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<h1 class="noindent">Description of Photometric Calibration</h1>
<p class="abstract">
<em>D R A F T</em>
</p>


<pre>


Photometrical calibration with Munipack


For theoretical background see the book

Astronomy Methods: A Physical Approach to Astronomical Observations
 by Hale Bradt.


We are starting from that property of
CCD detectors which are photon detectors.
That mean that these detect and stores directly
photons.

A count of taken photons in a pixel can de direved
directly from digital signal provided by the control
electronics. These devices gives a signal in data numbes
dn (sometimes in units ADU or DN). With knowledcke
og gain factor of electonic, the count of captured
photons cts is

cts = dn [ADU] * gain [e-/ADU]  [e-, photons]

Plack's proved that every photon carried energy

 e = h ν  [J]

and also we know that light are electromagnetics
waves. Every wave is a sum of many of waves which
the single wave carries also e= hν energy. The
detected waves carries energy protoprtional
of number of waves

 E = c h ν [J]

The beauty of photon detectors is that the c == cts
and we can directly measure of the cts.

Unfortunatelly the CCD are non-perfect detectors.
Not every photon is captured nad detected, the
optics is unperfect, we are observing via athmosthpehe,
and the count of detected and passed to a 1m2 above
athmostpehe will different about the efficiency

 E = η c h ν   [J]

where E is energy carried by elmag wave.

The energy will depends on exposure time T,
filter and area A of out relescope. For better
comparison results and simply for that we know
kalibration data in a normlaized form, the
energy is unified on to flux rate

 F = η c h ν / (T A)   [W/m2]

Note that flux is generally vector and its
direction is given by direction of wave.
Here we using the flux magnitude only.


The spectral sensitivity of a device (limited
by a filter) is generally a peak with effective
frequency (or wavelength) λeff and with
half-with Δλ. The tabulated values of fluxes
are given in a spectral flux normalized per
unit frequency or wavelength

  f0 [W/m2/Hz]


Because, we are measure in a filter and we have
a photon detector, it is better to compute
effective number of photons at effective wavwelength
(per unit time and area)

  n = f0 * Δλ / (h λeff) = f0 * Δν / (h νeff)

The ratio of

   c / n = η    (0 ≤ η ≤ 1)

and gives us the efficinecy of our aparature.

The physical mean of the formula is that we are
compute area included in filter and the area
is recomputed on the unit frequency interval.
The numbers of photons at λeff than corresponds
to an effective energy oh photons passet througnout
a filter.

More importnat is that the efficinecy η can
be appreciated also as the calibration factor
from our instumental count of photons to
a calibrated numbers of photons.

From numerical pouint of view, the values of η
will usually of order of tenths (or promiles
for an extremly bad observation conditions)
and the calibration will numericaly well determined.
Also errors distributions will for the ratio
simple.

The calibration parameter will depend on many factors:

* observation conditions (extinction)
* CCD device
* aparature (optical)

Usually the extinction will reduce about 0.5,
the CCD's eficinecy about 0.8 (80% quantum
response) and about 0.5 for filter + optics.
Therefore the value about 0.1 - 0.5 may be expected.



Once we knows η, the derivation of calibrated quantities
is easy. The calibrated photon flux from measured couns c:

  c(cal) = 1/η(cal) c / (T A)    [photons/s/m2]

and flux spectral density

  f(cal) = c(cal)  (h νeff) / Δν   [W/m2/Hz]

(note [W] = [J/s]). (add errors determination,
c has Possion and othesr gaussian distributions).


The flux is used to describe total flux of objects
(point sources as stars or quasars are easy, but
the total light of Sun, Moon or a galaxy can be
important also). When teh source can be resolved,
we could need also distribution of the light over
an area. The quantity to describe it is the intensity
I. Intensity is integral over cone Ω an therefore
for elemental cones (areas of the sky)

  F = I ΔΩ

The calibrated intensity is

  i(cal) = f(cal)/ΔΩ  [W/m2/Hz/sr]

where the steradian [sr] is reccomened by SI.
In astronomy the area of sky 57°x57° is unpractical
and one square arcsecond is often used [W/m2/Hz/arcsec2].

The realtion to traditional magnitudes m is for fluxes
(the calibrated flux f0 corresponds to star of magnitude 0):

 m = -2.5 log10(f(cal)/f0)

and for intensity, surface magnitude μ is used:

 μ = -2.5 log10(i(cal)/(f0/ΔΩ))






The following FITS conventions are used by Munipack:

* The T is determined by EXPTIME (EXPOSURE) keyword.
  The A is determined by AREA keyword.

* The filter is determined by FILTER keyword. The photometric
  system is give by PHOTSYS. The keyword is used to search
  provided tables and get values of Δν and νeff. Both
  can be specified from command line.

* The reference id of the photometric catalogue is given by EXTNAME
  keyword in reference table. (optional, rewrite from coommand line).

* The reference catalogui contains magnitudes, the fluxes are computed
  as f = f0*10**(0.4*m) where f0 is flux per unit time, area and frequnecy
  and the provided calibratuion tables are searched for f0.

* The calibration parameter η(cal) is coded as FOTCAL
  (photon-calibrate rather than flux-calibrate prefering greek
   spelling).

* The physical constatnts h and c are hardcoded as SI recomended quantities.

* The transformation coeddfifients between instrumental and stantard
  system can be provided as the table. When the table is missing, unit
  matrix is used. The matrix can be determinedt by XXX utility or whatever
  else. The format of the table with the matrix is described in XXX.

* For determination of extiunction, the air mass is reqired. One is computed
  from LATITUDE and LONGITUDE keywords of geographycal coordinates.
  The astrometry is supposed by default.



Add:

* clibration of more filter simultaneously
* add extinction
* add color extinction
* picture of spectra of typical stars konvoluted with photometric filters
* Poiison statistics
* application for calibrating Halpha filters with known width

</pre>




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