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May 5

Making use of earth gravitational model

There are many gravitational models for earth that extend to high accuracy, but they all apparently describe a surface of equal gravitational potential energy relative to a reference ellipsoid, which seems only useful on the surface and in a reference frame that rotates with the earth. How would one use such a model to calculate the direction and magnitude (3D vector) of Newtonian gravity force for an arbitrary 3D point in space relative to the center of earth, assuming that the instantaneous orientation of the earth is known. I'm ultimately only looking for a fairly low order approximate (4 - 10 constants) and don't care about longitudinal variations, but it must be good enough to capture the effect of the equatorial bulge on the moon and artificial satellites. 102.65.153.81 (talk) 12:18, 5 May 2020 (UTC) Eon[reply]

The model specifies the spherical harmonics. Each harmonics (l,m) depends on radius as

~r^{-l-1}

. By summing the harmonics your can calculate the gravitational potential at any point outside the solid Earth. Ruslik_Zero 21:03, 5 May 2020 (UTC)[reply]

I'm not sure where the

~r^{-l-1}

comes from and it especially confuses me that there is an "l" in there. This is not apparent at all from the article on spherical harmonics. I understand them to precisely not dependent on radius and is therefore useful to describe things on a surface only, or in the far field. There must be some separate compensation for radius/height, but I don't know what this would be and whether height would be taken as away from the center of mass or perpendicularly away from the surface. I imagine the latter, but then since all these vectors pointing perpendicular to the surface don't meet at a point in the middle, I'm also not sure from where height would be measured for compensation purposes – for example if assuming that the compensation for potential is (1/h) as one moves away outside the surface, where is h=0? If I could get potential at every point like this, I could just do a derivative to get to gravity force/acceleration. 102.65.153.81 (talk) 08:27, 6 May 2020 (UTC) Eon[reply]

Except for a spherical rotationally homogeneous body, the gravitational field lines are not straight. Close to the surface, they are perpendicular to the equipotential surface, but far away they point to the centre of mass of the body. I suppose the

\ell

and

m

stand for the indices of Laplace's spherical harmonics

Y_{\ell }^{m}(\theta ,\varphi )

. I'm insufficiently familiar with this matter to tell how a given harmonic contributes to the gravitational force at a given point, but a Google search for [calculate|compute gravity "gravitational model" harmonics] digs up some results that look promising. --Lambiam 10:19, 6 May 2020 (UTC)[reply]

Spirals and Einsteins equations

Are the equations for spirals (like under the logarithmic spirals) post in any way related to the Einsteins Field Equations under that post? Since the spirals are very evident in nature and even in galaxies, I wonder if there is some relation. The math is beyond me. Tofflet (talk) 17:18, 5 May 2020 (UTC)[reply]

...There exists at least one way to relate the equations for spirals to the equations that describe the geometry of space-time.

...If we showed you such a way to relate them, but the math were beyond you, would it enlighten you in any way? Or are you simply seeking a confirmation from somebody else who you think might understand it better than you, in the hopes that their confirmation could bring you some kind of solace? Because that is a logical fallacy called the argument from authority, and if you wish to use mathematical ideas, one of the first skills you should formalize is your ability to work within a logical framework.

Nimur (talk) 17:31, 5 May 2020 (UTC)[reply]

I cannot figure out what you are referring to. What is "spirals post" and where is "under the logarithmic spirals"? I don't see anything Einstein-related in the equations in our articles Spiral and Logarithmic spiral, nor anything spiral-related in the Einstein field equations. --Lambiam 09:50, 6 May 2020 (UTC)[reply]