Ncert Exercises & Solutions

Q. 38 A satellite is in an elliptic orbit around the earth with an aphelion of 6R and perihelion of 2R where R = 6400 km is the radius of the earth. Find the eccentricity of the orbit. Find the velocity of the satellite at apogee and perigee. What should be done if this satellite has to be transferred to a circular orbit of radius 6R?

$[G = 6.67 \times 10^{- 11}$ SI unit and M= $6 \times 10^{24}$ kg]

Hint: Use Kepler's laws of planetary motion.

Step 1: Find the eccentricity.

Given,

$r_{p}$ = radius of perihelion = 2R

$r_{a}$ =radius of aphelion = 6A

Hence, we can write,

$r_{a}$ = a(1+e)=6R ...(i)

$r_{p}$ = a(1-e)=2R ...(ii)

Solving Eqs. (i) and (ii), we get, eccentricity, e

$= \frac{1}{2}$

Step 2: Find the velocity of the satellite at apogee and perigee.

By conservation of angular momentum, angular momentum at perigee = angular momentum at apogee

$∴ m v_{p} r_{p} = m v_{a} r_{a}$

$∴ \frac{v_{a}}{v_{p}} = \frac{1}{3}$

where m is the mass of the satellite.

Applying the conservation of energy, energy at perigee = energy at apogee

$\frac{1}{2} m {v_{p}}^{2} - \frac{G M m}{r_{p}} = \frac{1}{2} m {v_{a}}^{2} - \frac{G M m}{r_{a}}$ where M is the mass of the earth.

$∴ {v_{p}}^{2} (1 - \frac{1}{9}) = - 2 G M (\frac{1}{r_{a}} - \frac{1}{r_{p}})$ (by putting

$v_{a} = \frac{v_{p}}{3}$ )

$v_{p} = \frac{{[2 G M (\frac{1}{r_{p}} - \frac{1}{r_{a}})]}^{1 / 2}}{{[\frac{8}{9}]}^{\frac{1}{2}}} = 6.85 k m / s e c v_{a} = 2.28 k m / s e c$

Step 3: Find the velocity required to transfer the satellite in the orbit of radius 6R.

For a circular orbit of radius r,

$v_{c} =$ orbital velocity=

$\sqrt{\frac{G M}{r}}$

For

$r = 6 R, v_{c} = \sqrt{\frac{G M}{6 R}} = 323 k m / s$

Hence, to transfer to a circular orbit at apogee, we have to boost the velocity by

$∆$ v= (3.23— 2.28) = 0.95km/s. This can be done by suitable firing rockets from the satellite.

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