variations in the hot B subdwarf KPD 2109+4401. At least five periodicities are seen in the light curve, from 182.5 to 198.4 s. The largest oscillation has an amplitude ≈8.6 mmag in white light and a period 196.3 s. Our model atmosphere analysis of the time-averaged optical spectrum of KPD 2109+4401 indicates that this star has $T_{eff} \approx 31,200$ K and $log g \approx 5.84$. A comparison with pulsation periods computed from stellar models having similar atmospheric parameters implies that the observed brightness variations must be identified with low-order radial and nonradial (*p* and *f*) pulsation modes. The overall similarity of periods and derived stellar parameters shows that KPD 2109+4401 is a genuine member of the EC 14026 class, the most recent family of pulsators uncovered in the field of asteroseismology. We note, however, that KPD 2109+4401 is slightly but significantly cooler than the previously known members of this class, thus widening significantly the empirical instability strip. Interestingly, theory predicts that a cooler EC 14026 pulsator (of a given surface gravity) should show larger excited periods, and this is indeed what KPD 2109+4401 shows, with periods reaching values as large as 198 s, not seen previously in other pulsators of the class. We also report on the current status of our ongoing survey to search for additional EC 14026 pulsators.

*Subject headings:* stars: interiors — stars: oscillations — subdwarfs

1. INTRODUCTION

We have recently embarked on a survey to search for pul-
sating subdwarf B (dB) stars. This was initially motivated by
the theoretical prediction of Charpinet et al. (1996) that a sub-
class of dB stars should show rapid luminosity variations due
to pulsational instabilities caused by a κ mechanism associated
with iron ionization. Further impetus came from the inde-
pendent observational efforts at the South African Astronom-
ical Observatory, which led to the discovery of the first real
pulsating sdB stars (Kilkenny et al. 1997; Koen et al. 1997;
Stobie et al. 1997; O'Donoghue et al. 1997). The study of these
pulsators (dubbed EC 14026 stars, after the prototype) offers
the tantalizing possibility of exploiting the full potential of
asteroseismology to investigate the sdB phase of stellar evo-
lution, a phase that has remained relatively neglected until re-
cently (see Dorman 1995 for a review on this).

The goals of our survey are (1) to search for EC 14026
pulsators in the northern hemisphere, (2) to investigate the
extent of this phenomenon in the $T_{eff}-\log g$ plane, and (3) to
determine the degree of purity of the empirical instability strip
in that diagram. In order to carry out that task successfully, we
rely on a list of target stars for which atmospheric parameters
have been derived on the basis of model atmosphere fits to
optical spectra. While reliable estimates of the effective tem-
peratures of sdB stars can be obtained through optical photom-

etry or ultraviolet energy distributions, it is well known that
such data are insensitive to surface gravity. Instead, the best
technique to infer the surface gravities remains Balmer line
fitting, which also provides good estimates of the effective
temperatures. Our sample of target stars was therefore selected
on the basis of this technique, picking candidates with atmos-
pheric parameters provided in the spectroscopic analyses of
Allard (1986; see also Wesemael et al. 1997), Moehler, Heber,
& de Boer (1990), Saffer et al. (1994), and Saffer & Liebert
(1998).

Our ongoing survey has led recently to a first positive result
with the detection of *p*-mode pulsations in the sdB star PG
1047+003 (Billères et al. 1997; see also O'Donoghue et al.
1998). With atmospheric parameters $T_{eff} \sim 34,370$ K and
$\log g \sim 5.7$, and periods in the range 104.2–161.9 s, PG
1047+003 appears to be a typical EC 14026 pulsator, at least
as judged with respect to the four previously known objects
of the class. It differs from the others, however, in that it does
not have a known main-sequence late-type companion. We
believe that this difference is not essential, since the EC 14026
phenomenon has been shown to be intrinsic to the sdB star
itself (see Charpinet et al. 1996, 1997; Fontaine et al. 1997)
and is not related to the possible presence of a companion
(which is relatively frequent for sdB objects according to Allard
et al. 1987, 1994; see also Green, Liebert, & Saffer 1997).

During the course of a recent observing run, we detected
multiperiodic luminosity variations characteristic of an EC
14026 pulsator in the light curve of the sdB star KPD

<footnote n="1">¹ Also Observatoire Midi-Pyrénées, 14 Avenue E. Belin, F-31400, Toulouse, France.</footnote>