# Delta-V vs Time equation for interplanetary transfers

I'm working on a project for my high school math class and my research question is about time vs delta-v budgets for an interplanetary transfer (e.g. earth to mars). I understand this is a relatively complex topic for my skill set but I would love to know more about this. I think I understand the basics of the Hohmann transfer, but I'm struggling to understand how to map out the equation.

From what I understand, the best approach to this problem would be to change the initial burn velocity from the first body, with the ideal transfer window, which would also change.

Any help is appreciated as I'm very lost.

Thank you!

EDIT: I'm asking how the departure burn from earth affects the flight time to other planets

• Welcome! Please could you clarify in your question if you are looking for either a) the time between transfer opportunities with minimum delta-v from Earth to Mars or b) how delta-v of the Earth departure burn can have an impact on the interplanetary flight time to Mars? Dec 5, 2022 at 7:40
• Alex, is your question "how delta V depends on launch date"? If yes, than the thing you need is "porkchop plot". Dec 5, 2022 at 10:31
• Thanks for the edit - take a look at this answer for a similar question and see if this (simplified) approach could help : space.stackexchange.com/questions/57265/… Dec 5, 2022 at 16:26
• The problem is that Lambert solution is beyond the level of high school math. My suggestions: Variant 1. If you are good enough in math and coding you can play yourserf with Lambert solver (in Python, for example). Variant 2. You can read about as David Hammen advises in his ansver and make a descriptional report. Variant 3. You can make it simpler and use only "vis viva equation" - it's nice because the solutions are "by pencil", but for limited cases. Google for this and for "Hohmann trasfer". What way to choose - it's up for you. Dec 6, 2022 at 5:34

Suppose you want to send a spacecraft from Earth to Mars, leaving Earth at time $$t_0$$ and arriving at Mars at some later time $$t_1$$. Ignoring the gravitational fields of Earth, Mars, the Moon, and all of the other planets, the only gravitational field of concern is the Sun.

There is always at least one, and oftentimes more than one, Newtonian mechanics conic section that will take the spacecraft from Earth's position at time $$t_0$$ and Mars position at time $$t_1>t_0$$. If $$t_1$$ is $$t_0$$ plus one second, the Newtonian solution will require traveling faster than the speed of light, but that is not a constraint in Newtonian mechanics. (The $$\Delta V$$ needed to achieve that faster than light transfer orbit is a bit problematic. It might be better to look for a longer transfer time than one second.)

Solving for the conic sections needed to accomplish the transfer is Lambert's problem. There are lots of articles on the internet and sections in textbooks regarding solving Lambert's problem. The $$\Delta V$$ needed to accomplish the transfer is the sum of the $$\Delta V$$ needed to transfer from Earth's orbital velocity at time $$t_0$$ to the transfer orbit, and the $$\Delta V$$ needed to transfer from the transfer orbit to Mars's orbital velocity. For shortish transfers, between one second and a bit less than one orbit, there are typically two solutions. (180° degree transfers are problematic as there are an infinite number of solutions.) Ignoring those 180° transfers, one of the two solutions will be better than the other in terms of total $$\Delta V$$.

There are two variables to play with in this scenario, the Earth departure time and the Mars arrival time. There is a cost associated with this pair of times. A simple cost metric is the total $$\Delta V$$, $$\Delta V_{\text{tot}} = ||\Delta V_0|| + ||\Delta V_1||$$, where $$\Delta V_0$$ is the $$\Delta V$$ needed to transfer from Earth's orbital velocity to that of the transfer orbit at time $$t_0$$ and $$\Delta V_1$$ is is the $$\Delta V$$ needed to transfer from transfer orbit to Mars's orbital velocity at time $$t_1$$. More complex cost metrics exist. Whichever cost metric you decide to use, this becomes a "simple" matter of cost minimization for a non-linear two variable problem. (I wrote "simple" because optimizing a non-linear two variable problem that is subject to singularities such as a 180° transfer is not "simple".)

What people at JPL and elsewhere do is to create a pork chop plot. What you'll find for transfers from Earth to Mars is that there is a narrow window in time every other year where the cost is not phenomenal. Two sets of feasible solutions arise, one set involving a shortish transfer time and the other a longish transfer time. When plotted graphically as contour lines that depict cost, with departure time on one axis and arrival time on the other axis, the result looks a bit like a pork chop, hence the name pork chop plot.

There are several questions at this site and at the Physics.SE sister site regarding pork chop plots. I suggest you look into these and come back with more specific questions.

• You might need to search separately for porkchop plot (two words) and pork chop plot (three words). Dec 5, 2022 at 11:34