Gravitational Self-Energy Profiles

Gravitational self-energy represents the work required to assemble a mass distribution by bringing matter from infinity. For a sphere with mass enclosed M(r) = M·(r/R)^α, the full self-energy is finite if and only if α > 1/2. Smaller exponents lead to infinite self-energy as the mass is concentrated too steeply near the origin, even though the total mass remains finite.

Worked Example

Consider a sphere with Mass exponent α = 1, Core fraction ε = 0.1, Total mass M = 1, and Outer radius R = 1. We can calculate the work required to assemble the outer shell surrounding the core.

The core mass is M(ε) = M·(ε)^α = 1·(0.1)¹ = 0.1.
The outer mass is M - M(ε) = 1 - 0.1 = 0.9.
The full self-energy for α > 1/2 is E = -α/(2α - 1) · (M²/R). For α = 1, this evaluates to -(1)/(2(1)-1) · 1 = -1.0.
The work to assemble the outer shell (from r = εR to r = R) is calculated by integrating the self-energy contributions outside the core:W_outer = -α/(2α - 1) · (M²/R) · (1 - ε^{2α - 1})
For α = 1 and ε = 0.1, the outer-shell assembly energy is -(1)/(1) · (1) · (1 - 0.1¹) = -0.9.

Experiment

Interact with the parameters in the Gravitational Self-Energy Profiles module to see how energy diverges for different mass distributions:

Set the Mass exponent to 3 (uniform density sphere). Observe that the full self-energy is -0.6 and finite.
Decrease the Mass exponent to 1. The mass is now more centrally concentrated, and the full self-energy becomes more negative (-1.0).
Set the Mass exponent to 0.5. Note how the outer-shell assembly energy approaches logarithmic divergence (-1.151... at ε = 0.1), while the full self-energy becomes "Divergent negative".
Change the Core fraction to 0 at α = 0.5 to verify the full self-energy diverges for an infinitely small core.

Limitations

The continuous power-law models M(r) ∝ r^α explored here are idealized mathematical profiles intended to illustrate divergent integrals, not real stellar equations of state. Physical objects cannot reach infinite density at their cores; fundamental physics limits (e.g., neutron degeneracy pressure or event horizons) take over long before a true mathematical divergence occurs. Additionally, this formulation uses normalized gravitational units (G = 1) for mathematical clarity rather than standard SI units.

References

The concepts underlying gravitational self-energy are fundamental in astrophysics and classical mechanics.

Feynman Lectures on Physics, Vol I, Chapter 13. Discusses gravitational potential energy and the energy of a sphere.
Ohio State University Astronomy Lecture 4. Details gravitational binding energy in stellar contexts.

Gravitational Self-Energy Profiles