Hydrostatic equilibrium is the condition in which the internal pressure gradient within a self-gravitating body exactly balances the inward pull of gravity at every layer. This balance maintains the structural stability of stars, planets, and other gaseous celestial bodies.

Consider a thin spherical shell of gas with infinitesimal thickness \( dr \), located at a distance \( r \) from the center of a star. At radius \( r \), the pressure is \( P(r) \) and the density is \( \rho(r) \).

The inner surface of the shell experiences a pressure \( P_1 = P(r) \), while the outer surface experiences \( P_2 = P(r + dr) \). The infinitesimal pressure difference across the shell is

$$ dP = P_2 - P_1. $$

The net outward force on the shell due to this pressure difference is

$$ F_P = -A \, dP, $$

where \( A = 4\pi r^2 \) is the surface area of the shell and the outward direction is defined as positive. The negative sign ensures that when \( dP < 0 \) (pressure decreases outward), the resulting \( F_P \) is outward.

The gravitational force acting inward on this same shell of thickness \( dr \) is

$$ F_G = -\frac{G \, M(r) \, \rho(r) \, A \, dr}{r^2}, $$

where \( G \) is the gravitational constant and \( M(r) \) is the mass enclosed within radius \( r \). The negative sign indicates that the force acts toward the center (decreasing \( r \)).

For hydrostatic equilibrium, the sum of the pressure and gravitational forces must vanish:

$$ F_P + F_G = 0. $$

Substituting the two forces and canceling \( A \), we obtain

$$ \frac{dP}{dr} = -\frac{G \, M(r) \, \rho(r)}{r^2}. $$

This is the differential equation of hydrostatic equilibrium. It expresses that, at every radius \( r \), the pressure gradient \( \frac{dP}{dr} \) balances the gravitational pull per unit volume.

It is often convenient to define the local gravitational acceleration

$$ g(r) = \frac{G \, M(r)}{r^2}, $$

so that the equation can be written as

$$ \frac{dP}{dr} = -\rho(r) \, g(r). $$

This equation shows that, at each point within a star, the outward pressure gradient balances the inward gravitational force. If the pressure gradient were smaller in magnitude, the star would collapse under gravity; if larger, it would expand. Thus, hydrostatic equilibrium defines the stable structure of stars and planets.

The mass function \( M(r) \) is related to the density by

$$ M(r) = \int_0^r 4\pi {r'}^{2} \rho(r') \, dr', $$

where \( r' \) is a dummy variable of integration. Mass outside the radius \( r \) does not contribute to the gravitational force at \( r \), as follows from Gauss’s law for gravity (or the shell theorem).

The hydrostatic equilibrium equation is one of the four fundamental equations of stellar structure, along with those of mass continuity, energy conservation, and energy transport.

1. Explain how the equation of hydrostatic equilibrium encodes the balance between the pressure gradient and gravity inside a star.
2. Why does only the mass enclosed within \( r \) contribute to the gravitational force at that radius in a spherically symmetric body?
3. What happens to a star if its internal pressure gradient becomes smaller in magnitude than required for hydrostatic support?