Astron. Astrophys. 325, 360-366 (1997)

3. The model of data acquisition

The model hereafter is a simplified summary of the part of the Tycho data reduction that is related to photometry. It consists of a mathematical description of the treatment of the transit observations obtained for a star with magnitudes and .

3.1. The transit detection

When the star is crossing a slit group, the photon counts exhibits a peak within an interval of second samples (this does not concern the raw photon counts, but the non-linearly folded photon counts presented in Bässgen et al. 1992). In the T -channel, the number of photons recorded during the transit within the interval of second, , obeys a Poisson distribution, with the parameter :

where and are the average numbers of background photons recorded in the interval in the channels and ; they are estimated from the photon counts around the transits. In practice, the total background, , is 50 counts on average, but it ranges from about 20 to 133 counts. The average numbers of photons received from the star, and , are related to the and magnitudes by the equations:

where and are calibration terms depending on the instrumental parameters of the transit (slit group, field, and part of the slit). These terms were derived in the photometric calibration; ranges from 13.4 to 13.9 mag and ranges from 13.1 to 13.5 mag. Therefore, the distribution function of is determined. This is slightly unrealistic, however, since is an integer number in the model, whereas the actual data reduction provided estimations of photon counts obeying a continuous distribution.

The detection of the transit depends on its signal-to-noise ratio (). Since the signal-to-noise ratio of the signal n is generally defined as , the signal-to-noise ratio of the transit in the T channel is:

The transit is detected when is larger than 1.5. In a conservative estimation, for stars with , and assuming a moderate total background of 50 counts, this threshold corresponds to or 11.4 mag, according to the calibration terms; these limits are 0.5 mag brighter when the maximum background is assumed. In reality, the cut-off at is not sharp, because the signal is estimated with a fast routine providing only an approximate value. However, this is ignored in the model.

When a transit is detected, the next step is the estimation of the and magnitudes, that is presented in Sect. 3.3. When is below 1.5, since the transit was predicted, a non-detection is recorded. The failure to detect transits with less than 1.5 is called the "detection censoring" hereafter.

3.2. The spurious non-detections

The non-detections are not only the censored transits. Transits with larger than 1.5 were sometimes omitted, for two reasons. The first one is that the predicted epoch used in the detection process may be wrong, since they were based on a preliminary determination of the attitude of the satellite. In such cases, the transit should be discarded, but, in practice, the information that the epoch was wrong was obtained only when a background transit was detected by chance; the transit was considered as valid otherwise (this concerns about half of the erroneous predictions). The second reason was that, in photometry, the detections were considered to be due to the stars only when the distance between the star and the position of the slit was less than 0.6 arcsec. As a consequence, even the stars bright enough for being detected each time they were crossing a slit group get some non-detections. These non-detections are not censored transits, since they are not due to signals below the detection threshold, but they are referred to as "spurious non-detections".

It appears from the bright standard stars that 6 % of the transits are spurious non-detections. This rate is assumed to be constant.

3.3. The determination of the magnitude

When a transit is detected, and photons are counted in the channels and respectively. The numbers of photons received from the star, and are estimated as the differences between the total numbers and and the mean backgrounds and . The measured magnitudes are then:

and

However, this calculation was actually not possible when the total numbers of photons were close to the mean background. In practice, a minimum signal-to-noise ratio of 0.5 is assumed in the model. When the is below this limit in one of both channels, the corresponding magnitude cannot be derived; in reality, the cut-off is rather unsharp, but, when a magnitude was derived although the is less than 0.5, the measurement is considered as doubtfull and it is ignored. The failure to measure the magnitude when the is less than 0.5 is called "the magnitude censoring" hereafter. In practice, since its threshold is much smaller than the detection threshold, the magnitude censoring only concerns faint stars with very small or very large color indices.

The calibration terms and were derived from the bright standard stars, as explained in Großmann et al (1995). Due to the censoring, it was then not verified if they were valid also for faint stars. This point is investigated hereafter. In order to avoid the bias due to the detection censoring, the measurements of the faint red standard stars are considered: since these stars have much fainter than , their detections depend essentially on , and the statistic of is affected only by the magnitude censoring. Therefore, the theoretical distribution of the signals is derived by assuming that obeys a Poisson distribution with the parameter , and with a cut-off on the left side when .

The distribution of the measured thus reveals an excess of bright magnitudes, as shown in Fig. 2 for the inclined slit group, and for and within small intervals. In fact, this is due to a non-linearity of the relation between the number of photons coming from the star and the signal estimated in the data processing. The medians of the actual measurements and of the theoretical distributions are considered for estimating the correction to apply to the signal. For instance, it comes from Fig. 2 that, for the inclined slit group, the excess is 4 counts when the measured signal is 13 and when the average background is 10 counts; in that case, this corresponds to an excess of 0.4 mag for a measured of 10.8 mag. All the measurements of the red photometric standard stars are used to determine the correction as a function of the slit group, of the background, , and of the measured signal, called . When , is larger than 13.3 counts (ie when the amplitude of the signal is larger than 2 counts per sample), it appears the true signal, , is given by the equation:

 Fig. 2. The distribution of the signals in , derived from measurements of red standard stars with fulfilling the following conditions: transit of the inclined slit system, between 3.3 and 13.3 photon counts (corresponding to amplitudes of 0.5 and 2 counts/sample respectively), and between 6.7 and 13.3 counts (ie a flux between 1 and 2 counts/sample). In order to take into account the magnitude censoring, the measurements with are discarded. The dotted line refers to the signals estimated in the photometric reduction, and the thin solid line to the corrected signal. The distribution derived from the model is plotted as a thick line for comparison; it contains gaps and peaks because the total photon count in , , is assumed to be an integer number.

with counts for the transits of the vertical slit group, and 13.3 counts for the inclined slits. Other relations exist for smaller values of . When these corrections are applied, the statistical properties of the magnitude measurements are fairly well fitting the model (see Fig. 2). The same relations are used to correct the estimations of , since they were obtained with the same algorithm.

The corrections of and does not affect the detection threshold , since the detection was performed with another routine for estimating the signal.

© European Southern Observatory (ESO) 1997

Online publication: May 5, 1998