Orbital Characteristics of Binary Stars and Extra-Solar Planets

Objective

To investigate the effect that changes in mass, separation, eccentricity, and inclination angle have on the characteristics of the radial velocity or photometric light curves of binary stars and stars with extra-solar planets.

Introduction

The most important property of a star is its mass, but stellar masses are much harder to measure than luminosities or surface temperatures. The most dependable method for "weighing" a star relies on Newton's version of Kepler's third law. This law allows us to calculate the masses of orbiting objects by measuring both the period and average distance (semimajor axis) of their orbit. We must be able to determine the orbital properties of the two stars in order to derive the masses and radii of the system.

There are three main classes of binary systems: visual binary, eclipsing binary, and spectroscopic binary. In this exercise, we examine the characteristics of a) spectroscopic binaries by noting the effects changes in mass, separation, eccentricity, and inclination angle have on the shapes of the radial velocity curves and the Doppler shifts of the lines in their spectra; b) eclipsing binaries by comparing the changes in the light curves depending on the masses of the two stars, their spectral type, and the inclination angle; and c) the orbits of stars with suspected Jupiter-like planets and compare them to "normal" binary systems.

1. First We Play Around to Get a Feel for How Binary Stars WORK

For an orbital animation where you can readily see and experiment with changing the masses of the two bodies, their separation, and the eccentricity of the orbit, go to: http://csep10.phys.utk.edu/, click on "sample chapter" link shown in left-hand frame, click on "The Modern Synthesis," "Universal Gravitation," "Kepler's 3rd Law." Scroll down a bit in this sub-window, and see a side-bar that says "Effect of the Center of Mass." You should see a link to a Java applet that demonstrates Newton's modified form of Kepler's third law. Play around with the eccentricity of the orbit, the semi-major axis, and the masses of the bodies. Make a mental note of how the period of the orbit is changed, and how the changes in the eccentricity really demonstrates Kepler's Third Law. Just below the link to the "effect of the center of mass" applet, you will find a link to a "Kepler's laws calculator." Follow that link and test the program out for a few moments, noting the changes in the period of the orbits as you manipulate the variables.

Also check out:

Questions for this first part:

1. For the Kepler's 3rd Law demonstration, what struck you as something "obvious" that perhaps did not seem so obvious before when you studied just the equations?
2. Let's say that you are instructing someone else on the "Newton version of Kepler's 3rd Law." Do you see any use for the "calculator"? Explain why or why not.
3. For the demonstration of the wavelength shifts for 57 Cygni, these two stars are both B5 main sequence stars, probably around 6 - 10 times the mass of the Sun, although one must be slightly more massive. How do I know that? The period is 2.8548 days. What is their separation in Astronomical Units? The simulation you worked with above will give you a maximum separation (the applet cannot go as close together as these two stars are), but given that Mercury's distance from the Sun is 0.42 AU's, even an approximation gives an incredibly small separation.
4. For the UZ Draconis system, what do you note about the graphics of the two stars? In your opinion, how will this affect the evolution of each star? What does it mean when we "phase the data," that is, what does that kind of a graph tell us?

2. Eclipsing Binary Simulation

You've been given a handout containing the phased data for changes in the magnitudes of 4 eclipsing binary stars (seemingly a biased selection as all of the curves indicate contact binaries). Work with the following java applet demonstrating Eclipsing Binaries, and try to duplicate this curves by manipulating the inclination angle, separation, and spectral types of the stars.

Note: I tried to figure out the units for "L" in the program, trying to translate them to a change in magnitude for a direct comparison with the actual light curves. I was not successful and there seems to be no explanation at all on the web site itself. The corresponding lecture seems to indicate that this is apparent magnitude, only upside down for astronomers.

Questions for this part:

1. Which combinations of angle, separation, and spectral types worked best for the XX Leo and beta Lyrae light curves?
2. Try other combinations of the 3 variables. Write down the "one you thought most impressive," and list the values of the variables and make a rough sketch of the light curve. Explain briefly why the light curve looks like it does.

3. Spectroscopic Binary Simulation

[The simulations used are Java applets written by Professor Terry Herter of Cornell University, and are used with his permission.]

Take a look at the image to the right of each of the following definitions and find the corresponding parameters on the simulation:

M1 or M2 The mass of each of the two stars. The distance between the two stars in solar radii. Eccentricity of the orbit Angle of the orbital plane of the stars to our line-of-sight. 0o = face on 90o = edge on This is the opposite from the standard notation. Angle of the major axis as measured in the orbital plane

General Instructions

The simulations used here are Spectroscopic Binary Simulation

Do some trial investigation to see how you can adjust each of the parameters for the simulation:

• Adjust each of the star parameters -- masses, separation, eccentricity, inclination, and node angle.

• Click "enter" to update the simulation parameters.

• Use "pause" to start and stop the simulation, if desired.

• If the picture is messed up at anytime, use "enter"to redraw it.

• The number between the "<=" and "=>"buttons, is the rough time (in seconds) it takes the simulation to complete an orbit. Make this number larger or smaller by clicking the "arrow" buttons.

Questions for this part:

See the extrasolar exercise below.

4. Using our knowledge of binary stars to detect extrasolar planets

Now for a real astronomical treat! Visit the California & Carnegie Planet Search web page. On this home page you will note the distribution of extrasolar planets as a function of distance from the primary star, and as a function of the metallicity of the primary star. If this stuff is new to you, be sure to take some time to study these results. They are significant.

Click on the Almanac of Planets link. There you will see the entire list of extrasolar planets with the primary star name, planet mass (times sin i), peiod of the orbit, semi-major axis, eccentricity, and semi-amplitude of the radial velocity variations of the primary star. Pick 5 stars to investigate further. Try to get a wide range of planet masses, semi-major axes (a), and eccentricities (e ). What we are going to do is try to duplicate these radial velocity curves with the "spectroscopic binary" java simulation to better understand how astronomers determine all of the parameters of these star-planet systems. Fill in the following table for the "questions for this part."

Questions for this part:

Star ID Planet
Mass
a e Values for Simulation Variables
M1 M2aeiw
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .

Comment on these results (pretty much anything that comes to mind).

Last updated on: