Let’s Put a Sailboat on Titan

The large moons orbiting the gas giants in our solar system have been getting increasing attention in recent years. Titan, Saturn’s largest moon, is the only natural satellite known to house a thick atmosphere. It’s surface, revealed in part by the Cassini probe, is sculpted by lakes and rivers. There is interest in exploring Titan further, but this is tricky from orbit because seeing through the thick atmosphere is difficult. Flying on Titan has been discussed around the web (sometimes glibly), and this was even one of the subjects treated by the immensely popular comic, XKCD.

However, there remains the problem of powering propulsion. The power requirements for flight are quite minimal on Titan, so solar wings might work. But Titan also presents an alternative: sailing.

Images from the Cassini mission show river networks draining into lakes in Titans north polar region. Credit: NASA/JPL/USGS.
Images from the Cassini mission show river networks draining into lakes in Titans north polar region. Credit: NASA/JPL/USGS.

With all those lakes and rivers, exploring Titan with a surface ship might be a great way to see much of the moon. The vehicle wouldn’t be sailing on water, though. The lakes on Titan are composed of liquid methane. The challenge is therefore making the vessel buoyant: liquid methane is only 45% as dense as liquid water. This means we would need a lot of displacement. A deep, hollow hull could do this, however, and it turns out that the liquid methane has an advantage that helps make up for the low density: it is much less viscous than water.

Reynolds number is proportional to the ratio of density to viscosity, and it turns out that friction drag on a hull is inversely proportional to Re. While Titan’s seas and lakes have only 45% the density of water, they also have only 8% of the viscosity. This means that the Titan sailing vessel would only experience about 26% of the friction drag as its Earth equivalent. [Yacht designers have found that the friction drag is about equal to 0.075/(log(Re)-2)^2)]. That leaves us room to make the hull deeper (important to compensate for the density as above), and longer (if we want a longer waterline, which will make the bow waves longer and improve maximum speed).

The sail itself would get less wind, on average, on Titan than Earth. Average wind speeds on Titan seem to be about 3 meters/s, according to Cassini, though it might be higher over the lakes. Average wind speed over Earth oceans is closer to 6.6 meters/s. But, the Titan atmosphere is also about 4x denser than Earth’s, and both lift and drag are proportional to fluid density. All told, this means that the total fluid force on the sail will be about 83% of what you’d get on Earth, all else being equal, which could be sufficient. There would be a premium on sail efficiency and size, and so we might have to take advantage of the low-friction hull to examine shapes with more stability that can house a larger, taller (and presumably high aspect ratio) sail.

This is all quite speculative, of course, but it provides a fun exercise and perhaps provides inspiration as we imagine tall-sailed robotic vessels silently cruising the lakes of Titan.

Titan Mare Explorer. Image credit: NASA/JPL
Titan Mare Explorer. Image credit: NASA/JPL

One concept for a boat on Titan has already been proposed: the Titan Mare Explorer (TiME) would send a floating high-tech buoy to land in a methane sea on this moon of Saturn to study its composition and its interaction with the atmosphere. But this Discovery-class mission concept was nixed in favor of sending the InSight lander to Mars.

But with all the recent discoveries on Titan by the Cassini spacecraft — things like lakes, seas, rivers and weather and climate patterns that create both fog and rain — a mission like this will be given more consideration in the future.

What’s the Best Design for a Flying Mars Robot?

Building a flying vehicle for Mars would have significant advantages for exploration of the surface. However, to date, all of our surface exploring vehicles and robotic units on Mars have been terrestrial rovers. The problem with flying on Mars is that the Red Planet doesn’t have much atmosphere to speak of. It is only 1.6% of Earth air density at sea level, give or take. This means conventional aircraft would have to fly very quickly on Mars to stay aloft. Your average Cessna would be in trouble.

But nature may provide an alternative way of looking at this problem.

The fluid regime of any flying (or swimming) animal, machine, etc. can be summarized by something called the Reynolds Number (Re). The Re is equal to the characteristic length x velocity x fluid density, divided by the dynamic viscosity. It is a measure of the ratio of inertial forces to viscous ones. Your average airplane flies at a high Re: lots of inertia relative to air stickiness. Because the Mars air density is low, the only way to get that inertia is to go really fast. However, not all flyers operate at high Re: most flying animals fly at much lower Re. Insects, in particular, operate at quite small Reynolds numbers (relatively speaking). In fact, some insects are so small that they swim through the air, rather than fly. So, if we scale up a bug-like critter or small bird just a bit, we might get something that can move in the Martian atmosphere without having to go insanely fast.

A potential design of an 'Entomopter' from Georgia Tech Research Institute
A potential design of an ‘Entomopter’ from Georgia Tech Research Institute
We need a system of equations to constrain our little bot. Turns out that’s not too tough. As a rough approximation, we can use Colin Pennycuick’s average flapping frequency equation. Based on the flapping frequency expectations from Pennycuick (2008), flapping frequency varies roughly as body mass to the 3/8 power, gravitational acceleration to the 1/2 power, span to the -23/24 power, wing area to the -1/3 power, and fluid density to the -3/8 power. That’s handy, because we can adjust to match Martian gravity and air density. But we need to know if we are shedding vortices from the wings in a reasonable way. Thankfully, there is a known relationship, there, as well: the Strouhal number. Str (in this case) is flapping amplitude x flapping frequency divided by velocity. In cruising flight, it turns out to be pretty constrained.

Our bot should, therefore, end up with a Str between 0.2 and 0.4, while matching the Pennycuick equation. And then, finally, we need to get a Reynolds number in the range for a large living flying insect (tiny insects fly in a strange regime where much of propulsion is drag-based, so we will ignore them for now). Hawkmoths are well studied, so we have their Re range for a variety of speeds. Depending on speed, it ranges from about 3,500 to about 15,000. So somewhere in that ballpark will do.

There are a few ways of solving the system. The elegant way is to generate the curves and look for the intersection points, but a fast and easy method is to punch it into a matrix program and solve iteratively. I won’t give all the possible options, but here’s one that worked out pretty well to give an idea:

Mass: 500 grams
Span: 1 meter
Wing Aspect Ratio: 8.0

This gives an Str of 0.31 (right on the money) and Re of 13,900 (decent) at a lift coefficient of 0.5 (which is reasonable for cruising). To give an idea, this bot would have roughly bird-like proportions (similar to a duck), albeit a bit on the light side (not tough with good synthetic materials). It would, however, flap through a greater arc at higher frequency than a bird here on Earth, so it would look a bit like a giant moth at distance to our Earth-trained eyes. As an added bonus, because this bot is flying in a moth-ish Reynolds Regime, it is plausible that it might be able to jump to the very high lift coefficients of insects for brief periods using unsteady dynamics. At a CL of 4.0 (which has been measured for small bats and flycatchers, as well as some large bees), the stall speed is only 19.24 m/s. Max CL is most useful for landing and launching. So: can we launch our bot at 19.24 m/s?

For fun, let’s assume our bird/bug bot also launches like an animal. Animals don’t take off like airplanes; they use a ballistic initiation by pushing from the substrate. Now, insects and birds use walking limbs for this, but bats (and probably pterosaurs) use the wings to double as pushing systems. If we made our bots wings push-worthy, then we can use the same motor to launch as to fly, and it turns out that not much push is required. Thanks to the low Mars gravity, even a little leap goes a long way, and the wings can already beat near 19.24 m/s as it is. So just a little hop will do it. If we’re feeling fancy, we can put a bit more punch on it, and that’ll get out of craters, etc. Either way, our bot only needs to be about 4% as efficient a leaper as good biological jumpers to make it up to speed.

These numbers, of course, are just a rough illustration. There are many reasons that space programs have not yet launched robots of this type. Problems with deployment, power supply, and maintenance would make these systems very challenging to use effectively, but it may not be altogether impossible. Perhaps someday our rovers will deploy duck-sized moth bots for better reconnaissance on other worlds.