Let’s say you’ve got nothing else going on, and you decide to launch a mission to Mars, as you do. Technically, assuming you had the rocket and the launch pad and the range clearance, the time and money, etc, you could do that tomorrow some time. BUT, this is a little like Steven Wright’s assertion that “Everywhere is walking distance if you have the time.”

No, it stands to reason that unless you REALLY want to burn a lot of time and money, you wait for that period where the geometrical arrangement between Earth and Mars is most efficient. The desired path your spacecraft will take is an elliptical orbit around the Sun (the Sun sits at one of the foci of the ellipse), where Earth is at one vertex (end of the major axis), and Mars WILL be at the other when you arrive. There are lots of subtle variations in the orbits between Earth and Mars that put some devilish details into these calculations – but in gross terms, this is the situation you want. A big ellipse with Earth at one end, and Mars at the other. This is a classic minimum-energy orbital maneuver called a Hohman transfer.

Now, orbital mechanics being what they are, an orbit of this particular size and shape takes a particular speed profile (or, more correctly, a particular energy level that trades kinetic energy in speed for potential energy in distance to fall back toward the Sun). While higher mass equates to more energy, and thus more fuel burned to get to this condition, the actual SPEED depends ONLY on the shape of your orbit, and the mass of the Sun, NOT the mass of your spacecraft (your own mass conveniently drops out of the equations). Engineers tend to speak in terms of “Delta-V”, or change in velocity, as they focus on target speeds. Fuel burn required to achieve this change is certainly higher for a space station than a cubesat, but the target speed is the same.

If you travel too fast, you’ll overshoot Mars’ orbit. Too slow, and you won’t get out far enough before the Sun pulls you back in. You CAN elect to carry a little extra speed to give you more room for error, and plan on a few maneuvers to slow you down and enter orbit around Mars – but every little bit of extra energy you’re carrying requires burning more fuel (and more money) TWICE – once to get moving, and again to slow down – so you don’t want to overdo it. Knowing how fast you have to go (and knowing you’ll launch WITH Earth’s orbital motion, and WITH its rotation to give you as much extra oomph as possible) you can calculate the time it will take for your spacecraft to cover this specific elliptical path. Again, there are some devilish details in differing orbital speeds during certain times of the year, as both Mars and Earth have elliptical orbits themselves, but if we approximate circular orbits…. It turns out that covering that 180 degrees of orbital transition between Earth and Mars takes 8 months or so (actually, 259 days).

So, the only question remaining is, where does Mars have to be NOW so that it and my spacecraft will be in the same place after its seven-month transit? Let’s see here, a year on Mars is 687 Earth days. The 259 days of travel is 37.7% of a Martian year, or about 136 degrees out of a circle. So Mars has to cover 136 degrees in the time it takes Earth to cover 180… ergo, when Mars is 44 degrees AHEAD of Earth, start the countdown!!

NASA JPL, Instructional material on Mars transfer geometries
(Public Domain in the USA)

It is interesting to note that the path your spacecraft takes to make this rendezvous will not be the shortest distance, nor will it necessarily take the shortest amount of time. You can visualize a light speed craft making the trip more or less when the two planets are at their closest approach, and then taking only 20 minutes or so to make the trip. But…a soft landing under these conditions might be difficult, to say nothing of the required advancements in rocket technology to get it moving that fast. It is ALSO interesting to note that in the 8 months of travel time, Earth will move MORE than 180 degrees in its orbit (i.e. more than half a year). So you launch when Mars is ahead, and then during the journey Earth PASSES Mars, and when your spacecraft reaches Mars, it is now ~75 degrees BEHIND Earth.

With all that as background, I can finally get to the POINT of this little story. It turns out, you see, that everybody can do this same math, and that everybody is (almost) equally concerned with money and the availability of rocket fuel.

The last time Mars was 44 deg ahead, it was July 2020. During that time, we had an international flurry of Mars missions being launched.

First up was the Hope Orbiter, launched by the United Arab Emirates Space Agency on July 19, 2020 from the Tanegashima launch complex in Japan. Hope is the first Mars mission launched by an Arab country, and has become a symbol of national pride for the UAE. The Hope orbiter contains an array of broad-spectrum sensors that will seek to gather data on why and how Mars lost its atmosphere.

Next, China launched its Tianwen-1 lander mission on July 23, 2020 (4 days later). Tianwen-1 consists of an orbiter AND a lander, and will study geology, the Martian atmosphere and ionosphere.

Finally, the USA launched Mars 2020, carrying the Perseverance Rover, on July 30, 2020 (A week after Tianwen-1). Perseverance will land at Jezero Crater on Mars via a sky-crane technique (proven with the Curiosity lander in 2012), and will collect soil samples for a future return mission, continuing the search for remnants of life on Mars. Perseverance also carries a helicopter called Ingenuity, which will perform the first free-flight unmanned surveillance of the planet from within the Martian atmosphere.

Artist’s Rendering of Perseverance Rover, and Ingenuity Helicopter, NASA (Public Domain in the USA)

All three of these missions took off within the same general launch window, and are all hurtling toward Mars at the same time. In the next few weeks, they will all arrive, at roughly the same time. Hope will begin an orbital insertion on Feb 9. Tianwen will settle into orbit the very next day, Feb 10. And finally, Perseverance will go straight to landing Perseverance on Feb 18.

It’s a veritable traffic jam up there! And we’re going to have a very exciting few weeks, watching each of these missions (hopefully) achieve safe arrival and insertion into the Martian environment.

And as far as the NEXT opportunity goes…. some extension of the math described above reveals that by the time Earth zips around its faster inner orbit and catches up to Mars again, placing it in that 44 deg-ahead window, approximately 2.2 Earth years have to go by. That means that the next Mars mission launch window will occur between late August and early October, 2022. During that window, if all goes well, a joint effort between European Space Agency (ESA) and Russia’s Roscosmos will launch the ExoMars mission carrying the Rosalind Franklin lander on another Hohmann transfer flight.

Math is great, isn’t it?

Get Out There

Troy

http://www.flying-squirrel.org

One thought on “Interplanetary Superhighway (Math, Geometry, and Mars Missions)

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