Newtonian gravitational potential
Gravitational potential is potential energy per unit test mass. For a point mass, or outside a spherical mass, choosing zero potential at infinity gives φ = −GM/r.
The negative sign reflects the chosen reference: energy must be supplied to move a test mass from a bound finite distance to infinity. Potential is a scalar, unlike gravitational field.
How to calculate gravitational potential
- Choose the unknown: Select potential, source mass, or distance.
- Use the stated reference: Potential inputs must use zero at infinity and are non-positive for a positive isolated mass.
- Enter center distance: Use distance from the source center, not altitude alone.
- Review applicability: Confirm the point-mass or exterior spherical-body Newtonian model is suitable.
Formula and variables
The calculator uses the 2022 CODATA gravitational constant G = 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻² and the zero-at-infinity convention.
φ = −GM/r- φ — Gravitational potential
- Potential energy per unit test mass (J/kg)
- G — Gravitational constant
- Newtonian constant of gravitation (m³ kg⁻¹ s⁻²)
- M — Source mass
- Point or spherical source mass (kg)
- r — Center distance
- Distance from the source center (m)
Potential near Earth radius
Use M = 5.972 × 10²⁴ kg and r = 6.371 × 10⁶ m.
- Source mass
- 5.972 × 10²⁴ kg
- Center distance
- 6.371 × 10⁶ m
- φ = −(6.67430 × 10⁻¹¹)(5.972 × 10²⁴)/(6.371 × 10⁶)
Result: Potential is approximately −6.2563 × 10⁷ J/kg.
Relative to zero at infinity, a test mass at that distance has negative gravitational potential energy per kilogram.
Understanding your results
Potential depends on its reference
Only potential differences directly determine changes in potential energy.
- A more negative value is deeper in the ideal gravitational potential well.
- As distance approaches infinity, potential approaches zero from below.
- Potential energy for a test mass m is U = mφ.
- Gravitational field is related to the spatial gradient of potential, not the potential value alone.
Assumptions
- Zero gravitational potential is defined at infinity.
- Newtonian gravity is sufficient.
- The source is a point mass or the evaluation point is outside a spherically symmetric mass.
Limitations
- Does not add potentials from multiple bodies or model nonspherical and interior mass distributions.
- Does not calculate relativistic spacetime effects, escape energy, or orbital energy directly.
- Does not propagate uncertainty in G, mass, or distance.
Common mistakes
- Dropping the negative sign while using zero at infinity.
- Confusing gravitational potential in J/kg with potential energy in joules.
- Entering altitude rather than center distance.
- Comparing potentials that use different zero references.
Practical use cases
Potential-well estimates
Compare ideal external gravitational potential at different radii.
Energy preparation
Calculate potential per unit mass before forming a potential-energy difference.
Frequently asked questions
Why is gravitational potential negative?
With zero chosen at infinity, a positive test mass at finite distance is in a bound state and requires added energy to reach infinity.
Is gravitational potential the same as potential energy?
No. Potential is energy per unit mass in J/kg; multiply by the test mass to obtain potential energy in joules.
Can gravitational potential be zero?
Under this convention it approaches zero at infinity, and it is zero everywhere if the source mass is zero.
Sources and review
- 2022 CODATA Recommended Values of the Fundamental Physical Constants — National Institute of Standards and Technology. Accessed 2026-07-13.
- Chapter 13 Key Equations — OpenStax University Physics Volume 1. Accessed 2026-07-13.
Reviewed 2026-07-13.