The Great Conjunction#

On December 21, 2020, Saturn and Jupiter appeared the closest together in the night sky that they have in the last 400 years (the “Great conjunction”). Assuming that both Jupiter and Saturn make circular orbits of a stationary sun and have paths unaffected by the masses of each other and other planets,

Part 1#

Write Newton’s second law for Jupiter in terms of $$\omega_J$$.

You can use the following variables: $$G$$, $$M\_{\odot}$$, $$M_J$$, $$M_S$$, $$R\_{\odot J}$$, $$R\_{\odot S}$$, $$\omega_S$$ and $$\omega_J$$.

(Here $$S$$ stands for Saturn, $$J$$ stands for Jupiter, and $$\odot$$ for the sun; $$R\_{\odot S}$$ is the distance from the sun to Saturn, $$R\_{\odot J}$$ is the distance from the sun to Jupiter; $$\omega_S$$ is the angular speed of Saturn around the sun, and $$\omega_J$$ is the angular speed of Jupiter around the sun. )

Note that it may not be necessary to use every variable. Use the following table as a reference for each variable:

For

Use

$$G$$

G

$$M_J$$

M_J

$$M_S$$

M_S

$$R\_{\odot J}$$

R_sunJ

$$R\_{\odot S}$$

R_sunS

$$\omega_S$$

o_S

$$\omega_J$$

o_J

Part 2#

Write Newton’s second law for Saturn in terms of $$\omega_J$$.

You can use the following variables: $$G$$, $$M\_{\odot}$$, $$M_J$$, $$M_S$$, $$R\_{\odot J}$$, $$R\_{\odot S}$$, $$\omega_S$$ and $$\omega_J$$.

(Here $$S$$ stands for Saturn, $$J$$ stands for Jupiter, and $$\odot$$ for the sun; $$R\_{\odot S}$$ is the distance from the sun to Saturn, $$R\_{\odot J}$$ is the distance from the sun to Jupiter; $$\omega_S$$ is the angular speed of Saturn around the sun, and $$\omega_J$$ is the angular speed of Jupiter around the sun. )

Note that it may not be necessary to use every variable. Use the following table as a reference for each variable:

For

Use

$$G$$

G

$$M_J$$

M_J

$$M_S$$

M_S

$$R\_{\odot J}$$

R_sunJ

$$R\_{\odot S}$$

R_sunS

$$\omega_S$$

o_S

$$\omega_J$$

o_J

Find $$\frac{\omega_S}{\omega_J}$$ if $$M_S = 5.68\times 10^{26}$$ kg, $$M_J = 1.898 \times 10^{27}$$ kg, $$R\_{\odot J}$$ = 5.2 AU and $$R\_{\odot S}$$ = 9.5 AU, where 1 AU is the Earth-sun distance.