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 1.8 × 107 m and its rotation period to be 22.3 hours. You have previously determined that the planet orbits 3.3 × 1011 m from its star with a period of 402 earth days. Once on the surface you find that the acceleration due to gravity is 93.4 m/s2. What are the mass of (a) the planet and (b) the star? (G = 6.67 × 10-11 N · m2/kg2)

1. (a) 2.1 × 10^26 kg, (b) 1.8 × 10^31 kg 2. (a) 1.1 × 10^26 kg, (b) 1.0 × 10^31 kg 3. (a) 2.1 × 10^26 kg, (b) 1.0 × 10^31 kg 4. (a) 1.1 × 10^26 kg, (b) 1.8 × 10^31 kg

Explanation / Answer

we can find the mass of the planet from g=GM/R^2 where G is newtonian grav cst, M the mass of the planet, R the radius and g thelocal accel due to grav

solving for M:

M = g*R^2/G

M = 93.4*(9x10^6)^2/6.67x10^(-11)
M = 1.13x10^26 kg.

find the mass of the star from Kepler's 3rd law

equate gravitational 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 r/time
v=2 x 3.3 x 10^11/402days x 86,400sec/day
v = 5.9666 × 10^4 m/s
M = (5.9666x10^4)^2*3.3x10^11/6.67x10^(-11)
M = 1.761 x 10^31 kg

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