Appendix F: Propulsion Formula Reference

This appendix collects the propulsion equations of the book into one scannable place, grouped by topic: the rocket equation and staging, thrust and specific impulse, nozzle thermodynamics, and electric propulsion. Each entry gives the formula, a one-line statement of what it computes or when to reach for it, and a link to the chapter where it is derived and its symbols are given physical meaning. Every symbol appears in Appendix A; the constants you feed these formulas (standard gravity $g_0 = 9.80665\ \text{m/s}^2$, the universal gas constant $R_u = 8.314\ \text{J mol}^{-1}\text{K}^{-1}$, engine and vehicle numbers) are in Appendix B and Appendix H. This is the propulsion companion to the orbital-mechanics formula reference (Appendix E); the delta-v "prices" these equations turn into propellant are mapped in Appendix G.

How to read these formulas. Nearly every relation here is the ideal form — perfectly expanded, isentropic, loss-free. Real engines fall a few percent short (friction, heat loss, non-frozen chemistry) and real ascents add gravity, drag, and steering losses; the chapters say where. A few reused letters need care, exactly as in the text: $c$ is the effective exhaust velocity (the $v_e$ of the rocket equation); $\gamma$ is the ratio of specific heats (not a flight-path angle); $T$ in $T/W$ is thrust (not temperature); $\varepsilon$ is the structural coefficient while $\epsilon$ (a different glyph) is the nozzle expansion ratio; $\mathcal{M}$ is molar mass while $M$ is the Mach number; and $R$ means the stage mass ratio in §F.1 but the specific gas constant in §F.3. Units are SI unless a field-standard unit is noted ($I_{sp}$ in seconds).

F.1 The rocket equation and staging

The master equation of spaceflight. It contains no thrust, no time, and no size — only exhaust velocity and the ratio of full mass to empty mass.

$$ \Delta v = v_e\,\ln\!\left(\frac{m_0}{m_f}\right) $$

The delta-v (m/s) a stage delivers, given effective exhaust velocity $v_e$ (m/s) and initial/final masses $m_0, m_f$ (any consistent unit). Derived in Ch. 3.

Watch the constraint. In the stage-sizing and payload-ratio formulas the denominator $1 - \varepsilon R$ must stay positive: ask a stage for a mass ratio above its ceiling $1/\varepsilon$ and $m_{\text{stage}}$ and $\pi$ go negative — the stage is physically impossible, the same failure that rules out single-stage-to-orbit with payload.

F.2 Thrust and specific impulse

Thrust is the momentum thrown per second plus a pressure correction at the nozzle mouth. Everything else in this section is bookkeeping on top of it.

$$ F = \dot m\,v_{\text{ex}} + (p_e - p_a)\,A_e $$

Thrust $F$ (N): momentum thrust $\dot m\,v_{\text{ex}}$ plus pressure thrust $(p_e - p_a)A_e$, where $\dot m$ is mass flow (kg/s), $v_{\text{ex}}$ the actual exhaust speed (m/s), $p_e$/$p_a$ the exit/ambient pressure (Pa), and $A_e$ the exit area (m²). Derived in Ch. 16.

Here $g_0 = 9.80665\ \text{m/s}^2$ is standard gravity as a defined constant (not the local gravity), which is exactly why $I_{sp}$ comes out the same in metric or imperial units.

F.3 Nozzle flow and thermodynamics

The de Laval nozzle turns hot, slow, high-pressure chamber gas into cold, fast, low-pressure exhaust. Its performance rests on three "hero" relations and a handful of supporting ones.

Ideal exhaust velocity — the number the rocket equation ultimately wants, out of chamber conditions:

$$ v_e = \sqrt{\frac{2\gamma}{\gamma - 1}\,\frac{R_u\,T_c}{\mathcal{M}}\left[\,1 - \left(\frac{p_e}{p_c}\right)^{\frac{\gamma - 1}{\gamma}}\,\right]} $$

Exit velocity (m/s) from chamber temperature $T_c$ (K), exhaust molar mass $\mathcal{M}$ (kg/mol), specific-heat ratio $\gamma$, and pressure ratio $p_e/p_c$. Equals the effective exhaust velocity for a perfectly expanded nozzle. Note the dominant scaling $v_e \propto \sqrt{T_c/\mathcal{M}}$ — light exhaust wins, which is why LOX/LH2 beats LOX/RP-1. Derived in Ch. 19; the chamber $T_c$ and $\mathcal{M}$ come from combustion, Ch. 18.

Choked mass flow — the flow rate locked in once the throat reaches Mach 1:

$$ \dot m = A_t\,p_c\,\sqrt{\frac{\gamma}{R\,T_c}}\left(\frac{2}{\gamma + 1}\right)^{\frac{\gamma + 1}{2(\gamma - 1)}}, \qquad R = \frac{R_u}{\mathcal{M}} $$

Mass flow (kg/s) set by throat area $A_t$ (m²) and chamber pressure $p_c$ (Pa) alone — nothing downstream. To throttle, change $p_c$. Derived in Ch. 19.

Area–Mach relation / expansion ratio — the geometry that fixes the exit state:

$$ \epsilon = \frac{A_e}{A_t} = \frac{1}{M_e}\left[\frac{2}{\gamma + 1}\left(1 + \frac{\gamma - 1}{2}M_e^2\right)\right]^{\frac{\gamma + 1}{2(\gamma - 1)}} $$

Expansion (area) ratio $\epsilon$ from the exit Mach number $M_e$; choosing $\epsilon$ fixes $M_e$ and hence the exit pressure. Derived in Ch. 19.

Two standard figures of merit package the same physics by splitting an engine into a chamber part and a nozzle part. The book develops their ingredients ($\dot m$, $F$, $c$, $p_c$, $A_t$) in Chapters 16 and 19 rather than these composite symbols, but they are worth keeping in a reference:

F.4 Electric propulsion

Electric thrusters decouple the energy source from the propellant, lifting the chemical ceiling on exhaust velocity — but the electrical power available becomes the new limit.

$$ F = \frac{2\,\eta\,P}{v_e} $$

Power-limited thrust (N): for electrical power $P$ (W) and total efficiency $\eta$ (fraction of $P$ that becomes beam kinetic energy, $\approx 0.5$–$0.7$), a higher exhaust velocity $v_e$ buys less thrust. Derived in Ch. 20.

The trade in one sentence: at fixed power, raising $v_e$ (and thus $I_{sp}$) lowers thrust, so electric propulsion sits at the high-efficiency, low-thrust end — days-to-months of gentle firing rather than minutes of chemical violence.

A closing note on scope and honesty

These are the ideal forms — the right place to start, and close enough that the answers are trustworthy, but not the last word. Real nozzles lose a few percent to friction and heat; real ascents add gravity, drag, and steering losses that the rocket equation never sees (that is why surface-to-LEO costs $\sim 9.4\ \text{km/s}$, not the $\sim 7.8\ \text{km/s}$ of orbital speed alone — see Ch. 4); and specific impulse itself rises with altitude as $p_a \to 0$. Use the ideal formula to get the shape and size of a design, then apply the efficiencies, margins, and altitude corrections the chapters describe. Every symbol here is defined in Appendix A, every constant in Appendix B, and representative engine numbers to test these equations against in Appendix H. When a formula and a real engine disagree, the chapters have taught you to trust neither blindly: find the assumption — perfect expansion, frozen chemistry, a calorically perfect gas — that the real hardware broke.