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Here $g$ is the gravitational field, $G$ is the gravitational constant, and $M$ is the total mass in the volume $V$. I wonder if this formula holds for any unit system. That is, does the coefficient $4\pi$ hold true for any unit system such SI or cgs? The maxwell equations change their form with the unit system given. So I am confused.

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The integral form of Newtonian gravitation as well as the differential form follows from Newton’s force law of gravity. The 4 pi factor comes from the integration over a solid angle and is therefore independent of the units used. The only time the equation would seem to change in form is if natural units are used. In that case G would become 1.

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  • $\begingroup$ In gravitational physics research, it is actually just as common (if not more common) to set $8 \pi G = 1$, which greatly simplifies the Einstein equation. $\endgroup$ – Michael Seifert Feb 24 '18 at 23:38

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