MASS DETERMINATIONS of POPULATION II BINARY STARS Kathryn E

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 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 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. Pikaia, developed by Charbonneau baix’s solutions, and comparing the estimated errors to (2002), is an optimization code that uses genetic algo- true differences from the presupplied orbital parameters rithms to find the maximum of an n-dimensional, user- provided a check of the accuracy of Pourbaix. supplied function. Genetic algorithms are designed to 5. RESULTS mimic evolutionary biology in seeking an optimum solution. A “population” of initial guesses are evaluated at 5.1. HD 157948 random places in the search space, so a good initial guess HD 157948 is a quadruple system with an estimated is not necessary. The best results are “bred” to produce metallicity of [m/H]=-0.75. Our primary target is the 3 spectroscopic binary near the center of this system. Ta- Table 2 shows the Pourbaix solutions for the inner pair ble 1 compares the Pourbaix solution for the inner pair compared to those obtained by Saia et al. (2006). In this to those obtained by Horch et al. (2006) and Saia et al. case, all three of the angles that describe the orbit appear (2006). The Pourbaix solutions are consistent and the er- inconsistent with previous work. The inclination error ror bars overlap for most parameters with the exception bar is high, but it does overlap Saia et al.’s (2006) value. of the position angle of the ascending node, Ω. Incon- The longitude of the periastron and the position angle sistencies of 180◦ in the angles that describe the orbit of the ascending node both vary without overlapping er- are merely due to differences in observation conventions ror bars. However, from the results of the previous two and do not affect the reliability of any other parameter. stars, we have confidence in trusting the reported mass Additionally, inconsistencies other than 180◦ do not ap- estimates and errors. pear to significantly affect the mass estimates returned With the individual mass determinations for the inner by Pourbaix, which is the most important parameter for two stars, we can calculate the mass of the third compo- this study. nent from Kepler’s equation: Figure 6 shows the visual orbit of HD 157948 with a3 only the astrometry points plotted, and Figure 5 shows P 2 = (1) the radial velocity curve of HD 157948 plotted over one M1 + M2 period. The addition of the Keck spectroscopy revealed Choosing M to be the combined mass of the primary and a linear change in the system radial velocity of 0.00076 1 secondary components, M gives the mass of the outer km s 1. Normalizing the radial velocity of each−observa- 2 − component as 0.239M . This is the first time a reliable tion to zero provides a well defined radial velocity curve estimate has been mad#e for the third component. These with the advantage of leaving the calculated orbital pa- low masses are consistent with the composite spectral rameters unaffected. The only parameter that should type. be affected by this change should be the system radial Figure 7 shows the visual orbit of the inner pair of HD velocity V ; however, the systemic velocity calculated 0 200580 obtained from the Pourbaix parameters with the by Pourbaix did not recognize this normalization, and five astrometry points plotted. the radial velocity plot was shifted from the data. Re- Figure 8 shows the radial velocity curve for the in- jecting the Pourbaix velocity and manually entering the ner pair of HD 200580. In this case, the curve for the systemic velocity as zero forced the calculated curve to secondary star was obtained with only the Keck spec- shift such that it matched the data. It is unclear as to troscopy, as earlier attempts were unable to obtain the why the Pourbaix code does not recognize the velocity double-lined spectrum. The different amplitudes reflect normalization, and it is a subject to be investigated fur- the different mass estimates returned by Pourbaix. ther. Figure 5 shows the radial velocity curve of HD 157948 calculated by Pourbaix with the forced normalization. The normalized Goldberg et al. (2002) and Keck 6. CONCLUSIONS spectroscopy points with error bars are also plotted. This research provides accurate masses with accompa- nying error estimates for three dwarf Population II bi- 5.2. HD 195987 nary systems – HD 157948, HD 195987 and HD 200580. HD 195987 has an estimated metallicity of [m/H] = These masses can effectively contribute to both a Mass- 0.83. Studied in depth by Torres et al. (2002), it pro- Luminosity Relationship for Population II stars and a vides− the best consistency check for our study. The Pour- Population II main sequence evolutionary track. The baix solution shows good agreement with the Torres et Monte Carlo analysis of our simultaneous least-square al. (2002) solution, again with the exception of the posi- adjustment of spectroscopic and astrometric data indi- tion angle of the periastron, but also the inclination. And cates that Pourbaix mass estimates are self-consistent again, the mass estimates are very consistent, a good sign and accurate. It also shows that the reported errors for the confidence level of our mass estimates and error appropriately mimic real-life errors. Furthermore, the estimates. Pourbaix mass estimates are unaffected by the the incon- Figures 2 and 3 compare the visual orbit obtained with sistencies in the angles that describe the geometry of the Torres et al.’s (2002) parameters to that obtained with orbit. Despite these inconsistencies, the mass estimates our Pourbaix parameters. The Torres et al. (2002) po- of this project corroborate those obtained by others. sition angle of the ascending node as reported in Table Additionally, there is a systematic difference between 3 has been shifted by 270◦ for the plot to allow better the parallax values calculated with Pourbaix and those comparison. This is due to a quadrant ambiguity in the given in the Hipparcos Catalogue. The following table conventional analysis of orbital angles. The Saia et al. shows that the Pourbaix parallax is consistently higher (2006) astrometry points are plotted on both orbits. than the Hipparcos parallax on the order of a few mil- Figure 4 shows the radial velocity curve for HD 195987 liarcseconds. with the Keck and Goldberg et al. (2002) data points plotted. The theoretical fit matches the data quite well, Parallax Estimates (mas) and the similar amplitudes of the curves reflect the sim- star Hipparcos This Project ilar masses returned by Pourbaix. HD 157948 19.78 22.84 5.3. HD 200580 HD 195987 44.99 57.4 HD 200580 17.83 21.21 HD 200580 is a triple system with a metallicity of [m/H]=-1.01. Again, our primary target is the close spectroscopic binary at the center of this triple system. 7. FUTURE WORK 4

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 Fig. 1.— An example of a function whose global extremum would Pourbaix, D. 1994, A&A, 290, 682 be easy to miss. Pourbaix, D. 1998, A&AS, 131, 377 Saia, M., 2006 M.S. Thesis, U. Mass Dartmouth The broad goal to be obtained with the accurate Torres, G., Boden, A.F., Latham, D.W., Pan, M., masses from this project is the determination of a Pop- Stefanik, R.P. 2002, AJ, 124, 1716 ulation II Mass-Luminosity Relationship and a Popula- tion II main sequence. This will require a more com- plete sampling of Population II masses, as this was only three systems of Horch et al.’s (2006) original sample of 5

TABLE 1

Parameter Horch et al. (2006) Malinda Saia(2006) This Project P (yrs) 1.22521 0.00033 1.22565 0.00031 1.22433 0.00034 a (mas) 32.17± 0.87 31.2± 2.1 31.3± 1.4 i (deg) 94.2 ± 2.0 99.7 ± 3.1 100.7± 3.7 Ω (deg) 51.5 ± 1.6 228.0± 2.3 141.4 ± 2.8 ± ± a ± T (Bess. year) 1986.380 0.010 1986.373 0.0095 1986.369 0.013∗ e 0.146 ±0.007 0.1503 ±0.0074 0.1474 ±0.0091 ω (deg) 179.4± 3.0b 179.2± 2.9 177.6± 3.8 1 ± ± ± V0(kms− ) +3.516 0.083 ...... 3.066110 0.12 π (mas) 23.65 ± 0.69 22.63 1.6 22.84 ±1.1 a (AU) 1.3600 ±0.0021 ...... ±.... 1.3708 ±0.004 Mass of A (M ) 0.887 ± 0.030 0.928 0.41 0.917 ±0.049 Mass of B (M#) 0.788 ± 0.021 0.813 ±0.032 0.801 ± 0.054 1# ± ± ± K1 (km s− ) 15.68 0.16 ...... 15.45 0.43 1 ± ± K2 (km s ) 17.65 0.25 ...... 17.69 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) 1.0094 0.0005 1.03398 0.0005 a (mas) 24.92± 0.66 25.3 ± 2.4 i (deg) 113.4± 1.3 102.3± 11 Ω (deg) 13.8 ±1.3 116.6 ± 6.5 T (Bess. year) 2002.078± 0.025 2002.041 ± 0.033a e 0.0798 ±0.0065 0.09535 ± 0.015 ω (deg) 42.0± 8.9 68.6 ±13b 1 ± ± V0(kms− ) ...... 1.719 0.18 π (mas) 22.63 1.6 21.21± 2.4 a (AU) ....±... 1.1910 ±0.0086 Mass of A (M ) ...... 1.106± 0.17 Mass of B (M#) ...... 0.474 ±0.066 1# ± K1 (km s− ) ...... 10.01 0.15 1 ± K2 (km s ) ...... 23.56 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) 0.157046 0.0000007 0.156937 0.0000015 a (mas) 15.378± 0.027 19.1± 0.82 i (deg) 99.364 ± 0.080 81.6± 6.7 Ω (deg) 334.960± 0.070 52.9 ± 2.5 T (Bess. year) 1999.478 ± 0.0001 1999.476 ±0.00020a e 0.30626 ±0.00057 0.3095 ± 0.0029 ω (deg) 357.40± 0.29 357.3 ± 0.58b 1 ± ± V0(kms− ) ...... -5.860 0.067 π (mas) 46.08 0.27 57.4 ± 2.7 a (AU) ...±.... 3.326 ±0.0014 Mass of A (M ) 0.844 0.018 0.835± 0.043 Mass of B (M#) 0.6650 ± 0.0079 0.658 ± 0.034 1# ± ± K1 (km s− ) ...... 28.95 0.11 1 ± K2 (km s ) ...... 36.73 0.14 − ±

a Epoch shifted by an integral number of periods for better comparison. b Shifted by 180◦. 6

Fig. 2.— Visual orbit of HD 195987 obtained with Pourbaix.

Fig. 3.— Visual orbit of HD 195987 obtained with Pourbaix. 7

Fig. 4.— Radial velocity curve of HD 195987 plotted over one period.

Fig. 5.— Radial velocity curve for HD 157948 plotted over one period and normalized to zero. 8

HD 157948

-0.03 x (")

0.02 -0.03 y (") 0.03

Fig. 6.— Visual orbit of the inner component of HD 157948.

HD 200580

-0.03 x (")

0.03 -0.03 y (") 0.03 Fig. 7.— Visual orbit of HD 200580 with the 5 astrometry points plotted. 9

Fig. 8.— Radial velocity curve of inner component of HD 200580 plotted over one period.