#contour plot

[instructorspockie]

Now that I work at a high performance computing center I can submit Mathematica notebooks to the cluster.

[instructorspockie]

Point charge potential on hexagons.

[cesarsul]

Confidence Intervals

Our N-body simulations show that the innermost planet of the GJ 581 system (planet E) needs to have an eccentricity below .2. Any higher and planets E and B get so close to each other that they can interact and make the system unstable. Last time I posted some chi^2 maps that show where the best fits to the radial velocity data are. 

It all boils down to this: The simulations say eccentricity of planet E should be below 0.2 and the radial velocity measurements say an eccentricity of ~ 0.3 best explains the data. How far away from the best fit value can we go before we are being unreasonable?

Using an F-test we can compare different models where the only variation is the eccentricity of E. The F-Test will tell us how likely it is that the models are the same.

This plot tells us that it's about 95% likely that a fit with eccentricity of .3 and .2 are the same.

[cesarsul]

Systemic Scripting

After some talking with the maintainer of Systemic I got a script to make chi squared maps. What you see is a plot of how the chi squared value changes when you change orbital parameters of planet e.

I was also able to recreate the Eccentricity of E vs Chi^2, Stellar Jitter, and RMS plots in a much less tedious way!

They aren't the easiest to read so I'll repost larger versions as photos.

[cesarsul]

New set of data where the distribution of eccentricities looks like this:

The most important thing to notice is that E has been allowed to vary over a much larger range to get a better idea of how its eccentricity affects the system.

Since I'm only interested in the the eccentricities of planets B and E, I'll only include the contour plot for those two planets.

Here are the scatter plots:

r^2 = 0.825

r^2 = 0.392

The green line represents the time of a stable run.

Using this data set, it looks like increasing the eccentricity of B tends to make the system more stable but E seems to have the most say.

[instructorspockie]

Interpolating Polynomial

  Let's say you have a function that looks something like this:

You can create what's called an interpolating polynomial by picking some points along the curve and make a new curve that has to go through each of the points.

Sometimes the interpolating polynomial is a good approximation and sometimes it's not. Interpolating polynomials aren't limited to two dimensions.

   Sometimes the interpolating polynomial is a good approximation and sometimes it's not. 

What happens when you try this with images? I don't know yet but this contour plot is pretty cool!

 Those aren't ants on a tree stump, they're random samples!