Formation of hot jupiters by slingshot model Naoya Okazawa Department of Earth Sciences, Undergraduate school of Science, Hokkaido University
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1 Formation of hot jupiters by slingshot model Naoya Okazawa Department of Earth Sciences, Undergraduate school of Science, Hokkaido University Planetary and Space Group
2 1,.,,.,.,.,..,.,.,,.,.,,,.,.,.,.,,,..
3 i APPENDIX
4 (Hayashi et al. 1985)...,..,. 10,..,,.,., Peg b, 0.052AU (Marcy et al. 1997) (Schneider 2011),. (e. g. Schilling 1996)., 1., 3,,,.,, (Nagasawa et al. 2008).
5 , 2,..,,. 1.3, ,
6 ,..,,.,.,.,.,, , 2, 2. 1.,.,,. 1: (National Space Science Data Center) [m ] [AU]
7 2 4 1, 2,.,.., 0.1AU.,,. 2,.. 1, (a < 0.1 AU),. 3. 1: , 481 (Schneider 2011 )
8 2 5 2: , 514 (Schneider 2011 ) 2.3 (Weidenschilling & Marzari 1996)..,,,., (Lin et al. 1996) (Rasio & Ford 1996), (Bodenheimer et al. 2000).,,.,.,,...,, (Lin et al. 1996). 3.
9 ,. 1,,.,.,., 4,.,, (Nagasawa et al. 2008). 3.1 Chambers et al. (1996), Marzari & Weidenschilling (2002), Chatterjee et al (2008) , (Gladman 1993). 2 3r H(1,2) (3.1.1) 1 a 1, m 1, 2 a 2, m 2, M, r H(1,2). r H(1,2) = a 1 + a 2 2 ( ) m1 + m 1/3 2 (3.1.2), 3,., 1, 2 (Marzari & Weidenschilling 2002). 3M, a i+1 = a i + Kr H(i,i+1) (3.1.3) K, (Chatterjee et al 2008).,
10 3 7,, (Chatterjee et al 2008).. 3.2,.,.,,,.,,..,... ( ), ( ).,,. 2, 1, 1... m 1 m , E L. E = Gµ(m 1 + m 2 ) (3.2.1) 2a L = µ (1 e 2 )(m 1 + m 2 )ag (3.2.2) G, a, µ µ m 1 m 2 /(m 1 + m 2 ). (3.2.1), ȧ = Gm 1m 2 2E 2 Ė 2a 2 = Ė (3.2.3) Gm 1 m 2. Ė, Ė < 0. ȧ < 0.
11 4 8 (3.2.2), L = 0. ė = 1 e2 2ae ȧ (3.2.4) (3.2.3), ė = 1 (1 e)(1 + e)a Ė (3.2.5) Gm 1 m 2 e q = a(1 e) e 1 (3.2.5). ė = Ė < 0 ė < 0. 2q Gm 1 m 2 Ė (3.2.6) a fin a ini e ini, (3.2.2) a fin = a ini (1 e 2 ini) = q ini (1 + e ini ) (3.2.7). a fin = q fin, e ini 1, q fin 2q ini, 2. q ini (1 + e ini )/(1 e ini ) 2q ini. 1,.,, , 1, 2, 3, 4. j,. m j d 2 r j dt 2 = i j Gm i m j r(i,j) 3 r ( i, j) (4.1.1) r j j, m i i, G, r (i,j) i j..
12 , ,,. x. dx dt = f(x, t) (4.2.1) t, x, f x t. t, x(t + t). x(t + t) = x(t) + dx(t) t + 1 d 2 x(t) dt 2 dt 2 t 2 + (4.2.2) ( t) 2, t i+1 = t + t x i+1. x i+1 = x i + f(x i, t i ) t (4.2.3)., t = t i, t = t i + t.. k 1 = f(x i, t i ) (4.2.4) k 2 = f(x i + k 1 t/2, t i + t/2) (4.2.5) k 3 = f(x i + k 2 t/2, t i + t/2) (4.2.6) k 4 = f(x i + k 3, t i + t) (4.2.7) x i+1 = x i + (k 1 + 2k 2 + 2k 3 + k 4 ) t (4.2.8) 6 k 1 t = t i, k 2 k 1 t/ d 2 x = g(x, t) (4.2.9) dt2
13 x 2 (4.2.9). g. t = t i a i 1, x p, v p. x pi = x i + v i t + t2 2 a i + t3 6 v pi = v i + a i t + t2 2 da i dt da i dt (4.2.10) (4.2.11) x i, v i t = t i. da i /dt g(x, t),, a i = Gm k r k r 3 k da i dt = Gm k [ vk r 3 k 3(r ] k v k )r k rk 5 (4.2.12) (4.2.13) m k k, r k, v k.,, (4.2.12), (4.2.13). t = t i + t a i+1 da i+1 /dt. a i, da i /dt, a i+1, da i+1 /dt d 2 a i /dt 2, d 3 a i /dt 3 (7.5 ). d 2 a i dt 2 = d 3 a i dt 3 = x i+1, v i ( ) 6(a i a i+1 ) t 4 da i dt + 2 da i+1 dt t 2 (4.2.14) ( 12(a i a i+1 ) + 6 t dai dt + da i+1 dt t 3 (4.2.15) x i+1 = x pi + t4 24 v i+1 = v pi + t3 6 ) d 2 a i dt 2 + t5 d 3 a i 120 dt 3 (4.2.16) d 2 a i dt 2 + t4 d 3 a i 24 dt 3 (4.2.17) 2 xy 3,.,.,, 2.
14 4 11, v 1 = v i m i /m 1, r 1 = r i m i /m 1. E( ) E i = 1 ( 2 m i 1 + m ) i (v i v cen ) 2 Gm 1m i (4.3.1) m 1 r (1,i). v cen 2.,., L, ( L i = m i 1 + m ) i (r i (v i v cen )) (4.3.2) m 1. 2,. 2, µ = m 1m 2 m 1 +m 2 a e (7.3.2 ). a i = Gµ(m 1 + m i ) (4.3.3) 2E i L 2 i e i = 1 µ 2 (4.3.4) (m 1 + m i )ag, 10 3 M k AU (k = 1, 5), 2.5 AU k 10 3 M (k = 1, 5), 3 30., M, 2 AU. 1, 1.. (Marzari & Weidenschilling 2002). 1.
15 semi-major axis [AU] time [year] 3: 3, 4. 4,. Chatterjee et al :, 5. 5,.
16 4 13 5:, AU, (a 5 AU).,. 6:,, q q < 0.05 AU (Rasio & Ford 1996). 0.5 AU,. 1, 2,.,,.
17 ,.. E = Gµ(m 1 + m 2 ) + 1 2a 2 I ω 2 (5.1.1) L = µ (1 e 2 )(m 1 + m 2 )ag + I ω (5.1.2) I, ω. (5.1.1), Ω p, Ė = Gµ(m 1 + m 2 ) 2a 2 ȧ + I ω ω., (5.1.2), = 1 2 µω2 paȧ + I ω ω (5.1.3) L = 1 2 µ G(m1 + m 2 ) a 3 aȧ + I ω = 1 2 µω paȧ + I ω (5.1.4). L = 0 (5.1.4). (5.1.5) (5.1.3), I ω = 1 2 µω paȧ (5.1.5) Ė = 1 2 µω paȧ(ω Ω p ) (5.1.6). Ė < 0 ȧ (ω Ω p ).,., (ω < Ω p ) (Ω p < ω ). 7.,,...
18 5 15 7: ( 2003 ).,.. 2.3,.,,.,. 5.2,.,.,., ( a < 0.1 ).. m, a 0 a 1, e 0 e 1
19 5 16. M L, ( ) L = m GMa fin (1 e 2 fin ) GMa ini (1 e 2 ini ) (5.2.1) (7.3.1 ). ω, ω = L I (5.2.2) I. 10. a ini = 2.5 AU( ), e ini = 0.98, a fin = 0.02 AU, e fin = 0, P rot,. ( ) I ( ) MR M 1/2 ( ) m 1 P rot (5.2.3) ,,. 8,. 1. M m J 8: ( Mamajek & Hillenbrand 2008, Baliunas 1996, Noyes et al ), (Lanza 2010 )
20 ,,.,.,,... T- 8 (Bouvier et al. 1993).. 8,.,. r, M, T, Ω p Ω p = GM/r 3, Ω Ω = 2π/T r c. r c = ( ) T 2/3 (GM) 1/3 (5.3.1) 2π.. 5.4,,. 9. a < 0.04 AU , a 0.04 AU 4.92, AU., τ a τ a (m 1/2 /m p ) (Murray & Dermott 1999) AU, ( ) 1/2 / ( ) 1/2 /.,,. 9
21 5 18.,.,,.,. 9: (Schneider 2011 ) 10: a < 0.04 AU, ( ) 1/2 / (Schneider 2011 )
22 6 19 6,,.,.,. 3,, 1.,,.,.,.,.,.,,,.,.,.,.,.,.,.,.,,,.
23 7 APPENDIX 20 7 APPENDIX 7.1,. r, R, M, M p, Ω K. a m.,. GM m (R a) 2 = GM pm a 2 + m(r a)ω 2 K (7.1.1) Ω 2 K = GM r 3, (r a) 2, GM m (r a) 2 = GM pm a 2 + Gm(r a)m r 3 (7.1.2) 1 (r a) 2 1 r 2 + 2a r 3 (7.1.3), M r 2 + 2aM r 3 a = r = M p + M a 2 ( Mp 3M r 2 ) 1 3 am r 3 (7.1.4) (7.1.5)
24 7 APPENDIX : ( 2007 II ( ) ).,.. F P r, FF P r = 2a r, FF C (7.2.2) (7.2.1) r 2 = r 2 + (2ae) 2 + 2(2ae)r cos ϕ (7.2.1) r(1 + e cos ϕ) = a(1 e 2 ) (7.2.2) a(1 e 2 ) = l (7.2.3) r = l 1 + e cos ϕ (7.2.4),., l, e, (a, e),.,.
25 7 APPENDIX 22,. E. E = m 2 (v2 r + vθ 2 ) GMm r = m ( ) dr 2 ( + r dθ ) 2 GMm 2 dt dt r (7.3.1) m, v r r, v θ θ. L = mr 2 dθ dt, E = m 2 ( ) dr 2 + L2 dt 2mr 2 GMm r (7.3.2) d dt = dθ dt d dθ = L mr 2 d dθ, u 1/r, E = m 2 ( ) L dr 2 mr 2 + L2 dθ 2mr 2 GMm r (7.3.3) cos l ( ) E = L2 du 2 + L2 u 2 GMmu (7.3.4) 2m dθ 2m ( ) 2mE du 2 ) 2 L 2 = + (u GMm2 dθ L 2 G2 M 2 m 4 L 4 (7.3.5) du dθ = ± ( ) (7.3.6) 2 2mE u GMm2 L 2 L + G 2 M 2 m 4 2 L 4 dx (a 2 x 2 ) = arccos ( x a) θ = ± arccos cos θ = r = u GMm2 L 2 (7.3.7) 2mE + G2 M 2 m 4 L 2 L 4 u GMm2 L 2 (7.3.8) 2mE + G2 M 2 m 4 L 2 L 4 L 2 GMm EL2 cos θ G 2 M 2 m 3 L2, e 1 + 2EL2 GMm 2 G 2 M 2 m 3 (7.3.9) r = l 1 + e cos θ (7.3.10)
26 7 APPENDIX 23.,. l, e,., a = l l, e 1 e 2, e 1 + 2EL2 G 2 M 2 m 3 E = GMm 2a (7.3.11) L = m (1 e 2 )agm (7.3.12) 7.3.2,.,., m 1, r 1,, m 2, r 2, R, r. R = m 1r 1 + m 2 r 2 m 1 + m 2 (7.3.13) r = r 1 r 2 (7.3.14),. r 1, r 2 (7.3.13), (7.3.14) L m 2 r 1 = r 1 R = r (7.3.15) m 1 + m 2 r 2 m 1 = r 2 R = r (7.3.16) m 1 + m 2 L = r 1 dr 1 m 1 dt + dr r 2 2 m 2 dt m 1 m 2 = r dr m 1 + m 2 dt = µr dr dt E. (7.3.13), (7.3.14) (7.3.17) m 2 r 1 = R + r (7.3.18) m 1 + m 2 m 1 r 2 = R r (7.3.19) m 1 + m 2 E, V E = 1 2 (m 1ṙ m 2 ṙ 2 2) + V (r 1 r 2 ) (7.3.20)
27 7 APPENDIX 24 (7.3.18), (7.3.19) ṙ1 2 = Ṙ2 + 2m 2 m 2 2 Ṙṙ + m 1 + m 2 (m 1 + m 2 ) 2 ṙ2 (7.3.21) ṙ2 2 = Ṙ2 + 2m 1 m 2 1 Ṙṙ + m 1 + m 2 (m 1 + m 2 ) 2 ṙ2 (7.3.22) E = m 1 + m 2 Ṙ 2 + m 1m 2 )ṙ2 Gm 1m 2 2 (m 1 + m 2 r (7.3.23) Ṙ = 0. µ m 1m 2 /(m 1 + m 2 ) E = µṙ 2 Gm 1m 2 r = µ 2 (v2 r + vθ 2 ) Gm 1m 2 r = µ ( ) dr 2 ( + r dθ ) 2 Gm 1m 2 2 dt dt r (7.3.24). v r r, v θ θ. E L a, e.,,.. E = Gµ(m 1 + m 2 ) (7.3.25) 2a L = µ (1 e 2 )(m 1 + m 2 )ag (7.3.26) 7.4 x(t + t), x(t + t) = x(t) + t dx(t) dt t n = n! n=0 + t2 2 d 2 x(t) dt 2 + t3 6 d 3 x(t) dt 3 + (7.4.1) d n x(t) dt n (7.4.2),.., t 2 O( t 2 ) O( t). 1., 4 4., 5, 4.
28 7 APPENDIX ,. k 1, k 2, k 3, k 4 t, x, f, k 1 = f(x) (7.4.3) ( k 2 = f x + 1 ) 2 k 1 t = f (x + 12 ) f(x) t ( 1 = f(x) + f (x) + 1 ( 1 6 f (x) 2 f(x) t ) 2 f(x) t + 1 ( 1 2 f (x) 2 f(x) t ) 3 + O( t 4 ) = f(x) f(x)f (x) t f 2 (x)f (x) t f 3 (x)f (x) t 3 + O( t 4 ) (7.4.4) ( k 3 = f x + 1 ) 2 k 2 t ( ) 1 = f(x) + f (x) 2 k 2 t + 1 ( ) f (x) 2 k 2 t + 1 ( ) f (x) 2 k 2 t + O( t 4 ) = f(x) f (x) (f(x) + 12 f(x)f (x) t + 18 ) f 2 (x)f (x) t 2 t + 1 ( 8 f (x) f(x) + 1 ) 2 2 f(x)f (x) t t = f(x) + 1 [ 1 2 f(x)f (x) t + + [ 3 16 f (x)f (x)(f(x)) k 4 = f(x) + f(x)f (x) t + + ) 2 48 f (x)(f(x)) 3 t 3 + O( t 4 ) 4 f(x)(f (x)) ] 8 f 2 (x)f (x) t 2 ] 48 f 3 (x)f (x) t 3 + O( t 4 ) (7.4.5) [ 1 2 f(x)(f (x)) ] 2 f 2 (x)f (x) t 2 [ 1 4 f(x)(f (x)) f 2 (x)f (x)f (x) f 3 (x)f (x) x(t + t). x(t + t) = x(t) + (k 1 + 2k 2 + 2k 3 + k 4 ) t 6 ] t 3 + O( t(7.4.6) 4 ) = x(t) + f(x) t f(x)f (x) t [(f(x))2 f (x) + f(x)(f (x)) 2 ] t [(f(x))3 f (x) + 4(f(x)) 2 f (x)f (x) + f(x)(f (x)) 3 ] t 4 +O( t 5 ) (7.4.7)
29 7 APPENDIX 26. d dt f(x(t)) = f dx dt = ff d 2 dt 2 f(x(t)) = d dt (ff ) = f(f ) 2 + f 2 f d 3 dt 3 f(x(t)) = d dt (f(f ) 2 + f 2 f ) = f(f ) 3 + 4f 2 f f + f 3 f (7.4.8) (7.4.8) (7.4.7), x(t + t) = x(t) + f(x) t + 1 df(x) t 2 2 dt + 1 d 2 f(x) 6 dt 2 t d 3 f(x) 24 dt 3 t 4 + O( t 5 ) (7.4.9). t f(x). S i (x) = A 0 + A 1 x + A 2 x A n x n (7.5.1) [a, b] [x i 1, x i ] (i = 1, 2, n), x 0, x 1,, x n. f(x), A 0, A 1, A n. f(x). f(x) f (x), 2n+2, A 2n a i (t i ) 3 S i. S i (t i ) = A 0 + A 1 t i + A 2 t 2 i + A 3 t 3 i (7.5.2) t i = i t. (7.5.2) a i (t i ). (7.5.2), (7.5.3) i = 0, 1, S i (t i ) = a i (t i ) (7.5.3) Ṡ i (t i ) = ȧ i (t i ) (7.5.4) a 0 = A 0 (7.5.5) a 1 = A 0 + A 1 t + A 2 t 2 + A 3 t 3 (7.5.6)
30 7 APPENDIX 27 (7.5.2), (7.5.4) i = 0, 1, ȧ 0 = A 1 (7.5.7) ȧ 1 = A 1 + 2A 2 t + 3A 3 t 2 (7.5.8). (7.5.5), (7.5.6), (7.5.7), (7.5.8) A 0, A 1, A 2, A 3,. A 0 = a 0 (7.5.9) A 1 = ȧ 0 (7.5.10) A 2 = 3(a 0 a 1 ) t(2ȧ 0 + ȧ 1 ) t 2 (7.5.11) A 3 = 2(a 0 a 1 ) + t(ȧ 0 + ȧ 1 ) t 2 (7.5.12), a(t + t) O( t 3 ), t = 0,, a(t + t) a(t) + da(t) t + 1 d 2 a(t) dt 2 dt 2 t d 3 a(t) 6 dt 3 t 3 (7.5.13) a( t) a 0 + da 0 dt t + 1 d 2 a 0 2 dt 2 t2 + 1 d 3 a 0 6 dt 3 t3 (7.5.14) d 2 a 0 dt 2 = 2A 2 = 6(a 0 a 1 ) t(4ȧ 0 + 2ȧ 1 ) t 2 (7.5.15) d 3 a 0 dt 3 = 6A 3 = 12(a 0 a 1 ) + 6 t(ȧ 0 + ȧ 1 ) t 3 (7.5.16)
31 7 APPENDIX 28,.,.,. 3,,.,,.,.
32 7 APPENDIX 29 Baliunas, S., D. Sokoloff, and W. Soon Magnetic Field and Rotation in Lower Main-Sequence Stars: an Empirical Time-dependent Magnetic Bode s Relation?. Astrophysical Journal Letters L99 Bouvier, J., S. Cabrit, M. Fernandez, E. L.Martin, and J. M.Matthews Coyotes-I - the Photometric Variability and Rotational Evolution of T-Tauri Stars. Astronomy and Astrophysics Chambers, J. E., G. W. Wetherill, and A. P.Boss The Stability of Multi- Planet Systems. Icarus Chatterjee, S., E. B. Ford, S. Matsumura, and F. A. Rasio Dynamical Outcomes of Planet-Planet Scattering. The Astrophysical Journal Lanza, A. F Hot Jupiters and the evolution of stellar angular momentum. Astronomy and Astrophysics A77 Lin, D. N. C, P. Bodenheimer, and D. C. Richardson Orbital migration of the planetary companion of 51 Pegasi to its present location. Nature Mamajek, E. E., and L. A. Hillenbrand Improved Age Estimation for Solar- Type Dwarfs Using Activity-Rotation Diagnostics. The Astrophysical Journal Marcy, G. W., R. P. Butler, E. Williams, L. Bildsten, J. R. Graham, A. M. Ghez, and J. G. Jernigan The Planet around 51 Pegasi. Astrophysical Journal Marzari, F., and S. J. Weidenschilling Eccentric Extrasolar Planets: The Jumping Jupiter Model. Icarus Nagasawa, M., S. Ida, and T. Bessho Formation Of Hot Planets By A Combination Of Planet Scattering, Tidal Circularization, And The Kozai Mechanism. The Astrophysical Journal Noyes, R. W., L. W. Hartmann, S. L. Baliunas, D. K. Duncan, and A. H. Vaughan Rotation, convection, and magnetic activity in lower main-sequence stars. Astrophysical Journal Rasio, F. A., and E. B. Ford Dynamical instabilities and the formation of extrasolar planetary systems. Science
33 7 APPENDIX 30 Schilling, G Hot Jupiters Leave Theorists in the Cold. Science Weidenschilling, S. J., and F. Marzari Gravitational scattering as a possible origin for giant planets at small stellar distances. Nature ,, HARP 1. - HPC). Vol No 72 Jean Schneider The Extrasolar Planets Encyclopaedia. National Space Science Data Center II ( ) 10 ebisawa/teaching/2007univtokyo.html
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