A satellite of mass m (where m ≪ Me) is initially in a circular orbit around the Earth at a height of 410 km above the Earth’s equator. Its operators would like to move it into a geosynchronous orbit using a Hohmann transfer orbit. Assume a spherical Earth with radius 6371 km.
(a) Sketch the satellite’s Hohmann transfer orbit.
(b) Find the satellite’s initial (circular) orbital speed according to an inertial observer.
(c) Find the maximum height of the satellite above the Earth’s surface when it is on its Hohmann transfer orbit.
(d) How much does the satellite’s initial speed need to be increased to place it on its Hohmann transfer orbit?
PART A
the initial speed can be calvulated with the hep of the formula
Vo = orbital velocity
G = universal gravitational constant = 6.67 x 10 -11 Nm2kg-2
M = mass of earth (standarded value approximate 6.0 x 10 24 kg)
r = radius of orbit from the center on the earth
for the question it is given that the
radius of earth 6371 km
height or orbit from earth surface = 410 km
so radius of orbit r = 6371+410 = 6781 km
for part B initial orbital velocity Voi =
Voi = 7682.30 m/s
part c the maximum height = ?
maximum height is equal to the height or radius of geosynchronous orbit
for identifing the maximum height
as per the defination of geosynchonous orbit the angular velocity of geosynchronous is equal to the angular velocity of earth on its axis.
the angular velocity of eath is equal to angular velocity = 2π / T
T = 24 hrs
angular velocity of earth = 7.3 x 10-5 rad/sec
so the velocity of geosynchronous Vgy = radius x angular velocity(ω)
velocity formula of a orbit =
Vgy =
by putting all value of G, M and angular velocity we will get =42549 km (approx)
so the maximum height of the satallite from the surface of earth is = 42549-6371= 36178 km
for part D we have to known the preigee and apogee
as the shown in figure we gave to calculate the velocity at pregee and apogee
The satellite travels at a speed of 7682m/s (part A) around the small circle. We need to speed the satellite up to 10104.5m/s (pregee) when it reaches the perigee so it will transfer onto the ellipse. We then need to slow the satellite down to 1593.6m/s at the apogee, and then speed it up again to 3100m/s (Vgy) to transfer it to the higher circle that orbits at 42549 km above the Earth’s surface.
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