Homework assignment 4

Please show all your work, written as neatly as possible. Hand in your papers to me directly or place them into the plastic mail folder outside my office door, anytime up to and including the class meeting on Wednesday, Jan 28.

This week's problem is all about the orbit of the double star 70 Ophiuchi.

I have created some tables with the measured position angle and separation of this binary star. The brighter star, the primary, is assumed to be at (0, 0) at all times, and the tables give the position of the secondary star in radial coordinates. The units of position angle are degrees away from North, and separation is measured in arcseconds.

You will note that there is a gap in the coverage of the orbit. You can find additional measurements made recently which fill this gap by looking at a paper by Germain et al., published in 1999. Its title is "Speckle Interferometry at the US Naval Observatory. II." Go to the ADS service to look up this paper. When you find it, go to the "On-line data" listed, and search for Target "70 Oph".

Create a plot of the observed orbit of 70 Oph, using data from all three sources.
Does Kepler's Second Law work on a projected orbit? Calculate the area swept out between 1844 and 1861, and the area swept out between 1885 and 1900.
Make a copy of your observed orbit, and use it (plus as many additional pieces of paper as you need) to draw the auxiliary ellipse associated with the orbit.
Use the auxiliary ellipse to determine the true eccentricity e, the true semimajor axis a (in arcsec), and the angle of inclination i for the orbit.
(optional) Continue with the analysis, as shown in the lecture, to mark the line of nodes on the true orbit, and find the argument of perihelion.
Look up the distance to 70 Oph, as measured by the Hipparcos satellite. What is the distance? What is the uncertainty?
Calculate the semimajor axis of the true orbit in AU. Assuming that all your graphical measurements and calculations are perfect, what is the uncertainty in the semimajor axis?
Estimate the period of the orbit from your data. What is a decent estimate of the uncertainty in the period?
Calculate the total mass of the 70 Oph system in terms of the solar mass. What is the uncertainty?
Radial velocities for both stars have been measured. Use this small table:
- radial velocity measurements
Note that these measurements contain both the overall radial velocity of the center of mass, plus the velocities of each star around the center of mass. Determine the ratio of masses: secondary as a fraction of primary.
What is the mass of each component of 70 Oph? Compare your values to any numbers from the literature you can find.