MASS DETERMINATIONS OF POPULATION II BINARY STARS

Transcription

1 MASS DETERMINATIONS OF POPULATION II BINARY STARS Kathryn E. Williamson Department of Physics and Astronomy, The University of Georgia, Athens, GA James N. Heasley Institute for Astronomy, University of Hawaii ABSTRACT Accurate mass determinations of Population II stars are essential to the inclusion of metallicity effects in stellar models and evolution theories. Currently, no Mass-Luminosity Relationship exists for these old, metal poor stars that reside mainly in the halo of the galaxy. This research contributes accurate mass estimates with corresponding errors for three dwarf Population II binary systems HD , HD and HD Results were obtained via a simultaneous least-squares adjustment of spectroscopic and astrometric data to find the best fit orbital parameters and masses with error estimates. Monte Carlo simulations of theoretical data sets were used to test the consistency and accuracy of the optimization techniques in order to gauge the reliability of results. These theoretical data are designed to match orbital parameters that likely describe the three binary systems of this study. The results of the Monte Carlo analysis imply reported mass estimates and error bars are indeed reliable for each particular orbit and given set of observation times. Subject headings: binaries: spectroscopic stars: evolution stars: individual(hd , HD , HD ) stars: Population II 1. INTRODUCTION The current stellar Mass-Luminosity Relationship (MLR) is based mainly on Population I stars that are more massive than the sun. Population II stars have been more difficult to study in terms of mass and luminosity because they mainly reside far away in old globular clusters in the halo of the galaxy. They can be identified by their characteristic high orbital inclinations from the galactic plane and high proper motions. The low metallicity of population II stars ([m/h] typically less than -0.70) indicates that they formed early in the history of the galaxy before nucleosynthesis populated the interstellar material with heavy elements. Studying these metal-poor stars will allow the development of a Mass- Luminosity Relationship to include metallicity effects. In addition to contributing to a Population II MLR, accurate mass determinations will provide an improved understanding of the Population II main sequence. Because mass determines how a star evolves, determining masses of these old, metal-poor stars will allow us to improve our understanding of the Population II evolutionary track. Dwarf Population II stars are of particular interest for determining population II evolution because their low masses cause them to age slowly, indicating their current observables are close to their initial evolutionary stage and that they are still on their main sequence track. Mass estimates will also contribute to forming a more detailed comparison with Population II evolutionary tracks. Determining the orbital parameters of binary systems is currently the best way to deduce stellar masses. Visual observations offer positional data in either polar or cartesian coordinates of the secondary star with respect to the primary and the orientation of the orbit with respect to the plane tangential to the celestial sphere. Spectroscopic data offer radial velocity measurements. Using Kepler s equations of motion, we can combine these measurements to deduce other information about the orbit and, ultimately, the mass. Individual masses of the constituent stars in a system can be obtained only if the physical scale of the orbit is known. The physical scale can be calculated with either a parallax value, such as those given in the Hipparcos Catalogue, or a combination of astrometry and double-lined spectroscopy. The latter combination was used for this project, providing an independent parallax estimate to compare to the Hipparcos Catalogue value. 2. BACKGROUND 2.1. Data An extensive set of spectroscopic data was obtained by Goldberg et al. (2002), along with preliminary orbital solutions for 34 binaries. The Astrometry needed to refine these solutions was published by Horch et al. (1999). Horch et al.(1999) chose 13 high proper motion binaries from Goldberg et al. s (2002) sample that could likely be resolved with the Hubble Space Telescope s Fine Guidance Sensors s (FGS) 10 mas resolution capabilities. From previous estimates of metallicity and propermotion in the literature, these binaries are believed to be population II dwarfs. The FGS data can be reduced to astrometry positional data via various methods, including Fourier transformations and S Curve deconvolutions. Saia et al. (2006) applied S Curve deconvolutions to the Horch et al. (1999) FGS data to determine the astrometry of five of the 13 binary systems. Using the Hipparcos parallax, the FGS astrometry, and the singlelined spectroscopic data that was available at the time, Saia et al. (2006) improved orbital solutions and mass determinations. Heasley has obtained spectroscopy for the second component of three of Saia s five stars using the Keck telecope, providing the data necessary to obtain the most accurate orbital solutions. For this project, the Keck spectroscopy was combined with Goldberg et

2 2 al. (2002) spectroscopy and Saia et al. (2006) astrometry to determine reliable estimates of individual masses. These three systems are HD , HD , and HD , and they are important to our study for various reasons that are discussed later Orbital Parameters In order to describe the orbit of a binary star, we must solve for specific orbital parameters. The following are the seven orbital parameters given by double-lined spectroscopy. P = Period T = Time of Periastron passage ω = Longitude of periastron in the plane of the true orbit e = Eccentricity V 0 = System radial velocity K 1 = Amplitude of primary radial velocity K 2 = Amplitude of secondary radial velocity Visual data provides three additional parameters to completely describe the orbit: i = Inclination Ω = Position angle of ascending node d = Distance to system (parsecs) To find the orbital parameters that best fit the observations in a least-squares sense, we define a 10-dimensional nonlinear fitness function from the difference of observed and calculated positions and velocities. The goal then is simply to minimize the fitness function, for which there are many different strategies. This seems simple enough, but optimization of such a large number of paramters can be quite difficult. How much confidence should we have in the solution returned by the computer program? What if we have a function similar to that in Figure 1 where the program could easily miss the narrow global extremum in favor of a broader local extremum? These are issues that optimization programs attempt to overcome. 3. OPTIMIZATION TECHNIQUES Three optimization methods were investigated for this project. BINARY was developed by Gudehus (2001) and is specifically tailored for applications to binary star parameters. It uses a homegrown gradient search method that is quite powerful when given a good initial guess; however, it can be quite sensitive to small perturbations in this guess. Pikaia, developed by Charbonneau (2002), is an optimization code that uses genetic algorithms to find the maximum of an n-dimensional, usersupplied function. Genetic algorithms are designed to mimic evolutionary biology in seeking an optimum solution. A population of initial guesses are evaluated at random places in the search space, so a good initial guess is not necessary. The best results are bred to produce the new generation of test solutions to be evaluated. This continues until some preset number of generations is reached, and the fittest result is supplied. We developed our own code that incorporates Pikaia to deduce the most accurate orbital parameters from spectroscopic 1 and astrometric data by maximizing the statistical χ 2 value. A third optimization code developed by Pourbaix (1998) implements a simultaneous least-squares adjustment of visual and spectroscopic observations via simulated annealing minimization. Simulated annealing mimics the metallurgical process of annealing in which substances are able to reach low energy states by cooling slowly. The program defines the analog of a temperature that regulates the rate of annealing. The program then slowly converges to the general neighborhood of the solution, and an additional hill climbing scheme fine-tunes the results. 4. METHOD 4.1. Theoretical Data Generation Program In order to test the reliability of the optimization programs to return self-consistent results with accurate errors estimates, we compared sets of data with known orbital parameters to the parameters reported by the optimization codes. This involved development of a theoretical data generation program from a set of pre-supplied orbital parameters. This program uses a set of input orbital parameters to calculate theoretical positional and velocity data, and it assigns this data to the real time epochs of our real data. Although real observation times are not evenly distributed along the orbit, this keeps the simulation as similar to a real life situation as possible. The theoretical data generation program then implements a random number generator and a Gaussian deviation function to add noise to the data such that its standard deviations mimic real life errors. For the spectroscopic data we chose σ V = 1.0 kms 1, and for the astrometric data we chose σ θ = 0.64 degrees, σ ρ = arcseconds, σ x = arcseconds, and σ y = arcseconds. The simulation program then outputs the positional and velocity data with these errors in a format to be used with any of the above optimization programs Self-Consistency and Accuracy of Errors Due to BINARY s sensitivity to initial guesses and because our Pikaia code is still under development, we investigate the reliability of Pourbaix s (1998) code. We implement a Monte Carlo analysis to check the ability of Pourbaix to return self-consistent results and accurate error estimates. Running 15 theoretical data sets in Pourbaix for each binary system allowed us to test the sensitivity of Pourbaix to each particular sampling of observations for a given orbit. The Monte Carlo analysis of these results provided a self-consistency check of Pourbaix s solutions, and comparing the estimated errors to true differences from the presupplied orbital parameters provided a check of the accuracy of Pourbaix. 5. RESULTS 5.1. HD HD is a quadruple system with an estimated metallicity of [m/h]= Our primary target is the

3 3 spectroscopic binary near the center of this system. Table 1 compares the Pourbaix solution for the inner pair to those obtained by Horch et al. (2006) and Saia et al. (2006). The Pourbaix solutions are consistent and the error bars overlap for most parameters with the exception of the position angle of the ascending node, Ω. Inconsistencies of 180 in the angles that describe the orbit are merely due to differences in observation conventions and do not affect the reliability of any other parameter. Additionally, inconsistencies other than 180 do not appear to significantly affect the mass estimates returned by Pourbaix, which is the most important parameter for this study. Figure 6 shows the visual orbit of HD with only the astrometry points plotted, and Figure 5 shows the radial velocity curve of HD plotted over one period. The addition of the Keck spectroscopy revealed a linear change in the system radial velocity of km s 1. Normalizing the radial velocity of each observation to zero provides a well defined radial velocity curve with the advantage of leaving the calculated orbital parameters unaffected. The only parameter that should be affected by this change should be the system radial velocity V 0 ; however, the systemic velocity calculated by Pourbaix did not recognize this normalization, and the radial velocity plot was shifted from the data. Rejecting the Pourbaix velocity and manually entering the systemic velocity as zero forced the calculated curve to shift such that it matched the data. It is unclear as to why the Pourbaix code does not recognize the velocity normalization, and it is a subject to be investigated further. Figure 5 shows the radial velocity curve of HD calculated by Pourbaix with the forced normalization. The normalized Goldberg et al. (2002) and Keck spectroscopy points with error bars are also plotted HD HD has an estimated metallicity of [m/h] = Studied in depth by Torres et al. (2002), it provides the best consistency check for our study. The Pourbaix solution shows good agreement with the Torres et al. (2002) solution, again with the exception of the position angle of the periastron, but also the inclination. And again, the mass estimates are very consistent, a good sign for the confidence level of our mass estimates and error estimates. Figures 2 and 3 compare the visual orbit obtained with Torres et al. s (2002) parameters to that obtained with our Pourbaix parameters. The Torres et al. (2002) position angle of the ascending node as reported in Table 3 has been shifted by 270 for the plot to allow better comparison. This is due to a quadrant ambiguity in the conventional analysis of orbital angles. The Saia et al. (2006) astrometry points are plotted on both orbits. Figure 4 shows the radial velocity curve for HD with the Keck and Goldberg et al. (2002) data points plotted. The theoretical fit matches the data quite well, and the similar amplitudes of the curves reflect the similar masses returned by Pourbaix HD HD is a triple system with a metallicity of [m/h]= Again, our primary target is the close spectroscopic binary at the center of this triple system. Table 2 shows the Pourbaix solutions for the inner pair compared to those obtained by Saia et al. (2006). In this case, all three of the angles that describe the orbit appear inconsistent with previous work. The inclination error bar is high, but it does overlap Saia et al. s (2006) value. The longitude of the periastron and the position angle of the ascending node both vary without overlapping error bars. However, from the results of the previous two stars, we have confidence in trusting the reported mass estimates and errors. With the individual mass determinations for the inner two stars, we can calculate the mass of the third component from Kepler s equation: P 2 = a 3 M 1 + M 2 (1) Choosing M 1 to be the combined mass of the primary and secondary components, M 2 gives the mass of the outer component as 0.239M. This is the first time a reliable estimate has been made for the third component. These low masses are consistent with the composite spectral type. Figure 7 shows the visual orbit of the inner pair of HD obtained from the Pourbaix parameters with the five astrometry points plotted. Figure 8 shows the radial velocity curve for the inner pair of HD In this case, the curve for the secondary star was obtained with only the Keck spectroscopy, as earlier attempts were unable to obtain the double-lined spectrum. The different amplitudes reflect the different mass estimates returned by Pourbaix. 6. CONCLUSIONS This research provides accurate masses with accompanying error estimates for three dwarf Population II binary systems HD , HD and HD These masses can effectively contribute to both a Mass- Luminosity Relationship for Population II stars and a Population II main sequence evolutionary track. The Monte Carlo analysis of our simultaneous least-square adjustment of spectroscopic and astrometric data indicates that Pourbaix mass estimates are self-consistent and accurate. It also shows that the reported errors appropriately mimic real-life errors. Furthermore, the Pourbaix mass estimates are unaffected by the the inconsistencies in the angles that describe the geometry of the orbit. Despite these inconsistencies, the mass estimates of this project corroborate those obtained by others. Additionally, there is a systematic difference between the parallax values calculated with Pourbaix and those given in the Hipparcos Catalogue. The following table shows that the Pourbaix parallax is consistently higher than the Hipparcos parallax on the order of a few milliarcseconds. Parallax Estimates (mas) star Hipparcos This Project HD HD HD FUTURE WORK

4 4 Fig. 1. An example of a function whose global extremum would be easy to miss. The broad goal to be obtained with the accurate masses from this project is the determination of a Population II Mass-Luminosity Relationship and a Population II main sequence. This will require a more complete sampling of Population II masses, as this was only three systems of Horch et al. s (2006) original sample of thirteen. It will also require accurate luminosity measurements taken at several wavelengths, which involves a thorough understanding of the FGS instrumentation sensitivity to color. Additional information about stellar energy distributions in the near-infrared also can be obtained by closer study of the Keck solutions for the velocities of each component. 8. REFERENCES Charbonneau, P. 2002, NCAR/TN-450+IA Goldberg, D., Mazeh, T., et al. 2002, AJ 124, 1132 Gudehus, D.H. 2001, AAS, 33, 850 Horch, E.P. Cycle 10 Hubble Space Telescope General Observer Proposal Horch, E.P., Franz. O.G and Wasserman, L.H., Heasley, J.N. 2006, AJ, 132, 836 Latham, D.W., Mazeh, T., Carney, B.W., et al. 1988, AJ, 96, 567 Latham, D.W., et al. 1992, AJ, 104, 774 Pourbaix, D. 1994, A&A, 290, 682 Pourbaix, D. 1998, A&AS, 131, 377 Saia, M., 2006 M.S. Thesis, U. Mass Dartmouth Torres, G., Boden, A.F., Latham, D.W., Pan, M., Stefanik, R.P. 2002, AJ, 124, 1716

5 5 TABLE 1 Parameter Horch et al. (2006) Malinda Saia(2006) This Project P (yrs) ± ± ± a (mas) ± ± ± 1.4 i (deg) 94.2 ± ± ± 3.7 Ω (deg) 51.5 ± ± ± 2.8 T (Bess. year) e ± ± ± a ± ± ± ω (deg) ± 3.0 b ± ± 3.8 V 0 (kms 1 ) ± ± 0.12 π (mas) ± ± ± 1.1 a (AU) ± ± Mass of A (M ) ± ± ± Mass of B (M ) ± ± ± K 1 (km s 1 ) ± ± 0.43 K 2 (km s 1 ) ± ± 0.20 a Epoch shifted by an integral number of periods for better comparison. b Shifted by 180. TABLE 2 Parameter Saia et al. (2006) This Project P (yrs) ± ± a (mas) ± ± 2.4 i (deg) ± ± 11 Ω (deg) 13.8 ± ± 6.5 T (Bess. year) e ± ± ± a ± ω (deg) 42.0 ± ± 13 b V 0 (kms 1 ) ± 0.18 π (mas) ± ± 2.4 a (AU) ± Mass of A (M ) ± 0.17 Mass of B (M ) ± K 1 (km s 1 ) ± 0.15 K 2 (km s 1 ) ± 0.54 a Epoch shifted by an integral number of periods for better comparison. b Shifted by 180. TABLE 3 Parameter Torres et al. (2002) This Project P (yrs) ± ± a (mas) ± ± 0.82 i (deg) ± ± 6.7 Ω (deg) ± ± 2.5 T (Bess. year) e ± ± ± a ± ω (deg) ± ± 0.58 b V 0 (kms 1 ) ± π (mas) ± ± 2.7 a (AU) ± Mass of A (M ) 0.844± ± Mass of B (M ) ± ± K 1 (km s 1 ) ± 0.11 K 2 (km s 1 ) ± 0.14 a Epoch shifted by an integral number of periods for better comparison. b Shifted by 180.

6 6 Fig. 2. Visual orbit of HD obtained with Pourbaix. Fig. 3. Visual orbit of HD obtained with Pourbaix.

7 7 Fig. 4. Radial velocity curve of HD plotted over one period. Fig. 5. Radial velocity curve for HD plotted over one period and normalized to zero.

8 8 HD x (") y (") Fig. 6. Visual orbit of the inner component of HD HD x (") y (") Fig. 7. Visual orbit of HD with the 5 astrometry points plotted.

9 Fig. 8. Radial velocity curve of inner component of HD plotted over one period. 9