NVERSE-SQUARE LAW AND MAGNITUDE FORMULAE 
 -----------------------------------
 John Pazmino
 NYSkies Astronomy Inc
 nyskies@nyskies.org
www.nyskies.org
 2015 December 3 initial 
 2018 November 28 current

Introduction 
 ---------- 
    The upbringing of home astronomers includes many concepts about 
light and luminous output of celestial objects. Almost always they are 
treated as distinct subjects with little hint that some are actually 
mutations of a one single concept. 
    That concept is the behavior of light radiating equally in all 
directions from a point source. Since celestial objects can often be 
treated as point sources as seen from far away and, in the absence of 
specific knowledge, they shine uniformly to all quarters of their 
world, understanding how light behaves is crucial for the astronomer. 
    An other prime consideration is that the emission is unmolested on 
its way to Earth. We assume, in absence of specific information to the 
contrary, that the intervening space is free of diffusing, absorbing,  
dispersive, reflective material. 
    A newer consideration is that the geometry of space is 'flat' over 
the path of the radiation. For extremely remote sources the radiation 
passes thru space that during the light travel is continuously 
expanding. ThIS expansion impresses a distortion into the way  these 
sources illuminate Earth. 
    I found several topics relating to luminous emission that here are 
discussed as facets of a one single concept, the Inverse-Square Law of 
radiation. I say 'radiation' because we can explore the universe in 
just about all wavelengths, or frequency, of electromagnetic energy, 
not only that in the luminous range of the spectrum. 

Light 
 ---
    Recall that 'light' is not itself a physical substance. It can not 
be objectively measured without recourse to human perception. If there 
were no humans, the universe would be 'dark' in that it would NOT be 
filled with luminous sources, as some sci-fi scenarios imagine. In 
fact the universe would be filled with radiant energy ALL of which is 
INVISIBLE simply because we humans aren't there to see it. 
    Light is the human physiological response to a limited range of 
wavelength of radiation incident into the eye-brain mechanism. 
Radiation within about 360nm and 720nm wavelength excites the eye-
brain to yield the the sensation of vision. Incident radiation outside 
these limits do not produce vision. 
    It took until the mid 20th century for astronomers to fully 
appreciate the distinction between light and general radiative 
emission. Photometers and light-meters ended up merely approximating 
the eye-brain response to radiation. Photography also was such an 
effort, leading to film chemistry that mimicked the human vision 
process. 
    In history, only light was available for exploring the universe 
form Earth. Our atmosphere filtered out large segments of the incoming 
radiation. Certain other segments simply were never imagined to exist. 
    The Sun was treated as a source of 'light' and 'heat', altho both 
were part of the one flux of radiation with different responses by 
humans. These were  vision and warmth. 
    Other radiations from celestial objects were hinted at in the 19th 
century, such as infrared studied by Herschel and radio eaves detected 
by Tesla. They were poorly studied for the crude instruments of the 
day. It wasn't until the 1930s when Jansky and Reber made deliberate 
investigations of radio-band noise from beyond Earth that we realized 
the potential range of other radiations emitted by celestial targets. 
    The opening of astronomy to the full range of celestial radiation 
began in the 1960s with astrophysical satellites. By the 1980s just 
about the entire electromagnetic spectrum was open for exploration 
from space. 

Inverse-Square Law 
 ----------------
    One of the most lousy explanations in astronomy is for one of the 
most basic ideas of astronomy! We all see the diagram of rays from a 
point diverging as they recede outward. They pass thru screens with 
one, four, nine, maybe sixteen squares. The trick is to see that as 
the distance from the source increases, the density of the rays 
received on the screens decreases as the square of that distance. 
    Altho very technicly this picture is correct, altho sometimes 
incorrectly drawn!, it is hardly a useful mechanism for furthering the 
home astronomer's career.
    Let's see what REALLY happens to the radiant energy. Place a 
sphere, of any desired radius, centered on the source. All radiation 
from the source must intercept this sphere, which in maths is called a 
Gauss sphere. The amount of radiation over the sphere is the same for 
all such spheres of what ever radius we choose.
    This is the key to understanding the inverse-square law. 
    We define irradiation as the radiant energy, or energy, the flow 
or flux, passing a unit area of the sphere. If the power output of the 
source is P, in watts, and the radius of the sphere is r, in meters, 
the irradiation, I, is cited in watt/meter2 at radius r from the 
source. The radiant energy is captured by the whole area of the 
sphere, which is 4*pi**(r^2), so. 

                  I = P / (4 * pi * (r ^ 2)) 

And that is it! We derived the Inverse-Square Law in a simple, 
elegant, mature, wholesome manner. 

    +---------------------------------+
    | INVERSE-SQUARE LAW OF RADIATION | 
    |                                 | 
    | I = P / (4 * pi * (r ^ 2))      | 
    +---------------------------------+ 

    The sphere enclosing the source does not have to a sphere, nor 
does it have to be centered on the source. Any closed surface, with no 
holes or tears, is valid. it may have folds, so the radiation enters 
and leaves thru them. It can be shown that in all cases the Inverse-
Square law holds true. The difference is the complexity of tracing the 
rays thru irregular enclosing surfaces.  We used a centered sphere for 
the simplicity of the maths. 

Watts and lumens 
 -------------- 
    We could have stopped here if this was an article within ordinary 
physics. The radiant power of the source is in watts. The distance, 
radius of the Gauss sphere, is in meters. The irradiation is then is 
watt/meter2.
    As a separate branch of science we played with 'light' which we 
treated as a luminous flux from the source. We dimensioned this flux 
in lumans and the incidence of this flux on the Gauss sphere was in 
lumens.meter2. Lumen is the analogy in light to the watt in total 
radiation. 
    The Inverse-Square law for light is identical to the general one, 
except for the units of measure. The lumems received per unit area is 
the illumination of that area. There still seems to be no definite 
word for the luminous output of a source. Lumwnpower is a common one. 
    On and off physicists tried to determine the equivalence of watts 
and lumen. A  lamp bulb consumes electric in watts and emits light in 
lumens of light. The effort was frustrated by the weak knowledge of 
the hums vision mechanism and the tenet that light was really a 
physical substance. Since watt is a measure of mechanical poert, the 
equivalence was sometimes called the mechanical equivalent of light. 

Try this experiment
 ----------------- You need a light-meter or other device that 
measures incident illumination in lux or foot-cadle. Recall that for 
all intents and purooses one foot-candle equals ten luc. Or, in 
essences, the 'metric foot-candle' is 10 lux. 
    In any rectangle room at home place a bare-bulb lamp in any 
convenient spot, like an end table. it does not have to centered in 
the room but in a place you can work all around it. 
    Measure the dimensions of the room and work out its surface area, 
all in square meters. All other lighting in the room are now turned 
off, leaving only the test lamp turned on. it may be easiest to do 
this experiment at night with window shades pulled down. 
    With the light-meter take illumination readings within each square 
meter of the room, including floor and ceiling. You may need a step-
stool to reach the ceiling and work around immobile furniture to get 
at certain parts of the wall. 
    tabulate the readings, m2-by-m2, and strike their sum. You in 
principle must multiply each by the area associated with it, because 
you already did the measurements in square-meter cells, the multiply 
is '1m2' for each reading. You may may have to do the explicit 
multiply for fractional square-meters sections of the room. 
    The sum should be equal to the lumen rating of the lamp bulb, 
which should be marked on the bulb or its package. A tolerance of 10-
15 percent would be quite a good success. 
    You actually surrounded the lamp with a Gauss surface, the room, 
to capture all of the lumens sent out by the lamp. You collected the 
illumination all over the room, summed them, to [more or less] equal 
that lumen output. You had to work in sections because the 
illumination over the room is very uneven. If the room was a sphere 
centered on the lamp, you would need only one reading, knowing it was 
the same all over the room. This assumes, as is largely true, the bulb 
radiates uniformly in all directions. 

Apparent magnitude 
  ---------------
    By history astronomers did not care about the physical measure of 
illumination. They used, as invented by Hipparchus, a scale of 
'brightness' for the stars. From the Greek era until the mid 1800s 
this scale was informal, based on eyeball assessments of the stars. 
Hipparchus himself assigned rank 1 to the brightest stars, about 15 of 
them above the horizon of Greece. The fainter stars, in 'steps' or 
'shades', earned ranks 2 thru 6. Since the brilliance of a star was 
its greatness, the ranking was called 'magnitude'.A star of Hipparchus 
rank 3 was a star of magnitude 3. 
    Until the 19th century magnitude assigned to star was sometimes 
erratic among astronomers. For telescopic stars, with no prior history 
of magnitude ranking, the scale diverged severely. 
    Pogson in the mid 1800s formalized the scale with a logarithmic 
sequence. He measured by crude photometers that stars five magnitude 
apart were almost 100 time different in illumination. He set the ratio 
to exactly 100 and made each magnitude step equal to 5root(100), about 
2.512. This scale preserved the bulk of ratings of stars already 
catalogued and provided a rational way to rate telescopic stars. 
    This eyeball ranking is still used by home astronomers for 
monitoring variable stars, comets,  meteors. The target is compared to 
field stars having designated magnitude ratings. 
    The logarithm scale of magnitude is

    +-----------------------------+ 
    | HIPPARCHUS-POGSON MAGNITUDE | 
    |                             | 
    | M - M0 = -2.5 * log(I/I0)   | 
    +-----------------------------+ 

    M0 and I0 are the zeropoint values for magnitude and illumination. 
The logarithm is on base 10, the common or Briggs scale. The minus 
signum forces the ranks to INCREASE in value for fainter stars, 
matching the trend of the Hipparchus scale. 
    newcomers sometimes find this relation confusing because a 
brighter star 'should' jave a greater value. Tink of magnitude as a 
rank or class in a company or camp or club. The lower-level members 
have a hifher rank number. The importnat members have the lower 
number, such as 'fist vice-chair'. As a member progresses and improves 
his status he moves to a higher class, with a lower numaaber. 

Newer developments 
 ------    -----
    At first we banked off of Polaris, assigning it magnitude 2.0. In 
the 1890s electrophotometry made magnitude assessments more objective, 
less personalized, and allowed for a link to the physics photometry 
system. The I0 for Polaris, with M0 = 2.0, was that electric current 
generated by its starlight.  Effort was spent to filter the incoming 
radiation to accept that range of wavelengths of human vision. To at 
least some degree the incoming watt/meter2 was approximately 
proportional to lumens/meter2. 
    An immediate benefit of electrophotometry was the greaater 
resolution of magnitude assessment in an objectrtive verfiable 
process. By eyeball, stars were assessed only to  a whole or half 
unit, which could not be objectively verified. 
    This objective nonhuman-based method moved the study of variable 
stars to a solid scientific level. By end of the 19thcentury a 
complete electrophotmetry of the sky's naked eye stars was done, 
issued as the Harvard Photometry. A refined edition came out in the 
19-ohs, the Harvard Revised Photometry. This catalog would be the 
standard for magnitude values until the 1960, when the Johnson 
photometry came into use. 
    Electrophotometry showed that some of the first magnitude stars 
were too bright for that rank. They were given the prolongated ranks 
of 0 and -1. In addition, planets usually are so bright that their 
magnitudes were in the negative range, too. Venus is typicly -4 
magnitude; Jupiter, -2. 
    In the1920s-30s we realized that the physical photometry system 
was out of whack. We were, without fully appreciating it, using 
laboratory illuminants ultimately based on combustion of hydrocarbon 
fuel. Until we learned more about blackbody radiation and the 
spectrosensitivity of human vision, we practiced photometry under 
false premises. 
    New illluminants were devised employing noncombustive processes, 
such as fluorescent and neon lamps,  and phosphorescence. Their 
luminous output in no way resembled the spectral profile of combustive 
sources. It took decades to shift physicists to a more fundamental 
scheme of photometry. 
    We dogged on with 'light' as some real physical feature that would 
yield to objective instrumental study. What saved us from catastrophe 
was the resilience of human vision to errors of measurement and 
application of photometry. We really didn't get out of 19th century 
photometric mindset until the millennium crossing. 
    On the astronomy side we in the 1960s were establishing major new 
observatories in the southern hemisphere, where Polaris is out of the 
local sky. We moved the magnitude zeropoint to Vega, visible from 
most reasonable southern latitudes. Vega was defined as magnitude 0.0. 
For narrow-band photometry the  illumination, or irradiation, from  
Vega within the photometer's bandwidth is defined as magnitude 0.0. 
   An other cause to let go of Polaris was its discovery as a delta 
Cephei variable star! Its illumination altered over a several day 
cycle by 2/10 magnitude. This was a totally intolerable swing of some 
23% in illumination. Polaris just ws no longer a stable reference 
standard. 
    Vega itself in the 1980s came under question when we found it had 
a circumstellar dust ring. Could the dust pass over the star to vary 
the light it sends us? So far it appears that the dust ring is too 
inclined to our line of sight. We do keep careful watch on it. 
 
Linked at last!
 -------------
    What is the zero-point to initialize the magnitude scale? This 
number is maddingly trough to find in astronomy litterature! It is 
vaguely mentioned as a special application of photometry. 
    When the International System of metric units was built in the 
1960s, we finally linked the photometries of physics to astronomy. In 
addition, within a few more years we developed electronic and digital 
means of recording incident radiation, enabling us to manipulate data 
about light in ways never before possible. 
    One significant result was the equation of magnitude and 
irradiation within the visual band of the spectrum. 

    +-----------------------------------+ 
    | MAGNITUDE-ILLUMINATION            | 
    |                                   | 
    | Mapp(0.0) = 2.56e-6 lumen/meter2  | 
    +---------------------------------
\ 
    The initial measurements of this equivalence were done at stations 
under clean dry air, as best as could be found, to minimize distortion 
of the incoming starlight by haze and moisture. In the Space Age the 
equivalence is assessed from above the atmosphere. 
    Values wander a bit among authors because the photometer 
spectrosensitivity profile may not match that of human vision. Vision 
varies widely across astronomers, as much as with any other human 
faculty In as much as vision is entirely an internal processing of 
incident irradiation by the eye-brain system, it is impossible to know 
just 'how bright' a given person see a target star. It is equally 
impossible to teach a person for 'how bright' he should see that 
target star.  
    Please keep in mind that 'light' as equated to irradiation is the 
STIMULUS entering the human eye-brain. It is NOT -- ad can not be -- 
the RESPONSE of the eye-grain. 

Angular magnitude 
 ---------------
    In all of astronomy, until the late 20th century, the magnitude 
system was applied only to point sources, like stars, as observed 
 by eye. It was applied occasionally to extended sources provided they 
remained angularly small, like the  Moon and planets. Large areas such 
as nebulae, comets, aurora, were not treated to the magnitude ranking. 
    Since the magnitude system is ganged to photometric units, it 
should be feasible to collect light from the entire angular extent of 
a large target and divide it by the target's angular area. 
    The figure is lumnen/(meter2.arcmin2) or lumnen/(meter2.arcsec2). 
This is next converted into magn/arcmin2 ot magn/arcsec2, the very 
angular magnitude of the target. 
    Be careful. The total illumination is NOT first turned to 
magnutude and THEN divided by the area! This would yield a 
ridiculously lw dim brightness per unit area. 
    A crucial point to mind is that angular area applies to sources 
either nebular in texture or not resolved into stars, like a star 
cluster seen by bare-eye. If the target resolves into stars, under 
magnification, there is no 'area' sending illumination to us. The 
light comes from separate point sources and the total magnitude 
procedure is required. 

Extravisual magnitude 
 -------------------
   Magnitude is specificly defined for luminous radiation within the 
optical spectral band. For the most part home astronomers observe only 
by light but equipment is entering the market to observe beyond the 
optical range. 
    Astronomers like to continue visualizing targets by a magnitude 
rating, even when the radiation is outside of the visual spectrum. 
They cite the 'magnitude of an ultraviolet or infrared source. The 
illusion may be to imagine its brightness if somehow human vision is 
sensitive to such radiation. A common basis of extravisual magnitude 
ratings is to compare the radiation to that from Vega in the same 
bandwidth. The irradiation from Vega is set at magnitude 0.0. 
    Since Vega is a blackbody radiator and most nonoptical radiation 
is not, the comparison can prodice insane values for the 'magnitude' 
of a given target. Vega, for example raidates weakly in the ar 
infrared, making a target's infrared radiation seem enormous, with a 
humongous magnitude rating. 
    In actuality the concept of magnitude as a measure of visual 
brightness is nonsense outside of the visual range of wavelengths. 
The entire premise of magnitude is to rank illumination as perceived 
by human vision. Radiation beyond the bandwidth of human vision 
produces no illumination or sensation of brightness. 
    Astronomers working outside the optical band use magnitude as a 
convenient logarithmic scale of relative irradiations. The one is 
compared  against an other. In this sense magnitude is like decibels. 
In no way would a radio technician claim that a signal of a given 
decibel strength sounds as loud as a note of the same decibel value if 
human hearing could hear the signal. 
    Under the assumption that a star is a pure blackbody emitter we 
tried to account or the limited range of irradiation we captured as 
light. From spectrometry we figure out the temperature of the star and 
generate a blackbody radiation curve for it, using standard procedures 
rom thermdynamics. of the entire spectral range of this curve e work 
out the portion falling within the optical range. The remainder, 
outside the optical band, is expressed as an increment of magnitude 
applied to the optical magnitude. The sum, always brighter than the 
optical magnitude, is the bolometric magnitude. The increment, 
bolometric correction, is a function only of temperature and can be 
tabulated for handy reference. 
    The Hertzsprung-Russell Diagram  is plotted by the optical 
magnitude, since it is only in recent times we could measure other 
regions of radiation from stars. Some astronomers suggest  to revise 
the HRD to plot bolometric magnitude  Such a HRD can easily e plotted 
because the normal one embeds the star temperature, none caught on. 

Parallax 
 ------
    Parallax is the swing of our sightline from Earth to the target as 
Earth orbits the Sun. It is also the angular radius of Earth's orbit 
as seen from the target. 
    Since stars are awfully far away, their light taking years and   
decades and centuries to reach us, the parallax angle is incredibly 
tiny. No known star has a parallax so large as one full arcsecond. 
Most stars making up constellations  have angles in the hundredths of 
an arcsecond. 
    The star distances can not be reasonably cited in terrestrial 
units like kilometers or Earth radii. Even the Earth-Sun distance, the 
astronomical unit, is way too small a unit. For example, the first 
star to yield a positive parallax, in the 1830s, was 61 Cygni with a 
distance of some 650,000 AU. Since this is one of the closer stars, 
other distances will be in millions of AU, a quantity that can not be 
easily visualized. 
    The lightyear came into use in the mid 1800s for popular astronomy 
litterature and it mainstreamed in the profession by about 1900. But 
the lightyear has to be calculated from an other measure of distance. 
It is NOT a timing of the light as it travels from the target. No 
kidding, I see authors making this claim! 
    With such minuscule angles, the parallax angle is inversely 
proportional to the distance by applying the small-angle rule of 
maths. The parallax angle is at the apex of the long slender triangle 
with the Earth orbit radius as base. For a given parallax the long 
sides, both essentially equal , have a definite length which can be be 
calculated. 
    This length is 206,265 AU for a parallax of one arcsecond. With 
this length, distance to the star, inversely proportional  to 
parallax, the one-arcsec length is a new, the 'parsec' from 'PARallax-
SECond'. For quick work we can round this to 200,000 AU and 5 parsec = 
1 million AU. 
    In terms of lightyears, one parsec is 3.26 lightyears. Many 
astronomers simply use parsecs without switching to lightyears. 
    The parsec has the simplicity of being identicly the reciprocal of 
the measured parallax. A star of 0.01 arcsecond parallax stands 1/0.01 
= 100 parsec away. If you insist, that's 326 lightyears. 

    +--------------------------------------+ 
    | PARALLAX-PARSEC RELATION             | 
    |                                      | 
    | parsec distance = 1/(parallax) | 
    +--------------------------------------+ 

Comparing illuminations 
 ---------------------
    The Inverse-Square Law can compare illuminations from different 
sources to find either the distance or the radiation output. We have 
two sources of equal power P. One is at a given distance r0; the 
other, at unknown distance r. We write out the illumination received 
from both. 

    I0 = P / (4 * pi * (r0 ^ 2)) 

    I = P / (4 * pi * (r ^ 2)) 

Divide the lower one by the upper: 

    I / I0 = (P / P) / (r ^ 2) / (r0 ^ 2) 
           = 1 / (r ^ 2) / (r0 ^ 2) 
           = (r0 ^ 2) / (r ^ 2) 

The received illumination from the equal sources is the inverse square 
ratio of their distances. We know the standard distance r0 and we 
solve for the unknown r 

    I / I0 = (r0 ^ 2) / (r ^ 2)  

     r ^ 2 = (I / I0 ) / (r0 ^ 2) 

    We can also have two sources of equal distance r but one has a 
known power P0 and the other has unknown power P 

    I / I0 = (P / P0) / (r ^ 2) / (r ^ 2) 
           = (P / P0) / (1) 
           = P / P0 

Solve for unknown P

    P = P0 / (I0 / I)
    P = P0 * I / I0

Absolute magnitude 
 ---------------- 
    This is an attempt to standardize the illumination system of stars 
by artificially setting the stars at a one distance from us. The 
distance is 10 parsecs, chosen for a good maths reason. For that 
distance, and the actual distance and illumination, or magnitude, of 
the star, a new fake magnitude is computed. The usual explanation is 
that it's the apparent magnitude the star would shine if it somehow 
was placed 10 parsecs away. 
    This computed magnitude is the absolute magnitude, a misleading 
name which we may for ever more be stuck with. Better names would have 
been 'normalized magnitude' or 'reduced magnitude'. There is nothing 
'absolute' about absolute magnitude and it is not even a property of 
the star. 
    To obtain the absolute magnitude we must have in hand the apparent 
magnitude and the distance to the star. Both are observed data from 
the star. 
    Absolute magnitude was first used in the 1910s when we accumulated 
databases of parallax and apparent magnitude of stars. The data were 
captured by the then-new incorporation of photographic astrometry and 
electrophotometry. 
    The absolute magnitude comes recta mente from the Inverse-Square 
Law and definition of magnitude. We compare the same star, with output 
P, at two distances. One is the actual distance r; the other, the 
standard one of 10pc, r0. 

    I / I0 = (P / P0) / (r ^ 2 / r0 ^ 2) 
           = (P / P) / (r ^ 2 / r0 ^ 2) 

I left out the 4*pi factor since it immediately cancels out in the 
denominator: ((4*pi*r^2)/(4*pi*r0^2)) -> *r^2/r0^2).
    Also P0 is P because we are working with one source moved between 
tow distances. 
 
    I / I0 = 1 / (r ^ 2 / r0 ^ 2) 
           = r0 ^ 2 / r ^ 2 

Because we be astronomers we work with the magnitude scale, not the 
raw photometric scale. We first take the log of both sides
 
    log(I / I0) = log(r0 ^ 2 / r ^ 2) 

    We now apply the definition of magnitude. 

    log(I) - log(I0) = log(r0 ^ 2) - log(r ^ 2) 
                     = 2 * (log(r0) - 2 * log(r)) 

We apply the magnitude definition to both sides. 

    -2.5 * (log(I) - log(I0)) = -2.5 * 2 * (log(r0) - log(r)) 

    Mapp - Mabs = -2.5 * 2 * (log(r0) - log(r)) 

Mapp is the apparent magnitude recorded for the star at its real 
distance r. Mabs is the artificial, absolute, magnitude of the star if 
it was 10pc away. 

    Mapp - Mabs = -5 * log(r0) - (-5 * log(r)) 
                = -5 * log(r0) + 5 * log(r)  

    Now comes the trick. We purposely picked 10pc as the distance for 
absolute magnitude BECAUSE the log of 10 is one! Astronomers hate 
maths as much as any one else. 

    Mapp - Mabs = -5 * log(10) + (5 * log(r)) 
                = -5 + 5 * log(r))  
                = 5 * log(r) - 5 

    Typicly we know the distance and Mapp and solve for Mabs 

    Mapp - Mabs = 5 * log(r) - 5 

    -Mabs = 5 * log(r) - 5 - Mapp 

     Mabs = -5 * log(r) + 5 + Mapp 
          = Mapp - (5 * log(r)) + 5 

    Recall that the distance in parsec is merely 1/parallax, where 
parallax is the actual observed parameter of the star. 

    Mabs = Mapp - (5 * log(r)) + 5 
         = Mapp - (5 * log(1/pi)) + 5 
         = Mapp + (5 * log(pi)) + 5 

    +---------------------------------+ 
   | ABSOLUTE MAGNITUDE FORMULA       | 
   |                            |     | 
   | Mabs  = Mapp + (5 * log(pi)) + 5 | 
   |                                  | 
   | Mabs  = Mapp - (5 * log(r)) + 5  | 
  + ----------------------------------+ 

where I give both versions, for distance and for parallax.
    By this formula and stating the Earth-Sun distance in parsecs, a 
very small number, we find that the Sun's absolute magnitude is +4.8. 
If our Sun was removed to 10pc, a modest remoteness for a star, it 
would be among the mediocre    tars in the sky. This helps to show how 
insignificant our Sun is on the scale of even nearby stellar reealm. 
    Because most stars are beyond 10 parsecs from Earth, their Mabs is 
much brighter, algebraicly smaller, than their Mapp. Their is no 
special meaning attached to the numerical value of Mabs since it 
drives from the arbitrary choice of the 10pc standard distance. 
    The derivation here joining the Inverse Square law to magnitude is 
almost neglected in the normal astronomy tuition. The two are treated 
as unrelated / separate topics. 

Luminosity 
 --------
    Altho 'absolute magnitude' is a poor choice of words for the 
normalized magnitude on a 10-parsec distance, it is a handy way to 
compare the luminous output of stars. With the Sun as a unit emitter 
of light, the relative output of any other star, in solar units, is 
the star's luminosity. This is NOT the star's full radiant output 
because luminosity ignores radiation beyond the visual spectrum. 
    Stars in general emit the bulk of their radiation in the visual 
range, by the blackbody mechanism. In the era when we could not 
observe beyond the optical band, we had no confident accounting for 
the extravisual radiation. We let luminosity equal radiopower and live 
with ay discrepancy.
     Stars placed the same distance away shine with magnitudes 
consonant with their luminous emission. That is 
    From the magnitude definition 

    m - m0 = -2.5 * log(L / L0) 


where L is the luminosity, luminous output, in place of P, the full 
radiation output. 
    Set m0 to the absolute magnitude of the Sun, +4.8, and I0 to the 
solar unit of luminosity 

    m - +4.8 = -2.5 * log(L / 1) 
             = -2.5 * log(L) 

When m is set to the absolute magnitude of a star, the luminosity 
ratio falls out 

    Mabs - +4.8 = -2.5 * log(L) 

    log(L) = (Mabs - +4.8)) / -2.5 
          = -0.4 * (Mabs - +4.8)  

    L = 10 ^ (-0.4 * (Mabs - +4.8))  

    +----------------------------------------+ 
    | ABSOLUTE MAGNITUDE-LUMINOSITY RELATION | 
    |                                        | 
    | L = 10 ^ (-0.4 * ( Mabs - +4.8))        | 
    |                                        | 
    | Mabs = -2.5 * log(I) + 4.8             | 
    +----------------------------------------+ 

    Star catalogs generally list either absolute magnitude or 
luminosity for its stars, It happens that ou may need the other 
figure. The Mabs or I formulae can be put into computr code for easier 
passage between the two. 
    Star Deneb, alpha Cygni, is 1.3 magnitude and about 900 parsecs 
away. This is uncertain due to possible filtering by interstellar 
medium along the Milky Way. How much more luminous than Sun is Deneb?  

     Mabs = Mapp - (5 * log(r)) + 5 
          = +1.3 - (5 * log(900)) + 5 
          = +6.3 - 5 *  log(900) 
          = +6.3 - 5 * 2.9542 
          = +6.3 - 14.7712 
          = -8.4710 

This is brilliant! it approximates the brightness of a half Moon.

   L = 10 ^ (-0.4 * ( Mabs - +4.8)) 
     = 10 ^ (-0.4 * (-8.4710 - +4.8)) 
     = 10 ^ (-0.4 * -13.2710) 
     = 10 ^ (5.3085) 
     = 203,400 

r, rounded because of the uncertainty of distance, 200,000 time more 
luminous than the Sun. Deneb is, in fact, among the most luminous 
stars visible in our sky. 
     Mind well the distinction between Mabs and I. Mabs is an 
artificial parameter while luminosity is a part of the radiation 
output of the star. We see this distinction by imagining we are at a 
abase on an exoplanet. Our catalog with absolute magnitude is 
worthless while that with luminosities remains  valid. 
    (Mapp-Mabs) by itself is the 'distance modulus', a function only 
of the target's distance. It is routinely employed for galactic 
studies. Under 2010s methods we can not measure the parallax of 
extragalactic objects. We wok only with distaances in parsec. Distance 
modulus is rough, with values commonly cite to only the whole 

Extrasolar planets 
 ----------------
    With the incandescent interest in extrasolar planets and there 
being, in 2015, some 90 bare-eye planetary stars over the whole 
celestial sphere, a new use for the absolute magnitude equation sprang 
up. When we see in out sky a planetary star, we can ask: 'How bright 
is out Sun in that star's sky?'
    This amounts to figuring out the apparent magnitude of the Sun at 
the star's distance, given the absolute magnitude of the Sun. That's 
+4.8. We shuffle the distance modulus formula into an apparent 
magnitude form 

   Mapp - Mabs = - 5 + 5 * log(r)   

   Mapp = Mabs - 5 + 5 * log(r)   

    +------------------------------+ 
    | APPARENT MAGNITUDE FORMULA   | 
    |                               | 
    | Mapp = Mabs - 5 + 5 * log(r) | 
    +------------------------------+ 

For the specific case of the Sun seen from a planetary star this 
collapses to a very simple form

    Mapp = Mabs - 5 + 5 * log(r) 
         = +4.8 - 5 + 5 * log(r) 
         = -0.2 + 5 * log(r) 
         = 5 * log(r) - 0.2 
    +--------------------------+ 
    | SUN'S APPARENT MAGNITUDE | 
    |                          | 
    | Mapp = 5 * log(r) - 0.2  | 
    +--------------------------+ 

    This is a very simple formula! Remember that r is in parsec, not 
lightyear. 
    By applying his formula to a few planetary stars, we find that our 
Sun would be among the dimmer stars in the planet's sky. This is based 
on human vision, of course. That's because stars in our sky tend more 
to be luminous than the Sun. As example to a star just 20 parsecs away 
the Sun would be a 6.3 magnitude star. This is the utter threshold of 
detection for good eyesight in a dark sky. 
    For example, planetary star Hamal, alpha Arietis, is 20 parsecs 
away. How bright is the Sun in its planet's sky?

    Mapp = 5 * log(r) - 0.2 
         = 5 * log(20) - 0.2 
         = 5 * (1.3010) - 0.2 
         = 6.5050 - 0.2 
         = 6.3050
 
From Hamal's planet our Sun is a 6.3 magnitude star, at the threshold 
of human vision, in the local sky. In Earth's sky Hamal is a 2.0 
magnitude star. The disparity of brightness, 4.3 magnitudes, translates 
into hamal being some 50 times more luminous than the Sun. 
 
 Total magnitude 
 -------------
    When two or more stars are angularly so close that they blend into 
a single point, their separate illuminations add to a total single 
value. This total illumination yields a total magnitude for the set 
of stars. This total is always brighter than the group's brightest 
star. 
    Total magnitude is almost always treated only for double stars, 
where the illuminations of two stars are added.
    In the old days, before calculettes, textbooks commonly had tables 
of two-star magnitude. A variation was a table of magnitude 
difference between the stars versus magnitude increment for the 
brighter one. 
    The summation method applies to any set of close stars, like an 
open cluster, tight conjunction, compact asterism. If by bare eye or 
low power the group merges into a single source, the method works. 
    First, the magnitude of each star is converted into illumination. 
The illuminations are summed. The sum is converted into the total 
magnitude. That may be: 

    +---------------------------------------------------+ 
    | TOTAL MAGNITUDE OF A CLOSE SET OF STARS           | 
    |                                                   | 
    | (tot magn) = -2.5 * log(sum(10 ^ (-0.4 * magnX))) | 
    +---------------------------------------------------+ 

Where magnX is the magnitude of each star in the group. 
    In the Pazmino CLuster the trapezium of four brightest stars gives 
the bulk of the cluster's illumination. Little more is added by the 
decorative dim stars. The trapezium stars are about 7.5, 7.6, 7.7, 
and 7.8 magnitude, varying slightly among authors. What is the total 
magnitude of the Pazmino Cluster? 

 (tot magn) = -2.5 * log((10 ^ (-0.4*7.5)) + ... + 10 ^ (-0.4*7.8)) 
            = -2.5 * log(1.000e-3 + 9.210e-4 + 8.318e-4 + 7.586e-4) 
            = -2.5 * log(3.511e-3)
            = -2.5 * (-2.455) 
            = 6.136 

    This is within the casual estimates from deepsky observers. The 
 magnitude in observing litterature is 6 to 6-1/2.                               
    More than four stars may be best handled by a computer program. It 
asks for the number of stars and then for the magnitude of each in 
turn. It outputs the total magnitude of the group. 
    In the two-star case, many instances can be passed up when the 
magnitude difference is more than 2-1/2 magnitude. The contribution of 
illumination by the dimmer star is less than 0.1 magnitude and the 
total magnitude is substantially that of the brighter component. 

delta Cephei star 
 ---------------
    If we had some way to independently know the absolute 
magnitude of a source and we record its apparent magnitude we can 
immediately find the distance to the source. delta Cephei, also 
Cepheid, stars were first applied to star distances in about 1910. We 
when we found that the absolute magnitude of a Cepheid star is a 
monotonic function of its period of oscillations of brightness. The 
magnitude is the mean between maximum and minimum luminous emission. 
    his is expressed in the Period-Luminosity Relation. The name 
reflects the use of absolute magnitude as luminosity. discovered in 
    delta Cephei stars are very luminous, letting us see them in other 
galaxies. Measuring their period is a matter of monitoring them for a 
few complete cycles of oscillation. Cepheids have a unique profile of 
light variation, not shared by any other kind of variable star. This 
makes it feasible to pick them out from a crowd of other variable 
stars. 
    With the period in hand we read out the absolute magnitude for the 
star. The monitoring also captures the apparent magnitude, also of the 
mean between max and min illumination. 
    We solve the absolute magnitude equation for r and insert the 
known values of Mapp and Mabs 

    Mapp - Mabs = 5 * log(r) - 5 

    Mapp - Mabs + 5 = 5 * log(r) 

   (Mapp - Mabs +  5) / 5 = log(r) 

    +-----------------------------------+ 
    | DELTA CEPHEI DISTANCE FORMULA     | 
    |                                   | 
    | log(r) = (Mapp - Mabs + 5) / 5    | 
    +-----------------------------------+ 

`   zeta Geminorum is a delta Cephei star varying between +3.6 and 
+4.2 magnitude in a 10.148 day period. How far away is the star?  The 
absolute magnitude of a delta Cephei star is either read off of a 
Period-Luminosity graph or calculated from a formula fitted to that 
graph. We use here the formula, which is one of several variations

    Mabs = -2.78 * log(period) - 1.43 
         = -2.78 * log(10.148) - 1.43 
         = -2.78 * (1.0064) - 1.43 
         = -2.7978 - 1.43 
         = -4.2278

    This formula is a curve-fit against a plotted P-L graph and is 
valid only for 'classical', delta Cephei, stars. It does not apply to 
W Virginis or RR Lyrae stars. 
    The Mabs is the average between maximum and minimum 
brilliance. The Mapp of zeta Geminorum is the average of its maximum 
and minimum illumination, O3.6+4.2)/2 = 3.9. Then 

    log(r) =  (Mapp - Mabs + 5) / 5 
           = (+3.9 - -4.2278 + 5) / 5 
           = 13.1218 / 5 
           = 2.6256
          
    r = 422.2706
     -> 422 parsec

 Type-Ia supernova 
 ---------------
    Certain supernvae are members of a binary star. In the course of 
the star's life it siphons from its companion to gradually increase in 
mass. Eventually the mass crosses the Chandrasekar limit, the largest 
mass a star can have before it by its own gravity collapses into a 
supernova.  Because the increase is gradual, it seems that all such 
stars trip into supernova at about the same limiting mass and erupt 
into about the same luminance or absolute magnitude. 
    There are many kinds on supernova but only the Type-Ia has this 
unique property of a uniform peak magnitude. other supernova processes 
generate unpredictable peak brilliance. 
    In addition, the Type-Ia star has a unique spectrum and light 
output profile, This lets us recognize a Type-Ia star if we miss 
catching it at peak emission. We fit the observed profile to ones from 
previous supernovae and read out the apparent magnitude it had at peak 
luminance. 
    This Mabs is -19.3, as best we know in the 2010s. Some astronomers 
suggest there is a leeway, maybe +/- 0.4 magnitude, due to chemical 
composition of the star and binary orbit dynamics. 
    We massage the magnitude-distance formula: 
 
    (Mapp - Mabs + 5) / 5 = log(r) 

    (Mapp - (-19.3) + 5) / 5 = log(r) 

    (Mapp + 19.3 + 5) / 5 = log(r) 

    (Mapp + 24.3) / 5 = log(r) 

    +------------------------------------+ 
    | TYPE-Ia SUPERNOVA DISTANCE FORMULA | 
    |                                    | 
    | log(r) = (Mapp + 24.3) / 5         | 
    +------------------------------------+ 

    So brilliant are these stars that home astronomers can spot them 
in the closer galaxies. It happens commonly that we can not see the 
diffuse patch of the galaxy but only the pinpoint of the star itself. 
With several scores of galaxies within reach of small scopes in New 
York City, we may observe a Type-Ia star once per decade or so. 
    We must apply two major corrections to Mapp. Firs, the star may be 
dimmed by the interstellar medium of the host galaxy. 
    The other is that beyond around 500 million parsecs the spacetime 
distortion effects of Hubble expansion must be considered. 
    For  galaxies which home astronomers can observe, the formula 
given here may be used as is. Hubble expansion is negligible and we 
usually have no data for interstellar dimming. 
    In 2011 a Type I-a supernova erupted in galaxy M101. Its maximum 
apparent magnitude was +10.0. How far off is M82? 

    log(r) = (Mapp + 24.3) / 5 
           = (10.0 + 24.3) / 5 
           = 34.3 / 5 
           = 6.8600

    r = 7,244,360   
     -> 7,200,000 parsec 

Comet magnitude 
 -------------
    We now come to an obsolescent case, predicting the apparent 
magnitude of a comet. It was developed in the 1930s, so far as I know, 
when home astronomers took over much of the comet finding and 
observing work from campus astronomers. 
    In the old days we knew far too little about how comets shine. We 
did suss out that a comet shines by reflected sunlight and self-
luminance induced by solar radiation. The portion for reflected light 
was handled by the Inverse-Square law to factor in the Earth-comet and 
Sun-comet distances. We had no decent model for the induced luminance. 
In spite of this lack, we include the solar-induced light into the 
comet magnitude equation in a simple recognition factor. 
    We had no good theory or model for the extent of a comet's tail or 
coma. We didn't try to add them into the comet's brightness. The 
equation applies only to the comet's head, ignoring extended coma and 
tail. 
    When a comet is discovered it is assigned an absolute magnitude 
and a magnitude gradient, or slope, factor. Both are often guesses 
taken from similarity of the instant comet to previous ones. 
    The absolute magnitude of a comet is the apparent magnitude when 
the comet is in equilateral triangle with Earth and Sun. Each side is 
1 AU. Phase effect is neglected because head radiates in all 
directions with no shadowed side. 
    As the Earth-comet side varies, the comet's magnitude changes 
according as the Inverse-Square Law. 
    When the comet-Sun side varies the comet changes its reflected 
light by the Inverse-Square Law AND ALSO its luminous output induced 
by interaction with solar radiation. This additional light is 
governed by an other power law, not in general an inverse square. 
    What is this other law? With no decent comet model until the 
explorations by spacecraft we could only cut-&-try a value for this 
other power. Choice was taken from experience with prior comets which 
behaved similarly to the instant one. 
    Crashing together these factors and jumping directly into 
logarithms, we have the comet magnitude formula: 

    +---------------------------------------------+ 
    | COMET MAGNITUDE FORMULA                     | 
    |                                             | 
    | Mapp = Mabs + 5 * log(r) + 2.5 * K * log(R) | 
    +---------------------------------------------+ 
  
    r is the comet-Earth distance in AU; R,comet-Sun distance in AU. 
    K is the slope or gradient parameter. It is composed of 2 for the 
reflected light and some other number for the solar-induced light. 
    The symbols differ widely among authors with no longer a strong 
effort for standardize them. One other common statement is  

    m1 = m0 + 5 * log(DELTA) + 2.5 * n * log(r) 

The symbols here line up with those in the boxed equation. The 
equation's circumstant text should explain the symbols in each 
instance. 
    The value of K can range from 2 for a dead worn-out comet to 4 or 
5 for a vigorous active comet jived by the Sun. One practice for 
brand-new comets is to assign K = 4 and hope for the best downrange. 
    Some astronomers combine K and the 2.5 factor into a one 
parameter. In such situations the slope parameter may range from 5 to 
8 or 10. 
    As the comet does its round thru the solar system, its behavior 
may depart form the current equation. News for the comet may contain 
revised values of Mabs and K to keep pace with the comet's current 
activity. In such news the 2.5 and K are commonly combined into a 
single number, ranging from 5 to 10 or 12.5. 
    Since the mid 1990s as we learned more about comets from 
spacecraft visits and better comet models, the comet magnitude 
equation is in a steady decline. Astronomers who today hear abut it  
often must examine old comet litterature for information. 

Conclusion
 --------
    Astronomers, on and off campus, learn of these various magnitude 
formula as separate topics with almost no attempt to correlate them. 
Here we see that the one Inverse-Square Law and the definition of 
'magnitude' are the root of all these formulae. They are simple 
permutations of each other. 
    Because the formulae are part of topics that can be scattered 
along the tuition of astronomy, it may be clumsy to collect them into 
a single lesson. Perhaps near the end of a course, when the separate 
forms are in hand, a summation can be offered..