Launch Vehicle Trajectory Simulation:
Multi-Stage Burn Modeling and Orbital Insertion

1. Introduction

Orbital launch vehicle trajectory simulation is fundamental to mission design, range safety analysis, and vehicle performance verification. While high-fidelity tools like POST II and GMAT exist for mission-level design, there remains a need for rapid, transparent trajectory models that can be validated, inspected, and iterated quickly.

We present launch-veh-sim, a Go-based trajectory simulator that models the complete ascent-to-orbit sequence for eleven launch vehicles spanning American, European, and Chinese space programs. The simulator implements a multi-burn second-stage architecture with coast phases and apogee-restart circularization, a capability essential for realistic modeling of modern upper stages.

The key contribution of this work is twofold: first, a validated physics model for multi-stage ascent with aerodynamic drag, altitude-dependent thrust/ISP, and spherical gravity; and second, a systematic identification of the pitch-table guidance regime boundary, demonstrating that stages with TWR < 0.35 require closed-loop guidance to achieve stable orbit insertion.

2. Vehicle Catalog

2.1 Overview

The simulator models eleven launch vehicle configurations, ranging from medium-lift vehicles like Atlas V 551 to super-heavy lifters like Starship. The following table summarizes the fleet:

2.2 Stage Specifications

First Stages

Second Stages

Boosters

3. Physics Models

3.1 Gravity Model

The simulation uses a spherical gravity model with the standard gravitational parameter $\mu = 3.986004418 \times 10^{14}$ m³/s² and equatorial radius $R_E = 6{,}378{,}137$ m. The gravitational acceleration at radius $r$ from Earth's center is:

The simulation accounts for Earth's rotation by initializing tangential velocity as $V_{t,0} = \Omega R_E \cos(i)$, where $\Omega = 7.2921159 \times 10^{-5}$ rad/s and $i$ is the launch inclination. This provides the free ride from Earth's rotation, which at 28.5° contributes approximately 408 m/s of horizontal velocity.

For orbital lifetime estimates, J2 perturbation ($J_2 = 1.08263 \times 10^{-3}$) should be considered for long-term altitude decay, though it is not modeled in the current propagation step.

3.2 Aerodynamic Drag

Atmospheric drag is modeled using a piecewise exponential atmosphere with five altitude regimes:

Drag force is computed relative to the co-rotating atmosphere:

where V_rel accounts for Earth's rotation (atmosphere co-rotating), and the effective drag area A is specified per vehicle via the DragArea field. The simulation uses C_D = 1 implicitly (absorbed into A). Drag is applied below 80 km altitude only, decomposed into radial and tangential components.

3.3 Thrust and Mass Flow

Engine thrust and specific impulse vary with altitude due to ambient back-pressure on the nozzle. The simulator linearly interpolates between sea-level and vacuum values:

Mass flow rate is derived from thrust and altitude-adjusted $I_{\text{sp}}$:

where $g_0 = 9.80665$ m/s². This is the Tsiolkovsky-consistent formulation that preserves the rocket equation $\Delta V = I_{\text{sp}} g_0 \ln(m_0/m_f)$ for constant $I_{\text{sp}}$.

The integration step uses dt = 0.1 s, which provides sufficient temporal resolution for capturing pitch transitions and burn phase boundaries while remaining computationally efficient.

4. Multi-Burn Trajectory Architecture

The simulator implements a five-phase trajectory model for the second stage, reflecting modern orbital insertion practice:

For vehicles without multi-burn capability (e.g., Delta IV M+ in the current configuration), the second stage fires continuously until propellant depletion, and no circularization occurs.

5. S2 Guidance: Pitch Tables vs. Closed-Loop

5.1 First-Stage Guidance: Pitch Tables

During first-stage flight, the vehicle follows a pre-programmed pitch schedule defined as a series of (time, pitch angle) waypoints. The pitch angle is the angle from local horizontal: 90° = vertical, 0° = horizontal. Between waypoints, pitch is linearly interpolated. Above 60 km, the pitch smoothly blends toward the velocity vector (prograde direction), with the blend fraction increasing linearly over the 60–140 km altitude range.

This approach is standard for first stages because their high TWR and the dominance of gravity losses make them relatively insensitive to small pitch errors. The pitch table serves primarily to shape the gravity turn rather than target a specific orbit.

5.2 Second-Stage Guidance: Adaptive Logic

For the second stage, the simulator implements a conditional guidance scheme:

if TWR >= 0.35:
    if Vr > 200 m/s (climbing):
        pitch = max(0, atan2(Vr, Vt) * 0.5)   // half-steer toward prograde
    elif Vr < -200 m/s (falling):
        pitch = asin(min(0.95, net_g / a_max))  // gravity-loss-compensating steer
    else (near-apogee):
        pitch = 0  // burn prograde
else:
    pitch = 0  // low-TWR: always burn prograde

The gravity-loss-compensating steer for $V_r < -200$ m/s is a key innovation: it computes the net downward gravitational acceleration not offset by centripetal acceleration, then determines what pitch angle would use the available thrust to counteract this sink rate. This prevents the vehicle from losing altitude during low-TWR burns.

5.3 The TWR Boundary

Analysis of the simulation results reveals a clear TWR boundary for pitch-table guidance effectiveness:

The pattern is clear: pitch-table guidance fails for stages with TWR < 0.35 that also lack a sufficiently long coast phase or appropriate burn fraction. Long March 5 succeeds despite low TWR because its second stage has a high $I_{\text{sp}}$ (442.6 s) and the burn fraction/coast trigger combination happens to produce a stable trajectory. The failures of Vulcan and Delta IV, and the overshoots of Atlas V and Ariane 6, all stem from the same root cause: the prograde-only guidance during S2 burn cannot adapt to the trajectory state.

6. Booster Modeling

Six of the eleven vehicles carry solid rocket boosters (SRBs) that fire in parallel with the first stage. The simulator models boosters as additional thrust sources with their own mass flow, separate from the first stage. Key modeling decisions:

• Parallel burn: Booster thrust and mass flow are added to the first stage. The combined acceleration determines the trajectory during the booster phase.
• Mass accounting: Booster propellant and dry mass are tracked separately. At burnout, both are jettisoned instantaneously, providing a discrete mass step that models separation.
• ISP consistency: Boosters use a single ISP value (sea-level) since SRBs operate predominantly in the lower atmosphere. Mass flow is computed as $T / (I_{\text{sp}} \times g_0)$.
• No altitude-dependent thrust: Unlike liquid engines, SRBs are modeled with constant thrust throughout their burn, which is a reasonable approximation for the composite-case motors used on all modeled boosters.

The separation event is logged as a milestone, and the sudden mass reduction often produces a visible acceleration increase in the trajectory data.

7. Orbital Insertion Analysis

7.1 Verified Orbits

Six vehicles achieve correct orbital insertion with the current guidance model:

All verified orbits have eccentricity < 0.005 after circularization, which is within the typical insertion accuracy for LEO missions. The circularization delta-V values (99–118 m/s) are consistent with theoretical predictions for the altitude range and confirm that the multi-burn architecture correctly computes the apogee-raise maneuver.

Falcon 9 Block 5 Trajectory Profile

The Falcon 9 trajectory is the most extensively validated, producing a 445 × 385 km orbit with a 92.9-minute period. Key milestones:

The approximately 3,900-second coast between S2 Burn 2 end and circularization start represents the transit from the transfer orbit perigee to apogee. The 99.6 m/s circularization burn at 445.5 km altitude completes the orbit insertion.

7.2 Problem Orbits

8. Validation Results

The following table compares simulated orbit parameters against published reference values where available:

Summary: 6/11 vehicles produce correct orbits (55% validation rate). 2/11 crash (Vulcan Centaur, Delta IV). 3/11 overshoot (Falcon Heavy, Atlas V, Ariane 6). All failures trace to the same root cause: lack of closed-loop altitude-targeting guidance for low-TWR upper stages.

8.1 Dynamic Pressure Analysis

Max Q values across the fleet are consistent with published data:

All verified vehicles show max Q around 24–41 kPa at approximately 12–13 km altitude, which is consistent with real-world launch vehicle data (typically 25–40 kPa). The higher max Q for Falcon Heavy (41 kPa) reflects its triple-core configuration providing higher initial acceleration.

9. Future Work

The following improvements are planned to address the identified guidance deficiencies:

9.1 Closed-Loop Guidance (Priority: High)

Implement a target-altitude guidance law that steers toward a desired orbital altitude rather than following fixed pitch tables. The most promising approach is a variant of Powered Explicit Guidance (PEG), which computes the required pitch angle to reach a target state given current position, velocity, and remaining propellant. This would resolve all five problematic vehicles.

9.2 TargetAltKm Parameter

Add a configurable TargetAltKm field to the vehicle configuration, allowing each simulation run to specify a desired circular orbit altitude. The guidance law would then compute the appropriate burn durations and pitch profiles to achieve this target.

9.3 Improved Coast Triggers

The current coast-phase termination logic uses hard thresholds ($|V_r| < 100$ m/s, alt > 80 km, timeout 1800 s). A more physical trigger would use orbital energy: compute the current semi-major axis and transition to S2 Burn 2 when the apogee target is reachable from the current state.

9.4 J2 Perturbation Propagation

While the gravitational model is currently spherical, adding J2 perturbation (secular drift in $\Omega$ and $\omega$) would improve long-term orbit prediction and enable realistic RAAN/argument-of-perigee targeting.

9.5 Delta IV Multi-Burn

Enable multi-burn mode for Delta IV, which currently fires its second stage continuously. The real Delta IV upper stage (DCSS) does perform multi-burn operations; enabling this in the simulator (with appropriate guidance) should resolve the impact failure.

9.6 Booster Landing Modeling

Add support for modeling booster flyback and landing, particularly for Falcon 9 and Falcon Heavy core/boosters. This would enable recovery cost analysis and trajectory planning for reentry burns.

9.7 Neutron Vehicle

Add the Rocket Lab Neutron configuration, which is included in the vehicle catalog but not yet validated with simulation data.

10. Conclusion

We have presented a numerical trajectory simulation framework for eleven launch vehicles that correctly models multi-stage ascent with aerodynamic drag, altitude-dependent engine performance, and multi-burn second-stage architecture. The simulator achieves verified orbital insertion for 6 of 11 vehicles, with circularization eccentricities below 0.005 and consistent max Q values.

The key finding is that pitch-table guidance, while adequate for first stages and high-TWR second stages (TWR > 0.35), systematically fails for low-TWR upper stages. Vehicles with Centaur-class second stages (TWR ≈ 0.08–0.14) require closed-loop guidance that can target specific orbital parameters. The current prograde-only fallback for low-TWR stages produces either impact (Vulcan Centaur, Delta IV) or significant overshoot (Atlas V, Ariane 6), depending on the energy state at S2 ignition.

The multi-burn architecture with coast phases and apogee-restart circularization correctly models the operational practice of modern hydrogen-fueled upper stages, and the simulation framework provides a solid foundation for implementing closed-loop guidance. The identified failure modes map directly to the absence of altitude-targeting, making the path to 11/11 validation clear: implement PEG or equivalent guidance for S2 phases.

The simulator's Go implementation provides sub-second execution time for all vehicles, enabling rapid iteration for trajectory design studies. The JSON output format supports automated validation pipelines and visualization tools.

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