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# Exoplanets Surface Gravity Distribution

A histogram showing the distribution of the surface gravity (lower limits) of the confirmed exoplanets in a subset of the sample that have both measured mass lower limits and radius measurements (sample snapshot from 5 August, 2012). It is important to remember that most exoplanet masses are minimum values, so consequently the surface gravity is also a minimum value (it cannot be measured directly for exoplanets but has to be inferred from the planet's mass and radius). Currently, the bulk of exoplanets do not have both mass and radius measurements, and on that date only 31% of confirmed exoplanets had measurements of both parameters. Plotted is the percentage of the subsample of exoplanets in each surface gravity interval. Surface gravities are shown as ratios of the exoplanet surface gravity to the surface gravity Earth (which is 9.80665 m s-2 - source: solarsystem.nasa.gov). Data are from the Extrasolar Planets Encyclopedia. For comparison, the surface gravities of the planets in our solar system and of Pluto are marked as Me, V, E, Ma, J, S, U, N, P corresponding to Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune and Pluto, respectively (numerical values are from solarsystem.nasa.gov). Earth's Moon would be positioned at about 0.16 (on the horizontal axis) in this diagram.

You can get instant access to the book Exoplanets and Alien Solar Systems:

For convenience, I have made the surface gravity 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.

In order to determine the weight of an object on a planet, simply multiply the planet's surface gravity relative to Earth by the objects weight on Earth. In other words, take a number on the horizontal axis and multiply it by the object's weight on Earth. Another way of putting it is that the numbers shown on the horizontal axis (surface gravity relative to Earth) respresent the fraction of an object's weight on Earth that would be measured on the surface of another planet. Notice how similar the surface gravity is for five out of nine of the solar system objects, and except for Pluto, the remainder have a surface gravity within a factor of 3 of that of Earth.

The peak of the surface gravity distribution for the sample of exoplanets is between 1 to 3 times the surface gravity of the Earth. As it happens, the distribution shows that the bulk of the exoplanets in the subsample have a surface gravity between a third and three times that of Earth. However, a significant fraction has a surface gravity greater than 3 times that of Earth.

One must take into account possible selection effects when interpreting the surface gravity distribution. A more detailed discussion, along with comprehensive references to the scientific literature can be found in the book Exoplanets and Alien Solar Systems.

The surface gravity distribution histogram was made by downloading the overall basic data file at Extrasolar Planets Encyclopedia as an excel (.csv) file, filtering rows that have both mass and radius measurements, followed by operating on the mass and radius columns as described in the technical section below.

## Calculation of Surface Gravity of Exoplanets

This section requires some knowledge of elementary physics.

The surface gravity for a spherical mass, $M$, is given by

$g \ = \ \frac{GM}{R^{2}}$

where $G$ is Newton's gravitational constant, and $R$ is the radius of the sphere. If we want to express the surface gravity as a fraction of the value for Earth ($g_{\rm E}$), we get

$\frac{g}{g_{\rm E}} \ = \ \left(\frac{M}{M_{\rm E}}\right) \left(\frac{R_{\rm E}}{R}\right)^{2}$

or

$g \ = \ 9.80665 \left(\frac{M}{M_{\rm E}}\right) \left(\frac{R_{\rm E}}{R}\right)^{2} \ \ {\rm m \ s^{-2}} \ .$

However, this equation assumes that $M$ and $R$ are in the same units as $M_{\rm E}$ and $R_{\rm E}$ respectively. The data table at Extrasolar Planets Encyclopedia has mass and radii in units of Jupiter masses and radii respectively, so we need to apply a transformation. We multiply by

$\left(\frac{M_{\rm J}}{M_{\rm J}}\right) \left(\frac{R_{\rm J}}{R_{\rm J}}\right)^{2} \ \equiv \ 1$

where $M_{\rm J}$ is the mass of Jupiter and $R_{\rm J}$ is the radius of Jupiter. The masses and radii of exoplanets in the data file are given as $(M/M_{\rm J})$ and $(R/R_{\rm J})$. Therefore,

$\frac{g}{g_{\rm E}} \ = \ \left(\frac{M_{\rm J}}{M_{\rm E}}\right) \left(\frac{R_{\rm E}}{R_{\rm J}}\right)^{2} \left(\frac{M}{M_{\rm J}}\right) \left(\frac{R_{\rm J}}{R}\right)^{2} .$

Now from solarsystem.nasa.gov we have

$\left(\frac{M_{\rm J}}{M_{\rm E}}\right) \ = \ 317.828,$ $\left(\frac{R_{\rm E}}{R_{\rm J}}\right)^{2} \ = \ \left(\frac{1}{10.9733}\right)^{2}$

so that

$\frac{g}{g_{\rm E}} \ = \ 2.63948 \ \left(\frac{M}{M_{\rm J}}\right) \left(\frac{R_{\rm J}}{R}\right)^{2}$

or

$g \ = \ 25.8845 \ \left(\frac{M}{M_{\rm J}}\right) \left(\frac{R_{\rm J}}{R}\right)^{2} \ \ {\rm m \ s^{-2}} \ .$

Bear in mind that the surface gravities calculated in this way are necessarily only nominal estimates because a perfect, uniform sphere is assumed. The surface gravity of Jupiter calculated using the above equation is higher than the measured value by just over 4%. For exoplanets, at the moment the simple estimate is the best that can be done because it is not yet possible to make more detailed measurements than a mass and radius. A larger source of error is uncertainties in the masses and radii. The masses are generally only lower limits, and the radii typical have uncertainties of tens of percent or even higher.

It is also interesting to express the surface gravity in terms of the average density, $\rho$, by using $M = (4/3)\pi R^{3} \rho$:

$g \ = \ 4\pi R \rho$

and, for the ratio of the surface gravity relative to the value for Earth,

$\frac{g}{g_{E}} \ = \ \left(\frac{R}{R_{E}}\right) \left(\frac{\rho}{\rho_{E}}\right) \ .$

The last equation means that of you grow a sphere at constant density, its surface gravity increases linearly with the radius because the increase due to mass is more than the decrease due to radius (size).

Note: The data in the surface gravity distribution are from a time when the total number of confirmed exoplanets was 777, and these resided in 623 alien solar systems, 105 of which were multiplanet systems. The subsample that had both mass and radius measurements was composed of 241 exoplanets.
File under: What is the surface gravity distribution of exoplanets? What is the surface gravity distribution of extrasolar planets? What are the surface gravities of exoplanets? How strong is the surface gravity on exoplanets?