Cold Core Model of Earth's Structure

[martillo]

Moderation note: this post and replies to it were moved to 20339, because they are about a different subject than this thread’s

 

Forgive me If I distract your (all in this thread) attention for a little but I would like you to consider (just for a little) some very unexpected possibility for Earth's core.

 

Some times we don't find just what we are not looking for.

 

I know it would be hard to accept mainly at a first view but became to make sense someway to me since some time ago.

 

Here it goes:

 

What if the Universe is much more alive than we curently believe and stars are not just a "ball of fire" and planets and moons are not just "balls of some earth".

What if some kind of what we could call "natural supercomputing system"

...

[modest]

[[ Can somebody please whip out a plot of the actual "real" force of gravity as you approach the core? One with constant density, and one with a linearly increasing density? ]]

 

<snip>

 

...for linearly increasing density I can find no source so I'll attempt to work out a plot...

 

<snip>

 

It plots as:

which I plotted at

this website

.

 

While looking up some stuff for the other hydrogen core thread I found the real data of a realistic model (PREM model) which doesn't assume linearly increasing density. It's in table form:

 

Google book -- Allen's astrophysical quantities

 

I put it in Excel and plotted it:

 

 

File attached.

 

For what it's worth, this does match ColcCo's gvhot plot:

 

 

~modest

[Cold-co]

Pyrotec:

Sorry I didn't catch your post from a week ago. Did you get the book I sent? If you did its appendix explains how the figures were derived and what they signify. If you want to discuss my calculations by phone, you can get my phone number the Phoenix directory.

[Cold-co]

Modest:

Thank you for verification that my calculations match those of the PREM model that was calculated by Professor Adam M. Dziewonski (Harvard) and a grad student named Anderson. Their model shows up in most geophysics textbooks

[modest]

Is that where you got your gvhot data—in a geophysics textbook? Or, did you calculate it by some method which you could make reproducible for us?

 

~modest

[Cold-co]

Modest:

Applying gravity’s elastic nature to gram masses inside the earth, suggests gravity’s horizontal vectors work in a manner similar to the pull exerted by molecules in the skin of a rubber balloon. It seemed reasonable then that the strength of pull (packing effect) by a gram mass at any depth within an orb would be obtained by rerunning calculations similar to the ones Newton used to prove that an orb’s total mass can be considered to be located at the orb’s center—a very tedious trigonometric calculation.

To analysis the packing vectors at work within the earth, I used three different models—cold-core, hot-core, and uniform 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 above, density is concentrated in the core in the hot-core model. All models have a radius of 6371 km and all have the same total mass.

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 gravity vectors. By moving the radius at which the gram-mass is located and employing an iterative process, I solved for the vertical and horizontal gravity vectors produced by each individual division. Resultant gravity vectors for the radius selected for the location of the gram mass are shown in a previous post. Values for vertical gravity in my hot-core model match well with values obtained by Dziewonski. This increases my confidence that my trigonometric approach is equivalent to his way of calculating vertical gravity for various levels within the earth.

Pyrotex has been reviewing my trigonometric matrix but has yet to figure our that it takes an itterative process of moving the location of the gram mass to get the final results that I posted earlier.

[modest]

I used a series of trigonometric functions to solve for horizontal, as well as vertical gravity vectors. By moving the radius at which the gram-mass is located and employing an iterative process, I solved for the vertical and horizontal gravity vectors produced by each individual division...

 

Values for vertical gravity in my hot-core model match well with values obtained by Dziewonski. This increases my confidence that my trigonometric approach is equivalent to his way of calculating vertical gravity for various levels within the earth.

 

That sounds very interesting. You have my utmost curiosity in how exactly you got the gvhot results. I'm hoping that you will be able to explain well enough to make the process reproducible. That is to say: I'd like to understand how you got the gvhot plot well enough that I would be able to do it myself.

 

To simplify things, I've built a toy model which would be much simpler than working with all of earth's shells and densities:

Each change in density happens at an additional radius of 1,000 km and each shell is less dense than the one below it by 1 g/cm³ (this is not meant to represent the earth—only to be a simple example we can work on). Do you think it would be possible to show step by step how you would get vertical acceleration for this mock-up?

 

Hopefully, when done, your method will give results similar to:

at r = 1000 km, acceleration = 1.4 m/s

²

at r = 2000 km, acceleration = 2.31 m/s

²

at r = 3000 km, acceleration = 2.80 m/s

²

at r = 4000 km, acceleration = 2.87 m/s

²

which I get using the normal Newtonian method.

 

Do you think we could work through this? I realize it will probably take more than one post, but I'm very interested in what you've done and I think seeing it done might be the only way to understand it.

 

~modest

[Cold-co]

Modest:

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 you can solve the g forces produced by that shell trigonometrically. I used two degree increments. The odd degrees were the points where mass was located.

I'm working on breaking up my work into segments that I can post where they are available for review. I'll be a while doing it though.

[Cold-co]

Modest:

The hot-core calculations you are interested in are posted at http://members.cox.net/nchristianson4/hot-corecalculations.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.

 

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=R1RI+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**.5RI+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**.5RI+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]

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]

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”.

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.