Cold Core Model of Earth's Structure

[Cold-co]

Modest:
The hot-core calculations you are interested in are posted at http://members.cox.net/nchristianson...lculations.xls
Have fun with them.

[Cold-co]

Modest:
You may want review the post made by Stereologist on page 3. It gives the schematic of how the forces were trigonometrically calculated.

[modest]

Thank you Cold-co. Please allow me a few days to look it over.

~modest

[Cold-co]

Modest:
When you download the matricies into your system and start to manipulate the location of the gram mass, be sure to highlight the column before employing the remove and replace function. I learned the hard way that numbers elsewhere in the matricies and you don't want to replace those numbers. Also, I found it convenient to manipulate the numbers and then let the matrix return to the original where the gram mass is on the surface.
I'm sure you will realize that the matricies can be be modified to reflect the cold-core model and the average density model simply by changing densities. Be sure the densities work out the add up to the mass of the earth.

[modest]

Ok, Cold-Co. I don't know what to make of your spreadsheet. It is 2,000 lines of unnotated numbers. There are no comments, no physics formulas shown, no units, etc. I need you to meet me halfway here.

Originally Posted by Cold-co

The model you lay out is basically correct, but to get it to work you need to individually isolate each shell then slice it across its vertical axis into rings then rotate the total mass of each ring around to a common hemi-circumference. Once you have that for a specific shell...

I can't be sure what you mean. Here is the toy model:

Assuming a "shell" is the mass between each change in density then I could interpret your quote above like so:

I guess the shells have now been split down the center, rotated around to a "hemi-circumference" so that there now appears to be 2 dimensional half-shells. Is this correct and what's next?

As a recap, I'm looking for your method of finding vertical acceleration in a mass of changing density.

~modest

[CraigD]

Though I’ve not read it in detail, Dave Typinski’s “Earth’s Gravity” paper, which UncleAl linked to a while back , shows a rather complicated (but I’m guessing correct) integral calculus method of calculating the acceleration of gravity at different depths in the Earth.

In the tradition of simple arithmetic and brute force as an alternative to higher math , I wrote a short program that represents Earth (or a toy version) as about a million point masses, their masses based on their distance from the center, and calculates the acceleration of a test point at various depths by calculating and summing the gravity acceleration between it and each of the million points. To avoid being thrown off by a high acceleration from a nearby point mass, it ignores points within a fixed radius “bubble” around the test point.

Here’re the accelerations, in gs, for depths of 0, 0.1, 0.2 ... 0.9 radii, and depth-mass data from various sources
(Preliminary Reference Earth Model (PREM) (Dziewonski & Anderson, 1981))
1.0000 1.0153 0.9904 0.9995 1.0472 1.0350 0.8164 0.6402 0.4597 0.2495

(Modest’s “5,4,3,2” toy model)
1.0000 0.9633 0.9513 0.9108 0.8399 0.8070 0.6518 0.5197 0.3880 0.2060

(the toy model with “31,23.9,11.7,2.09”)
1.0000 0.8257 0.6372 0.4703 0.3433 0.1761 0.1248 0.0517 0.0046 0.0041

My program only roughly agrees with Typinski’s graph or the PREM’s acceleration data.
(here it is, in the 0, 0.1 ... format)
1.0000 1.0197 1.0126 1.0173 1.0503 1.0235 0.8377 0.6591 0.4483 0.2227

Increasing the number of point masses would, I think, improve the agreement, but as on the clunky old laptop, and quick but inefficient interpreted language (MUMPS)I’m using, it takes about 30 sec to calculate the acceleration of a single test point with one million points masses, so this’ll take a while.

Here’s the code:

s FX=0,X=R1\RI+1*RI f X=-X:RI:X s X2=X*X,Y=R12-X2,DX=TX-X,DX2=DX*DX i Y>0 s Y=Y**.5\RI+1*RI f Y=0:RI:Y s Y2=Y*Y,XY2=Y2+X2,Z=R12-XY2,DXY2=Y2+DX2 i Z>0 s Z=Z**.5\RI+1*RI f Z=0:RI:Z s Z2=Z*Z,XYZ2=Z2+XY2,DXYZ2=Z2+DXY2 i R12>XYZ2,DXYZ2>R22 s R=XYZ2**.5,DR=DXYZ2**.5,RM=$o(RM(R)),FX=DX/DR*RM(RM)/DXYZ2+FX ;XFX
k RM f RM=1:1:$l(R,",") s RM(R1-$p(R,",",RM))=$p($p(R,",",RM),":",2) ;XRM
s R1=+$g(R1,6378137),(R2,RI)=+$g(R2,R1**3*4/3*$zpi/1e6**(1/3)\1) w "R1: ",R1,"/ " r R,! s:R]"" R1=R,RI=R1**3*4/3*$zpi/1e6**(1/3)\1 w "RI: ",RI,"/ " r R,! s:R]"" (R2,RI)=R w "R2: ",R2,"/ " r R,! s:R]"" R2=R s R12=R1*R1,R22=R2*R2 ;X0

x X0 f  r !,R,! q:R=""  x XRM f TX=R1:-R1/10:0 s H20=$p($h,",",2) w TX,"  " x XFX w FX,"  ",$p($h,",",2)-H20,!

[stereologist]

I thought MUMPS was long gone. I did not look over the code but wanted to mention that:

One of the issues that you need to consider is the distribution of points in your work. The simple approach is to distributing points on a sphere results in points concentrated at the poles. You want to evenly distribute the points throughout a volume.

[Pyrotex]

Originally Posted by Cold-co

Modest:
... I learned the hard way that numbers elsewhere in the matricies and you don't want to replace those numbers.

...I found it convenient to manipulate the numbers and then let the matrix return to the original where the gram mass is on the surface.

...Be sure the densities work out the add up to the mass of the earth.

ColdCo,
I have gone over the data you sent to me (twice) by parsel mail, and I have to echo Modest's observation.
The matrix is just a mass (mess) of numbers.
There are no explanations of where the numbers came from.
No instructions for interpreting the numbers.
No clues as to what functions you used.
No clues as to how you calculated anything.

I'm really sorry, but you wasted a couple of bucks in postage.

I'm no closer now than ever to understanding what it is you have done.

Oh, and the 3 sentences I quoted from your post are obviously missing words and are unintelligible.

Pyro

[CraigD]

Originally Posted by stereologist

I thought MUMPS was long gone.

As an ANSI-document focused language standards organization – the old “M Development Committee”, it for all practical purposes is long gone in N. America (the last language document was in 1995). I’ve heard, but don’t much credit rumors of a functioning standards body in Europe.

As the underlying language of a lot of major, high-volume systems, primarily in healthcare and banking, it’s alive and well, though for the most part living under the proprietary alias “Cache object script”. An even superficial telling of this tale would be long and terribly off this thread’s topic.

The posted MUMPS code is a minimalistic kind that avoids the use of the language’s traditional “routine” program structures – in MUMPS terms, it’s “xecute code only”.

Originally Posted by stereologist

One of the issues that you need to consider is the distribution of points in your work. The simple approach is to distributing points on a sphere results in points concentrated at the poles. You want to evenly distribute the points throughout a volume.

The code takes almost the simplest approach I could think of for distributing the point masses: a cubic “crystalline structure” covering the cubic region (-R,-R,-R) – (R,R,R). Each point mass is therefore exactly equidistant from its 6 closest neighbors, like the centers of stacked dice.

In a fit of instinctive coding efficiency, I avoid finding the mass of points outside of a radius R sphere centered at (0,0,0) by “cutting the corners” of the inner 2 loops (Y and Z), and cheat by only allowing the test point to have coordinates (x,0,0), and calculating only for points in the positive Y,Z quadrant – that is, really considering only (-R,0,0) – (R,R,R). The code would work perfectly well, but be slower, without one or both of these tricks, as point masses outside of the radius R sphere (the Earth) would be given a zero mass and have no effect on the test particle.

I’ll play some more with variations on this approach, despite my misgivings that this sort of thing, and the marvelous machines that make it possible, threaten to make calculus-ignoramuses of us all.

[Cold-co]

Modest:
I like your toy model, but to make it work trigonometrically you have to slice each orb into rings (annuluses) across the vertical axis you have shown. Then rotate the mass of each ring around the vertical axis to a chosen circumference. Now you have the masses of the rings in a position where you can deal with them trigonometrically. I chose to use 18 separate shells of different densities. Hence, I needed to have 18 different spreadsheets. I know the concept is hard to grasp, but it is the same concept that Newton used to prove all of an orb's mass can be considered to be located at the orb's center.
In column G the matrices are set for a gram mass on earth's surface. When you want to change its location to a new radius just highlight column G and do a remove and replace operation for the radius at which you want to find the gravitational forces. The trigonometrics will take care of the rest. It is a tedious itterative operation.