Pad to Orbit

The velocity a stage can add is set by its exhaust velocity (specific impulse \(I_{sp}\)) and how much of its mass is propellant — the Tsiolkovsky rocket equation:

Because \(\Delta v\) grows only with the logarithm of the mass ratio, brute-forcing orbit with a single stage is wildly inefficient. Staging — dropping dead tankage mid-flight — is what makes orbit practical, and is why every vehicle here sheds stages as it climbs.

Each engine here is modelled by its sea-level thrust and \(I_{sp}\); mass flow follows from

As ambient pressure \(p\) falls with altitude, thrust rises toward its vacuum value — which is why a first stage visibly gains punch as it leaves the atmosphere.

Density and pressure use an exponential isothermal model with scale height \(H\approx 8.5\,\text{km}\):

Crucially, drag and dynamic pressure use velocity relative to the co-rotating atmosphere — the vehicle lifts off already carrying ~407 m/s of eastward Earth-rotation speed. Max Q, the structural design driver, typically peaks here at 25–60 kPa, consistent with real launchers.

Guidance shapes the climb against the air-relative flight-path angle (so the rocket reads as vertical on the pad, not horizontal), commanding

so the nose falls smoothly from vertical toward the horizon as apoapsis approaches target. Once apoapsis is reached the vehicle latches into insertion, regulating vertical speed to zero and throttling to pin apoapsis while the near-horizontal burn lifts periapsis. Insertion is declared when periapsis and apoapsis both clear the atmosphere.