# The shape of the earth$\ldots$

....is an oblate spheroid because centrifugal force stretches the tropical regions to a point farther from the center than they would be if the planet did not rotate. So we all learned in childhood, and it seems perfectly obvious. However...

I am at $45^\circ$ north latitude. Does that mean

• An angle with vertex at the center of the earth and one ray pointing toward the equator at the same longitude as mine, and one ray pointing toward me, is $45^\circ$ (that would mean I'm closer to the north pole than to the equator, measured along the surface, as becomes obvious if you think about really extreme oblateness); or
• The normal to the ground where I stand makes a $45^\circ$ angle with the normal to the ground at the equator at the same latitude (this puts me closer to the equator than to the north pole); or
• something else?

If for the sake of simplicity we assume the earth is a fluid of uniform density, it seems one's potential energy relative to the center of the earth would be the same at all points on the surface.

• Would the force of gravity at my location, assuming no rotation, be directly toward the center? Would it be just as strong as if the whole mass of the earth were at the center and my location is just as far from the center as it is now?
• Would the sum of the force of gravity (toward the center or in whichever direction it is) and the centrifugal force (away from the axis) be normal to the surface at my location?
• Given all this, how does one find the exact shape?
• How well does that shape in this idealized problem match that of the actual earth?
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This belongs on physics, although I would still suggest to edit it to be a single question. – Phira Oct 30 '11 at 18:27
@Phira If I wondered whether the Cartesian vortex theory or Newton's theory is closer to the truth, that would certainly belong to physics and not to mathematics. But if you accept Newton's physics, all that remains to answer the questions above is mathematics. (BTW, this problem was the "crucial test" of the Cartesian-versus-Newtonian theories. The former predicted an oblong rather than an oblate earth. Geodetic measurements in 1733, paid for by the French taxpayers, showed it was indeed oblate, and the Cartesian theory did not survive that blow.) – Michael Hardy Oct 30 '11 at 18:39

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@MichaelHardy If you accept all of theoretical physics, then most problems of theoretical physics "are mathematics". But it is much more likely that a physicist knows physics. – Phira Oct 30 '11 at 18:47

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@MichaelHardy What makes you think that you will get a better answer here than on the physics site? – Phira Oct 30 '11 at 18:48
– Qmechanic♦ Oct 30 '11 at 20:21

## 2 Answers

1, At 45 deg (N) latitude you are closer to the North pole, to picture this just draw the Earth as a much more extreme oblate spheroid.

2, The shape of the Earth is set by the outward rotational force exactly balancing the inward gravitational force at every point (except for local geography). So the overall potential is always down (except for local geology)

There is a useful intro to the difference between mean sea level and the Earth's surface at ESRI (makers of the popular GIS/mapping software)

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+1 for answering some of the questions and adding the link.  Most of the question remains unanswered. – Michael Hardy Nov 1 '11 at 15:26

It is not popular to begin an answer with a question but I would do that: How do you know that you are at 45° north latitude?

There are several latitudes defined. If you got the latitude from a map, Google Earth or maps, of read it at your GPS receiver, that its is geographic or geodetic latitude. In such a case 45° is the angle between normal to reference ellipsoid and equatorial plane. The reference ellispoid (for example WGS84 used in the Google and GPS) is a mathematical construction, an easily managed approximation to the actual shape of Earth.

If you got the latitude by your own direct measurement of height of the north pole (in the sky) above horizon or its distance from zenith (supposing you took into account refraction, abberation, nutation etc.), than it is astronomic latitude. In that case 45° is the angle between normal to geiod and the equatorial plane. Geoid is equipotential surface passing through your point of observation.

Finally, there is also geocentric latitude - angle between the equatorial plane and a line passing through your place and Earth's center.

If Earth is a perfect non rotating sphere and distribution of its mass has spherical symmetry than all three latitudes has the same value at every point of globe.

See Coordinate systems by R. Knippers for additional details.

If you require that gravitational force at every point of the globe point towards the Earth's center than mass distribution must have sperical symmetry. Since rotation is axially symmetric you cannot get the required result unless mass distribution is so special (and unphysical) that it compensates for effect of rotation added.

Only if sphape of Earth closely follows equpotential surface (for example is covered by ocean) than neat force (gravity plus centrifugal one) is perpendicular to the surface at every place.

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