The Changing Period of the Eclipsing Binary TZ Bootes

Zach Heinen

2004 June 9



Abstract

The W UMa binary TZ Bootes was observed on 2004 May 4 with the 0.25 m Great Ohio Telescope. The observations covered one entire cycle of the eclipse, therefore allowing a full light curve to be plotted. The changing period of the system could then be observed by plotting an O-C (observed minus calculated) curve.

1. Introduction

   Most of the stars we see are in binary systems. In these systems, two stars orbit around the common center of mass. Depending on inclination and their position relative to the earth the stars will pass in front of one another in their orbits. This causes a dip in the light curve (a plot of magnitude versus time). This is known as an eclipse. From these light curves, the period of the eclipsing binary system can be determined. The type of binary of interest is called a W UMa binary. This type of system is where the stars are so close that they are actually in contact with each other (also known as contact binaries). TZ Bootes is an example of such an object, with an orbital period of 0.297 days (Qian and Liu, hereafter QL).

   Since its discovery in 1927, TZ Bootes has been the subject of numerous observations. The reason this system is so interesting is that it has a changing period (Hoffmann 1978). Hoffmann proposed an ephemeris (Hoffman 1980) to predict where a minimum in the light curve would appear. Observations of the system were made and the location of the minimum was obtained. From this data, an O-C curve (observed minus calculated) was plotted. The O-C curve is a way of checking if the proposed ephemeris is correct. If the O-C curve is a constant, then it shows that the ephemeris is correct, and the period is not changing. However, it has been shown (QL) that the period of TZ Bootes is indeed changing. So they proposed a new ephemeris that is quadratic in nature.

   It is thought (QL) that the decrease in period is due to mass transfer from the more massive component to the less massive component. From the change in period, a mass-transfer rate can be obtained. This can be done using the equation


dP/P = 3(M1/M2-1)dM1/M1

   The main goal of this project was to verify that the period of TZ Bootes has changed since 1980. Another goal (and more interesting goal) in this project was to see if a new point could be added to the O-C curve of (QL) and to see if it lies along their curve. Finally, an attempt to calculate the mass-transfer rate was made.

2. Observations

   TZ Bootes (RA(2000) = 15:08:09.2, Dec(2000) = 39:58:12) was observed with the 0.25 meter Great Ohio Telescope (GOT) on 2004 May 4 (UT). The field was set up so that the target and the two comparison stars were in the same frame. Conditions were ideal in that there were no clouds, though it was not photmetric. The moon was almost at full phase, but it didn't affect the results. Six flats in V band were taken in the evening and four flats were taken in the morning. Nine zero frames were taken and 3 x 120s and 3 x 30s darks were also taken. For this project, the minimum needed to be observed. The magnitude of the object changed by about .6 magnitudes. This requires the errors in the observations to be within one percent so there is no mistake as to where the minimum is. Signal to noise calculations determined the exposure time to be used. The exposure time originally chosen was thirty seconds. After hooking up the auto guider, however, it was seen that it was possible to get better data with 120s exposures and not over-expose the target. 18 x 120s exposures were taken on the target between 0300 and 0740 UT.

At 0800 UT, a power fluctuation caused the autoguider on the telescope to be turned off. So, it became impossible to observe the target for 120s without getting streaks as it moved across the sky. After some trial and error, it was seen that 30s exposures were long enought to give adequate data, but short enough so that the stars didn't have time to move across the frame during the exposure. Signal to noise calculations allowed 30s exposures. 27 x 30s exposures were taken between 0800 and 0943 UT.
Table 1. V observations of TZ Bootes, after averaging frames. Corrected magnitude means variable star magnitude - comparison star magnitude.

3. Reductions

   The data was reduced following the procedure in Massey (1997) and Massey and Davis (1992). The zeros were combined into one zero frame and the darks were combined into two dark frames (one for the two minute darks and one for the thirty second darks). The flat fields were then dark and zero corrected, as were the actual object frames. After looking at the flats for any weird anomalies, the flats were combined into one flat field. The object fields were then divided by the flats to produce nice, clear frames.

   To find the magnitudes of TZ Bootes and the two comparison stars, the task phot from the package noao.digiphot.daophot in IRAF was used. This task gives the flux, magnitude, and error in magnitude for the target and the comparison stars. The full width at half maximim (FWHM) was measured for each star and an average of about 4 pixels was obtained. The aperture was then set to about four times the average, or around 15 pixels.

   Since the data from the second half of the night was much more noisy, frames from the end of the night were combined. To do this, the flux from four frames was added up and from this the magnitude was calculated. The noise was decreased dramatically after doing this. Errors were then combined and propagated through using the usual techniques. Had the frames not been combined, the error would have been much higher.

4. Results

   The light curve of TZ Bootes is presented in Figure 1.
Figure 1. TZ Bootes' light curve. Magnitude is variable star - comparison stars.

Once the lightcurve was obtained, a best fit curve had to be applied. This was done in the graphing program "xmgr". This was done as opposed to fitting a model for the lightcurve because no such model could be found in the literature. The best fit curve was fitted doing the following. It was assumed that the form of the fit would be a polynomial, since it is not sinusoidal. The program xmgr lets the user choose what rank the polynomial should have. After looking at all the fits through rank 10, it was determined that the best fit curve was of order seven with the coefficients found in table 2.
Table 2. Table of the coefficients of the best fit curve. A[0] corresponds to the constant and A[i] refers to the ith power of x.

The curve's R-squared value determines the error in the polynomial. This allows the minimum with uncertainty to be found. This error was then propagated through during the process of finding the minimum. The time of primary minimum was then found to be
HJD = 2453129.87±0.07 days

   Comparing the data to that of (QL), a calculation of the epoch was made. This calculation stems from the last available data point in the QL paper. The epoch (or cycle number as Qian and Liu call it) was calculated to be
E = 45420±.235

Using the ephemeris of Hoffmann (1980)
MinI = 2439632.8418 + 0.d2971620 x E

where E is the cycle number, the time of primary minimum was calculated to be
MinI = 2453129.93 ± .02

Therefore, the O-C (observed - calculated) value for this epoch (45420) is
O-C (present) = -.09 ± .02

5. Discussion

With the O-C value for the new data calculated, a new point was added to QL's O-C curve (Figure 2).
Figure 2. Original O-C curve taken from Qian and Liu (2000) along with the new data point. The new point is the one with the error bars. All other data points on this curve are taken from previous works including Hoffmann (1980). The solid line is a quadratic fit through the original data from Qian and Liu.

Recall that the QL paper used Hoffmann's ephemeris. It's obvious that it is incorrect in predicting the time of primary minimum as can be seen from the non-linearity of the curve. A new ephemeris that is quadratic in nature is then proposed. The new ephemeris is then
MinI = 2439632.8525+0.29716063 x E - 4.8 x 10-11 x E 2

This was obtained through a second-order least-squares solution of the original (O-C) curve.

Looking at the data, the new data point does not fit the curve proposed by QL. There are two possible reasons:

  1. The period of TZ Bootes is changing even more than originally thought
  2. The current estimate of the O-C data point is wrong.
The first reaction is to automatically assume that the current study is wrong. But first, a discussion of TZ Bootes' period change is in order. In the paper by QL, there have been three jumps in the period in the years between 1948 and 1995. In other words, the period is assumed to have undergone a steady decrease, and then "jump" back up. This can be seen in figure 3.
Figure 3. Residuals from the quadratic ephemeris for TZ Bootes and their description by several linear ephemerides. This figure was taken from the paper by Qian and Liu.

The values in this figure suggest the variations in the (O-C) curve are not continuous. It cannot be determined from the available data whether or not this would explain why the current study is in error. It does, however, suggest an alternate reason as to why the data point is off. Another reason the data might be off, is that the point used to calculate the primary minimum may, in fact, not have been the primary minimum. The primary minimum might actually have occured around HJD = 3129.70. There is, however, no data near that point, so a precise calculation cannot be made at this time. If this project were to be done again, a constant seven hour observing window would be needed for accurate calculations.

The final goal of this project, a calculation of the mass-transfer rate, could not be accomplished with the present data. The calculation requires a rate of change of the period, which the current data does not give. The primary goal of observing a period change, however, was accomplished.

References

Hoffmann, M., 1978, A&As 33,63
Hoffmann, M., 1980, A&As 40, 263
Massey, Philip, 1997, A User's Guide to CCD Reductions with IRAF
Massey, P., Davis, L., 1992, A User's Guide to Stellar CCD Photometry with IRAF
Qian, S., Liu, Q., 2000, Astron. Astrophys. 355, 171