Orbital Insertion Satellite Delta-V Thruster Burn (Aerospace Engineering)

An aerospace mission analyst at the UNLV Howard R. Hughes College of Engineering is calculating the terminal orbit-raising propulsion burn for a communications satellite being inserted into geostationary transfer orbit (GTO). The satellite utilizes two bi-propellant reaction control thrusters aligned along orthogonal vehicle axes.

To minimize mechanical structural fatigue and tank pressure drop, mission controllers seek to minimize a quadratic propellant energy cost metric:

where $x$ and $y$ represent directed thrust velocity changes ($\Delta v_x$ and $\Delta v_y$, in kilometers per second, $\text{km/s}$).

To achieve orbital capture and rendezvous with the required orbital plane, the velocity components must strictly satisfy the orbital mechanics flight constraint:

1. Formulate the gradient vectors $\nabla f(x, y)$ and $\nabla g(x, y)$ in component bracket form.
2. State the vector equation of Lagrange multipliers $\nabla f = \lambda \nabla g$ and decompose it into individual scalar component equations.
3. Eliminate $\lambda$ to establish a linear relationship between the required velocity burns $x$ and $y$.
4. Substitute this relationship into the flight constraint to solve for the optimal velocity increments $(x_{\text{opt}}, y_{\text{opt}})$ and the multiplier $\lambda$.
5. Compute the minimum propellant burn cost $f_{\min}$.
6. Interpret the physical and mission engineering meaning of the Lagrange multiplier $\lambda = \frac{d f_{\text{opt}}}{d k}$.