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Just what the title says, I'm trying to understand how ships would be built for seas on low-gravity worlds. I've searched around a little for an identical question but surprsingly turned up empty-handed. Questions like, How would lower gravity affect motion?, Designing vehicles for different gravities, and How would it feel to sail a Rocheworld ocean? either focus on different aspects of low-gravity and seas or different types of vehicles like surface & air.


Wave properties & gravity

Intuitively, it takes more energy to create a wave of equal height on a planet with higher gravity than it does lower gravity. $U=mgh$, the higher the wave, the more potential energy, and potential energy depends on $g$. If $g$ goes down, $h$ goes up, when all else is constant. According to Wiki's page on wave energy, here, the relationship between mean energy of a wave (per unit area), $E$ [J/m^2], and its height, $H$ [m], is: $$E=\frac{1}{16}dgH^{2}, \quad H=\sqrt{\frac{16E}{dg}}$$

where $d$ [kg/m^3] is fluid density and $g$ [m/s^2] is gravitational acceleration. ($H$ is the height from crest to trough, not amplitude; $a=\frac{1}{2}H$) From the 2nd equation, you can see that $H\propto\frac{1}{\sqrt{g}}$. Assuming water's density is the same, the energy of a 1 m tall wave on Earth is ~612 J/m^2. Using the rearranged equation to solve for $H$, plugging in the same wave energy for 1/3 gravity, we get a wave height of ~1.7 m. The same imparted energy yields a taller wave.

Using Airy wave theory, for waves whose amplitudes are negligible compared to water depth, we can find these relationships: $$\omega\propto\sqrt{g}, \quad H\propto\frac{1}{\sqrt{g}}, \quad c\propto\sqrt{g}$$

The wave angular frequency, $\omega$, (inversely proportional to wave period) and wave speed, $c$, are proportional to the square root of gravity (grows smaller as gravity grows smaller), while the wave height is inversely proportional to gravity (grows larger as gravity grows smaller). The overall trend is that lower gravity means longer, taller, and slower waves.


Types of ships

There really isn't a general, spherical-cow approximation of ship. There are many shapes and sizes of sea-going vessels, designed for different cargos, speeds, and voyages. I'm interested in two rough categories of vessel: large barge-types, traveling large distances under engine-power, of large cargo capacity for transporting bulk containers of goods; and smaller, lighter cutters, single-mast, traveling short distances under wind-power, carrying a light cargo & crew complement.
Assume any modern construction materials and design principles are available.

With the information I have here, I reason that large barge-types would be taller and have a lower center of gravity to account for the on-average larger wave heights, which are able to impart more of their wave energy to the hull surface area. I've heard of something called "parametric rolling", in which wave conditions affect the stability of a ship when the wave encounter frequency is near the ship's roll frequency, inducing a kind of resonance which sometimes causes ships to lose containers over the side. I'm not quite sure how this effect would change under lesser gravities.
As for smaller cutters, a lower center of gravity keeping the ship more upright in various wind & wave conditions makes sense, I think. Or, maybe not. Cutters would be, in a sense, proportionally smaller to the waves they're facing. There's often a trend of things being taller in lower-gees: taller people, taller trees, etc. I'm just not sure what would change. I have very little personal experience with oceans and ships and I hope somebody here has a better idea than I do lol.


If it matters, the gravity + atmospheric properties of my specific world are:

  • $g$ = 3.34 m/s^2
  • $P_{sealevel}$ = 101,325 Pa
  • $d_{sealevel}$ = 1.26 kg/m^3
  • $T_{sealevel}$ = 280 K (note, mean ocean temp. will be cooler)

I ran the numbers to see if the sea water density would be drastically different under the different planetary conditions, and it wasn't. ~998 kg/m^3.

5th order polynomial approx., T°C: $$d_{T}\left(T\right)=999.83311+\sum_{n=1}^{5}a\left[n\right]T^{n},$$ $$a=\left[0.0752,-0.0089,7.36413\cdot10^{-5},4.74639\cdot10^{-7},1.34888\cdot10^{-9}\right].$$

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    $\begingroup$ (a) What's the point of using the spoiler function? (b) Why haven't you included an analysis of water friction? (c) Why is, "they'd be wider and heavier" not a sufficient answer? You've provided a breathtaking level of specific detail (thanks for that!) to support asking a pretty general question. $\endgroup$
    – JBH
    Commented Jul 17, 2022 at 22:18
  • $\begingroup$ @JBH (a) I didn't want the clutter of unnecessary math, (b) what's water friction? (c) I don't know, you tell me :) I'm not sure whether answer is simply "wider and heavier". From my short calculation in the Q, waves on my planet can be nearly 2x as tall, for the same mass of water. It feels on the threshold of being a fundamentally different environment. A good answer could perhaps address which aspects of the ships would change, why, and in what fashion. $\endgroup$
    – BMF
    Commented Jul 17, 2022 at 22:29
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    $\begingroup$ Friction, drag. Looks like the professionals like to call it hydrodynamic drag. It's the tendency for water to not appreciate things moving through it. Boat stability is complex math, but not complex ideas. Minimal width to minimize drag, deep enough keel to guarantee rudder control. Long enough to handle the waves. You'll have the ability to use wider (more stable) boats due to the low gravity, but they'll be longer due to the waves, and heavier to guarantee sufficient rudder to turn the boat. $\endgroup$
    – JBH
    Commented Jul 17, 2022 at 22:43
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    $\begingroup$ @fishinear I calculated density to be similar to Earth's at sea level, ~1.26 kg/m^3. To find it, I took the derivative of pressure over a small measurement of height and divided by gravity: $d=-\frac{1}{g}\frac{d}{dz}P$, [N/m^2]/m = [(kg*g)/m^3]/g = kg/m^3 $\endgroup$
    – BMF
    Commented Jul 18, 2022 at 16:59
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    $\begingroup$ @RobbieGoodwin Ah, we're talking about readers, here. I was talking about conveying info to people who read this Q. Yeah, no, readers will get none of the back-end, lest I scare them all away. $\endgroup$
    – BMF
    Commented Jul 18, 2022 at 22:56

5 Answers 5

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One parameter for ship design is hull speed or displacement speed which also depends on gravity.

$v_{hull}=\sqrt{\frac{L_{WL}g}{2\pi}}$

Where $L_{WL}$ is the length of the waterline in meters. You get more drag when the bow wave is equal to the waterline length of the vessel. Basically as you increase speed you get more drag from the interference of waves. Hull speed was known for a long time, the more modern description naval architects use is the Froude number. A smaller gravity gives smaller hull velocity or a larger Froude number.

Froude Number from Wikipedia

I think the implication is that you could think about having more hydrofoils or craft where you plane the hull. Basically minimize the area of the hull in contact with the water. Planing the hull can give a significant speed increase.

The other alternative is to make the ships longer to have them go faster. There are techniques to strengthen longer ships so they don't break, (for example the bow and stern supported by the waves, the middle unsupported) but with less gravity and the same strength of materials that may not be as much of a problem.

Ship stability is an issue, the period at which the ship would recover from a roll should get longer and recovery from being tilted would take a longer time with less gravity. So I am guessing that rolling, yawing and pitching would be more extreme. So with bigger waves and that maybe more sea sickness.

You might want to think through the center of gravity, center of buoyancy and metacentric height for the different ship designs you consider if you start changing the dimensions of the hulls significantly. In general the righting moment will be less with less gravity. Another stability criteria is the free surface effect if you have liquids on board. In lower gravity it seems like they could slosh around more.

Proportionally, it seems like the wind forces could remain the same for a given wind speed, and with less of a righting moment and bigger waves could make the dynamics more exciting. Even ships with out sails can get pushed around by the wind if they have enough area out of the water.

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    $\begingroup$ Regarding ship rocking, which would be the major trouble of all listed IMHO, I say trimarans would be more profitable in such a world, both in terms of resisting roll/pitch and speed, as three hulls could be made thinner in case of a fast ship while retaining carrying capacity. $\endgroup$
    – Vesper
    Commented Jul 18, 2022 at 11:56
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    $\begingroup$ A fully foiled vessel would completely sidestep the hull-speed problem. Given that the force required to fully lift the hull out of the water (like an AC72) is proportional to the local gravitational flux, a low grav environment could see foiling craft made of simple material, such as wood composites or even steel... $\endgroup$
    – Aron
    Commented Jul 19, 2022 at 3:00
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Submarines

With more dangerous waves we can do three things. Either take them head on, making ships that battle the wave. Accept them, using ships that can ride the waves. Evade them, going over or under the waves and never having to deal with them.

As your question doesn't include air transport in a conveniently lower gravity, we choose submarines. They can travel under the waves and be practically immune to weather and wave changes.

As the density of water seems to be similar we have more advantages. The water pressure doesn't increase as fast with depth thanks to the lower gravity. That helps us make submarines that are much larger. Cargo ship size perhaps.

You might think to take advantage of the lower gravity to make cargo ships that put the real life versions to shame. Yet in real life we have set maximum sizes for a reason. We can already build larger ships, but harbours have no are barely any way to accommodate them. And as we see they are even in this Earthly climate susceptible to the weather, losing cargo or even whole ships. In short, there is no reason to build larger boats. But the lower gravity might make cargo submarines affordable and much safer than wave cresting alternatives.

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  • $\begingroup$ Are the waves more dangerous? It seems to me like you'd get an exaggeration of the low-frequency components of ocean waves (for lack of a better term), as low-energy, low-frequency waves can achieve greater amplitude for the same wave energy in lesser gravity. It's not clear to me whether that's more dangerous, or just the same danger at a different scale. You have a good point, though. The best solution to a problem is avoiding the problem in the first place. $\endgroup$
    – BMF
    Commented Jul 18, 2022 at 14:31
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    $\begingroup$ @BMF my reasoning was in part on what you wrote, in part induction. The waves are not just higher, but steeper and slow. That means they have more time to get the ship to roll. More energy against the hull and possibly more energy from high waves on the ship itself. The weather also makes increasingly high waves. Waves of 5-8m at 8 Beaufort would become 8,5 to 13,5m by your numbers (didn't plug it into the formula. Need to double check). From this perspective it seems more dangerous, especially with same speed boats that still encounter about the same density water. $\endgroup$
    – Trioxidane
    Commented Jul 18, 2022 at 14:52
  • $\begingroup$ that sounds about right. You could possibly also think about it as the wave-energy for high-frequency terms being shifted "down" to low-frequency terms, (think of water like a spring and gravity as a spring constant) which UVphoton in their answer shows that low-frequency waves approaching 1.5 ship lengths, the "hump speed", are the worst to operate in, and may as a result be more common. $\endgroup$
    – BMF
    Commented Jul 18, 2022 at 15:07
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TV tropes has a trope about the hardness of science fiction. BMF should ask themself how "hard" they want their story to be.

https://tvtropes.org/pmwiki/pmwiki.php/SlidingScale/MohsScaleOfScienceFictionHardness

Here is a frame challenge:

Can a world with a surface gravity of only 3.34 meters per second per second have an atmospheric presssure of 101,325 Pa, or one Earth atmosphere?

Earth has a surface gravity of 9.80665 meters per second per second (1 g), and a sea level atmospheric pressure of 101,325 Pa, or one Earth atmosphere. So your world has a surface gravity of 0.34 g.

Actually the amount of time that a world can retain whatever atmosphere it may produce or acquire depends primarily on its escape velocity, and not on its surface gravity. Since the escape velocity of a world is calculated using a different formula than its surface gravity, the diiference in surface gravity between two worlds will usually not be the same as the difference in their escape velocities.

I note that Mars, and Mercury, have low surface gravities and atmospheres much less dense than 1 Earth Atmosphere.

Mars has a surface gravity of 3.72076 meters per second per second (0.379 g), and Mercury has a surface gravity of 3.7 meters per second per second (0.377 g). Both are somewhat more than your desired world with 0.34 g.

The escape velocity of Earth is 11.186 kilometers per second. The escape velocity of Mars is 5.027 kilometers per second (0.449 that of Earth) and Mercury's is 4.25 kilometers per second (0.349 that of Earth). In those cases Mars has a higher relative escape velocity than surface gravity, and Mercury has a lower relative escape velocity than surface gravity.

So it is possible to design a planet with an escape velocity relatively higher than its surface gravity.

Habitable planets for Man Stephen H. Dole, 1964, discusses the escape velocity necessary for a planet to retain a dense atmosphere for geological eras of time and for oxygen to accumulate in its atmosphere.

https://www.rand.org/content/dam/rand/pubs/commercial_books/2007/RAND_CB179-1.pdf

Planets lose atmosphere when it either combines with other materials to be come solid instead of gas, or escapes into outer space. Gases escape into space from the topmost layers of the palntary atsmsphere, their exospheres, where gases are very thin and very hot. The temperatures in the exospheres of worlds are usually several times the temperatures at the sufaces of the worlds. And of course the hotter the gas particles are, the faster they move, and the more likely they are to move faster than the escape velocity of their world and so be lost into space.

ONpages 34 and 35 Dole discusses the ratio between the planetary escape velocity and the root-mean-square velocity of a gas in the exosphere. How long it takes the original amount of that gas in the atmosphere to be reduce to 1/e, or 0.3678796 of the original amount. depends on the ratio of the escape velocity divided by the root-mean-square of the gas's exosphere velocity.

Table 5 on page 35 shows that if the ratio is 1 or 2, the time to reduce the atmosphere by that amount is zero.

If the ratio is 3, the time to reduce the atmosphere is a few weeks.

If the ratio is 4, the time to reduce the atmosphere is several thousand years.

If the ratio is 5, the time to reduce the atmosphere is about ahundred million years.

If the ratio is 9, the time to reduce the atmosphere is approximately infinite.

Thus a comparatively minor change in the ratio between the escape velocity of a world and the root-mean-square of gas in its exosphere can make the difference between the planet loosing all its atmosphere almost instantly and retaining its atmosphere almost infinitely.

Of course in some cases atmospheric gaess can be replaced by sources on the planet as fast as they are lost into space. But obviously it would be a lot more probable for natural forces to replace atmosphere as fast as it is lost if it takes 100 million years for the atmosphere to be reduced to 0.3678 of its orginal amount than if it takes only 100 million seconds or about 3.2 years.

And the stellar wind can cause a planet to lose atmosphere if the planet doesn't have a magnetosphere to divert charged particles away from the exosphere.

But the escape velocity is much more important.

On page 54 Dole noted that the temperatures in Earth's exosphere are between 1000 K and 2000 K. Dole said that if the temperaturs in the exosphere of a planet never exceedes 1000 K, and the planetary surface is still warm enough for liquid water, a planet could retain a long term oxygen rich atmosphere if its escpe velocity was at least 6.25 kilometers per second (5 times 1.25 kilometes per second).

According to Dole's figures, based on his formula for relating the mass, density,a nd radius of a planet, a planet with an escape vleocity of 6.25 kilometer per second would have a mass 0.195 that of Earth, a radius of f.63 Earth radius, and a surface gravity of 0.49 g, considerably higher than the 0.34 g you desire.

If there can be considerable variation in the overall density of a habitable planet, it might be possible to design a planet which as a surface gravity as low as 3.34meters per second per second (0.34 g), and an escape velocity as high as 6.25 kilometers per second, 0.5587 that of Earth.

But there are limitations to how high or low you can make the average density of a planet which has a solid surface partially covered by oceans and partially above the oceans, as on Earth.

Earth has an average density of 5.514 grams per cubic centimeter.

Our solar system was four terrestrial type planets, of which Earth is the densest, and four giant planets which are largely composed of hydrogen and helium gases and so have low average densities.

Neptune has the highest average density of the known giant planets, at 1.638 grams per cubic centimeter, 0.297062 that of Earth. And Neptune doesn't have anything like a solid surface.

imagine a world with the same average density as Neptune but the same mass as Earth. To have the same mass as Earth with 0.297062 the density it woudl have to have 3.3663006 times the volume of Earth. 1.49872 is the cube root of 3.3663673, which is close enough. Since the average radius of Earth is 6,671.0 kilometers, the average radius of such a planet would be about 9,548.3451 kilometers.

According to this online surface gravity calculator:

https://philip-p-ide.uk/doku.php/blog/articles/software/surface_gravity_calc

The planet would have a surface gravity of 0.45 g, higher than you desire.

According to this online escape velocity calculator, the planet would have an escape velocity of 9.137 kilometers per second, about 0.8168 that of Earth.

Suppose that a planet has an average density of about 2.757 grams percubic centimeter, about half that of Earth. If such a planet had the mass of Earth, it would have twice the volume of Earth. Since 1.26 is the cube root of 2.00376, that is close enough. That planet would have a radius of about 8,027.46 kilometers.

Such a planet would have a surface gravity of 0.63 g, and an escape velocity of 9.965 kilometers per second.

So it looks like you need to use a planet less dense. But I am not sure that a world less dense than that could exist without having a lot of liquid, enough liquid that its solid surface would be covered by hundreds of kilometers or miles of liquid. If you want native land dwelling people on that planet to build boatsout of wood from trees on land, that would be no good.

There have been a number of other posts in various places where I have tried to calculate worlds which surface gravities as low as possible and escape velocities high enough to retain atmospheres for billions of years.

And maybe I can link to them.

Part Two: A titanic solution.

There is a world in our solar system which has an atmmosphereic surface pressure even higher than that of Earth, despite having a very low escape velocity, much lower than an habitable world should have. It is Titan, the largest moon of Saturn. Titan receives only about 1 percent as much energy from the Sun as Earth does, so it is much colder in Titan's exosphere than in Earth's exposphere, so Titan's low escape velocity of 2.639 kilometersper second, 0.2359 that of Earth, is enough to retain its atmosphere.

Of coure that means the surface of Titan is very cold, so water is frozen rock hard. The temperaturea and pessure on Titan is near the triple point of methane, so methan can be, and is, solid, liquid, and gaseous on Titan. Titan is believed to have many methane lakes of various sizes; The largest, Kraken Mare, is about 900 kilometers long, and even the smallest known ones are large enough for boating.

I note that the surface gravity of Titan is 0.632 kilometers per second per second, less than you desire, and that liquid methane may have different properties than liquid water. But hums owuld able to survive on Titan with breathing apparatus and temperature suits, and it might possibly be that their are lifeforms on Titan and/or similar worlds in other star systems, lifeforms which use liquid methane as their solvant instead of liquid water.

Part Three: The smallest possible habitable worlds.

Since Earth type life forms use liquid water, and liquid water requires a dense anough atmosphere, a planet has to be able to retain a dense enough atmospehre for geological eras of time to retain liquid water and thus have water oceans and lakes for the life forms. And you also want a planet with water oceans and lakes to sail on.

Recently some scientists have imagined how some types of planets could retain atmosphere andthus liquid water despite having much lower mass and escape velocity than previously believed.

Under some conditions, a planet as small as 0.0268 Earth mass could retain an atmosphere and liquid water for geological eras of time, according to their calculations.

https://earthsky.org/space/small-rocky-exoplanets-can-still-be-habitable/

But what's the difference from worlds used in earlier calculations, and what is the catch?

The discussion is about water worlds of low mass.

The orginal article: https://iopscience.iop.org/article/10.3847/1538-4357/ab2bf2

Says:

We assume that the low-gravity waterworld has a pure water vapor atmosphere and a water reservoir fixed at 40% of the planet's total mass.

So the solid surfaces of those planet and moons are likely to be at the bottoms of oceans hundreds or thousands of kilometers deep. Such planets would not have native land dwelling intelligent beings to build boats out of wood, and there wouldn't be any trees to build old fashioned boats out of.

But space travelers from other worlds could bring their own boats and ships to such worlds to sail in their oceans. The study assumed that the atmospheres would be all water vapor from the oceans.

But ultaviolent light would beak up water vapor molecules into hydrogen and oxygen atoms. The hydrogen atoms would escape much faster than the oxygen atoms, so oxygen might accululate in the atmosphere and which might eventually become breathable.

And possibly the oceans of that world might develop floating islands of some type where land plants and animals might evolve, possibly including people who make baot sout of tree equivalents.

Part Four: Worlds With Roofs.

A world which has too low an escape velocity to retain an atmosphere naturally might be terraformed to have an artificial atmosphere and the terraformers might put a roof over the world to hold the atmosphere in.

There are several different hypothetical types of "shellworld" maegastructures. Two theoretical types of shellworlds are:

An inflated canopy holding high pressure air around an otherwise airless world to create a breathable atmosphere.5 The pressure of the contained air supports the weight of the shell.

Completely hollow shell worlds can also be created on a planetary or larger scale by contained gas alone, also called bubbleworlds or gravitational balloons, as long as the outward pressure from the contained gas balances the gravitational contraction of the entire structure, resulting in no net force on the shell. The scale is limited only by the mass of gas enclosed; the shell can be made of any mundane material. The shell can have an additional atmosphere on the outside.[5][6]

The first type of shell world could have lakes and oceans on its surfice beneath the shell, while the second type could have water as well as atmosphere within its shell. Enough mass of water would form a sphere surrounded by the gas.

Part Five: Conclusion.

Asking for a world with surface water to sail on, an Earth like atmosphere, and a surface gravity of 3.34 meters pr second per second, causes some problems in designing the world which the OP didn't anticipate.

I hope that my suggested solutions to the problems will be helpful.

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    $\begingroup$ I knew something like this was coming. It always comes, when I give out extra details, and I can't say why. I should start throwing in a disclaimer or something. Look, I appreciate your effort, but the world-specific details I threw into the Q are accurate for my world; however, you know nothing else about it. In the Worldbuilding scene, it's not a safe assumption that all worlds are naturally forming and have withered in geological abandon for eons. I didn't include too far extraneous details about the actual structure of the world because I could not see them being relevant. $\endgroup$
    – BMF
    Commented Jul 18, 2022 at 7:05
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    $\begingroup$ I will say, you have a useful, competent, and well-cited answer, but I'm not sure your frame challenge really exists here. As an aside I threw in air conditions at sea level, and you attached those conditions to a hypothetical planet and analyzed the likelihood of its natural formation and persistence through geological time. I feel like it's out of place here, not in the spirit of my original Q. $\endgroup$
    – BMF
    Commented Jul 18, 2022 at 7:18
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    $\begingroup$ I don't want to see it deleted though. It's useful for the question it sets out to answer, but I feel like it belongs to a better-suited Q. $\endgroup$
    – BMF
    Commented Jul 18, 2022 at 7:20
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    $\begingroup$ Another reason why I think your frame challenge doesn't exist is that, even if you're right and I "didn't anticipate" the problems in my world, it doesn't change the Q in any way, which is "How would low-gravity seas affect ship design?" A frame challenge addresses the premise of the Q. $\endgroup$
    – BMF
    Commented Jul 18, 2022 at 7:35
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    $\begingroup$ @BMF science-based tag attracts people who like to calculate, so whenever you give them data, they go fiddle with it with quite special outcomes, often resulting in a described fictitional world to violate some laws of physics, sometimes sinking the entire proposed configuration. This is normal here, it will come. $\endgroup$
    – Vesper
    Commented Jul 18, 2022 at 12:44
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Waves will be larger, but not steeper. So wavelength will still be 7 to 10 times wave height. (~7 is where waves start to break.

Since a wave will be taller for a given energy, and wave height is a factor in how fast wind energy is transfered to wave energy, I suspect that waves will kick up a lot faster, and that dangerous waves will form on a shorter reach (distance the wind has over the water)

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  • $\begingroup$ Oh, hmm, that's a good point. Air density is close to Earth's, so I'm probably looking at close to ordinary wind forces but transferring more energy over more wave surface area. That's good to know, thanks $\endgroup$
    – BMF
    Commented Jul 23, 2022 at 23:51
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Well, sailboats are out because there isn't much righting moment to resist the heeling moment of the sails. Or the boat's have to be huge. There is a scaling law where RM scales with size a little bit faster than heeling moment does.

Sea wave size and shape and velocity are are all totally different. Linear wave theory, which is pretty useful on Earth, won't work as well. Boat wakes will also be very different. There won't be any "mill ponds", all the gravitational free fluid surfaces will look like a the inside of a washing machine.

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    $\begingroup$ Correction, KEEL-boats are out. There are many different configurations that aren't keel boats. For example, a daggerboard would generate righting moment dynamically using hydrodynamics. Out riggers could provide righting moment using "buoyancy". A canting keel could even be used to make a keel boat, maybe... $\endgroup$
    – Aron
    Commented Jul 19, 2022 at 3:06
  • $\begingroup$ Catamarans. Twin hulls are more stable, but cargo storage is a PITA, and there are BIG twisting forces on the deck. $\endgroup$ Commented Jul 26, 2022 at 2:24

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