A STUDY OF THE PHOTOMETRIC VARIABILITY OF THE PECULIAR MAGNETIC WHITE DWARF WD 1953−011

Taking the above arguments into account, we measured the EW and r_c of the Hβ spectral line in observations with ZTSH and Hα in all other cases. In measurements of the EWs we used all parts of the profiles including the satellite sigma components within the window of ±50 Å from the Balmer line cores. In order to minimize selection effects due to the use of different telescopes, spectrographs, and different spectral lines, all the measurements within individual groups of observations with a given instrument and spectral line were normalized by their mean values, averaged over a full rotational cycle of the star. For example, normalized EW values of the Hα profile obtained with the VLT mean that the measured EWs were normalized by their averaged value obtained in the frame of all spectral observations of WD 1953−011 exclusively with FORS1. The same was done for the results obtained with the other telescopes. Because in all observing runs with the different telescopes the observations were distributed more or less homogeneously over the star's rotational cycle, such a normalization provided essential unification of the data. Thus, hereinafter, when mentioning r_c and EW we assume their normalized values. This normalization makes it possible to consider all measurements from the different spectral lines together. Results of the measurements are presented in Table 1. Here we also present measurements from spectral material obtained in the previous studies with the Anglo-Australian 4 m telescope (AAT; Maxted et al. 2000), from the 8 m European telescope, and the 6 m Russian telescope (Valyavin et al. 2008).

The V-band CCD photometric observations were carried out in the standard manner. For the comparison stars we used the same targets as used by Brinkworth et al. (2005). Calibrating measured fluxes from WD 1953−011 by fluxes from the standard stars, we finally obtained a series of m_V values in stellar magnitudes at each of the individual short-duration (a few minutes) exposures. The characteristic uncertainty of the measurements is about 0.02 stellar magnitudes.

The rotation period of WD 1953−011 (P ≈ 1.45 days) and a preliminary inspection of the series of the m_V values allowed us to rule out a variability timescale shorter than a few thousand seconds. Hence, we decided to consider a series of the magnitude determinations where each individual value was obtained from the weighted average of several consecutive points within equivalent exposure bins of about 1500 s with two exceptions at JD2454329.351 and JD2454329.442.¹¹ Error bars have been obtained as standard deviations from the averaged means. The measurements are listed in Table 2.

The new observables (m_V, r_c, and EW) obtained for WD 1953−011 from photometry and spectroscopy together with photometric data from Brinkworth et al. (2005) make it possible to revise the degenerate's rotation period on a time base of about 10 years. All the observables used are variable with the star's rotation.

To determine the rotation period, we applied the Lafler–Kinman method (Lafler & Kinman 1965) with user interface programmed by V. Goransky (2004, private communication). Power spectra of variation of the observables are presented in Figure 1. The first plot (from top to bottom) presents the periodogram of m_V obtained from our photometric observations. The second plot is the periodogram obtained from our photometric observations together with those obtained by Brinkworth et al. (2005). The third plot illustrates the power spectrum of normalized Hα and Hβ EWs. The fourth plot is the power spectrum of the normalized residual intensity of the Hα and Hβ lines obtained from all available spectroscopic observations of WD 1953−011. As can be seen, all the spectra exhibit a single strong peak at a frequency of about 0.69 day⁻¹ (P ≈ 1.45 days).

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Detailed study of the periodograms has shown that the most significant and sinusoidal signal common to all observables corresponds to a period of P = 1.441788(6) days. This period is very close to P = 1.441769 days found by Brinkworth et al. (2005) from their photometry. Due to this agreement, here we choose this period for phasing of all the data with the following ephemeris (for which JD0 corresponds to the maximum of the light variation):

$$\begin{eqnarray*} &&{\rm JD} = 2,454,329.872 + 1.441788(6)\,E. \end{eqnarray*}$$

In Figure 2 we illustrate the phase variations of the photometric magnitude (the upper two panels on the figure), EW of the Balmer lines (the third panel from top), and residual intensity r_c of the lines. All the data are phased with the ephemeris described above. The lower two plots present the phase variation of the degenerate's global surface magnetic field (B_G: mean field modulus integrated over the disk) with the same ephemeris and the projected fractional area S of the strong-field area on the disk (given in percentage of the full disk area). These data are taken from Valyavin et al. (2008) and re-phased with the current ephemeris. The two vertical dashed lines in Figure 2 correspond to those phases (ϕ ≈ 0.5; 1.5) when the star exhibits the minimum brightness, and therefore maximum projection of the dark area onto the disk.

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As can be seen from Figure 2, the spectroscopic observable r_c also tends to have the smallest values at the phases of the light minimum. According to the above explanations this suggests a direct relationship between the dark area and strong-field spot. However, in comparison with the behavior of the residual intensity, the variation of EW demonstrates a non-sinusoidal shape with its absolute minimum shifted by Δϕ ≈ −0.05 relative to the phase of the light minimum (ϕ = 0.5). Averaging EW in the phase bins 0.41 < ϕ < 0.47 and 0.5 < ϕ < 0.55 and calculating the gradient of the data between these two phases, we have the formal absolute minimum of the EW change to be at ϕ ≈ 0.44 ± 0.017. Further, phased results of direct measurements of the projected area S (for details, see Section 4.2 and Table 3 in Valyavin et al. 2008) of the strong-magnetic spot on the disk seem to have the similar tendency, shifted in phase by this value. Because of this we shall discuss this phenomenon in the present study in more detail.

This phase shift was first observed by Wade et al. (2003) in their study of WD 1953−011 where they also estimated the "visibility" of the strong-magnetic area via analysis of Hα EWs. This shift could be produced by the presence of some morphologic difference in the distribution of the field and photometric flux intensities within the corresponding spots. Although the origin of the shift could be artificial due to uncertainties in our simplifying assumption that the variation of EWs of the Balmer line profiles is attributed to the strong-magnetic area only. The shift is, however, weak. And the general behavior of all the observables related to the dark and strong-field magnetic areas exhibits correlated behavior suggesting their geometric relationship.

Taking all of the above into account, we suppose that the dark and magnetic spots are physically connected. In order to illustrate this conclusion, below we model the dark spot from photometry comparing it with the geometry of the strong-field area obtained by Valyavin et al. (2008).

In our model we assume that the star's flux variation is due to surface inhomogeneity in the temperature distribution. Assuming a black body relationship between temperature and energy flux we have

$$\begin{equation} m_V - \overline{m_V} = -2.512 \cdot lg \frac{F_V}{\overline{F_V}} = -2.512 \cdot lg \frac{e^{ \frac{h c}{\lambda _v k \overline{T_{{\rm eff}}}}} -1 }{ e^{ \frac{h c}{\lambda _v k {T_{{\rm eff}}}}} -1 }, \end{equation} \tag{ 1 }$$

where h, c, and k are the Plank constant, speed of light, and Boltzmann constant; λ_v is wavelength centered at the V-band filter; F_V and $\overline{F_V}$ are individual and averaged (over the entire rotational cycle) photometric fluxes, while T_eff and $\overline{T_{{\rm eff}}}$ are individual and averaged effective temperatures, respectively. Taking $\overline{m_V} = 13\hbox{$.\!\!^{\rm m}$}632$ from our data, $\overline{T_{{\rm eff}}} = 7920$ K (Bergeron et al. 2001), assuming that $e^{ \frac{h c}{\lambda _v k {T_{{\rm eff}}}}} - 1 \approx e^{ \frac{h c}{\lambda _v k {T_{{\rm eff}}}}}$ and weighting the visible T_eff by the line-of-sight flow of radiation as an average mean of the local surface temperatures T(θ, ϕ), we obtain the following approximation equation:

$$\begin{eqnarray} T_{{\rm eff}} &=& \frac{\int\!\! \int T(\theta, \phi) \overline{n} I(u,\overline{n}) \sin \theta \, d \theta \, d \phi }{\int\!\! \int \overline{n} I(u,\overline{n}) \sin \theta \, d \theta \, d \phi }\nonumber\\ &=& \frac{28926.765}{m_V - 9.98} = F(m_V), \end{eqnarray} \tag{ 2 }$$

where θ and ϕ are the spherical angles relative to the rotational axis of the star, $\overline{n}$ is the cosine of the angle between the surface element normal and the line of sight, and u is the limb darkening. For every square element, this cosine can be determined via the inclination i of the rotational axis with respect to the line-of-sight coordinates θ, ϕ of the square, and rotation phase f as follows:

$$\begin{equation} \overline{n} = \cos i \cdot \cos \theta + \sin i \cdot \sin \theta \cdot \cos (\phi + f). \end{equation} \tag{ 3 }$$

In the integration procedure we assume negative $\overline{n}$ to be zero in order to integrate only over the visible hemisphere. Assuming a linear limb-darkening law and dividing the sphere by the elementary angles Δθ, Δϕ, Equation (2) can be rewritten as

$$\begin{equation} \Sigma \Sigma T_{l,n}\cdot A_{l,n,k} = F(m_V)_k, \end{equation} \tag{ 4 }$$

where

$$\begin{eqnarray*} &&A_{l,n,k} = \frac{\overline{n_{l,n,k}} I_{l,n} \sin \theta _l\,}{\Sigma \Sigma \overline{n_{l,n,k}} I_{l,n} \sin \theta _l\, }. \end{eqnarray*}$$

Here, index k corresponds to an observed rotation phase, and l, n run latitudinal and longitudinal positions of the elementary squares in the star's rotational coordinate system.

Applying all available photometric observations covering the full rotation cycle, Equation (2) can be solved by using any of the nonlinear least-square methods if the angle i is known. In our model we used an inclination of i = 18° as determined by Valyavin et al. (2008). With the limb darkening u = 0.45, the temperature distribution up to 60 surface elements was calculated as presented below.

A general application of Equation (4) to the analysis of nearly sinusoidal signals provides more than a single solution, including physically unrealistic ones (negative surface temperatures, for example). In order to restrict the range of possible solutions for surface temperatures, we modified the temperatures to have the following parameterized form

$$\begin{equation} T(\theta, \phi) = T_0 + F(x[\theta, \phi ]), \end{equation} \tag{ 5 }$$

where F(x) is a positive function of the parameter x and T₀ is a physically reasonable lower temperature limit (say at the center of the coolest area). To choose a reasonable F(x) we tested several power, polynomial, and exponential functions. Finally, the simplest case F(x) = x² was chosen. This case provides satisfactory convergence of Equation (4) if T₀ is fixed.

In such a formulation Equation (4) was evaluated using the Marquardt χ² minimization method (Bevington 1969). In our solution we minimize the function:

$$\begin{equation} \chi ^2 = \Sigma (F(m_V)_k - \Sigma \Sigma T_{l,n}\cdot A_{l,n,k}) ^ 2. \end{equation} \tag{ 6 }$$

The used algorithm in FORTRAN-77 is presented by Press et al. (1992).

In Figure 3 we illustrate the tomographic portraits (left panels) of the resulting smoothed temperature distribution patterns in comparison to the strong-magnetic area. Dark areas correspond to low temperatures and brighter areas are higher temperatures. In these plots, the vertical axis is latitude, the horizontal axis is longitude. The longitude is presented from −180° to +180°. Middle panels are the corresponding phase light curves (solid lines) obtained from the model fits of the photometric observations (filled circles): vertical axis is m_V, horizontal axis is rotation phase. Right panels illustrate in gray scale the temperature distribution in the tomographic portraits. Five examples of the star's temperature distribution with different T₀ are illustrated (from top to bottom T₀ = 6500 K, 6750 K, 7000 K, 7100 K, 7120 K). The strong-field spot is schematically illustrated in Figure 3 by the elliptical area bordered by a white solid line.

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All the illustrated solutions demonstrate two characteristic features: the presence of a temperature gradient from the rotation pole to the equator as well as the presence of a large low-temperature spot (the shadow) covering 15%–20% of the star's hemisphere. The size of the shadow corresponds well to the size of the strong-field area estimated by Valyavin et al. (2008). Three solutions at the lower temperature limits T₀ = 6500 K, T₀ = 6750 K, and T₀ = 7000 K (the upper three plots in Figure 3) present the most stable fits of the data revealing temperature changes around the latitude of the strong-field area location (θ ≈ 67°). Higher temperature limits shift the shadow to upper latitudes and provide less robust fits to the photometric data (see the lower two plots in Figure 3). From consideration of all the solutions, we give the characteristic value to the lower temperature limit (the temperature at the center of the shadow) to be between 6000 K and 7000 K.

In order to find simpler solutions for the temperature distribution we also examined i as a parameter. The most stable and physically reasonable solutions are found within the range 10° ⩽ i ⩽ 20°. Within this range the tomographic portraits demonstrate similar features and temperature contrasts as discussed above.

The phase shift discussed previously (ϕ ≈ 0.05–0.1) might be produced by a longitudinal shift between the dark and magnetic spots of about 20°, and certainly not higher than 40°. However, even in these extreme cases the dark and strong-magnetic areas are overlapped by more than 60%. We therefore suggest that the dark and magnetic spots are indeed physically connected.

The small difference in phasing of the corresponding observables (m_V and S in Figure 2) can also be explained by the following obvious fact. In our previous investigation of WD 1953−011 (Valyavin et al. 2008) where we determined the rotationally modulated projection S of the strong-field area to the line of sight, we assumed a constant temperature everywhere over the surface. Taking into account that the center of the low-temperature region may have an effective temperature about 20% below the mean effective temperature of the photosphere, we may suspect that these coolest parts of the surface may be undetectable spectroscopically in the integrated hydrogen spectrum of the degenerate. This could affect our analysis of the position of the strong-field area and produce a small artificial shift in the position of the strong-magnetic area.

The presence of the global gradient between the rotation pole to equator is also a remarkable result. We find this gradient from practically all tested T₀. The cases of T₀ ≈ 7100 K provide smaller values of the gradient. Lower values of T₀ increase it. Also, testing smaller i (11° ⩽ i ⩽ 14°) we found some solutions with the smallest, but nevertheless non-zero gradient. In all cases, the presence of the gradient makes the shadow area have more diffusive edges, giving a smaller contrast in temperatures between the shadow and neighboring areas. In this paper, however, we are not making conclusions about the existence of the gradient, but only discuss the implications.