Orbit Plane Tracker (2D)

Two-body motion with thrust

The spacecraft obeys r'' = -mu r / r^3 + F/m with mu = 398600.44 km^3/s^2. It is integrated with a fourth-order Runge-Kutta scheme whose step is limited to 2% of the local dynamical time, so eccentric orbits stay accurate. The osculating orbit (dashed) comes from the instantaneous state: specific energy eps = v^2/2 - mu/r, semi-major axis a = -mu/(2 eps), eccentricity vector e = ((v^2 - mu/r) r - (r.v) v)/mu, periapsis a(1-e), apoapsis a(1+e) and period 2 pi sqrt(a^3/mu).

Propellant, delta-v and efficiency

Thrust F burns propellant at F/(Isp g0); delta-v spent is the integral of the thrust acceleration. The energy efficiency is the specific energy gained from thrust divided by v x delta-v; it is 1 for a burn exactly along the velocity and falls toward 0 for radial thrust and below 0 for retrograde thrust. Burning only near periapsis, where v is highest, puts the same delta-v where it gains the most energy (the Oberth effect).

Hohmann transfer

The planner applies the impulsive burn dv1 = sqrt(mu(2/r1 - 2/(r1+r2))) - sqrt(mu/r1) at the start, lets the spacecraft coast to apoapsis, then circularises with dv2 = sqrt(mu/r2) - v_apo. LEO 300 km to GEO costs about 3.9 km/s.

Numerical checks on this page

(1) Specific energy drift over ten eccentric coasting orbits. (2) The integrated period against Kepler's third law. (3) The integrated LEO to GEO transfer against the closed-form total delta-v and apoapsis radius.

Honest simplifications

Point-mass spherical Earth: no J2, drag, third bodies or eclipse; planar motion; thrust and specific impulse are constant (throttle is a simple scale), thrust is allowed to be continuous and large for visibility; finite-burn gravity losses are real in the continuous modes but the Hohmann planner uses ideal impulses. Atmospheric re-entry is not modelled: the orbit is flagged "decaying" below 120 km periapsis and an impact stops the run below the surface.

• Run Continuous prograde and watch the orbit spiral outwards; efficiency stays near 1.
• Compare with Burn at periapsis only: the orbit becomes an ellipse with a rising apoapsis for the same throttle.
• Retrograde: decay lowers periapsis until the status turns to decaying and then impact.
• Hohmann to GEO shows both impulsive burns and their delta-v.
• Radial thrust changes eccentricity but gives little energy: efficiency is about zero.