A satellite is in an elliptical orbit around the earth with aphelion of `6 R` 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 xx 10^(-11) SI` units and `M = 6 xx 10^(24) kg]`

Explanation

Given, `r_(p)=` radius of perihelion `=2R`
`r_(a)=` radius of apnelion `=6 R`
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=1/2`
By conservation of angular momentum, angular momentum at perigee = angular momentum at apogee
`:." "mv_(p)r_(p)=mv_(a)r_(a)`
`:." "v_(a)/v_(p)=1/3`
where m is mass of the statellite.
Applying conservation of energy, energy at perigee = energy at apogee
`1/2 mv_(p)^(2)-(GMm)/r_(p)=1/2 mv_(a)^(2)-(GMm)/r_(a)`
Where M is the mass of the earth.
`v_(p)^(2) (1-1/9)=-2GM (1/r_(a) - 1/r_(p))=2GM (1/r_(p)-1/r_(a))" "("By putting "v_(a)=v_(p)/3)`
`v_(p)=([2GM (1/r_(p)-1/r_(a))]^(1//2))/([1-(V_(a)//V_(p))^(2)]^(1//2))= [((2GM)/R (1/2-1/6))/((1-1/9))]^(1//2)`
`=((2//3)/(8//9) (GM)/R)^(1//2)=sqrt(3/4 (GM)/R)=6.85 km//s`
`v_(p)=6.85` km/s, `v_(a)= 228` km/s
For circular orbit of radius r,
`v_(c)=` orbital velocity `=sqrt((GM)/r)`
For `r=6 R, v_(c)=sqrt((GM)/(6R))=323` km/s.
Hence, to transfer to a circular orbit at apogee, we have to boost the velocity by `Delta = (3.23-2.28)=0.95` km/s. This can be done by suitably firing rockets from the satellite.