# How the Archive Calculates Values in the Composite Planet Data Table

The Composite Planet Data table utilizes data and empirically derived relationships from the literature to create a near-comprehensive database of key physical parameters for all confirmed exoplanets and their host stars. While the full set of parameters for a given system may not be internally consistent, as parameter values are often drawn from multiple references, this table allows users to explore overarching demographic properties. Here we describe the underlying logic and formulae used to generate this Composite Planet Table.

## Table Population Using Empirical Data from the Literature

The population of the Composite Planet Table using empirical data from the literature differs in one significant way from that of the Confirmed Planet Table. In the Confirmed Planet table, all parameters with reported values for a given planet are drawn from the same literature reference. While it is thus possible that for some planets there may be parameters with measured values that are not reported in the Confirmed Planet Table, the parameter set that is reported is guaranteed to be internally and physically self-consistent. By contrast, the goal of the Composite Planet Table is to provide users with a table that is as complete as possible. This indicates that the Composite Planet Table will necessarily include parameters from a variety of references for a given planet, indicating that the set of reported parameter values may not be internally or physically self-consistent.

## General Algorithm for Selection of Empirical Parameter Values

The decision tree for populating a given parameter for a given planet in the Composite Planet Table with a value from the literature is as follows. If, for a given parameter for a given planet...

1. ...a value exists and that value is attributed to the default reference in the Confirmed Planet Table, use that value.
2. ...there does not exist a value attributed to the default reference in the Confirmed Planet Table, use the value for that parameter that has the smallest absolute uncertainty. In the case of asymmetric uncertainties, average the uncertainties to estimate the absolute uncertainty.
3. ...there does not exist a value attributed to the default reference in the Confirmed Planet Table and multiple parameter values have equal absolute uncertainties, use the parameter value from the most recent publication.

## Algorithm for Selection of Optical and Near-Infrared (NIR) Magnitude Empirical Values

### Optical Magnitude

1. If a V-band (Johnson) magnitude value exists, use that value.
2. If no V-band (Johnson) magnitude value exists, use the Kepler magnitude value.
3. If no V-band (Johnson) magnitude value exists and no Kepler magnitude value exists, leave the cell empty ("null").

### NIR Magnitude

1. If a KS-band (2MASS) magnitude value exists, use that value.
2. If no KS-band (2MASS) magnitude value exists, use the J-band (2MASS) magnitude value.
3. If no KS-band (2MASS) magnitude value exists and no J-band (2MASS) magnitude value exists, leave the cell empty ("null").

## Table Population using Physically Motivated Mathematical Relationships

Here we describe and explicitly list the formulae used to calculate values for certain planetary or stellar parameters, using the empirically determined values of other parameters reported in the Composite Planet Table as input. But first, a few notes about our parameter value calculations:

• We do not calculate a given parameter value if at least one of the input parameter values required for the calculation is a limit and not a central value.
• We do not report uncertainties for any parameter value that has been calculated.
• The associated reference column for a parameter value that has been calculated reads "Calculated Value," as it is not tied to a literature reference.

### Planet Radius $R_{p}$

For planets that do have an empirically determined value for the planet mass $M_{p}$ but do not have an empirically determined value for the planet radius $R_{p}$, regardless of whether the planet has an empirically determined planet density $\rho_{p}$, we invoke the mass-radius relation of Chen & Kipping (2017) to derive the planet radius $R_{p}$. Chen & Kipping (2017) define a four-term piecewise power law function that spans the masses of all planetary objects in the NASA Exoplanet Archive. The form for deriving a planet radius $R_{p}$, given a planet mass $M_{p}$, is the following:

$$\mathcal{R} = \mathcal{C} + \mathcal{M} \times \mathcal{S},~{\rm where}$$ $\mathcal{R}$ = log$_{10}$($R_{p}$/$R_{\oplus}$), $\mathcal{C}$ = a constant term (in log$_{10}$ units), $\mathcal{M}$ = log$_{10}$($M_{p}$/$M_{\oplus}$), $\mathcal{S}$ = the slope of the power-law relation, $R_{\oplus}$ represents the radius of the Earth, and $M_{\oplus}$ represents the mass of the Earth. We use the following parameters, which are provided in their Table 2 (or derived thereof), to compute a planet radius $R_{p}$ given an input, empirically determined planet mass $M_{p}$: $$\{\mathcal{C};~\mathcal{S}\} = \begin{cases} \{0.00346; \hspace{3.5mm} 0.2790\} & \text{if}~~~M_{p} < 2.04 M_{\oplus} \\ \{-0.0925; \hspace{2.5mm} 0.589\} & \text{if}~~~2.04 \le M_{p} / M_{\oplus} < 132 \\ \{1.25; \hspace{6.2mm} -0.044\} & \text{if}~~~132 \le M_{p} / M_{\oplus} < 26600 \\ \{-2.85; \hspace{6.2mm} 0.881\} & \text{if}~~~M_{p} \ge 26600 M_{\oplus}. \end{cases}$$

### Planet Mass $M_{p}$

For planets that do have an empirically determined value for the planet radius $R_{p}$ but do not have an empirically determined value for the planet mass $M_{p}$, regardless of whether the planet has an empirically determined planet density $\rho_{p}$, we rearrange the form of the mass-radius relation presented in Chen & Kipping (2017) (and reproduced above) to provide a radius-mass relation. The form for deriving a planet mass $M_{p}$, given a planet radius $R_{p}$, is the following:

$$\mathcal{M} = (\mathcal{R} - \mathcal{C}) / \mathcal{S}.$$

However, it is crucial to note that, since the third term of the mass-radius relation has a negative slope (i.e., $\mathcal{S} < 0$), there exists a regime of planet radius $R_{p}$ for which a given planet radius $R_{p}$ does not uniquely map into a single planet mass $M_{p}$. This inherently degenerate regime spans approximately ${11.1 \le R_{p} / R_{\oplus} \le 14.3}$. We thus abstain from calculating a planet mass $M_{p}$ in this regime, which spans multiple orders of magnitude (in planet mass $M_{p}$). The resulting piecewise function is thus:

$$\{\mathcal{C};~\mathcal{S}\} = \begin{cases} \{0.00346; \hspace{3.5mm} 0.2790\} & \text{if}~~~R_{p} < 1.23 R_{\oplus} \\ \{-0.0925; \hspace{2.5mm} 0.589\} & \text{if}~~~1.23 \le R_{p} / R_{\oplus} < 11.1 \\ \{-2.85; \hspace{6.2mm} 0.881\} & \text{if}~~~R_{p} \ge 14.3 R_{\oplus}. \end{cases}$$

### Planet Density $\rho_{p}$

For planets for which the archive has both the planetary radius $R_{p}$ and the planetary mass $M_{p}$, but for which the archive does not have a planet density $\rho_{p}$, the density is calculated using the standard mass-radius-radius volume relationship (assuming the planet is a sphere) to calculate the planet density $\rho_{p}$. The explicit functional form is:

$$\rho_{p} = 3M_{p} / (4\pi R_{p}^{3}).$$

### Stellar Luminosity $L_{*}$

For host stars for which the archive has both the stellar radius $R_{*}$ and the stellar effective temperature $T_{\rm eff}$, but for which the archive does not have a stellar luminosity $L_{*}$, we use the Stefan-Boltzmann Law. The explicit functional form is:

$$L_{*} = 4\pi R_{*}^{2} \sigma T_{\rm eff}^{4}.$$

## References

Chen, J., & Kipping, D. 2017, ApJ, 834, 17