Gravitational Potential Calculator

Introduction

Gravitational potential is a compact way to describe gravity using energy rather than force. In Newtonian physics, the gravitational potential V at a point is defined as the potential energy per unit mass. Its SI unit is joules per kilogram (J/kg). Conceptually, it answers a practical question: how much energy per kilogram would you need to supply to move a tiny test mass from this location out to infinity, assuming it starts and ends at rest?

This calculator solves for one missing quantity in the classic point-mass, or spherically symmetric, model: potential V, central mass M, or distance r from the mass center. It is useful for quick checks in orbital mechanics, escape energy estimates, comparing gravitational depth around different bodies, and sanity-checking homework problems. Because potential is a scalar, it often gives a cleaner big-picture view than force does. Instead of asking only how strongly gravity pulls, you can ask how deep the local energy well is and how much energy a spacecraft or projectile would need to climb back out.

How to use the calculator

1. Use consistent SI units: mass in kilograms (kg), distance in meters (m), potential in J/kg.
2. Fill in exactly two fields and leave the third field blank.
3. Click Compute Missing Quantity. The result appears below the form.

Scientific notation is accepted, so values like Earth’s mass can be entered as 5.97e24. The distance r is measured from the center of the mass, not from its surface. If you know altitude above a planet, use r = planet_radius + altitude. If you are working with a spacecraft far from a planet, r is simply the planet-center distance at that moment. That center-based measurement is one of the most common places where otherwise careful calculations go wrong.

Formula (Newtonian gravitational potential)

Newton’s inverse-square law for gravitational force between two masses is: $F=\frac{GMm}{r^2}$ From this, the gravitational potential, meaning potential energy per unit mass, for a point mass, or for a spherically symmetric body when you are outside the body, is:

Formula: V = − (G M) / r

$V=-\frac{\mathrm{G\; M}}{r}$

Where G is the universal gravitational constant, M is the mass creating the field, and r is the distance from the mass center. The calculator uses: $G=6.67430\times {10}^{-11}{m}^3·{\mathrm{kg}}^{-1}·s^{-2}$ (implemented numerically as 6.6743e-11).

Rearrangements used by the calculator are straightforward. If V is blank, it computes V = −GM/r. If M is blank, it rearranges to M = −Vr/G. If r is blank, it uses r = −GM/V. The signs matter. Because the zero of potential is usually set at infinity, gravitational potential near an attracting mass is negative in this convention.

Worked example (ISS altitude)

Find the gravitational potential at the altitude of the International Space Station, about 400 km above Earth. Use Earth’s mass M = 5.97e24 kg and Earth’s mean radius R ≈ 6.37e6 m. The distance from Earth’s center is r = R + h = 6.37e6 + 4.00e5 = 6.77e6 m.

Plug the numbers into V = −GM/r and you get approximately −5.9e7 J/kg. The value is negative because the reference is defined as V = 0 at infinity, and gravity is attractive. In plain language, a kilogram of mass sitting at that altitude is still in a deep gravitational well. It would need tens of megajoules of energy per kilogram to get all the way out to infinity with no leftover speed.

To reproduce this with the calculator, enter Mass M = 5.97e24 and Distance r = 6.77e6, leaving Potential V blank. If you compare that value with Earth’s surface potential, you will notice that the ISS is only a little less negative than the surface value. That is a useful reminder that low-Earth orbit is still very much inside Earth’s gravity well.

Assumptions and limitations

This page uses the standard Newtonian point-mass formula, which is extremely useful but not universal. It works exactly for a point mass and for a spherically symmetric body when the point of interest lies outside the body. The calculator is therefore excellent for most introductory astronomy, orbital, and planetary examples, but it intentionally keeps the model simple.

• Point mass or spherical symmetry: V = −GM/r is exact for a point mass and for a spherically symmetric body outside the body. Inside a planet, the mass distribution matters.
• Distance is from the center: Using surface distance instead of center distance is a very common source of error.
• Newtonian model: Relativistic effects are ignored. For most solar-system calculations this is fine; near very compact objects it is not.
• Sign convention: This calculator uses the common physics convention where potential is negative and approaches 0 as r → ∞.
• Single source mass: It does not sum potentials from multiple bodies. In Newtonian gravity, potentials add linearly, so multi-body situations require adding contributions from each mass.

Interpretation and practical notes

Gravitational potential is a scalar field that often simplifies problems. Instead of tracking vector forces directly, you can compare potential values and use energy conservation. The specific mechanical energy of a moving object is: $\epsilon =\frac{v^2}{2}+V$ Bound orbits have negative total specific energy, while escape corresponds to reaching zero total energy at infinity.

Near Earth’s surface, small changes in potential can be approximated by ΔV ≈ g h for modest height differences, where g ≈ 9.81 m/s². This is consistent with the full −GM/r expression when h is small compared with Earth’s radius. If you are comparing two nearby altitudes, the gh approximation is usually more convenient than computing two large negative potentials and subtracting them. The full formula matters most when the change in distance is not tiny compared with the body’s radius, as in satellite motion, interplanetary travel, or escape-speed reasoning.

Reference values (for scale)

The table below lists approximate surface potentials for several bodies. These values are meant for intuition and quick comparisons. Your computed results may differ slightly depending on the mass and radius values you use, but the scale is what matters most. A deeper negative value means a deeper gravity well and a larger energy cost per kilogram to climb out.

The magnitudes help explain why escaping massive bodies is difficult. Earth’s surface potential of roughly −62 MJ/kg means a 1 kg object must gain about 62 MJ of energy per kilogram to reach infinity with zero speed. In orbit, objects are still in a negative potential well, but their kinetic energy partially offsets the potential energy. This is why orbital motion can be analyzed cleanly with energy methods: the balance between kinetic energy and potential energy determines whether a trajectory is bound, parabolic, or hyperbolic.

Common pitfalls (and how to avoid them)

Many incorrect results come from unit or interpretation mistakes rather than from the formula itself. If your output looks surprising, check the following items before assuming the physics is wrong.

• Mixing kilometers and meters: If you enter 6371 for Earth’s radius, you are using kilometers. Convert to meters: 6.371e6.
• Using altitude as r: Altitude above the surface is not the same as distance from the center. Use r = R + h.
• Forgetting the sign: In this convention, V is negative. If you enter a positive potential, the computed distance may come out negative or otherwise nonphysical.
• Leaving the wrong field blank: The calculator expects exactly one blank field. If you fill all three, it cannot know which one to solve for; if you fill only one, it is underdetermined.
• Interpreting V as potential energy: This calculator uses specific potential, meaning per unit mass. To get potential energy for a mass m, multiply: U = mV.

Related quantities you can compute from potential

Once you have gravitational potential, you can connect it to several other useful quantities. These are not computed automatically here, but the relationships help you interpret the number you get and see why this calculator is a useful starting point instead of an isolated formula.

• Potential energy: For a test mass m, the gravitational potential energy is U = mV. Example: if V = −6.26e7 J/kg at Earth’s surface, then a 2 kg mass has U ≈ −1.25e8 J relative to infinity.
• Escape speed: Setting total specific energy to zero gives v_escape = sqrt(−2V). At Earth’s surface, this yields about 11.2 km/s.
• Orbital speed (circular orbit): For a circular orbit at radius r, v = sqrt(GM/r). Combining with V = −GM/r shows how orbital speed and potential are tied to the same ratio.
• Gravitational acceleration: The magnitude of the gravitational field is g(r) = GM/r². Potential is related to how gravity changes with radius, which is why potential is so useful in energy-based reasoning.

FAQ

Why is gravitational potential negative?

The zero of potential is chosen at infinity. Because gravity is attractive, you must do positive work to separate masses to infinity, so the potential at finite r is lower than zero. The negative sign in V = −GM/r encodes that convention.

Can I use this for locations inside a planet?

Not directly. Inside a planet, the potential depends on how mass is distributed with radius. The outside-the-body formula can still be used for points above the surface, and it is exact for a spherically symmetric body when r is greater than the body’s radius.

Does this include the Moon’s effect on Earth or the Sun’s effect on satellites?

No. This calculator models a single source mass. In Newtonian gravity you can add potentials from multiple bodies, but you must compute each contribution with its own distance and then sum them.

Is this the same as gravitational potential energy?

It is closely related but not the same quantity. Potential V is energy per unit mass. Potential energy is U = mV. Using specific quantities is convenient because it removes the test mass from the equation.

Privacy note

This calculator runs entirely in your browser. No values are sent to a server, and the computation happens locally when you press the button. If you want a hands-on feel for the same idea, the optional mini-game below turns the formula into a fast visual challenge: diving inward makes the well deeper because r shrinks, and swapping to a heavier world makes the whole field more negative because M grows.