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On a porkchop plot for a given departure date (X axis), why are there two local energy minimums along the arrival date (Y-axis) for a transfer between Earth and Mars? The Hohmann transfer orbit seems to have one optimal solution, but the plot's two minimums seem fairly distinct in time but similar in energy; 15.0 to 15.5 and 15.5 to 16.0 in terms of C3L (something I only vaguely understand; KSP mostly deals in delta-V)

Earth to Mars porkchop plot

A NASA page about sending spacecraft to Mars refers to "Type 1" and "Type 2" transfer orbits, which clashes with a perhaps simple view that there should be a singular optimal Hohmann transfer orbit; the smallest ellipse tangent to two orbits. Granted, the planets' orbits are not perfectly circular, but the different transit times (red contours) for local minimums differ by almost a factor of two! (about 200 days and about 400 days) I'm not sure if I'm reading the plot correctly though, as there is a disconnect between the given Earth-Mars plot, and the prose on the page as well, which talks about 7 month (~200 day) or 10 month (~300 day) transfers.

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    $\begingroup$ "porkchop plots"? +1 $\endgroup$ – Danu Jul 4 '14 at 2:56
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    $\begingroup$ "Porkchop plots" are what NASA/JPL (and other agencies) use to plan gravity assists and more mundane transfers. $\endgroup$ – David Hammen Jul 4 '14 at 2:58
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    $\begingroup$ A link to the nasa.gov page that contains the referenced diagram: mars.jpl.nasa.gov/spotlight/porkchopAll.html . Nick T, it would be good to add this link to your question. Sorry, but I can't take the time to answer this question. I'm on vacation as of now, but the real start will be at 3AM when a long drive commences. $\endgroup$ – David Hammen Jul 4 '14 at 3:03
  • $\begingroup$ Paper on some of the details of earth-mars porkchop plots: trs.jpl.nasa.gov/bitstream/handle/2014/44336/13-0679_A1b.pdf $\endgroup$ – BowlOfRed Jul 26 '18 at 18:59
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From a previous revision to this answer:

This is not quite a full answer. I need some sleep before I start on a long drive at 3 AM.

I started on that long drive six months ago but then I never completed the answer. This question recently briefly reappeared on the front page. It's well past time to make my answer complete.

What are porkchop plots?

When the goal is to send a spacecraft from the Earth to another planet, it's not enough to reach the target planet's orbit. The vehicle has to meet the target planet itself. The amount of energy needed to accomplish this varies widely depending on departure and arrival dates. A porkchop plot is a graphical interplanetary mission planning tool that depicts as a contour plot the required energy as a function of departure date and arrival date. The energy needed by the launch vehicle is key in determining the feasibility of such a mission. A mission plan that requires more energy than a launch vehicle could possibly provide is not feasible. In addition to feasibility, a porkchop plot aids in planning key mission operations and in planning the optimal trajectory between the two planets. The plot shown in the question and replicated below show launch energy. It does not show the change in energy needed at arrival. Other porkchop plots do show this as a second set of contour lines.

The original post asks three key questions:

  • Why is there a gap in porkchop plots?
  • Why don’t they use an optimal Hohmann transfer?
  • Why are there two local energy minima?

Before answering the above, it will help to explain how a porkchop plot is constructed.

How are porkchop plots constructed?

As a mission planning tool, a porkchop plot makes certain simplifying assumptions with regard to reaching the target planet. Later on, more detailed analyses address those simplifying assumptions. The key simplifying assumptions used in making a porkchop plot are the patched conic approximation and impulsive maneuvers.

These assumptions reduce the problem to one of finding Keplerian orbits about the Sun that take the spacecraft from the vicinity of the Earth to the vicinity of the target planet in the requisite amount of time. Finding such transfer orbits is the subject of Lambert's problem. A number of such transfer orbits might exist. I’ll denote the angle subtended by the departure point, the Sun, and arrival point as $\theta$, with $\cos \theta = \frac{\vec r_1 \cdot \vec r_2} {r_1 r_2}$. The principal value of this angle will be between 0° and 180°, inclusive. For now I’ll ignore cases where $\theta$ is 0° or 180°. This means that the transfer plane is well-defined and that the number of solutions is finite.

Lambert's problem does not have closed form solutions; a number of iterative techniques have been developed to find solutions. One solution, “the short way”, or “Type 1” transfers, has the change in true anomaly equal to $\theta$ as described above. Another solution, the “long way”, or “Type 2” transfers, has the change in true anomaly equal to 360°-$\theta$. Other solutions may exist as well. For example, one way to transfer from Earth to Mars in 2.5 years is to make more than a complete orbit during the transfer. Porkchop plots typically only show the Type 1 and Type 2 solutions, and typically only show at most one of these two solutions for a given pair of departure and arrival dates. If one of the two solutions is close to optimal, the other solution will inevitably follow a retrograde path and thus will involve huge expenditures of energy. There’s no reason to show these highly sub-optimal solutions.

Why is there a gap in porkchop plots?

The plot can be cleaned up further by removing cases where the better of the two solutions still involves huge energy expenditures. Huge energy expenditures are obviously going to result when the transfer time is very short or very long. A not so obvious place where this happens is when the angle subtended between the line from the Earth and Sun at departure and the target planet and Sun at arrival is nearly 180°. That the Earth and target planet have slightly different orbital planes means that the transfer plane will be nearly orthogonal to the planetary orbital planes when the transfer angle is close to but not equal to 180°. This makes the approach of having a maneuver at departure and a maneuver at arrival extremely expensive for those transfers that are close to 180°. Removing those very expensive transfers from view is what creates the gap in the porkchop plot.

This excessive cost for near 180° transfers is to some extent an artifact of the approach used to create a porkchop plot. Adding a third maneuver enables the use of much smaller in-plane maneuvers at the start and end, with a small plane change somewhere along route. There’s a problem, however. This mid-course plane change would necessarily mean thrust from the spacecraft itself. This is undesirable. Tspacecraft itself provides very little of the energy with the two burn approach. The energy for the Earth departure comes from the launch vehicle, and in the case of Mars, most of the energy for Mars arrival comes from aerobraking. It’s better to fold that plane change into the maneuvers at departure and arrival so as to keep the thrust needed by the spacecraft down to a minimum.

Why don’t they use an optimal Hohmann transfer?

An optimal Hohmann transfer doesn’t exist. Hohmann transfers are in-plane maneuvers that transfer from one circular orbit to another that share a common orbital plane. Planetary orbits are slightly inclined with respect to one another and are elliptical rather than circular. Generalizing the concept of a Hohmann transfer to that of a 180° transfer, in most cases that 180° transfer doesn’t exist. When it does, that the orbits are not coplanar and that the orbits are not circular means that this 180° transfer is no longer optimal.

Why are there two local energy minima?

There aren’t just two local energy minima. There are a countable infinite number of local minima. A porkchop plot only shows the first two. The other solutions take even longer than Type 1 and Type 2 transfers and are more sensitive to errors.

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  • $\begingroup$ I understand that high energy contours aren't shown because they're basically useless, but I don't intuitively get that if there is a lowest-energy trajectory (Hohmann), why it would give (effectively) two answers. Are they both a Hohmann trajectory? Is it because the orbits aren't perfectly circular? Is it because the plot reflects the energy to both intercept and match the destination's speed? If so, could one of the solutions be practically much lower due to aerobraking, or would that apply to both? $\endgroup$ – Nick T Jul 4 '14 at 5:09
  • $\begingroup$ There is no such thing as a circular orbit in reality. Circular orbits are a "spherical cow" idealization. Neither Earth (e=0.01671123) nor Mars (e=0.093315) has a circular orbit. The minimum energy transfer will sometimes be less than 180° ("short way"), sometimes more ("long way"). If the orbits were circular, the "short way" and "long way" would coincide, both being exactly 180° transfers. $\endgroup$ – David Hammen Jul 4 '14 at 7:55
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Found the answer on this site: http://hopsblog-hop.blogspot.com/2013/01/deboning-porkchop-plot.html

It's because of the plane change required to get from the first body's orbit to the other's. The gap in the middle represents the necessary delta-v to do a 180 degree plane change. You'll notice that they only show up when doing ballistic transfers directly from one body to the other. Porkchop plots for transfers that include a mid-course plane change don't have the ridge between the two local minima.

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