The translational and rotational states of a flight vehicle are propagated via a 14-element state vector $\mathbf{x} = [\mathbf{r}, \mathbf{v}, \mathbf{q}, \boldsymbol{\omega}, m]^T$:

$$\dot{\mathbf{x}} = \begin{bmatrix} \dot{\mathbf{r}} \ \dot{\mathbf{v}} \ \dot{\mathbf{q}} \ \dot{\boldsymbol{\omega}} \ \dot{m} \end{bmatrix} = \begin{bmatrix} \mathbf{v} \ \frac{1}{m}\left(\mathbf{T}_I + \mathbf{D}_I + \mathbf{L}_I\right) + \mathbf{g}(\mathbf{r}) \ \frac{1}{2} \mathbf{\Omega}(\boldsymbol{\omega}) \mathbf{q} \ \mathbf{I}^{-1}\left(\mathbf{M}B - \boldsymbol{\omega} \times (\mathbf{I}\boldsymbol{\omega})\right) \ -\frac{|\mathbf{T}|}{I{\text{sp}} g_0} \end{bmatrix}$$

where the skew-symmetric quaternion rate matrix avoids gimbal lock singularities:

$$\mathbf{\Omega}(\boldsymbol{\omega}) = \begin{bmatrix} 0 & -\omega_x & -\omega_y & -\omega_z \ \omega_x & 0 & \omega_z & -\omega_y \ \omega_y & -\omega_z & 0 & \omega_x \ \omega_z & \omega_y & -\omega_x & 0 \end{bmatrix}$$

Dynamic mass depletion directly updates the instantaneous inertia tensor $\mathbf{I}(t) = \text{diag}(I_{xx}, I_{yy}, I_{zz})$ and center of mass $x_{\text{CM}}(t)$.

Compressible gas dynamics through converging-diverging de Laval nozzles govern propulsion efficiency. The cross-sectional area ratio to throat area ($A / A^*$) is related to Mach number $M$ and specific heat ratio $\gamma$:

$$\frac{A}{A^*} = \frac{1}{M}\left[\frac{2}{\gamma+1}\left(1 + \frac{\gamma-1}{2}M^2\right)\right]^{\frac{\gamma+1}{2(\gamma-1)}}$$

Stagnation-to-static thermodynamic ratios throughout the expansion:

$$\frac{T_0}{T} = 1 + \frac{\gamma-1}{2}M^2, \quad \frac{p_0}{p} = \left(1 + \frac{\gamma-1}{2}M^2\right)^{\frac{\gamma}{\gamma-1}}$$

Total effective vacuum thrust accounts for exit momentum and pressure differential across exit plane area $A_e$:

$$F = \dot{m} v_e + (p_e - p_a) A_e = C_F \cdot p_c \cdot A_t$$

where $C_F$ is the thrust coefficient and $p_a$ is local ambient barometric pressure evaluated via the 1976 US Standard Atmosphere.