Lets say we have a satellite wanting to go to Mars. Do satellites such as MAVEN inject into a Hohmann transfer orbit while in Earth's orbit, or do they go out of Earth's Sphere of Influence into solar orbit, and then inject into a Hohmann transfer orbit? The reason I'm asking this is because it requires a long burn time to obtain the necessary $\Delta V$ and by that time, the satellite can be in the wrong position as satellites move fast in Earth's orbit (LEO specifically). However this can all be avoided while in solar orbit.
Do satellites inject into a Hohmann transfer orbit in Low Earth Orbit or in Solar orbit?
In general all spacecraft will try and do most of their acceleration as low in a gravity well as possible. This gets you the maximum benefit from your fuel, something referred to as the Oberth effect. For this reason an interplanetary mission will generally inject directly into transfer orbit to its (first) destination planet from Low Earth Orbit, or even directly from launch without entering LEO.
The trajectory design will include allowing for the actual time the rockets burn for and the movement of the spacecraft during that time. There is also generally an allowance for a few small trajectory corrections midflight.
$\begingroup$ Do you have any sources? Such as a mission report of a satellite, and etc.? $\endgroup$– Star ManJul 24, 2019 at 16:49
$\begingroup$ This is quite an interesting pair of examples planetary.org/blogs/emily-lakdawalla/2013/… In both cases the burns are all done as close to Earth as possible. $\endgroup$ Jul 24, 2019 at 18:44
A "Hohmann orbit" is an ellipse connecting orbits around the starting and destination objects, and in this case, it's an eccentric orbit around the Sun which touches (is tangent to) the initial Earth orbit and possibly the final Mars orbit.
You have to exceed Earth's escape velocity to even be in such an orbit, so the delta-v is the difference between your orbital velocity around the Earth and the transfer orbital velocity. Don't forget the different reference frames! (see answer(s) to How to best patch my conics?)
The technique of building up a plan from transferring from one conic orbit to another is called the patched conic approximation and has been a staple of trajectory design due to its simplicity, speed of computation and search for optimal design, and it usually provides relatively good initial accuracy.
$\begingroup$ I added a few supporting links and a bit more background, I hope you don't mind. Feel free to roll back or edit further. Welcome to Space! $\endgroup$– uhohJul 24, 2019 at 0:59
$\begingroup$ So the burn is executed in Earth's SOI, based on the patched conic approximation? $\endgroup$– Star ManJul 24, 2019 at 18:25
$\begingroup$ @StarMan the patched conic approximation is just the first step in planning a space mission. That plan will be refined in stages using a series of more precise (but more computationally expensive) techniques that allow for finite burn duration, simultaneous gravitational attraction of multiple bodies and so on. $\endgroup$ Jul 24, 2019 at 19:30
$\begingroup$ Nice edit, thanks for that ! I'm usually on mobile, so links and lots of typing is at best inconvenient. $\endgroup$ Jul 24, 2019 at 22:44