How Gravity Scales with Radius at Fixed Density

Start with mass at fixed density

For a uniform spherical body of density $\rho$ and radius $R$, mass is not independent of size:

$$M=\frac43\pi\rho R^3.$$

Substituting this relationship into familiar gravity formulas reveals which quantities depend on radius and which depend only on density.

Surface gravity grows linearly with radius

The surface gravitational acceleration is

$$g_s=\frac{GM}{R^2} =\frac43\pi G\rho R.$$

Among bodies with the same mean density, doubling the radius doubles surface gravity. The mass grows by a factor of eight, but the inverse-square distance contributes a factor of one quarter, leaving a factor of two.

Escape speed also grows linearly

The escape speed from the surface is

$$v_{\mathrm{esc}}=\sqrt{\frac{2GM}{R}} =R\sqrt{\frac{8\pi G\rho}{3}}.$$

At fixed density, doubling radius therefore doubles escape speed. This is a different scaling from a comparison at fixed mass, where increasing radius would reduce escape speed. Stating what remains fixed is essential before making a scaling claim.

The rotational shedding threshold depends on density

At the equator of a cohesionless rotating body, surface contact is lost when the required centripetal acceleration reaches the inward gravitational acceleration:

$$\omega^2R=g_s.$$

Using the fixed-density expression for $g_s$ gives

$$\omega_{\mathrm{crit}}=\sqrt{\frac{4\pi G\rho}{3}}.$$

Radius cancels. Two ideal spherical bodies with the same density have the same critical angular speed and critical rotation period, even though the larger one has stronger surface gravity and a larger escape speed.

These results assume spherical symmetry, uniform density, and no material cohesion. Real rubble-pile asteroids, differentiated planets, and irregular bodies can depart from the simple scaling, and the first loss of surface contact occurs at the equator in this idealized model.