Determining the launch period for Hohmann transfer

Question

How do you determine the day of a specific launch? I know there are launch windows but they only refer to the time at which a spacecraft must be launched.

For a LEO it is apparently 365 days, but what if I'll be using the LEO to transfer to a GSO and then do a Hohmann transfer to some other (dwarf) planet such as Ceres, clearly I cant launch any day of the year as ceres must be at apogee when the spacecraft arrives?

I thought of using GSO as a parking orbit since I have little information on the orbital parameters of parking orbits (aphelion, perihelion distances etc) usually utilized. I am new to orbital mechanics and so any help would be appreciated, thank you.

Answer 1

Hohmann transfers only describe transfers between 2 circular orbits. So if you're looking to find when would be a good time to leave Earth to get to Ceres using Hohmann transfers, you don't take into account the gravity of the Earth, you only imagine transferring from one circular heliocentric orbit to another circular heliocentric orbit, like in this case from Earth to Mars:

You can calculate the time of flight from a Hohmann transfer from the semi-major axis of the transfer, which is calculated using the two circular orbits:

$sma_{transfer} = \dfrac{r_{\text{Earth}} + r_{\text{Mars}}}{2}$

$t_{transfer} = \pi \sqrt{\dfrac{a_{\text{transfer}}^3}{\mu_{\text{sun}}} }$

Where the equation for the transfer time is half of the period of the elliptical orbit transfer and $\mu_{\text{sun}} = GM_{\text{sun}}$.

For finding a good time, you also want to know the synodic period between 2 circular orbits, which is calculated by:

$T_{synodic} = \dfrac{2\pi}{| n_2 - n_1 |} = \dfrac{T_1 T_2}{| T_1 - T_2 |}$

Where n is the mean motion of an orbit, and T is the period.

Outside of Hohmann transfers, there also exist Lambert's problem, where you can calculate the trajectory from one position vector to another given a time of flight.

Doing this a bunch of times then gets you porkchop plots, which are used to calculate when is a good time (from a delta V perspective) to leave Earth and arrive at Mars.

This is a delta V porkchop plot, but they are often split up into 2 burns (Earth departure and Mars arrival delta Vs).

From the Lambert problem solution you are then able to calculate how much excess delta V you need to escape from Earth and get on the trajectory to get to Mars, and how much excess velocity you have when you arrive at Mars. So from this you can calculate the delta V required to get you on that Mars trajectory given some Earth parking orbit that you're at. So you can then compare how much delta V it would take to do a transfer to Ceres from LEO, or any other orbit. And as far as when to launch, you have to align the outbound hyperbolic asymptote (since from Earth's perspective you are in a hyperbolic orbit meaning you will escape Earth's sphere of influence) to the direction of the velocity vector that you get from solving Lambert's problem. But those calculations are a bit more complex