Extrasolar planets and their properties

This is an excerpt from a worksheet on extrasolar planets for second year undergraduate students. The concepts I aim to teach are:

selection bias
methods of detection of extrasolar planets
data filtering
simple plotting

Question 1

Plot the planetary mass ($M_p \sin i$) vs orbital separation ($a$) on a log-log scale for the following cases:

all extrasolar planets
highlight the Hot Jupiters in this plot
planets detected by the radial velocity method
planets detected by the transit method



In [3]:

    
# Get the exoplanet data from the URL into the variable called 'all'
# This file has over 3000 lines, so this step can take a minute 
#    depending on your connection

all <- read.csv(url("http://exoplanet.eu/catalog/csv/"), header = TRUE)

# Let's start by listing the columns in this data file
colnames(all)









    





	'X..name'
	'planet_status'
	'mass'
	'mass_error_min'
	'mass_error_max'
	'mass_sini'
	'mass_sini_error_min'
	'mass_sini_error_max'
	'radius'
	'radius_error_min'
	'radius_error_max'
	'orbital_period'
	'orbital_period_error_min'
	'orbital_period_error_max'
	'semi_major_axis'
	'semi_major_axis_error_min'
	'semi_major_axis_error_max'
	'eccentricity'
	'eccentricity_error_min'
	'eccentricity_error_max'
	'inclination'
	'inclination_error_min'
	'inclination_error_max'
	'angular_distance'
	'discovered'
	'updated'
	'omega'
	'omega_error_min'
	'omega_error_max'
	'tperi'
	'tperi_error_min'
	'tperi_error_max'
	'tconj'
	'tconj_error_min'
	'tconj_error_max'
	'tzero_tr'
	'tzero_tr_error_min'
	'tzero_tr_error_max'
	'tzero_tr_sec'
	'tzero_tr_sec_error_min'
	'tzero_tr_sec_error_max'
	'lambda_angle'
	'lambda_angle_error_min'
	'lambda_angle_error_max'
	'impact_parameter'
	'impact_parameter_error_min'
	'impact_parameter_error_max'
	'tzero_vr'
	'tzero_vr_error_min'
	'tzero_vr_error_max'
	'k'
	'k_error_min'
	'k_error_max'
	'temp_calculated'
	'temp_calculated_error_min'
	'temp_calculated_error_max'
	'temp_measured'
	'hot_point_lon'
	'geometric_albedo'
	'geometric_albedo_error_min'
	'geometric_albedo_error_max'
	'log_g'
	'publication_status'
	'detection_type'
	'mass_detection_type'
	'radius_detection_type'
	'alternate_names'
	'molecules'
	'star_name'
	'ra'
	'dec'
	'mag_v'
	'mag_i'
	'mag_j'
	'mag_h'
	'mag_k'
	'star_distance'
	'star_distance_error_min'
	'star_distance_error_max'
	'star_metallicity'
	'star_metallicity_error_min'
	'star_metallicity_error_max'
	'star_mass'
	'star_mass_error_min'
	'star_mass_error_max'
	'star_radius'
	'star_radius_error_min'
	'star_radius_error_max'
	'star_sp_type'
	'star_age'
	'star_age_error_min'
	'star_age_error_max'
	'star_teff'
	'star_teff_error_min'
	'star_teff_error_max'
	'star_detected_disc'
	'star_magnetic_field'
	'star_alternate_names'



In [4]:

    
plot(all$semi_major_axis,all$mass_sini,xlab= "a [AU]", xlim = c(1e-3,100), ylim = c(1e-3, 50),
     pch = 20, ylab="M sin i [Mjup]", frame.plot = TRUE, cex.lab = 1.4, log = "xy", cex = 1.0)

The plot command is smart enough to remove the points that have missing data, but if we wanted to filter these out ahead of time, we could do:



In [7]:

    
# Load dplyr, which makes sorting dataframes a breeze

library(dplyr, lib.loc="/Library/Frameworks/R.framework/Versions/3.3/Resources/library")

# Keep only the two columns of interest, and filter out all the NAs
all %>%
select(mass_sini, semi_major_axis) %>%
na.omit() -> ma_array

# Now plot these filtered data 
plot(ma_array$semi_major_axis, ma_array$mass_sini, xlab= "a [AU]", xlim = c(1e-3,100), ylim = c(1e-3, 50),
     pch = 20, ylab="M sin i [Mjup]", frame.plot = TRUE, cex.lab = 1.4, log = "xy", cex = 1.0,
    main = "All Extrasolar Planets")

# Let's show where in the plot we'll find the Hot Jupiters by placing a partly transparent blue box over
rect(0.01, 0.1, 0.1,100, col = "skyblue", border = "transparent")

# The rectangle covers the data points, so let's plot them again, over the top
points(ma_array$semi_major_axis, ma_array$mass_sini, pch = 20)

The blue rectangle highlights the region of the plot we'll get the so-called "Hot Jupiters" -- Jupiter-like planets orbiting very near to their parent stars.

b) Now we want to filter for only those planets detected through the radial velocity method.



In [8]:

    
all %>%
filter (detection_type == "Radial Velocity") %>%
select(mass_sini, semi_major_axis) %>%
na.omit() -> radvels

# Now plot these filtered data 
plot(radvels$semi_major_axis, radvels$mass_sini, xlab= "a [AU]", xlim = c(1e-3,100), ylim = c(1e-3, 50),
     pch = 20, ylab="M sin i [Mjup]", frame.plot = TRUE, cex.lab = 1.4, log = "xy", cex = 1.0, 
     main = "Planets detected through Radial Velocity Searches")

c) Now we want to filter for only those planets detection through the transit method



In [9]:

    
all %>%
filter (detection_type == "Primary Transit") %>%
select(mass_sini, semi_major_axis) %>%
na.omit() -> transits

# Now plot these filtered data 
plot(transits$semi_major_axis, transits$mass_sini, xlab= "a [AU]", xlim = c(1e-3,100), ylim = c(1e-3, 50),
     pch = 20, ylab="M sin i [Mjup]", frame.plot = TRUE, cex.lab = 1.4, log = "xy", cex = 1.0, 
     main = "Planets detected through Transits")

Question 2

Looking at the plots above, do you think the Hot Jupiters are an observational bias (selection effect) or a real planetary population?

Question 3

The High Accuracy Radial Velocity Planet Searcher (HARPS) has found over one thousand exoplanets using the radial velocity search method. It has an accuracy of 1 m/s in radial velocity.

How does this translate into limits on the masses and orbits of planets you can detect with HARPS orbiting a $1 M_\odot$ star?

From our lecture notes, we express Kepler's equation for the star-planet system as follows: $$ v_* \sin i = 28.4\left(\frac{P}{1\mathrm{yr}}\right)^{-1/3}\left(\frac{M_p\sin i}{M_J}\right)\left(\frac{M_*}{M_\odot}\right)^{-2/3}\mathrm{ms}^{-1} $$ where $v_*$ is the velocity of the star, $P$ is the orbital period of the planet around the star (in years), $M_J$ is the mass of Jupiter, $M_*$ is the mass of the parent star in $M_\odot$.

Using $M_* = 1 M_\odot$, and Kepler's harmonic law $P^2 = a^3$ (remember, this works in this specific choice of units), we can rewrite the equation above as:

$$ \frac{M_p\sin i}{M_J} = \frac{v_*\sin i}{28.4\,\mathrm{m/s}}\left(\frac{a}{\mathrm{AU}}\right)^2 $$

With HARPS's accuracy of 1 m/s, we'd probably believe a planetary detection if it had a radial velocity greater than 3 m/s, so for this case we set $v_*\sin i$ = 3 m/s. We now plot this as the pink line on the graph below:



In [10]:

    
# Make the original plot
plot(ma_array$semi_major_axis, ma_array$mass_sini, xlab= "a [AU]", xlim = c(1e-3,100), ylim = c(1e-3, 50),
     pch = 20, ylab="M sin i [Mjup]", frame.plot = TRUE, cex.lab = 1.4, log = "xy", cex = 1.0)

# Now overplot the line in the equation above
curve(3/28.4 * x^2, add=TRUE, col = "magenta", lwd = 3.0)

HARPS will be insensitive to planets orbiting at distances to the right of the pink line and to planets with masses below the pink line.



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