Question

1. The plant neptune travels around the sun with a period of 165 years. show that the radius of its orbit is approximately 30 times that of earth's orbit, both being considered as circular.

2. The planet of mass m moves around the sun of mass M in an elliptical orbit. the maximum and the minimum distance of the planet from the sun are r1 and r2 respectively . find the relation for the time period of the planet in terms of r1 and r2.

Answer 1

$$\begin{gathered} \mathit{P}\mathit{A}\mathit{R}\mathit{T}\mathbf{\hspace{0.17em}}\mathbf{1} \\ Fromkeplerslawoftimeperiods: \\ {T}^2\propto R^3 \\ IfweputTinyearsthenwehavetoputRinastronomicalunits\left(AU\right) \\ \frac{{T_1}^2}{{T_2}^2}\mathbf{}=\mathbf{}\frac{{{\mathbf{R}}_{\mathbf{1}}}^{\mathbf{3}}}{{{\mathbf{R}}_{\mathbf{2}}}^{\mathbf{3}}} \\ \frac{ \\ 1}{{165}^2}=\frac{{{\mathrm{R}}_1}^3}{{{\mathrm{R}}_2}^3} \\ {\mathrm{R}}_2={\mathrm{R}}_1\times \sqrt[3]{{165}^2}=30.083{\mathrm{R}}_1 \\ \mathrm{or},\mathrm{nearly}30\mathrm{times}\mathrm{that}\mathrm{of}\mathrm{the}\mathrm{earth}. \\ \mathbf{PART}\mathbf{}\mathbf{2}\mathbf{:} \\ \mathrm{For}\mathrm{an}\mathrm{elleptical}\mathrm{orbit}\mathrm{the}\mathrm{length}\mathrm{of}\mathrm{the}\mathrm{semi}-\mathrm{major}\mathrm{axis}\mathrm{is}\mathrm{given}\mathrm{by} \\ \mathrm{R}=\frac{{\mathrm{R}}_1+{\mathrm{R}}_2}{2} \\ \mathrm{now}, \\ \mathrm{keeping}\mathrm{the}\mathrm{sun}\mathrm{at}\mathrm{the}\mathrm{focus}\mathrm{the}\mathrm{kepler}\text{'}\mathrm{s}\mathrm{law}\mathrm{of}\mathrm{periods}\mathrm{gives}, \\ {\mathrm{T}}^2\propto {\mathrm{R}}^3 \\ {\mathrm{T}}^2=\frac{4{\mathrm{\pi }}^2}{\mathrm{GM}}\times {\mathrm{R}}^3 \\ \mathrm{T}=2\mathrm{\pi R}\sqrt{\frac{\mathrm{R}}{\mathrm{GM}}} \\ \mathrm{T}=2\mathrm{\pi }\left(\frac{{\mathrm{R}}_1+{\mathrm{R}}_2}{2}\right)\times \sqrt{\left(\frac{{\mathrm{R}}_1+{\mathrm{R}}_2}{2}\right)\times \frac{1}{\mathrm{GM}}} \end{gathered}$$