If Jupiter and Saturn are both in line with each other and the Sun and located o

Question

If Jupiter and Saturn are both in line with each other and the Sun and located on the same side with respect to the Sun, calculate the location of the center of mass of this system of three bodies with respect to the center of the Sun. Express your answer in terms of the Sun’s radius.

We can approximate Jupiter’s and Saturn’s distance from the Sun as aJ =7.791011m,aS =1.431012m . The Sun’s equatorial radius is 6.96108m . The average radii of the two planets are RJ = 6.99 107 m, RS = 5.82 107 m . The Sun’s mass is 1.9891030kg, while MJ =1.89861027kg,MS =5.6841026kg, are Jupiter’s and Saturn’s mass, respectively.

On the way to solving the problem, please express the Sun’s and Jupiter’s mass in units of Saturn’s mass (i.e. how many times more massive is each than Saturn). Express the distances and the radii of Jupiter and Saturn in terms of the Sun’s radius. You answer must be expressed in terms of the Sun’s radius. Be sure to show a sketch.

Explanation / Answer

Ms = mass of sun = 1.98 x 1030 kg

MJ = mass of jupiter = 1.8986 x 1027kg

Msat = mass of saturn = 5.684 x 1026kg

assume sun at the origin

X1 = distance of jupitor from sun = 7.79 x 1011 m

X2 = distance of saturn from sun = 1.43 x 1012 m

center of mass is given as

Xcm = (Ms (Xo) + MJ X1 + Msat X2 )/ (Ms + MJ + Msat)

Xcm = ((1.98 x 1030 ) (0) + (1.8986 x 1027) (7.79 x 1011 ) + (5.684 x 1026 ) (1.43 x 1012 )) / ((1.98 x 1030 ) + (1.8986 x 1027) + (5.684 x 1026 ))

Xcm = 1.16 x 109 m

mass of sun = (1.98 x 1030 ) / (5.684 x 1026 ) )mass of saturn = 3483.5 times mass of saturn

mass of jupitor = 1.8986 x 1027 / (5.684 x 1026 ) )mass of saturn = 3.34 times

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