Reviewed by: JJ Ben-Joseph

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Gravity is often introduced through the force it exerts on masses, encapsulated by Newton's inverse-square law $F=\frac{\mathrm{G\; M\; m}}{r^2}$. Equally important, yet sometimes overlooked in early studies, is the concept of gravitational potential. Defined as the potential energy per unit mass, gravitational potential tells us how much work is required to bring a test particle from infinitely far away to a point in space. For a spherically symmetric mass distribution the potential is $V=-\frac{\mathrm{G\; M}}{r}$, where $G$ is the universal gravitational constant. This calculator allows you to compute $V$, $M$, or $r$ given the other two quantities.

The gravitational potential concept provides a scalar field that simplifies many problems. Instead of dealing with vector forces, one can often calculate differences in potential and use energy conservation to deduce motion. Potential is particularly useful when analyzing escape velocity, orbital mechanics, or the behavior of satellites and planets. It also plays a role in general relativity, where potential helps describe how time flows in a gravitational field. Our calculator uses the classical formula, which is adequate for most everyday and solar system situations where relativistic corrections are tiny.

To use the calculator, fill in any two of the three fields. Enter a mass and distance to find the potential; specify potential and mass to solve for the radius corresponding to that potential; or input potential and distance to determine the mass responsible. The script ensures that exactly one field is blank and then rearranges the equation $V=-\frac{\mathrm{G\; M}}{r}$ accordingly. The gravitational constant is taken as $G=6.67430\times {10}^{-} \mathrm{m³\cdot kg^-1\cdot s^-2}$. Results are displayed with full precision in joules per kilogram for potential, kilograms for mass, and meters for distance.

Gravitational potential is negative because the force is attractive: work must be done against gravity to separate masses, while gravity does positive work when they fall together. At infinite separation the potential is defined to be zero. As you move closer to a mass, the potential decreases, approaching negative infinity at the center of an ideal point mass. For extended objects like planets, the formula holds outside the body as if all mass were concentrated at its center. Inside, one must account for the mass distribution, but near Earth's surface the potential difference between two heights $h$ apart approximates $\mathrm{\Delta V}=gh$, where $g$ is the gravitational field strength.

The table below lists sample potentials around various celestial bodies to provide a sense of scale.

Body	Mass (kg)	Radius (m)	Surface Potential (J/kg)
Earth	5.97×1024	6.37×106	−6.26×107
Moon	7.35×1022	1.74×106	−1.54×106
Jupiter	1.90×1027	6.99×107	−2.73×108
Sun	1.99×1030	6.96×108	−1.91×1011

The magnitudes reveal why it is so difficult to escape massive bodies. Earth's surface potential of roughly −62 MJ/kg means that a kilogram mass must gain that much energy per kilogram to reach infinity. Rockets achieve this by accelerating propellant to high velocities. In orbit, objects still reside in a negative potential well, but their kinetic energy balances the gravitational potential energy so that total mechanical energy becomes negative for bound orbits and zero for escape trajectories.

Gravitational potential also offers insight into tides. The Moon's potential varies slightly across Earth's diameter because the near side is closer than the far side. This differential potential creates tidal bulges in the oceans. In a similar way, Earth's potential variation across a satellite contributes to tidal locking, gradually synchronizing the satellite's rotation with its orbital period. The potential concept thus illuminates a variety of phenomena beyond simple free fall.

In advanced contexts, potential helps solve the Poisson equation $\partial V^2=-4\pi \mathrm{G\rho }$ for mass density $\rho$. Potential theory leads naturally to Laplace's equation in vacuum regions, enabling expansion in spherical harmonics to describe nonspherical mass distributions. While this calculator targets the simple point-mass case, understanding the basics prepares students for these more elaborate analyses encountered in astrophysics and geophysics.

Operating entirely within your browser, the calculator requires no network access and preserves privacy. The JavaScript function checks the number of provided inputs and warns if the configuration is ambiguous. Scientific notation is accepted, so you can enter values like 5.97e24 for Earth's mass. The output message clearly states the computed quantity with appropriate units.

As an example, suppose you wish to know the gravitational potential 400 km above Earth's surface, the altitude of the International Space Station. Enter $M=5.97\times {10}^{24}$ and $r=6.77\times {10}^6$ to obtain $V=-5.90\times {10}^7$ J/kg. This is slightly less negative than the surface value, reflecting the weaker gravitational grip at altitude. If you instead know the potential and distance, the calculator can infer the mass; for instance, using $V=-2.73\times {10}^8$ J/kg at $r=6.99\times {10}^7$ returns Jupiter's mass of 1.90×10²⁷ kg.

Gravitational potential underpins energy methods in orbital mechanics. The total specific mechanical energy of an orbiting body is $\epsilon =\frac{v^2}{2}+V$. For bound orbits, $\epsilon$ is negative and equals $-\frac{\mathrm{G\; M}}{\mathrm{2\; a}}$, where $a$ is the semi-major axis. Understanding potential allows you to derive orbital speeds, periods, and transfer energies with elegant energy conservation arguments instead of detailed force analyses.

Beyond classical physics, gravitational potential influences time itself. General relativity predicts that clocks run more slowly in deeper gravitational wells. Although our calculator operates within Newtonian theory, the potential it computes approximates the gravitational redshift formula $\frac{\mathrm{\Delta t}}{t}\approx \frac{V}{c^2}$ for weak fields. For GPS satellites, engineers must account for both special relativistic and gravitational potential differences to maintain timing accuracy.

Whether you are tackling homework, planning a space mission, or simply curious about the invisible energy landscape sculpted by massive objects, this tool offers a convenient and educational aid. Experiment with masses ranging from asteroids to stars, vary the distance, and observe how the potential changes. By grasping gravitational potential, you gain deeper insight into orbital dynamics, energy conservation, and the interplay between mass and spacetime.