Potato-shaped Earth

[lemur]

The following map depicts Earth as having a topography of variable gravity: http://www.bbc.co.uk...onment-12911806

 

1) Why does gravity not correspond with altitude? It is weaker in the Himalayas and the oceans than many other places on land.

 

2) Could Earth or another planet be non-spherical and still appear to be because of gravity? E.g. if Earth was potato-shaped, wouldn't water still disperse across the surface in a way that the surface would appear flat to a sea-bound observer? At what altitude, then, would the odd-shape begin to become apparent or would it at all?

 

edit: Could an odd-shaped mass have a gravitational field with the same shape such that a satellite "free-falling" through its orbital geodesics would always maintain more or less equal distance from the surface, or rather from the same level of gravitation?

Edited April 5, 2011 by lemur

[Janus]

The shape of the Earth is due to its gravity and spin. It is a slightly oblate spheroid. Since the Earth's shape is shaped by these forces, are the oceans. the Oceans surface is is all at equal gravitational potential (not equal gravitational force). If the Earth spun faster, its shape would become more oblate and so would its oceans. An extreme example of this would be the world described in the article "Whirligig World" by Hal Clement, A world who's equatorial diameter is some 2.4 times its polar diameter.) Even with this world, the oceans follow the shape of the planet.

 

The only way that would get an ocean that didn't follow the shape of the planet is if the planet was so small that its gravity is to small to have a dominant effect on the planet itself. In such a case, the oceans would still follow an equal potential surface, but necessarily the shape of the world. However, a body small enough to hold an irregular shape against the force of gravity would not have enough gravity to hold on to an "ocean" for very long.

[SMF]

Janus. I am pretty sure that Clement's novel "Mission of Gravity" was based on the Whirligig idea. This was a really fun book and Clement's science fiction was a factor in getting me interested in science a very long time ago, so I am going to have to dip into deep storage to find the novel. SM

[Janus]

Janus. I am pretty sure that Clement's novel "Mission of Gravity" was based on the Whirligig idea. This was a really fun book and Clement's science fiction was a factor in getting me interested in science a very long time ago, so I am going to have to dip into deep storage to find the novel. SM

 

Yep, he even includes the original article as an afterward for the novel. Hal really took the concept of hard SF to heart. Mission of Gravity has an honored place on my bookshelf.

[imatfaal]

The following map depicts Earth as having a topography of variable gravity: http://www.bbc.co.uk...onment-12911806

 

1) Why does gravity not correspond with altitude? It is weaker in the Himalayas and the oceans than many other places on land.

 

2) Could Earth or another planet be non-spherical and still appear to be because of gravity? E.g. if Earth was potato-shaped, wouldn't water still disperse across the surface in a way that the surface would appear flat to a sea-bound observer? At what altitude, then, would the odd-shape begin to become apparent or would it at all?

 

edit: Could an odd-shaped mass have a gravitational field with the same shape such that a satellite "free-falling" through its orbital geodesics would always maintain more or less equal distance from the surface, or rather from the same level of gravitation?

 

Link to the European Space Agency which has released results from its satellite GOCE - link here it clearly demonstrates the variations in gravity/geoid.

 

In general it seems that "Gravitational acceleration at Earth's surface is about 9.8 m/s², varying from a minimum of 9.788 m/s² at the equator to a maximum of 9.838 m/s² at the poles." from the same site. But there is a lot of regional variation - some of which we cannot explain other parts of which are due to local variations in rock density. From the maps available on the above site I guess you are lightest in Southern India/Sri Lanka and heaviest in Indonesia/Papua New Guinea link here

I think you are right - a small dense potato shape surrounded by fluid would appear to have close to spherical surface (geoid). But on the orbit - no, I think that's not possible; in classical physics orbits are calculated with all the mass of a body at the centre of mass - you would orbit in an elliptical orbit around the centre of mass

 

 

Edited April 6, 2011 by imatfaal

[lemur]

I think you are right - a small dense potato shape surrounded by fluid would appear to have close to spherical surface (geoid). But on the orbit - no, I think that's not possible; in classical physics orbits are calculated with all the mass of a body at the centre of mass - you would orbit in an elliptical orbit around the centre of mass

I guess what confuses me is if you're sailing across a gravitational gradient in water, how do you know whether the gradient is also an altitude change or not? Does the surface shape of any body of water always behave as the surface of a perfect sphere relative to the center of gravity, regardless of the actual shape and gravitational oddities of the planet/body it's on? Or can it vary topographically according to varying gravity levels?

[Schrödinger's hat]

I guess what confuses me is if you're sailing across a gravitational gradient in water, how do you know whether the gradient is also an altitude change or not? Does the surface shape of any body of water always behave as the surface of a perfect sphere relative to the center of gravity, regardless of the actual shape and gravitational oddities of the planet/body it's on? Or can it vary topographically according to varying gravity levels?

 

Tidal and other wave effects aside, the surface of the water should be a surface of constant potential in the (rotating) frame of the surface of the planet.

So if you had, say, a planet sized cylindrical object then the water would form a roughly cylindrical ocean (it would bulge in the middle somewhat as gravity there would be much stronger).

The ocean surface would be fairly smooth the whole way so other than the horizon looking more curved in one direction than the other you wouldn't really be able to tell (unless the planet was very small/dense).

The faster the object rotates the more stretched the ocean will be in that direction.

[lemur]

Tidal and other wave effects aside, the surface of the water should be a surface of constant potential in the (rotating) frame of the surface of the planet.

So if you had, say, a planet sized cylindrical object then the water would form a roughly cylindrical ocean (it would bulge in the middle somewhat as gravity there would be much stronger).

The ocean surface would be fairly smooth the whole way so other than the horizon looking more curved in one direction than the other you wouldn't really be able to tell (unless the planet was very small/dense).

The faster the object rotates the more stretched the ocean will be in that direction.

But what happens when you go from the round surface of the cylinder to one of the ends? Or maybe an even more blatant example would be sailing over the corner of a cube? I suppose a cube corner would stick out of the ocean like a mountain, though, so maybe it's a bad example. Actually, maybe any water on the end of the cylinder would cascade toward the center and try to form a sphere as close to the center of gravity as possible. But then what if you had a oddly-shaped hollow planet where the gravity was not due to the core but to very dense crust? I guess this is getting a little far-fetched, but I'm still confused about how different levels of gravity on Earth can deviate from relative altitudes. I always assumed elevation correlated 1-to-1 with gravitation, controlling for the centrifuge effect of rotation.

Edited April 11, 2011 by lemur

[swansont]

I guess this is getting a little far-fetched, but I'm still confused about how different levels of gravity on Earth can deviate from relative altitudes. I always assumed elevation correlated 1-to-1 with gravitation, controlling for the centrifuge effect of rotation.

 

The mass distribution is not uniform. This was mentioned in the article.

[lemur]

The mass distribution is not uniform. This was mentioned in the article.

I vaguely remember this but I don't think it mentioned in what sense exactly it isn't uniform. Was it due to density/consistency or different plate-thickness. If the core/mantle is molten and therefore fluid, I don't see why it wouldn't form a sphere and exert uniform gravity to the crust. Or is gravity more complex in that it compounds through the whole diameter so that, e.g. the himalayas' mass causes higher gravity on the side of the Earth opposite the Himalayas (or something like that)?

[swansont]

F = GMm/r^2