AS 3004 Stellar Dynamics 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

AS 3004 Stellar Dynamics Angular momentum of the orbit Angular momentum vector J, defines orbital plane –J = m 1 L 1 + m 2 L 2 and L 2 =  k l = G(m 1 + m 2 )a(1-e 2 ) and L 1 2 =  k l = Gm 2 3 /(m 1 + m 2 ) 2 a 1 (1-e 2 ) and a 1 /a = m 2 /(m 1 + m 2 ) –same for L 2 –hence therefore and the final expression for J is

AS 3004 Stellar Dynamics 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 3004 Stellar Dynamics Orbit in Space –N: ascending node. projection of orbit onto sky at place of maximum receding velocity –angles Ø is the true anomoly,  longitude of periastron and , the longitude of the ascending node

AS 3004 Stellar Dynamics Elements of the orbit using angles i, inclination and , we have L x = Lsin(i)sin(  ), L y = -Lsin(i)cos(  ), L z = Lcos(i) hence, to define orbit in plane of sky, we have –quantities (a,e,i, , ,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.

AS 3004 Stellar Dynamics Applications to Spectroscopic Binary Systems star at position P 2, polar coords are (r,  +  ) –project along line of nodes: r cos(  +  ) and –Perpendicular to line of nodes, r sin(  +  ) –project this along line of sight: z = r sin(  +  ) sin(i) radial velocities along line-of sight is then using and Kepler’s 2 nd Law

AS 3004 Stellar Dynamics 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(  ) = 0 and (  ) = , hence –if e=0, V rad is a cosine curve –as e > 0, velocity becomes skewed.

AS 3004 Stellar Dynamics Radial Velocities Radial velocity for two stars in circular orbit –with K 1 =100 km/s and K 2 = 200 km/s q=m 2 /m 1 = 0.5

AS 3004 Stellar Dynamics Radial Velocities Radial velocity for two stars in eccentric orbit –with e=0.1,  = 45 o

AS 3004 Stellar Dynamics Radial Velocities Radial velocity for two stars in eccentric orbit –with e=0.3,  = 0 o

AS 3004 Stellar Dynamics Radial Velocities Radial velocity for two stars in eccentric orbit –with e=0.6,  = 90 o

AS 3004 Stellar Dynamics Radial Velocities Radial velocity for two stars in eccentric orbit –with e=0.9,  = 270 o

AS 3004 Stellar Dynamics Minimum masses –From

AS 3004 Stellar Dynamics Minimum masses II In typical astronomical units –solar masses, km/s for velocities, and days for periods a 1,2 sin(i) = x 10 4 (1-e 2 ) 1/2 K 1,2 P km –projected semi-major axes m 1,2 sin 3 (i) = x (1-e 2 ) 3/2 (K 1 +K 2 ) 2 K 2,1 P solar masses minimum mass only attainable for double-lined spectroscopic binaries (know both K 1 and K 2 ) –known as SB2s –otherwise have mass function 1/(2  G) = x when measuring in solar masses