This Exercise Set has been submitted for peer review, but it has not yet been accepted for publication in the PICUP collection.
Interactive modeling of exoplanets
Developed by Jay Wang
Exoplanets  extrasolar planets outside our own solar system, are currently being discovered at a rapid rate. To date (August 2017), there are 3,502 confirmed exoplanets, with many candidates being actively researched to determine their authenticity and properties such as mass and orbital parameters.
The radial velocity (RV) method is responsible for most of the earliest discovered and confirmed exoplanets. In this method, star wobbling velocities due to exoplanets are measured from Doppler shift. By fitting the observational data, one can extract important parameters of exoplanets including orbital period, eccentricity, and velocity amplitude. The lower mass limit can be estimated from theses parameters.
In this exercises set, students will interactively model RV datasets to extract these parameters. They will manipulate parameter sliders to visually explore the large parameter space, observe the effects of each parameter, and fit the RV datasets to obtain accurate parameter values despite the number of free parameters space (five). In addition, the students will also be able to generate mystery RV datasets, exchange them between teams, and confirm the parameters with each other.
Subject Areas  Mechanics, Mathematical/Numerical Methods, and Astronomy/Astrophysics 

Levels  Beyond the First Year and Advanced 
Learning Objectives 
Students who complete these exercises will be able to:
 describe the parameter space in radial velocity modeling and estimate the orderofmagnitude of radial velocity (**Exercise1**)
 model the RV dataset of a known planet  earth (**Exercise 2**), and to
 isolate and observe the effect of each parameter
 obtain best visual fit by interactively adjusting five parameters
 compute the lower mass limit
 infer inclination angle from known earth mass
 determine the properties of HD 3651 b from fitting observational data, and compare with confirmed research results (**Exercise 3**)
 engage in team work, reverse the process, and model RV datasets generated themselves (**Exercise 4**)

Time to Complete  120 min 
### Exercise 1: Estimating general properties of radial velocity
Imagine observing the solar system from a fixed point in space.
Thinking about the wobbling velocity of the sun due to orbiting planets like the earth, list the parameters that could affect that velocity, such as masses, distance, etc.
Discuss your list with your neighbors or team members.
Given that the distance of the earth to sun at $\sim 10^{11}$ m and a mass ratio of $\sim 10^{6}$, what is the order of magnitude of the wobbling velocity of the sun?
 A: $10^6$ m/s
 B: $10^1$ m/s
 C: $10^{1}$ m/s
 D: $10^{6}$ m/s
### Exercise 2: Modeling the RV dataset of earth
Start the program (`RV_model.py` or `RV_model.ipynb`), making sure to use the RV dataset file `RVdatafile='earth.txt'` and that the file is in the same directory as the program. Drag the slider to change the value of one parameter at a time, observing the effect of each without regard to the goodness of fit for now.
Check the extreme values and see how they affect the RV curve. For instance, what is the shape of the curve if eccentricity is zero? Why?
Next, vary the five parameters to obtain the best visual fit. When the fit become close, adjust the parameters in small increments. For this set of 'clean' data, you should be able to have the fitted curve passing through all the data points.
Read out the velocity amplitude. Is it comparable to your estimation in **Exercise 1**?
What is the eccentricity? Is it as expected from the data?
Using the best fit values for the RV amplitude $\mathcal{V}$ and eccentricity $e$, compute the lower mass limit $m\sin i$ from Eq. (\ref{mass}). Because this is in earth mass, the value should be less than one. Calculate the inclination angle $i$, and compare with the answer.
Also, compare the eccentricity with the actual value.
### Exercise 3: Determining the mass and orbital properties of HD 3651b
The observational radial velocity data for the exoplanet orbitting a Henry Draper star HD 3651, labeled HD 3651b, is stored in the file `hd3651b.txt`. Run the program with this file, making sure to change the period to `T=62.23` (days). Note the scatter in the data points and the associated error bars. Adjust the parameters as you did earlier to obtain a best overall fit.
Next, using the best fit parameters:
 compute the lower mass limit from Eq. (\ref{mass}), and compare with the currently accepted value.
 calculate the semimajor axis $a$ from Kepler's third law, $T^2=4\pi^2 a^3/GM$ where $G$ is the gravitational constant and $M$ is the mass of the host star (0.8 solar mass). Compare this with the currently accepted value and with the semimajor axis of earth's orbit.
Given the scatter in the data, discuss the uncertainties in these computed values. Also speculate the causes of the scatter. Can you give a reasonable range for the eccentricity?
### Exercise 4: Generating one's own RV datasets and working in teams to model them
Here is your chance to be in the seat of an observational astronomer and to generate the RV dataset of your dream, albeit virtual, exoplanet and to work with your teammates to decipher their properties. Follow instructions from the instructor, including `template_datagen.py` if you are to write the data generator yourself. Generate an RV dataset according to your own parameters. Swap the dataset with a teammate without revealing the parameters. Model the dataset to obtain the best fit parameters, and compare with the parameters used to generated the dataset. Discuss any discrepancies.
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Credits and Licensing
The instructor materials are ©2017 Jay Wang.
The exercises are released under a Creative Commons AttributionNonCommercialShareAlike 4.0 license