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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.

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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.

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    $\begingroup$ First of all remove the references to Earth and Mars and allow to use the custom mu. The second, you calculated the magnitude of velocity but lost the direction. And also I need a phase when I need to make the impulse. $\endgroup$
    – Robotex
    Commented Dec 16, 2021 at 12:11

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