This section presents the complete mathematical formulation underlying the Apogee simulator. All equations are derived from first principles and mapped directly to the implementation.
Overview¶
The rocket ascent problem combines:
- Classical Mechanics - Newton's laws in rotating reference frames
- Atmospheric Physics - US Standard Atmosphere 1976
- Orbital Mechanics - Two-body Keplerian dynamics
- Hybrid Systems - Event-driven phase transitions
Core Topics¶
State Vector¶
Definition of the 5-dimensional state space in planar polar coordinates:
Covers coordinate system choice, physical interpretation, and state initialization.
Equations of Motion¶
Complete derivation of the governing ODEs:
Step-by-step derivation from Newton's Second Law in path coordinates.
Atmosphere Model¶
US Standard Atmosphere 1976 implementation:
Density and speed of sound as functions of altitude.
Orbital Mechanics¶
Two-body orbital diagnostics at engine cutoff:
Derivation of specific energy, angular momentum, and Keplerian elements.
Flight Phases¶
Hybrid dynamical system with event-driven transitions:
- Vertical Ascent → Pitchover
- Gravity Turn → Stage Separation
- Coast → Stage 2 Ignition
- Stage 2 Burn → Orbit Insertion
Mathematical Conventions¶
Throughout this documentation:
| Symbol | Meaning | Units |
|---|---|---|
| \(r\) | Geocentric radius | m |
| \(\lambda\) | Downrange central angle | rad |
| \(v\) | Speed magnitude | m/s |
| \(\gamma\) | Flight-path angle | rad |
| \(m\) | Vehicle mass | kg |
| \(T\) | Thrust | N |
| \(D\) | Drag force | N |
| \(\alpha\) | Steering angle | rad |
| \(\mu\) | Earth gravitational parameter | m³/s² |
| \(R_E\) | Earth radius | m |
LaTeX Source¶
The complete mathematical formulation is available in the LaTeX source file located at docs/nm_final_project.tex in the repository root.
All equation numbers in this documentation reference the corresponding equations in the LaTeX report.