Rocket Mass

Notation

• Initial Mass: $m_{\text{o}}$
• Payload Mass: $m_{\text{pl}}$
• Propellant Mass: $m_{\text{pr}}$
• Inert Mass: $m_{\text{in}}$

Relationships

• Basic:
  • $$m_{\text{o}}=m_{\text{pl}}+m_{\text{pr}}+m_{\text{in}}$$
• Inert Mass Fraction:
  • $$\delta \equiv \frac{m_{\text{in}}}{m_{\text{o}}}=\frac{m_{\text{pr}}}{m_{\text{pl}}+m_{\text{pr}}+m_{\text{in}}}$$
• Payload Fraction:
  • $$\lambda \equiv \frac{m_{\text{pl}}}{m_{\text{o}}}=\frac{m_{\text{pl}}}{m_{\text{pl}}+m_{\text{pr}}+m_{\text{in}}}$$
• Parametric Mass Ratio:
  • $$r=\lambda +\delta$$
• Stage Inert Mass Fraction:
  • $$ϵ\equiv \frac{m_{\text{in}}}{m_{\text{in}}+m_{\text{pr}}}$$
• Propellant Mass Fraction:
  • $$\text{PMF}\equiv 1-ϵ=\frac{m_{\text{pr}}}{m_{\text{in}}+m_{\text{pr}}}$$
• (whatever $\rho$ is called):
  • $$\rho \equiv \frac{m_{\text{pl}}}{m_{\text{in}}+m_{\text{pr}}}$$
  • $$\rho =\frac{r-ϵ}{1-r}$$
  • $$r=\frac{\rho +ϵ}{\rho +1}$$
• Given $\Delta v$ and $I_{\text{sp}}$:
  • $$r=e^{-\frac{\Delta v}{g{I}_{\text{sp}}}}$$
• Given $\delta$:
  • $$\lambda =r-\delta ⟹\boxed{m_{\text{o}}=\frac{m_{\text{pl}}}{\lambda }}$$
• Given $ϵ$:
  • $$m_{\text{stage}}=\left(\frac{1-r}{r-ϵ}\right)m_{\text{pl}}⟹\boxed{m_{\text{o}}=\left(\frac{1-ϵ}{r-ϵ}\right)m_{\text{pl}}}$$