Quote of the Day
Politics is a way of ruling divided societies without undue violence.
This is an exciting time for astronomy -- we are just now beginning to obtain spectra from exoplanets. It seems as if new exoplanet discoveries are being announced every week.
I was reading an article on Space.com about some great work on obtaining the spectra from planets orbiting HR 8799 (Figure 1). While looking at the image, I thought it would be interesting to see if I can duplicate some of their orbital calculations.
This analysis will be very approximate. I will apply some simple orbital mechanics. There are four planets in the article's image. To estimate their orbital radii and periods, I need to make a few assumptions.
- The orbits are centered on the middle of the dark region.
This is equivalent to saying that the star is much more massive the planets. A planet and its star rotate about a foci of the orbital ellipse called the barycenter. With the star being much more massive than the orbiting planets, the barycenter of the orbit is very near the star (maybe even inside the star's diameter).
- The orbits are perfect circles.
Another assumption that makes my analysis simple. This assumption allows me to measure the orbital radii by measuring from the middle of the dark region to the center of the planets.
- The orbital plane is perpendicular to our line of sight.
I really have no idea as to the plane angle relative to our line of sight. I will simply make the simplest assumption. Again, this means that we are looking at a perfect circle.
My analysis consists of two parts: (1) determination of the orbit size, and (2) determination of the orbital period.
Orbit Size Determination
Figure 2 shows how I estimated the orbital radii of the four planets. I put the image into Visio and I just started adding some dimensions. I then used the Wikipedia radius of 68 AU for planet e as a reference value to use to estimate the radii of the other planets.
Figure 3 shows my scaling calculations.
My radii estimates are within 6% of the published values. Not bad considering the assumptions that I had to make.
Orbit Period Determination
Figure 4 shows how I estimated the orbital periods. Note how close my estimated values are to published values.
My results are within a few percent of the published results. Again, not bad considering my assumptions.
I feel like I understand the orbital basics of this exoplanet system with just a little bit of algebra.