Methods

Chapter IV: On The Shoulders of (Some) Giants: The Relationship Between

4.3 Methods

The primary goal of this work is to measure planet occurrence, particularly of small close-in planets and cold gas giants. Many studies of RV or transit surveys use the

intuitive occurrence measurement method known as “inverse detection efficiency”

(Howard, Marcy, Bryson, et al.,2012; Petigura, Howard, et al., 2013). According to this procedure, one estimates occurrence in a region of parameter space by counting up the planets found in that region, with each planet weighted by the search completeness in that region. We measured the search completeness map of our survey by injecting many synthetic signals into each dataset, and computing the fraction of signals in a given region that are recovered by our search algorithm, RVSearch. Inverse detection efficiency as defined in Foreman-Mackey, Hogg, and Morton,2014 is actually a specific case of a Poisson likelihood method, in which one models an observed planet catalog as the product of an underlying Poisson process and empirical completeness map.

Following the analysis in Fulton, Rosenthal, et al., 2021, we used the Poisson likelihood method to model the occurrence of planets. Given a population of observed planets with orbital and 𝑀sin𝑖 posteriors {ω}, and associated survey completeness map𝑄(ω), and assuming that our observed planet catalog is generated by a set of independent Poisson process draws, we can evaluate a Poisson likelihood for a given occurrence modelΓ(ω|θ), where 𝜽is a vector parameterizing the rates of the Poisson process. The observed occurrence ˆΓ(ω|θ) of planets in our survey can be modeled as the product of the measured survey completeness and some underlying occurrence model,

Γ(ωˆ |θ) =𝑄(ω)Γ(ω|θ). (4.1)

The Poisson likelihood for an observed population of objects is

L =𝑒⁻

∫ Γ(ω|θ)ˆ 𝑑ω 𝐾

Ö

𝑘=1

Γ(ωˆ 𝑘|θ), (4.2)

where 𝐾 is the number of observed objects, and ω𝑘 is the 𝑘th planet’s orbital pa- rameter vector. The Poisson likelihood can be understood as the product of the probability of detecting an observed set of objects (the product term in Equation 2) and the probability of observing no additional objects in the considered parameter space (the integral over parameter space). Equations 1 and 2 serve as the foundation for our occurrence model, but do not take into account uncertainty in our measure- ments of planetary orbits and minimum masses. In order to do this, we use Markov Chain Monte Carlo methods to empirically sample the orbital posteriors of each

system (Foreman-Mackey, Hogg, Lang, et al.,2013a; Fulton, Petigura, Blunt, et al., 2018; Rosenthal, Fulton, et al.,2021). We can hierarchically model the orbital pos- teriors of each planet in our catalog by summing our occurrence model over many posterior samples for each planet. The hierarchical Poisson likelihood is therefore approximated as

L ≈𝑒⁻

∫ Γ(ω|θ)ˆ 𝑑ω 𝐾

Ö

𝑘=1

1 𝑁_𝑘

𝑁_𝑘

Õ

𝑛=1

Γ(ωˆ ^𝑛

𝑘|θ) 𝑝(ω^𝑛

𝑘|α), (4.3)

where𝑁_𝑘is the number of posterior samples for the𝑘th planet in our survey, andω^𝑛

𝑘

is the𝑛th sample of the𝑘th planet’s posterior. 𝑝(ω|α)is our prior on the individual planet posteriors. We placed uniform priors on ln(𝑀sin𝑖) and ln(𝑎). We usedemcee to sample our hierarchical Poisson likelihood, and placed uniform priors on𝜃. Approach to planet multiplicity

We want to evaluate the link between the presence of any inner small planets and the presence of any cold gas giants. Therefore, for all combinations of the presence or absence of these two planet types, we are interested in estimating the probability that a star hosts at least one planet. This quantity is distinct from the number of planets per star, both because many stars host more than one small planet (Howard, Marcy, Bryson, et al., 2012; Fang and Margot, 2012; He, Ford, Ragozzine, and Carrera, 2020) and because the probability of hosting at least one planet must be less than 1. We attempt to resolve this issue with two constraints on our model.

First, we place a hard-bound prior on the integrated occurrence rate, so that it has an upper limit of one planet per star. Second, in the case of planetary systems that contain multiple detected planets in the class of interest, we only count the planet that was first detected by our search algorithm. We also report expected number of planets per star in Table4.2, by including all companions in multi-planet systems.

The resulting estimate of the probability that a star hosts at least one planet de- pends on the search completeness in the mass and semi-major axis range of each individual planet. This biases our sample towards planets with greater RV semi- amplitudes, which tend to be closer-in and higher-mass. These planets are in higher- completeness regions and therefore will usually be detected first by iterative search algorithms. Figure4.2shows the observed multiplicity of the detected small planets in our sample. Note that this distribution is not corrected for search completeness, so it cannot be interpreted as the true underlying multiplicity distribution. Rather, it

is showing how many multi-planet systems we detect with respect to systems where we only detect one small planet.

1 2 3 4 5

Observed Small Planet Multiplicity 0

5 10 15 20

Number of hosts

Figure 4.2: A histogram of observed small planet multiplicity in our sample. This is not corrected for search completeness, so it should only be interpreted as the multiplicity of detected planets, not as the underlying multiplicity distribution.

There are 719 total stars in the CLS, around 29 of which we have detected small planets.