Gravitational Self-Energy Profiles
Build a sphere shell by shell and find when finite mass still has divergent self-energy.
Free theory articleBuild a sphere shell by shell and find when finite mass still has divergent self-energy.
Concept & interpretation
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.
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.