You have been hired as a consultant for the new Star Trek TV series to make sure

Question

You have been hired as a consultant for the new Star Trek TV series to make sure that the science in the show is correct. In one episode, the crew of the Enterprise goes into standard orbit around a newly discovered planet. The plot requires that the planet be hollow and contain the underground cities of a lost civilization. From orbit the science officer determines that the radius of the planet is one-fourth that of Earth. The first officer beams down to the surface of the planet and measures that his weight is only one-half of his weight on Earth. How does the mass of this planet compare with the mass of the Earth? If it were hollow, its density would be less than Earth. Are the measurements consistent with a hollow planet?

(a) What is the mass of this planet in terms of the mass of the Earth (M)?
(b) What is the density of this planet in terms of the density of the Earth (D)?

Explanation / Answer

the radius of the planet is assumed to be Rp
Rp=1/4 Re

mass of earth be Me, and the mass of the planet be Mp

and let the mass of the first officer be m

Weight of the first officer on earth = force by which earth attracts the man (gravitational force )

We=-G(Me)m/(Re)2

and Wp=-G(Mp)m/Rp2 G here is the universal constant of gravitation

We=2Wp

-G(Me)m/(Re)2 =-2*G(Mp)m/Rp2

putting in values we get, Me=32*Mp

Mp=Me/32

Density of earth De= Me/[(4/3)Re3], density of planet Dp (assuming it to be a filled sphere) is De=(Me/32)/([(4/3)(Re/4)3])

which gives us that De=0.5 Dp

now this calculation is done using planet to be spherical. So if the planet is assumed to be hollow the density of planet will be still more because the mass would be concentrated around a smaller volume. So the measurements are consistent with a hollow planet

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