(a) Express in terms of and . · OlympiadBench #985

Physics · mechanics

Problem

(a) Express

\alpha

in terms of

M_{1}

and

M_{2}

. 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}

Context question: (b) Find the period

T

and angular frequency

\omega

of the orbital motion. Context answer:

\omega=\sqrt{\frac{G\left(M_{1}+M_{2}\right)}{\left|\overrightarrow{r_{2}}-\overrightarrow{r_{1}}\right|^{3}}}

T=2 \pi \sqrt{\frac{\left|\overrightarrow{r_{2}}-\overrightarrow{r_{1}}\right|^{3}}{G\left(M_{1}+M_{2}\right)}}

Extra Supplementary Reading Materials: Circular restricted three-body problem Let us transform to a non-inertial frame rotating with angular velocity

\vec{\omega}=(0,0, \omega)

about an axis normal to the orbital plane of masses

M_{1}

and

M_{2}

, with the origin at their center of mass. In this frame,

M_{1}

and

M_{2}

are stationary at the Cartesian coordinates

(-\alpha R, 0,0)

and

((1-\alpha) R, 0,0)

respectively. The third mass

m

is not stationary in this frame; in this non-inertial frame its position is

\vec{r}(t)=(x(t), y(t), 0)

. The masses satisfy

M_{1}, M_{2} \gg m

. Consider

m

to be so small that it does not affect the motion of

M_{1}

M_{2}

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