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Two-stage launcher placing 22 t in a 400 km orbit, then a Hohmann transfer to geostationary orbit: rocket equation, plane change and propellant budget

Tool	Orbital and Rocket Delta-v
Source	Martin Company (1963). Space Flight Handbooks, Volume 1: Orbital Flight Handbook, Part 2, Mission Sequencing Problems. NASA SP-33 Part 2, National Aeronautics and Space Administration, Washington, DC. Page VI-48 (the bi-elliptic ratios); Rickman, S. L. (2020). Introduction to Orbital Mechanics and Spacecraft Attitudes for Thermal Engineers. NASA Engineering and Safety Center. Charts 95 and 152 to 157 (geostationary altitude, Hohmann transfer).
Level	Undergraduate

Problem

An illustrative two-stage launch vehicle (no particular vehicle) lifts a 22,000 kg payload from latitude 28.5 degrees to a 400 km circular orbit about Earth. Find the ideal velocity of the vehicle and compare it with what the ascent needs for entered losses; then find the two burns of a Hohmann transfer from that orbit to geostationary altitude, the cost of the 28.5 degree plane change, and the propellant a 5,400 kg spacecraft with a 320 s engine needs for the transfer.

Inputs to Enter

Stage 1: dry 25,000 kg, propellant 400,000 kg, I_sp 300 s, thrust 7,600 kN; stage 2: dry 4,500 kg, propellant 100,000 kg, I_sp 348 s, thrust 980 kN; payload 22,000 kg
Parking orbit 400 km, launch latitude 28.5 deg, gravity loss 1.5 km/s, drag and steering loss 0.2 km/s (entered estimates)
Final altitude 35,786 km, plane change 28.5 deg, bi-elliptic intermediate altitude 200,000 km
Budget: 2,398 m/s (1 %), 1,824 m/s (1 %), 50 m/s (10 %); end mass 5,400 kg, I_sp 320 s
Earth: GM 398,600.4355 km³/s² (JPL DE440), equatorial radius 6,378.137 km, rotation period 23.9345 h (NASA fact sheet); g_0 = 9.80665 m/s²

Open the Orbital and Rocket Delta-v tool (loaded with this example)

Answers

Stage 1: m_0/m_f = 551,500 / 151,500 = 3.6403, Δv = 300 · 9.80665 · ln 3.6403 = 3.8012 km/s, T/W 1.405, burn 154.8 s; stage 2: 126,500 / 26,500 = 4.7736, Δv 5.3344 km/s, T/W 0.790, burn 348.2 s
Ideal velocity 9.136 km/s; lift-off mass 551,500 kg; payload fraction 3.989 % (4.112 % at the optimum split of the same velocity)
Required: v_c 7.6686 + 1.5 + 0.2 − 0.4087 (site speed) = 8.960 km/s; reserve 0.176 km/s
Hohmann 6,778.1 → 42,164.1 km: Δv_1 = 10.0660 − 7.6686 = 2.3975 km/s, Δv_2 = 3.0747 − 1.6182 = 1.4565 km/s, total 3.8540 km/s, transfer time 19,048.6 s = 5.291 h
With the 28.5 deg plane change: 4.2215 km/s all at the second burn, 4.1963 km/s with 2.23 deg taken at the first burn; a separate plane-change burn would make it 5.3676 km/s
Radius ratio 6.22 is below 11.939, so the Hohmann transfer beats any bi-elliptic one (4.3492 km/s through 200,000 km altitude); stationary radius 42,164.2 km, altitude 35,786.1 km, speed 3.0747 km/s
Budget 4,319.2 m/s with margins: mass ratio 3.9605, start mass 21,386.7 kg, propellant 15,986.7 kg; 613 kg inside the 22,000 kg payload

Notes

The tool's default inputs are this case; press Submit. The vehicle and the losses are illustrative, not a printed example. Checked against an independent numerical solution (integration of the two-body equations for the transfer and of the thrust equation for the burns, constrained optimisation for the staging split) and against printed values of public documents: the bi-elliptic ratios 11.939 and 15.582, the geostationary altitude 35,786 km and others listed in test_orbital_rocket_delta_v.php (1,479 checks).