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2 What are the transit data?

2.1 Availability of the data

  The TD are contained on a CD-ROM (Disk 6) in Vol. 17 of the Hipparcos and Tycho Catalogues. A formal description of the TD, including detailed format specifications of the CD-ROM files, is found in Vol. 1, Sect. 2.9, of the Hipparcos and Tycho Catalogues. The TD come from an intermediate step of the data reductions performed by NDAC. In previous publications, the equivalent intermediate NDAC data were referred to as "Case History Files" (Söderhjelm et al. 1992). The data represent the scans (transits) of a selected number of targets from the Hipparcos Catalogue (HIP), comprising 37$\,$368 systems with 38$\,$535 different HIP entries. (Some systems have two or three HIP entries; cf. Fig. 2.) All Hipparcos stars that were classified as double, multiple or suspected non-single are included in the TD. Also, stars having problematic solutions in the Hipparcos Catalogue are included in the data set. In order to provide reference objects for calibrations or comparisons, several thousand ($\sim
5\,000$) "bona fide" single stars were also included in the TD.

2.2 The modulated detector signal

  The physical layout of the Hipparcos optical instrument was a telescope with two viewing directions in a plane perpendicular to the satellite's spin axis. The two different fields of view were combined onto a single focal surface by a special combining mirror. The light from each program star within either field of view was focused onto the modulation grid, which was located in the focal surface.

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 ($\simeq 7$ 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 $\simeq 130$ ms the IFOV would cycle through the programme stars located within the $0\hbox{$.\!\!^\circ$}9~\times~0\hbox{$.\!\!^\circ$}9$ 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 semi-circular, 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 ($\simeq 350$ 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 0.3-0.4 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 wavelength-dependent 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 well-defined instrument to the object.

\includegraphics [width=5cm,height=7cm,clip]{ds1699f1.eps}\end{figure} 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

2.3 The reference point

  One of the key points of the TD is that the modulation phase in each scan is expressed with respect to a well-defined reference point on the sky. The astrometric parameters for the reference point (different for each target) are also specified in the TD. Usually the reference point corresponds to the data given in the Hipparcos Input Catalogue (HIC; Turon et al. 1992). However, the phase calibration of the TD was made as if the coordinates of the reference point were expressed in the Hipparcos reference frame (nominally coinciding with the International Celestial Reference System, ICRS; Feissel & Mignard 1998). The phase offset of the TD from the reference point thus gives the differential correction to the position from the reference point in the ICRS system.

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 $p=n2\pi$, 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 ($p \simeq 2$ rad).

The raw detector signal consists of a sequence of photon counts, Nk, $k=1,2,~\dots\,$, 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  
I_k = b_1 + b_2\cos{p_k} + b_3\sin{p_k} + b_4\cos{2p_k}
+ b_5\sin{2p_k}\end{displaymath} (1)
where b1-b5 are the Fourier coefficients given in the TD. Ik is the expected stellar count rate in sample k (expressed in photons per sample of 1/1200 s) and pk is the reference phase of the sample. The reference phase is defined by the following two conditions (cf. Fig. 1): (1) that pk=0 if a slit is exactly centred on the reference point at the mid-time of sample k; and (2) that pk is increasing from 0 to $2\pi$ over the time it takes the grid to move one grid period (1.2074 arcsec) over the reference point.

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 point-spread function in image processing. The rectification mentioned in Sect. 2.2 meant that the expected signal from a point source of magnitude $\mbox{\it Hp}=0$ (in the Hipparcos photometric system) located at the reference point is: 0pt  
I_k = K[1+M_1\cos{p_k}+M_2\cos{2p_k}].\end{displaymath} (2)
Here K=6200 counts per sample, M1=0.7100 and M2=0.2485 are constants chosen not far from the actual mean calibration values at mid-mission (see Figs. 14.4, 14.3 and 5.6 in Vol. 3 of ESA 1997).

2.4 Spatial frequencies

  As illustrated by Fig. 1 the resulting signal will depend on the direction of scanning, even in the simple case of a point source with a fixed position relative to the reference point. The direction of scanning across the reference point can be specified by a position angle $\theta$ defined in the usual way, i.e. with $\theta=0$ for a scan where the grid is moving in the direction of increasing $\delta$ ("North"), and $\theta=90^\circ$ for a scan in the direction of increasing $\alpha$ ("East"). The spatial frequency of the grid, f (expressed in radians of modulation per radian on the sky), is nominally $(1296000~\mbox{arcsec})/(1.2074~\mbox{arcsec}) \simeq
1\,073\,400$. In reality it varies slightly, mainly because of differential stellar aberration, which changes the apparent scale by up to $\pm 0.01$ per cent depending on the barycentric velocity vector of the satellite. For the production of the TD data, the apparent scale and orientation of the grid, as projection onto the sky in the vicinity of the reference point, were strictly calculated for each transit, taking into account aberration as well as the satellite attitude and the calibrated field-to-grid geometric transformation. The result is expressed as the rectangular components of the spatial-frequency vector in the tangent plane of the sky:  
f_x = -f\sin\theta, \qquad f_y = -f\cos\theta.\end{displaymath} (3)
The minus signs mean that the spatial-frequency vector ($\vec{f}$)points in the opposite direction to the motion of the grid across the reference point. This is purely a matter of convention, which was adopted for historic reasons.

2.5 The phase of an arbitrary point source

  Consider now a point source of magnitude Hp which, in a particular scan, is displaced by (x,y) from the reference point, where x is measured positive towards increasing $\alpha$ and y towards increasing $\delta$. Given the spatial frequency components of the scan, fx and fy, the expected signal is:  
I_k = 10^{-0.4{\it Hp}}K [1 + M_1\cos(p_k+\phi) + 
M_2\cos 2(p_k+\phi)]\end{displaymath} (4)
\phi = f_x x + f_y y\end{displaymath} (5)
is the phase shift on the grid caused by the positional offset. [That $\phi$ is added to pk in Eq. (4), rather than subtracted, is consistent with the sign convention adopted in the definition of fx and fy.]

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 $\rho$ of the reference point, the neglected non-linear terms are generally of order $\rho^3$, or <0.1 mas for $\rho<160$ arcsec. Since the TD for a given target are normally confined to an area set by the size of the IDT sensitive spot ($\sim 30$ 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 time-dependent combination of the differences between the astrometric parameters of the star and of the reference point. The differences in $\alpha$, $\delta$, $\mu_{\alpha*}$ and $\mu_\delta$ 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 ($\vec{s}$) of the satellite relative to the solar system barycentre at the time of observation. To facilitate this calculation, a third spatial frequency component, $f_{\rm p}$, is supplied with the TD. This is basically just the scalar product $-\vec{s}'\vec{f}$, where the satellite position $\vec{s}$ is in astronomical units and $\vec{f}$ is the previously defined spatial-frequency vector of the grid. With this definition it is found that a parallax difference of $\Delta\pi$ relative to the reference point causes the additional phase shift $\Delta\phi = f_{\rm p}\Delta\pi$ in Eq. (5). The complete expression for the phase of a star with astrometric parameters $(\alpha,\delta,\pi,\mu_{\alpha*},\mu_{\delta})$ is therefore  
\phi = f_x(\Delta\alpha{*}+t\Delta\mu_{\alpha*}) 
+ f_y(\Delta\delta+t\Delta\mu_{\delta}) + f_{\rm p}\Delta\pi\end{displaymath} (6)
where t is the time of the transit, expressed in (Julian) years from J1991.25, and $\Delta\alpha{*}=(\alpha-\alpha_0)\cos\delta_0$,$\Delta\delta=\delta-\delta_0$, $\Delta\pi=\pi-\pi_0$, $\Delta\mu_{\alpha*}=\mu_{\alpha*}-\mu_{\alpha*0}$, $\Delta\mu_{\delta}=\mu_{\delta}-\mu_{\delta0}$ are the differences with respect to the astrometric parameters of the reference point, $(\alpha_0,\delta_0,\pi_0,\mu_{\alpha*0},\mu_{\delta 0})$.

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 "grid-step ambiguity" is normally resolved by the combination of scans from a variety of directions in the course of the mission.

2.6 Changing the reference point

  The choice of reference point for the Transit Data is arbitrary, as long as it is sufficiently near the target for linearization errors to be negligible ($\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
$\displaystyle ...  arcsec; see Sect. 2.5). As mentioned previously, the adopted reference point usually corresponds to the HIC values. The user may ask why the final astrometric parameters in the Hipparcos and Tycho Catalogues were not used instead. There were several reasons for this. First of all, the TD were derived directly from intermediate files of the NDAC reduction (Sect. 2.1), which were compiled in their final version almost a year before the astrometric parameters of the Hipparcos Catalogue became available. Secondly, most of the objects are double or multiple, and it is not evident which point to use in such cases. Finally, TD are available also for cases where no valid astrometric solution is provided in the Hipparcos Catalogue. In those cases it would have been necessary to use something like the HIC data anyway. Although it would have been possible, using the formulae below, to transform the TD to some other, possibly more natural or desirable reference point, such a process would in the end have been largely arbitrary. In addition, since any manipulation of the data always entails some risk of error, however small, it was felt better to leave the reference points as defined in the NDAC intermediate data files. The only modification applied was the transformation from the intermediate NDAC reference frame into the ICRS system, which was deemed essential; this was achieved by rotating the positions and proper motions of the reference points, rather than modifying the TD coefficients. (This explains why the position and proper motion of the reference point never coincide exactly with the HIC values.)

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 pre-Hipparcos 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, non-zero 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 fx, fy, $f_{\rm p}$ provided for each transit. Let $\Delta\alpha*=(\alpha_0'-\alpha_0)\cos\alpha_0$,$\Delta\delta=\delta_0'-\delta_0$, 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
b_1' = b_1, \\  \nonumber \end{displaymath}   

b_2' = b_2\cos\phi - b_3\sin\phi,\\  \nonumber\end{displaymath}   

b_3' = b_2\sin\phi + b_3\cos\phi,\\  \nonumber\end{displaymath}   

b_4' = b_4\cos 2\phi - b_5\sin 2\phi,\\  \nonumber\end{displaymath}   
b_5' = b_4\sin 2\phi + b_5\cos 2\phi.\end{displaymath} (7)
An example of the change of reference point is shown in Figs. 6 and 7.

2.7 Multiple pointings and target positions

  One complication in the Hipparcos satellite operation and data reductions was caused by double and multiple stars having separations roughly in the range 10 to 30 arcsec. As already mentioned, the main Hipparcos detector had a sensitive area (IFOV) of about 30 arcsec diameter. This area could be directed towards any pre-defined point on the sky currently within the telescope field of view. The IFOV pointing, calculated from the real-time knowledge of the satellite attitude and the celestial position of the target, typically had errors of 1 to 2 arcsec rms. Ideally, no star should be observed while its image was close to the edge of the IFOV, where the guiding errors might produce a distorted signal. For double and multiple systems with separations less than about 10 arcsec this could be achieved by centering the IFOV somewhere in the middle of the system, so that all components remained within the flat-topped central part of the IFOV sensitivity profile. For systems with separations greater than about 30 arcsec the individual components (or subsystems with separations below 10 arcsec) could be observed as single stars, again avoiding signal distortion from the IFOV edges.

However, systems with intermediate separations ($\simeq 10$ 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 two-pointing system with a separation of about 15 arcsec. When pointing at the brighter star (HIP 71, $Hp\simeq 8.4$), the other component (HIP 70, $Hp\simeq 10.6$)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 (ground-based) 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 three-pointing 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 $(\Delta\alpha*,\Delta\delta)$ 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 multiple-pointing 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 real-time attitude determination (normally 1-2 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.

Table 1: Summary of the contents of the TD index and data files on Disk 6 of the Hipparcos Catalogue. Data fields in the Transit Data File are separated by the character "|". In the header record there are 13 fields designated JH1 through JH13, etc.

\multicolumn{2}{@{\hspace{0pt}}l}{\it Transit Data I...
 ...mas)\\ JT19 & computed (0) or assumed (1) standard errors \\ \hline\end{tabular}

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