↑ orb3e

◭ Gravity turn

A launch ascent where gravity alone pitches the vehicle over — thrust stays along the velocity vector. Defaults to a two-stage vehicle: watch stage 1 drop away, and the circularization Δv fall to near zero once the upper stage does its job. Set stage 2's propellant to 0 to see how much more a single stage would need.

Inputs

An eastward launch gains ω·R·cos(lat) of free velocity — max at the equator (0°), none at the poles (90°). 28.5° ≈ Cape Canaveral.
Stage 1
Stage 2
Set to 0 to fly single-stage and see how much lower it reaches.
Frontal area; the drag coefficient Cd is computed from the local Mach number. 0 = vacuum. Earth uses the US Standard Atmosphere 1976; Moon & Mercury are airless.

Visualization

Set a target orbit altitude and the tool solves the pitch-over angle that lofts the ascent to that apogee (shown as "pitch kick"). Altitude vs downrange over the curved Earth; gravity bends the flight-path angle down as horizontal speed builds. The circularization Δv is what it then costs to turn that apogee into a circular orbit.

apogee—pitch kick—staging—final burnout—v@burnout—downrange—circularize Δv—rotation assist—energy to orbit—max-Q—energy efficiency—

About the gravity turn

A gravity turn (or zero-lift turn) is how essentially every rocket flies to orbit. Just after lift-off, moving slowly and pointing straight up, the vehicle pitches over by a small angle — the pitchover kick. From then on it holds its thrust aligned with its velocity and lets gravity, not the control system, bend the trajectory downrange: gravity pulls the velocity vector, and with it the vehicle, steadily from vertical toward horizontal as speed builds.

Two things make this the efficient ascent. Keeping thrust along the velocity means near-zero angle of attack, so aerodynamic loads stay low through the dense lower atmosphere. And using gravity to do the turning avoids “steering losses” — Δv spent thrusting sideways rather than adding speed. The competing penalty is gravity loss, the Δv lost to holding the vehicle up against gravity, which rewards getting horizontal sooner; the ascent profile balances the two (plus drag loss lower down).

Reaching orbit is a two-part job, and this tool now reports both. The ascent gets the vehicle up to an apogee, but at that apogee it is on a ballistic ellipse — moving slower than a circular orbit at that altitude — so it would fall back without a second, prograde circularization burn, shown here as the Δv from the apogee speed up to circular. The energy to orbit is the specific mechanical energy (ε = v²/2 − μ/r) the vehicle must gain versus sitting on the pad; it works out to ~31–63 MJ/kg for any low orbit, dominated by the kinetic energy of orbital speed — a useful reminder that reaching orbit is about going fast sideways, not just going up. The energy-efficiency readout puts a number on that cost. It divides the payload's mechanical energy actually delivered by the ascent — its kinetic plus potential energy at apogee, measured relative to the pad (½·v² − μ/r) — by the jet kinetic energy the burned propellant carried, ½·Σ(m·v_e²), where the exhaust velocity is v_e = Isp·g₀. It is always well under 100%: most of that jet energy is flung away with the spent exhaust, and more is lost to drag and gravity, so only a slice reaches the payload. (The denominator is the ideal jet energy, not the propellant's raw chemical energy — which is higher — so the true chemical efficiency is lower still.) Staging, by shedding dead mass, measurably raises the fraction: the default two-stage runs several points more efficient than the same vehicle flown single-stage. This tool integrates the planar equations (Runge–Kutta) over a selectable central body (Earth, Moon, Mars, Venus, Mercury). Earth's drag uses the layered US Standard Atmosphere 1976 (density and speed of sound by altitude layer) with a Mach-dependent drag coefficient — flat subsonic, the transonic drag rise peaking near Mach 1.1, then falling off supersonically — applied to a reference area; Mars and Venus use exponential atmospheres, and the Moon and Mercury are airless. The peak dynamic pressure "max-Q" (and the Mach it occurs at) falls out of this, the structural design driver for a real vehicle. Past apogee the animation shows both outcomes: circularize and you stay in orbit (green), or coast ballistically and fall back to the surface (red). It models multi-stage vehicles — each spent stage is jettisoned at separation, which is what lets a rocket shed dead mass and reach orbit — and the central body's rotation: an eastward launch starts with the pad's surface velocity ω·R·cos(latitude), free orbital velocity that shrinks the circularization burn (maximal at the equator, zero at the poles — which is why launch sites hug the equator and rockets fly east, never west). It still omits engine throttling (so max-Q reads a little high versus a real vehicle that throttles down through it) and, being planar, the Coriolis curvature the rotating ground adds to the track.

References