2 Observations and data reduction

2.1 XMM-Newton EPIC

DP Leo was observed using XMM-Newton on 22 of November 2000 for a net exposure time of 19 949 s. DP Leo was detected in all three EPIC detectors (Turner et al. 2001; Strüder et al. 2001). The thin filter was used and the CCDs were read out in full window mode.

Before extracting source photons, the data were processed using the current release of the XMM-Newton Science Analysis System (version 5.1). Standard procedures of data screening (creation of an image and a background light curve) revealed time intervals with enhanced particle background. These intervals were excluded from the subsequent timing analysis using an approach described below (see Sect. 2.3). This reduces the accepted exposure time to 15 827 s with the EPIC-PN detector. The observations were performed without any interruption, i.e. full phase-coverage of the $P_{\rm orb} = 5388$ s binary was achieved with an average exposure of $\sim$150 s per 0.01 phase unit.

Figure 1: Phase-averaged X-ray light curves of the ROSAT and the XMM-Newton observations. Phase bins have a size of 0.01 phase units of the $P_{\rm orb} = 5388$ s binary.

2.2 ROSAT PSPC

The field of DP Leo was also observed with the ROSAT PSPC on May 30, 1992 (ROR 300169, PI: Cordova) and on May 30, 1993 (ROR 600263, PI: Petre) for a net exposure time of 8580 s and 23 169 s, correspondingly. Results of the 1992 observations were presented by Robinson & Cordova (1994), the results of the much more extended observations of 1993 are unpublished.

Although the net exposure time of the two ROSAT observations was larger than the binary period, in neither case was complete phase coverage achieved due to the close proximity of the periods of the satellite and of the binary.

There are further X-ray observations reported by Biermann et al. (1985), and Schaaf et al. (1987), respectively, with the EINSTEIN and EXOSAT satellites. We make use of the timing of the X-ray eclipses detected with Einstein, the EXOSAT data have a too low count-rate and are not used further in this paper.

   
2.3 Timing analysis with a Bayesian change point detection method

In order to study the abrupt changes of the X-ray count rate particularly at eclipse ingress and egress we performed a timing analysis of the datasets using a Bayesian change point detection method developed by Scargle (1998, 2000). This method is well suited for a statistical examination when the arrival times of individual X-ray photons are registered (see Hambaryan et al. 1999). It is superior to methods which work on binned data, since it requires no a priori knowledge of the relevant time-scale of the structure which will be investigated.

The method is applicable to data that are known to originate from a nearly ideal Poisson process, i.e. a class of independent, identically distributed processes, having zero lengths of dead time. The data gathered in XMM-Newton EPIC-PN and ROSAT PSPC observations allow the measurement of arrival times of individual X-ray photons with a resolution of 73.3 ms and 0.1 ms, respectively, a resolution much smaller than the ingress and egress time scale which is of the order of seconds. The EPIC-MOS data cannot be used for the study of the eclipse length, since they provides a resolution of only 2.6 s.

Scargle's (1998, 2000) method decomposes a given set of photon counting data into Bayesian blocks with piecewise constant count-rate according to Poisson statistics. Bayesian blocks are built by a Cell Coalescence algorithm (Scargle 2000), which begins with a fine-grained segmentation. It uses a Voronoi tessellation of data points, where neighboring cells are merged if allowed by the corresponding marginal likelihoods (see Scargle 2000).

We repeat here the essential parts of the method, expanding upon particular modifications of the original method as used in the present application. Assume that during a continuous observational interval of length T, consisting of m discrete moments in time (spacecraft's "clock tick''), a set of photon arrival times D ( t_i, t_i+1,...,t_i+n) is registered. Suppose now that we want to use these data to compare two competing hypotheses, The first hypothesis is that the data are generated from a constant rate Poisson process (model M₁) and the second one from two-rate Poisson process (model M₂). Evidently, model M₁ is described by only one parameter $\theta$ (the count rate) of the one rate Poisson process while the model M₂ is described by parameters $\theta_1$, $\theta_2$ and $\tau$. The parameter $\tau$is the time when the Poisson process switches from $\theta_1$ to $\theta_2$ during the total time T of observation, which thus is divided in intervals T₁ and T₂.

By taking as a background information (I) the proposition that one of the models under consideration is true and by using Bayes' theorem we can calculate the posterior probability of each model by (the probability that M_k(k=1,2) is the correct model, see, e.g., Jaynes 1997)

$$% Pr(M_k\vert D,I)=\frac{Pr(D\vert M_k,I)}{Pr(D\vert I)}Pr(M_k\vert I)$$ (1)

where Pr(D|M_k,I) is the (marginal) probability of the data assuming model M_k, and Pr(M_k|I) is the prior probability of model $M_k\; (k=1, 2)$. The term in the denominator is a normalization constant, and we may eliminate it by calculating the ratio of the posterior probabilities instead of the probabilities directly. Indeed, the extent to which the data support model M₂over M₁ is measured by the ratio of their posterior probabilities and is called the posterior odds ratio

$$% O_{21}\equiv\frac{Pr(M_2\vert D,I)}{Pr(M_1\vert D,I)}=\left... ...\right]\left[\frac{Pr(M_2\vert I)}{Pr(M_1\vert I)}\right]\cdot$$ (2)

The first factor on the right-hand side of Eq. (2) is the ratio of the integrated or global likelihoods of the two models and is called the Bayes factor for M₂ against M₁, denoted by B₂₁. The global likelihood for each model can be evaluated by integrating over nuisance parameters and the final result for discrete Poisson events can be represented by (see, for details, Scargle 1998, 2000; Hambaryan et al. 1999)

 

$$% B_{21}=\frac{1}{B(n+1,m-n+1)}\sum{B(n_1+1,m_1-n_1+1)} B(n_2+1,m_2-n_2+1)\Delta \tau ,$$ (3)

where B is the beta function, n_j and m_j, (j=1,2), respectively are the number of recorded photons and the number of "clock ticks'' in the observation intervals of lengths T₁ and T₂. $\Delta \tau$ is the time interval between successive photons, and the sum is over the photons' index.

The second factor on the right-hand side of Eq. (2) is the prior odds ratio, which will often be equal to 1 (see below), representing the absence of an a priori preference for either model.

It follows that the Bayes factor is equal to the posterior odds when the prior odds is equal to 1. When $B_{21}\;>\;1$, the data favor M₂over M₁, and when $B_{21}\;<\;1$ the data favor M₁.

If we have calculated the odds ratio O₂₁, in favor of model M₂over M₁, we can find the probability for model M₂ by inverting Eq. (2), giving

$$% Pr(M_2\vert D,I)=\frac{O_{21}}{1+O_{21}}\cdot$$ (4)

Applying this approach to the observational data set, Scargle's (1998, 2000) method returns an array of rates, $(\theta_1, \theta_2,..., \theta_{\rm cp})$, and a set of so called "change points'' $(\tau_1, \tau_2,...., \tau_{\rm cp-1})$, giving the times when an abrupt change in the rate is determined, i.e. a significant variation. This is the most probable partitioning of the observational interval into blocks during which the photon arrival rate displayed no statistically significant variations.

We determined the timing accuracy of these change points through simulations. We generated 1000 data sets (photon arrival times) with one change point each. The data in the two segments obeyed Poisson statistics. Each simulated data set had approximately the same characteristics as the observed data in terms of number of registered counts, spanned time, characteristic time scales of expected variations, and was analyzed exactly in the same way. The standard deviation of the distribution of change points was found to be $\Delta t_{{\rm cp}} = \pm (2$-3) s, if phase-folded data are used. The uncertainty was larger, when data in original time sequence were used due to the smaller total number of photons involved. We adopted an uncertainty of 2.5 s for the observationally determined change points which were used to derive the eclipse length.

Figure 2: Determination of the eclipse length for the ROSAT 1993 (upper two panels) and the XMM-Newton observations (lower panel) of DP Leo. The upper panel shows the distribution of the reciprocals of the time intervals between neighbouring photons, the two lower panels binned X-ray light curves (bin size 5 s). Vertical lines indicate the change points in the Poisson process, the space between them is our measured eclipse length.

In Fig. 2 we visualize the outcome of the process for the ROSAT observations performed in 1993 and for the XMM-Newton observations (EPIC-PN data only). The top panel shows the distribution of the reciprocal of the time interval between neighbouring photons (1993 data), in the two lower panels the X-ray light curves binned in intervals of 5 s are plotted. The change points determined by our method are indicated by vertical lines.

  
2.3.1 XMM-Newton EPIC PN

As a first step, we applied the change point detection method to a background region free of any X-ray source. This allowed us to determine time intervals where the background showed no significant variation. These were regarded as good time intervals and further used for the timing analysis of DP Leo.

We extracted $\sim$3180 EPIC-PN photon events from the source, whose arrival times were corrected to the solar system barycenter using the "barycen'' task, as implemented in SAS version 5.1.

We used phase-folded data and data in original time sequence in order to determine different quantities. The length of the bright phase and the eclipse length were measured in phase-folded data, the times of individual eclipses (for a period update) were measured in original time sequence.

The mean bright-phase count-rate, the eclipse length, bright phase center and length of bright phase are listed in Table 1, whereas the times of mid-eclipse of the individual eclipses are listed in in Table 2. The times given there are barycentric Julian ephemeris days, i.e. they take into account the 14 leap-seconds introduced between the first and the last data point. Since leap-seconds were omitted by seemingly all authors in the past (all timings in the literature are given in HJD only), we computed for all eclipse times we found in the literature the leap-second correction and include those times in the table for consistency and future work.

 

 

Table 1: Features of the X-ray light curve derived from ROSAT and XMM-Newton observations of DP Leo. For a given epoch and instrument we list the mean count rate in the phase interval 0.80-0.90, the length of the eclipse $\Delta t_{\rm ecl}$, the phase of the center of the bright phase interval  $\phi _{\rm C}$, and the length of the bright phase $\Delta \phi _{\rm B}$. The center of the bright phase is interpreted as accretion spot longitude.

Epoch	Mission/Det	CR	$\Delta t_{\rm ecl}$	$\phi _{\rm C}$	$\Delta \phi _{\rm B}$
		[s-1]	[s]
1992.4	ROSAT/PSPC	0.35	$237\pm5$	$0.006\pm 0.006$	-
1993.4	ROSAT/PSPC	0.50	$233\pm5$	$0.013\pm 0.006$	0.57
2000.9	XMM-Newton/EPIC	0.25	$237\pm5$	$0.067\pm 0.006$	0.57

2.3.2 ROSAT PSPC

We also performed a timing analysis of the ROSAT PSPC observations. In both cases we used a circular region with $30\hbox{$^{\prime\prime}$ }$ radius to extract source photons. In total 2705 and 8385 source counts were extracted, respectively. The radius chosen encompasses 85% of the events in the ROSAT point spread function. Photon arrival times were corrected to the solar barycenter, as implemented in the Extended Scientific Analysis Software System (Zimmerman et al. 1998).

These data were treated in the same manner as the XMM-Newton data and the corresponding results are also listed in Tables 1 and 2.

  
2.4 HST observations

Ultraviolet spectra of DP Leo were obtained on 1991 October 31 on three consecutive HST orbits. The RAPID mode was chosen to provide time resolution of 1.6914 s per low-resolution spectrum. In each orbit one complete eclipse was covered. Due to the close proximity of the orbital periods of the spacecraft and of the binary star, almost the same binary phase was covered during the three HST-orbits and full-phase coverage was not achieved. The data were published by Stockman et al. (1994). They modeled the observed spectra at certain phases with black-body spectra. Eclipse times were derived from the maxima of the time derivatives of the light curve. The data are re-analysed in the context of this paper, in order to derive a consistent set of epochs for the times of conjunction of the white dwarf. The eclipse part of the phase-folded light curve used for our analysis is shown in Fig. 3. It was derived from the original data by averaging the spectral range 1340-2400 Å, which was found free of emission lines.

   
2.5 OPTIMA observations

DP Leo was re-observed with the photon-counting camera OPTIMA (Straubmeier et al. 2001) attached to the 3.5 m telescope at Calar Alto on January 10, 2002. The observations started UT 02:54:57 and lasted 156 min, i.e. two eclipses were completely covered. The observing conditions were good with no significant seeing or transparency variations. The original data were binned in chunks of 1 s effective integration time and were performed in white light, i.e. without any filter. In the context of this paper we are concentrating on the eclipse properties only. The light curve, binned in 1 s time intervals of the first eclipse (cycle 56 307) is shown in Fig. 3, too. Since the first eclipse seemingly has the somewhat better noise properties, we are using in our analyse only those data. The second eclipse is consistent with the first, the results are essentially unchanged when data from the second eclipse are included.

Figure 3: HST/FOS and OPTIMA light curves of DP Leo obtained in October 31, 1991, and January 10, 2002, respectively. The FOS data shown are mean fluxes in the wavelength range 1300-2400 Å in units of 10-14 erg cm-2 s-1, OPTIMA data were obtained in white light (units are arbitrarily scaled count rates). Phase bins are 2.7 s (FOS) and 1 s (OPTIMA), respectively. The dotted line in the lower panel indicates the fit to the stream component (see text for details). The data were phased according to the final ephemeris of Eq. (5).