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Rocket Launch Phase Simulator

The purpose of this project is to simulate a launch vehicle from liftoff, occurring at the equatorial line, up to the possible achievement of orbit. This project required the coherent coupling of aerodynamics, thermodynamics, and orbital mechanics within a numerically stable scheme.

This simulator, developed in C++20 with SFML, aims to model a physically consistent ascent while avoiding purely kinematic simplifications.

The project repository can be viewed and downloaded by clicking the button below.

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How to Use the Program

The considerable complexity of the program required the use of several input parameters that the user must provide at the start of the simulation.
To prevent the software from becoming inaccessible to non-expert users and to simplify data entry, at startup the user is asked whether to use the base engine model, a simplified version with few parameters, or an advanced version that requires knowledge of many parameters.

These parameters can be conveniently edited by the user in the file:

assets/input_data/simulation_params.json

For convenience, reasonable default values are already set.

Below is a brief description of all configuration parameters.
Warning: if advanced engines are selected, modifying the base engine section will have no effect and viceversa.

Configuration Parameters

Parameter Description
Rocket  
rocket.name Identifier name of the launch vehicle.
rocket.stage_num Total number of stages (1 initial solid stage + N liquid stages).
rocket.mass_structure_kg Main structural mass of the vehicle, excluding propellant and tanks.
rocket.upper_area_m2 Equivalent frontal area used in aerodynamic drag calculations.
rocket.solid_propellant_mass_kg Total propellant mass of the solid propellant stage.
rocket.solid_container_mass_kg Structural mass of the solid propellant booster, discarded at separation.
rocket.solid_engine_count Number of solid propellant engines firing simultaneously.
rocket.liquid_propellant_masses_kg List of propellant masses for each liquid propellant stage.
rocket.liquid_container_masses_kg List of structural tank masses for each liquid propellant stage.
rocket.liquid_engines_per_stage Number of liquid propellant engines active in each stage.
Base Engines  
engine.base.isp_s Specific impulse expressed in seconds.
engine.base.cm Mass flow coefficient.
engine.base.chamber_pressure_pa Combustion chamber pressure (Pa).
engine.base.burn_area_m2 Effective propellant burning area in the base model.
Advanced Solid Engines  
engine.advanced_solid.burn_area_m2 Initial burning surface area of the solid grain.
engine.advanced_solid.nozzle_throat_area_m2 Nozzle throat area of the solid propellant engine (controls mass flow).
engine.advanced_solid.nozzle_exit_area_m2 Nozzle exit area of the solid propellant engine (affects gas expansion).
engine.advanced_solid.chamber_temperature_k Combustion chamber gas temperature (K).
engine.advanced_solid.grain_dimension_m Characteristic grain dimension affecting burn evolution.
engine.advanced_solid.grain_density_kg_m3 Solid propellant density.
engine.advanced_solid.burn_rate_a Solid propellant burn rate coefficient.
engine.advanced_solid.burn_rate_n Pressure exponent in the solid propellant regression law.
engine.advanced_solid.propellant_molar_mass_g_mol Molar mass of solid propellant engine combustion products (g/mol).
Advanced Liquid Engines  
engine.advanced_liquid.chamber_pressure_pa Combustion chamber pressure of the liquid engine (Pa).
engine.advanced_liquid.nozzle_throat_area_m2 Nozzle throat area of the liquid propellant engine.
engine.advanced_liquid.nozzle_exit_area_m2 Nozzle exit area of the liquid propellant engine.
engine.advanced_liquid.chamber_temperature_k Combustion chamber gas temperature of the liquid propellant engine (K).
engine.advanced_liquid.propellant_molar_mass_g_mol Molar mass of combustion gases in the liquid propellant engine (g/mol).

The user will also be asked to specify the altitude at which the rocket will attempt orbital insertion.
This altitude must be greater than 60,000 m.

Main File Organization

  • Rocket → rocket management, dynamics, and kinematics
  • Engine → physics of all selectable engine types
  • Atmosphere → simulation of atmospheric parameters at different altitudes

Core Dynamic Implementation: rocket.cpp

The file rocket.cpp represents the dynamic and kinematic core of the simulator:
it integrates propulsion, aerodynamics, gravity, stage management, and numerical integration, thus governing the time evolution of the simulation.

Choice of Reference Frame

The simulation uses an inertial frame centered at the Earth’s center, expressed in polar coordinates:

\[(r, \psi)\]

where:

  • ( r ) = distance from Earth’s center
  • ( $\psi$ ) = polar angle
  • ( $v_r$ ) = radial velocity
  • ( $v_t$ ) = tangential velocity

Why not a Cartesian system?

A Cartesian system would require:

  • Explicit handling of Earth curvature
  • Continuous normalization of the gravity vector
  • Greater numerical instability at high altitude
  • Increased difficulty in graphical visualization of the rocket position

Numerical Integration: Runge–Kutta 2 (Midpoint Method)

The file uses an RK2 (midpoint method) integrator to update velocity and position.

Why not explicit Euler?

The classical Euler integrator:

\[x_{n+1} = x_n + v_n \Delta t\]

introduces:

  • Poorer energy conservation during time evolution
  • Instability in orbital regimes
  • Amplified errors in centrifugal terms ( $\frac{v_t^2}{r}$ )

Implemented Equations

In inertial polar coordinates:

\[\dot{v_r} = \frac{F_r}{m} - \frac{v_t^2}{r}\]

and

\[\dot{v_t} = \frac{F_t}{m} + \frac{v_r v_t}{r}\]

The term:

\[\frac{v_t^2}{r}\]

is the geometric centripetal acceleration.

The term:

\[\frac{v_r v_t}{r}\]

is the geometric coupling term.

Advantages of RK2

  • Reduces error during time evolution ($O(\Delta t^3)$ compared to $O(\Delta t^2)$ for Euler)
  • Maintains good orbital stability
  • Balanced compromise between accuracy and computational cost

Stage Management

The program can handle stage separation. However, note that due to the current design, the solid fuel stage must be detached before the first liquid stage.

Separation is handled by predicting how much propellant will be consumed in the next iteration and attempting to use as much propellant as possible.

In the current version, it is not possible to manually control the thrust level of each engine during the simulation.
This makes orbit insertion significantly more challenging and reduces interactivity.

A future version is planned in which the user will have greater control over the launch phase.

Thrust Direction Control

To achieve an optimal pitch angle evolution, rather than defining a regression law analytically, real mission telemetry data was used as reference.

Specifically, telemetry data from several missions, especially Falcon 9, was analyzed, and an average pitch profile as a function of altitude was derived.

improve_theta(...)

Reads an altitude–angle guidance file.

Design choice:

  • Data-driven pitch profile
  • Complete separation between guidance and dynamics

Challenges encountered:

  • Need to store file_pos
  • Avoid continuous file rewind
  • Stabilize angle changes above 20 km

Mach-Dependent Aerodynamics and Transonic Regime

Drag force is modeled as:

\[F_d = \frac{1}{2} \rho v^2 C_d(M) A\]

The main complexity lies in the dependence of the drag coefficient \(C_d(M)\) on Mach number.

Drag Divergence

Near Mach 1:

  • Shock waves form
  • Wave drag increases sharply
  • $C_d$ becomes strongly nonlinear

A piecewise definition introduces undesirable numerical discontinuities.

Smooth Regime Transitions

To avoid numerical artifacts, the following function is implemented:

double Cd_from_Mach(double M);

The function uses smooth transitions based on hyperbolic tangents:

\[C_d(M) = C_{sub} * \frac{C_{trans} - C_{sub}}{2} \left(1 + \tanh(k_1(M - M_1))\right) * \frac{C_{sup} - C_{trans}}{2} \left(1 + \tanh(k_2(M - M_2))\right) * \dots\]

This guarantees:

  • At least $C^1$ continuity
  • Smooth acceleration profiles
  • Numerical stability

The modeled regimes are:

  • Subsonic
  • Transonic
  • Supersonic
  • Hypersonic

Modeling of Main Atmospheric Parameters simulation.cpp

A simple exponential decay of density is insufficient for a realistic ascent simulation. Aerodynamic drag and propulsion performance critically depend on the local thermodynamic state.

International Standard Atmosphere (ISA)

The simulator implements the ISA model, computing atmospheric properties layer by layer:

  • Troposphere
  • Tropopause
  • Stratosphere (up to ~51 km, extendable)

Each layer is modeled using:

Hydrostatic equilibrium: $\frac{dP}{dh} = -\rho g$

Ideal gas law: $ P = \rho R T $

From these relations, altitude-dependent quantities are derived:

  • Pressure $P(h)$
  • Density $\rho(h)$
  • Temperature $T(h)$
  • Local speed of sound $a(h)$

For more technical details, see: ISA model.

Engine Modeling (engine.cpp)

The file engine.cpp implements different levels of propulsion complexity:

  1. Base Engine – constant specific impulse model
  2. Advanced Solid Engine – full internal ballistics coupled with chamber pressure
  3. Advanced Liquid Engine – choked flow model with isentropic relations

The architecture is polymorphic: each engine exposes the same public interface, allowing the user complete freedom of choice:

  • delta_m(dt) → mass lost consumption
  • eng_force(pa, time, theta) → thrust vector

For the mathematical modeling, the following references were used:

Base Engine

This model assumes:

  • Constant chamber pressure
  • Constant mass loss
  • Constant specific impulse

Mass Flow Rate

\[\dot{m} = p_0 \cdot A_{burn} \cdot c_m\]

where:

  • ( $p_0$ ) = nominal combustion chamber pressure
  • ( $A_{burn}$ ) = burning surface area
  • ( $c_m$ ) = empirical mass loss coefficient

Fuel consumption per timestep:

\[\Delta m = \dot{m} \Delta t\]

Thrust

\[F = \dot{m} I_{sp} g_0\]

where:

  • ( $I_{sp}$ ) = specific impulse
  • ( $g_0$ ) = standard gravity acceleration (m/s²)

Advanced Solid Engine

This model implements the internal physics of a solid propellant rocket motor.

Propellant Regression Law (Saint-Robert’s Law):

\[\dot{r} = a P_c^n\]

where:

  • ( a ) = burn rate coefficient
  • ( n ) = pressure exponent
  • ( $P_c$ ) = chamber pressure

Generated Mass Flow Rate

\[\dot{m}_{gen} = \rho_p A_b \dot{r}\]

where:

  • ( $rho_p$ ) = propellant density
  • ( $A_b$ ) = burning area

Mass Flow Through the Nozzle (Choked Flow)

\[\dot{m}_{noz} = \frac{P_c A_t}{c^*}\]

where:

  • ( $A_t$ ) = throat area
  • ( $c^*$ ) = characteristic velocity

Mass Balance

\[\frac{dP_c}{dt} = \frac{R_{spec} T_c}{V_c} (\dot{m}_{gen} - \dot{m}_{noz})\]

where:

\[R_{spec} = \frac{R}{M}\]

Characteristic Velocity

\[c^* = \sqrt{ \left(\frac{2}{\gamma + 1}\right)^{\frac{\gamma + 1}{\gamma - 1}} R_{spec} T_c }\]

Exit Mach Number Calculation

Numerical solution of the area–Mach equation:

\[\frac{A_e}{A_t} = \frac{1}{M_e} \left( \frac{2}{\gamma+1} \left(1 + \frac{\gamma-1}{2} M_e^2\right) \right)^{\frac{\gamma+1}{2(\gamma-1)}}\]

The equation is solved numerically using the Newton–Raphson method to obtain the supersonic exit Mach number ( $M_e$ ).

Exit Pressure

\[\frac{P_e}{P_c} = \left( 1 + \frac{\gamma-1}{2} M_e^2 \right)^{-\frac{\gamma}{\gamma-1}}\]

Exhaust Velocity

\[v_e = \sqrt{ \frac{2\gamma}{\gamma - 1} R_{spec} T_c \left( 1 - \left(\frac{P_e}{P_c}\right)^{\frac{\gamma - 1}{\gamma}} \right) }\]

Total Thrust

\[F = \dot{m} v_e + (P_e - P_a) A_e\]

The second term represents the pressure contribution due to the difference between exit pressure and ambient pressure.

Advanced Liquid Engine

Assumptions:

  • Constant chamber pressure
  • Choked flow at the throat
  • Isentropic expansion

Mass Flow Rate

\[\dot{m} = \frac{P_c A_t}{c^*}\]

Exit Mach Number

Determined by numerically solving the same area–Mach equation:

\[\frac{A_e}{A_t} = \frac{1}{M_e} \left( \frac{2}{\gamma+1} \left(1 + \frac{\gamma-1}{2} M_e^2\right) \right)^{\frac{\gamma+1}{2(\gamma-1)}}\]

Exit Pressure

\[P_e = P_c \left( 1 + \frac{\gamma - 1}{2} M_e^2 \right)^{-\frac{\gamma}{\gamma - 1}}\]

Exhaust Velocity

\[v_e = \sqrt{ \frac{2\gamma}{\gamma - 1} \frac{R T_c}{M} \left( 1 - \left(\frac{P_e}{P_c}\right)^{\frac{\gamma - 1}{\gamma}} \right) }\]

Thrust

\[F = \dot{m} v_e + (P_e - P_a) A_e\]

Differences Between Models

Model Complexity Pressure Evolution Realism
Base Low Constant Approximate
Advanced Solid High Dynamic High
Advanced Liquid Medium/High Constant (assumed) Realistic

The file engine.cpp represents the most sophisticated modeling layer of the simulator.

The objective is not merely to compute thrust,
but to reproduce the coupling between:

  • Combustion gas thermodynamics
  • Chamber pressure dynamics
  • Compressible flow in the nozzle
  • Interaction with ambient pressure

Conclusion

The project evolved from a simple ascent simulator into a structured implementation of:

  • Layered atmospheric thermodynamics
  • Internal propulsion modeling
  • Orbital mechanics
  • Numerical integration

Particular importance was given to the rigorous separation between propulsion, aerodynamics, and kinematics through polymorphism, ensuring improved architectural scalability.

The result is a numerically consistent representation of the real engineering and physical challenges involved in reaching orbit.

This project has been one of the most exciting challenges I have ever undertaken, a codebase of size and complexity completely new to me and full of unexpected difficulties.
It taught me the importance of organizing code as cleanly as possible and carefully verifying the impact of every small modification.

It brought sleepless nights and extraordinary moments shared with my trusted collaborator, whom I thank once again for his invaluable contributions.

👥 Contributors

This project was made possible thanks to the contribution of: