John Pazmino
 NYSkies Astronomy Inc 
 2017 January 13 

    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. 

Your own HR diagram 
    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 
    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 
spreadsheet's graph functions. 
    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 
 ----------   ----
    If done neatly with clear clean lettering your homemade HR diagram 
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, 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 
outside this article. 
    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 
    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 
to produce this radiance is 
    (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 

    |                                           | 
    | 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   | 

    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 
    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. 

Radiation Laws 
    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 
centigrade, Celsius, leading to wrong results. For hotter stars it 
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 

    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) ^ (+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 
    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 
    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 a steady-state radius. Without this pressure the star would fall 
into itself, as actually happens in certain situations in a star's 
    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) = 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 
ultimately dying. We skip this phase in this article. 
    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) 
    |                                    | 
    | (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 
    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 
    (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 

        |                                    | 
        | 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 
distance modulus formula is misleading. 

    The Main Sequence features discussed in this article are scattered 
in the astronomy litterature. Home astronomers learn of them one-by-
one, often losing their interrelations.. Presenting these features 
here brings out the interconnection among them. There is some maths to 
work thru, only ordinary algebra handled by a sci/tech calculette. 
    For the newcomer to our profession it may seem bizarre that a 
graph developed quite a hundred years ago held its secrets hidden for 
so many decades. In the 1910s we didn't know just what a star ws, what 
is was made of, and how it lived and died. We knew almost nothing 
about atomic and nuclear science. Einstein's mass-energy physics was a 
brand-new theory.