= Newtonian gravitational potential
{c}
{title2=$\Phi$}
The <Newtonian gravitational potential> $\Phi$ is <potential energy> per unit <mass> of a <test particle>. Its <gradient> determines the <Newtonian gravitational field>:
$$
U=m\Phi,\qquad \mathbf g=-\nabla\Phi,\qquad \mathbf F=-\nabla U=-m\nabla\Phi.
$$
In <SI>, $U$ has units $\mathrm J$, while $\Phi$ has units $\mathrm{J\,kg^{-1}}=\mathrm{m^2\,s^{-2}}$; its <gradient> has units of <acceleration>, not <force>. Adding a constant to $\Phi$ leaves the <Newtonian gravitational field> unchanged. For an isolated <point mass>, the convenient reference is $\Phi(\infty)=0$. https://ocw.mit.edu/courses/16-07-dynamics-fall-2009/0bb19205a6316ba88f88284bf7257ec6_MIT16_07F09_Lec16.pdf[Peraire and Widnall, MIT Dynamics, Lecture 16: potential per unit mass]{ref}
Increasing the radius from $r_0$ to $r_1>r_0$ raises the <Newtonian gravitational potential energy>:
$$
\Delta U=GMm\left(\frac1{r_0}-\frac1{r_1}\right)>0.
$$
An external agent supplies this positive <work> if the initial and final <kinetic energies> agree; gravity does <work> $-\Delta U$. For $r_1=2r_0$, the required <work> is $GMm/(2r_0)$. <University Physics Volume 1 by William Moebs and others/13.3 Gravitational Potential Energy and Total Energy>{ref}
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