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Fields from Spherical Masses

Level 1 - Physics topic page in Gravitation.

Principle

For spherical mass distributions, the gravitational field depends on how much mass is enclosed by the chosen radius.

Notation

\(M\)
total spherical mass
\(\mathrm{kg}\)
\(R\)
sphere radius
\(\mathrm{m}\)
\(r\)
distance from the center
\(\mathrm{m}\)
\(M_{\mathrm{enc}}\)
mass enclosed inside radius \(r\)
\(\mathrm{kg}\)
\(\rho\)
uniform mass density
\(\mathrm{kg\,m^{-3}}\)
\(\vec g\)
gravitational field
\(\mathrm{N\,kg^{-1}}\)
\(g(r)\)
gravitational field magnitude at radius \(r\)
\(\mathrm{N\,kg^{-1}}\)

Method

Derivation 1: Outside the sphere

Outside a spherically symmetric body, the entire mass acts as if it were concentrated at the center.

Outside field magnitude
\[g(r)=G\frac{M}{r^2}\qquad(r\ge R)\]
Outside field vector
\[\vec g=-G\frac{M}{r^2}\hat e_r\]

Derivation 2: Inside a uniform sphere

At an interior point, only the mass enclosed inside radius \(r\) contributes to the inverse-square result. For a uniform density, the enclosed mass grows like volume.

Enclosed volume
\[V_{\mathrm{enc}}=\frac43\pi r^3\]
Enclosed mass
\[M_{\mathrm{enc}}=\rho\frac43\pi r^3\]
Inside field magnitude
\[g(r)=G\frac{M_{\mathrm{enc}}}{r^2}=\frac43\pi G\rho r\]
Rewrite using total mass
\[g(r)=G\frac{M}{R^3}r\qquad(r\le R)\]
Inside a uniform sphere, the field grows linearly with radius.

Derivation 3: Check the surface match

The inside and outside formulas should agree at the surface.

Inside at \(r=R\)
\[g(R)=G\frac{M}{R^3}R=G\frac{M}{R^2}\]
Outside at \(r=R\)
\[g(R)=G\frac{M}{R^2}\]

The graph shows the full piecewise behavior for a uniform sphere. The field rises linearly inside, peaks at the surface, then falls as \(1/r^2\) outside.

012345600.250.50.751rrelative ginsideoutsidesurface
The surface is where the linear interior model meets the inverse-square exterior model.

The change of shape at the surface matters physically: near the center of a uniform sphere the field is small because the enclosed mass is small.

Rules

These are the compact results from the method above.

Outside sphere
\[\vec g=-G\frac{M}{r^2}\hat e_r,\qquad g(r)=G\frac{M}{r^2}\quad(r\ge R)\]
Inside uniform sphere
\[g(r)=G\frac{M}{R^3}r=\frac43\pi G\rho r\quad(r\le R)\]
Surface field
\[g(R)=G\frac{M}{R^2}\]

Examples

Question
A planet is modeled as a uniform sphere. What is the field magnitude at radius
\[R/2\]
in terms of the surface field
\[g_s=GM/R^2\]
?
Answer
\[g\left(\frac{R}{2}\right)=G\frac{M}{R^3}\left(\frac{R}{2}\right)=\frac12\frac{GM}{R^2}=\frac12 g_s\]

Checks

  • Outside a spherical mass, the field behaves exactly like a point mass at the center.
  • Inside a uniform sphere, the field is zero at the center and increases linearly with radius.
  • The inside and outside formulas must agree at the surface.
  • The linear inside law does not apply to arbitrary non-uniform density profiles.

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Kepler Models

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Apparent Weight