# What is a reasonable day length for a planet with 4 times earth radius?

The fictional planet has a mass of 18 times Earth's and four times earth's equatorial radiusreference. This gives it a gravitational acceleration of 11.02m/s^2 (1.29G) and an escape velocity of 23.72km/s (1.9 times Earth's 11.18km/s). The surface area is 16 times that of our planet.

This is probably a former mini-Neptune with high metallicity and a rocky core which had the lighter gases stripped away and became a super-earthreference. It has oceans and an atmospheric composition comparable to Earth's now.

I'm afraid that, due to its massive size, a faster rotational speed would cause massive winds or water currents because of oceanic tides and the Coriolis effect.

Assuming it is in the goldilocks zone for its star and is inhabited by human-like sentient life and earth-like life, what is a reasonable day length for a planet this size, in Earth hours?

• This may be a duplicate of this question Aug 18, 2022 at 5:02
• @DawoodibnKareem the Answers there do not address the need of a gentler weather to support life. The linked question is broader and more generic. Aug 18, 2022 at 12:20
• @Mindwin If the planet has 4 times the radius of Earth it will have 64 times the volume. Aug 18, 2022 at 14:28
• @Mindwin. Have you calculated the density of your world? It would probably have ot have so much water that at Earth like temperatures it would totally covered by a world wide ocean tens, hundreds, or thousands of kilometers deep. it wouldn't have any land surface except for hypothetical floating islands or continents. Aug 18, 2022 at 14:41

You might wan tto decide what score you want your story to have on the Sliding Scale of Science Fiction Hardness.

If you are content with your story having a low score and very implausible science, then you don't have to read the rest of my answer.

But if you want your story to have as few scientific implausibilities and impossibilites as possible, you should read my discussion of the planet you have described in your question.

Even though your planet could have a rotation period which was good for life, other aspects of the planet you describe might make it have a global ocean with no land surface, amking it hard for intelligent beings who resemble humans to evolve there.

Part One:

I don't know what effect a planet's rotation speed will have of wind patterns and ocean currents.

Part Two:

A planet's rotation period can have other effects on its habitability.

Habitable Planets for Man, Stephen H. Dole, 1964, was a scientific study of the requirements for a planet to be habitable for human beings (and of course also for beings who hve the exact same enviromental reqirements). Other discussions of planetary habitability seem to be the more general case of habitabiity for some type of light water using life in general instead for humans in particular.

The planet in the question is supposed to be inhabited by human like sentient life, and so it should probably be habitable for humans.

On pages 41 to 46 Dole discusses how the rotation rate of planet can affect its shape and the surface agravity at different lattitudes. On pages 58 to 61 Dole discusses the effets of rotation rates on human habitability.

Dole believed that if a planet rotated too fast, the surface gravity at the equator would fall so low that the planet would be unstable. And Dole also believed that the longer the planet's rotatin period was, the hotter it would get in the days and the colder it wuld get in the nights. Eventually the daily temperature differences would get too great for habitabiity, and plants might die during the night for lack of light. On page 60 Dole writes:

just what extremes of rotation rate are compatible with habitability is difficult to say. Those extremes, however, might be estimated at, say, 96 hours (4 Earth days) per revolution at the lower end of the scale and 2 to 3 hours per revolution at the upper end. or at angular velocities where the shape becomes unstable because of the high rotation rate.

The planet is described in the question:

The fictional planet has a mass of 18 times Earth's and four times earth's equatorial radius. This gives it a gravitational acceleration of 11.02m/s^2 (1.29G) and an escape velocity of 23.72km/s (1.9 times Earth's 11.18km/s). The surface area is 16 times that of our planet.

If the planet has 4 times the radius of Earth it will have 16 times the surface area and 64 times the volume. With 18 times the mass in 64 times the volume, the planet will have an overall density of 0.28125 that of Earth. 0.28125 times Earth's overall density of 5.414 grams per cubic centimeter is 1.5508125 grams per cubic centimeter. That will be important later.

Earth's average radius is 6,371 kilometers, but its radius at the equator is 6,378.137 kilometers. So the planet would have an equatorial radius of 25,512.548 kilometers.

According to this online escape velocity calculator, https://www.calctool.org/astrophysics/escape-velocity the planet's escape velocity would be 23.73 kilometers per second. And its first cosmic velocity would be 16.78 kilometers per second. The first cosmic velocity is the orbital velocity, which varies wth distance from the planet.

Using this orbital velcoity calculator, https://www.omnicalculator.com/physics/orbital-velocity I find that the orbital velocity at the surface should be 16.77 kilomeers per second.

So objects on the surface travelling between 16.77 and 23.73 kilometers per second would be travelling faster than orbital velocity and rise out of the atmosphere and fall back into the atmosphere, heating it up. And the heat from the equatorial regions would spread over the planet, and probably make it too hot of liqud water using life. Objects at the equator travelling more than 23.73 kilometers per second would escape from the planet.

The planet would have an equatorial circumference of 160,299.93 klometers.

So if matter at the equator traveled at 16.77 kilometers per second it would take the planet 9,558.7316 seconds, or 159.31219 minutes, or 2.652031 hours, to make one rotation.

If matter at the equator traveled at 23.73 kilometers per second, it would take the planet 6,755.1592 seconds, or 112.58598 minutes, or 1.876433 hours, to make one rotation.

Of course if the planet rotated very fast, its shape would change as it became more and more oblate. So my calculations are not very accurate, but do give a rough indication of the shortest day lengths compatable with the planet's stabiity.

And the giant planets in our solar system rotate much faster than Earth, but much slover than that limit.

On pages 45 and 46, Dole discusse a mathematical relationship between the masses of the planets in our solar system and their rotation rates. HOwever, since then there have been major discoveries about the rotation rates of Mercury and Venus, and the masses and rotation rates of PLuto, other dwarf planets, moons of the giant planets, asteroids, trans Neptunian objects (TNOs), and exoplanetsorbiting other stars.

Most minor planets have rotation periods between 2 and 20 hours.1 As of 2019, a group of 887 bodies – most of them are stony near-Earth asteroids with small diameters of barely 1 kilometre – have an estimated period of less than 2.2 hours.

https://en.wikipedia.org/wiki/List_of_fast_rotators_(minor_planets)

The periods given in this list are sourced from the Light Curve Data Base (LCDB),3 which contains lightcurve data for more than 15,000 bodies. Most minor planets have rotation periods between 2 and 20 hours.1 As of 2019, a group of approximately 650 bodies, typically measuring 1–20 kilometers in diameter, have periods of more than 100 hours or 41⁄6 days. Among the slowest rotators, there are currently 15 bodies with a period longer than 1000 hours.1 According to the Minor Planet Center, the sharp lower limit of approximately 2.2 hours is due to the fact that most smaller bodies are thought to be rubble piles – conglomerations of smaller pieces, loosely coalesced under the influence of gravity – that fly apart if the period is shorter than this limit. The few minor planets rotating faster than 2.2 hours, therefore, can not be merely held together by self-gravity, but must be formed of a contiguous solid.2

https://en.wikipedia.org/wiki/List_of_slow_rotators_(minor_planets)

If the larger the mass, the faster the rotation, It would be imposible for tiny asteroids to have such variagle roations periods, some must faster than even the giant panets, and some much slower than even the slowest roatatioing planets. Even if the pass of planets has an effect on their rotation periods it cannont be the only factor that influences their rotation periods.

So I don't think the mass/rotation rate forumula Dole used is valid. As I remember, one theory about planetary rotation rates is that they are largely determined by relative accident, by various large impacts between the planets and other large objects.

So you don't have to assume that your planet would have to rotate either too fast or too slaw to be habitable, based on its mass.

Part Three:

Could a habitable planet have the density described in the question?

The fictional planet has a mass of 18 times Earth's and four times earth's equatorial radius. This gives it a gravitational acceleration of 11.02m/s^2 (1.29G) and an escape velocity of 23.72km/s (1.9 times Earth's 11.18km/s). The surface area is 16 times that of our planet.

And as I wrote above, such a planet with 18 times the mass of Earth in 64 times the volueme of Earth would have 0.28125 times Earth's average over all density of 5.514 grams per cubic centimeter. Thus it would have an average overall density of 1.5508125 grams per cubic centimeter.

When a planetary mass body or planemo becomes massive enough that matter in its interior is compressed and heated up to liquid form for millions or billions of years until the planet finishes cooling off, the liquid matter becomes separated by weight, so that that most of the heavier matter sinks to the center of the planet.

Thus the solid or liquid matter at he center of a planemo will have have a higher average density than the matter at the surface, which will give the planet an average overall density intermediate between that of the surface material and that of the core material.

So if two planets with the same overall mass have different proportions of low density and high density materials, the planet made out of a lower density mix of materials will be less dense overall and have a larger volume than the planet with a denser mixture of materials, which will have a smaller volume and be more dense.

And another factor influences the overall density of a planet. The more massive the planet o rother type of planemo, the stronger its gravity will be, and the more the materials in its core will be compressed by the weight of hundreds or thousnds of kilometers of rock above to a greater density.

Obviously the giant planets will have lower density than terrestrial planets like Earth, because they formed in colder regions of the solar system where there was much more light weight volitile materials for their higher masses, escape velocities, and surface gravities, to attract and capture. So giant planets will have much lower average densities than terrestrial type planets.

On pages 27 to 33 Dole considered the relationship between mass and radius (and thus volueme and density) in the terrestrial planets in the solar system,plotting themin figures 7 and 8. Dole had only five such planemos to plot on his graphs, Mercury, Venus, Earth, Mars, and the Moon.

And you will see that on chart 8 there is a large shaded area instead of a dot for Mercury, because its volume and mass was little known in the early 1960s.

The Moon has 0.0123 the mass of Earth, 0.2727 the radius, 0.02 the volume, and 0.606 the density (3.334 grams per cubic centimeter).

Mars has 0.107 the mass of Earth, 0.532 the radius, 0.151 the volume, and 0.713 the density (3.9335 grams per cubic centimeter).

Venus has 0.815 the mass of Earth, 0.9499 the radius, 0.857 the volume, and 0.983 the density (5.423 grams per cubic centimeter).

Earth has 1.000 the mass of Earth, 1.000 the radius, 1.000 the volume, and 1.000 the density (5.514 grams per cubic centimeter).

Thus the more massive objects get compressed more and more, so that their radius, volume, and densities do not increase proportionally to their mass.

Mercury has 0.055 the mass of Earth, 0.3829 the radius, 0.056 the volume, and 0.984 the density (5.427 grams per cubic centimeter).

So Mercury has a mass, radius, and volume between the Moon and Mars, but a density bebween Venus and Earth.

Several theories to explain Mercury's unusually high density are mentioned at:

https://en.wikipedia.org/wiki/Mercury_(planet)#Internal_structure

If some process could make a small planet unusually dense, possibly some process could make a large planet have an unusually low density.

Part Four: Could such a low density planet have any solid surface area?

The density of the planet as described would be 1.5508125 grams per cubic centimeter. The four terrestrial planets and The Moon all have solid surfaces, although 70 percent of Earth's solid surface is covered by a few kilometers of water.

And their densities range from 3.334 (the Moon) to 5.514 (Earth) grams per cubic centimeter.

The densities of the five large terrestrial type planetary mass objects or planemos range between 2.1498 and 3.5555 the density of the planet as described in the question.

The four giant planets have massive amospheres of ultralight hydrogen and helium ranging from 0.70 grams per cubic centimeters (Saturn) to 1.76 grams per cubic centimer (Neptune). So the calculated density of your planet is within the range of giant planet densities.

There are other planemos in the solar system with low densities comparable to the calculated density of 1.5508125 grams per cubic centimeter for your planet as described.

Those bodies are farther from the Sun than the terrestrial planets and are mixtures in various amounts of solid rocks and solid frozen liquids, mostly water, with ammonia and methane.

The five recognized dwarf planets and five best candidates for the status of dwarf planets have densities ranging from 1.50 grams per cubic centimeter (Orcus and Salacia) to 2.43 grams per cubic centimeter (Eris).

The nineteen planetary mass objects among the natural satellites in the solar system have densities ranging from 0.95 grams per cubic centimeter (Tethys) to 3.528 grams per centimeter (Io).

So the calculated density of your planet would be in the ranges of the densities of outer solar system bodies which are largely composed of ices. With a density of 1.5508125 gramspercubic centimeter, your wurld would be denser than Dione at 1.48 grams per cubic centimeter and less dense than Enceladus at 1.61 grams per cubic centimeter.

Based on its density, Dione’s interior is likely a combination of silicate rock and water ice in nearly equal parts by mass.[18]

https://en.wikipedia.org/wiki/Dione_(moon)#Physical_characteristics_and_interior

Initial mass estimates from the Voyager program missions suggested that Enceladus was composed almost entirely of water ice.[58] However, based on the effects of Enceladus's gravity on Cassini, its mass was determined to be much higher than previously thought, yielding a density of 1.61 g/cm3.6 This density is higher than those of Saturn's other mid-sized icy satellites, indicating that Enceladus contains a greater percentage of silicates and iron.

So a world with teh calculated density of your woorld would be almost 50 percent water ice by mass.

And what happens when a world with ahigh percentage of water is close enough to its star to have surface temperatures suitable for Earth life? The water melts, and forms vast world wide oceans coveringthe solid surface with water tens, hundreds, or thousands of kilometers deep.

So there would be no islands or continents for land living lifeforms likehumanoid aliens on your planet, unless it somehow had natural or artifical floating islands or continents.

So possibly you could ask a question about land masses floating on water and see what suggestions you get.

• Thanks for the in-depth answer and the frame challenge. I loved the idea of floating continents. Aug 19, 2022 at 17:22
• @Mindwin Then maybe you can come up with a good idea for floating continents. when I search for floating islands I keep finding discussions about island floating in the air. I suppose if someone comes up with a good way to make continents float in the air, making them float on water should be easier. Aug 20, 2022 at 0:55

If you are afraid of winds and currents, you can have a slower rotating planet. Why not give it an rotation rate comparable to Earth?

As far as I know there isn't a correlation between mass and rotation rate. See Supplementary Figure 5. While the figure shows only larger mass objects, I don't think there is a plausibility issue with having low rotation rate for your planet, after all, there are many mechanisms that can alter rotation rates (the paper mentions planet-planet scattering, collisions, disk migration, and tides imposed by exomoons). At the end of the day it's your planet.

How does 4 * 24 hours sound?

• This is a practical answer. We're constantly learning new things about the universe - and they often upset our belief about what the limits to celestial mechanics really are. There's nothing unbelievable about so massive a planet having a 24 hour rotational period.
– JBH
Aug 17, 2022 at 22:54

There are four terrestrial planets in the solar system:

• Earth, 6,371 km radius and a 24 hour day
• Venus, 6,052 km radius and a 5,832 hour day
• Mars, 3,390 km radius and a 25 hour day
• Mercury, 2,440 km radius and a 1,408 hour day

The gas giants have between 10 and 17 hour days, and Pluto has a 153 hour day.

Note that Venus and Earth are almost identically sized and Venus has a day 200 times as long. Based on the data points we have available, I'd say just about any day length is reasonable.

We lack much data about what forms sentient life may take. Given that this is 56 times the mass of Earth, and thus 3 times the mass of Neptune, I'd be more concerned, hard science wise, that it would be a gas giant and not an Earth-like planet at all, and thus we know nothing about if and under what conditions sentient life may form on it.

Edit: Given the change to make this human-like, humans can't live on gas giants, no matter what the day length.

• You do not address the need for the planet to be habitable. Aug 17, 2022 at 19:11
• @Mindwin Most of that is new to the question. Aug 17, 2022 at 19:15
• @Mindwin, you got it the other way around: it's life to adapt on the planet where it develops, not the planet to adapt to the life it wants to hos
– L.Dutch
Aug 17, 2022 at 19:18
• @Mindwin As I added, hard science wise, your concern is that it should be a gas giant and not a Earth like planet. Aug 17, 2022 at 19:25
• @Mindwin Venus doesn't vary much during night, and even if it does, life may just adapt to it. Even on Earth, the temperature deep under water doesn't vary much at all no matter what the surface conditions, nor do the winds and tides matter. Even if winds and tides are stronger, it's hard to say what's too strong. Aug 17, 2022 at 19:30

As an amateur Worldbuilder: I would be inclined to keep the linear velocity at the surface about the same as Earth's, which is c.1670kph at the equator. Let's say 2000kph for your planet, and then it's equatorial circumference will be c.160,000kms.

That gives you a very reasonable 80hr day. Comfortably within the 4 day limit suggested by Dole and quoted by @Golding. Despite its size your planet shouldn't experience odd sub-orbital phenomena. With a low average density you might consider a location in the far goldilocks zone which, combined with a slow rotational speed, will allow for plenty of warmish liquid water prone to icebergs.

These icebergs in higher latitudes and having a very deep global ocean could grow to very large calving sizes from the polar icecaps. Perhaps 100's of Kms across, have great depth and hence altitude too. They might therefore move slowly, having great interia, and by held in the cool higher latitudes by quite violent 'orbital' winds occurring in the convergence zones either side of the plant's tropics. This would cause them not to melt away for perhaps tens of thousands of years.

Allowing for the environmental dynamism which is believed to contribute to the evolution of complex life forms. Add to this mix areas of the planet in which submarine faults have built up many highly active caldera whose peaks have broken the surface of the global ocean, perhaps at the poles and close to the equator.

The ash from these having built up deep deposits of 'loess' type soil from primordial times. Coating vast areas of the ice shelves and the icebergs broken off from them. It is so deep that lower layers have compacted to provide a base rock layer between the ice and the surface. The base rock layer protects the underlying ice from rapid from weathering and also allows old coastlines to develop morphologies that , in many places, slope down to and under the water of the surrounding oceans.

It is these shallows which in the distant past history of the planet air breathing life was able to develop and gain an evolutionary foothold. The volcanism would have made the oceans brackish, but subterranean ice together with the inevitable precipitation provides abundant fresh water on the ice islands and continental shelves.

The planet could have marine organisms, phytoplanktons, which feed oxygen into the atmosphere but also fix the sulphur as well as the carbon emissions from the vulcanism... so that the air is sweet and has allowed carbon and oxygen based life to become dominant in its seas and on its 'lands'.

So within the parameters given for the proposed planet as regards density, and despite its size and the need for it to allow the evolution of sentient humanoid-type life forms, 3.35 earth days per planet day would appear to be reasonable. Or at least, given the level of our ignorance on cosmic matters, it does not require a very large stretch of the imagination!