Calculating Earth to Mars launch windows

[Adam Reynolds]

I was trying to find out exactly what date The Martain would have to take place. We know that it must be a time when Mars it at its opposition point right around November(because a major plot point was that they had fresh potatoes for Thanksgiving).

According to author Andy Weir, he left this as an exercise for the reader and there actually is a valid date within the 21st century that would allow this to occur.

How exactly would you calculate when this would be? I haven't found a list of opposition dates that would go into the 2030s and beyond that would allow one to find this without doing a large amount of complicated mathematics.

[SpottedKitty]

Here's a Google search that comes up with a variety of ways of looking at essentially the same kind of data. The "Cosmic Train Schedule" page has a table of links including Earth-to-Mars, which gives launch windows for about the next 300 years.

I haven't read the book, so I'm a bit dubious about the "fresh potatoes" thing. A standard minimum-energy coasting transfer orbit from Earth to Mars will always take the same time, about 8-9 months with launch opportunities about every 26 months. The only way round that is a spaceship with a constant-thrust drive, e.g. Ion or Nuclear Thermal; is this gone into at all in the book? The only way to get fresh food on a trip like that is to grow it on the way out.

Incidentally, this has interesting consequences for mission planning for any manned Mars trip — when you get there, you must either leave almost at once, or wait until the next Mars-to-Earth window opens.

[Parallax]

I do believe that in the film, the main character actually grows said potatoes once he's stranded on Mars.

[Eternal_Freedom]

He grows them from fresh vaccuum-sealed potatoes they'd brought with them for Thanksgiving, so that gives you an approximate end date for their planned 30-day mission.

[Simon_Jester]

I haven't read the book, so I'm a bit dubious about the "fresh potatoes" thing. A standard minimum-energy coasting transfer orbit from Earth to Mars will always take the same time, about 8-9 months with launch opportunities about every 26 months.

The only way round that is a spaceship with a constant-thrust drive, e.g. Ion or Nuclear Thermal; is this gone into at all in the book?

Yes; it is explicitly stated that the NASA spacecraft Hermes which delivers the astronauts to (and from) Mars is a constant-thrust vessel. Although total delta-V appears to still be limited, as illustrated by the limitations on the Rich Purnell maneuver listed later in the story.

The only way to get fresh food on a trip like that is to grow it on the way out.

In this case, the potatoes don't have to be fresh- they just have to be frozen and then capable of sprouting again when left out in the open.

Incidentally, this has interesting consequences for mission planning for any manned Mars trip — when you get there, you must either leave almost at once, or wait until the next Mars-to-Earth window opens.

The Ares missions (Watney's mission in the book/movie is the third) fit this profile; the astronauts are supposed to spend thirty days on the surface. This is why it is so difficult for Watney to survive when his crew leaves him for dead- he's left with only the supplies needed to support a six-person expedition for thirty days, and needs to somehow survive for years.

[Eternal_Freedom]

If the normal low-energy transfer orbit takes 8-9 months, why does the film keep referring to them spending 400+ days in flight? And that's before they try the Purnell Maneuver that is.

[Simon_Jester]

If you mean the proposed Iris probe launches to send a resupply package to Watney...

Probably because they are NOT launching those at the ideal launch windows? The most optimal time for a fast, low-energy launch is during the window. At other times, you can still send a probe to Mars- but it will take longer, or require a lot more delta-V to get there in the same amount of time.

[SpottedKitty]

Probably because they are NOT launching those at the ideal launch windows? The most optimal time for a fast, low-energy launch is during the window. At other times, you can still send a probe to Mars- but it will take longer, or require a lot more delta-V to get there in the same amount of time.

Yes, the links I gave upthread were all for optimum minimum-energy transfer orbits (IIRC called Hohmann orbits). Go too far outside the launch windows, and the only way to get there from here "right now" is to use one or more gravitational slingshots to give your velocity a good hard kick at each planetary encounter. 400+ days sounds like a credible guesstimate for just past the end of an Earth-Mars window, using a single Venus or Venus-Earth encounter to catch up. And then the whole thing has to be recalculated for the return trip, because all the planets have moved, and you're even further out of the optimum Mars-Earth window.

Beyond that, things get complicated. It is Rocket Science in this case...

[Eternal_Freedom]

Ah, I'd forgotten that part, my apologies.

It's funny really, but I always had the impression that the launch windows for the Hohmann were the only times you could launch a mission, rather than just the ideal ones. Learn something new every day I guess.

[Simon_Jester]

If you can launch a rocket fast enough, you can go from Earth to Mars any time you like... eventually. It's just that going quickly and efficiently requires a great deal of timing.

[Eternal_Freedom]

Yeah I get that. I suppose my brain just latched onto "launch windows" and assumed they were fixed and you could use them and only them. Oh well.

[Adam Reynolds]

So I actually found the answer to my question indirectly. Promotional material for the film indicates that Whatney and Johanssen both joined NASA in 2032, indicating that the mission had to be in either 2033 or 2036. 2033 is the logical date as the ideal point is in December, closer to the November window than 2036 which would wind up in February. The slightly later date would also justify the Purnell maneuver as it would mean that Mars was a closer distance for that to work.

[Simon_Jester]

Yeah I get that. I suppose my brain just latched onto "launch windows" and assumed they were fixed and you could use them and only them. Oh well.

Space is big and empty; you can launch stuff to Mars whenever you like if you have enough delta-V and don't care about efficiency.

Realistically it's going to be a looong time before we don't care about efficiency...

[Guardsman Bass]

Yep. IIRC the Delta-V to push straight off from Earth and make the crossing to Mars when they're at their closest is particularly high, around 250 km/s.

[Simon_Jester]

Although if you have that kind of delta-v to throw around, well... at a casual estimate, the trip time would be... Mars has an orbital radius of 1.6 AU, so at opposition* that's 0.6 times 93 million miles is about 56 million miles apart... divide by five and multiply by eight to get about 89 million kilometers... round to 90...

90 million kilometers, with most of the trip taken at a speed of 125 km/s (you need half to speed up and half to slow back down)... 1000 kilometers per eight seconds, one million kilometers per eight thousand seconds, which is approximately two and a quarter hours. A little over ten million kilometers a day, therefore...

...aaaand you'd make the trip in about nine days.

IF you had a rocket capable of high-acceleration burns with a total delta-v budget of 250 km/s, which is far, far, far in excess of anything I can imagine doing without, oh, high-end nuclear propulsion systems.


*For anyone who doesn't know, that's when Mars and the Sun are on opposite sides of the sky- i.e. when Mars and Earth are closest.

[Guardsman Bass]

It'd be fantastic. If it's some type of crazy-powerful nuclear-electric system, you might even be able to simulate gravity the whole way by accelerating/decelerating at a fraction of a g the whole trip (although you'd then have to redesign the crew habs to flip around after the halfway point, since the "direction" of the "gravity" would switch).

And of course, you'd be looking at drastically shorter trip times to the outer solar system as well - weeks and months instead of years. It would actually make a crewed mission out there potentially feasible without running the mission on a five-year-time-scale (or longer).

[Simon_Jester]

Well, an acceleration burn imparting 125 km/s would actually only translate to roughly 12500 seconds at one gravity. So the engine burns would compose only about one day out of your nine day flight- which means you could travel significantly faster than that if your engine performance were that good.

[Eternal_Freedom]

How long would it take for an Earth/Mars flight at opposition if you could maintain a 1g acceleration?

[Imperial528]

For the entire trip? Just under five days at opposition, ignoring the sun. Assuming I did my math correctly.

That is, modeling as a Braichistochrone trajectory, we get t=2*SQRT((401E+9m)/9.806m/s) which means t=~404E+3s, or about five days.

[Zeropoint]

although you'd then have to redesign the crew habs to flip around after the halfway point, since the "direction" of the "gravity" would switch

Well, technically, you would need to flip the habitat sections around. Since spaceships usually have only one set of main engines (to save on mass), you'd need to flip your engines around, too.

The generally accepted way to accomplish both of these goals at once is to turn the whole ship around.

[Guardsman Bass]

I'm confused on that one. The ship's accelerating outwards, pushing you towards the back of it. Then it flips, starts decelerating - are you being pushed against the "back" again, or towards the "front" in that situation?

[Eternal_Freedom]

You'll still be pushed towards the engine end, because that's where the thrust is. Even if you are slowing down, the force of acceleration is acting the same way as when you're speeding up.

[SpottedKitty]

I'm confused on that one. The ship's accelerating outwards, pushing you towards the back of it. Then it flips, starts decelerating - are you being pushed against the "back" again, or towards the "front" in that situation?

Acceleration is acceleration. It feels the same when you're on the ship, but when you're looking at it from outside you can see it's pointing in the opposite direction after turning over.

Think of it as two arrows drawn outwards from the ship. Before turnover, the small "acceleration" arrow is constant and the big "velocity" arrow is growing longer, and both are pointing in the same direction; forwards. After turnover, the "acceleration" arrow is the same length but it's swung round to point in the opposite direction, and the "velocity" arrow is shrinking.

[TimothyC]

I was trying to find out exactly what date The Martain would have to take place. We know that it must be a time when Mars it at its opposition point right around November(because a major plot point was that they had fresh potatoes for Thanksgiving).

According to author Andy Weir, he left this as an exercise for the reader and there actually is a valid date within the 21st century that would allow this to occur.

How exactly would you calculate when this would be? I haven't found a list of opposition dates that would go into the 2030s and beyond that would allow one to find this without doing a large amount of complicated mathematics.

Weir has, on facebook, stated that the window used is the 2035 window (SOL 1 is November 7th, 2035). This fits with the Hermes not fitting a perfect Hohmann Transfer because of the long thrust periods.
Facebook link.