5 Optical spectral index map

Firstly, we consider how to derive a map of the spectral index $\alpha_{\rm RU}$ from the presented images, at a common effective beam size of 0 $.\!\!^{\prime\prime}$2.

   
5.1 Definition of optical spectral index

Independently of whether a spectrum actually does follow a power law over any range of frequencies, a local two-point spectral index can be defined between any two surface brightness measurements B₁, B₂ at frequencies $\nu_1$ and $\nu_2$ (with $\nu_2 < \nu_1$), respectively, as

$$% \alpha = \frac{\ln\frac{B_1}{B_2}}{\ln \frac{\nu_1}{\nu_2}}\cdot$$ (1)

The error on the spectral index is computed from the values of the noise $\sigma_1, \sigma_2$ in the respective input images:

$$% \sigma_{\alpha} = \frac{1}{\ln\frac{\nu_1}{\nu_2}} \sqrt{\frac{\sigma^{2}_1}{B^{2}_1} + \frac{\sigma^{2}_2}{B^{2}_2}}\cdot$$ (2)

This formula shows that the spectral index error depends on the S/N of the input images and the "baseline'' between the wavelengths at which the observations are made. In the following, we will use $\nu_1 = \nu(U)$ for the UV and $\nu_2 = \nu(R)$ for the red, together with B₁ = B_U, B₂ = B_R. With the mean wavelengths of the filters, 2942.8Å for F300W and 6162.8Å for F622W (Biretta et al. 2000), a cut on the signal-to-noise ratio $S/N > S/N_{{\rm cut}}$ of the input images limits the spectral index error to at most

$$% \sigma_{\alpha_{RU}} \leq { \frac{1}{\ln \frac{6162.8}{2942... ...{\frac{2}{S/N_{{\rm min}}}} = \frac{1.9}{S/N_{{\rm min}}}\cdot$$ (3)

In our case, the UV-band S/N is always much inferior to that in the red band, so the error will actually be dominated by the UV noise and hence smaller than the maximum from Eq. (3) for the most part. The noise in the input images is calculated from the calibrated images and includes shot noise due to both observed photons and dark current as well as the read noise, but excludes systematic errors, whose expected magnitude we now examine.

   
5.2 Systematic errors for a spectral index determination

We consider the systematic errors which may be introduced when combining two images taken through different filters, or by different instruments and telescopes. The main danger in a determination of spectral gradients lies in a misalignment between the images which would introduce spurious gradients; referring the flux to different effective beam sizes will lead to wrong spectral index determinations as well. Following considerations given in Sect. A.1, we deduce that a 5% limit on the flux error due to misalignment requires aligning the images to better than 10% of the effective PSF full width. The error in the PSF determination is negligible when, as in our case, the smoothing Gaussian is much wider than the PSF. The 0 $.\!\!^{\prime\prime}$2 effective resolution aimed for thus requires knowing the relative alignment of all images in the data set to better than 20mas or 0.44 PC pixels. The absolute telescope pointing does not need to be known for this purpose as we tie all positions to the quasar core as origin. After detailed investigations of all issues related to relative alignment of HST images (see Sect. A.2), we used the following procedure: as the first step, all exposures through one filter and within one set of observations are summed up using the relative positions from the "jitter files'' provided as part of the observing data package (typical error 5 mas). The exposures through F300W were distributed over three different sets of observations, so we obtain three intermediate images. For aligning these with each other, we use the average of the positional shifts of the quasar and three astrometric stars (observed on the Wide Field chips) measured on the intermediate image or the short exposure in each set of observations (error 10 mas). The field flattener windows inside WFPC2 introduce a wavelength dependence of the plate scale, which has to be removed prior to combination of the images taken through different filters. We therefore sample all four intermediate images (three UV and one red) onto a grid of one tenth of their average plate scale with a pixel size of 0 $.\!\!^{\prime\prime}$0045548. We sum the UV intermediate images to give a UV final image. Because of the scale change, we have to align this to the red image using the quasar position only (estimated error 10 mas-15 mas). We then rebin both images to the final pixel size of 0 $.\!\!^{\prime\prime}$045548. Adding up all errors in quadrature, the error margin of 20 mas is just kept.

Figure 3: Optical spectral index $\alpha _{RU}$ ( $f(\nu ) \propto \nu ^{\alpha }$) at 0 $.\!\!^{\prime\prime}$2 resolution. The contours show the red-band image and are logarithmic with a factor $\sqrt {2}$ from 0.07 $\mu$Jy/beam to 0.8 $\mu$Jy/beam.

Note that the rebinning has only matched the scales of the two images, but not removed the geometric distortion of the focal plane. The next-order wavelength-dependent term in the geometric distortion solution is at least two orders of magnitudes smaller than the scale difference. Both the red and the ultraviolet image can thus be assumed to have identical distortion solutions. We therefore ignore the geometric distortion for the remainder of this paper, but caution that it must be taken into account when comparing these data with data obtained at other instruments.

   
5.3 Calculation of the optical spectral index map

We have now aligned the images in the two filters to better than 0 $.\!\!^{\prime\prime}$02. When comparing flux measurements on these images, they have to be referred to a common beam size. The maps are therefore smoothed to a common 0 $.\!\!^{\prime\prime}$2 using Gaussians with a width matched to the effective FWHM of the PSF in each image. This operation retains the original pixel size. As few point sources are available, the effective FWHM is determined on PSF models generated using the TinyTIM software (Krist 1999), resulting in 0 $.\!\!^{\prime\prime}$08 on the red and 0 $.\!\!^{\prime\prime}$06 on the UV-band image. Because of sampling effects, the measured widths are larger than the size of the Airy disk of the HST's primary at the corresponding wavelengths. Figure 3 shows the spectral index map obtained from the images (Figs. 1 and 2) according to Eq. (1). Only those points are shown in the map which have an aperture S/N of at least 5 on both images. This limits the error in the spectral index to 0.4 (Eq. (3)). The error is less than 0.1 inside all knot regions, so that a colour change indicates statistically significant variations of the spectral index. Figure 4 shows the spectral index along a tracing of the jet obtained from a rotated map with the jet's mean position angle of 222 $.\!\!^\circ$2 along the horizontal.

Figure 4: Run of the red-band brightness and optical spectral index along the outer half of the jet in 3C273, for a 0 $.\!\!^{\prime\prime}$2 beam. $\alpha _{RU}$ was determined from Figs. 1 and 2, while $\alpha _{BRI}$ for a 1 $.\!\!^{\prime\prime}$3 beam is taken from Röser & Meisenheimer (1991). For comparison, we show $\alpha _{{\rm fit}}$, the corresponding spectral index obtained from synchrotron spectra fitted by Meisenheimer et al. (1996a) and Röser et al. (2000). While the observed spectral index agrees with older data, it is now clear that the fit is inadequate for the optical part of the spectrum. The steeper spectral index of the fit may either be due to contamination of the infrared flux by a "backflow'' component around the jet, or because of the presence of a second high-energy electron population in the jet which is not included in the fit (see Sect. 6.3.1 for a discussion of the discrepancies).

   
5.4 Colours of the extensions

The outer extension is not bright enough to show up on the map in its entirety. The variation of $\alpha _{RU}$ across it are consistent with a spiral galaxy. The inner extension's two knots have markedly different colours on the spectral index map. Knot In2 shows a spectral index gradient, $\alpha \approx -0.9$ to -0.3 roughly parallel to the jet and outwards from the quasar position. Knot In1 has a much steeper spectral index of about -1.5 and shows no gradient. Interestingly, the southern extension S has the flattest spectrum ( $\alpha_{RU} \approx -0.4$) of all regions on the map.

   
5.5 Spectral index of the jet

The optical spectral index declines globally outwards from -0.5 near the onset of the optical jet at A to -1.6 in D2/H3. This trend does not continue into the hot spot, again stressing the physical distinction between jet and hot spot. The general steepening is in agreement with previous determinations of the knots' synchrotron spectrum which showed a decline of the cutoff frequency outwards (Meisenheimer et al. 1996a; Röser et al. 2000). The optical spectral index $\alpha _{BRI}$ determined at 1 $.\!\!^{\prime\prime}$3 resolution (Röser & Meisenheimer 1991) agrees very well with our new determination of $\alpha _{RU}$ at the much higher resolution (Fig. 4). The only discrepancy arises in C2/D1; we defer a discussion to Sect. 6.3.1. The smooth variations of the spectral index along the jet show that the physical conditions in the jet change remarkably smoothly over scales of many kpc. There is no strict correlation between red-band surface brightness and spectral index like that found in the jet in M 87 (Meisenheimer et al. 1996a; Perlman et al. 2001). There is a marginally significant flattening of the spectrum in the transitions A-B1, B1-B2, C1-C2, and moving out of C2, consistent with the absence of losses. In any case, the overall steepening of the spectrum (from region A down do D2/H3) is less rapid than that within individual regions (e.g., A and B2). There is no significant steepening from D1 out to the bridge between H3 and H2, despite large variations in surface brightness. The criss-cross morphology in C1, C2, B1, B2 and D1 is reflected on the spectral index map as a band of one colour crossing a second one. This is most clearly seen in knot C1 which has a green band of $\alpha \approx -1.1$ across an orange region of $\alpha \approx -1.4$, supporting the interpretation of two emission regions appearing on top of each other. The spectral index near the hot spot shows a flip from flat (-1.0) to steep (-1.5).