WS 11.1, Problem 3: What's in a Light Curve?

This blog post is loosely modeled after problem 3 on worksheet 11.1.  I'm not going to answer the specific questions from the problem to generalize to all sets of data, but I am going to follow the path that the problem sets to discover what can be learned from a transit light curve.  This should be fun.

The problem gives us three pieces of information.

• The system's period is 3.1 days (not necessary since I'm not actually answering the parts of the problem, but now we know we can find a lot of the system's properties with its period).  
• This light curve

• And this radial velocity graph

The first part of the problem asks us to qualitatively describe the brightness distribution of the star.  We can tell this by looking at the bottom of the light curve.  The dots along the curve represent the amount of light in each image taken of the system.  The dots stay fairly close to the line, so the star's brightness is relatively evenly distributed.  But because the dip is more of a curve than a straight line, so we know that the center of the star is slightly more luminous than the edges.  This is to be expected due to the limb darkening effect, which basically just says that the center of the star will appear brighter.  (Yay for self-explanatory scientific terms!)

The second part asks us to find the ratio of the radii of the planet and star.  This is a little bit trickier than the first part because it involves actual math, but it's not that bad.  The theory behind the solution to this problem is that that the depth of the light curve depends on the ration of the areas of the two objects.  Actually solving the problem is a pretty logical process.  If the star is really big, the light curve will be pretty shallow.  In the same vein, the curve will be shallow if the planet is small (because it's not blocking a lot of the star's light).  So the depth of the curve goes as 

$\frac{R_P^2}{R_*^2}$

When the problem asks us to find the ratio of the objects' radii, we can find the average depth of the light curve and take its square root.  

The third part asks us to find the ratio of the planet's semimajor axis with the star's radius 

$\frac{a_P}{R_*}$

To solve this problem, we look at the duration of the transit.  If you believe that velocity is distance over time (which I do), then you can also believe that time is distance over velocity.  The distance that the planet has to travel to complete a transit is twice the star's radius, and it's velocity is $v_P$.  Lucky for us, the velocity of the planet is dependent upon its semimajor axis in the following way:

$v_P=\frac{2\pi a_P}{P}$

which means the ratio of the semimajor axis to the star's radius should be 

$\frac{a_P}{R_*}=\frac{P}{\pi t}$

where t is the duration of the transit.

It's important to realize that this equation is only true for equatorial transits (transits that happen along the equator of the star).  A more general equation would account for the transit time of a transit that travels along any chord on the star.

$T=T_{equatorial}(1-b^2)^{1/2}$

where b is called the impact parameter, which is basically the place on the star where the planet transits.  For equatorial transits, b is 0.  For transits where the planet just barely grazes the star--so it doesn't really cause any dip in the light curve at all--b is 1.  

$b=1-\delta^{1/2}\frac{T}{\tau}$

where $\tau$ is the time of egress--the time from the moment the planet first touches the star to the moment the last point of the planet crosses the first edge of the star--and delta is the depth of the transit.  

One of the parts of the question asks us to show that the ratio found above is related to density.  I started with the equation for density

$\rho_*=\frac{M_*}{4R_*^3}$

assuming that $\pi$ is 3.

But we can simplify this further by using the mass-radius scaling factor found in this post, which says that mass scales proportionally with radius.  

$\rho_*=\frac{1}{R_*^2}$

Because of this relationship, we can say that the planet's semimajor axis is related to the star's density in the following way:

$a_P=\frac{PR_*}{\pi t}=\frac{P}{\pi t sqrt{\rho_*}}$

If we know the radius of the planet (which we do, because we know the ratio of the radii of the two objects and the radius of the star), we can also find its density based on the 

$\rho_*=\frac{1}{R_*^2}$

equation.  

Here we have it.  Armed with three simple (I say simple, but in actuality, those graphs are not too easy to get) pieces of information, we can find the following properties

• ratio of the radii of the two objects in the system
• semimajor axis of the planet (which we can then use along with the mass ratios to find the semimajor axis of the star)
• the densities of both the star and the planet
• the impact parameter

Like I said at the beginning of this post, light curves are fun!