Example: Hohmann Transfer

The Geostationary Operational Environmental Satellite system encompasses a set of spacecraft to perform imaging of the earth from GEO. The most recently launched satellite is GOES-17, which had a launch mass of 5,192 kg.

Assume that the launch vehicle placed GOES-17 into a circular LEO with an altitude of 250 km. Determine the \(\Delta v\) and propellant mass required for a Hohmann transfer orbit to GEO.

The total \(\Delta v\) requirement is the sum of the change required to go from LEO onto the transfer orbit, and the change required to go from the transfer orbit to GEO. First, let’s calculate the velocity on the circular orbit.

import math as m

mu = 3.986E5  # km**3/s**2
R_E = 6378  # km
r_leo = 250 + R_E  # km

v_leo = m.sqrt(mu / r_leo)

The LEO velocity is \(v_{\text{LEO}} =\) 7.755 km/s. Next, let’s calculate GEO altitude and velocity. We know for GEO that the satellite must be orbiting with the same angular velocity as the surface of Earth.

sidereal_day = 86164.0905  # s
r_cubed = mu * sidereal_day**2 / (4 * m.pi**2)
r_geo = r_cubed ** (1 / 3)
v_geo = m.sqrt(mu / r_geo)

The GEO radius is \(r_{\text{GEO}} =\) 42164.154 km and the velocity is \(v_{\text{GEO}} =\) 3.075 km/s.

The transfer ellipse will have a semi-major axis length, \(a\), equal to half the total distance between the two circular orbits. In addition, the perigee radius of the transfer orbit will be equal to \(r_{\text{LEO}}\) and the apogee radius will be equal to \(r_{\text{GEO}}\). Using this information we can calculate the orbital angular momentum and the velocities.

r_p = r_leo
r_a = r_geo
h_t = m.sqrt(2 * mu * r_a * r_p / (r_a + r_p))
v_tp = h_t / r_p
v_ta = h_t / r_a

This gives a transfer orbit perigee velocity of \(v_{t,p} =\) 10.195 km/s and apogee velocity of \(v_{t,a} =\) 1.603 km/s. Finally, we can calculate the \(\Delta v\) and the propellant mass \(\Delta m\). The Centaur rocket stage that served as the second stage for the GOES-17 mission has an \(I_{sp}\) of 450.5 s.

Delta_v = abs(v_geo - v_ta) + abs(v_tp - v_leo)
I_sp = 450.5
goes_mass = 5_192  # kg
Delta_m = goes_mass * (1 - m.exp(-Delta_v / (I_sp * 9.81E-3)))

Finally, the transfer velocity change is \(\Delta v =\) 3.912 km/s and the propellant mass is \(\Delta m =\) 3049.671 kg.