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Exoplanets: Detection via Doppler Shift (Wobble method)

Developed by Deva O'Neil - Published November 27, 2023

The orbits of a star and an exoplanet is modeled using the Euler-Cromer method. Students compute the motion of the star and calculate the Doppler shift of the observed Hydrogen spectrum. The Bohr model is used to compute the spectral lines of Hydrogen.
Subject Areas Modern Physics, Waves & Optics, Astronomy, and High Energy / Nuclear / Particle First Year and Beyond the First Year Glowscript Students will be able to: * Apply Newton's laws to find the planet's force on the star (Exercise 1) * Use the Bohr model to calculate the expected frequency of a Hydrogen absorption line (Exercise 2) * Apply the Doppler shift formula for a moving non-relativistic source to determine the frequency observed as a result of the star's motion (Exercise 3) 60 min

These exercises are not tied to a specific programming language. Example implementations are provided under the Code tab, but the Exercises can be implemented in whatever platform you wish to use (e.g., Excel, Python, MATLAB, etc.).

Astrophysicists use several methods to detect the presence of planets around stars outside our solar system. These exoplanets are difficult to observe directly; it is usually easier to detect the light from the stars that they orbit. The light emitted by the stars gives hints of the planet's presence. In this activity, we will calculate how the frequency of the light emitted changes as a result of the planet's pull on the star. *Exercise 1* A planet's orbit around a star is a visible result of its gravitational interaction with the star. But forces are reciprocal: a star that exerts a force on a planet must have an equal and opposite force acting on it due to the planet. Although not as obvious, a star with a planet in its system exhibits a wobble due to the pull of the planet. This wobble provides a way to detect an exoplanet's presence. The template provided exhibits the force on a planet (mass $m$) due to a star (mass $M$). The planet is chosen to be unrealistically large, one-tenth the mass of the star, to make the wobble of the star easily visible. The code does not update the star's motion to reflect the planet's presence; that is your task in this exercise. Each time through the loop, 1. Determine the acceleration of the star 2. Use the acceleration to update the velocity of the star 3. Use the velocity of the star to update the position of the star If you are successful, your graph will exhibit the z component of the star's velocity. _Plausibility Checks_ Verify the following: 1. The planet and the star should trace out circular orbits around a common center-of-mass that does not move. Rotate the viewing angle to verify this. 2. In your code, check that Newton's first and second law are satisfied: * The force of the planet on the star should be equal in magnitude, but opposite, to the force of the star on the planet * Newton's second law, $F_{net} = ma$, requires that the acceleration be inversely proportional to the mass of the object. Check that you are applying Newton's 2nd law with the correct mass for the star. *Exercise 2* In the following exercises, we will view the orbital plane from the side, so that the star's motion will have a components towards or away from the observer. Before we calculate the effect this motion has on the wavelength of the light, we will first calculate the wavelength of light that the observer would expect to measure in the absence of motion. As written, the code template inserts a random wavelength of light to be observed. In reality, the wavelengths are observed based on the star's absorption lines. A star's composition is dominated by Hydrogen. As electrons in Hydrogen transition to higher energy levels, they absorb energy from the blackbody radiation of the star. This leaves dark lines in the observed emission spectrum of the star. For Hydrogen, the energy level transitions that result in absorption lines in the visible part of the spectrum occur from initial values of $n = 2$. Fix the frequency in the code to correspond to one of the visible transitions in Hydrogen (for example, the $n = 2$ to $n = 4$ transition). You might recall the Rydberg formula, $1/ \lambda = R_H \left(1/n_i^2 - 1/n_f^2\right)$, where $R_H = 1.097*10^{7} ~m^{-1}$. *Exercise 3* Due to the Doppler effect, the star's velocity should have an effect on the frequency of light observed. Assuming that the observer is stationary and viewing the orbital plane from the side, insert the correct expression for the Doppler shift of the starlight $\Delta f$. * When finished, check the plausibility of your answer. When the star is coming towards the observer, is the frequency shifted higher or lower? * The passage of the planet in the front of the star suggests another way of detecting the planet's presence. What will happen to the brightness of the star's spectrum when the planet passes between it and the observer?