Determination of Size of the Emitting Region in Eclipsing Cataclysmic Variable Stars
DDETERMINATION OF SIZE OF THE EMITTING REGION IN ECLIPSING CATACLYSMIC VARIABLE STARS
I.L. Andronov , K.D. Andrych Department “High and Applied Mathematics, Odessa National Maritime University, Mechnikova Str., 34, 65029, Odessa, Ukraine, tt_ari @ ukr.net Department of Astronomy, Odessa National University, Shevchenko Park, 65014, Odessa, Ukraine
ABSTRACT. The dependencies of the phase of eclipse of the white dwarf’s centre and the durations of the ascending and descending branches of the light curve on the binary system’s parameters were computed using the spherically- symmetric approximation and the more accurate model of the elliptical projection onto the celestial sphere of the com-panion (red dwarf) that fills its Roche lobe. The parameters of eclipses in the classical eclipsing polar OTJ 071126+440405 = CSS 081231:071126+440405 were esti-mated.
Keywords:
Stars: variable – stars: binary – stars: cata-clysmic
1. Introduction
Interacting binary stars are natural laboratories with ex-treme conditions that are inaccessible in terrestrial laborato-ries. There are many different processes that allow to deter-mine physical characteristics of binary systems using meth-ods for mathematical modeling of observations. Especially useful are studies of eclipsing variables, par-ticularly, eclipsing cataclysmic variables, which allow to determine some physical parameters from photometric ob-servations only. On the New Year night 31.12.2008, Denisenko and Korotkiy (2009) discovered the unique eclipsing polar (OTJ 071126 + 440405) in the Camelopardalis constellation. The light curve shows that the system has a very short duration of the ascending and descending branches of the primary mini-mum, what indicates a very small size of the radiation source. Our theoretical work has been done for interpreting the observational data.
2. Effective Dimensions of the Stars
The simplest classical approach is to apply a model of spherical stars not only in Algol-type systems (Shulberg, 1971; Tsessevich, 1980; Chinarova, 2006), but also in cata-clysmic systems (Shafter A., 1984; Horne, 1985; Downes et al., 1986; Garnavich et al., 1990 and, more recently, Aung-werojwit et al., 2012), assuming that the red dwarf (which fills its Roche lobe and thus is tidally distorted) is spherical. Typically, the radius of this sphere R is defined as VR where V is the volume of the Roche lobe, which may be es-timated from a suitable approximation of Eggleton (1983) ,)1ln( qDq CqAR where q = M / M is the mass ratio, M – the mass of the com-pact primary, M – the mass of the secondary which fills its Roche lobe. The values of the coefficients were adopted to be C =0.49 and D =0.6. The asymptotical approximations for this formula are R / A = Cq for q <<1 and R / A = C / D for q >>1. This expression is a good approximation also for intermedi-ate values of q . However, the Roche lobe is definitely not spherical, hav-ing the largest size along the line of centers, and the smallest in the “polar” direction (along the rotational axis). For our task – modeling of entry/exit time (duration of the descending/ascending branch of the light curve) at the eclipse as functions of the parameters of the binary system (including the size of the white dwarf). We have chosen two models: the popular model of "spherical" red dwarf, and much more ac-curate model of it’s elliptical projection onto the celestial sphere (there is assumed that red dwarf filed it’s Roche lobe)
3. Calculating the Size of the Emitting Area
Let us consider a cataclysmic binary system, in which red and white dwarf obscure each other. Because these systems are very close (have a small orbital period), they have circular orbits, and this facts greatly simplifies the calculation. For convenience, we consider a coordinate system with its center in the red dwarf, so the white dwarf rotates around it. The center of the white dwarf during its orbital motion shows an ellipse. The coordinates x and y can be expressed as: sin 2 , x A cos2 , y B cos B A i , where is the phase expressed in units of the orbital period P , thus a multiplier of 2 is needed for conversion to radians. It rises to unity that mean the system made full rotation. Here A and B correspond to minor and major axis of el-lipse, which is the projection of the circular orbit of a radius A on the picture area (an area that passes through the center of the first stars perpendicularly to the line of sight). Inclina-tion of the orbit i is the angle between the line of sight and the rotation axis of the binary system. The distance between the centers of the stars can be cal-culated by the Pythagorean theorem. Taking into account the above formulas, it may be written in the following form: R = A (1 – cos ·sin i ). After some transformations, we get the following form of the same equation, which is convenient to determine the phase corresponding to the projected distance R : iAR sin1arccos2 We consider that, at the time of internal contact, the dis-tance between the centers equals to the difference in the stel-lar radii, and at point of external contact – to sum of their radii. Obviously, for convenience, one can use the relative values of the stellar radii. The phases of contacts were calculated as well as the phase difference between external and internal contacts, which corresponds to the length of ascending (or descend-ing) branch of the eclipse and is measured in units of the orbital period. Results are presented in Fig.1 and Fig.2, where the radius of the white dwarf and the inclination of the orbit were fixed, respectively. As in cataclysmic binary systems one of the stars – the red dwarf – fills its Roche lobe, it is deformed. To within a few hundredths of a percent profile stars can be ap-proximated to an ellipse (Andronov, 1992). The limb of the red dwarf indicated in Fig. 3 by a red line. The ellipse – like curves N and N are the locuses of points distant from the limb of the red dwarf by the radius of the white dwarf. Coordinates of these points can be calculated by these formulas, respectively. Figure 1: Phase of the external contact and the duration of the as-cending branch of the light curve for different pairs of values of the relative radii r and r , varying in increments of 0.05 in the range r ≤ r ≤ (1- r ). Inclination i = 90 o . Figure 2: Phase of the external contact and the duration of the as-cending branch of the light curve for different pairs of values of the relative radius r , which varies in increments of in the range r ≤ r ≤ (1- r ) , and inclination of the orbit from 30 o to 90 o in steps of 5 o . The parameter r =0.05. Figure 3: Scheme of internal and external contacts in eclipse.
For the calculation of the contact points, it is necessary to solve the following system of equations:
21 21
BnRyy AnRxx yx Let the phase of crossing the center of the white dwarf by the limb of the red dwarf that is е . This phase can be calcu-lated respectively to these formulas for the elliptical and spherical approximations. In the case, if r е ( in + ex )/2 iab iAb e
222 2222 coscos2sin If b = a = r e : i iAr ee sincos2sin Andronov (1992) got coefficients for determining the ra-dius of the object in the orbital ( a / A ) and polar ( b / A ) plane (that is the plane passing through the center line and the axis of rotation of the system), following the Eggleton’s (1983) form. In his work, they are denoted as sin (0 o ) ( C =0.4990, D =0.5053) and sin (90 o ) ( C =0.4394, D =0.5333), respec-tively. The elliptic approximation for other angles is correct within 0.2% and 0.5%, respectively. For better accuracy, we used linear interpolation for the ratio of precise/fit values. We adopted the mass ratio of q =0.3 in this system. Then we obtain the following values of the parameters: r e / A =0.28103, a / A =0.27216, b / A =0.26219. In Fig. 4, the duration of the eclipse depending on the in-clination for both models is shown. Figure 4: Duration of the ascending/descending branch - (in seconds) as a function of inclination i for the models of circle (blue) and ellipse (red). The vertical lines show limiting values of i . Figure 5: The dependence of the phase е of the eclipse of the white dwarf center (expressed in seconds) on inclination for the models of circle (blue) and ellipse (red). The horizontal line shows the observed value. o determine the duration of the entrance/exit of the eclipse, е and the corresponding errors for all parameters, we write the system of conditional equations: VUEPCTT VUEPCTT VUEPCTT VUEPCTT
104 103 102 101 where T is an initial epoch, С – correction to the initial epoch, P – period, E – number of cycle, U – the time between crossing the center of the red dwarf limb by the white dwarf and the middle of eclipse, V – half the length of the ascending / descending branches of the light curve. From the observations, we have the following values: Т , Т , Т , Т , Е , Т Moments of contacts were obtained from the observa-tions obtained by Dr. Sergey V. Kolesnikov using the 2.6 m telescope named after G.A.Shajn in the CrAO.
Table 1: The moments of eclipse contacts.
BJD, 2400000+ Types of con-tacts BJD, 2400000+ Types of contacts 54946.19335 1 54949.20429 1 54946.19340 2 54949.20435 2 54946.19836 3 54949.20932 3 54946.19842 4 54949.20937 4
The parameters were determined in the Excel using the method of the least squares. We have determined the period of the system: P =117.18292 i =79.1177 o ) and the spherical approximations ( i =77.1231 о ). According to the observations, we obtained the duration of the entrance/exit of the eclipse (descending/ascending branches of the light curve): 4.752 M =0.543 M , corresponding to q =0.3 and M =0.163 M . From the dependence "mass – radius" for white dwarfs (Andronov and Yavorskiy 1990), we obtained the radius of the white dwarf. Using our calcula-tions for inclination of the orbit, we obtain the duration of the entry/exit (time between external and internal contact) of 63.2 seconds for the elliptical approximation, which is by 13.3 times more than the expected value for the white dwarf. The Fig. 6 shows two models: the red dwarf, white dwarf at the phase of eclipsing center and the emitting region. Red line shows the elliptical model for the red dwarf, the violet line – circular model. The visible trajectories of the center of the white dwarf for different values of inclination of the orbit are shown. Vertical lines show different values of the pa-rameter е , including what is observed. From the cross point of the ellipse, one may determine inclination. Also the white dwarf and the emitting area are shown in the same scale. Figure 6: The scheme of eclipse preserving the scale for the numeri-cal values of the model system OTJ 071126 + 440405.
4. Conclusions The dependencies of the phase of eclipse of the white dwarf’s centre and the durations of the ascending and de-scending branches of the light curve on the binary system’s parameters were computed using the spherically – sym-metric approximation. Similar computations were performed with the more accu-rate model of the elliptical projection onto the celestial sphere of the companion (red dwarf) that fills its Roche lobe. The parameters of eclipses in the classical eclipsing polar OTJ 071126+440405 were estimated. The duration of entering/exiting the eclipse is shown to be 13.3 times shorter than the theoretical predictions. Hence, the emitting region is markedly smaller (~1300 km) as compared to the white dwarf’s diameter. That is supposed to be the “hot spot” region.
Acknowledgements.
We would like to sincerely thank Dr. S.V. Kolesnikov for his observations that initiated our research and Dr. V.I. Marsakova for helpful discussions.
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