Determining where no correlation exists I need to determine at what value of a variable mictrb there would be almost no correlation between two other variables: Equivalent width and Abundance (description and graphs below).
I have been told that when mictrb = 0.9, then there is no correlation between Equivalent Width and Abundance. The first chart below, I've been told, shows this. The next two charts show different values of 'mictrb' and the correlations seen between Equivalent width and Abundance.
The technique often used here is to adjust mictrb to remove any trend in abundance as a function of Equivalent width.
Could someone describe what I'm looking at here? What is it about the first chart which indicates there is no correlation between the two axes, whilst the other two charts don't show this?
mictrb is the micro-turbulence present in the atmosphere of a star.
Equivalent width is the width of an element's waveform, think sine-wave (x-axis)
Abundance is the amount of that element present in the atmosphere (y-axis).




 A: I think someone is confused here.  Looking at the figures you provide, it is clear that there is a correlation between Equivalent-width and Abundance when mictrb=.9.  The top figure plainly states that the correlation is -.43, and the probability of getting a value that far from 0, if 0 were the true value, is .01; looking at the scatterplot, that seems correct, the points tend to get lower as you move from left to right.  On the other hand, as @MansT points out, the last figure reports that when mictrb=0, the correlation is -.2, and the p-value is .27.  (I should note in passing that I'm not sure if the data look bivariate normal, and that the true relationship appears somewhat curvilinear in all three plots.)  
In general, you can think of this as a multiple regression scenario in which you want to describe how Abundance is related to Equivalent-width, micro-turbulence, and the interaction between them:
$$
\text{log}_{10}A=\beta_0+\beta_1\text{mictrb}+\beta_2w_\lambda\text{[A]}+\beta_3\text{mictrb*}w_\lambda\text{[A]}
$$  

(Sorry, I can't figure out the mathjax for Angstrom, maybe I'll edit it later.)  

A first approximation of the level of mictrb where there is no correlation between Equivalent-width and Abundance would be to solve for the value of mictrb that causes the last two terms to cancel each other out.  For instance, if $\beta_2=1$ and $\beta_3=2$, then there should be no relation between Abundance and Equivalent-width when mictrb = -.5 (I suspect it doesn't make any sense to have a negative Micro-turbulence, this is just an example, and that would imply that they are never truly uncorrelated.)
