How to study galactic open clusters with some excel + visual basic for application programmed spreadsheets.

Valter Arnò

Abstract:

In order to minimize the time required to perform a photometric analysis and on the other hand, to improve the complete derivation and determination of fundamental parameters for every photometric open cluster study, it's shown how this result can be achieved using some Excel+ Vba programmed spreadsheet.

The described software has allowed the author, to win 2003 edition of the Guido Horn D’Arturo Prize supported, under the auspices of the Italian Astronomical Society, by:

Before starting, I wish to thank all the readers for their patience during the reading of my English.

Preface

With the introduction, also in amateur field, of photoelectric photometer and even more after ccd, it has become possible to make observations, hardly realizable only with the previous photographic support.

Between the various fields where it's possible, for an amaterur, to give a serious contribution to the astronomical research, there is surely, the one of the open star clusters.

In dealing this argument, we will proceed giving for granted the knowledge of the UBV photometric system [3],[4],[31],[35],[37] and of the techniques of acquisition and elaboration of astronomical images that are necessary for these kind of studies.

The previous restriction, even if disagreeable, seems however necessary to be able to talk about the remarkable amount, of other astronomical research fields, that we find in these studies.

The problem we will deal here with, regards the study of galactic open clusters using Excel electronic sheet particularly concerning acquisition, automatic reduction and interpretation of the photometric values obtained from UBV measurements.

In fact the possession of photometric values for the members of a galactic open cluster, represents only the first step of the deductive job, that must follow the measurements phase.

In order to determine the fundamental parameters of an open galactic cluster, elaborating UBV measurements in the Johnson & Morgan system [35], it will be necessary to have the following performances:

 1. Good ability in calculation.

 2. One complete libray of mathematical and statistical functions.

 3. Possibility of representing solutions graphically.

In our PC, the above mentioned activities are normally developped by the electronic sheet and the author’s attempt is to show how using VBA (visual basic for application) [23] [25] [29] Excel can be programmed, in order to transform it in a program devoted to study of the open clusters.

A short survey will give, to the reader, an idea of how much can be obtained from one photometric analysis using the proposed software. Those who wish to know some more can obtain entire package from the author in freeware distribution.

Fig. 1: the Hr Trace start page with Internet link.

The Hr Trace software

This program, using the techniques connected to the photometric diagrams [3] [4] [6] [10] allows us to obtain, for the members of studied cluster, parameters like: distance, age, metallicity, absolutes magnitudes, bolometric magnitudes, intrinsic colours, radii, effective temperatures, brightness and various other data.

Naturally, in order to begin, it will be necessary to decide how to obtain the photometric values, wether from measurements carried out to the telescope, or from technical literature.

At this purpose it's available, in the program, one appropriate interface for their introduction see fig.2, plus some links to greater astronomical databases  as: ADS, CDS, WEBDA, from which it's possible to get and download the required data via Internet.

Fig. 2: Hr Trace IC 2581 UBV data input interface.

All we know on formation and evolution of stars, derives from colour - magnitude diagrams study of galactic clusters.

If for these objects, it can be reasonably presumed that the constituent stars have been formed at the same time and from an identical chemical mixture, then the evolutionary stages eventually reached, will depend only on the mass of several members.

Therefore plotting on HR diagram different clusters sequences having various age like in fig. (9), we can see the infuence that, this last factor, has on stellar evolution.

The mass plays a fundamental role in the stellar evolution, in a certain way, is a clock that determines the duration of every evolutionary phase, from pre-main sequence (Pms) to main sequence (Zams) until evolutionary ends.

Substantially this role contributes to differentiate the times of permanence in every evolutionary phase, so that the mainly massive stars have at their disposal shorter times comparing to the less massive ones.

Thinking to a star of solar type, we can say that the time of permanence on the main sequence is regulated from the following relationship:

Time of permanence = (Amount of H available/Speed of fusion of H in He).

In other words, if we consider the mass like the available amount of H and the brightness like the fusion speed of H in He, the previous one can be written also:

Time of permanence = (Mass/Brightness).

Indicating then, with the relationship t/t_S the time permanence on zams comparing with the sun, that is of the order of t_S∝10¹⁰ years, using the relation L∝ M³ we can write:

t/t_S = (M/M_S) / (L/L_S) = (M/M_S) / (M/M_S)³ = 1 / (M/M_S)²

Therefore for a star of 100 solar masses the order of magnitude will be:

t = 10¹⁰ × (1/100)²  = 1 × 10⁶.

The Reduction Procedure

From the first jobs of Johnson & Morgan [35], we have learned that the methodology to obtain the fundamental parameters of a galactic cluster using the UBV photometry, for those members that are found on the zams, is the following:

1)Calculate the medium colour excess <E(B-V)> for the cluster.

2)Calculate the individual colour excess for every cluster member using:

E(B-V) = (B-V) - (B-V)o

3)Once determined the amount of individual E(B-V) for every photometric member, we can derive the intrinsic colour characterizes them using:

(B-V)o = (B-V) - E(B-V)

4)Now from the existing relation between the colour excesses E(U-B) and E(B-V) defined like  E(U-B) = 0,72E(B-V) + 0,05E(B-V)²  we can find (U-B)o using:

(U-B)o = (U-B) – E(U-B)

5)At this point, it is possible to obtain the values of apparent individual magnitudes, without the effects of the interstellar absorption with the following relation:

Vo =V - RE(B-V)

6)Here, V represents the apparent magnitude in the visual and R is the relationship between total and selective absorption, R = A_V / E(B-V), whose value generally are included between 3 and 3,25 medium value 3,15. The value of R, however, depends on the direction and from several studies carries out it's known that it can assume, in particular regions of the galaxy, remarkably greater values. High values of R, often, are observed in clusters many young that have small E(B-V) values (Johnson-Borgmann).

7)The empirical relation between (B-V)o colour index and absolute magnitude [1] [5] [6] [17] [37] [41] [50] [53] [64], relation also known like (zero age main sequence), allows us to obtain, for every photometric member where (B-V)o is known, the value of absolute magnitude M_V.

8)Once we know the absolute magnitude we can calculate individually, for every photometric member of the cluster, its distance modulus using the following equation:

Vo-M_V = V - RE(B-V) - M_V

9)Mediating later on all the modules we can obtain the medium true distance modulus <Vo-M_V> for the cluster.

10)Now we only have to move from CM diagrams to theoretical HR diagram. On the last one, we can compare our results with the evolutionary tracks and isochrones, obtained from theoretical stellar models calculation.  

Particularly Hr Trace uses, for plot sequence and post-sequence, a mix of tracks from G. Schaller, D. Schaerer, A. Maeder and G. Meynet [49] stellar models, in the range 0,9 ÷ 120 solar masses. While for Pms phase, the tracks are from F. D’antona and I. Mazzitelli [16] stellar models, in the range 0,5 ÷ 2,5 solar masses.

To program an Excel automation for the execution of the ten previous points, is not particularly difficult and can represent a didactically useful exercise, since it introduces us to the use of programming techniques in Windows.

Photometric analysis of IC 2581 with Hr Trace

We begin concretely along the 10 previous points trying to illustrate the operation rheology of some interfaces of the program, applying them to southern open cluster IC 2581.

Through Nasa/Ads link, we obtain the photometric values (see fig. 2), that are contained in the following jobs:

[40] LLOYD EVANS T.:“The open cluster IC 2581“Mon.Not.Roy.Astr.Soc. 1969, 146, 101

[52] TURNER D.G.:   “Differential reddenig in the open cluster IC 2581”Astronomical Journal 1973, 78, 597

[56] TURNER D.G.:   “The value of R in IC 2581”Astronomical Journal 1978, 83, 1081

In order to determine colour excess E(B-V) it will be necessary, first of all, to possess an intrinsic colours calibration [19] [50].

At this purpose numerous researchers have worked, in the second half of the past century, to a series of empirical calibrations [11] [19] [26] [27] [28] [32] [37] [39] [50], obtained from the study of statistically rappresentative unreddened star samples of several spectral types. The tables 1a and 1b in appendix (I) and (II), show intrinsic colour calibration for the stars that belong to the several luminosity classes as determined by Schmidt-Kaler 1982 [50].

One two colour diagram (CC) is obtained placing the colour index (B-V) versus (U-B).

Arranging then, on such diagram, the nearest stars in order to avoid the undesired effects of interstellar absorption, the turning out curve  will be the one of intrinsic colours, that have a characteristic shape in UBV system.

Fig. 3: Two colour diagram and Schmidt-Kaler luminosity class V intrinsic colour calibration.

Such shape, see fig. 3, depend on the distribution of energy in star spectra, and in particular, the observed minimum at (B-V) = 0,0, is associated to a strong depression caused by ultraviolet hydrogen continuous absorption in stars with this (B-V) colour index.

When the distance of observed stars is not too large, typically smaller of 100 parsec, the colour indices are independent from the distance, but when this is larger, the effects produced from the interstellar absorption on colours cannot be neglected any more.

In particular the effects on the light that come to us from the stars will be: one attenuation of the brightness and a change of the true colour of the star observed towards the red one.

In other words, if interstellar absorption is present, the stars will appear weaker and much red.

On the base of things just said, when we plot the observed colour indices on a CC diagram, the effect of the interstellar absorption will be shown placing the observed values more at right and lower, in relation to the slope given from the ratio E(U-B)/E(B-V) and respect to the position of the curve that represents intrinsic colours.

Here, the distance between the position of the points that represent the observed indices and the intrinsic colours curve, calculated along the slope given by the ratio E(U-B)/E(B-V), will supply the <E(B-V)> medium colour excess.

The interstellar absorption, if not resolved, becomes an indetermination factor on the value of the absolute magnitudine and therefore on the distance like expressed from (1):

M_V  = m_V + 5 + 5 Log d – A_V      (1)

Where coefficient A_V represents the total contribution of the interstellar absorption, and is connected to the selective absorption as showed below (2):

A_V  = RE(B-V)     (2)

Naturally as consequence of (2) we have:

R = A_V / E(B-V)    (3)

An other very important geometric locus that can be defined on CC diagram is the reddening line here and after RL.

The RL, is the locus where are founded the stars of the same spectral class, stars that have therefore the same intrinsic colours, but subordinates to various degrees of interstellar absorption (selective).

The fundamental property of the RL is to intersect the two colours curve in the point correspondent to the intrinsic colour indices of the stars that are found on it.

Now since RL slope is known, becomes possible to determine the intrinsic colours of an observed star in UBV system.

From a rigorous point of view, the RL is not just a straight line, even if its curvature turns out negligible. In fact, the correct expression for the RL in UBV system [63] is of the type:

E(U-B) = XE(B-V) + YE(B-V)²   (4)

Where X is the RL slope and Y is its curvature. The values commonly accepted for X and Y are: X = 0,72 and Y = 0.05.

On these grounds to determine the reddening on diagram CC will consist in a system resolution, formed from the expressions of the RL and the two colours curve, whose mutual intersection will represent the intrinsic or de-reddened colours. Considering however, than the analytics expression of the two colours curve in its complex is not surely linear, as deductible from the observation of fig. 3, but indeed can be well described from a six order polinomials, it’s clear that the solution of the system can involve some difficulties.

Such difficulties, can be resolved taking in consideration that the majority of stars contained in a galactic open cluster are Early Type, therefore stars of population I, blue and young, of the early spectral types.

As we know from MK classification, the stars of the early spectral types of population I belong to the classes O and B and therefore, with reference to the fig. 3, we can take in consideration only the single feature of two colours curve that represent these spectral classes.

If we make attention to geometry of the considered feature (drawn yellow in fig. 3), we notice that this feature is, with great approximation, rectilinear and that in the case of the Schmidt-Kaler intrinsic colours tabulation, can be expressed from the folowing regression line:

(U-B)o = 3,7082(B-V)o + 0,04    (5)

Going back now to the RL definition: E(U-B) = 0,72E(B-V) + 0,05E(B-V)² we can, neglecting the curvature term 0,05E(B-V)² because effectively very small, reducing our problem to determine the intrinsic colours from the observed ones, to a system of two straight lines whose common solution represents the searched values.

Therefore calculate the values of intrinsic colours for programmed stars in our open cluster,  will consist in constructing a series of reddenig lines with slope E(U-B)/E(B-V), that connecting the observed positions with the intrinsic colours curve allow us to read the intersection points coordinates.

The only attention we must lend using this technique, is to avoid its application, on CC diagram, where ambiguous solutions can be introduced (multiple intersection), like explained in fig. 4.

The determination of individual and medium colour excess with Hr Trace

In the Hr Trace software the determination of individual and medium colour excess E(B-V) happens in CC diagram, using for default, the Schmidt-Kaler empirical calibration [50] even if the program has various other calibrations as those of: Becker [5], Eggen [17], Fitzgerald [19], Johnson [37] and Mermilliod [41].

For the graphical curves construction on CC diagram, Hr Trace uses tabular values from several authors, representing them through six order interpolating polimomials.

The necessity of using an high degree for the polinomials, derives from the complex shape of the two colours curve in UBV system.

Fig. 4 Search of intrinsic colours using reddenig lines.

The general analytics shape for the interpolating polinomials is:

(U-B)o  =  a (B-V)o⁶ + b (B-V)o⁵ + c (B-V)o⁴ + d (B-V)o³ + e (B-V)o² + f (B-V)o + g    ( 6 )

Where the previus equation (6) represent the intrinsic colours locus and the values of polinomials regression coefficients for several authors, are those in table 2.

Table 2  Polinomials regression coefficients for intrinsic colours locus.

The applicability field of the six orders calibrations is the one of table 3:

Table 3  Applicability field of polinomial regression (6).

The derivation of <E(B-V)> happens using the sliding fit technique, that consists in moving two colour calibration towards right and the bottom, in relation to the RL slope sets up, until fitting the distribution of observed open cluster members in study.

Fig. 5 < E(B-V) > Interface determination initial situation for IC 2581.

In fig 5 we can see the starting situation for IC 2581, while in fig. 6 it can be seen the determination of <E(B-V)> = 0,42 using the sliding fit technique method on CC diagram.

Fig. 6 Obtaining of < E(B-V) > = 0,42 for IC 2581.

The stars included between the two lines ± 0,03 magn. around the empirical calibration are automatically considered main sequence photometric members, once obtained the best-fit. At the same time in background the code calculates, for every captured photometric member, the following values: individual E(B-V), individual E(U-B), individual (B-V)o, individual (U-B)o, individual Vo.

In this phase, for all the other measured stars that are not Early Type, the code assign to their colour excesses the average calculated for the Early Type stars.

For default this interface operates with the reddenig lines method, but various other methods for individual E(B-V) calculation are available like: the Morgan & Harris method, the Clarià method [15], the Gutierrez-Moreno method [26] and the Johnson & Morgan Q method [35].

Before passing beyond this argument it's necessary to spend two words for the not infrequent case in which, studying an evolved cluster, it's not succeeded to find Early Type stars. The fact that does not find young stars does not mean that the cluster originally did not contain any, but only that the age of what we observe, has been sufficient to allow, the early type stars, to turn themselves in red giants. In these cases, possible difficulties are exceeded from the software that uses for for the determination of <E(B-V)>, one different interface to select.

Immediately after <E(B-V)> parameter determination and automatic capture of Early Type present in the photometric sequence, the code leads user to the general analysis menù of fig. 7.

Fig. 7 general analysis Menù.

Some considerations on capture algorithm for the Early Type stars

When we explained how the code performs the capture of Early Type stars in the photometric sequence, we have said that, once obtained the best-fit between intrinsic colour calibration and the observed sequence, the code calculates, in background, the individual values for captured stars.

Even if what we have said, corresponds exactly to the procedure obtained by the code, it is necessary in any case to make some clarification.

The intrinsic colours in the first column of Tab. 1a and 1b that you see in appendix (I) and (II), refer to luminosity class V and are valid for main sequence stars.

However, in an open galactic cluster are found also evolved stars, that belong to various other luminosity classes.

Obviously, to various luminosity classes, will correspond different intrinsic colours, whose diplacement from the value of class V can be very remarkable especially for classes II, Ia, Iab and Ib.

Observing carefully the tables of appendix (I) and (II) we can see, relating to spectral types the existing variation between the luminosity classes V, III, II, Ib, Iab, Ia and the colour indices (B-V)o and (U-B)o respectively.

Quantitatively the intrinsic colours values for the luminosity classes V and III differs very little and therefore to exchange one normal giant with a main sequence star, during the intrinsic colours search procedure using the reddenig lines method on CC diagram,  will correspond to carry out a small and practically much negligible error for Early Type stars.

The same confusion between one giant or main sequence star and one bright-giant of luminosity class II, can involve one error ranging between 0.01 to 0.04 magnitudes for O and B Early Type.

Even bigger will be the error if one normal giant or main sequence star is confused with one supergiant belonging to one of Ib, Iab, Ia luminosity classes. Such an error will be comprised in a range between 0.06 and 0.07 magnitudes for O and B Early Type.

It's useless to say, that errors of this entity can lead to determine wrong intrinsic colours compared with those really observed.

We can see an optimal example of this type of errors, just in the photometric sequence of the cluster that we are studying.

The most luminous star of this cluster is one A7 supergiant belonging to Ia-O luminosity class Turner [56].

For this star, HD 90772, Cousins and Stoy of the South African Astronomical Observatory have determined the following photometric UBV values:

V = 4,64 (B-V) = 0,52 (U-B) = - 0.01      (9)

As far as we know, from spectroscopic survey, that HD 90772 is one A7 Ia-O Supergiant using the Schmidt-Kaler 1982 [50] tabulation we can derive the HD 90772 intrinsic colours as follows:

(B-V)o  = 0,13  e  (U-B)o = 0,09     (10)

If tracing the RL with slope E(U-B)/E(B-V) passing through the position of this star on CC diagram, we try the intersection with the luminosity class V feature, rather than with the equation that represents the intrinsic colours for the Ia luminosity class, defined from the following (11):

(U-B)o  = 2,2115(B-V)o – 0,5679     (11)

we will find wrongly a B6-B7 star, rather than the more corrected A7.

This because in the first case, (intersection of the RL with the luminosity Class V intrinsic sequence), we will determine the following of intersection point:

(B-V)o = - 0,135  and  (U-B)o = - 0,49       (12)

While in the second case, (intersection of the RL with the luminosity class Ia intrinsic sequence), we have:

(B-V)o = 0,13  e  (U-B)o = 0,067      (13)

It's necessary at this point to specify that Hr Trace, during calculation of the intersection point as far as the luminosity class V and III use, in order to determine E(B-V) and E(U-B), one linear approximation to intrinsic colours as defined from (5).

While for luminosity classes II, Ia, Iab, Ib, uses a series of linear approximations as (11), in order to determine the E(B-V) values and after all deriving the intrinsic colour:

(B-V)o = (B-V) - E(B-V).

Moreover using the (B-V)o value, just determined, inside one 6° order polinomial approximation to the intrinsic colours, for the considered luminosity class, obtain (U-B)o.

It appears clearly therefore that the de-reddening algorithm, beyond calculating individual E(B-V) values must also guarantee, when differential extinction allows it, the correct interpretation of luminosity classes on CC diagram.

The rightness of interpretation of luminosity classes concurs to isolate with certainty, the main sequence stars class V, that the code must capture and use to determine, not only the medium E(B-V) value, but also the medium distance modulus for this stars group.

In fig. 8 we see the Hr Trace best-fit mapping and caputure indication for main sequence Early Type stars.

Fig. 8 Hr Trace mapping and Early Type capture on two colour diagram for IC 2581.

Fig the 8 also show the Ia intrinsic colour calibration and can be very clearly seen that HD 90772 belongs to this Luminosity class.

How to obtain the true distance modulus <Vo-MV> with Hr Trace

In determining the distance of a galactic cluster for photometric way, the method followed by Hr Trace, through appropriated interfaces, is the Zams fitting one.

As we have already said at point 5 of the procedure, the several empirical calibrations existing are dued to various authors: Becker [5], Blaauw [6], Eggen [17], Johnson [37], Mermilliod [41], Schmidt-Kaler [50], Turner [53] etc, that put in relation the de-reddened  (B-V)o colour index, with the absolute magnitude of one star. The theoretical foundations on which the previous calibrations stand can be summarized in the following way:

1) The position of a star on HR diagram is determined by its mass, chemical composition and age (Wogt-Russel theorem).

3) The position on the Zams, is determined by the mass.

4) In the Zams phase, the stars transform hydrogen into helium.

5) The Zams permanence time is given by the Mass/Luminosity ratio.

6) The Zams is a mass sequence.

7) The stars during their evolution drop in and go out from the Zams.

8) The Zams is the low evolutionary luminosity locus.

If we arrange on a colour-absolute magnitude diagram the stars belonging to various cluster, we notice that they seem to detach from an envelop curve, under which there are only the white dwarf.

This curve envelope, is called Zams (Zero age main sequence) and the separation points have been call Turn-off points.

According to the theory of the stellar evolution, after a relatively short phase of gravitational contraction, when the nucleus reach up a sufficient temperature to start the fusion process of hydrogen into helium, the stars catch up the Zams.

The initial position on zams is determined from the mass, that also fix the values of effective temperature, colour and absolute magnitude.

The stars remain on the Zams, until about 10 - 12% of available hydrogen is exausted in the nucleus.

While the hydrogen is getting exausted, the main sequence stars exit from the Zams and move, in HR diagram, towards the zone occupied from the red giants. It appears therefore clearly that the masses of the stars that are found on the Turn-off point, will be as smaller as the oldest one is the cluster (see fig. 9).

In the same way, taking in consideration the existing  relation between mass and brightness, we will also be able to define the cluster age from the brightness of its Turn-off point, that will be as small as old is the cluster.

Fig. 9 Composite colour-magnitude diagram X = (B-V)o, Y = Mv.

We try to eliminate, with the previous methods, the interstellar absorption from the photometry of a neighbor cluster for which it's also possible to determine, with geometric methods (ex.: parallax and/or moving cluster etc), the distance of its members.

But if we know the distance, we also know the absolute magnitude and since we have eliminated from the photometry the interstellar absorption, we also know the intrinsic colours of cluster members.

In these conditions it's possible to determine one empirical relation between intrinsic colours and absolute magnitude.

This relation, once derived, will constitute a zero calibration and we can use it in order to obtain, for comparison, the distance of other galactic clusters.

Now to obtain quantitatively the distance modulus of any cluster, we will have to compare our calibration with the cluster distribution in study on (B-V)o, Vo diagram and afterwards calculate the distance between calibration and distribution.

This last operation can be olso obtained graphically, making sliding towards the bottom our calibration until we meet the cluster distribution. The above described technique it's known as Zams fitting.

The program Hr Trace uses, for default, the Schmidt-Kaler 1982 empirical zams [50], that alouds to obtain the mathematical best-fit from the code.

As for CC interface the Zams lines have been constructed with six order polinomial interpolations of tabular values supplied from several the authors, the coefficients are those of table 4, while in table 5 we find the applicability field of the various empirical calibrations.

It's worth also in this case the same previous polinomial equation, used this time, in order to calculate the value of the absolute magnitude as a function of the intrinsic colour index (B-V)o.

Mv  =  a (B-V)o⁶ + b (B-V)o⁵ + c (B-V)o⁴ + d (B-V)o³ + e (B-V)o² + f (B-V)o + g      (7)

the (7), represent the Zams analytics expression.

Tab. 4 Several authors Polinomial regression Zams.

Tab. 5 Validity fied for Tab 4 coefficients.

With mathematical best-fit procedure the program calculates, for the captured photometric members with CC interface, the Early Type true individual distance modulus (Vo-MV) and subsequently mediating over all the modules, obtains the medium true distance modulus <Vo-MV> for cluster.

For all other empirical calibrations (B-V)o, M_V available in the program it's always possible to obtain one graphical fitting. In these cases, however, it will be necessary to proceed searching the best-fit to the cluster distribution with an eye to the dispersion of the points and try to fit the less luminous envelop, from an evolutionary point of view, for the members found on the Zams.

We see practically how it's possible to determine the distance of our cluster using the Hr Trace software.

Observing in fig.10 the (B-V)o, Vo plane, we can see that the IC 2581 main sequence is very well defined in the interval from (B-V)o = -0,32 to (B-V)o = 0,0 while the blue Turn-off point is found in correspondence of the (B-V)o value -0,25.

The observed dispersion points for IC 2581 in Vo is very little and this leaves us to preview a very low diplacement in the best-fit calculations (better adaptation to the points).

Before starting the Fitting Wizard procedure available in Hr Trace for the automatic best-fit calculation, it will be necessary to select the (B-V)o interval where we wish to find the best-fit of the empirical calibration with the IC 2581 data.

It will be also necessary to indicate, always selecting in the appropriate window, the read value of blue Turn-off to Hr Trace, so that the code can calculate the cluster age, using A. Maeder, G. Meynet and C. Mermilliod [42] calibration.

In order to obtain this, it will be sufficient to carry the cursor on the turn-off point and automatically read, from the diagram, the (B-V)o value, or in lack of an obvious turn-off point, on the final part of the Zams from the blue side of the sequence.

Fig. 10 Starting situation for the distance modulus search of IC 2581 with Hr Trace.

Fig 11 Best-fit obtained from (B-V)o = -0,30 to (B-V)o = 0,20 is 12,35 magn.

Introducing in Hr Trace the selected values and pressing Fitting Wizard command, we obtain the situation in figure 11, where the true distance modulus value turns out to be: 12,35 magn.

Fig. 12 Starting true distance modulus serach on the (U-B)o, Vo plane.

As forseen from the W. Becker procedure [4] [5], it will be necessary to obtain the distance modulus also in (U-B)o, Vo plane and afterwards to carry out the average of the values obtained on the two planes.

The starting and final situation, for IC 2581, is shown in figures 12 and 13.

Fig. 13 the obtained best-fit from (U-B)o = -1,20 to (U-B)o = 0,01 is 12,34 magn.

The practically identical values on the two planes, show as the calculations, for the search of colour excess, obtained in CC interface, have occured in a right way.

Hr Trace, after this phase, supplies in a particular interface, the elaborations as it's showed at in fig. 14.

Fig. 14 final distance modulus summary for IC 2581, < Vo-MV > = 12,345 ± 0,15.

It's now necessary, at this point, to explain the value of the modulus that appears, as data archive, in fig 14. The value for IC 2581 is <Vo-MV> = 11,65. This Turner's value [52], is based on the photometry of Lloyd Evans [40]. The value of the modulus has been gained from Turner, considering that R in this area can reach the value of 5,5. However  Moffat has subsequently criticized such value of R [43], suggesting that random errors in the photometry, as well as uncorrected identification of the members, can produce a wrong determination of the intrinsic colours.

Two years later Whittet, van Breda & Glass 1976 have determined, from infrared observations that HD 90706 is a supergiant member of IC 2581, with normal value of R = 3.1 ± 0,2. Always Turner in 1977 has found, from photometric and spectroscopic observations a value of R 3,05 ± 0,23 for cepheids VY Carinae region lies only 2° degrees from IC 2581.

All these indications have brought Turner [56] to reconsider the previous determination of R and from new spectroscopic measurements of 22 stars in the field of IC 2581, assigning absolute magnitudes with the Walborn calibration 1972, 1973 Turner determines, using the variable extinction diagram, the following values for association OB around to IC 2581:

R  =  3,11 ± 0,18  and  <Vo–MV>  =  12,29 ± 0,10

To extend the same value of R also to IC 2581, Turner, has subsequently compared the values found from the variable extinction diagram for IC 2581 in Turner [54], with those obtained from the analysis on the OB association around to cluster in Turner [56]. The results have show that many points of IC 2581 are perfectly compatible with the relation obtained for the OB association.

All this implies that it IC 2581 as well as OB association are at the same distance, and in particular that they are not subordinates to anomalous extinction value.

Therefore the previous result: R = 5,5 and <Vo-MV> = 11,65 can probably depend on the random errors combination arranged in the Llyod Evans photometry [40].

From this interface we can always obtain, using the Walker's criteria, those objects with elevated memberschip probability, that in our case are the stars included between the dotted lines in fig. 15

Fig 15 Memberschip for IC 2581 on the fitting interval with the Walker's criteria.

Fig 16 colour-absolute magnitude graph for IC 2581 with indications for luminosity classes and cepheids instability strip.

After have determined the distance modulus, the code can calculate the absolute magnitudes and plot the colour-absolute magnitudes (B-V)o, MV graph as it can be seen in fig. 16.

The simple observation of the fig. 16 diagram, shows that IC 2581 is a young galactic cluster composed by young stars of blue and white-blue colour, with superficial temperatures ranging between 8000° K and 40000° K.

Even the photometric radii are considerable, the values calculate by Hr Trace using Wesselink [65] calibration ranging in the interval between R/R_S = 2,2 and R/R_S = 280.

Figure 21 in appendix (III), shows the Hr Trace absolute magnitude table and other data for IC 2581.

The maximum values calculate by Hr Trace are those of HD 90772 and HD 90706 two supergiants of very high absolute magnitude  Mv= -8,96 and Mv = -7,16 but with similar bolometric magnitude Mbol= -8,93 and -8,60.

It's also interesting to observe the relationship between the photometric radii of these two stars, that turn out to be (HD 90772 / HD 90706) = 3,67.

The relationship between the radii of two supergiants, must be considered as a function of the effective temperature of two stars, that it's calculated by Hr Trace in 8033°K for HD 90772 and 14327°K for HD 90706.

Evidently the colder body HD 90772, needs a huge radiating surface of warmer body HD 90706, in order to reach the absolute magnitude of Mv = - 8,96.

This situation is very well explained by the Stefan-Boltzmann law(8):

L = 4pR² sT⁴       (8)

Shortly (8) affirms that: for two stars of same brightness if one  has a temperature of 16000 °K and the other a temperature of 8000 °K we will have: L/L = 1 = (R/r)² × (16000/8000)⁴ =  (R/r)² × 16 therefore, the warmer star will be approximately 1/4 of the colder star.

HR theoretical diagram Log L/L_S, Log T_eff.

We have used the effective temperature tabulations of Bohm-Vitense [8], H.L. Johnson [39] and P. Flower [20] [21] [22] to convert the observed intrinsic colour values in order to pass on theoretical plane.

For these calibrations Hr Trace uses the following polinomial coefficients:

Tab. 7 Polinomial regression coefficients for colour index versus effective temperature calibrations.

As usual, in the following table we find coefficients applicability field:

Tab. 8 Field of validity for table 6 coefficients.

The polinomial equation used to obtain the Teff  values from intrinsic colour index is similar to the previus ones  (6) and (7) :

Log T_eff  = a (B-V)o⁶ + b (B-V)o⁵ + c (B-V)o⁴ + d (B-V)o³ + e (B-V)o² + f (B-V)o + g      ( 14 )

In particular for the P. Flower calibration, it has been necessary to express a 7° degree term like:

Log T_eff  = a (B-V)o⁷ + b (B-V)o⁶ + c (B-V)o⁵ + d (B-V)o⁴ + e (B-V)o³ + f (B-V)o² +  g (B-V)o +  h    ( 15 )

The calculation of the amount Log L/L_S is on the contrary obtained through (16):

Log L / L_S   =    4,72 – ( Vo + BC – DM ) / 2,5       ( 16 )

Where BC is the bolometric correction and DM is the true distance modulus <Vo-MV>.

To go from an observative to theoretical point of wiew, that’s to say from the colour-magnitudine (CM) diagram to HR diagram, it's necessary to transform the colour index in effective temperature and absolute magnitude in bolometric magnitude.

In both cases, the rheology followed by the code to make calculations is always the same, therefore we will only limit ourselves to list the calibrations used for the conversion between absolute magnitudes and bolometric magnitudes.

The conversion between absolute magnitudes and bolometric magnitudes is the following:

M_bol  = Mv  +  Bc           ( 17 )

For the calculation of the bolometric correction Hr Trace uses the H.L. Johnson [39] and P. Flower [22] (B-V)o, BC tabulations.

Once obtained the conversions, we go to one new interface in order to compare our calculations with a series of stellar models as it's showed in fig. 17.

The superposition of the evolutionary tracks from stellar models on IC 2581 sequence, clearly shows that the majority of the members of this cluster is formed by stars founded in the range between 2 and 15 solar masses.

Only HD 90772 and HD 90706 have been found in the range between 30 and 40 solar masses, as we could expect considering the highest absolute magnitude of these stars.

The yellow-red triangle on the diagram, shows for comparison, the solar position.

In the same interface it's also calculated the cluster age from the blue turn-off value, using the A. Maeder, G. Meynet and C. Mermilliod calibration [42].

Afterwards without exit from this interface we can overlap, on the sequence of the points that represent our star cluster, one series of isochrones from  V.Castellani, A.Cheffi and O.Straniero [14] stellar models in the range between 30 Myr and 10 Gyr.

Finally we arrive to the end of the study, passing to Teff, Mbol diagram of fig. 18, where we observe HD 90772 and HD 90706 to reach one bolometric magnitude of Mbol =-8,96 and -8,60 respectively.

Fig. 17 IC 2581 on theoretical HR diagram  Log T_eff, Log L/L_S.

Fig. 18 the diagram Log T_eff, M_bol for IC 2581.

Moreover, on Log Teff, Mbol diagram it's possible to compare the clusters with tracks from the stellar models of Geneva group Schaller for metallicity Z = 0,02 and Z = 0,001 and of Padova group Bertelli, for metallicity Z = 0,02, Z=0.008 and Z=0.004.

Particular attention has been put, to software level, for the analyses of objects found in the Cepheid instability strip. In fig. 19 you can see the Ngc 129 cepheid member DL Cass, at his third crossing of instability strip.

Fig. 19 Ngc 129 Cepheid Member DL Cass. found at his third crossing of instability strip.

For multiple comparison between various clusters one appropriate interface is available for this type of analysis see fig.20.

Fig. 20 Ngc 457 and IC 2581 on Hr Trace multiple comparisons interface

Many other performances are available for the analysis of intermediate and advanced age clusters, or to obtain parameters when these objects are observed through particularly reddened sky fields where it's impossible to get individual E(B-V) value only in the photometric way.

It's chosen to comment IC 2581, because an important series of photometric studies carried out in relatively recent times, are present in literature.

On the other hand, the presence of some very luminous supergiants in this cluster, has allowed us to illustrate the technique on how the code discriminates between the luminosity classes and to see how the software carries out main sequence Early Type search and capture.

For northern observers, the IC 2581 boreal counterpart is NGC 457 in Cassiopeia. The two clusters are in comparison in fig. 19.

Ngc 457 is morphologically very close to IC 2581 and like this last one is dominated by the very luminous F0 Ia supergiant f Cass, close to HD 90772 in fig.19.

Our survey is over, but I hope I have gained some readers to open cluster study and, why not, Excel programming also.

References and Bibliography

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Appendix (I): (B-V)o colour index for several luminosity classes from Schmidt – Kaler

Spectral Type	V	III	II	Ib	Iab	Ia
O5	-0,33	-0,32	-0,32	-0,32	-0,31	-0,31
O6	-0,33	-0,32	-0,32	-0,32	-0,31	-0,31
O7	-0,32	-0,32	-0,32	-0,31	-0,31	-0,31
O8	-0,32	-0,31	-0,31	-0,29	-0,29	-0,29
O9	-0,31	-0,31	-0,31	-0,28	-0,27	-0,27
B0	-0,30	-0,29	-0,29	-0,24	-0,23	-0,23
B1	-0,26	-0,26	-0,26	-0,20	-0,19	-0,19
B2	-0,24	-0,24	-0,23	-0,18	-0,17	-0,16
B3	-0,20	-0,20	-0,20	-0,14	-0,13	-0,12
B5	-0,17	-0,17	-0,16	-0,10	-0,10	-0,08
B6	-0,15	-0,15	-0,14	-0,08	-0,08	-0,06
B7	-0,13	-0,13	-0,12	-0,06	-0,05	-0,04
B8	-0,11	-0,11	-0,10	-0,04	-0,03	-0,02
B9	-0,07	-0,07	-0,07	-0,03	-0,02	0,00
A0	-0,02	-0,03	-0,03	-0,01	-0,01	0,02
A1	0,01	0,01	0,01	0,03	0,02	0,02
A2	0,05	0,05	0,03	0,04	0,03	0,03
A3	0,08	0,08	0,07	0,06	0,06	0,05
A5	0,15	0,15	0,11	0,09	0,09	0,09
A7	0,20	0,22	0,16	0,13	0,12	0,12
A8	0,25	0,25	0,18	0,14	0,14	0,14
F0	0,30	0,30	0,25	0,19	0,17	0,17
F2	0,35	0,35	0,30	0,23	0,23	0,22
F5	0,44	0,43	0,38	0,33	0,32	0,31
F8	0,52	0,54	0,58	0,56	0,56	0,56
G0	0,58	0,65	0,71	0,76	0,76	0,75
G2	0,63	0,77	0,81	0,86	0,87	0,87
G5	0,68	0,86	0,89	1,00	1,02	1,03
G8	0,74	0,94	0,99	1,14	1,14	1,17
K0	0,81	1,00	1,08	1,20	1,25	1,25
K1	0,86	1,07	1,14	1,24	1,32	1,32
K2	0,91	1,16	1,29	1,33	1,36	1,36
K3	0,96	1,27	1,40	1,46	1,46	1,46
K5	1,15	1,50	1,49	1,59	1,60	1,60
K7	1,33	1,53	1,57	1,61	1,63	1,63
M0	1,40	1,56	1,58	1,64	1,67	1,67
M1	1,46	1,58	1,59	1,68	1,69	1,69
M2	1,49	1,60	1,59	1,68	1,71	1,71
M3	1,51	1,61	1,60	1,69	1,69	1,69
M4	1,54	1,62	-	1,75	1,76	1,76
M5	1,64	1,63	-	-	1,80	-
M6	1,73	1,52	-	-	-	-
M7	1,80	1,50	-	-	-	-
M8	1,93	1,50	-	-	-	-

Tab. 1a: (B-V)o colour index between luminosity classes V, III, II, Ib.Iab.Ia from S. Kaler

Appendix (II) (U-B)o intrinsic colour index for several luminosity classes from Schmidt – Kaler

Spectral Type	V	III	II	Ib	Iab	Ia
O5	-1,19	-1,18	-1,17	-1,17	-1,17	-1,17
O6	-1,17	-1,17	-1,16	-1,16	-1,16	-1,16
O7	-1,15	-1,14	-1,14	-1,14	-1,14	-1,14
O8	-1,14	-1,13	-1,13	-1,13	-1,13	-1,13
O9	-1,12	-1,12	-1,12	-1,13	-1,13	-1,13
B0	-1,08	-1,08	-1,08	-1,07	-1,06	-1,05
B1	-0,95	-0,97	-1,00	-0,99	-1,00	-1,00
B2	-0,84	-0,91	-0,92	-0,92	-0,93	-0,96
B3	-0,71	-0,74	-0,82	-0,82	-0,83	-0,85
B5	-0,58	-0,58	-0,69	-0,70	-0,72	-0,76
B6	-0,50	-0,51	-0,62	-0,67	-0,69	-0,72
B7	-0,43	-0,44	-0,54	-0,60	-0,63	-0,68
B8	-0,34	-0,37	-0,44	-0,53	-0,55	-0,62
B9	-0,20	-0,20	-0,32	-0,46	-0,49	-0,58
A0	-0,02	-0,07	-0,20	-0,33	-0,83	-0,44
A1	0,02	0,07	-0,12	-0,24	-0,29	-0,37
A2	0,05	0,06	-0,05	-0,20	-0,25	-0,30
A3	0,08	0,10	0,02	-0,10	-0,14	-0,20
A5	0,10	0,11	0,08	0,00	-0,08	-0,10
A7	0,10	0,11	0,10	0,09	0,00	0,09
A8	0,09	0,10	0,11	0,11	0,11	0,12
F0	0,03	0,08	0,12	0,15	0,15	0,15
F2	0,00	0,08	0,14	0,19	0,18	0,18
F5	-0,02	0,09	0,16	0,27	0,27	0,27
F8	0,02	0,10	0,24	0,40	0,41	0,43
G0	0,06	0,21	0,32	0,50	0,52	0,52
G2	0,12	0,39	0,42	0,62	0,63	0,65
G5	0,20	0,56	0,60	0,81	0,83	0,82
G8	0,30	0,70	0,78	1,05	1,07	1,08
K0	0,45	0,84	0,95	1,15	1,17	1,18
K1	0,54	1,01	1,07	1,21	1,28	1,28
K2	0,64	1,16	1,33	1,35	1,32	1,32
K3	0,80	1,39	1,58	1,56	1,60	1,60
K5	1,08	1,81	1,74	1,79	1,80	1,80
K7	1,21	1,83	1,79	1,84	1,84	1,84
M0	1,22	1,87	1,91	1,90	1,90	1,90
M1	1,21	1,88	1,93	1,92	1,90	1,90
M2	1,18	1,89	1,94	1,95	1,95	1,95
M3	1,16	1,88	1,77	1,98	1,95	1,95
M4	1,15	1,73	-	2,12	2,00	2,05
M5	1,24	1,58	-	-	-	-
M6	1,32	1,16	-	-	-	-
M7	1,40	-	-	-	-	-
M8	1,53	-	-	-	-	-

Tab. 1b: (U-B)o colour index between luminosity classes V, III, II, Ib.Iab.Ia from S. Kaler

Appendix (III): IC 2581 absolute magnitudes Tab and other data from Hr Trace.

Fig. 21 Absolute magnitudes Tab and other data for IC 2581.

 

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