SEQUENCING THE STARS

ROBERT J. VANDERBEI

Using images acquired with modern CCDcameras, amateur astronomers can makeHertzsprung-Russell diagrams from theirown images of clusters. In this way, we candemonstrate for ourselves that globularclusters are old and far whereas openclusters tend to be young and near.

It is easy to forget that less than 100 years ago,no one knew the process by which stars producelight. Today, we understand quite well that stars,which are composed mostly of hydrogen,produce energy by nuclear fusion. Theenormous pressure created by the mass of astar causes pairs of hydrogen nuclei in its core tofuse into a helium nucleus thereby converting asmall amount of mass into energy. Hence, wehave starlight.

For most of its lifetime, a star’s brightness andcolor depend almost exclusively on a singlevariable—how much hydrogen was present at itsbirth, when a cloud of hydrogen coalesced underthe influence of gravity to form the star. In otherwords, brightness and color depend on mass.Stars significantly more massive than our Sunare hot, bright, and blue—the more massive, thehotter, brighter, and bluer. Stars less massivethan our Sun are fainter, cooler, and redder.Hence, if we make a scatter plot of color versusabsolute brightness (luminosity) most points inthis scatter plot fall roughly on a line, called theMain Sequence. Such plots were firstconstructed by Ejnar Hertzsprung and HenryNorris Russell in 1910 (one hundred years ago)and are therefore called Hertzsprung-Russelldiagrams, or HR-diagrams for short.

The most difficult part in constructing anHR-diagram is to find the actual luminosity ofeach star. For this, we need to know how faraway the star is. If we know that, we can convertits apparent magnitude (the magnitude itappears to us in the night sky) to its absolutemagnitude (the magnitude it would appear to an

observer at a standard distance from the star,say, 10 parsecs). Absolute magnitude isessentially equivalent to luminosity. It tells ushow bright a star actually is. But, knowing thedistance is key. Without knowing distance, it isdifficult to relate apparent magnitude to absolutemagnitude. For this reason, it was really quite anachievement when Hertzsprung and Russell firstconstructed their diagrams. However, clusters(both open and globular) give us a way tocircumvent the distance issue. The stars in acluster are all at about the same distance fromus. Hence, for such stars, we can use apparentbrightness as a surrogate for luminosity.

To illustrate, I recently took a sequence ofshort-exposure pictures of the Beehive Cluster(M44). This cluster is large, listed in cataloguesas 95 × 95 arcminutes, and so I opted to do atwo-frame mosaic. My mosaic doesn’t cover theentire cluster but I captured a big central part ofit. I actually made three separate mosaics, usingshort exposures (0.08 seconds), mediumexposures (8 seconds), and long exposures (80seconds). Then, I used the freeware programSOURCE EXTRACTOR, commonly referred to asSExtractor, to identify all the stars in theseimages and for each one an estimate of its colorand brightness. SExtractor produces a plain textfile listing data for each star, one star per line. Iculled from the lists any star whose flux wassuch that the brightest pixels would havesaturated the CCD chip. Then I normalized thefluxes to represent what I would have gotten froma 1 second exposure and I combined the threelists into one master list. I imported this masterlist into a plotting program (MATLAB) and made aplot of color versus luminosity. That plot is shownhere. The main sequence is plainly evident! Aswith any wide-field image, there are several starsthat do not belong to the object under study. Inthis case, these are fainter background starswhose colors are largely uncorrelated withapparent brightness because they are at varyingdistances and therefore apparent brightness tellsus nothing about absolute brightness.

The main sequence is only part of the story.When a star gets old, generally after billions ofyears, it exhausts the hydrogen available forfusion in its core. At this point, which marks thebeginning of a prolonged death process (lasting

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about a million years or so), other fusionreactions take over. No longer is there a simplerelationship between mass, color, and luminosity.In the first stage of this death dance, the stargrows in size, cools, and becomes redder. Suchold stars form a distinct branch in an HR-diagramcalled the red giant branch. Stars that started outwith plenty of mass, burn rapidly and die young(after merely hundreds of millions of years)whereas stars that started out with less massspend billions of years as main sequence starsbefore eventually becoming red giants. Ofcourse, the final stage of stellar death also

depends on the mass of the star—massive starseventually explode and are seen for a short timeas supernovae whereas less massive starsexpire more gracefully by sloughing off theirouter shell, which we see as planetary nebulae,eventually leaving behind just a faint remnant ofa star called a white dwarf.

White-dwarfs are far from red-giants on the HRdiagram. How does the evolution from one to theother look on such a diagram? As a red giantages, it eventually begins to fuse helium intocarbon. At this point, the star turns bluer, movingto the left on the HR diagram. These stars form a

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Open Cluster M44 (Beehive). Image taken by the author.

second prominent branch called the horizontalbranch. After that, at the last stage, the starrapidly diminishes in brightness and becomes avery faint white dwarf. This last transition takes

place over a relatively short time interval andtherefore we see very few stars making thistransition.

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The Main Sequence is clearly visible.

Globular clusters are known to be veryold—roughly as old as our galaxy itself. In fact,all of the stars in a globular cluster formed atabout the same time billions of years ago. Thereare no remaining bright, hot, blue stars inglobular clusters. They have all lived, loved, anddied billions of years ago. All that remains arethe less massive, cooler, stars. There arehundreds of thousands of them in a globularcluster. The brightest stars one sees in theseclusters have left the main sequence and are inthe late red-giant phase of their evolution.Hence, the bright stars in an HR-diagram of aglobular cluster are red giants. The brighter thestar, the redder and “gianter” it is. Take a look atthe illustration showing the HR-diagram for M13constructed from the professional catalogs in thepublicly accessible VizieR database(http://vizier.u-strasbg.fr/viz-bin/VizieR). This plotis typical for a globular cluster. The brightest

stars are red-orange. The red-giant branch formsa narrow curve that arcs up and to the right. Wealso see a branch consisting of mediumbrightness blue stars. These stars are on abranch called the horizontal branch (even thoughit looks more vertical than horizontal in thisparticular case). Stars on the horizontal branchare more massive than red giants and aretherefore further along in their stellar life cycle.Normally, the horizontal branch connectshorizontally over to the red giant branch. Thewhitish stars on this horizontal branch are thefamous RR-Lyrae stars. These are variable starsmeaning that their magnitude varies with time(over the course of hours). Hence, they don’thave one particular magnitude to plot on thevertical axis and I believe for this reason mayhave been purged in the VizieR database fromwhich the chart shown here was made.

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Globular Cluster M13. Image taken by the author.

I have taken photographs of many globularclusters over the past several years. Recently, Iconstructed HR diagrams for them. A few areshown here.

The first thing to notice is that my HR diagram forM13 looks similar to the one shown before.There are, of course, some differences. I’mshowing more stars. The VizieR database that Iused for the “professional” HR diagram excludesstars that are crowded by nearby stars making ithard to obtain accurate color and luminosityinformation. My program is not so cautious. Also,the VizieR diagram uses the accepted method ofcomputing star color, which is to subtract theso-called visual magnitude (V) from the bluemagnitude (B). This is called the B-V color-index.There is a standard specification for the B and Vfilters that one is to use. My pictures were takenwith the filters I have, namely red (R), green (G),and blue (B) filters. It seemed reasonable to useB-R magnitudes in place of B-V. Hence, for each

star, I assigned a slightly different numericalvalue for its color than in the VizieR plot.

My plots show some stars of various colors thatseem to be too bright. These are nearby starsthat don’t belong to the globular cluster but justhappen to be in the field of view.

I like globular clusters so much that I tend torevisit my favorites each year and retake theirpicture. Hence, I can compare pairs ofHR-diagrams of the same cluster taken atdifferent times. In general, such a pair ofdiagrams should look the same. But, the RRLyrae stars are variable and therefore can beexpected to have slightly different brightnesseswhen imaged at two different times (even if takenjust hours or days apart). To illustrate theRR-Lyrae stars, I modified my simple plottingprogram to read in two images of a single clusterand produce on one chart a combinedHR-diagram. If a given star has about the samebrightness in the two images, I simply plot a

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All stars in a globular cluster formed at the same time but with various initial masses. The arrows show theprogression from lower mass stars to more massive ones.

single point using the average of the twobrightnesses and the average of the two colors.But, if the brightnesses are significantly different,I plot both points and connect them with a yellowline. Such an HR-diagram for M5 is shown here.The RR Lyrae stars are exactly where we expectthem to be—on the horizontal branch. Some ofthe very faint stars, magnitude 18 and fainter,also get yellow lines. In these cases, however,the spread is much more likely a result of noisein the image data—these are, after all, very faintstars. The estimate of brightness and color isbound to be quite imprecise. Finally, one brightstar not on the red-giant branch appears to bevariable. It is probably a field star and perhaps itreally is variable.

My full collection of HR-diagrams and thecomputer program I used to produce them can

be found athttp://www.princeton.edu/∼rvdb/images/NJP/HRdiagMatlab.html.As explained earlier, star colors and luminositiescan be extracted from an image file using thefreely available software program calledSExtractor(http://www.astromatic.net/software/sextractor).Two other programs that will automaticallyextract all stars in an image are AstroArt(http://www.msb-astroart.com/) and Mira(http://www.mirametrics.com/). SExtractor andAstroArt produce plain text files listing data foreach star.

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Hertzsprung-Russell diagram created using the CCD images that produced the picture shown on theprevious page.

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