In classical mechanics, the gravitational potential at a location is equal to the work (energy transferred) per unit mass that would be needed to move an object to that location from a fixed reference location. It is analogous to the electric potential with mass playing the role of charge. The reference location, where the potential is zero, is by convention infinitely far away from any mass, resulting in a negative potential at any finite distance.
In mathematics, the gravitational potential is also known as the Newtonian potential and is fundamental in the study of potential theory. It may also be used for solving the electrostatic and magnetostatic fields generated by uniformly charged or polarized ellipsoidal bodies.
The gravitational potential (V) at a location is the gravitational potential energy (U) at that location per unit mass:
where m is the mass of the object. Potential energy is equal (in magnitude, but negative) to the work done by the gravitational field moving a body to its given position in space from infinity. If the body has a mass of 1 kilogram, then the potential energy to be assigned to that body is equal to the gravitational potential. So the potential can be interpreted as the negative of the work done by the gravitational field moving a unit mass in from infinity.
In some situations, the equations can be simplified by assuming a field that is nearly independent of position. For instance, in a region close to the surface of the Earth, the gravitational acceleration, g, can be considered constant. In that case, the difference in potential energy from one height to another is, to a good approximation, linearly related to the difference in height:
The gravitational potential V at a distance x from a point mass of mass M can be defined as the work W that needs to be done by an external agent to bring a unit mass in from infinity to that point:
where G is the gravitational constant, and F is the gravitational force. The product GM is the standard gravitational parameter and is often known to higher precision than G or M separately. The potential has units of energy per mass, e.g., J/kg in the MKS system. By convention, it is always negative where it is defined, and as x tends to infinity, it approaches zero.
The gravitational field, and thus the acceleration of a small body in the space around the massive object, is the negative gradient of the gravitational potential. Thus the negative of a negative gradient yields positive acceleration toward a massive object. Because the potential has no angular components, its gradient is
where x is a vector of length x pointing from the point mass toward the small body and is a unit vector pointing from the point mass toward the small body. The magnitude of the acceleration therefore follows an inverse square law:
The potential associated with a mass distribution is the superposition of the potentials of point masses. If the mass distribution is a finite collection of point masses, and if the point masses are located at the points x1, ..., xn and have masses m1, ..., mn, then the potential of the distribution at the point x is
If the mass distribution is given as a mass measure dm on three-dimensional Euclidean space R3, then the potential is the convolution of −G/|r| with dm. In good cases[ clarification needed ] this equals the integral
where |x − r| is the distance between the points x and r. If there is a function ρ(r) representing the density of the distribution at r, so that dm(r) = ρ(r) dv(r), where dv(r) is the Euclidean volume element, then the gravitational potential is the volume integral
If V is a potential function coming from a continuous mass distribution ρ(r), then ρ can be recovered using the Laplace operator, Δ:
This holds pointwise whenever ρ is continuous and is zero outside of a bounded set. In general, the mass measure dm can be recovered in the same way if the Laplace operator is taken in the sense of distributions. As a consequence, the gravitational potential satisfies Poisson's equation. See also Green's function for the three-variable Laplace equation and Newtonian potential.
The integral may be expressed in terms of known transcendental functions for all ellipsoidal shapes, including the symmetrical and degenerate ones.These include the sphere, where the three semi axes are equal; the oblate (see reference ellipsoid) and prolate spheroids, where two semi axes are equal; the degenerate ones where one semi axes is infinite (the elliptical and circular cylinder) and the unbounded sheet where two semi axes are infinite. All these shapes are widely used in the applications of the gravitational potential integral (apart from the constant G, with 𝜌 being a constant charge density) to electromagnetism.
A spherically symmetric mass distribution behaves to an observer completely outside the distribution as though all of the mass was concentrated at the center, and thus effectively as a point mass, by the shell theorem. On the surface of the earth, the acceleration is given by so-called standard gravity g, approximately 9.8 m/s2, although this value varies slightly with latitude and altitude. The magnitude of the acceleration is a little larger at the poles than at the equator because Earth is an oblate spheroid.
Within a spherically symmetric mass distribution, it is possible to solve Poisson's equation in spherical coordinates. Within a uniform spherical body of radius R, density ρ, and mass m, the gravitational force g inside the sphere varies linearly with distance r from the center, giving the gravitational potential inside the sphere, which is
which differentiably connects to the potential function for the outside of the sphere (see the figure at the top).
In general relativity, the gravitational potential is replaced by the metric tensor. When the gravitational field is weak and the sources are moving very slowly compared to light-speed, general relativity reduces to Newtonian gravity, and the metric tensor can be expanded in terms of the gravitational potential.
The potential at a point x is given by
The potential can be expanded in a series of Legendre polynomials. Represent the points x and r as position vectors relative to the center of mass. The denominator in the integral is expressed as the square root of the square to give
where, in the last integral, r = |r| and θ is the angle between x and r.
(See "mathematical form".) The integrand can be expanded as a Taylor series in Z = r/|x|, by explicit calculation of the coefficients. A less laborious way of achieving the same result is by using the generalized binomial theorem. The resulting series is the generating function for the Legendre polynomials:
valid for |X| ≤ 1 and |Z| < 1. The coefficients Pn are the Legendre polynomials of degree n. Therefore, the Taylor coefficients of the integrand are given by the Legendre polynomials in X = cos θ. So the potential can be expanded in a series that is convergent for positions x such that r < |x| for all mass elements of the system (i.e., outside a sphere, centered at the center of mass, that encloses the system):
The integral is the component of the center of mass in the x direction; this vanishes because the vector x emanates from the center of mass. So, bringing the integral under the sign of the summation gives
This shows that elongation of the body causes a lower potential in the direction of elongation, and a higher potential in perpendicular directions, compared to the potential due to a spherical mass, if we compare cases with the same distance to the center of mass. (If we compare cases with the same distance to the surface, the opposite is true.)
The absolute value of gravitational potential at a number of locations with regards to the gravitation from [ clarification needed ] the Earth, the Sun, and the Milky Way is given in the following table; i.e. an object at Earth's surface would need 60 MJ/kg to "leave" Earth's gravity field, another 900 MJ/kg to also leave the Sun's gravity field and more than 130 GJ/kg to leave the gravity field of the Milky Way. The potential is half the square of the escape velocity.
|Location||with respect to|
|Earth's surface||60 MJ/kg||900 MJ/kg||≥ 130 GJ/kg|
|LEO||57 MJ/kg||900 MJ/kg||≥ 130 GJ/kg|
|Voyager 1 (17,000 million km from Earth)||23 J/kg||8 MJ/kg||≥ 130 GJ/kg|
|0.1 light-year from Earth||0.4 J/kg||140 kJ/kg||≥ 130 GJ/kg|
Compare the gravity at these locations.
In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a planet, moon, asteroid, or Lagrange point. Normally, orbit refers to a regularly repeating trajectory, although it may also refer to a non-repeating trajectory. To a close approximation, planets and satellites follow elliptic orbits, with the center of mass being orbited at a focal point of the ellipse, as described by Kepler's laws of planetary motion.
In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as
In fluid mechanics, hydrostatic equilibrium is the condition of a fluid or plastic solid at rest, which occurs when external forces, such as gravity, are balanced by a pressure-gradient force. In the planetary physics of Earth, the pressure-gradient force prevents gravity from collapsing the planetary atmosphere into a thin, dense shell, whereas gravity prevents the pressure-gradient force from diffusing the atmosphere into outer space.
An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.
In physics, work is the energy transferred to or from an object via the application of force along a displacement. In its simplest form, for a constant force aligned with the direction of motion, it equals the product of the force strength and the distance traveled. A force is said to do positive work if when applied it has a component in the direction of the displacement of the point of application. A force does negative work if it has a component opposite to the direction of the displacement at the point of application of the force.
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols , , or . In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form. Informally, the Laplacian Δf (p) of a function f at a point p measures by how much the average value of f over small spheres or balls centered at p deviates from f (p).
Geopotential is the potential of the Earth's gravity field. For convenience it is often defined as the negative of the potential energy per unit mass, so that the gravity vector is obtained as the gradient of this potential, without the negation. In addition tothe actual potential, a hypothetical normal potential and their difference, the disturbing potential, can also be defined.
Stellar dynamics is the branch of astrophysics which describes in a statistical way the collective motions of stars subject to their mutual gravity. The essential difference from celestial mechanics is that the number of body
In mathematical physics, scalar potential, simply stated, describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in traveling from one position to the other. It is a scalar field in three-space: a directionless value (scalar) that depends only on its location. A familiar example is potential energy due to gravity.
A classical field theory is a physical theory that predicts how one or more physical fields interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called quantum field theories. In most contexts, 'classical field theory' is specifically intended to describe electromagnetism and gravitation, two of the fundamental forces of nature.
In classical mechanics, the shell theorem gives gravitational simplifications that can be applied to objects inside or outside a spherically symmetrical body. This theorem has particular application to astronomy.
In mathematics, a multiple integral is a definite integral of a function of several real variables, for instance, f(x, y) or f(x, y, z). Integrals of a function of two variables over a region in are called double integrals, and integrals of a function of three variables over a region in are called triple integrals. For multiple integrals of a single-variable function, see the Cauchy formula for repeated integration.
Spherical multipole moments are the coefficients in a series expansion of a potential that varies inversely with the distance R to a source, i.e., as 1/R. Examples of such potentials are the electric potential, the magnetic potential and the gravitational potential.
In physics, the Green's function for Laplace's equation in three variables is used to describe the response of a particular type of physical system to a point source. In particular, this Green's function arises in systems that can be described by Poisson's equation, a partial differential equation (PDE) of the form
In physics, the Laplace expansion of potentials that are directly proportional to the inverse of the distance, such as Newton's gravitational potential or Coulomb's electrostatic potential, expresses them in terms of the spherical Legendre polynomials. In quantum mechanical calculations on atoms the expansion is used in the evaluation of integrals of the inter-electronic repulsion.
In physics, Gauss's law for gravity, also known as Gauss's flux theorem for gravity, is a law of physics that is equivalent to Newton's law of universal gravitation. It is named after Carl Friedrich Gauss. It states that the flux of the gravitational field over any closed surface is equal to the mass enclosed. Gauss's law for gravity is often more convenient to work from than is Newton's law.
In general relativity, a point mass deflects a light ray with impact parameter by an angle approximately equal to
In geophysics and physical geodesy, a geopotential model is the theoretical analysis of measuring and calculating the effects of Earth's gravitational field.
A Jacobi ellipsoid is a triaxial ellipsoid under hydrostatic equilibrium which arises when a self-gravitating fluid body of uniform density rotates with a constant angular velocity. It is named after the German mathematician Carl Gustav Jacob Jacobi.