Gravity Turn Trajectory Estimator
Numerically simulate a gravity turn ascent trajectory and estimate velocity, altitude, and delta-v losses for any rocket on Earth or Mars.
🚀 What is a Gravity Turn Trajectory?
A gravity turn trajectory is the standard launch profile used by most orbital rockets. After a brief vertical phase immediately after liftoff, the vehicle makes a small pitch maneuver called the kickover, and then gravity itself curves the flight path from vertical toward horizontal. Because the velocity vector continuously aligns with the thrust direction throughout this turn, the rocket experiences zero angle of attack, minimizing structural bending loads and eliminating the need for active aerodynamic pitch control during the ascent arc.
The technique matters for three reasons. First, it is propellant-efficient: an unconstrained gravity turn accumulates less gravity loss than a constant-pitch trajectory for the same burnout conditions. Second, it is structurally benign: without angle of attack, aerodynamic side loads are near zero, allowing lighter vehicle structures. Third, it is naturally stable: small perturbations from wind or engine mismatch are self-correcting because the vehicle angle-of-attack tends to return to zero. This combination of efficiency and structural simplicity is why every modern expendable launcher, from Falcon 9 to Ariane 5 to ISRO PSLV, flies a gravity turn ascent.
In practice, launch vehicles fly a modified or pitch-programmed gravity turn rather than a pure one. The guidance computer continuously adjusts the pitch rate to match a planned angle-versus-time profile, targeting a precise orbital injection condition (altitude, velocity, and flight path angle). The pure gravity turn that this calculator simulates is the idealized, analytically tractable version. It is an excellent educational model and a useful first approximation for preliminary design studies before detailed trajectory optimization.
The trajectory divides into two phases. The initial vertical phase lasts from liftoff until the kickover altitude, during which the vehicle climbs straight up to clear terrain and build speed. After kickover, the gravity turn phase begins. The vehicle pitches a few degrees from vertical and then holds zero angle of attack. Gravity accelerates the horizontal velocity component, which in turn causes the flight path angle to decrease further, which causes even more horizontal acceleration. This positive feedback loop is the gravity turn mechanism, and it continues until engine cutoff (MECO) or until the vehicle reaches a horizontal flight path at orbital altitude.
📐 Formula
Net Δv = Isp × g0 × ln(m0 / mf) − Δvgrav − Δvdrag
Isp = specific impulse (s)
g0 = 9.80665 m/s² (standard gravity)
m0 = initial wet mass (kg)
mf = burnout dry mass = m0 − (F / (Isp × g0)) × tburn
Δvgrav = ∫ g(t) sin γ(t) dt (gravity loss, m/s)
Δvdrag = ∫ D(t)/m(t) dt (drag loss, m/s)
γ = flight path angle from horizontal (°)
dγ/dt = −(g cos γ)/v + v cos γ / (R + h) (gravity turn equation)
D = CD × A × (0.5 × ρ × v²) (drag force, N)
ρ = ρ0 × exp(−h / H) (exponential atmosphere)
g(h) = g0 × (R / (R + h))² (inverse-square gravity)
Example: Isp 310 s, m0 70 t, mf 20.6 t: ideal Δv = 310 × 9.807 × ln(3.40) = 3,715 m/s
📖 How to Use This Calculator
Steps
1
Choose body and set rocket parameters - Select Earth or Mars as the launch body. Enter thrust in kN, liftoff mass in tonnes, and specific impulse (Isp) in seconds. The initial thrust-to-weight ratio is displayed immediately.
2
Set drag area and burn parameters - Enter the drag area (Cd times reference area in m²), the total powered burn time in seconds, and the kickover altitude in km where the gravity turn begins.
3
Run the trajectory simulation - Click Calculate to run the Euler integration. Results show MECO velocity, altitude, and downrange distance along with gravity losses, drag losses, and max dynamic pressure.
4
Switch to Delta-V Budget mode - Use the Delta-V Budget tab to enter your rocket's total available delta-v and separate gravity, drag, and steering loss estimates to check whether a target orbit altitude is achievable.
💡 Example Calculations
Example 1 — Medium Earth Launcher, 100-Second Burn
1500 kN thrust, 70-tonne liftoff mass, Isp 310 s, Cd x A = 3 m², 100 s burn, 2 km kickover
1
Propellant flow rate: 1,500,000 / (310 x 9.807) = 494 kg/s. Propellant consumed in 100 s: 49,400 kg. Burnout mass: 70,000 - 49,400 = 20,600 kg.
2
Ideal delta-v: 310 x 9.807 x ln(70,000/20,600) = 3,040 x 1.223 = 3,718 m/s. Initial TWR: 1,500,000 / (70,000 x 9.807) = 2.18.
3
The simulator integrates the trajectory. Gravity losses accumulate at roughly 9.8 m/s per second during the vertical phase and less during the turn. Drag losses are highest near Max-Q at low altitude.
Results: MECO velocity near 3.0 km/s, altitude 50 to 70 km, with gravity losses approximately 600 m/s and drag losses approximately 130 m/s
Try this example →Example 2 — High-Thrust Earth Launcher, 80-Second Burn
2500 kN thrust, 80-tonne liftoff mass, Isp 320 s, Cd x A = 4 m², 80 s burn, 1.5 km kickover
1
Propellant flow rate: 2,500,000 / (320 x 9.807) = 795 kg/s. In 80 s: 63,600 kg burned. Burnout mass: 80,000 - 63,600 = 16,400 kg.
2
Ideal delta-v: 320 x 9.807 x ln(80,000/16,400) = 3,138 x 1.585 = 4,974 m/s. Initial TWR: 2,500,000 / (80,000 x 9.807) = 3.19 (very high). Burn ends in just 80 s, limiting gravity loss.
3
High TWR means the vehicle passes through the dense lower atmosphere very quickly, so Max-Q is reached early and drag losses are moderate. The short burn time also keeps gravity losses lower than Example 1.
Results: MECO velocity near 4.5 km/s, altitude 40 to 60 km, with gravity losses below 500 m/s due to the short burn and high TWR
Try this example →Example 3 — Earth LEO Delta-V Budget Check
Available DV 9400 m/s, gravity loss 1100 m/s, drag loss 300 m/s, steering loss 150 m/s, target 400 km LEO
1
Total losses: 1,100 + 300 + 150 = 1,550 m/s. Net delta-v: 9,400 - 1,550 = 7,850 m/s.
2
Circular orbit velocity at 400 km: sqrt(3.986e14 / (6,371,000 + 400,000)) = sqrt(3.986e14 / 6,771,000) = 7,672 m/s.
3
Mission margin: 7,850 - 7,672 = 178 m/s. Loss fraction: 1,550 / 9,400 = 16.5%. Status: Orbit Achievable with 178 m/s reserve.
Net DV = 7,850 m/s, Required = 7,672 m/s, Margin = +178 m/s, Loss fraction = 16.5%
Try this example →Example 4 — Mars Ascent Vehicle Budget
Available DV 5500 m/s, gravity loss 500 m/s, drag loss 30 m/s, steering loss 100 m/s, target 300 km Mars orbit
1
Mars atmosphere is very thin (surface density 0.020 kg/m^3), so drag losses for a Mars ascent vehicle are negligible. Gravity on Mars is 3.72 m/s^2, so gravity losses accumulate more slowly than on Earth.
2
Total losses: 500 + 30 + 100 = 630 m/s. Net delta-v: 5,500 - 630 = 4,870 m/s.
3
Circular orbit velocity at 300 km Mars: sqrt(4.283e13 / (3,389,500 + 300,000)) = sqrt(4.283e13 / 3,689,500) = 3,407 m/s. Margin: 4,870 - 3,407 = 1,463 m/s (very comfortable).
Net DV = 4,870 m/s, Required = 3,407 m/s, Margin = +1,463 m/s, Status: Orbit Achievable
Try this example →❓ Frequently Asked Questions
What is a gravity turn trajectory in rocketry?
A gravity turn trajectory is a launch ascent path where the rocket makes one small pitch maneuver (the kickover) at low altitude and then holds zero angle of attack for the remainder of powered flight. Gravity continuously rotates the velocity vector from vertical toward horizontal without any active pitch control. The technique minimizes aerodynamic structural loads, avoids gravity turn drag, and typically achieves good propellant efficiency compared to a purely vertical trajectory. It is the dominant launch strategy for expendable orbital rockets worldwide.
What does MECO stand for and why does MECO velocity matter?
MECO stands for Main Engine Cutoff. It marks the end of the first stage burn. MECO velocity is the rocket's speed at that instant and is the key output of a first-stage trajectory analysis. If the vehicle has a single stage, MECO velocity must exceed orbital velocity minus upper atmosphere deceleration. For a two-stage vehicle, MECO velocity plus the second-stage delta-v must together exceed the orbital injection requirement. Typical MECO-1 velocities for heavy launch vehicles are 2 to 3 km/s at 60 to 80 km altitude.
How are gravity losses calculated in a trajectory simulation?
Gravity loss is integrated numerically as the sum of g(t) times sin(gamma(t)) times dt over each time step of the simulation. During the vertical phase (gamma = 90 degrees), gravity loss accumulates at the full gravitational acceleration rate. As the gravity turn progresses and gamma decreases toward zero, gravity loss rate falls proportionally. Total gravity loss for a typical Earth launch is 800 to 1,500 m/s, depending on trajectory shape and burn duration. Low-TWR vehicles that burn slowly accumulate more gravity loss because they spend more time at high flight path angles.
What is the optimal kickover altitude for minimizing total losses?
The optimal kickover altitude balances gravity loss against drag loss. Kicking over early (0.5 to 1 km) reduces gravity losses because the vehicle turns horizontal sooner, but it flies fast at low altitudes where air is dense, increasing drag losses and Max-Q. Kicking over late (3 to 5 km) reduces drag losses but allows gravity to act downrange longer. For typical Earth launches, kickover at 1 to 2 km provides near-optimal total losses. On Mars, with its thin atmosphere, earlier kickover (0.5 to 1 km) incurs negligible drag penalty and more than compensates in reduced gravity loss.
Why does a higher thrust-to-weight ratio reduce gravity losses?
Gravity loss equals the integral of g times sin(gamma) over the entire burn. A vehicle with TWR 2.0 burns propellant twice as fast as one with TWR 1.0 (for the same Isp), reaching the same velocity in half the time. Since gravity loss accumulates with time, the higher-TWR vehicle accumulates roughly half the gravity loss. However, higher TWR requires larger, heavier engines, which reduce propellant fraction and ideal delta-v. The economically optimal TWR for first stages is typically 1.3 to 1.8, balancing reduced gravity losses against the engine mass penalty.
What causes max dynamic pressure (Max-Q) and how do rockets handle it?
Dynamic pressure q = 0.5 times rho times v^2 increases as speed builds during ascent and decreases as air density drops with altitude. The maximum value (Max-Q) occurs where these effects balance, typically at 10 to 15 km altitude and Mach 1.0 to 1.5. At Max-Q, aerodynamic bending and shear loads on the launch vehicle are at their peak. Most launch vehicles briefly throttle down their engines near Max-Q (the throttle bucket) to limit structural loads. After passing Max-Q, the vehicle can throttle back up because the rapidly decreasing air density reduces loads even as speed continues to increase.
What is the delta-v loss budget for an Earth-to-LEO mission?
A 400 km circular orbit has an orbital velocity of about 7,672 m/s. Total delta-v from Earth's surface to that orbit is approximately 9,200 to 9,600 m/s when including losses. The breakdown is typically: orbital velocity 7,672 m/s plus gravity losses 1,000 to 1,300 m/s plus drag losses 100 to 400 m/s plus steering losses 50 to 200 m/s. Efficient vehicles with high TWR and good aerodynamic design target the low end of this range. Less efficient trajectories with low TWR or large drag areas can exceed 10,000 m/s total delta-v requirement.
How does atmospheric density affect the trajectory simulation?
This calculator uses an exponential atmosphere: density = rho0 times exp(-h / H), where rho0 is sea-level density and H is the scale height. For Earth: rho0 = 1.225 kg/m^3, H = 8,500 m. For Mars: rho0 = 0.020 kg/m^3, H = 11,100 m. This model accurately represents the atmospheric profile to about 60 km altitude, covering the region where most aerodynamic drag occurs. Above that, drag becomes negligible in the simulation regardless of model accuracy. The Mars atmosphere model explains why Martian drag losses are so small: surface density is only 1.6% of Earth's.
What is flight path angle and how does it change during ascent?
Flight path angle (gamma) is the angle between the velocity vector and the local horizontal plane. At liftoff, gamma = 90 degrees (purely vertical). During the gravity turn, gamma decreases continuously toward 0 degrees (purely horizontal). At orbital injection, gamma should be 0 degrees for a circular orbit or slightly positive for an elliptical orbit with the apogee elsewhere. The gravity turn equation d(gamma)/dt = -(g cos gamma) / v + v cos gamma / (R + h) governs this evolution. The first term represents the gravitational pitch-down rate; the second term is a small centrifugal correction for low altitudes.
Can this calculator be used to design a real rocket trajectory?
This calculator is suitable for educational analysis, preliminary concept evaluation, and understanding the sensitivity of trajectory performance to rocket parameters. It uses simple Euler integration, an exponential atmosphere, and a pure gravity turn assumption. Production launch vehicle trajectories require full 6-DOF simulations with complete atmospheric models (winds, turbulence, 1976 US Standard Atmosphere), real engine performance maps, structural load constraints, mission-specific injection targets, and guidance law optimization. Results from this calculator typically agree with full simulations to within 5 to 15% for velocity and altitude at MECO, which is useful for early trade studies.
What is steering loss and how large is it for a typical mission?
Steering loss occurs when the thrust vector is tilted relative to the velocity vector to follow a guidance-commanded trajectory. The component of thrust perpendicular to velocity does no useful work. For a well-designed gravity turn trajectory, steering losses are small: 50 to 200 m/s. They are larger for vehicles flying dog-leg maneuvers to avoid overflight of populated areas, for trajectories targeting highly inclined orbits that require significant azimuth changes, or for vehicles with poor aerodynamic stability requiring large gimbaled corrections. The Delta-V Budget mode lets you include an estimated steering loss in your total mission delta-v accounting.
How does the gravity turn trajectory estimator handle propellant depletion?
The simulator tracks propellant consumed at each integration step using mdot = Thrust / (Isp times g0). If the specified burn time would consume more than 99% of the liftoff mass in propellant, the calculator returns an error before running the simulation, prompting you to reduce burn time or increase liftoff mass. This prevents physically impossible cases where a rocket burns more propellant than it carries. During the simulation itself, the calculation stops when the burn time elapses or the propellant is exhausted, whichever comes first.