Presets
Playback
Mission
Altitudes and the plane change to remove.
Burn split
Fraction of the plane change done at the apogee burn.
Engine and clock
Layers
Model checks
The science
Circular and transfer speeds. v_c = sqrt(mu / r). The Hohmann ellipse between r1 and r2 has a = (r1 + r2) / 2, perigee speed v_p = sqrt(mu (2/r1 - 1/a)) and apogee speed v_a = sqrt(mu (2/r2 - 1/a)); the coast takes t = pi sqrt(a^3 / mu).
Burn with a plane change. A burn that also turns the velocity by an angle di needs, by the law of cosines, dv = sqrt(v_before^2 + v_after^2 - 2 v_before v_after cos(di)). Turning the plane where the speed is lowest is cheapest, which is why the apogee is attractive. But the apogee burn must also raise the speed to circular, and the two requirements share one burn. The two inclination angles obey di1 + di2 = di; the total dv1(di1) + dv2(di2) has a single minimum. For a 28.5 degree start the optimum puts about 2.2 degrees at perigee and the rest at apogee (a commonly quoted figure; the simulator computes it).
Propellant. m_prop / m_0 = 1 - exp(-dv / (I_sp g_0)) with g_0 = 9.80665 m/s2. A 100 m/s saving is a few percent of the launch mass in orbit.
Honest simplifications. Impulsive burns (instant dv; real burns last minutes and lose some efficiency), spherical Earth without J2, no Moon or Sun perturbation, circular start and end orbits, both burns exactly at the line of nodes, no engine thrust-to-weight limit, no phasing or launch window. The satellite and Earth are drawn out of scale so they are visible; the orbit lines are to scale.
What to try
- Load Cape Canaveral to GEO and drag the split slider from 0 to 1: the total delta-v dips to a minimum near 92 % at apogee, not at either end.
- Pick Kourou to GEO (5.2 deg): the optimum moves towards doing almost all of the turn at apogee, and the saving shrinks.
- Raise the target altitude from 20 000 km to 40 000 km and watch the optimum split change.
- Switch the engine Isp to 450 s and see how little propellant a 100 m/s saving represents.