How does mass affect orbital period?

Question

32189 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-04-19T07:32:42

When one object orbits another due to gravity (i.e. planet around a sun) we say that the centripetal force is brought around by the force of gravity:

$\frac{m v^{2}}{r} = \frac{G M m}{r^{2}}$ $(mv^2)/r=(GMm)/r^2$

$\frac{v^{2}}{r} = \frac{G M}{r^{2}}$ $v^2/r=(GM)/r^2$

$v = \frac{2 π r}{t}$ $v=(2pir)/t$

$\frac{4 π^{2} r^{2}}{2 r t^{2}} = \frac{G M}{r^{2}}$ $(4pi^2r^2)/(2rt^2)=(GM)/r^2$

$t^{2} = \frac{2 π^{2} r^{3}}{G M}$ $t^2=(2pi^2r^3)/(GM)$

$t = \sqrt{\frac{2 π^{2} r^{3}}{G M}}$ $t=sqrt((2pi^2r^3)/(GM))$

An increase in the mass of he orbited body causes a decrease in the orbital period.