Consider two stars with the same mass M orbiting each other in a circular orbit with diameter (separation between the stars’ centers) a. Kepler’s laws tell us that the stars’ orbital angular velocity is

Assume that each star’s mass is concentrated near its center, so that everywhere except near a star’s center the gravitational potential, in an inertial frame, is Φ = −GM/r₁ − GM/r₂ with r₁ and r₂ the distances of the observation point from the center of star 1 and star 2. Suppose that the two stars are “tidally locked”: tidal gravitational forces have driven them each to rotate with rotational angular velocity equal to the orbital angular velocity Ω. (The Moon is tidally locked to Earth, which is why it always keeps the same face toward Earth.) Then in a reference frame that rotates with angular velocity Ω, each star’s gas will be at rest, v = 0.

(a) Write down the total potential Φ + Φ_cen for this binary system in the rotating frame.

(b) Using Mathematica, Maple, Matlab, or some other computer software, plot the equipotentials Φ + Φ_cen = const for this binary in its orbital plane, and use these equipotentials to describe the shapes that these stars will take if they expand to larger and larger radii (with a and M held fixed). You should obtain a sequence in which the stars, when compact, are well separated and nearly round. As they grow, tidal gravity elongates them ultimately into tear-drop shapes, followed by merger into a single, highly distorted star. With further expansion, the merged star starts flinging mass off into the surrounding space (a process not included in this hydrostatic analysis).

Transcribed Image Text:

Ω = {2GM/a3.