What is the mass of Planet X in solar masses?
Planet X is observed to have a small moon. This moon is observed to orbit the planet once per month at a distance of 15 Planet X diameters. What is the mass of Planet X in solar masses?
Through previous problems, I have found that the diameter of Planet X is 1.2×10^8 meters and has a semi-major axis of 45 AU with a period of 300 Earth years.
Is the answer:
A. 2.5 × 10?13 solar masses
B. 7.4 × 10?8 solar masses
C. 2.5 × 10?4 solar masses
D. 8.4 × 10?4 solar masses
E. 2.4 × 10?3 solar masses
F. 390 solar masses
Answer by Zardoz
M + m = 4?²r³/GP², since m is small
M = 4?²r³/GP², since 4?²/G is a constant with a value of 5.92[11]
M = 5.92[11](r³/P²).
If we use units of seconds and meters then the mass will be returned in kilograms. To get M? we’ll need to divide by 1.99[30] kg. 5.92[11]/1.99[30] = 2.97[-19]; ergo,
M = 2.97[-19](r³/P²).
A period of “one month” isn’t a particularly exacting standard. For the sake of calculation let’s call it ? 30 days ? 2.6[6] seconds.
And r = 15 • 1.2[8] m = 1.8[9] m
M = 2.97[-19](1.8[9]³/2.6[6]²)
M = 2.97[-19](5.83[27]/6.76[12])
M = 2.97[-19] • 8.62[14]
M = 2.6[-4] M?
ANSWER C. 2.5[-4] solar masses.
Edit: I thought it referenced Nubiru at first, but then since values were given I realized it was a generic planet X.
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