# Specific potential energy

Specific potential energy is potential energy of an object per unit of mass of that object. In a gravitational field it is the acceleration of gravity times height, $e_{u}=gh$.

## Mathematical form

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:
$V(\mathbf {x} )={\frac {W}{m}}={\frac {1}{m}}\int _{\infty }^{x}\mathbf {F} \cdot d\mathbf {x} ={\frac {1}{m}}\int _{\infty }^{x}{\frac {GmM}{x^{2}}}dx=-{\frac {GM}{x}},$ 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 $\mathbf {a} =-{\frac {GM}{x^{3}}}\mathbf {x} =-{\frac {GM}{x^{2}}}{\hat {\mathbf {x} }},$ where x is a vector of length x pointing from the point mass toward the small body and ${\hat {\mathbf {x} }}$ is a unit vector pointing from the point mass toward the small body. The magnitude of the acceleration therefore follows an inverse square law: $\|\mathbf {a} \|={\frac {GM}{x^{2}}}.$
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 x$_1$, ..., x$_n$ and have masses m$_1$, ..., m$_n$, then the potential of the distribution at the point x is $V(\mathbf {x} )=\sum _{i=1}^{n}-{\frac {Gm_{i}}{\|\mathbf {x} -\mathbf {x} _{i}\|}}.$
If the mass distribution is given as a mass measure dm on three-dimensional Euclidean space R$^3$, then the potential is the convolution of −G/|r| with dm. In good cases this equals the integral $V(\mathbf {x} )=-\int _{\mathbb {R} ^{3}}{\frac {G}{\|\mathbf {x} -\mathbf {r} \|}}\,dm(\mathbf {r} ),$ 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 $V(\mathbf {x} )=-\int _{\mathbb {R} ^{3}}{\frac {G}{\|\mathbf {x} -\mathbf {r} \|}}\,\rho (\mathbf {r} )dv(\mathbf {r} ).$
If V is a potential function coming from a continuous mass distribution ρ(r), then ρ can be recovered using the Laplace operator, Δ: $\rho (\mathbf {x} )={\frac {1}{4\pi G}}\Delta V(\mathbf {x} ).$ 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.