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We can go to Mars every year, doing a delicate ballet with gravity bowls

If you are into colonizing Mars, you have heard that we can launch only every 26 months during a small window of time. That is because only certain trajectories to Mars work.

This throws a spanner into planning for your colony resupply missions and manned missions alike. As we saw in the movie ‘The Martian’ with Matt Damon playing astronaut-botanist Mark Watney, missing your flight home and waiting for rescue could become a slightly tedious affair filled with self-doubt, potatoes and disco.

The ability to send missions every year and not need exotic propulsion nor suffer other penalties would be a real breakthrough. It appears we possibly could.

William Tell on steroids…

The current approach essentially consists of aiming very, very precisely and shooting an arrow into a speeding apple 400 million kilometers away. Marksmanship has indeed seen progress.

But because the apple has an atmosphere, moves at an angle and has gravity, you also have to slow your craft down, with the right force, at the right angle to be captured by its gravity.

Miss the angle or the burn, and your vehicle either sees a fiery death (too steep), or skips off the atmosphere like a pebble (too shallow) flying out beyond Mars without any chance to ever reach it. Tricky,  but it has been successfully used for decades. Most of the time.

Hohmann Transfer Orbit Hohmann transfer orbit, labelled 2, from Earth orbit (1) to a higher orbit (3) e.g MARS (Credit: Leafnode – Own work based on image by Hubert Bartkowiak – (CC BY-SA 2.5))

This trajectory, the best we knew, is called a Hohmann transfer orbit. Because the two planets have to be relatively close to each other, the launch window opens only every 26 months. It closes only a handful weeks later, when the orbits of Mars and Earth move out of alignment. Once launched, a standard rocket needs 9 months to reach Mars. Faster transfers are possible if you use either a bigger rocket or a smaller payload. This is because you trade a higher change in velocity (or Delta-V) against payload.

This brief launch window, is why NASA, ESA,  ISRO, all scramble to get their MARS missions launched with a precise cadence of about every other year. When a science experiment has a last-minute malfunction before launch, this could mean your launch window closes and you have to wait another two full years. This head ache happened to the INSIGHT mission which was originally planned to launch in  March 2016. Unfortunately, because a mission critical Seismometer destined to hunt for Mars-quakes (SEIS, originally delivered by the French CNES), sprung a small leak (it lost its vacuum), INSIGHT will now launch May 5 2018.

A More Flexible Option

But are we tied to this 2 year interval? For decades the conventional wisdom said yes.  Recently however, a new approach has been found allowing for more planning flexibility.


Edward Belbruno, a mathematician, artist, Princeton University scientist specialized in celestial mechanics had the intuition, already in the ’80-ies when he worked at NASA JPL, that one might do better. He looked at the problem and thought there might be a different way to get from one gravity bowl, into another one, which is what essentially happens when you go from planet to planet.

Brute Force

The Hohmann transfer can be most likened to a brute force approach. You shoot out in almost a straight line and once at Mars, you violently push the brakes not to overshoot your destination. You must do so, because you essentially have two intersecting trajectories. The probe, on its voyage, is flying so fast that the gravity of MARS simply is not strong enough to capture it in its gravity bowl. The probe will indeed deflect from its course, but without a burn you will leave Martian orbit (your trajectory remains hyperbolic).


Compared to this, the Ballistic approach, also called low energy trajectory, is more akin to a delicate ballet. Imagine those two gravity bowls as being real bowls and put them side by side. In reality both are moving. Now imagine that you are trying to push a grape out of one bowl with as little energy as possible, balancing it on the (gravity) rim of both bowls, and then letting it slide effortlessly into the other bowl, ideally tipped over only by the force of gravity.  Before the grape settles down in the middle, it will follow a slightly chaotic path. This is also the case in this Low energy trajectory.

What the pictures below cannot convey, is that the rims of the gravity bowls extend and touch each other ever so slightly. The trajectories are also more three-dimensional than shown. Still, they get the point across. The space that connects them can be thought of as a small river, or a balancing rope, where the gravity of the bodies cancel out. If one uses slow but efficient ion propulsion, the goal then is to spiral out from Earth to eventually follow them and spiral in at the destination, but if you use chemical propulsion, only the arrival part of the trajectory is different from a Hohmann transfer.

You essentially shoot out towards MARS, along this sweet spot, in the same manner as you would on a Hohmann transfer. But this time you aim for a space far in front of MARS, a couple of million kilometers, making sure the heliocentric orbit of the probe closely matches the eccentricity of the ellipse of MARS around the sun, and have MARS catch up with you. To make small adjustments on this last bit, and slide into the gravity bowl, you only need to sip propellant. The result can be a 25% reduction on the propellant use for the capture burn.  Hence, the name low energy trajectory.

Translated back to the William Tell analogy: You shoot the arrow far in front of the apple, seemingly hover in place, and wait for the apple to catch the arrow.

“The fact that one is allowed to move to a place  far from Mars has an implication on the launch period from the Earth to get to Mars. For the case of a Hohmann transfer, there
is a small launch period of a few days that must be satisfied when the Mars and Earth line up. This is because a point, i.e. Mars, has to be directly targeted. If this is missed for any reason, a large penalty in cost may occur since launch may not be possible. This problem would be alleviated if the launch period could be extended. By targeting to this location out in front rather than to Mars, it is not necessary to wait every two years, but rather, depending on how far the arrival point is separated from Mars, the time of launch could be extended significantly. This is because an orbit is being targeted, rather than a single point in the space.” (2014 paper, p. 16)

To sum up, the trajectory comes with a couple of benefits

  • If you want, you can have a 25 % reduction on the capture burn but the route is quite slow. If you speed it up you lose the propellant advantage, but the duration matches that of a Hohmann transfer in most cases.
  • Hence, you can find trajectories that match the Hohmann transfer at different times of the orbital dance.
  • If you do use less fuel, you save USD 50.000 per kg of propellant. The difference could mean using a smaller, less expensive rocket, creating additional savings.
  • Conversely, you could opt to carry more payload on the same spacecraft.
  • To be captured you only need to end up in a region in the vicinity of Mars comparable to the rim of that bowl (this is called the weak stability boundary).
  • The trajectory works for slow efficient cargo missions and for manned missions.
  • But most importantly, you can launch to MARS at least once every year.

Concrete benefits for Martian settlements

Imagine being able, instead of launching once every 26 months, launch payloads every 6 months, or for a large slice of this 26 period.  Supplying a colony with food, parcels would become possible on a predictable, regular schedule.

  • Just like in the old days, knowing exactly when mail arrives and having the anticipation grow to see it drop from the sky (seeing that Tesla finally land) will be a reason for even the most depressed Martian colonist to celebrate and break up the routine a bit.
  • Regular resupply also enable Mars colonies (or others) to be resupplied with smaller vehicles, as you do not need to plan for every contingency in the next 26 months. Because you can call for supplies of specific items and get them a year or more sooner than you would in a Hohmann cadence, not everything needs to be shipped with multiple redundancy on a big rocket. It is not perfect. But, as any pregnant woman will tell you, waiting 9 months is better than 26 months (the partner nods).
  • If trading propellant for payload is an option, you might get the time down to three to four months.
  • This cadence also falls in line with yearly reporting cycles for commercial businesses. The latter is quite important as it makes both expenditures and profits predictable. Seemingly small things like this are no longer trivial. CEO’s know they need to keep (institutional) investors with tightly managed risk port folio’s well-informed and happy.


Still here? You want to know more?

When one goes into detail, it turns out the math gets pretty complicated. Belbruno, et al., made it work for the Moon (Hiten 1991, JAXA), but at MARS, the math is more contrived. (He received the Humboldt Price in mathematics)

To use another metaphor, it really is comparable to trying to dance on a tight rope while being pulled from all sides. In this case planets are doing the pulling and the math is a many body problem, requiring fuzzy chaotic orbits.

Also, the visual metaphors in this article are not an accident. Belbruno strongly feels that his artistic paintings helped him to get a feel, intuition for the aesthetics and process of  the math.

Luckily Belbruno is a very persistent mathematician and, cooperating with other universities, has already found some solutions to the complex math, iterating in backward time, needed to calculate a region called the weak stability boundary,  the spacecraft needs to end up in.

Ballistic trajectory_Belbruno_weak stability boundary
Calculated shape of  weak stability boundary around MARS – the ship needs to remain in the blue area to enable capture for little or no energy compared to Hohmann  (Screenshot from Belbruno’s presentation below)

However, to extend the solutions to distances farther away from Mars, or arriving earlier in the orbit, more detailed simulations are required. In their 2014 papers they just desired to prove that what worked for the Moon in 1991, in principle also works for Mars.

Are you the one?

If you are math inclined, or would like to help tackle the challenge to calculate a good  regular ferry schedule for the next century, you can get a nice introduction in both the Hohmann transfer and the low energy capture from the inventor himself (below). Or you can follow this link to his 2014 paper he coauthored with Francesco Topputo from the Milan Politecnic University (*)

If you do so, please inform our crew, Belbruno and all others of the results. There are  prizes of many natures to be won.


(*) Earth–Mars Transfers with Ballistic Capture, Francesco Topputo, Edward Belbruno, DOI: 10.1007/s10569-015-9605-8;  as:arXiv:1410.8856,

For more information, Belbruno has a personal website:


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