M071 · Spherical gravity

Gravity Field & Shell Theorem

Inside a hollow shell, where does the shell's gravity go? Nowhere; it cancels. Inside a uniform spherical shell, the near side pulls harder but has less mass, the far side pulls more weakly but has more, and the two balance exactly. The field is zero everywhere inside. The potential is not zero, though: it is constant, equal to its value at the shell's surface.

Set the densities of three spherical layers, move a probe along the radius to read the field, and drop a particle through a tunnel along a diameter. Compare a hollow cavity with a uniform sphere, then weigh the source from accelerations measured outside it.

Selected time $\text{Ready}$

Probe radius $\text{Ready}$

Probe inward acceleration $\text{Ready}$

Probe enclosed mass $\text{Ready}$

Probe specific potential $\text{Ready}$

Total source mass $\text{Ready}$

Tunnel particle position $\text{Ready}$

Tunnel particle speed $\text{Ready}$

Gravity clock $\text{Ready}$

Maximum specific-energy error $\text{Ready}$

Endpoint refinement difference $\text{Ready}$

Mass fitted from raw exterior scan $\text{Ready}$

Acquired field stations $\text{Ready}$

Set the density, then follow orbits and their averages

Inside a hollow shell, where does the shell's gravity go?

Sphere radius

R

6371.00\,km

Core boundary fraction

R_c/R

0.35\,

Core density

\rho_c

5500.00\,\mathrm{kg\,m^{-3}}

Middle-layer density

\rho_m

5500.00\,\mathrm{kg\,m^{-3}}

Outer-layer density

\rho_o

5500.00\,\mathrm{kg\,m^{-3}}

Radial field probe

r_p/R

0.60\,

Rest-release position

x_0/R

1.00\,

Observation window in gravity clocks

T/t_g

7.00\,

Integration step over characteristic clock

\Delta t/t_g

0.0050

Selected time fraction

t/T

0.28\,

Sensor noise gain

\eta

0.00\,

The sphere has three layers of constant density, and the middle layer ends at four fifths of the radius. The densities are adjustable for teaching; they do not model Earth's interior.

A tunnel with no mass and negligible width runs along a diameter, so the source stays spherical. Rotation, drag, the mass dug out for the tunnel and the test mass's pull back on the sphere are left out.

The blue probe and the gold falling particle move independently. All acceleration arrows share one scale. The cutaway marks where the density changes; it does not mean mass is missing from the source.

Specific potential, the potential energy per unit mass, is zero at infinity. A scan simulates acceleration measurements, and the mass fit uses only readings outside the sphere at calibrated radii. The noise is the noise gain times one hundredth of a metre per second squared.

The Newtonian radial field is prescribed in advance. Numerical errors are shown, and the limits of each approximation are stated.

Start with the essentials

Focus question: Inside a hollow shell, where does the shell's gravity go?
One-sentence intuition: Zero field does not imply zero potential.

Core mathematical model

Mass enclosed within a radius

M(<r)=4\pi\int_0^r\rho(s)s^2\,ds

The layer densities set how much mass lies inside each radius. Shells outside that radius exert no net force there.

Field and potential

\begin{aligned}g(r)&=\frac{GM(<r)}{r^2}\\\Phi(r)&=-G\left[\frac{M(<r)}r+4\pi\int_r^R\rho(s)s\,ds\right]\end{aligned}

Outer shells exert no force inside their cavity but still add a constant to the potential there. At the centre the limiting value is used, so nothing is divided by zero.

Motion inside a uniform sphere

\begin{aligned}g(r)&=\frac{4\pi G\rho}{3}r\quad(r\le R)\\x(t)&=x_0\cos\left(\sqrt{\frac{4\pi G\rho}{3}}\,t\right)\end{aligned}

A particle released from rest inside a uniform sphere oscillates harmonically. The force stays finite as it passes the centre.

Energy per unit mass

\varepsilon=\frac12\dot x^2+\Phi(|x|)

In this idealization the tunnel runs along a diameter and removes none of the source mass.

Weighing the source from outside

g_i=GM\,r_i^{-2}

Fit a line through zero to the raw accelerations measured outside the source, using the calibrated radii. The fit is never given the source mass.

Common difficulties

What the model assumes

Typical misconceptionAn ideal model like this automatically describes a real planet or a relaxed star cluster.

Better mental modelThe source is fixed in advance, the test particle does not pull back on it, and the effects left out are listed.

Run the experiment

01
Move the field probe
Drag the blue handle from the centre to the exterior. Compare the solid field with the dashed exterior point-mass reference.
What to observe: Inside the cavity the acceleration is zero while the potential remains negative.
02
Change the source
Try Uniform sphere, Dense core and Rest inside a hollow shell, and change all three densities.
What to observe: Where the density changes affects the field inside, but outside the sphere the field depends only on the total mass.
03
Send a particle across the cavity
Compare Rest inside a hollow shell with Cross the empty cavity, and drag the gold time handle.
What to observe: A particle that enters the field-free cavity keeps the velocity it had on entry.
04
Weigh the source from readings
Acquire a radial field scan, add noise and export the raw CSV.
What to observe: The mass fitted from the outside readings is shown separately from the exact source mass.