The PostNewtonian Effects of General Relativity on the Equilibrium of Uniformly Rotating Bodies. II. The Deformed Figures of the Maclaurin Spheroids
Abstract
The equations of postNewtonian hydrodynamics are solved appropriately for a uniformly rotating homogeneous mass with symmetry about the axis of rotation. The postNewtonian figure is obtained as a deformation of the Newtonian Maclaurin spheroid (with semiaxes and , say) by a Lagrangian displacement proportional to (2) =a'12( 2Xl 2x2,  ), where w denotes the distance from the axis of rotation and x1 and x2 are the Cartesian coordinates in the equatorial plane. It is shown that the equation defining the boundary of the postNewtonian configuration is of the form 2 2 R 5 4 2+2t2S2t(e)422 =0, ai a3 a1 a1 a1 a3 (e) is a determinate function of the eccentricity e of the Maclaurin spheroid and R5 (= 2CM/c2) is the Schwarzschild radius. The function S t(e) is tabulated in the paper. Further, the angular velocity of rotation of the postNewtonian configuration differs from that of the Maclaurin spheroid by an amount which is also tabulated. The solution of the postNewtonian equations exhibits a singularity at a certain eccentricity e*(= 0.985226) of the Maclaurin spheroid. The origin of this singularity is that at e* the Maclaurin spheroid allows an infinitesimal neutral deformation by a displacement proportional to (2); and the Newtonian instability of the Maclaurin spheroid at e* is excited by the postNewtonian effects of general relativity.
 Publication:

The Astrophysical Journal
 Pub Date:
 January 1967
 DOI:
 10.1086/149003
 Bibcode:
 1967ApJ...147..334C