Staging Problem

General Considerations

• Consider a staged launch vehicle:
  1. Stage 1:
     • Thrusts and expends propellant, reaching speed $V_1$
     • Stage 1 drops off
  2. Stage 2:
     • Thrusts and expends propellant, adding incremental speed $V_2$
     • We are ignoring gravity losses here (bad)
• For Stage 2:
  • $${\lambda }_2=\frac{M_L}{M_{0,2}-M_L}$$
  • $${ϵ}_2=\frac{M_{S,2}}{M_{0,2}-M_L}$$
  • $$M{R}_2=\frac{1+{\lambda }_2}{{ϵ}_2+{\lambda }_2}$$
• For Stage 1:
  • $${\lambda }_1=\frac{M_{0,2}}{M_{0,1}-M_{0,2}}$$
  • $${ϵ}_1=\frac{M_{S,1}}{M_{0,1}-M_{0,2}}$$
  • $$MR=\frac{1+{\lambda }_1}{{ϵ}_1+{\lambda }_1}$$
• Generally, payload of $i^{\text{th}}$ stage is initial mass of the ${\left(i+1\right)}^{\text{th}}$ stage

Why the push for Single Stage to Orbit (SSO)?

• With more stages:
  • Increases complexity
  • Increases cost
  • More things can go wrong (increases risk)
• Advances in material technology made SSO feasible
  • SSO was still not found to be advantageous
  • Other tech advances won out

Staging Basics

• Assume $u$ is the same for each stage (same technology throughout the rocket)
• Assume $ϵ$ is the same also (simplifying assumption)
• $$\frac{\Delta V_{\text{total}}}{u}=\sum _i\left(\log \left(\frac{1+{\lambda }_i}{ϵ+{\lambda }_i}\right)\right)$$
• $$\begin{array}{rl}\Delta V_i& =u\log {\left(\frac{M_0}{M_F}\right)}_i\\ & =u\log \left(\frac{1+{\lambda }_i}{ϵ+{\lambda }_i}\right)\end{array}$$
• To maximize $\Delta V_{\text{total}}$, we must optimize ${\lambda }_i$

IMPORTANT

Constraint Equation

• $$\frac{M_L}{M_{0,1}}=\prod _{i=1}^N\left(\frac{{\lambda }_i}{1+{\lambda }_i}\right)$$
• Optimize through gradient descent