◠ Spiral

Continuous-thrust trajectory design. Set two circular orbits, a plane change, and a thruster; see the Δv, transfer time, propellant, and the true trajectory. v0.2 — numerically propagated (RK4) with guidance that circularizes at arrival. Start and end are circular orbits; the path between is the real, generally-eccentric motion. Too much thrust and it can't circularize — it escapes.

Transfer

From the body's center — Earth surface ≈ 6,371 km (LEO ≈ 6,778). The perigee if you set an apogee below.
Leave blank for a circular start; set above the perigee for an ellipse (GTO apogee ≈ 42,164).
Leave blank for a circular target; set above the perigee for an elliptical target (e.g. Molniya).
Edelbaum combined altitude + inclination change.
Adds Earth's equatorial-bulge perturbation — watch the orbit plane precess (nodal regression) over many revolutions.

Trajectory

The real trajectory, integrated step by step. Green settles into a circular orbit; red means the thrust is too high to circularize and the orbit escapes. Press play to fly it — the amber arrow is the thrust heading the vehicle points. Drag the view to rotate and tilt, scroll or pinch to zoom, double-click to reset.

t— r— a— e— i— thrust—

Result

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Δv
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Transfer time
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Propellant
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Final mass
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Revolutions
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Initial accel

About continuous-thrust transfers

Electric propulsion — ion and Hall thrusters — produces thrust thousands of times smaller than a chemical engine but at five to ten times the exhaust velocity, so it sips propellant. The price is time. Because the thrust is a tiny fraction of local gravity, the vehicle cannot make a sharp burn; it thrusts continuously along its velocity and eases outward (or inward) over a slowly-expanding spiral of hundreds or thousands of near-circular revolutions.

That makes a low-thrust transfer behave nothing like the clean two-burn arc of an impulsive Hohmann. A real all-electric LEO-to-GEO climb is on the order of 1,500–2,000 revolutions over months. The orbit is never quite circular during the climb — thrusting continuously pumps up a small eccentricity the guidance must keep nulling, and if the thrust is set too high relative to gravity the eccentricity runs away and the orbit escapes instead of circularizing. This tool integrates the motion directly (Runge–Kutta) so that failure is visible rather than assumed away.

Low thrust is now mainstream: NASA's Dawn visited Vesta and Ceres on ion engines, ESA's SMART-1 spiraled to the Moon, BepiColombo cruises to Mercury under solar-electric power, and most modern communications satellites raise themselves from the launcher's drop-off orbit to GEO with electric thrusters, trading months of spiral for a large cut in launch mass. Plane changes cost more the faster you are moving, which is why they are done high and slow; an optional J2 term adds the nodal precession the real, oblate Earth produces.

References