Rocket Equation

Inputs

Method

The propellant a known dry mass needs for a Δv.

Solve for

Spacecraft and mission

Delta-v

m/s

Dry mass

Propulsion

The thruster sets the Isp; its thrust sets how long the burn takes.

Propulsion type

Thrust

Margins

Propellant loaded beyond what the Δv burns. Neither is burned in the nominal case, so both ride along to burnout.

Propellant margin

Residuals

Results

Solved

Loaded propellant 12.4kg

Wet mass 162.4kg

Mass budget

Dry mass 150kg

Usable propellant 10.9kg

Margin propellant 1.09kg

Residual propellant 0.37kg

Burnout mass 151.5kg

Performance

Delta-v 150m/s

Isp 220s

Exhaust velocity 2,157m/s

Mass ratio 1.072

Propellant mass fraction 7.6%

Burn

Mass flow rate 1.85g/s

Burn time 98min

Method

Single-stage propellant sizing after SMAD: the rocket equation solved for whichever quantity is unknown, with the propellant margin and residuals carried as mass the burn has to push.

Δv = I_sp g₀ ln(m₀ / m_f) The Tsiolkovsky rocket equation: the Δv an engine gives is its exhaust velocity times the log of how much lighter the spacecraft ends than it began. Every method below is this one equation solved for a different unknown.

Derivation

Load the margins The tanks hold more than the Δv burns: a margin on the usable propellant, and residuals the tanks cannot deliver. Neither is burned in the nominal case, so both are still aboard at burnout, and the engine pushes them through the whole Δv. k = (1 + m) / (1 − r) − 1, m_p = m_u(1 + k), m_f = m₀ − m_u

Propellant from dry mass Solving the equation for the usable propellant with the carried margins folded in. As kE approaches 1 the load runs away: past that Δv the margin propellant costs more to carry than it adds, and no load closes. m_u = m_d E / (1 − kE), E = e^Δv/vₑ − 1

Propellant from wet mass With the launch mass fixed, the equation gives the usable propellant directly; the dry mass is what is left once the loaded propellant is taken out. The same limit applies: at it, nothing is left. m_u = m₀(1 − e^−Δv/vₑ), m_d = m₀ − m_p

Δv from masses Take the margins out of the tank’s load to find what the burn can use, then read the Δv straight off the mass ratio. m_u = m_p / (1 + k), Δv = v_e ln(m₀ / m_f)

Required Isp The same mass ratio, solved for the exhaust velocity instead. Compare it against a thruster’s datasheet: below the answer, the tank is too small for that thruster. I_sp = Δv / (g₀ ln(m₀ / m_f))

Time the burn Thrust over exhaust velocity is the propellant flow, and the usable propellant over the flow is how long the engines fire in total: minutes for chemical propulsion, weeks or months for electric. ṁ = F / v_e, t_b = m_u / ṁ

Symbols

Input	Description	Unit
Δv	Velocity change the propellant must deliver	m/s
m_d	Dry mass: everything but propellant	kg
m₀	Wet mass: fueled, at separation	kg
m_p	Loaded propellant, margin and residuals included	kg
I_sp	Specific impulse of the thruster	s
F	Thrust	N
m	Propellant margin, as a share of usable propellant	%
r	Residuals, as a share of loaded propellant	%

Derived	Description	Unit
v_e	Effective exhaust velocity: Isp·g₀	m/s
k	Propellant loaded beyond the usable, per kg usable
m_u	Usable propellant: what the Δv burns	kg
m_f	Burnout mass: dry mass plus margin and residuals	kg
m₀/m_f	Mass ratio
ṁ	Mass flow rate: F / vₑ	kg/s
t_b	Burn time: mᵤ / ṁ	s

Reference values

First-order figures for sanity-checking inputs.

Propulsion	Used s	Typical s
Cold gas (N₂)	65	50–75
Hydrazine monopropellant	220	200–230
Green monopropellant	235	230–250
Bipropellant (MMH/NTO)	310	300–325
Hall thruster	1,500	1,200–2,000
Gridded ion	3,000	2,500–3,500

Maneuver	Δv m/s
LEO drag make-up, 500 km	< 25 per year
Deorbit from 500 km to a 50 km perigee	~130
GTO to GEO circularization	~1,500
GEO north-south station-keeping	~50 per year
GEO disposal to the graveyard orbit	~11

This is the ideal rocket equation after SMAD: one stage, one engine, one Isp for the whole Δv, and impulsive burns, with no gravity or steering losses, which a long, low-thrust burn or a finite burn near a planet adds on top. The margin and residuals are carried to burnout, which is conservative for the margin: propellant spent from it lightens the spacecraft for the rest of the mission. Tank, pressurant and propulsion hardware belong in the dry mass. Treat these results as first-pass sizing, not a propulsion subsystem design.