Transit Duration and Depth of an Exoplanet?

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Homework Statement

Hi, I'm a senior taking a course on exoplanets right now. I have a question regarding their transit across a star. I understand how it works in theory, but I had some questions regarding the equation to use. Here's the problem:

A 0.05 solar radius planet orbits a main-sequence K star of radius = 0.8 solar radii and mass = 0.8 solar masses at an orbital radius of a = 0.5 AU. If the inclination of the planet's orbit to the plane of the sky is i = 89.8^o, calculate the duration and depth of the observed transit. What is the maximum orbital radius at which a transit would still be detected?

I'm checking into ask what units I'm supposed to use for this sort of problem, and to make sure that I'm using the correct equations.

Homework Equations

I'm using the equation found here: https://www.paulanthonywilson.com/e...tion-techniques/the-exoplanet-transit-method/

The page has helped me understand the transit method better, and I am using the equations it lists for transit duration, impact parameter and flux ratio.

The Attempt at a Solution

The first thing I did was convert the solar radii values to AU. Solving the problem this way gives an impact parameter of .469, which is a fine number, but when I attempt to solve for transit duration I end up with an imaginary number in the term for transit length since the star plus planets' radii squared is so much smaller than 5 AU. This is the reason I've come to this forum for help. Was converting the units a mistake? Am I using the wrong formulas? Or does my error lie elsewhere? Finding the flux ratio and maximum orbital radius shouldn't be too hard, since the listed site gives a formula for flux ratio and for the maximum orbital radius I can set b to 1 and then solve for a. Thank you for you help, and please let me know if you're able to spot any errors I've committed.

 

[TSny]

It appears that the link has an error in figure 1. The side of the triangle shown as $b$ in the figure should actually be $bR_*$ due to the way $b$ is defined. Thus, the figure should be as below.

In the formulas for $2l$ and the transit time, you will need to use $bR_*$ in place of $b$.