Characteristic velocity

Short description: Measure of rocket performance

Characteristic velocity or [math]\displaystyle{ c^* }[/math], or C-star is a measure of the combustion performance of a rocket engine independent of nozzle performance, and is used to compare different propellants and propulsion systems. c* should not be confused with c, which is the effective exhaust velocity related to the specific impulse by: [math]\displaystyle{ I_s = \frac{c}{g_0} }[/math]. Specific Impulse and effective exhaust velocity are dependent on the nozzle design unlike the characteristic velocity, explaining why C-star is an important value when comparing different propulsion system efficiencies. c* can be useful when comparing actual combustion performance to theoretical performance in order to determine how completely chemical energy release occurred. This is known as c*-efficiency.

## Formula

[math]\displaystyle{ c^* = \frac{p_1A_t }{\dot{m}} }[/math]

* [math]\displaystyle{ c^* }[/math] is the characteristic velocity (e.g. m/s, ft/s)
* [math]\displaystyle{ p_1 }[/math] is the chamber pressure (e.g. Pa, psi)
* [math]\displaystyle{ A_t }[/math] is the area of the throat (e.g. m2, in2)
* [math]\displaystyle{ \dot{m} }[/math] is the mass flow rate of the engine (e.g. kg/s, slug/s)

[math]\displaystyle{ c^* = \frac{I_sg_0}{C_F} = \frac{c}{C_F} =\sqrt{\frac{RT_1}{k} \Bigl(\frac{k+1}{2}\Bigr)^{\frac{k+1}{k-1}}} }[/math]

* [math]\displaystyle{ I_s }[/math] is the specific impulse (e.g. sec)
* [math]\displaystyle{ g_0 }[/math] is the gravitational acceleration at sea-level (e.g. m/sec^2)
* [math]\displaystyle{ C_F }[/math] is the thrust coefficient
* [math]\displaystyle{ c }[/math] is the effective exhaust velocity (e.g. m/sec)
* [math]\displaystyle{ k }[/math] is the specific heat ratio for the exhaust gases
* [math]\displaystyle{ R }[/math] is the gas constant per unit weight (e.g. J/kg-K)
* [math]\displaystyle{ T_1 }[/math] is the chamber temperature (e.g. K)

## References

* Rocket Propulsion Elements, 7th Edition by George P. Sutton, Oscar Biblarz
* Rocket Propulsion Elements, 9th Edition by George P. Sutton, Oscar Biblarz

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