GG450 Lecture 3 Gravity -2: Gravity Potential Jan 13, 2006

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GRAVITY POTENTIAL One of the most powerful tools in gravity analysis is the gravity potential. While the acceleration is a vector field where the effects of different bodies must be calculated by 3-d vector addition, the potential is a scalar field whose value at any point depends only on the masses and distances to all bodies considered.

To obtain the gravitational acceleration at any point, all you need to do is calculate the gradient of the potential. The acceleration in any particular direction is the derivative of the potential in that direction. Thus, we can sum up all the effects of the bodies of interest and then take the derivative in the vertical direction to obtain the gravity anomaly caused by those bodies that would be measured by a gravimeter. So, what IS the potential? For a POINT MASS: (2.5)

The gravitational acceleration is then the gradient of the potential, or: If we only care about the acceleration in the vertical direction - which is the case for gravity measurements taken at the surface of the earth - then What makes this so interesting is that these values are easy to calculate in Matlab, and the effects of each mass are independent of all other masses, so we can easily add and remove masses to generate models without having to worry about the directions to those masses.

Consider the gravity field generated (in 2 dimensions) by three point masses: % gravpot % gravity potential % calculates 3-body potential and calculates the gradient in two dimensions clear all % mass values and locations: pointmass=[ ];% masses of bodies xy=[ ; ; ]; % calculate potentials on 10x10 grid pot=zeros(100,100); %100x100 grid X=[1:100]; Y=X;

for ii=1:100 I=ii; % available for scaling for jj=1:100 J=jj; % available for scaling for kk=1:3 % calculate distance from mass to test point r=sqrt((xy(kk,1)-I)^2+(xy(kk,2)J)^2); % calculate potential (ommiting gamma for ease) pot(ii,jj)=pot(ii,jj)+pointmass(kk)/r; end potlog=log10(pot); % get log of potential for contours contour(X,Y,potlog,20) % potential contour plot

These contours show lines of equal potential from three point masses. They are called equipotentials. Whenever you move along an equapotential, you do no work. A ball rolling along an equipotential will not accelerate. The “up” direction is perpendicular to the equipotential lines. Why plot the log of the potential?

The following commands calculate the gradient and plot directions for the gradient: hold on [accelx accely]=gradient(pot,10000); % get gradient of the potential % go through next loop to normalize for constant-length arrows for ii=1:100; for jj=1:100; atn=atan2(accely(ii,jj),accelx(ii,jj)); accely(ii,jj)=sin(atn); accelx(ii,jj)=cos(atn); end quiver(X,Y,accelx,accely,.5); % plot directions of acceleration axis equal

The arrows show the directions of acceleration at each point on a grid. Note the “zero” point between the two masses.

The GEOID: The geoid is the shape of the earth's surface defined by an equipotential surface - or a surface where the gravity potential is a constant. On earth, sea level closely approximates the geoid. What will happen to sea level above a big hole in the ocean? Until recently, the shape of the geoid was unknown and extremely difficult to measure, since the direction "up" is defined to be perpendicular to the geoid, it is difficult to measure the geoid topography - much like it was difficult to figure out that the earth isn't flat.

Satellites are strongly affected by gravity, and small changes in their orbits are used to show changes in the geoid.

Short wavelength geoid undulations reflect ocean depths and tectonic variations, while the source of some of the very long wavelength undulations are more likely related to undulations in the core-mantle boundary.

It is often easy to obtain a good estimate of the vertical component of gravity by obtaining the difference in potential between two points separated by a small vertical distance: See "Gravity Above a Sphere" Excel file for g calculated directly and using potential. This method works particularly well for calculating anomalies from large numbers of discrete bodies or shapes.