= Newtonian gravitational field
{c}
{title2=$\mathbf g$}
The <Newtonian gravitational field> is the gravitational <force> per unit <mass> of a <test particle>. For a <point mass> $M$ at the origin,
$$
\mathbf g(\mathbf r)=\frac{\mathbf F}{m}=-\frac{GM}{r^3}\mathbf r,\qquad g(r)=|\mathbf g|=\frac{GM}{r^2}.
$$
The <test particle>'s <mass> cancels: different small <masses> have the same <gravitational acceleration> at the same <position>. In <SI>, the <Newtonian gravitational field> has units $\mathrm{N\,kg^{-1}}=\mathrm{m\,s^{-2}}$. The same exterior expression holds for a spherically symmetric source, by the <spherical shell theorem>.
Choose a reference radius $r_0>0$, set $g_0=GM/r_0^2$, and use <nondimensionalization> with $z=r/r_0$. Then $g/g_0=z^{-2}$: at $z=1,2,4$, the magnitudes are $1,1/4,1/16$. This makes the spatial variation visible without specifying a planet. https://openstax.org/books/university-physics-volume-1/pages/13-1-newtons-law-of-universal-gravitation[OpenStax: inverse-square attraction and spherical sources]{ref}
\Image[/gravity-force-and-potential.png]
{title=Normalized Newtonian gravitational field and point-mass potential}
{description=Doubling the radius quarters the magnitude of the <Newtonian gravitational field>, but halves the magnitude of the negative <Newtonian gravitational potential>. Both curves approach zero as radius increases; the inward <gravitational acceleration> is minus the <gradient> of the <Newtonian gravitational potential>.}
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