Question

A satellite is geostationary in a particular orbit. It is allowed to go to another orbit having an orbital radius 2 times that of the earlier orbit from the centre of the earth. The time period in the second orbit is
A. 48 hours
B. 24 hours
C. $48\sqrt 2 \,{\text{hours}}$
D. $24\sqrt 2 \,{\text{hours}}$

Answer 1

Use Kepler’s third law to determine the orbital period of the satellite in the second orbit. For a geostationary satellite, the orbital period of the satellite in orbit is the same as the rotation period of the earth.

Formula used:
${T^2} = \dfrac{{4{\pi ^2}}}{{GM}}{a^3}$
Here, G is the universal gravitational constant, M is the mass of earth and a is the semimajor axis of the orbit of satellite.

Complete step by step answer:
According to Kepler’s third law, the period of satellite moving in a orbit of semimajor axis a is given as,
${T^2} = \dfrac{{4{\pi ^2}}}{{GM}}{a^3}$
Here, G is the universal gravitational constant, M is the mass of earth and a is the semimajor axis of the orbit of satellite.
For the first orbit, the period of the satellite in its orbit will be,
$T_1^2 = \dfrac{{4{\pi ^2}}}{{GM}}a_1^3$ …… (1)
For the second orbit, the period of the satellite in that orbit will be,
$T_2^2 = \dfrac{{4{\pi ^2}}}{{GM}}a_2^3$ …… (2)
Divide equation (2) by equation (1).
$\dfrac{{T_2^2}}{{T_1^2}} = \dfrac{{\dfrac{{4{\pi ^2}}}{{GM}}a_2^3}}{{\dfrac{{4{\pi ^2}}}{{GM}}a_1^3}}$
$\Rightarrow \dfrac{{T_2^2}}{{T_1^2}} = {\left( {\dfrac{{{a_2}}}{{{a_1}}}} \right)^3}$
$\therefore {T_2} = {\left( {\dfrac{{{a_2}}}{{{a_1}}}} \right)^{\dfrac{3}{2}}}{T_1}$
For a geostationary orbit, the orbital period is 24 hours.
We have given, the radius of the second orbit is twice of the first orbit. Therefore, substitute $2{a_1}$ for ${a_2}$ and 24 hours for ${T_1}$ in the above equation.
${T_2} = {\left( {\dfrac{{2{a_1}}}{{{a_1}}}} \right)^{\dfrac{3}{2}}}\left( {24\,{\text{hours}}} \right)$
$\Rightarrow {T_2} = {\left( 2 \right)^{\dfrac{3}{2}}}\left( {24\,{\text{hours}}} \right)$
$\Rightarrow {T_2} = 2\sqrt 2 \left( {24\,{\text{hours}}} \right)$
$\therefore {T_2} = 48\sqrt 2 {\text{ hours}}$

So, the correct answer is “Option C”.

Note:
As the radius of the orbit increases, the period of the satellite also increases.
To check whether your answer is right or wrong refer to the above conclusion.