Gravitational pull based on density variations

J

jwpaine

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This is a message that I posted to my "Dynamic Earth" class, (we have to make two posts a week about our reading) I think I might have made a fool out of myself, and if someone could comment on my second paragraph.
If I collected the density data under a given point on Earth at a specific depth, would I be able to approximate the gravitational pull for a certain body at that point using the method that I described? Sometimes I get these ideas and then once I think of it in a different way, I see that what I said is false. My hypothesis would be that I would not be able to find the gravitational pull by isimply ntegrating newton's universal gravitation equation for changing depth at several density differences, since there are an infinite number of points within the earth between 90 < t < 0 degree's off vertical to the Earth's core.... so I would have to account for changes in density for mass pulling on me from various angles... right? Not just what is directly under me.

Gravity is the weakest of the four fundamental forces. Einstein's theory of general relativity (space-time curvature) and quantum field theory (interaction of gravitons) are two theories which attempt to explain this force which is exhibited between any two bodies. "The magnitude of the force is proportional to the product of the masses; it is inversely proportional to the square of their distances (pg 135)" For anyone who has taken physics, the force between two bodies can be calculated using Newton's universal gravitation equation: F = G[ (m1 * m2 ) / r^2 ] where F is the gravitational pull between the two masses, m1 and m2 are the two masses of each body, respectively, r is the distance between the center of each body and G is the universal gravitational constant (which is the same no matter where you go in the universe). The Earth does not have a uniform density, and because density is the distribution of mass in a given volume, and Newton's universal gravitation equations says the attraction is dependent on mass (and distance) than that means that different locations around Earth will exhibit a stronger gravitational pull.

In today's lecture, Dr. Johnson mentioned that what you feel when standing on the surface of the Earth is the total integrated gravitational pull from the dirt for which you stand on, all the way down to the core. If you have taken integral Calculus than you will know that this scenario is a great application for a definite integral, as the total gravitational pull can be approximated by the sum of tiny changes in distance (and density) as we proceed from the core to Earth's surface. The only challenging part is that not only are we working with changing distance, but we are working with density variations as we proceed deeper into the mantle. We don't have perfect data for the density of material all the way down to the core, so instead scientists use tools such as gravimeters (page 135) to get a /very/ close approximation of the gravitational pull, at different points around Earth. If I have some free time, I might do some research online on the density at various points of depth, and split each up into a separate partition (core, outer core, 3 or 4 places between crust and outer core) and integrate the total gravitational pull for each partition relative to the surface. It would be interesting to see the % error of doing the math myself by researching the required data, and actual calculations done at same points around Earth, using gravimeters.

Detailed maps have been created to show changes in Earth's gravity due to topographic effects and density variations. Scientists have been able to calculate the elevation (above sea level) for specific points for there to be a gravitational equalibrium. Example: Point A on earth (at sea level) has a greater gravitational pull on the same object placed at point B, due to variations in density below point A and B. Therefore, we can calculate the elevation above point A where the grativational pull is equal to point B. This is why certain places around Earth have naturally higher sea levels (concave down) then other places around Earth, for water is being pulled into these points which lie above points of higher density.
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Newton's theory of gravitation is based on "action-at-a-distance". Thus for the gavitational pull of the earth can be calculated as a point mass at the center.

If you consider earth as a sphere - and start to dig inside - you can show that the mass of the earth pulling can be replaced by the mass of the sphere you are standing on. I believe Gauss showed that for electrostatic charge in a non-conducting sphere. That is the ramification of inverse square law - as long as the density is uniform on each surface.
 
The density variations effect Earth's gravity at different spots - that is how scientists have plotted the Geoid (map of density variations based on gravimeter readings)

Below is a picture of earth's gravitational field anomalies due to variations in density.

Geoids_sm.jpg


I want to research data about the density at various depths and integrate accordingly to try and see how it related to actual gravimeter readings. Would my second paragraph make sense?
 
Yes it does - but that is because earth is not a perfect spehere - with its material uniformly distributed. In otherwords:
\(\displaystyle \frac{d\rho}{d\theta}\, \ne \, 0\)

Then ofcourse you'll see such effects.

However if you assume

\(\displaystyle \frac{d\rho}{d\theta}\, \ = \, 0\)

and

\(\displaystyle \frac{d\rho}{dr}\, \ne \, 0\) [that is density varies only in the radial direction]

Then my statement in the post above is true.
 
Jwpaine wrote: “My hypothesis would be that I would not be able to find the gravitational pull by simply integrating Newton's universal gravitation equation for changing depth at several density differences, since there are an infinite number of points within the earth between 90 < t < 0 degree's off vertical to the Earth's core.... so I would have to account for changes in density for mass pulling on me from various angles... right? Not just what is directly under me.”
Click to expand...

I believe you are exactly correct Mr. Paine, and I would add that the calculation you propose is a case of putting the cart before the horse. The only reason we know of the density changes is exactly because of our gravimetric measurements. The dynamics of the fluid portions of the Earth’s interior are not well understood (for example, the mechanism that results in magnetic pole reversals). There are movements within these regions, including upwellings, that result in a variety of manifestations (such as continental drift, i.e., plate tectonics). Concurrent with these movements are density fluctuations (though some of these would only be evident on geologic time scales). The “solid ground” we stand on is really more like pond scum – the least dense material of which the Earth is composed, which naturally floats on its surface. We can only form our theories of the Earth’s interior from limited data – data gleaned from gravimetric measurements (greatly enhanced now by satellite measurements, I believe) and from earthquake events that send shock waves through the Earth and are measured at various points around the Earth. The point being that we only know of the density fluctuations after we take measurements; we cannot use calculations to predict them. You wanted to “to research data about the density at various depths and integrate accordingly”. We can, in fact, only do the opposite: take our measurements, then do our calculations to say what density might be where.
 
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