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How Much Fuel Does It Take to Reach Geostationary Orbit?

🛰️ Orbital mechanics
How Much Fuel Does It Take to Reach Geostationary Orbit?
ABS-3A departs the Falcon 9 upper stage after launch, 2 Mar 2015 - the first commercial satellite to reach its working orbit on ion engines alone, over the following months (SpaceX, CC0).

Question

A one-tonne satellite wants its slot in the geostationary belt. The speed bill is 3.86 km/s - what does it cost in propellant, and what does the engine you pick have to do with it?

Run the rocket equation backwards and it becomes a price list: propellant = payload × (e^(Δv/vₑ) − 1). The exhaust speed vₑ sits in the denominator of an exponent, so the engine does not merely tint the bill - it chooses the currency. With a storable-propellant upper stage of the AJ10 family (the engine that braked Apollo into lunar orbit), the same tonne costs roughly 2,540 kg of propellant. With a liquid-hydrogen stage like KVD-1 or RL10, about 1,340 kg. With an ion engine throwing xenon at ~29 km/s, about 140 kg - eighteen times less. What ion engines buy with mass they pay back in time: a chemical kick burns for minutes to hours, xenon thrusts gently for months - ABS-3A, the first all-electric commercial satellite, raised itself to geostationary orbit over roughly half of 2015. Real launches ride a rocket's own burn most of the way (that is the transfer orbit's burn 1), leaving the satellite's motor the last ~1.5 km/s - cheaper again, same exponential.

The math

Propellant = m·(e^(Δv/vₑ) − 1), with Δv = 3.86 km/s for the full 400 km-to-belt transfer (see the transfer card). Per 1,000 kg payload: storable vₑ = 3.05 km/s (AJ10, Isp 311 s): e^1.26 = 3.54 → ~2,540 kg. Hydrolox vₑ = 4.53 km/s (KVD-1, Isp 462 s): e^0.85 = 2.34 → ~1,340 kg. Ion vₑ = 29.4 km/s (NSTAR, to 3,000 s): e^0.13 = 1.14 → ~140 kg. From a standard GTO the satellite's own bill is the apogee circularisation, Δv = 1.46 km/s: ~610 / ~380 / ~51 kg. For perspective, orbit itself is cheap energy: v²/2 + GM·(1/R − 1/r) at 400 km = 29.4 + 3.7 = 33 MJ per kg - a tonne holds ~33 GJ, the burn-value of ~770 kg of kerosene (43.1 MJ/kg). The expense is never the energy; it is carrying the propellant.
Interactive price list: set a payload and read the propellant bill for three engine classes - per tonne, the full climb costs ~2,540 kg of storable propellant, ~1,340 kg of liquid hydrogen, or ~140 kg of xenon, at burn times of minutes, minutes, and months (ABS-3A raised itself on ion engines alone over half of 2015). Already riding a launcher's transfer orbit, the satellite's own motor bills only the last 1.46 km/s.

Why it matters

Tsiolkovsky's logarithm on one card, this its cash register: because the mass ratio is exponential in Δv/vₑ, the entire industry optimises the ratio first and everything else second - which is why upper stages are the engineering showpieces, why ABS-3A's self-raise in 2015 was a landmark (the saved propellant becomes payload or launch price), and why constellations now drift to altitude on ion thrusters. Launch prices are commercial secrets and market moods; kilograms and months are physics, recomputable by anyone with this card's constants. The compendium can honestly price a transfer - and that is exactly the kind of honest number this collection is for.

Related

How Do You Reach a Higher Orbit? The Rocket Equation How Fast Is 'Orbit'? Konstantin Tsiolkovsky Newton's Cannon: the Thought Experiment That Did 270 Years of Field Work

Sources

  1. Wikipedia - 'Rocketdyne AJ10' (storable Isp 311 s = 3.05 km/s; Apollo service module engine) [1]
  2. Wikipedia - 'KVD-1' (hydrolox upper stage, Isp 462 s) [2]
  3. Wikipedia - 'Deep Space 1' (NSTAR ion engine, Isp 1,000-3,000 s) [3]
  4. Wikipedia - 'ABS-3A' (first all-electric commercial comsat; launched 2 Mar 2015, handed over 31 Aug 2015) [4]
  5. Wikipedia - 'Geostationary transfer orbit' (typical GTO; apogee kick by the satellite) [5]
  6. Wikipedia - 'Kerosene' (lower heating value 43.1 MJ/kg) [6]
  7. Wikipedia - 'Tsiolkovsky rocket equation' (Δv = vₑ·ln(m₀/mf)) [7]