# Description of Photometric Calibration

D R A F T

    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