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I am writing a science-fiction story, which contains a 'crazy' planet. I would like some input into the consequences of this physics according to standard physics.

The planet in question is Earth-sized and it orbits a Sun-like star. Within the story, this planet has been taken over by a malignant 'force' and has been hollowed out, so that there is one mile of basalt crust forming a spherical shell, and the inside is a void filled with nitrogen. This force keeps the planet intact, but does not affect the planet in any other way (i.e. any standard-physics problems regarding the integrity of this spherical shell can be ignored, as the story explains them in other ways). The force has no mass.

Given this setup, what would the gravity be like at the center of the planet? In the story, the protagonist is able to fly around in a large bubble of zero-gravity at the middle. Would this be possible? (Assume that the protagonist can survive the atmosphere.) Could he, assuming he has a suitable raft, get around by blowing?

Emilio Pisanty
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Robert Goddard-Wright
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    The question is more suitable for world building stackexchange – tryst with freedom Dec 15 '20 at 06:39
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    Possible duplicates: https://physics.stackexchange.com/q/150238/2451 and links therein. – Qmechanic Dec 15 '20 at 09:42
  • @Buraian I disagree: this question only asks about physical possibilities and phenomena. The fact that it regards a fictional planet does not automatically mean it is only suited for worldbuilding.se – Noumeno Dec 15 '20 at 09:50
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    Fair @Noumeno, then I'd suggest OP boils the question down to purely the physics. – tryst with freedom Dec 15 '20 at 10:14
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    The core question here is about standard physics, so it belongs here instead of [worldbuilding.se]. However, the question does need to be very explicitly asked about the standard physics. The science-fiction background is relevant but incidental, and it needs to be phrased in that way. (For more information see this and this threads on Meta.) – Emilio Pisanty Dec 15 '20 at 12:35
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    Does this answer your question? Gravitational field intensity inside a hollow sphere – Quillo Dec 15 '20 at 13:06
  • In addition to the concerns on the structural integrity of the basalt shell, the assumption that the interior can be plausibly filled with nitrogen (with a flat density profile) is extremely suspect. You basically have two physically-realistic choices: (1) either the cloud of nitrogen contracts under its self-gravity (either to a pure-gas cloud, or condensing in the center, depending on the total amount of gas) with no gas at the edge, or (2) there is nonzero pressure at the edge, so you need an external atmosphere to hold it down or an airtight seal at the shell. – Emilio Pisanty Dec 15 '20 at 16:00

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If you neglect the gravity of the nitrogen gas, there is no net gravitational force anywhere inside. This is a result of the famous “Shell Theorem” for inverse-square forces. At any point, you are attracted by all the atoms in the shell, but the vector sum of all these forces — in different directions, and having different magnitudes because they are caused by atoms at various distances from you — turns out to be zero. This assumes a perfectly spherical shell of uniform mass per unit area.

If you want to take the nitrogen gas into account, at radius $r$ from the center you are attracted by the mass of all the nitrogen in a sphere of radius $r$ around the center, as if this mass were concentrated in a point at the center. The nitrogen at larger radii doesn’t attract you for the same reason that the shell doesn’t.

G. Smith
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    You should possibly mention that you have neglected the total mass of the nitrogen inside the void. All mass in the "sphere below you" at any location within the planet will still attract you with a resultant gravitational force (while all mass in "the shell above you" cancels out gravitationally as you describe). This total mass purely from nitrogen gas might be significant or might not depending on the size of the planet, I recon. – Steeven Dec 15 '20 at 06:55
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    @Steeven Thanks for the suggestion. I’ve edited my answer. – G. Smith Dec 15 '20 at 06:57
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    Perhaps you could mention that the planet's crust would not be perfectly spherical, implying there would be some nonzero force. – Daddy Kropotkin Dec 15 '20 at 12:38
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    Suddenly I feel the urge work out the density gradients in a self-gravitating sample of nitrogen gas at temperature $T$ (or would an adiabatic model be more realistic), and about the motion of an object falling in this gas volume including air resistance. Dammit, I've got work to do today. – Michael Seifert Dec 15 '20 at 15:50
  • For an earth-diameter body with a shell a mile thick of material like earth's basalt, I come up with a total basalt mass of 2.4×10¹⁸ kg; that's half of a millionth of Earth's mass. To fill it with N₂ at an arbitrary 20⁰C at 1 atm average pressure would take over 10²¹ kg of gas, which overwhelms the basalt mass by a factor of 1000, but which would still only have half of a thousandth of earth's gravity. Tough question is how to get the basalt to stick together under the rotational force and the gas pressure. – CCTO Dec 15 '20 at 22:13
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To add to @GSmith comment, if you take into account the nitrogen inside, you would only feel the gravitational pull of the nitrogen between you and the center of the planet (as explained). It is easy to see that anyone who falls inside will end up right at the centre of the planet:

  1. You begin to fall towards the centre of the planet due to the pull of the nitrogen inside.
  2. Since you can go through nitrogen (its not solid but gas!) you will go past the centre out towards the other end of the planet.
  3. This oscillatory motion would continue for ever would it not be for the air resistance of the nitrogen which keeps substracting speed/energy from you. This means that with every swing about the centre you reach a lower altitude each time (just like in a kid's swing) until eventually you end up in centre. Think of this motion like a real pendulum which ends up stopping at the middle but in a 3D kind of way. What a way to die!
FriendlyLagrangian
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    Probably not. The nitrogen would get denser as you go towards the center - much like a gas giant planet. I would guess that it would become denser than the human body at some level before the center, so you would float at that level. – jamesqf Dec 16 '20 at 00:05
  • Very close to the centre then. Even if the gas is denser than the body, it would be still gas like (unless extremely dense) and a body with such velocity could potentially end up quite far inside it. – FriendlyLagrangian Dec 16 '20 at 13:05
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Ok let's go:

  1. It is not physically possible to have a situation like you described: since your planet has the size of the earth but it's hollow it would surely collapse into itself, crushed by its own gravity, and it would form a ball with much smaller radius. This is why you don't see large hollow bodies in nature. There isn't a material strong enough to form a shell like you described and not crush under gravity, I am pretty sure about that. You would have to invent some sort of implausible sci-fi material to form the shell, really implausible material.
  2. You mention a force, but it's not clear how this force is produced or how it acts. You also state "the force has no mass".. this is ridiculus: a force is not an object, talking about the mass of a force is meaningless.
  3. But let's ignore this force you mention for now, and let's suppose also that in some way your shell planet does not collapse: then we can apply a famous theorem of calculus, usually used in the context of electrostatics but it will also work here: the shell theorem; using it we can prove that the gravitational force inside your shell planet must be zero everywhere not only in the center of it. (Of course this is true with some possible corrections, in the sense that if you have a massive mountain on the shell then the symmetry is partially broken and you will have some light gravitational pull inside your planet).
  4. In a context where the force acting on the protagonist is zero and there is little drag yes, I think it would be possible to move by blowing. But this would be a very inconvenient method of locomotion to say the least.
  5. Sci-fi is not an easy topic. You really should study some physics before attempting to write a book on it, otherwise you will likely make a joke of yourself, especially when it's time to publish it.
Noumeno
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  • Let me point out that the word force was likely meant not in the physical sense, but to denote a mighty entity that is able to significantly affect the world, e.g. "an armed force". It is also stated that it is this "force" that had hollowed out the planet and is keeping it intact despite the fact that the shell would likely collapse on its own. In this context, the statement that "the force has no mass" is meaningful, since according to the question this mighty entity is still there. Despite this, it does not exert any gravitational pull (or this pull is negligible). – FlashCactus Dec 15 '20 at 16:32
  • This is clear now, but it was not clear before; the question has been edited after my answer and I suspect because of my answer, so I think I shouldn't modify what I have written. – Noumeno Dec 15 '20 at 16:36
  • Ah, okay. If I were you, though, I'd still modify the answer. From the perspective of SE as a knowledgebase it doesn't really matter that your answer reflects the original wording of the question: someone coming here a year from now will expect the answer to correspond to the question as written at that time, not the outdated original wording. In fact, given the lack of other answers that mention this force thing, you could suggest an edit to the question itself, replacing the offending word with some other less ambiguous term, like entity, which would add clarity to the whole thread. – FlashCactus Dec 15 '20 at 16:57
  • I would like to see the math that proves that such an object is impossible. And forces do have mass, at least according to some theories, e.g. https://en.wikipedia.org/wiki/Higgs_mechanism – jamesqf Dec 15 '20 at 17:04
  • @jamesqf the question's topics are clearly all treatable with Classical Mechanics, I don't see the need to appeal to Quantum Field Theory in this context. And indeed Classicaly forces have no mass – Noumeno Dec 15 '20 at 19:02
  • @Noumeno, Classically, perhaps, but (like relativity) that's perhaps because in our normal conditions, the mass of a field is insignificant. – jamesqf Dec 16 '20 at 00:03