In the context of radio interferometry and aperture synthesis, we
may identify with the complex visibility function
associated with the source brightness distribution and single-antenna reception pattern
(Thompson et al. 1994). The visibility
function is usually expressed in terms of coordinates (u,v)
which give the projection of the interferometer baseline on the sky
plane and are expressed in wavelengths. The relation to the TD
spatial frequency components is simply
The distribution of the observations in the uv plane is all-important for the possibility to reconstruct complicated images from the measured visibilities V(u,v). Unfortunately the Hipparcos scanning law and the use of a modulating grid with just a single period seriously limit the uv coverage of the TD. According to Eqs. (3) and (12) the coverage is limited to the central point (u,v)=(0,0) and two concentric rings with radii and wavelengths. Moreover, for objects in the ecliptic region of the sky (ecliptic latitude ) the scanning law constrains the scan angle such as to produce a gap of `missing' scans roughly in the east-west direction. At there is instead a surplus of scans in the east-west direction (cf. Fig. 3). For high-latitude objects, finally, the coverage is usually more uniform in .
In continuing the analogy with radio interferometry, we will discuss in the next section how images can be produced using the Transit Data. Utilizing the experience developed for aperture synthesis imaging, we use only publicly available software for producing and deconvolving these images.
We have written and made publicly available (Sect. 6) a concise Fortran program which reads TD from the CD-ROM format (hip_j.idx and hip_j.dat) for any given object and produces an output file in the UV-FITS format. In the following we describe some of the features of the UV-FITS format and how the TD were adapted to it.
The basic FITS (Flexible Image Transport System; Wells et al. 1981; NOST 1993), well known in optical astronomy, was designed to transport digital data in the form of n-dimensional regular arrays, such as CCD images, with associated information on coordinates, dates, scales, units, etc. given in an ASCII header. However, aperture synthesis visibility data do not come in regular arrays, at least not in all axes, and thus an amendment to the basic FITS was required allowing the definition of ordered sets of small arrays (Greisen & Harten 1981). In the following we assume that the reader has some rudimentary knowledge about the basic FITS format.
In a UV-FITS file each random group contains the visibility data associated with a particular point in uv space and time. The group consists of a set of parameters followed by a regular array of measurements. The parameters are, for instance, the uv coordinates and date of the measurements. The measurement array may be multi-dimensional with, for instance, the different frequencies and Stokes (polarization) components marked along two of the axes. In the UV-FITS header the use of random groups is signified by having a first axis of length zero () and by setting the keyword (true). Visibility data are stored as three values in the measurement array, namely the real part of the visibility, the imaginary part, and an associated weight. In the array, this corresponds to an axis of length 3 and type .
For the Hipparcos TD there are three visibility measurements per transit, corresponding to the spatial frequencies , and . The total number of random groups is therefore . parameters specify each group, namely the Fourier coordinates (u,v,w), the baseline (defining which pair of antennae that formed the interferometer), and the date of the observation (split in two numbers containing the integer and fractional parts of the Julian date). For the TD the w coordinate is always zero.
The UV-FITS format requires that the (u,v) coordinates are expressed in seconds, while in Eq. (12) they are dimensionless. The conversion factor requires the specification of a reference wavelength, for which we arbitrarily adopted nm. The corresponding scale factor for the (u,v) coordinates in Eq. (12) is then , where c is the speed of light. For consistency, the frequency associated with each TD observation must then be given as .
The measurement array in each group is 5-dimensional (, since the first axis has zero length for group data). Its size is ,where the axes are of type , ,, and , respectively. The values on each axis are specified in the header by for ; in particular the frequency is given by and the reference position by and .The three values in the measurement array are, as mentioned before, the real and imaginary parts of V(u,v) and an associated weight. For the TD the weight is always equal to 1.
The aperture synthesis programs also require the names and geocentric positions of the antenna stations to be specified, although this is rather pointless in our case. We formally specify six stations and identify a different pair with each spatial frequency. The stations are arbitrarily named to form the acronym "HIPUVF", which will appear on some plots.
An apparent limitation of the FITS format is the lack of keywords for proper motion and parallax. Until such keywords become standard, the proper motions and parallax values of the reference point are given in the ASCII header as comments.
|Figure 3: UV coverage for HIP 97237. For every point (u,v) marked in this diagram there is a measurement of the complex visibility V representing the amplitude and phase of the detector signal during a certain transit. Each transit provides five equidistant visibilities (including the origin) oriented along a straight line through the origin. The celestial orientation of the line is normal to the slits of the Hipparcos modulation grid at the transit. u is the spatial frequency in the east-west direction, expressed in modulation periods ("wavelengths") per radian on the sky; v is the north-south component of the spatial frequency in the same unit|
Figure 3 shows the UV-coverage of HIP 97237. It is seen that the object received rather many scans, although predominantly in the (ecliptic) east-west direction. The "dirty beam" (Fig. 4) reflects this anisotropy as a characteristic pattern along the east-west section, reminiscent of the basic light modulation curve in Fig. 1.
|Figure 4: The dirty beam (point spread function) associated with the UV coverage of HIP 97237 shown in Fig. 3|
|Figure 5: The dirty map of HIP 97237, obtained by an inverse Fourier transform of all the measured complex visibilities|
The "dirty map", basically obtained as the inverse Fourier transform of the complex visibilities, is shown in Fig. 5. Already from this image it is obvious that the object was offset from the expected (HIC) position by about 5 arcsec. This is probably the reason why an acceptable solution was not found in the Hipparcos astrometric reductions. Deconvolution of the dirty image, using the dirty beam as kernel (point spread function), can be achieved by means of the CLEAN algorithm (Högbom 1974) implemented in Difmap. One result of this process (which depends on several parameters selectable in Difmap) is shown in Fig. 6. Both components of the double star are now clearly seen. The offset of the primary component from the reference point, estimated from the cleaned image, is arcsec. The position of the secondary component relative to the primary is approximately 0.9 arcsec towards position angle . The power in the primary peak of the cleaned map is 0.0455 units, where 6200 units corresponds to magnitude Hp=0 (Sect. 2.3); the magnitude of the primary can thus be estimated at .
Figure 7 illustrates the change of reference point for the TD described in Sect. 2.6. The astrometric parameters of the primary component in HIP 97237, relative to the reference point, were estimated by means of the model fitting procedure described in Sect. 5. The approximate results were (cf. Fig. 9) mas, mas, mas, mas yr-1, mas yr-1. Applying the corresponding phase shifts to the TD, according to Eqs. (6) and (7), effectively changes the reference point to coincide with the primary component. Performing the image synthesis on the modified TD gives the cleaned image in Fig. 7. As expected, the primary now appears at the centre of the map. The shift in parallax and proper motion of the reference point improves the relative phasing of the superposed scans, resulting in a slightly increased peak power (from 0.0455 to 0.0457 units).
|Figure 6: The cleaned map of HIP 97237, obtained by deconvolution of Fig. 5 with the beam in Fig. 4. The cleaned map clearly reveals the two components of the double star|
|Figure 7: The cleaned map of HIP 97237 resulting from modified TD, in which the astrometric parameters of the reference point were shifted to coincide with the astrometric parameters of the primary component. determined using the model fitting procedure described in Sect. 5|
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