What is the mass, in M Sun, of a star observed to have a planet with an orbital

Question

What is the mass, in MSun, of a star observed to have a planet with an orbital radius of 4.00 AU and a period of 18.00 years?

Hint: Newton's version of Kepler's third law states that P2=42R3G(MStar+Mp)P2=42R3G(MStar+Mp), where PP is the orbital period, RR is the average orbital radius, MSunMSun is the mass of the star, and MpMp is the mass of the planet. As shown in your book, when MpMSunMpMSun (the planet is much less massive than the star), the mass of the star can be determined by MStar=42R3GP2MStar=42R3GP2. You can save work on unit conversions if you treat this as a ratio problem comparing this system to the Earth-Sun system.

Explanation / Answer

given
orbital radius of planet = R = 4 AU
Period, T = 18 Y
now

from kepler's third law
T^2 = 4*pi*pi*a^3/GM
now for T in earth years, a in AU
1 = 4pi^2/GMs ( Ms is mass of sun)
Ms = 4*pi^2/G

hence for the given situation
18^2 = 4*pi^2*4^3/GM
M = Ms*(4^3/18^2) = 0.19753Ms

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