Editing Antimatter rocket (section)

===Modified relativistic rocket equation===
The loss of mass specific to antimatter annihilation requires a modification of the relativistic rocket equation given as<ref name=AIAA-98-3403>[http://trs-new.jpl.nasa.gov/dspace/bitstream/2014/20238/1/98-1132.pdf ''Evaluation of Propulsion Options for Interstellar Missions''] {{webarchive|url=https://web.archive.org/web/20140508061651/http://trs-new.jpl.nasa.gov/dspace/bitstream/2014/20238/1/98-1132.pdf |date=2014-05-08 }} Robert H. Frisbee, Stephanie D. Leifer, AIAA Paper 98-3403, July 13–15, 1998.</ref>

{{NumBlk|:|<math>\frac {M_0}{M_1} = 
\left(\frac{1+ \frac{\Delta v}{c}}{1- \frac{\Delta v}{c}}\right)^{\frac{c}{2 I_{\text{sp}}}}
</math>|{{EquationRef|I}}}}

where <math>c</math> is the speed of light, and <math>I_{\text{sp}}</math> is the specific impulse (i.e. <math>I_{\text{sp}}</math>=0.69<math>c</math>).

The derivative form of the equation is<ref name=AIAA-2003-4696/>

{{NumBlk|:|<math>\frac {dM_{\text{ship}}}{M_{\text{ship}}}=
\frac {-dv ( 1 - I_{\text{sp}} \frac {v}{c^2})}
{(1 - \frac {v^2}{c^2})(-\frac {I_{\text{sp}} v^2}{c^2} +  (1 - a) v + a I_{\text{sp}})}
</math>|{{EquationRef|II}}}}

where <math>M_{\text{ship}}</math> is the non-relativistic (rest) mass of the rocket ship, and <math>a</math> is the fraction of the original (on board) propellant mass (non-relativistic) remaining after annihilation (i.e., <math>a</math>=0.22 for the charged pions).

{{EquationNote|Eq.II}} is difficult to integrate analytically. If it is assumed that <math>v \sim I_{\text{sp}}</math>, such that <math>(1 - \frac {I_{\text{sp}} v}{c^2}) \sim (1 - \frac {v^2}{c^2})</math> then the resulting equation is

{{NumBlk|:|<math>\frac {dM_{\text{ship}}}{M_{\text{ship}}}=
\frac {-dv}{(-\frac {I_{\text{sp}} v^2}{c^2} + (1 - a) v + a I_{\text{sp}})}
</math>|{{EquationRef|III}}}}

{{EquationNote|Eq.III}} can be integrated and the integral evaluated for <math>M_0</math> and <math>M_1</math>, and initial and final velocities (<math>v_i = 0</math> and <math>v_f = \Delta v</math>).
The resulting relativistic rocket equation with loss of propellant is<ref name=AIAA-2003-4696/><ref name=AIAA-98-3403/>

{{NumBlk|:|<math>\frac{M_0}{M_1}=\left(\frac{(-2I_{\text{sp}}\Delta v/c^2+1-a-\sqrt{(1-a)^2+4aI_{\text{sp}}^2/c^2})(1-a+\sqrt{(1-a)^2+4aI_{\text{sp}}^2/c^2})}{(-2I_{\text{sp}}\Delta v/c^2+1-a+\sqrt{(1-a)^2+4aI_{\text{sp}}^2/c^2})(1-a-\sqrt{(1-a)^2+4aI_{\text{sp}}^2/c^2})}\right)^{\frac{1}{\sqrt{(1-a)^2+4aI_{\text{sp}}^2/c^2}}}
</math>|{{EquationRef|IV}}}}