[gnuastro-commits] master a228f0d 2/2: Book: further clarifications on the surface brightness

[Mohammad Akhlaghi]

branch: master
commit a228f0df164c1c204bc220ed01347cf23597fff1
Author: Mohammad Akhlaghi <mohammad@akhlaghi.org>
Commit: Mohammad Akhlaghi <mohammad@akhlaghi.org>

    Book: further clarifications on the surface brightness
    
    Following Raul's correction in the previous commit, I noticed that I had
    confused dimension-less units with unitless quantities! We have many units
    that are also in logarithmic scale, for example the decibel. For more see
    https://en.wikipedia.org/wiki/Logarithmic_scale#Logarithmic_units. This
    confusion (on my side) was partly caused by the Wikipedia page for
    astronomical magnitude, that said its is "unitless".
    
    On the other hand, Radians and Steradians (which are actually ratios) are
    standard SI "units", they are just dimensionless.
    
    So the term "unitless" has been changed to dimensionless and generally, I
    started discussing the coverage of an object on the sky in the standard
    Steradians, and went onto describe why we use arcsec^2 instead. Also, to
    argue why brightness should be divided by area (instead of magnitude), I
    used the logarithmic vs. linear scale.
---
 doc/gnuastro.texi | 31 +++++++++++++++++++++----------
 1 file changed, 21 insertions(+), 10 deletions(-)

diff --git a/doc/gnuastro.texi b/doc/gnuastro.texi
index 8b61a45..9ba81b3 100644
--- a/doc/gnuastro.texi
+++ b/doc/gnuastro.texi
@@ -17195,9 +17195,9 @@ Therefore discussing brightness directly will involve a 
large range of values wh
 So astronomers have chosen to use a logarithmic scale to talk about the 
brightness of astronomical objects.
 
 @cindex Hipparchus of Nicaea
-But the logarithm can only be usable with a unit-less, and always positive 
value.
-Fortunately, in theory brightness is always positive.
-To remove the units, we divide the brightness of the object (@mymath{B}) by a 
reference brightness (@mymath{B_r}).
+But the logarithm can only be usable with a dimensionless value that is always 
positive.
+Fortunately brightness is always positive (atleast in theory@footnote{In 
practice, for very faint objects, if the background brightness is 
over-subtracted, we may end up with a negative brightness in a real object.}).
+To remove the dimensions, we divide the brightness of the object (@mymath{B}) 
by a reference brightness (@mymath{B_r}).
 We then define a logarithmic scale as @mymath{magnitude} through the relation 
below.
 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).
 
@@ -17208,26 +17208,37 @@ The @mymath{-2.5} factor in the definition of 
magnitudes is a legacy of the our
 @noindent
 @mymath{m} is defined as the magnitude of the object and @mymath{m_r} is the 
pre-defined magnitude of the reference brightness.
 One particularly easy condition is when the reference brightness is unity 
(@mymath{B_r=1}).
-This brightness will thus summarize all the hardware specific parameters 
discussed above into one number as the reference magnitude which is commonly 
known as the @emph{Zero point} magnitude (because when @mymath{B=Br=1}, the 
right side of the magnitude definition above will be zero).
+This brightness will thus summarize all the hardware-specific parameters 
discussed above (like the conversion of pixel values to physical units) into 
one number.
+That reference magnitude which is commonly known as the @emph{Zero point} 
magnitude (because when @mymath{B=Br=1}, the right side of the magnitude 
definition above will be zero).
 Using the zero point magnitude (@mymath{Z}), we can write the magnitude 
relation above in a more simpler format:
 
 @dispmath{m = -2.5\log_{10}(B) + Z}
 
+@cindex Steradian
+@cindex Angular coverage
+@cindex Celestial sphere
 @cindex Surface brightness
-Another important concept is the distribution of brightness over the object's 
detected area.
-For this, we define the @emph{surface brightness} which is defined as the 
magnitude of an object's brightness divided by its angular-coverage (or 
``area'' in the sky, usually in units of arcsec@mymath{^2}).
-Note that since this ``area'' is an angular measurement of the size of the 
astronomical object (projected on the celestial sphere), it is also unit less.
-A common mistake is to divide the magnitude by the area, but this is wrong 
because magnitudes don't have units.
-It is the brightness that should be divided by area, and the magnitude of that 
ratio is defined to be the surface brightness.
+@cindex SI (International System of Units)
+Another important concept is the distribution of brightness over its area.
+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 brightness is measured over an area of A 
arcsec@mymath{^2}, then the surface brightness becomes:
 
 @dispmath{S = -2.5\log_{10}(B/A) + Z = -2.5\log_{10}(B) + 2.5\log_{10}(A) + Z}
 
 @noindent
-In other words, the surface brightness (in units of 
magnitude/arcsec@mymath{^2}) is related to the object's magnitude (@mymath{m}) 
and area (@mymath{A}, in units of arcsec@mymath{^2}) through this equation:
+In other words, the surface brightness (in units of mag/arcsec@mymath{^2}) is 
related to the object's magnitude (@mymath{m}) and area (@mymath{A}, in units 
of arcsec@mymath{^2}) through this equation:
 
 @dispmath{S = m + 2.5\log_{10}(A)}
 
+A common mistake is to follow the mag/arcsec@mymath{^2} unit literally, and 
divide the object's magnitude by its area.
+But this is wrong because magnitude is a logarithmic scale while area is 
linear.
+It is the brightness 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.
+
 @node Profile magnitude, Invoking astmkprof, Brightness flux magnitude, 
MakeProfiles
 @subsection Profile magnitude