Low-Thrust Spiral Orbit Transfer Calculator
Astrodynamics & Space Mission Design: Determine required velocity increment ($\Delta V$), thrust duration, and propellant consumption for continuous low-thrust electric propulsion orbital raising via Edelbaum's analytical model.
Initial & Target Orbital Parameters
Spacecraft Mass & Thruster Specs
Spiral Transfer Mission Output
Edelbaum Low-Thrust Continuous Transfer Mechanics (1961)
Continuous electric propulsion spirals enable satellites to raise orbits from Low Earth Orbit (LEO) to Geostationary Orbit (GEO) with superior mass efficiency.
1. Edelbaum Analytical Delta-V Equation
ΔV = √[ V_0² - 2 · V_0 · V_f · cos( (π/2) · Δi ) + V_f² ]
2. Rocket Mass Ratio & Thrust Duration
m_final = m_0 · exp( -ΔV / (I_sp · g_0) ) Δt_thrust = ( m_0 - m_final ) / [ Thrust / (I_sp · g_0) ] [seconds]
Frequently Asked Questions
How does continuous low-thrust orbital transfer differ from a Hohmann impulsive transfer?
A Hohmann transfer applies two instantaneous high-thrust rocket burns at apsides, transitioning along an intermediate Keplerian ellipse. In contrast, low-thrust electric thrusters (Hall or Ion engines generating milliNewtons of thrust) fire continuously over weeks or months, creating a gradual quasi-circular outward spiral trajectory described by the Edelbaum equation.
Why does an orbital plane change combine efficiently with orbit raising in the Edelbaum model?
The velocity cost of plane change is proportional to local orbital velocity: $\Delta V_{plane} = 2 V \sin(\Delta i / 2)$. In high orbit (e.g. near GEO where $V \approx 3.07\,\text{km/s}$ vs $7.6\,\text{km/s}$ in LEO), changing inclination is dramatically cheaper. Edelbaum's optimal steering law smoothly steers the thrust yaw angle out-of-plane toward the end of the transfer, saving enormous delta-V.
Why are all-electric propulsion satellites dominating commercial geostationary orbits?
Chemical apogee kick engines have an $I_{sp}$ of only $320\,\text{seconds}$, requiring $50\% \sim 60\%$ of launch vehicle payload mass to be propellant. High-efficiency electric thrusters ($I_{sp} = 1500\sim 3000\,\text{seconds}$) consume one-fifth the propellant mass, allowing operators to launch a communications satellite twice as capable on the same commercial rocket.