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<a class="headtitle" href="munipack.html">Munipack</a>
<a class="headsubtitle" href="munipack.html">A general astronomical image processing software</a>
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<h1>Artificial Frames</h1>
<p class="abstract">
This tool is designed to create the artificial, nevertheless photometric
realistic, frames. They are intended for modelling of various observation
effects as well as for testing purposes.
</p>
<h2>Synopsis</h2>
<code>
munipack artificial [.. parameters ..]
</code>
<h2>Description</h2>
<p>
Artificial frames are created by artificial photons having origin
in physical principles and mathematical formulas rather than in nature.
This approach can be very useful for understanding
of true nature of many effects which can be else encountered
only after long period of an observational experience.
</p>
<p>
There is list of currently implemented features:
<!-- The models describes impact of these effects:-->
</p>
<ul>
<li>Quantum efficiency of detector, telescope parameters.</li>
<li>Atmospheric extinction (monochromatic)</li>
<li><a href="https://en.wikipedia.org/wiki/Astronomical_seeing">Astronomical Seeing</a>
</li>
<li>Point spread functions (PSF): Gaussian, Moffat and seeing.</li>
<li>Background noise</li>
<li>Light curves: plain wave, Algol-like, δ Cep and user defined</li>
<li>Field stars: by a catalogue or generated randomly.
<!--
<li>Appearance of clouds</li>
-->
</ul>
<p>
The frames are generated:
</p>
<ul>
<li>on base of catalogue stars which simulates nearly real field or</li>
<li>on base of randomly generated positions and magnitudes of stars.</li>
</ul>
<h3>Modelling of Telescope Properties</h3>
<p>
The properties of star images can be easy simulated for
various quantum efficiency <i>η</i> of a detection apparatus (including
detector itself), an exposure duration and a telescope diameter.
</p>
<p>
Both angular resolution and collecting area of a telescoped
are proportional to diameter.
The parameter <samp>--airy</samp> can set it directly, but
better way is set diameter (via <samp>--diameter</samp> in meters)
which sets both diffraction limit and input area.
</p>
<h3>Modelling of Extinction</h3>
<p>
An observed light intensity <i>I(X)</i> passing the clear (no clouds) Earth's
atmosphere is attenuated proportionally of air-mass <i>X</i> according
to the formula
</p>
<p>
<i>I(X) = I<sub>0</sub> e<sup>-k(X-1)</sup></i>
</p>
<p>
where <i>I<sub>0</sub></i> is an extra-atmospheric intensity.
The extinction coefficient <i>k</i> depends on a spectral band.
</p>
<p>
The sky brightness (see below) is modelled by the same way
(<i>b<sub>Z</sub></i> is its zenit value)
</p>
<p>
<i>b(X) = b<sub>Z</sub> e<sup>-k(X-1)</sup></i>
</p>
<h3>Modelling of seeing</h3>
<p>
Turbulent motion in atmosphere creates large cells
which slightly reflect light rays. The nature of turbulent
motion is totally chaotic. There is a formula describing
effect of chaotic moving of picture of a star due atmosphere
during long exposures. It modelled star profile as function
of air mass.
</p>
<p>
By <a href="https://en.wikipedia.org/wiki/Fried_parameter">Mr.Fries theory</a>
(very nice description has <a href="http://link.springer.com/book/10.1007/978-3-540-76583-7/page/1">Electronic Imaging in Astronomy</a> by Ian S. McLean),
the seeing radius is
<a href="http://www.astro.auth.gr/~seeing-gr/seeing_gr_files/theory/node17.html">modelled</a> as
</p>
<p>
<i>r'<sub>0</sub> = r<sub>0</sub> X<sup>0.6</sup></i>.
</p>
<p>
<i>r'<sub>0</sub></i> is an actual radius of stars at air-mass <i>X</i>,
while <i>r<sub>0</sub></i> is the one at zenith.
</p>
<h3>Modelling of Star Profiles</h3>
<p>
These profiles known as point spread functions (PSF) are available:
</p>
<ul>
<li>Seeing spread function which convoluted diffraction pattern of
telescope aperture and Gaussian spread by seeing. It is the best
method for simulation of any star natural profile.
<!--The spread
(convolution) can be computed by direct method (repeatelly
shift and add difraction pattern to the profile) which is very
slow, but gives non-symatetric profiles) and by Fourier convolution
method which is fast and results are perefctly smooth.
-->
</li>
<li>
<a href="http://adsabs.harvard.edu/abs/1969A%26A.....3..455M">Moffat profile</a>
which is commonly used for rough description of observed profiles.
</li>
<li>
Gaussian profiles <i>exp(-r<sup>2</sup>/r<sup>2</sup><sub>0</sub>)</i>
which is the basic estimator. It is ideal for
theoretical work, the practical use is limited on extra-atmospheric
applications, such as spacecraft or Moon base observatories, without
image perturbations by the seeing.
</li>
</ul>
<table>
<tr>
<td class="blank">
<figure>
<img src="GAUSS_PSF.jpeg" alt="GAUSS_PSF.jpeg" title="Gauss PSF">
<figcaption>Gauss</figcaption>
</figure>
</td>
<td class="blank">
<figure>
<img src="MOFFAT_PSF.jpeg" alt="MOFFAT_PSF.jpeg" title="Moffat PSF">
<figcaption>Moffat</figcaption>
</figure>
</td>
<td class="blank">
<figure>
<img src="SEEING_PSF.jpeg" alt="SEEING_PSF.jpeg" title="seeing PSF">
<figcaption>seeing</figcaption>
</figure>
</td>
</tr>
</table>
<p>
The plain diffraction pattern, representing response of an input aperture
of a telescope without atmosphere seeing, can be generated as seeing spread
with zero seeing parameter. It is difficult to expose it, even in laboratory,
due to limited dynamical range of detectors.
</p>
<p>
The distribution of seeing is supposed as Gaussian and convoluted
with actual diffraction profile. There are two methods convolution
implemented:
</p>
<ul>
<li>Direct convolution -- diffraction profile is randomly shifted
in both directions and the result is accumulated. This slow method
gives slightly asymmetrical frames and is intended for
simulating of short exposures.</li>
<li>Fourier convolution -- this method generates both diffraction profile
and Gaussian with the Fried's width. The profiles are convoluted
with help of <a href="https://en.wikipedia.org/wiki/Convolution">convolution theorem</a> and
<a href="https://en.wikipedia.org/wiki/Fast_Fourier_transform">FFT</a>. This method is fast and results are perfectly smooth.
</ul>
<p>All stars, on the image, are modelled with the same profile.</p>
<h3>Modelling of Background</h3>
<p>
Background <i>B(x,y)</i> is modelled as a plane
</p>
<p>
<i>b(x,y) = B<sub>0</sub> + ΔB<sub>x</sub> (x - x<sub>0</sub>) +
ΔB<sub>y</sub> (y - y<sub>0</sub>)</i>
</p>
<p>
<i>
B(x,y) = N( b(x,y), Δb)
</i>
</p>
<p>
where <i>B<sub>0</sub></i> is a mean level (derived
from <samp>--skymag</samp>)
at centre of the picture <i>x<sub>0</sub>, y<sub>0</sub></i>,
<i>N(b, Δb)</i> is a function which makes a noise with
<a href="https://en.wikipedia.org/wiki/Normal_distribution">Normal
distribution</a>.
The parameters are mean level <i>b</i> and standard deviation <i>Δb</i>,
which is computed as <i>Δb = √ B<sub>0</sub></i>.
</p>
<p>
The optional gradient in background in counts per pixels is given
by terms <i>ΔB<sub>x</sub>, ΔB<sub>y</sub></i>
(<samp>--sky-grad-x, --sky-grad-y</samp>). It can do modelling of
a light pollution (by Moon). The common values for gradients are
of order 0.0001 … 0.001 (negative values means negative slope of
the plane in given direction).
</p>
<table>
<caption>Common sky brightness in zenit</caption>
<tr><th>value [mag/arcsec2]</th><th>light conditions</th></tr>
<tr><td>above 22</td><td>natural sky, high-altitude observatory</td></tr>
<tr><td>22</td><td>dark place, excellent night</td></tr>
<tr><td>21</td><td>countryside sky, average night</td></tr>
<tr><td>20</td><td>suburban sky, poor night</td></tr>
<tr><td>18</td><td>urban sky, full Moon</td></tr>
<tr><td>under 17</td><td>city sky, twilight</td></tr>
</table>
<p>
Values in the table are rought and determined by my observing
experiences
(<a href="https://en.wikipedia.org/wiki/Bortle_scale">wiki page⤴</a>).
</p>
<h3>Star fields</h3>
<p>
Field stars are generated by a random number generator,
if no <samp>-c</samp> option (a catalogue) is presented.
Catalogue star coordinates and magnitudes are used otherwise.
</p>
<p>
Rectangular (without projection) coordinates of the stars has
<a href="https://en.wikipedia.org/wiki/Uniform_distribution_(continuous)">
uniform distribution</a> in ranges given by a frame size.
The standard Fortran random number generator, with no initial seed, is used.
</p>
<p>
Magnitude distribution, meaning a probability to found number
of stars in given magnitude interval per square degree, is
modelled as an exponential function with a cut-off to prevent
<a href="https://en.wikipedia.org/wiki/Olbers%27_paradox">Olbersʼ
paradox</a>.
I derived a rough approximation
of the observed (surface) distribution of stars by brightens
from <a href="https://www.cosmos.esa.int/web/gaia/dr1">DR1</a>
by <a href="https://www.cosmos.esa.int/web/gaia">Gaia mission</a>:
</p>
<p>
<i>
Σ(m) = 10<sup>-4</sup> · e<sup>1.1m</sup> [□°],
m < m<sub>0</sub>
</i>
</p>
<p>
The cut-off <i>m<sub>0</sub></i> is given by <samp>--maglim</samp>
parameter.
Large values of <i>m<sub>0</sub></i> and large field of view
can effectively exhaust the computer memory.</p>
<p>
The distribution approximation must be considered as an effective description.
Stars are distributed over our sky very non homogeneously. Standard galaxy
textbooks (like Binney's & Merrifield's
<a href="http://adsabs.harvard.edu/abs/1998gaas.book.....B">Galactic Astronomy</a>)
gives more extensive description with references.
</p>
<h3>Models of light curves</h3>
<p>
The artificial tool has three models of light curves included. The model
of the pulsating star δ Cep (data computed by <a href="http://cds.aanda.org/component/article?access=bibcode&bibcode=2015A%252526A...584A..80M">Mérand et al.</a>)
and the eclipsing binary star Algol (observed by <a href="http://adsabs.harvard.edu/abs/1989ApJ...342.1061K">Kim, Ho-Il</a>). Both the curves has been approximated by
Fourier series. The last model is a simple cosine wave.
</p>
<figure>
<img class="figure" src="lcapprox.svg" alt="lcapprox.svg" title="LC">
<figcaption>The approximation of light curves of δ Cep and Algol.</figcaption>
</figure>
<p>
There is also possibility to define own
<a href="https://en.wikipedia.org/wiki/Fourier_series">Fourier series</a>
approximation of light curves which is suitable for wave-like patterns.
Do create a FITS table with format similar to <a href="fourier.lst">fourier.lst</a>.
The table contains real Fourier coefficients <i>a<sub>n</sub>, b<sub>n</sub></i>.
If the coefficients are estimated from measured data with noise,
any appearance of high-order terms indicates inappropriateness of Fourier approximation.
I can't recommend use on curves with sharp edges as Algol-like objects has.
Note. The Fourier approximation gives very good approximation
of Cepheids but is not very suitable for Algoids. The spline approximation
can be recommended in that case.
</p>
<p>
Universal flux-time pattern is implemented as a time series table.
It can represent a periodic object like Algol star or exoplanet transition,
but it can also model an non periodic behaviour like a supernova explosion
or an optical afterglow. The usage is described in
<a href="artific.html#lc">Light Curves</a> section of Overview.
</p>
<h3>Watermark</h3>
<p>
Every generated frame is marked by a symbol which indicates
its artificial origin to prevent potential confusion.
The mark can not be hide.
</p>
<h2>Parameters</h2>
<dl>
<dt><samp>--psf [SEEING|MOFFAT|GAUSS]</samp></dt><dd>
Selects PSF function: SEEING (the best, slow, default), MOFFAT (commonly used),
GAUSS (basic model, fast)
</dd>
<dt><samp>--spread [AUTO|FFT|RANDOM]</samp></dt><dd>
The method used to spread diffraction pattern (useful with --psf SEEING only):
FFT (fast, smooth), RANDOM (slow, natural look) or AUTO (the RANDOM
is selected for exposures shorter than 3 seconds)
</dd>
<dt><samp>--hwhm hwhm</samp></dt><dd>
Half with of half of maximum of spread Gaussian in pixels. It represents
half of seeing parameter. The parameter is common for all PSF.
</dd>
<dt><samp>--airy airy</samp></dt><dd>
Radius of Airy spot in pixels. It is useful for seeing PSF.
This parameter can be also set by <samp>--diameter</samp>.
</dd>
<dt><samp>--beta β</samp></dt><dd>
Value of β exponent of Moffat profile (<i>β > 0, β ≠ 1</i>).
</dd>
<dt><samp>--maglim m<sub>0</sub></samp></dt><dd>
Sets the most faint stars on frame for random generated field stars.
This option is active only when star catalogue (by <samp>-c,--cat</samp>)
is not used.
</dd>
<dt><samp>--lc-model [Cepheid|Algol|wave]</samp></dt><dd>
The light curve model: `Cepheid' means light curve of δ Cep, `Algol' means β Per,
`wave' means cosine function.
This parameter is mutually exclusive with <samp>--lc-table, --lc-fourier</samp>.
</dd>
<dt><samp>--lc-table file</samp></dt><dd>
The light curve model is determined by an smoothing spline
interpolation in a table. The table is represented
by the passed FITS file (see <a href="dataform_tmseries.html">
Time Series Table</a> for format description).
This parameter is mutually exclusive with <samp>--lc-model, --lc-fourier</samp>.
</dd>
<dt><samp>--lc-fourier file</samp></dt><dd>
The light curve model is given by the coefficients
of Fourier series listed in the FITS file.
This parameter is mutually exclusive with <samp>--lc-model, --lc-table</samp>.
</dd>
<dt><samp>--lc-mag mag, --lc-amp amp</samp></dt><dd>
The mean magnitude of the variable source.
The relative amplitude of light curve. For example, 0.1 (10%) means
cca 0.1 magnitude. The amplitude is ignored with conjunction of
<samp>--lc-table</samp>.
</dd>
<dt><samp>--lc-jd0 jd0, --lc-per period </samp></dt><dd>
Time elements of the light curve: jd0 is a reference time in
Julian date (exact meaning depends on the kind of the curve), period in
days. Don't use modified JD.
</dd>
<dt><samp>--lc-ra α, --lc-dec δ</samp></dt><dd>
A Right Ascension and Declination of the variable source. If the coordinates
are undefined, the centre of projection (as <samp>--rcen, --dcen</samp>)
is used.
</dd>
<dt><samp>--sky-mag skymag</samp></dt><dd>
Set sky brightness in magnitudes per square arcsecond.
Default value is 21 meaning good observing conditions.
</dd>
<dt><samp>--sky-grad-x xmag, --sky-grad-y ymag</samp></dt><dd>
Background change in magnitues per square arcsecond and pixel
in given direction.
The order of common values is in the interval 0.0001 … 0.001.
It simulates a light pollution (by Moon).
</dd>
<dt><samp>--area area</samp></dt><dd>
The detection area of a simulated telescope in square meters.
It is 1m² by default (see <samp>--diameter</samp> option).
This is equivalent area for 56 cm (22 inch) diameter telescope.
</dd>
<dt><samp>--diameter diameter</samp></dt><dd>
Sets diameter (twice of radius) of a telescope in meters. If set,
the area (<samp>--area</samp>) and Airy radius (<samp>--airy</samp>)
is (re-)defined.
</dd>
<dt><samp>--exptime time</samp></dt><dd>
The exposure time of generated frames in seconds.
</dd>
<dt><samp>--qeff η</samp></dt><dd>
The quantum efficiency of a whole apparatus (<i>0 ≤ η ≤ 1</i>).
</dd>
<dt><samp>--atmosphere</samp></dt><dd>
Switch-on modelling of the atmosphere. The model includes
both extinction and seeing.
</dd>
<dt><samp>--extk k</samp></dt><dd>
Monochromatic extinction coefficient. Setting on zero effectively
suppress of the extinction determination.
</dd>
<dt><samp>--long λ, --lat φ</samp></dt><dd>
Geographic coordinates of a station in degrees (-east, +north).
Brno has coordinates <samp>--long -16.6, --lat +49.2</samp>.
</dd>
<dt><samp>--date YYYY-MM-DD, --time HH:MM:SS</samp></dt><dd>
Date as YYYY-MM-DD and time as HH:MM:SS of (initial) frame.
</dd>
<dt><samp>--count #</samp></dt><dd>
Total count of generated frames.
</dd>
<dt><samp>--timestep time</samp></dt><dd>
Time delay between simulated exposures in seconds.
</dd>
<dt><samp>-f, --filter filter</samp></dt><dd>
Simulated filter.
</dd>
<dt><samp>-c, --cat file</samp></dt><dd>
Input catalogue with star positions and magnitudes.
</dd>
<dt><samp>--fov fov</samp></dt><dd>
Field of view in degrees.
</dd>
<dt><samp>--rcen α, --dcen δ</samp></dt><dd>
Centre of the field of view in Right Ascension and Declination
in degrees.
</dd>
<dt><samp>--scale s</samp></dt><dd>
Scale of the frame in degrees per pixel.
</dd>
<dt><samp>--angle φ</samp></dt><dd>
Angle of rotation of the frame around the centre in degrees.
Clockwise direction is positive.
</dd>
<dt><samp>--width width, --height height</samp></dt><dd>
Dimensions of output frame in pixels.
</dd>
</dl>
<h2>Examples</h2>
<pre>
$ munipack artificial
$ xmunipack artificial.fits
</pre>
<p>
<a href="artific.html">Overview</a> comes with more useful examples.
</p>
<h2>Light curve table</h2>
<p>The table has similar format as the time-series table.
It contains magnitudes (optionally, with errors)
with columns: TIME,MAG,MAGERR.
Records in the table must be sorted in time-increasing
order. <a href="artific.html">Overview</a> describes its application.
</p>
<h2>Table of Fourier coefficients</h2>
<p>
This table contains two valid columns with
<a href="https://en.wikipedia.org/wiki/Fourier_series">Fourier series</a>
approximation of a light curve. The columns are: first is the real
and second the imaginary part of Fourier complex coefficient.
The text file <a href="fourier.lst">fourier.lst</a> demonstrates
an instance of data (NAXIS2 must be updated when modified). A FITS file,
required by <samp>--lc-fourier</samp>, will be created as
</p>
<pre>
$ munipack fits --restore fourier.lst
</pre>
<h2>See Also</h2>
<p>
<a href="artific.html">Overview of Artificial sky</a>,
<a href="http://physics.muni.cz/~hroch/artific.pdf">Artificial Sky …</a> (seminary talk),
<a href="man_com.html">Common options</a>.
</p>
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