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Astron. Astrophys. 346, 892-896 (1999)

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Appendix A: fitting of the barycentre

In this appendix, we demonstrate that fitting an unresolved image of several point sources (i.e. the maximum separation between point sources is much less than the size of the PSF) by a single blurred point source yields a position that is the barycentre of the point sources. We also show that the resulting barycentre coordinates are not biased by a limited field of view or some removed bad data points (which is not the case for a simple barycentre).

If the observed object is constituted by N point sources, the measured data at position [FORMULA] is:

[EQUATION]

where [FORMULA] is the noise (residual), [FORMULA] is the assumed point spread function, hereafter PSF, [FORMULA] and [FORMULA] are the brightness and position of the [FORMULA] source. It is worth noting that the PSF can account for the detector response. For our data, since the detector oversamples the telescope plus turbulence PSF (the pixel size was [FORMULA] of the seeing), we chose a gaussian PSF model that neglects detector effects such as pixelation.

If the size of the object (i.e. the maximum separation [FORMULA]) is much smaller than that of the PSF, then the object is unresolved and the whole data can be fitted by a model consisting in a single blurred point source:

[EQUATION]

where the total brightness F and mean position µ are the only parameters of the model. These parameters are customarily obtained by a weighted least square fit, that is by minimizing:

[EQUATION]

where [FORMULA]. The parameters F and µ of the model are therefore solution of:

[EQUATION]

For sake of simplicity, we consider a frame approximatively centered at the object mean position so that all the positions [FORMULA], ..., [FORMULA] and µ are negligible with respect to the size of the PSF. We Taylor expand the PSF at each pixel position around [FORMULA]:

[EQUATION]

where [FORMULA] and [FORMULA]. From that Taylor expansion, the data and model Eqs. (A1-A2) are approximated by:

[EQUATION]

Under these approximations, the parameters to fit are solution of a linear system and read:

[EQUATION]

where

[EQUATION]

Since the PSF is approximately symmetrical (an even function), its gradient is an odd function and [FORMULA] is negligible (unless the field of view is strongly shifted with respect to the barycentre of the observed object); then:

[EQUATION]

where the expected values of the bias terms (two last right hand side terms in the above expressions) are zero. This demonstrates that, under the considered approximations, the expected value of the fitted position is the true barycentre [FORMULA] of the object. It is worth noting that [FORMULA] must not be exactly the true PSF: it is sufficient that the mean weigthed residuals [FORMULA] have a negligible value.

Eq. (A4) shows that the expected value [FORMULA] of the fitted position does not depends on the completeness of the data set: [FORMULA] even if some bad data points are removed or if the field of view is truncated. This is not the case of the simple barycentre of the data:

[EQUATION]

which may be severely biased by the background noise and/or the limited field of view.

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© European Southern Observatory (ESO) 1999

Online publication: June 17, 1999
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