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THE ASTROPHYSICAL JOURNAL, 541:841È848, 2000 October 1
2000. The American Astronomical Society. All rights reserved. Printed in U.S.A.(
THE MAXIMUM AGE OF TRAPEZIUM SYSTEMS
HELMUT A. ABT
Kitt Peak National Observatory,1Box 26732, Tucson, AZ 85726-6732; abt=noao.edu
AND
CHRISTOPHER J. CORBALLY, S.J.
Vatican Observatory Research Group, University of Arizona, Tucson, AZ 85721; ccorbally=as.arizona.edu
Received 1999 December 29; accepted 2000 April 10
ABSTRACT
We sought to determine the maximum age of Trapezium systems by studying possible trapezium
systems that were selected independently of their occurrence in H II regions. We started with the unpub-
lished catalog by Allen, Tapia, & Parrao of all the known visual systems having three or more stars in
which the maximum separation is less than 3.0 times the minimum separation. Their catalog has 968
such systems whose most frequent primary type is F, which does not describe young systems. With a
CCD on the Kitt Peak 0.9 m telescope we obtained UBV frames for 265 systems accessible with our
equipment on Kitt Peak. The frames were used to obtain UBV photometry for about 1500 stars with an
accuracy of ^0.04 mag between V\7 and 14 mag. Also these frames were used to obtain astrometry
with an accuracy of in position angle and in separation. For the brightest star in each^0¡.015 ^0A.01
system we obtained a spectral type to determine the distance and reddening to the system. The measures
were used to determine physical membership from stars that (1) Ðt a single color-magnitude diagram, (2)
Ðt a common color-color diagram, and (3) show no astrometric motion compared to visual measures
made (mostly) a century ago. Combining the results with spectroscopic data for 20 additional Allen et al.
systems by Abt, we found that 126 systems had only optical companions to the primaries, 116 systems
contained only a single physical pair, 13 were hierarchical systems with 3È6 members and having separa-
tion ratios of more than a factor of 10, two were small clusters, and only 28 Ðtted the criteria of Tra-
pezium systems. However, as shown by Ambartsumian, about 9% of the hierarchical systems should
appear to be Trapezium systems in projection. Those, like other hierarchical systems, have a broad dis-
tribution of primary spectral types. We isolated 14 systems that seem to be true Trapezium systems.
They have primary types of B3 or earlier, indicating a maximum age of about 5 ]107yr. This upper
limit is consistent with the estimate made by Allen & Poveda for an age of several million years for these
dynamically unstable systems. These Trapezia are also large with a median radius of 0.2 pc and a
maximum radius of 2.6 pc. We asked why the sample of 285 possible Trapezium systems yielded only 14
true ones, despite the attempt made by Allen et al. to eliminate optical companions with a ““ 1% Ðlter,ÏÏ
i.e., demanding that each companion have less than a 1% chance of being a Ðeld star of that magnitude
within a circle of its radius from the primary. The explanation seems to be that the double star catalogs
are based mostly on BD magnitudes that, fainter than V\12 mag, are systematically too faint by 1
mag.
Subject headings : binaries : visual È open clusters and associations : general È
stars: fundamental parameters
On-line material: additional Ðgures, machine-readable tables
1.INTRODUCTION
Trapezium systems are physical systems of three or more
stars with roughly equal separations. An arbitrary working
rule is that the largest separation in a Trapezium system is
no more than 3 times the smallest separation. In contrast,
hierarchical systems have factors of at least 10 between the
largest and the smallest separations. Typically, a hierarchi-
cal system will consist of a close pair and a distant third
star, or a close pair and distant close pair.
Of course we see these multiple-star systems only in pro-
jection against the plane of the sky. Multiple-star systems
tend to be spherical, rather than coplanar (Worley 1967).
Therefore, a hierarchical system consisting of two close
pairs widely separated could, if they lay nearly along our
line of sight, simulate a Trapezium system. Ambartsumian
1Operated by AURA, Inc., under contract with the NSF.
(1954) called those ““ pseudo-Trapezium systems ÏÏ and esti-
mated in a statistical analysis that about 9% of a sample of
hierarchical systems would appear to be such pseudo-
Trapezium systems.
Trapezium systems are dynamically unstable: the orbits
of their component stars are not closed, and they will evolve
into hierarchical systems or disperse. Through extensive
numerical computer simulations, Allen & Poveda (1974,
1975) have estimated that their maximum ages should be a
few million years.
Trapezium systems, of course, were named after the Tra-
pezium in the Orion Nebula Cluster. The reported expan-
sion (Strand 1957) of the Orion Nebula cluster proved to be
spurious. Allen & Poveda (1974) showed from observations
that no Trapezia in their sample of 33 systems showed any
measurable expansion.
Trapezia have been found in gaseous nebulae
(Ambartsumian 1954; Sharpless 1954). Ambartsumian pro-
841
842 ABT & CORBALLY Vol. 541
vided a catalog of 108 such systems. Their association with
gaseous clouds added evidence that Trapezia are young.
But there is a logical dilemma here. If one looks at young
gaseous nebulae and sees Trapezium systems, that does not
imply that all Trapezia are young. There are many apparent
Trapezia listed in the catalogs of visual double or multiple
stars that are not imbedded in gas clouds.
One goal of this study is to see if physical Trapezia can
occur among old stars that are not in gaseous clouds. What
we did was to study a large sample of possible Trapezium
systems gleaned from visual multiple star catalogs and to
see if there are physical Trapezium systems among old stars.
The second goal is to determine from observational data the
maximum age of Trapezium systems.
Part of this Trapezia search independent of age was done
by Allen, Tapia, & Parrao (1977). They searched through an
early version of the IDS Catalogue (Je†ers, van den Bos, &
Greeby 1963) that had 53,836 systems listed and isolated
968 possible Trapezia. Those were ones with the following
characteristics: each system had three or more stars with a
ratio of the largest to smallest separation within the system
of not more than 3.0. Also they attempted to eliminate
systems that were contaminated with optical components
through the use of a conservative ““ 1% Ðlter.ÏÏ That is, they
required that for each star in the system, the chances of
Ðnding a Ðeld star at that Galactic latitude and longitude
and of that magnitude within the area of the system is less
than 1%.
That catalog of 968 possible Trapezia was never
published because of the unexpected characteristics of those
systems, e.g., a most frequent primary type of F. They rea-
lized that a detailed study of those systems needed to be
made before the reality of all the 968 possible Trapezia
could be believed. This project describes such a study.
We studied as many of the 968 systems as could be
observed from our location and with certain limitations of
magnitudes and separations. To establish the physical
reality of each system we required that each of the following
criteria be met: (1) astrometric measurements with CCD
frames compared to older visual measures must show no
relative proper motions (beyond those that would be com-
patible with orbital motions); (2) the members of each
system should Ðt a single color-magnitude diagram for stars
at the same distance; (3) the stars should Ðt a single color-
color diagram for stars with the same reddening. These tests
required CCD frames in three colors (U,B, and V); those
frames could also be used for the astrometry. We also
needed at least one MK spectral type in each system to
determine the distance and reddening to the system.
2.OBSERVATIONS AND DATA
Of the 968 possible Trapezium systems, we eliminated
those south of [20¡ declination because their minimal
zenith distances of 52¡ would be too large for good photo-
metric measurements. Our CCD equipment with a 0.9 m
telescope had a minimum shutter duration of 1 s, so stars
brighter than V\7 saturated the system. Therefore, we
eliminated all systems containing stars brighter than that
limit. For systems in which all the stars were fainter than
V\11 mag, we felt that the older observations might not
be very reliable. Because the seeing tended to be 1AÈ2A,we
eliminated systems having more than one separation
smaller than 2A. For some systems it was obvious from the
published visual measures that the components were mostly
optical ones, so those optical systems were not measured.
Finally, 31 of these systems had been studied earlier (Abt
1986) with MK spectra of each of the components, so it was
not felt to be necessary to obtain the CCD measurements
for those. We observed 265 systems (see Fig. 1).
The UBV observations were made with the No. 1 0.9 m
telescope at the Kitt Peak National Observatory. At the
f/13.5 focus an RCA 512 ]300 pixel CCD chip gave a Ðeld
of (E-W) by (N-S), which is a convenient size for most4@.1 2@.5
visual multiple star systems. The scale was pixel~1.We0A.48
used 10 moonlit nights distributed during nine months.
Eight of the nights were fully or partially photometric. A red
and a blue equatorial standard (Landolt 1973, 1983) was
observed every hour. Generally, each night such a pair of
stars was observed from low to high air mass and one from
high to low air mass. Both the standard stars and the Tra-
pezia Ðelds were observed sequentially through the U,B,
and VÐlters. Exposure times ranged from 1 s to several
minutes (for the Uframes).
The journal of observations is given in Table 1. The large
number of CCD frames obtained each night shows both the
efficiency of this program and the need for a batch approach
to the reductions. The last column of Table 1 gives the sky
quality.
2.1. Photometric Data
Aperture photometry was performed on each CCD frame
with the IRAF2data reduction package APPHOT. This
requires an input of starting star coordinates and the
FWHM of the images. The latter was determined by plot-
2IRAF is distributed by the National Optical Astronomy Observa-
tories, which is operated by Aura, Inc., under contract with the National
Science Foundation.
TABLE 1
JOURNAL OF OBSERVATIONS
SFWHMT
Night UT Date (pixels) Frames Conditions
1 ....... 1987 Sep 6 4.0 66 Clouds early
2 ....... 1987 Sep 7 4.6 145 Clear
3 ....... 1987 Sep 8 4.6 174 Clear
4 ....... 1987 Dec 3 4.0 159 Clear
5 ....... 1987 Dec 6 4.2 105 Clouds Ðrst half
6 ....... 1987 Dec 7 3.2 208 Clear
7 ....... 1988 May 27 5.0 123 Clear
8 ....... 1988 May 28 5.0 136 Windy, var. seeing
9 ....... 1988 May 29 5.5 76 First half only; poor seeing
10...... 1988 May 31 4.5 133 Clear
No. 2, 2000 MAXIMUM AGE OF TRAPEZIUM SYSTEMS 843
FIG. 1.ÈCopies of the Vmagnitude CCD frames for 266 Trapezium Ðelds. The stellar designations or our arbitrary ones correspond to the Ðrst column
entries in Table 3. A few Ðelds, e.g., Trap 156, have moonlight contamination. Additional panels of this Ðgure appear in the electronic edition of The
Astrophysical Journal.
ting a selection of stellar image proÐles and settling on a
mean FWHM for each. That mean is a measure of the
average seeing and the values are listed in Table 1. Then
APPHOT computed accurate centers, sky background
values, and magnitudes and errors, whose computational
details are in the package speciÐcations (Davis 1987). A
series of apertures in units of the FWHM were used and a
corresponding series of magnitudes were generated. We
determined that an aperture of 1.5 times the FWHM was
the best compromise between the goals of maximizing the
number of star pixels processed while separating the
occasional crowded images. Some images were still too
blended to be separable. This had to be accepted because
procedures such as DAOPHOT (Stetson 1987), which use
the point spread function to deblend images, either would
be too labor intensive or lacked sufficient images to be
applied.
The aperture of choice, 1.5 times the FWHM, proved to
give magnitudes just a factor of 1.0042 ^0.0015 fainter than
a wide aperture of 4 times the FWHM. Therefore, a zero-
point correction from a Ðnite to an inÐnitely wide aperture
was not necessary for our desired accuracy. This decision is
conÐrmed by Massey et al. (1989), who found that a Ðxed-
size CCD aperture, extended beyond a certain size, merely
844 ABT & CORBALLY Vol. 541
excluded the same fraction of light. The chosen aperture
was also insensitive to air mass between 1.1 and 2.0 at about
half this level (1.0019 ^0.0007). This implies that di†erences
in seeing resulting from di†erent air masses would have
negligible e†ects on the photometry. These conclusions
were conÐrmed during the reduction of the magnitudes to
the UBV system when residuals from di†erent apertures
were compared. Somewhat surprisingly, when taking the
Ðrst three nights and all three colors together, the mean of
the residuals doubled from ^0.013 mag for the 1.5 FWHM
aperture to ^0.026 mag for the 3.0 FWHM one. The
increasing sky counts (in moonlight) with the larger aper-
ture may have o†set the increased star counts.
The reduction of instrumental magnitudes to the UBV
system was accomplished by Peter StetsonÏs CCD photo-
metry calibration package, CCDCAL, whose characteristics
are described by Massey et al. (1989) and by Cook &
Aaronson (1989). Because the programs were devised for a
single Ðeld, we modiÐed them for the multiple Ðelds of the
Trapezium systems while retaining the integrity of the cal-
culations. P. Massey (1999, private communication)
expressed some caution with using the residuals calculated
in the program that produces the transformation equations,
CCDSTD. Therefore, the same standard stars were run
through the program to process the program stars,
CCDCAL. Only a minor disparity was detected in the
residuals from either program, one which grew for nights
when fewer standards were observed. The programs have
two special features: one is to reduce the observational data
in the form of magnitudes, rather than photometric indicies,
which suits CCD integration times; the other is to leave the
forms of the transformation equations open to deÐnition,
and so di†erent e†ects can be explored. The general form of
the transformations were
u\U]A0]A1(U[B)]A2X]A3T]A4T2, (1)
b\B]B0]B1(B[V)]B2X]B3T]B4T2, (2)
v\V]C0]C1(B[V)]C2X]C3T]C4T2, (3)
where Xis the air mass and Tis the time of the observation
(in decimal hours) since the start of the night.
Initially, the time of observation was not used as a
parameter in the transformation equations. However, on
seven nights the calculated residuals showed a drift with
time, often with a more rapid drift at the beginning of the
night. This has been noticed at Kitt Peak by Stetson &
Harris (1988). First- or second-order terms in time were
signiÐcant in improving the transformations for seven
nights.
Massey (1985) has described the error in the timing of
CCD shutters which will a†ect the short exposure frames.
For KPNO shutters that error is about 1% for our shortest,
1sVexposures. We also explored this during our
reductions by including a term for exposure time but that
gave negligible improvements.
The SIMBAD database was searched for UBV photo-
metry of stars within the Ðelds of the observed Trapezium
systems. While the SIMBAD data are of variable quality
and authorships, our agreement with the published photo-
metry gives an external check. The di†erences (CCD-
SIMBAD) are [0.014 ^0.043 in V,]0.013 ^0.034 in
B[V, and [0.034 ^0.064 in U[B. The systematic di†er-
ences are smaller than the errors and the errors should be
considered as the sums (added as squares) in our photo-
metry plus that in SIMBAD.
Another way to estimate the photometric accuracy is to
compare the 89 program stars that were measured more
than once. Such a comparison is independent of errors in
SIMBAD. They show that for stars brighter than V\14,
the dispersions in Vare independent of brightness at
*V\^0.043 ; for B[Vthey are nearly constant at
*(B[V)\^0.022 and U[Bthey are nearly constant at
*(U[B)\^0.041. A sample graph is shown in Abt &
Corbally (1997). However, fainter than V\14 the errors
grow rapidly to ^0.2 at V\15. Therefore, in the data
listed in Table 3 (below), we have quoted values to only one
decimal place between V\14.0 and 16.0, and we have
deleted all photometric data for stars fainter than V\16.0.
2.2. Astrometric Data
The APPHOT reduction process produced (x,y) coordi-
nates in terms of pixel positions in each frame and color. We
searched the SIMBAD database for 20 wide visual pairs
having various accurate separations (range of 11AÈ82A) and
orientations; these were observed fairly evenly through the
observing runs. These allowed us to determine the scale and
orientation of the pixel frames. We used the method of
Bertiau & De Graeve (1967) as summarized in Bertiau &
Fierens (1977). The resulting scale was 0A.482 ^0A.011
pixel~1 for all three Ðlters, all nights, and all separations.
We used two ways to determine the orientation angle of
the frames with respect to the celestial axes. First, during
two runs we double exposed a Trapezium Ðeld with a right
ascension shift between the exposures. Pairs of images for
the same star deÐne a chord perpendicular to the decli-
nation axis, providing that the telescope is accurately
aligned. The second method involved using the same
material as for the scale determination. The Ðrst method
gave an error of ^0.002 radians or It was found that^0¡.1.
this Ðrst method was about 10 times more sensitive than the
second method.
With the pixel scale and rotation angles determined, the
position angles and separations of all images in our Ðelds
were calculated. These were done for the components listed
in the IDS so that changes in position angles and separa-
tions would be apparent. We generally averaged the results
from all three Ðlter frames, unless the stellar brightness
(generally for faint stars) through one of the Ðlters gave
deviant results.
For 69 pairs of stars that were observed on two nights, we
compared the di†erences in position angles and separations.
For stars brighter than V\10, the mean error in position
angle was it was less than to V\14. The^0¡.015; ^0¡.1
mean error in separation was for stars brighter than^0A.01
V\11 and remained less than to V\14. The error^0A.05
dependence upon stellar magnitudes is shown in Abt &
Corbally (1997).
We compare in Table 2 the accuracy of these CCD
astrometric measures with those of experienced double-star
observers. We selected a random sample of 20 double stars
with separations between 3Aand 12Ameasured visually 81
times by Van Biesbroeck (1974). His mean accuracy was
in position angle and in separation. We also^1¡.5 ^0A.10
collected data for 20 pairs with separations of 3Ato 7Aand
measured visually 63 times by Worley (1989). His accuracy
in position angle was and in separation it was^0¡.99
For repeated visual measures in BurnhamÏs com-^0A. 08.
No. 2, 2000 MAXIMUM AGE OF TRAPEZIUM SYSTEMS 845
TABLE 2
ACCURACIES OF ASTROMETRIC MEASUREMENTS
ACCURACY
Position Angle Separation
SOURCE TECHNIQUE (deg) (arcsec)
Van Biesbroeck 1974 ...... Visual ^1.5 ^0.10
Worley 1989 ............... Visual ^0.99 ^0.08
Burnham 1906 ............. Visual ^0.67 ^0.41
Josties et al. 1978 .......... Photographic ^0.053 ^0.069
This study .................. CCD ^0.015 ^0.01
pilation (Burnham 1906) by various observers for stars with
separations greater than 3Abut with an average of 36A, the
accuracy in position angle is and in separa-^0¡.67 ^0A.41
tions. Finally, we note the accuracy of photographic mea-
sures by Josties et al. (1978). Their mean accuracy were
in position angle and in separation. The^0¡.053 ^0A. 069
purpose of this comparison is not simply to show that the
CCD measures were 10È100 more accurate than the visual
measures or 3È7 times more accurate than the photographic
measures, but to determine to what extent we can trust
older astrometric measures with which we compare the new
ones.
2.3. Spectral ClassiÐcation
The spectral types for the stars brighter than B\11 were
observed photographically by the Ðrst author with the Kitt
Peak 2.1 m telescope and Cassegrain spectrograph as
described by Abt (1986). The early-type standard stars were
selected from Morgan, Abt, & Tapscott (1978) and the late-
type ones by Morgan & Keenan (1973) and Keenan &
McNeil (1976). The classiÐcation accuracy was about ^1
spectral subclass and ^0.7 luminosity classes.
The stars fainter than B\12 were observed by the
second author with a CCD on the Steward Observatory 2.3
m telescope and Cassegrain spectrograph. He compared
visual scans with those of standard stars from the same
sources. Between B\11 and 12, at least eight stars were
observed with both telescopes and the small di†erences in
results were reconciled.
3.FINAL DATA AND INTERPRETATIONS
Our procedure involved classifying at least one star in
each group to obtain the approximate distance and the
reddening to each group. We compared for those stars our
photometry with BlaauwÏs (1963) absolute magnitudes and
FitzGeraldÏs (1970) colors for stars of the same MK types.
The distance moduli are good only to roughly ^1 mag
because of the intrinsic width of the main sequence (or
scatter in absolute magnitudes for each luminosity class).
The reddening values are probably valid to ^0.03 mag.
Then with those distance moduli and reddening, we Ðtted
the observed color-magnitude diagram [Vvs. and(B[V)0]
color-color diagram vs with those of[(U[B)0(B[V)0]
FitzGerald (1970). Stars that were more than 1.5 mag low
(or high) in the color-magnitude diagrams were considered
to be background (or foreground) optical companions.
Stars that fell more than ^0.1 mag o† in the color-color
diagrams were considered to have signiÐcantly di†erent
reddening and were thought to be background or fore-
ground optical companions. Illustrations of such sample
diagrams are shown in Abt & Corbally (1993). Finally, we
compared the current position angles and separations with
the published ones in the WorleyÏs Washington Catalog
(US Naval Observatory), the on-line successor to the IDS.
Those measurements were typically made a century ago.
Stars that showed a motion more than 1¡ in position angle
or more than in separation were considered to be non-0A.5
members except for very nearby systems where the orbital
motions in a century may be that large.
The data and results for each group are given in Tables
3A and 3B, which are given in full only in the electronic
edition of this journal. These tables are complex and need
explanation. In Table 3A we give for each group the group
name (e.g., Trap. 1) and the 1900 right ascension and decli-
nation of the central star (e.g., and 43¡46@). The00h06m.0
assumed values are taken from the Washington Catalog or
IDS. That is followed by the ADS number (e.g., ADS 137), if
it has one. That is followed by the conclusions about the
group, e.g., components 1 and 2 form a physical pair, the
reddening mag, and the dis-R\(B[V)[(B[V)0\0.06
tance modulus mag.d\V0[M\8.8
The data in Table 3B contain the following information.
First is the Trapezium group number with the star number
and the identifying letter in the ADS catalog or elsewhere.
Then we list our value of the visual magnitude to two sig-
niÐcant Ðgures. For stars between V\14.0 and 16.0 we
include only one decimal place, and no photometry is given
for stars fainter than V\16.0. Then we give our B[Vand
U[Bcolors, again truncating to one decimal place for stars
fainter than V\14.0. Following that is our MK classi-
Ðcation. That is followed by the date of the published
astrometric data where the separation is given in arcseconds
and the position angle in degrees. Each set of measures refer
to the star designated on that line relative to the Ðrst star or
the one marked ““ A,ÏÏ e.g., in Trap 1 star 2 \Bhas a separa-
tion of from A and at a position angle of 332 deg. A ““ vÏÏ9A.6
following any entry indicates that the old measures show
that quantity to be variable. That information is followed
by the new measures where the time is in decimal years. The
measures of separations in arcseconds and position angles
in degrees are given to two decimal places each. The Ðnal
entry gives our conclusion about the physical association of
each component with, usually, the Ðrst star or the one
marked ““ A.ÏÏ If the star deviated in the color-magnitude
diagrams, we could identify it as a background (““ backgr.ÏÏ)
or foreground (foregr.ÏÏ) optical companion. If it deviated in
the astrometric measures or color-color diagrams, we called
it a ““ nonmember.ÏÏ A few marginal cases were called pos-
sible members (““ poss memÏÏ).
846 ABT & CORBALLY Vol. 541
TABLE 3A
CONCLUSIONS ABOUT THE POSSIBLE TRAPEZIUM SYSTEMS
Trap Number R.A. (1900) Decl. (1900) ID Conclusions Reddening Distance
Trap 1 ........ 00 06.0 43 46 ADS 137 1, 2 physical pair 0.06 8.8
Trap 2 ........ 00 06.3 29 15 ADS 141 1, 2 physical pair 0.06 5.0
Trap 3 ........ 00 06.6 19 19 BD]19 15 1, 2 poss phys pair 0.09 10.2
Trap 4 ........ 00 09.9 59 13 ADS 192 1, 2 physical pair 0.54 10.5
Trap 6 ........ 00 17.4 61 41 ADS 307 All optical 0.40 10.5
Trap 12 ...... 00 40.1 31 06 HD 4279 1, 2 physical pair 0.23 7.9
Trap 13 ...... 00 47.0 56 05 ADS 719 1A, 1B, 2, 3 trap system 0.36 10.9
Trap 17 ...... 00 59.9 12 18 ADS 893 1, 2 physical pair 0.03 6.6
Trap 20 ...... 01 05.2 51 46 ADS 970 1, 2, 7 hierarchical 0.25 6.6
Trap 21 ...... 01 06.4 62 07 ADS 984 All optical 0.36 8.9
Trap 23 ...... 01 10.9 13 12 HD 7604 All optical 0.01 9.0
Trap 25 ...... 01 17.1 [26 16 CD[26445 1, 2 physical pair [0.08 11.6
Trap 27 ...... 01 18.9 27 04 ADS 1119 1, 3 physical pair 0.03 4.4
Trap 29 ...... 01 23.4 05 43 HD 8989 All optical 0.60 6.1
Trap 31 ...... 01 23.8 24 30 BD]24 221 All optical 0.15 9.0
Trap 38 ...... 01 40.5 [02 54 BD[03 251 1, 2 physical pair 0.02 5.7
NOTE.ÈTable 3A is published in its entirety in the electronic edition of T he Astrophysical Journal. A portion is shown here
for guidance regarding its form and content.
To these results for 265 systems, we will add the results
from Abt (1986) that used no modern photometry or new
astrometric measures, but that used MK classiÐcations for
all the stars in each group. Of the 31 stars in that program,
11 were measured this time but we include the results for 20
other systems in the Allen et al. (1977) catalog that included
stars that were too bright to measure with the CCD system.
Nearly half (44%) of the 285 systems were ones in which
the primary star had no physical companions (““ all
optical ÏÏ), although it is possible that there was a physical
pair among the fainter stars; we did not search carefully for
such possible pairs. The reason why so many of these pro-
posed Trapezium systems turn out to be optical systems,
despite the use of the 1% Ðlter, will be discussed below.
Nearly half (41%) of the remaining systems have only a
single physical pair (or possible physical pair (““ poss phys
pair ÏÏ) and therefore also are not Trapezium systems. The
results for all 285 systems are as follows: 126 systems have
only optical companions to the primaries, 116 systems
contain only one physical pair, although one (Trap. 619) has
two pairs at very di†erent distances, 13 systems are hierar-
chical ones of 3È6 members with separations ranging over
more than a factor of 10, two systems are small clusters with
8 and 12 members, and 28 systems are apparent Trapezium
systems according to the above deÐnition.
The primaries of the 126 optical systems have a distribu-
tion of primary types (without regard to luminosity classes)
that resembles that of the Henry Draper stars (Allen 1973)
within the errors involved. That is to be expected because
these are random stars that, by chance, happened to have
three or four background or foreground stars along the
same lines of sight. The mean spectral type is F4 ^1 for the
optical primaries compared with about F7 for the HD stars.
The primaries of the 116 systems having only one physi-
cal pair have a distribution of primary types that again
resembles the HD distribution; the mean type is F4 ^1.
TABLE 3B
OBSERVATIONAL DATA
Star VB[VU[BType Old Dates Sep. P.A. New Date Sep. P.A. Member ?
Trap11A...... 9.26 0.23 0.13 A5 III(n) 1893 1938 1987.683
Trap12B...... 11.04 0.28 0.10 1893 1938 9.6 332.0 1987.683 9.80 332.78 Member
Trap13C...... 13.89 0.50 0.08 1893 1938 BC 4.8 349.0 1987.683 5.58 350.57 Backgr.
Trap 1 4......... 12.98 0.31 0.08 1893 1938 1987.683 79.20 207.70 Backgr.
Trap21A...... 10.04 0.68 0.05 G1 V 1926 1935 1987.683
Trap22B...... 10.33 0.73 0.13 1926 1935 14.5 359.0 1987.683 14.23 359.03 Member
Trap23C...... 12.60 0.55 [0.16 1926 1935 BC 38.4 351.0 1987.683 37.28 344.28 Backgr.
Trap24D...... 13.49 0.81 [0.15 1926 1935 CD 5.2 141.0 1987.683 6.11 139.61 Backgr.
Trap31A...... 9.86 1.62 2.02 K7 III 1912 1926 1987.683
Trap32B...... 11.58 1.08 0.75 1912 1926 12.5 236.0 1987.683 12.24 237.81 Member
Trap33C...... 14.4 0.6 0.1 1912 1926 14.0 50.0 1987.683 15.64 46.78 Nonmem.
Trap 3 4......... 13.97 0.32 0.04 1912 1926 1987.683 56.20 249.18 Backgr.
Trap41A...... 8.06 0.63 0.32 A4 II 1876 1928 1987.686
Trap42B...... 11.62 0.45 0.26 1876 1928 20.9 125.0 1987.686 21.77 125.63 Member
Trap 4 3......... 13.48 0.94 0.01 1876 1928 BC 10.0 333.0 1987.686 10.26 226.84 Foregr.
Trap 4 4......... 13.08 0.98 [0.29 1876 1928 1987.686 78.86 147.66 Foregr.
NOTE.ÈTable 3B is published in its entirety in the electronic edition of T he Astrophysical Journal. A portion is shown here for guidance regarding its
form and content.
No. 2, 2000 MAXIMUM AGE OF TRAPEZIUM SYSTEMS 847
Similarly, the primary types for the 13 hierarchical systems
and two clusters again has the same distribution, although
the sample size is rather small for statistics, as the HD stars
and a mean type of A9 ^4. However, the 28 apparent Tra-
pezium systems di†er markedly in that two-thirds of the
primaries are of types O and B, and the mean is A1 ^3.
Table 4 gives a list of the 28 apparent Trapezium systems
arranged in order of their primary types. One star (Trap.
837A) lacks a spectral type but its colors and reddening
indicate that it is of type A2.
However, we must remember that some hierarchical
systems will appear to be Trapezium systems in projection.
Ambartsumian (1954) estimated that 9% of a sample of
hierarchical systems will be such ““ pseudo Trapezium
systems.ÏÏ For the 285 proposed systems minus the 126
optical systems, the sample of 159 should have 14 pseudo
Trapezium systems. But which 14 of the stars listed in Table
4 are the pseudo systems and which are the true Trapezium
systems?
As we found above for other hierarchical systems, the 14
pseudo systems should have a spectral type distribution
similar to the Henry Draper stars. Thus, we would expect
zero O stars, two Bs, three As, three Fs, two Gs, four Ks,
and zero Ms. In fact, a random sampling of 2200 HD stars
shows that nearly all the listed B stars are B8 or B9; sta-
tistically the two B stars should be late Bs. Within the sta-
tistical accuracy of these small numbers, the pseudo
Trapezium systems easily account for all the systems listed
in Table 4 with primaries later than B plus three of the B
stars, and those should be the late B stars.
The last Ðve B star systems in Table 4 are o† the main
sequence. Let us trace them backward to their main-
TABLE 4
APPARENT TRAPEZIUM SYSTEMS
Trap. Number Primary Spectral Type Number of Members
13.............. O5 V 4
857 ............ O7 V 4
900 ............ O8 V 6
870 ............ O9 V 3
761 ............ B0 Ib 3
593 ............ B0.5 III 3
177 ............ B1 V 4
748 ............ B1 Vn 6
754 ............ B1 Vn 3
49.............. B2 Vn 3 ]6 hier.
511 ............ B2 V 5
951 ............ B3 V 7
133 ............ B6 IIIp 5
657 ............ B8.5 IbÈII 4
600 ............ B9 Ia 5
120 ............ B9 IIa? 4
239 ............ B9 II 3 ]3 hier.
305 ............ A0È2 III 3 ]2 hier.
130 ............ A2 Vb 3
837 ............ (A2) 3
655 ............ A5 V 4
357 ............ Am(H :F0) 3
840 ............ F2 III 4
687 ............ F2 V 3
110 ............ G7 III 3
320 ............ G8 III 4
351 ............ G9 III 3
756 ............ K1 III 3
sequence origins by assuming the absolute magnitudes by
Blaauw (1963) and assuming 1.5 mag brightening in bolo-
metric magnitude since the main sequence. Then the orig-
inal main-sequence types were B7, B3, B1, B4, and B4,
respectively. Therefore, by deleting the three latest B stars
(B7, B4, and B4), we Ðnd that all of the true Trapezium stars
have main-sequence types of B3 or earlier. The list of true
Trapezium systems are the Ðrst 12 systems in Table 4 plus
Trap. 657 and 600.
Thus, the main-sequence age of a B3 star is the maximum
observed age of a Trapezium system. The Trapezia may be
younger because the B3 V primaries may be near the zero-
age main-sequence, rather than at the maximum age for
main-sequence B3 stars.
What is the maximum age of a B3 V star? An early
calibration (Sandage 1958) gave 3 ]107yr. For the alpha
Persei cluster with an earliest type of B3 V, Mermilliod
(1981) derived 5 ]107yr and Prosser (1992) derived
8]107yr. The models by Maeder & Meynet (1988) and
Bertelli et al. (1994) give ages of 3È7]107yr. A reasonable
maximum age for a B3 V star seems to be about
5^2]107yr. This maximum age is consistent with the
age of a few millions years derived by Allen & Poveda (1974,
1975) from numerical simulations of the dynamical lifetime
of these unstable systems. We know of no more recent or
more detailed simulations for Trapezia.
In summary, we have looked at many multiple-star
systems that looked like Trapezia, rather than hierarchical
systems, and showed that the observations are consistent
with the expectation from numerical simulations that these
systems must be young.
These Trapezium systems are large in size as we can
expect from their relatively faint apparent brightnesses, high
intrinsic brightnesses for OB stars, and systems with dimen-
sions of the order of 1@. The 14 true Trapezium systems have
a median radius to the farthest outlying member of 40,000
AU \0.2 pc and a maximum radius of 535,000 AU \2.6
pc. This maximum is approximately the dimension of an
open cluster. These dimensions are also consistent with
values derived earlier (Abt 1988).
Finally, why did we Ðnd only 14 true Trapezium systems
out of an original sample of 285 systems proposed by Allen
et al. (1977), despite their serious attempt to eliminate
optical companions with their 1% Ðlter? The answer seems
to be that the Bonner Durchmusterung visual estimates of
magnitudes used in the Washington Catalog are systemati-
cally in error.
A comparison for the Ðrst 300 stars between our CCD V
magnitudes and the BD Vmagnitudes shows systematic
TABLE 5
SYSTEMATIC ERRORS IN BONNER DURCHMUSTERUNG MAGNITUDES
STANDARD ERROR
VCCD VBD Per Star Per Mean NUMBER OF STARS
7.52 ....... 7.12 ^0.37 ^0.10 14
8.51 ....... 8.38 ^0.36 ^0.06 34
9.54 ....... 9.58 ^0.51 ^0.08 42
10.48 ...... 10.76 ^0.46 ^0.07 51
11.49 ...... 12.08 ^0.75 ^0.11 49
12.45 ...... 13.35 ^0.86 ^0.11 64
13.42 ...... 14.60 ^1.06 ^0.17 39
14.29 ...... 16.21 ^1.10 ^0.45 7
848 ABT & CORBALLY
di†erences that have been graphed by Abt & Corbally
(1997). The numerical values are given in Table 5. The table
shows that for magnitudes 7 and 8, the BD magnitudes
average several tenths of a mag too bright. At V\9 the
magnitudes are correct but with a large scatter of ^0.5
mag. However, for fainter magnitudes the BD values are
about 1 mag too faint. That means that when Allen et al.
(1977) used contemporary statistics on the numbers of stars
of, say, V\13, within a given area of the sky to apply their
1% Ðlter, they were allowing too many companions of
to get through.VBD \14
We thank C. Allen and A. Poveda for providing a copy of
their unpublished catalog and for suggesting that we
explore the nature and ages of an unbiased sample of Tra-
pezium systems. We appreciate much help with the photo-
metric reductions from Lindsey Davis, Phillip Massey, and
Nigel Sharpe.
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