Gravity Turn Trajectory Estimator

A gravity turn is a launch trajectory where the rocket's velocity vector rotates continuously toward the horizontal under the influence of gravity alone, with no aerodynamic lift needed. After a brief vertical phase, the vehicle pitches slightly and then lets gravity curve the path. Because the velocity and thrust vectors remain aligned throughout, structural bending loads are minimized and no active pitch control is needed during the turn phase. This is the trajectory naturally flown by most expendable launch vehicles and is the reason rockets appear to arc gracefully downrange after liftoff.

MECO stands for Main Engine Cutoff, the moment the first stage engine stops firing after propellant exhaustion or a planned cutoff command. MECO velocity is the rocket's speed at that instant, measured in km/s in an Earth-centered inertial frame. For a Falcon 9 first stage, MECO-1 occurs at about 2.5 km/s at roughly 65 to 70 km altitude. The second stage then ignites to continue accelerating toward orbital velocity (about 7.7 km/s for LEO). The gravity turn estimator computes MECO velocity, altitude, and downrange distance after a single powered burn.

Gravity loss is the delta-v wasted overcoming gravity during powered flight. It equals the integral of g(t) times sin(gamma(t)) dt over the burn, where gamma is the flight path angle from horizontal. Flying vertically (gamma = 90 deg) accumulates gravity loss at the full local gravity rate. Flying horizontally (gamma = 0 deg) accumulates no gravity loss. A typical Earth launch accumulates 800 to 1,500 m/s of gravity loss depending on burn duration, TWR, and trajectory shape. Low-TWR vehicles burn longer and accumulate more gravity loss.

Drag loss is the velocity decrement due to aerodynamic drag: the integral of D/m dt over the burn, where D = 0.5 x rho x v^2 x Cd x A. Most drag loss occurs in the first 30 to 50 km where atmospheric density is high and the rocket is still accelerating through Mach 1. Drag losses for typical orbital rockets are 100 to 500 m/s. To minimize drag losses: use a slender fairing to reduce Cd x A (drag area), throttle back slightly near Max-Q, and avoid flying too fast too low.

Dynamic pressure q = 0.5 x rho x v^2 (Pa) is the local aerodynamic pressure on the vehicle's windward side. Max-Q is the peak value during ascent, reached when the atmospheric density is still significant but the vehicle is moving fast. At Max-Q, structural loads on the fairing, payload, and stage interfaces are at their maximum. Falcon 9 reaches Max-Q at about 78 seconds and 13 km altitude. The vehicle throttles back slightly to reduce bending and shear loads. Staying below the vehicle's Max-Q design limit is a key ascent constraint.

In a pure gravity turn, the vehicle makes one small attitude change (the kickover) near the start of ascent and then holds zero angle of attack. Gravity curves the path with no further active pitch control needed. A pitch-programmed trajectory continuously adjusts pitch rate to follow a pre-planned angle-versus-time profile, allowing the guidance computer to optimise for minimum propellant or a specific injection condition. Modern launch vehicles use pitch programs (not pure gravity turns) but the gravity turn concept remains the dominant physical mechanism that shapes the ascent arc.

Reaching a 400 km circular orbit requires about 9,400 m/s of total delta-v from the ground. The breakdown is approximately: orbital velocity 7,669 m/s at 400 km plus gravity losses 1,100 to 1,300 m/s plus drag losses 100 to 400 m/s plus steering losses 50 to 200 m/s. The total including all losses is 9,100 to 10,000 m/s depending on trajectory efficiency and launch site latitude. The Delta-V Budget mode in this calculator lets you subtract actual loss estimates from any available delta-v to check whether your rocket can close the mission.

Mars surface gravity is 3.72 m/s^2 (38% of Earth). Lower gravity means the rocket needs less TWR to lift off, gravity losses accumulate more slowly, and the vehicle can fly a shallower initial arc without falling back. A Mars ascent vehicle reaching a 300 km orbit needs about 3,400 m/s orbital velocity and typically 400 to 600 m/s of gravity and drag losses. The thin Martian atmosphere (surface density about 0.020 kg/m3 versus Earth's 1.225 kg/m3) contributes negligible drag loss despite the slower vehicle velocity needed to fly through it.

Kickover altitude is the height at which the rocket makes its initial pitch maneuver away from vertical, triggering the gravity turn. A lower kickover (0.5 to 1 km) causes the rocket to arc toward horizontal sooner, reducing gravity losses but increasing drag losses because it flies fast at low altitudes. A higher kickover (3 to 5 km) keeps the vehicle more nearly vertical through the dense atmosphere, reducing drag losses but allowing gravity to act downrange for longer, increasing gravity losses. Real vehicles typically kick over at 1 to 2 km.

Higher TWR means the vehicle accelerates faster, burning propellant in a shorter time and spending less time fighting gravity. A TWR of 2.0 halves the effective burn duration compared to TWR 1.0 for the same delta-v, approximately halving gravity losses. The trade-off is that higher TWR requires larger, heavier engines, which reduce the payload fraction. Most first stages operate at TWR 1.3 to 1.8 at liftoff (increasing as propellant is consumed). Second stages at TWR 0.7 to 1.2 accept higher gravity losses in exchange for lighter, more propellant-efficient designs.

The simulator uses Euler numerical integration at 0.5-second steps with an exponential atmosphere model, inverse-square gravity, and simple drag. It captures the dominant physics of a gravity turn ascent and produces results accurate to within 5 to 15% of full 3-DOF trajectory simulations for typical launch vehicles. Limitations include: no atmospheric wind model, no vehicle attitude control, no throttling, and a simplified atmosphere (standard exponential versus full 1976 US Standard Atmosphere with stratosphere inversion). Use results for educational estimates and preliminary feasibility checks, not for flight-critical trajectory design.

Steering loss (also called guidance or pitch loss) is the delta-v lost because the thrust vector is not perfectly aligned with the velocity vector during trajectory corrections. When a guidance system commands the vehicle to fly a path that differs slightly from straight-line thrust, some thrust is directed perpendicular to the velocity, doing no useful work. For well-designed trajectories, steering losses are small: 50 to 200 m/s for a typical ascent. They are largest for vehicles with poor aerodynamic stability that require large gimbaled corrections, or for missions with aggressive dog-leg maneuvers to reach a specific inclination.