Question

As planets with a wide variety of properties are being discovered outside our solar system, astrobiologists are considering whether and how life could evolve on planets that might be very different from earth.One recently discovered extrasolar planet, or exoplanet, orbits a star whose mass is 0.70 times the mass of our sun.This planet has been found to have 2.3 times the earth's diameter and 7.9 times the earth's mass.For planets in this size range, computer models indicate a relationship between the planet's density and composition. Observations of this planet over time show that it is in a nearly circular orbit around its star and completes one orbit in only 9.5 days. How many times the orbital radius r of the earth around our sun is this exoplanet's orbital radius around its sun? Assume that the earth is also in a nearly circular orbit

Answer 1

R₁ = 14.44 R₂

Step-by-step explanation:

For sun-earth system

GMm/R² = mω²R

GM =ω²R³

GM₁ =ω₁²R₁³

For star - planet system

GM₂ =ω₂²R₂³

Dividing

M₁ / M₂ = ω₁²R₁³ / ω₂²R₂³

R₁³ /R₂³ = M₁ω₂² / ω₁²M₂

= M₁T₁²/T₂²M₂

Given

M₁ / M₂ = 1 / 0.7

T₁ = 365 days

T₂ = 9.5 days

R₁³ /R₂³ = M₁²365² /9.5² (.7M₁ )²

(R₁ /R₂)³ =365² / 9.5² X .7²

= 3012

R₁ /R₂

= 14.44

R₁ = 14.44 R₂