Energy of Orbit

1. Energy of Orbit • Energy of orbit is E = T+W; (KE + PE) • where V is the speed in the relative orbit Hence the total Energy is is always negative. Binding Energy = -E > 0

2. Angular momentum of the orbit • Angular momentum vector J, defines orbital plane • J = m1L1 + m2L2 and L2 =k l = G(m1 + m2 )a(1-e2) and L12 =k l = Gm23 /(m1 + m2 ) 2 a1 (1-e2) and a1/a = m2 /(m1 + m2 ) • same for L2 • hence therefore and the final expression for J is

3. 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

4. Orbit in Space • N: ascending node. projection of orbit onto sky at place of maximum receding velocity • angles Ø is the true anomoly, w longitude of periastron and W, the longitude of the ascending node

5. Elements of the orbit • using angles i, inclination and W, we have • Lx = Lsin(i)sin(W), Ly = -Lsin(i)cos(W), Lz = Lcos(i) • hence, to define orbit in plane of sky, we have • quantities (a,e,i,w,W,T) are called the elements of the ellipse • provide size, shape, and orientation of the orbit in space and time! • N.B. difference between barycentre and relative orbits • radial velocity variations give information in the barycentre orbits • light curves give information in terms of the relative orbit.

6. Applications to Spectroscopic Binary Systems • star at position P2, polar coords are (r, q+w) • project along line of nodes: r cos(q+w) and • Perpendicular to line of nodes, r sin(q+w) • project this along line of sight: z = r sin(q+w) sin(i) • radial velocities along line-of sight is then using and Kepler’s 2nd Law

7. Spectroscopic Orbital Velocities • The radial velocity is usually expressed in the form • where, K, is the semi-amplitude of the velocity defined as K has maximum and minim values at ascending and descending nodes when(q+p) = 0 and (q+p) = p, hence • if e=0, Vrad is a cosine curve • as e > 0, velocity becomes skewed.

8. Radial Velocities • Radial velocity for two stars in circular orbit • with K1=100 km/s and K2 = 200 km/s q=m2/m1 = 0.5

9. Radial Velocities • Radial velocity for two stars in eccentric orbit • with e=0.1, w= 45o

10. Radial Velocities • Radial velocity for two stars in eccentric orbit • with e=0.3 , w= 0o

11. Radial Velocities • Radial velocity for two stars in eccentric orbit • with e=0.6 , w= 90o

12. Radial Velocities • Radial velocity for two stars in eccentric orbit • with e=0.9 , w= 270o

13. Minimum masses • From

14. Minimum masses II • In typical astronomical units • solar masses, km/s for velocities, and days for periods • a1,2 sin(i) = 1.3751 x 104 (1-e2) 1/2 K1,2 P km • projected semi-major axes • m1,2 sin3(i) = 1.0385 x 10-7 (1-e2)3/2 (K1+K2)2 K2,1 P solar masses • minimum mass only attainable for double-lined spectroscopic binaries (know both K1 and K2 ) • known as SB2s • otherwise have mass function 1/(2pG) = 1.0385 x 10-7 when measuring in solar masses