4 Spectral analysis

Unfortunately, the total RXTE PCA data set of 684 ks is not uniform in energy calibration, because the gain settings on the PCA instrument have been changed a number of times since the launch of RXTE. For the derivation of the total pulsed spectrum of PSR B0540-69 we selected therefore the observations performed in the third and longest gain epoch (April 15, 1996, 23:05-March 22, 1999, 17:37), to obtain a dataset with a constant channel-to-energy relation with total exposure of 462 ks. For each observation in this data set we derived the pulse profile and the response corrected for the offset angle. If an observation contained one or more time intervals in which the number or the order of active PCUs was different, then these intervals were treated as separate data blocks. All offset-corrected response files were summed using weighing factors (f_i) as calculated in Eq. (1), where  $N_{{\rm pcu},i}$ is the number of PCUs on in data block i, t_i is the exposure of data block i and T is the total exposure of the selected (third epoch) dataset.

$$f_{i} = \frac{N_{{\rm pcu},i} t_{i}}{5 T}\cdot$$ (1)

Pulse profiles in 14 differential energy ranges between 3.4 and 30.2 keV for the 462 ks data set were derived following the same procedure as explained in the timing analysis.

In order to determine the excess counts in the broad pulse profiles for all 14 differential energy ranges, we fitted these profiles with the shape of the best-model fit shown in Fig. 1. Again, we verified that the model fits were statistically in agreement with the data for all energy intervals. Using the numbers of excess counts and the derived response values we can derive the PSR B0540-69 pulsed spectrum measured by the PCA.

  

Table 3: Flux values derived from the ROSAT PSPC, RXTE PCA and RXTE HEXTE data (see Fig. 4). In this table we give 2$\sigma$ upper limits.

$$\begin{tabular}{r @{--} l r} \hline\hline\noalign{\smallskip ... ...6 \times 10^{-7}$ \\ \noalign{\smallskip }\hline \end{tabular}$$

For the ROSAT and RXTE HEXTE data we performed a consistent spectral analysis. Also for the profiles measured by these instruments we determined the pulsed excess counts by making fits with the empirical model. The resulting total pulsed X-ray spectrum of PSR B0540-69 is shown in Fig. 4; the flux values are given in Table 3. From the plot it is evident that the X-ray spectrum has not the shape of a single power-law. A single absorbed power-law fit to just the ROSAT PSPC and RXTE PCA data renders a photon index of  $1.845~ \pm~ 0.004$, fixing the hydrogen column density ( $N_{{\rm H}}$) at  $4.6 \times 10^{21}$ cm^-2 (Kaaret et al. 2001). This value is consistent with earlier measurements: $1.83\pm 0.13$ (Kaaret et al. 2001), $1.94 \pm 0.03$ (Mineo et al. 1999) and  $1.3 \pm 0.5$ (Finley et al. 1993). However, our reduced $\chi^2$ is unacceptably high with a value of 4.75 with 20 d.o.f., rejecting this simple shape. Therefore we have fitted the data with an absorbed power-law model with an energy dependent index:

$$F(E_{\gamma}) = {\rm e}^{-{N}_{{\rm H}} \cdot \sigma} \alpha E_{\gamma}^{-(\beta + \gamma \ln E_{\gamma})}.$$ (2)

In the fit the parameters $\alpha$, $\beta$ and $\gamma$ were kept free and  $N_{{\rm H}}$ was fixed again at  $4.6 \times 10^{21}$ cm^-2.

This resulted in a $\beta$ of  $1.360 \pm 0.005$ and $\gamma$ of  $0.143 \pm 0.003$, which means that the photon index softens at higher energies (for example at 1 keV: $\sim$1.35, 2 keV: $\sim$1.56, 4 keV: $\sim$1.76 and 8 keV: $\sim$1.95). With a $\chi ^2_{\rm r}$ of 31.0/19 the model fits reasonably well to the measured spectrum. For energies between 20 and 50 keV, the measured shape is poorly defined, and the flux values of the PCA and HEXTE are also consistent with a power-law with index 2.

Figure 4: Total pulsed spectrum of the Crab pulsar and PSR B0540-69 for X-ray energies above 0.01 keV. Crab (top): Flux measurements taken from Kuiper et al. (2001). PSR B0540-69 (bottom): Flux measurements and upper limits from ROSAT PSPC (open squares, 0.01-2.5 keV), RXTE PCA (open circles, 3.4-30 keV) and RXTE HEXTE (open diamonds, 14-300 keV) are shown, as well as upper limits at high-energy $\gamma$-rays from CGRO EGRET (Thompson et al. 1994). The sizes of the arrows on the left indicate the shift in flux ($\pm$1$\sigma$) to put the Crab pulsar at the LMC distance. We used $2.0 \pm 0.5$ kpc for the Crab and $49.4 \pm 3.4$ kpc for the PSR B0540-69 (LMC) distance.