A&A 374, 243-249 (2001)
DOI: 10.1051/0004-6361:20010728
M. Wolf 1 - R. Diethelm 2 - K. Hornoch 1
1 - Astronomical Institute, Charles University Prague,
180 00 Praha 8, V Holesovickách 2, Czech Republic
2 - Astronomical Institute, University of Basel,
Venusstrasse 7, 4102 Binningen, Switzerland
Received 6 March 2001 / Accepted 21 May 2001
Abstract
Several new times of minimum light recorded with
photoelectric means have been gathered for two bright eccentric
eclipsing binaries
TV Cet (
)
and
V451 Oph (
).
Analysis of all available eclipse timings of TV Ceti has revealed
a small motion of the line of apsides of
deg cycle-1,
corresponding to an apsidal period of
years.
The contribution from the general relativity effects is dominant
(
).
In this system, the third body on an eccentric orbit with a period
of 28.5 years is also predicted.
The more precise values for the apsidal motion elements were computed
for V451 Oph, where apsidal motion with a period of
years was
confirmed. The corresponding internal structure constants log k2 were
derived.
Key words: stars: binaries: eclipsing - stars: individual: TV Cet - stars: individual: V451 Oph - stars: fundamental parameters - relativity
Eclipsing binaries are excellent laboratories for studying a wide variety of processes in stellar astrophysics. Their usefulness extends far beyond their textbook role in the determination of stellar masses and radii. The study of apsidal motion in detached eclipsing binary systems with eccentric orbit is known as an important source of information for the stellar internal structure as well as the possibility of verification of general relativity. Suitable objects for this research were recently collected by Giménez (1994).
In this paper, we report new results for our observational project initiated in 1993 with the main purpose of monitoring eclipsing binaries with eccentric orbits, see also Wolf et al. (1999) or Wolf (2000). In this paper, we compare observational data of apsidal motion rates for two eclipsing systems with significant relativistic contribution. The two relatively bright equatorial objects, TV Cet and V451 Oph, are analysed here. V451 Oph is a well studied early-type eclipsing binary, whose orbit has been known to be eccentric and to exhibit apsidal motion. On the other hand, TV Ceti, due to its rather long orbital period, is a rarely investigated system of similar brightness.
System |
JD Hel.- | Error | Epoch | Method | Reference |
2400000 | [day] | Filter | Observatory | ||
TV Cet | 46687.8088* | 0.0093 | 549.5 | pe, R | Caton & Hawkins (1987) |
47147.58576 | 0.00022 | 600.0 | pe, V | Caton et al. (1989) | |
47566.338 | 0.001 | 646.0 | pe, B | Diethelm (1989) | |
48135.2339 | 0.0020 | 708.5 | pe, H | Hipparcos | |
48622.3207 | 0.0020 | 762.0 | pe, B | Diethelm (1992) | |
50470.2868* | 0.0004 | 965.0 | CCD, - | Agerer & Hübscher (1998) | |
51421.5165 | 0.0003 | 1069.5 | pe, UBV | this paper, SAAO | |
51776.544 | 0.001 | 1108.5 | CCD, R | this paper, Ondrejov | |
51899.5004 | 0.0001 | 1122.0 | CCD, R | this paper, Lelekovice | |
V451 Oph | 46210.530 | 0.002 | 147.5 | pe, B | Diethelm (1985) |
46298.392 | 0.002 | 187.5 | pe, B | Diethelm (1985) | |
47307.73282 | 0.00035 | 647.0 | pe, V | Caton et al. (1989) | |
47437.3303 | 0.000 | 706.0 | pe | Isles (1991) | |
47654.79386 | 0.00028 | 805.0 | pe, V | Caton et al. (1989) | |
47696.5275 | 0.000 | 824.0 | pe | Isles (1992) | |
48108.3891 | 0.0005 | 1011.5 | pe, B | Diethelm (1990) | |
48476.3240 | 0.000 | 1179.0 | pe, BV | Isles (1997) | |
48499.3838 | 0.0008 | 1189.5 | pe | Blättler (1992) | |
48655.342 | 0.002 | 1260.5 | pe, H | Hipparcos | |
49560.3397 | 0.0004 | 1672.5 | pe, UBV | Müyesseroglu et al. (1996) | |
51415.3614 | 0.0003 | 2517.0 | pe, BV | this paper, SAAO | |
51773.4089** | 0.0006 | 2680.0 | CCD, - | this paper, Metzerlen | |
51774.5105 | 0.0001 | 2680.5 | CCD, R | this paper, Lelekovice | |
51806.3563 | 0.0001 | 2695.0 | CCD, R | this paper, Lelekovice |
In order to enlarge the number of times of minimum light, new observations for both systems were carried out. Our new photoelectric photometry was performed at four observatories with the aim of securing several well-covered primary and secondary minima for each variable:
The CCD measurements in Ondrejov and Lelekovice were done using
the standard Cousins R filter. Flat fields for the reduction of the
CCD frames
were routinely obtained from exposures of regions of the sky taken
at dusk or dawn. Several comparison stars were chosen in the same frame
as the variables. During the observations, no variations in the brightness
of these stars exceeding the possible error of measurements
(typically
0.005 mag) were detected. No correction was allowed
for differential extinction, due to the proximity of the comparison stars
to the variable and the resulting negligible differences in the air mass.
Photoelectric observations at SAAO were obtained with the modular photometer utilizing a Hamamatsu GaAs R943-02 photomultiplier during two weeks in August 1999. Each observation of an eclipsing binary was accompanied by observation of a local comparison star. The photoelectric measurements were done in the UBV filters of the Johnson photometric system with 10 s integration time. All observations were reduced to the Cousins E-region standard system (Menzies et al. 1989).
The new times of primary and secondary minimum and their errors were determined using the least squares fit of the data, by the bisecting cord method or the Kwee-van Woerden algorithm. Only the lower part of the eclipse was used. These times of minimum are presented in Table 1. Some of the published moments were newly determined using only the lower part of its observed light curve.
The apsidal motion in both systems was studied by means of an O-C diagram analysis. We have collected all reliable times of minimum light gathered from the literature as well as from current databases of BAV and BBSAG observers or from the Besançon Double and Multiple Star Database. All photoelectric times of minimum were used with a weight of 10 in our computation. The current less precise measurements were weighted with a factor of 5, while the earlier visual and photographic times of minimum were given a weight of 1 or 0 due to the large scatter in these data.
The detached eclipsing binary TV Cet
(also HD 20173, BD
502, SAO 111068,
HIP 15090, PPM 146367, AN 270.1934, FL 266;
,
,
mag; Sp. F2+F5)
is a rarely investigated binary with an eccentric orbit
(e = 0.055) and relatively long orbital period of about 9.1 days.
It was discovered to be a variable by Martynov (1951),
who derived the first light elements
All photoelectric times of minimum light published in Meyer (1972), Jørgensen (1979), Caton & Hawkins (1987), Caton et al. (1989) as well as Agerer & Hübscher (1998) were incorporated in our analysis. A total of 20 times of minimum light were used in our analysis, with 9 secondary eclipses among them.
![]() |
Figure 1: O-C graph for the times of minimum of TV Cet. The continuous and dashed lines represent predictions for primary and secondary eclipses, respectively. The individual primary and secondary minima are denoted by circles and triangles, respectively. Larger symbols correspond to the photoelectric measurements which were included in calculations with higher weight. |
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The eclipse timings listed partially in Table 1 allow the determination
of linear ephemerides indepedently for primary and secondary eclipses
with the following results(numbers between parentheses indicate errors
in the last digits and E is the number of cycles):
Adopting the orbital inclination derived from the light curve
solution of
(Jørgensen 1979), the apsidal motion
elements can be computed. The parameters found and their internal
errors of the least squares fit (in brackets) are given in Table 2.
In this table
denotes the sidereal period,
the anomalistic
period, e represents the eccentricity and
is the rate
of periastron advance (in degrees per cycle or in degrees per year).
The zero epoch is given by T0 and corresponding position of
the periastron is represented by
.
The relation between the sidereal and the
anomalistic period,
and
,
is given by
![]() |
Figure 2: O-C2 diagram for the times of minimum of TV Cet after substraction the terms of apsidal motion. The curve represents a light-time effect for the third body orbit with a period of 28.5 years and an amplitude of about 0.003 days. The individual primary and secondary minima are denoted by circles and triangles, respectively. |
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Subtracting the influence of apsidal motion, the O-C2 diagram in Fig. 2 can be plotted. The sinusoidal variation of these values are remarkable and could be caused by a light-time effect. A preliminary analysis of the possible third body orbit gives the following parameters:
P3 (period) | =
![]() |
= 28.5 years | |
T3 (time of periastron) | = JD
![]() |
A (semiamplitude) | =
![]() |
e3 (eccentricity) | =
![]() |
![]() |
=
![]() |
These values were obtained together with the new
mean linear ephemeris
The acceleration of the rate of apsidal motion
caused by the presence of the third body
(Martynov 1973) is
![]() |
(1) |
![]() |
(2) |
More precise non-linear light elements of the eclipsing pair including
the term of the light-time effect with the circular orbit are
Element | TV Cet | V451 Oph |
T0 [HJD] | 24 41685.5839 (3) | 24 45886.5310 (2) |
![]() |
9.10328692 (8) | 2.19659703 (12) |
![]() |
9.10329452 (8) | 2.19667490 (12) |
e | 0.0545 (4) | 0.0120 (5) |
![]() ![]() |
0.00030 (8) | 0.0128 (4) |
![]() ![]() |
0.012 (3) | 2.12 (6) |
![]() |
100.9 (0.2) | 253.4 (0.5) |
U [years] | 30000 (8000) | 170 (5) |
The double-lined detached eclipsing binary V451 Oph
(also HD 170470, BD
3528, SAO 103766,
HIP 90599, PPM 134790, AN 104.1935, FL 2483;
,
,
mag; Sp. B9-A0) is a relatively bright and
well-known binary with a slightly eccentric orbit (e=0.0125).
It was discovered to be variable by Hoffmeister (1935).
Colacevich (1953) obtained the first photoelectric light curve,
determined the photomeric elements and found the correct period
of the system
.
The first spectroscopic study was presented by Heard & Morton (1962),
who derived the absolute parameters of this system
(
,
,
,
).
Next spectroscopic study was carried out by Popper (1971), who obtained
the radial velocity curves with semiamplitudes
K1 = 129.3 kms-1 and
K2 = 152.3 kms-1.
Apsidal motion in V451 Oph was first announced by Plavec et al. (1960).
Several studies have been made of the apsidal motion rate of V451 Oph
since its discovery as an eccentric eclipsing binary.
Other photometric study has been published by Giuricin et al. (1980).
Khaliullin & Kozyreva (1984) obtained four-colour WBVR photometry
at the Tian-Shan Observatory, finding an apsidal motion with a period
of U = 170 yr, the internal structure constant
and an eccentricity of e = 0.012.
Complete simultaneous Strömgren four-colour photometry of V451 Oph
was obtained in 1983-84 at Sierra Nevada in Spain (Clausen et al. 1987).
The most detailed analysis is that by Clausen et al. (1986) using
those data.
They obtained separately way and using different methods, apsidal
motion rates of
deg/cycle, with the period of
yr and a value for orbital eccentricity of
,
in good agreement with each other given their expected mean errors.
The internal structure constant log
was also calculated.
In that paper the following linear ephemerides were obtained for
primary and secondary eclipses, respectively:
![]() |
Figure 3: O-C diagram for V451 Oph. See legend for Fig. 1. |
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The system is known, moreover, to have a large relativistic contribution to the observed apsidal motion rate. This made V451 Oph a case of special interest from the point of view of the orbit rotation. Since the mentioned papers, several new times of minima have been measured and thus allow us to decrease the involved observational uncertainties.
Recently, the relativistic apsidal motion in this binary was discussed by Claret (1997), who found excellent agreement of observational data with theoretical prediction based on stellar theoretical models.
All photoelectric times of minimum light given in
Pohl & Kizilirmak (1970, 1975),
Kizilirmak & Pohl (1971, 1974),
Batistini et al. (1974),
Khaliullin & Kozyreva (1984) and
Clausen et al. (1986, their Table 2)
were incorporated in our calculation.
From the Hipparcos photometry (Perryman 1997), we were able to determine
one additional time of minimum light.
A total of 67 photoelectric times of minimum light were used in our
analysis, with 29 secondary eclipses among them.
The orbital inclination was adopted to be
based
on the photometric analysis (Clausen et al. 1986). The computed
apsidal motion elements and their internal errors of the least squares
fit are given in Table 2. The O-C diagram is shown in Fig. 3.
More precise non-linear light elements including one term in
e are
![]() |
Figure 4: O-C2 diagram for the times of minimum of V451 Oph after substraction the terms of apsidal motion. The individual primary and secondary photoelectric minima are denoted by circles and triangles, respectively. |
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Parameter | Unit | TV Cet | V451 Oph |
M1 | ![]() |
1.39 (5) | 2.78 (6) |
M2 | ![]() |
1.27 (4) | 2.36 (5) |
r1 | 0.05872 (13) | 0.2155 (20) | |
r2 | 0.05029 (15) | 0.1655 (20) | |
Source | Jørgensen | Clausen et al. | |
(1979) | (1986) | ||
![]() |
deg
![]() |
0.00024 | 0.0010 |
![]() |
% | 80.2 | 7.8 |
![]() |
deg
![]() |
0.00006 | 0.0118 |
log
![]() |
-2.03 (15) | -2.477 (45) | |
log
![]() |
-2.23 | -2.48 |
Observations of binary systems allow us to determine the internal structure constant, k2, which is related to the variation of density within the star and is an important parameter of stellar evolution models. It is best studied in binary systems with eccentric orbits that show apsidal motion. The period of rotation of the periastron in eccentric eclipsing binaries does not allow us to derive the individual internal stellar constant of the component stars.
System | Spectral | e | P | U | e3 | P3 | Source |
type | [days] | [years] | [years] | ||||
HS Her | B4+A4 | 0.021 | 1.637 | 72.4 | 0.7 | 74.1 | Wolf (2001) |
U Oph | B5+B5 | 0.003 | 1.677 | 21.2 | 0.22 | 38.7 | Kämper (1986) |
DR Vul | B0+B0.5 | 0.095 | 2.251 | 36.3 | 0.73 | 62.8 | Wolf et al. (1999) |
YY Sgr | B6+B5 | 0.159 | 2.628 | 290.4 | 0.44 | 44.3 | Wolf (2000) |
RU Mon | B7+B7 | 0.396 | 3.585 | 347 | 0.46 | 73.3 | Wolf et al. (1999) |
TV Cet | F2+F5 | 0.055 | 9.103 | 30000 | 0.25 | 28.5 | this paper |
The observational average value of
is given by the relation
![]() |
(3) |
Taking into account the value of the eccentricity and the masses
of the components, one has to subtract from
a relativistic correction
(Levi-Civita 1937;
Giménez 1985)
![]() |
(4) |
![]() |
(5) |
The values of
and resulting mean internal
structure constants
are given in Table 3.
Theoretical values
according to available theoretical models
for the internal stellar structure along the main sequence, computed by
Claret & Giménez (1992) for a variety of masses and chemical compositions,
are also given.
We derive updated apsidal motion elements for two eccentric eclipsing binaries by means of an O-C diagram analysis. Both systems analysed here present important relativistic contributions to the total apsidal motion.
Concerning TV Cet, we have detected apsidal motion at a rate
deg cycle-1.
A substantial fraction of this (
80%) is due to the
contribution from general relativity and our measurement
is consistent with theory.
The difference between theoretical and observed values
of the internal structure constants is caused probably by
the relatively high uncertainty of the observed rate of apsidal
motion.
In the case of V451 Oph, our resulting orbital eccentricity and the period of apsidal motion is in a good agreement with elements previously obtained by Khaliullin & Kozyreva (1984) and Clausen et al. (1987). A few more times of minimum added over the next decade will make V451 Oph one of the best investigated systems in the observational check of stellar structure models. Nevertheless, the scatter of O-C values seems to be relatively high for the detached eclipsing system. Substracting the influence of apsidal motion, the O-C2 diagram in Fig. 4 can be plotted. From this diagram, no other phenomenon (i.e. presence of a third body in the system) can be simply derived. The differences from the zero line are substantially larger than the standard errors of many observed times of minimum.
A small group of triple eccentric eclipsing binaries presenting apsidal motion and light-time effects, is given in Table 4. These excellent labratories of celestial mechanics deserve a continuous monitoring. More high-accuracy timings of these systems are necessary in the future in order to enlarge the time span for a better analysis of the apsidal motion, especially in TV Cet, where the light-time effect is only poorly determined.
Acknowledgements
This investigation was supported by allocation of SAAO observing time, in part by the Grant Agency of the Czech Reublic, grant No. 205/99/0255 and by the research plan J13/98: 113200004 Investigation of the Earth and the Universe. MW wishes to thank Dr. David Kilkenny, SAAO, for help with the photometric data reductions. We are also thankful to Mr. Wilhelm Kleikamp, BAV, for sending his original data of TV Ceti, and to Dr. Valentina Kozyreva, SAI, Moscow, for supplying us with her photoelectric measurements of V451 Oph. RD wishes to thank the Emilia Guggenheim-Schnurr Foundation for financial support. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France, and of NASA's Astrophysics Data System Bibliographic Services.