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GrapesOfWrath
2001-Dec-08, 02:23 PM

During precession, does the axis of the Earth ever get perpendicular to the Ecliptic plane? (http://image.gsfc.nasa.gov/poetry/ask/a10973.html) says that the angle of the Earth to the ecliptic is 24.5 degrees, but that it is 23.5 degrees in Where on the earth can the Ecliptic lie on the horizon? (http://image.gsfc.nasa.gov/poetry//ask/a10837.html)

What is the 'Geodetic Altitude' of an object above the ground? (http://image.gsfc.nasa.gov/poetry/ask/a11776.html) says that the acceleration of gravity is constant on the surface of the geoid. Do you weigh differently at the North Pole than what you do at the equator? (http://image.gsfc.nasa.gov/poetry//ask/a11511.html) points out that things weigh less at the equator.

He goes so far to say in Is the Earth a perfect sphere? (http://image.gsfc.nasa.gov/poetry//ask/a11024.html) that the Earth is pear-shape, rather the oblate spheroid flattened at the poles that it actually is. He calls it a "geode" in Exactly how round is the Earth? (http://image.gsfc.nasa.gov/poetry//ask/a11818.html), but he does note that the biggest effect is flattening at the poles. However, he says that the pear-shape of the earth is on the order of a kilometer, when it actually is about twenty meters. That's a thousand times less than the flattening.

Ducost
2001-Dec-08, 03:25 PM
What is the 'Geodetic Altitude' of an object above the ground? (http://image.gsfc.nasa.gov/poetry/ask/a11776.html) says that the acceleration of gravity is constant on the surface of the geoid. Do you weigh differently at the North Pole than what you do at the equator? (http://image.gsfc.nasa.gov/poetry//ask/a11511.html) points out that things weigh less at the equator.

It was my understanding that things weigh less at the equator do to Earth's rotation not any difference in the pull of gravity.

Kaptain K
2001-Dec-08, 06:53 PM
It was my understanding that things weigh less at the equator do to Earth's rotation not any difference in the pull of gravity.

Actually, it's both.

Kaptain (adding to the confusion) K

2001-Dec-08, 09:12 PM
The first one (24.5 vs 23.5) may be a simple typo. The angle does change with time, however.

The gravity at the surface of a geoid is constant. The Earth rotates, so the totalforce on a mass changes with latitude.

The Earth is not a perfect oblate spheroid. There are higher order terms to the shape, and it is roughly pear shaped. The more resolution you use, the weirder the shape gets. To first approximation, it's a sphere. To second, it's oblate. To third, it's a pear. After that, well, do a web search. /phpBB/images/smiles/icon_wink.gif

GrapesOfWrath
2001-Dec-09, 03:18 AM
On 2001-12-08 16:12, The Bad Astronomer wrote:
The first one (24.5 vs 23.5) may be a simple typo. The angle does change with time, however.Typos happen. /phpBB/images/smiles/icon_smile.gif

The gravity at the surface of a geoid is constant. The Earth rotates, so the totalforce on a mass changes with latitude.The gravity on a geoid is not constant at all, even if the earth were not rotating. The surface of a geoid is equipotential, but not equi-acceleration. An object at the equator is twenty kilometers farther from the center of the Earth than an object at the pole, without any consideration of the rotation of the Earth, and so the acceleration due to gravity will be less. The centrifugal force will lessen it further.

The Earth is not a perfect oblate spheroid. There are higher order terms to the shape, and it is roughly pear shaped. The more resolution you use, the weirder the shape gets. To first approximation, it's a sphere. To second, it's oblate. To third, it's a pear. After that, well, do a web search. /phpBB/images/smiles/icon_wink.gif You mean this? (http://mentock.home.mindspring.com/agu1992.htm) /phpBB/images/smiles/icon_smile.gif As the Space Scientist points out, the Earth's oblateness transferred to a basketball would be a depression of 1/32 of an inch. The depressions due to the so-called pear-shape is hundreds of times less than that. The Earth can not be called pear-shaped--in fact, as I mentioned at that link, the third-order shape is a tetrahedron, not a pear.

Wiley
2001-Dec-10, 04:58 PM
On 2001-12-08 22:18, GrapesOfWrath wrote:
You mean this? (http://mentock.home.mindspring.com/agu1992.htm) /phpBB/images/smiles/icon_smile.gif As the Space Scientist points out, the Earth's oblateness transferred to a basketball would be a depression of 1/32 of an inch. The depressions due to the so-called pear-shape is hundreds of times less than that. The Earth can not be called pear-shaped--in fact, as I mentioned at that link, the third-order shape is a tetrahedron, not a pear.

Hey GAG,

I'm a little confused. How does one get non-smooth surface (a tetrahedron) with from smooth functions (spherical harmonics)? If I recall correctly, the Y_{3,2} mention in your abstract looks the donut equivalent of siamese twins.

GrapesOfWrath
2001-Dec-10, 05:26 PM
I probably should have said tetrahedral instead of tetrahedron. The Y-(3,2) shape has four highs spaced equidistantly on the sphere, with four lows spaced similarly--that is, like a tetrahedron would if it were fitted into the sphere. The pear-shape is Y_(3,0), with a high, a band of low, a band of high, and a low.

Still, that third-degree undulation is miniscule compared to the second-degree. It is possible to combine the degrees, but they are essentially independent--changes in the third-degree components don't affect the second-degree. So, a pear has a strong third -degree component, but a relatively small second-degree component, and that's why it is pear-shaped. The Earth has a large second-degree component and a small third-degee component--it is an oblate spheroid, not pear-shaped. Even more, the Earth's third-degree component is not even really pear-shaped.

Wiley
2001-Dec-10, 05:40 PM
Thanks Grapes,

I found a website that shows the spherical harmonics. So between you and the website, I understand - I think /phpBB/images/smiles/icon_smile.gif.

Another query: Do you know the axis with the minimum number of significant coefficients a priori? If so how? Or do you rotate the axis after you have calculated some arbitrary axis?

GrapesOfWrath
2001-Dec-10, 05:47 PM
Hey, don't hold back, let's see that spherical harmonic website!

On 2001-12-10 12:40, Wiley wrote:
Another query: Do you know the axis with the minimum number of significant coefficients a priori? If so how? Or do you rotate the axis after you have calculated some arbitrary axis?
The initial axis is the rotational axis of the Earth, and the spherical harmonic coefficients of the Earth's gravity field are computed to that reference frame. The sum of the squares for each degree is constant under change of axis, and there are straightforward formulae for converting from one axis to another. The program I wrote in qbasic twelve years ago to find the optimum axis took only a couple seconds on a 8088 PC.

Wiley
2001-Dec-10, 06:56 PM
On 2001-12-10 12:47, GrapesOfWrath wrote:
Hey, don't hold back, let's see that spherical harmonic website!

Here's the website of the spherical harmonics java viewer (http://wwwvis.informatik.uni-stuttgart.de/~kraus/LiveGraphics3D/java_script/SphericalHarmonics.html).

The initial axis is the rotational axis of the Earth, and the spherical harmonic coefficients of the Earth's gravity field are computed to that reference frame. The sum of the squares for each degree is constant under change of axis, and there are straightforward formulae for converting from one axis to another. The program I wrote in qbasic twelve years ago to find the optimum axis took only a couple seconds on a 8088 PC.

Thanks!

GrapesOfWrath
2001-Dec-10, 08:03 PM
On 2001-12-10 13:56, Wiley wrote:
Here's the website of the spherical harmonics java viewer (http://wwwvis.informatik.uni-stuttgart.de/~kraus/LiveGraphics3D/java_script/SphericalHarmonics.html).
Cool. Just remember that the wave functions are the square of the spherical harmonic function. So Y_(3,2) will show up at that site with eight peaks (4 negavite lows, 4 positive highs, all squared). (1 (http://hades.ph.tn.tudelft.nl/Internal/PHServices/Documentation/MathWorld/math/math/s/s578.htm)),(2 (http://www3.uniovi.es/~quimica.fisica/qcg/harmonics/harmonics.html)),(3 (http://www.ngs.noaa.gov/GEOID/research.html)--the geoid.)

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<font size=-1>[ This Message was edited by: GrapesOfWrath on 2001-12-10 15:07 ]</font>