DETERMINATION OF THE DISTANCE
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?
in Taurus, 400 ly away
in Hercules, 25000 ly away
|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 )
Fig. 4: The typical HRD
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.
Determine the distance and estimate the age of the open cluster NGC 6025.
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.
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.