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Investigating the properties of the near-contact binary system TW CrB.

Introduction

Close binary star systems are classified according to the shape of their combined lightcurve, as well as the physical and evolutionary characteristics of their components. Key factors in their classification are the position of the components relative to the barycentre of the system, their colour, magnitude and in particular the degree to which their Roche lobes have been filled. Eclipsing binaries have orbital planes close to the observer's line of sight so that components eclipse each other with a consequent periodic change in the brightness of the system.

TW CrB is such an eclipsing binary star. Its short period of less than one day, plus the degree to which the components fill their respective Roche lobes, results in a classification of near-contact binary that may possess close evolutionary connections with the W UMa systems.1 In these systems the onset of eclipses is difficult to pinpoint exactly from their lightcurves due to the component stars' ellipsoidal shapes, resulting from their mutually strong gravitational interaction.

TW CrB has been studied since 1946, but to our knowledge no spectroscopic radial velocity measurements have been published. The first ephemeris for TW CrB was compiled in 1973 by V. P. Tsesevich of the Astronomical Observatory, Odessa State University, Ukraine, based upon fourteen datasets. Extensive data have been subsequently published by BBSAG (the Swiss Astronomical Society) and other authors leading to a review of this binary system by Zhang & Zhang in 2003.1 The analysis by Zhang & Zhang spanned the period 1974 to 2001 and provided a revised ephemeris which indicated that the system is a detached near-contact binary with rapidly increasing period. Their analysis also suggested that the primary component was a slightly evolved main sequence star of spectral type F8 with an under-massive secondary.

More recently Caballero-Nieves et al. reported that their observations using multiband photometry indicated a combination of an early A0 and late K0 spectral type. (2) In addition there was a strong suggestion that one or both of the components exhibited a hot spot with at least one of these objects filling its Roche lobe.

In 2010 and 2011 we performed CCD photometry on TW CrB and these times of minima, together with all known earlier timings, have allowed us to construct new lightcurves and re-compute the ephemeris based upon 200 datasets spanning the period 1946 to 2011. In this paper we present these observations together with a revised photometric and simulation analysis of this binary system.

Observations

Using frames taken on 2010 May 25 & 30 at the remotely operable Sierra Stars Observatory, California, we measured the position of target star TW CrB with Astrometrica using the UCAC3 (US Naval Observatory Astrograph CCD, 2009) catalogue. (3) The position was measured as: RA 16h 06m 50.679 [+ or -] 0.007s, Dec +27[degrees]16' 34.62" [+ or -] 0.08" (2000). The Tycho catalogue lists the parallax of TW CrB as 31.1 milli-arcsec (mas), equivalent to a distance of 32.2 parsec, with a proper motion (pmRA) of -27.5 mas/yr and pmDE of -4.2mas/yr.

Our observations of TW CrB were carried out during May and June of 2010 and 2011. These observations were made from the four different locations listed in Table 1.

Throughout our work all recorded times were corrected to heliocentric Julian dates (HJD). Early observing sessions concentrated on capturing as much information from the target as possible using Johnson B, V and R filters. Analysis of the data from these early sessions was then used to plan later observing sessions. At the start of every observing session we took dusk flats using the filters scheduled for use in the evening's observing session. Bias frames and dark frames covering the exposures planned for the session were also taken. Frames were selected at random for assessing image quality which included checking saturation levels, monitoring SNR ratios, looking for signs of star trails and other irregularities.

Photometric analysis

Reference stars and differential photometry

We carried out our initial photometric analysis using the MaxIm DL software package employing differential aperture photometry to generate lightcurves for each band in order to determine the system's minima. (4) Using the Aladin Sky Atlas, (5) a star known to be of constant magnitude was selected as a reference star (Table 2a), which is then compared with the changing magnitude of the target star using the software package. Other non-variable stars (Table 2a) were also selected that were of similar magnitude to the target star. These check stars were not used in the analysis, but were used to ensure that there was no variability in the reference star. The standard deviation of the check star lightcurves and the SNR values of the target star were used to calculate uncertainties. The phase folded lightcurves obtained are shown in Figure 1.

Catalogue-based magnitudes of TW CrB were derived using the photometry software package Canopus. (6) The results of this analysis were used to generate phase and normalised flux values compatible with the Binary Maker 3 system modelling programme. (7) Canopus V10 makes available the magnitudes taken from the Carlsberg Meridian Catalogue 14 (CMC 14) and the Sloan Digital Sky Survey (SDSS) catalogue and transforms the J-K magnitude of 2-MASS and the r' magnitude of SDSS to BVRI magnitudes. The values obtained from these sources are consistent to within 0.02 magnitudes when using a calibration method involving the Canopus add-on, Comparison Star Selector, which picks comparison stars of similar colour and magnitude to the target star.

[FIGURE 1 OMITTED]

Four comparison stars were used for the Canopus photometry and the same comparison stars were used for each image. Table 2b shows the locations of these comparison stars and their magnitudes. Using the average derived magnitudes of the target star from each comparison star and the standard deviation of the average, the final value for the target star was obtained for each frame. The standard deviation incorporates the uncertainty in the measurement of the target and comparison stars, taken from the SNR values of the target and the reference star, the uncertainty in the catalogue value and the uncertainty in the correction for colour difference.

In order to reduce scatter and to enable smoothing of the phased light curves, adjacent data points up to a maximum interval of four minutes were binned and averages computed. A normalised flux lightcurve was then obtained for each of the three bands using a Fourier transform fit. This binning had virtually no effect on the Fourier coefficients (to the third decimal place) nor to the normalised phased and flux values used in Binary Maker 3 modelling.

Qualitative lightcurve analysis

Visual inspection of the lightcurve provided basic information about this binary system. We conclude that: (i) the system does not have a relatively flat 'out-of-eclipse' lightcurve; (ii) the system shows significant differences in eclipse depths indicating differences in temperatures for the two components; (iii) the lightcurves vary continually, even when not eclipsing, because the components' visible cross-sectional areas are continually changing--this implies that the surfaces of these two components are in close proximity to the critical Roche equipotential containing the inner Lagrangian point, and their shapes are being distorted into ellipsoids because of gravitational and tidal forces; (iv) at minima, the light curves are not flattened so the eclipses are not total; (v) the short orbital period of the system, less than 0.6 days, is indicative of their close proximity and (vi) in the combined band phase plots, Figure 1, it can be seen that the system becomes redder during the eclipses as can be seen by the B lightcurve fading more than the V and R lightcurves.

We can conclude from the lightcurve that the two stars are unlikely to be in contact, or over contact, although the possibility cannot be ruled out.

Visual inspection of the lightcurves, once data misalignment has been excluded, indicates the presence of the O'Connell effect whereby the two out-of-eclipse maxima of the lightcurves are unequal. (8,9) The maxima are expected to be equally high, because the observed luminosity of an eclipsing binary system when the two components are side by side should be equal to the luminosity in the configuration half an orbital period later when they have switched positions. The O'Connell effect is an area of ongoing research. This particular feature was not reported by earlier observers which suggests that this is an active system that may have been in a period of quiescence. (1,2)

Period analysis

All known times of minima for TW CrB are listed in Table 3. This includes the 112 timings contained in the Zhang & Zhang analysis, (1) and a further 91 timings taken from other related sources identified in Table 3. The datasets include 12 times of minimum obtained from our CCD measurements using The Open University's remotely-operable PIRATE facility (2010), (10) and Sierra Stars Observatory, California (2010 and 2011). These timings were analysed using the Kwee-van Woerden methodology, (11) contained within the Peranso period analysis software. (12)

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

All pre-1974 timings are from photographic imaging and post-1998 timings are from either photoelectric (pe) or CCD imaging. With the exception of one photoelectric observation (E = -3,058.5) all timings recorded between 1974 and 1998 are visual. The timings of minima span of 65 years permits a detailed investigation into the long term variation in period of TW CrB. When calculating the ephemerides we have assigned weightings of 1 to visual minima; 2 to photographic minima and 10 to CCD and pe minima. This was in line with the weighting applied by Zhang & Zhang. (1)

Linear and second order polynomial regression analysis was applied to the 203 timings to generate new linear and quadratic ephemerides. The plot of the O-C residuals led us to eliminate three datasets (E = -2,335.0; -2,323.0; + 1,484.0) whose O-C values deviated by more than 4 sigma, for their observing methodology, from the quadratic fit. The resulting ephemerides, with standard errors in parenthesis, calculated from the remaining 200 timings are:

HJD [Min.sub.lin] = 2,451,273.4740(2) + 0.58887492(2)E [1] HJD [Min.sub.quad] = 2,451,273.4701(1) + 0.58887562(2)E+ 5.37(11) x [10.sup.-11][E.sup.2] [2]

Our calculated linear O-C residuals are listed in Table 3 and displayed in Figure 2, together with the quadratic curve for the elements above. This curve suggests that the period of TW CrB is increasing with time and consistent with secular mass transfer between the binary components. The average rate of period increase, taken from the quadratic ephemeris, equates to 1.07(2) x [10.sup.-10] days per cycle or 6.66(14) x [10.sup.-8] days per year.

Lightcurve simulation

We simulated the observed lightcurves using the software package Binary Maker 3. (7) This program is similar to, but not a derivative of, the Wilson-Devinney code; (13) it uses the same nomenclature but does not have the same error handling capability. To compensate for this we used a range of values in our simulation. The program allows the adjustment of a number of system parameters and then allows comparison of the calculated lightcurve with the observed lightcurve.

[FIGURE 4 OMITTED]

[T1.sub.eff] was determined to be 5700K [+ or -] 200K. This was derived from Allen's Astrophysical Quantities using the (J-K) = 0.41 derived from the 2MASS catalogue entry for TW CrB. (14) This suggests a spectral type for TW CrB components of early G and early K. All other parameter values were derived by minimising the difference between the observed lightcurve and the lightcurve calculated with Binary Maker 3.

Since V was the middle passband of those in which we observed TW CrB we used this band to obtain the non-wavelength-dependent parameters. We then adjusted only the wavelength-dependent parameters to obtain a match with the R and B passband lightcurves.

[FIGURE 5 OMITTED]

As [T1.sub.eff] is 5700K [+ or -] 200K this means that both [T1.sub.eff] and T[2.sub.eff] are well below 7500K and implies that both components are fully convective, hence we are justified in setting the gravity brightening coefficients g1 and g2 to 0.32. Similarly the reflection coefficients Alb1 and Alb2 can be set to 0.5. The limb darkening coefficients were derived from the Van Hamme table. (15) Using this information T[2.sub.eff] was determined to be 5400K [+ or -] 150K.

Not having the spectroscopic data on this system needed to fix the mass ratio, q, this was estimated by minimising mass ratio over a range of values for [T1.sub.eff] and T[2.sub.eff]. This led to a mass ratio of 0.725 [+ or -] 0.010 (see Figure 3).

In order to achieve a best fit it was necessary to set the fillout factors to -0.03281 and -0.0252, respectively. These factors indicate the degree to which the stars' physical surfaces are inside their Lagrangian surfaces. These numbers are small and negative, indicating that both stars are nearly filling their Roche lobes.

Visual inspection of the lightcurves indicates that the amplitude of the B lightcurve is greater than either the R or V. This preponderance was proven in statistical analysis performed by the authors but not presented here. Figure 4 shows the observed and calculated lightcurve for the B filter.

The increase in amplitude of the blue lightcurve, taken with the secondary being close to filling its Roche lobe, led to the introduction of a hot spot on the primary. This is consistent with the allusion to mass transfer by Zhang & Zhang and the conclusions reached by Caballero-Nieves et al., who noted the same phenomena in their colour difference analysis of the system. (1,2)

In order to further improve the fit on the lightcurve shoulders it was necessary to introduce two starspots on the secondary and the combined effect can be seen in Figure 5 when compared with Figure 4. Figures 6 and 7 show the lightcurves, with starspots and hotspot, for the V and R passbands. This is shown graphically in Figure 8.

Table 4 lists the derived system parameters.

Discussion

Orbital period behaviour

Examination of Figure 2 for epochs greater than zero shows that there is a systematic error with most (O-C) residuals lying above the quadratic curve for epochs up to about 4,000 and below the curve for epochs greater than 4,000. This systematic error suggests that a model of secular mass transfer between the binary components is incomplete or inappropriate for TW CrB. Possible alternative explanations considered are (i) abrupt (episodic) mass ejections or transfers; or (ii) cyclical changes possibly due to either magnetic effects or the presence of a third body.

[FIGURE 6 OMITTED]

Abrupt (episodic) changes

Abrupt or episodic mass transfers can cause step changes in the orbital period of a binary system. This can be represented by a series of linear ephemerides and the corresponding (O-C) plot will consist of straight lines, each reflecting an interval of constant orbital period. Examination of Figure 2 suggests that two straight lines can be fitted to the (O-C) residuals as shown in Figure 9, with the first abrupt change occurring prior to epoch 32,061 and the second between epochs -10,951.5 and -6,791.0. The corresponding linear ephemerides are:

HJD [Min.sub.lin1] = 2,451,273.4470 (1) + 0.58887329 (8)E (for E < -10.951.5) [3]

HJD [Min.sub.lin2] = 2, 451,273.4706 (1) + 0.58887584 (2)E (for E > -6,791.0) [4]

However this model does not resolve all issues, particularly: (i) it does not fully explain the systematic errors in the (O-C) values seen in Figure 2 for epochs greater than zero. This is more clearly illustrated in the centre plot of Figure 10; (ii) abrupt or episodic changes would require the period to remain constant between each mass ejection. Figure 11 shows the results of an analysis of the period of TW CrB measured over 13 year intervals from 1946, which clearly shows the orbital period to be continually changing; (iii) although TW CrB is a relatively close binary system, recorded at 32.2 pc, there appears to be no supporting observational evidence for episodic mass ejections; (iv) our lightcurve simulation of TW CrB requires the presence of a hotspot which lends support to some form of continuous mass transfer model and not to an episodic event.

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

Cyclical changes

Some of the issues raised with the abrupt change model can be resolved by introducing a sinusoidal term into the quadratic ephemeris of Eqn [2]. This takes the form of Eqn [5] which combines the secular orbital period increase with a superimposed cyclical change to the orbital period:

[HJD.sub.min] = A + BE + [CE.sup.2] + D sin([omega]E + [pi]) [5]

Solutions to Eqn [5] can be found iteratively using the Levenberg-Marquardt technique, which has been implemented with Origin-Lab software. (16) In this implementation, the coefficients A, B and C were taken from the quadratic ephemeris of Eqn [2]. These coefficients were held constant whilst varying D, [omega] and [pi] to find the best fit to the restricted data set of E [greater than or equal to] 0. This range of E (epoch) was chosen as it makes use of only the more precise CCD and pe timings. The parameters derived from the solution to Eqn [5] are listed in Table 5.

Applegate (17) has proposed that binary period modulation can occur when there is magnetic activity in one of the stars of a binary system. Such activity can lead to changes of oblateness and angular momentum and, through gravity coupling, to orbital period modulation. Our calculated modulation period for TW CrB is consistent with Applegate's findings, but constraining the data to the more precise CCD and pe timings restricts this analysis to approximately 45% of one modulation period. Further timings during the predicted modulation period will be necessary to confirm this explanation. Equation [5] also makes the assumption that the modulation period of the binary is a constant. This is not necessarily the case when magnetic effects are present and variations in the binary modulation period may be observed.

[FIGURE 9 OMITTED]

The (O-C) residuals for the three approaches are drawn in Figure 10 for E [greater than or equal to] 0. The upper plot for the quadratic residuals is de rived from Equation [2] which clearly shows the systematic error for epochs greater than zero. The middle plot is for episodic changes derived from Equation [4] and shows that some systematic errors remain. The bottom plot is for the sinusoidal residuals derived from Equation [5] and indicates that this is the best fit of the three models, i.e. it is the closest to the zero residual horizontal line in the plot.

[FIGURE 10 OMITTED]

Another possible explanation for the O-C systematic errors of Figure 2 would be light-travel-time effects driven by the presence of a circumbinary sub-stellar companion. A similar analysis has been conducted by Kim, Jeong et al. (18) on YY Eridani, which is a W UMa binary. Equation [5] would need to be modified to include the orbital parameters of the third body. Also more timings spanning the modulation period would be needed for a meaningful analysis to be undertaken on TW CrB.

A previous analysis of TW CrB has suggested that the period growth could be attributed to mass transfer between the components of the binary system at a rate of 2.74 x [10.sup.-7] [M.sub.[??]]/yr. In this analysis Zhang & Zhang (1) assumed that the primary component was a main sequence star of mass 1.19 [M.sub.[??]]. Using the same assumption for primary star mass together with the conservative mass transfer equation derived by Kwee (19) ([DELTA]P/P = 3([M.sub.1]/[M.sub.2] - 1)[DELTA][M.sub.1]/[M.sub.1]) and our calculated underlying change of orbital period of 6.66(14) x [10.sup.-8] days/yr and binary mass ratio 0.73, we find an average mass transfer rate of 1.21(3) x [10.sup.-7] [M.sub.[??]]/yr. This value is approximately half that estimated by Zhang & Zhang. (1)

Lightcurve simulation

The lightcurve simulation led to two interesting features being identified; a hotspot and two starspots.

The modelling process suggested a system that has two components each of which is nearly filling their Roche lobe; that in turn supports the possibility of mass transfer with the generation of a hotspot. This is consistent with the work of other researchers: Zhang & Zhang, and Caballero-Nieves et al. It is also consistent with our findings of variation in period described elsewhere in this paper.

In addition we modelled two starspots to enhance the curve fit. These too are consistent with cyclic variations found in the period of the system and described elsewhere in this paper. A possible explanation for this is an Applegate type electromagnetic mechanism. The chromospheric activity implied by the starspots make this a very likely X-ray source. The ROSAT Bright Star Catalogue confirms that TW CrB is an X-ray source. (20) Also TW CrB is identified in SIMBAD with X-ray source 2XMM J160650.6 + 271634. (21) However, our data do not allow us to investigate this further.

[FIGURE 11 OMITTED]

Conclusions

We have calculated a new ephemeris for the near-contact binary system TW CrB based on all the available timings going back to 1946, and we have revised the average rate of change of the period of this system. During our investigation we found evidence that the period change is slowing and that the change may be cyclical, but there is insufficient ephemeris data to make a judgement on the mechanism causing this possible variation.

It is clear from the lightcurves and from the photometric solution that there is an increase in amplitude in the blue band with respect to the other two bands. This is likely to be caused by a hotspot on the primary companion, which considering that the secondary has nearly filled its Roche lobe, implies evidence of mass transfer.

Clearly future research is needed to confirm the potential long-term cyclical behaviour of the period and thus be able to indicate the mechanism underpinning this behaviour. This would be a very long term project and it is hoped that investigators would take up this task in the future. Also radial velocity measurements would be needed to confirm the mass ratio and other parameters.

Acknowledgments

We would like to thank Dr U. Kolb, Dr L. McComb and Dr F. Vincent of The Open University for their assistance in this work, and the University for making the PIRATE observatory available as part of The Open University module S382, 'Astrophysics'. We extend our thanks to N. Cornwall, A. Grant, M. Hajducki, N. Smith and M. Treasure who assisted with some of the PIRATE observations. We would also like to thank Dr David Boyd, past President of the BAA, for his valuable suggestions, and the referees for their constructive comments that enhanced the paper.

Addresses: DP: 2 Springside Walk, St Leonards on Sea, East Sussex, TN38 0QF [astro@davidpulley.co.uk]

GF: Pillar Box House, South Stoke, Oxfordshire, RG8 0JS [gfaillace3@aol.com].

CO: 46 Little Lullaway, Basildon, Essex, SSI5 5JH [carlosfandangos@hotmail.com]

DS: Beechwood House, Cryers Hill Lane, High Wycombe, Bucks., HPI5 6AA [astro@beechwoodhouse.me.uk]

References

(1) Zhang X.-B. & Zhang R.-X., ApJ, 125, 1431 (2003): http://iopscience. iop.org/1538-3881/125/3/1431/pdf/202421.web.pdf

(2) Caballero-Nieves S. M., Smith E. & Strelnitski V., Bull.AAS, 36, 1348 (2004): http://adsabs.harvard.edu/abs/2004AAS ... 205.0904C

(3) Raab H., Astrometrica: http://www.astrometrica.at/

(4) Maxim DL, Astronomical imaging software, licensed by Diffraction Ltd: http://www.cyanogen.com

(5) Aladin Sky Atlas, Centre de donnees astronomiques de Strasbourg: http:/ /aladin.u-strasbg.fr/aladin.gml

(6) Warner B. D., MPO Canopus software v.10: http:// www.minorplanetobserver.com/MPOSoftwareMPOSoftware.htm

(7) Bradstreet D., Binary Maker v.3.0: http://www.binarymaker.com/

(8) Qing-Yao Liu & Yu-Lan Yang, Chin. J. Astron. Astrophys. 3, 142 (2003): http://www.chjaa.org/2003/2003_3_2p142.pdf

(9) Wilsey N. J. & Beaky M. M., Soc. Astr. Sci. 28th Ann. Sympos. on Telescope Science, 2009: http://adsabs.harvard.edu/abs/2009SASS ... 28..107W

(10) Lucas R. J. & Kolb U., J. Brit. Astron. Assoc., 121(5), 265 (2011)

(11) Kwee K. K. & van Woerden H., Bull.Astr.Inst.Nlds., XII, 327 (1956): http://articles.adsabs.harvard.edu//full/1956BAN....12..327K/ 0000327.000.html

(12) Vanmunster T., Peranso v.2.50: http://www.peranso.com/

(13) Wilson R. & Devinney E., ApJ, 166, 605 (1971): http://adsabs. harvard.edu/full/1971ApJ ... 166 ... 605W

(14) Cox A. N. (ed.), Allen's Astrophysical Quantities, 4th edn., Springer, New York, 2001. Table 7.6 p. 151

(15) Van Hamme W., AJ., 106, 2096 (1993): http://adsabs.harvard.edu/ abs/1993AJ. ... 106.2096V

(16) OriginLab data analysis and graphing software: http:// www.originlab.com/

(17) Applegate J. H., ApJ., 385, 621 (1992): http://articles.adsabs. harvard.edu/full/1992ApJ ... 385 ... 621A

(18) Kim C.-H., Jeong J. H. et al., ApJ., 114, 2753 (1997): http:// adsabs.harvard.edu/full/1997AJ. ... 114.2753K

(19) Kwee K. K., Bull.Astr.Inst.Nlds., XIV, 131 (1958): http://articles. adsabs.harvard.edu//full/1958BAN. ... 14 ... 131K/0000131.000.html

(20) The ROSAT All-Sky Survey bright source catalogue: http://vizier.ustrasbg.fr/viz-bin/VizieR?-source=IX/10

(21) SIMBAD astronomical database: http://simbad.u-strasbg.fr/simbad/

Received 2011 June 9; accepted 2012 March 28
Table 1. Our observations, 2010/2011 May & June

Observation site/               Start time (JD)   Observing team
instrumentation

Observatorio                    2455329.339       D.Smith, Faillace,
  Astronomico de Mallorca
  07144 Costrix, Spain                            M.Treasure, A.Grant
  (PIRATE)
Celestron 14 telescope;         2455332.342       N.Smith, Grant
3910mm FL @ f/11,               2455335.353       Pulley, Treasure
SBIG camera STL 1001E,          2455338.353       Owen, Pulley
1024x1024 pixels @ 24Lim,       2455341.347       N.Smith, M.Hajducki
22x22 arcmin field of view.     2455344.426       D.Smith, Pulley
                                2455347.400       Faillace, Pulley
                                2455350.380       Owen, Treasure
                                2455353.366       Faillace, N.Cornwall
Sierra Stars Observatory,       2455323.791       Faillace
  Markleeville,                 2455337.732       Faillace
  California, USA
Nighthawk CC06 telescope;       2455341.777       Faillace
6100 mm FL @ f/10,              2455346.749       Faillace
Finger Lakes Instrumentation    2455681.770       Faillace
  Pro Line camera,
Kodak KAF-09000                 2455684.719       Owen
  3056x3056 pixel CCD,
21x21 arcmin fov.               2455705.928       Pulley
                                2455708.873       Owen
                                2455737.728       Pulley
                                2455738.906       D.Smith, Pulley
GRAS--New Mexico, USA           2455328.858       Faillace
Deep Space: Takahashi           2455336.894       Faillace
  Mewlon telescope,
3572mm FL @ f/11.9,             2455339.712       Faillace
FLI IMG 1024 Dream Machine
1024x1024 pixels @ 24Lim,
23.6x23.6 arcmin fov.
South Stoke, United Kingdom     2455739.451       Faillace
11" SCT f/5.0. Camera:          2455742.411       Faillace
  SBIG ST 9EXE

Table 2a. Coordinates of stars used in Maxim DL
differential photometry

Photometry       Catalogue no.     RA (J2000)     DEC (J2000)
identity                           (2MASS)        (2MASS)

Target star      TYC 2038-1478-1   16 06 50.703   +27 16 34.58
Reference star   TYC 2038-1347-1   16 06 25.203   +27 18 29.97
Check star       GSC 02038-01473   16 06 23.262   +27 17 16.64
Check star       GSC 02038-01381   16 06 38.770   +27 16 16.32
Check star       GSC 02038-01346   16 07 15.050   +27 11 26.31
Check star       GSC 02038-01353   16 07 12.366   +27 20 00.15

Table 2b. Coordinates and derived B, V and R magnitudes
of comparison stars used for absolute photometry in the
Canopus software package

Catalogue no.      RA (J2000)    DEC (J2000)      B       V
                    (2MASS)        (2MASS)

GSC 02038-01473   16 06 23.262   +27 17 16.64   13.46   12.75
GSC 02038-01577   16 06 27.648   +27 17 04.25   14.64   13.90
GSC 02038-01270   16 07 07.748   +27 18 00.06   13.69   13.16
GSC 02038-01672   16 07 07.170   +27 20 04.75   15.53   15.00
                                   Average:     14.33   13.70

Catalogue no.       R      r'     B-V    V-R

GSC 02038-01473   12.35   12.58   0.71   0.40
GSC 02038-01577   13.48   13.69   0.74   0.42
GSC 02038-01270   12.85   13.05   0.54   0.31
GSC 02038-01672   14.68   14.99   0.54   0.31
                  13.34   13.58   0.63   0.36

Table 3. Times of minima of TW CrB

HJD             Method   E           O-C           Ref
(2,400,000+)                         (see below)

32,061.4530     ph       -32,625.0    0.0234       1
34,092.4800     ph       -29,176.0    0.0208       1
34,926.3290     ph       -27,760.0    0.0229       1
35,957.4380     ph       -26,009.0    0.0119       1
36,037.5350     ph       -25,873.0    0.0219       1
37,080.4200     ph       -24,102.0    0.0094       1
37,191.7200     ph       -23,913.0    0.0121       1,8
37,202.3200     ph       -23,895.0    0.0123       1,8
37,402.5380     ph       -23,555.0    0.0128       1,8
37,789.4240     ph       -22,898.0    0.0080       1,8
37,898.3720     ph       -22,713.0    0.0142       1,8
38,502.5520     ph       -21,687.0    0.0085       1,8
38,935.3760     ph       -20,952.0    0.0094       1,8
40,764.4100     ph       -17,846.0   -0.0021       1,8
42,200.3840     vis      -15,407.5    0.0004       1,2,8
42,201.5520     vis      -15,405.5   -0.0093       1,2,8
42,202.4390     vis      -15,404.0   -0.0057       1,2,8
42,212.4580     vis      -15,387.0    0.0025       1,2,8
42,214.5130     vis      -15,383.5   -0.0036       1,2,8
42,215.3980     vis      -15,382.0   -0.0019       1,2,8
42,220.4020     vis      -15,373.5   -0.0033       1,2,8
42,221.5740     vis      -15,371.5   -0.0091       1,2,8
42,258.3870     vis      -15,309.0   -0.0008       1,2,8
42,288.4180     vis      -15,258.0   -0.0024       1,2,8
42,296.3570     vis      -15,244.5   -0.0132       1,2,8
42,296.3710     vis      -15,244.5    0.0008       1,2,8
42,337.2920     vis      -15,175.0   -0.0050       1,2,8
42,404.7150     vis      -15,060.5   -0.0082       1,2,8
42,404.7180     vis      -15,060.5   -0.0052       1,2,8
42,455.6620     vis      -14,974.0    0.0011       1,2,8
42,491.5790     vis      -14,913.0   -0.0032       1,2,8
42,493.6410     vis      -14,909.5   -0.0023       1,2,8
42,509.5410     vis      -14,882.5   -0.0019       1,2,8
42,516.6110     vis      -14,870.5    0.0016       1,2,8
42,524.5550     vis      -14,857.0   -0.0042       1,2,8
42,568.4320     vis      -14,782.5    0.0016       1,2,8
42,570.4860     vis      -14,779.0   -0.0055       1,2,8
42,606.4100     vis      -14,718.0   -0.0029       1,2,8
42,616.4120     vis      -14,701.0   -0.0117       1,2,8
42,621.4340     vis      -14,692.5    0.0048       1,2,8
42,716.2340     vis      -14,531.5   -0.0040       1,2,8
42,780.7110     vis      -14,422.0   -0.0088       1,2,8
42,791.6160     vis      -14,403.5    0.0020       1,2,8
42,836.6630     vis      -14,327.0    0.0001       1,2,8
42,837.5480     vis      -14,325.5    0.0017       1,2,8
42,840.4860     vis      -14,320.5   -0.0046       1,2,8
42,858.4510     vis      -14,290.0   -0.0003       1,2,8
42,870.5180     vis      -14,269.5   -0.0053       1,2,8
42,878.4690     vis      -14,256.0   -0.0041       1,2,8
42,882.6000     vis      -14,249.0    0.0048       1,2,8
42,886.4170     vis      -14,242.5   -0.0059       1,2,8
42,905.5530     vis      -14,210.0   -0.0083       1,2,8
43,177.6200     vis      -13,748.0   -0.0015       1,2,8
43,254.4650     vis      -13,617.5   -0.0047       1,2,8
43,295.3930     vis      -13,548.0   -0.0035       1,2,8
43,358.4010     vis      -13,441.0   -0.0051       1,2,8
43,581.5840     vis      -13,062.0   -0.0057       1,2,8
43,734.3900     vis      -12,802.5   -0.0128       1,2,8
43,765.3100     vis      -12,750.0   -0.0087       1,2,8
43,770.3160     vis      -12,741.5   -0.0081       1,2,8
44,022.3550     vis      -12,313.5   -0.0076       1,2,8
44,085.3700     vis      -12,206.5   -0.0022       1,2,8
44,123.3490     vis      -12,142.0   -0.0057       1,2,8
44,382.4520     vis      -11,702.0   -0.0076       1,2,8
44,437.5100     vis      -11,608.5   -0.0094       1,2,8
44,502.2890     vis      -11,498.5   -0.0067       1,2,8
44,701.6230     vis      -11,160.0   -0.0068       1,2,8
44,711.6370     vis      -11,143.0   -0.0037       1,2,8
44,824.4090     vis      -10,951.5   -0.0013       1,2,8
47,274.4240     vis      -6,791.0    -0.0004       1,2,8
47,304.4440     vis      -6,740.0    -0.0130       1,2,8
47,665.4230     vis      -6,127.0    -0.0143       1,2,8
47,695.4670     vis      -6,076.0    -0.0029       1,2,8
47,728.4330     vis      -6,020.0    -0.0139       1,2,8
47,741.4010     vis      -5,998.0    -0.0012       1,2,8
47,754.3510     vis      -5,976.0    -0.0064       1,2,8
47,996.3850     vis      -5,565.0     0.0000       1,2,8
48,013.4530     vis      -5,536.0    -0.0094       1,2,8
48,016.4080     vis      -5,531.0     0.0012       1,2,8
48,069.4010     vis      -5,441.0    -0.0045       1,2,8
48,086.4690     vis      -5,412.0    -0.0139       1,2,8
48,089.4230     vis      -5,407.0    -0.0043       1,2,8
48,099.4170     vis      -5,390.0    -0.0211       1,2,8
48,125.3540     vis      -5,346.0     0.0054       1,2,8
48,132.4120     vis      -5,334.0    -0.0031       1,2,8
48,357.3690     vis      -4,952.0     0.0036       1,2,8
48,358.5240     vis      -4,950.0    -0.0191       1,2,8
48,404.4650     vis      -4,872.0    -0.0103       1,2,8
48,407.4020     vis      -4,867.0    -0.0177       1,2,8
48,442.4530     vis      -4,807.5    -0.0048       1,2,8
48,460.4150     vis      -4,777.0    -0.0035       1,2,8
48,480.4350     vis      -4,743.0    -0.0052       1,2,8
48,758.3760     vis      -4,271.0    -0.0132       1,2,8
48,768.3940     vis      -4,254.0    -0.0061       1,2,8
48,788.4140     vis      -4,220.0    -0.0078       1,2,8
49,116.4270     vis      -3,663.0     0.0019       1,2,8
49,166.4700     vis      -3,578.0    -0.0095       1,2,8
49,176.4820     vis      -3,561.0    -0.0084       1,2,8
49,212.4100     vis      -3,500.0    -0.0017       1,2,8
49,472.3941     pe       -3,058.5    -0.0059       1,2,8
49,550.4190     vis      -2,926.0    -0.0070       1,2,8
49,560.4260     vis      -2,909.0    -0.0108       1,2,8
49,570.4250     vis      -2,892.0    -0.0227       2,8
49,570.4290     vis      -2,892.0    -0.0187       2,8
49,570.4350     vis      -2,892.0    -0.0127       1,2,8
49,570.4370     vis      -2,892.0    -0.0107       2,8
49,570.4420     vis      -2,892.0    -0.0057       2,8
49,580.4480     vis      -2,875.0    -0.0106       1,2,8
49,878.4240     vis      -2,369.0    -0.0053       1,2,8
49,895.5060     vis      -2,340.0    -0.0007       1,2,8
49,898.4187     vis      -2,335.0    -0.0323       8
49,905.4864     vis      -2,323.0    -0.0311       8
49,915.5060     vis      -2,306.0    -0.0224       1,8
49,918.4660     vis      -2,301.0    -0.0068       1,2,8
50,189.3450     vis      -1,841.0    -0.0102       1,2,8
50,193.4696     vis      -1,834.0    -0.0078       1,8
50,193.4738     vis      -1,834.0    -0.0036       1,8
50,193.4740     vis      -1,834.0    -0.0034       10
50,200.5399     vis      -1,822.0    -0.0040       1,8
50,209.3700     vis      -1,807.0    -0.0070       1,2,8
50,239.4020     vis      -1,756.0    -0.0076       1,2,8
50,249.4170     vis      -1,739.0    -0.0035       1,2,8
50,312.4320     vis      -1,632.0     0.0019       1,2,8
50,515.5865     vis      -1,287.0    -0.0054       1, 8
50,557.3940     vis      -1,216.0    -0.0081       1,2,8
50,570.3560     vis      -1,194.0    -0.0013       1,2,8
50,660.4448     vis      -1,041.0    -0.0104       1,8
50,660.4461     vis      -1,041.0    -0.0091       1,8
50,660.4500     vis      -1,041.0    -0.0052       1,2,8
50,660.4517     vis      -1,041.0    -0.0035       1,8
50,660.4531     vis      -1,041.0    -0.0021       1 8
50,673.3950     vis      -1,019.0    -0.0154       1,2,8
50,696.3670     vis      -980.0      -0.0096       1,2,8
50,956.0700     vis      -539.0      -0.0004       1,8
51,273.4705     pe       0.0         -0.0035       1,3,8
51,274.0590     CCD      1.0         -0.0039       10
51,283.7764     CCD      17.5        -0.0029       1,4,8
51,311.7468     CCD      65.0        -0.0040       1,4,8
51,659.4790     pe       655.5       -0.0025       1,8
51,664.1902     CCD      663.5       -0.0023       1,5
51,665.0746     CCD      665.0       -0.0012       1,5
51,671.2583     CCD      675.5       -0.0007       1,5
51,672.1405     CCD      677.0       -0.0018       1,5
51,675.0837     CCD      682.0       -0.0030       1,5
51,680.3839     pe       691.0       -0.0026       1,8,9
52,009.5655     CCD      1,250.0     -0.0021       1,8,9
52,147.3621     CCD      1,484.0     -0.0023       1,8,9
52,147.3747     CCD      1,484.0      0.0103       1,8,9
52,352.2921     pe       1,832.0     -0.0007       1,8
52,360.5358     pe       1,846.0     -0.0013       1,8
52,373.4913     pe       1,868.0     -0.0010       1,8,9
52,510.4053     CCD      2,100.5     -0.0005       1,8,9
52,692.6613     CCD      2,410.0     -0.0012       1,8,9
52,721.5164     pe       2,459.0     -0.0010       1,8,9
52,741.5387     pe       2,493.0     -0.0005       1,8,9
53,107.5251     pe       3,114.5      0.0002       1,8,9
53,165.5295     pe       3,213.0      0.0004       1,8,9
53,388.7151     CCD      3,592.0      0.0024       1,8,9
53,448.1887     CCD      3,693.0     -0.0004       1,8,9
53,463.4983     pe       3,719.0     -0.0015       1,8,9
53,473.8044     CCD      3,736.5     -0.0007       1,8,9
53,492.3532     CCD      3,768.0     -0.0015       1,8
53,493.5331     CCD      3,770.0      0.0007       1,8
53,499.4206     CCD      3,780.0     -0.0006       1,8
53,502.3644     CCD      3,785.0     -0.0012       1,8
53,503.5423     pe       3,787.0     -0.0010       1,8,9
53,550.0647     pe       3,866.0      0.0003       1,8
53,747.3379     pe       4,201.0      0.0004       1,8
53,801.5142     CCD      4,293.0      0.0002       1,8
53,859.5170     pe       4,391.5     -0.0012       1,8
53,900.4458     CCD      4,461.0      0.0008       1,8
54,172.5058     CCD      4,923.0      0.0006       1,8
54,172.5060     CCD      4,923.0      0.0008       8
54,172.5063     CCD      4,923.0      0.0011       8
54,185.4615     CCD      4,945.0      0.0010       8
54,185.4617     CCD      4,945.0      0.0012       1,8
54,199.5934     pe       4,969.0     -0.0001       8
54,213.4330     CCD      4,992.5      0.0010       8
54,556.4529     CCD      5,575.0      0.0012       8
54,556.4530     CCD      5,575.0      0.0013       8
54,556.4531     CCD      5,575.0      0.0014       8
54,556.4534              5,575.0      0.0017       10
54,911.5452     CCD      6,178.0      0.0019       8
54,924.5009     CCD      6,200.0      0.0024       8
54,930.3888     CCD      6,210.0      0.0015       8
54,950.4115     CCD      6,244.0      0.0025       8
54,950.4116     CCD      6,244.0      0.0026       8
54,960.1276     CCD      6,260.5      0.0022       8
55,269.8772     CCD      6,786.5      0.0036       8
55,293.4322     CCD      6,826.5      0.0036       8
55,332.5919     CCD      6,893.0      0.0031       7
55,335.5364     CCD      6,898.0      0.0032       7
55,338.4804     CCD      6,903.0      0.0028       7
55,341.4246     CCD      6,908.0      0.0026       7
55,349.0802              6,921.0      0.0029       10
55,681.7951     CCD      7,486.0      0.0034       7
55,684.7395     CCD      7,491.0      0.0035       7
55,705.9397     CCD      7,527.0      0.0042       7
55,708.8842     CCD      7,532.0      0.0043       7
55,737.7390     CCD      7,581.0      0.0042       7
55,738.9165     CCD      7,583.0      0.0040       7
55,739.5053     CCD      7,584.0      0.0039       7
55,742.4501     CCD      7,589.0      0.0043       7

Note: The values in italics (E= -2,335.0; -2.323.0;
1,484.0) were not used to calculate the ephemerides.
Vis = visual; pe = photoelectric; ph = photographic.

References: (1) Krakow [http://www.as.up.
krakow.pl/o-c/]; (2) BBSAG Bulletins [http://
www.astroinfo.ch/bbsag/bbsag_e.html] (3)
Agerer & Hubscher 2000; (4) Diethelm 2001; (5)
Zhang & Zhang 2003; (6) Baldinelli & Maitan
2002; (7)This study; (8) Lichtenknecker BAV; (9)
IBVS [http://www.konkoly.hu/IBVS/IBVS.
html]; (10) GSV Search Gateway; Czech Astronomical
Society

Table 4. Binary Maker 3 derived parameters

Wavelength independent parameters

Mass ratio (q)[M.sub.2]/[M.sub.1]    0.725 [+ or -] 0.010
Inclination                          89.6 [+ or -] 0.1
[T1.sub.eff] H                       5700K [+ or -] 200K
[T2.sub.eff]                         5400K [+ or -] 150K
g1 = g2: assumed                     0.32
Alb1 = Alb2: assumed                 0.5
Omega1                               3.33 [+ or -] 0.09
Omega2                               3.2575 [+ or -] 0.0075
Fillout1                             -0.03281
Fillout2                             -0.0252
Lagrangian L1                        0.53234
Lagrangian L2                        1.646104

Primary hotspot

  Co-latitude ([degrees])            93.5
  Longitude ([degrees])              92.8
  Radius ([degrees])                 12

Secondary starspot 1

  Co-latitude ([degrees])            57.1
  Longitude ([degrees])              309
  Radius ([degrees])                 13

Secondary starspot 2

  Co-latitude ([degrees])            82
  Longitude ([degrees])              310
  Radius ([degrees])                 13

Wavelength dependent parameters
                                               Filter
                                     R         V        B

Centre wavelength ([Angstrom])       7000      5500     4450
Luminosity 1                         0.6232    0.6361   0.6481
Luminosity 2                         0.3768    0.3639   0.3519
X1 (limb darkening)                  0.481     0.584    0.72
X2 (limb darkening)                  0.504     0.618    0.774

Temp factor (%)

  Primary hotspot                    126       95       117
  Secondary starspot 1               95        68       98
  Secondary starspot 2               94        69       99

(*) Allen's Astrophysical Quantities14

Table 5. The fitted parameters
to eqn [5]

Parameter   Final value                Unit

A           2451273.4701(1)            day
B           0.58887562(2)              day
C           5.37 x [10.sup.-11] (11)   day
D           9.3 x [10.sup.-4] (5)      day
[omega]     3.6 x [10.sup.-4] (3)      rad/cycle
[phi]       1.3(2)                     rad
Pmod        28.1(2.4)                  yr
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Author:Pulley, D.; Faillace, G.; Owen, C.; Smith, D.
Publication:Journal of the British Astronomical Association
Article Type:Report
Geographic Code:4EUUK
Date:Apr 1, 2013
Words:7363
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