A newly discovered planet is in a circular orbit around a distant star with an o

Question

A newly discovered planet is in a circular orbit around a distant star with an orbital period of 350 Earth days.

A newly discovered planet is in a circular orbit around a distant star with an orbital period of 350 Earth days. The planet also rotates on its axis, making one full rotation every 4.50 Earth days. The radius of the planet is rp 4.50 x 106 m and the radius of the planet's orbit about the star is rs 7.00 x 1011 m. Determine the ratio of the radial acceleration, due to the rotation of the planet, of an object at the equator of the planet (acp) to the radial acceleration of the planet in its orbit about the star (acs) ace = acs How is the radial acceleration of the object related to its orbital period and radius?

Explanation / Answer

vp = 2*pi*rp/T so acp = vp^2/rp = 4*pi^2*rp/T^2

Now for the orbital bit, let T0 = 350 days (convert to seconds). The planet travels the circumference of its orbit in T0 so

Vo = 2*pi*rs/T0 and acs = vo^2/rs = 4*pi^2*rs/T0^2

the acp/acs = rp*T0^2/(rs*Tp^2) = 4.5*10^6*350^2/7*10^11*4.5^2 = 0.039

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