## What is knowable?

I have been thinking about the limits of what is "knowable" or "measurable." Specifically, I'm looking at this paper: Dickey et al., "Recent Earth Oblateness Variations: Unraveling Climate and Postglacial Rebound Effects", Science, 298, 6 December 2002, 1975-1977. This is a well-known study by a JPL group of the oblateness of the earth based on satellite data. From the abstract:

We have determined that the observed increases in J2 are caused primarily by a recent surge in subpolar glacial melting and by mass shifts in the Southern, Pacific, and Indian oceans. When these effects are removed, the residual trend in J2 (-2.9 x 10^(-11)/year) becomes consistent with previous estimates of PGR from satellite and eclipse data.
So they are saying melting glaciers are helping make the earth fatter near the equator. What's J2, you ask?

J2 is the coefficient of degree 2, order 0 of the non-dimensional spherical harmonic representation of the mass distribution of the Earth system. It is directly related to the diagonal elements of the inertia tensor of the Earth by J2=-(Ixx+Iyy-2Izz)/(2Ma^2) where the z-axis is orientated along the rotation axis, M is the total mass, and a the mean radius.
Ok, I agree that is a reasonable measure of fatness.The paper's data indicate that J2 has varied between -20 and 20 x 10^(-11) between 1984 and 2002. The image shows the graphed data.

Now my question... 10^(-11)??? Is such a tiny dimensionless quantity really measurable like this? Using sattelite data? Suppose I gave you some nearly spherical object, and asked you to measure how different from one the ratio of the longest to shortest axis is, i.e. Along/Ashort-1, a unitless number, not unlike what the JPL group is measuring. You may use whatever measuring device you want, in any lab you like. Is there any measurement system that would give you such an accuracy? 10^(-11) is one in a hundred billion. It's astounding to me.