You are the science officer on a visit to a distant solar system. Prior to landi

Question

You are the science officer on a visit to a distant solar system. Prior to landing on a planet you measure its diameter to be 1.8x10^7m and its rotation period to be 22.3 hours. You have previously determined that the plant orbits 2.2x10^11 m form its star with a period of 402 earth days. Once on the surface you find that the free-fall acceleration is 12.2 m/s^2

PLEASE ANSWER BOTH QUESTIONS!!!!!!!!!!!!!!

A) What is the mass of the planet (answer should be in kg and SHOULDN'T depend on x10)

B) What is the mass of the star (answer should be kg and SHOULDN'T depend on x10)

Explanation / Answer

The mass of the planet from g=GM/R^2

where G is newtonian gravity constant , M the mass of the planet, R the radius and g the local acceleration due to gravity.

solving for M:

M=gR^2/G

M=12.2*(9x10^6)^2/6.67x10^(-11)
M=1.48x10^25kg

M= 14815592203898050974512743.628186 kg

The mass of the star from Kepler's 3rd law (or the procedure that derives the law)

equate grav force to centripetal force

GMm/r^2=mv^2/r
M=mass of star
m=mass planet
r=size of orbit
v=orbital speed

this becomes M=v^2r/G

v=circumference of orbit/time = 2 pi r/time
v=2 pi x 2.2x10^11/402days x 86,400sec/day
v=39798.13800152907410066459049207 m/s

M= (39798.13800152907410066459049207)^2*2.2x10^11/6.67x10^(-11)
M=5224230786289738724310697017760.30 kg

it is approx 2.6 times the mass of the sun

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