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# Exoplanets Density Distribution

A histogram showing the distribution of the average density of the confirmed exoplanets in a subset of the sample that have both measured radii and mass lower limits (as of 5 August, 2012). Since the masses are lower limits, the densities are also lower limits. Note that a small fraction of exoplanets have densities that are greater than 1.2 times that of Earth and are not shown in this histogram in the interest of clarity (but all the densities are shown in the plot of density versus semimajor axis).

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Plotted is the percentage of the subsample of exoplanets in each density interval. Densities are shown as ratios of the exoplanet densities to the density of the Earth (i.e., densities are shown in units of the density of the Earth). The densities are calculated from radii and mass lower limits, which are from the Extrasolar Planets Encyclopedia. For comparison, the densities of the planets in our solar system are marked as Me, V, E, Ma, J, S, U, N, corresponding to Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, and Neptune, respectively (numerical values of radii and masses used to calculate the densities are from solarsystem.nasa.gov).

For convenience, I have made the data available in an EXCEL file so that you can play around with it yourself and make your own plots and/or do your own analysis. It should also be useful for educational purposes, in terms of assignments, tests, demonstrations, etc.

Possible selection effects should be considered when interpreting the exoplanets density distribution. Read more about exoplanets, and the mass distribution with Exoplanets and Alien Solar Systems, which includes comprehensive references to the scientific literature, and discussion of selection effects.

### Calculation of the Exoplanet Densities (Advanced)

This section requires some knowledge of basic physics.

The Extrasolar Planets Encyclopedia has mass and radius values listed in units of the mass and radius of Jupiter respectively. The densities relative to Earth were calculated by filtering the data table for rows that have measured values of both mass and radius and then applying the formula derived below. Here $M$, $R$, and $\rho$ are exoplanet masses, radii, and densities respectively, and the subscripts J and E indicate quantities pertaining to Jupiter and Earth respectively.

\begin{align} \frac{\rho}{\rho_{\rm E}} \ & = \ \frac{M}{(4/3)\pi R^{3}} \ \div \ \frac{M_{\rm E}}{(4/3)\pi R_{\rm E}^{3}} \\ & = \ \left(\frac{M}{M_{\rm E}}\right) \left(\frac{R_{\rm E}}{R}\right)^{3} \\ & = \ \left(\frac{M_{\rm J}}{M_{\rm E}}\right) \left(\frac{R_{\rm E}}{R_{\rm J}}\right)^{3} \left(\frac{M}{M_{\rm J}}\right) \left(\frac{R_{\rm J}}{R}\right)^{3} \end{align}

Now from solarsystem.nasa.gov we have

$\left(\frac{M_{\rm J}}{M_{\rm E}}\right) \ = \ 317.828, \\ \left(\frac{R_{\rm E}}{R_{\rm J}}\right)^{3} \ = \ \left(\frac{1}{10.9733}\right)^{3}$

so that

$\frac{\rho}{\rho_{\rm E}} \ = \ 0.24054 \left(\frac{M}{M_{\rm J}}\right) \left(\frac{R_{\rm J}}{R}\right)^{3} \ .$

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Note: The histogram utilizes data from 241 exoplanets, taken from a snapshot (on 5 August, 2012) when there were 777 confirmed exoplanets in total (residing in 623 alien solar systems, 105 of which harbored more than one exoplanet).
File under: What is the density distribution of exoplanets? What is the density distribution of extrasolar planets? What are the densities of exoplanets? How dense are exoplanets?