## Velocity at perihelion

We are now then primarily interested in calculating the velocity needed by the spacecraft to enter the correct transfer orbit at perihelion, that is, when it leaves the earth's orbit.

At perihelion the distance to the Sun r will coincide with the average radius of the terrestrial orbit, that is, what we have called dEarth. Substituting r for dEarth in expression (2) :

Vperihelion={2*G*Ms*[1/dEarth - 1/(dEarth+dMars)]}^1/2

Simplifying :

Vperihelion={2*G*Ms*dMars /[dEarth *(dEarth+dMars)]}^1/2 (3)

As we will use the Earth's orbital velocity to boost up our own transfer orbit we are really more interested in the difference in velocity between the transfer orbit and the terrestrial orbit.

Expression 1 can be applied to the Earth's supposed circular orbit, as initially the spacecraft will be traveling along with the Earth in its orbit around the Sun. For the terrestrial orbit, the semimajor axis of the orbit will then be dEarth and r, the distance from the body in motion to the Sun will also be constant and equal to dEarth:

1/2*m*V^2 - G * Ms*m / dEarth = - G*Ms*m/(2dEarth)

Simplifying :

Vearth orbit = (G*Ms/dEarth)^1/2 (4)

The required difference in velocity (which will eventually be supplied by the rocket burn from the propulsion system) will be obtained by subtracting expressions (3) and (4)

DeltaV = Vperihelion - Vearth orbit

Replacing the values,

DeltaV = 2942 m/s or approx. 3 km/s