Doc: Added description on FLUX_PETRO in Photometry section. Re-wrote part of...

Isophotal flux: :param:`FLUX_ISO`

---------------------------------

:param:`FLUX_ISO` is computed simply by integrating pixels values :math:`p_i` within the detection footprint, with the additional constraint that the background-subtracted, filtered value of detection image pixels must exceed the threshold set with the ``ANALYSIS_THRESH`` configuration parameter:

:param:`FLUX_ISO` is computed simply by integrating the background-subracted pixels values :math:`p_i` from the measurement image within the detection footprint, with the additional constraint that the background-subtracted, filtered value of detection image pixels must exceed the threshold set with the ``ANALYSIS_THRESH`` configuration parameter:

.. math::

 :label: fluxiso

Fixed-aperture flux: :param:`FLUX_APER`

---------------------------------------

:param:`FLUX_APER` estimates the flux above the background within a circular aperture.

:param:`FLUX_APER` estimates the flux from the measurement image above the background inside a circular aperture.

The diameter of the aperture in pixels is defined by the ``PHOTOM_APERTURES`` configuration parameter.

It does not have to be an integer: each "regular" pixel is subdivided in :math:`5\times 5` sub-pixels before measuring the flux within the aperture.

If :param:`FLUX_APER` is provided as a vector :param:`FLUX_APER[n]`, at least :math:`n` apertures must be specified with the ``PHOTOM_APERTURES`` configuration parameter.

-------------------------------------------

:param:`FLUX_AUTO` provides an estimate of the “total flux” by integrating pixel values within an adaptively scaled aperture.

|SExtractor|’s automatic aperture photometry routine is inspired by Kron’s “first moment” algorithm :cite:`1980ApJS_43_305K`.

|SExtractor|’s automatic aperture photometry routine derives from Kron’s “first moment” algorithm :cite:`1980ApJS_43_305K`:

#. An elliptical aperture is :ref:`defined by the second order moments of the object’s light distribution <shape_iso_def>`, with semi-major axis :math:`a={\tt A\_IMAGE}`, semi-minor axis :math:`b={\tt B\_IMAGE}`, and position angle :param:`THETA_IMAGE`.

   The ellipse's major and minor axes are multiplied by 6 (which corresponds roughly to twice the size of the isophotal footprint on each axis).

#. Within this elliptical aperture :math:`{\cal E}`, Kron's "first moment" is computed:

#. An elliptical aperture is :ref:`defined by the second order moments of the object’s light distribution <ellipse_iso_def>`, with semi-major axis :math:`a={\tt A\_IMAGE}`, semi-minor axis :math:`b={\tt B\_IMAGE}`, and position angle :param:`THETA_IMAGE`.

#. The ellipse's major and minor axes are multiplied by 6 (which corresponds roughly to twice the size of the isophotal footprint on each axis).

#. Inside this elliptical aperture :math:`{\cal E}`, an analog of Kron's "first moment" is computed:

.. math::

  :label: kron_r1

  r_1 = \frac{\sum_{i\in\cal E} d_i\,p_i}{\sum_{i\in\cal E} p_i},

  :label: kron_radius

where :math:`d_i` and :math:`p_i` are respectively the Euclidean distance to the (isophotal) centroid and the pixel value above the local background at image pixel :math:`i`.

:cite:`1980ApJS_43_305K` and :cite:`1987AA_183_177I` have shown that for stars and galaxy profiles convolved with Gaussian seeing, :math:`\ge 90\%` of the flux is expected to lie within a circular aperture of radius :math:`k r_1` if :math:`k = 2`, almost independently of the magnitude.

Experiments have shown :cite:`1996AAS_117_393B` that this conclusion remains unchanged if one replaces the circular aperture with a "Kron ellipse" :math:`{\cal K}`, which is simply the ellipse :math:`{\cal E}` above, scaled such that the principle axes become :math:`\sqrt{a/b}\,k r_1` and :math:`\sqrt{b/a}\,k r_1`.

  r_{\rm Kron} = \frac{\sum_{i\in\cal E} r_i\,p^{(d)}_i}{\sum_{i\in\cal E} p^{(d)}_i},

:param:`FLUX_AUTO` is the flux above the background summed over the Kron ellipse:

where :math:`p^{(d)}_i` is the pixel value *in the detection image*. :math:`r_i` is what we shall call the "reduced pseudo-radius" at pixel :math:`i` 

.. math::

  :label: flux_auto

  :label: reduced_radius

  {\tt FLUX\_AUTO} = \sum_{i\in\cal K} p_i.

  r_i \equiv \sqrt{{\tt CXX\_IMAGE} \times \Delta x_i^2 + {\tt CYY\_IMAGE} \times \Delta y_i^2 + {\tt CXY\_IMAGE} \times \Delta x_i \Delta y_i},

The quantity :math:`kr_1` is known as the *Kron radius*, and is provided in |SExtractor| by the :param:`KRON_RADIUS` catalog parameter in units of :param:`A_IMAGE` (or :param:`B_IMAGE`):

where :math:`\Delta x_i` and  :math:`\Delta y_i` are the pixel coordinates relative to the detection centroid:

.. math::

  :label: kron_radius

  {\tt KRON\_RADIUS} = \frac{1}{\sqrt{{\tt A\_IMAGE}\times{\tt B\_IMAGE}}}\,k\,r_1.

  \begin{aligned}

  \Delta x_i & = x_i - {\tt X\_IMAGE}\\

  \Delta y_i & = y_i - {\tt Y\_IMAGE}.

  \end{aligned}

:cite:`1980ApJS_43_305K` and :cite:`1987AA_183_177I` have shown that for stars and galaxy profiles convolved with Gaussian seeing, :math:`\ge 90\%` of the flux is expected to lie inside a circular aperture of radius :math:`k r_{\rm Kron}` with :math:`k = 2`, almost independently of the magnitude.

Experiments have shown :cite:`1996AAS_117_393B` that this conclusion remains unchanged if one replaces the circular aperture with the "Kron elliptical aperture" :math:`{\cal K}` with reduced pseudo-radius :math:`k r_{\rm Kron}`.

:param:`FLUX_AUTO` is the sum of pixel values from the measurement image, subtracted from the local background, inside the Kron ellipse:

.. math::

  :label: flux_auto

  {\tt FLUX\_AUTO} = \sum_{i\in\cal K} p_i.

The quantity :math:`k r_{\rm Kron}`, known as the *Kron radius* (which in |SExtractor| is actually a "reduced pseudo-radius") is provided by the :param:`KRON_RADIUS`.

:math:`k = 2` defines a sort of balance between systematic and random errors.

By choosing a larger :math:`k = 2.5`, the mean fraction of flux lost drops from about 10% to 6%, at the expense of |SNR| in the measurement.

Very noisy objects may sometimes end up with a Kron radius being too small, even smaller that the isophotal footprint of the object itself. For this reason, |SExtractor| imposes a minimum size for the Kron radius, which must be larger than :math:`\sqrt{ab}\, r_{\rm min}`

The user has full control over the parameters :math:`k` and :math:`r_{\rm min}` through the ``PHOT_AUTOPARAMS`` configuration parameters. ``PHOT_AUTOPARAMS`` is set by default to ``2.5,3.5``.

Very noisy objects may sometimes end up with a Kron ellipse being too small, even smaller that the isophotal footprint of the object itself. For this reason, |SExtractor| imposes a minimum size for the Kron radius, which cannot be less than :math:`r_{\rm Kron,min}`.

The user has full control over the parameters :math:`k` and :math:`r_{\rm Kron,min}` through the ``PHOT_AUTOPARAMS`` configuration parameters. ``PHOT_AUTOPARAMS`` is set by default to ``2.5,3.5``.

..

   .. figure:: ps/simlostflux.ps

  Figure [figphot] shows the mean loss of flux measured with isophotal (threshold 24.4 magnitude.arsec\ :sup:`-2`), corrected isophotal and automatic aperture photometry for simulated galaxies on a typical Schmidt-survey B\ :sub:`J` plate image.

  The automatic adaptive aperture photometry leads to the lowest loss of flux.

.. _flux_petro_def:

Petrosian aperture flux: :param:`FLUX_PETRO`

--------------------------------------------

Similar to :param:`FLUX_AUTO`, :param:`FLUX_PETRO` provides an estimate of the “total flux” by integrating pixel values within an adaptively scaled elliptical aperture. :param:`FLUX_PETRO`\ 's algorithm derives from Petrosian’s photometric estimator :cite:`1976ApJ_209L_1P,2001AJ_121_2358B,2001AJ_122_1104Y`:

#. An elliptical aperture is :ref:`defined by the second order moments of the object’s light distribution <ellipse_iso_def>`, with semi-major axis :math:`a={\tt A\_IMAGE}`, semi-minor axis :math:`b={\tt B\_IMAGE}`, and position angle :param:`THETA_IMAGE`.

#. The ellipse's major and minor axes are multiplied by 6 (which corresponds roughly to twice the size of the isophotal footprint on each axis).

#. Within this elliptical aperture :math:`{\cal E}`, the *Petrosian ratio* :math:`R_{\rm P}(r)` is computed:

.. math::

  :label: petrosian_ratio

  R_{\rm P}(r) = \frac{\sum_{0.9r < r_i < 1.1r} p^{(d)}_i}{\sum_{r_i < r} p^{(d)}_i} \frac{N_{r_i < r}}{N_{0.9r < r_i < 1.1r}},

where :math:`p^{(d)}_i` is the pixel value *in the detection image*. :math:`r_i` is the "reduced pseudo-radius" at pixel :math:`i` as defined in :eq:`reduced_radius`.

The *Petrosian ellipse* :math:`{\cal P}` is the ellipse with reduced pseudo-radius :math:`N_{\rm P}r_{\rm P}`, where :math:`r_{\rm P}` is the *Petrosian radius* defined by

.. math::

  :label: petrosian_radius

  R_{\rm P}(r_{\rm p}) \equiv 0.2

:math:`r_{\rm P}` is provided in |SExtractor| by the :param:`PETRO_RADIUS` catalog parameter.

The Petrosian factor :math:`N_{\rm P}` is set to 2.0 by default.

Very noisy objects may sometimes end up with a Petrosian ellipse being too small.

For this reason, |SExtractor| imposes a minimum size for the Petrosian radius, which cannot be less than :math:`r_{\rm P,min}`.

The user has full control over the parameters :math:`N_{\rm P}` and :math:`r_{\rm P,min}` through the ``PHOT_PETROPARAMS`` configuration parameters. ``PHOT_PETROPARAMS`` is set by default to ``2.0,3.5``.

The Petrosian flux is the sum of pixel values from the measurement image, subtracted from the local background, inside the Petrosian ellipse:

.. math::

  :label: flux_petro

  {\tt FLUX\_PETRO} = \sum_{i\in\cal P} p_i.

Photographic photometry

-----------------------

  adsnote = {Provided by the SAO/NASA Astrophysics Data System}

}

@ARTICLE{2001AJ_121_2358B,

   author = {{Blanton}, M.~R. and {Dalcanton}, J. and {Eisenstein}, D. and 

	{Loveday}, J. and {Strauss}, M.~A. and {SubbaRao}, M. and {Weinberg}, D.~H. and 

	{Anderson}, Jr., J.~E. and {Annis}, J. and {Bahcall}, N.~A. and 

	{Bernardi}, M. and {Brinkmann}, J. and {Brunner}, R.~J. and 

	{Burles}, S. and {Carey}, L. and {Castander}, F.~J. and {Connolly}, A.~J. and 

	{Csabai}, I. and {Doi}, M. and {Finkbeiner}, D. and {Friedman}, S. and 

	{Frieman}, J.~A. and {Fukugita}, M. and {Gunn}, J.~E. and {Hennessy}, G.~S. and 

	{Hindsley}, R.~B. and {Hogg}, D.~W. and {Ichikawa}, T. and {Ivezi{\'c}}, {\v Z}. and 

	{Kent}, S. and {Knapp}, G.~R. and {Lamb}, D.~Q. and {Leger}, R.~F. and 

	{Long}, D.~C. and {Lupton}, R.~H. and {McKay}, T.~A. and {Meiksin}, A. and 

	{Merelli}, A. and {Munn}, J.~A. and {Narayanan}, V. and {Newcomb}, M. and 

	{Nichol}, R.~C. and {Okamura}, S. and {Owen}, R. and {Pier}, J.~R. and 

	{Pope}, A. and {Postman}, M. and {Quinn}, T. and {Rockosi}, C.~M. and 

	{Schlegel}, D.~J. and {Schneider}, D.~P. and {Shimasaku}, K. and 

	{Siegmund}, W.~A. and {Smee}, S. and {Snir}, Y. and {Stoughton}, C. and 

	{Stubbs}, C. and {Szalay}, A.~S. and {Szokoly}, G.~P. and {Thakar}, A.~R. and 

	{Tremonti}, C. and {Tucker}, D.~L. and {Uomoto}, A. and {Vanden Berk}, D. and 

	{Vogeley}, M.~S. and {Waddell}, P. and {Yanny}, B. and {Yasuda}, N. and 

	{York}, D.~G.},

    title = "{The Luminosity Function of Galaxies in SDSS Commissioning Data}",

  journal = {AJ},

   eprint = {astro-ph/0012085},

 keywords = {Galaxies: Fundamental Parameters, Galaxies: Photometry, Galaxies: Statistics},

     year = 2001,

    month = may,

   volume = 121,

    pages = {2358-2380},

      doi = {10.1086/320405},

   adsurl = {http://adsabs.harvard.edu/abs/2001AJ....121.2358B},

  adsnote = {Provided by the SAO/NASA Astrophysics Data System}

}

@ARTICLE{2013AA_554A_101B,

   author = {{Bouy}, H. and {Bertin}, E. and {Moraux}, E. and {Cuillandre}, J.-C. and {Bouvier}, J. and {Barrado}, D. and {Solano}, E. and {Bayo}, A.},

    title = "{Dynamical analysis of nearby clusters. Automated astrometry from the ground: precision proper motions over a wide field}",

  adsnote = {Provided by the SAO/NASA Astrophysics Data System}

}

@ARTICLE{1976ApJ_209L_1P,

   author = {{Petrosian}, V.},

    title = "{Surface brightness and evolution of galaxies}",

  journal = {ApJL},

 keywords = {Astronomical Models, Galactic Evolution, Luminous Intensity, Red Shift, Surface Properties, Big Bang Cosmology, Cosmology, Hubble Diagram, Iterative Solution, Point Sources},

     year = 1976,

    month = oct,

   volume = 209,

    pages = {L1-L5},

      doi = {10.1086/182253},

   adsurl = {http://adsabs.harvard.edu/abs/1976ApJ...209L...1P},

  adsnote = {Provided by the SAO/NASA Astrophysics Data System}

}

@ARTICLE{1856MNRAS_17_12P,

   author = {{Pogson}, N.},

    title = "{Magnitudes of Thirty-six of the Minor Planets for the first day of each month of the year 1857}",

month={Sep}

}

@ARTICLE{2001AJ_122_1104Y,

   author = {{Yasuda}, N. and {Fukugita}, M. and {Narayanan}, V.~K. and {Lupton}, R.~H. and 

	{Strateva}, I. and {Strauss}, M.~A. and {Ivezi{\'c}}, {\v Z}. and 

	{Kim}, R.~S.~J. and {Hogg}, D.~W. and {Weinberg}, D.~H. and 

	{Shimasaku}, K. and {Loveday}, J. and {Annis}, J. and {Bahcall}, N.~A. and 

	{Blanton}, M. and {Brinkmann}, J. and {Brunner}, R.~J. and {Connolly}, A.~J. and 

	{Csabai}, I. and {Doi}, M. and {Hamabe}, M. and {Ichikawa}, S.-I. and 

	{Ichikawa}, T. and {Johnston}, D.~E. and {Knapp}, G.~R. and 

	{Kunszt}, P.~Z. and {Lamb}, D.~Q. and {McKay}, T.~A. and {Munn}, J.~A. and 

	{Nichol}, R.~C. and {Okamura}, S. and {Schneider}, D.~P. and 

	{Szokoly}, G.~P. and {Vogeley}, M.~S. and {Watanabe}, M. and 

	{York}, D.~G.},

    title = "{Galaxy Number Counts from the Sloan Digital Sky Survey Commissioning Data}",

  journal = {AJ},

   eprint = {astro-ph/0105545},

 keywords = {Cosmology: Observations},

     year = 2001,

    month = sep,

   volume = 122,

    pages = {1104-1124},

      doi = {10.1086/322093},

   adsurl = {http://adsabs.harvard.edu/abs/2001AJ....122.1104Y},

  adsnote = {Provided by the SAO/NASA Astrophysics Data System}

}

@INPROCEEDINGS{2002ASPC_281_228B,

   author = {{Bertin}, E. and {Mellier}, Y. and {Radovich}, M. and {Missonnier}, G. and