How to fly from Earth to Mars if the differences between phase angles is too big for Hohmann transfer but we have fuel for other transfers?

Hohmann transfer is possible when Earth and Mars positions are appropriate for this https://ai-solutions.com/_freeflyeruniversityguide/interplanetary_hohmann_transfe.htm

This happens one time in two years.

But what can we do, if positions do not satisfy that condition but we need to fly here?

What transfers can be used in this situation (for i.e. Mars is in opposite side of Sun)? I need two solutions: fastest and fuel-efficient.

• You can transfer between planets at any time, the efficiency and cost just varies a lot if you do so without a precise Hohmann transfer. Look to en.wikipedia.org/wiki/Porkchop_plot , which is the tool that shows you the effort and time needed for a transfer, on a 2D+ plot of transfer duration vs. departure time. Oct 11, 2021 at 12:37
• @PcMan But maybe there are the more effective transfers in that case? Oct 11, 2021 at 12:39
• Without involving other planets, the Hohmann transfer is the cheapest way to move between planets. The Porkchop plot is exactly the tool that shows you what other options you have if you choose to deviate from this optimal minimum. But all alternatives will require more deltav to get there than the minimal case of the Hohmann. Oct 11, 2021 at 12:42
• More deltav is ok for me. Oct 11, 2021 at 12:43
• The problem is that outside the launch window the delta-v quickly climbs to beyond what we can do with chemical rockets. You need something way beyond chemical if you want to skip the launch window. Oct 12, 2021 at 0:06

You're looking for porkchop plots (and Lambert's problem).

These plots are created by brute force solving Lambert's problem for a range of departure times and arrival times.

note: The numbers on the slanted cyan lines give the duration of the trajectory in days.

For example, Mars 2020 launched on 2020 July 30, which is 30 on the x-axis, and arrived on 2021 February 18, which is ~109 on the y-axis. These coordinates give a delta V that is inside of the 6km/s contour, which is the smallest on this plot. To clarify, this plot was created by the sum of the v-infinity vectors when departing Earth and arriving at Mars. In a lot of cases these two values are split up in porkchop plots, but I personally find this one easier to read (at least to get an initial idea of where is good to look for dates).

The gap in the middle represents a ~180 degree change in true anomaly in the transfer, which is (kind of) the Hohmann solution (not exactly since the planets' orbits are not exactly circular or coplanar)

Below the gap are transfers with less than 180 degrees of true anomaly change:

And above the gap is greater than 180 degrees of true anomaly change:

Note that these are for direct transfers. Gravity assists trajectories use different types of analysis (starting with v-infinity matching)

• Do the numbers on the cyan lines give the duration of the trajectory in days? Oct 11, 2021 at 19:13
• @PM2Ring Yes they are my apologies for not including that in the explanation Oct 11, 2021 at 22:34
• You mean you don't burn Hohmann transfer on departing earth and make the 1.8 degrees of plane change when you cross Mars' plane in deep space? Oct 11, 2021 at 23:29
• @Joshua For the hohmann transfer part I mean that Hohmann assumes a transfer between 2 circular and coplanar orbits with 180 degrees change in true anomaly of the transfer orbit. Earth and Mars are not in circular orbits (close though) and not coplanar (again close). Lambert's problem takes in 2 position vectors and a time of flight between them to calculate the trajectories. the delta V in this case refers to the magnitude of the difference of the departing velocity vector and Earth's velocity vector, and the arriving velocity vector and Mars' velocity vector Oct 12, 2021 at 0:32
• @Joshua this type of analysis does not take into account trajectory correction / deep space maneuvers, its only used as a first order look at what are good times to launch / arrive Oct 12, 2021 at 0:35

I need two solutions: fastest and fuel-efficient.

There is actually one in the fuel-efficient case, which involves doing a gravity assist by Venus.

It has about the same delta-v cost as a regular Hohmann transfer to Mars, but can potentially be used when the angle isn't right since it has other requirements for planetary alignment.

Requiring 3 planets in the right place instead of just 2, the opportunity occurs only about every 11th year. It's however notable in that it allows a "fast" roundtrip to Mars, the "Crocco Grand Tour" after Gaetano Crocco. It's faster than a Hohmann transfer back and forth, since it can use Venus to bypass one of the "bad" planetary alignments.

But other than that, your options are just patience or big thanks of rocket propellant.

• "Requiring 3 planets in the right place instead of just 2" I'm trying to find a universal solution to make the ships in my game be able to travel to any place any time. Oct 12, 2021 at 8:02
• But I want to add additional gravitational accelerations if we a lucky and planets on route in a good positions. Oct 12, 2021 at 8:03
• Yeah, this is just a lucky opportunity once in a while. Oct 12, 2021 at 8:43
• @Robotex While it is not a "universal solution" and may need to be looked at with a grain of salt, have you studied Project Rho? Oct 17, 2021 at 2:03
• @ikrase What is it? Oct 18, 2021 at 7:55