Science Forums 2

[Cold-co]

To All:
Thank you for your responses, but I feel you have missed the point I made in my initial post. If there are other models of earth that can be considered then they must meet the moment of inertia found by the flattening equation. The low moment of inertia derived from that equation dictates most of earth's mass be located in her inner core. That is why I started looking into the components of the equation itself.
To analysis the packing accelerations at work within the earth, I used three different models—cold-core, hot-core, and average density. Each model uses the same eighteen divisions of seismically known shells: crust, lithosphere, asthenosphere, 1st bonded shell, 1st transition (phase change), 2nd bonded shell, 2nd transition, five divisions of the 3rd bonded shell, four divisions of the outer core and two divisions of the inner core. Except for the average model, which has the same density for each of its shells, density is proportional to seismic wave speeds in the cold-core model and, as required for the standard hot-core model, density is concentrated in the core. All models have a radius of 6371 km and all have the same total mass. See physical characteristics shown below.
To fill in a mental picture of what goes on with respect to gravitational accelerations, I used an adaptation of Newton’s model of Thin Spherical Shells. The model he used to prove gravitational forces acting on a small body (gram-mass) external to earth’s surface can be considered to be located at earth’s center. That model effectively rotates the total mass of an annulus (ring) around to a single point where the gravitational accelerations merge into a single acceleration. This merged acceleration can then be broken into two accelerations; a vertical acceleration (v) and a horizontal acceleration (h). Then, using trigonometric functions the values for h and v can be determined.
Just as Newton did, I set up my model’s eighteen separate divisions as individual spherical shells of zero thickness. Ninety annulus-masses for a selected shell-radius rotate around to concentrate at odd (1, 3, 5 ... 177, 179) degree points. After creating spreadsheets for each shell, I used a series of trigonometric functions to solve for horizontal, as well as vertical gravitational accelerations. By moving the radius at which the gram-mass is located and employing an iterative process, I solved for the vertical and horizontal gravitational accelerations produced by each individual division. Resultant gravitational accelerations for the radius selected for the location of the gram mass are shown below. Values for vertical accelerations in my hot-core model match well with values obtained by Dziewonski. This increased my confidence that my trigonometric approach is equivalent to his way of calculating vertical gravity for various levels within the earth.
My question for this forum is, “Have I done something wrong mathematically?”
I'm sorry but I cannot figure out how to insert the results of my calculations.