Ion Engine Orbit-Raising: Direct Two-Body Integration (2D)

Ion Engine

Thrust 120 mN

Specific impulse I_sp 3000 s

Spacecraft wet mass 900 kg

Time acceleration 6.0 days/s

RK4 Trajectory (Cartesian)

Radius r (instant.)

6771 km

Orbital period

92.6 min

Semi-major axis a

6771 km

RK4 vs analytic Δa

0.000%

Eccentricity e

0.0000

Mission time

0.0 d

Δv used

0.00 km/s

RK4 sub-steps/frame

Propellant: 100%— kg left

Integrating full 2-body equations of motion, step by step.

How it works (2D model)

The 3D companion tracks the orbit-raising spiral by assuming a circular orbit at every instant and analytically differentiating the specific orbital energy ε = −μ/(2a) for continuous tangential thrust. This 2D simulator instead makes no circular assumption: it integrates the raw Newtonian equations of motion for position r = (x, y) and velocity v = (v_x, v_y) directly in the inertial frame, with thrust applied along the instantaneous velocity direction every step:

d r/dt = v
d v/dt = −μ r/|r|³  +  (F/m)·(v/|v|)
ṁ = −F / (I_sp · g₀)

Each step is advanced with a classical 4th-order Runge-Kutta integrator (RK4), with the sub-step size scaled to a small fraction of the current orbital period so the fast circular motion stays numerically stable even while the semi-major axis itself grows over simulated weeks. After every step the code also extracts the osculating orbit elements directly from the instantaneous state — the same way real orbit-determination software does — using the vis-viva energy and the specific angular momentum h = x·v_y − y·v_x:

ε = |v|²/2 − μ/r        a = −μ/(2ε)
e = √(1 + 2εh²/μ²)      T = 2π√(a³/μ)

In parallel, the same analytic da/dt = 2(F/m)√(a³/μ) formula the 3D view relies on is integrated forward from the identical starting orbit. Because low-thrust orbit-raising only works at all when the orbit stays very nearly circular (e stays tiny), the two independent methods — one from raw forces in Cartesian coordinates, one from an energy-rate shortcut — should track each other closely. The live "RK4 vs analytic Δa" readout is exactly that check: it turns the 3D simulator's assumption into a number you can watch, rather than take on faith.

Thrust — engine force in millinewtons; higher thrust raises the orbit faster and also increases the (still small) eccentricity excited per orbit.
I_sp — exhaust velocity in units of g₀; higher I_sp means less propellant burned per unit of Δv.
Wet mass — total spacecraft mass including propellant; a lighter craft accelerates outward faster for the same thrust.
Time acceleration — simulated days per real second; the RK4 sub-step count is recomputed every frame to stay accurate regardless of how fast time is running.

RK4 spiral trajectory Earth Spacecraft