Sylvana Yelda |
REU in Astrophysics - 2005 | |

University of Michigan | Advisor: Robert Mathieu | |

syelda@umich.edu | University of Wisconsin - Madison |

This research was done as part of the NSF sponsored *Research Experience for Undergraduates* at the University of Wisconsin-Madison Astronomy department . Thank you to UW-M and NSF for a great summer! I would also like to thank Dr. Robert Mathieu for being such a wonderful mentor.

NGC 6819 is an intermediate-age (~2.4 Gyr) open cluster located ~300 pc above the Galactic plane, appearing in the constellation Cygnus. It is a very rich, concentrated cluster with approximately solar metallicity. The mean radial velocity of the cluster is 2.17 km/s. Here is a CMD for NGC 6819 that I created while at UW. The highlighted marks represent the binaries for which we have found orbital solutions. In this study, I was only interested in binaries on the main sequence. So only binaries with B-V < 0.8 were included.

Before determining whether or not a star was a binary, we first had to make sure that it was a probable member of this cluster. If an object's radial velocity fell within 3 standard deviations of the cluster mean radial velocity (2.17 km/s), it was considered a likely member. We then considered objects as variable stars if the radial velocity root-mean-square was greater than 2 km/s. The next step was to look at the spectra for each variable star and search for spectroscopic binaries.

Spectroscopic binaries are found by examining the cross-correlation function for the object and some reference spectrum. In this case, we used the solar spectrum as the reference. For a cross-correlation function, the object spectrum is shifted with respect to the reference spectrum, pixel by pixel, and the strength of the correlation between the two is plotted against pixel shift (this is all done in IRAF). We see a peak where the correlation was the strongest. A Gaussian is then fit to the peak and the center of the Gaussian is the measured radial velocity. (Actually, the value of the pixel shift at the center of the Gaussian is converted to a radial velocity). Cross-correlation heights less than 0.50 were excluded from any analyses.

If one star in a binary system is much fainter than the other, we will get only one peak in the cross-correlation function. This is called a single-lined spectroscopic binary, or SB1. If the two stars in the system are of comparable brightnesses, we will see two peaks in the cross-correlation, one for each star. This is known as, you guessed it, a double-lined spectroscopic binary, or SB2. We can then fit a Gaussian to each of these peaks and get out a radial velocity for each star.

The next step was to find an orbital solution for these variable stars. By looking at the radial velocities and corresponding julian dates, we can usually guess an approximate orbital period. If we couldn't get a good estimate, we plotted the velocities against julian dates in IDL. This usually gave us a better idea about the period. Once we had a ballpark figure for the period, we used a program called SB1 to find an orbital solution. This found solutions for the brighter star in the system (the one we have radial velocities for). If we also had data for the fainter star (in an SB2), we could use a second program, called SB2, in addition to SB1, and come out with orbital solutions for both components of the system.

SB1 searched for orbital periods within the interval that we input (which we estimated based on the velocities and dates). The search gave a best period (based on the lowest chi-squared value). The search was repeated several times while narrowing the interval down on the best period found previously. Once an acceptable period was found, the orbit was plotted up along with several orbital parameters listed (i.e., eccentricity, amplitude). Here are examples of orbital solutions we found. The first and second are examples of SB1s (one eccentric, one circular), and the third is a SB2.

The circular orbit above is the shortest-period circular orbit found in NGC6819 with a period of only 1.85 days! This corresponds to an orbital radius of 0.03 AU!

Now knowing orbital parameters for several binaries, we created an eccentricity-log(Period) diagram for stars on the main sequence (B-V < 0.8 in CMD above). This is shown below:

Notice that there is a lack of long-period circular orbits and relatively few short-period eccentric orbits. This can be explained by tidal circularization.

The orbital parameters of a binary system constantly change due to the tidal interactions between the two stars until a stable equilibrium is reached. Although energy is dissipated in this system, the total angular momentum is conserved. As the stars tidally interact, the rotational angular momentum is transferred to orbital angular momentum. Thus, the eccentricity of the orbit decreases (becomes more circular), causing the stars to "spin down," or, rotate at a slower rate.

How exactly does this happen? Because the tidal force goes as 1/(*distance*)^3, closer binaries will experience a much greater tidal force than will binaries with larger separations. The tidal interactions cause tidal bulges on the stellar surfaces. When these bulges are misaligned with respect to the line joining the centers of the stars, a torque is produced. If there is a torque, there is a change in angular momentum (T=dL/dt). As mentioned above, the rotational angular momentum from one star is generally transferred to the orbital angular momentum of the other star. An increase in orbital angular momentum corresponds to a decrease in eccentricity (the orbit becomes more circular). The closer binaries (or those with shorter periods) will therefore circularize first. So short period binaries will tend to have circular orbits, while those with longer periods will have a distribution of nonzero eccentricities. With time these eccentric binaries will circularize, and the observed circularization period (see next paragraph) will increase. Then theoretically, we should be able to predict a period at which binary orbits become circular for a population of a given age.

The tidal circularization period mentioned above is a measure of the degree of tidal circularization in a coeval population. It is the period at which a typical binary has evolved to an eccentricity consistent with zero in the age of the binary population. The typical binary is assumed to have had an initial eccentricity of *e*~0.3-0.4 (Mathieu 2005). The older the population is, the more time it has had to circularize. Because shorter period orbits will circularize first, an older cluster will have a greater circularization period than will a younger cluster.

It is important to note that determining the circularization period is not straightforward since there is a distribution of initial eccentricities, stellar masses, and angular momenta. In the past, researchers have measured what is called the "tidal circularization cutoff period". This period is taken to be that of the longest period circular orbit. Several problems arise with this method. Not only does this take into account only one orbit in the entire cluster, but there may be several eccentric binaries with periods shortward of this cutoff period. In other words, this cutoff period is not a good representation of the circularization period of the entire population. Meibom & Mathieu (2005) have developed a diagnostic for determining the circularization period by taking into account all binary orbits found in a given cluster. A function is fit to the data in an e-log(P) diagram such that the deviations of the data from this function are at a minimum (see figure below). Details about this function can be found in Meibom & Mathieu (2005).

Using this technique, Meibom & Mathieu (2005) found circularization periods for several clusters, as shown in the plot below. With the exception of Hyades/Praesepe, there is a trend of increasing circularization period with age.

Plot from Meibom & Mathieu (2005)

The horizontal gray band, the dashed gray curve, and the solid curve each represent predictions based on different theories. As can be seen, these theories cannot explain the observed distribution of tidal circularization period with population age.

After fitting the circularization function of Meibom & Mathieu (2005) to NGC6819 data, we find a circularization period (CP) of only 1.60 days. The e-log(P) diagram with the circularization function is shown below. The total absolute deviation between the observed and calculated eccentricities as a function of step location is also shown. The minimum deviation defines the circularization period, and in this case, it appears to be anywhere between 1.4 and ~5 days. Thus, the choice of CP=1.6 days is arbitrary and does not appear to be accurate.

The fit to the data above does not take into consideration the error on the eccentricities. Several highly-eccentric, short period (~10 days) binaries in the plot above seem to be pulling the function upwards quickly. If we look at the error bars, however, we can see that the eccentricities for the circular orbits have relatively smaller errors than do those with greater eccentricities. If we fit the circularization function to the data after giving more weight to the orbits with smaller eccentricity errors, we get a different picture:

We now see a much more clearly defined minimum deviation between the observed and calculated eccentricities. The circularization period, factoring in eccentricity error, is now 7.5 days.

If we were to now place NGC 6819 into the CP-log(Age) diagram above with all of the other clusters, it would appear to follow the observed trend nicely:

The circularization periods for the clusters in this plot, however, were not found by weighing the eccentricity errors. Once the errors are taken into account, we may have a better idea regarding the ongoing tidal circularization in coeval binary populations.

The orbital parameters of binary systems are always changing due to gravitational interactions between the stars. Binary orbits with the shortest periods tend to be circular, while those with longer periods are noncircular. This is explained by the fact that the tidal force goes as 1/(distance)^3, and so stellar separation is very important.

The degree of tidal circularization in NGC6819 is consistent with results found in other coeval populations. No circular orbits were found beyond P~25 days, and only few eccentric orbits were found at relatively short periods. The tidal circularization period was measured at ~7.5 days using the technique of Meibom & Mathieu (2005). With an age of ~2.4 Gyr, NGC6819 provides additional evidence supporting the trend of increasing tidal circularization period with population age. As the population evolves, binaries with longer and longer periods become circularized.

References:

Mathieu, R. D. 2005, in ASP Conf. Ser. 333, Tidal Evolution and Oscillations in Binary Stars, ed. A. Claret, A. Gimenez & J.-P. Zahn (Granada: ASP), 26

Meibom, S. & Mathieu, R. D. 2005, ApJ, 620, 970