While looking through the Mt. Palomar telescope, you discover a large planetary

Question

While looking through the Mt. Palomar telescope, you discover a large planetary object orbited by a single moon. The moon orbits the planet every 8.10 hours with the centers of the two objects separated by a distance roughly 2.45 times the radius of the planet. Fellow scientists speculate that the planet is made of mostly iron. In fact, the media has dubbed it the "Iron Planet" and NASA has even named it Planet Hephaestus after the Greek god of iron. But you have your doubts. Assuming the planet is spherical and the orbit circular, calculate the density of Planet Hephaestus.

Explanation / Answer

Consider mass of planet is M and radius is r.

then distance d = 2.45r

now for circular motion,

Gravitational force = centripetal force

G M m / d^2 = m w^2 d

G M = w^2 d^3

and w = 2 pi / T = (2 pi ) / (8.10 x 3600 s ) =2.15 x 10^-4 rad/s

6.67 x 10^-11 x M = (2.15 x 10^-4 )^2 (2.45 r)^3

M / r^3 = 10191.76 kg / m^3

Density = mass / volume

= M / (4 pi r^3 / 3) = (3 /4pi ) ( M / r^3)

= (3 / 4pi ) (10191.76) = 2433.1 kg /m^3

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