AGE, METALLICITY AND DISTANCE OF M13 ABSTRACT

AGE, METALLICITY AND DISTANCE OF M13 SEAN LINDEN, ABBOTT VELDHUIZEN, AND EMILIE DUNHAM Department of Astronomy, Case Western Reserve University, Cleveland, OH 44106; stl19@case.edu, amv47@case.edu, etd12@case.edu ABSTRACT We investigate the properties of galactic globular cluster M13. We use imaging of the cluster in Sloan g and r bands to construct a color-magnitude diagram in g vs. g-r. Our color-magnitude diagram reproduces several known features of M13, including the RR Lyrae gap in the horizontal branch and the long blue horizontal branch tail. We compare our color-magnitude diagram to that of Johnson et al. 1998 and find generally good agreement. We then fit isochrones from the Padua isochrone package for Sloan from Girardi et al. 2004 to our colormagnitude diagram and derive from the fit [Fe/H] = -1.6, R = 7.3 kpc, and T = 10.25 Gyr. We compare our results to those from the literature and find that while our metallicity is in good agreement with previously derived values, we underestimate the distance and age of the cluster; we propose that our isochrone package systematically underestimates these quantities. We briefly discuss the second parameter problem of globular clusters and conclude that our results could support either of two competing hypotheses. 1. INTRODUCTION The study of globular clusters (GCs), and in particular galactic globular clusters (GGCs), is an important component of research on the Milky Way and on the universe itself. As some of the oldest known objects in the galaxy, they represent valuable laboratories for studying stellar populations. Determining the clusters ages, and how those ages relate to the clusters metallicities, kinematics, and locations, gives us a great deal of information on galactic formation history. Sarajedini and King (1989) were among the first to study the age-metallicity relation (AMR) in GGCs (Dotter et al. 2011). By measuring the ages of GGCs both inside and outside a Galactocentric distance of 15 kpc, they found that GCs closer than 15 kpc have a shallower AMR than those further away, suggesting different formation histories; more recent work has upheld this bifurcation (see, e.g., Chaboyer et al. 1996, Marín-Franch et al. 2009). The shapes of the horizontal branches (HBs) of GCs is also of interest. It has long been known (Sandage and Wallerstein 1960; van den Bergh 1967; Sandage

and Wildey 1967) that the morphology of the HB in GGCs cannot simply be a function of metallicity; there must be a second parameter involved. In order to obtain data on the relative age, etc. of a cluster, astronomers fit an isochrone, a theoretical CMD, to their observed cluster CMD. Isochrones are generated from a combination of stellar evolution models, which predict luminosities and effective temperatures as functions of time and chemical content ([Fe/H], [α/fe], and helium abundance Y), and stellar atmosphere models, which transform luminosity and effective temperature into observables such as color and magnitude (An et al. 2009). Thus, the parameters of the best-fit isochrone to a given CMD hold much information about the stars that comprise the CMD, including derived quantities such as distance and age. The globular cluster M13 is of particular interest among GGCs because of the shape of its HB; it is very long and blue and is a prototype for long blue tail structures (Sandage 2010). M13 is among the most-cited pieces of evidence for the second parameter problem and comprises one half of the M13-M3 second parameter pair. Comparing M13 to M3, another GC with nearly identical metallicity (<[Fe/H]> = -1.53 for M13 versus <Fe/H]> = -1.45 for M3; Sneden et al. 2004), M3 has a substantially redder HB. M3 and M13 were one of the earliest second parameter pairs identified by astronomers. Here, we present findings on the age and metallicity of M13. We construct a CMD for the cluster from data taken from the Sloan Digital Sky Survey. We compare our CMD to another CMD constructed by data collected by Jennifer Johnson (OSU) and discuss major differences between our findings. We then fit a Padua isochrone to the Sloan data and derive from it the age and metallicity of the cluster. 2. DATA ACQUISITION In order to find the age, metallicity, and distance to the M13 globular cluster, we used a Sloan Digital Sky Survey (SDSS) g-band and r-band image of the cluster. The SDSS 2.5 meter Cassegrain reflecting telescope is located in Apache Point Observatory, New Mexico. The images were taken with the 120-megapixel CCD attached to the telescope, and it has a field of view of 1.5 degrees (http://www.sdss.org/). It was necessary for us to gather information from 2 different filters to create a color-magnitude diagram. SDSS has 5 filters (ugriz), but we used only the g and r filters. The g-band and r-band average wavelengths are 4686 Angstroms and 6165 Angstroms respectively M13 is located at approximately an RA of 16h41m41.6s and a Dec of +36d27m41s (J2000). The exact coordinates of the individual stars are exhibited in figure 1 below. The g-band and r-band images we used display the center of

the cluster off to the left corner, leaving us with stars in the outer part of the cluster to examine. 3. METHODS To transform our two raw images of M13 into useful data, we used many tools. The most important was IRAF. This program found the magnitudes of around 5,000 stars (after cuts) for us. The photometry was then used to create a colormagnitude diagram. We had to make cuts because some star magnitudes had large errors, other stars were elongated due to the telescope, and a few were too close to the center of the cluster for IRAF to measure magnitudes accurately. After the cuts, our final diagram was fit with isochrones from Padua for SDSS data. With those models, we estimated the metallicity, age and distance to M13. 3.1 IRAF The packages we used to get the photometry for stars in M13 were noao, digiphot, apphot, daophot and daofind. The parameters inserted into daophot included the averaged FWHM and sigma of the sky value of 10 unsaturated stars as well as read noise and gain. The FWHM and sigma we used are 2.705 and 8.94 respectively. The read noise and gain that we used can be found in table 1 and are different for both the g and r band. We also set initial limits of the sharpness and roundness of stars for IRAF to find. In order to find all the stars to do photometry on, we used the IRAF package daofind. Stars were found in x-y coordinates of the image. We had to transform these coordinates into RA and Dec so that we could match the g and r band images accurately. The 2 images are offset, so they are not the same in x-y coordinates. Once the images were matched, we continued on to finding the magnitudes of the same stars in both bands. We used the apphot package to do photometry. Before running the package, we had to change the annulus, dannulus, aperture and airmass parameters. We set the aperture to be 3, the annulus to be 8 and the dannulus to be 2. These numbers allow enough of the star s light to be measured without collecting much noise from the sky. Airmass was calculated using the following equation: x = sec (90 altitude) We found the altitude that the images were taken at in the header and used them to calculate that x(g-band)=1.0043 And x(r-band)= 1.0049. With these parameters, apphot found the photometry for around 10,000 stars. But because the results are only an instrumental magnitude, they need to be converted using the following equation:

FIG. 1: This is a plot of the position in RA and Dec of Johnson s data (black) and our data (red) overplotted. The plot shows the RA and Dec coordinates of the stars we found photometry of in M13. It also shows the position cuts we made. The green box surrounds stars we did not include in our color magnitude diagram due to being irresolvable. Finally, the plot compares our data to Johnson s data. It shows that we took less points overall than she did when calculating our photometry. g r Zero Point SDSS -24.39982-23.96686 Airmass Coefficient (k) Gain (electrons per ADU) Read Noise (variance in ADU 2 ) 0.246473 0.16686 4.0 4.76 1.96 1.32 Galactic extinction (r) 0.055 0.038 TABLE 1: Constants for in the Sloan g and r filters. The zero point and airmass coefficient numbers come from many stars that have been calibrated by the SDSS final calibration pipeline (SDSS Final Calibration Pipeline). The gain and read noise numbers were found in Gunn, 1998. The galactic extinction values were found for M13 in NED (http://ned.ipac.caltech.edu/forms/byname.html)

m = m inst zp iraf zp SDSS kx r where m inst are the magnitudes the apphot gives, zp iraf is 25, zp SDSS, k, and r are found in table 1, and x is the airmass found above. 3.2 CUTS The methods above gave us the magnitudes of about 10,000 stars in the g-band and r-band (these magnitudes will from now on be called g and r). To clean up our CMD and to reduce scatter, we made cuts in sharpness, roundness, error, and location in the cluster. We started by including stars with sharpness between 0.4-0.8 and roundness between -1 and +1. This introduced a lot of scatter closer to the g colors. Our first attempt at cutting was extreme; we set the sharpness to go from 0.5 to 0.65 and the roundness to go from 0 to -0.8 and ran the photometry again. This cut got rid of too many stars that should have been included and biased the sample too much. More than half of our original number of stars disappeared. Because of this we decided to try more modest values. We ran the photometry again with sharpness between 0.4 and 0.7, and roundness between 0.25 and -1.0. This cut eliminated about 3,000 stars, and most of them were the ones we did not want. Below is figure 2 that shows the sharpness and roundness of the original stars and which ones we cut. The top left plot in figure 3 shows these cuts in a CMD. Next we wanted to cut out stars that had a large error in their magnitudes. Generally dimmer stars have a larger error because they are harder to see, as not much of their light reaches us. When the error is less than 0.02, the data is really good, when it s between 0.02 and 0.05, the data is okay, and when it s over 0.05, the data is bad. We tried a few cuts to determine which was best. A cut of star magnitudes with errors above.05 resulted in 23 magnitude stars to remain visible, while with a cut of errors above.04, we could still see 22.5 magnitude stars. Also as we cut stars with smaller errors, more lower main sequence stars disappeared (they are fainter, so have a larger error). Because we know stars exist here, we did not want to get rid of too many. Our final decision was to cut stars that had a magnitude error of more than 0.05. These cuts can be seen in the top right plot in figure 3. The daofind package had trouble finding individual stars in the center of the M13 globular cluster. They were too densely packed to resolve, so the magnitudes of stars in the center of the cluster were likely incorrect. A position cut was necessary. We tried several methods for doing position cuts, and found that there was no tangible difference between doing a position cut as a semi-circle, rectangle, or square. Thus we went with the simplest method for doing our position cuts in order to remove some of the stars in the densest regions of the cluster. We removed stars inside the x-y coordinates (1600, 400), this cut can also be seen as RA and Dec coordinates in figure 1. We tried larger cuts, but

decided that they eliminated too many stars. The final position cuts are shown in the lower left plot in figure 3. FIG. 2: This plot shows the sharpness and roundness of the stars we found the positions of in our initial photometry of M13. We found that our data was fairly well constrained but that the roundness had a large spread. We can see from this graph that the atmosphere and the telescope naturally distort the shapes of these stars because the mean roundness is around -0.5. The black box shows the cuts that we made and then went back and re ran our photometry. 3.3 ISOCHRONES To find the age, metallicity and distance to M13, we fit isochrones to our data. The models are Padua isochrones and are specific to SDSS data (Girardi et al. 2004). Isochrones give magnitudes of a stellar population for a given metallicity and age. The Padua isochrones ranged from.0001 to.01 dex in Z, and 10 million to 10.25 billion years. We needed to convert the absolute magnitudes (M) to apparent (m), to compare the isochrone accurately to our M13 apparent magnitudes. To do this, we used the following equation: m = M + 5 log(distance) 5 r

Where r is the galactic extinction for M13 referenced in table 1 (corrects for foreground reddening). Using the g and r band apparent magnitudes, we plotted the isochrones on a CMD at different ages. We fit the isochrones to our data on a color magnitude diagram by changing its distance and age. FIG. 3: In the top left graph we see show the color magnitude diagram before (red) and after (black) sharpness roundness cuts. In the top right graph we show the color magnitude before error uncertainty cuts but with sharpness roundness cuts (black) and then with error cuts and sharpness roundness cuts (black). In the bottom left graph we add position cuts (black) and compare it to a color magnitude diagram with all the cuts previously mentioned but without cutting based on location in the cluster (red). The bottom right graph shows our final color magnitude graph with position, error in magnitude and sharpness and roundness cuts made to the data. We believe that these cuts were justifiable and did not bias our data. The key components of the CMD, being the turn off point, the lower branch, and the Horizontal branch are all well constrained even after making data cuts.

There are many different isochrone models to choose from depending on the metallicity of the cluster. So, to start, we had to choose isochrones with an appropriate metallicity to match our data. Instead of trying all of them, we began by using a previously measured metallicity. The published [Fe/H] value we found was -1.58 (VandenBerg 2013). But because the different isochrone choices quote metallicity values as Z, we had to do a conversion given in Carraro et al. 1994: Log(Z) = 1.03 [ Fe H ] 1.698 Using this equation, we calculated that Z is about 0.0004. So, we used the isochrones with Z=0.0004 to start with. Once we chose a reasonable age and distance, the fit was very good. But, to make sure that we did not start out with the incorrect metallicity, we tried a few other isochrones. With Z=0.004, the isochrones never fit the giant branch very well. No matter what values we inserted for age and metallicity, the isochrones overestimated the red giant branch. With Z=.001, the isochrones also did not fit the giant branch or the turn-off. The Z=0.0001 isochrones fit the horizontal branch better than the Z=0.0004 isochrones, but it did not fit the main sequence or the turn-off very well. We thought it was very important for the isochrones to fit the main sequence and turn off because they are the main features in a color magnitude diagram. We chose to use a Z=0.0004 isochrone because, although it deviates slightly from the first red giant branch, it fits the horizontal branch, the lower main sequence, and the giant branch turn-off very well. 4. RESULTS The M13 globular cluster has a metallicity of -1.6, an age of 10.25 billion years, and is 7,300 pc away from us, according to our computations and isochrone fitting. The results are shown in figure 4 below. The errors in our magnitudes are shown in figure 5. Although these errors are not specific to age, metallicity or distance, magnitude errors affect the results significantly. If our magnitudes were too far off, a different isochrones may fit to on the color magnitude diagram giving different results.

FIG. 4: An isochrone fit (green) to our color magnitude (black) plotted alongside Johnson s color magnitude diagram (red).

FIG. 5: This plot shows us that the uncertainty in our magnitudes decreased as we went to brighter stars. The g-band uncertainties (black) are the lower than r-band uncertainties (red) and have the smallest deviation. This is most likely because we did our calibration of the star coordinates in the g-band and converted to the r-band. There is a characteristic fall off in error as we went to lower magnitude (brighter) stars, which is what we would expect. Our method of photometry seemed to have a larger effect in the redder band. 5. DISCUSSION 5.1 CMD Properties When comparing our CMD to others in literature we look first at the work of Johnson et al. 1998. For this we were able to do a direct comparison using the same g vs g-r colors and magnitudes from Sloan that they used to calibrate their CMD of M13. Figure 6 shows our data plotted with Johnson et al. The first thing one notices is that there are many more faint stars on their lower main sequence than we have. This is due mostly to the fact that many of those stars in our photometry had uncertainties in their magnitudes too large to be considered accurate. Johnson et al. used the DAOGROW and DAOPHOT II routines to work their instrumental magnitudes (Johnson et al. 1998). We used a less sophisticated photometry package, and thus our uncertainties are systematically higher than their errors, particularly when looking at the lower main sequence. Our position, sharpness, and roundness cuts undoubtedly removed data from our sample that was good. That is another reason our lower

branch seems sparse and more trimmed than Johnson et al. One thing that is important to note about the two CMD s is that the shapes and evolutions of the red giant branch (RGB) and horizontal branch (HB) branch are identical. Even though we present results with much less data (18,533 stars vs 4,180 stars) the overall morphology of M13 remains unaffected. We also see that the turn off point of M13 is identical in both CMD s. It is important that these features are identical because we can extend our analysis of M13 to how the turn off, RGB, and HB affect the cluster s metallicity and age (two fundamental properties one wants to know about a globular cluster). Our comparison with Johnson et al. both justified our data cuts, and reinforced our confidence in our photometry. FIG. 6: shows a plot of Jennifer Johnson s color magnitude (in red) vs ours (in black). 5.2 HORIZONTAL BRANCH MORPHOLOGY GC CMDs generally show horizontal branches that have a prominent gap in the HB. This gap in the CMD incorrectly suggests that the cluster has no stars in this region of its CMD (Sandage et al 2006). The stars that live in this region are pulsating horizontal-branch stars are known as RR Lyrae variable stars (Cho et al. 2005). This omission often results in the RR Lyrae gap seen in many published globular cluster CMDs (Sandage 2006). It is important that our CMD shows this gap because if we had a significant amount of data in that region it would suggest that our simple photometrical routine was capable of accurately

measuring stars we know that we shouldn't be able to measure due to the difficulty in measuring a variable star s flux. Thus the gap we see is consistent with what we would expect. Looking at Figure 6 we can see the HB of our cluster extends farther into the blue than we might have predicted from our comparison of a globular cluster with similar metallicity like M3 (Meszaros et al. 2009). Paltrinieri et al. 1998 notes that the HB of M13 extends almost 1 magnitude fainter than the turn off point for the cluster. This is something that we do not see, most likely due to our data reduction methods, but nevertheless we do see the long blue tail feature that Platinieri et al. talks about. This tail extends just below the RR Lyrae gap down to about g~ 18.5. Sweiget et al. 1997 pointed out that deep mixing in the cluster could increase the envelope abundance of helium which could lead to enhanced mass loss along the RGB. Buonanno et al. (1997) examined the role of stellar density in the morphology of the HB and suggested that clusters with higher central densities are more likely to populate the bluest extremes of the HB (see also Fusi Pecci et al. 1993). This mass loss in turn can affect the HB morphology, and could prove to be one of the driving forces behind the HB gaps mentioned previously, the instability gap, and the long blue tail. 5.3 METALLICITY The metallicity we derived for M13 was Z=.0004. Using the conversion from Carraro et al. 1994 we get an [Fe/H] of -1.60 which agrees very well with the Vandenberg et al. 2013 value of -1.58. This is encouraging because this paper along with many others like Johnson et al 1998 use a method of determining ages based on differential magnitudes. We however determined our ages strictly using the Padua isochrones for the Sloan filter system. This provides a consistency check on both the result of the metallicity of the cluster as well as a confirmation that neither method of age determination greatly biases the results for relatively young clusters, which M13 is. 5.4 AGE AND DISTANCE We reported the age of M13 from the isochrone analysis to be 10.25 Gyr. Vandenburg et al 2013 and Alves et al. 2004 both cite the age of M13 to be around 11.4 Gyr. Our model is underestimating the age of M13, but it is systematic because we did not have isochrones that went beyond 10.25 Gyr. This is something we could have inferred from the fit because our isochrone fit deviates from our model in the RGB and thus even though we model the turn off well we could have done a better job modeling the upper RGB and pushing our model further down the HB. The isochrones we used were from the Girardi et al. 2004 basic set. This set was derived by using the Girardi et al. (2000) evolutionary tracks together with previous Padova tracks for massive stars and lower ages of less than 10 8 yr (from Bressan et al. 1993 & Bertelli et al. 1994). If we had chosen to use a different set of Sloan isochrones we might have been

able to push our fit up to 11 Gyr but as Girardi et al. 2004 describes, this induces more error and uncertainties in the fitting that would have also skewed our results. We determined then that the isochrones we used were reasonable ones, and ones that we still feel accurately model M13. Reid et al. 1997 cites the distance modulus to M13 as 14.48. This corresponds to a distance of roughly 7800 pc. From our fit we got a distance to M13 to be 7300. Again we are underestimating the distance, for what we believe to be the same reason as above. 5.5 SECOND PARAMETER PROBLEM A long-standing problem with comparing globular clusters is the second parameter problem. Johnson et al 1998 points out that this problem has been studied at length by several groups including Lee et al. 1994, Buonanno et al. 1997, Stetson, VandenBerg & Bolte 1996, and Catelan & de Freitas Pacheco 1995. The second parameter problem is one where two globular clusters of relatively the same metallicity have very different HB morphologies. Lee et al. argued that the second parameter to describe this difference was the difference in age of the two clusters. Johnson et al. used a relative age dating method to conclude that the age difference between M3 and M13 (the two clusters used to talk about this problem) is only about 2 Gyr. This is not enough of an age difference to explain the differences in HB morphology (Johnson et al. 1998 & Meszaros et al. 2009). Johnson et al. points out that helium abundance can explain the variations in HB morphology and should be considered a viable candidate for the second parameter, something that Sweigert et al points out as well, but not specifically looking at M13. This is important because if the helium abundance can explain the differences in HB morphology than we do not need to rely on an age estimation to derive key differences in clusters. Contrary to the findings of Johnson et al. and Sweigert et al., Rey et al. 2001 reports that with an updated calibration on the age differences between M3 and M13, as well as un updated model for how age affects HB morphology they report that roughly the same difference in age reported by Johnson et al. can actually account for the difference in HB morphology and thus can be thought of as the global second parameter that Lee et al. had discussed. This of course is still a widely debated topic that our photometry and data analysis cannot shed much light onto. It is just important to note that our findings indicate the possibility for either model to be correct, and more detailed follow up models would need to be applied to our data in order to determine if either the helium abundance, age, or some combination of the two could be the missing second parameter for determining key cluster properties. 6. CONCLUSION

We created a CMD in Sloan g vs. g-r of the globular cluster M13 and investigated its properties by fitting Padua isochrones to it. We derived values for its age, metallicity, and distance in large part consistent with values from the literature, although our results contain systematic errors from our photometric and isochrone packages. Our findings are also in generally good agreement with those of Jennifer Johnson. Who did what: Sean Linden, Emilie Dunham, and Abbott Veldhuizen all contributed to the literature search. Sean Linden was responsible for making the plots and writing section 5, along with spearheading the iraf component of the project. Emilie Dunham is responsible for writing sections 2, 3, and 4 and helping with the data reduction in iraf. Abbott Veldhuizen wrote sections 1, 6, and the abstract, along with leading the final editing and drafting of the paper. ACKNOWLEDGEMENTS We would like to thank Professor Chris Mihos (CWRU) for his extensive and ongoing guidance regarding this project and for not making us do this whole shebang in LaTeX. We also thank Jennifer Johnson (OSU) and the Sloan Digital Sky Survey for allowing us to use their datasets. REFERENCES Paltrinieri, B., Ferraro, F. R., Fusi Pecci, F., & Carretta, E. 1998, MNRAS, 293, 434 Johnson, J. A., & Bolte, M. 1998, AJ, 115, 693 Buonanno, R., Corsi, C., Bellazzini, M., Ferraro, F. R., & Fusi Pecci, F. 1997, AJ, 113, 706 Fusi Pecci, F., Ferraro, F. R., Bellazzini, M., Djorgovski, S., Piotto, G., & Buonanno, R. 1993, AJ, 105, 1145 Cho, D.-H., Lee, S.-G., Jeon, Y.-B., & Sim, K. J. 2005, AJ, 129, 1922 Sandage A, Tammann, Gustav A., 2006, ARAA, 44, 93-140 Lee, Y.-W., Demarque, P., & Zinn, R. 1990, ApJ, 350, 155 Stetson, P. B., VandenBerg, D. A., & Bolte, M. 1996, PASP, 108, 560 Catelan, M., & de Freitas Pacheco, J. A. 1993, AJ, 106, 1858 Rey, Soo-Chang, Yoon, Suk-Jin, et al. 2001 ApJ, 122, 3219-3230 Meszaros, Sz., Dupree, A. K., et al. 2009 ApJ, 137, 42828-4295 Alves, D. R., Cook, K. H. et al., AAS, 36, 1425 Reid, Neil, I., 1997, AJ, 114, 1 Girardi, L. et al., 2004, arxiv.astro-ph/0404358 (style for preprints before April 2004) Gunn et al., 1998, 116, 3040 VandenBerg, D., et al., 2013, ApJ, 775, 134 Carraro G.,1994, A&A, 287, 761 Sneden, C., et al., 2004, AJ,127, 2162 Sandage, A. 2010, ApJ, 722, 79 An, D., Pinsonneault, M. H., Masseron, T., et al. 2009, ApJ, 700,523 Dotter A., Sarajedini A., Anderson J., 2011, ApJ, 738, 74 Mar ın-franch, A., et al. 2009, ApJ, 694, 1498 Chaboyer, B., Demarque, P., & Sarajedini, A. 1996, ApJ, 459, 558 Sandage, A., & Wallerstein, G. 1960, ApJ, 414, 580 Sandage, A. & Wildey, R. 1967, ApJ, 150, 469 van den Bergh, S. 1967, AJ, 72, 70