# Specific impulse

The specific impulse (commonly abbreviated Isp) of a propulsion system is the impulse (change in momentum) per unit of propellant.

Depending on whether the amount of propellant is expressed in mass or in weight (by convention weight on the Earth) the dimension of specific impulse is that of speed or time, respectively, differing by a factor of g, the gravitational acceleration at the surface of the Earth.

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## General considerations

Essentially, the higher the specific impulse, the less propellant is needed to gain a given amount of momentum. In this regard a propulsion method is more fuel-efficient if the specific impulse is higher. This should not in any way be confused with energy-efficiency, which can even decrease as specific impulse increases, since many propulsion systems that give high specific impulse require high energy to do so.

In addition it is important that thrust and specific impulse not be confused with one another. The specific impulse is a measure of the thrust per unit of propellent that is expelled, while thrust is a measure of the momentary or peak force supplied by a particular engine. Therefore it is quite possible to have a propulsion system with a very high specific impulse (such as an ion thruster: 3,000 seconds), but producing a low thrust due to low propellent flow (micrograms per second).

When calculating specific impulse only propellant that is carried with the vehicle before use is counted. For a chemical rocket the propellant mass therefore would include both fuel and oxidizer, for air-breathing engines only the mass of the fuel is counted, not the mass of air passing through the engine.

## Examples

### Specific impulse of various propulsion technologies

 Engine "C" eff. exhaust velocity(N·s/kg or m/s) Specific impulse(s) Fuel mass(kg) Energy expended(GJ) Jet engine 30,000 3,000 50,000 2135 Solid rocket 1,000 100 190,000 95 Bipropellant rocket 5,000 500 8,200 103 Ion thruster 50,000 3,000 620 775 VASIMR 300,000 30,000 100 4,500

An example of a specific impulse measured in time is 459 seconds, or, equivalently, an effective exhaust velocity of 4500 m/s, for the Space Shuttle Main Engines when operating in vacuum.

An air-breathing engine typically has a much larger specific impulse than a rocket: a jet engine may have a specific impulse of 2000-3000 seconds or more.

In some ways, comparing specific impulse seems unfair in the case of jet engines and rockets. However in rocket or jet powered aircraft, specific impulse is approximately proportional to range, and rockets do indeed perform much worse than jets.

The highest specific impulse for a chemical propellant ever test-fired in a rocket engine was lithium, fluorine, and hydrogen (a tripropellant): 542 seconds (5320 m/s). However, the combination is impractical, see rocket fuel.

Nuclear thermal rocket engines differ from conventional rocket engines in that thrust is created strictly through thermodynamic phenomena, with no chemical reaction. The nuclear rocket typically operates by passing hydrogen gas over a superheated nuclear core. Testing in the 1960s (http://www.lascruces.com/~mrpbar/rocket.html) yielded specific impulses of about 850 seconds (8340 m/s), about twice that of the Space Shuttle engines.

A variety of other non-rocket propulsion methods, such as ion thrusters, give much higher specific impulse but with much lower thrust; for example the Hall effect thruster on the Smart 1 satellite has a specific impulse of 1640 s (16100 m/s) but a maximum thrust of only 68 millinewtons.

## Specific impulse in seconds

For all vehicles specific impulse (impulse per unit weight-on-Earth of propellant) in seconds can be defined by the following equation:

[itex]\mathrm{Thrust}=I_{\rm sp} \cdot \frac{dm} {dt} \cdot g_{\rm 0} \,[itex]

where:

Thrust is the thrust obtained from the engine

Isp is the specific impulse measured in seconds

[itex]\frac {dm} {dt} [itex] is the mass flow rate, which is minus the time-rate of change of the vehicle's mass, since fuel is being expelled.

g0 is the acceleration at the Earth's surface

This Isp in seconds value is somewhat physically meaningful - it is the number of seconds a unit weight of fuel would last if the engine would apply a unit force (if an engine could be scaled proportionately). As such it is a value that can be used to compare engines; much like 'miles per gallon' is for cars.

The advantage that this formulation has is that it may be used for rockets, where all the reaction mass is carried onboard, as well as aeroplanes, where most of the reaction mass is taken from the atmosphere. In addition, it gives a result that is independent of units used (provided the unit of time used is the second).

## Rocketry - specific impulse in seconds

In rocketry, where the only reaction mass is the propellent, an equivalent way of calculating the specific impulse in seconds is also frequently used. In this sense, specific impulse is defined as the change in momentum per unit weight-on-Earth of the propellent:

[itex]I_{\rm sp}=\frac{v_{\rm e}}{g_{\rm 0}}[itex]

where

Isp is the specific impulse measured in seconds

[itex]v_{\rm e}[itex] is the average exhaust speed along the axis of the engine

g0 is the acceleration at the Earth's surface

It may seem odd that the acceleration or weight at the Earth's surface is in the definition, while the rocket may be far from the Earth. However, accelerations are often measured in terms of g0; for example, astronauts should not be subjected to an acceleration more than a few times this value. Additionally, in Imperial units the relationship between force and mass is defined to involve the acceleration due to gravity. Thus pounds (force) and pounds (mass), both used in rocketry, when divided, must be additionally multiplied by g0 to get the acceleration in more usual units. The official Imperial unit of mass the slug, which is not popular for obvious reasons, was introduced to make Imperial units more like the SI units and avoid this multiplication. This, the common use of pounds for both force and mass, is in fact the chief reason g0 enters so often into rocketry definitions, and is likely the reason two definitions of specific impulse are in common use.

When expressed in units of seconds, the specific impulse can be interpreted in the following ways:

• the impulse divided by the sea-level weight of a unit mass of propellant
• the time one kilogram of propellant lasts if a force equal to the weight of one kilogram is produced, for example for a hypothetical hovering over the Earth (imagine the fuel to be supplied from outside, so that the mass on which the thrust is applied does not reduce by spending fuel)
• the time one pound mass of propellant lasts if a force of one pound is produced, for example for a hypothetical hovering vehicle over the Earth (imagine the fuel to be supplied from outside, so that the mass on which the thrust is applied does not reduce by spending fuel)
• alternatively, for engines that can not produce a large thrust: approximately the time one kilogram of propellant lasts if an acceleration of 0.01 g of a mass of one 100 kilogram is produced
• 100 times the time an acceleration g can be produced (i.e. a thrust equal to the weight on Earth of the current mass) with a propellant mass of 1 % of the current total mass (100 times the time it takes in this case to reduce the total mass by 1 %)
• the time an acceleration g can be produced with a propellant mass of 63.2 % of the initial total mass (the time it takes in this case to reduce the total mass by a factor e, to 36.8 %)
• twice the net power to produce an acceleration of 1 m/s2 to a mass which at Earth has a weight of 1 N (i.e. a mass of 102 grams)

For example for hydrogen/oxygen, with a specific impulse of 4500 m/s or 460 seconds:

• in m/s:
• the effective exhaust speed is 4,500 m/s
• the impulse produced per unit mass of propellant used is 4,500 N·s per kg
• the thrust is 4,500 N if the propellant mass flow rate is 1 kg/s
• the delta-v that can be produced with a propellant mass of 1 % of the current total mass (the delta-v that reduces the mass by 1%) is 45 m/s
• the delta-v that can be produced with a propellant mass of 63.2 % of the initial total mass (the delta-v that reduces the total mass by a factor e, to 36.8 %) is 4,500 m/s
• the power-thrust ratio is 2,250 W/N
• in s:
• one kilogram of propellant lasts 460 seconds if an acceleration g of a mass of one kilogram is produced
• one kilogram of propellant lasts 460 seconds if an acceleration of 0.01 g of a mass of 100 kilogram is produced
• it takes 4.6 seconds to reduce the total mass by 1 % if an acceleration g is produced
• an acceleration g during 460 seconds can be produced with a propellant mass of 63.2 % of the initial total mass (it is the time it takes in this case to reduce the total mass by a factor e, to 36.8 %)
• the net power to produce an acceleration of 1 m/s2 to a mass of 102 grams is 230 W.

## Rocketry - specific impulse as a speed

In rocketry the specific impulse as the impulse per unit mass of propellant used is simply the effective exhaust velocity:

[itex]I_{\rm sp}=v_{\rm e} \,[itex]

where

Isp is the specific impulse, as defined above, and measured in metres per second (in the U.S. feet/second).

ve is the effective exhaust velocity measured in metres per second.

It is related to the thrust, or forward force on the rocket by the equation:

[itex]\mathrm{Thrust}=I_{\rm sp} \cdot \frac {dm} {dt} \,[itex]

where

[itex]\frac {dm} {dt} [itex] is the mass flow rate, which is minus the time-rate of change of the vehicle's mass, since fuel is being expelled.

A rocket must carry all its fuel with it, so the mass of the unburned fuel must be accelerated along with the rocket itself. Minimizing the mass of fuel required to achieve a given push is crucial to building effective rockets. Using Newton's laws of motion it is not difficult to verify that for a fixed mass of fuel, the total change in velocity (in fact, momentum) it can accomplish can only be increased by increasing the exhaust velocity.

A spacecraft without propulsion follows an orbit determined by the gravitational field. Deviations from the corresponding velocity pattern (these are called delta-v) are achieved by sending exhaust mass in the direction opposite to that of the desired velocity change.

Due to the law of conservation of momentum, to change the speed of the spacecraft by an amount equal to 1% of the exhaust speed, approximately requires an exhaust mass equal to 1% of the mass of the spacecraft, including the fuel that has not yet been spent.

As a useful rule of thumb the delta-v that can be produced with a propellant mass of 63.2 % of the initial total mass is equal to the exhaust velocity (see Rocket equation.)

The speed is also approximately twice the power per unit thrust

For a delta-v that is much smaller than the specific impulse, the fuel required is approximately proportional to the delta-v. For a delta-v that is larger than the specific impulse, this requirement of carrying the fuel and spending much of the fuel on accelerating the fuel, gives rise to an exponential increase in fuel requirement (and larger tanks which also add to the mass). See spacecraft propulsion calculations and Tsiolkovsky rocket equation for details.

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