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Question 970839: Kepler's third law of planetary motion states that the square of the orbital period of any planet is proportional to the cube of the mean orbital radius of that planet, with the constant of proportionality being the same for all planets.
Using the fact that the orbital period of the earth is 1 year and its mean
orbital radius is 150 ×10^6 km, calculate the mean orbital radius for Venus, given that its orbital period is 7.38 months.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! this is basically a question in direct variation.
the problem is not difficult, but the arithmetic is messy.
the formula for direct variation is y = k*x.
if p = orbital period and m = mean orbital radius, then the equation of y = k*x becomes:
p^2 = k * m^3
this means that the orbital period squared is directly proportional to the mean orbital radius cubed.
with eath, we'll set 1 year as 12 months to keep the values consistent with each other.
the equation for earth becomes:
(12)^2 = k * (150 * 10^6)^3
we solve for k to get:
k = (12)^2 / (150 * 10^6)^3
now we use that value of k to solve for the mean orbital radius of venus.
the formula of p^2 = k * m^3 becomes:
(7.38)^2 = (12)^2 / (150 * 10^6)^3 * m^3
solve for m^3 to get:
m^3 = (7.38)^2 / (12)^2 * (150 * 10^6)^3
this is the messy part.
it's best to just use your calculator to solve it.
you will get:
m^3 = 1.276509376 * 10^24
take the cube root of both sides of this equation to get:
m = 108477916.7 km.
this is equivalent to 108.4779167 * 10^6 km.
this solution is consistent with what i researched for the mean orbital radius of venus and of earth.
earth is 150 * 10^6 and venus is approximately 108 * 10^6.
this confirms the solution is correct.
here's what i found for mean orbital radius of the earth.
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The radius of the earth's orbit around the sun (assumed to becircular) is 1.50 10^8 km, and the earth travels around this orbit in 365 days
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1.50 * 10^8 km is the same at 150 * 10^6 km.
here's what i found for mean orbital radius of venus.
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Venus orbits the Sun at an average distance of about 108,000,000 km and completes an orbit every 224.65 days.
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108,000,000 is equivalent to 108 * 10^6.
224.65 days is equivalent to 7.38 months.
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