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Question

If earth be at half of its present distance from sun how many days will there be in a year?

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Solution

From Kepler's 3rd laws of planetary motion we know that
T^2= kR^3
where
K is a constant
T is orbital period = 365 days for earth.
R is semi-major axis of orbit.

Taking the ratio,
(T1/T2)^2 = (R1/R2)^3

​​Putting, T1 = 365 days, R1=R, R2 =R/2
We get
(365/T2)^2 = (R/(R/2))^3
(365/T2)^2 = 2^3 = 8
​​​​​T2^2 = 365^2 / 8
​​​​​​T2=129.04
Therefore, Number of days in one year will be about 129 days.

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