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O raspredelenii bol'shih poluosei orbit vnesolnechnyh planet
<< 4. Funkcii raspredeleniya | Oglavlenie | 6. Zaklyuchenie >>

5. Chislennye rezul'taty

Issleduem oblast' obnaruzhimosti . Ee harakternyi razmer zadaetsya parametrom , vhodyashim v uravnenie (3). Pereidem v vyrazhenii dlya k bezrazmernym massam , , , gde ,  - massy Solnca i Yupitera sootvetstvenno. Poluchaem

(14)

Zdes'  - geliocentricheskaya krugovaya skorost' tela nulevoi massy na rasstoyanii . Za poslednee razumno vzyat' astronomicheskuyu edinicu, togda  m/s (s izbytochnoi tochnost'yu). Issleduem oblast' znachenii parametra v zavisimosti ot mass i . Porogovuyu velichinu luchevoi skorosti polozhim ravnoi 10 m/s, chto sootvetstvuet tochnosti sovremennyh nablyudenii. Zametim, chto dlya bol'shinstva otkrytyh k nastoyashemu vremeni planet luchevye skorosti prevyshayut  m/s. V tabl.1 dany znacheniya privedennoi massy (sverhu) i parametra obnaruzhimosti (snizu) dlya razlichnyh (pervyi stolbec, v skobkah privedeny nazvaniya planet Solnechnoi sistemy, imeyushih blizkie k massy) i s ukazaniem sootvetstvuyushego spektral'nogo klassa dlya zvezd glavnoi posledovatel'nosti (verhnyaya stroka). Iz tabl.1 vidno, chto na bol'shom (neskol'ko a. e.) udalenii ot central'noi zvezdy v nastoyashee vremya mozhno nablyudat' tol'ko planety s massami . Esli ogranichit' zonu, v kotoroi mogut nahodit'sya planety, radiusom 100 a. e., to statisticheski znachimoe raspredelenie bol'shih poluosei orbit mozhno poluchit' tol'ko dlya planet s massami . Chtoby postroit' analogichnoe raspredelenie dlya planet s massami Yupitera, neobhodimy nablyudeniya s  m/c. V etom sluchae, kak sleduet iz (14), znacheniya uvelichatsya v 9 raz po sravneniyu s privedennymi v tabl. 1. No etogo nedostatochno dlya postroeniya raspredeleniya planet s massami Saturna. Planety s massami Urana i Neptuna mozhno obnaruzhit' tol'ko v neposredstvennoi blizosti ot poverhnosti zvezdy (radius Solnca - 0.005 a. e.). Planety zemnogo tipa dannym metodom zafiksirovat' nevozmozhno.

Tablica 1. Znacheniya privedennoi massy planety i kriticheskogo rasstoyaniya (a.e.)
v zavisimosti ot i pri m/s


  0.25 0.50 0.75 1.00 1.25 1.50
  M9 M0 K0 G2 F6 F1
10
  3000 2000 1000 800 600 500
1
(Yupiter) 30 20 10 8 6 5
0.3
(Saturn) 3 1 1 0.7 0.6 0.5
0.05
(Uran, Neptun) 0.08 0.04 0.03 0.02 0.02 0.01
0.003
(Venera, Zemlya)
0.0002
(Merkurii)


Privedennaya massa  - velichina nenablyudaemaya. Neposredstvenno iz nablyudenii vyvoditsya

V predpolozhenii, chto vse orientacii vektora ploshadei ravnoveroyatny, srednee znachenie massy planety ravno

Obratim vnimanie, chto ispol'zovanie vmesto mnozhitelya

nekorrektno i privodit k sistematicheskomu zanizheniyu mass na 20%:

V tabl. 2 privedeny svedeniya o 76 izvestnyh na 18 oktyabrya 2001 g. vnesolnechnyh planetah, dvizhushihsya vokrug zvezd glavnoi posledovatel'nosti. Dannye o massah planet, bol'shih poluosyah i ekscentrisitetah ih orbit, amplitudah kolebanii luchevoi skorosti zvezdy [5]

(15)

gde  - period obrasheniya planety, vzyaty iz kataloga [6]. Svedeniya o massah zvezd bralis' iz rabot [6,7]. Dlya zvezd, massy kotoryh ne udalos' naiti v ukazannyh istochnikah, velichiny byli vychisleny s pomosh'yu tret'ego zakona Keplera na osnove dannyh [6] o periodah i bol'shih poluosyah orbit planet. V tablice eti dannye otmecheny kursivom. Na osnove svedenii o massah i byli vychisleny privedennye massy , rasstoyaniya :

(16)

i veroyatnosti obnaruzheniya planet. Znacheniya privedeny dlya porogovyh velichin luchevoi skorosti i 3 m/s, sootvetstvuyushih tochnosti sovremennyh nablyudenii i blizkih k predel'no realizuemoi tochnosti metoda. Parametr pokazyvaet, na kakom maksimal'nom udalenii ot zvezdy mozhet byt' zaregistrirovana planeta, dvizhushayasya po krugovoi orbite, pri zadannom . Veroyatnosti vychisleny dlya  m/s. Analiz tabl. 2 pokazyvaet, chto bol'shie poluosi prakticheski vseh otkrytyh planet sushestvenno men'she dazhe pri  m/s. Veroyatnosti otkrytiya dlya podavlyayushego bol'shinstva planet prevyshayut 0.90. Isklyuchenie sostavlyayut HD 4208 (),  Eri (), HD 83443 c (), 47 UMa c (), HD 16141 (). Eto legko ob'yasnit'. Vo-pervyh, v nablyudaemyh sistemah znacheniya luchevoi skorosti pochti vsegda prevoshodyat rassmatrivaemye znacheniya v neskol'ko raz, a inogda i na poryadki. V takih sistemah planety dolzhny byt' massivnymi i (ili) dvigat'sya na nebol'shom rasstoyanii ot zvezdy. Obnaruzhenie zvezd s malymi pozvolit naiti bolee dalekie planety. Kak vidno iz tabl. 2, kogda znacheniya i blizki, raznosti mezhdu i maly (naprimer, HD 83443 c, HD 16141, 47 UMa c,  Eri) - planety nablyudayutsya na sravnimyh s rasstoyaniyah. Iz 76 vnesolnechnyh planet tol'ko u 22 privedennaya massa men'she privedennoi massy Yupitera . Dlya bol'shinstva iz etih planet . Ispol'zuya tabl.1, legko ocenit' maksimal'nye rasstoyaniya, na kotoryh eti planety mogli byt' otkryty: oni budut v 9-81 raz men'she ukazannyh v tablice. Dlya planety s massoi Yupitera eto rasstoyanie sostavit 0.1-0.9 a. e., chto soglasuetsya s dannymi tabl. 2. Vo-vtoryh, sushestvennym selektivnym faktorom yavlyaetsya vremya, zatrachennoe na nablyudenie konkretnoi zvezdy. Postepenno etot faktor budet oslabevat', no spustya 7 let posle nachala regulyarnyh nablyudenii prezhdevremenno ozhidat' otkrytiya planet s bol'shimi poluosyami 20-30 a. e. i periodami orbital'nogo dvizheniya neskol'ko desyatkov i soten let. V nastoyashee vremya nel'zya sdelat' nadezhnye statisticheskie vyvody o nablyudaemom raspredelenii elementov orbit vnesolnechnyh planetnyh sistem, poskol'ku tochnost' sovremennyh nablyudenii
nedostatochna dlya obnaruzheniya planet s massoi na rasstoyaniyah, bol'shih neskol'kih astronomicheskih edinic. Effekt selekcii, opredelyaemyi v pervuyu ochered' tochnost'yu nablyudenii, privodit k tomu, chto planety otkryvayutsya na sravnitel'no nebol'shih rasstoyaniyah - ot zvezd (sm. tabl. 2). Primer Solnechnoi sistemy pokazyvaet, chto maksimal'nye radiusy orbit planet mogut prevyshat' . Esli zadat' raspredelenie , to mozhno svyazat' mezhdu soboi srednie znacheniya radiusov planetnyh orbit:  - poluchaemoe iz nablyudenii i sootvetstvuyushee plotnosti ;  - istinnoe, opredelyaemoe raspredeleniem , a takzhe , , poluchaemye otbrasyvaniem vseh planet s .

(17)


Tablica 2. Parametry vnesolnechnyh planetnyh sistem. Rasstoyaniya dany v a.e., skorosti - v m/s


Zvezda
HD 83443 b 0.82 0.34 0.038 0.08 57.0 1.1 12.6 0.98
HD 46375 1.00 0.25 0.041 0.02 35.2 0.5 5.6 0.96
HD 179949 1.24 0.93 0.045 0.00 112.0 5.7 62.8 1.00
HD 187123 1.00 0.54 0.042 0.01 72.0 2.4 26.2 0.99
Boo 1.20 4.14 0.047 0.02 474.0 115.7 1285.6 1.00
BD 1.10 0.48 0.046 0.05 60.6 1.7 18.9 0.99
HD 75289 1.05 0.46 0.048 0.00 54.0 1.6 18.1 0.99
HD 209458 1.03 0.63 0.046 0.02 82.0 3.1 34.7 0.99
51 Peg 0.98 0.46 0.052 0.01 55.2 1.7 19.4 0.99
And b 1.10 0.68 0.059 0.02 70.2 3.4 37.8 0.99
HD 68988 1.20 1.90 0.071 0.14 187.0 24.4 270.8 1.00
HD 168746 0.94 0.24 0.066 0.00 28.0 0.5 5.5 0.93
HD 217107 0.96 1.29 0.072 0.14 139.7 14.0 156.0 1.00
HD 162020 0.70 13.73 0.072 0.28 1813.0 2174.3 24158.8 1.00
HD 130322 0.79 1.15 0.092 0.05 115.0 13.6 150.7 1.00
HD 108147 1.06 0.35 0.098 0.56 37.0 0.9 10.4 0.95
HD 38529 1.40 0.77 0.129 0.27 53.6 3.4 38.2 0.98
55 Cnc 0.90 0.93 0.118 0.03 75.8 7.8 86.5 0.99
HD 13445 = GJ 86 0.79 4.23 0.117 0.04 379.0 183.5 2038.6 1.00
HD 195019 0.98 3.55 0.136 0.01 271.0 104.2 1157.5 1.00
HD 6434 1.00 0.48 0.154 0.30 37.0 1.9 20.8 0.96
HD 192263 0.75 0.81 0.152 0.22 68.2 7.1 78.7 0.99
HD 83443 c 0.79 0.17 0.174 0.42 14.0 0.3 3.3 0.64
GJ 876 c 0.32 0.56 0.130 0.27 81.0 7.9 88.2 0.99
CrB 1.00 0.99 0.224 0.07 61.3 7.9 88.2 0.99
HD 74156 b 1.05 1.55 0.276 0.65 108.0 18.5 205.4 0.99
HD 168443 b 0.84 7.64 0.295 0.53 470.0 562.9 6254.5 1.00
GJ 876 b 0.32 1.89 0.207 0.10 210.0 90.4 1004.8 1.00
HD 121504 1.02 0.89 0.317 0.13 45.0 6.3 70.1 0.97
HD 178911 b 0.90 6.46 0.326 0.14 343.0 373.9 4154.6 1.00
HD 16141 0.99 0.22 0.351 0.28 10.8 0.4 4.4 0.34
HD 114762 0.82 10.96 0.351 0.33 615.0 1186.7 13185.5 1.00
HD 80606 0.90 3.43 0.438 0.93 414.0 106.3 1180.7 1.00
70 Vir 1.10 7.42 0.482 0.40 316.2 405.5 4505.1 1.00
HD 52265 1.13 1.14 0.493 0.29 45.4 9.3 103.5 0.97
HD 1237 0.98 3.45 0.505 0.51 164.0 98.4 1093.2 1.00
HD 37124 0.91 1.13 0.547 0.31 48.0 11.4 126.3 0.98
HD 82943 c 1.05 0.88 0.728 0.54 34.0 6.0 66.5 0.94
HD 8574 1.10 2.23 0.756 0.40 76.0 36.6 406.5 0.99
HD 169830 1.40 2.95 0.823 0.34 83.0 50.3 559.1 0.99
And c 1.10 2.05 0.828 0.24 58.0 30.9 343.9 0.99
HD 12661 0.81 2.84 0.795 0.19 89.1 80.7 896.3 1.00
HD 89744 1.40 7.17 0.883 0.70 257.0 297.5 3305.2 1.00
HD 202206 0.90 14.68 0.768 0.42 554.0 1936.3 21514.9 1.00
HD 134987 1.05 1.58 0.810 0.24 50.2 19.3 214.0 0.98
HD 17051 = Hor 1.03 2.98 0.970 0.16 80.0 69.8 776.0 0.99
HD 92788 1.06 3.88 0.969 0.28 113.0 115.1 1278.3 1.00
HD 142 1.10 1.36 0.980 0.37 40.0 13.6 151.3 0.96
HD 28185 0.90 5.59 1.000 0.06 168.0 281.3 3125.0 1.00
HD 177830 1.15 1.24 1.100 0.40 34.0 10.8 120.3 0.95
HD 4203 1.06 1.64 1.090 0.53 51.0 20.6 228.4 0.97
HD 27442 1.20 1.42 1.180 0.02 34.0 13.6 151.2 0.96
HD 210277 0.92 1.29 1.120 0.45 39.1 14.7 162.8 0.96
HD 82943 b 1.05 1.63 1.160 0.41 46.0 20.4 227.0 0.97
HD 19994 1.29 1.83 1.260 0.20 42.0 21.0 233.0 0.97
HD 114783 0.92 0.99 1.200 0.10 27.0 8.6 95.9 0.93
HD 222582 1.00 5.18 1.350 0.71 179.6 217.4 2415.2 1.00
HD 23079 1.10 2.76 1.480 0.03 62.0 56.1 623.3 0.99
HD 141937 1.00 9.69 1.480 0.40 247.0 763.3 8481.6 1.00
HD 160691 1.08 1.99 1.650 0.62 54.0 29.7 330.0 0.97
HD 213240 0.95 3.75 1.600 0.31 91.0 120.1 1334.4 0.99
16 Cyg B 1.00 1.68 1.690 0.68 50.0 22.9 254.0 0.96
HD 4208 0.93 0.81 1.690 0.04 18.3 5.7 63.5 0.84
HD 10697 1.10 6.08 2.120 0.11 114.0 272.2 3024.8 1.00
47 UMa b 1.03 2.54 2.090 0.06 49.3 50.7 563.8 0.98
HD 190228 1.20 5.01 2.250 0.43 96.0 170.0 1888.4 0.99
HD 50554 1.10 4.91 2.380 0.42 95.0 177.6 1973.7 0.99
And d 1.10 4.29 2.560 0.31 70.4 135.5 1505.9 0.99
HD 33636 0.99 7.71 2.620 0.39 148.0 486.4 5404.6 1.00
HD 106252 1.05 6.81 2.610 0.54 139.0 356.4 3959.7 1.00
HD 168443 c 0.84 16.96 2.870 0.20 289.0 2774.0 30821.9 1.00
14 Her 0.85 4.05 3.170 0.45 70.4 156.3 1736.9 0.99
HD 39091 1.10 9.94 3.500 0.67 194.0 727.6 8084.8 1.00
HD 74156 c 1.05 7.46 3.470 0.40 121.0 427.9 4753.9 1.00
Eri 0.80 0.88 3.360 0.60 19.0 7.8 87.1 0.76
47 UMa c 1.03 0.76 3.730 0.10 11.1 4.5 50.5 0.42


Primer.

Ravnomernoe raspredelenie:

(18)

Vychislim plotnosti veroyatnosti i . Iz (6) s uchetom (18) i (4)

gde . Otsyuda, sleduya (5) i (7), nahodim

Ispol'zuya (12) ili (13), poluchaem

Srednie rasstoyaniya, soglasno (17), ravny

Sootnosheniya mezhdu srednimi rasstoyaniyami sleduyushie:

(19)

Iz (19) sleduet, chto pri malyh nablyudaemoe raspredelenie radiusov orbit planet mozhet sil'no otlichat'sya ot deistvitel'nogo. Dlya planet s massoi poryadka Yupitera i men'she vypolnenie etogo usloviya garantirovano. Deistvitel'no, dlya Solnechnoi sistemy sleduet vzyat' , (Yupiter), (Saturn). Poetomu dlya Yupitera , . Dlya Saturna , . Pohozhie rezul'taty my poluchili dlya stepennogo i pokazatel'nogo raspredelenii.
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