DETERMINATION OF THE DISTANCE AND THE AGE OF A STAR CLUSTER Rainer C. Gaitzsch "EAAE Summerschools" Working Group Teacher's Academy of Bavaria (Germany)

Abstract

Examples will be shown how to get spectra of bright stars, easily made with ordinary material available at school. In order to obtain a Hertzsprung-Russell diagram (HRD) of certain stars, spectra have to be analysed.

By means of data-banks of several star clusters on given papers, the participants for themselves will work out an HRD of a certain star cluster. In this diagram the apparent magnitude is plotted against the surface-temperature (resp. the color-index B-V). The group will then discuss their results. By comparing their own HRDs with the standard main-sequence of stars given in absolute magnitudes, the participants will derive the distance of their own cluster and learn about how to estimate its age.

Special computer simulations with the participants being involved will illustrate some questions to phenomena concerning the evolution of stars, i.e.: What happens in the star's core? Where do the cluster-stars shift their location in the HRD during the process of aging? Why do all open star clusters disperse at last?

• Stars are spread all over the sky. Some of them are standing alone and some of them are embedded in star clusters. But not one single star starts its existence individually; every star was born in a cluster. At the nightsky we can see two different kinds of clusters: the open star clusters with some hundred members which are located in the disk area of our galaxy, mostly in the spiral arms, and the globular clusters containing some hundred thousands of stars, located in the halo of our galaxy.
  How can we know something about those very far-away objects? We get all our information by reading their spectra.

   

  Fig. 1 a: Open star cluster M45 ("Plejades") in Taurus, 400 ly away

  1 b: Globular cluster M13 in Hercules, 25000 ly away

• It is possible to get spectra from some bright stars, only using ordinary equipment available in every physics department at high school. An example is shown to the participants where a student, 18 years of age, built a simple spectrograph and got some quite nice spectra which he then could analyse.
  The star creates its spectrum in a natural way by its own ordinary movement across the nightsky.
  
  Fig. 2 a: mounting of prism and camera     2 b: spectrum of Wega with H-lines

• In order to obtain a Hertzsprung-Russell-Diagram (HRD) of certain stars or of a star cluster, spectra have to be analysed professionally. One of the most important data a spectrum can reveal is the surface temperature of the star. This cannot easily be made at school. The star's visual apparent magnitude cannot be measured at school in an easy way either. But we can use a data bank of a certain star cluster we wish to explore. Instead of the star's temperature the color index (B-V) is often used in the diagram. There is a clear correlation between temperature T and color index (B-V). (W.J.Kaufmann: "Universe", or Gondolatsch u.a.: "Astronomie II"). If the star is hot it is bluish with an (B-V) index of less than zero. If a star is cool its (B-V) index is positive. The Sun's (B-V) index is about +0.62. After measuring a star's B and V mag., an astronomer can estimate the star's temperature from a graph like the one here.

  (Fig. 3: Blackbody temperature versus color index )

• Before we will work out our own HRD of an open star cluster we should know something about the meaning of an HRD in general.
  HRDs demonstrate that there are different kinds of stars. In those diagrams of star clusters the apparent magnitude is plotted against the surface temperature respectively the color index (B-V). Every dot represents a star whose brightness and tempearture have been measured. The fact that the data fall into three distinct regions means that there are three very different kinds of stars in the sky: ordinary main-sequence stars, red giants and white dwarfs.

  Fig. 4: The typical HRD

  Fig. 5: The evolutionary track of a sun-type star

  In the inner core of every main-sequence-star hydrogen is transformed into helium, so the nuclear fusion acts like this: H ® He.
  All those stars roughly obey the mass-luminosity relation L ~ M ³. That means:

  The more massive stars are developping much faster in comparison to our sun than the less massive ones do. For the average time of the evolution (t) we may set: t ~ M (storage of fuel), but also t ~ 1 / L (loss of energy by radiation). So in conclusion:

  t ~ M / L ~ 1 / M ² (roughly; the value for our sun is about t = 1× 10¹⁰ years).

  So if we can find the turnoff point in the HRD of a star cluster where the main-sequence-stars end their first phase of a star's life and track off into the region of the red giants, we can estimate its age.

  Fig. 6 a: a young open star cluster ("Hyades")

  Fig. 6 b: an old globular star cluster (M3)

  The shape of the main sequence is the same for all star clusters of whatever age, with only minor variations. This fact provides a very important means of finding the distance of a star cluster otherwise unknown. We just have to compare the colour-magnitude-diagram we obtained from our cluster with the standard main sequence, where the absolute magnitude M is plotted against the color index.

  The difference between apparent mag. m and absolute mag. M (the so-called distance modulus) is given by: m - M = 5 × log (r / 10 pc), where r is the distance.
  The distance modulus is the shift in the vertical axis needed to bring its colour-mag-diagram into coincidence with the standard main sequence. The cluster may, on account of age, have lost its most luminous stars; however, stars which belong to the lower part of the main sequence are always present.

Exercise:

Determine the distance and estimate the age of the open cluster NGC 6025.

• Draw the colour-mag-diagram from the data given in table 1, in the form of apparent mag. m (=V) against (B-V). (appendix 1)
• Draw the standard main sequence from the data given in table 2, in the form of absolute mag. M against (B-V).
• Get the difference m - M and calculate the distance r of cluster NGC 6025.
• Use the turnoff point from your NGC 6025 -diagram as an age indicator and estimate the age of the star cluster. (Compare also with table 3)

  Hint: Since the average extension of the range for temperatures is about 3000 K to 30000 K and the average extension of the range of luminosity (in units of the sun's luminosity) is about 0.01 to 10000, it is a good estimation for L ~ T ⁶ (see also Fig. 4). With the law of Stefan-Boltzmann L ~ R ²×T ⁴ and the mass-luminosity relation L ~ M 3 one can estimate M ~ T 2 and for the time of evolution t ~ T ^-4 respectively t ~ L ^{- 2/3}.

• Try also to get values for the ages of the two clusters shown in Fig. 6a (Hyades) and Fig. 6b (M3).

Additional exercise:

• a) Calculate the distance and the radius of this cluster M13 in ly.
• b) Determine the abs. mag. of M13 and estimate its mass in units of sun-masses on condition that all star-members are sun-type stars (M = 4.8).

Table 1

table 2

table 3

Fig. 7 : HRD-Simulation program of the evolution of a star cluster;
time : 300 Mio. years.

Fig.8 : HRD, indicating the turnoff point of several star clusters. At those points the most massive stars leave the main sequence for the red-gainst region.