# Gravitational potential
The gravitational potential is the gravitational potential energy of a unit test mass at a point in a gravitational field, a scalar quantity from which the field and the motion of bodies can be calculated. Equivalently, it is the gravitational energy per unit mass of a particle.[1](https://astronomy.swin.edu.au/~gmackie/glossary.html) For a point mass \( M \), the potential at distance \( r \) is
\[ \Phi(r) = -\frac{GM}{r}, \]
with the zero of potential chosen at infinity, so the potential is negative everywhere and approaches zero as \( r \to \infty \).[2](https://www.feynmanlectures.caltech.edu/I_14.html) A test mass \( m \) placed in this potential has potential energy \( U = m\Phi \).[2](https://www.feynmanlectures.caltech.edu/I_14.html)
| Key fact | Detail |
|---|---|
| Definition | Gravitational energy per unit mass; potential energy of a test body is \( U = m\Phi \)[1](https://astronomy.swin.edu.au/~gmackie/glossary.html) • [2](https://www.feynmanlectures.caltech.edu/I_14.html) |
| Point mass | \( \Phi = -GM/r \), zero at infinity, negative everywhere[2](https://www.feynmanlectures.caltech.edu/I_14.html) |
| Governing equation | Poisson's equation \( \nabla^2 \Phi = 4\pi G\rho \); reduces to Laplace's equation \( \nabla^2 \Phi = 0 \) where the density is zero[3](https://www.phy.olemiss.edu/~luca/Topics/grav/newton.html) • [4](https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Celestial_Mechanics_%28Tatum%29/05%3A_Gravitational_Field_and_Potential/5.10%3A_Nabla_Gradient_and_Divergence) |
| Spherical shell | Outside: \( -Gm/r \), as if all mass were at the center; inside: constant \( -Gm/a \), so the field is zero[2](https://www.feynmanlectures.caltech.edu/I_14.html) |
| Uniform solid sphere | Inside: \( \psi = -\dfrac{GM}{2a^3}(3a^2 - r^2) \); outside: \( -GM/r \)[5](https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Celestial_Mechanics_%28Tatum%29/05%3A_Gravitational_Field_and_Potential/5.08%3A_The_Gravitational_Potentials_Near_Various_Bodies/5.8.09%3A_Solid_Sphere) |
| Binding criterion | Total energy zero or greater: the object escapes; total energy negative: the object is gravitationally bound[6](https://openstax.org/books/university-physics-volume-1/pages/13-3-gravitational-potential-energy-and-total-energy) |
| Escape speed | \( v_{\mathrm{esc}} = \sqrt{2GM/r} \), the speed at which total energy is zero[7](https://www.youtube.com/watch?v=3edB1t6s4WM) • [6](https://openstax.org/books/university-physics-volume-1/pages/13-3-gravitational-potential-energy-and-total-energy) |
## Definition and relation to potential energy
The potential is defined so that the potential energy of an object at a point \( (x, y, z) \) is \( m \) times the potential \( \Psi \) at that point. Richard Feynman, the Caltech physicist who co-developed quantum electrodynamics, presents this in his Lectures as a matter of convenience: instead of writing three components of a vector field, one gives a single scalar function, and because the potential is a scalar, contributions from many masses are added without worrying about direction.[2](https://www.feynmanlectures.caltech.edu/I_14.html) For point masses \( m_1, m_2, \ldots \), the potential at a point \( p \) is the sum of the individual contributions,
\[ \Psi(p) = \sum_i -\frac{Gm_i}{r_{ip}}. \]
[2](https://www.feynmanlectures.caltech.edu/I_14.html)
Only differences in potential energy are physically meaningful; the zero can be placed anywhere, and changing the reference point adds a constant to the potential.[2](https://www.feynmanlectures.caltech.edu/I_14.html) For an object near Earth's surface, where the field is approximately uniform, the potential energy is \( U(y) = mgy \) plus an arbitrary constant, and lifting a mass through a height \( h \) stores energy \( mgh \) in the object-Earth system.[8](https://openstax.org/books/college-physics/pages/7-3-gravitational-potential-energy) For separations comparable to Earth's radius, the \( 1/r \) form must be used instead; lifting a 9000 kg Soyuz vehicle from Earth's surface to the 400 km altitude of the ISS requires \( \Delta U = 3.32 \times 10^{10} \) J.[6](https://openstax.org/books/university-physics-volume-1/pages/13-3-gravitational-potential-energy-and-total-energy)
**Field from the potential.** The gravitational field is recovered from the potential by differentiation: the force component in one direction is the negative partial derivative of the potential energy with respect to that coordinate, \( F_x = -\partial U/\partial x \), taken with the other coordinates held constant.[2](https://www.feynmanlectures.caltech.edu/I_14.html) Where the potential is constant, there is no field, and where the potential energy is constant there is no force.[2](https://www.feynmanlectures.caltech.edu/I_14.html)
## Poisson's equation
Because the gravitational field is the negative gradient of the potential and its divergence equals \( -4\pi G \) times the local density, the potential obeys [Poisson's equation](https://www.edgepedia.org/poissons-equation),
\[ \nabla^2 \Phi = 4\pi G\rho. \]
[3](https://www.phy.olemiss.edu/~luca/Topics/grav/newton.html) • [4](https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Celestial_Mechanics_%28Tatum%29/05%3A_Gravitational_Field_and_Potential/5.10%3A_Nabla_Gradient_and_Divergence) At any point where the local density is zero this becomes [Laplace's equation](https://www.edgepedia.org/laplaces-equation), \( \nabla^2 \Phi = 0 \); no matter how complicated the mass distribution, the potential must satisfy these equations.[4](https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Celestial_Mechanics_%28Tatum%29/05%3A_Gravitational_Field_and_Potential/5.10%3A_Nabla_Gradient_and_Divergence) The curl of a gravitational field is zero, which is what allows a scalar potential to exist at all.[4](https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Celestial_Mechanics_%28Tatum%29/05%3A_Gravitational_Field_and_Potential/5.10%3A_Nabla_Gradient_and_Divergence)
Far from any bounded mass distribution, the potential can be expanded in multipoles:
\[ \Phi(r) = -\frac{GM}{r} - \frac{G\,\mathbf{D}\cdot\mathbf{r}}{r^3} - \frac{1}{2}G\,Q_{ij}\,\frac{r^i r^j}{r^5} + \cdots, \]
where \( \mathbf{D} \) is the dipole moment, always zero when measured about the center of mass, and \( Q_{ij} \) is the quadrupole moment, which vanishes for a spherical mass distribution.[3](https://www.phy.olemiss.edu/~luca/Topics/grav/newton.html) A perfectly spherical body therefore produces a pure \( 1/r \) potential at large distances.
## Potentials of extended bodies
**Spherical shell.** Adding the contributions from all parts of a spherical shell of radius \( a \) and mass \( m \) gives a potential that is zero at infinity, varies as \( 1/r \) down to the radius \( a \), and is constant inside the shell. Outside, the potential is \( -Gm/r \), exactly the same as if all the mass were located at the center; inside, it is \( -Gm/a \) everywhere.[2](https://www.feynmanlectures.caltech.edu/I_14.html) Because the potential inside is constant, the field there is zero: moving an object between any two points inside the shell requires no work.[2](https://www.feynmanlectures.caltech.edu/I_14.html) This is the content of Newton's [shell theorem](https://www.edgepedia.org/shell-theorem).[9](https://www.sparknotes.com/physics/gravitation/potential/section3/)
**Solid sphere.** The potential outside a solid sphere is the same as if all the mass were concentrated at a point in the center, and this holds even for non-uniform density as long as it is spherically distributed.[5](https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Celestial_Mechanics_%28Tatum%29/05%3A_Gravitational_Field_and_Potential/5.08%3A_The_Gravitational_Potentials_Near_Various_Bodies/5.8.09%3A_Solid_Sphere) Inside a uniform sphere of radius \( a \) and mass \( M \), the potential at distance \( r \) from the center is found by adding the contribution from the mass interior to the point, \( -GM_r/r \) with \( M_r = r^3 M / a^3 \), to the integrated contributions of the shells above it:[5](https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Celestial_Mechanics_%28Tatum%29/05%3A_Gravitational_Field_and_Potential/5.08%3A_The_Gravitational_Potentials_Near_Various_Bodies/5.8.09%3A_Solid_Sphere)
\[ \psi = -\frac{GM}{2a^3}\left(3a^2 - r^2\right). \]
Unlike the shell, the potential inside a solid sphere is not constant; it is steepest in magnitude at the center, where \( \psi = -3GM/2a \), and the field grows linearly with distance from the center before falling off as \( 1/r^2 \) outside.[5](https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Celestial_Mechanics_%28Tatum%29/05%3A_Gravitational_Field_and_Potential/5.08%3A_The_Gravitational_Potentials_Near_Various_Bodies/5.8.09%3A_Solid_Sphere) • [10](https://refreshkid.com/learn/ap-physics-c-mechanics/unit-2/spherical-gravity/)
## Binding energy and escape
The sign of the total energy, kinetic plus potential, determines whether a body escapes. As \( r \to \infty \), \( U \to 0 \). If the total energy is zero, the object comes to rest infinitely far away; it has just escaped. If the total energy is positive, kinetic energy remains even at infinity. In either case the object is not gravitationally bound. For a negative total energy E in the potential U = -GMm/r, the kinetic energy need not reach zero, but nonnegative kinetic energy requires r ≤ GMm/|E|, so the object remains gravitationally bound.[6](https://openstax.org/books/university-physics-volume-1/pages/13-3-gravitational-potential-energy-and-total-energy) Because energy is a scalar, this conclusion holds for any direction of motion, and a bound system can maintain orbital motion rather than falling together.[6](https://openstax.org/books/university-physics-volume-1/pages/13-3-gravitational-potential-energy-and-total-energy)
[Escape velocity](https://www.edgepedia.org/escape-velocity) is the speed at any position such that the total energy is zero.[6](https://openstax.org/books/university-physics-volume-1/pages/13-3-gravitational-potential-energy-and-total-energy) Setting the kinetic energy \( \tfrac{1}{2}mv^2 \) equal to the magnitude of the potential energy \( GMm/r \) gives
\[ v_{\mathrm{esc}} = \sqrt{\frac{2GM}{r}}. \]
[7](https://www.youtube.com/watch?v=3edB1t6s4WM) The formulas for point objects apply to finite-sized, spherically symmetric objects as well, provided \( r \) stays greater than the sum of the radii of the two objects; below that, they collide, and neither positive nor negative total energy precludes finite-sized masses from colliding.[6](https://openstax.org/books/university-physics-volume-1/pages/13-3-gravitational-potential-energy-and-total-energy)
## Beyond Newtonian gravity
In cosmological structure formation, the standard approach assumes a homogeneous expanding background and encodes the gravitational effect of inhomogeneities entirely in a Newtonian potential, though how completely this captures the dynamics of general relativity remains an active question in precision cosmology.[11](https://arxiv.org/abs/1711.06681)
## References
1. [Glossary entry: gravitational potential, Swinburne University COSMOS](https://astronomy.swin.edu.au/~gmackie/glossary.html)
2. [The Feynman Lectures on Physics, Vol. I Ch. 14: Work and Potential Energy](https://www.feynmanlectures.caltech.edu/I_14.html)
3. [Topics: Newtonian Gravitation, University of Mississippi](https://www.phy.olemiss.edu/~luca/Topics/grav/newton.html)
4. [5.10: Nabla, Gradient and Divergence, Physics LibreTexts (Jeremy Tatum, University of Victoria)](https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Celestial_Mechanics_%28Tatum%29/05%3A_Gravitational_Field_and_Potential/5.10%3A_Nabla_Gradient_and_Divergence)
5. [5.8.9: Solid Sphere, Physics LibreTexts (Jeremy Tatum, University of Victoria)](https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Celestial_Mechanics_%28Tatum%29/05%3A_Gravitational_Field_and_Potential/5.08%3A_The_Gravitational_Potentials_Near_Various_Bodies/5.8.09%3A_Solid_Sphere)
6. [13.3 Gravitational Potential Energy and Total Energy, University Physics Volume 1, OpenStax](https://openstax.org/books/university-physics-volume-1/pages/13-3-gravitational-potential-energy-and-total-energy)
7. [Escape velocity derivation, Science Ready (video)](https://www.youtube.com/watch?v=3edB1t6s4WM)
8. [7.3 Gravitational Potential Energy, College Physics, OpenStax](https://openstax.org/books/college-physics/pages/7-3-gravitational-potential-energy)
9. [Gravitation: Potential, Newton's Shell Theorem, SparkNotes](https://www.sparknotes.com/physics/gravitation/potential/section3/)
10. [Gravity inside a sphere and a shell, AP Physics C: Mechanics Unit 2, Refresh Kid](https://refreshkid.com/learn/ap-physics-c-mechanics/unit-2/spherical-gravity/)
11. [Comparing Fully General Relativistic and Newtonian Calculations of Structure Formation, arXiv:1711.06681](https://arxiv.org/abs/1711.06681)
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*Topic: Encyclopedia › Physical world and mathematics › Physics › Classical physics › Mechanics › Momentum, energy, and work › Mechanical energy › Potential energy › Gravitational potential and Newtonian potential energy*
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