Rocket launch - 2D trajectory and vehicle

What is modelled

A generic two-stage kerosene and liquid-oxygen launcher flies from the pad to low Earth orbit. The first stage climbs straight up, a small pitch-over kick tips it over, and from then on gravity alone bends the path (a gravity turn). When the first stage is empty it separates; after a short coast the upper stage lights and steers to the target altitude. It stops burning when its orbit no longer dips below that altitude.

Equations

• Motion in the orbital plane over a non-rotating Earth, in polar form: dr/dt = v sinγ, dθ/dt = v cosγ / r, dv/dt = (T cosα − D)/m − g sinγ, v dγ/dt = T sinα/m − (g − v²/r) cosγ with g = μ/r². The angle of attack α is zero for stage 1 (pure gravity turn) and is chosen by a simple altitude-capture rule for the upper stage.
• Thrust follows the ambient pressure: Isp(p) = Ispvac − (Ispvac − Ispsl) p/p0, T = ṁ g0 Isp. Mass falls at the constant rate ṁ, so the rocket equation Δv = Isp g0 ln(m0/mf) sets the ideal budget.
• Drag D = ½ ρ v² Cd(M) A with density, pressure and the speed of sound from the layered International Standard Atmosphere (exponential tail above 86 km) and a simple transonic drag curve. Dynamic pressure q = ½ ρ v² peaks at max-q.
• Orbit from the state: ε = v²/2 − μ/r, a = −μ/(2ε), h = r v cosγ, e = √(1 + 2εh²/μ²), perigee a(1 − e), apogee a(1 + e). The fairing (1.7 t) is dropped at 215 s.

Checks run in the page

• A vacuum, gravity-free burn of each stage, integrated with RK4, reproduces the Tsiolkovsky delta-v.
• The air density from the ISA code matches the standard-atmosphere table from sea level to 70 km.
• A 400 km circular orbit integrated for one period closes on itself, conserves energy and has the period 2π√(a³/μ).
• An ellipse started at 200 km perigee reaches the apogee given by the orbit-element formulas and conserves angular momentum.
• In the real flight code, the upper-stage delta-v accumulated by the integrator equals the rocket equation, and the default flight ends in orbit.

Honest simplifications

• The vehicle is generic: its masses, thrust and Isp are kerosene-and-LOX-class numbers I chose, not those of any real rocket, and no real company or launcher is represented.
• The Earth does not rotate (a real launch gains up to 465 m/s from the equator), there are no winds, no thrust-vector limits, no throttling, no boil-off, and the flight is in one plane.
• The drag coefficient curve is a rough shape, the guidance rule is a simple feedback and not a real ascent guidance law, and the cutoff is "perigee reaches the target" rather than an optimised burn.
• Stage separation, the fairing and the first stage after separation are drawn as free bodies moving under gravity only; re-entry and landing are not modelled.
• The trajectory plot is altitude against downrange distance with the Earth drawn flat; the 3D page shows the same flight over a globe.

What to try

• Choose Kick too small and Kick too big and compare the trajectories: the first climbs steeply and overshoots, the second flattens in thick air and falls back.
• Choose Heavy payload and see the upper stage run out of propellant before reaching the target orbit.
• Watch max-q: dynamic pressure peaks about a minute after liftoff because the air thins faster than the rocket speeds up.