The US Military WGS (Wideband Global SATCOM) system of communication satellites have been launched into GEO. Each satellite offers 4,875 GHz of instantaneous switchable bandwidth, covering simultaneously 19 geographical areas around the globe. These satellites can also change orbits slightly using a mix of 10 and 22-Newton thrusters.
The satellites are controlled and monitored by the 3rd Space Operations Squadron stationed at Shriver Air Force Base
What does "GEO" stand for?
Given: (not ALL of these facts below may be useful, so need to judge and choose)
• G = 6.67 x 10-11 Nm2/kg2
• gE at sea level = 9.81 N/kg
• Earth mass = M = 5.972 x 1024
• WGS Satellite mass = m = 5,987 kg
• Earth Radius = RE = 6,371 km
• GEO from Earth Equator = rS = 35,786 km
• GEO from Earth Center = rC = 42,164 km
(a) Calculate the period and angular velocity of a WGS satellite in GEO (in appropriate units, SF, notation)
(b) Orbital velocity of WGS satellite in GEO
(c) Gravitational Potential Energy of Satellite in GEO
(d) Earth's Gravitational Field at GEO
(e) Kinetic Energy of satellite in GEO
(f) Total Mechanical Energy of Satellite at GEO
(g) Escape velocity of any satellite, IF one wanted to boost it into deep space, beyond the gravitational boundaries of the solar system.
(a) Calculate the period and angular velocity of a WGS satellite in GEO:
The period (T) of a satellite in a circular orbit is given by:
T = 2π/ω, where ω is the angular velocity
Angular velocity (ω) is given by:
ω = √(GM/r^3), where G is the gravitational constant, M is the mass of the Earth, and r is the distance from the center of the Earth to the satellite
Substitute the values:
ω = √[(6.67 x 10^-11 Nm^2/kg^2) * (5.972 x 10^24 kg) / (42164 km * 1000)^3] = 7.292 x 10^-5 rad/s
Now, calculate the period:
T = 2π / (7.292 x 10^-5 rad/s) = 86,164 seconds
(b) Orbital velocity of WGS satellite in GEO:
Orbital velocity (v) is given by:
v = r * ω, where r is the distance from the center of the Earth to the satellite
Substitute the values:
v = (42164 km * 1000) * 7.292 x 10^-5 rad/s = 3074 m/s
(c) Gravitational Potential Energy of Satellite in GEO:
Gravitational potential energy (U) is given by:
U = -GMm / r, where m is the mass of the satellite
Substitute the values:
U = -(6.67 x 10^-11 Nm^2/kg^2) * (5.972 x 10^24 kg) * 5987 kg / (42164 km * 1000) = -3.201 x 10^10 J
(d) Earth's Gravitational Field at GEO:
The gravitational field (g) at GEO is given by:
g = GM / r^2
Substitute the values:
g = (6.67 x 10^-11 Nm^2/kg^2) * (5.972 x 10^24 kg) / (42164 km * 1000)^2 = 0.224 m/s^2
(e) Kinetic Energy of satellite in GEO:
Kinetic energy (K) is given by:
K = 1/2 * m * v^2, where v is the orbital velocity
Substitute the values:
K = 1/2 * 5987 kg * (3074 m/s)^2 = 2.218 x 10^10 J
(f) Total Mechanical Energy of Satellite at GEO:
The total mechanical energy (E) is given by:
E = K + U
Substitute the values:
E = 2.218 x 10^10 J - 3.201 x 10^10 J = -9.83 x 10^9 J
(g) Escape velocity of any satellite:
Escape velocity (Ve) is given by:
Ve = √(2GM / r)
Substitute the values:
Ve = √(2 * 6.67 x 10^-11 Nm^2/kg^2 * 5.972 x 10^24 kg / (42164 km * 1000)) = 1.117 x 10^4 m/s