Gravitational Binding Energy Sphere Calculator

Inputs

Input Mode

Choose any two properties; the third is derived.

Mass

Use scientific notation when needed.

Radius

Radius must be positive.

Average Density

Uniform-density model; real bodies vary.

Note: This model assumes a uniform-density sphere. For layered planets or stars, the true binding energy differs, often by tens of percent or more.

Formula Used

For a uniform-density sphere of total mass M and radius R, the gravitational binding energy is:

U = −(3/5) · G · M² / R

The negative sign indicates energy must be added to disperse the sphere to infinity. This calculator also reports the magnitude |U| for easy comparisons.

How to Use This Calculator

Select an input mode (choose any two of mass, radius, density).
Enter values and choose appropriate units for each field.
Click Calculate to display results above this form.
Use Download CSV or Download PDF to export results.
Compare |U|, escape speed, and specific binding energy across objects.

Example Data Table

Sample values (uniform sphere approximation). Results are approximate.

Object	Mass (kg)	Radius (m)	\|U\| (J)	v_esc (m/s)
Earth (rough)	5.972×10²⁴	6.371×10⁶	2.242×10³²	1.119×10⁴
Sun (rough)	1.988×10³⁰	6.963×10⁸	2.274×10⁴¹	6.174×10⁵
Neutron star (1.4 M☉, 12 km)	2.784×10³⁰	1.200×10⁴	2.586×10⁴⁶	1.759×10⁸

Try these rows in the calculator to reproduce similar magnitudes.

Notes and Interpretation

Scaling: Doubling mass increases |U| by about four times, if radius stays fixed.
Compactness: Smaller radius at the same mass makes the object much more tightly bound.
Limits: Extremely compact objects require relativistic models, not this approximation.

Gravitational Binding Energy Sphere Guide

1) Purpose of this calculator

This tool estimates the gravitational binding energy of a uniform sphere. Binding energy is the energy you must supply to separate the body completely. It is useful for planetary formation, stellar structure, and impact analysis.

2) What binding energy means physically

Gravity stores negative potential energy in a bound object. A larger magnitude means the body is harder to disrupt. In many problems, |U| sets the scale for deformation and catastrophic breakup.

3) Model and constant values

The calculation uses the uniform-density sphere result, U = −(3/5)GM²/R. The gravitational constant is G = 6.67430×10⁻¹¹ m³·kg⁻¹·s⁻². The density shown is the mean density implied by your inputs.

4) Input options and units

You can provide mass and radius, density and radius, or mass and density. The third property is derived using M = (4/3)πR³ρ or R = (3M/4πρ)^1/3. Unit menus support common planetary and stellar scales for faster comparisons.

5) What the calculator reports

Results include U, |U|, and the specific binding energy |U|/M in J/kg. Surface potential Φ = −GM/R is also shown in J/kg. Escape speed v_esc = √(2GM/R) helps validate compactness.

Because |U| scales with M²/R, small input changes can matter a lot. If you increase mass by 10% while holding radius constant, |U| rises by about 21%. If you shrink radius by 10% at fixed mass, |U| rises by about 11%. These sensitivities help when estimating uncertainties from observational data. This makes the tool valuable for quick scenario screening work. When comparing objects, keep units consistent and use scientific notation for clarity.

6) Practical data points for context

For Earth-like inputs (M≈5.97×10²⁴ kg, R≈6.37×10⁶ m), the uniform-sphere estimate gives |U|≈2.24×10³² J and v_esc≈11.2 km/s. For the Sun (M≈1.99×10³⁰ kg, R≈6.96×10⁸ m), |U| is about 2.27×10⁴¹ J, with v_esc≈617 km/s.

7) Compact objects and energy scales

A neutron star is extremely compact, so |U| becomes enormous. Using 1.4 solar masses and a 12 km radius gives |U|≈2.59×10⁴⁶ J. The calculator also expresses energy in kWh and in foe, where 1 foe = 10⁴⁴ J.

8) Accuracy, limits, and best practices

Real bodies are not uniform; cores and envelopes change the true value. Relativistic effects matter for very compact objects, especially neutron stars. Use this tool for fast estimates, then refine with detailed structure models.

FAQs

1) Why is the binding energy negative?

Gravity lowers potential energy as matter assembles. The negative sign indicates a bound configuration. You must add energy equal to |U| to disperse it completely.

2) Which input mode should I use?

Use the two properties you know best. Mass and radius are common for planets and stars. Density and radius work well for laboratory spheres and material models.

3) What does specific binding energy tell me?

It is |U| divided by mass, measured in J/kg. Higher values mean each kilogram is more tightly bound. It is useful for comparing bodies of very different sizes.

4) Is the uniform-density assumption realistic?

It is an approximation. Differentiated planets and stars have varying density profiles. The true binding energy may differ substantially, but the scaling trends remain useful.

5) How do I sanity-check my result?

Check escape speed and mean density. If v_esc seems too high for your object, recheck units. Compare your magnitude to known bodies for plausibility.

6) What does “foe” mean in the output?

It is a common astrophysics energy unit. One foe equals 10⁵¹ erg, which is 10⁴⁴ joules. It helps compare results to supernova-scale energies.

7) Can I use this for very compact stars?

Use it for rough estimates only. White dwarfs and neutron stars can require relativistic corrections. For precision, use an equation-of-state and a structure solver.