Gravitational Potential Calculator

Introduction to gravitational potential

Gravitational potential is the energy landscape around a mass, and this calculator evaluates that landscape with the Newtonian point-mass model. The quantity V represents potential energy per unit mass, so its SI unit is joules per kilogram (J/kg). A more negative value means a deeper gravitational well, which is why satellites, planets, and stars are often compared through their potentials rather than through force alone.

This calculator solves for one missing quantity in the relationship between gravitational potential V, central mass M, and center-to-point distance r. Use it when you need a quick outside-the-body estimate for a planet, moon, or star; when you want the mass implied by a known potential; or when you need the distance that corresponds to a particular energy depth. The sign convention is the standard one from introductory physics: zero potential at infinity and negative potential near the attracting body.

How to use the gravitational potential calculator

1. Enter values in SI units so the gravitational potential calculator can work directly with kilograms, meters, and 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, which makes large planetary masses and orbital distances easier to enter. Remember that r is measured from the center of the body, not from the surface. If you start from an altitude, convert it first with r = radius + altitude; that center-to-center distance is the value the Newtonian formula needs. This is especially important for spacecraft, because a surface height that looks small on paper can still change the computed potential enough to matter.

Formula for Newtonian gravitational potential

Newton’s inverse-square law for gravitational force between two masses is: 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

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: (implemented numerically as 6.6743e-11).

The calculator rearranges the same relation whichever field you leave blank. If V is blank, it solves V = −GM/r. If M is blank, it solves M = −Vr/G. If r is blank, it solves r = −GM/V. Because potential is defined relative to infinity, the negative sign is not an error; it is the convention that makes gravitational wells read correctly.

Worked example: gravitational potential 400 km above Earth

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 a large upward push to escape Earth entirely.

To reproduce this with the calculator, enter Mass M = 5.97e24 and Distance r = 6.77e6, leaving Potential V blank. Comparing that value with Earth’s surface potential shows why low Earth orbit is only a modest step above the ground in potential terms: the orbit is high enough to matter, but it is nowhere near the zero reference at infinity.

Assumptions and limitations of V = −GM/r

This gravitational-potential calculator uses the standard Newtonian point-mass formula, which is the right tool for many school, engineering, and astronomy problems but not for every mass distribution. It is exact for a point mass and for any spherically symmetric body when the point of interest is outside the body. That makes it ideal for planets, moons, and stars treated at a distance, while keeping the model intentionally 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.

Interpreting gravitational potential values

Gravitational potential is a scalar field, which means it is often easier to reason with than force when you are tracking energy changes. The specific mechanical energy of a moving object is: Bound orbits have negative total specific energy, while escape corresponds to reaching zero total energy at infinity.

Near Earth’s surface, small height changes can be approximated by ΔV ≈ g h for modest separations, where g ≈ 9.81 m/s². That approximation matches the full −GM/r formula when h is tiny compared with Earth’s radius. If you are comparing two close altitudes, the gh shortcut is usually easier than subtracting large negative potentials. Use the full relation when the change in radius is not negligible, as in satellite motion, interplanetary transfers, or escape-speed questions.

Reference gravitational potential values

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 specific energy to reach infinity with zero speed. In orbit, objects are still in a negative potential well, but their kinetic energy partly offsets that depth. That is why energy methods are so useful: the balance between kinetic and potential energy tells you whether a trajectory is bound, parabolic, or hyperbolic.

Common gravitational-potential pitfalls (and how to avoid them)

Many incorrect results come from unit or interpretation mistakes rather than from the formula itself. If the output looks strange, 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 derived from gravitational 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 show why the calculator is useful as a starting point rather than 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 about gravitational potential

Why is gravitational potential negative in this calculator?

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 for this gravitational potential calculator

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 physics, the optional mini-game below turns the same V = −GM/r relationship into a visual challenge: move inward to deepen the well because r shrinks, and switch to a heavier world to deepen it because M grows.