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Course 12.425. Problem Set 2. Due 2 Oct. 2007.
1. The Stellar Magnitude System.
a. Derive the stellar magnitude formula given below, using the fact that five magnitudes is a factor of 100.
Fb
mb − ma = −2.5 log10 . (1)
Fa
b. If the planet-star flux ratio is 10−3, what is the difference in magnitude?
2. Planet-Star Center of Mass. The HD 209458 star and planet system have the following properties: M=
∗
1.01M�, Mp = 0.69MJ , P = 3.525 days, a = 0.045 AU, and e = 0.
a. Find a1, the star’s distance to the planet-star common center of mass. How does this compare to the radius
of the star HD 209458A (1.12R)?
b. What is a2, the planet’s distance to the planet-star common center of mass? How does this compare to the
semi-major axis a?
Problems 3 through 6: Radial Velocity Fits to Real Data Sets.
The section lead by TA on Tuesday September 25 will introduce radial velocity fits to data sets as relevant to
this problem set. We will be using computers to try some fits. If you have a laptop, please bring the laptop to
section on Tuesday September 25. If you do not have a laptop, you can share with a friend,
The rest of this homework set involves orbital fits to radial velocity data sets. The fits correspond to model
planetary systems. The goal of the fitting procedure is to find a planetary model that does the best possible
job of fitting the data. We will use the Systemic console to obtain the orbital fits. The console has all of the
features that professional astronomers use to find planets, and indeed, planets such as Gliese 581 c have been
found by users of the Systemic console before they were announced by professionals. It is possible that you can
discover a previously unknown planet by doing the best possible job on problem 6.
For each planet you fit, be sure to print out either 1) a screen shot of your best fit, including the χ2 value or
2) the data plot including your fit, and a list of χ2, period, mass, mean anomaly, eccentricity, and longitude of
perihelion. See the attached sample printout. If a residual periodogram is needed in a given problem, please
print and submit that periodogram as well.
Please make sure to upload your fits for problems 3, 4b, 5a,b and 6 to the Systemic backend. Also, be sure to
save all of your work on your desktop.
3. Getting Going.
a. Download the console from http://oklo.org/?page id=86 and install it on your computer. The directions are
given on the download page.
b. Register as a new user on the Systemic backend at http://www.oklo.org/php/login.php. The first nine
characters of your username should be mit12 425. The last characters should be initials.
c. Follow the first two tutorials using the console. They are available at www.oklo.org.
d. Following the instructions on the backend, upload the fit for Upsilon Andromedae that you will make using
tutorial 2. Please make sure that your χ2 value is as close to unity as possible.
4. 51-Pegged. In 1995, two Swiss astronomers shocked the world by announcing that they had discovered a
bizarre planet orbiting the Sun-like star 51 Pegasi. It is interesting to find this planet in the radial velocity data
sets, which are available on the console under 51Peg B06L.
a. Using the periodogram to get started, find the best 1-planet fit to the data set. What period and mass does
the planet have?
b. Using the periodogram of the residuals, find a 2-planet fit to the data. You must submit this fit following the
instructions on the backend, as you did for the previous problem. Do not submit your results for part (a)! What
is the value of the F-test statistic for your 2-planet fit in comparison to your 1-planet fit? Does this indicate a
strong chance that you have discovered a new planet?
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