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

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4. The de-censoring procedure

4.1. The transit selection

Only the transits with background flux in the T channel smaller than [FORMULA] counts/second are used in photometry. Moreover, it is verified that no other star, among the half a million brightest ones, is crossing a slit system simultaneously; the transit is discarded otherwise.

Each transit is characterized by several parameters:

  • Flags referring to the part of slit system that was crossed by the star. The calibration terms in Eq. 2, [FORMULA] and [FORMULA] are determined by these flags.
  • The background fluxes in the [FORMULA] and in the [FORMULA] channels. These fluxes provide the mean numbers of background photons collected during the transit, [FORMULA] and [FORMULA].

The selected transits all contribute to the determination of the [FORMULA] and [FORMULA] magnitudes of the star, but they have not the same quality: the background is variable from 1700 to [FORMULA] counts/second, and, moreover, the sensitivity of the vertical slit group is 0.3 mag better than that of the inclined one. In order to take into account the observation conditions, any transit receives two weights, one for each channel. Each weight is defined as the square of the average signal-to-noise ratio, which is derived from the background and from the average signal computed with Eq. 2. Therefore, the weights do not depend on the result of the transit, but only on the conditions, and on the [FORMULA] and [FORMULA] magnitudes of the star that are assumed in the iteration step, as explained in Sect. 2.

4.2. The detected transits

A transit is detected when a detection is found closer than 0.6 arcsec from the position of the slit. When a magnitude measurement is available, the number of photons received from the star is corrected with Eq. 6. Next, if the signal-to-noise ratio of the measurement is larger than 0.5 after the correction, the corrected signal is accepted. It is considered as affected by the magnitude censoring otherwise.

When a measurement is missing due to magnitude censoring, the mean total number of photons below the censoring limit is derived from the model. The mean number of photons received from the star is then imputed to the measurement.

4.3. The non-detected transits

The mean numbers of photons corresponding to a transit not detected are simultaneously computed for both channels, taking into account two possible causes: the detection censoring, and the spurious non-detections.

The treatment of the detection censoring is not trivial, since this effect depends on the T-channel. Pairs of photon counts [FORMULA] are generated by simulation; those with [FORMULA] below the detection limit are taken into account to compute the mean numbers of photons received from the star when the transit occurred. This simulation is also used to derive the probability of getting a censored detection, when the prediction of the transit was reliable.

The mean numbers of photons corresponding to a spurious non-detection, [FORMULA] and [FORMULA], are directly derived from Eq. 2, assuming the values of [FORMULA] and [FORMULA] used in the iteration step.

The mean numbers of stellar photons corresponding to a non-detected transit are then derived by combining these two origins, assuming the 6 % proportion of spurious non-detections.

4.4. The mean [FORMULA] and [FORMULA] magnitudes of the star

In on-ground photometry, the mean magnitude of a star is computed as the average of the magnitude measurements. This method is used since the distribution of the logarithm of the photon counts obeys then a Gaussian law, due to scintillation (Sterken & Manfroid, 1992). This is not true for Tycho photometry, and the mean magnitudes are derived from the average intensities, as explained hereafter.

In the calculations above, all transits received a mean signal for each channel. These photon counts are transformed into intensities by the equation:

[EQUATION]

where M refers to the channel [FORMULA] or [FORMULA]. The average intensities [FORMULA] and [FORMULA] are then derived from all the selected transits, taking their weights into account. The mean [FORMULA] and [FORMULA] magnitudes of the star are finally:

[EQUATION]

The next iteration is then based on these values, until the calculation has converged.

4.5. Estimation of the errors

The errors are derived from the variance of the average intensity of the star in each channel, ie:

[EQUATION]

where [FORMULA] is the weight of a transit in the M channel (ie [FORMULA] or [FORMULA]). The problem is to derive the variances of the intensities for each transit. When the intensity comes from an actual measurement, the calculation is simple. Since [FORMULA], it is derived from Eq. 7 that:

[EQUATION]

The variance of the mean intensities imputed to the censored data are much more difficult to estimate; they are related to the accuracy of the model, including that of the assumed magnitudes, but this is the very result that is searched. Moreover, a fast computation is required, since the de-censoring process must be applied to a very large number of stars. It appears finally that the simplest way to solve this problem is just to ignore the contribution of the censored data in the calculation of the errors. This approximation is quite acceptable in practice, as shown in the next section.

The errors of the magnitudes are derived from the variances of the intensities. The probability distribution of the derived magnitudes is not symmetrical, however, and the "error on the bright side" must be distinguished from "the error on the faint side". These errors are respectively:

[EQUATION]

and

[EQUATION]

The exact calculation of the errors of the colour index [FORMULA] is also not trivial, since the colour index is a function of the ratio of the average intensities [FORMULA] and [FORMULA]. When the errors are defined as corresponding to the percentiles 16 % and 84 %, they are calculated by the equations:

[EQUATION]

and

[EQUATION]

with c and t coming from

[EQUATION]

and

[EQUATION]

[FORMULA] is the error on the "blue side", ie so that the probability that the actual [FORMULA] is less than the computed one minus [FORMULA] is 16 %. On the other side, [FORMULA] refers to the "red side" (in practice, it appears that the formulae: [FORMULA] and [FORMULA] are good approximations of Eq. 13 and 14, although they are much simpler).

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© European Southern Observatory (ESO) 1997

Online publication: May 5, 1998

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