3 Timing analysis

For all selected photons measured with the RXTE PCA with energies above 2 keV barycentric corrections to their arrival times have been derived using the FTOOLS program fasebin (Blackburn 1995) and a source position of $\alpha_{2000} = 05^{\rm h}40^{\rm m}11\hbox{$.\!\!^{\rm s}$ }221$ and $\delta_{2000} = -69{\hbox{$^\circ$ }}19{\hbox{$^\prime$ }}54\hbox{$.\!\!^{\prime\prime}$ }98$ (Kaaret et al. 2001). The ephemeris used to derive the pulse profile is presented in Table 2. This ephemeris predicts accurately the instantaneous rotation frequency of the source for the epochs of our observations. However, it is not a phase connected timing solution.

For each observation contributing to our total data set of 684 ks we derived a pulse profile. Because the ephemeris is not a phase connected solution for long timescales, we cannot combine the individual profiles directly. Therefore, we correlated all pulse profiles first with a profile template based on the longest data block and later again with the result of the aligned combination to obtain the final profile. The statistical uncertainties in the estimates of the phase shifts varied approximately over the range 0.6-3%.

 

 

Table 2: Ephemeris of PSR B0540-69 determined by Kaaret et al. (2001).

Parameter	Value
Val. range (MJD)	46 992-51 421
t0 (MJD)	47 700.0
$\nu_{0}$ (Hz)	19.8593584982(40)
$\dot{\nu}_{0}$ (10-10 Hz s-1)	-1.8894081(7)
$\ddot{\nu}_{0}$ (10-21 Hz s-2)	3.7425(43)
$\alpha_{2000}$	$05^{\rm h}40^{\rm m}11\hbox{$.\!\!^{\rm s}$ }221$
$\delta_{2000}$	$-69{\hbox{$^\circ$ }}19{\hbox{$^\prime$ }}54\hbox{$.\!\!^{\prime\prime}$ }98$

Note: Numbers in parentheses are 1$\sigma$ errors in the last quoted digits.

Figure 1: Master pulse profile of PSR B0540-69 derived from the total 684 ks RXTE PCA data set for energies 2-20 keV. The broken line shows the best fit empirical model consisting of two symmetric Gaussian profiles separated $\sim$0.2 in phase. The lower figure shows the distribution of the residuals in number of $\sigma$'s.

Figure 1 shows the resulting pulse profile (2-20 keV) of PSR B0540-69 for the total screened exposure of 684 ks. The statistical accuracy of this profile is at least an order of magnitude higher than for earlier published profiles (e.g. Eikenberry et al. 1998; Mineo et al. 1999; Hirayama et al. 2002) and the genuine profile becomes more clearly visible: A single broad pulse with a clear dip at the top. Such a shape could be the result of two narrower pulses separated $\sim$0.2 in phase. To investigate this further, we attempted to describe the shape of the profile as the sum of two symmetrical Gaussians on top of a flat background, by making a fit to the measured profile with as free parameters the widths, positions and amplitudes of the gaussians. The resulting best fit is shown in Fig. 1 (positions:  $p_{{\rm 1}} = 0.330 \pm 0.004$, $p_{{\rm 2}} = 0.531 \pm 0.003$; widths: $\sigma_{{\rm 1}} = 0.102 \pm 0.003$, $\sigma_{{\rm 2}} = 0.079 \pm 0.002$) with underneath the fit residuals expressed in number of $\sigma$'s. The overall fit looks good, particularly for the component with maximum at phase 0.531, but there are a few phase intervals in which there appear significant deviations from this empirical model ( $\chi ^2_{\rm r}$ of fit: 167.3/93 ; $P(\chi^2 > 167.3 \mid 93) = 3.65 \times 10^{-6}$ (4.7$\sigma$ dev.)). The most significant positive excess is around phase 0.3 (near the position of one of the Gaussians) reaching a significance of 5.4$\sigma$, proving that the top of the broad profile has significant fine structure. The positive excesses far in the wings at phases around 0.1 and 0.7 reach significances of 3.6 and 4.0$\sigma$, respectively. We do not consider these latter deviations significant, because there was no a priori reason to select these phases (no single trial). Hirayama et al. (2002) discussed a hump in the ASCA profile at a phase consistent with our excess around phase 0.1, as a possible separate interpulse in analogy to the Crab profile. With our better statistics we do not confirm the detection of this hump. Finally, the negative excursion at phase 0.24, together with the positive excess at phase 0.3 suggest that the leading wing of the profile is steeper than fitted with the Gaussian profile.

Figure 2: Pulse profiles of PSR B0540-69 from the optical window to hard X-rays. a) Optical profile as observed by Boyd et al. (1995) with Hubble's High Speed Photometer (HSP). b)-c) ROSAT PSPC, energy ranges 0.02-1.3 and 1.3-2.5 keV. d)-j) RXTE PCA, energy ranges 2-4, 4-6, 6-8, 8-12, 12-16, 16-20 and 20-30 keV. k)-l) RXTE HEXTE, energy ranges 14-25 and 34-48 keV. The RXTE HEXTE lightcurve for energies 25-34 keV is not shown since it does not give a significant signal due to high instrumental background. The ROSAT and HST profiles have been aligned in phase to the RXTE profiles by cross correlation.

The high counting statistics in the PCA data allow us to construct significant pulse profiles in differential energy windows from 2 keV up to 30 keV (Figs. 2d to j), showing the first profiles above 10 keV. Up to 12 keV the statistical accuracy is very high and some fine structure on top of the broad profile seems to be consistently present in successive energy windows, most notably the sharp maximum at phase $\sim$0.3.

Figure 2 also shows pulse profiles we derived from RXTE HEXTE data at higher energies and from ROSAT data at lower energies. We could construct the HEXTE profiles of PSR B0540-69 by following the same analysis steps for the same observation windows as was performed for the PCA timing analysis, applying the phase shifts determined in the correlation analysis of the high-statistics PCA profiles to the low-statistics HEXTE profiles. This allowed us to reveal the PSR B0540-69 pulse profile up to 48 keV. The ROSAT profiles are derived using the data reported earlier in Sect. 2.3, and the phase alignment was determined by cross correlating the PCA master profile with the total ROSAT profile. Finally, the optical PSR B0540-69 pulse profile from the High Speed Photometer (HSP) aboard the Hubble Space Telescope (HST) (Boyd et al. 1995) is shown. The optical profile shape is strikingly similar to the PCA X-ray profile. Therefore, we choose to align also the optical profile by cross correlation. We note, however, that Ulmer et al. (1999) report in a preliminary communication that there is a phase shift of 0.17 between the optical and the X-ray pulse, the optical arriving earlier. We would like to see this result confirmed.

At a first glance it is not obvious, due to the varying statistics, whether the pulse shape varies from the optical to the hard X-ray windows. Therefore, we fitted now all profiles in Fig. 2 with the empirical model of two Gaussian profiles with the positions and the widths fixed to the best-fit values shown in Fig. 1 and leaving the normalisations free. We verified that all fits are good with reduced $\chi^2$ values ranging between 0.93 and 1.32 (98 d.o.f.), except for the optical profile (18 d.o.f.). Only this optical profile appears to be somewhat broader than the X-ray profiles. Figure 3 shows the ratio of the counts in the two Gaussian components from the optical regime to the hard X-rays. This ratio appears to be surprisingly constant, underlining the stability of the profile shape and showing that the spectra of the two empirical components are within the statistics the same. Therefore, we cannot conclude that the broad and structured pulse profile of PSR B0540-69 is the sum of two distinct components with different spectra.

Figure 3: Ratio of the pulsed (excess) counts assigned to the two Gaussians components of the model fit to the profiles shown in Fig. 2. This distribution is consistent with being flat with a P1/P2 value of  $1.63 \pm 0.01$ with a  $\chi ^2_{\rm r}$ of 21.66 / 11 resulting in a random probability of 2.7%. Data points are derived from the optical HST HSP (square), ROSAT PSPC (circle), RXTE PCA (diamond) and RXTE HEXTE (triangle) observations.