(b) Find the period and angular frequency of the orbital motion. · OlympiadBench #984

Physics · mechanics

Problem

(b) Find the period

T

and angular frequency

\omega

of the orbital motion. Given constants: 3. The circular restricted three-body problem In general, there is no exact solution of the three-body problem, in which three masses move under their mutual gravitational attraction. However, it is possible to make some progress by adding some constraints to the motion. Two-body problem Let's start with the motion of two masses,

M_{1}

and

M_{2}

. Assume both masses move in circular orbits about their center of mass. Consider the inertial frame whose origin coincides with the center of mass of the system. Context question: (a) Express the equations of motion of

M_{1}

and

M_{2}

in terms of the gravitational constant

G

and the position vectors

\overrightarrow{r_{1}}

and

\overrightarrow{r_{2}}

which point from the origin to

M_{1}

and

M_{2}

, respectively. Context answer:

\begin{aligned} & \frac{d^{2}}{d t^{2}} \overrightarrow{r_{1}}=\frac{G M_{2}}{\left|\overrightarrow{r_{2}}-\overrightarrow{r_{1}}\right|^{3}}\left(\overrightarrow{r_{2}}-\overrightarrow{r_{1}}\right) \\ & \frac{d^{2}}{d t^{2}} \overrightarrow{r_{2}}=\frac{G M_{1}}{\left|\overrightarrow{r_{1}}-\overrightarrow{r_{2}}\right|^{3}}\left(\overrightarrow{r_{1}}-\overrightarrow{r_{2}}\right) \end{aligned}

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