Gravitational Equipotential Surfaces

Gravitational Equipotential Surfaces

Edouard Louis Roche, a 19th century mathematician, studied equipotential surfaces arising from an external point-mass on a fluid mass. To apply his investigations to stars, one needs to assume that stars are centrally condensed objects. During and after Roche's lifetime, the astronomical community believed that stars were homogeneous, incompressible, self-gravitating objects. In the early twentieth century, astronomers concluded that stars were not homogeneous objects, but rather centrally condensed objects (Chandrasekhar 1933). In the Roche model, the stellar components are assumed to be centrally condensed objects that rotate synchronously in circular orbits around the center of mass. Therefore, the equipotential surfaces, or Roche surfaces, can approximate the surfaces of constant density around each component. The following expression describes the equipotential surfaces around two point-mass objects

$$-\Omega = \frac{1}{2}(x^2+y^2) + \frac{(1-\mu)}{r_1} + \frac{\mu}{r_2}.$$ (4.15)

Combining Eqn. 4.3 and Eqn. 4.15 together yields

$$V^2 = -2\Omega - C.$$ (4.16)

Therefore, the equipotential surfaces coincide exactly with the zero-velocity curves in the restricted three-body problem ($-\Omega=U$ of Eqn. 4.10). These two surfaces coincide because surfaces of gravitational equilibrium between two finite masses would exert zero net force on a particle placed there, in effect, allowing the particle to have zero velocity in the rotating frame of reference.

Zdenek Kopal (1955, 1959, 1978) extensively explains the mathematics developed to apply equipotential surfaces to the study of binary stars. The critical equipotential surfaces around each component is the Roche limit or critical Roche lobe of that star. The critical Roche surfaces are the outermost closed equipotential surfaces around each star. These surfaces connect at a point between the stars along the axis of the center of gravity. This point is called the L1 point. Kuiper (1941) utilized the Roche model to explain the spectroscopic anomalies observed in $\beta$ Lyrae. He explained that the L1 point provides ``a route for transferring mass from one star to the other, while L2 and L3 control the flow of mass from the system to the outer space.'' The L1, L2 and L3 points are locations of low gravitational potential, or saddle points, as compared to the surrounding regions (Sahade & Wood 1978). (Refer to Fig. 4.2 for schematic diagram of the Lagrangian points). Therefore, at these location, small external forces that are exerted on these particles will easily perturb them into different orbits away from these locations.