Calculator Inputs
Formula Used
For a uniform sphere, the magnitude of gravitational binding energy is:
U = (3/5)GM²/R
For a custom internal structure model, the calculator uses:
U = αGM²/R
For two masses separated by distance r, the calculator uses:
U = GM₁M₂/r
Here, G is the gravitational constant, M is mass, R is radius, and α is a dimensionless structure coefficient. Additional outputs use E = mc², average density ρ = M / ((4/3)πR³), and compactness 2GM/Rc².
How to Use This Calculator
- Select a calculation mode based on your physical model.
- Enter masses using the unit selectors for direct conversion.
- Enter radius for a sphere or separation for a two-body system.
- Use α = 0.6 for a uniform sphere approximation.
- Set display precision to control scientific notation detail.
- Click the calculate button to show results above the form.
- Review energy, mass deficit, density, compactness, and velocity outputs.
- Use the CSV or PDF buttons to export the results.
Example Data Table
| Object | Mass (kg) | Radius (m) | Model | Approx. Binding Energy (J) |
|---|---|---|---|---|
| Earth | 5.972e24 | 6.371e6 | Uniform sphere | 2.241759e32 |
| Jupiter | 1.898e27 | 6.9911e7 | Uniform sphere | 2.063497e36 |
| Sun | 1.9885e30 | 6.9634e8 | Uniform sphere | 2.273981e41 |
| Earth-Moon Pair | 5.972e24 and 7.348e22 | 3.844e8 separation | Two-body system | 7.619234e28 |
Frequently Asked Questions
1. What does gravitational binding energy represent?
It is the energy needed to separate a bound object or system completely against gravity. Larger mass and smaller size usually increase the binding energy magnitude significantly.
2. Why does the uniform sphere formula use 3/5?
That factor comes from integrating gravitational potential energy through a sphere with constant density. Real stars and planets often have layered density profiles, so their true factor can differ.
3. When should I use the custom structure coefficient option?
Use it when you already know a better structural coefficient from theory, simulation, or literature. It helps model centrally condensed objects more realistically than a simple constant-density sphere.
4. Is the two-body result the same as total system binding energy?
It gives the magnitude of the pair’s gravitational potential energy at the chosen separation. Full dynamical binding can depend on orbital kinetic energy and whether the orbit is truly bound.
5. What does mass-equivalent deficit mean here?
It converts binding energy into an equivalent mass using E = mc². This value shows how much rest-energy corresponds to the calculated gravitational binding energy.
6. Why include compactness in the output?
Compactness measures how relativistically concentrated the mass is. Small values indicate Newtonian conditions, while larger values suggest stronger relativistic effects and caution with simple formulas.
7. Can I use this for neutron stars or black hole scales?
You can estimate trends, especially with the compactness output. However, strongly relativistic objects may require equations of state or general relativity instead of simple Newtonian expressions.
8. What units can I enter in this calculator?
Masses can be entered in kilograms, grams, Earth masses, Jupiter masses, or solar masses. Lengths can be entered in meters, centimeters, kilometers, radii, or astronomical units.