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The Two-Body Problem

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The Two-Body Problem The Two-Body Problem • Newtonian gravity • 2 point masses – good approx for stars r m2 r = r1 - r2 m1 R rˆ = r / r r1 r2 O total mass : M = m1 + m2 m1 r1 + m2 r 2 – O: origin centre of mass : R = M AS 4024 Binary Stars and Accretion Disks Two-Body motion • Newtons laws of motion + law of gravitation Equations of motion : - Gm m F = m r&& = 1 2 rˆ F = m r&& = -F 1 1 1 r 2 2 2 2 1 add the equations m1 r&&1 + m2 r&&2 = 0 integrate dt m1 r&1 + m2 r&2 = A again m1 r1 + m2 r 2 = A t + B definition of R M R = A t + B •\ Centre of mass moves with constant velocity – ( unless acted on by an external force ) AS 4024 Binary Stars and Accretion Disks Relative Motion • Important for eclipses • Subtract two equations of motion -G M r&& = &r& -r&& = rˆ 1 2 r 2 • Multiply by m = m1 m2 / M = “reduced mass” - G M m - G m m m &r& = rˆ = 1 2 rˆ r 2 r 2 • Relative orbit is as if: – orbiter has reduced mass m = reduced mass – stationary central mass is M = total mass. AS 4024 Binary Stars and Accretion Disks The Relative Orbit m = m1 m2 M r&(t) r(t) M = m1 + m2 Important for Eclipses AS 4024 Binary Stars and Accretion Disks 2 EllipticalBarycentric Orbits Orbits R2 (t) R1(t) Important for Radial Velocities AS 4024 Binary Stars and Accretion Disks Barycentric Orbits • Important for radial velocity curves m1 R1 + m2 R2 = 0 centre of mass frame m1 + m2 M M r = R1 - R2 = R1 = R1 = - R2 m2 m2 m1 equations of motion : G m G m R&& = - 2 r R&& = - 1 (- r) 1 r 3 2 r 3 3 – Eliminate r using r = f(R1) and r = f(R1) : 3 R 3 R && G m2 1 && G m1 2 R1 = - 2 3 R2 = - 2 3 M R1 M R2 – the acceleration of each star relative to the centre of mass AS 4024 Binary Stars and Accretion Disks Relative vs Barycentric orbits – ( a, e, P, v ) relative orbit – ( a ,e, P ,v )1,2 barycentric orbits – m1 and m2 on straight line thru C P1 = P2 = P e1 = e2 = e a1 = a m2 / M a2 = a m1 / M a = a1 + a2 M = m1 + m2 a1 : a2 : a = V 1 :V 2 :V = m2 : m1 : M Kepler : p 2 M 3 / 2 3 / 2 4 = = m2 M = m1 M 2 3 3 3 GP a a1 a2 AS 4024 Binary Stars and Accretion Disks Orbital Speed orbital speed : V 2 = r& 2 + r 2q& 2 l conic section : r = 1 + e cos q d é l ù l 1 + e cos q = Þ -e ( sin q ) q& = - r& dt ëê r ûú r 2 specific angular momentum : r 2q& = L r 2q& L \ r& = e sin q = e sin q l l L L r q& = = (1 + e cos q ) r l AS 4024 Binary Stars and Accretion Disks Orbital Speed 2 æ L ö 2 V 2 = ç ÷ [e2 sin 2q + (1+e cos q ) ] è l ø L 2 2 l = æ L ö G M = ç ÷ [e2 + 1+ 2e cos J ] L G M è l ø = 2 l L æ L ö 2 = ç ÷ [2(ecos q +1)+e - 1 ] è l ø 2 2 2 L é 2 1- e ù V = ê - ú l ë r l û AS 4024 Binary Stars and Accretion Disks Orbital Speed 2 2 2 L é 2 1 - e ù V = ê - ú l ë r l û ellipse : l = a(1 - e 2) L 2 = GM l é 2 1 ù V 2 = GM - ëê r a ûú 2 2 é 2 1 ù ellipse e < 1 l = a(1 - e ) V = GM ê - ú ë r a û GM circle e = 0 l = a V 2 = a 2GM parabola e = 1 V 2 = r é 2 1 ù hyperbola e > 1 l = a(1 - e 2) V 2 = GM + ëê r a ûú AS 4024 Binary Stars and Accretion Disks Energy of Orbit Kinetic energy : 1 1 m m KE = T = m V 2 + m V 2 = 1 2 V 2 2 1 1 2 2 2 2 M æ 2 1 ö orbital speed : V 2 = G M ç - ÷ è r a ø Potential energy: ¥ G m1 m2 G m1 m2 PE = W = -ò 2 dr = - r r r G m m Total energy : E = T +W = - 1 2 < 0 2 a Binding energy : - E > 0 AS 4024 Binary Stars and Accretion Disks Angular momentum of the orbit • Angular momentum vector J, defines orbital plane 2 2 – J = m1L1 + m2L2 and L = G M a (1-e ) 2 3 2 ) 2 and L1 = G (m2 /M a1 (1-e ) and a1/a = m2 /M – same for L2 – hence m 2 m 2 L = 2 L; L = 1 L 1 M 2 2 M 2 therefore G m 2 m 2 J 2 = 1 2 a (1- e 2 ) M and the final expression for J is 2p a 2 m m J = 1 2 1- e 2 P M AS 4024 Binary Stars and Accretion Disks Orbital Angular momentum • Given masses m1,m2 and Energy E, – the angular momentum J determines the shape of the orbit – ie the eccentricity (or the conic section parameter l ) • For given E, – circular orbits have maximum J – J decreases as e -> 1 – orbit becomes rectilinear ellipse • relation between E, and J very important in determining when systems interact mass exchange and orbital evolution AS 4024 Binary Stars and Accretion Disks.
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