A small satellite of mass m moves in an elliptical orbit with perigee r1 and apo

ID: 1998544 • Letter: A

Question

A small satellite of mass m moves in an elliptical orbit with perigee r1 and apogee r2 from the Earth's center.

Calculate:

a) The total energy E of the satellite's orbit

b) the satellite's speed at perigee and apogee

c) the orbital angular momentum l

d) the orbit's semimajor axis alpha and eccentricity epsilon

e) by how much energy does the energy E need to be changed to put the satellite in a circular orbit with radius r1

Write your answers in terms of the perigee and apogee radii r1, r2, gravitational constant G, earth mass Me, and the satellites mass m.

Let 2a is the length of the major axis of the elliptical orbit .

here since perigee r1 and apogee r2 are given from the center of earth ,

rmax = apogee =r2

rmin = r1 = perigee

2a =rmax + rmin = r1 +r2

a = (r1 +r2 ) /2

total energy of satellite ,E = - k /2a

where k = G Me m

E = -G Me m /(r1 +r2 )

semimajor axis = a = (r1 + r2 )/2

l /r = 1+ cos

epsilon ,eccentricity = (rmax - rmin) / (rmax + rmin ) =( r2 -r1) / (r2 +r1 )

orbital angular momentum L = [ mk2(2 - 1) /2E ] 1/2

L = [ m G2Me2m2[ ((r2 - r1)2 /(r2 +r1)2 ) -1 ] / 2 x -GMem /(r2 +r1) ]1/2

((r2 - r1)2 /(r2 +r1)2 ) -1 = -4r2r1/(r2 +r1 )2

substituting in L and solving,

L = m [ 2G Mer1r2 / (r1 +r2) ]1/2

velocity at perigee vp= L /mrmin = [ 2G Mer1r2 / (r1 +r2) ]1/2 /r1

velocity at apogee va = L /mrmax = [ 2G Mer1r2 / (r1 +r2) ]1/2 /r2

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