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Table 5
Zonal harmonic coefficients γ2n and residuals ΔR(θ)/R(θ)
M = 5 M⊙ (ZAMS) | ΔR(θ)/R(θ) = [R(θ)model − R(θ)δn] /R(θ)model × 104 | ||||||||||||||
β | Ωo/Ωcr | γ 2 | γ 4 | γ 6 | γ 8 | 0° | 5° | 15° | 30° | 40° | 50° | 60° | 70° | 80° | 90° |
2 | 1.0 | −0.119 × 10-2 | −0.595 × 10-3 | 0.486 × 10-4 | −0.26 × 10-3 | 9 | 10 | 12 | 20 | 30 | 38 | 43 | 45 | 48 | 49 |
2.0 | 0.149 × 10-2 | −0.318 × 10-3 | 0.274 × 10-3 | −0.104 × 10-3 | -8 | -7 | 3 | 17 | 29 | 35 | 39 | 42 | 46 | 47 | |
3.0 | 0.308 × 10-2 | 0.829 × 10-4 | 0.305 × 10-4 | −0.729 × 10-5 | -3 | -4 | -5 | 6 | 17 | 25 | 31 | 35 | 37 | 39 | |
3.5 | 0.302 × 10-2 | −0.986 × 10-4 | 0.103 × 10-4 | −0.497 × 10-6 | -115 | -107 | -45 | 12 | 21 | 26 | 38 | 44 | 47 | 48 | |
3.9 | 0.321 × 10-2 | −0.868 × 10-4 | 0.153 × 10-5 | −0.112 × 10-7 | -2651 | -2416 | -907 | 242 | 19 | 133 | 128 | 127 | 126 | 127 | |
4 | 1.0 | −0.148 × 10-2 | −0.757 × 10-3 | 0.660 × 10-4 | −0.374 × 10-3 | 1 | 1 | 13 | 20 | 31 | 40 | 44 | 45 | 48 | 49 |
2.0 | 0.761 × 10-3 | −0.525 × 10-3 | 0.165 × 10-3 | −0.138 × 10-3 | 2 | 3 | 9 | 20 | 31 | 38 | 42 | 45 | 47 | 49 | |
3.0 | 0.360 × 10-2 | −0.398 × 10-3 | 0.260 × 10-3 | −0.525 × 10-4 | 20 | 17 | 1 | 19 | 30 | 36 | 40 | 44 | 46 | 48 | |
4.0 | 0.579 × 10-2 | −0.277 × 10-3 | 0.189 × 10-3 | −0.317 × 10-4 | -54 | -49 | -17 | 16 | 28 | 33 | 37 | 41 | 43 | 44 | |
5.0 | 0.636 × 10-2 | −0.771 × 10-4 | 0.101 × 10-4 | −0.125 × 10-5 | -12 | -13 | -9 | 5 | 18 | 26 | 31 | 35 | 37 | 38 | |
6.2 | 0.615 × 10-2 | −0.172 × 10-3 | 0.337 × 10-5 | −0.280 × 10-7 | -3801 | -3358 | -1152 | 277 | 51 | 63 | 57 | 54 | 53 | 52 | |
6 | 1.0 | −0.170 × 10-2 | −0.682 × 10-3 | −0.716 × 10-5 | −0.409 × 10-3 | 12 | 12 | 13 | 20 | 31 | 40 | 44 | 46 | 48 | 49 |
2.0 | 0.146 × 10-3 | −0.564 × 10-3 | 0.574 × 10-4 | −0.164 × 10-3 | 7 | 7 | 12 | 21 | 31 | 39 | 43 | 46 | 48 | 49 | |
4.0 | 0.549 × 10-2 | −0.429 × 10-3 | 0.246 × 10-3 | −0.309 × 10-4 | -32 | -27 | -1 | 21 | 32 | 37 | 42 | 45 | 48 | 49 | |
5.0 | 0.773 × 10-2 | −0.403 × 10-3 | 0.230 × 10-3 | −0.359 × 10-4 | -70 | -63 | -18 | 19 | 31 | 36 | 40 | 44 | 47 | 48 | |
7.0 | 0.905 × 10-2 | −0.284 × 10-3 | 0.789 × 10-4 | −0.923 × 10-5 | -82 | -76 | -30 | 14 | 17 | 19 | 37 | 44 | 48 | 49 | |
8.4 | 0.885 × 10-2 | −0.233 × 10-3 | 0.404 × 10-5 | −0.297 × 10-7 | -6993 | -6158 | -2102 | 512 | 54 | 23 | 8 | 0 | 5 | 7 | |
8 | 1.0 | −0.180 × 10-2 | −0.760 × 10-3 | −0.140 × 10-4 | −0.417 × 10-3 | 12 | 12 | 14 | 20 | 31 | 40 | 44 | 46 | 48 | 49 |
3.0 | 0.199 × 10-2 | −0.556 × 10-3 | 0.223 × 10-3 | −0.138 × 10-3 | -1 | 0 | 10 | 22 | 32 | 39 | 43 | 46 | 49 | 50 | |
5.0 | 0.720 × 10-2 | −0.453 × 10-3 | 0.260 × 10-3 | −0.332 × 10-4 | -45 | -38 | -4 | 22 | 33 | 39 | 43 | 47 | 50 | 50 | |
7.0 | 0.112 × 10-1 | −0.459 × 10-3 | 0.139 × 10-3 | −0.170 × 10-4 | -116 | -106 | -41 | 18 | 32 | 37 | 42 | 46 | 49 | 50 | |
9.0 | 0.113 × 10-1 | −0.331 × 10-3 | 0.101 × 10-4 | −0.201 × 10-6 | -258 | -228 | -63 | 17 | 21 | 31 | 36 | 41 | 44 | 45 | |
10.0 | 0.106 × 10-1 | −0.351 × 10-3 | 0.974 × 10-5 | −0.126 × 10-6 | -2562 | -2316 | -712 | 180 | 19 | 23 | 29 | 42 | 47 | 49 | |
M = 5 M⊙ (TAMS) | |||||||||||||||
2 | 1.0 | −0.276 × 10-2 | −0.818 × 10-3 | 0.147 × 10-3 | −0.413 × 10-3 | 11 | 11 | 14 | 24 | 37 | 47 | 52 | 54 | 58 | 60 |
2.0 | −0.231 × 10-2 | −0.433 × 10-3 | 0.325 × 10-3 | −0.104 × 10-3 | -13 | -11 | 3 | 21 | 34 | 41 | 45 | 49 | 53 | 54 | |
3.0 | −0.839 × 10-3 | −0.114 × 10-3 | 0.603 × 10-4 | −0.564 × 10-5 | -96 | -88 | -34 | 17 | 28 | 46 | 53 | 58 | 60 | 61 | |
3.5 | −0.341 × 10-3 | 0.311 × 10-4 | −0.104 × 10-5 | 0.133 × 10-7 | 328 | 284 | 80 | -9 | 77 | 96 | 104 | 108 | 110 | 111 | |
4 | 1.0 | −0.278 × 10-2 | −0.928 × 10-3 | 0.920 × 10-4 | −0.487 × 10-3 | 13 | 13 | 15 | 24 | 38 | 48 | 53 | 55 | 58 | 60 |
3.0 | −0.186 × 10-2 | −0.506 × 10-3 | 0.276 × 10-3 | −0.359 × 10-4 | -28 | -23 | 1 | 22 | 34 | 40 | 44 | 48 | 51 | 52 | |
4.0 | −0.709 × 10-3 | −0.413 × 10-3 | 0.184 × 10-3 | −0.191 × 10-4 | -131 | -113 | -27 | 23 | 32 | 44 | 48 | 50 | 52 | 53 | |
5.0 | −0.589 × 10-3 | 0.131 × 10-3 | −0.831 × 10-5 | 0.193 × 10-6 | 475 | 398 | 76 | 22 | 58 | 70 | 76 | 81 | 84 | 85 | |
5.5 | 0.141 × 10-3 | 0.651 × 10-5 | −0.129 × 10-6 | 0.726 × 10-9 | 789 | 682 | 128 | 86 | 108 | 116 | 120 | 122 | 123 | 124 | |
5.7 | 0.417 × 10-3 | −0.225 × 10-5 | 0.604 × 10-8 | −0.609 × 10-11 | -3913 | -3268 | -518 | 80 | 92 | 106 | 105 | 105 | 104 | 104 | |
6 | 1.0 | −0.280 × 10-2 | −0.938 × 10-3 | 0.377 × 10-4 | −0.502 × 10-3 | 14 | 14 | 16 | 25 | 38 | 49 | 53 | 55 | 58 | 60 |
3.0 | −0.206 × 10-2 | −0.674 × 10-3 | 0.211 × 10-3 | −0.518 × 10-4 | -9 | -6 | 9 | 24 | 35 | 43 | 48 | 52 | 54 | 55 | |
4.0 | −0.149 × 10-2 | −0.507 × 10-3 | 0.253 × 10-3 | −0.166 × 10-4 | -48 | -39 | -1 | 23 | 33 | 39 | 43 | 47 | 49 | 50 | |
5.0 | −0.504 × 10-3 | −0.482 × 10-3 | 0.207 × 10-3 | −0.206 × 10-4 | -160 | -136 | -24 | 24 | 32 | 43 | 47 | 49 | 51 | 52 | |
6.0 | −0.489 × 10-3 | 0.129 × 10-3 | −0.609 × 10-5 | 0.605 × 10-7 | 116 | 89 | 9 | 25 | 46 | 55 | 61 | 65 | 68 | 69 | |
7.0 | 0.504 × 10-3 | 0.222 × 10-5 | 0.156 × 10-6 | 0.144 × 10-8 | 297 | 256 | 39 | 44 | 59 | 64 | 66 | 68 | 68 | 69 | |
8 | 1.0 | −0.282 × 10-2 | −0.899 × 10-3 | −0.708 × 10-4 | −0.480 × 10-3 | 16 | 16 | 17 | 25 | 37 | 49 | 54 | 56 | 59 | 60 |
3.0 | −0.219 × 10-2 | −0.757 × 10-3 | 0.174 × 10-3 | −0.118 × 10-3 | -0 | 2 | 12 | 25 | 37 | 45 | 50 | 53 | 56 | 57 | |
5.0 | −0.119 × 10-2 | −0.505 × 10-3 | 0.259 × 10-3 | −0.194 × 10-4 | -72 | -57 | -3 | 23 | 33 | 39 | 43 | 46 | 48 | 49 | |
6.0 | −0.306 × 10-3 | −0.508 × 10-3 | 0.216 × 10-3 | −0.214 × 10-4 | -199 | -164 | -23 | 25 | 33 | 43 | 47 | 49 | 50 | 51 | |
8.0 | 0.434 × 10-4 | 0.886 × 10-4 | −0.445 × 10-5 | 0.727 × 10-7 | 932 | 755 | 74 | 38 | 54 | 61 | 65 | 69 | 71 | 71 | |
9.0 | 0.757 × 10-3 | −0.292 × 10-5 | 0.526 × 10-8 | −0.346 × 10-11 | -14389 | -11830 | -1088 | -162 | -153 | -150 | -150 | -150 | -150 | -150 | |
M = 15 M⊙ (ZAMS) | |||||||||||||||
2 | 1.0 | −0.276 × 10-3 | −0.598 × 10-3 | 0.139 × 10-3 | −0.325 × 10-3 | 9 | 9 | 11 | 19 | 30 | 38 | 42 | 44 | 47 | 49 |
2.0 | 0.438 × 10-2 | −0.309 × 10-3 | 0.240 × 10-3 | −0.877 × 10-4 | -8 | -6 | 3 | 17 | 29 | 35 | 39 | 43 | 46 | 47 | |
3.0 | 0.743 × 10-2 | −0.638 × 10-4 | 0.320 × 10-4 | −0.766 × 10-5 | -4 | -5 | -4 | 7 | 18 | 26 | 33 | 37 | 40 | 41 | |
3.5 | 0.664 × 10-2 | −0.201 × 10-3 | 0.147 × 10-4 | −0.773 × 10-6 | -98 | -90 | -35 | 12 | 20 | 22 | 32 | 41 | 45 | 46 | |
4.0 | 0.587 × 10-2 | −0.152 × 10-3 | 0.250 × 10-5 | −0.170 × 10-7 | -602 | -5395 | -2079 | 609 | 389 | 342 | 337 | 336 | 336 | 337 | |
4 | 1.0 | −0.876 × 10-3 | −0.666 × 10-3 | 0.103 × 10-3 | −0.408 × 10-3 | 10 | 10 | 12 | 20 | 31 | 39 | 43 | 45 | 47 | 49 |
2.0 | 0.295 × 10-2 | −0.478 × 10-3 | 0.218 × 10-3 | −0.163 × 10-3 | 1 | 2 | 9 | 20 | 31 | 38 | 42 | 45 | 47 | 49 | |
4.0 | 0.118 × 10-1 | −0.478 × 10-3 | 0.190 × 10-3 | −0.329 × 10-4 | -50 | -46 | -17 | 16 | 29 | 35 | 39 | 43 | 46 | 47 | |
5.5 | 0.128 × 10-1 | −0.444 × 10-3 | 0.177 × 10-4 | −0.489 × 10-6 | -190 | -168 | -51 | 19 | 18 | 25 | 32 | 37 | 40 | 41 | |
6.2 | 0.110 × 10−1 | −0.416 × 10-3 | 0.137 × 10-4 | −0.213 × 10-6 | -2217 | -210 | -688 | 208 | 89 | 8 | 30 | 41 | 47 | 49 | |
6.4 | 0.107 × 10-1 | −0.337 × 10-3 | 0.753 × 10-5 | −0.731 × 10-7 | -5925 | -5253 | -2038 | 562 | 240 | -50 | -11 | 1 | 8 | 10 | |
6 | 1.0 | −0.120 × 10-2 | −0.729 × 10-3 | 0.195 × 10-4 | 0.375 × 10-3 | 11 | 11 | 13 | 20 | 31 | 39 | 43 | 45 | 47 | 48 |
3.0 | 0.621 × 10-2 | −0.473 × 10-3 | 0.281 × 10-3 | −0.943 × 10-4 | -9 | -7 | 6 | 21 | 32 | 39 | 43 | 45 | 48 | 50 | |
5.0 | 0.151 × 10-1 | −0.666 × 10-3 | 0.232 × 10-3 | −0.375 × 10-4 | -66 | -60 | -19 | 19 | 33 | 39 | 42 | 47 | 49 | 51 | |
7.0 | 0.187 × 10-1 | −0.805 × 10-3 | 0.421 × 10-4 | −0.161 × 10-5 | -268 | -233 | -67 | 24 | 26 | 33 | 40 | 46 | 49 | 51 | |
8.0 | 0.167 × 10-1 | −0.722 × 10-3 | 0.296 × 10-4 | 0.623 × 10-6 | -1477 | -1329 | -450 | 130 | 54 | 23 | 34 | 41 | 45 | 47 | |
8.4 | 0.155 × 10-1 | −0.620 × 10-3 | 0.205 × 10-4 | −0.319 × 10-6 | -3411 | -3002 | -1074 | 303 | 137 | 7 | 30 | 40 | 45 | 47 | |
8 | 1.0 | −0.144 × 10-2 | −0.726 × 10-3 | 0.267 × 10-4 | −0.402 × 10-3 | 11 | 11 | 13 | 20 | 31 | 39 | 43 | 45 | 47 | 49 |
2.0 | 0.107 × 10-2 | −0.598 × 10-3 | 0.167 × 10-3 | −0.300 × 10-3 | 7 | 8 | 12 | 21 | 32 | 40 | 43 | 45 | 48 | 49 | |
4.0 | 0.922 × 10-2 | −0.464 × 10-3 | 0.261 × 10-3 | −0.413 × 10-4 | -20 | -17 | 4 | 22 | 33 | 40 | 44 | 47 | 50 | 51 | |
6.0 | 0.177 × 10-1 | −0.790 × 10-3 | 0.237 × 10-3 | −0.358 × 10-4 | -81 | -72 | -23 | 21 | 36 | 41 | 46 | 50 | 53 | 54 | |
8.0 | 0.229 × 10-1 | −0.110 × 10-2 | 0.781 × 10-4 | −0.450 × 10-5 | -243 | -219 | -64 | 24 | 33 | 40 | 46 | 53 | 57 | 58 | |
9.6 | 0.219 × 10-1 | −0.106 × 10-2 | 0.488 × 10-4 | −0.119 × 10-5 | -1380 | -1237 | -394 | 119 | 56 | 32 | 43 | 51 | 56 | 58 | |
M = 15 M⊙ (TAMS) | |||||||||||||||
2 | 1.0 | 0.833 × 10-2 | 0.230 × 10-2 | −0.373 × 10-3 | 0.111 × 10-2 | -33 | -33 | -44 | -73 | -113 | -144 | -158 | -167 | -176 | -182 |
2.0 | 0.797 × 10-2 | 0.105 × 10-2 | −0.102 × 10-2 | 0.344 × 10-3 | 39 | 32 | -11 | -71 | -116 | -1412 | -157 | -171 | -183 | -189 | |
3.0 | 0.657 × 10-2 | −0.890 × 10-3 | 0.950 × 10-5 | 0.539 × 10-5 | -176 | -141 | -12 | -11 | -82 | -120 | -141 | -156 | -166 | -169 | |
3.5 | 0.260 × 10-2 | −0.138 × 10-3 | 0.343 × 10-5 | −0.319 × 10-7 | -3014 | -2614 | -735 | -79 | -78 | -76 | -78 | -80 | -82 | -83 | |
4 | 1.0 | 0.836 × 10-2 | 0.251 × 10-2 | −0.670 × 10-4 | 0.123 × 10-2 | -39 | -40 | -48 | -74 | -112 | -144 | -159 | -166 | -175 | -180 |
2.0 | 0.820 × 10-2 | 0.205 × 10-2 | −0.872 × 10-3 | 0.628 × 10-3 | -1 | -6 | -34 | -78 | -117 | -144 | -157 | -168 | -178 | -183 | |
3.0 | 0.774 × 10-2 | 0.125 × 10-2 | −0.101 × 10-2 | 0.169 × 10-3 | 102 | 84 | -1 | -79 | -121 | -144 | -158 | -171 | -182 | -186 | |
4.0 | 0.708 × 10-2 | 0.197 × 10-3 | −0.545 × 10-3 | 0.780 × 10-4 | 330 | 304 | 105 | -67 | -122 | -136 | -150 | -163 | -174 | -177 | |
5.0 | 0.518 × 10-2 | −0.515 × 10-3 | 0.240 × 10-4 | −0.414 × 10-6 | -3170 | -2691 | -481 | -53 | -90 | -103 | -115 | -126 | -132 | -135 | |
5.5 | 0.173 × 10-2 | −0.236 × 10-4 | 0.145 × 10-6 | −0.323 × 10-9 | -9898 | -8248 | -1336 | -187 | -175 | -173 | -174 | -175 | -177 | -177 | |
6 | 1.0 | 0.836 × 10-2 | 0.270 × 10-2 | −0.959 × 10-4 | 0.142 × 10-2 | -41 | -41 | -48 | -74 | -113 | -145 | -159 | -166 | -174 | -179 |
3.0 | 0.805 × 10-2 | 0.175 × 10-2 | −0.837 × 10-3 | 0.176 × 10-3 | 41 | 29 | -27 | -82 | -119 | -143 | -158 | -169 | -178 | -182 | |
4.0 | 0.765 × 10-2 | 0.128 × 10-2 | −0.105 × 10-2 | 0.134 × 10-3 | 181 | 151 | 10 | -84 | -125 | -146 | -159 | -171 | -181 | -186 | |
5.0 | 0.736 × 10-2 | 0.417 × 10-3 | −0.650 × 10-3 | 0.889 × 10-4 | 433 | 384 | 97 | -79 | -126 | -138 | -152 | -165 | -175 | -179 | |
6.0 | 0.676 × 10-2 | −0.873 × 10-3 | 0.531 × 10-4 | −0.115 × 10-5 | -1319 | -1044 | -112 | -63 | -97 | -115 | -132 | -147 | -157 | -160 | |
7.0 | 0.258 × 10-2 | −0.515 × 10-4 | 0.467 × 10-6 | −0.153 × 10-8 | -10115 | -8462 | -852 | -213 | -209 | -212 | -214 | -219 | -222 | -223 | |
8 | 1.0 | 0.840 × 10-2 | 0.276 × 10-2 | −0.536 × 10-4 | 0.145 × 10-2 | -42 | -42 | -49 | -73 | -113 | -145 | -160 | -166 | -174 | -179 |
3.0 | 0.818 × 10-2 | 0.200 × 10-2 | −0.749 × 10-3 | 0.346 × 10-3 | 14 | 5 | -37 | -82 | -118 | -142 | -157 | -168 | -177 | -181 | |
5.0 | 0.768 × 10-2 | 0.120 × 10-2 | −0.104 × 10-2 | 0.136 × 10-3 | 267 | 218 | 20 | -87 | -126 | -146 | -159 | -171 | -181 | -186 | |
6.0 | 0.757 × 10-2 | 0.398 × 10-3 | −0.615 × 10-3 | 0.797 × 10-4 | 499 | 441 | 88 | -84 | -126 | -138 | -152 | -165 | -175 | -179 | |
7.0 | 0.725 × 10-2 | −0.891 × 10-3 | 0.435 × 10-4 | −0.301 × 10-6 | -893 | -679 | -40 | -73 | -101 | -120 | -138 | -153 | -164 | -167 | |
8.5 | 0.342 × 10-2 | −0.704 × 10-4 | 0.645 × 10-6 | −0.211 × 10-8 | -5137 | -12194 | -990 | -271 | -273 | -278 | -284 | -290 | -294 | -295 |
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