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## HR diagrams of open clusters, calculation of the distance to the cluster

###### Introduction

The HR diagram (Hertzsprung-Russell) represents in a 2D graph the relation between the luminosity (Y axis) and the temperature of the stars or the equivalent of the spectral type (X axis), representing to the left the hottest stars and to the right the coldest. The diagram graphically represents the evolution of the stars, distinguishing the area called the main sequence, young stars that are still burning hydrogen.

Using photometry techniques, an HR diagram can be made of a group of stars, for example those belonging to a cluster, representing on the Y axis the apparent magnitude measured with a filter V and on the X axis the difference between the magnitude measured with a filter B and the measurement with the filter V. The value of B-V is directly related to the color or spectral type of the star: small B-V (for example, smaller than 0.5) correspond to hot stars and colder stars have a B-V value greater than 2.

Since in a cluster all the stars are of the same age, an interesting aspect of the HR diagrams calculated with V vs B-V is that they allow to calculate the distance of the cluster if the difference between apparent magnitude and the absolute magnitude is known. We can obtain this difference by means of the distance between where we observe that the isochronous curve fits in apparent magnitude (curve where the stars are located in the HR diagram for the same age, different masses) and the theoretical isochron in absolute magnitude. This gap (m-M), is called distance module and allows us to know the distance to the cluster by the formula m-M = 5*log (r / 10). Later we will see that this formula requires a small modification due to interstellar absorption.

Additionally, the age of the cluster can be verified by checking that the shape of the theoretical isochron corresponds to the distribution of stars found in the HR diagram.

In short, the realization of an HR diagram by an amateur astronomy is a very interesting exercise that allows to learn and practice concepts of theoretical physics.

Given that hundreds of stars are in a cluster, the HR diagram can be done manually, but it is preferable to automate it. Certain programs of easy access are required, some of them free or in any case economic, and although there are undoubtedly many ways to reach equivalent results, I describe below the process that I use.

The following table shows a summary of the diagrams made and the distances obtained:

###### Used tools

Capture of the photos and processed: Maxim DL

Astrometric calculations: Astroart, specifically the option of astrometry and star list, since it allows to obtain a text file with the list of stars identified and data of right ascension (RA), declination (DEC), luminosity (ADU) and relationship noise signal (S / N). The rest of the data provided by the file are not used.

Identification of stars: WCSTools, utility package and library available for free on the network. In particular, the scat utility allows you to search for objects in a catalog in a certain area. The Tycho 2 catalog is used since it contains a sufficient number of stars (2.5 million, 99% of those with a magnitude of V 11). An example to locate which star is the one in our Astroart list has RA = 82.131522 and DEC = 35.823931 would be:

scat.exe -c ty2 -shn 10 -r -10 82.131522 35.823931 J2000

where you are looking for a maximum of 10 stars in a radius of 10 arcs of second environment to the position. The result gives us the name of the star in the catalog Tycho2 and the corresponding magnitude V and B.

Diagram drawing: GNUPLOT tool for drawing using online commands.

Calculation of isochrones: using a CMD (Color Magnitude Diagram) utility that allows downloading a text file with isochron data based on age and metallicity:

Obtaining data from open clusters: from the web page WEBDA, database of observations of open clusters. It allows obtaining and verifying data of age, metallicity, distance, etc.

I use all the software tools under Windows 7. Astroart is directly compatible with Windows and both WCSTools and GNUPlot have been installed under Cygwin (similar to a Linux distribution under Windows).

In addition to the tools used, it is necessary to process the downloaded files of Astroart (list of stars with their position and brightness values ​​in V and B), as well as the file downloaded from CMD (isochronous values ​​for absolute magnitude) because it is necessary they have an adequate format for GNUPlot. For processing I use a simple program in C, but it could be done with any language.

Processing is summarized in the following steps:

Input files:

• V.sta, generated with Astroart with the list of stars captured with the V filter

• B.sta, generated with Astroart with the star list captured with the B filter

• iso.dat, generated with CMD with isochron points.

Process steps:

• Reading of B.sta and association of each star to that of list V by proximity of RA and DEC. Stars that do not match because they are not in both files with RA and DEC close enough are discarded.

• Calculation of the apparent magnitude from the data of V and B of a reference star located with WCSTools

Output files:

• Writing of file V vs B-V for each star with valid format for GNUPlot

• Writing the file M vs B-V for the isochronous format valid for GNUPlot

###### Formulas used

The following formulas and criteria are used:

• For the calculation of the apparent magnitude of each star based on the apparent magnitude and brightness (ADU) of a known star:

• Calculation of the distance module: it is calculated by means of the separation between the theoretical isochron and the isochron that fits the star position in the diagram, for example for B-V = 1

• Calculation of the distance. From the module of distance and interstellar absorption:

• Distance in parsec = 10 ^ (distance module + 5 - interstellar absorption) / 5)

• Interstellar absorption = 3.1 * E (B-V) where E (B-V) is the extinction difference between the wavelengths of B and V (reddening).

###### Cluster M38 / NGC1912

M38 is an open cluster in the constellation Auriga. Photographs of 20 seconds have been taken with filters V and B and processed with Maxim DL. The procedure for obtaining the HR diagram and distance calculation is described in the previous sections.

In this case, they have identified:

• 600 stars in the photo with V filter

• 379 stars in the photo with filter B

• 346 stars agree on RA and DEC on both photos and are the ones depicted in the diagram. The main cause of stars with filter V not having pairing with filter B is that the exposure was equal for both filters and in the photo with B below the magnitid V = 15.3 the corresponding star is not seen.

• It has not been discriminated by signal-to-noise ratio.

M38 in V resulting from stacking 6 shots of 20 seconds. LX200 8" tube on EQ8 has been used to primary focus with STx8ME camera.

For the photo in B, the same configuration has been used.

Field of view: 22x14 minute arcs

The blue line represents the theoretical isochron for a metallicity Z = 0.019 and an age of 8.45 (in terms of the logarithm of the years) and a "reddening" of 0.248.

The green line represents the same isochrone displaced by a distance module of 10.91, which correctly locates it in the distribution of stars. It is best appreciated in the following enlarged photo.

For the distance module of 10.91 it means a distance to the cluster of 1067 parsec.

In this graph, stars are represented in a range of B-V from 0 to 2, together with the isochrone already displaced for a distance module of 10.91.

The diagram coincides quite well with isochrone in the main sequence zone and is compatible with the theoretical isochronic form as a function of age and metallicity.

###### Open Cluster NGC884

NGC884 is an open cluster in the constellation of Perseus. We have made 5 photographs of 20 seconds with a V filter and 5 seconds of 40 seconds with a B filter and processed with Maxim DL (bias, flat, dark and summed). The procedure for obtaining the HR diagram and distance calculation is described in the previous sections.

In this case, they have identified:

• 300 stars in the photo with filter V, rejecting S / N <20

• 184 stars in the photo with filter B, rejecting S / N <20

• 174 stars match on RA and DEC on both photos.

Photo of NGC844 in V resulting from stacking 5 shots of 20 seconds. LX200 8" tube on EQ8 has been used to primary focus with STx8ME camera.

For the photo in B there have been 5 shots of 40 seconds with the same equipment.

Field of view: 22x14 minute arcs

The blue line represents the theoretical isochron for a metallicity Z = 0.019 and an age of 7.05 (in terms of the logarithm of the years) and a "reddening" of 0.560.

The green line represents the same isochrone displaced by a distance module of 13.6, which correctly locates it in the distribution of stars. It is best appreciated in the following enlarged photo.

For the distance module of 13.6 it means a cluster distance of 2360 parsec.

In this graph the stars are represented in a range of B-V from 0 to 2, together with the isochrone already displaced for a distance module of 13.6.

The diagram coincides quite well with isochrone in the main sequence zone and is compatible with the theoretical isochronic form as a function of age and metallicity.

###### Open Cluster M37

M37 is an open cluster in the constellation Auriga. I have done 5 photographs of 30 seconds with filter V and 3 of 90 seconds with filter B and worked with Maxim DL (bias, flat, dark and summed) were made. The procedure for obtaining the HR diagram and distance calculation is described in the previous sections.

In this case have been identified astrometrically:

• 300 stars in the photo with filter V, refusing stars in the processing with S / N <20

• 300 stars in the photo with filter B, rejecting in the process the stars with S / N <20

• 280 stars agree on RA and DEC in both photos and are the ones that are represented in the diagram

• The reference star for the photometry of the group has been the tyc 2410-1293-1 of magnitude 10.02 in V and 10.65 in B

M37 in V resulting from staking 5 shots of 30 seconds. LX200 8" tube on EQ8 has been used to primary focus with STx8ME camera.

For the photo in B have been 3 shots of 90 seconds with the same equipment.

Field of view: 22x14 minute arcs

The blue line represents the theoretical isochron for a metallicity Z = 0.019 and an age of 8.54 (in terms of the logarithm of the years) and a "reddening" of 0.302.

The green line represents the same isochrone displaced by a distance module of 11.6, which places it correctly in the distribution of stars. It is best appreciated in the following enlarged photo.

For the distance module of 11.6 ti means a distance to the cluster of 1357 parsec.

In this graph the stars are represented in a range of B-V from 0 to 2, together with the isochrone already displaced for a distance module of 11.6.

The diagram coincides quite well with isochrone in the main sequence zone and is compatible with the theoretical isochronic form as a function of age and metallicity.

A group of stars that can be red giants are distinguished towards the center of the graphic.

###### Open Cluster M67

M67 is an open cluster in the constellation of Cancer, of the oldest known. We have made 10 photographs of 60 seconds with a V filter and 10 of 180 seconds with a B filter and processed with Maxim DL (bias, flat, dark and summed). The procedure for obtaining the HR diagram and distance calculation is described in the previous sections.

In this case they have been identified astrometrically:

• 300 stars in the photo with V filter, refusing stars in the processing with S / N <10

• 268 stars in the photo with filter B, rejecting in the process the stars with S / N <10

• 245 stars agree on RA and DEC in both photos and are the ones that are represented in the diagram

• The reference star for the photometry of the group has been the tyc 814-1795-1 of magnitude 10.07 in V and 10.02 in B

M67 in V resulting from stacking 10 shots of 60 seconds. LX200 8 "tube on EQ8 has been used to primary focus with STx8ME camera.

For the photo in B (not shown) 10 shots of 180 seconds have been made with the same equipment.

Field of view: 22x14 minute arcs

The blue line represents the theoretical isochron for a metallicity Z = 0.019 and an age of 9.4 (in terms of the logarithm of the years) and a "reddening" of 0.059.

The green line represents the same isochrone displaced by a distance module of 9.9, which places it correctly in the distribution of stars. It is best appreciated in the following enlarged photo.

For the distance module of 9.9 there is a distance value to the cluster of 877 parsec.

In this graph the stars are represented in a range of B-V from 0 to 2, together with the isochrone already displaced for a distance module of 9.9.

The diagram coincides quite well with isochrone in the main sequence zone and is compatible with the theoretical isochronic form as a function of age and metallicity.

A group of stars that can be red giants are distinguished towards the center of the graph and in the prolongation zone of the main sequence other area with a significant number of potential lagged blue stars. These stars are studied in the detection section of special stars in the HR diagrams.

###### Open Cluster M36

M36 is an open cluster of very young stars in the Auriga constellation. I have made 6 photographs of 30 seconds with a V filter and 5 of 90 seconds with a B filter and processed with Maxim DL (bias, flat, dark and summed) were made. The procedure for obtaining the HR diagram and distance calculation is described in the previous sections.

In this case they have been identified astrometrically:

• 234 stars in the photo with filter V, refusing to process the stars with S / N <20

• 165 stars in the photo with filter B, rejecting in the process the stars with S / N <20

• 153 stars agree on RA and DEC in both photos and are those that are represented in the diagram

• The reference star for the photometry of the group was the tyc 2412-1383-1 of magnitude 9.01 in V and 9.04 in B

M36 in V resulting from stacking 6 shots of 30 seconds. LX200 8 "tube on EQ8 has been used to primary focus with STx8ME camera.

For the photo in B (not shown) 5 shots of 90 seconds have been made with the same equipment.

Field of view: 22x14 minute arcs

The blue line represents the theoretical isochron for a metallicity Z = 0.019 and an age of 7.46 (in terms of the logarithm of the years) and a "reddening" of 0.222.

The green line represents the same isochrone displaced by a distance module of 11.1, which correctly locates it in the star distribution. It is best appreciated in the following enlarged photo.

For the distance module of 11.1 there is a distance value to the cluster of 1208 parsec.

In this graph the stars are represented in a range of B-V from 0 to 2, together with the isochrone already displaced for a distance module of 11.1.

The diagram coincides quite well with isochrone in the main sequence zone and is compatible with the theoretical isochronic form as a function of age and metallicity.

It is appreciated that practically all the stars are in the main sequence, as can be expected for a very young cluster.

###### Open Cluster M44

M44, also known as Praesepe, is an open cluster in the constellation Cancer. 10 photographs of 10 seconds with a V filter and 10 of 30 seconds with a B filter and processed with Maxim DL (bias, flat, dark and summed) were made. The procedure for obtaining the HR diagram and distance calculation is described in the previous sections.

In this case they have been identified astrometrically:

• 43 stars in the photo with filter V, refusing to process the stars with S / N <20

• 36 stars in the photo with filter B, rejecting in the process the stars with S / N <20

• 34 stars agree in RA and DEC in both photos and are the ones that are represented in the diagram

• The reference star for the photometry of the group has been the tyc 1395-2711-1 of magnitude 7.31 in V and 7.51 in B

M44 in V resulting from adding 10 shots of 10 seconds. LX200 8 "tube on EQ8 has been used to primary focus with STx8ME camera.

For the picture in B (not shown) 10 shots of 30 seconds have been made with the same equipment.

Field of view: 22x14 minute arcs

The blue line represents the theoretical isochron for a metallicity Z = 0.019 and an age of 8.86 (in terms of the logarithm of the years) and a "reddening" of 0.009.

The green line represents the same isochrone displaced by a distance module of 6.2, which places it correctly in the distribution of stars. It is best appreciated in the following enlarged photo.

For the distance module of 6.2 there is a distance value to the cluster of 171 parsecs. This small distance explains its high brightness and large apparent size.

In this graph the stars are represented in a range of B-V from 0 to 2, together with the isochrone already displaced for a distance module of 6.2.

The diagram coincides quite well with isochrone in the main sequence zone and is compatible with the theoretical isochronic form as a function of age and metallicity.

###### Open cluster NGC752

NGC752 is an open cluster in the Andromeda constellation. We have made 5 photographs of 15 seconds with a V filter and 5 of 45 seconds with a B filter and processed with Maxim DL (bias, flat, dark and summed). The procedure for obtaining the HR diagram and distance calculation is described in the previous sections.

In this case they have been identified astrometrically:

• 82 stars in the photo with filter V, rejecting stars in the processing with S / N <10

• 63 stars in the photo with filter B, rejecting in the process the stars with S / N <10

• 58 stars agree on RA and DEC in both photos and are those that are represented in the diagram

• The reference star for the photometry of the group has been the tyc 2816-0177-1 of magnitude 10.11 in V and 10.54 in B

NGC752 in V resulting from stacking 5 shots of 15 seconds. LX200 8 "tube on EQ8 has been used to primary focus with STx8ME camera.

For the photo in B (not shown) 5 shots of 45 seconds have been made with the same equipment.

Field of view: 22x14 minute arcs

The blue line represents the theoretical isochron for a metallicity Z = 0.019 and an age of 9.05 (in terms of the logarithm of the years) and a "reddening" of 0.034.

The green line represents the same isochrone displaced by a distance module of 8.4, which places it correctly in the distribution of stars. It is best appreciated in the following enlarged photo.

For the distance module of 8.4 there is a cluster distance value of 456 parsec.

In this graph the stars are represented in a range of B-V from 0 to 2, together with the isochrone already displaced for a distance module of 8.4.

The diagram coincides quite well with isochrone in the main sequence zone and is compatible with the theoretical isochronic form as a function of age and metallicity.

###### Open cluster NG1528

NGC1528 is an open cluster in the constellation Perseus. 5 photographs of 40 seconds with V filter and 5 of 80 seconds with filter B and processed with Maxim DL (bias, flat, dark and summed) were made. The procedure for obtaining the HR diagram and distance calculation is described in the previous sections.

In this case they have been identified astrometrically:

• 300 stars in the photo with V filter, refusing stars in the processing with S / N <10

• 300 stars in the photo with filter B, rejecting in the process the stars with S / N <10

• 257 stars agree on RA and DEC in both photos and are the ones that are represented in the diagram

• The reference star for the photometry of the group has been the tyc 3340-0429-1 of magnitude 10.37 in V and 10.83 in B

NGC1528 in V resulting from stacking 5 takes of 40 seconds. LX200 8 "tube on EQ8 has been used to primary focus with STx8ME camera.

For the picture in B (not shown) 5 shots of 80 seconds have been made with the same equipment.

Field of view: 22x14 minute arcs

The blue line represents the theoretical isochron for a metallicity Z = 0.019 and an age of 8.568 (in terms of the logarithm of the years) and a "reddening" of 0.258.

The green line represents the same isochron displaced by a distance module of 10.20, which correctly locates it in the distribution of stars. It is best appreciated in the following enlarged photo.

For the distance module of 10.20 there is a distance value to the cluster of 758 parsec.

In this graph the stars are represented in a range of B-V from 0 to 2, together with the isochrone already displaced for a distance module of 10.2.

The diagram coincides quite well with isochrone in the main sequence zone and is compatible with the theoretical isochronic form as a function of age and metallicity.

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