Analog of Gauss' law in gravity

[LagrangeEuler]

Gauss law in case of sphere which has charge $q$ is $\oint \vec{E}\cdot d\vec{S}=\frac{q}{\epsilon_0}$

Is there some anologone for case of sphere with mass $m$ such that
$\oint \vec{G}\cdot d\vec{S}=4\pi \gamma m$ and what is $\vec{G}$ in that case?

 

[Conformal Bear]

Yes - it is Gauss' law for gravitation. Note that it differs from Gauss' law in electrostatics by the presence of a minus sign on the right:
$$\oint_{\partial V} \vec{g} \cdot d\vec{S} = -4\pi Gm$$
This is because gravitation is strictly attractive, while the electrostatic force can be either attractive or repulsive. The field $\vec{g}$ is the gravitational field, defined completely analogously to the electric field whereby the force experienced by a particle of mass $\mu$ in the field is $\vec{F}_{g} = \mu \vec{g}$.

 

[Dr. Courtney]

https://en.wikipedia.org/wiki/Gauss's_law_for_gravity

Gauss's law for gravity is not nearly as well known. I recall some years ago my wife gave a talk in a physics department and referred to it.

There were plenty of physicists in the room who were not familiar with the idea, but everyone quickly grasped it with a 30 second explanation.