Question

if the distance between the earth nad the sun is reduced to half of its present value, what would be the number of days inone year?

Answer 1

From Kepler's 3rd laws of planetary motion we know that
$T^2=k{R}^3$
Where
K is a constant
T is orbital period = 365 days for earth
R is semi-major axis of orbit
$$\begin{gathered} \Rightarrow {\left(\frac{T_1}{T_2}\right)}^2={\left(\frac{R_1}{R_2}\right)}^3 \\ Putting \\ {T}_1=365days \\ {R}_1=R \\ {R}_2=\raisebox{1ex}{$R$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \\ Weget \\ {\left(\frac{365}{T_2}\right)}^2={\left(\frac{R}{\raisebox{1ex}{$R$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)}^3=8 \\ \Rightarrow T_2^2=\frac{{365}^2}{8}\Rightarrow T_2=129.04Days \end{gathered}$$
Therefore, Number of days in one year will be about 129 days