[gnuastro-commits] master 7d61e871: Book: further edits on brightness, flux and magnitude section

[Mohammad Akhlaghi]

branch: master
commit 7d61e87186a53a4d7eb2d08961f74d22d2032efd
Author: Mohammad Akhlaghi <mohammad@akhlaghi.org>
Commit: Mohammad Akhlaghi <mohammad@akhlaghi.org>

    Book: further edits on brightness, flux and magnitude section
    
    Until now, the various important physical concepts (and their units) were
    blended into the paragraph texts, making them hard to notice for a fast
    access to one definition. However, later for the optical astronomy
    terminology, we had divided each in a separate "table".
    
    With this commit, the physical concepts are also discussed in a similar
    format to the optical astronomy terms. The text was also edited to fit into
    this structure.
---
 doc/gnuastro.texi | 109 ++++++++++++++++++++++++++----------------------------
 1 file changed, 53 insertions(+), 56 deletions(-)

diff --git a/doc/gnuastro.texi b/doc/gnuastro.texi
index 61067857..21505c7b 100644
--- a/doc/gnuastro.texi
+++ b/doc/gnuastro.texi
@@ -29371,55 +29371,52 @@ It might even be so intertwined with its processing, 
that adding new columns mig
 @node Brightness flux magnitude, Quantifying measurement limits, Detection and 
catalog production, MakeCatalog
 @subsection Brightness, Flux, Magnitude and Surface brightness
 
-@cindex Flux
-@cindex Luminosity
+After taking an image with your smartphone camera (or a large telescope), the 
value in each pixel of the output file is a proxy for the amount of 
@emph{energy} that accumulated in it (while it was exposed to light).
+In astrophysics, the centimetre–gram–second (cgi) units are commonly used so 
energy is commonly reported in units of @mymath{erg}.
+In an image, the energy of a galaxy for example will be distributed over many 
pixels.
+Therefore, the collected energy of an object in the image is the total sum of 
the values in the pixels associated to the object.
+
+To be able to compare our scientific data with other data, optical astronomers 
have a unique terminology based on the concept of ``Magnitude''s.
+But before getting to those, let's review the following basic physical 
concepts first:
+
+@table @asis
+@item Brightness (@mymath{erg/s})
 @cindex Brightness
-@cindex Quantum efficiency
-The @emph{brightness} of an object is defined as its @emph{measured} power 
(energy in units of time).
-In astrophysics cgi units are commonly used so brightness is reported in units 
of @mymath{erg/s}.
-In an image, the brightness of a nearby galaxy for example will be distributed 
over many pixels.
-Therefore, the brightness of an object in the image, is the total sum of the 
values in the pixels (proxy for the energy collected in that pixel) associated 
to the object, divided by the exposure time.
-
-Since it is @emph{measured}, brightness is not an inhrenet property of the 
object: at different distances, the same object will have different brightness 
measures.
-Brightness is also affected by many other factors, which we can summarize as 
@emph{artifacts}.
-Here are some examples (among many others):
-@itemize
-@item
-The brightness of other background/foreground sources in the same line of 
sight may be added to it.
-@item
-Photons may be absorbed or scattered on their way to the detector (which 
reduce its brightness).
-@item
-The distortion or point spread function of the optical system may change its 
shape.
-@item
-Electronic issues in the detector may cause different measurements of the same 
object in different pixels.
-@end itemize
-Through various data reduction and data analysis methods that are implemented 
on the raw observations, we (try to!) remove all these artifacts to have the 
``pure'' brighntess of each object in the image.
-But dealing with the artifacts is beyond the scope of this section, so let's 
assume the reduction and analysis were done perfectly and we don't have 
artifacts.
+To be able to compare with data taken with different exposure times, we define 
the @emph{brightness} which is the measured power (energy divided by time).
+
+@item Flux (@mymath{erg/s/cm^2})
+@cindex Flux
+To be able to compare with data from different telescope collecting areas, we 
define the @emph{flux} which is defined in units of brightness per 
collecting-area.
+This area is historically reported in units of @mymath{cm^2}.
 
 @cindex Luminosity
-The @emph{flux} of an object is defined in units of 
energy/time/collecting-area.
-The collecting area is the telescope aperture (usually in units of 
@mymath{cm^2}).
-Therefore, flux is usually written in astrophysics literature as 
@mymath{erg/s/cm^2}.
 Knowing the flux (@mymath{f}) and distance to the object (@mymath{r}), we can 
define its @emph{luminosity}: @mymath{L=4{\pi}r^2f}.
-Therefore, while flux and luminosity are intrinsic properties of the object, 
the brightness we measure depends on our detecting tools (hardware and 
software).
+This is the total power, in @mymath{erg/s}, the object emits in all directions.
+Luminosity has the same units as brighntess, but as shown above its 
intepretations is very different: unlike brightness (a measured property), 
luminosity is an inherent property of the object that is calculated from the 
combination of multiple measurements (flux and distance).
+Our focus in this section is on direct measurements of electromagnetic energy, 
not position-related measurements, so we do not use or describe luminosity any 
more in this section and have not allocated a separate item in this list for it.
 
-@cindex Jansky (Jy)
+@item Spectral flux density (@mymath{erg/s/cm^2/Hz} or @mymath{erg/s/cm^2/\AA})
 @cindex Spectral Flux Density
 @cindex Frequency Flux Density
 @cindex Wavelength Flux Density
 @cindex Flux density (spectral)
-An important fact that we have ignored until now is the wavelength (or 
frequency) range that the incoming brightness is measured in.
+To take into account the spectral coverage of our data, we define the 
@emph{spectral flux density}, which is defined in either of these units (based 
on context): @mymath{erg/s/cm^2/Hz} (frequency-based) @mymath{erg/s/cm^2/\AA} 
(wavelength-based).
+
 Like other objects in nature, astronomical objects do not emit or reflect the 
same flux at all wavelengths.
 On the other hand, our detector techologies are different for different 
wavelength ranges.
 Therefore, even if we wanted to, there is no way to measure the ``total'' (at 
all wavelengths) brightness of an object with a single tool.
-It is therefore important to account for the wavelength (or frequency) range 
of the light that we are measuring.
-For that we have @emph{spectral flux density}, which is defined in either of 
these units (based on context): @mymath{erg/s/cm^2/Hz} (frequency-based) 
@mymath{erg/s/cm^2/\AA} (wavelength-based).
+To be able to analyze objects with different spectral features, it is 
therefore important to account for the wavelength (or frequency) range of the 
photons that we measured through the spectral flux density.
 
-A ``Jansky'' is an existing unit of frequency-based spectral flux density 
(commonly used in radio astronomy), such that 
@mymath{1Jy=10^{-23}erg/s/cm^2/Hz}.
+@item Jansky (@mymath{10^{-23}erg/s/cm^2/Hz})
+@cindex Jansky (Jy)
+A ``Jansky'' is a certain value of frequency flux density (commonly used in 
radio astronomy).
 Janskys can be converted to wavelength flux density using the 
@code{jy-to-wavelength-flux-density} operator of Gnuastro's Arithmetic program, 
see the derivation under this operator's description in @ref{Unit conversion 
operators}.
+@end table
 
-Having clarified, the basic physical concepts above, let's review the 
terminology that is used in optical astronomy to refer to them.
+Having clarified, the basic physical concepts above, let's review the 
terminology that is used in optical astronomy.
 The reason optical astronomers don't use modern physical terminology is that 
optical astronomy precedes modern physical concepts by thousands of years.
+As a result, once the modern physical concepts where mature enough, optical 
astronomers found the correct conversion factors to better define their own 
terminology (and easily use previous results) instead of abandoning them.
+Other fields of astronomy (for example X-ray or radio) were discovered in the 
last century when modern physical concepts had already matured and were being 
extensively used, so for those fields, the concepts above are enough.
 
 @table @asis
 @item Magnitude
@@ -29427,20 +29424,19 @@ The reason optical astronomers don't use modern 
physical terminology is that opt
 @cindex Flux to magnitude conversion
 @cindex Astronomical Magnitude system
 The spectral flux density of astronomical objects span over a very large 
range: the Sun (as the brightest object) is roughly @mymath{10^{24}} times 
brighter than the fainter galaxies we can currently detect in our deepest 
images.
-Therefore discussing spectral flux density directly will involve a large range 
of values which can be inconvenient and hard to visualize/understand/discuss.
-Optical astronomers have chosen to use a logarithmic scale for the spectral 
flux density of astronomical objects.
+Therefore the scale that was originally used from the ancient times to measure 
the incoming light (written by Hipparchus of Nicaea; 190-120 BC) can be nicely 
parametrized as a logarithmic function of the spectral flux density.
 
 @cindex Hipparchus of Nicaea
 But the logarithm can only be usable with a value which is always positive and 
has no units.
 Fortunately brightness is always positive.
 To remove the units, we divide the spectral flux density of the object 
(@mymath{F}) by a reference spectral flux density (@mymath{F_r}).
 We then define a logarithmic scale through the relation below and call it the 
@emph{magnitude}.
-The @mymath{-2.5} factor in the definition of magnitudes is a legacy of the 
our ancient colleagues and in particular Hipparchus of Nicaea (190-120 BC).
+The @mymath{-2.5} factor is also a legacy of our ancient origins: was 
necessary to match the used magnitude system of Hipparchus which was used 
extensively in the centuries after.
 
 @dispmath{m-m_r=-2.5\log_{10} \left( F \over F_r \right)}
 
 @noindent
-@mymath{m} is defined as the magnitude of the object and @mymath{m_r} is the 
pre-defined magnitude of the reference brightness.
+@mymath{m} is defined as the magnitude of the object and @mymath{m_r} is the 
pre-defined magnitude of the reference spectral flux density.
 For estimating the error in measuring a magnitude, see @ref{Quantifying 
measurement limits}.
 
 The equation above is ultimately a relative relation.
@@ -29450,16 +29446,17 @@ To tie it to physical units, astronomers use the 
concept of a zero point which i
 @cindex Zero point magnitude
 @cindex Magnitude zero point
 A unique situation in the magnitude equation above occurs when the reference 
spectral flux density is unity (@mymath{F_r=1}).
-In other words, the increase in brightness that produces to an increment in 
the detector's native measurement units (usually referred to as ``analog to 
digital units'', or ADUs, also known as ``counts'').
-The word ``increment'' was intentionally used because ADUs are discrete and 
measured as integer counts.
+In other words, the increase in spectral flux density that produces to an 
increment in the detector's native measurement units (usually referred to as 
``analog to digital units'', or ADUs, also known as ``counts'').
+
+The word ``increment'' above is used intentionally: because ADUs are discrete 
and measured as integer counts.
 In other words, a increase in spectral flux density that is below @mymath{F_r} 
will not be measured by the device.
 The reference magnitde (@mymath{m_r}) that corresponds to @mymath{F_r} is 
known as the @emph{Zero point} magnitude of that image.
 
-The increase in brightness (from an astrophysical source) that produces an 
increment in ADUs depends on all hardware and observational parameters that the 
image was taken in.
-These include the quantum efficiency of the dector, the detector's coating, 
the transmission of the optical path, the filter transmission curve, the 
atmospheric absorption (for ground-based images; for example thin high-altitude 
clouds or at low altitudes) and etc.
-The zero point therefore allows us to summarize all these ``observational'' 
(non-astrophysical) factors into a single number and compare different 
observations from different instruments at different times (critical to do 
science).
+The increase in spectral flux density (from an astrophysical source) that 
produces an increment in ADUs depends on all hardware and observational 
parameters that the image was taken in.
+These include the quantum efficiency of the dector, the detector's coating, 
the filter transmission curve, the transmission of the optical path, the 
atmospheric absorption (for ground-based images; for example observations at 
different altitudes from the horizon where the thickness of the atmosphere is 
different) and etc.
 
-Using the zero point magnitude (@mymath{m_r=Z}), we can write the magnitude 
relation above in a simpler format (recall that @mymath{F_r=1}):
+The zero point therefore allows us to summarize all these ``observational'' 
(non-astrophysical) factors into a single number and compare different 
observations from different instruments at different times (critical to do 
science).
+Defining the zero point magnitude as @mymath{m_r=Z} in the magnitude equation, 
we can write it in simpler format (recall that @mymath{F_r=1}):
 
 @dispmath{m = -2.5\log_{10}(F) + Z}
 
@@ -29467,14 +29464,13 @@ Using the zero point magnitude (@mymath{m_r=Z}), we 
can write the magnitude rela
 @cindex Magnitude, AB
 The zero point is found through comparison of measurements with pre-defined 
standards (in other words, it is a calibration of the pixel values).
 Gnuastro has an installed script with a complete tutorial to estimate the zero 
point of any image, see @ref{Zero point estimation}.
-
-Having the zero point of an image, you can convert its pixel values to the 
same physical units as the reference that the zero point was measured on.
-Historically, the reference was defined to be measurements of the star Vega 
(with the same instrument and same environment), producing the @emph{vega 
magnitude} sysytem where the star Vega had a magnitude of zero (similar to the 
catalog of Hipparchus of Nicaea).
+Historically, the reference was defined to be measurements of the star Vega, 
producing the @emph{vega magnitude} system.
+In this system, the star Vega had a magnitude of zero (similar to the catalog 
of Hipparchus of Nicaea).
 However, this caused many problems because Vega itself has its unique spectral 
features which are not in other stars and it is a variable star when measured 
precisely.
 
 Therefore, based on previous efforts, in 1983 Oke & Gunn 
@url{https://ui.adsabs.harvard.edu/abs/1983ApJ...266..713O,proposed} the AB 
(absolute) magnitude system from accurate spectroscopy of Vega.
 To avoid confusion with the ``absolute magnitude'' of a source (at a fixed 
distance), this magnitude system is always written as AB magnitude.
-The equation below was defined such that a star with a flat spectra around 
@mymath{5480\AA} have a similar magnitude in the AB and Vega-based systems, 
where @mymath{F_\nu} is the frequency-based spectral flux density (in units of 
@mymath{erg/s/cm^2/Hz}):
+The AB magnitude zero point (when the input is frequency flux density; 
@mymath{F_\nu} with units of @mymath{erg/s/cm^2/Hz}) was defined such that a 
star with a flat spectra around @mymath{5480\AA} have a similar magnitude in 
the AB and Vega-based systems:
 
 @dispmath{m_{AB} = -2.5\log_{10}(F_\nu) + 48.60}
 
@@ -29489,14 +29485,13 @@ $ astarithmetic sdss.fits 22.5 counts-to-jy
 @cartouche
 @noindent
 @strong{Verify the zero point usage in from new databases:} observational 
factors like the exposure time, the gain (how many electrons correspond to one 
ADU), telescope aperture, filter transmission curve and other factors are 
usually taken into account in the reduction pipeline that produces high-level 
science products to provide a zero point that directly converts pixel values 
(in what ever units) to Janskys.
-But some reduction pipelines may not account for some these for special 
reasons: for example not account for the gain or exposure time.
-To avoid annoying strange results, when using a new database, verify that the 
zero points they provide directly converts pixel values to Janskys (is an AB 
magnitude zero point), or not.
-If not, you can follow steps described below.
-This information is usually in the documentation of the database.
+But some reduction pipelines may not account for some of these for special 
reasons: for example not account for the gain or exposure time.
+To avoid annoying strange results, when using a new database, verify (in the 
documentation of the database) that the zero points they provide directly 
converts pixel values to Janskys (is an AB magnitude zero point), or not.
+If they not, you need to apply corrections your self.
 @end cartouche
 
 Let's look at one example where the given zero point has not accounted for the 
exposure time (in other words it is only for a fixed exposure time: 
@mymath{Z_E}), but the pixel values (@mymath{p}) have been corrected for the 
exposure time.
-One solution would be to first multiply the pixels by the exposure time, use 
that zero point and delete the temporary file.
+One solution would be to first multiply the pixels by the exposure time, use 
that zero point to get your desired measurement, and delete the temporary file.
 But a more optimal way (in terms of storage, execution and clean code) would 
be to correct the zero point.
 Let's take @mymath{t} to show time in units of seconds and @mymath{p_E} to be 
the pixel value that would be measured after the the fixed exposure time (in 
other words @mymath{p_E=p\times t}).
 We then have the following:
@@ -29513,13 +29508,16 @@ From the properties of logarithms, we can then derive 
the correct zero point (@m
 @cindex Celestial sphere
 @cindex Surface brightness
 @cindex SI (International System of Units)
-An important concept is the distribution of an object's brightness over its 
area.
+The definition of magnitude above was for the total spectral flux density 
coming from an object (recall how we mentioned at the start of this section 
that the total energy of an object is calculated by summing all its pixels).
+The total flux is (mostly!) independent of the angular size of your pixels, so 
we didn't need to account for the pixel area.
+But when you want to study extended structures where the total magnitude is 
not desired (for example the sub-structure of a galaxy, or the brightness of 
the background sky), you need to report values that are independent of the area 
that total spectral flux density was measured on.
+
 For this, we define the @emph{surface brightness} to be the magnitude of an 
object's brightness divided by its solid angle over the celestial sphere (or 
coverage in the sky, commonly in units of arcsec@mymath{^2}).
 The solid angle is expressed in units of arcsec@mymath{^2} because 
astronomical targets are usually much smaller than one steradian.
 Recall that the steradian is the dimension-less SI unit of a solid angle and 1 
steradian covers @mymath{1/4\pi} (almost @mymath{8\%}) of the full celestial 
sphere.
 
 Surface brightness is therefore most commonly expressed in units of 
mag/arcsec@mymath{^2}.
-For example, when the spectral flux density is measured over an area of A 
arcsec@mymath{^2}, then the surface brightness becomes:
+For example, when the spectral flux density is measured over an area of A 
arcsec@mymath{^2}, the surface brightness is calculated by:
 
 @dispmath{S = -2.5\log_{10}(F/A) + Z = -2.5\log_{10}(F) + 2.5\log_{10}(A) + Z}
 
@@ -29533,14 +29531,13 @@ But this is wrong because magnitude is a logarithmic 
scale while area is linear.
 It is the spectral flux density that should be divided by the solid angle 
because both have linear scales.
 The magnitude of that ratio is then defined to be the surface brightness.
 
-One usual application of this is to convert an image's pixel values to surface 
brightness, when you know its zero point.
 Besides applications in catalogs and the resulting scientific analysis, 
converting pixels to surface brightness is usually a good way to display a FITS 
file in a publication!
 See @ref{FITS images in a publication} for a fully working tutorial on how to 
do this.
 @end table
 
 @cartouche
 @noindent
-@strong{Do not warp or convolve magnitude or surface brightness images:} 
Warping an image involves calculating new pixel values (of the new pixel grid) 
from the old pixel values.
+@strong{Do not warp or convolve magnitude or surface brightness images:} 
Warping an image involves calculating new pixel values (of the new pixel grid) 
from the input grid's pixel values.
 Convolution is also a process of finding the weighted mean of pixel values.
 During these processes, many arithmetic operations are done on the original 
pixel values, for example, addition or multiplication.
 However, @mymath{log_{10}(a+b)\ne log_{10}(a)+log_{10}(b)}.