Binary Star Orbits

Greater Mass = Stronger Gravitational Field More than half of stars are members of binary systems.

Kepler’s three laws determine the motion of the planets.

  1. Relative orbit is a conic section with one of the objects at a focus
  2. Line connecting two bodies sweeps out equal areas in equal times
  3. Product of the square of the period and mass of the system is proportional to the cube of the mean seperation

Shape of Orbit

Binary stars have closed orbit, typically elliptical. Explanation from Wolfram of constructing one from the Foci: Ellipse

Ellipse has a semi-major axis, and a semi-minor axis. Eccentricity is OF/OA where OF is origin to focus, and OA is the semi major axis radius.

Eccentricity $$ e = OF/OA $$

Area $$ \Pi a b $$ $$ \Pi a^2 (1 - e^2)^{\frac{1}{2}} \tag{Based on Eccentricity}$$

Also: $$ b = a \sqrt{1 - e^2} $$

Periastron is the point where stars are closest (along the semi-major axis).

First Law

One star is at the focus, the other follows the elliptical path. That is relative, although in practice they are both orbiting around a center of mass. Center of Mass

a, the semi-major axis is the ‘mean’ seperation of the orbiting bodies.

Second Law

TBD - Orbital momentum is conserved. Constant area per unit time. $$ r^2 \omega = \frac{2 \Pi a^2 \sqrt{1 - e^2}}{P} $$

Third Law

Square of the period is proportional to the cube of the mean seperation P is period, a mean seperation, M Masses $$ \frac{G}{4 \Pi ^2} P^2 (M_1 + M_2) = a^3 \tag{Eq. 7.5} $$

Using Solar Masses: M Masses in solar units, P years, a in astronomical units $$ (M_{\odot 1} + M_{\odot 2}) = \frac{a^3}{p^2} \tag{Eq. 7.6} $$

Radial Velocity

Calculated by red shift or blue shift as moving radially away from observer $$ \varDelta \lambda = \lambda - \lambda_0 $$

Hence velocity: $$ v_r = \frac{\varDelta \lambda}{\lambda_0} $$

Only works if star is bright enough for spectrum to be obtained.
\(\varDelta \lambda\) can only be measured to \(\plusmn 0.001nm\), so need to use short orbital period.

If star is bright enough, it works for even far stars.

Proper Motion

Movement across the plane of the sky. u is arcseconds/year, d is parsecs $$ V_t = 4.74 \mu d \tag{km/sec}$$

logo