Design a spiral transfer matched to the "high Isp, low thrust" character of ion engines and Hall thrusters, using the Edelbaum approximation and the Tsiolkovsky rocket equation. Vary spacecraft mass, thrust, Isp and the initial / target orbit altitudes and compare the required ΔV, propellant mass and burn time against a chemical Hohmann transfer in real time.
Parameters
Spacecraft mass m₀
kg
Initial (wet) mass, including propellant
Thrust T
mN
Electric thrusters are typically 10–500 mN
Specific impulse Isp
s
Hall 1500–2500, ion 3000–4500 as a guide
Initial orbit altitude
km
Altitude above the Earth surface; 500 km is a typical LEO
Target orbit altitude
km
35786 km is geostationary (GEO)
Thrusting profile
Continuous, eclipse-limited duty, or optimized steering
Results
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ΔV (Edelbaum) (m/s)
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Propellant mass (kg)
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Propellant fraction (%)
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Burn time (day)
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Acceleration (μm/s²)
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Hohmann ΔV ratio (%)
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Orbit transfer animation (Earth-centred)
The blue disc is Earth. The spacecraft (white dot) follows the yellow spiral from the inner blue initial orbit out to the green target orbit, completing many revolutions.
Altitude vs time — spiral altitude evolution
Hohmann vs spiral — ΔV, propellant and time comparison
Edelbaum approximation. For coplanar, tangential thrusting the required ΔV is simply the difference between the two circular orbit speeds. μ is the Earth gravitational parameter, r is the orbital radius (Earth radius + altitude).
Tsiolkovsky rocket equation. A higher Isp shrinks the exponent, so the propellant m_p needed for the same ΔV drops exponentially. g₀ = 9.81 m/s².
$$\dot m = \frac{T}{g_0\,I_{sp}},\quad t_{\text{burn}} = \frac{m_p}{\dot m}$$
Mass-flow rate ṁ and burn time t_burn. Even with low thrust T, ṁ is correspondingly small because of the high Isp, so the burn is long.
Low-Thrust Spiral Trajectory Transfer — Electric Propulsion Mission Design
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How is "electric propulsion" actually different from chemical rockets? I keep hearing about ion engines and Hall thrusters.
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Quick version: chemical = big thrust for a short time, electric = tiny thrust for a very long time. A chemical engine produces kilonewtons but at Isp ≈ 300 s. An ion engine produces only tens to hundreds of millinewtons but at Isp 3000–4500 s. Isp is roughly "how long 1 kg of propellant can hold up 1 g of weight" — an efficiency number — and a 10× higher Isp cuts the propellant you need for the same ΔV by an enormous factor.
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With such a small thrust you can't do a big chemical-style burn, right? Then how do you actually climb from LEO to GEO?
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That's where the spiral trick comes in. You keep firing all the time and let the orbit gradually expand while it stays almost circular. There is no Hohmann-style "kick at periapsis, kick at apoapsis"; instead a continuous tangential thrust slowly opens the orbit out in a spiral. With the defaults (2 t, 100 mN, Isp 3000 s) LEO→GEO needs about 4538 m/s of ΔV, 286 kg of propellant and roughly 974 days of firing.
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974 days! That's nearly three years. A Hohmann burn would arrive in hours — why would anyone deliberately pick something this slow?
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Look at the "Hohmann ΔV ratio" card on the right. The spiral does cost about 15–20% more ΔV than Hohmann. But once you put it into the rocket equation, the story reverses. A chemical Hohmann at Isp 300 s would burn nearly 1.5 t of propellant on a 2 t satellite. The spiral at Isp 3000 s only needs about 300 kg. In other words, the same launcher delivers a far larger "useful" satellite to GEO. That is exactly why Boeing 702SP and many of the European Eutelsat satellites have adopted Electric Orbit Raising (EOR).
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So you pay with time to save propellant. Can electric propulsion be used to reach the Moon or Mars too?
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It can — and already is. ESA's SMART-1 reached the Moon from GTO in 2003 using only a Hall thruster and took about 13 months. NASA's Dawn is the only spacecraft ever to orbit two extraterrestrial bodies (Vesta and Ceres) using ion propulsion, with cumulative firing past five years. Mars-class missions need a 2–3 year budget, but the payload mass per ton of launcher is in another league. Crewed missions still favour higher-thrust NTP/NEP because radiation dose and crew time become the limiting constraint.
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When I drag the Isp slider down to 1000 s, the propellant fraction shoots up. Is that the engine getting "more chemical"?
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Exactly — the exponential in the rocket equation is biting back. Halving the Isp roughly doubles the negative exponent, and the propellant fraction grows fast. At LEO→GEO ΔV ≈ 4.5 km/s, the fraction is about 14% at Isp 3000 s, 37% at Isp 1000 s, and well above 60% at Isp 500 s. But just cranking Isp up forever isn't free either: the burn gets so long that solar-array degradation and operations cost dominate. Real design is a joint optimization of ΔV, Isp, thrust and operating duration.
Frequently Asked Questions
It is the transfer strategy used by electric propulsion (ion or Hall thrusters), which combines a very high specific impulse with a very small thrust. Because the thrust cannot deliver an impulsive ΔV the way a chemical engine can, the spacecraft fires continuously and slowly changes its orbit radius over many revolutions, drawing a spiral path. This tool uses the Edelbaum approximation ΔV = |v₁ − v₂| to estimate the ΔV, propellant mass and burn time for a coplanar circular-to-circular spiral.
Edelbaum's solution assumes that the thrust-to-mass ratio is small compared with the circular orbit velocity, that the thrust is always tangent to the velocity (optimal steering), and that the spacecraft stays nearly circular while the radius changes slowly. Under those assumptions the required ΔV is simply the difference between the two circular speeds |v₁ − v₂| with v = √(μ/r). Extended Edelbaum solutions also include inclination changes, but this tool restricts itself to coplanar transfers. For LEO→GEO the spiral ΔV is larger than Hohmann, yet the propellant mass drops sharply thanks to the high Isp.
Propellant mass follows the Tsiolkovsky equation m_p = m₀·(1 − exp(−ΔV/(g₀·Isp))), so a higher Isp shrinks the exponent and m_p drops dramatically. Chemical engines sit at Isp ≈ 300 s, Hall thrusters reach 1500–2500 s and ion engines 3000–4500 s. Even if LEO→GEO ΔV goes from 3.8 to 4.5 km/s, a 10× higher Isp cuts the propellant fraction of launch mass to roughly one eighth, so a same-class rocket can deliver a much larger 'useful' spacecraft. Boeing 702SP and recent commercial GEO satellites use this Electric Orbit Raising (EOR) in production.
The burn time is t_burn = m_p / ṁ, and the mass-flow rate ṁ = T/(g₀·Isp) is a small number set by the thruster. A standard LEO→GEO case (a 2 t spacecraft, 100 mN, Isp 3000 s) takes roughly half a year to a year, lunar transfers take 1–2 years and NASA's Dawn spacecraft accumulated over five years of ion firing across the Vesta–Ceres mission. A duty cycle (eclipses, operations) extends the wall-clock time further. For electric missions, payload mass typically wins over transit time.
Real-World Applications
Electric Orbit Raising of commercial GEO communications satellites: Boeing 702SP, SES-15, Eutelsat 172B and the "EP-only" buses launched on Falcon 9 / Falcon Heavy reach GEO from GTO using only Xenon Hall thrusters. Dropping the chemical apogee kick motor cuts launch mass by 30–40%, so the same launcher class carries more useful payload. The price paid is a three- to six-month orbit raising before commercial service starts.
Lunar, asteroid and deep-space exploration: ESA's SMART-1 (2003) was the first lunar transfer with a Hall thruster, and NASA's Deep Space 1 (1998) and Dawn (launched 2007) are the canonical ion-propulsion deep-space missions. Dawn is the only spacecraft ever to orbit two extraterrestrial bodies (Vesta and Ceres), with a cumulative ΔV around 11 km/s — completely out of reach for chemical propulsion. Japan's Hayabusa and Hayabusa 2 used the μ10 and μ20 ion engines for the Itokawa and Ryugu return missions.
Small-satellite and CubeSat applications: Sub-millinewton electric propulsion modules from companies like ENPULSION and Busek now make orbit change and station-keeping practical for 12U–100 kg spacecraft. Every Starlink V2 mini satellite carries a Hall thruster used for initial orbit raising, constellation maintenance and end-of-life deorbit. Electric propulsion is also key to meeting the 25-year debris mitigation rule.
North-south station-keeping (NSSK): A geostationary satellite needs around 50 m/s/yr of north-south ΔV to fight luni-solar perturbations. Over a 15-year design life that becomes hundreds of kilograms of chemical propellant; a 1800 s Hall thruster cuts it by an order of magnitude. NSSK is the oldest and most universal commercial application of electric propulsion.
Common Misconceptions and Pitfalls
The first trap is treating Edelbaum as a 5% answer in every case. The ΔV = |v₁ − v₂| used here assumes that the acceleration is small compared to the orbital velocity (typically a/v < 10⁻⁴). For high-power (kW-class) electric propulsion on a low-mass spacecraft with small initial radius, that approximation breaks down and the true ΔV drifts by several percent. On top of that, low Earth orbits feel atmospheric drag, GEO transfers feel luni-solar perturbations, and deep-space missions feel solar radiation pressure — Edelbaum alone can be 10–20% off in those regimes. Detailed mission design therefore uses numerical integrators like GMAT, STK or JAXA Orbita.
The second trap is assuming "more Isp is always better". Higher Isp cuts propellant mass, but for the same electrical power, thrust drops in inverse proportion (T = 2·η·P/(g₀·Isp)). Going from 3000 s to 5000 s at the same 5 kW input drops thrust to roughly 60% and lengthens the burn proportionally. A long burn directly translates into solar-array radiation damage, PPU and cathode wear life, and operator hours, so the real optimization is over ΔV, available power and operating duration jointly. Commercial GEO buses often land around Isp 1600–2000 s as the practical sweet spot.
The third trap is reading the "974-day burn time" output as the actual mission duration. The number returned here is cumulative thrusting time; it does not include eclipse blackouts, PPU thermal limits, ground-shift operations or science modes. Real EOR campaigns typically achieve 60–80% duty, with SMART-1 averaging about 65% during its lunar climb. The "Duty cycle 80%" option here is only a coarse correction; serious mission planning uses session schedules and an eclipse model. Add in commissioning, low-power modes and contingencies, and most programmes budget 1.5–2× the raw burn time as the planned operations window.
How to Use
Enter spacecraft dry mass (kg) and thrust level (mN) for your ion or Hall thruster system.
Input specific impulse (seconds) and initial orbit altitude (km) to define baseline orbital parameters.
Run simulation to compute spiral trajectory ΔV via Edelbaum equation, propellant mass, burn duration, and acceleration profile in microgravity regime.
Compare results against equivalent Hohmann transfer to assess low-thrust efficiency penalty.
Worked Example
Xenon Hall thruster mission: spacecraft mass 850 kg, thrust 150 mN, Isp 1600 s, LEO 400 km. Simulation yields ΔV (Edelbaum) = 3420 m/s, propellant mass = 234 kg, burn time = 18.7 days, acceleration = 176 μm/s². Hohmann equivalent requires 3100 m/s, so low-thrust spiral penalty is +10.3%. Propellant fraction reaches 27.6%, typical for ion-drive GEO insertions.
Practical Notes
Edelbaum ΔV penalty increases with lower thrust-to-weight ratio; below 0.5 mN per tonne, expect 12–18% margin over Hohmann.
Spiral duration scales inversely with acceleration; 100 mN thruster cuts burn time by half relative to 50 mN, critical for lunar or Mars trajectory windows.
Isp trade: bumping Xenon (1600 s) to Krypton (900 s) halves propellant mass but doubles thrust demand for same trajectory duration.