Wednesday, August 25, 2021

How to calculate the mass and size of an exoplanet?

In this article we will try to explain and discuss how to calculate the mass and size of an exoplanet using different methods used for detecting them. So let us start!

Calculating mass of an exoplanet


First of all we will see how to calculate the mass of an exoplanet. The mass of an exoplanet can be calculated from the radial velocity curve of the star due to its motions induced by the exoplanet using the formula



Here P is the orbital period of exoplanet around its host star and G is the universal gravitational constant. However, this equation will only give us the mass of the exoplanet if it is orbiting in the same plane as that of the star. If the planet is aligned at some angle with the plane of the star, then the velocity V_star will change according to the formula



So accordingly the mass of exoplanet will also change like this



This is the reason why the observed masses of exoplanets are written as MPlanet sin i or just M sin i. But how the astronomers determine the inclination of the exoplanet. The inclination of the host star’s rotational axis can be measured through spectroscopic measurements of its projected rotational velocity and from knowing its rotational period and radius. If the planetary orbit is assumed to lie in the same plane as the star’s equator, just like planets of the solar system and the Sun, then sin i may be calculated and the actual mass of the exoplanet can be easily determined.


However, according to some recent studies of transiting exoplanets, it was found that about one third of exoplanet systems have a significant variation between the star’s spin and the exoplanet’s orbit. However, such exoplanet systems are found mainly for the massive stars with large planets. Therefore, the above method of determining star’s rotation axis inclination to determine the rotation axis of exoplanet’s orbit is only used for lower-mass host stars.


Interestingly there is another method of determining the inclination of exoplanets orbit and that is using the transit method. Here you can see the transit curve of an exoplanet
Credit :-NASA


The inclination of the exoplanet can be calculated using the formula



here TD is the duration of the transit in seconds, P as the orbital period of both star and planet, a is the semi-major axis of the exoplanet’s orbit in meters and Rstar is the star’s radius. The orbital period P can be calculated using the formula



And the semi-major axis of the exoplanet’s orbit can be calculated using the formula



Now we know how to calculate the mass of exoplanet and the inclination of exoplanet’s orbit. Let us see how to calculate the size of the exoplanet. If assume that at the center of the transit whole disk of the exoplanet is silhouetted against the surface of its host star, then the exoplanet’s radius can be calculated using the formula



here TA is the amplitude of the transit variation which is expressed as a fraction of the star’s brightness outside of the transit.

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