International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 127030 | https://doi.org/10.5402/2011/127030

W. H. Elsanhoury, M. A. Hamdy, M. I. Nouh, A. S. Saad, S. M. Saad, "Luminosity Function of Some Open Clusters", International Scholarly Research Notices, vol. 2011, Article ID 127030, 9 pages, 2011. https://doi.org/10.5402/2011/127030

Luminosity Function of Some Open Clusters

Academic Editor: G. Chernov
Received11 Jan 2011
Accepted21 Feb 2011
Published26 Apr 2011

Abstract

We investigated the luminosity function (LF) and initial mass function (IMF) of some open clusters having different ages. To calculate the LF, we followed the classical definition by van Rhijn (1936). Statistical investigation of the dispersion around a range of magnitudes concerning what is called Wielen dip revealed that the dip is unreal. To confirm the unreality of the dip, we computed the IMF for these open clusters, the statistical investigation of the IMF confirmed the results obtained using the LF, that is, there is no dip for these open clusters under study.

1. Introduction

Open clusters constitute an important part of a process transforming gas and dust into stars. They are observed as the most prominent parts in the regions of active star formation, or as tracers of the ceased star formation process in the general Galactic field. However, the role they are playing in this process has still not been fully understood. In spite of their prominence, there are indications that classical open clusters contribute only 10% or even less input [1–3] to the total stellar population of the Galactic disc [4].

The LF is among the most important tools to analyze the composition and evolution of stellar systems and is especially effective in studies of open star clusters, since the absolute magnitudes and ages of their constituent stars can be determined fairy accurately.

Wielen [5] derived the LF (πœ™) for solar neighborhood stars with aid of Gliese's catalogue [6, 7] by counting Gliese's stars in appropriate volumes of space. In this work, the unit of πœ™ is stars per unit magnitude interval in a complete sphere of radiusπ‘Ÿ=20 pc.

Miller and Scalo [8] presented a comparison between McCuskey [9], Luyten [10], and Wielen [5] LFs along the LF adopted by him. The LF of Wielen and McCuskey shown a small dip around 𝑀𝑉=7. Miller and Scalo [8] attach there is no physical significance of this dip because (a) its size is within the errors of the determination of the LF and (b) this dip is not found by all investigators.

Lee and Chun [11] selected some of open the clusters (22 open clusters) and 3 associations on the following criteria: (i) correct photometric observations, (ii) easy selection of member stars, (iii) wide range of magnitude and mass, and (iv) large number of MS stars, taking into account the brightening effect of bright MS stars. Using these objects, they derived the LF and mass function MF. The results obtained were consistent with solar neighborhood stars by Wielen [5].

Lee and Chun [11] indicated the presence of a dip at π‘€π‘‰βˆΌ7 mag. or logβ„³βˆΌβˆ’0.2 in open clusters, which were studied in his paper, similar to what was called Wielen dip which appears in solar neighborhood stars in spite of the fact that Wielen, in his study [5] using solar neighborhood stars did not mention the existence of any dip. Moreover, Lee and Chun [11] mentioned that conclusive confirmation of the existence of this dip in the cluster initial mass function IMF must be made by using many LFs of open clusters older than Hyades because the possibility of the turnover of cluster LF at low mass (β„³<1β„³βŠ™) has been suggested [12, 13].

Lee and Sung [14] studied the LF for the Pleiades cluster, showing what they called Wielen dip at 𝑀𝑉=7 mag. Also, Lee et al. [15] studied the LF for the Hyades and Praesepe clusters. According to this study, a Wielen dip was clearly seen at 𝑀𝑉=9 mag. (more precisely at𝑀𝑉=8.5∼9 mag.). This dip is about 2 mag. fainter than the case of the Pleiades cluster whose Wielen dip position is consistent with that for the solar neighborhood field stars.

Belikov et al. [16] studied the fine structure of Pleiades LF and premain sequence evolution, they found that there are three features (dips) in the observed cluster LF in a magnitude range 𝑀𝑉=5 mag.–12 mag. The first dip together with the adjacent maximum has a pre-MS nature. In their study of the theoretical LFs construction and comparing with the observations, they argued that since the LF depends on both IMF and mass-luminosity relation MLR derivative and since MLR of pre-MS stars differs from that of MS stars, the pre-MS branch (in the cluster where it is present) should affect the LF. Especially, a sharp contrast is expected at (or just below) the turn on the point at the transition between MS and pre-MS branches. According to Piskunov et al. [17], this transition produces a local β€œbump” in the LF followed by a dip (herein they call this β€œbump” as β€œH-peak” and the effect of peak and dip existence as β€œH-feature”: they chose this abbreviation to stress that the feature is related to the beginning of hydrogen burning as a star approaches the ZAMS) while the two dips (at 𝑀𝑉=7.5 mag. and 𝑀𝑉=9.5 mag.) are assumed to be field LF features: Wielen and Kroupa et al. (WK for short) dips, into which theoretical models fail to reproduce them.

In the study of the low-luminosity stellar MF, Kroupa et al. [18] discussed possible mechanisms causing the WK dips. They proposed that both details of the LF are the result of mass-luminosity relation fine structure generated by the equation of state and the opacity law. They proposed this idea by causing low-mass ZAMS models and showed that the Kroupa et al. dip is caused by the influence of H2 molecules on the equation of state. They attributed the Wielen dip to the increasing importance of Hβˆ’ ions as an opacity source. Belikov et al. in [16] using the more sophisticated models of D’Antona and Mazzitelli [19], concluded that these dips did not appear in their results.

In the present paper, we tried to study the LF of the some open clusters of different ages and magnitudes, with concentration on magnitude range 6≀𝑀𝑉≀9 (the range of the so called Wielen dip) in order to(i)test the reality of the presence of this dip (i.e., Wielen dip) in open cluster,(ii)to find the correct position of that dip, if present, and its dependence on cluster age.

2. Observational Materials

If the number of stars is increased in the study of luminosity function of open clusters, the statistical weight can be increased, reducing the possible selection effort. This general approach can be achieved only if the each cluster has statistically enough member stars. Accordingly, the criteria of the selection process of the open clusters to be included in this study to meet the goals that must be achieved were as follows: (a) the open cluster must include enough numbers of most probable members that cover the range of mag. (6≀𝑀𝑉≀9), and (b) the most probable member stars in the cluster are determined from proper motion and position.

In our study of the local population of Galactic open clusters, we obtained data that now allow a reliable construction of their luminosity and mass functions. Our cluster sample identified in the all-sky compiled catalogue ASCC-2.5 [20]. For each star projected on a cluster area, a membership probability was determined in an iterative that takes spatial, photometric, and proper motion distributions of stars into account within the corresponding area on the sky [21]. At the end of the iterations for each cluster, we obtained new coordinates of the cluster center, the cluster size, the mean proper motion, the distance from the Sun, reddening, and age [22, 23]. These parameters were determined with data on the most probable members, for example, stars having kinematic membership probability higher than 50%. The results are included in the catalogue of open cluster data (COCD) and its extension [22, 23]. This completeness limit corresponds to a range of magnitude6≀𝑀𝑉≀9. Table 1 shows the basic data of 9 open clusters selected according to the above criteria.


Cluster nameRA 2000 h. m. s.Dec 2000 d. m. s.Range of magnitudesRange of distances (pc) 𝑁 (Known member stars)

Stock 202 14.4+59 16βˆ’3.115–12.7094.371–500085
NGC 86902 19.1+57 09βˆ’7.114–14.5023.0185–10000152
NGC 88402 22.0+57 08βˆ’4.338–14.5023.0185–2631.579146
NGC 216806 09.1+24 212.3018–11.8835.30814–181.818234
NGC 226406 41.0+09 53βˆ’3.383–11.01712.361–4347.826128
NGC 251607 58.1βˆ’60 45βˆ’2.611–11.1919.1491–3333.33345
NGC 263208 40.4+19 41βˆ’0.815–12.225.96338–1492.537363
NGC 353211 06.0βˆ’58 44βˆ’2.436–11.686.32511–709.2198650
Ruprecht 14719 16.7βˆ’16 180.16625–10.89937.2674–413.22326

3. Method of Analysis

The LF in our analysis was determined using the classical method of Van Rhijn [24]. It is defined as the number of stars in certain magnitude interval per unit volume. The volume and magnitude elements were determined by dividing the cluster into shells with certain width depending on the number of the stars in each cluster.

Double frequency tables of the number of stars in each radius interval (shell) corresponding to each magnitude interval were established. In order to determine the LF from the double frequency table, we counted the number of stars which fall in the magnitude interval corresponding to each distance interval, then we divided the counts in each shell by the volume of this shell (i.e. density) and constructed a table representing the mean absolute magnitude and densityπœ™(𝑀𝑉). Table 2 represents a sample of double frequency table for Stock 2 as an example, into which the number of members is 85, distributed into equal magnitude intervals (i.e., absolute magnitude) and distance intervals (measuring in parsec unit).


Dis. (pc)
Mag (absolute mag.). 4.37–546.223 546.223–1088.07 1088.07–1629.93 1629.93–2171.78 2171.78–2713.63 2713.63–3255.48 3255.48–3797.34 3797.34–4339.19 4339.19–4881.04 4881.04–5422.89

βˆ’3.115–1.39912
βˆ’1.399–0.31731
0.317–2.03471
2.034–3.7506
3.750–5.46713
5.467–7.18313
7.183–8.89922
8.99–10.6169
10.616–12.3326
12.335–14.0491

It was found that 80 members are included in the distance interval (4.370; 546.223 pc), the remaining 5 members were distributed in other intervals as shown in the table, according to the statistical distribution of the data. Using the double frequency table, we calculated the LF as it is represented in Table 3 according to Van Rhijn’s [24] definition of LF.


Mean magnitude  Density (LF) πœ™ ( 𝑀 𝑉 ) l o g πœ™ ( 𝑀 𝑉 )

βˆ’2.257 1 . 3 3 1 Γ— 1 0 βˆ’ 1 1 βˆ’10.876
βˆ’0.5417 2 . 6 8 4 Γ— 1 0 βˆ’ 9 βˆ’8.571
1.176 6 . 0 9 7 6 Γ— 1 0 βˆ’ 9 βˆ’8.215
2.892 5 . 1 2 1 Γ— 1 0 βˆ’ 9 βˆ’8.291
4.608 1 . 1 0 9 5 Γ— 1 0 βˆ’ 8 βˆ’7.955
6.325 1 . 1 0 9 5 Γ— 1 0 βˆ’ 8 βˆ’7.955
8.041 1 . 8 7 6 Γ— 1 0 βˆ’ 8 βˆ’7.727
9.758 7 . 6 8 1 Γ— 1 0 βˆ’ 9 βˆ’8.1146
11.474 5 . 1 2 1 Γ— 1 0 βˆ’ 9 βˆ’8.2906
13.191 8 . 5 3 4 Γ— 1 0 βˆ’ 1 0 βˆ’9.069

4. Results

4.1. The Luminosity Function

Depending on the previously mentioned statistical method, the LF of the studied clusters of our interest were calculated. The LF of the clusters studied are tabulated in Tables 4, 5, 6, 7, 8, 9, 10, 11, and 12.


𝑀 𝑉 βˆ’2.257βˆ’0.54171.1762.8924.608
l o g πœ™ ( 𝑀 𝑉 ) βˆ’10.876βˆ’8.571βˆ’8.215βˆ’8.291βˆ’7.955

𝑀 𝑉 6.3258.0419.75811.47413.191
l o g πœ™ ( 𝑀 𝑉 ) βˆ’7.955βˆ’7.727βˆ’8.1146βˆ’8.2906βˆ’9.069


𝑀 𝑉 βˆ’6.237βˆ’4.484βˆ’2.731βˆ’0.977βˆ’0.7762.529154.282
l o g πœ™ ( 𝑀 𝑉 ) βˆ’12.264βˆ’9.317βˆ’9.106βˆ’8.858βˆ’8.687βˆ’8.643βˆ’8.451

𝑀 𝑉 6.0367.7909.54211.29613.04914.802
l o g πœ™ ( 𝑀 𝑉 ) βˆ’8.297βˆ’8.342βˆ’8.150βˆ’8.255βˆ’9.121βˆ’9.597


𝑀 𝑉 βˆ’3.558βˆ’2βˆ’0.44βˆ’1.122.6784.2385.797
l o g πœ™ ( 𝑀 𝑉 ) βˆ’8.759βˆ’8.629βˆ’8.165βˆ’7.921βˆ’7.301βˆ’6.756βˆ’6.641

𝑀 𝑉 7.3568.91510.47412.03413.59315.152
l o g πœ™ ( 𝑀 𝑉 ) βˆ’6.59βˆ’6.465βˆ’6.448βˆ’6.804βˆ’7.846βˆ’7.846


𝑀 𝑉 3.0814.6416.27.7599.31810.87712.437
l o g πœ™ ( 𝑀 𝑉 ) βˆ’7.244βˆ’6.081βˆ’5.553βˆ’4.362βˆ’4.768βˆ’5βˆ’5.467


𝑀 𝑉 βˆ’2.0230.6993.426.1418.86311.584
l o g πœ™ ( 𝑀 𝑉 ) βˆ’9.8βˆ’9.118βˆ’9.039βˆ’9.118βˆ’9.04βˆ’9.817


𝑀 𝑉 βˆ’1.5830.4752.5324.5906.6478.70510.763
l o g πœ™ ( 𝑀 𝑉 ) βˆ’8.335βˆ’8.541βˆ’8.345βˆ’8.047βˆ’8.269βˆ’8.093βˆ’8.445


𝑀 𝑉 0.0061.6483.2904.9336.5758.2179.85911.501
l o g πœ™ ( 𝑀 𝑉 ) βˆ’7.016βˆ’6.744βˆ’6.734βˆ’7βˆ’6.793βˆ’6.851βˆ’7.395βˆ’7.094


𝑀 𝑉 βˆ’1.4380.5592.5554.5516.5488.54410.54012.537
l o g πœ™ ( 𝑀 𝑉 ) βˆ’8.046βˆ’7.602βˆ’7.59βˆ’6.314βˆ’5.995βˆ’6.092βˆ’6.217βˆ’6.995


𝑀 𝑉 1.1993.2655.3307.3969.46111.527
l o g πœ™ ( 𝑀 𝑉 ) βˆ’7.729βˆ’7.034βˆ’6.254βˆ’5.828βˆ’6.032βˆ’6.129

NGC 2264, 2168, and Praesepe had been studied before by Lee et al. in 1984, the data used were selected on basis of photometric observations and proper motion studies. The LF and MF of these clusters are different with cluster age. On the other hand, other clusters (except Ruprecht 147) had been also studied by Lee and Chun in 1988. According to these studies, the combined LFs of open clusters and associations, the initial luminosity function ILF, and initial mass function IMF of these clusters were derived. They show slight differences in the bright and faint parts as compared with the field star ILF and IMF. However, the general shapes of the cluster ILF and IMF agree well with those for field stars in the solar neighborhood. We note that, all of these above studies [11, 25] depend on counting the number of stars within certain magnitude interval, regardless of the classical definition adopted by Van Rhijn [24].

Recently, Kalirai et al. [26] studied luminosity functions for NGC 2168: the LF rises until 𝑀𝑉=5 and then dips down at 𝑀𝑉=7 and rises again. This dip was most likely the result of poor statistics or cluster stars being subtracted off in the blank field and not physical. In that sense, Kalirai et al. [26]: stated that we do note that the luminosity function given in Barrado Y NavascuΓ©s et al. [27] does not show a dip at this magnitude. This was constructed by using the I band LF in Table 3 of Barrado et al. [27] and correcting to V by using the empirical fiducial in Table 1 of that paper.

Sanner and Geffert [28] studied the fields of nine open clusters (e.g., Stock 2, Pleiades, and Praesepe), based on Tycho-2 catalogue. They determined membership probabilities for the stars in the cluster fields from the stellar proper motions and used the Tycho-2 photometry to compute the initial mass function (IMF) for the clusters from the main sequence turn-off point down to approximately. 1π‘€βŠ™.

In this work, the computed LFs of the studied open clusters are represented in Figure 1. From the general appearance of the curves in this figure, it is so difficult to conclude whether there is a dip or not.

In order to have an accurate decision for the presence of the dip, we calculated the dispersion (i.e., standard deviation) of LF of the whole curve and the dispersion around the region of the dip which was suggested by Lee et al., 1997.

The results of these calculations are shown in Table 13. From this table, it is clear that the dispersion around the region where Wielen dip position was suggested is much smaller than the dispersion of the whole cluster, which assures the absence of any dip in the region of 7 mag. or 9 mag., except in Praesepe where the dispersion around the dip was slightly larger than the dispersion of the whole curve. The results obtained in the present work are in agreement with those obtained by Miller and Scalo [8], where they concluded that there is no physical significance of this dip because its size is within the errors of the determination of the LF, and the dip is not found by all investigators.


Cluster nameDispersion of the LF curve of the whole clusterDispersion around the dip region

Stock 20.9120.21
NGC 869 (h-Per)1.0760.1
NGC 884 (Ο‡-Per)0.8370.095
NGC 2168 (M35)0.9540.605
NGC 2264 S Mon0.3790.046
NGC 25160.180.176
Praesepe NGC 2632 (M44)0.2230.332
NGC 35320.8070.111
Ruprecht 1470.7300.212

The absence of any dip in the LF of the studied clusters in the present work makes us more interested to use Lee’s original data [15] for calculating the general dispersion of the figure and the dispersion around the suggested magnitude intervals of the dip.

We redraw Lee's data of 1997 presented in Table 14 as shown in Figure 2 for the three open clusters, Hyades (closed triangle), Praesepe (opened square), and Pleiades (opened circle) with solar neighborhood (closed circle), keeping Lee’s scale in our drawings. Comparing Figure 2 with Lee’s Figure 3 which is copied from the paper published by Lee et al. (1997). It was found that there is a mistake in his Figure 3 regarding to the Pleiades cluster. Into which the LF of the Pleiades cluster lies above the LF of the solar neighborhood stars and below the LF of the Hyades cluster while in Figure 2 the LF of the Pleiades cluster is above all the LF curves of the other open clusters in most parts, and it lies between Hyades and Praesepe clusters starting from 𝑀𝑉≃15 mag.


𝑀 𝑉 βˆ’6βˆ’5βˆ’4βˆ’3βˆ’2βˆ’1012
34567891011
121314151617181920
Hyades πœ™ ( 𝑀 𝑉 ) 419
303233494845344986
1816296796672310
Praesepe πœ™ ( 𝑀 𝑉 ) 320
23596367827868114200
221148146946
Pleiades πœ™ ( 𝑀 𝑉 ) 111935
41587377626995137182
1981826410
Field star Β· πœ™ ( 𝑀 𝑉 ) Γ— 3 0 0.2190.712.0285.22811.61820.47030.48733.42936.402
48.43166.54674.83869.20258.90174.15281.306109.678195.039
263.100138.07797.75153.71835.49130.91224.5549.3352.885

Β·Solar neighborhood field stars.

We calculate the dispersions of the LF curve of the whole cluster and the region around the suggested dip for these three open clusters given by Lee et al. in 1997. The results are shown in Table 15, from which it is clear that the dispersion around the region of the dip suggested by them is much smaller than the dispersion in the LF curve of the whole cluster, also its amplitude was approximately measured based on shape of the curve around the dip, it is clear that these amplitude values are within the dispersion around the dip region as shown in Table 15. This makes us on the side of the nonreality of the suggested dip. Especially, if we know that Lee et al. have included white dwarfs in their LF calculations, and LF as defined in the literature was not followed exactly by them since the volume was not considered in their calculations of LF.


Cluster nameDispersion of the LF curve of the whole clusterDispersion at the dip regionAmplitude due to dip region

Hyades0.4310.2460.1587
Praesepe0.5250.2140.10701
Pleiades0.40.1270.0464

4.2. The Present Day Mass Function (PDMF) and IMF

The PDMF πœ™ms(logβ„³) is defined as the number of main sequence stars per unit logarithmic mass interval per square parsec in the solar neighborhood. The PDMF of main sequence field star is related to the LF of field stars πœ™(𝑀𝑉) [8] byπœ™ms𝑀(logβ„³)=πœ™π‘‰ξ€Έ||||𝑑𝑀𝑉||||𝑀𝑑logβ„³2𝐻𝑉𝑓ms𝑀𝑉,(1) where 𝑑𝑀𝑉/𝑑logβ„³ is the slope of the [(absolute magnitude 𝑀𝑉, mass β„³-) relation] and converts the LF to an MF. The term 2𝐻(𝑀𝑉) is the result of integrating the LF perpendicular to the plane of the Galaxy, assuming an exponential distribution with scale height 𝐻(𝑀𝑉). The factor 𝑓ms(𝑀𝑉) gives the fraction of stars at a given magnitude which is on the MS. The values of the absolute magnitudes, masses, 𝑑𝑀𝑉/𝑑logβ„³, scale heights, and the fraction 𝑓ms adopted in the present investigations were given by Miller and Scalo [8].

The definition of the field star IMF in terms of all stars ever formed may seem confusing at first. However, in the case of a time-constant IMF (as assumed here), the IMF at any given time has the same shape as the IMF at any other time and, therefore, the same shape as the IMF of all stars ever formed. Note that the field star IMF as defined here and the IMF of stars in open clusters are similar quantities in that both give the mass distribution of stars at birth.

Stars with MS lifetimes greater than the age of the Galaxy will be found on the MS today regardless of when they were formed. For these stars (e.g., open clusters), the PDMF (πœ™ms(logβ„³)) and IMF (πœ‰(logβ„³)) are identical [8, 15]:πœ™ms(logβ„³)=πœ‰(logβ„³),𝑇msβ‰₯𝑇0.(2)

Now, we aimed to study the IMF of the above list of clusters (where the data were collected from ASCC-2.5 catalogue) to show if there is a Wielen dip at region logβ„³β‰ˆβˆ’0.25(β„³β‰ˆ0.56β„³βŠ™) for Praesepe cluster and logβ„³β‰ˆβˆ’0.12(β„³β‰ˆ0.76β„³βŠ™) for other open clusters [15]. Now, referring to (1) to calculate IMF for the above list of clusters, we get the data in Tables 16, 17, 18, 19, 20, 21, 22, 23, and 24.


𝑀 𝑉 l o g ( β„³ / β„³ βŠ™ ) l o g πœ‰ ( l o g β„³ )

βˆ’2.2571.007βˆ’2.496
βˆ’0.5410.761βˆ’1.402
1.1760.528βˆ’0.459
2.8920.3070.331
4.6090.0980.971
6.325βˆ’0.0991.459
8.041βˆ’0.2841.795
9.758βˆ’0.4571.980
11.474βˆ’0.6172.013
13.191βˆ’0.7651.895


𝑀 𝑉 l o g ( β„³ / β„³ βŠ™ ) l o g πœ‰ ( l o g β„³ )

βˆ’6.2371.623βˆ’5.616
βˆ’4.4841.344βˆ’4.141
βˆ’2.7311.077βˆ’2.824
βˆ’0.9770.823βˆ’1.666
0.7760.581βˆ’0.665
2.5290.3520.177
4.2820.1360.861
6.036βˆ’0.0671.387
7.789βˆ’0.2581.755
9.542βˆ’0.4361.965
11.296βˆ’0.6012.017
13.049βˆ’0.7541.910
14.802βˆ’0.8931.646


𝑀 𝑉 l o g ( β„³ / β„³ βŠ™ ) l o g πœ‰ ( l o g β„³ )

βˆ’3.5581.201βˆ’3.426
βˆ’1.9990.969βˆ’2.322
βˆ’0.4400.747βˆ’1.342
1.1190.535βˆ’0.488
2.6790.3340.241
4.2380.1420.846
5.797βˆ’0.0401.325
7.356βˆ’0.2121.679
8.915βˆ’0.3741.908
10.475βˆ’0.5252.012
12.034βˆ’0.6671.991
13.593βˆ’0.7981.845
15.152βˆ’0.9201.574


𝑀 𝑉 l o g ( β„³ / β„³ βŠ™ ) l o g πœ‰ ( l o g β„³ )

3.0810.2830.409
4.6410.0940.981
6.200βˆ’0.0851.428
7.759βˆ’0.2551.750
9.318βˆ’0.4141.947
10.877βˆ’0.5632.019
12.437βˆ’0.7021.965


𝑀 𝑉 l o g ( β„³ / β„³ βŠ™ ) l o g πœ‰ ( l o g β„³ )

βˆ’2.0230.973βˆ’2.338
0.6990.592βˆ’0.706
3.4200.2410.544
6.141βˆ’0.0791.414
8.863βˆ’0.3681.902
11.584βˆ’0.6272.010


𝑀 𝑉 l o g ( β„³ / β„³ βŠ™ ) l o g πœ‰ ( l o g β„³ )

βˆ’1.5830.909βˆ’2.048
0.4750.622βˆ’0.826
2.5320.3520.178
4.5900.1000.965
6.648βˆ’0.1351.533
8.705βˆ’0.3521.884
10.763βˆ’0.5522.018


𝑀 𝑉 l o g ( β„³ / β„³ βŠ™ ) l o g πœ‰ ( l o g β„³ )

0.0060.686βˆ’1.085
1.6480.466βˆ’0.227
3.2900.2570.493
4.9330.0591.075
6.575βˆ’0.1271.517
8.217βˆ’0.3021.821
9.859βˆ’0.4671.986
11.501βˆ’0.6202.012


𝑀 𝑉 l o g ( β„³ / β„³ βŠ™ ) l o g πœ‰ ( l o g β„³ )

βˆ’1.4380.888βˆ’1.955
0.5590.610βˆ’0.781
2.5550.3490.188
4.5510.1040.952
6.548βˆ’0.1241.511
8.544βˆ’0.3361.865
10.540βˆ’0.5312.014
12.537βˆ’0.7101.958


𝑀 𝑉 l o g ( β„³ / β„³ βŠ™ ) l o g πœ‰ ( l o g β„³ )

1.1990.525βˆ’0.448
3.2650.2600.483
5.3300.0131.194
7.396βˆ’0.2161.686
9.461βˆ’0.4281.959
11.527βˆ’0.6222.012

Again, we computed the dispersion of IMF and also the dispersion around the region of the dip (as a function of mass) which was suggested by Lee et al. [15].

The results of these calculations are shown in Table 25. Here, it is clear that the dispersion around the region of the Wielen dip which was suggested by Lee et al. [15] is much smaller than the dispersion of the whole cluster, which assure the absence of any dip in the region of logβ„³β‰ˆβˆ’0.25(β„³β‰ˆ0.56β„³βŠ™) for Praesepe cluster and logβ„³β‰ˆβˆ’0.12(β„³β‰ˆ0.76β„³βŠ™) for other open clusters. It is worthy to note that the dispersion of IMF of whole Praesepe cluster is much larger than the dispersion around the dip.


Cluster nameDispersion of the IMF curve of the whole clusterDispersion around the dip region

Stock 21.5870.264
NGC 869 (h-Per)2.5680.286
NGC 884 (c-Per)1.8130.305
NGC 2168 (M35)0.6070.262
NGC 2264 (S Mon)1.7080.318
NGC 25161.5200.250
NGC 2632 (Praesepe, M44)1.1360.238
NGC 35321.4580.258
Ruprecht 1470.9680.387

5. Conclusion

Finally, we concluded that the examination of IMF revealed the same results obtained with the LF, for example, there is no dip in the distribution of both LFs and IMFs. Also, Wielen [29] stated that most people explain the so-called Wielen dip in the LF of field stars as a consequence of a special property of the relation between stellar mass and stellar luminosity in the region of the dip. If the IMF would be universal and valid for clusters too, then we should see the dip also in the LF of clusters. However, there are strong indications that the IMF in clusters differ from that of the field stars. In this case, the dip may not be strongly visible in the cluster LF.

Acknowledgment

The authors would like to thank Professor Dr. R. Wielen, for his communication and comments on this paper.

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Copyright © 2011 W. H. Elsanhoury et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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