How to calculate an initial impulse for escape from planet's sphere of influence and lay on a specific orbit around the Sun?

Question

I have a spacecraft parked on orbit around the planet. I want to make interplanetary transfer to another planet. I can find the orbit around the Sun that I can use to move from the end of departure planet's sphere of influence to the start of arrival planet SOI for the given amount of time. So, I have two velocities vectors: one is the velocity which I need to have on the border of the first SOI and the second that I will have at the second SOI when I arrive.

I need to calculate impulse (and moment when start it) needed for transfer from current orbit to have the needed velocity at given point (hyperbolic V_infinity). And also the second impulse at arrival planet to go to parking orbit.

Answer 1

You need the "orbital Pythagoras"

$$v^2 = v_e^2 + v_{\infty}^2$$

That is, your current velocity inside a SOI $v$, the current escape velocity $v_e$, and the hyperbolic excess velocity $v_{\infty}$

Example:

We're in LEO, and want to do get into a Mars transfer orbit. The perihelion velocity of an elliptic orbit touching both the orbit of Earth and Mars is 32.73km/s. The Earth itself is travelling at 29.78km/s, so a the velocity we need at the edge of Earth's SOI is 2.94km/s. ($v_{\infty}$)

In LEO, the escape velocity is 11.01km/s. So the velocity we need to have is:

$$v^2 = v_e^2 + v_{\infty}^2$$

$$v = \sqrt{v_e^2 + v_{\infty}^2}$$

$$v = \sqrt{(11.01km/s)^2 + (2.94km/s)^2}$$

$$v = 11.40km/s$$

In LEO, we're already travelling at 7.78km/s, so the impulse is 11.40km/s - 7.78km/s = 3.62km/s.