A&A 373, 447-458 (2001)
DOI: 10.1051/0004-6361:20010593

HST optical spectral index map of the jet of 3C273[*]

S. Jester 1 - H.-J. Röser 1 - K. Meisenheimer 1 - R. Perley 2 - R. Conway 3


1 - Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany
2 - NRAO, PO Box 0, Socorro, NM 87801, USA
3 - Jodrell Bank Observatory, Macclesfield, Cheshire, SK11 9DL, UK

Received 26 March 2001 / Accepted 23 April 2001

Abstract
We present HST images at 622nm and 300nm of the jet in 3C273 and determine the run of the optical spectral index at 0 $.\!\!^{\prime\prime}$2 along the jet. The smoothness of spectral index changes shows that the physical conditions are varying smoothly across the jet. There is no correlation between the optical flux and spectral index, as would be expected for relativistic electrons suffering strong cooling due to synchrotron emission. We find no evidence for localized acceleration or loss sites. This suggests that the spectral shape is not changing much throughout the jet. We show that relativistic beaming and/or sub-equipartition magnetic fields cannot remove the discrepancy between light-travel time along the jet and the lifetime of electrons emitting optical synchrotron radiation. We consider this further evidence in favour of a distributed electron acceleration process.

Key words: galaxies: jets - quasars, individual: 3C273


   
1 Introduction

While radio jets are a common feature of radio galaxies and quasars, optical emission has to date been observed from only about 15 extragalactic jets. As shown by polarimetric observations (starting with Baade 1956 for M 87), both the radio and optical emission is synchrotron continuum radiation. While information on the source's magnetic field structure may be obtained from the polarisation structure, the diagnostic tool for the radiating particles is a study of the synchrotron continuum over as broad a range of frequencies as possible, i.e., from radio to UV or even X-ray wavelengths, and with sufficient resolution to discern morphological details. The radio and optical emission observed from hot spots in radio jets can be well explained by first-order Fermi acceleration at a strong shock in the jet (the bow shock) (Meisenheimer & Heavens 1986; Heavens & Meisenheimer 1987; Meisenheimer et al. 1989, 1997). But it is not clear that the optical synchrotron emission from the jet body, extending over tens of kiloparsecs in some cases, can be equally well explained by acceleration at strong shocks inside the jet. As is well known from standard synchrotron theory, electrons with the highly relativistic energies required for the emission of high-energy (optical and UV) synchrotron radiation have a very short lifetime which is much less than the light-travel time down the jet body in, e.g., 3C273. Observations of optical synchrotron emission from such jets (Röser & Meisenheimer 1991; 1999) as well as from the "filament'' near Pictor A's hot spot (Röser & Meisenheimer 1987; Perley et al. 1997) suggest that both an extended, "jet-like'' and a localized, "shock-like'' acceleration process are at work in these objects in general and 3C273's jet in particular (Meisenheimer et al. 1997). The extended mechanism may also be at work in the lobes of radio galaxies, where the observed maximum particle energies are above the values implied by the losses within the hot spots (Meisenheimer et al. 1996) and by the dynamical ages of the lobes (Blundell & Rawlings 2000). The fundamental question is thus: how can we explain high-frequency synchrotron emission far from obvious acceleration sites in extragalactic jets? Although most of the known optical jets are very small and faint (Scarpa & Urry 2000), there are a few jets with sufficient angular size and surface brightness to be studied in detail: those in M 87 (a radio galaxy), PKS  0521-365 (an elliptical galaxy with a BLLac core), and 3C273 (a quasar). We have embarked on a detailed study of the jet in 3C273 using broad-band observations at various wavelengths obtained with today's best observatories in terms of resolution: the VLA (in combination with MERLIN data at $\lambda$6cm) and the HST. Using these observations, we will derive spatially resolved (at 0 $.\!\!^{\prime\prime}$2) synchrotron spectra for the jet. 3C273's radio jet extends continuously from the quasar out to a terminal hot spot at 21 $.\!\!^{\prime\prime}$5 from the core, while optical emission has been observed only from 10 $^{\prime \prime }$ outwards[*]. On ground-based images, the optical jet appears to consist of a series of bright knots with fainter emission connecting them. So far, synchrotron spectra have been derived for the hot spot and the brightest knots using ground-based imaging in the radio (Conway et al. 1993), near-infrared $K^{\prime}$-band (Neumann et al. 1997) and optical I, R, B-bands (Röser & Meisenheimer 1991) at a common resolution of 1 $.\!\!^{\prime\prime}$3 (Meisenheimer et al. 1996a; Röser et al. 2000). The radio-to-optical continuum can be explained by a single power-law electron population leading to a constant radio spectral index[*] of -0.8, but with a high-energy cutoff frequency decreasing from $10^{17}\,$Hz to 1015Hz outwards along the jet. The aim of the study is both the determination of the spectral shape of the synchrotron emission, and by fitting synchrotron spectra according to Meisenheimer et al. (1989), deriving the maximum particle energy everywhere in the jet. The observed spectra can then be compared to predictions from theoretical work.
 

 
Table 1: Passbands and limiting magnitudes for the observations. Point source: $10\sigma $ detection limit, prediction by the exposure time calculator. Extended source: $5\sigma $ per pixel detection limit, determined from the background noise measured on reduced frames. Magnitudes are Vegamags referred to the corresponding HST filter band.
Filter Mean $\lambda$ FWHM Exposure Point source Extended source
  nm nm s mag $\mu $Jy mag/ $\ifmmode\hbox{\rlap{$\sqcap$ }$\sqcup$ }\else{\unskip\nobreak\hfil \penalty50\h... ...$ } \parfillskip=0pt\finalhyphendemerits=0\endgraf}\fi\hbox{$^{\prime\prime}$ }$ $\mu $Jy/ $\ifmmode\hbox{\rlap{$\sqcap$ }$\sqcup$ }\else{\unskip\nobreak\hfil \penalty50\h... ...$ } \parfillskip=0pt\finalhyphendemerits=0\endgraf}\fi\hbox{$^{\prime\prime}$ }$
F300W 301 77 35500 26.1 0.04 20.8 5.2
F622W 620 92 10000 27.7 0.04 23.0 2.8


As an intermediate result of our study, we present HST WFPC2 images of the jet in 3C273 in a red (F622W) and near-UV (F300W) broadband filter. The near-UV observations constitute the highest-frequency detection of synchrotron emission from 3C273 so far. (Extended X-ray emission has also been observed, but it is unclear at present whether this, too, is due to synchrotron radiation, Röser et al. 2000; Marshall et al. 2001; Sambruna et al. 2001.) From these images, we construct an optical spectral index map at 0 $.\!\!^{\prime\prime}$2 resolution. After a description of the observations and the data reduction in Sects. 2 and 3, we examine the direct images in Sect. 4. The creation and description of the spectral index map follow in Sect. 5. We analyse the map in Sect. 6 and conclude in Sect. 7. Details regarding the alignment of HST images are found in Appendix A.

   
2 Observations

Observations were made using the Planetary Camera of the second Wide Field and Planetary Camera (WFPC2) (Biretta et al. 2000) on board the HUBBLE SPACE TELESCOPE[*] in three sets of observations on March 23rd and June 5th/6th, 1995, for 35500s through filters F300W (ultra-violet, U) and 10000s through F622W (roughly $R_{\rm C}$ in the Kron-Cousins system). The resulting limiting magnitudes are listed in Table 1. The jet was imaged onto the center of the Planetary Camera (PC) chip which has 800 by 800 pixels with a nominal size of 0 $.\!\!^{\prime\prime}$0455 projected on the sky. The telescope was oriented such that the position angle of the PC chip's y-axis was $\approx$ $154\hbox{$^\circ$ }$. This way, the quasar itself is also mapped on the PC, while each of three neighbouring astrometric reference stars (see Table 3 in Röser & Meisenheimer 1991) was observed on one of the three Wide Field chips. This choice gives distortion-free images of the jet, while allowing the use of the quasar and the reference stars for image alignment. The total exposure time was split into individual exposures of about 2500s or one HST orbit, executed in three separate sets of observations. Each of these exposures was done at a telescope pointing slightly offset from all others (by up to 1 $^{\prime \prime }$) to facilitate the correction for chip artifacts. One short exposure was obtained in each set of observations and filter to measure the positions of the quasar and reference stars, most of which are saturated on the long exposures. Our study of 3C273's jet will be conducted employing a 0 $.\!\!^{\prime\prime}$2 effective beam size. This results in a typical signal to noise ratio (S/N) per resolution element in the red band of around 100 in the brightest regions and 30-40 in inter-knot regions. Because both the jet flux and the WFPC2 and telescope throughput decrease towards the UV, the UV-band S/N is only about 40 in the brightest regions, while the inter-knot regions are barely detected.

   
3 Data reduction

The data were re-calibrated under IRAF with the STSDAS package provided by STScI, using the "most recent'' calibration reference files as of December 1998, because some of the files had changed since the original observations. The values of background noise measured on the calibrated frames agree well with the values expected from photon statistics, as calculated from the expected read noise, dark current and sky background level. One of the PC chip's charge traps (Whitmore & Wiggs 1995; Voit et al. 1998) lies inside the jet image, in Col. 339. This has no observable effect on the faint UV image, but the effect had to be corrected on the well-exposed red-band images. This was done by replacing the affected portion of each image by the corresponding pixels from an offset image, as a correction according to Whitmore & Wiggs (1995) unduly increased the noise in the corrected part of the image. The images were initially registered using the commanded offsets to the nearest pixels. This alignment is sufficient for the rejection of cosmic rays as these only affect a small number of adjacent pixels. Cosmic rays were rejected using a standard $\kappa$-$\sigma$ algorithm, rejecting all pixel values deviating more than $4 \sigma$ from the local (low-biased) median in a first pass, and neighbouring pixels with more than $2.5 \sigma$ deviation in a second pass. The number of pixels treated this way agrees with the expected cosmic ray hit rate for the images. A model of the sky background and "horizontal smear'' (increased pixel values in rows containing saturated pixels from the quasar's core, Chap. 4 of Biretta et al. 2000) was fitted in the part of the image containing the jet using second-order polynomials along rows. The coefficients of the polynomials were then smoothed in the perpendicular direction. The conversion from photon count rate to physical flux units used the throughput information provided by STScI which is valid after the 1997 SYNPHOT update.

  \begin{figure} \includegraphics[width=11cm,clip]{ms1311f1.eps} % \end{figure} Figure 1: The jet in red light (620nm) after background subtraction. Logarithmic grey-levels run from 0 to 0.04$\mu $Jy/pixel, 0 $.\!\!^{\prime\prime}$08 effective beam size, 0 $.\!\!^{\prime\prime}$045 pixel size. The quasar core lies 10 $^{\prime \prime }$ to the northeast from A. The labelling of the jet features as introduced by Lelièvre et al. (1984) and extended by Röser & Meisenheimer (1991), together with the hot spot nomenclature from Flatters & Conway (1985) is also shown. Note that the labelling used by Bahcall et al. (1995) is slightly different.
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  \begin{figure} \includegraphics[width=11cm,clip]{ms1311f2.eps} % \end{figure} Figure 2: The jet in UV light (300nm) after background subtraction. Logarithmic grey-levels run from 0 to 0.014 $\mu $Jy/pixel, 0 $.\!\!^{\prime\prime}$06 effective beam size, 0 $.\!\!^{\prime\prime}$045 pixel size.
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4 Maps of the optical brightness

The calibrated images are presented in Figs. 1 and 2. The morphology of the jet is identical in both images and appears rather similar to the morphology in high-resolution radio maps (Conway et al. 1993; Bahcall et al. 1995). The exception to this is the radio hot spot, being the dominant part in the radio but fairly faint at high frequencies. Our images show structural details of the optical jet which were not discernible on earlier, shallower and undersampled HST WF images of 0 $.\!\!^{\prime\prime}$1 pixel size (Bahcall et al. 1995). Based on our new maps, the term "knots'' seems inappropriate for the brightness enhancements inside the jet, as these regions are resolved into filaments. The higher resolution necessitates a new nomenclature for the jet features (Fig. 1). For consistency with earlier work (Lelièvre et al. 1984; Flatters & Conway 1985; Röser & Meisenheimer 1991), our nomenclature is partly at variance with that introduced by Bahcall et al. (1995). The jet is extremely well collimated - region A has an extent (width at half the maximum intensity) of no more than 0 $.\!\!^{\prime\prime}$8 perpendicular to the average jet position angle of $\sim$ $222\hbox{$^\circ$ }$ (opening angle $\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil $\displaystyle ... $5\hbox{$^\circ$ }$). Even region D2/H3 is only 1 $^{\prime \prime }$ wide (opening angle $\approx$ $2.5\hbox{$^\circ$ }$). The optical jet appears to narrow towards the hot spot, in the transition from H3 to H2. Region A is now seen to extend further towards the core than previously known. It may be noteworthy that Lelièvre et al. (1984) reported the detection of an extension of knot A towards the quasar, whose existence at the reported flux level was not, however, confirmed by later work. The criss-cross pattern visible in regions C1 and C2, and less clearly in B1-2 and D1, is reminiscent of a (double?) helical structure (Bahcall et al. 1995), but could also be explained by oblique double shocks (Hardee & Norman 1989). The jet has three "extensions'' (Fig. 1), none of which has been detected at radio wavelengths. The morphology of the outer extension supports the classification as a galaxy based on its colours made by Röser & Meisenheimer (1991). The nature of the other two extensions, however, remains unknown even with these deeper, higher resolution images. The northern inner extension was already resolved into two knots (In1, In2) on a FAINT OBJECT CAMERA image (Thomson et al. 1993). The two knots are extended sources and clearly connected to each other. The southern extension is featureless and an extended source. Comparing the direct images, we can immediately estimate that the jet's colour slowly turns redder outwards from region A. The similarity of the jet images in both filters shows that there are no abrupt colour changes within the jet. For a quantitative assessment of the extensions' and the jet's colour in the following, we derive an optical spectral index map.

   
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

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

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

 \begin{displaymath}% \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 \end{displaymath} (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 = BU, B2 = BR. 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

 \begin{displaymath}% \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 \end{displaymath} (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.
  \begin{figure} \resizebox{11cm}{!}{\includegraphics{ms1311f3.eps}} % \end{figure} 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.
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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.
  \begin{figure} \resizebox{11cm}{!}{\includegraphics{ms1311f4.ps}} % \par\end{figure} 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).
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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).

   
6 Discussion

   
6.1 The jet's extensions

Before analysing the jet emission proper, we consider the relation of the extensions to the jet. Our new images confirm the outer extension's nature as spiral galaxy. The spectral index map gives us few hints about the nature of the emission mechanism for the inner and southern extensions. We have therefore measured their integral fluxes at 300nm and 620nm (this paper) and at 1.6 $\mu $ (HST NICMOS camera 2 imaging[*], in prep.). Figure 5 shows the resulting spectral energy distributions. In1, with the steepest $\alpha _{RU}$, is also the brightest in the infrared. The infrared flux points of both extension S and of In2 lie on the continuation of the power law with index -0.4, determined on the optical spectral index map. If this power law were valid up to radio wavelengths, it would predict a flux at $\lambda 6$cm of about 50 $\mu $Jy for the northwestern knot In1, and of about 45 $\mu $Jy for the southern extension S. This is a factor of order 100 below the flux from the jet at this wavelength (Conway et al. 1993), and any emission at this level would not be detected even on our new VLA maps with an rms noise of $\approx$0.8mJy. There is no similarly obvious extrapolation of In1's SED to radio wavelengths. A full comparison will be presented in a future paper. The inner extension shows a polarisation signal on unpublished ground-based maps made by our group and on polarimetric images obtained with the Faint Object Camera on board HST (Thomson et al. 1993). A polarisation signal could be due to scattered quasar light (Röser & Meisenheimer 1991) or, together with possible radio emission, synchrotron radiation. We aim to clarify the nature of the inner extensions by planned polarimetric and spectroscopic observations with the VLT.
  \begin{figure} \resizebox{6.8cm}{!}{\includegraphics{ms1311f5.eps}} \end{figure} Figure 5: Broad-band flux points for the radio-quiet extensions to the jet from HST imaging at 1.6$\mu $, 620nm and 300nm.

   
6.2 The hot spot

The optical counterpart to the radio hot spot (H2) appears very faint on both images, and the spectral index map shows a flip from flat to steep there. Both features are explained by the Meisenheimer & Heavens (1986) hot spot model: the lifetime of electrons emitting in the optical is quenched by the strong magnetic field in the hot spot. The spectral index flip is expected if there is an offset between the emission peaks at different frequencies. This offset is predicted by the theory by Meisenheimer & Heavens (1986) and is clearly seen when comparing optical and radio images (Röser et al. 1997; a detailed comparison of all data will be presented in a future paper). By its spectrum and morphology, the hot spot appears exactly as expected for a high-loss synchrotron emission region downstream of a localized strong shock with first-order Fermi acceleration. Finally, we note that although the term "hot spot'' is historically in use for this part of the jet, the term is ill-defined and may have become inadequate, but we defer a discussion of its adequacy to a future paper.

   
6.3 Main body of the jet

Our HST images (Figs. 1 and 2) show a close coincidence of the jet's morphology over a factor of 2 in frequency (optical and UV). The spectral index map (Fig. 3) shows amazingly smooth variations in the physical conditions over the entire jet. The electrons' synchrotron cooling leaves little imprint on the spectral index along the jet, contrary to expectations.
   
6.3.1 The spectral index at HST resolution

  \begin{figure} \resizebox{8.2cm}{!}{\includegraphics{ms1311f6.eps}} \end{figure} Figure 6: Optical spectral index against red-band brightness for the outer half of the jet in 3C273 (data as in Fig. 4). There is no strict correlation like that observed in the jet of M 87.
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The two striking features of our spectral index map are the smooth variation of the spectral index over the entire jet (Fig. 4), and the lack of a strong correlation between red-band brightness and spectral index (Fig. 6) like that observed in the jet of M 87 (Meisenheimer et al. 1996a,b; Perlman et al. 2001). There are large but smooth variations of the spectral index, while the surface brightness remains fairly constant over the jet's projected extent of about 10 $^{\prime \prime }$. Conversely, there are large local variations of surface brightness without strong changes in the spectral index. We do not observe abrupt changes of the spectral index inside the jet like we do in the hot spot. The run of the spectral index is thus consistent with the complete absence of energy losses over scales of many kiloparsecs, in spite of the large observed synchrotron luminosity, which indicates the need for re-acceleration of particles inside the jet. The mismatch between synchrotron cooling scale and extent of the jet even led Guthrie & Napier (1975) to conclude that the jet's emission could not be synchrotron radiation. The same lack of apparent synchrotron cooling is implied by the surprising overall correspondence between $\alpha _{BRI}$ and $\alpha _{RU}$ (Fig. 4) which have been determined at vastly different resolutions (1 $.\!\!^{\prime\prime}$3 and 0 $.\!\!^{\prime\prime}$2, corresponding to 4.2h60kpc and 640h60pc, respectively). The small differences (strongest in regions A and C2/D1) can be explained at least in part by the different beam sizes: the spectral index determined by smoothing the HST images to 1 $.\!\!^{\prime\prime}$3 makes the discrepancy smaller. In addition, the emission region becomes wider towards longer wavelengths, an effect already noticeable in the near-infrared $K^{\prime}$ band (Neumann et al. 1997) and attributed to a "backflow'' (Röser et al. 1996). The presence of such a steep-spectrum, diffuse component more extended than the jet channel might also explain the discrepancy. The general agreement indicates a constancy of the shape of the spectrum even at the scales resolved by HST. To first order, the appearance of the jet at all wavelengths can be explained through such a constant spectral shape consisting of a low-frequency power law and a high-frequency curved cutoff. The assumed spectral shape is shifted systematically in the $\log\nu-\log S_{\nu}$ plane, passing different parts of the curved cutoff through the optical flux point. This leads to similar morphological features at all wavelengths, in accord with observations (Conway et al. 1993; Bahcall et al. 1995). Moving the spectral shape through a constant optical flux point leads to a correlation of steeper optical spectral index with higher radio flux. This correlation reproduces the observed tenfold increase of the radio surface brightness from A to D2/H3 (Conway et al. 1993) and the overall steepening of the optical spectral index. However, this steepening happens on much longer time scales than expected from synchrotron cooling alone. Assuming an electron energy distribution with a fairly sharp cutoff, drastic jumps in $\alpha _{RU}$ would be expected at the locations of the shock fronts if the jet knots were (strong) shocks like the hot spot without extended re-acceleration acting between them. The absence of strong cooling makes it impossible to pinpoint localised sites at which particles are either exclusively accelerated or exclusively undergo strong losses. Any acceleration site must therefore be considerably smaller than the beam size we used ( 640h60-1pc), and these acceleration sites must be distributed over the entire jet to explain the absence of cooling. This, together with the low-frequency spectral index of $\approx$-0.8, corresponds to the jet-like acceleration mechanism proposed by Meisenheimer et al. (1997). Meisenheimer et al. (1996a) have presented fits of synchrotron continua to the observed SEDs of knots A, B, C, D and the hot spot H at 1 $.\!\!^{\prime\prime}$3 which have also been used by Röser et al. (2000). It is noted that the optical spectral index predicted from these spectra ( $\alpha _{{\rm fit}}$ in Fig. 4) is always steeper than the observed spectral index, that is below $\alpha _{BRI}$ and $\alpha _{RU}$. This indicates that the fitted spectrum is not fully adequate at the highest frequencies. This may again be due to contamination of the near-infrared flux by the same "backflow'' component mentioned above. The contamination would make the IR-optical spectral index (which dominates the run of the spectrum at high frequencies) steeper than the optical spectral index, as observed. Alternatively or additionally, there could be deviations of the optical-UV spectral shape from a standard synchrotron cutoff spectrum which may be interpreted as the first observational hint towards the existence of a second, higher-energy electron population producing the X-ray emission (Röser et al. 2000). Detailed statements about the adequacy of model spectra have to be deferred to a future paper considering the full radio, infrared and optical data set.
   
6.3.2 Can beaming account for the lack of cooling?
Heinz & Begelman (1997) proposed that sub-equipartition magnetic fields combined with mildly relativistic beaming could explain the lack of cooling in the jet of M 87 - which is, however, ten times shorter than that of 3C273. As an alternative to postulating re-acceleration, we consider whether low magnetic field values and beaming could lead to electron lifetimes sufficient to allow electrons to be accelerated at region A to illuminate the entire jet down to the hot spot over a projected extent of 32 h60-1kpc (the argument will become even more stringent by demanding acceleration in the quasar core). We consider the electron lifetime against synchrotron and inverse Compton cooling off cosmic microwave background photons; the synchrotron self-Compton process is negligible for electrons in the jet (Röser et al. 2000), as is Compton scattering off the host galaxy's star light. The total energy loss rate of an electron with energy E due to synchrotron radiation and inverse Compton scattering, averaged over many pitch-angle scattering events during its lifetime, is

 \begin{displaymath}% -\frac{{\rm d}E}{{\rm d}t} = \frac{4}{3} \sigma_{{\rm T}} c U_{\rm tot} \beta^2 \left(\frac{E}{m_{\rm e}c^2}\right)^2, \end{displaymath} (4)

where $U_{\rm tot} = U_{{\rm CBR}}(z) + U_{\rm mag}$ is the sum of the energy densities of the background radiation and magnetic field, respectively, and $\sigma_{{\rm T}}$ is the Thomson cross-section (Longair 1994). We integrate this equation from $E=\infty$ at t=0 to E(t), assuming $\beta = 1$ (appropriate for the highly relativistic electrons required for optical synchrotron radiation) and substitute for the electron's energy $E = \gamma m_{\rm e}c^2$. Inverting yields the maximum time that can have elapsed since an electron was accelerated, given its Lorentz factor $\gamma$ (van der Laan & Perola 1969):

 \begin{displaymath}% t(\gamma) = \frac{m_{\rm e}c^2}{ \frac{4}{3} \sigma_{{\rm T}} c U_{\rm tot} \gamma}\cdot \end{displaymath} (5)

This is the "electron lifetime'', inversely proportional to both the energy density in which the electron has been "ageing'', and the electron's own energy. As most of the electron's energy is radiated at the synchrotron characteristic frequency $\propto \gamma^2 B$, we can substitute for $\gamma$ in Eq. (5) in terms of the observing frequency and the magnetic field in the source. Hence, Eq. (5) becomes (in convenient units)

 \begin{displaymath}% t_{{\rm life}} = \frac{51\,000\,{\rm y}}{B_{-9,{\rm IC}}(z... ...left(\frac{B_{-9,{\rm jet}}}{\nu_{15}} \right)^{\frac{1}{2}}, \end{displaymath} (6)

where $B_{-9,{\rm jet}}$ is the magnetic flux density in nT of the jet field, the background radiation energy density has been expressed in terms of an equivalent magnetic field, $B_{-9,{\rm IC}} = (1+z)^2\times 0.45\,{\rm nT}$, and $\nu_{15}$ is the observing frequency in 1015Hz (van der Laan & Perola 1969). Note that as the substituted $\gamma \propto B^{-\frac{1}{2}}$, setting $B_{-9,{\rm jet}}=0$ is now meaningless. To be fully adequate for electrons in a relativistic jet at cosmological distances, the equation needs to be modified. Firstly, the frequency local to the source is (1+z) times the observing frequency because of the cosmological redshift. Furthermore, the radiating electron may be embedded in a relativistic flow with bulk Lorentz factor $\Gamma$ with three consequences: relativistic time dilation and Doppler shift, and boost of the background radiation energy density. The relativistic time dilation enhances the electron lifetime in the jet frame by a factor $\Gamma$. The Doppler shift between the emission frequency $\nu_{{\rm int}}$ in the jet frame (equal to the characteristic frequency) and the observation frequency $\nu_{{\rm obs}}$ is given by $\nu_{{\rm int}} = \nu_{{\rm obs}} {\cal D}^{-1}$, where ${\cal D}(\Gamma,\theta) = [\Gamma(1-\beta_{{\rm jet}} \cos \theta)]^{-1}$, the Doppler boosting factor for an angle $\theta$ to the line of sight (see Hughes & Miller 1991, e.g.). A relativistic flow perceives the energy density of the background radiation field boosted up by a factor $\Gamma^2$ (Dermer 1995). Inserting these yields

 \begin{displaymath}% t_{{\rm life}}({\cal D},z) = \frac{\Gamma {\cal D}^{\frac{... ...\frac{B_{-9,{\rm jet}}}{(1+z)\nu_{15}} \right)^{\frac{1}{2}}. \end{displaymath} (7)

The electron lifetime attains a maximum value at a certain value of the jet's magnetic field (Fig. 7) (van der Laan & Perola 1969). On either side of the maximum, the lifetime is decreased by larger losses suffered before it is observed: for a higher magnetic field in the jet, the synchrotron cooling is more rapid ( $t \propto 1/U_{\rm tot}$, Eq. (5)).
  \begin{figure} \par\resizebox{6.7cm}{!}{\includegraphics{ms1311f7.eps}} \end{figure} Figure 7: Maximum age against synchrotron and inverse Compton cooling (off microwave background photons at 3C273's redshift) of an electron radiating at observed UV wavelengths plotted against the jet magnetic field. Solid line, no beaming; dashed line, relativistic Doppler boosting with $\Gamma =10$ and a line-of-sight angle $\theta =45\hbox {$^\circ $ }$.
Open with DEXTER

A lower jet field requires an electron of higher Lorentz factor for emission at the given frequency, which also suffers more rapid losses ( $t \propto 1/\gamma$, Eq. (5)). By differentiation of Eq. (7), the value of the maximum lifetime is

 \begin{displaymath}% t_{{\rm max}}(B_{{\rm IC}}, {\cal D}, z) = \sqrt{\frac{\cal... ...\frac{3}{2}} \left[(1+z) \nu_{15}\right]^{\frac{1}{2}} }\cdot \end{displaymath} (8)

Note that the largest possible value for the factor $\sqrt{{\cal D}/\Gamma}$ is $\sqrt {2}$. $t_{{\rm max}}$ is a firm upper limit for the lifetime of a synchrotron-radiating electron from a source at redshift z in a flow of bulk Lorentz factor $\Gamma$, whatever the magnetic field strength in the source. It is deduced only from the fact that synchrotron emission is observed at a certain frequency, and that electrons which can radiate at this frequency suffer drastic energy losses either by synchrotron or by inverse Compton cooling between the time of acceleration and the time of emission. The only additional assumptions are rapid pitch-angle scattering and a homogeneity of conditions throughout the electron's lifetime.

The VLBI jet close to the core has a line-of-sight angle near 10$^\circ$ and a bulk Lorentz factor near 10 (Abraham & Romeo 1999). A line-of-sight angle $\theta \approx 45\hbox{$^\circ$ }$ has been inferred for the flow into the hot spot from independent considerations of the jet's polarisation change there and the hot spot's morphology (Conway & Davis 1994; Meisenheimer et al. 1997). We have plotted the lifetime of an electron responsible for emission from the jet in 3C273 observed at 300nm as function of the jet's magnetic field in Fig. 7 for the extreme cases of no beaming in the optical jet and beaming identical to that in the VLBI jet with $\Gamma=10, \theta = 10\hbox{$^\circ$ }$, and for an intermediate case with $\Gamma=10, \theta=45\hbox{$^\circ$ }$ (though note that $\Gamma=10, \theta = 10\hbox{$^\circ$ }$ is unrealistic as there is a difference in position angle between the VLBI jet at 244$^\circ$ and the arcsecond jet at 222$^\circ$). The equipartition flux densities derived for the jet lie in the range of 15nT (Neumann 1995) up to 67nT for regionA (Röser et al. 2000), leading to maximum ages of 100-800 years, less than the light travel time from one bright region to the next. In the absence of beaming effects, the largest possible lifetime for electrons in 3C273 from Eq. (8) is about 58000y, again short of the required values. The "boosted lifetime'' can be at most $\sqrt {2}$ larger than this. There is thus no combination of $\Gamma, \theta$ which enhances the electron lifetime to the 100000y required for illumination of the entire jet in 3C 273 by UV-radiating electrons. Thus, the invocation of mild or even drastic beaming and/or sub-equipartition fields cannot resolve the discrepancy between the synchrotron loss scale and the extent of the optical jet of 3C273, as has been possible for the jet in M 87 (Heinz & Begelman 1997). As another alternative to invoking quasi-permanent re-acceleration, the existence of a "loss-free channel'' in which electrons can travel down a jet without synchrotron cooling has been proposed by Owen et al. (1989). As an extreme version of this case, we assume that the electron travels along the jet in zero magnetic field and is observed as soon as it enters a filament with magnetic field $B_{{\rm fil}}$. The energy loss between acceleration and synchrotron emission is then only due to inverse Compton scattering. The lifetime in 3C273 is then 130000y $\times \Gamma^{-1} \sqrt{B_{{\rm -9,fil}} {\cal D}/\nu_{15}}$. Again, if the jet flow in 3C273 is highly relativistic, the electrons suffer heavy inverse Compton losses and the lifetime mismatch persists. In any case, it remains to be shown that the "loss-free channel'' is a physically feasible configuration of an MHD jet.

   
7 Conclusion

We have analysed deep HST images at 300nm and 600nm (Figs. 1 and 2) of the jet in 3C273 and constructed an optical spectral index map at 0 $.\!\!^{\prime\prime}$2 resolution (Fig. 3). The optical spectral index varies smoothly over the entire jet, indicating a smooth variation of the physical conditions across the jet. Unlike in M 87 (Meisenheimer et al. 1996a; Perlman et al. 2001), there is no strong correlation between optical brightness and spectral index (Figs. 4 and 6). The spectral index map thus shows no signs of strong synchrotron cooling at any location in the jet. Particle acceleration at a few localised sites in the jet is not sufficient to explain the absence of strong cooling. This does not preclude the possibility that the enhanced-brightness regions are shocks - but even if they are, re-acceleration between them is necessary to explain the observed spectral index features. We have further shown that relativistic effects cannot lead to significant enhancements of the electron lifetime in 3C273's jet, whatever the bulk Lorentz factor (Sect. 6.3.2), strengthening previous electron lifetime arguments. The need for a continuous re-acceleration of electrons emitting high-frequency synchrotron radiation in the jet of 3C273 is thus evident. Mechanisms have been proposed which can explain the apparent lack of cooling by distributed re-acceleration. These include acceleration by reconnection in thin filaments (Lesch & Birk 1998) and turbulent acceleration (Manolakou et al. 1999). Both processes manage to maintain the injection spectrum over distances much larger than the loss scales, although the latter so far only maintains cutoff frequencies in the range of $10^{12}\,$Hz-$10^{13}\,$Hz, i.e., below the values observed in 3C273. We note that for those jets which are bulk relativistic flows at high Lorentz factors, the increased inverse Compton losses form a further sink of energy that has to be filled by re-energization processes inside the jets. This requirement becomes more severe at higher redshifts. As has been suggested previously (Celotti et al. 2001), inverse Compton scattering off cosmic microwave background photons might explain the so far unaccounted-for X-ray flux from 3C273's jet (Röser et al. 2000; Marshall et al. 2001) and from Pictor A's jet and hot spot (Wilson et al. 2001). If this is true, the outward-decreasing X-ray flux from the jet in 3C273 indicates that the jet is still highly relativistic near region A and slows down towards the hot spot. Recently, the detection of extended X-ray emission from the jet in PKS 0637-752 by the new X-ray observatory CHANDRA has been reported and a similar explanation has been brought forward (Chartas et al. 2000; Schwartz et al. 2000; Tavecchio et al. 2000; Celotti et al. 2001). Like for 3C273, in situ re-acceleration is required in PKS 0637-752 to explain the mismatch between de-projected extent of the jet (>1Mpc) and inverse-Compton loss scale (10kpc) (Tavecchio et al. 2000). In order to understand the physical conditions in extragalactic jets like that in 3C273, it is necessary to detect even the most subtle variations in the parameters describing the synchrotron spectrum (that is, cutoff frequency, break frequency and especially low-frequency spectral index), which requires the deepest images at the highest resolutions and in many wavelength bands to detect variations at all. A theory of the physical processes at work in this jet initially has to explain both the simple overall spectral shape, as well as its constancy over scales of many kiloparsec. The details of this physical process will be constrained by the subtle deviations from the simple spectral shape, such as those tentatively identified in Sect. 6.3.1. We aim to find these deviations with the full data set, including new radio and near-infrared data in addition to the presented optical and UV images.

Acknowledgements
  We are grateful to D. van Orsow for his assistance with the HST observations. This research has made use of NASA's Astrophysics Data System Abstract Service.

   
Appendix A: Image alignment

   
A.1 Flux errors from misalignment

Consider two images which are not registered correctly and which have slightly different PSF widths. When performing photometry on these images, we assume that they are registered perfectly and have identical, known beam sizes. This amounts to making a flux measurement in a certain aperture in one image, but in a slightly offset aperture of slightly different size in the second image. The error is largest in the steepest gradients in the image, which are the flanks of the PSF of width $\sigma$, smoothed to the desired effective beam size $\sigma_{\rm eff}$, with $\sigma_{\rm eff}^2 = \sigma^2 + \sigma_{\rm smooth}^2$. Its magnitude can be assessed by considering the PSF as a Gaussian at given position and of given width, and as smoothing filter a second Gaussian slightly displaced from the PSF and of width slightly different from that achieving the desired effective beam area. The result of the performed wrong flux measurement is then proportional to the integral

\begin{displaymath}\int\int_{-\infty}^{\infty} \exp \left(-\frac{x^2 + y^2}{2 \... ...{\rm smooth}} + \delta \sigma)^2} \right) {\rm d}x {\rm d}y, \end{displaymath}

where $\delta x$ is the offset of the aperture from the correct position and $\delta \sigma$ is the error in the determination of the PSF width. The correct measurement is obtained by setting $\delta x=0$ and $\delta \sigma=0$, and from this one obtains the fractional flux error as a function of the two errors. The fractional flux error $\Delta f$ for a misalignment is
  
$\displaystyle % \Delta f$ = $\displaystyle 1 - {\rm e}^{\frac{\delta x^2}{2\sigma_{\rm smooth}^2}}$  
  $\textstyle \approx$ $\displaystyle \frac{\delta x^2}{2\sigma_{\rm smooth}^2} {\rm\ if\ } \frac{\delta x^2}{2\sigma_{\rm smooth}^2} \ll 1$ (A.1)
  $\textstyle \Leftrightarrow$ $\displaystyle \frac{\delta x}{{FWHM_{\rm eff}}} = \sqrt{\frac{\Delta f}{4 \ln(2)}} \approx 0.6 \sqrt{\Delta f}.$ (A.2)

Hence, the relative flux error is better than 5% if the misalignment $\delta x < 10\%$. Similarly, for a wrong PSF width, the fractional error is

\begin{displaymath}% \Delta f = \frac{2\delta\sigma}{\sigma_{\rm smooth}} \frac... ...{\rm eff}} {\rm\ if\ } \delta\sigma \ll \sigma_{\rm smooth}. \end{displaymath} (A.3)

This error is negligible if the desired effective PSF is much larger than the intrinsic PSF of the input images, as is the case here.

   
A.2 Refined image alignment


  \begin{figure} \resizebox{8.5cm}{!}{\includegraphics{ms1311f8.eps}} \end{figure} Figure A.1: The separation of a number of stars was measured both on a F336W (UV) and a F673W (red) image of R136. This plot shows the difference in separation between the frames, $\Delta S$, plotted against the separation on the "red'' image, S. The best-fitting straight line is shown. $\Delta S$ grows systematically with S, indicating a differing scale between the frames. The slope of the line is $(1.14\pm .04)\times 10^{-3} $, the intercept with the ordinate is $(0.3 \pm 8)\times 10^{-2}$. The slope is not changed within the errors by forcing the line to pass through the origin.

The two obvious ways to determine the relative shift between any two images are measuring the positions of point sources on the various images, or using the engineering files (also termed jitter files) provided as part of the observing data package. If the telescope pointing was known to be precise to better than 0 $.\!\!^{\prime\prime}$02, we could simply rely on the commanded shifts. The HST has a pointing repeatability within a single telescope visit[*] of about 5 milli-arcsecond (mas). The offsets between individual exposures are accurate to 15 mas, leading to a total error of about 16 mas. This is already comparable to the demanded accuracy. The situation is expected to be worse for relating exposures in different visits, when the telescope has been pointing elsewhere in the mean time. Unfortunately, measuring the positions of only a few (four, in our case) astrometric reference stars does not immediately lead to accurate measurements of the telescope's pointing - especially since in our case, each star lies on a different chip. The undersampling of the telescope PSF by the WFPC2 pixels and the so-called sub-pixel scattering of the WFPC2 detectors lead to an additional scatter of the centroid positions of point sources, approximately uniformly distributed between +0.25 and -0.25 pixels and in excess of statistical uncertainties (Lallo 1998, priv. comm.). With only a small number of centroidable point sources available, the centroiding errors are of the same magnitude as the intrinsic pointing errors of the telescope. It is therefore worth considering the pointing error sources en détail to ensure that the alignment is at the required 0 $.\!\!^{\prime\prime}$02 level[*]. Alignment errors can be caused by roll or pointing errors and less obviously by a scale difference between exposures using different filters in the same camera. The importance of the various alignment error sources can be estimated by considering the effect they have on the hot spot location if the quasar images are assumed to coincide. The hot spot is separated by 22 $^{\prime \prime }$ from the quasar, corresponding to about 480 PC pixels. The roll repeatability of the HST is at the 10 $^{\prime \prime }$ level. The engineering data provided with HST exposures record roll angle differences of about 6 $^{\prime \prime }$. This is well below the rotation of 3 $.\mkern-4mu^\prime$5 which would produce a 0.44 pixel difference over 480 pixels. The telescope roll differences can thus be neglected. The engineering files record the telescope pointing in three-second intervals and can be used to calculate the offsets. Their accuracy is only limited by the so-called "jitter'', vibrations due to thermal effects. The jitter was below 10 mas in all exposures, and below 5 mas in most. There is an additional uncertainty from the transformation between the telescope's focal plane and the detector: the location of a camera inside the telescope may change slightly over time (shifts, rotations, or both). This uncertainty is irrelevant for relative positions as long as the location and orientation of WFPC2 and the Fine Guidance Sensors (FGS, these perform the guiding observations) inside the telescope is stable, which is the case for the employed shifts of about 1 $^{\prime \prime }$ and for the timescales between the visits. Within a single visit, the engineering file information is used to obtain relative offsets, with a typical 5 mas error. The values differ from the commanded shifts by a few milli-arcseconds at most. All of these offsets are by an integer number of PC pixels. Between the various visits, the telescope has been pointing to a different part of the sky. One should therefore not assume that the relative shifts between various visits as determined from the jitter files are as accurate as shifts within one visit. We therefore measured the positions of the four astrometric reference stars on each of the short exposures (in fact, one of the stars is very faint in the UV, so the position was determined on a long exposure for this one). The shifts determined from the four point sources' positions have a typical standard deviation (accuracy) of 15 mas in each coordinate. On the three visits' sum images aligned this way, the scatter of the quasar image position is less than 5 mas and 7 mas in x and y, respectively. This means that although the measured shifts have a fairly large scatter, the resulting value is precise to about 10 mas. Finally, we note that the observed shifts of the stellar positions and those obtained from the jitter files agree to better than 0 $.\!\!^{\prime\prime}$02 in all cases, with an rms value of 0 $.\!\!^{\prime\prime}$01. There are, however, systematic differences between these and the commanded values. Hence, we do not blindly rely on the latter. Because of differential refraction in the MgF2 field flattener windows employed in WFPC2 (Trauger et al. 1995), the pixel scales of images taken through the F622W and F300W filters differ by about 0.1%. This alone is enough to eat up the alignment error budget of 0.44 pixels over 480 pixels separation. The wavelength dependence was expected from ray-tracing studies of the WFPC2 optics. Its presence and magnitude were experimentally confirmed by comparing archival images[*] of the star cluster R136 taken through similar filters as those employed in the present work (F336W and F672N) (Fig. A.1). The scale difference has to be removed before combining the images to a spectral index map. This was done in the following manner: the plate scale of each image was calculated using the parameters in Trauger et al. (1995). All images were resampled to a grid with pixel size of 0 $.\!\!^{\prime\prime}$0045548, which is one tenth of the average of the original scales, using bilinear interpolation. The result was then binned in blocks of $10\times 10$ pixels to a common pixel size of 0 $.\!\!^{\prime\prime}$045548. The scale of the two images is then identical to better than 1 part in 10000. The resampling required the use of the quasar image as common reference point between the two filters. The QSO's position can be determined to about 10-15 mas by centroiding routines on the unsaturated images.

References

 
Copyright ESO 2001