Revealing the parameters of the open cluster
Ruprecht 58 ⋆
E. Giorgi a,1 , R. A. Vázquez a,1 , G. R. Solivella a,1 ,
R. B. Orellana a ,
a Facultad
de Ciencias Astronómicas y Geofı́sicas UNLP, IALP-CONICET, Paseo
del Bosque s/n, 1900, La Plata, Argentina
and
J. Nuñez b
b Facultad
de Ingenierı́a, CIBA, Universidad Nacional de Jujuy, Gorriti 237, 4600,
S. S. de Jujuy, Argentina
Abstract
We present results of a study that combines U BV I photometry, MK spectral classification and proper motions in the area of the, up to now unknown, open cluster
Ruprecht 58 at the Puppis region. Star counts from the 2MASS data catalog together with the analysis of CCD U BV I photometry demonstrate that it is a real
open cluster with 9′ size approximately. The cluster is placed at a distance of 3.9
kpc and is about 250 Myr old with mean reddening E(B−V ) = 0.33 mag. Proper
motions confirm Ruprecht 58 is a real cluster with mean absolute proper motions
µα cosδ = −2.77±0.45 mas/yr and µδ = 4.54±0.45 mas/yr in the magnitude range
13.5 < V < 14.5 and µα cosδ = −2.70 ± 0.32 mas/yr and µδ = 3.19 ± 0.32 mas/yr
in the range 14.5 < V < 16.0. The computation of the cluster mass spectrum slope
yielded x = 1.8 in the mass range from ≈ 1.4 to ≈ 4m⊙ .
Key words: Open clusters and associations; individual: Ruprecht 58 –
methods: statistical - stars: kinematics – stars: luminosity function,
mass function - stars: Hertzsprung-Russell (H-R) and C-M diagrams
⋆ Based on observations collected at the Complejo Astronómico El Leoncito
(CASLEO), Argentina.
1 Visiting astronomer, Complejo Astronómico El Leoncito operated under agreement between the Consejo Nacional de Investigaciones Cientı́ficas y Técnicas de la
República Argentina and the National Universities of La Plata, Córdoba and San
Juan
Preprint submitted to Elsevier Preprint
21 February 2007
1
Introduction
The detection of the Canis Major (CMa) over-density by Martin et al. (2004)
produced a renaissance of interest for the Third Galactic Quadrant (180◦ <
l < 270◦ ) of the Milky Way. The vivid debate in the last year (Momany
et al. 2004, Bellazzini et al. 2004) on this over-density clearly demands a
better picture of the Galaxy structure in this region. Young open clusters are
recognized as ideal spiral arm tracers (Becker & Fenkart 1970, Feinstein 1994)
as they are near the spiral arm in which they formed. Surprisingly, the shape
and extent of the Perseus and Cygnus-Norma arms in the Third Quadrant are
far from being clear and settled. Russeil (2003) using star forming complexes
finds that both the Perseus and Norma-Cygnus arms are not visible at all
in the Third Quadrant, confirming previous results by May et al. (1997).
Nevertheless, they could confirm previous suggestions about the shape and
location of the Galactic warp and show how bridges of material are present in
a few anti-center directions. To date, no study has probed the spiral structure
of the Third Galactic Quadrant using young star clusters which are vital to
better trace the spiral pattern in largely overlooked region of the Galaxy.
Additionally, open clusters are not only the main source of stellar enrichment
in our galaxy but also the kind of objects through which the theories of star
formation and evolution can be adequately addressed; they offer a chance to
know the probable spatial variations in the star formation rates (Scalo 1986)
and give information regarding the history and evolution of the galactic disk
(e.g. Janes & Adler 1982; Massey et al. 1995).
For the last several years our group has been conducting a systematic, homogeneous and accurate U BV RI photometric survey of open clusters in this
part of the Galaxy [see Moitinho (2001), Giorgi et al. (2005, and references
therein), Baume et al. (2004), Carraro et al. (2005a, and references therein)]
with special emphasis in Vela-Puppis. Our target on this occasion is Ruprecht
58, an open cluster that WEBDA sets at α2000 = 8h 14m 7s and δ2000 = −31◦ 57′
(l = 250.4◦ , b = +1.6◦ ) but for which we found no information in any existing data bases. Lindoff (1968) reports BV photographic photometry down
to V = 15.2 mag which could not be compared to ours because of no stellar
identification in his work. Anyway, Lindoff’s conclusion is that no open cluster
exists down to his limiting magnitude.
In Sect. 2 we describe the observation routines and the reduction processes.
The analysis of the photometric and of spectroscopic data, the reddening discussion, membership, distance and age can be found in Sect. 3. The proper
motion analysis is presented in Sect. 4. The cluster mass spectrum is determined in Sect. 5 and the conclusions are given in Sect. 6.
2
2
Observations
2.1 Photometry
CCD observations in the U BV I system were carried out in the field of Ruprecht
58 on the nights of December 24 and 25, 2003, at the Complejo Astronómico
El Leoncito, CASLEO, Argentina, using the 2.15-m telescope equipped with
a CCD ROPER 1300B, 1340 × 1300 and 0.226 ′′ /pix scale, covering 4.2′ on a
side. Exposure times were of 300 seconds in U ; 300, 50 and 5 seconds in B;
300, 30 and 3 seconds in V , and 120, 60 and 3 seconds in I under seeing values
ranging from 1.3′′ to 2.1′′ .
Fig. 1. Finding chart of Ruprecht 58 adapted from a DSS image. The circle is the computed angular size of the cluster 9′ approximately (centered at
α2000 = 8h 14m 50s , δ2000 = −31◦ 56′ 29′′ ); the squares show the four science frames
observed at CASLEO. North is at top and East at left. The plus sign is the cluster
center as given in WEBDA
Instrumental magnitudes were obtained by means of the point spread function (PSF) fitting using the DAOPHOT (Stetson 1987) package within IRAF 2
and tied to the standard U BV I system with several standard stars of Landolt (1992) list and secondary standards in the cluster NGC 2571 previously
calibrated with Landolt standards as well. A description of the reduction procedure can be found in Giorgi et al. (2002). Mean extinction coefficients at
CASLEO, 0.49 in U , 0.27 in B, 0.12 in V and 0.03 in I, were applied; the
2
IRAF is distributed by NOAO, which are operated by AURA under cooperative
agreement with the NSF
3
transformation equations to the standard system were of the form:
u = U + u1 + u2 × X + u3 × (U − B)
b = B + b1 + b2 × X + b3 × (B − V )
v = V + v1 + v2 × X + v3 × (B − V )
i = I + i1 + i2 × X + i3 × (V − I)
where u2 , b2 , v2 and i2 are the extinction coefficients for the U BV I bands
respectively, X the air masses of each exposure and u1 , b1 , v1 , i1 , u3 , b3 , v3 ,
and i3 the fitting constants values.
The area covered in the current survey is shown in Fig. 1 where each square
superposed onto the Digitized Sky Survey plate, DSS, represents the CCD
frame. Table 1 lists the typical DAOPHOT errors found at different V magnitude ranges.
Table 1
Daophot errors as function of the V magnitude
V range
ǫV
ǫ(B−V )
ǫ(U −B)
ǫ(V −I)
10-14
0.02
0.02
0.03
0.02
14-16
0.02
0.03
0.04
0.03
16-18
0.02
0.03
0.05
0.09
18-20
0.02
0.03
0.10
-
> 20
0.04
0.08
-
-
Table 2 (available at CDS) contains the x-y star positions and the photometric
output including V magnitudes and B − V colors for 1117 stars, U − B for 654
stars, and V −I for 1075 stars. For a reduction of the uncertainties level, no star
with DAOPHOT errors larger or equal than 0.10 mag have been considered
in any further analysis. An abbreviated version of Table 2 is given below.
2.2 Spectroscopy
Spectral data for 13 of the brightest cluster stars were collected at the 2.15m telescope of CASLEO (Argentina) during two observing runs in March
11-14 and April 1-4, 2004. The spectra were obtained with the REOSC–DS
Cassegrain spectrograph and a Tek 1024×1024 detector using a 600 l/mm grating in the first order. The spectra have a wavelength range of 3900-5500 Å(the
traditional spectral region for classification from CaII K line to Hβ, allowing
a precise MK type), 2.5 Å/pixel dispersion (≈ 1800 resolution) and were reduced using standard procedures with IRAF. For the clasification purposes we
used MK standard stars taken with the same configuration at CASLEO and
4
Table 2
Photometry, astrometry, proper motions and membership for stars in the field of
Ruprecht 58 (abbreviated version)
ID
x
y
α(2000)
δ(2000)
V
B-V
U-B
V-I
µα
µδ
(mas/y)
(mas/y)
Rem.
P(%)
1
880.92
-386.42
123.72478360
-32.01410950
10.10
0.90
0.93
1.03
-4.10
-2.70
2
1895.48
399.01
123.65309450
-31.96508480
11.43
0.06
0.32
0.02
-6.60
1.90
0.00
0.00
3
100.80
1027.29
123.78266270
-31.92884730
11.72
0.23
0.16
0.17
-4.30
-1.40
0.00
4
1474.87
543.16
123.68346560
-31.95677750
11.56
0.39
0.20
0.42
0.30
3.00
0.00
5
1158.19
786.31
123.70649620
-31.94233560
11.90
1.10
1.07
1.26
-8.10
4.30
0.00
6
1052.24
903.29
123.71425950
-31.93531280
12.42
0.28
0.13
0.35
-0.40
-4.40
0.00
7
44.14
1081.63
123.78679360
-31.92560030
12.62
0.16
0.01
0.13
-0.40
2.50
0.00
8
1108.14
1807.24
123.71152770
-31.88016200
12.73
1.32
1.32
1.50
-2.50
10.60
0.00
-8.10
3.20
0.00
9
880.11
1239.97
123.72706980
-31.91499840
12.79
0.71
0.22
0.87
10
971.58
561.10
123.72178947
-31.95647382
12.90
1.00
0.75
1.26
11
764.10
124.81
123.73386920
-31.98309870
12.91
0.70
0.34
0.77
-31.70
10.00
0.00
12
618.70
674.39
123.74502950
-31.94975560
13.00
1.05
0.76
1.27
-2.70
5.30
0.00
13
1812.40
252.92
123.65884560
-31.97406980
13.02
0.46
0.07
0.60
-7.00
10.10
0.00
14
1524.46
-582.18
123.68308780
-32.02507050
13.07
0.56
0.11
0.63
15
1805.79
143.99
123.66339442
-31.98150036
13.25
0.52
0.08
0.67
16
1302.21
391.29
123.69864540
-31.96666266
13.28
1.07
0.86
1.34
17
318.79
556.76
123.76638830
-31.95726670
13.50
0.38
0.22
0.45
-4.10
3.20
0.00
18
1106.17
1082.41
123.71063890
-31.92434230
13.55
0.50
0.05
0.59
-8.10
6.50
0.00
19
1097.09
-362.09
123.70930950
-32.01235750
13.64
1.35
1.79
1.53
-3.70
-0.80
20
672.28
200.23
123.74053330
-31.97858920
13.66
0.29
0.17
0.36
2.30
-1.50
0.00
m
0.06
m
0.62
21
605.00
-79.74
123.74501770
-31.99571230
13.70
1.15
1.23
1.34
6.00
-0.10
22
1086.30
221.64
123.71086210
-31.97681370
13.74
0.29
0.15
0.34
-0.50
1.20
0.00
23
1314.29
416.52
123.69779980
-31.96514898
13.74
0.94
0.74
1.15
24
596.07
-770.69
123.74807520
-32.03638110
13.80
0.69
0.26
0.83
25
1344.73
858.54
123.69566860
-31.93862748
13.86
0.26
0.09
0.35
26
1080.61
1308.93
123.71283360
-31.91056500
13.86
0.56
0.09
0.71
-3.60
2.90
27
1541.37
-53.59
123.67786180
-31.99304390
13.90
1.02
0.86
1.21
-4.50
3.90
28
969.86
1301.44
123.72076210
-31.91114390
13.93
0.30
0.12
0.43
-0.70
6.00
29
1590.77
644.44
123.67529650
-31.95047140
13.93
0.23
0.09
0.29
5.50
3.00
m
0.05
30
1238.27
1305.48
123.70152270
-31.91059810
13.94
0.28
0.11
0.40
-0.50
3.80
m
0.76
31
1009.34
2132.93
123.71914620
-31.86216420
13.96
1.02
0.89
1.19
32
2210.73
46.93
123.62998560
-31.98617980
14.00
0.52
0.14
0.68
-4.70
7.80
33
795.72
2112.69
123.73409940
-31.86337860
14.11
0.46
0.17
0.58
m
0.77
0.00
m
0.72
0.21
34
34.18
1275.52
123.78740750
-31.91360880
14.17
0.30
0.18
0.33
m
35
1454.71
1478.02
123.68797030
-31.90145862
14.18
0.28
0.11
0.39
m
36
881.73
1971.47
123.72807900
-31.87185180
14.19
0.80
0.53
0.91
37
377.72
716.87
123.76237270
-31.94743730
14.24
0.28
0.18
0.33
-5.80
3.70
38
108.46
751.40
123.78173800
-31.94564590
14.24
0.68
0.26
0.78
-17.60
5.50
39
154.97
1160.24
123.77894590
-31.92068060
14.24
1.34
1.48
1.60
-3.30
4.80
40
341.78
1067.31
123.76543740
-31.92612840
14.26
0.26
0.16
0.29
-11.50
-5.10
m
0.54
0.00
0.62
m
0.00
Note: star coordinates are in degrees and proper motions in mas/y. (::) denotes photometric
errors ǫphot > 0.1. The proper motion µα is given at the cluster declination.
Rem: m shows probable cluster members according to the photometric criterion explained
in Sect. 3.
Last column gives the star probability according our proper motions analysis.
Some stars were labeled PME (probable member by extension) and NME (non-member by
extension) from an extrapolation procedure using the parameters derived in Sect. 4. The
complete table can be found in the electronic edition of New Astronomy too
the Digital Spectra Classification of R.O. Gray 3 and the MK Standard 4 .
3
http://nedwww.ipaccaltech.edu/level5/Gray
5
Fig. 2. Spectra of 13 stars grouped by our classification given in Table 3
Spectra and the respective spectral classification are shown in Fig. 2. It is
worth mentioning that for a number of stars we found discrepancies in the
intensities of the K CaII lines and the spectral types assigned according to
the H lines in the sense that K CaII lines show intensities corresponding to
spectral earlier than according to the H lines. These stars compose the 38%
(two of them are giants; other three of the main sequence) of the total stars
with spectral types and we assume them peculiar stars; the others constitute
the 61% and are are of giant types. The results and description of peculiarities
are given in column 9 of Table 3. We are confident that our classification is
not wrong for more than one sub-type and that the luminosity class is precise.
3
Data Analysis
3.1 Cluster size
The reliability of Ruprecht 58 and its spatial extension were analyzed using
2MASS data of all stars included inside a 30′ × 30′ box centered in the cluster published coordinates. The technique is based in star counts that were
performed in a series of successive rings 1′ width from the cluster center [as
done in Baume et al. (2004), Carraro et al. (2005a)]. Counts in each ring were
divided by the ring area to construct the density profile. During this process
we noticed some sort of count fluctuations that we assume are due to the presence of patches of dust across the cluster surface. So, several attempts had to
4
http://stellar.phys. appstate.edu/Standards
6
Fig. 3. The star density as a function of the distance to the √
center of the cluster
computed with 2MASS as explained in the text. Bars are the N counts
be done until a clean star density peak was found at α2000 = 8h 14m 50s and
δ2000 = −31◦ 56′ 29′′ . That sets a new cluster center slightly shifted from the
original one given in WEBDA. The density profile in Fig. 3 reveals an evident
stellar over-density that extends up to r = 4.5′ ± 1′ where it merges with the
stellar density of background. This value is adopted here as the cluster radius.
Comparing to the area covered by our photometric survey shown in Fig. 1 we
are confident that we have surveyed ≈ 75 % of the whole cluster.
3.2 Cluster members, reddening, distance and age of Ruprecht 58
Fig. 4. Left panel: TCD of all stars inside the cluster radius. The continuous lines
show the intrinsic loci for main sequence stars (Schmidt-Kaler 1982) in the normal
position and the dashed one shifted by E(B−V ) = 0.33 and E(U −B) = 0.29 (§3.2).
Right panel: Idem for the comparison field stars
Figures 4 and 5 show the U − B/B − V two-color (TCD) and the V /(B − V ),
7
Fig. 5. Upper panel: The V /B − V , V /U − B, and V /V − I diagrams of all stars
inside the cluster radius. The Schmidt-Kaler (1982) ZAMS and the intrinsic locus
of the V /V − I sequence from Cousins (1978) adapted from the finding of §3.2
are shown as continuous lines. Lower panel: The same for the comparison field
diagram
V /(U − B) and V /(V − I) color-magnitude diagrams, (CMD), respectively,
of all stars inside the cluster limits (< 4.5′ ); stars outside the cluster limit
(field stars) were plotted too in a similar but not scaled diagrams. All figures
show that Ruprecht 58 is a true cluster since the TCD shows a star sequence
from 0.2 < B − V < 0.6 sharing a common reddening value as indicated by
the intrinsic line (Schmidt-Kaler, 1982) displaced EB−V = 0.33 ± 0.02 and
EU −B = 0.29 ± 0.04. These stars, mainly late B- and A-type stars, constitute
the upper main sequence of the cluster which extends from V ≃ 13.5 to ≃ 17.
The U − B/B − V field star diagram does not show such star sequence but
a few stars of late A-types with similar reddening values what would indicate
that for r > 4.5′ it is still possible to find some cluster members. Something
similar happens in the CMDs for the cluster and the comparison areas where
the potential cluster members clearly emerge for V < 17. The visual inspection
reveals too a number of stars in the CMDs that probably belong to an evolved
8
field star population.
To assess memberships in Ruprecht 58 we consider that the TCD shows reddening values for the bright probable members going from EB−V = 0.25 to
EB−V = 0.41 approximately. Therefore, all the stars falling outside that reddening limits are assumed cluster non members. The stars inside the reddening
limits were then fitted using the Schmidt-Kaler (1982) ZAMS and mean reddening values EB−V = 0.33 and EU −B = 0.29. A careful comparison of the
position of every star in each diagram was made again to reject out those
ones located 1.3 mag above the ZAMS and 0.3 mag below it. This way we
excluded evolved late-type and/or main sequence field stars of foreground
without eliminating probable binaries. This method works reasonably well for
main sequence stars brighter than V ≈ 17 but it becomes uncertain longwards
that magnitude where the contamination by field interlopers may be of some
relevance. Despite this, they will cause no alteration of the cluster parameters
when computed with stars down to V ≈ 17. Some bright yellow/red stars at
the top of the CMDs are not easy to reject out as they could be late evolved
cluster members. To clarify this point we need first to derive the cluster distance and then to apply the spectroscopic parallax method using the spectral
classification we have already done.
Table 3
Spectral types of stars in the area of Ruprecht 58
No.
S.T.
V
(B − V )
(U − B)
10.09
0.89
0.92
E(B−V )
MV
d⊙ [pc]
1
G8III
(0.00)
0.8
721
2
B9IV
11.43
0.06
4
F0IIIp∗
11.56
0.38
0.31
0.13
-0.2
1762
0.20
0.18
1.5
5
G8III
11.89
1.10
1.07
794
0.16
0.8
1312
6
A0III
12.42
0.27
0.12
0.31
0.0
1986
9
G2III
12.78
0.70
0.22
(0.00)
0.9
2377
Comments to S.T.
k A7V-H F0III
13
F8Vp∗
13.02
0.46
0.07
(0.00)
4.0
637
k F0V-Gband F5V-H F8V
15
G0Vp∗
13.25
0.52
0.08
(0.00)
4.4
589
k F2V-Gband F5V-H G0V
16
G5III
13.28
1.07
0.86
0.21
0.9
2218
17
F0Vp∗
13.50
0.38
0.22
0.08
2.7
1288
18
F8Vp∗
13.55
0.50
0.05
(0.00)
4.0
813
20
A1IV
13.66
0.29
0.17
0.28
0.7
2618
21
G8III
13.70
1.15
1.23
0.21
0.8
2818
k A3V-H F0V
k F0V-Gband F5V-H F8V
Note: E(B−V ) and MV come from the color index and absolute magnitude according to the star spectral type
(Schmidt-Kaler 1982); for peculiar stars MV and E(B−V ) correspond to the spectral type classification using
hydrogen lines
To compute the cluster distance, fitting the Schmidt-Kaler (1982) ZAMS, we
need to analyze first the extinction law, R = AV /E(B−V ) , towards the cluster
using the (B −V ) vs (V −I) plots of all the probable cluster members as shown
in Fig. 6. It is simple to see that cluster members tightly follow the galactic
reddening vector E(V −I) /E(B−V ) = 1.24 (Dean et al. 1978) suggesting then
that the extinction law is a normal one with R = 3.1. If R is normal the mean
visual absorption in the zone turns out to be [AV = 3.1× < E(B−V ) >= 1.02 ±
0.18]; the reddening-free colors and V0 magnitudes of cluster members can be
immediately obtained by correcting the observed colors with the average color
9
Fig. 6. The cleaned TCD, B − V vs V − I, of stars in the zone of Ruprecht 58.
Intrinsic lines for luminosity classes, V –continuous– and III –dashed–, and the
relation given by Dean et al. (1978) are also indicated
excesses given above, and the computed visual absorption, AV . The ZAMS
fitting shown in Fig. 5 was carried out on the three CMDs [for the V0 /V − I
intrinsic locus it has been adapted from the correspondence of colors and
spectral types given by Cousins (1978) and Schmidt-Kaler (1982)]. The mean
of the fittings yielded a reddening-free distance modulus of V0 − MV = 12.96
which locates the cluster at 3.9 kpc from the Sun. If we assume a 0.2 error of the
fitting (from eye-inspection) and it is added quadratically to the uncertainties
in color excess and visual absorption, the distance to the cluster may be wrong
by no more than ±400 pc.
We can now apply the spectroscopic parallax method using the intrinsic color–
spectral type correspondence as given in Schmidt-Kaler (1982) to secure the
membership of the bright red/yellow stars mentioned above. Some of these
stars show a few hundredths of negative EB−V (indicated with brackets in
Table 3) which are explained by spectral sub-type uncertainties and the few
amount of reddening undergone by them. Values given in column 8, Table
3, show that these stars are not cluster members. This is, the spectroscopic
classification shows that bright yellow/red stars are foreground to the cluster.
A question arises about stars No. 20 (A1IV) and 21 (G8III) that are the
farthest stars (d > 2.6 kpc) and could still be members of the cluster in case
of a probable variation of the stellar sub-type. We find this hard to happen
as for the giant G8III and the A1IV stars, the absolute magnitudes would
vary no more than 0.2–0.3 respectively. These variations are not important
enough to get these stars close to the cluster. Column 12, Table 2, remarks
the photometric membership that are adopted in this article.
As for the cluster age, we measured in Fig. 7 the locations of the red and blue
“turn-off”, TOs –see Figs. 5b, 5d, and 5e of Meynet et al. (1993)– which were
set at MV ≈ 0.0...0.4 and (B − V )0 ≈ 0.0...0.03 respectively. They give an age
of 250 ± 120 Myr. We also superposed isochrones from a set of evolutionary
10
Fig. 7. The cleaned CMDs of the cluster. Small open circles are the probable cluster
members from photometry and spectroscopy. Big open circles and numbers indicate
stars with spectral types. Continuous lines are the isochronous curves for 250 and
300 Myr, from above to below respectively
tracks computed with mass loss and overshooting by Girardi et al. (2000);
some spread at the upper main sequence stars makes it difficult to assign a
unique age to this cluster. Therefore the cluster age ranges from 250 to 300
Myr. It is to be noticed from Fig. 7 that the number of binaries among cluster
members could be high producing thus an artificial age spread and making
this cluster younger than it is.
4
Confirming the nature of Ruprecht 58
Another way to confirm the cluster reliability comes from UCAC2 (Zacharias
et al. 2004) proper motion catalog containing 1859 stars in a 20′ radius from
the center of Ruprecht 58. The catalog precision in the star positions goes
from about 20 mas for stars in the range 10 < V < 14 to about 70 mas at
V ≈ 16 while proper motion errors go from 1-3 mas/yr down to V ≈ 12 and
4-7 mas/yr at fainter magnitudes. Fig. 8 (left panel) shows the proper motionvector point diagram of 518 stars in a 10′ radius where 126 of them –shown
with filled squares– have photometry obtained by us. The right panel of the
figure shows the V distribution of stars with photometry and proper motions
contrasted with stars having only photometry. Since longwards V > 15.5,
the catalog shows evident signs of incompleteness, our analysis is seriously restricted to the bright stars of the sample. A detailed description of the method
11
Fig. 8. Left: The proper motion-vector point diagram of 518 stars in the cluster
area. Filled squares are stars with proper motions and photometry. Small open
circles for the remaining stars. Right: The magnitude distribution of all stars in
our sample (hatched histogram) and of stars with proper motions and photometry
(white histogram)
we are using can be found in de Elia (2004). In short, we determine cluster
memberships using a variation of the Vasilevskis et al. (1958) method who proposed a mathematical model where an elliptical bivariate normal frequency
distributions and another circular one, for field and cluster stars respectively,
describe the problem entirely. Since the parameters of the distributions are
function of star magnitudes and the angular size of the selected field, the
membership probabilities may be overestimated for stars far from the cluster
center and underestimated for stars near it while underestimated for bright
stars and overestimated for the faint ones. Jones & Walker (1988) improved
the method using an exponential function to describe the areal star density for
the cluster stars according to Van den Bergh & Sher (1960). As shown below,
the method requires that nine unknown parameters be simultaneously fitted
using the method of maximum likelihood (Sanders 1971). Both distributions
are described by:
(µxi − µxc )2 + (µyi − µyc )2
ρc (Vi , ri )
exp
−
,
Φci (µxi , µyi , Vi , ri ) =
2πσ 2
2σ 2
"
#
for cluster stars and
ρf (Vi )
(µx − µx )2 (µyi − µy0 )2
Φfi (µxi , µyi , Vi ) =
,
exp − i 2 0 −
2πσx σy
2σx
2σy2
"
12
#
for field stars; where µx0 , µy0 denote the centroid of the field stars; µxc , µyc the
centroid of the cluster stars; σx , σy eliptical dispersions; σ circular dispersion;
µxi , µyi ,ri , Vi are the proper motions, the distance from the cluster center
and the magnitude of the i-th star. Besides,
−r
ρc (Vi , ri ) = ρ0 (Vi )e
ri
0 (Vi )
,
with ρc (Vi , ri ) to describe the areal cluster star density as a function of Vi and
ri ; ρ0 (Vi ) is the central surface density, r0 (Vi ) the characteristic radius; the
areal density for the field stars is described by ρf (Vi ) = f0 (Vi ) where f0 (Vi )
only depends on magnitudes. Nc , the number of cluster members, is obtained
from Nc = 2πρ0 (Vi )r0 (Vi )2 . The dynamical membership probabilities for i-th
star can be calculated as follow:
Pi =
Φc i
Φci + Φfi
The star sample was subdivided into four groups of V , V1 < 13.5, 13.5 <
V2 < 14.5, 14.5 < V3 < 16.0 and V4 > 16.0. The criterion to group stars this
way is that each sub-sample must be enough populated for a better statistical
analysis. Finally, we stated that stars with Pi > 0.40 are probable dynamical
members. The result of the method applied to groups V2 and V3 is given in
Table 4; the last column of Table 2 gives the computed probabilities for each
of the 126 stars.
Table 4
Cluster and field star parameters
Cluster
Vi
Nc
r0 [arcmin]
ρ0 [arcmin−2 ]
µxc [mas/y]
µyc [mas/y]
σ[mas/y]
V2
13
1.67
0.73
-2.77
4.54
1.64
V3
47
3.54
0.60
-2.70
3.19
2.16
Field
Vi
f0 [arcmin−2 ]
µx0 [mas/y]
σx [mas/y]
µy0 [mas/y]
σy [mas/y]
V2
0.25
-4.34
5.44
3.02
4.21
V3
0.65
-4.17
5.18
1.91
4.19
It is worth mentioning that the characteristic radius r0 (Vi ), increases with the
magnitude; therefore, bright cluster members are much more concentrated
than the faint ones. The proper motion dispersion also increases for faint
magnitudes, but not so clearly as r0 does.
For group V1 , as seen in Table 2, the cluster has no dynamical members
brighter than V = 13.5. This result is entirely consistent with the spectrophotometric result that shows no member above V = 13.5. The situation is, on
the other hand, completely uncertain for V4 star group [V > 16.0]. A reading
of Table 2 gives: twenty one stars are simultaneously dynamical and photometric members; eighteen stars are photometric members but not astrometric
13
members; fifty eight stars are neither photometric nor astrometric members
and twenty five stars that are dynamical members were found non-members
from the photometric analysis.
Table 4 shows that at 1σ the field and cluster distributions do not differ
substantially. However, it is worth mentioning that the study of proper motions
and photometry/spectroscopy show a definite coincidence for the brightest
stars which have been found not to be cluster members. Certainly, errors in
the proper motion measures and the huge un-completeness of the sample make
the dynamical membership estimate rather unstable below V = 13.5...14.
Notwithstanding, star counts and the proper motion study give both a similar
–increasing towards faint magnitudes– cluster size. That is why, in the next
section, the cluster mass spectrum will be determined only with photometric
members.
5
The cluster mass spectrum
We shall build now the cluster mass spectrum, defined as the number of stars
found per mass interval. In particular, for all the stars below the cluster TO,
the mass spectrum reflects the initial mass function, IMF, if we assume all
the star formed at a same time. In the present case bright (most massive)
members are easy to identify; but it is not a simple task to find all the faint
(less massive) members as they are mixed with field interlopers. However, field
interloper effect has been minimized in the best possible way so that we assume the apparent luminosity function, LF, of the cluster [constructed with
only photometric members] is trustable down to V ≈ 17 and can be converted
into the cluster mass spectrum.
Table 5
Luminosity function and mass spectrum of Ruprecht 58
∆MV
N
log m
log( ∆ dN
)
log m
∆MV
N
log m
log( ∆ dN
)
log m
-0.5 .. 0.0
6
0.59
1.90
2.5 .. 3.0
21
0.19
2.57
0.0 .. 0.5
7
0.51
1.99
3.0 .. 3.5
23
0.14
2.63
0.5 .. 1.0
13
0.44
2.27
3.5 .. 4.0
10
0.09
2.28
1.0 .. 1.5
10
0.38
2.18
4.0 .. 4.5
11
0.03
2.36
1.5 .. 2.0
19
0.31
2.46
4.5 .. 5.0
6
-0.01
2.12
2.0 .. 2.5
25
0.25
2.62
5.0 .. 5.5
2
-0.05
1.68
The apparent LF was converted into the true LF using the cluster distance
modulus and the visual absorption given above to obtain individual MV s. The
final step was to apply the mass-luminosity relation from Scalo (1986) to each
luminosity bin in the true LF (see Table 5) to derive the stellar mass distribution shown by the mass points in Fig. 9. This is a probed and straightforward
method to transform the LF into the mass spectrum for all stars below the
cluster TO when the range of star formation time is less than the cluster age
14
(Phelp & Janes 1993, 1994). Since the mass spectrum is defined by the number of stars counted in the mass range m ±∆m/2, the slope x of the mass
distribution can be computed by:
x=−
log(dN/∆logm)
log m
assuming the mass spectrum is represented by a power law.
Fig. 9. The mass function of Ruprecht 58. Error bars depict
given mass bin. Solid line is the slope x = 1.8 of the fitting
√
N of the counts at a
To derive the slope of the mass spectrum we applied a statistical-weighted
least squares fit to all the mass points in Fig. 9 (covering the magnitude range
13.5 < V < 17.25 ) except for the last four bins that surely are affected
by incompleteness. Notice that this apparent magnitude range is the one of
maximum certainty on memberships from a photometric and spectroscopic
point of view. The fitting yielded a slope x = 1.8 ± 0.2, a value higher than
the standard one determined by Salpeter (1955) but of the same order of the
typical values found by Tarrab (1982) for clusters of about 300 Myr. Using
“only” stars located in the cluster main sequence the mass spectrum slope
is free from evolutionary effects. No matter the presence of interlopers, the
construction of the mass spectrum of any cluster may undergo other type of
strong uncertainties such as the presence of accretion disks (probable of no importance in clusters this age), differential reddening and unresolved multiple
systems that spoil the results; but no doubt, the net effect due to unresolved
binaries (Scalo 1986) is the most difficult to assess: if the mass spectrum of
the secondary stellar components is similar to that of primary stars then their
influence on the computed slope is negligible (Vanbeveren 1982); however, detailed analysis made by Sagar & Richtler (1991) demonstrate that undetected
binaries can rise the slope of a mass function by 0.25 for a binary percentage
of 50%. In the present case, allowing for fitting errors, the slope we found is
15
close to the upper limit of the x range, −1.7 ± 0.5 to −1.3 ± 0.5, valid for the
mass range from 1 -10 m⊙ as described by Scalo (1998).
6
Concluding remarks
For the first time a CCD U BV I photometric survey plus spectral classification have been done in the area of Ruprecht 58 to reveal its fundamental
parameters. Our results find additional support in a proper motion analysis
performed with data taken from UCAC2.
The cluster angular size from star count is close to 9′ corresponding to a linear
size of 10 pc approximately. The mean cluster reddening values are E(B−V ) =
0.33 and E(U −B) = 0.29 . The absorption law towards it is normal with a mean
visual absorption of AV = 1.02 . The cluster is then located at a distance of
3.9 kpc and its age ranges from 250 to 300 Myr according to Girardi et al.
(2000) isochrone curves. It is placed in a galactic zone containing about 60
more clusters with distances between 1 and 5 kpc, with a peak from 1 to
3 kpc. There are also several Hii regions detected in the optical and radio, and
7 Wolf-Rayet stars mostly of C-type (van der Hucht 2001) – all in the first 4
kpc from the Sun. In particular, 2◦ eastwards Ruprecht 58, two Hii regions,
RCW 19 and 20 (H109α source 253.6-0.20) appear at kinematic distances of
3.1 kpc (Georgelin & Georgelin 1976), i.e. in front of the cluster and therefore
with no spatial relation to it. All this would indicate that the cluster is placed
along the Local-arm, probably in the conjunction with the internal
edge of the Perseus spiral-arm (Carraro et al. 2005b; Moitinho et
al. 2006).
The slope of the cluster mass spectrum is x = 1.8, steeper than the typical
mass spectrum of field stars (Salpeter 1955) but comparable and even flatter
than the average of mass spectrum slopes found by Tarrab (1982) for cluster
with ages ranging from 300 to 600 Myr and reasonably close to the slope values
proposed by Scalo (1998) for the mass range 1 to 10 m⊙ .
This publication makes use of data from the Two Micron All Sky Survey funded
by the National Aeronautics and Space Administration and the National Science
Foundation. This article is partially based in the Digitized Sky Survey.
EG, RAV, GRS and RBO acknowledge the financial support from the PIP 5970
(CONICET) and to the IALP-CONICET. Special thanks are given to the CASLEO
staff for the technical support.
16
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