MASS DETERMINATIONS OF POPULATION II BINARY STARS Kathryn E. Williamson Department of Physics and Astronomy, The University of Georgia, Athens, GA 30602-2451 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 157948, HD 195987 and HD 200580. Results were obtained via a simultaneous least-squares adjust- ment 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 157948, HD 195987, HD 200580) — stars: Population II 1. INTRODUCTION surements to deduce other information about the orbit The current stellar Mass-Luminosity Relationship and, ultimately, the mass. Individual masses of the con- (MLR) is based mainly on Population I stars that are stituent stars in a system can be obtained only if the more massive than the sun. Population II stars have physical scale of the orbit is known. The physical scale been more difficult to study in terms of mass and lumi- can be calculated with either a parallax value, such as nosity because they mainly reside far away in old globular those given in the Hipparcos Catalogue, or a combination clusters in the halo of the galaxy. They can be identi- of astrometry and double-lined spectroscopy. The latter fied by their characteristic high orbital inclinations from combination was used for this project, providing an inde- the galactic plane and high proper motions. The low pendent parallax estimate to compare to the Hipparcos metallicity of population II stars ([m/H] typically less Catalogue value. than -0.70) indicates that they formed early in the his- tory of the galaxy before nucleosynthesis populated the 2. BACKGROUND interstellar material with heavy elements. Studying these 2.1. Data metal-poor stars will allow the development of a Mass- An extensive set of spectroscopic data was obtained Luminosity Relationship to include metallicity effects. by Goldberg et al. (2002), along with preliminary or- In addition to contributing to a Population II MLR, bital solutions for 34 binaries. The Astrometry needed accurate mass determinations will provide an improved to refine these solutions was published by Horch et al. understanding of the Population II main sequence. Be- (1999). Horch et al.(1999) chose 13 high proper motion cause mass determines how a star evolves, determining binaries from Goldberg et al.’s (2002) sample that could masses of these old, metal-poor stars will allow us to likely be resolved with the Hubble Space Telescope’s Fine improve our understanding of the Population II evolu- Guidance Sensors’s (FGS) 10 mas resolution capabili- tionary track. Dwarf Population II stars are of particu- ties. From previous estimates of metallicity and proper- lar interest for determining population II evolution be- motion in the literature, these binaries are believed to be cause their low masses cause them to age slowly, indi- population II dwarfs. The FGS data can be reduced to cating their current observables are close to their initial astrometry positional data via various methods, includ- evolutionary stage and that they are still on their main ing Fourier transformations and S Curve deconvolutions. sequence track. Mass estimates will also contribute to Saia et al. (2006) applied S Curve deconvolutions to forming a more detailed comparison with Population II the Horch et al. (1999) FGS data to determine the as- evolutionary tracks. trometry of five of the 13 binary systems. Using the Determining the orbital parameters of binary systems Hipparcos parallax, the FGS astrometry, and the single- is currently the best way to deduce stellar masses. Visual lined spectroscopic data that was available at the time, observations offer positional data in either polar or carte- Saia et al. (2006) improved orbital solutions and mass sian coordinates of the secondary star with respect to determinations. Heasley has obtained spectroscopy for the primary and the orientation of the orbit with respect the second component of three of Saia’s five stars using to the plane tangential to the celestial sphere. Spec- the Keck telecope, providing the data necessary to ob- troscopic data offer radial velocity measurements. Using tain the most accurate orbital solutions. For this project, Kepler’s equations of motion, we can combine these mea- the Keck spectroscopy was combined with Goldberg et 2 al. (2002) spectroscopy and Saia et al. (2006) astrome- the new “generation” of test solutions to be evaluated. try to determine reliable estimates of individual masses. This continues until some preset number of “generations” These three systems are HD 157948, HD 195987, and HD is reached, and the “fittest” result is supplied. We de- 200580, and they are important to our study for various veloped our own code that incorporates Pikaia to deduce reasons that are discussed later. the most accurate orbital parameters from spectroscopic and astrometric data by maximizing the statistical 1 χ2 2.2. Orbital Parameters value. In order to describe the orbit of a binary star, we must A third optimization code developed by Pourbaix solve for specific orbital parameters. (1998) implements a simultaneous least-squares adjust- The following are the seven orbital parameters given ment of visual and spectroscopic observations via simu- by double-lined spectroscopy. lated annealing minimization. Simulated annealing mim- ics the metallurgical process of annealing in which sub- P = Period stances are able to reach low energy states by cooling • slowly. The program defines the analog of a temperature T = Time of Periastron passage • that regulates the rate of annealing. The program then slowly converges to the general neighborhood of the so- ω = Longitude of periastron in the plane of the lution, and an additional hill climbing scheme fine-tunes • true orbit the results. e = Eccentricity • 4. METHOD V0 = System radial velocity 4.1. Theoretical Data Generation Program • In order to test the reliability of the optimization pro- K1 = Amplitude of primary radial velocity • grams to return self-consistent results with accurate er- K2 = Amplitude of secondary radial velocity rors estimates, we compared sets of data with known or- • bital parameters to the parameters reported by the opti- Visual data provides three additional parameters to mization codes. This involved development of a theoret- completely describe the orbit: ical data generation program from a set of pre-supplied orbital parameters. This program uses a set of input or- i = Inclination bital parameters to calculate theoretical positional and • velocity data, and it assigns this data to the real time Ω = Position angle of ascending node • epochs of our real data. Although real observation times d = Distance to system (parsecs) are not evenly distributed along the orbit, this keeps • the simulation as similar to a real life situation as pos- To find the orbital parameters that best fit the observa- sible. The theoretical data generation program then im- tions in a least-squares sense, we define a 10-dimensional plements a random number generator and a Gaussian nonlinear fitness function from the difference of observed deviation function to add noise to the data such that its and calculated positions and velocities. The goal then is standard deviations mimic real life errors. For the spec- 1 simply to minimize the fitness function, for which there troscopic data we chose σV = 1.0 kms− , and for the are many different strategies. This seems simple enough, astrometric data we chose σθ = 0.64 degrees, σρ = 0.002 but optimization of such a large number of paramters arcseconds, σx = 0.002 arcseconds, and σy = 0.002 arc- can be quite difficult. How much confidence should we seconds. The simulation program then outputs the posi- have in the solution returned by the computer program? tional and velocity data with these errors in a format to What if we have a function similar to that in Figure 1 be used with any of the above optimization programs. where the program could easily miss the narrow global 4.2. Self-Consistency and Accuracy of Errors extremum in favor of a broader local extremum? These are issues that optimization programs attempt to over- Due to BINARY’s sensitivity to initial guesses and be- come. cause our Pikaia code is still under development, we in- vestigate the reliability of Pourbaix’s (1998) code. We 3. OPTIMIZATION TECHNIQUES implement a Monte Carlo analysis to check the ability Three optimization methods were investigated for this of Pourbaix to return self-consistent results and accu- project. BINARY was developed by Gudehus (2001) and rate error estimates. Running 15 theoretical data sets in is specifically tailored for applications to binary star pa- Pourbaix for each binary system allowed us to test the rameters. It uses a homegrown gradient search method sensitivity of Pourbaix to each particular sampling of ob- that is quite powerful when given a good initial guess; servations for a given orbit. The Monte Carlo analysis of however, it can be quite sensitive to small perturba- these results provided a self-consistency check of Pour- tions in this guess.