Calculating Exoplanet Properties

After an exoplanet has been identified using a given detection method, scientists attempt to identify the basic properties of the planet which can tell us what it might be made of, how hot it might be, whether or not it contains an atmosphere, how that atmosphere might behave, and finally, whether the planet may be suitable for life. It is often useful to first determine basic properties of the parent star (such as mass and distance from the Earth). This is then followed by the use of planetary detection methods to calculate planetary mass, radius, orbital radius, orbital period, and density. The density calculation will provide clues as to what the planet is made of and whether or not it contains a significant atmosphere.

Mass and Distance of Parent Star

The mass and distance of an exoplanet's parent star must often be calculated first, before certain measurements of the exoplanet can be made. For example, determining the star's distance is an important step in determining a star's mass (see below). Knowing the mass of a star then allows the mass of the planet to be measured, for example when using the Radial Velocity Method.

Distance

Astronomers have several methods to calculate stellar distances; when searching for exoplanets, only the nearest stars are searched, and in this case, the parallax method is the most simple and effective.

The image below illustrates how the parallax method is used to calculate stellar distances. As Earth orbits the Sun, the image of a nearby star will appear to travel across the image of background stars much farther away. The maximum value of this angle, measured at two opposite ends of the orbit, can be used along with the distance from the Earth to Sun to calculate the distance of the earth to the target star using simple geometry.

http://www.le.ac.uk/ph/faulkes/web/stars/r_st_overview.html

Mass

Once the distance of the nearby star is calculated, its mass can then be determined. First, the apparent luminosity of the star, or how bright it is as seen from the Earth, is measured. Using this measurement along with the stellar distance that was calculated previously, the actual brightness of the star can be determined. Since brightness, or luminosity, is directly related to mass for a given star type (see the Hertzsprung-Russel diagram), the calculated brightness can be used to determine its mass. An example of using the mass-luminosity relationship to determine stellar mass is illustrated in the image below. From the image, it can be seen that stars which are twice as massive as the sun, such as Sirius, are more than eight times as bright. That is, we can say that there is a linear relationship between the logarithm of the actual luminosity of a star and the logarithm of its mass.

http://csep10.phys.utk.edu/astr162/lect/binaries/masslum.html

Characterization of the Exoplanet

Once the properties of the parent star are known, certain properties of the exoplanet can be characterized, such as orbital radius, planet radius, and mass.

Orbital Radius

The first calculation comes from Kepler's Third Law (shown below), where 'G'  is Newton's Gravitational Constant.The period, 'P', is the orbital period of the exoplanet, and comes directly from the measured period using, for example, the transit or radial velocity detection methods (Detection Methods page). The mass of the star, 'M', was calculated above using the mass-luminosity relationship of stars. Finally, the mass of the exoplanet, 'm', in the equation can be ignored, since it is much smaller than the mass of the parent star. As an example, since the Sun is about three hundred thousand times heavier than the Earth, ignoring the mass of the Earth in this calculation woud introduce an error of less than 0.001%. The equation can be solved for the only remaining variable which is the orbital radius, 'a'.

https://www.courses.psu.edu/astro/astro001_pjm25/gravitation2.html

Planet Mass

Once the orbital radius has been determined, the mass of the planet can be calculated using Newton's Law of Gravitation shown below. Here, 'G' is once again the Gravitational constant, 'm1' is the mass of the parent star, 'r' is the orbital radius (this was 'a' in the equation above), and 'Fg' is the force of gravity between the parent star and the exoplanet. The force of gravity can be determined from the Doppler shift measured using the radial velocity method. The equation can be solved for the final remaining variable, 'm2', which is the mass of the exoplanet.

http://physics.about.com/od/classicalmechanics/a/gravity.htm

Planet Radius

The remaining properties to be determined are radius and density. To determine the planet radius, the brightness drop of the parent star that occurs during a planetary transit is measured. This brightness drop is directly related to the ratio of the planet radius to the radius of its parent star, as shown in the image below. Note that the calculation to determine the radius of the parent star has not been shown, but it is relatively easily calculated using, for example, the Stefan-Boltzmann Law.

http://www.rmg.co.uk/upload/pdf/KS4_Radius_and_Volume_of_Exoplanets.pdf

Density

Calculating the average density of the exoplanet is a simple matter of dividing the mass by the volume, where the volume is determined using the radius calculated above. 

Drawing Conclusions from Exoplanet Properties

What sort of conclusions can be made about an exoplanet given these properties?

First, the orbital radius plays a key role in determining whether or not the planet can support life. If the planet is too close to its parent star, the planet will be tremendously hot; the planet's molecules will be travelling so quickly, and with so much energy, that most of the chemistry that is seen on Earth will not be possible. Indeed, it is unlikely that any molecules capable of supporting life will form, and certainly not liquid water. Planets that are sufficiently close to their star will also likely be stripped of any atmosphere that they might have had. On the other hand, if the planet is too far away from the parent star, it will be too cold for liquid anything, and liquids are thought to be essential for life. Many scientists assume that liquid water is necessary for life, and if this is true, then there are lower and upper bounds on the orbital radius which would allow the exoplanet to harbour life. These bounds define what astronomers call the 'Habitable Zone' because the temperature is just right for life as we know it.

Second, the planet's mass can determine whether or not the planet can support an atmosphere. Some planet masses are too low, and their gravity is too weak, to hold on to an atmosphere. On a planet without an atmosphere, it is unlikely that complex life can evolve. Other planets are so massive, and their gravity so large, that life may not be possible; if it does exist on these planets, it will likely be quite different than any life seen on Earth, so scientists are not focusing their efforts on these massive planets in their search for life.

Finally, the density can give clues about the planet's composition and whether or not it might have a significant atmosphere. For example, for medium to large sized planets which are not very dense, it is possible to determine that they likely have an atmosphere, or perhaps that their composition is mostly ice and gas. On the other hand, dense planets are more likely to be rocky, and some may even be composed of more exotic materials such as graphite or even diamond.