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LIFE OF STARS ON THE MAIN SEQUENCE
--------------------------------
John Pazmino
NYSkies Astronomy Inc
nyskies@nyskies.org
www.nyskies.org
2017 January 13

Introduction
----------
The NYSkies Astronomy Seminar on 2017 February 3 discusses
features of the Hertzsprung-Russell diagram. From the complexity of
this diagram, the session handles only the Main Sequence section.
Other parts of the diagram are handled in future Seminar sessions.
As I was assembling material for this session during the Civil
Rights Day weekend I found that functions of the Main Sequence are
treated in dispersed domains of astronomy litterature. Home
astronomers often learn of these functions piece-by-piece. The
interconnection of the many  Main Sequence features may be missed in
the usual tuition of home astronomy.
With most of the Seminar items in hand, to be compiled into the
takeaways, I consolidate them here in a useful  handy reference for
home astronomers.

Supplemental  HR diagram
----------------------
As a supplement for this article, you should have to hand an HR
diagram with its axes and major sections labeled. Any good astronomy
text, in either paper or digital form, will do. Stay away from
diagrams of a gaudy or cartoon style.
Is there a 'standard' or 'official' HR diagram?
No, there isn't. Each author builds his own, some with lots of
symbols and color, others with plain lettering and lines. Even the
axis scales may vary,, according as the author's taste.
The diagram you need should have for the vertical axis both
'absolute magnitude' and 'luminosity'. Usually one is dimensioned on
the left edge; the other, right. The horizontal scales should display
the 'spectral class' and 'temperature'. A extra bonus is a scale for
'color index', often called 'B -V'. Usually one scale is at the bottom
edge; the other, top.
It's helpful but not required that prominent stars in the sky are
plotted. Certainly the Sun should be marked.

-----------------
One of the exercises home astronomers work thru in their
upbringing is building a Hertzsprung-Russell diagram for themselfs. In
the old days, before computers, this was done with graph paper and a
printed catalog of stars. The usual catalog was that in the RASC
Observer's Handbook, covering about 350 stars down to +3-1/2
magnitude. An other available catalog was that in the USNO
Astronomical  Ephemeris and Nautical Almanac. It covers stars to 5th
magnitude.
To fill in the bottom of the HR diagram, additional stars were
plotted from the Handbook's table of nearest stars. These stars are
for the most part too faint for the main catalog.
When computers diffused thru home astronomy, HR diagrams were made
with digital catalogs of stars, like extracted from planetarium
software, and a spreadsheet program. After a bit of arrangement of the
catalog data in the spreadsheet, the HR diagram was generated thru the
The catalogs routinely gave the absolute magnitude and spectral
class of the stars, ready to plot on the diagram. A few gave
luminosity but in early works this was discordant from the absolute
magnitude. Some of us did a sanity check for several stars using the
formulae for absolute magnitude and luminosity.

Recent developments
----------   ----
could carry you for the rest of your astronomy career far better than
many  published ones. There was one disruption for elder astronomers
with diagrams constructed before the mid 1990s. In thee late 1990s the
HIPPARCOS astrometric spaceprobe produced a new catalog of stars with
new absolute magnitudes. These derived from the greatly improved
distances the craft obtained for the stars. The HIPPARCOS data was
issued in digital files thru the Web. By then most home astronomers
had computers and softwares to generate an updated HR diagram.
We now wait for 2018 when catalogs from the Gaia spacecraft are
issued. The extra facility is that the HR diagram may be generated on
screen thru Web applications with deep customizing features. While
there may be little improvement in the brighter stars, there can by
substantial revision of data for the faint  stars. These are thousands
of lightyears away, out to the interstellar fag limit, beyond reach of
HIPPARCOS.. It could be feasible from Gaia data to build an HR diagram
for an defined block of deep space around our spiral arm, and not be
constrained to a few hundred lightyears around the Sun.

Stars
----
Stars, real ones and not merely 'celestial orbs' are globes of
some 3/4 by mass of hydrogen, 1/4 of helium, and a couple percent of
all of the other chemical elements. By convention, elements other than
hydrogen and helium are called 'metals' and 'heavies'. Of course, many
of these elements are not anything like metal, durable shiny,and all
that. Many are not at all heavy, with large nuclei. Occasionally these
other elements are given as a group the made-up chemical symbol 'Hv'
and their fractional content in a star is called 'metallicity'.
The metals are already mixed into the star during its formation.
They were embedded in the parent nebula, put there by previous
generations of stars. They were created in older stars by methods we
pass over here but involve a star's activity after it runs its life on
the Main Sequence.
In the Main Sequence life no more heavies are made, only some helium
from the fusing of hydrogen. Nuclear processes, like the carbon cycle
employ preexisting metals in the star, not new atoms created in it.
For home astronomers mastering the hydrogen-burning process, with
peripheral knowledge of nuclear physics, is a massive adjunct in the
future astronomy career. Of special importance here is that the entire
function of the Main Sequence in the HR diagram flows recta mente from
the hydrogen fusion energy production.

Main Sequence
-----------
The bulk of stars in a sample plot in the HR diagram on a diagonal
band from lower right to upper left. This band is the Main Sequence.
This distribution is characteristic of a random set of stars in space,
such as a block of the Milky Way. Besides the domination of the Main
Sequence, there can be stars plotted else where in the diagram but
they are usually a small fraction of the stars in the sample.
It's possible that a given sample of stars has a small, thin, weak
Main Sequence, due to peculiar circumstances of the sample. We are
alerted to such conditions recta mente from the thin Main Sequence.
This sample deserves closer attention.
From the first presentation of the HR diagram in the 1910s  until
the eve of World War II the Main  Sequence was a mysterious feature.
It could not be explained before we learned how stars shine. In the
late 1930s nuclear physics advanced to suggest that stars produce
energy by a nuclear fusion process.
When the War broke out all nuclear work was impounded into
military projects. After the War, application of revived nuclear
physics showed that the Main Sequence is the locus of stars generating
energy by fusing hydrogen into helium. A star in the hydrogen fusion
process must settle in luminosity  and temperature only on the Main
Sequence, no where else in the HR diagram.
Stars off of the Main Sequence produce energy by nuclear process
other than hydrogen burning. These stars and energy methods are
The dominance of stars in the Main Sequence comes from the greater
fraction of their life spent there. A solar-like star spends about 9
billion years on the Main Sequence and only 1 billion in other parts
of the HR diagram. As a good approximation, the life of a star is the
time it resides on the Main Sequence, plus a fluff-factor of 10%.

Hydrogen to helium
----------------
I give here a 'black box' explanation of how hydrogen in a MS star
is converted into helium. There are several sets of nuclear reaction
that end up burning hydrogen into helium, according as the starr's
mass and portion of heavies in the star. Some sets of reaction include
nuclei of heavies, already in the star from the parent nebula.
All the reactions, with the details hidden inside the black box,
take in four hydrogen nuclei, protons, and let out one helium nucleus,
an alpha particle. In the process radiant energy and neutrinos are
emitted. These leave the star, escaping into space.
The radiant energy is electromagnetic waves, photons, some of
which on the way to Earth are detected by humans as light and heat.
The neutrinos are almost inactive against matter, passing thru the
overlying bulk of the Sun. They weren't positively detected until the
1970s.
The hydrogen-to-helium burning is

+------------------+
p + p + p + p -->| internal details |--> He + photons + neutrinos
| of black box     |
+-------------------+

The radiant energy comes from the conversion of some mass of the
ingredient protons via the Einstein equation, E=mc2. Four protons have
a mass slightly more than the one helium.  The difference shows up as
emitted energy equivalent to the mass difference.

mass of 4 protons = (4) * (1.673e-27kg) = 6.692e-27kg
mass of 1 alpha or helium               = 6.645e-27kg
-----------
difference, released as energy          = 0.047e-27kg, ~0.7%

energy emitted = (0.047e-27kg) * (3e8m/s) ^ (2)
=  4.23e-12 joule

How does this work for the Sun, a typical MS star? The radiance of
the Sun is some 3.90e26joule/second as assessed from measurements from
ground labs and from geophysical satellites.. The number of reactions

(3.90e26j/s) / (4.23e-12 j/reaction) = (9.22e37 reaction/s)

The mass of these reactions lost as radiation is

(Sun mass loss) = (9.22e37 reaction/s) * (0.047e-27kg/reaction)
= (4.33e9kg/s) -> ~4-1/3 million ton/second

A sanity check is obtained from the radiance of the Sun, coming
from mass conversion thru the Einstein equation. This was probably
realized by astronomers when in the 1920s they tried Einstein physics
and just could not conceive of any process to produce such immense
energy output. Nuclear physics was still too crude to apply to the
stars. The incredible loss of mass was a new mystery that wasn't
solved until the late 1930s and elaborated after world War II. That is

(Sun mass loss) = (3.90e26j/s) / (3e8m/s) ^ (2)
= 4.33e9kg/s -> ~4-1/3 million ton/s

Wouldn't such an immense loss of mass be noticed over the couple
centuries we studied the Sun? Perhaps by mutation of Earth's orbit as
the solar gravity weakens? This loss is

mass of Sun     = (1.99e30kg)
Sun's mass loss = (4.33e9kg/s)
/1/    ratio        = (2.18e-21 part/second) -> 6.87e-12 part/century

This is beyond detection, even with the careful monitoring of planet
orbits since the 1700s.
How long can the Sun last? Nuclear reactions occur only in the
core of the Sun such that over its entire lifespan only 10%, at most,
of all the constituent hydrogen is consumed. In addition, the Sun is
only about 80% hydrogen, the rest being native helium and other
elements from the parent nebula. And of all the py available hydrogen,
only 0.7% is actually lost mass. The life of the Sun on the Main
Sequence, estimated from these considerations, is

(Sun MS life) = (0.1) * (0.8) * (0.007) * (1.99e30kg)
/ (4.33e9kg/s)
=  2.58r17s -> 8.2 million years

I collect here as a handy reference some parameters for the H-He
process and the Sun

+-------------------------------------------+
| SOME SOLAR DATA FOR MAIN  SEQUENCE        |
|                                           |
| mass of Sun = 1.999e30kg                  |
| radiance of Sun = 3.90e26j/s              |
| fraction of hydrogen in Sun = 80%         |
| fraction hydrogem in H-He process = 10%   |
| fraction hydrogen turned to helium = 0.7% |
| mass of proton = 1.67e-27kg               |
| mass of alpha, helium = 6.645e-27kg       |
| mass loss per H-He = 0.047e-27kg          |
| energy per H-He =  4.23e-12 joule         |
| number of H-He per second = 9.22e37       |
| mass loss of Sun = 4.33e9kg/s            |
| estimated Sun's MS lifespan = 8.2e9 yr   |
+-------------------------------------------+-

Luminosity
---------
Everything in the HR diagram concerns only the luminous or
optical or visual part of the star's radiation output. It wasn't until
the 1950s that we could explore stellar radiation beyond the visual
range of the spectrum. Before then, all information about a star was
collected only from its light output.
Some attempts were made to expand the HR diagram to include all
wavelengths emitted by stars, bring that we realized the full output
was a blackbody profile. These efforts didn't catch on.
A middle expansion was to apply a fluff-factor to the light output
to fill out the entire Planck curve. This is the 'bolometric
correction', BC, expressed as a magnitude. It always increases the
'brightness' of the star because it adds in radiation not visible as
light.
On the whole stars emit radiation with a blacckbody or Planck
distribution which is a function of temperature. For most stars this
Planck distribution happens to contain most of its energy within the
visual range of wavelengths.
Very cool stars emit substantial energy in the infrared with a
lesser portion sent out as light. Yet they plot on the HR diagram for
only their visual output.
Luminosity is cited in solar units or the equivalent in absolute
magnitude.  For most purposes the Sun's absolute magnitude is +4.8 or
even just +5.  A quick method of translating between luminosity and
absolute magnitude is to recall that -5 magnitude difference equals a
+100 ratio of luminosity and that -2.5 magnitude equals +10 ratio.

------------
The various laws of radiation in physics typicly apply to the
entire wavelength range and not just that in the visual band. It is a
lucky feature of nature that most stars do emit the bulk of radiation
within the optical band, allowing many radiation laws to work with
tolerable error. This method fails for cool stars because a large and
maybe greater portion of their radiation is outside the optical band.
One useful law is the Stefan-Boltzmann law for the radiation given off
per unit area of the star for a given temperature. Technicly it gives
the watt/meter2 of output, NOT only lumen/meter2 for light.
Stars have no 'surface' or 'ground'. When we speak of surface or
other topographic features, we are looking at the photosphere of the
star. The surface temperature of the Sun , as example, is 6,000K,
meaning the temperature of the photosphere. Stars on the Main Sequence
have sharply defined photospheres, delimiting the stars as globes and
letting us treat them as 'surfaces'.
Very large stars, the giant and supergiant stars, are so tenuous
they have an extended depth of photosphere, causing dispersion in
results from applying the S-B formula. Images of nearby supergiants
show  diffusely defined spheres with high 'surface relief'. Some also
go thru pulsations and turbulence in the photosphere.
The law is

(watt/meter2) = (sigma) * (temperature) ^ 4

where temperature is in Kelvin degrees. A common mistake is to use
doesn't matter much because the two scales are offset by 273 degrees,
which for a temperature of tens of thousands of Kelvin is a small
error. it's best to keep to Kelvin and not look for exceptions.
sigma is the Stefan-Boltzmann constant to fix up the units on both
sides of the formula. It is 5.67e-8 watt/(meter2.Kelvin4) or 5.67e-5
erg/(sec.cm2.Kelvin4).

Mass-Luminosity
-------------
To add population to the HR diagram we plotted stars in binary
systems, whose mass we knew from the system's orbital behavior. We
realized that the stars plotted along the Main Sequence were ordered
by mass.  Binary components plotted off of the Main Sequence didn't
show an obvious pattern for mass.
Mass for stars is virtually always in solar units. Among known
stars the mass ranges from about 1/10 Sun to about 40 Suns. The order
is low mass at the bottom of the Main Sequence to high mass at the
top. The Sun, mass = 1 and luminosity = 1, is conventionally placed in
the middle of the Main Sequence. The luminosity or absolute magnitude
scale is slided to put luminosity = 1 or absolute magnitude = +5 mid
way up the graph.
Stars less than 1/10 Sun do not compress their hydrogen hot and
dense enough to ignite nuclear fusion. They shine by gravity  heating
as 'brown dwarfs'. Because they are so cool they emit little visual
radiation, leaving them out of the HR diagram.
Stars more than 25 Suns live for only tens of millions of years
are mostly all gone today from the general stellar population.
This ordering of stars by mass along the Main Sequence derives from
models of energy production.  A star of given mass put thru the model
ends up with a specific temperature and luminosity on the Main
sequence. This leads to the mass-luminosity curve, a one-to-one
correspondence between star mass and luminosity.
Over the entire Main Sequence, in solar units,

(mass) = (luminosity) ^ (+0.3), or

(luminosity) = (mass) ^ (+3.3)

+---------------------------------------+
| MASS-LUMINOSITY RULE ON MAIN SEQUENCE |
|                                       |
| (mass) = (luminosity) ^ (+0.3), or    |
|                                       |
| (luminosity) = (mass) ^ (+3.3)        |
+---------------------------------------+

The exponent of mass increases upward  along the Main Sequence,
from about +2.5 near the bottom end to about +4 near the top. A middle
refinement is that the exponent is +4 for stars more than 5 Sun mass
and +2.5 for less than 1/2 Sun.  The range of the luminosity exponent
is +0.4 to +0.25.
Altho this rule was found from components of binary stars, it
works for single stars. Historicly we studied binary stars with
distinctly separate members, with no interaction, other than gravity,
between them. They, except for orbital involvement, behaved as single
stars.
One common error is to apply the M-L relation  all over the
Hertzsprung-Russell diagram. The rule is fundamentally the consequence
of hydrogen burning, which occurs only on the Main Sequence. There is
no simple M-L rule else where on the HR diagram.

Energy escape from Sun
--------------------
The gamma ray, photon, produced by fusion in the Sun's core is NOT
the photon that leaves the photosphere into space. Soonest the photon
is created it is intercepted by a free electron in the solar plasma,
is absorbed, and emitted as a brand-bew photon. This photon in turn is
captured by ann other electron, and so on, all thruout the body of the
Sun.
Each emitted photon is sent out from the electron in any
direction, not straight toward the photosphere. The new photon is
emitted by the intercepting electron in any direction. A vastly larger
path is covered by the chain of photon-electron encounters to reach
the surface.
This is an application of the 'random walk' situation. There are
many methods of the random walk, with many different ways to model it
and all with fiddly maths.
The path between steps in the photon chain is computed as a
gradient from the dense center of the Sun to the almost vacuum of the
photosphere. For this piece a rough average value is one centimeter.
Authors cite step lengths of one millimeter to ten centimeters. The
photon takes time to travel to the next step, at lightspeed. a
centimeter away. This is 3.33e-11 second.
It turns out by eyeballing several articles on solar random walk
that the number of one-centimeter steps in by the chain of photons and
also the total length of that chain is (solar radius)^(2.5), also in
centimeters. This value varies widely among authors according as the
model they apply.

(photon chain length) = (Sun radius) ^ (2.5)
= (6.96e10cm) ^ (2.5)
= (1.278e27cm)

Since each step is one cm long, this is also the number of steps on
the way to the solar surface. This is multiplied by the photon travel
time for each step.

(photon chain time) = (1-cm  steps) * (light-time)
= (1.278e27) * (3.33e-11s)
= (4.257e16s)
-> 1,349,000 years

This is an awfully long time, even tho this method of calculating
it is crude. Estimates of the actual time for a photon to escape from
the Sun range from a few hundred thousand years to about ten million
years. The point to realize is that the photon leaving the photosphere
now is the last one of a chain of photon-electron encounters that
began long before humans sprang forth on Earth.
It is this last photon, free from further interactions, that takes
the eight or so minutes to reach Earth. Any information about the
activity in the core were erased eons ago on the way to the
photosphere and we have no direct knowledge of it  by the arriving
flux of photons.
The fusion reactions also release neutrinos. These particles are
almost  inert against matter and pass straight thru the Sun without
interacting with electrons. They travel at some 95% lightspeed,
reaching Earth in a;most nine minutes. Because they are coming from
activity in the solar core now, study of these neutrinos is crucial in
probing the Sun's energy production process.

Star Size
-------
Each encounter of a photon with an electron imparts a momentum to
the electron and itself losses a bit of energy. The net momenta of all
the photons leaving the core is upward toward the photosphere and
constitutes a photon pressure on the plasma outside the core. This
plasma is lofted out against the gravity infall and settles the star
into itself, as actually happens in certain situations in a star's
end-game.
This radius combined with the Stefan-Boltzmann law has the
sufficient surface area to radiate away the entire energy output of
the star. There is a correspondence between the luminosity and
temperature of the star and its radius.
The temperature, thru the Stefan-Boltzmann law, gives the
watt/meter2 radiant output of the star. The surface area is then

(area*) = (luminosity*) / ((sigma) * (temperature* ^ 4))

Invariably both luminosity and size are cited in solar units, not
raw watts and meters. This simplifies the equation. The area, in solar
units, is merely the ratio of the above formula for star and Sun,

(area* / area0) =  (lum* / lum0) / (temp* / temp0) ^ (4)
=  (lum* / lum0) * (temp0 / temp*) ^ (4)

(area) =  (luminosity) * (temp0 / temp*)  ^ (4)

While this is correct, it is far more usual to state 'size' by
radius or diameter in solar units, than by surface area. The area is
proportional to the square of the radius,

(radius) ^ (2) = (luminosity) * (temp0 / temp*) ^ (4)

Take the square root of both sides,

(radius) = sqrt(luminosity) * (temp0 / temp*) ^ (2)
= sqrt(luminosity) * (6,000 / temp*) ^ (2)

+----------------------------------------------------+
|  RADIUS OF STAR FROM LUMINOSITY AND TEMPERATURE     |
|                                                    |
| (radius) = sqrt(luminosity) * (6,000 / temp*) ^ (2) |
+-----------------------------------------------------+

This method works all over the HR diagram and is the basis of
calling certain stars 'giants' and 'dwarfs'. A star with radius much
larger than Sun's is a giant star; less, dwarf.
The trend of radius is from small in the lower left of the HR
diagram to large at the upper right. The range is from dwarfs of about
1/100th Sun to supergiants of over 100 Sun.
Along the MS the trend is weaker because temperature and
luminosity increase together. Stars near the bottom of the MS are
around 1/10 solar radius while those near the top are some ten times
larger than the Sun.

Star Life
-------
A MS star has a finite supply of fuel, hydrogen, to burn into
helium. This is contained in the core of the star where the density and
temperature are severe enough to support the fusion process. Hydrogen
away from the core never enters into fusion. Some could be circulated
into the core by convection and then ignite. Main Sequence stars are
quiet globes of gas with minor convection thru the core. Of the star's
total hydrogen, only 10% at most is turned into helium.
The hydrogen fusion is a steady and stable energy source for stars
on the Main Sequence, generating a constant flow of energy.  The star
maintains about the same temperature and luminosity for its entire
stay on the Main Sequence.
Main Sequence stars do not store or bank their generated energy
within them. All generated energy must radiate from the star as
luminosity. The observed luminosity equals the rate of energy
production. Eventually  the hydrogen gives out and hydrogen burning
shuts off. The star then begins other energy generating methods but
they give the star a luminosity and temperature that do not plot on
the Main Sequence.
The star migrates to other sections of the HR diagram. It spends
about 10% more of its full lifespan in this end-game situation before
The residence on the Main Sequence is approximately, in solar
units, the mass divided by the luminosity, That is, it's the supply of
fuel divided by the rate of consumption.

(MS life) = (mass) / (luminosity)

The Sun's life on the Main Sequence is nominally 9 billion years, so

(MS years) = (mass) * (9 billion years) / (luminosity)

Recalling the Mass-Luminosity formula, we can write the star's MS
life, in solar units,  as

a  (MS life) = (mass) ^ (-2.3) = (luminosity) ^ (-0.7)

+------------------------------------+
| MAIN SEQUENCE LIFE OF A STAR       |
|                                    |
| (MS life) = (mass) / (luminosity)  |
|                                    |
a  | (MS life) = (mass) ^ (-2.3)        |
|           = (luminosity) ^ (-0.7)  |
+------------------------------------+

Zero Age Main Sequence
--------------------
The Main Sequence in a typical HR diagrams is a broad band of
stars, not a thin line. Stars in the general population have a wander
on the MS for several reasons. One is that during MS residence a star
does slightly increase its temperature and luminosity by a couple
percent. The other is that the fraction of 'metals' modify the star's
nuclear processes. They yield slightly different temperature-
luminosity values for a given mass of star.
For a homogeneous sample, like from an open cluster, the locus of
stars on the Main Sequence is a narrow band.
It is not practical to start counting a star's age from its actual
birth. There is no definite moment before the MS phase of life that
signals the birth of a star. Since stars condense from nebulae, they
may be obscured from view until long after they are consolidated
globes.
We let a star be 'born' when it arrives on the Main sequence and
count the star's life before then as negative time.It's like some farm
or sport animals who are all declared to be one year old on the next
New Year's Day after actual birth.
The MS for these stars is called the Zero Age Main Sequence, ZAMS.
it is sometimes drawn on HR diagrams as a reference line, specially if
the sample of stars has weak MS representation.

Age of a group
------------
If we have a group of stars, all at the same distance away, we can
use the MS to estimate the age of that group. This technique helped
prove that stars condensed from nebulae. We plot an HR diagram of the
group, like an open cluster, from spectra and photometry of the
individual stars. Chances are that the graph has a definite Main
Sequence section.
We find that the diagram commonly has only the lower part of the
MS. There is no upper part. The upper cut-off end is at a certain
luminosity or mass.
The group contains stars of assorted mass, which we may allow were
created more or less simultaneously, over a few million years. That
is, the stars arrived on the Main Sequence more or less ot once.
As te group lives, high mass stars eventually finish their
hydrogen fusion while low mass stars are still running it. The high
mass stars, no longer able to stay on the MS, shift away, causing the
interruption in the MS curve. These stars plot above and right of the
MS,, which now has a turn-off point at its upper end.
The Main Sequence life corresponding to the mass or luminosity at
the turn-off is conventionally taken as the 'age' of the cluster, the
elapsed time since it was formed.

Color Magnitude Diagram
---------------------
When the stellar group, maybe a dwarf galaxy, is at an unknown
distance away we can not plot a true HR diagram. We do not have the
absolute magnitude or luminosity of the member stars. In some cases we
can not capture good spectra of the separate stars for their spectral
class due to crowding or faintness.
In such a case we take a picture of the group in a blue and a
yellow filter, of astronomical photometric specifications.  The yellow
filter approximates the visual aspect of the group. On the pictures we
assess the blue and visual (yellow) apparent magnitude of the stars.
The difference, blue minus visual, is the 'color index', often stated
as 'B - V'.
A graph is  plotted similar to the HR diagram with apparent visual
magnitude against color index. This is the Color-Magnitude diagram,
CMD. Stars in the group deploy in this graph in the same pattern as if
they were plotted on a true HR diagram.
In particular, the Main Sequence in the CMD is the same as on a
true HR diagram.. The distinction is that it plots against apparent,
not absolute, magnitude.

Distance of a group
-----------------
For a group of stars of unknown distance  a Color-Magnitude
diagram is plotted. A point on the CMD Main Sequence is selected and
its color index and apparent magnitude is read out. A point is found
in a general-purpose HR diagram's MS with the SAME color index. Its
absolute magnitude is taken off.
Most HR diagrams have spectral class or temperature as the
horizontal scale, not color index. Color index has a one-to-one
equivalence for either other scale.
These two magnitudes relate the normalized illumination at 10
parsecs to that at the unknown distance of the group. The astronomy
version of the inverse-square law is applied to get the distance. This
is
(Mapp - Mabs) = (5) * log(d) - (5)

where the logarithm is on base 10. This equation is the 'distance
modulus' equation and is nothing but a disguised form of the inverse-
square law of point-source radiation. The (Mapp - Mabs) is the very
distance modulus, sometimes denoted by Greek letter mu.
Solving for distance, d, skipping intermediate shuffling

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

The inverse or anti log is the very distance in parsecs. To keep
things right way round, Mapp is renamed to Mcmd; Mabs, Mhrd

+------------------------------------+
| DISTANCE MODULUS FOR MAIN SEQUENCE |
|                                    |
| log(d) = (Mcmd - Mhrd + 5) / (5)   |
+-------------------------------------+

Mind well that the '-' in the distance modulus is a negative
operator and not a signum for Mabs or Mhhrd. The Mabs is just about
always a minus value because the stars must be very luminous to be
observed from Earth. The negative operator '-' and the signum '-' must
never be confused. Do proper algebra in the formula.
For extremely remote targets, where the Hubble redshift is
significant, many astronomers skip solving for distance and keep the
distance modulus as is for an index of distance. In realms of
substantial Hubble redshift the radiation is emitted thru a spacetime
of one profile and arrives at us in spacetime of an other. This
distorts the inverse-square relation and a distance obtained from the

Conclusion
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