SECTION 6: Origin of the Elements Current understanding of the origin of the chemical elements, or nucleosynthesis, can be traced to the classic paper of Burbidge, Burbidge, Fowler and Hoyle (Rev. Mod. Phys. 9, 547 (957). In it, they laid out the framework for synthesizing the elements during the steps of stellar evolution as a star develops from a simple Main Sequence star such as our Sun to the spectacular explosions of Supernovae. Fowler s later work on stellar evolution earned him the Nobel Prize. In the context of creating our Universe from first principles, the process can be described as the interaction of the fundamental particles under the influence of the basic forces, controlled by Nature s conservations laws, as illustrated below. Fundamental Particles (quarks, gluons, leptons, photons, neutrinos, etc.) + Basic Forces (gravity, electromagnetic, nuclear) Interactions Conservation Laws Universe: Sea of Stability Elemental Abundances Present-day observables for testing this theory are found in the abundances of the chemical elements (Fig. 6.) and the sea of nuclear stability (Fig..). Fig.6. Abundances of the elements relative to Si = 0 6 as a function of mass number.
The primary sources of nucleosynthesis are believed to be creation in the Big Bang, the subsequent evolution of stars, and the interaction of cosmic rays with interstellar material. Cosmological Nucleosynthesis in the Big Bang The concept of the Big Bang was first proposed by George Gamov to explain the synthesis of all the elements in one cosmic event. Although experiments later showed that such a scenario could not generate elements beyond helium, the Big Bang aspect of the theory was supported and reinforced by several pieces of evidence. The Red Shift Since the early analysis of Hubble, studies of the spectra from all distant galaxies has been shown to be Doppler-shifted to the red, implying that everything is moving away from our solar system. Two important conclusions arise from this now highly documented observation. First, the Universe is expanding. And second, all matter has a common origin. By correlating the amount of Red-Shift with galactic distances, it is estimated that the age of the Universe is 3 ± 0 9 years. Universal Black-body Radiation -- Penzias and Wilson were awarded the Nobel prize for their observation of an isotropic Black-body radiation spectrum that does not come from our galaxy. This spectrum has a temperature profile that is consistent with a Blackbody radiation spectrum of T =.75 ± 0.00 o K for the present Universe, for which the average density is 0 3 g/cm. This radiation is believed to be the remnant of a primordial explosion, the Big Bang. Abundances of the Light Elements H and He The elements hydrogen and helium account for 98% of the elements in Nature, indicating that the Universe must have been formed from the simplest particles. Studies of the isotopes H, H, 3 He and 4 He, especially in old halo stars where stellar evolution is in its early stages, show that there is little variation in these isotopic abundances across the cosmos, with the important ratio He/H = 0.3 ± 0.0 This result suggests a common origin for these two elements, which is identified with the Big Bang. The above observations give rise to the STANDARD MODEL, which postulates that the Universe originated in a hot, dense explosion involving the simplest particles the Big Bang. The conditions at which this explosion occurred can be inferred from the temperature and density of the Universe at the present time, extrapolated back 3 billion years. Some of the basic assumptions of the standard model are: Only known particles and forces are included. Matter versus energy dominance: energy drives the expansion and gravity (mass) constrains it; i.e. E = Mc is relevant. Temperature is a function of density. The universe cools as it expands (remembering that <E> = (3/) kt ; k = 0.86 0 0 MeV/K.
The chronology of the Big Bang can be traced as follows: () Elementary Particle Phase Prior to the first 0-6 s of the Big Bang most of the mass-energy of the Universe is in the form of ENERGY. At this stage the temperature is T > 0 3 o K, which is a mass-energy equivalent greater than the mass of a nucleon, the necessary building blocks for nuclei. Any nucleons or complex particles that might form in this heat bath would quickly dissolve. () Hadron Phase As the Universe expands and cools through the time scale 0 6 s to ~ s, the temperature (0 3 K > T > 0 0 o K) become low enough for neutrons and protons to form. Formation of the simplest complex nucleus, H, is inhibited at these temperatures because of its low binding energy of. MeV. At this stage an equilibrium forms involving the following reactions: p + e n + v n + e + p + Reaction rates determine the p/n ratio, which needs to be ~ : to form H. (3) Nucleosynthesis Phase At a time roughly 3 minutes after the initial expansion, when the temperature has dropped to ~ 0 9 o K and the density is ~ 0. g/cm 3, nucleus formation begins through the following sequence of reactions: 0 H + n H 0 3 3 3 4 7 3Li 4 He EC 3 4 7 He + He Be H + n H H H H + He H + H He He n 3 3 0 A more complete picture of the reaction network is shown in App.6. The reaction chain is essentially complete when 4 He is reached, although a small amount (~0 - ) of 7 Li is produced. The synthesis of heavier elements is strongly attenuated by the effects of nuclear shell structure and very short lifetimes for nuclei just beyond the doubly magic 4 He nucleus. The theory predicts the abundances of H, H, 3 He, 4 He and 7 Li quite well. (4) Cooling Phase Expansion and cooling continue, allowing the neutrons to decay into protons. Nuclear reactions now cease as the temperature is too low to overcome the Coulomb barrier for the proton-proton reaction. Matter now dominates the Universe. With their.8 minute half-life, neutrons decay into protons and are no longer available as reactants. (5) Chemistry Phase After ~ 0 5 years have elapsed, the temperature dropped to less than 0 5 o K, comparable to the electron binding energies of hydrogen, helium 3
and lithium atoms. At this point the first atoms are formed, accompanied by a huge photon burst that generates the microwave background mentioned above. H + + e H + At this point the cosmos continues to expand and cool. Were it not for the force of gravity, nucleosynthesis would cease and we would be left with a very dull Universe. Stellar Nucleosynthesis As the Universe expands, localized inhomogeneous regions develop and subsequent gravitational attraction sets the stage for galaxy formation. As a result, the density begins to increase, reheating matter in the core of the gravitational field. This process sets the stage for the development of stars, the simplest of which are Main Sequence stars. Main Sequence Stars Hydrogen Burning Approximately 90% of the stars in the Universe are classified as Main Sequence stars, of which our Sun, with a mass M = 0 33 g, is a typical example. These are the first stars that form from the primordial Big Bang debris. As gravitational forces contract matter in a first-generation star, the increase in density causes the temperature in the core to rise and ionize the medium. (No neutrons are present, as in the case of the Big Bang). Unless the star s mass is greater than about one-half that of the Sun, electrostatic repulsion inhibits nuclear reactions and further contracts the star, but not to sufficient densities to promote nuclear burning. For more massive stars, gravitational forces dominate and heat the core to temperatures of order 0 7 o K. At this point proton burning is ignited in the high-energy tail of the Maxwell-Boltzmann distribution, as shown in Fig. 6.. T N(E) T T > T N(E) E e E/kT E p V Coul Fig 6. Comparison of proton kinetic energy distributions at two temperatures, T > T. Reactions occur only for the most energetic protons. When conditions in the core reach T - 0 7 K and a density 00 g/cm 3, hydrogen burning commences. This density is large with respect to that of the Universe ( 0-3 g/cm 3 ) and hydrogen gas at STP (~ 0-4 g/cm 3 ) but is much smaller 4
that the density of a nucleus (~0 4 g/cm 3 ). The fundamental reaction that is the rate-determining step in hydrogen burning is the fusion of two protons to form deuterium H H H + 0 -. Note that this reaction involves the creation of an anti-lepton and a lepton. Therefore it is governed by the WEAK FORCE and proceeds very slowly. For this reason Main Sequence stars such as our Sun can survive billions of years, depending on their mass. This is in sharp contrast to the formation of H in neutron-proton collisions during the several-second lifetime of the Big Bang, which involves the strong nuclear force. The neutrino is the only particle that can escape from the core of a star and during the past forty years great progress has been made in detecting solar neutrinos and testing theories of stellar burning. These experiments are described in Appendix 6.. Once H is formed the following chain of exothermic reactions occurs, called the ppi chain: H H H + 0 - H H He + 3 3 4 He He + H 4 NET: 4 H He + 6. 7 MeV. This reaction chain, which is the principal energy source for our solar system, is the first step in stellar evolution. It is also the concept behind the nuclear fusion reactor, discussed in the section on nuclear power (section 8). Depending on the mass and elemental composition, in later-generation stars there are other reaction chains that produce the same result; i.e. the conversion of protons into 4 He nuclei. Each of these uses intermediate nuclei as catalysts. These cycles include: ppii: 7 Li catalyst ppiii: 7 Be catalyst CNO: C catalyst Once hydrogen burning begins, the heat evolved counterbalances gravitational attraction and a star exists in a state of quasi-equilibrium. A temperature-mass profile of the Sun is shown in Fig. 6.3. As far as element composition is concerned, hydrogen burning adds a small amount of additional 4 He to that from the Big Bang, but no heavier elements. 5
The lifetime of a star with the mass of our Sun is expected to be about 0 0 years, so we are presently about half-way through its life-expectancy. Since the Coulomb Barrier for the 4 He - 4 He reaction is four times higher than for proton-proton interactions, 4 He cannot undergo nuclear burning at the core temperature of a Main Sequence star. Thus, 4 He continues to accumulate in the core as hydrogen continues to burn in the surrounding envelope. Fig. 6.3 Temperature and mass profile of our sun from the core outward The eventual fate of a Main Sequence star depends upon its mass. If the mass is less than M ~ 0 33 g, gravitational pressure is insufficient to heat the core further and the next step in stellar evolution is blocked. At this stage the star becomes a White Dwarf, burning away its remaining hydrogen as it enters the stellar graveyard. For more massive stars such as the Sun, evolution continues and the star enters the Red Giant phase. Astronomers trace stellar evolution in terms of a Hertzsprung-Russell diagram, which correlates the luminosity of a star with its temperature, discussed further in Appendix 6.3. Red Giant Stars Helium Burning As 4 He continues to accumulate in the core of a star of mass M > 0 33 g, gravitational pressure continues to compress and heat the material in the core. The heating of the core produces a significant expansion of the star s oouter envelope, creating a nascent Red Giant star (Fig. 6.4). When the temperature has increased to T ~ 0 8 K and a corresponding density of 0 5 g/cm 3, conditions for the nuclear burning of 4 He develop. However, the elements Li, Be and B are very weakly bound nuclei and therefore burn up as soon as they are formed, thus blocking the pathway to carbon and heavier elements. 6
Nature subverts this problem by means of the 3 reaction in which three 4 He nuclei react on a very short time scale (<0-6 s) to form C a two-step reaction that could only occur at the very high densities that exist in the core of a star. 4 4 He He 4 8 Be* ; ~0 6 s 8 Be* + 4 He C + ; E =7.65 MeV (O+) This reaction is exothermic, thereby stabilizing the star against gravitational attraction. Because of the three-body nature of the 3 reaction, the reaction rate is slow, resulting in a relatively long lifetime of 0 7 0 8 years for a typical Red Giant star. Our sun will eventually become a Red Giant and then die away to a White Dwarf due to insufficient mass. The elemental mix that evolves in the core of a Red Giant enables further nucleosynthesis, leading to a richer chemical composition of the star. Two important reactions are 6 6 4 0 C He O + and 8 O He Ne +, 4 6 8 both of which are exothermic. This favored reaction path accounts for the peaks in the abundances for C, 6 O and 0 Ne in Fig. 6.. In addition, secondary reactions involving C, 6 O and 0 Ne nuclei can produce isotopes of these elements as well as the odd-z elements N and F. Thus, in a first generation Red Giant the 0 Fig. 6.4 The Red Giant Betelgeuse, compared with the orbits of earth and Jupiter. 7
biologically important elements carbon, nitrogen and oxygen, as well as fluorine and neon, have been added to Nature s inventory. Synthesis of heavier elements in Red Giants is attenuated by the increasing Coulomb Barriers of the helium-burning reaction products, so that once again the star remains in quasi-equilibrium as its 4 He fuel is consumed in the core. Hydrogen continues to burn in the surrounding envelope, causing the star to expand dramatically and reach giant status. As with Main Sequence stars, if the mass of the Red Giant is too small, it degenerates into a White Dwarf. For more massive stars, approximately five times the mass of the Sun, stellar evolution continues to its next stage, which now proceeds at a much faster rate. Explosive Nucleosynthesis When the concentration of carbon and oxygen in the core of a massive Red Giant becomes sufficiently high, gravitational pressure further condenses and heats the core. At a temperature of order of T ~ 5 x 0 8 o K and density ~ 5 x 0 5 g/cm 3, the conditions enable the most energetic carbon and oxygen nuclei to exceed the Coulomb Barriers and begin to react. The initial stage involves Carbon and Oxygen Burning and is characterized by exothermic fusion reactions such as C + C 0 Ne + 4 He or 3 Na + H C + 6 O 4 Mg + 4 He or 7 Si + n 6 O + 6 O 8 Si + 4 He. These reactions proceed very rapidly as they all involve strong, two-body interactions, unlike the weak force that controls hydrogen burning and the threebody nature of helium burning. The rapid evolution of energy causes the star to develop conditions that under some circumstances can lead to Nova outbursts. At the same time, the nuclide composition is enriched by secondary reactions which can form odd-z elements and odd-a isotopes; e.g. C( 6 O, n) 7 Si 7 Al and 4 Mg(n, ) 5 Mg. Coulomb repulsion suppresses fusion reactions at this stage, diverting element synthesis along a new path called Silicon Burning, or the e-process (e for equilibrium). This process kicks in at temperatures T ~ 5 x 0 9 o K and densities ~ 5 x 0 6 g/cm 3. The e-process involves an equilibrium between (, ) and (, ) reactions that are generated in the heat bath of the star s core. Thus a chain of 8
successive reactions occurs that emphasizes the production of alpha particle nuclei, i.e. nuclei with even Z and N and mass number A = 4n. 8 Si() 4 Mg 8 Si() 3 S() 36 Ar() 40 Ca() 44 Ti ----- > 56 Ni Although the reaction path goes in both directions, nuclear material is processed in the direction of 56 Ni, which then undergoes beta decay to form 56 Fe, Nature s most stable nucleus. 56 8 Ni Co EC/ EC 56 56 7 A schematic structure of a massive star that has evolved through the sequence of burning stages that produces an 56 Fe core is shown in Fig. 6.5. Fe Fig. 6.5 Schematic diagram of the burning envelopes for a star that has developed an iron core. Corresponding temperatures and pressures are indicated on the right. 9
Synthesis of the elements beyond iron is strongly inhibited by the large Coulomb barriers and the fact that the Q-values for fusion reactions are usually negative due to the role of 56 Fe as Nature s most stable nucleus. As a result, the thermal pressure due to nuclear reactions in the core no longer resists gravitational pressure, leading to a destabilization that results in a massive implosion as the star undergoes gravitational collapse. Supernova Explosions the r-process Gravitational collapse is believed to occur on a time scale of 0-000 seconds, producing conditions in the core of order T ~ 5 x 0 0 o K and densities ~ 0 8 g/cm. This rapid compressional heating triggers an explosive a shock wave that permeates the star, producing a massive stellar explosion, or supernova. Fig.6.6 shows the Crab Nebula, the remnants of a supernova that exploded in 054 A.D. an was observed and catalogued by Chinese astronomers. Fig.6.6 The Crab Nebula, remnant of a supernova that exploded in 054 A.D. In the hot, dense environment of the central core, nuclear reactions are triggered in two separate zones. First, in the core, where temperatures are highest, nuclei are dissolved in the ambient heat bath. Schematically, the reactions are summarized as: Fe ~4 MeV 3 He + 4 n 56 4 6 H + n H + e n + 0
i.e. 56 Fe nuclei are decomposed into alpha particles, which subsequently break up into their component neutrons and protons. Finally, electrons are captured by the protons to form additional neutrons and neutrinos. The net result is the copious production of neutrons and neutrinos in the iron core of the star. The high concentration of neutron in the core has two important implications for the fate of the supernova. First, in the most central region, gravitational pressure may compress the neutron gas to densities of 5 0 4 g/cm 3, well in excess of normal nuclear density. At this point the neutrons are believed to condense to form a neutron star, a stellar object with a mass comparable to the Sun, but with a radius of only ~0 km. Pulsars are believed to be neutron stars and indeed there is a pulsar in the center of the Crab Nebula. More recently, supernova 987a was observed to explode and has been followed closely to test theories of supernova behavior. One supporting piece of evidence was that there was a spike in the number of neutrinos observed in the Solar Neutrino detectors (App 6.) in coincidence with the visible observation of the event, Second, the interaction of neutrons with the cooler 56 Fe envelope surrounding the hot, dense core provides a mechanism for circumventing the high Coulomb barriers and negative Q-values that inhibit fusion reactions involving charged particles and heavier nuclei. In this environment, the elements up to uranium are synthesized in a matter of seconds via rapid neutron capture on 56 Fe seed nuclei, the r-process (r for rapid). Nuclear reactions in the r-process increase the mass number via neutron capture; e.g. 56 57 58 69 Fe n Fe n Fe etc. As more and more neutrons are added, a point is eventually reached where the neutron binding energy for a given element becomes so small that the neutrons are no longer bound and mass buildup is terminated. At that point the neutron excess isotopes undergo negatron decay, resulting in an increase in atomic number Z. Thus, negatron decay produces an increase in Z along the reaction chain. Fe 59 6 Fe - 69 7 Co + - + 70 + n 7Co n 7 7 Co - 7 8 Ni + - + Through this sequence of mass A increase via neutron capture and charge Z increase via beta decay, the elements up to uranium and beyond are synthesized. The r-process populates highly neutron excess isotopes that eventually decay to the first stable isotope in the beta decay chain. Thus the r-process preferentially produces the heavier isotopes of an element. The terminal step in the path to
higher masses is imposed by nuclear fission. When atomic numbers of Z > 90 are reached in the r-process chain, neutron-induced fission reactions become increasingly probable, splitting the heavy products into lighter nuclei that are then recycled through the r-process. This situation has been substantiated in atmospheric hydrogen-bomb tests during the 950s, in which the elements promethium (Z = 6) and Es, Md and Fm (Z = 99-0) were discovered in the debris of the explosion. 56 Fe n heavy elements f fission products n Stellar Evolution: A Cyclic Process The burning cycles described thus far pertain to a schematic first-generation star that arises from the debris of the Big Bang, as illustrated in Fig 6.7. Fig. 6.7 Life cycle of a first-generation star, beginning with colescence from the remnants of the Big Bang through the supernova stage. However, the stars that we observe today, including the Sun, have gone through multiple burning cycles. Evidence for this is provided by the spectral lines that are
found in the light emitted from the Sun (Fig.6.8), which show the existence of elements up to uranium. Fig. 6.8. Visible spectrum of light from the Sun. Lines indicate emission lines from all the stable elements. In later-generation stars there is a richer mix of nuclei, introducing the possibility for secondary reactions that contribute to the elemental and isotopic abundances. First-generation stars cannot adequately account for many odd-z and odd-a nuclei, as well as the lighter isotopes of the elements beyond iron and the elements lithium, beryllium and boron. The s-process : Neutron Capture on Slow Time Scale In later-generation Red Giants, secondary reactions on iron and heavier r-process residues produce heavy nuclei via neutron-capture reactions over the several million year lifetime of such stars, thus slowly. This process is known as the s- process (s for slow). The neutrons originate in secondary reactions between the abundant 4 He nuclei in the core of Red Giants with the C, O and Ne nuclei formed in previous generations, e.g. 3 C(,n) 6 O ; 7 O(,n) 0 Ne ; Ne(,n) 4 Mg As with the r-process, the s-process increases the A and Z of seed nuclei via sequential neutron capture reactions and negatron decay. The difference is that because the neutrons are captured over a very long time scale (A increases), betaunstable nuclei will decay (Z increases) before another neutron is captured. Thus, the s-process path adheres closely to the line of beta-stability, as in the following case involving the stable isotopes 56,57,58 Fe 56 6 57 6 58 6 Fe + n Fe n Fe n Fe (45 d, - ) 59 6 59 Co + + 7 The s-process path is superimposed on the chart of the nuclides in Fig. 6.8. 3
NUMBER OF NEUTRONS Fig 6.8 s-process path (solid line) superimposed on the chart of the nuclides for Z = 45 55. Note that beta decay occurs whenever an unstable nucleus is encountered. The exception is for 07 Pd, for which the half-life is 7 x 0 6 years, much longer than the average neutron capture time of ~000 years. The dashed arrows in the lower right of the plot show the beta-decay path followed by the neutron-excess r-process residues. Note that Sn can only be made by the r- process and Te can only be made by the s-process. Sb can be made by both. Termination of mass buildup in the s-process occurs at 09 Bi, the heaviest stable nucleus in the chart of the nuclides. Additional neutron captures produce unstable species that only recycle 09 Bi nuclei according to the following series of reactions. 09 83 Bi + n - Po 06 Pb n 07 Po 0 0 83Bi 84 ---> 09 Bi The s-process is particularly amenable to experimental studies because nuclear reactors provide a similar environment for the relevant reactions. Globally, the s- and r-processes account for most of the elemental abundances of the stable nuclei beyond iron successfully. The exceptions are those neutron deficient isotopes such as Sn, which have very low abundances in nature. These isotopes are believed to be synthesized in secondary reactions such as the rp-process (rapid proton capture), which is believed to occur in hot rapidly-evolving stellar environments. 4
GCR + ISM Nucleosynthesis The stellar nucleosynthesis scenario described thus far omits three light elements Li, Be and B (LiBeB). Although a small amount of 7 Li can be made in the Big Bang, the remaining isotopes 6 Li, 9 Be and 0, B are too fragile to survive in stellar interiors and require a cold, dilute mechanism to account for their formation. The clue to the LiBeB mechanism is found in the abundance curve for galactic cosmic rays compared with that for the solar system, shown in Fig.6.9. Fig. 6.9 Elemental abundance curve for solar system material (black line) with that for galactic cosmic rays (red curve). Galactic cosmic rays (GCR) are energetic nuclei that bombard the solar system from elsewhere in the galaxy, perhaps supernova explosions. The primary GCR constituents are H, He, C, N and O. When these nuclei collide with the interstellar medium (ISM) in the solar system, the LiBeB isotopes are formed. Since the compositon of the GCR and ISM are well-known, nuclear reaction cross section measurements have permitted a quantitative test of the GCR + ISM mechanism. The model accounts for 6 Li, 9 Be and 0, B very well, but needs an additional source of 7 Li, which is presumably provided by the Big Bang. 5
Appendix 6. The Hydrogen-Burning network The reaction network involved in the ppi, ppii and ppiii mechanisms of Hydrogen Burning is illustrated below. Appendix 6. The Solar Neutrino Experiments The measurement of neutrinos emitted from Hydrogen-Burning in the core of the Sun has led to new understanding of solar physics. The spectrum for the various neutrino- 6
producing reactions is shown above. Over the past 50 years, several detector systems have been constructed to measure solar neutrinos The detector and the relevant reaction include: Homestake, SD: 37 7 37 8 0-37 8 37 7 Cl + Ar + e - ; Ar + e - Cl GALLEX (Italy) SAGE (Russia) 7 3 Ga Ge + e - 7 3 Kamiokande (Japan) +e - v +e SNO (Canada/US) + H n + p + v (0 3 tons of D O) See also: http://www.hep.anl.gov/ndk/hypertext/solar_experiments.html GALLEX: http://www.mpi-hd.mpg.de/nuastro/gallex/detector.htm As an example of how these detectors function, the GALLEX experiment is described here.the experimental procedure for GALLEX is as follows: 30.3 tons of gallium in form of a concentrated GaCl 3 -HCl solution are exposed to solar neutrinos. In GaCl 3 -HCl solution, the neutrino induced 7 Ge atoms (as well as the inactive Ge carrier atoms added to the solution at the beginning of a run) form the volatile compound GeCl 4, which at the end of an exposure is swept out of the solution by means of a gas stream (nitrogen). The nitrogen is then passed through a gas scrubber where the GeCl 4 is absorbed in water (see figure ). The GeCl 4 is finally converted to GeH 4, which together with xenon is in troduced into a proportional counter in order to determine the number of 7 Ge atoms by observing their radioactive decay. These experiments have found only 30 50% of the expected neutrino flux, a result that is explained in terms of neutrino oscillations during the ten minute journey from the Sun to the earth. App. 6. 3 Hertzsprung Russell Diagram The evolution of stars is summarized in terms of a Hertzspsrung-Russell diagram in which the luminosity of a star compared to the Sun is plotted against its surface temperature. Since stars are essentially black bodies,we know that Hotter things are brighter. o Energy radiated per unit time per unit area is proportional to T 4, o so bigger T means more energy radiated Bigger things are brighter. o Energy radiated per unit time per unit area is proportional to T 4, o so bigger surface area means more energy radiated. 7
Putting this together, we have [Taken from http://zebu.uoregon.edu/~soper/stars/hrdiagram.html For Main Sequence stars (90% of the stellar inventory) the observations show the following evolution as a function of stellar mass. When compared with all stars, the following schematic classification emerges from the Hertzsprung-Russell diagram. 8
9
0