Kepler\'s Law states that the relationship between the period of a planet (defin
ID: 3093416 • Letter: K
Question
Kepler's Law states that the relationship between the period of a planet (defined as the length of time it takes for a planet to complete one revolution around the sun) and the planet's mean distance from the sun (i.e. the "orbit") is a power model. The following are astronomical data taken from our solar system. Find the equation which gives P as a function of d by fitting a power model between each pair of consecutive points and then take an average to find a reasonable value for the power and the co-efficient. 1 Estimate the number of years it takes the planet Uranus to revolve around the sun once, given that it is 2.87 times 10 12 meters from the sun. It takes Pluto 247.7 years to revolve around the sun one time. Estimate its distance from the sun.Explanation / Answer
a) assume a power model like P = d to estimate and , take log of both sides ln P = ln + ln d So if you plot ln P (on the y-axis) versus ln d (x-axis) forvarious planets you should get a straight line. When i plotted it, i got (by fitting with a straight line), ln = -32.693 >> = 6.33*10^-15 = 1.4999 (theoretically it should be 1.5) so the power model looks like P = 6.33*10-15 d1.4999 b) P(uranus) = 6.33*10-15 (2.87*10^12)1.4999= 30689 days c) to get d from P, we have to rewrite the model as d = (P/6.33*10-15)^(1/1.4999) d (pluto) = (247.7*365/6.33*10-15)^(1/1.4999) =5.89*10^12
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