# A planet rotating twice per second; what mass is needed to hold it together, and how oblate would it be?

I want to create a planet that has a frequency of two rotations per second, thus making a "Planet without Night". The planet will be like a flattened disc, due to inertia causing the planet to be an oblate spheroid. Gravity on the planet should also be very different at the "equator" than at the poles. The effects this planet would have on the civilization would be quite intriguing.

My question is a two-parter: firstly, I would like the mass of the planet to be great enough to create a gravitational force that is greater than the tangential speed created by the spin of the planet. Secondly, I need to know how oblate the planet would be with the mass and given rotational speed. An approximate graph (like y = x^2) would be nice for visualization. The gravity at the "equator" would optimally be "low" such that someone probably couldn't accidentally jump off the planet, but is lighter than the moon (if that's possible). The gravity at the pole would optimally be greater than or equal to earth's gravity.

It would seem to me that the mass would affect how oblate the planet is, which will affect the distance from the center of the planet, thus affecting the tangential speed. If the resultant tangential speed was greater than the resultant gravity, the mass would have to be changed. I can't figure out how to figure out how to find these factors, partially because I can only think of doing a trial-and-error method and don't know a "proper" way to find them, and because I don't know how to determine how oblate the planet is based on mass and speed.

• Comments are not for extended discussion; this conversation has been moved to chat. – Monica Cellio Jan 18 '17 at 5:00
• "Gravity on the planet should also be very different at the "equator" than at the poles" - that's actually not really correct. A rapidly rotating object becomes flattened precisely because that balances the net gravitational and inertial forces at the surface. There would be variations due to varying density of rock as on Earth, but you wouldn't expect huge variations in surface gravity due to the rapid rotation. – Nathaniel Jan 18 '17 at 10:17
• You should split your concept into several questions, to flesh out your concept. You are proposing something that is not real unless magic is involved. And I believe the effects on civilization to be quite intriguing, indeed: every person is flung into space the moment they step outdoors. – Mindwin Jan 18 '17 at 10:35
• But if you are willing to go the magic way, check diskworld. – Mindwin Jan 18 '17 at 10:35
• This has many similarities to Mesklin, a fictional hard-science planet. "each Mesklin day is 17.75 minutes long ". – David Cary Jan 18 '17 at 16:08

# Your planet is a pulsar

From Wikipedia, the acceleration caused by centrifugal force of a rotating object is $$\omega\times(\omega\times r).$$ Since the direction is known to be tangential to both the direction of rotation and the axis of rotation, and surface gravity acts in the opposite direction, we can just use the magnitude $\omega^2 r$.

The acceleration of surface gravity on a sphere is $$g = \frac{4}{3}\pi G\rho r.$$

On Earth, the angular velocity of rotation is about $7.29\times10^{-5}$ radians per second, and the radius is 6371 km, which gives a centrifugal acceleration of $$\left(7.29\times10^{-5}\right)^2 \cdot 6371000 = 0.034 \text{ m/s}^2.$$ Surface gravity is, of course, 9.81 m/s2 (as calculated in the second link), so is much more significant than centrifugal acceleration.

Lets determine how dense a spherical planet must be for gravity to hold it together against a given centrifugal acceleration. We can set the two forces equal to each other. $$\omega^2 r = \frac{4}{3}\pi G\rho r.$$ We can cancel the radii and plug in constants to get $$\omega^2 = 2.10\times10^{-10}\rho.$$

If we plug in your planet's rotation (2 rotations or $4\pi$ radians per second) and solve for density, we find that $$\frac{(4\pi)^2}{2.10\times10^{-10}} = \rho = 7.51\times10^{11} \text{ kg/m}^3.$$ The good news is that this density is a million times less dense than a neutron star. The bad news is that it is a million times more dense than the densest elements, and denser than white dwarfs and electron degenerate matter. Also bad news, is that this is the minimum density required just to keep your planet together, actual density would have to be higher in reality.

I'm not explicitly interested in doing further calculations, but I believe that the incredible density will keep the planet in a sphere due to its gravity. What you are actually describing is more or less a pulsar, which are neutron stars that have rotational periods as small as milliseconds.

• Comments are not for extended discussion; this conversation has been moved to chat. – HDE 226868 Jan 19 '17 at 16:28

A more useful approach may be a multi-star system, which is fairly common in nature. I believe that it's possible for a planet in a multi-star system to have an orbit that can take it between the two stars. A trinary system or even quaternary system may be useful. You could also include ice moons that would be much more reflective than Earth's moon, or a light-scattering atmosphere.

• That seems to make sense for making the planet lit all the time. Though, even if a faster rotational speed won't work for constant lighting, it would be interesting to have the planet have differential gravitaion due to an oblong shape created by a rotational speed that, while fast, is within the realm of reality. Is there an equation that can be used to accurately predict the shape based on the various factors? – Iter Jan 17 '17 at 19:32
• Make sure not to put too many large planets into your star system. Otherwise, every few years, the 3 suns are eclipsed, resulting in the bioraptors escaping and killing your crew. – Aron Jan 18 '17 at 2:47
• @Aron Or the civilization colapsing and entering a fire-seeking frenesi from being plainly scared from the night. – T. Sar Jan 18 '17 at 13:18
• Also its impossible to get orbital stability except by a considerable separation of scale (3-body problem, chaotic instability). You can have a small star orbiting a large star (binary star systems are common) and a planet orbiting close to the small star. But you can't get no night at all that way. – nigel222 Jan 18 '17 at 14:49
• Reminds me of Isaac Asimov's Nightfall story en.wikipedia.org/wiki/Nightfall_(Asimov_novelette_and_novel) – TrevorWiley Jan 18 '17 at 16:41

You might be able to get a planetary environment without night by having your planet tidally locked to its sun, so there is a hemisphere of perpetual day and one of perpetual night. The latter would be too cold for life, or perhaps not quite that bad (think the South Pole on Earth). The problem is that a likely consequence would be that all the water ends up frozen on the dark side, and then all the atmosphere condenses on the dark side, so there's no life and no story.

Could weather and/or ocean currents distribute enough heat from light side to dark side to stop this happening? If you write a good story, suspension of disbelief can probably be achieved. I don't know the hard science answer to this, or whether its an open question.

• This is an interesting idea. I expect a thick atmosphere would help. – Charles Jan 18 '17 at 15:21
• So might a lot of geothermal heating, arising from more radioactive elements than Earth or a handwaved phase change going on in the planet's core. – nigel222 Jan 18 '17 at 15:28

I want to create a planet that has a frequency of two rotations per second, thus making a "Planet without Night".

A planet without night doesn't really need such a rapid rotation, it just needs light reaching it from more than one direction.

Someone mentioned a binary system as an example.

You might also consider a planet orbiting close to a larger body ( a gas giant ? ) in such a way that it's was usually in a line with between the gas giant and the star. The gas giant could reflect a lot of light back onto what would otherwise be the dark side of the planet. This would light your planet continually, while allowing a reasonable rotation period. There would be a definite difference between daylight and "no so much daylight" as well, which is potentially useful.

A planet orbiting a gas giant might also lead to some spectacular aurora-like displays providing additional illumination.

• Maybe at the L4 or L5 Lagrange points between a gas giant and the star? They're stable (unlike the L1 Lagrange point, which would otherwise be ideal). At 60 degrees away from the larger bodie's orbit, there would be some night. :/ – Peter Cordes Jan 18 '17 at 23:53