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Colour Magnitude Diagram
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Introduction: 

 

The Hertzsprung-Russell (HR) diagram provides the relationship between the absolute magnitude,Mv, and the color index of stars. In any particular cluster we expect that a large number of stars would lie on the main sequence. However below a certain magnitude the stars would have branched off to become giants. Recall that more luminous stars have lower magnitude and have shorter life spans. Hence the apparent magnitude as a function of color index, called the color-magnitude plot, would show a sudden turn at small values of mv. Star with smaller values of mv have left the main sequence in this cluster.

The relationship between the apparent magnitude mv and absolute magnitude Mv is given as follows :

                                                                             «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»m«/mi»«mi»v«/mi»«/msub»«mo»-«/mo»«msub»«mi»M«/mi»«mi»v«/mi»«/msub»«mo»=«/mo»«mn»5«/mn»«mi mathvariant=¨normal¨»log«/mi»«mfenced»«mrow»«mi»r«/mi»«mo»/«/mo»«mn»10«/mn»«mi»p«/mi»«mi»c«/mi»«/mrow»«/mfenced»«mo»+«/mo»«msub»«mi»A«/mi»«mi»v«/mi»«/msub»«/math»

Here r is the distance to the cluster and Av is the correction due to extinction. The difference mv-Mv is called the distance modulus . Once we know this difference we can deduce the distance r. For simplicity, here we shall set Av to zero. The distance modulus of a cluster can be determined by considering the stars on the main sequence. This can be done by comparing the color-magnitude diagram of the cluster with the HR diagram of a reference cluster. By adding a suitable number mv of all stars you will find that a subset of stars aligns with the main sequence on the HR diagram. This gives a measure of the difference mv-Mv. Use this to determine r.

Having done this we can plot the cluster data on the HR diagram. Next we determine the mass of the most luminous star in this cluster which is still on the main sequence. It's position on the diagram gives us it's absolute magnitude. Compare this with the absolute magnitude of Sun, which is 4.8 and deduce it's luminosity relative to solar luminosity, i.e. (L/Lsun). This is related to the mass of star by the formula (Source-Wikipedia)

                                                                                «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mi»L«/mi»«msub»«mi»L«/mi»«mrow»«mi»S«/mi»«mi»u«/mi»«mi»n«/mi»«/mrow»«/msub»«/mfrac»«mo»=«/mo»«msup»«mfenced»«mfrac»«mi»M«/mi»«msub»«mi»M«/mi»«mrow»«mi»S«/mi»«mi»u«/mi»«mi»n«/mi»«/mrow»«/msub»«/mfrac»«/mfenced»«mrow»«mn»3«/mn»«mo».«/mo»«mn»5«/mn»«/mrow»«/msup»«/math»

The lifetime of a star on main sequence is given by the formula:

                                                                            «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨left¨ rowspacing=¨0¨»«mtr»«mtd»«mi»T«/mi»«mo»=«/mo»«msup»«mn»10«/mn»«mn»10«/mn»«/msup»«mi»y«/mi»«mi»r«/mi»«mi»s«/mi»«mfenced»«mfrac»«mi»M«/mi»«msub»«mi»M«/mi»«mrow»«mi»S«/mi»«mi»u«/mi»«mi»n«/mi»«/mrow»«/msub»«/mfrac»«/mfenced»«mfenced»«mfrac»«msub»«mi»L«/mi»«mrow»«mi»S«/mi»«mi»u«/mi»«mi»n«/mi»«/mrow»«/msub»«mi»L«/mi»«/mfrac»«/mfenced»«/mtd»«/mtr»«mtr»«mtd»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»=«/mo»«msup»«mn»10«/mn»«mrow»«mn»10«/mn»«mi»y«/mi»«mi»r«/mi»«mi»s«/mi»«/mrow»«/msup»«msup»«mfenced»«mfrac»«msub»«mi»L«/mi»«mrow»«mi»S«/mi»«mi»u«/mi»«mi»n«/mi»«/mrow»«/msub»«mi»L«/mi»«/mfrac»«/mfenced»«mfrac»«mrow»«mn»2«/mn»«mo».«/mo»«mn»5«/mn»«/mrow»«mrow»«mn»3«/mn»«mo».«/mo»«mn»5«/mn»«/mrow»«/mfrac»«/msup»«/mtd»«/mtr»«/mtable»«/math»

Hence the ratio of the luminosity of the star relative to solar luminosity gives us the time spent on the main sequence. This is equal to the age of the cluster.

Plot the apparent magnitude as a function of color for a particular cluster. Also plot the absolute magnitude as a function of colour for a reference cluster. This is essentially the Hertzsprung-Russell plot for this cluster.

 

 

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