LIFE OF STARS ON THE MAIN SEQUENCE -------------------------------- John Pazmino NYSkies Astronomy Inc email@example.com 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. 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 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 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 ---- 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 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 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 +-------------------------------------------+ | 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. 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 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 a steady-state radius. Without this pressure the star would fall 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 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) +------------------------------------+ | 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 distance modulus formula is misleading. Conclusion -------- 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.