INVERSE-SQUARE LAW AND MAGNITUDE FORMULAE
-----------------------------------
John Pazmino
NYSkies Astronomy Inc
nyskies@nyskies.org
www.nyskies.org
2015 December 3 initial
2020 April 25 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))
I :: 1 / (r ^ 2)
And that is it!
We derived the Inverse-Square Law in a simple,
e legant, 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 power, 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-candle. Recall that for all intents and
purposes one foot-candle equals ten lux. Or, in essences, the 'metric
foot-candle' is 10 lux. The foot-candle had an oldstyle definition but
almost universally we slided it into the metric system as being 10
lumen/meter2.
In any rectangle room at home place a bare-bulb lamp in any
convenient spot, like an end table. it does not have to be centered in
the room but in a place where you can work all around it.
Measure the dimensions of the room and work out its surface area,
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 closed. This lessens the
influx of lighting from outside.
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 have to do the explicit
multiply for fractional square-meter sections of the room.
The sum approximates 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.
Limits of ISL
-----------
ISL is defined for point sources. In the experiment with the lamp and
room, the lamp is not a point source. If the room is several meters
long and wide from any point n its surface the lamp is a small annular
extended source. Is ISL valid for such sources?
There are no truly point sources, not even in labs devoted to
photometry. A blackhole is a geometrical point but it does not
radiate. Radiation associated with a blackhole comes from outside its
event horizon, which is not a point source.
When a source has angular extent as seen at the enclosing surface,
light rays from the source arrive at sthe receiving surface in
overlapping radial paths. Each point of the source overlays its own
ISL effect on the surface.
the inverse-square law is a limiting situation. As a source
appears angularly smaller, by shrinking or diaphragm or distance, is
approximates a point and ISL becomes a better means of assessing the
illumination fro it.
As the source enlarges angularly, ISL becomes a worse and worse
illumination method. How large can a source be to throw doubt on use
of the inverse-square law?
It by good luck for astronomy, celestial light sources are overall
angularly so small that they can be treated as point sources without
sensible error. Stars, planets, galaxies, most nebulae and clusters,
even the very Sun and Moon, are point sources for use of the inverse-
square law.
Large targets like huge comets, the Nubeculae, Milky Way band,
aurorae, clouds, twilight glow produce erroneous results under ISL.
We look at two examples of extended light sources, a long linear
source like a fluorescent or neon tube and a flat area like panel-
lucent ceiling or open ground.
The linear source is of indefinite length to avoid complications
from the ends. We surround a unit length of the line with a cylinder,
of radius r and centered on the line. This is for geometric simplicity
like the sphere for a point sources. Light rays flow from the line to
illuminate the cylinder.
The illuminated area is the circumference of the cylinder times
the unit length along the linear source. We have
I - P / (2 * pi * r)
I :: 1 / r
From a linear source the light falls off as the inverse radius, NOT
radius2.
For a planar source, taking a unitt area. rays flow orthongally to
the plane. The plane is idefinitely large to avoid vomplication from
the edges.
We surround the unit area of the source with a flat surface
parallel to the source plane, again for geometric ease. Lumens arrive
at the receiving surface in parallel streams with no divergence . The
surface at all distances from the source get the same illumination
I = P / (unit area)
. I - constant
An experiment to try is taking illumination readings of the ground
from various elevations above it. a cloudy day is best because the
ground is then radiating an even light with no shadows and highlights.
Elevation is easiest obtained by observing from high-rise
buildings, aerial highways, elevated transit lines. Other options are
a ride in a balloon or glider.
The readings should all be the same for all elevations.
Apparent magnitude
---------------
By history astronomers did not apply 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 class, tier, 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 in
Latin 'magnitudo', 'magnitude' in English. A star of Hipparchus rank
3 was a star of magnitude 3.
Until the 19th century magnitude value of a stars was sometimes
erraticly assigned. For telescopic stars, with no prior history of
magnitude ranking, the scale diverged severely among observers.
Pogson in the mid 1800s formalized the scale with a logarithmic
sequence. He measured by crude photometers that stars five magnitude
steps apart were almost 100 times 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.
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 algebraic value for fainter
stars, matching the trend of the Hipparchus scale.
Newcomers sometimes find this relation confusing because a
brighter star 'should' have a greater value. Think of magnitude as a
rank or class in a social hierarchy. The lower-level members have a
higher rank number. The important 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 number. The highest class,
rank, tier, is #1.
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 greater
resolution of magnitude assessment in an objective verifiable 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 19th century a
complete electrophotometry of the bare-eye stars was 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 1st magnitude stars were
too bright for that rank. They were given the extended 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.
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.
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 |
| |
| uMapp(0.0) = 2.5351e-6 lumen/meter2 |
| |
| Mapp = -13.99 + (-2.5 * log(Illum)) |
+-------------------------------------+
The second equation can be solved for Illum if given an apparent
magnitude. For an illumination of 1 lumen/meter2 the apparent
magnitude is -13.99. This is close to that of the full Moon, which
shines about 1/2 lux onto the ground when overhead.
The value cited here comes from a review of observations from
space and high-elevation stations, to remove atmospheric effects and
span more of the star's spectral range. an older value is still in
circulation, Maap(0.0) = 2.56e-6 lumen/meter2. This is within 1/2% of
the current value and can be used without sesnisble error for most
astronomy photometry. it was developed from ground stations with
correction and factors for atmosphere ans spectral constraint.
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 lumen/(meter2.arcmin2) or lumen/(meter2.arcsec2).
This is next converted into magn/arcmin2 to magn/arcsec2, the very
angular magnitude of the target.
Be careful. The total illumination is NOT first turned to
magnitude 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 produce insane values for the 'magnitude'
of a given target. Vega, for example raidates weakly in the far
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
from 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 unit of distance, 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 = 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 realm.
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(L) + 4.8 |
+----------------------------------------+
Star catalogs generally list either absolute magnitude or
luminosity for its stars. It happens that you may need the other
figure. The Mabs or L formulae can be put into computer 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 work only with distances in parsec. Distance
modulus is rough, with values commonly cited to only the whole or one-
half magnitude.
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 |
| |
| Maapp= 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 more luminous than the Sun.
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 as a smaity check for the maths.
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 here applies to any set of close stars, like
a small open cluster, tight conjunction, compact asterism. If by bare
eye or low power the group merges into a single source, the method
works.
This method works for globular clusters and galaxies. The
number of stars is then so huge to make its use impractical.
First, the magnitude of each star is converted into illumination.
The illuminations are summed. The sum is converted into a new
magnitude, the total magnitude.
A simplification is that the shift from magnitude to illumination
is based on magnitude 0.0. It skips going to actual lumen/m2. A star
of magnitude +6.0 converts to 0.01, for being 1/100 the illumination
from a 0.0 star. The conversion to total magnitude also skips actual
illumination. units.
That may be:
+---------------------------------------------+
| TOTAL MAGNITUDE OF A CLOSE SET OF STARS |
| |
| Mtot = -2.5 * log(sum(alg(-0.4 * magnX))) |
+---------------------------------------------+
where magnX is the magnitude of each star in the group. alg is the
inverse log function, alg(N) = 10^(N).
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?
Mtot = -2.5 * log((alg(-0.4*7.5)) + ... + alg (-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 special case of a double star it is easier to put the
brighter star as unit illumination. First take the difference in
magnitude m magn(dimmer)-magn(brighter). This is a positive number.
Then do the magn-to-illum conversion on this difference. Add '1',
the brighter star's illumination,to sum the illumination of both
stars.
Finally convert the sumed illumination to total magnitude. This is
always brighter than the brighter star.
+-----------------------------------------+
| TOTAL MAGNITUDE FOR A DOUBLE STAR |
| |
| Mtot = -2.5 * log(1 + alg(-0.4 * Mdiff)) |
+------------------------------------------+
Many instances can be passed up when the magnitude difference is
more than 2-1/2. The contribution of illumination by the dimmer star
is less than 0.1 magnitude. The total magnitude is substantially that
of the brighter component.
delta Cephei star
---------------
delta Cephei, also Cepheid, stars were first applied to star
distances in about 1915. 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.
This is expressed in the Period-Luminosity Relation. The name
reflects the use of absolute magnitude as luminosity.
delta Cephei stars by good fortune are very luminous, letting us
see them in other galaxies. They wee the first means of mapping the
universe beyond our Milky way.
Measuring their period is a matter of monitoring them for a few
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 kinds of variable
star.
With the period in hand we read out the absolute magnitude for the
Cepheid 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
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 in the litterature.
+-----------------------------------+
| CEPHEID PERIOD-MAGNITUDE FORMULA |
| only for classical epheid star |
| |
| Mabs = -2.78 * log(period) - 1.43 |
+------------------------------------+
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 Gaia astrometric spaceprobe in 2018 determined a distance to
Polaris, the nearest Cepheid, as 131.1 parsecs. This is within the
range previously assessed for calibrating the P-L relation. It matches
the P-M formula here.
The Mapp of zeta Geminorum for Cepheid work is the average of its
maximum and minimum illumination, (3.6+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
Main-Sequence fitting
------------------
In the Hertzsprung-Russell Diagram stars that shine by the
hydrogen-helium energy process align along a narrow band, the Main
Sequence. A star on the MS has a specific absolute magnitude and
spectral class corresponding to its mass.
The laws of nature work the same every where such that in an
aggregate of stars, like a cluster or galaxy, the MS is the same as
that for nearby stars.
Unless we previously know the distance to the cluster we can not
plot its stars on an HRD by their absolute magnitude. ,
In the stead we plot the stars by their apparent magnitude. The
cluster's MS is displaced verticly from the MS of a standard HRD.
This displacement is measured as (Mapp - Mabs) at a given spectral
class. Values are taken from several points along the two MS curves
and an average is worked up.
This magnitude displacement gives directly the distance of the
cluster.
Note that the target's MS is always fainter than, graphicly under,
the standard MS. Else the cluster would be close enough for direct
distance measurement.
+--------------------------------+
| MAIN-SEQUENCE FITTING FORMULa |
| |
| log(r) = (Mapp - Mabs + 5) / 5 |
+--------------------------------+
For an example, the sigma Orionis cluster has a MS shifted 8.2
magnitude fainter than the standard MS. (Mapp - Mabs) = +8.2. The
separate Mapp and Mabs aren't needed because the shift is scaled
directly off of the HRD in magnitude units.
log(r) = (Mapp - Mabs + 5) / 5
= ((Mapp - Mabs) + 5) / 5
= (+8.2 + 5) / 5
= 13.2 / 5
= 2.6400
r = 436.5 parsec
A collateral method is applied to single stars, not part of a
cohaerent group. If the star can be spectrometricly placed on the
standard Main Sequence, its absolute magnitude is read out. This is
subtracted from the star's apparent magnitude to get (Mapp - Mabs).
The above formula yields the star's distance. This method is commonly
called spectrometric distance0or spectrometric parallax.
Spectrometric distance is weak for stars off of the Main Sequence.
Such stars do not have unique plots on the HRD, and a distance modulus
can not be confidently calculated.
Type-Ia supernova
---------------
Certain supernvae are members of a binary star. In the course of
the star's life it pulls off 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:
log(r) = (Mapp - Mabs + 5) / 5
= (Mapp - (-19.3) + 5) / 5
= (Mapp + 19.3 + 5) / 5
= (Mapp + 24.3) / 5
+------------------------------------+
| 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. First, 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 Ia supernova erupted in galaxy M101. Its maximum
apparent magnitude was +10.0. How far off is M101?
log(r) = (Mapp + 24.3) / 5
= (10.0 + 24.3) / 5
= 34.3 / 5
= 6.8600
r = 7,244,360
-> 7,200,000 parsec
Asteroid magnitude
----------------
Home astronomers observe asteroids almost exclusively when they
are near opposition. They are then closest to Earth and present a
full-phase disc to us. These ae the traditional asteroids in the Main
Belt region of the solar system between Mars and Jupiter.they follow
planetary orbits of greater rexcentricity and inclination than
planets.
Specifications for an asteroid include an 'absolute magnitude'.
This is the apparent magnitude of the asteroid when in equilateral
triangle formation with Earth and Sun and assumed to be in full phase.
The sides of this triangle are each 1 AU.
The geometry is not a plausible one for a Main Belt asteroid,
which typicly stays at least 2-1/2 AU from Sun. Even for a highly
excentric asteroid such as an Apollo class asteroid, the phase in
triangular arrangement is not full at all, but gibbous.
But this artificial absolute magnitude does allow us to work out
the expected magnitude near opposition.
An asteroid shines only by reflected light from the Sun.
Illumination from the Sun is governed by the ISL and the asteroid's
distance from Sun, R. The asteroid's illumination on Earth is an ISL
function of its distance from us, r. The magnitude formula becomes
Mapp = Mabs + 5 * log(r) + 5 * log(R)
Note well the final illumination on Earth is produced by two ISL
applications, first from Sun to asteroid and then asteroid to Earth.
This formula fails dismally for an asteroid away from opposition, it
then has less than full disc illumination. There is no consistent
practice to factor in phase, as a percent of full-disc, into a
magnitude estimate for asteroid far from opposition.
+---------------------------------------+
| ASTEROID MAGNITUDE FORMULA |
| valid only near opposition |
| |
| Mapp = Mabs + 5 * log(r) + 5 * log(R) |
+---------------------------------------+
r is the asteroid-Earth distance in AU; R, asteroid-Sun. Phase is
ignored here because we are examining an asteroid near opposition when
it has full disc illumination from the Sun.
When an asteroid is discovered its parameters are determined, such
as Mabs.
A special case is a near_Earth asteroid flying by near
A special case is an asteroid during a close flyby of Earth. it the
proximity is really close, a few hundredth AU, the R, solar distance,
is close to 1 AU. The magnitude formula reduces to
+----------------------------------+
| ASTEROID FLYBY MAGNITUDE FORMULA |
| valid for poopostion flybys |
| |
| Mapp = Mabs + 5 * log(r) |
+----------------------------------+
This assumes full phase, like when proximity occurs on the outward
side of Earth, with asteroid in the midnight sky. For proximity any
where else around Earth, this formula fails terribly.
We estimate the apparent magnitude of asteroid 1998-OR2 that flues
past Earth in April 2020. proximity is on the 29th at 0.04 AU. By luck
this takes place near the asteroid's opposition, with assumed full
disc. the absolute magnitude from the asteroid's specs is +15.8
Mapp = Mans + f * log(r)
= +1538 + 5 * log(0.04)
= +15.8 + 5 * (-1.3979)
= +15.8 + (-6.9897)
= +8.8103
-> +8.8
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.
This is 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
want, we include the solar-induced light into the comet magnitude
prediction 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 the 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 ISL, To this is added to its luminous output induced by
interaction with solar radiation. This additional light is governed by
an other power function, not in general an inverse square.
What is this other factor? With no decent comet model until the
explorations by spacecraft we could only cut-&-try a value for this
other factor. The factor was taken from experience with prior comets
which behaved like 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, 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 to 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 ranges from 2 for a dead worn-out comet to 5 or 6
for a vigorous active comet jived by the Sun. For K = 2 the comet
formula collapses to the asteroid formula because a dead comet acting
like an asteroid. One practice for brand-new comets is to assign K = 4
and hope for the best as the comet prcedes 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
10 or 12.
As the comet does its round thru the solar system, its behavior
may depart form the current equation. Observations of the comet may
call for revised values of Mabs and K to keep pace with the comet's
current activity. Mabs and K are not stable parameters during a
comet's apparition.
Since the mid 1990s as we learned more about comets from
spacecraft visits and better comet models, the comet magnitude
equation is in a slow obsolescence. it is still in wide use, like for
comet-tracking software.
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
…long 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..