= Newton's law of universal gravitation
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{title2=$\mathbf F=-GMm\mathbf r/|\mathbf r|^3$}
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<Newton's law of universal gravitation> gives the attractive <force> between two <point masses>. With source <mass> $M$ at the origin and relative <position> $\mathbf r$ of <mass> $m$, write $r=|\mathbf r|>0$:
$$
\mathbf F_{M\to m}=-\frac{GMm}{r^3}\mathbf r,\qquad |\mathbf F_{M\to m}|=\frac{GMm}{r^2}.
$$
Here $G$ is the <gravitational constant>. The minus sign points toward $M$; the magnitude follows an <inverse-square law>, so doubling the separation quarters the <force>. The opposite <force> on $M$ satisfies <Newton's third law>. https://openstax.org/books/university-physics-volume-1/pages/13-1-newtons-law-of-universal-gravitation[OpenStax: mutual gravitational forces]{ref}
Both <masses> accelerate. Applying <Newton's second law> to each in the isolated <two-body problem> gives
$$
\ddot{\mathbf r}=-\frac{\mu}{r^3}\mathbf r,\qquad \mu=G(M+m).
$$
The relative <standard gravitational parameter> includes both <masses>; replacing it by $GM$ requires the <test particle> approximation $m\ll M$.
Historically, Newton's Book III, Proposition VII argues that gravitational attraction applies to all bodies and scales with their <masses>; its second corollary states the <inverse-square law> for individual particles. https://en.wikisource.org/wiki/The_Mathematical_Principles_of_Natural_Philosophy_(1729)/Proposition_7:_Gravity_tending_to_all_Bodies[Newton, Principia, Book III, Proposition VII, Motte's 1729 translation]{ref}
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