Extrasolar Wobbles

My notes as I look for planets orbiting GJ 581.

[cesarsul]

Mercury Runs with Mutual Inclinations 4 Planet Model

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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.

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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]

I'm trying to figure out how to write a script to make a chi squared map using the systemic console. There is an example script that does just that but life doesn't ever make things that easy. It took me forever to find this so I'm posting it here so I don't forget.

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[cesarsul]

I was poking around on systemic and I saw that it has something called Markov Chain Monte Carlo in the View menu. I don't know what it does exactly but it looks like you can use the Annealing panel to randomly vary parameters and compare the chi sqr values.

I started from the best fit I could get in systemic chi sqr ~ 2.7 and let the eccentricity, mean anomaly, and longitude of periastron vary. Below are the plots:

[cesarsul]

I've been working on figuring out some way to calculate how radial velocity fits would change given different orbital parameters (that doesn't involve painstakingly typing and recording numbers) and I cam across this. It goes over how where the fit models come from and how to derive them. My life is now better.

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Eccentricity Shifts and Survival Time

Sometimes during simulations the eccentricities of the planets will shift. I wanted to see if the change in eccentricity had anything to do with how long the simulation would last.

Here are the plots:

For the most part the planets didn't change their eccentricities during their simulated lifetime. I filtered out all the runs with no change:

Doesn't look like eccentricity shifting has anything to do with anything.

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Had a group meeting today and we talked about what a high eccentricity is. I made this histogram out of curiosity. The data comes from exoplanets.org, a place that keeps data on exoplanets.

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How close do the planets in the 4 planet model get to planet e and does it affect the simulation lifetime?

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A Strange Triangle

Last time I had some curious data that looked like it lived inside a triangle. I don't really know what to make of it but here is basically what I've done since. The first thing I did was put the data into bins. The gridlines show the bin edges.

I then kept track of the number of points in each bin and got a kind of two dimensional histogram.

The colors represent the count. This plot is kind of like a density plot. I thought I could find a spot where the density starts to drop quickly drop and just call that an edge. It turns out that was really hard to do but here are some cool plots I made:

The dumbest thing I could think of to do was to use Mathematica's Manipulate function to graph a line and eyeball the right slope and intercept.

I put in some stuff to try to find the triangle that included the most points while using the least area. This is the best line I could come up with.

This plot looks very similar to one I made a while ago about allowable eccentricities. In that case the slope had to do with the ratio of semi major axes of the orbits. I'm not sure what this line represents or if it is even significant so I think I'm just going to start spending my time on something else.

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I wanted to see if the minimum orbit separation between planets E and B in the 4 planet model had any affect on how long the simulations ran.

That is distance is shown below in red.

After a few Mathematica commands I came up with this

The further apart planets E and B are from each other, the longer the system tends to live. What's most interesting is that weird cut off at about 0.006 AU. If the planets are separated by at least this distance, they get to live much longer lives.

To get a better idea of what's going on, I picked out those special Mercury runs and put them on top of a contour plot.

I'm still new at tumblr so I'm not sure how to make these plots bigger but I'll re-upload them as pictures so it's easier to see them. 

Here is a contour plot I posted earlier.

Just as a reminder, the colors represent how long the Mercury runs lasted . Here is the same contour plot with those weird data.

All these runs seem to live inside a triangle. It's very similar to the plots I made about allowable eccentricities. Hmmm... I wonder what's the deal with that edge in the scatter plot and if that slope is significant in any way.

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I'm just reposting these tables so it's easier to read them.

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Moving to the five planet model

GJ 581 could have between 4 and 6 planets so I've started running simulations based on the 5 planet solution proposed by Vogt et al. 

I started out by putting the 5 planet circular fit parameters from Table 9 into Systemic, which is a tool that can be used to fit radial velocity data. The orbits were allowed to be eccentric and I got the resulting parameters:

I set up some Mercury runs where the eccentricities were allowed to vary 50% from this table. I've done 1000 runs so far and most of them made it to the end (no close encounters).

I noticed that the eccentricities were really low, which is probably why so many of the runs survived. I started playing with the fits in Systemic and got a new one where the eccentricities are higher.

I started some new runs based on this table last night and as of this posting 12 out of a thousand are still going, which is more like what I was getting when I was running 4 planet simulations.

The time it takes for the Mercury runs to finish has almost doubled. It makes sense because adding a new planet SHOULD require more computer time but today I'm going to recheck to make sure I haven't typed something wrong.