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Challenge Team Interim Report

 Team Number: 056 School Name: Cuba High School Area of Science: Earth and Space Project Title: Determining the Light Curves of Eclipsing Binary Star Systems
 Abstract Interim Final Report

Problem Definition

Binary star systems are composed of two stars that revolve around the same center of gravity. Binary systems are seen as a single point of light, which is an illusion created by the distance between the observer and the system. A binary system becomes an eclipsing binary system, when, relative to an observer, one star passes in front of the other. This causes the periodic brightening and dimming of the star. When the stars are alongside one another, the star is at its maximum brightness. The primary minimum occurs when the brighter star is eclipsed by the dimmer star, and the secondary minimum occurs when the dimmer star is eclipsed by the brighter star. When magnitude is graphed as a function of time, a light curve showing the varying brightness of the system is created.

The objective of this program is to write a C++ program that computes the light curve of an eclipsing binary star system taking the following into consideration: mass of stars, radii of stars, magnitude, luminosity, spectral types, orbital speed, and orbital inclination.

Research

Binary stars are two stars that revolve around a central point of gravity. They can be classified as visual, spectroscopic, and eclipsing binaries. Visual binaries are visually separable. Many visual binaries have been discovered. Most of these systems have not yet been characterized because it takes long period of time to calculate their orbits (…DSTARS...).

Spectroscopic binaries cannot be separated optically. This is because of the relative closeness of the two stars. These stars are identifiable only by the superposition of their spectra (Petit 169).

The third type of variable stars is eclipsing binaries. Eclipsing binaries are not in their own class, but a special case of spectroscopic binaries. The eclipsing occurs when one of the stars passes in front of the other. When the principle star is eclipsed, there is a decrease in the luminosity of the system and this is called the principal minimum. When the secondary star is eclipsed, there is a decrease in luminosity, but not as great. This is called the secondary minimum (170).

A light curve is a plot of brightness versus time. This plot produces a curve that shows the intensity of the light at certain times (…sfasu.edu…). Important information can be obtained by looking at the light curve. Characteristics of the stars, such as mass, relative size, shapes, various levels of surface brightness, and orientation information, can all be obtained by looking at the light curve (Wilson 3).

Because a great variety of eclipsing binaries exist, light curves of eclipsing binary star systems differ. The three classes of eclipsing binaries are EA, EB, and EW. The first class, EA is the most common. This group consists of two stars that are a large distance apart. The eclipses occur with a deep principal minimum, a shallow secondary minimum and relative stability outside the eclipses (Petit 171).

Class EB, are star systems that consist of giants or super giants of low density, and different sizes. Because of the attraction between them, they take on an ovoid shape, and the light curve is rounded, compared to the sharpness of class EA (172).

The last class, EW, consists of two stars, usually of equal size and brightness. This group is very similar to the group EB, in the relative distance between the stars of the system. However, since the stars are so close, the stars take on a highly askew appearance. On the light curve, the primary and secondary eclipses appear to be alike, and well rounded (171).

Computational Plan

Program inputs:

The program will determine the light curve of eclipsing binary star systems according to the program inputs which are as follows:

i = Orbital inclination
M1, M2 = Masses of the Stars
L1, L2, = Luminosities of the Stars
R1, R2, = Radii of the Stars
R = Distance between the Stars Centers

Program outputs:

The output of the C++ program will be the brightness of the eclipsing binary star system graphed against the phase of the system. To produce this output, formulas to find the coordinates of the two stars and the flux or the amount of energy radiated per unit area per unit time will be used in the program.

Formulas:

The following variables and equations will be used in the C++ program to determine the coordinates of star 1 (x1, y1, z1) and star 2 (x2, y2, z2) assuming the orbit of the stars is circular.

Q = M2 / M1
R = Distance between the Stars Centers
q = Azmuth Angle = 2p*Phase
Phase = (Time since Primary Eclipse)/(Orbital Period)
x= R sin (q)
y= R cos (i)cos (q)
z= sin (i) cos (q)

The coordinates of star 1 are: x1= -x / 1 + (1/Q), y1= -y / 1 + (1/Q), and z1= -z / 1 + (1/Q).
The coordinates of star 2 are: x2= x / 1 + Q, y2= y / 1 + Q, and z2= z / 1 + Q.

We are currently working on the formulas for determining the brightness of the binary star system based on flux and the position of the stars.

Results Expected

We expect that our program will determine the light curves of different eclipsing binary star systems according to what the user inputs. Our hope is that in using our program, people can understand how the masses, radii, and luminosities of the two stars in the system, affect the shape of the light curve. We also want to show how changing the orbital inclination can affect shape of the light curve.

Progress to Date

We have obtained research about how mass, luminosity, orbital inclination, and other factors affect the light curve of eclipsing binary stars. Several formulas for the actual C++ program have been found. A formula to find the position of stars in a circular orbit has been found. It is hoped that formulas to determine coordinates of stars with elliptical orbits can be found, since most binary star systems have elliptical orbits. We have obtained a formula to find the brightness of the system, however we have yet to find formulas that take limb darkening into consideration. We would like to obtain formulas that take the changing velocities of stars with different masses into consideration. The model we are starting with is a simple model, but we hope that after finding all the equations we can improve the accuracy of the results of the program. We are fortunate to have a mentor, Kenneth Luedeke, a retired Air Force engineer, available to help us. Mr. Luedeke collects variable star data for the AAVSO (American Association of Variable Star Observers).

Bibliography

Bruton, Dan. "Eclipsing Binary Stars." Online. December 19, 1995. September 7, 1999. Http://www.physics.sfasu.edu/astro/ebstar/ebstar.html.

Guchez, Michele. "Double Stars." Online. December 1998. September 7, 1999. http://www.astro.oma.be/D2/DSTARS.html#TYPES.html

. Petit, Michael. Variable Stars. New York: John Wiley & Sons, 1987.

Wilson, R.E. "Understanding Binary Stars Via Light Curves." I.A.P.P.P. Communication. No. 55 (1994): p. 3.

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