Aerospace & Defense Calculators
Atmosphere, compressible flow, and flight-side math.
- Frequency ⇄ WavelengthWavelength for any frequency, with velocity factor for cables.λ = (c · VF) / f
- Standard AtmosphereISA temperature, pressure, density, and speed of sound at altitude.ISA layers: T = T₀ + λ·Δh, P from the hydrostatic + gas-law integration per layer
- VSWR / Return Loss / ΓEnter any one of VSWR, return loss, or reflection coefficient — get all of them plus mismatch loss.Γ = (VSWR−1)/(VSWR+1) RL = −20·log₁₀|Γ|
- Free-Space Path LossFSPL between isotropic antennas — the first line of every link budget.FSPL(dB) = 32.45 + 20·log₁₀ f(MHz) + 20·log₁₀ d(km)
- RF Link BudgetReceived power and margin from TX power, gains, distance, and losses.Pr = Pt + Gt + Gr − FSPL − L
- Noise Figure ⇄ Noise TemperatureConvert between NF in dB and equivalent noise temperature in kelvin.Te = (F − 1) · 290 K
- Doppler ShiftFrequency shift for a moving target — one-way or radar (two-way).fd = 2·v·f/c (radar) fd = v·f/c (one-way)
- Radio / Radar HorizonLine-of-sight distance from antenna heights, with standard refraction.d(km) ≈ 4.12 (√h1 + √h2) (4/3-earth refraction)
- Gain from BeamwidthsApproximate directive gain from the two −3 dB beamwidths.D ≈ 41253 / (θaz · θel) (degrees)
- RF Power DensityFar-field power density from EIRP and distance — the exposure math.S = EIRP / 4πd² E = √(377·S)
- Bolt Torque from PreloadTightening torque for a target preload — the K-factor shortcut everyone actually uses.T = K · F · d
- Linear Thermal ExpansionΔL = αLΔT for common structural materials.ΔL = α · L · ΔT
- Euler Column BucklingCritical buckling load Pcr = π²EI/(KL)² with standard end conditions.Pcr = π²·E·I / (K·L)²
- Flywheel Kinetic EnergyStored rotational energy E = ½Iω² from inertia and speed.E = ½ · I · ω²
- Kinetic EnergyKE = ½mv² — the energy of anything moving in a straight line.KE = ½ · m · v²
- Projectile Range (No Drag)Range, max height, and flight time on flat ground — vacuum ballistics.R = v²·sin 2θ / g h = v²·sin²θ / 2g t = 2v·sinθ / g
- Mass Moment of Inertia (Shapes)Rotational inertia for the four shapes that cover most machine parts.I_disc = ½mr² I_tube = ½m(r₁²+r₂²) I_sphere = ⅖mr² I_rod = mL²/12
- Isentropic Flow RatiosT/T₀, P/P₀, ρ/ρ₀, and A/A* from Mach number — the compressible-flow table, minus the table.T₀/T = 1 + (γ−1)/2·M² P/P₀ = (T/T₀)^(γ/(γ−1)) A/A* = f(M, γ)
- Normal Shock RelationsJump conditions across a normal shock: M₂ and the pressure, density, temperature, and total-pressure ratios.P₂/P₁ = 1 + 2γ/(γ+1)·(M₁²−1) M₂² = (1 + (γ−1)/2·M₁²) / (γM₁² − (γ−1)/2)
- Oblique Shock (θ-β-M)Weak-shock angle β and downstream conditions for a wedge at supersonic speed — with detached-shock detection.tan θ = 2·cot β·(M₁²sin²β − 1) / (M₁²(γ + cos 2β) + 2)
- Mach Number from Airspeed & AltitudeTrue airspeed plus ISA altitude → Mach number and the local speed of sound.M = V / a, a = √(γ·R·T), T from ISA layers
- Dynamic Pressure (q)q = ½ρV² from velocity plus either an ISA altitude or a direct density.q = ½·ρ·V²
- Reynolds NumberRe = ρVL/µ with one-click air or water properties, or your own fluid.Re = ρ·V·L / µ = V·L / ν
- Bernoulli Pressure ChangeDownstream pressure from an upstream state, two velocities, and an elevation change.p₁ + ½ρv₁² + ρgz₁ = p₂ + ½ρv₂² + ρgz₂
- Drag ForceF = ½ρV²·C_d·A — aerodynamic drag and the power it costs.F_D = ½ · ρ · V² · C_d · A
- Lift ForceL = ½ρV²·C_L·S — lift from speed, wing area, and lift coefficient.L = ½ · ρ · V² · C_L · S
- Terminal VelocitySteady fall speed where drag balances weight: √(2mg / ρ·C_d·A).V_t = √( 2·m·g / (ρ·C_d·A) )
- Tsiolkovsky Rocket EquationΔv from specific impulse and mass ratio — the tyranny, quantified.Δv = I_sp · g₀ · ln(m₀ / m_f)
- Circular Orbit Velocity & PeriodSpeed and period of a circular Earth orbit from altitude.v = √(µ/r) T = 2π·√(r³/µ), µ_Earth = 3.986×10¹⁴ m³/s², r = R_E + h
- Escape Velocity√(2µ/r) for Earth, Moon, or Mars — from the surface or from altitude.v_esc = √(2µ / r) — exactly √2 × circular orbital speed
- Hohmann TransferTwo-burn transfer between circular Earth orbits: Δv₁, Δv₂, total, and time of flight.Δv₁ = √(µ(2/r₁ − 1/a)) − √(µ/r₁), a = (r₁+r₂)/2, t = π√(a³/µ)
- Specific ImpulseI_sp and effective exhaust velocity from thrust and propellant mass flow.I_sp = F / (ṁ · g₀) v_e = F / ṁ, g₀ = 9.80665 m/s²
- Mass & Force Converterkg, lb, oz, slug — and their weights in N, kgf, lbf at standard gravity.1 lb = 0.45359237 kg (exact, 1959) F = m·g₀, g₀ = 9.80665 m/s²
- Angle ConverterDegrees, radians, gradians, arcminutes, mrad — and the NATO mil.1 rad = 180/π ≈ 57.29578° 6400 NATO mil = 360° (so 1 mil ≈ 0.98175 mrad)
- Speed Converterm/s, km/h, mph, knots, ft/s — with the sea-level Mach number thrown in.1 kt = 1852 m/h = 0.5144444 m/s 1 mph = 0.44704 m/s a₀(ISA, SL) = 340.294 m/s
- Fresnel Zone ClearanceFirst-zone radius at mid-path and the 60% clearance a microwave link actually needs.r₁ = 17.32·√(d/(4f)) at mid-path, d in km, f in GHz, r in m
- Basquin S-N Fatigue LifeHigh-cycle fatigue life from the Basquin power law — enter stress amplitude to get cycles, or cycles to get allowable stress.σa = σ'f · (2Nf)ᵇ Nf = ½ · (σa / σ'f)^(1/b)
- Paris Law Crack Growth RateFatigue crack growth per cycle from the stress-intensity range using the Paris power law.da/dN = C·(ΔK)ᵐ, ΔK = Y·Δσ·√(π·a)
- Fracture Toughness — Critical Stress / Crack SizeSolve linear-elastic fracture mechanics for critical stress, crack length, or the stress-intensity factor given K_IC.K = Y·σ·√(π·a) , σ_c = K_IC / (Y·√(π·a)) , a_c = (K_IC / (Y·σ))² / π
- Larson-Miller Creep LifeTrade time against temperature for creep-rupture — solve the LMP, remaining life in hours, or allowable metal temperature.LMP = T · (C + log₁₀ t) t = 10^(LMP/T − C) T = LMP / (C + log₁₀ t) — T in K (or °R), t in hours, C ≈ 20 for most steels
- Thermal Shock Resistance (Kingery R′)First and second thermal-shock parameters for brittle ceramics — the max temperature drop a part survives, and how conductivity buys margin.R′ = σ_f·(1−ν) / (E·α) , R‴ = R′·k
- Hydrogen Embrittlement Threshold StressEstimate the sustained-load threshold stress for a high-strength steel from its notched fracture strength and an F519-style threshold ratio.NTS = Ftu · Kt , σ_th = NTS · R_th
- Forman Crack Growth Rate (R-ratio)Fatigue crack growth per cycle with mean-stress (R-ratio) and fracture-toughness effects — the equation under NASGRO and AFGROW.da/dN = C·ΔKⁿ / [ (1−R)·Kc − ΔK ] , Kmax = ΔK / (1−R)
- Choked Nozzle (Mach 1 Throat)Choked mass flow, throat conditions, and critical pressure ratio for a converging nozzle running sonic at the throat.ṁ = A*·P₀·√(γ/(R·T₀))·(2/(γ+1))^((γ+1)/(2(γ−1))), P*/P₀ = (2/(γ+1))^(γ/(γ−1)), T*/T₀ = 2/(γ+1)
- Drag Divergence Mach NumberEstimate the Mach number where transonic wave drag takes off, from critical Mach or Korn's sweep/thickness relation.Korn: M_dd·cosΛ = κ − (t/c)/cosΛ − C_l/(10·cos²Λ) , M_dd ≈ M_cr + 0.08
- Oswald-Efficiency Induced DragInduced drag coefficient and force from lift, aspect ratio, and Oswald efficiency.C_Di = C_L² / (π · e · AR), D_i = C_Di · q · S
- Propeller Actuator-Disk ThrustIdeal thrust, induced velocity, and Froude efficiency for a propeller or rotor from disk loading and flight speed.T = 2·ρ·A·(V₀ + w)·w , A = πD²/4 , η = V₀/(V₀ + w) = 1/(1 + w/V₀)
- CG Shift from Fuel BurnFind the new center of gravity after burning a mass of fuel located at a known station.CG₁ = (W₀·CG₀ − W_f·CG_f) / (W₀ − W_f)
- Stagnation Temperature (Compressible Flow)Total (stagnation) temperature from static temperature and Mach number for a calorically perfect gas.T₀/T = 1 + (γ−1)/2 · M²
- Ballistic Coefficient β (Reentry)Compute a reentry body's ballistic coefficient from mass, drag coefficient, and reference area — the number that sets how deep it plunges before decelerating.β = m / (C_D · A)
- Bipropellant O/F RatioSize oxidizer and fuel flow rates from a mixture ratio and total propellant flow.r = ṁₒ / ṁ_f , ṁ_total = ṁₒ + ṁ_f , ṁ_f = ṁ_total / (1 + r)
- TVC Gimbal TorqueActuator torque needed to deflect a gimbaled rocket engine against thrust misalignment offset.τ = F·e·cos(δ) + J·α + k·δ , where e is thrust-axis offset from the gimbal pivot
- Plate Critical Buckling StressElastic critical buckling stress of a thin rectangular plate — pick the k for your edge conditions and load case, get σ_cr and load per unit width.σ_cr = k · π² · E / [12(1 − ν²)] · (t/b)²