Formula sheet

Aerospace Propulsion and Performance Formula Sheet

Aerospace propulsion formulas for Mach number, thrust, nozzle flow, sizing ratios, TSFC, Breguet range, rocket equation, electric propulsion, and margins.

Branch: Aerospace Engineering
Content: Formula sheet
Updated: May 29, 2026
Revision: v1.0.8 · reviewed

This formula sheet collects first-pass relationships for aerospace propulsion and flight performance. Use it for screening and consistency checks. Detailed aircraft and spacecraft work requires validated atmosphere models, engine decks, aerodynamic data, mass properties, control limits, thermal limits, uncertainty bounds, and mission rules.

Atmosphere and Mach number

Speed of sound for an ideal gas:

a=\sqrt{\gamma R T}

Mach number:

\displaystyle M=\frac{V}{a}

Dynamic pressure:

\displaystyle q=\frac{1}{2}\rho V^2

Reynolds number:

\displaystyle Re=\frac{\rho V L}{\mu}

State whether speed is true airspeed, equivalent airspeed, calibrated airspeed, ground speed, or exhaust velocity.

Lift and drag balance

Lift:

L=qSC_L

Drag:

D=qSC_D

Steady level flight approximation:

L\approx W

T\approx D

Lift-to-drag ratio:

\displaystyle \frac{L}{D}=\frac{C_L}{C_D}

These equations assume the aerodynamic coefficients apply to the Mach number, Reynolds number, configuration, and angle condition being analysed.

Sizing and Performance Ratios

Thrust-to-weight ratio:

\displaystyle \frac{T}{W}

Wing loading:

\displaystyle \frac{W}{S}

Power loading:

\displaystyle \frac{W}{P_A}

Specific excess power:

\displaystyle P_s=\frac{(T-D)V}{W}

Breguet range factor:

\displaystyle RF=\frac{V}{c}\frac{L}{D}

Sizing ratios should be tied to mission segment, altitude, Mach number, configuration, installed thrust, fuel state, and certification or mission constraints.

Mission Mass Fractions

Segment mass fraction:

\displaystyle MF=\frac{m_{end}}{m_{start}}

Fuel or propellant fraction for a segment:

\displaystyle f_{fuel}=1-\frac{m_{end}}{m_{start}}

Reserve fraction:

\displaystyle f_{res}=\frac{m_{reserve}}{m_{fuel,total}}

Mass fractions should state payload, trapped fuel, unusable propellant, reserves, contingency, and whether masses are wet, dry, launch, landing, or segment-specific.

Thrust from momentum

Simplified net thrust:

F\approx \dot{m}(V_e-V_0)+(p_e-p_0)A_e

where $\dot{m}$ is mass flow, $V_e$ is exit velocity, $V_0$ is incoming velocity, and the pressure term accounts for exit pressure mismatch.

Mass flow:

\dot{m}=\rho A V

Installed thrust:

T_{installed}=T_{uninstalled}-D_{installation}-L_{inlet}-L_{nozzle}

The installation term is conceptual; define all losses explicitly before using it.

Nozzle and Mass-Flow Screening

Ideal gas mass flow:

\dot{m}=\rho A V

Mach-area relation for a quasi-one-dimensional isentropic nozzle:

\displaystyle \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)}}

Choked ideal mass flow at the throat:

\displaystyle \dot{m}=A_t p_t\sqrt{\frac{\gamma}{RT_t}}\left(\frac{2}{\gamma+1}\right)^{\frac{\gamma+1}{2(\gamma-1)}}

Ideal nozzle exit velocity from total temperature:

\displaystyle V_e\approx \sqrt{2c_pT_t\left[1-\left(\frac{p_e}{p_t}\right)^{(\gamma-1)/\gamma}\right]}

Nozzle checks must state total conditions, throat area, pressure ratio, gas properties, choking state, losses, and whether the nozzle is underexpanded or overexpanded.

Propulsive power

Propulsive power:

P_T=TV

Power required:

P_R=DV

Excess power:

P_{excess}=P_A-P_R

Rate of climb:

\displaystyle ROC=\frac{P_{excess}}{W}

The maximum climb condition may differ from the minimum drag or maximum endurance condition.

Efficiency

Overall propulsion efficiency:

\eta_o=\eta_{th}\eta_p

Propulsive efficiency for an ideal jet expression:

\displaystyle \eta_p=\frac{2V_0}{V_e+V_0}

Thermal efficiency:

\displaystyle \eta_{th}=\frac{W_{net}}{Q_{in}}

Carnot limit:

\displaystyle \eta_{Carnot}=1-\frac{T_C}{T_H}

Real propulsion systems operate below ideal limits because of component losses, pressure drops, heat transfer, mixing, cooling, mechanical losses, and off-design operation.

Fuel consumption

Thrust-specific fuel consumption:

\displaystyle TSFC=\frac{\dot{m}_f}{T}

Fuel mass over time:

\displaystyle m_f=\int \dot{m}_f\,dt

If TSFC is approximately constant over a segment:

m_f\approx TSFC \cdot T \cdot t

Use consistent units. TSFC values depend on altitude, Mach number, throttle setting, fuel definition, and whether thrust is installed or uninstalled.

Breguet range

Jet aircraft screening range:

\displaystyle R=\frac{V}{c}\frac{L}{D}\ln\left(\frac{W_i}{W_f}\right)

where $c$ is thrust-specific fuel consumption in consistent units.

Endurance screening relation:

\displaystyle E=\frac{1}{c}\frac{L}{D}\ln\left(\frac{W_i}{W_f}\right)

Breguet equations are segment models. Add climb, descent, reserves, wind, speed schedule, alternate, and operational constraints separately.

Takeoff and stall screening

Stall speed:

\displaystyle V_s=\sqrt{\frac{2W}{\rho S C_{L,max}}}

Approximate acceleration during ground roll:

\displaystyle a_x\approx \frac{T-D-\mu(W-L)}{m}

where $\mu$ is rolling friction coefficient.

Takeoff calculations require configuration, runway slope, wind, density altitude, engine failure cases, rotation speed, obstacle clearance, and regulatory rules.

Jet engine cycle checks

Ideal Brayton-cycle thermal efficiency with pressure ratio $r_p$ :

\displaystyle \eta_{Brayton}=1-\frac{1}{r_p^{(\gamma-1)/\gamma}}

Compressor temperature ratio for ideal isentropic compression:

\displaystyle \frac{T_{t2}}{T_{t1}}=r_p^{(\gamma-1)/\gamma}

Component isentropic efficiency, compressor form:

\displaystyle \eta_c=\frac{T_{t2s}-T_{t1}}{T_{t2}-T_{t1}}

Component isentropic efficiency, turbine form:

\displaystyle \eta_t=\frac{T_{t3}-T_{t4}}{T_{t3}-T_{t4s}}

Real engine calculations require station definitions, maps, corrected flow, cooling flows, mechanical losses, surge margin, and nozzle state.

Specific impulse

Specific impulse:

\displaystyle I_{sp}=\frac{T}{\dot{m}g_0}

Equivalent exhaust velocity:

c_e=I_{sp}g_0

Rocket equation:

\displaystyle \Delta v=c_e\ln\left(\frac{m_0}{m_f}\right)

Mass ratio for a required ideal velocity increment:

\displaystyle \frac{m_0}{m_f}=e^{\Delta v/c_e}

Propellant mass fraction, using final mass after burn:

\displaystyle \zeta_p=\frac{m_0-m_f}{m_0}=1-\frac{1}{m_0/m_f}

Impulse from a finite burn:

\displaystyle I_{tot}=\int T\,dt

High specific impulse reduces propellant mass for a required impulse, but it does not imply high thrust or short burn time.

A real delta-v budget should include gravity losses, drag losses, steering losses, residual propellant, attitude-control propellant, dispersions, and operational reserves.

Electric propulsion

Ion-thruster beam power approximation:

\displaystyle P_b\approx \frac{1}{2}\dot{m}V_e^2

Thrust from exhaust velocity:

T=\dot{m}V_e

Power-limited thrust:

\displaystyle T\approx \frac{2\eta P_{in}}{V_e}

where $\eta$ is thruster efficiency and $P_{in}$ is input electrical power.

Total impulse:

\displaystyle I_{tot}=\int T\,dt

Electric propulsion trades low thrust for high exhaust velocity and long duration. Include power processing, thermal rejection, plume effects, neutralization, and spacecraft operations.

Attitude and yaw

Yaw rate:

r=\dot{\psi}

Yaw angle from rate integration:

\psi(t)=\psi_0+\int_0^t r(\tau)\,d\tau

Moment from thrust offset:

M=T l

where $l$ is moment arm.

Gyroscope bias causes integrated angle drift:

\Delta \psi_{bias}\approx b t

where $b$ is rate bias. Attitude calculations should state coordinate frame, sign convention, sensor alignment, bandwidth, and filtering.

Validation and uncertainty

Relative error:

\displaystyle e_{rel}=\frac{|x_{measured}-x_{model}|}{|x_{measured}|}

Propagated uncertainty, first-order independent form:

\displaystyle \sigma_y^2\approx \sum_i\left(\frac{\partial y}{\partial x_i}\right)^2\sigma_{x_i}^2

Performance margin:

M=P_{available}-P_{required}

Thrust margin:

M_T=T_{available}-T_{required}

Delta-v margin:

M_{\Delta v}=\Delta v_{available}-\Delta v_{required}

Fuel or propellant margin:

M_m=m_{prop,available}-m_{prop,required}

State operating point, instrumentation, calibration, atmosphere, configuration, installation, and confidence level when reporting performance margin.

REF

Disciplines