Download Period-Luminosity Relations for delta Scuti Stars

Pontificia Universidad Católica de Chile
Facultad de Fı́sica
Departamento de Astronomı́a y Astrofı́sica
Period-Luminosity Relations
for delta Scuti Stars
by
Camila Andrea Navarrete Silva
Practice report presented to the Physics Faculty
of Pontificia Universidad Católica de Chile,
as one of the requirements for the
Bachelor degree in Astronomy.
Supervisor :
Correctors :
Dr. Márcio Catelan (PUC, Chile)
Dr. István Dékány (PUC, Chile)
Dr. Radostin Kurtev (UV, Chile)
July, 2011
Santiago, Chile
c
2011,
Camila Andrea Navarrete Silva
Agradecimientos
Le quiero agradecer a mi profesor guı́a, Dr. Márcio Catelan, por su tiempo,
paciencia, sugerencias y por su apoyo durante todo el transcurso de esta práctica.
Al Dr. Rodolfo Angeloni por su constante apoyo y por los consejos que me dio para
mejorar este trabajo. A los Dres. István Dékány y Javier Alonso por sus útiles ideas
y comentarios. A Paloma Pérez por su tiempo y ayuda desinteresada.
A mi madre, Marina, por su amor, paciencia, confianza, apoyo, por creer en mı́
más que nadie y por incentivarme a estudiar lo que yo quisiera. A mi hermana
Carolina porque cada año su paciencia y empatı́a aumentan con lo, de a poco, se ha
vuelto una amiga.
Quiero agradecerle también a mis amigas Marı́a José, Alice, Natalia y Nicole por
todos sus buenos deseos, su inmenso apoyo y porque a pesar de los años y mi escasa
disponibilidad nuestra amistad sigue intacta. En especial le agradezco a Natalia por
su compañı́a y su fundamental ayuda durante el último mes de este trabajo.
A mis compañeros y hoy amigos: Felipe, Cristóbal, Pablo, Enrico, Matı́as, Johanna, Ignacio, Simón y Antonio por todo el tiempo y momentos especiales que
hemos pasado juntos. Un agradecimiento especial a Johanna por su ayuda y buena
voluntad siempre, pero sobre todo durante los últimos dı́as de ésta práctica. A Carol,
Karina y Rosario porque estos años no habrı́an sido los mismos sin compañeras como
ustedes. A Pedro Salas por ayudarme, de manera desinteresada, a aprender un poco
más de Python al igual que Néstor Espinoza a quien además le agradezco por sus
siempre útiles crı́ticas y comentarios.
Por último, agradezco el financiamiento del Proyecto Fondecyt Regular Nro.
1110326 y del Núcleo Milenio para la Vı́a Láctea (Programa Iniciativa Cientı́fica
Milenio, P07-021F).
i
Contents
Agradecimientos
i
Resumen
iv
Abstract
v
Introduction
1
1 Observations and data reduction
1.1 Observations . . . . . . . . . . .
1.2 Open clusters . . . . . . . . . .
1.2.1 NGC 7062 . . . . . . . .
1.2.2 NGC 1817 . . . . . . . .
1.2.3 NGC 6134 . . . . . . . .
1.3 Data reduction . . . . . . . . .
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9
10
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12
13
2 Photometric reductions
15
2.1 Photometry with DoPHOT . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Color-magnitude diagrams . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Light curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Period-Luminosity relations
26
3.1 Magnitude calibration . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2 PL relations for NGC 7062 and NGC 6134 . . . . . . . . . . . . . . . 28
3.3 NGC 1817: Fundamentalized periods . . . . . . . . . . . . . . . . . . 34
4 Theoretical approach
37
4.1 Visible bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
ii
CONTENTS
4.2
4.3
Infrared bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wavelength dependence of PL slope . . . . . . . . . . . . . . . . . . .
40
41
5 Summary and future work
45
A Tables
47
Bibliography
55
iii
Resumen
Se derivaron relaciones Perı́odo-Luminosidad (PL) en las bandas J, H y Ks del
infrarrojo cercano para estrellas δ Scuti de baja amplitud de tres cúmulos abiertos:
NGC 7062, NGC 1817 and NGC 6134. Se usaron observaciones de series de tiempo
y fotometrı́a diferencial para obtener las curvas de luz y perı́odos; sin embargo los
datos no tienen la precisión requerida para detectar la variabilidad intrı́nseca de las
estrellas por lo que series temporales de alta precisión son necesarias. No obstante
lo anterior, se pudieron calcular magnitudes promedio con baja dispersión para cada
estrella variable. Las magnitudes fueron transformadas al sistema fotométrico de
2MASS. Usando los periodos detectados en estudios previos, dos relaciones PL fueron
encontradas tanto para NGC 7062 como para NGC 6134. Para cada relación se
sugirió un correspondiente orden radial n. Para NGC 1817 no fue encontrada ninguna
relación porque las variables observadas tienen diferentes modos de pulsación. En
este caso, los perı́odos fueron fundamentalizados y las relaciones PL para el modo
fundamental fueron derivadas. Además, relaciones PL teóricas aproximadas fueron
calculadas para las bandas BVIRJHK usando valores tabulados de luminosidades y
periodos y correcciones bolométricas derivadas a partir de modelos. Comparando
los valores observacionales y teóricos de las pendientes, las relaciones PL para NGC
6134 parecen ser las más consistentes. Las relaciones teóricas muestran que mientras
la longitud de onda efectiva aumenta, la pendiente y la dispersión ed la relación
decrecen, como se ha señalado para las Cepfeidas por Madore & Freedman (1991) y
por Freedman et al. (2008). Esto sugiere que, en las bandas infrarrojas, la relación
PL es más bien una relación perı́odo-radio. Además, las relaciones PL teóricas y
observacionales derivadas en este trabajo tienen pendientes más negativas que las
calculadas previamente por King (1990).
iv
Abstract
Period-Luminosity (PL) relations in the near-infrared J, H and Ks bands were
derived for low-amplitude δ Scuti stars in the open clusters NGC 7062, NGC 1817
and NGC 6134. Time-series observations and differential photometry were used to
get light curves and periods, but the data do not have the precision required to
detect the intrinsic variability of the stars; high-precision time series are required.
Nevertheless, mean magnitudes with low dispersion were derived for each variable.
Magnitudes were transformed to the 2MASS photometric system. Using the periods
detected in previous studies, two PL relations were found for both NGC 7062 and
NGC 6134. For each relation, a radial order n was suggested. For NGC 1817 a
relation was not found because the variables have different pulsation modes. In this
case, the periods were fundamentalized and PL relations for the fundamental mode
were derived. Also, approximate theoretical PL relations were found for BVIRJHK
bands using tabulated values for luminosities and periods, and bolometric corrections
derived from models. Comparing observational and theoretical slope values, PL
relations for NGC 6134 seem to be the most consistent. Theoretical relations show
that while effective wavelength increases, the slope and dispersion of the relation
decrease, as noted for Cepheids by Madore & Freedman (1991) and by Freedman et
al. (2008). This suggests that the PL relation is rather a period-radius relation in
infrared bands. Also, both theoretical and observational PL relations derived in this
work have slopes that aremore negative than the ones derived previously by King
(1990).
v
Introduction
A very noticeable and useful property of pulsating stars is the existence of a
Period-Luminosity (PL) relation. The most common and used is the PL relation
for Classical Cepheids, which was derived for the first time by Henrietta Leavitt
(Leavitt, 1912) and since then has played a leading role in the establishment of the
scale of the Universe, because these objects are among the primary indicators of the
scale of extragalactic distance.
PL relations for delta Scuti stars have been established in the V band (since
Fernie, 1964, and lately by Petersen & Christensen-Dalsgaard, 1999, Santolamazza
et al., 2001, Pych et al., 2001 and Templeton et al., 2002), VR band (Garg et al.,
2010) and recently in VI-Wesenheit-index (Majaess et al., 2011). As for Cepheids, the
delta Scuti PL relation has proved to be a useful and reliable method for measuring
distances. Despite the fact that more precise PL relations are expected in infrared
bands than the ones derived in visible bands, only one attempt to derive the relation
in the infrared has been carried out to date (King, 1990).
The observational advantages of delta Scuti stars are their shorter periods compared to Cepheids, a consequence of their smaller radii, and also, as Breger (1979)
noted, they are the second most common and numerous group of pulsators in the
Galaxy, after the pulsating white dwarfs. A tight PL relation, in addition to these
characteristics, would allow us to compare observations with pulsation theory and
to determine luminosities and distances of stars, open and globular clusters, nearby
galaxies and the Galactic center where delta Scuti stars have been observed (McNamara et al., 2000, 2007; Poleski et al., 2010). This would lead to an improvement of
the galactic distance scale.
1
INTRODUCTION
Delta Scuti stars: General properties
Delta Scuti (δ Scuti or DSCT) are intrinsic variable stars of pulsating type. Due to
their kind of variability and their position in the instability strip, they are considered
to be part of the Cepheid family. Table 1 has the typical designation for the different
stars in this group, their range of periods and population type. Figure 1 shows the
position of the main types of the pulsating stars in the Hertzsprung-Russell diagram.
Table 1: Cepheid family. Adapted from Allen (2000).
IAU designation
DCEP
DSCT
SXPHE
RR
CW
Name
Population
Classical Cepheids
I
δ Scuti
I
SX Phoenicies
II
RR Lyrae stars
II
W Vir + BL Her stars
II
Period (days)
1.5-60
0.04-0.2
0.03-0.08
0.2-1
1-80
δ Scuti are stars of spectral type of A0 to F5 with masses between 1.5 M and
2.5 M for stars with solar metal abundances and between 1.0 and 2.0 solar masses
for metal-poor stars. They have luminosity classes between V (dwarf) and III (subgiant). Most of them belong to Population I but a few variables show metallicities
and space velocities typical of Population II. They have brightness fluctuations from
0.003 to 0.9 magnitudes in V, with a typical amplitude of 0.02 mag (Breger, 1979).
The excitation mechanism of δ Scuti stars is the κ mechanism, the same as that
for other stars in the classical instability strip. The driving zone is believed to be in
the He II and H ionization zones (Stellingwerf, 1979). The excitation of pulsation in
these zones, in particular the He II ionization zone with temperatures over 40000 K,
is enough to counterbalance the damping in the underlying layers (Breger, 2000).
To date, more than 600 δ Scutis have been found in our galaxy, including members
of clusters and field stars. An extensive catalogue and review of the statistical
properties of the known δ Scuti stars can be found in Rodrı́guez & Breger (2001).
2
INTRODUCTION
Figure 1: Schematic Hertzsprung-Russell diagram illustrating the location of several
classes of pulsating stars. The dashed line marks the zero-age main sequence, ZAMS,
where the Solar-like pulsating variables and γ Doradus stars are located, with the lowest
luminosities. Above them, the Cepheid Instability Strip is situated, delimited by the two
dashed lines that intersect with the ZAMS. Inside of it there are δ Scuti stars with the
subgroup of rapidly oscillating Ap stars (roAp) RR Lyrae stars and Cepheids. Crossing the
ZAMS the dot-dashed line marks the horizontal branch at the end of which the irregular
variables (Irr) and Mira stars are located. The dotted line corresponds to the cooling
curve of the white dwarfs, where one can find active planetary-nebula nuclei, PNNV,
DOV, commonly known as part of the group of GW Virginis stars, DAV or V777 Her and
DBV or ZZ Ceti stars. Adapted from Christensen-Dalsgaard (2003).
3
INTRODUCTION
Subgroups
Mainly, δ Scuti stars are located in the lower part of the classical Cepheid instability strip but some δ Scutis have been found among pre-main sequence stars
(Breger, 1979; Kurtz & Marang, 1995). In addition, some massive δ Scuti stars
(with more than 2 M ) evolve from the main sequence towards the giant region and
across the Cepheid instability strip at higher luminosities. Because of this, their
periods are longer than the average overlapping RR Lyrae periods, but the former
are distinguished from the latter due to their significant rotation, which RR Lyrae
stars do not have (Breger, 2000). In general, despite the exceptions described above,
it is common to determine two well-defined subgroups for δ Scuti stars:
• High-amplitude delta Scuti stars (HADS): First classified as AI Velorum
stars. They have V amplitudes ≥ 0.30 mag and simple light curves because
they are pulsating in the fundamental or first overtone modes. Due to that,
they have been used to estimate the distance to the LMC and to stellar clusters.
• SX Phoenicies (SX Phe) are δ Scuti stars of Population II, with shorter
periods and lower amplitudes. They have been found in globular clusters as
blue stragglers. From an evolutionary point of view they are unusual and they
have been proposed as the result of merged binary stars (Mateo et al., 1990).
The vast majority of δ Scuti stars are low-amplitude variables. This can be
predicted by pulsation theories and has been measured, as demonstrated in Figure 2
(taken from Rodrı́guez et al., 2000). Obviously, the highest-amplitude variables are
easier to detect than the lowest, therefore the distribution can be biased toward large
amplitudes. The second peak in the distribution, around 0.3 mag, is in agreement
with the amplitude range for the HADS group.
Pulsation modes
δ Scuti stars are known as multiperiodic variables. For example, the stars with
more frequencies achieved are FG Vir and 44 Tau with 79 and 29 frequencies, respectively (Breger et al., 2005, Antoci et al., 2006). Their pulsation modes can be radial
and nonradial, but the majority of δ Scuti stars are nonradial pulsators. They can
pulsate, simultaneously, in a large number of p-modes. In addition there are also
4
INTRODUCTION
Figure 2: Histogram taken from Rodrı́guez et al. (2000) with the distribution of their 636
δ Scuti as a function of the visual amplitude (∆ V).
(pure) radial pulsators. The nonradial pulsations found photometrically are loworder (n = 0 to 7) and low-degree (` ≤ 3) p-modes. However, using spectroscopy,
high-degree nonradial modes have been found with ` up to 20 (e.g., τ Peg). The
radial pulsators mainly pulsate in the fundamental mode and its first few overtones.
For radial modes, for a δ Scuti star with Tef f = 7800 K, 1.7 M , 15 L , Y =
0.28 and Z = 0.02, the predicted period, pulsating constant Q and the ratio between
periods is summarized in Table 2, adapted from Breger (1979). Periods and their
ratios for modes higher than the third one were taken from Hareter et al. (2008).
Table 2: Radial Q values and period ratios for a typical δ Scuti star.
Pulsation mode
Period Pi /Pi−1
Fundamental, F 0.07861
1st Overtone, 1H 0.05950 0.761
2nd Overtone, 2H 0.04846 0.810
3rd Overtone, 3H 0.04095 0.845
4th Overtone, 4H 0.03533 0.862
5th Overtone, 5H 0.03109 0.879
6th Overtone, 6H 0.02774 0.882
5
Pi /PF
1.000
0.757
0.617
0.521
0.449
0.396
0.353
Q (days)
0.0329
0.0251
0.0203
0.0172
—–
—–
—–
INTRODUCTION
Period-Luminosity relation
The Period-Luminosity (PL) relation, as mentioned previously, is a remarkable
property of pulsating stars. It relates a direct measurable parameter, the period,
and a relatively uncertain parameter, luminosity. The observational form of the
relation is between the mean absolute magnitude of the star and the logarithm of
its period. There are other forms for the relation considering a color term (PeriodLuminosity-Color, PLC relation) and/or a metallicity term (PLZ relation) that have
demonstrated to cause a decrease in the intrinsic scatter of the PL relations in the V
band (see McNamara, 1997; Petersen & Christensen-Dalsgaard, 1999; Santolamazza
et al., 2001).
It has been found that the different pulsation modes tend to separate the stars in
different, approximately parallel, PL relations which have quite similar slope values
(see the results of Tsvetkov, 1985 and Santolamazza et al., 2001). As does the color
or the metallicity term, the division of the PL relation according to the pulsation
modes reduces the dispersion.
A historical review
There is a great amount of studies about the δ Scuti PL relation for visible
bands. Some of them consider δ Scutis as a subgroup of the Cepheid variables,
others consider them and SX Phoenicis as a single group, while others use only
high-amplitude δ Scuti stars (HADS).
Table A.1 is a review with the different coefficients derived for the PL relation
in the V-band. The first study to discover that a separation due to different modes
exists was Dworak & Zieba, (1975), but the authors could not explain the separation
itself and attributed it to a separation between bright and faint stars. Tables A.2
and A.3 have the coefficients derived for PLC relations for BV bands and Strömgren
by bands, respectively.
Tables A.1, A.2 and A.3 are adapted from Table III of Tsvetkov (1985), including
the newest results after 1985. For more details about the relations prio to 1985, see
Tsvetkov (1985) and the references therein.
6
INTRODUCTION
Period-luminosity relation at infrared bands
PL relations at infrared wavelengths have more advantages than the ones derived
at visible bands: the amplitude of the variables is smaller, with which the mean
magnitudes have lower dispersion; the interstellar extinction is lower (AK /AV ∼ 1/10);
and the infrared luminosities are less sensitive to temperature changes (instability
strip is narrower), so PL relations are tighter, with low rms-dispersion.
The unique attempt to derive a PL relation at infrared wavelengths was carried
out by King (1990). He derived PL relations for field δ Scuti stars using absolute
magnitudes at 12 µm (IRAS magnitudes) and at J, H and K bands. Magnitudes were
transformed to absolute magnitudes using distance moduli derived from tabulated
values of absolute magnitudes, calculated using calibrations for ubvyβ photometry,
and apparent magnitudes in the V band. King found very tight PL relations considering that he did not separate the stars in their different modes. The slope values
found by him were -1.682, -1.537 and -1.736 for the J, H and K band, respectively,
and their associated dispersions were 0.044, 0.027 and 0.039 mag for each band.
The aim of this work is precisely to derive, as King did, PL relations at infrared
wavelengths for δ Scuti stars, but with some substantial differences: first, stars
in open clusters will be used, because in this case distance measurements are not
required to obtain the slope for the relations; furthermore, low-amplitude variables
were used, with the disadvantage that it is more difficult to detect variability but
with the advantage that the mean magnitude will be more accurately determined.
Finally, unlike King, different PL relations are expected if different pulsation modes
are present.
This practice report is subdivided in other five chapters (besides the Introduction). In chapter 1 the principal properties of the open clusters and δ Scuti observed
and used for derived PL relations will be presented. Chapter 2 has the description
of the photometric reductions and the color-magnitude diagrams obtained for the
clusters, with an analysis of the possibility to obtain light curves for low-amplitude
δ Scuti stars with the precision achieved from the data. Chapter 3 contains the PL
relations for the near-infrared bands, J, H and Ks , found for each cluster, and a discussion about the possible pulsation modes for the stars of the sample. A simplified
approach to derived PL relations from theoretical values, for Johnson-Cousins-Glass
bands, can be found in chapter 4. Finally, chapter 5 has the conclusions and the
possible future work.
7
Chapter 1
Observations and data reduction
1.1
Observations
VISTA Variables in the Vı́a Láctea (VVV) is an ambitious ESO Public Survey
aiming to perform the first ever IR variability monitoring of the entire galactic bulge
and of an adjacent portion of the disk. VVV survey will cover a sky area of 520 sq.
deg, amounting to a total of ∼ 109 point sources, ∼ 106 of which should be variables.
This very large number of light curves cannot be classified using traditional methods, but instead an automated classification scheme must be developed in order
to properly classify the detected variable stars. In this context, VVV Templates
Project1 has the main purpose of to build a large database of well-defined, highquality, near-IR light curves for variable stars of different types, which will form the
basis of the VVV automated classification algorithms. The details and current status
of the VVV survey and the VVV Templates Project are in Catelan et al. (2011).
As part of the data collected by the VVV Templates project, for this work timeseries data for three open clusters –NGC 7062, NGC 1817 and NGC 6134– in J,
H and Ks (K-short) bands were used. The first two were observed with the 1.52-m
Telescopio Carlos Sánchez (TCS) at Observatorio del Teide, Tenerife, Canary Islands.
The instrument used was the infrared camera CAIN-III. For NGC 6134 the telescope
used was the InfraRed Survey Facility (IRSF) 1.4-m telescope at Sutherland and the
infrared camera SIRIUS.
1
http://www.vvvtemplates.org/.
8
CHAPTER 1. OBSERVATIONS AND DATA REDUCTION
NGC 7062 was observed during 5 nights, since 27 of October until 31 of October,
2010. 25 images were taken in H, 40 images in J and 89 images in Ks . NGC 1817
was observed along 8 nights on January, 2011, from 4th to 11th, taking 16 H-band,
119 J-band and 122 Ks images. Between 4th April, 2011 and 20th April, 141 images
of NGC 6134 for each filter were taken.
A brief characterization of the instruments used is shown in Table 1.1. In the
next section the open clusters selected and their δ Scuti stars are described.
Table 1.1: Characteristics of the instruments used.
TCS + CAIN-III
FoV (arcmin × arcmin)
4.25 × 4.25
Gain (e-/ADU)
8.5
R.O.N. (e-)
70
Scale (”/pix)
1
Dithering (cycles)
9
1.2
IRSF + SIRIUS
7.7 × 7.7
5
30
0.45
10
Open clusters
The basic characteristics of the observed clusters can be found in Table 1.2,
whereupon each cluster is described in a separate subsection. Table data were taken
from the WEBDA web page2 , except the value of [Fe/H] for NGC 7062, which was
taken from Peniche et al. (1990).
Table 1.2: Basic data of the open clusters.
Cluster
NGC 7062
NGC 1817
NGC 6134
α2000
21 23 27
05 12 15
16 27 46
δ2000
Distance (pc)
+46 22 42
1480
+16 41 24
1972
-49 09 06
913
log t (yr)
8.465
8.612
8.968
[Fe/H]
-0.35
-0.26
+0.18
# δ Scuti
13
17
6
Finding charts constructed with Ks images of the clusters with the variables
marked are shown in Figures 1.1 and 1.2.
2
http://www.univie.ac.at/webda/
9
CHAPTER 1. OBSERVATIONS AND DATA REDUCTION
1.2.1
NGC 7062
Open cluster with Galactic coordinates l = 89.9◦ , b = -02.7◦ and a diameter
of 6 arcminutes. UBV photographic photometry was measured by Hassan (1973)
and Strömgren photographic photometry by Peniche et al. (1990). UBV CCD photometry was performed by Viskum et al. (1997) and more precise photometry by
Freyhammer et al. (2001, hereafter FAS01) using the 2.56-m Nordic Optical Telescope (NOT) at La Palma. The variable stars of the cluster have 2MASS infrared
mean magnitudes. They are listed in Table A.7.
NGC 7062 has 20 variable stars, 13 of which are δ Scuti according to FAS01.
Ten of them are fit into the CAIN-III field of view: V1 → V6 and V10 → V13.
However, V4 and V10 are not members of the cluster (according to FAS01) so they
were excluded from the PL relation determination. V12 was also rejected because it
is too faint, difficult to distinguish and it is close to the edge of images.
Table A.4 summarizes the basic properties of the δ Scuti members of the cluster
that are inside the field of view (FoV) of the images. Adapted from Table 2 of FAS01,
the first column has the ID, the second the GCVS name, the third and fourth the
equatorial coordinates, the fifth and sixth the frequency (in cd−1 ) and the associated
S/N (in the B band, but in the V band for V2), respectively, and, only as reference,
the V average magnitude and the corresponding semi-amplitude, in the seventh and
eighth columns respectively.
Most of the variables are multiperiodic. Only V6, V11 and V13 are double
or single-mode pulsators. However, mode identification for these variables is not
available in the literature.
1.2.2
NGC 1817
Relatively large cluster, with a diameter of 15 arcminutes. Its Galactic coordinates are l = 186.1◦ and b = -13.1◦ . The pursuit of variables in the cluster started
with 7 δ Scuti stars detected with the 80-cm telescope, IAC80, at Observatorio del
Teide, Tenerife, done by Frandsen & Arentoft (1998a) with CCD photometry. Later,
Strömgren and BV Johnson photometry was done by Arentoft et al. (2005) (hereafter A05) and Bouzid et al. (2006), detecting at least 12 δ Scuti. Some but not all
of the variables have 2MASS mean magnitudes, which can be found in Table A.8.
The latest CCD photometry was done by Andersen et al. (2009) using data taken
with the 1.5-m Danish Telescope, at La Silla, and the 2.56-m NOT. They found 8
10
CHAPTER 1. OBSERVATIONS AND DATA REDUCTION
Figure 1.1: Images taken with TCS + CAIN-III. Left: NGC 7062 Ks finding chart. Right:
Finding chart of NGC 1817 in Ks . In both images North is down and East is left.
new variables, in addition to the 18 variables detected by A05, thus reaching a total
of 26 variable stars. 17 of them (V1 → V12, V19 → V22 and V26) are δ Scuti stars
and two more are candidates (V13 and V17, which was also suggested to be a γ Dor
star). In addition, there are 2 confirmed eclipsing binaries, one of them, V4, with a
component that is also a δ Scuti star, 4 γ Dor stars and one without classification,
that is suggested to be a contact binary. This huge amount of variables makes the
cluster a very useful target for asteroseismology, but with the disadvantage that it
is very extended.
V1, V6, V8, V10 and V12 were classified as not cluster members according to
proper motions done by Balaguer-Núñez et al. (2004); however, A05 and Majaess et
al. (2011) disagree. They found that V1 to V12, except V10, are cluster members
according to their position in the color-magnitude diagram.
The field of view of CAIN-III fits 9 δ Scuti stars, inside of which 6 are possible
members of the cluster (V1, V2, V3, V6, V8 and V11). V4 was rejected due to the
fact that it is a binary star. Coordinates and pulsational data for these δ Scuti stars
and the candidate V17 are shown in Table A.5, adapted from Table 2 of A05.
A preliminary mode identification was done by A05 using theoretical models
for δ Scuti stars with masses between 1.6 to 2.0 M developed by J. ChristensenDalsgaard, which relate frequencies for radial modes (` = 0) and absolute V mag11
CHAPTER 1. OBSERVATIONS AND DATA REDUCTION
nitudes. The pulsation modes derived from their Figure 18 were added in the final
column of Table A.5. If the pulsation mode identification is accurate, the stars in
the cluster have different pulsation modes, and it will not be possible to derive PL
relations for each mode directly.
1.2.3
NGC 6134
Southern open cluster with Galactic coordinates l = 334.9◦ and b = -0.2◦ . With
6 arcmin of diameter, it is a very suitable open cluster for asteroseismology. The first
attempt to find variables was done by Kjeldsen & Frandsen (1989) with the Danish
1.5-m telescope at La Silla, when 3 δ Scuti (# 5, # 29, #40) were detected with
BV photometry. Later, Frandsen & Kjeldsen (1993) detected 2 more variables, and
more recently Frandsen et al. (1996, hereafter F96) corrected some frequencies and
detected one more δ Scuti, reaching a total of 6 in the cluster. Five of them have
2MASS mean magnitudes (see Table A.9).
As SIRIUS FoV is bigger than the dimensions of the cluster, all δ Scutis were
observed. The identification numbers used were those of F96. In Table A.6 the basic
pulsation data for δ Scutis are presented, adapted from Table 3 and 4 of F96. In
addition, GCVS names were added.
Figure 1.2: Ks image taken with IRSF + SIRIUS. Left: Whole image of the cluster and
a central part marked with a white box. Right: Zoom corresponding to the white box in
the image on the right, with the δ Scutis in the cluster. North is up and East is right.
Some attempts to perform mode identification were done by Frandsen & Kjeldsen
12
CHAPTER 1. OBSERVATIONS AND DATA REDUCTION
(1993) and Audard et al. (1995). In the first case, as an example of how modes for δ
Scuti in open clusters can be identified, they used models of Fitch (1981) and fitted
absolute magnitude and frequency (in µHz). They found two variables pulsating
in the fundamental mode (# 9 and # 87), and other three in the first overtone.
However, with the new data obtained by F96, some frequencies were rejected or
modified. In the second case, the Q value was used to determine possible pulsation
modes. They found that star # 87 could have (n,`) = (2,0) and # 348 (n,`) =
(3,0), while for # 9 a g-mode is suggested. However, the latter star has 4 very close
frequencies that probably are the result from splitting due to rotation. In both cases
the results are only tentative and based on some assumptions (the same effective
temperature for all the stars or not taking rotation into account) that could lead to
a different identification of the modes.
1.3
Data reduction
TCS images were reduced using pipelines developed and provided by the observatory. In the case of IRSF images, they were received already reduced. For
completeness, the reduction procedure is briefly described below.
The most important aspects that need to be taken into account when reducing
infrared images are flat-fielding and bad pixels. Due to the thermal variations in the
surroundings of the telescope and the instrument (within a time-scale of minutes) two
types of flat-field images are taken: bright and dark, usually using dome flat images
with the lights on and off, respectively. Combining the bright and dark images into
one master bright flat and one master dark flat, flat-field images are generated as
the subtraction between the master images. Then, science images are divided by the
flat-field images, as usual in visible photometry.
CAIN images have some bad pixels (dead or hot) and they should be corrected
using a mask that makes linear interpolations. However, the available mask does
not work satisfactorily, therefore this correction was dropped. A random error is
introduced due to the fact that, despite bad pixels having fixed positions, the pointing
accuracy of the telescope leads to significant changes in the pixel positions of the
stars from one image to another.
Another important aspect is the sky determination and substraction. Infrared observations have the disadvantage that atmospheric emission is high in this wavelength
range. Due to that, integration times should be short in order to avoid saturation.
13
CHAPTER 1. OBSERVATIONS AND DATA REDUCTION
To observe faint stars or to take deep images and properly correct sky counts, the
dithering technique is used. It consits in to take several exposures with the position
of the telescope slightly displaced each time, relative to the field center. If the field is
crowded, sky images are taken with shifts greater than the size of the stellar image.
If the field is not very crowded, the sky is determined in each dithered science image
in the same way as for visible images. A detail description of the infrared reduction
steps and other considerations can be found in Glass (1999).
Finally, after the subtraction of the median sky image from each dithered science
image, they are aligned, considering the telescope offsets, and combined into a final
science image.
14
Chapter 2
Photometric reductions
2.1
Photometry with DoPHOT
To derive the magnitudes of the stars in each frame, the DoPHOT crowded-field
photometry package was used, updated and rewritten by Javier Alonso1 , based on
the original photometry package developed by Schechter et al. (1993).
DoPHOT can detect and model particular objects as a star, double star, cosmic
ray, galaxy, etc. Just like other photometry packages, the objects are detected if they
are found above a specific threshold, and the stellar Point Spread Function (PSF) is
modeled by using an elliptical Gaussian function. It starts fitting a simple Gaussian
profile generated with the values of FWHM, mean sky, gain and R.O.N given by the
user for one image. Subtracting the modeled stars from the field a new threshold
is calculated and the process is repeated until there is no value above the lowest
threshold. In each pass, stars found during previous passes are put back on the
frame and model parameters are re-calculated. DoPHOT also calculates aperture
magnitudes in the last step.
For NGC 7062 and NGC 1817, aperture photometry was preferred because they
are not crowded fields. PSF photometry would be less accurate because the fields
do not have enough stars to obtain reliable star models. NGC 6134 is more crowded
and in this case, PSF photometry makes sense. In addition, there are some variable
stars that have a very near companion, so aperture photometry would have more
disadvantages than advantages.
An example of the photometric errors achieved for NGC 7062 in the JHKs bands is
1
This version can be obtained from [email protected].
15
CHAPTER 2. PHOTOMETRIC REDUCTIONS
shown in Figure 2.1. As it can be noted for magnitudes higher than 14, in each band,
the value of photometric errors is higher than 0.2 magnitudes, which is problematic
to detect millimagnitude variables. However, given the quality of the TCS images,
the errors could not be reduced beyond this level.
Figure 2.1: Photometric errors from DoPHOT for a NGC 7062 frame of each filter.
With the intention not to add more errors the images were not aligned. Instead,
a Python code was used to find the coordinates, magnitudes and their errors of
variable stars in each image, taking as input the photometric outputs of DoPHOT,
the coordinates of the star in the first image and the shifts between the images.
Obviously, this procedure could have problems when the coordinates or shifts are
not accurate, because it was based on the minimum distance between the variable
and the stars in the photometric files. To reject the matching errors, the output
coordinates were visually checked using tvmark over the corresponding image. If the
star was not found to be the variable, the magnitude was rejected. The matching
errors for a cluster in one band were, approximatively, a 5% of the total images, in
the best cases, and in the worst 50%, but in these cases there were other reasons like
stars are near the edges of the images, near bad pixels or in a blend with a nearby
star.
16
CHAPTER 2. PHOTOMETRIC REDUCTIONS
2.2
Color-magnitude diagrams
With the purpose of keeping the photometry without the errors and do not reduce
the effective size of the images, both consequences of the alignment process, the
IRAF2 task xyxymatch was used in order to cross-match two whole photometry files
for 2 filters, J and Ks , for each cluster. Using the pairs obtained with this task the
color-magnitude diagrams (CMDs) were obtained and are shown in Figures 2.2, 2.3
and 2.4 for NGC 7062, NGC 1817 and NGC 6134, respectively. In all of them δ
Scuti stars are plotted as blue circles.
Figure 2.2: Color-magnitude diagram of NGC 7062. There are some spurious stars,
probably due to erroneous results from the cross-match.
It is necessary to note that in the case of NGC 1817 the FoV of CAIN-III is
not wide enough to cover the whole cluster. In fact, only a small number of stars is
observed therefore its CMD is poor; however, an incipient main-sequence is glimpsed.
2
IRAF is distributed by the National Optical Astronomy Association (NOAO), which is operated
by the Association of Universities for Research in Astronomy (AURA), under cooperative agreement
with the National Science Foundation.
17
CHAPTER 2. PHOTOMETRIC REDUCTIONS
Figure 2.3: Color-magnitude diagram of NGC 1817. It has few stars due to the fact that
only a small region of the cluster was observed.
On the contrary, the CMD of NGC 6134 has more stars than expected. There
are some dispersed stars with (J-Ks ) colors higher than 2 magnitudes that are not
part of the main sequence. This contamination could be explained in two ways: (1)
SIRIUS camera has more than 7 arcmin per side and the cluster has only 6 arcmin
of diameter and (2) fore and background stars of the Galactic field, that are not
members of the cluster.
In the CMD shown in Figure 2.4, the stars in the field of view of the camera
but outside the size of the cluster were identified and omitted, using the scale of the
camera, and some stars still have redder colors than expected. Therefore, considering
also that the cluster is practically on the Galactic plane (Galactic latitude of only
0.2 deg.), the most plausible explanation is those stars are Galactic stars.
18
CHAPTER 2. PHOTOMETRIC REDUCTIONS
Figure 2.4: Color-magnitude diagram of NGC 6134. It has some unreliable stars in
the region of colors higher than 2 magnitudes, probably due to incorrect results from the
cross-match between the catalogs.
2.3
Light curves
To obtain light curves, differential photometry was used. First, 10 comparison
stars were selected. The criteria used were that they had approximately the same
magnitude as the variable star, were isolated and were always inside the images,
away from the edges. Evidently, non-variable stars were selected according to the
papers used as reference for each cluster.
A second criterion used with these 10 stars was that photometric errors were the
lowest possible. With that, the three best comparison stars were selected. Additionally, another star was used as a check. Light curves were calculated as the difference
between the variable and each of the three comparison stars, and the difference between comparison stars and check star were also calculated. The light curves were
phase-folded using periods from the literature.
Contrary to expectations, the variability of the stars could not be detected with
19
CHAPTER 2. PHOTOMETRIC REDUCTIONS
certainty. The main reason for this, in CAIN images, can be the photometric errors
achieved: in the best cases they were about 1 mmag, but this value can change from
image to image depending on the seeing, star position in the CCD or the presence of
clouds. On average, the photometric errors were about 0.007 magnitudes in Ks band,
so when the difference between the variable and the comparison star was calculated
the propagated error was in the order of 10 mmag, which is of the same order or
higher than the amplitude of the variables’ variations.
As an example of the precision obtained with the differential photometry three
light curves in Ks band, before phasing, are shown in Figures 2.5, 2.6 and 2.7, which
correspond to V2 of NGC 7062, V1 of NGC 1817 and # 9 of NGC 6134, the highestamplitude variables of each cluster, respectively (see Tables A.4, A.5 and A.6).
Figure 2.5: Top and middle: Differential light curve for V2 of NGC 7062. Bottom:
Difference between the two best comparison stars used. Black dotted lines delimit the
dispersion around zero. Blue dot-dashed lines delimit the amplitude of the variable in B.
20
CHAPTER 2. PHOTOMETRIC REDUCTIONS
Figure 2.6: Top and middle: Differential light curve for V1 of NGC 1817. Bottom:
Difference between the two comparison stars used. The dashed and dotted lines have the
same meaning that in Figure 2.5.
Each plot consists of two light curves, made with the best comparison stars, at
the top and middle, and at the bottom the difference between the two comparison
stars. Expected B-amplitudes were marked with blue dot-dashed lines and the 1-σ
dispersion of the data is delimited with black dotted lines. Light curves correspond to
the nights with more coverage of Julian Days. In the three cases it can be noted that
the dispersion of the light curve is the same or higher than the variation detected at
the B band. Considering that the lowest amplitudes are expected at infrared bands,
the photometric errors and the dispersion of the light curves make it impossible to
measure the infrared variation. However, it should be noted that the variations and
photometric errors for # 9 star of NGC 6134, Figure 2.7, are the lowest compared
with the other light curves. This is due to the fact that SIRIUS images have better quality than the CAIN images and the observations for this cluster had better
atmospheric conditions.
21
CHAPTER 2. PHOTOMETRIC REDUCTIONS
Figure 2.7: Top and middle: Differential light curve for #9 star of NGC 6134. Bottom:
Difference between the two comparison stars used. The dashed and dotted lines have the
same meaning that in Figures 2.5 and 2.6.
Another aspect that could hinder the measurement of variability is the multiperiodic nature of the variables. These have non-regular light curves, and because
of this, it is more difficult to distinguish between real and spurious variations. NGC
7062 V4 (δ Scuti non-member of the cluster) and V5 are examples of how nonregular
variables’ light curves could be; see Figures 2.8 and 2.9, taken from FAS01. Considering that for these detections the variables were monitored during more than 7
hours per night, twice the time observed for this cluster, and used a 2.56-m telescope
(1 meter larger than TCS) it is not surprising that the variability cannot be detected
with CAIN images.
The size of the telescope is another important aspect that could restrict the
possibility to detect a variable. The diameter of the telescope needed to detect
oscillations of δ Scuti stars and their amplitude and magnitude were related by
Frandsen & Arentoft (1998a). Their plot is shown in Figure 2.10 where, for 4 hours
22
CHAPTER 2. PHOTOMETRIC REDUCTIONS
Figure 2.8: Differential light curve for V4 in the B-band. Despite the fact that this is
the variable with the highest amplitude, the dispersion of the observations with respect
the frequency-solution can be seen. Taken from FAS01.
Figure 2.9: B-band differential light curve of V5. A beating of different modes can be
seen. Taken from FAS01.
of observations, V magnitude and diameter are related. The dotted line defines the
limit where scintillation noise is comparable to photon noise. The other lines, from
left to right, correspond to amplitudes of 0.5 mmag, 1 mmag and 2 mmag.
According to the plot, NOT, the 2.56-m telescope used to detect variables in NGC
7062 and NGC 1817 in previous works is enough for the magnitudes and amplitudes
of the variables. They can detect 1 mmag amplitudes and variables with 15 V-mag.
And it could detect, also, up to 9 different periods for a single variable (V1 of Table
A.4). With that, the 1.5-m telescope used to observe this cluster is not big enough
to measure the same label of variability at infrared wavelengths.
23
CHAPTER 2. PHOTOMETRIC REDUCTIONS
Figure 2.10: Relation between V magnitude and diameter, in cm, of the telescope required for measuring different magnitude variations. The dotted line represents the limit
where scintillation noise is comparable to photon noise. Dot-dashed lines are associated to
amplitudes of 0.5 mmag, 1 mmag and 2 mmag, from left to right. Taken from Frandsen &
Arentoft (1998b).
For NGC 6134 the photometry had better results but it was not possible to obtain
reliable light curves, despite the fact that the observations were more numerous
than CAIN-images, SIRPOL images do not have bad pixels and, furthermore, the
photometric errors are the lowest. The telescope used, the 1.4-m IRSF telescope, is
practically the same size or bigger than others used to detect variables in this cluster
(Kjeldsen & Frandsen, 1989, used the Danish 1.5-m telescope; Balona, 1995, a 1.0-m
telescope; and Frandsen et al., 1996, 1.0-m and 0.9-m telescopes). But, to detect the
variability of these stars in the near-infrared the most suitable solution to improve
the light curves could be to use a bigger telescope.
In addition to the above reasons, it is remarkable that in all the previous works
the magnitudes were decorrelated in order to improve the accuracy of the light curves,
i.e. the authors adjust a linear function for the magnitudes to separate the intrinsic
variations in the magnitude and the effects of seeing, FWHM, sky background and
position of the star on the CCD. This process also allows to combine data from
different telescopes. An example of how a light curve of NGC 6134 was improved
using decorrelation is shown in Figure 2.11, taken from Balona (1995). They adjust
24
CHAPTER 2. PHOTOMETRIC REDUCTIONS
a fit of V = α x + β, where x is the seeing in arbitrary units and found that the
coefficient α changes from star to star, with an average value of ∼ −100.
Figure 2.11: Bottom: Raw light curve obtained with differential photometry by Balona
(1995) using a 1-meter telescope. The Y axis corresponds to differential V magnitude.
Top: Decorrelated light curve using as fit V = α x + β, where x is the seeing in arbitrary
units.
Hence, bad pixels, photometric errors, the number of observations, photometric
conditions, telescope size and the non-decorrelated magnitudes did not allow to detect the variability of the stars. Clearly, an immediate future work would be focused
on decorrelating the magnitudes in order to improve the light curves. Despite the
foregoing, for the Period-Luminosity relation it is not necessary to have light curves.
In fact, low-amplitude δ Scuti stars are more convenient because it is easier to determine a mean magnitude. Furthermore, it is demonstrated in the case of Cepheids
that the dispersion of the slope is lower in the infrared compared to the visible bands
(see Madore & Freedman, 1991), which can be expected for δ Scuti stars too.
25
Chapter 3
Period-Luminosity relations
3.1
Magnitude calibration
The magnitudes of the variable stars in each image were transformed to the Two
Micron All Sky Survey 1 (2MASS, Cutri et al., 2003) photometric system, according
to
m2M ASS = mλ − kλ X − ZPλ ,
(3.1)
where kλ is the extinction coefficient for the respective filter, X the airmass of the
image and ZPλ the zero point.
Atmospheric extinction was corrected only for TCS images. From the photometry performed in the clusters observed it was not possible to derive a tight relation
between magnitudes and airmass principally because that not all the nights were
photometric. Hence, the extinction coefficients derived for the observatory location by Cabrera-Lavers et al. (2006) were used: kJ = 0.179 ± 0.011 mag/airmass,
kH = 0.148 ± 0.012 mag/airmass and kKs = 0.167 ± 0.013 mag/airmass. Clearly, to
adopt these values is an approximation, but it does not affect substantially the mean
magnitude, as required for PL relations.
For IRSF images, taken at Sutherland, the extinction coefficients at infrared
bands are negligible (less than 0.01 mag/airmass). In fact, for the H band a relation
was found and is shown in Figure 3.1, with an extinction coefficient kH = 0.0174 ±
0.0004 mag/airmass with σk = 0.004. However, for the other bands the relation is
less clear, thus the correction was dropped.
1
http://www.ipac.caltech.edu/2mass
26
CHAPTER 3. PERIOD-LUMINOSITY RELATIONS
Figure 3.1: Magnitude for different airmass values gives the atmospheric extinction in
the H-band for IRSF images.
Zero points were calculated as the difference between the extinction-corrected
magnitudes and 2MASS magnitudes. For that, 10 non-variable stars were used
in each cluster, with which a weighted average zero point was calculated for each
image in each filter. With that, any difference between images due to changes in
atmospheric conditions were considered.
Color terms were not used because, as it is mentioned by Cabrera-Lavers et al.
(2206), the transformations between TCS and 2MASS have a small dependence on
2MASS colors and, in some cases, for redder stars the corrections are smaller than
the internal accuracy of the survey. Furthermore, CAIN and SIRIUS images were
not taken simultaneously and the same number of images is not available in the three
filters, then the determination of the color terms could only be performed for average
colors, which is inaccurate.
27
CHAPTER 3. PERIOD-LUMINOSITY RELATIONS
3.2
PL relations for NGC 7062 and NGC 6134
For each variable star the period associated to the highest signal according to
the previous works (FA01 for NGC 7062 and F96 for NGC 6134) was used. It was
not possible to confirm or to determine new periods because the observation time
for each epoch is insufficient.
Mean magnitude for each variable was calculated as the weighted average of
the calibrated magnitudes. The weights were the inverse of the squared errors.
Each error was calculated as the propagated error with the contributions of the
photometric errors from DoPHOT, the error of the extinction coefficient when used,
and the error in the mean ZP. The dispersion of the mean magnitudes over the square
root of the number of considered data was used as error, as usual.
The mean magnitudes have, as it is expected, low dispersion values because lowamplitude variables were used. Considering that in some cases more than 100 values
for the magnitudes were used, the errors are practically insignificant. Despite that,
statistical errors are not representative of intrinsic errors because there are errors
that cannot be quantified such as the different positions of the stars in the CCD and
different sensitivities of the pixels.
Calibrated mean magnitudes were plotted against the logarithm of the period
(in days) for each variable. Some possible PL relations were found for NGC 7062
and NGC 6134 variables. At the first attempt each relation was fitted with leastsquares approach. The coefficients for the relations derived can be found in Tables
A.10 and A.11 for NGC 7062 and NGC 6134 variables, respectively. # 1 relation
is for the brightest stars and # 2 is for the faintest. In both cases the # 2 relation
seems more accurate than the # 1, especially in the case of NGC 6134 where only
two bright stars are available for the linear fit. Due to that the # 1 relations were
dropped and re-derived using the slope corresponding to # 2 and a new intercept.
This was calculated as the value with which the difference between the data and the
PL relation fit, with the slope from # 2 relation, is the minimum.
Figures 3.2, 3.3 and 3.4 show the PL relations found, the derived # 2 relation
and the new # 1 relation calculated as a shift of the former, for H, J and Ks bands,
respectively. Errors less than 0.1 magnitudes were not plotted because they have the
same size of the points.
28
CHAPTER 3. PERIOD-LUMINOSITY RELATIONS
For NGC 7062 the # 2 PL relations (for the faintest stars) found are:
MJ = (−4.335 ± 0.415) log P + 9.059(±0.399), rms = 0.141,
(3.2)
MH = (−4.564 ± 0.629) log P + 8.657(±0.060), rms = 0.021,
(3.3)
MKs = (−4.414 ± 0.346) log P + 8.634(±0.332), rms = 0.118.
(3.4)
And for NGC 6134:
MJ = (−3.207 log P ± 0.157) log P + 7.268(±0.168), rms = 0.085,
(3.5)
MH = (−3.259 log P ± 0.153) log P + 7.864(±0.165), rms = 0.083,
(3.6)
MKs = (−3.305 ± 0.159) log P + 7.785(±0.171), rms = 0.086.
(3.7)
With these relations the possible radial mode orders, n, could be estimated considering the period ratios associated to the two relations found. If they have the
same slope, a, then, for a given magnitude, the periods for two radial modes, Pn and
Pm , and their corresponding intercepts, bn and bm , are related by
∆ log P = log
Pm
Pn
=
(bn − bm )
.
a
(3.8)
Table 3.1 has the values of ∆ log P and Pm /Pn for NGC 7062 and NGC 6134 for
the relations in the three bands.
29
CHAPTER 3. PERIOD-LUMINOSITY RELATIONS
Table 3.1: Period ratios associated to the PL relations found for each cluster and band.
NGC 7062
∆ log P
-0.382
-0.363
-0.381
Pm /Pn
0.414
0.433
0.415
NGC 6134
Band
∆ log P
J
-0.230
H
-0.228
Ks
-0.222
Pm /Pn
0.588
0.591
0.599
Band
J
H
Ks
For NGC 6134 the period ratio approximately corresponds to the relations for
fundamental (for the faintest) and for the second overtone (the brightest) variables,
given the expected period ratio P2 /P0 = 0.616. These modes are not so different from
those derived by Audard et al. (1995). They found, as was mentioned in Chapter 2,
n = 2 for # 87 and g-mode for # 9 instead of the fundamental mode, and n = 3 for
# 348 in lieu of the second overtone.
For NGC 7062, the period ratio value is quite similar to the expected for fundamental and fourth overtone (P4 /P0 = 0.449). This means that V2, V11 and V13 are
pulsating in the fundamental mode while V1, V3, V5 and V6, the brightest, with
(n, `) = (4,0). Pulsation mode identification for the variables is not available, so an
independent pulsation mode identification is required to confirm these values.
30
CHAPTER 3. PERIOD-LUMINOSITY RELATIONS
Figure 3.2: PL relations in the H-band for NGC 7062 and NGC 6134 δ Scuti stars.
The bottom dashed lines for the faintest stars are the result of a least-squares fit. The
top dashed lines have the same slope as the one at the bottom but they were shifted to
constrain the brightest stars.
31
CHAPTER 3. PERIOD-LUMINOSITY RELATIONS
Figure 3.3: PL relations in the J-band for NGC 7062 and NGC 6134 δ Scuti stars. The
meaning of dashed lines is the same as in the H-band (see Figure 3.2).
32
CHAPTER 3. PERIOD-LUMINOSITY RELATIONS
Figure 3.4: PL relations in the Ks -band for NGC 7062 and NGC 6134 δ Scuti stars.
Dashed lines have the same meaning as for H and J bands (Figures 3.2 and 3.3).
33
CHAPTER 3. PERIOD-LUMINOSITY RELATIONS
3.3
NGC 1817: Fundamentalized periods
For NGC 1817, as expected from the mode identification done by A05, no relation
was found. The distribution of the variables in the MJ - log (P ) plane is shown in
Figure 3.5.
Figure 3.5: Mean magnitudes for NGC 1817 variable stars against their log (P ). As
expected, no PL relations can be constrained because most of the pulsating modes are
different, while these pulsating in the same mode have practically the same period.
Despite the foregoing, the different pulsation modes of NGC 1817 δ Scuti stars
can be reduced to the fundamental mode using Figure 18 of A05. The authors scaled
the frequencies of the radial modes with the mean density and thus with the masses,
or absolute magnitudes, of the stars. They derived the absolute V magnitude of
some variables considering the reddening and cluster distance, and using models
they derived a relation between the fundamental radial mode and more than 10
overtones. For each MV , the corresponding radial mode for each frequency detected
can be found.
Using these relations and the frequencies associated to each variable, the absolute
magnitude in V was derived, and with those, the respective fundamental frequency,
as presented in Table 3.2. With that, a PL relation for the fundamentalized periods
can be derived for the measured infrared mean magnitudes.
34
CHAPTER 3. PERIOD-LUMINOSITY RELATIONS
Table 3.2: Fundamentalized periods of δ Scuti in NGC 1817.
ID
MV
V1
V2
V3
V6
V8
V11
1.37
0.75
2.25
0.81
2.205
2.18
f0
(cd−1 )
7.82
5.35
13.35
5.56
12.96
12.79
log P0
(days)
-0.893
-0.728
-1.125
-0.744
-1.113
-1.107
The PL relations found are shown in Figure 3.6. Just as in other cases, errors
lower than 0.1 magnitudes were not plotted. The relations fitted for each band are:
MJ = (−3.806 ± 0.056) log P + 8.833(±0.049), rms = 0.042,
(3.9)
MH = (−3.857 ± 0.065) log P + 8.668(±0.059), rms = 0.025,
(3.10)
MKs = (−3.701 ± 0.087) log P + 8.679(±0.080), rms = 0.075.
(3.11)
35
CHAPTER 3. PERIOD-LUMINOSITY RELATIONS
Figure 3.6: PL relations for the fundamental mode for the δ Scuti in NGC 1817. V17
was not included because is probably a γ Doradus star (Arentoft et al., 2005).
36
Chapter 4
Theoretical approach
With the purpose of estimating the slope value, very simple theoretical PL relations were constructed using tabulated values for periods and luminosities for δ
Scuti stars. Luminosities and magnitudes can be related calculating the bolometric
magnitude and the bolometric correction, BC, for the given wavelength, with the
following relations:
Mbol = Mbol, − 2.5 log (L/L ),
Mλ = Mbol − BCλ ,
(4.1)
(4.2)
where Mbol, is the Sun’s bolometric magnitude, assumed as 4.74 magnitudes, and
L and L are the luminosities of the star and the Sun, respectively.
Bolometric corrections were taken from Girardi et al. (2002), who estimate BC
for several photometric systems, including the Johnson-Cousins (BVRI) and Glass
(JHK) systems, as function of Tef f , log (g) and [M/H]. The available models consider
two different values of [M/H]: +0.0 and -0.5.
The values used for derived BC corrections and bolometric magnitudes are listed
in Table 4.1 that corresponds to a part of Table 16.2 of Allen (2000). For each
value of gravity the associated range of temperatures and BC were selected from the
models and with them, as the authors recommended, linear interpolation was done
to obtain BCs for intermediate values of effective temperature. This procedure was
performed for each metallicity value.
37
CHAPTER 4. THEORETICAL APPROACH
Table 4.1: Physical and observational properties of δ Scuti stars. Part of Table 16.2 from
Allen (2000).
log (P0 ) MV
-1.4
+2.7
-1.0
+1.7
-0.7
+0.8
log (L/L )
0.90
1.22
1.58
log Tef f
3.91
3.88
3.86
log g
4.5
3.9
3.6
Q
0.039
0.037
0.037
With the bolometric corrections, absolute magnitudes for BVRIJHK bands were
derived for each luminosity and associated period, i.e., for each metallicity value
three magnitudes for a possible PL relation were derived. It is not a very accurate
method because it has few points and some approximations, but, roughly speaking,
this method can give an estimation of the slope value for different wavelengths.
4.1
Visible bands
For the two values of [M/H] different but quite similar PL relations were found
using least-squares fitting. Due to this similarity only the relations for [M/H] = +0.0
were plotted and they can be found in Figure 4.1. The corresponding PL relations
for this case are the following:
MB = (−2.36 ± 0.14) log P − 0.62(±0.15), rms = 0.078,
(4.3)
MV = (−2.47 ± 0.16) log P − 0.96(±0.17), rms = 0.078,
(4.4)
MR = (−2.58 ± 0.15) log P − 1.21(±0.16), rms = 0.076,
(4.5)
MI = (−2.67 ± 0.15) log P − 1.43(±0.16), rms = 0.075.
(4.6)
38
CHAPTER 4. THEORETICAL APPROACH
Figure 4.1: PL relations for the fundamental mode in the BVRI bands. They were derived
using theoretical values for periods and luminosities, which were converted into magnitudes
using bolometric corrections. For V band MV theoretical values, taken directly from Table
4.1, were also plotted as blue squares.
For [M/H] = -0.5 the relations derived are:
MB = (−2.36 ± 0.16) log P − 0.58(±0.17), rms = 0.078,
(4.7)
MV = (−2.46 ± 0.15) log P − 0.92(±0.17), rms = 0.078,
(4.8)
MR = (−2.57 ± 0.15) log P − 1.16(±0.16), rms = 0.076,
(4.9)
MI = (−2.65 ± 0.15) log P − 1.38(±0.16), rms = 0.077.
(4.10)
As can be noted, the slope values are practically the same for the different metallicity ratios and the rms-dispersions for each relation are very similar too.
39
CHAPTER 4. THEORETICAL APPROACH
Figure 4.1 also shows the values of MV , and their corresponding fit, taken directly
from Table 4.1. They are plotted as blue squares and their associated PL relation is:
MV = (−2.70 ± 0.14) log P − 1.06(±0.15), rms = 0.040
(4.11)
This PL relation is marginally different compared to the equations (4.4) and
(4.8) that were derived with the bolometric corrections. The difference, as Figure
4.1 shows, is due to the point at log (P ) = −1.4 that is slightly lower for the one
derived with bolometric correction compared with the value listed by Allen. However,
considering the errors, is not a substantial difference.
4.2
Infrared bands
The same procedure for the JHK bands gave the relations plotted in Figure 4.2
for [M/H] = +0.0, and the fitting results are:
MJ = (−2.82 ± 0.15) log P − 1.73(±0.16), rms = 0.074,
(4.12)
MH = (−2.93 ± 0.14) log P − 1.94(±0.16), rms = 0.072,
(4.13)
MK = (−2.94 ± 0.14) log P − 1.98(±0.16), rms = 0.072.
(4.14)
As in the visible bands, for [M/H] = -0.5 the slope values and intercepts are
practically the same as the ones found for [M/H] = +0.0:
MJ = (−2.81 ± 0.15) log P − 1.70(±0.16), rms = 0.074,
(4.15)
MH = (−2.94 ± 0.14) log P − 1.95(±0.16), rms = 0.072,
(4.16)
MK = (−2.95 ± 0.14) log P − 1.97(±0.16), rms = 0.072.
(4.17)
The above equations were derived from absolute magnitudes, an so only the slopes
can be compared with those derived using observational data for the fundamental
mode. For NGC 7062, equations (3.2), (3.3), (3.4) at J, H and Ks appear steeper,
in absolute value, than the theoretical ones, even considering the errors. For this
cluster the relations are less accurate than the others (they have higher rms and
errors for the slopes and intercepts), which may be because of the color corrections
40
CHAPTER 4. THEORETICAL APPROACH
Figure 4.2: PL relations in the near-infrared bands derived using bolometric corrections
and theoretical values for periods and luminosities.
having been dropped or due to one or more incorrect periods.
Equations (3.5), (3.6) and (3.7) for NGC 6134 are more consistent with the theoretical relations derived. For NGC 1817, and its associated PL relations, equations
(3.9), (3.10) and (3.11), the slopes are higher than the obtained with theoretical
values, but they are also the more accurate ones. The most probable reason for that
difference is that the fundamental periods derived for the variables are not accurate.
4.3
Wavelength dependence of PL slope
Madore & Freedman (1991, MF01), derived PL relations at visible and nearinfrared bands using self-consistent calibrating data sets, i.e. the same group of
stars (Cepheids on the Large Magellanic Cloud) was used. With their results they
claimed that the PL relation slope becomes more negative as the effective wavelength
increases while the rms-dispersion of the relation decreases. Freedman et al. (2008)
determined PL relations for LMC Cepheids at mid-infrared wavelengths (3.6, 4.5,
5.8 and 8.0 µm) and with these slopes and retaking the slopes of MF01 for the
41
CHAPTER 4. THEORETICAL APPROACH
visible bands and the improved slopes for the near-infrared derived by Persson et al.
(2004), they showed that they appear to be asymptotically converging the slope of
a period-radius relation.
Likewise, King (1991) commented that at infrared wavelengths a PL relation
may be expected to exist as a form of a period-radius relation. His argument was
that infrared luminosities are more indicative of the radii of δ Scuti stars taking into
account their ranges of masses. In addition, as Freedman et al. (2008) comment, in
the infrared the mean magnitudes of the stars are less sensitive to changes in temperature, which is caused by the much-decreased sensitivity of the infrared surface
brightness to the temperature width of the instability strip, which means that at
infrared wavelengths the PL relation is more like a period-radius relation at fixed
surface brightness.
A quick way to see this behavior is considering the pulsation constant, Q, defined
as
r
Q=P
ρ̄
,
ρ¯
(4.18)
where P is the period in days, ρ̄ and ρ¯ are the mean density and mean solar density,
respectively. Expressing the density in terms of mass and radius:
4M
⇒ Q = P M 1/2 R−3/2 ,
(4.19)
3πR3
where M and R are the mass and radius in solar units, respectively. Furthermore,
the period-radius relation from this is:
ρ≈
2
1
2
log P + log M − log Q.
(4.20)
3
3
3
Thereby, the slope of the period-radius relation is, at fixed mass, 2/3 ∼ 0.67.
log R =
42
CHAPTER 4. THEORETICAL APPROACH
It is important to notice that this value is valid for all the pulsating stars, and so
it is expected that the value of the slope of the period-radius relation has, in order
of magnitude, the same value for both Cepheids and δ Scuti stars. Translating this
value into a magnitude:
log R ≈ 0.67 log P + C1 ,
(4.21)
log R2 ≈ 1.34 log P + C2 ,
(4.22)
−2.5 log R2 ≈ −3.35 log P + C3 .
(4.23)
Gieren et al. (1999) derived a period-radius relation for Cepheids as log (R) =
0.68 log (P )+C, which corresponds to a slope, in magnitudes, of (0.680 ± 0.017)× -5
= (-3.40 ± 0.085) magnitudes. This value is in agreement with the one shown above.
Tsvetkov (1988) derived semi-empirical and theoretical period-radius relations for δ
Scuti and an empirical one for Classical Cepheids. He found a slope of (0.651 ±
0.002) for Cepheids and higher slopes for δ Scuti stars: a semi-empirical value of
(0.768 ± 0.008) and a theoretical value of (0.727 ± 0.002), both for the fundamental
mode, with which the slope, in magnitudes, could be between (-3.84 ± 0.04) and
(-3.64 ± 0.01) for δ Scuti stars.
With these considerations the slopes derived in the previous section were plotted
against the effective wavelength of each band and are shown at the top of Figure
4.3, whereas at the bottom the rms-dispersion for the PL relations is shown. In
both cases it is clear that while the wavelength increases, the slope and dispersion
decreases. It is clear, also, that as it occurs with Cepheids, for δ Scuti in the infrared
the slope is more negative.
Considering these results, the absolute value of the slopes derived for δ Scuti stars
in NGC 7062 (equations (3.3), (3.2) and (3.4)) appear overestimated. On the other
hand, the absolute values of slopes derived for NGC 6134 appear more reliable and
in good agreement with those expected (equations (3.5), (??) and (??)).
Slope values derived with the fundamentalized periods for δ Scuti stars in NGC
1817 (equations (3.10), (3.9) and (3.11)) also appear a little overestimated. In this
case, the source of major uncertainties appears to be the determination of the periods
for the fundamental mode.
43
CHAPTER 4. THEORETICAL APPROACH
Figure 4.3: Top: Wavelength dependence of the PL slope for the BVRIJHK bands.
Bottom: Dispersion dependence of the PL slope for the same bands. It is clear that slopes
and dispersions tend to decrease while the wavelength increases.
With all these arguments, the slope values of King (1990) (-1.682, -1.537 and 1.736 for J, H and K) appear seriously underestimated (in absolute value). However,
he obtained relations with low dispersion, which are very tight. The reason for this
discrepancy is still not clear. However, the sources of error in his work are, probably,
the determination of the distance modulus and the tabulated periods used. Also,
King mentioned that his slopes were in good agreement with the one derived for the
V magnitude by Dworak & Zieba (1975). However, as Table A.1 shows, the newest
values of the slopes for the V band are more negative than the one derived by them.
If King’s stars had 2MASS magnitudes it could have been interesting to derive a new
PL relation for those stars, but most of them lack 2MASS measurements.
44
Chapter 5
Summary and future work
Infrared observations and photometry were performed in order to derive PeriodLuminosity relations for low-amplitude δ Scuti variables from three open clusters:
NGC 7062, NGC 6134 and NGC 1817. It was not possible to detect the intrinsic
variability of the stars due to the fact that the amount and quality of the observations
were not good enough for this purpose. It was also not possible to detect or confirm
the periods previously detected for the variables because the observation time for
each epoch is insufficient for multiperiodic variables. Nevertheless, accurate mean
magnitudes, for H, J and Ks bands, for the 20 δ Scuti stars observed, were derived
with which Period-Luminosity relations were found.
Both for NGC 7062 and for NGC 1817, two PL relations were derived. Only one
of them, in each case, appears to be reliable and was used to fix the slope and to
recalculate the intercept of the second one. From the PL relations, pulsation modes
for the variables are suggested. For NGC 6134 they are similar to those derived
in previous works. For NGC 7062 no prior pulsation mode identification exists,
therefore pulsation mode identification is necessary to confirm these results. For
now, they are only tentative. For NGC 1817 a PL relation was not found because the
variables in the cluster have different pulsation modes. In this case, fundamentalized
periods were derived, with which a PL relation was calculated.
Very simple theoretical PL relations were derived for BVRIJHK bands using
periods, luminosities, effective temperatures, gravity values and metallicity ratios
tabulated for δ Scuti stars. Only 3 points were used to construct the relations.
Despite that, it was found that, while the effective wavelength of the band increases,
the slope value of the PL relations decreases. This is is a consequence of infrared
45
CHAPTER 5. SUMMARY AND FUTURE WORK
bands are less sensitive to temperature changes and interstellar extinction. As a
consequence, PL relations at long wavelengths approach a period-radius relation.
Both for theoretical and observational PL relations derived, the slope values are,
in absolute value, higher than the ones derived by King (1990), which is the only
study about PL relations at infrared bands for δ Scuti stars to date. The reason
for this disagreement is not clear, but, with the considerations described before, his
values are, probably, underestimated.
More detailed models and better photometric measurements are required to study
theoretical and observational Period-Luminosity relations for these variable stars.
High-amplitude delta Scuti stars would be more suitable for this purpose than lowamplitude ones because the former are easier to detect and they are pulsating in
the fundamental or the first radial overtones. However, low-amplitude δ Scuti stars,
as those studied in this work, pose a more interesting challenge for the theoretical
field (for asteroseismology studies), and for the observational efforts improve the
photometry.
46
Annex A
Tables
Table A.1 consists in Period-Luminosity relations with the form MV = a log P +b.
Table A.1: PL relations for delta Scuti stars from a number of sources. .
a
-1.35
-2.6
-3.06
-2.76
-2.28
-1.88 ± 0.24
-1.41 ± 0.39
0.48 ± 0.22
-1.66 ± 0.34
-2.932
-2.942
2.698
-3.74
-3.05 ± 0.05
-3.13 ± 0.04
-3.26 ± 0.04
-2.88 ± 0.17
b
σ Reference
0.33
*
2
C
*
3
-1.88 ± 0.25 *
4
-1.53 ± 0.26 *
-0.93
*
5
-0.46 ± 0.27 *
6
-1.36 ± 0.43 *
7
0.86 ± 0.25
-0.11 ± 0.44 *
10
-1.247
*
11
-1.588
*
-1.771
*
- 1.91
*
12
-1.32 ± 0.06 0.07
13
-1.88 ± 0.05 0.05
-2.43 ± 0.06 0.05
-0.77 ± 0.25 *
14
47
ANNEX A. TABLES
Table A.2 has Period-Luminosity-Color relations for visible filters. An asterisk is
included when dereddened color terms were used of the form
Mv = a log P + b(B − V ) + c
Table A.2: PLC relations for delta Scuti stars from a number of sources. .
a
-2.5
-2.06 ± 0.32
b
3.50
4.14 ± 1.32∗
c
Reference
- 1.51
1
-1.61 ± 0.61
10
Table A.3 has Period-Luminosity-Color relations for the Strömgren color term
(b-y). Dereddened color terms are marked with an asterisk. The form of the PLC
relations are:
MV = a log P + b(b − y) + c.
Table A.3: PLC relations for delta Scuti stars from a number of sources using Strömgren
color terms. .
a
-2.5
-3.23
-2.94
-2.65
-2.21 ± 0.23
-2.64
-3.052
b
5.5
5.6∗
5.1∗
6.75∗
5.44 ± 1.4∗
7.0
8.456
c
Reference
-2.0
3
-3.01 ± 0.22
4
-2.58 ± 0.25
-2.39
5
-1.68 ± 0.39
6
-2.48 ± 0.24
8
-3.121 ± 0.31
9
References: (1) Fernie (1964), (2) Frolov (1969, 1970), (3) Breger (1969), (4)
Leung (1970), (5) Valtier (1972), (6) Elliot (1974), (7) Dworak & Zieba (1975), (8)
Breger and Bregman (1975), (9) Breger (1979), (10) Frolov and Irkaev (1984), (11)
King (1991), (12) Høg & Petersen (1997), (13) Santolamazza et al. (2001), (14) Pych
et al. (2001). (1) ⇒ (10) were directly taken from Tsvetkov (1985).
48
ANNEX A. TABLES
Table A.4: Basic and pulsation data for δ Scuti variables of NGC 7062 inside CAIN’s
FoV and members of the cluster. Adapted from Freyhammer et al. (2001).
ID
GCVS
α2000
δ2000
f
(Cyg)
cd−1
V1 V2448 21 23 29.81 46 22 38.6 14.2
12.2
24.7
21.0
12.8
17.8
25.4
17.4
11.0
V2 V2449 21 23 30.62 46 21 38.4 4.6
2.6
7.4
V3 V2451 21 23 35.86 46 24 11.1 19.2
13.5
19.5
11.5
V5 V2444 21 23 21.72 46 25 12.4 20.2
21.0
22.4
V6 V2443 21 23 21.58 46 22 60.0 13.3
11.1
V11 V2450 21 23 33.50 46 22 8.0 11.6
11.1
V13 V2442 21 23 18.53 46 21 23.3 12.4
49
S/N
V
aV
mmag
16.7 13.94 2.3
12.6
1.6
9.5
8.3
19.7
2.3
7.5
5.1
6.2
11.3
21.7 13.38 10.4
6.1
4.7
5.5
2.8
9.6 14.41 3.6
9.9
2.8
4.6
4.6
7.6 14.47 1.8
7.1
6.4
7.0 13.41 1.0
4.7
1.0
7.1 15.61 1.6
6.5
7.0 14.92 1.8
ANNEX A. TABLES
Table A.5: Basic and pulsation data for δ Scuti of NGC 1817 inside CAIN’s FoV. Adapted
from Arentoft et al. (2005).
ID
GCVS
(Tau)
V1
V1182
V2
V1181
V3
V1179
V6
...
V8
...
α2000
δ2000
f S/N
cd−1
5 12 42.82 16 41 43 19.94
13.17
10.22
6.93
14.86
20.53
22.02
23.91
8.88
14.57
5 12 40.81 16 42 00 18.52
19.05
19.99
17.21
15.14
5 12 37.40 16 42 31 18.57
18.27
25.62
30.09
19.83
5 12 33.08 16 41 50 18.07
20.70
14.90
16.59
5 12 33.71 16 43 22 20.93
29.28
37.91
20.30
18.69
Continued on next page. . .
50
V
aV
mmag
16.8 13.49 1.3 - 5.0
n
5
2
18.6 12.87 0.8 - 2.9 7/8
7/8
8.8 14.37 2.2 - 4.1
1
1
6.6 12.93 0.7 - 1.0
7
8
11.1 14.32 0.9 - 2.3
2
4
ANNEX A. TABLES
Table A.5 – Continued
ID
GCVS
(Tau)
V11
...
V17
...
α2000
δ2000
f S/N
cd−1
V
aV
mmag
n
25.51
31.23
16.85
23.78
32.64
5 12 30.37 16 41 29 4.02 6.3 14.30 1.4 - 1.6 21.57
2
5 12 37.48 16 43 56 2.04 13.1 16.97 6.1 - 19.6
1.39
51
ANNEX A. TABLES
Table A.6: Basic and pulsation data for δ Scuti variables in NGC 6134. Adapted from
Frandsen et al. (1996). Declination of # 161 was corrected from the paper according to
GCVS coordinates for V357 Nor.
ID
9
GCVS
(Nor)
V356
α2000
δ2000
16 27 48.77 -49 10 43.8
87
V386
16 2740.04 -49 10 25.4
159
V355
16 27 39.87 -49 09 41.1
161
348
V357
V388
16 27 51.49 -49 09 20.3
16 27 49.16 -49 06 43.7
397
V387
16 27 43.14 -49 07 24.1
52
f
S/N
(µHz)
63.5 11.4
61.2
7.9
65.6
9.5
59.3
7.5
89.8
5.1
200.5 10.5
162.4 4.2
220.5 4.2
139.3 10.4
244.8 7.2
116.1 7.9
87.3 22.1
133.5 12.9
76.4
8.8
168.1 5.2
324.3 4.8
V
12.26
13.52
13.15
11.94
12.43
13.57
aV
(mmag)
9.7
6.7
8.1
6.4
4.3
2.0
0.8
0.8
5.9
4.1
4.5
6.2
6.2
4.2
1.7
1.6
ANNEX A. TABLES
Table A.7: 2MASS magnitudes for δ Scuti stars observed in NGC 7062.
ID J (mag) eJ (mag) H (mag) eH (mag) Ks (mag) eKs (mag)
V1 12.347
0.022
12.084
0.024
11.980
0.022
V2 11.938
0.022
11.689
0.023
11.603
0.019
V3 12.956
0.023
12.684
0.026
12.579
0.026
V5 13.208
0.020
13.024
0.026
12.911
0.036
V6 12.207
0.023
12.042
0.024
11.929
0.022
V11 13.619
0.032
13.313
0.043
13.161
0.040
V13 13.197
0.032
13.129
0.046
13.053
0.045
Table A.8: 2MASS magnitudes for δ Scuti stars observed in NGC 1817.
ID J (mag) eJ (mag) H (mag) eH (mag) Ks (mag) eKs (mag)
V1 12.521
0.023
12.390
0.026
12.315
0.025
V2 11.815
0.026
11.654
0.027
11.555
0.028
V3 13.258
0.023
13.083
0.026
12.979
0.032
Table A.9: 2MASS magnitudes for δ Scuti stars observed in NGC 6134.
ID J (mag) eJ (mag) H (mag) eH (mag) Ks (mag) eKs (mag)
9
10.621
0.024
10.319
0.024
10.198
0.023
87 12.147
0.033
11.855
0.034
11.729
0.034
161 10.397
0.029
10.110
0.028
9.999
0.026
348 10.954
0.024
10.681
0.024
10.588
0.025
397 12.141
0.024
11.881
0.022
11.805
0.025
53
ANNEX A. TABLES
Table A.10: PL relation coefficients for δ Scuti in NGC 7062. Relation # 1 corresponds
to V1, V3, V5 and V6 and # 2 is for V2, V11 and V13.
Band Relation
a
H
#1
-5.356
#2
-4.564
J
#1
-5.286
#2
-4.335
K
#1
-5.064
#2
-4.414
σa
0.827
0.109
0.508
0.720
1.217
0.600
b
5.927
8.657
6.232
9.059
6.142
8.634
σb
1.008
0.105
0.620
0.692
1.483
0.576
rms
0.092
0.021
0.057
0.141
0.136
0.118
Table A.11: PL relation coefficients for δ Scuti in NGC 6134. Relation # 1 corresponds
to # 348 and # 161 stars and # 2 is for # 9, # 87, # 159 and # 397.
Band Relation
a
σa
b
σb
rms
H
#1
-3.089
0
7.299
0
0
#2
-3.259 0.307 7.864 0.330 0.083
J
#1
-2.867
0
6.853
0
0
#2
-3.207 0.314 7.268 0.337 0.085
K
#1
-2.900
0
7.440
0
0
#2
-3.305 0.319 7.785 0.342 0.086
54
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