Week 8: WS14 #3

3. For the transit light curve and radial velocity time series, and assuming a 3.1 day orbit, 

a) What is the qualitative brightness distribution of the star's surface as viewed from the Earth? 

Normally, these plots are straight at the bottom, rather than curved. This plot indicates that the star in question is brighter in the center than at the edges, which means that the transiting planet makes the most difference when it's directly in front of the planet.

b) What is the planet's radius compared to the star's radius? 

When the planet is directly in front of the star, it cuts off on average about 2.7% of the star's light, based on the light curve. This should relate directly to their radii:
\(0.027=\frac{A_p}{A_\star}=\frac{\pi R_p^2}{\pi R_\star^2}\)
\(\frac{R_p}{R_\star}=0.164\)

c) What is the planet's semimajor axis compared to the star's radius? 

On a previous problem we determined that \(t_{transit}=\frac{2R_\star P}{2\pi a}\). It appears that the transit time (the time it takes for the planet's center to traverse the star) takes 180 minutes, or 10800 seconds. We know that the period is 3.1 days or 267840 seconds.
\(10800s=\frac{R_\star(267840s)}{\pi a}\)
\(\frac{a}{R_\star}=7.89\)

d) What is the size of the planet compared to Jupiter if the star has a radius of \(0.8R_\odot\)? 

\(\frac{R_p}{R_\star}=\frac{R_p}{0.8R_\odot}=0.164\)
\(R_p=\frac{0.1312R_\odot}{1}\frac{1R_{Jup}}{0.1R_\odot}=1.312R_{Jup}\)

e) Show that the scaled semimajor axis, \(a/R_\star\), is related to the stellar density. 

We can solve Kepler's third law for stellar mass:
\(P^2=\frac{4\pi^2a^3}{GM_\star}\)
\(M_\star=\frac{4\pi^2a^3}{GP^2}\)
Dividing both sides by the stellar volume will give you the density on the left:
\(\frac{M_\star}{(4/3)\pi R_\star^3}=\frac{4pi^2a^3}{GP^2}\frac{3}{4\pi R_\star^3}\)
\(\rho_\star=\frac{3\pi}{GP^2}(\frac{a}{R_\star})^3\)

f) What is the density of the star compared to the density of the Sun? 

\(\rho_\star=\frac{3\pi}{6.674\times10^{-8}cm^3/gs^2)(267840s)^2}(7.89)^3=0.968g/cm^3\)
This is smaller than the Sun's density, 1.408g/cm.

g) What is the density of the planet compared to the density of Jupiter? 

In a previous worksheet we determined the relation between the star's speed K and its mass, the planet's mass, and the period:
\(m_p=K(\frac{PM_\star^2}{2\pi G})^{1/3}\)
However, we need to find the mass of the star first. Since we know the density of the star and its size compared to the Sun, we can calculate its mass:
\(\frac{\rho_\star}{\rho_\odot}=\frac{0.968}{1.408}=\frac{(M_\star/R_\star^3)}{(M_\odot/R_\odot^3)}\)
\(0.687\frac{M_\odot}{R_\odot^3}=\frac{M_\star}{R_\star^3}\)
\(0.687\frac{M_\odot}{M_\star}=\frac{R_\odot^3}{R_\star^3}=(\frac{1}{0.8})^3\)
\(0.3517M_\odot=M_\star\)
\(M_\star\approx7\times10^{32}g\)
Now we can calculate the planet's mass:
\(m_p=(50000cm/s)(\frac{(267840s)(7\times10^{32}g)^2}{2\pi(6.674\times10^{-8}cm^3/gs^2)})^{1/3}=3.39\times10^{30}g\)
Now we can find the ratio between the planet's density and Jupiter's density:
\(\frac{\rho_p}{\rho_{Jup}}=\frac{\frac{3.39\times10^{30}g}{(4/3)\pi(1.312R_{Jup})^3}}{\frac{1.90\times10^{30}g}{(4/3)\pi(R_{Jup})^3}}=\frac{3.39\times10^{30}}{(1.90\times10^{30})(2.26)}=0.79\)
This planet is about 79% the density of Jupiter.