Transiting Planets and Stellar Radial Velocities

Ay 20 – Transiting Planets

Problem 1

Primary author:  Joanna Robaszewski

Secondary author:  Cassi Lochhaas

Abstract

This problem looks into one of the equation governing the relationship between the masses of a planet and a star orbiting their center of mass and the radial velocity of the star.

Introduction

As a planet orbits a star, the planet has some effect on the motion of the star.  The change in the motion of the star can be inferred through the Doppler effect.  When the star moves away from the observer, its emissions are measured to have a longer wavelength.  When the star moves toward the observer, its emissions have a shorter wavelength.  Measuring the differences caused by the planet in the radial velocity of the star allows observers to better estimate the mass of the planet, provided the star’s mass is already known.

Questions and Results

Below are the radial velocity time series for two star-planet systems:

Radial velocity is measured in meters/second on the y-axis and time is measured in years on the x-axis.  We assume that we are viewing the system edge on.  In the first plot, the star has a mass of 1.8 solar masses.  In the second plot, the star has a mass of 1.7 solar masses.  Using the equation:

where M_p is the mass of the planet, M_* is the mass of the star, V_* is the maximum radial velocity of the star found in the plots, P is the period of the planet around the star, and G is Newton’s gravitational constant, we want to find the mass of the planets of the two systems.

In the case of the 1.8 solar mass star we have:

For the 1.7 solar mass star, we find that the planet has a mass of:

Once again, my calculations don’t quite match up with what Cassi and I found in class.  This might just be from me estimating the period of the planet’s orbit from the graphs slightly differently than before.  While working this problem out in class, we found that both the planets had masses on the order of 10³⁰ grams, which is the mass of Jupiter.  This is reasonable because many exoplanets discovered are about Jupiter size.  In either case, the equation shows that the larger the ratio of the planet’s mass over the star’s mass, the larger the star’s maximum radial velocity.  The period also plays a part in determining the radial velocity of the star, but I’m not entirely sure as to why an increase in the period would also cause an increase in the radial velocity.  It may have something to do with the fact that the planet has more time to pull the star in one direction, because it is spending more time in one general direction from the star.  I’ll think about it.