Some of the expressions derived in this appendix also appear in Waller (1990). Here we have carried out a more detailed analysis of the procedures used for flux-calibrating broad- or narrow-band imaging data. We have also considered the case that the field stars used for the normalization of the continuum image have absorption features in their spectra and considered the presence of other emission-lines in the galaxy spectrum (e.g. the [NII] 6548, 6584 ÅÅ doublet in our H filters) within the narrow-band filter used.

We consider two images, one taken with a narrow-band filter ( 100 Å; NB hereafter) and another taken with a broad-band one ( 1000 Å; BB hereafter). If we would be observing an astronomical object having a line in emission within the wavelength range of these filters, the total fluxes obtained (in counts after the sky subtraction) would be

(A1) (A2) |

respectively for the NB and BB filters, where *g*_{NB} and
*g*_{BB} are the ratios between counts and flux in
ergs^{-1}cm^{-2} (including exposure time, gain, system
total efficiency, and atmospheric extinction),
*f*_{line,}
and
*f*_{cont,}
are the fluxes per unit wavelength (in units
of erg s^{-1} cm^{-2} Å^{-1}) emitted by
the object due to the line and the continuum, respectively, and
*S*_{line,}
and
*S*_{cont,}
are the normalized
response functions of the NB and BB (including both the filter and
detector efficiency), respectively.

Assuming that the continuum is approximately flat in the spectral region of both filters, we can write

(A3) (A4) |

where FWHM is the Full Width at Half Maximum of the corresponding filter.

If we now consider that the width of the emission-line is
significantly narrower that the NB filter (less that 1/10^{th}
the FWHM_{NB}) we can simplify these expressions to

(A5) (A6) |

where *f*_{line} is the total flux of the emission line (in
erg s^{-1} cm^{-2}) and *S*_{NB, line} and
*S*_{BB, line} are the normalized responses of the
filters at the wavelength of the emission line.

Then, defining

(A7) (A8) (A9) (A10) |

we obtain *f*_{line},
*f*_{cont,}, and the equivalent width of the line (EW) as

(A11) (A12) (A13) |

In order to solve these equations (and obtain the pure emission-line image of the object) we need to determine the , , , and parameters. The , , and parameters can be easily obtained from the normalized response function of the NB and BB filters given the rest-frame wavelength of the emission-line considered and the recession velocity of the object.

With regard to the
parameter
two approaches can be
followed. First, we can determine the value of the
parameter
if both the NB and BB images are flux-calibrated just dividing the
corresponding calibration factors. However, this parameter can be also
measured without flux-calibrating any of the two images. In this
sense, we can measure the total counts on both images for a large
number of objects with well-known spectral properties in the
wavelength range of interest. In the case of the line equivalent
width, it is not necessary either to calibrate any of the images since
Equation A13 does not depend on *g*_{NB} or
*g*_{BB}. On the other hand, if we are interested in deriving
the emission-line flux (or the continuum flux) of the object at least
one of the images should be flux-calibrated.

Thus, with regard to the derivation of
, if we
measure the
total counts on both the NB and BB for a number of *featureless*
objects (usually field stars) we get

(A14) (A15) |

which leads to

(A16) |

This is the most commonly used method for subtracting the continuum
contribution in narrow-band images. This method is particularly simple
and powerful because the *f*_{NB} / *f*_{BB} ratio
(= ×
) can be
used to scale the BB image to the NB one
and obtain a pure emission-line image without any previous knowledge
about the filters response functions.

In addition, in some cases it is not possible to find *featureless* stars usually because the emission-line of interest (or
other lines) are present in absorption (or emission) in their
spectra. In that case the Equation A16 has to be substituted by

(A17) |

where the sum is extended to the number of *features* present in
the field stars spectra (equivalent widths would be negative in
absorption and positive in emission).

The values determined for
,
(or the
×
product)
would allow to obtain the
continuum-subtracted image. However, if we intend to calibrate the
resulting image resolving *f*_{line} in
Equation A11 we have to previously derive *g*_{NB} or
*g*_{BB}. The observation of several spectrophotometric
standards stars would allow to obtain the following relation

(A18) |

where ZP is the zero point of the calibration,
*k*_{, line}
is the extinction coefficient for the NB filter, *t*_{NB}
is the exposure time of the NB image, and
*m*_{cont,} is the
monochromatic magnitude (see e.g. Hamuy et al. 1992), which is
related with the flux via

(A19) (A20) |

where
*f*_{cont,} is
expressed in erg s^{-1} cm^{-2} Hz^{-1},
*f*_{cont,}
in erg s^{-1} cm^{-2} Å^{-1}, and
is in Å.

Thus, once the ZP and *k*_{NB} coefficients are obtained from
the Bouger-line fit to the spectrophotometric standards data we can
obtain the conversion factor *g*_{NB} from
Equations A14, A18, and A20 as

(A21) |

and, then

(A22) |

On the other hand, in the case that only the broad-band image is flux calibrated we obtain

(A23) |

where
*k*_{BB} is the extinction coefficient for the BB filter,
*C* and *k*_{C} are the color and color coefficient for a
particular color term, and *t*_{BB} is the exposure time
for the BB image. Since *m*_{BB} is also expressed as

(A24) |

we derive *g*_{BB} using

(A25) |

which finally leads to the flux of the emission line

(A26) |

In those cases where both photometric and spectrophotometric
calibrators are available the comparison of the
*g*_{NB} / *g*_{BB} ratio with the right side of
Equation A16 should provide an additional test for the
reliability of the spectral response functions assumed for the NB and
BB filters. In our case the differences derived between these two
values were of the order of 5 per cent.

Finally, it is important to take into account the contribution that
other emission lines could have to the fluxes and equivalent widths
derived using Equations A1, A22, A26, and A13. Although the terms due to
these other lines (typically the doublet
[NII]
6548, 6584
ÅÅ for observations in the light of
H) are not included in
Equations A1 and A2 their contribution can be corrected by considering that
*f*_{line}, as it appears in these equations, can be
defined in a way that

(A27) |

where *f'*_{line} and *f'*_{j} are the
corrected fluxes for the
line of interest and those other lines included in the filter,
respectively, and the sum in the index *j* is extended to all
contaminating lines but not the line of study. If we now consider the
line ratios between the contaminating lines and the line of interest
given by spectroscopy observations,

(A28) |

we obtain

(A29) |

This expression and the corresponding correction of the
EW_{line} are valid as long as the contribution of the
contaminating lines to the flux within the BB filter is negligible.