Q. 38

# A satellite is in an elliptic orbit around the earth with aphelion of 6R and perihelion of 2 R where R = 6400 km is the radius of the earth. Find 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 × 10 ^{- 11} SI units and M = 6 × 10^{24} kg]

Answer :

Given

Radius of earth = R = 6400 km

Aphelion earth - satellite distance r_{a} = 6R

Perihelion earth - satellite distance r_{p} = 2R

We know for an ellipse

Where e is the eccentricity of the ellipse, and the semi major axis.

Now, the angular momentum of the satellite is conserved we then, we have

Where m is the mass of satellite and v_{p} and v_{a} are the velocities at the perigee and apogee respectively

Now from conservation of energy at the apogee and perigee

Where K and U are the kinetic and gravitational potential energy of the satellite

Where M is the mass of earth. Using the relation between v_{p} and v_{a} above we have

Putting the values of r_{p} and r_{a} we have

To transfer the satellite to a circular orbit of radius 6R the satellite’s velocity has to be maintained at a value of

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