Hydrostatic equilibrium and mass continuity

Define m(r) as the mass enclosed by radius r, and rho as mass per volume. A thin spherical shell has volume equal to its area times its thickness. This gives mass continuity without requiring a constant density profile. Shells outside a chosen radius exert no net gravitational force there under spherical symmetry. The inward gravitational acceleration therefore depends on enclosed mass, rather than the total stellar mass everywhere. Near a regular center, enclosed mass grows as the cube of radius.

The outward pressure force on a small element is minus the pressure gradient divided by density. Setting its acceleration to zero requires pressure to fall outward. Here P is total isotropic material plus radiation pressure when radiation is trapped sufficiently well to be treated locally. Adding a separate radiative force while also retaining its same contribution in total pressure would double count support. An equation of state must still relate pressure, density, temperature, and composition; the two mechanical equations alone do not close a stellar model.

The characteristic dynamical time follows by comparing a displacement of order the stellar radius with an acceleration of order its surface gravity. Hydrostatic evolution is appropriate when structural changes occur much more slowly than this time. It does not mean every fluid velocity vanishes: small convective motions can coexist with an averaged hydrostatic background. Strong pulsations require the acceleration term in the momentum equation, and relativistically compact objects require a different gravitational structure equation.

At the center, radius and transported luminosity vanish while pressure and temperature remain finite. At the exterior, an atmosphere supplies pressure and temperature conditions at a specified matching depth. Zero surface pressure is a useful mathematical idealization for our uniform-density sphere, but is not an exact photospheric condition. A genuine stellar calculation is a boundary-value problem constrained at both ends. Choosing central values arbitrarily and integrating outward need not produce the desired total mass or atmospheric match.

Let L(m) be the outward luminosity transported through a mass shell, excluding freely escaping neutrinos. Its derivative measures local net heating after allowing for changing internal energy and compression work. Nuclear reactions can supply heat, neutrinos can remove it, and contraction can release stored gravitational and thermal energy. Hydrostatic equilibrium removes acceleration from the momentum equation; it does not remove these thermal time derivatives. A contracting object can shine even when its local nuclear heating is negligible.

We define specific internal energy u without nuclear rest-mass energy and put the associated reaction energy in epsilon nuclear. Neutrino losses must be counted once using the same convention. At fixed composition, the thermal derivative combination is minus temperature times the specific-entropy derivative; changing composition requires consistent chemical terms or their inclusion in the reaction-energy convention. Thermal equilibrium sets the derivative combination to zero. Temperature transport and abundance evolution then supply the remaining equations needed to follow the star through time.