(a) Show that the embedding surface of Eq. (26.48) is a paraboloid of revolution everywhere outside the star.

(b) Show that in the interior of a uniform-density star, the embedding surface is a segment of a sphere.

(c) Show that the match of the interior to the exterior is done in such a way that, in the embedding space, the embedded surface shows no kink (no bend) at r = R.

(d) Show that, in general, the circumference/(2π) for a star is less than the distance from the center to the surface by an amount of order one sixth the star’s gravitational radius, M/3. Evaluate this amount analytically for a star of uniform density, and numerically (approximately) for Earth and for a neutron star.

Transcribed Image Text:

= S /0 Z= dr' [r'/(2m) - 1]ź (26.48)