Specific impulse: what it is, the formula, and rocket engine values

Specific impulse (Isp) is the thrust a rocket engine makes for each unit of propellant weight it burns per second. It is quoted in seconds. A higher number means the engine gets more push out of the same propellant, which is why it is the first figure anyone asks about a rocket engine after thrust.

This page covers the formula, why the unit is seconds, why the same engine has two values (sea level and vacuum), what physically sets the number, and what real engines achieve, with a source for every figure.

What is specific impulse?

Thrust tells you how hard an engine pushes. It says nothing about how much propellant the engine burns to do it. Two engines with the same thrust can differ a lot in how fast they empty the tanks, and the one that burns less carries the vehicle further on the same load.

Specific impulse puts that into one number: thrust divided by the weight flow rate of propellant. Put another way, it is the total impulse the engine extracts from each unit of propellant weight. If an engine has an Isp of 300 s, one kilogram of its propellant can produce a thrust equal to that kilogram's standard weight (9.80665 N) for 300 seconds.

The specific impulse formula

For an engine running steadily:

Isp = F / (ṁ · g₀)

• F is thrust, in newtons.
• ṁ is the propellant mass flow rate, in kg/s. For a liquid engine this is all the propellant the engine consumes, oxidizer and fuel together.
• g₀ is standard gravity, 9.80665 m/s² exactly. It is a defined constant, not the local gravity where the engine happens to be.

Over a whole burn, the same idea reads as total impulse divided by the weight of propellant consumed: Isp = I_t / (m_p · g₀).

Effective exhaust velocity

Drop the g₀ and you get a velocity:

c = F / ṁ = Isp · g₀

c is the effective exhaust velocity. It is the exhaust speed that would produce the measured thrust if all of it came from momentum. The real thrust equation has a pressure term as well:

F = ṁ · vₑ + (pₑ − pₐ) · Aₑ

where vₑ is the actual exhaust velocity at the nozzle exit, pₑ the exit pressure, pₐ the ambient pressure and Aₑ the exit area. So c = vₑ + (pₑ − pₐ) · Aₑ / ṁ. When the nozzle exit pressure matches the surroundings, the pressure term vanishes and c equals vₑ.

Isp and c carry the same information. Some textbooks, and much of the European and Russian literature, quote c in m/s (or N·s/kg) where American sources quote Isp in seconds. If you see "specific impulse of 4,400", the unit is m/s.

Why specific impulse is measured in seconds

Work through the units: N ÷ (kg/s × m/s²) leaves seconds. That alone is not the reason it caught on. The reason is that dividing by g₀ makes the number the same in every unit system.

In US customary units, thrust is in pounds-force and flow in pounds-mass per second. One pound-force is the weight of one pound-mass at standard gravity, so lbf ÷ (lbm/s) also comes out in seconds, with the same numerical value as the SI calculation. That is why Northrop Grumman's solid-motor catalog lists specific impulse in "lbf-sec/lbm" and the values read directly as seconds. An engine quoted at 311 s is 311 s on either side of the Atlantic, while its exhaust velocity is 3,050 m/s in one and about 10,000 ft/s in the other.

The g₀ is a conversion constant. An upper-stage engine firing between planets has the same specific impulse it had on the test stand in vacuum conditions. Nothing about local gravity enters it.

Why specific impulse matters

The ideal rocket equation ties vehicle performance directly to c, and so to Isp:

Δv = Isp · g₀ · ln(m₀ / m_f)

m₀ is the mass at ignition and m_f the mass at burnout. With a mass ratio of 5, an engine at 300 s gives about 4.73 km/s of velocity change. At 450 s the same mass ratio gives about 7.10 km/s. The engine with higher Isp does not need a better tank or a lighter structure to get there. It extracts more from each kilogram.

Isp is not the only thing that matters, though. Liquid hydrogen with liquid oxygen gives the highest Isp of the common propellant combinations, but hydrogen is very low in density, so its tanks are large and heavy, and high-Isp engines are not always high-thrust engines. That is why booster stages often use denser propellants and why hydrogen engines like the RL10 are so often found on upper stages.

Sea level versus vacuum specific impulse

An engine that fires inside the atmosphere usually has two quoted values, and vacuum is always the higher one. The thrust equation shows why.

Once the throat is choked, nothing downstream changes the mass flow or the flow at the nozzle exit, as long as the nozzle flows full. Ambient pressure enters only through the −pₐ · Aₑ term. At sea level the air pushes back on the whole exit area. In vacuum it does not. So for the same engine at the same operating point:

F_vac − F_sl = pₐ · Aₑ
Isp_vac − Isp_sl = pₐ · Aₑ / (ṁ · g₀)

A bigger exit area loses more at sea level, which creates the central conflict of nozzle design. A large expansion ratio (exit area divided by throat area) turns more of the gas's thermal energy into directed velocity, so in vacuum it raises Isp. At sea level the same nozzle overexpands: the exit pressure drops below ambient, the pressure term turns negative, and if the gap gets large enough the flow separates from the nozzle wall, with side loads that can damage the engine. The nozzle guide covers the shapes this produces.

So a first-stage engine gets a moderate expansion ratio and an upper-stage engine gets a very large one. Two published examples show the size of the effect:

• Merlin. SpaceX published the first-stage Merlin 1D at 282 s at sea level and 311 s in vacuum. The upper-stage Merlin Vacuum, which SpaceX describes as nearly identical but modified to operate in vacuum, most visibly with a much larger nozzle, is quoted at 348 s.
• STAR 48B. Northrop Grumman's STAR 48B solid motor comes with two nozzles on the same motor, propellant and chamber pressure. The short nozzle, at an initial expansion ratio of 39.6:1, gives an effective specific impulse of 286.0 s in vacuum. The long nozzle, at 54.8:1, gives 292.1 s. That is a 2% gain for nothing but more nozzle.

The diminishing return in the second example is typical. Each step in expansion ratio buys less than the one before it, while nozzle length and mass keep growing.

What sets a rocket engine's specific impulse

For an ideal nozzle, the exit velocity is:

vₑ = √( 2γ/(γ − 1) · (R · T_c / M) · [1 − (pₑ / p_c)^((γ − 1)/γ)] )

T_c is the chamber temperature, M the mean molar mass of the combustion products, R the universal gas constant, γ the ratio of specific heats, and p_c the chamber pressure. Three things in it decide most of an engine's Isp.

Chamber temperature over molar mass, T_c/M. Hot gas made of light molecules moves fast. That ratio is the main difference between propellant combinations. Hydrogen burned with oxygen makes water plus leftover hydrogen, both light. Kerosene adds carbon monoxide and carbon dioxide to the exhaust, which are much heavier. This is why the hydrogen-oxygen engines in the table below quote 452 to 465.5 s in vacuum and the kerosene engines 311 to 348 s.

It also explains why hydrogen engines run fuel-rich on purpose. The stoichiometric mixture ratio for hydrogen and oxygen is about 7.9 by mass, yet the RS-25 runs at 6.0 and the RL10B-2 at 5.88. Extra hydrogen lowers the combustion temperature a little, but it lowers the mean molar mass of the exhaust more, so T_c/M improves and Isp peaks on the fuel-rich side. Engines stop short of that peak because every extra kilogram of hydrogen needs a large tank, and a cooler chamber is easier on the walls as well. The combustion chamber guide covers what those walls have to survive.

Pressure ratio, p_c/pₑ. The bracket in the formula is the fraction of the available enthalpy the nozzle converts into velocity. A larger expansion ratio lowers pₑ and recovers more of it. A higher chamber pressure lets a sea-level nozzle expand further before it overexpands, which is one reason first-stage engines push chamber pressure so hard.

Losses. Real engines deliver less than the ideal formula. Incomplete combustion, exhaust that leaves at an angle instead of straight back, friction in the boundary layer along the nozzle wall and heat lost to the walls each take a share. The engine cycle can take more. A gas-generator engine burns some of its propellant just to drive the turbopump and dumps that gas at low velocity. A published engine Isp counts that propellant in ṁ, so the engine figure sits below what its main chamber achieves on its own.

Specific impulse of real rocket engines

Every figure in this table comes from the manufacturer's own published data, listed under Sources at the end of this page. Where a source quotes no sea-level value, the engine is an upper-stage or in-space design.

The c column is Isp × g₀, calculated here. The STAR 48B values are the catalog's effective specific impulse in vacuum. The Merlin 1D figures are the ones SpaceX published in 2013 for an engine rated at 147,000 lbf of sea-level thrust.

Read the RS-25 row against the sea-level formula above. Its 86-second gap between sea level and vacuum is large because its nozzle is large: a 69:1 expansion ratio on an engine that lights on the launch pad. Its 2,994 psia chamber pressure is part of what lets a nozzle that size run at sea level. The engine fires from liftoff until it is far above the atmosphere, so its nozzle is a compromise between the two conditions.

A worked example

Take an engine on a sea-level test stand. It makes 100 kN of thrust at a propellant flow of 35 kg/s, and its nozzle exit area is 0.1 m². Assume the nozzle flows full, with no separation.

1. Effective exhaust velocity at sea level: c = F / ṁ = 100,000 / 35 = 2,857 m/s.
2. Sea-level specific impulse: Isp = c / g₀ = 2,857 / 9.80665 = 291.3 s.
3. Ambient pressure force on the exit: pₐ · Aₑ = 101,325 Pa × 0.1 m² = 10,133 N.
4. Vacuum thrust: 100,000 + 10,133 = 110,133 N. The mass flow does not change, because the throat is choked.
5. Vacuum specific impulse: 110,133 / (35 × 9.80665) = 320.9 s.

The same engine gains about 30 s, roughly 10%, just by leaving the atmosphere. The calculation also runs in reverse: quote an RL10B-2 at 465.5 s and you have said its effective exhaust velocity is 465.5 × 9.80665 = 4,565 m/s.

How specific impulse shows up in Meco

I built Meco to let you push on numbers like these and watch what moves, and Isp is one of the numbers it reports.

Right-click a gas branch or a Thrust Chamber that exhausts into an outlet node, choose Add Sampler to Charts, then Thrust. After a run, the card shows thrust with Isp in seconds beneath it. The Isp divides thrust by g₀ times the propellant flow from every gas source in the model, a gas generator included, so a gas-generator engine pays for its turbine drive gas the same way a published engine figure does. The nozzle-only value is in the tooltip, and the card's menu lets you choose which sources count.

The ambient pressure is the pressure of that outlet node, which you set with the control parameter wired to it. So the pₐ in the thrust equation above is yours to choose. If that pressure pushes the flow off the nozzle wall, the card's Flow Separation row says so.

Isp is also a mission target. Mission 3, Cold Gas Reality Check, asks a nitrogen thruster for at least 77 s. The way there is warmer gas and a wider nozzle exit: the T_c/M and expansion ratio story from this page, at a scale where nothing burns.

Further reading

The rocket propulsion primer puts specific impulse beside the other things an engine design has to answer for. For the full derivations, chapters 2 and 3 of Rocket Propulsion Elements are the standard treatment, and the rocket engine books page says where it and the free NASA documents fit in a reading order.

Sources

1. NASA Glenn Research Center, Specific Impulse: the thrust equation, equivalent velocity and the definition of Isp.
2. NASA Glenn Research Center, Rocket Thrust Equations: choked throat flow, area ratio and exit conditions.
3. NIST, CODATA value: standard acceleration of gravity: g₀ = 9.80665 m/s², exact.
4. L3Harris, RS-25 engine: Isp at 109% power level (366 s sea level, 452 s vacuum), mixture ratio 6.0, area ratio 69:1, chamber pressure 2,994 psia.
5. Aerojet Rocketdyne, RL10 engine (archived copy): RL10B-2 specific impulse 465.5 s, mixture ratio 5.88:1.
6. SpaceX, Falcon 9 (archived copy, 2013): Merlin sea-level Isp 282 s, vacuum Isp 311 s, sea-level thrust 147,000 lbf, LOX and RP-1 propellants.
7. SpaceX, Falcon 9 (archived copy, 2015): Merlin Vacuum vacuum Isp 348 s.
8. Northrop Grumman, Propulsion Products Catalog, STAR 48B pages (102 and 103): effective specific impulse 286.0 s and 292.1 s, expansion ratios 39.6:1 and 54.8:1, propellant TP-H-3340.
9. George P. Sutton and Oscar Biblarz, Rocket Propulsion Elements, ninth edition, Wiley, chapters 2 and 3: the ideal exhaust velocity relation, the rocket equation and the sources of nozzle loss.

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