The modulation grid consisted of a series of opaque and transparent bands. As the satellite spun, the diffraction image of each star traversed the grid perpendicular to the bands, resulting in a periodic ( ms period) modulation of the light intensity behind the grid. The varying intensity was measured by the Image Dissector Tube (IDT), a photomultiplier with an electronically steerable sensitive spot ("Instantaneous Field of View", IFOV) of about 30 arcsec diameter. During each "interlacing period" of ms the IFOV would cycle through the programme stars located within the field of view. There were on average 4 to 5 programme stars within the field of view at any given time.
The telescope entrance pupil (for each of the two viewing directions) was semicircular, with diameter 0.29 m and with some central obscuration. The Airy radius of the (ideal) diffraction image was thus around 0.5 arcsec for an effective wavelength of 550 nm. The modulation grid had a basic period of 1.2074 arcsec, i.e. the separation between the centres of adjacent transparent bands (slits). The slit width was about 0.46 arcsec, well matched to the Airy disk size and representing a compromise between a sharp intensity maximum (requiring narrow slits) and high photon throughput (requiring wide slits).
Since the grid was periodic, the detector signal, being the convolution of the diffraction image with the grid transmittance, must also be a periodic function of the image centroid coordinate on the grid. Disregarding noise and variations in detector sensitivity, etc., the signal therefore consisted of a constant (DC) component plus modulated components having spatial frequencies that are integer multiples of the fundamental grid frequency. However, from the convolution theorem it follows that the signal cannot contain higher spatial frequencies than were already in the diffraction image. The maximum frequency in the diffraction image is given by the maximum separation of any two points in the pupil and the minimum detected wavelength ( nm). For a pupil diameter of 0.29 m this gives a spatial period of 0.25 arcsec. Thus the theoretically highest frequency in the signal is four cycles per grid period (the "fourth harmonic"). The fourth and third harmonics, with cycles of 0.30.4 arcsec, are however very strongly damped by the slit width (0.46 arcsec), which causes an averaging over more than one cycle for these components. As a result, the detector signal in practice contains only the first and second harmonics, in addition to the DC (mean intensity) component. Given the spatial frequency, the detector signal is therefore completely parametrized by five numbers, i.e. the DC term and the coefficients of the first two harmonics in the Fourier series representing the periodic signal. Furthermore, since Poisson (photon) noise is by far the dominating noise source for most Hipparcos observations, it can be shown that a proper estimate of these five Fourier coefficients constitutes a sufficient statistic for the further estimation of the photometric and geometric characteristics of the target, independent of its complexity. The TD contain precisely these Fourier coefficients along with the spatial frequencies and other ancillary data. From the viewpoint of statistical estimation, practically no information was therefore lost by compressing the raw photon counts (on average some 4500 bytes per transit and target) into the five Fourier coefficients (included in the TD file).
The detector signal thus measured certain components of the diffraction image moving over the modulation grid. The diffraction image, of course, had aberrations due to the imperfections of the telescope and chromatic effects caused by the wavelengthdependent diffraction. Also, the sensitivity of the IDT varied across the field and over the time. All of these factors affected the detector signal. The user of the TD need not worry about this, because the TD have been "rectified", which means all instrumental and colour effects have been removed as far as possible by means of the various calibrations produced in the data reductions. Therefore, the TD should represent the response of an idealized, constant, and welldefined instrument to the object.
Figure 1: This figure illustrates the definition of reference phase p (as in Eq. (1)) by means of the reference point R on the sky (with specified astrometric parameters). p=0 at the instant when a slit is exactly centred on the reference point. In the lower diagram the curves A and B show the detector signal, as function of reference phase, produced by the same point source S when scanned in different directions. The intensity maximum is displaced from p=0 depending on the separation of S and R as projected on the grid, modulo the grid period 1.2074 arcsec 
The reference point in general has its own values for proper motion and parallax, in addition to position. These values are contained in the TD "header record", see Sect. 3.2.1. Of course, the real star also has a position, proper motion, and parallax. These two points move independently on the sky, according to their respective proper motion and parallax. In a particular transit the phase is determined by the scan direction and the distance between the two stars perpendicular to the slits. Figure 1 illustrates how the phase varies in two different scans across the star (S) and reference point (R). In scan A the star is in phase with the reference point, producing a light maximum for p=0 (and more generally for , where n is an integer). Scan B has the star out of phase with the reference point, resulting in a light maximum about a third of a period later ( rad).
The raw detector signal consists of a sequence of photon counts, N_{k},
, obtained in successive samples of 1/1200 s integration
time. The counts represent an underlying deterministic intensity
modulation which (after background subtraction and various calibrations)
is modeled as
0pt
(1) 
Equation (1) is the most general representation of the signal
produced by any object. In order to interpret this signal in terms of
an object model it is necessary to know what the signal would be for a
point source of unit intensity; in other words, we need the equivalent
of the pointspread function in image processing. The rectification
mentioned in Sect. 2.2 meant that the expected signal from a
point source of magnitude (in the Hipparcos
photometric system) located at the reference point is:
0pt
(2) 
(3) 
(4) 
(5) 
Equations (4)(5) are based on a linearization of the transformation between spherical coordinates and their projection on the tangent plane of the sky through the reference point. Within angular radius of the reference point, the neglected nonlinear terms are generally of order , or <0.1 mas for arcsec. Since the TD for a given target are normally confined to an area set by the size of the IDT sensitive spot ( arcsec), or a few times this area (for multiple pointings), the linearization is always adequate.
The positional offset (x,y) is in general caused by
a timedependent combination of the differences between the astrometric
parameters of the star and of the reference point. The differences in
, , and are easily converted
to an offset which is a linear function of time^{}. The parallax
difference produces a shift which is more complicated to calculate, as
it depends on the position () of the satellite relative to the
solar system barycentre at the time of observation. To facilitate this
calculation, a third spatial frequency component, , is supplied
with the TD. This is basically just the scalar product
, where the satellite position is in
astronomical units and is the previously defined
spatialfrequency vector of the grid. With this definition it is
found that a parallax difference of relative to the
reference point causes the additional phase shift
in Eq. (5). The complete
expression for the phase of a star with astrometric parameters
is therefore
(6) 
The periodicity of the grid means that, in a given scan, positional differences that are multiples of the grid period in the direction of scanning will not produce measurable differences in the detector signal. This "gridstep ambiguity" is normally resolved by the combination of scans from a variety of directions in the course of the mission.
In some circumstances it may be desirable to change the reference point for a set of TD. This is particularly relevant for the construction of aperture synthesis images (Sect. 4), where the reference point defines the centre of the image. An object with a poorly determined preHipparcos position may be severely offset from the centre of the reconstructed map, possibly falling outside the map altogether or creating a false image due to aliasing from a position outside the map. Not only the positional offset, but also errors in the proper motion and parallax of the reference point may create problems for the image reconstruction. The effect of such errors will be a blurring of the image due to the relative motion between the reference point and the true object. The proper motion error produces a blurring along a straight line in the map while the parallax error causes an additional elliptical blurring. Most objects with large proper motions (> 100 mas yr^{1}) or parallaxes (> 100 mas) have reasonable, nonzero estimates of these quantities in the Input Catalogue, so the blurring effect is usually not very serious for qualitative evaluation of the images. However, for a quantitative analysis the effect needs to be considered.
Changing the astrometric parameters of the reference point requires
that the phase of each transit is adjusted to take into account
the apparent positional offset of
the new reference point from the
old one at the time of the transit. Fortunately, this is easily
done by means of the spatial frequency components f_{x}, f_{y},
provided for each transit. Let
,, etc., be the (small) differences
between the new (') and the old reference point
in terms of the five astrometric parameters. The required phase
shift for a particular transit is then given by Eq. (6).
The corresponding transformation of the Fourier coefficients in
Eq. (1) is
(7) 
However, systems with intermediate separations ( to 30 arcsec) could not be observed without some adverse effects of the IFOV edges. In order to allow at least some useful astrometric information to be extracted for such systems, each component (or subsystem) received a separate pointing. For example, HIP 70 and HIP 71 formed such a twopointing system with a separation of about 15 arcsec. When pointing at the brighter star (HIP 71, ), the other component (HIP 70, )would be just at the edge of the IFOV and a (variable) fraction of its signal was added to that of the brighter star. Conversely, when pointing at the fainter component, some fraction of the brighter star's signal would be added. Proper reduction of such systems must consider the mutual (and possibly distorted) influence of each component upon the other, as was indeed done in the Hipparcos data reductions. In some multiple systems three different pointings were needed.
The situation is further complicated by the fact that the targeted position of the IFOV was sometimes updated in the course of the mission, usually because the original (groundbased) position was found to be wrong by several arcsec. Knowledge of both the original and the updated position, and the time of updating, may then be necessary for proper interpretation of the observations.
In order to cope with two and threepointing systems as well as updated positions, the concept of "target positions" was introduced in the TD. For a set of TD referring to a particular double or multiple system, target positions are defined by their offset coordinates from the adopted reference point, rounded to the nearest arcsec. In most cases there is just one target position, coinciding with the reference point [offset coordinates =(0,0)]. For multiplepointing systems there is at least one target position for each pointing, with a different HIP number attached to each pointing. For objects whose coordinates relative to the reference point changed in the course of the mission, a new target position was introduced whenever the offset coordinates changed by more than 1 arcsec. To within the errors of the realtime attitude determination (normally 12 arcsec rms) it can therefore be assumed that the IFOV was pointed to the specified target position.
In the TD the results from different pointings pertaining to the same system have been collected together and expressed relative to a common reference point. This has been done for all systems deemed to be "difficult" in the sense explained above, but not for very wide systems where the mutual influence of the component signals was negligible. The TD moreover contains the bookkeeping data necessary to calculate the actual target positions in each transit.

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