n

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

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 no 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, 
reflective material.
    A newer consideration is that the geometry of space must remain 
'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 light emission that here are 
discussed as facets of a one single concept, the Inverse-Square Law of 
radiation. I say 'radiation' because by now we can explore the 
universe in just about all wavelengths, or frequency, of 
electromagnetic energy and not just that in the luminous range of the 
spectrum.
  
The concepts 
 ----------
    The concepts I work with here are 
        * Inverse-Square Law of radiation 
        * distance modulus or absolute magnitude 
         * luminosity 
        * extrasolar planets 
        * delta Cephei star 
        * Type-Ia supernova 
        * comet magnitude law 

    These are routinely examined as separate unrelated subjects in 
astronomy with no clue that all are interrelated into a single 
subject, the properties of radiant energy. 

Light 
 ---
    Recall that 'light' is not itself a physical substance. It can not 
be onjectively 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 phyiological response to a limited range of 
wavelength of incident radiation into the eye-brain mechanism. 
radiation within about 360nm and 720nm wavelength escites 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 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 visison process.
    In history, only light was available for exploring the universe 
form Earth. Out atmosphere filtered out large segments of the incoming 
radiation and others simply were never imagined to try looking for. 
The Sun was treated as a source o f'light' and 'heat', altho both were 
part of the one flux of electromagnetic radiation with different 
responces by humans, vision and warmth.
    Other radiations were hinted at in the late 1890s but poorly 
studied for the crude instruments of the day. Tesla and Birkeland were 
amost entirely ignored after their work was announced. It wasn't unti 
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 targerts.
    Full exposure of Earth to other radiation came in the 1960s with 
astrophysical satellites. By the 1980s essentially 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 that silly 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 dilution of the rays 
received decreases as the square of the 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 sphere of what ever radius e choose. And this is the key to 
understanding the inverse-square law. 
    We define irradiation as the radiant energy, or energy per time 
for a 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 illumination I is cited in watt/meter2 at radius r from 
the source. That is, 

    irradiation = radiation / (area of sphere) 
              I = P / (4 * pi * (r ^ 2))

The P/(4*pi) is a constant for all spheres, reducing the equation to 

    P = const1 / r ^ 2 

    I = const2 / 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 shell enclosing the source does not have to a sphee, nor does 
it have to be centered on the source. We use a centered sphere for the 
simplicity of the maths. A more complex analysis is possible for the 
irrgular excentric enclosure. 

Astronomy versus physics
 ----------------------- 
    We could have stopped here is this was an article within ordinary 
physics. The radiant power of the source is in watts, or lumens if 
confined to just luminous emission. The distance, radius of the 
enclsoing sphere, is in meters. The illumination is then is 
watt/meter2 or lumen/meter2. The latter has its own name, the lux, 
cinnibkt ysed in photometry, photography, videography. 
    By history astronomers did not care about the physical measure of 
illumnation. They, as invented by Hipparchus, a scale of 'brightness' 
for the stars. From the Greek era until the mid 1800s this scale was 
informal, baed on eyeball assessments of the stars. Hipparchus scheme 
assigned tank 1 to the brightest stars, about 15 of them. the fainter 
stars, in 'steps' earned ranks 2 thru 6. Since in ancient times the 
brilliance of a star was also its greatness, the ranking was called 
'magnitude'. Pogson in the mid 1800s formalized the scale with a 
logarithmic sequence. At first the ranks of other stars were eyeballed 
against Polaris, or other fundamental standards scattered across the 
sky. In deed, this eyeball ranking is still used by home astronomers 
for monitoring variable stars. The variable is compared to field stars 
with designated magnitude ratings. 
    The logarithm scale of magnitude is

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

    M0 and I0 are the zero-point values for magnitude and 
illumination. The logarithm, here and else where in this piece, are 
for 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 original scale. 
    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 linking to the physics photometry 
system. This 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. 

Newer developments 
 ------    -----
    In the early 20th century 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 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 many decades to shift physicists to a more 
fundamental scheme of photometry. We really didn't get out of 19th 
century photometric mindset until the millennium crossing. 
    On the astronomy side we in the 1930s were establishing major 
observatories in the southern hemisphere, where Polaris is out of the 
local sky. We moved the zero-point to Vega, visible from almost any 
reasonable southern-latitude location. South Pole stations came in the 
late 20th century. 
    An other cause to let go of Polaris was its discovery in about 
1900 as a delta Cephei variable star! Its illumination altered over 
several days by 2/10 of a magnitude. It 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 but we do keep careful watch on it. 
 
Linked at last!
 -------------
    What is the zero-point to initialize the scale. This factor is 
amazingly trough to find in astronomy litterature! It is noted as a 
special applicaton of photometry in physics works.                    
    When the current International System of the metric system 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 optical or visual band of the spectrum. 

    +---------------------------------------+ 
    | MAGNITUDE-ILLUMINATION                | 
    |                                       | 
    | magnitude 0.0 = 2.56e-6 lumen/meter2  | 
    +---------------------------------------+

    The first statement of this equivalence, in the 1950s conditioned 
it to prevail under clean dry air. The intent was to minimize  
intervening obscuration of starlight thru the atmosphere. Today, in 
the Space Age, the equivalence applies above the atmosphere.A spinoff 
of photometry in the Space Age is an improved model of atmospheric 
attenuation of starlight across the spectrum. 
    This equivalence seems still uncertain because in current 
litterature I see small variations. I suspect there may be 
discrepancies in the eye's spectrosensitivity function employed to 
derive this equivalence. 

Curious features 
 -------------
    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, aurorae, 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 source and convert it to a magnitude value. The result is 
magnitude/arcmin2 or magnitude/arcsec2. This was tried in the mid 20th 
century when home astronomers started to observe deepsky objects with 
large-aperture telescopes. 
    With no standard scheme of measuring  angular magnitudes, authors 
of deepsky litterature commonly concocted their own rankings. The 
observing litterature was filled with widely discordant magnitude 
values for each target. Such dispersion of values made it real tough 
for home astronomers to assess the chances of seeing the target. 
Often it was a matter of luck and fate. 
    An other feature of magnitude is that it is specificly defined 
only for luminous radiation. For the most part home astronomers 
observe only by light but equipment is slowly 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. 
    In actuality the concept is nonsense because the entire premise of 
magnitude is to rank illumination as perceived by human eye. For 
radiation beyond human perception the notion has no meaning. Radio 
astronomers and high-energy astronomers do not attach a magnitude 
system to their radiations. 

Parsec
 ----
    The parsec is a unit of distance, based on the parallax 
measurement of the target. Parallax is cited in arcseconds, the swing 
of our line of sight to the target as Earth orbits the Sun. Or it is 
the angular radius of Earth's orbit as seen from the target. 
    Since stars are really awfully far away, requiring their light 
to take 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 long slender triangle with the Earth orbit radius as base 
and a given parallax at the apex has a definite length which can be in 
itself a new unit of distance. 
    This length is 206,265 AU with the name 'parsec, from PARallax-
SECond. For quick work we can round this to 200,000AU and 5 parsec = 1 
millioon AU. 
    In terms of lightyears, one parsec is 3.26 lightyears. Many 
astronomers simply use parsecs without the switch 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 = !/(parallax angle) | 
    +--------------------------------------+ 







Comaparing 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 nboth. 

    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) / ( 
           = 1 / (r ^ 2) / (r0 ^ 2) 
           = (r0 ^ 2) / (r ^ 2) 

e received illumination from the equal soueces is the inverse square 
ratio of their distances. We know the standard distance r0 and we 
solve for the unkown one 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 magnitude the star would appear to 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 choices would 
have been 'normalized magnitude' or 'reduced magnitude'. There is 
nothing 'absolute' about it 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 accumulate 
databases of parallaxes and apparent magnitudes 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 / P) / (r ^ 2 / r0 ^ 2) 

I left out the 4*pi factor since it immediately cancels out in the 
denominator. 
 
    I / I0 = (1) / (r ^ 2 / r0 ^ 2) 

    I / I0 = (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) 

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

You should refresh yourself in log operations to understand the 
conversion from algebra to logs.
    We now apply the definition of magnitude. 
 
    log(I) - log(I0) = 2 * (log(r0) - log(r)) 

    -2.5 * log(I) - (-2.5 * 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. Mabs is the artificial, absolute, magnitude of the star if 
it was 10pc away. 

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

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

Distance modulus 
 -------------- 
    The Mapp-Mabs is the 'distance maodulus', symbol Greek mu. I never 
saw the practical value of this parameter except for guess-&-golly 
estimates of distance. It's used for galaxy distances with the 
unspoken notion that Mabs is the mgnitude the galaxy would appear if 
it was moved to 10 parsecs away. I wonder how big that galaxy is to 
remain condensed into a starlike point form so close. 
    We solve for Mabs 

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

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

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

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

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

    Provided that we know both the star's apparent magnitude and the 
istance, we can calculate the star's absolute magnitude. 
    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 away it would be among the mediiocre s    
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.

Caution!
 ------
    The formulae above are sometimes stated in a subtilely different 
form but are really the same equations. Some astronomers want to keep 
the observed data, parallax, without taking the inverse for patsec. 
Recalling that parsec = 1/parallax and the behavior of logarithms, we 
can rewrite the absolute magnitude equation 

    +-------------------------------+ 
    | ABSOLUTE MAGNITUDE DORMULA    | 
    |                               | 
    |  Mabs = Mapp + 5 + 5 * log(p) | 
    +-------------------------------+ 

where p is the parallax angle. The flip of signum is crucial! 
Newcomers to the profession, and a few seasoned astronomers, mix this 
up. I stay with distance r, not further working with parallax p. 
    An other mistake is to use lightyears, not parsecs. In both cases 
all downstram work is corrupted. 

Luminosity
 --------
    Absolute magnitude is founded on human arbitration, a practice 
deprecated generally in science. It would be far better to give the 
luminous output of a star by some property not dependent on humans. We 
In fact have such a property, luminosity, the luminous output in terms 
of the Sun's as a unit.
    Luminosity is sometimes explained by saying that some number of 
Suns rolled together equal the light of the star. This is not really 
correct, since only the Suns on the exterior surface of the ball of 
Sun would send light to us. The Suns inside would be blocked from view 
by the Suns above them. A more correct simulation is for the Suns to 
be laid over an armature large enough to be completely covered by 
them, like beads or tiles. Then all the Suns send radiation outward 
into space.
    The luminosity of the target is a straight derivation from the 
definition of magnitude 

        M - M0 = -2.5 * log(P / P0) 

M is the absolute magnitude of the target. MM0  is that of Sun, +4.8. 
P is the power of the target and P0 is that of Sun. P0 is set to 
unity. 

        (M - M0 / -2.5 = log(P / P0) 

        (M - (+4.8)) / -2.5 = log(P / 1) 

        (M - 4.8) / -2.5 = log(P) 

        (-0.4) * (M - 4.8) = log(P) 

        +-----------------------------+ 
        | LUMINOSITY FORMULA          | 
        |                             | 
        | (-0.4) * (M - 4.8) = log(P) |
        +-----------------------------+ 

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) 

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

    You may calculate this for the set of planetary stars you show to 
the public or newer astronomers.

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 fix the distance to the source. delta Cephei, also 
Cepheid, stars were first applied to star distances in about 1910. We 
before then found that the absolute magnitude of a Cepheid star is a 
monotonic function of its period of oscillations of brightness. This 
is expressed graphicly as the Period-Luminosity Relation. This 
relation was discovered in the 19-ohs. 
    delta Cephei stars are very luminous. We see them in other 
galaxies. Measuring their period is a matter of monitoring them for a 
couple weeks to capture 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. Since the star's radiation changes with time, we use either the 
peak emission or the mean of peak to valley in the oscillations. 
    While examining the star's apparent magnitude we may have to Maybe 
correct it for interstellar medium dimming. this can come from both 
from the Milky Way and the star's galaxy.
    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     | 
    |                                   | 
    | (Mapp - Mabs + 5) / 5 = log(r)    | 
    +-----------------------------------+ 

    A different distance modulus applies to each delta Cephei star 
because they have different absolute magnitudes according as their 
periods. 

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. If the supernova erupted 10 parsecs away it 
would shine like 1,000 full Moons! Some astronomers suggest there is a 
leeway, maybe +/- 0.4 magnitude, die 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 | 
    |                                     | 
    | (Mapp + 24.3) / 5 = log(r)          | 
    +------------------------------------+ 

    The immense Mabs makes the star visible at immense distances from 
us. We use the Type-Ia method to map out the farthest realms of the 
universe, finding them in very early galaxies close to the Bigbang 
event. Type-Ia supernovae occur once per century in a galaxy. With so 
many galaxies, we have always a good sample of stars to do a dense 
mapping. 
    So brilliant are these stars that home astronomers can sometimes 
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.  First is that the 
star may be dimmed by the interstellar medium of the host galaxy. 
    The other is that beyond 100 Mpc the spacetime distorion effects 
of Hubble expansion must be considered. For  galaxies ehich home 
astronomers can observe, the formula given here may be used as is. 







Comet magnitude 
 -------------
    We now come to an obsolescent case, predicting the apparent 
magnitude of a comet. It was developed in the 1930s, so far 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 srlf-
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 limonance. 
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 never tried to add their brightness into the equation. 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 if the 
comet is in equilateral triangle with Earth and Sun. Each side is 1 
AU. Phase effect is ignored as if the head is fully lighted. 
    As the Earth-comet side varies, the comet's magnitude changes 
according as the Inverse-Square Law. We ignore phase effects as the 
angular elongation from the Sun also changes. 
    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 light and heat. This additional light is 
govenrned by an other power law. 
    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 other comets whic 
behave similarly to the instant one. 
    Crashing 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 is the 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 
accompanying 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. 
    As the comet does it 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, this 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 formula 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.