Code

The larger question is whether we could find a certain sized companion planet to every system out to a given AU, maybe around 10 AU. The answer is almost certainly not, but it might have interesting implications for exoplanet research. To understand this problem a useful dispersion (maybe around RV 5, as 5 m/s is easy to get to at the 12th magnitude) and a useful sensitivity to planet mass (e.g. ~10 Jupiter masses, which is too large - maybe a tenth of that) had to be found. 

Initially, then, I found the relationship (a scaling relationship) for the mass of a planet given dispersion of its linear radial velocities (sigma), the minimum period 4*
T (taken from the baseline time from the observations) and the total number of observations N (the signal-to-noise ratio scales to the number of observations to the -1/2 power), given an assumed solar mass approximately equal to that of the sun, a circularized orbit, and a signal to noise ratio of S/N = 10. The sensitivity is below.

To better understand the theoretical considerations involved in determining the sensitivity. To do that we asked one of Caltech’s dynamicists, Konstantin Batygin. Since we are interested in learning physical things about planets, we are looking for cold friends that will give the Hot Jupiters some detectable eccentricity (which we can later on convert into a Love number). So, we are looking for eccentric outer planets. Luckily, there exists a relationship between the eccentricity ratio and the semi-major axis ratio of the pair of planets (see equation 19 of http://arxiv.org/pdf/1102.0274v1 - betas correspond to the eccentricities. 

e1/e2 ~ a1/a2

for tidally relaxed systems, or age of the system exceeding three circularization timescales given by eq.13 of http://arxiv.org/pdf/1102.0274v1. This suggests that s long as the outer companion is not much much less massive than the Hot Jupiter, the only thing that matters is the semi-major axis ratio. Consequently, larger a1/a2 is favorable for larger e1.

Bottom line:

• mass: as long as the companion is a gas giant/brown dwarf, we're good - not very sensitive to this 
• semi-major axis:1 AU is much better than 5AU. I wouldn't bother looking at 10 AU companions - e1/e2 is too small to do anything useful with.
• timescale: the Hot Jupiter itself should have had time to reach dynamical equilibrium i.e. star age >> 1 circularization timescale. 
• eccentricity: in order for the interior calculation trick to work, the outer planet must be pretty damn eccentric, so that has its own implications for the shape of the RV signal.

Our ultimate goal is to constrain the existence of x Jupiter mass planets, within y AU (maybe 2 or 5). I explored this idea with an IDL program, which inputs an RV amplitude we will be able to achieve a precision – which is dependent on the V mag of the star and the exposure time (i.e. the number of photons to reach the telescope), and the number of observations. By changing the precision, we will take a random sampling of RV measurements, and taking the population of these binned into a histogram, we can visualize the final distribution and how accurate the observations are to the actual radial velocity trend. I accomplished this in my rvmontecarlo.pro, but am trying to rewrite the code to do it backwards (more on that later).

I will try to do a monte carlo analysis, much simpler than that of in Crepp & Johnson 2011, drawing for such factors as the period of the orbit, the mass of the planet, eccentricity and others. This will have to fit with the aim of RV  4m/s and the exposure time of under 15 minutes per system. We have a total time of about 20 hrs per year, and the overhead per star is around 90 seconds/exposure, so ultimately the plan is to find stars that have interesting characteristics and that can be observed within the Keck time allotted.

It should be relatively straightforward to do this many times, assuming an observing window, and drawing randomly from an eccentricity distribution, and picking a random phase. I should try create some kind of mass-period efficiency plot for the CPS observations. This would be a really valuable exercise for the group and would constitute a real accomplishment for a SURF student.