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 gNB and
gBB are the ratios between counts and flux in
ergs-1cm-2 (including exposure time, gain, system
total efficiency, and atmospheric extinction),
fline,
and
fcont,
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
Sline,
and
Scont,
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 1/10th the FWHMNB) we can simplify these expressions to
![]() |
(A5) (A6) |
where fline is the total flux of the emission line (in erg s-1 cm-2) and SNB, line and SBB, line are the normalized responses of the filters at the wavelength of the emission line.
Then, defining
![]() |
(A7) (A8) (A9) (A10) |
we obtain fline,
fcont,, 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 gNB or
gBB. 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 fNB / fBB 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 fline in
Equation A11 we have to previously derive gNB or
gBB. 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, tNB
is the exposure time of the NB image, and
mcont,
is the
monochromatic magnitude (see e.g. Hamuy et al. 1992), which is
related with the flux via
![]() |
(A19) (A20) |
where
fcont, is
expressed in erg s-1 cm-2 Hz-1,
fcont,
in erg s-1 cm-2 Å-1, and
is in Å.
Thus, once the ZP and kNB coefficients are obtained from the Bouger-line fit to the spectrophotometric standards data we can obtain the conversion factor gNB 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 kBB is the extinction coefficient for the BB filter, C and kC are the color and color coefficient for a particular color term, and tBB is the exposure time for the BB image. Since mBB is also expressed as
![]() |
(A24) |
we derive gBB 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 gNB / gBB 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
fline, 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 EWline are valid as long as the contribution of the contaminating lines to the flux within the BB filter is negligible.