What are the sources of stellar energy? What processes support the "life" of stars? Give an idea about the evolution of ordinary stars and red giants, explain the processes taking place in their interiors. What is the outlook for the evolution of the Sun?
Like all bodies in nature, stars do not remain unchanged, they are born, evolve, and finally "die". To trace the life path of stars and understand how they age, it is necessary to know how they arise. Modern astronomy has a large number of arguments in favor of the assertion that stars are formed by the condensation of clouds of gas-dust interstellar medium. The process of formation of stars from this medium continues at the present time. The clarification of this circumstance is one of the greatest achievements of modern astronomy. Until relatively recently, it was believed that all stars were formed almost simultaneously, some billions of years ago. The collapse of these metaphysical ideas was facilitated, first of all, by the progress of observational astronomy and the development of the theory of the structure and evolution of stars. As a result, it became clear that many of the observed stars are relatively young objects, and some of them arose when there was already a person on Earth.
Central to the problem of the evolution of stars is the question of the sources of their energy. Indeed, where, for example, does the huge amount of energy necessary to maintain the solar radiation at approximately the observed level for several billion years come from? Every second the Sun radiates 4*10 33 ergs, and for 3 billion years it radiated 4*10 50 ergs. There is no doubt that the age of the Sun is about 5 billion years. This follows at least from modern estimates of the age of the Earth by various radioactive methods. It is unlikely that the Sun is "younger" than the Earth.
Advances in nuclear physics made it possible to solve the problem of sources of stellar energy as early as the end of the thirties of our century. Such a source is thermonuclear fusion reactions occurring in the interiors of stars at a very high temperature prevailing there (of the order of ten million degrees). As a result of these reactions, the rate of which strongly depends on temperature, protons are converted into helium nuclei, and the released energy slowly "leaks" through the interiors of stars and, finally, significantly transformed, is radiated into the world space. This is an exceptionally powerful source. If we assume that initially the Sun consisted only of hydrogen, which as a result of thermonuclear reactions completely turned into helium, then the released amount of energy will be approximately 10 52 erg.
Thus, to maintain radiation at the observed level for billions of years, it is enough for the Sun to "use up" no more than 10% of its initial supply of hydrogen. Now we can present a picture of the evolution of some star as follows. For some reasons (several of them can be mentioned), a cloud of the interstellar gas-dust medium began to condense. Pretty soon (of course, on an astronomical scale!) Under the influence of universal gravitational forces, a relatively dense, opaque gas ball is formed from this cloud. Strictly speaking, this ball cannot yet be called a star, since in its central regions the temperature is insufficient for thermonuclear reactions to begin. The pressure of the gas inside the ball is not yet able to balance the forces of attraction of its individual parts, so it will be continuously compressed.
Some astronomers previously believed that such "protostars" were observed in individual nebulae as very dark compact formations, the so-called globules. The success of radio astronomy, however, forced us to abandon this rather naive point of view. Usually not one protostar is formed at the same time, but a more or less numerous group of them. In the future, these groups become stellar associations and clusters, well known to astronomers. It is very likely that at this very early stage of the evolution of a star, clumps of smaller mass form around it, which then gradually turn into planets.
When a protostar contracts, its temperature rises, and a significant part of the released potential energy is radiated into the surrounding space. Since the dimensions of the contracting gaseous sphere are very large, the radiation per unit area of its surface will be negligible. Since the radiation flux from a unit surface is proportional to the fourth power of temperature (the Stefan-Boltzmann law), the temperature of the surface layers of the star is relatively low, while its luminosity is almost the same as that of an ordinary star with the same mass. Therefore, on the "spectrum-luminosity" diagram, such stars will be located to the right of the main sequence, i.e. they will fall into the region of red giants or red dwarfs, depending on the values of their initial masses.
In the future, the protostar continues to shrink. Its dimensions become smaller, and the surface temperature increases, as a result of which the spectrum becomes more and more "early". Thus, moving along the "spectrum - luminosity" diagram, the protostar "sits down" rather quickly on the main sequence. During this period, the temperature of the stellar interior is already sufficient for thermonuclear reactions to begin there. At the same time, the pressure of the gas inside the future star balances the attraction, and the gas ball stops shrinking. The protostar becomes a star.
It takes relatively little time for protostars to go through this very early stage of their evolution. If, for example, the mass of the protostar is greater than the solar mass, only a few million years are needed; if less, several hundred million years. Since the time of evolution of protostars is relatively short, it is difficult to detect this earliest phase of the development of a star. Nevertheless, stars in this stage, apparently, are observed. We are talking about very interesting T Tauri stars, usually immersed in dark nebulae.
Once on the main sequence and ceasing to shrink, the star radiates for a long time practically without changing its position on the "spectrum - luminosity" diagram. Its radiation is supported by thermonuclear reactions taking place in the central regions. Thus, the main sequence is, as it were, the locus of points on the "spectrum - luminosity" diagram, where a star (depending on its mass) can radiate for a long time and steadily due to thermonuclear reactions. A star's position on the main sequence is determined by its mass. It should be noted that there is one more parameter that determines the position of an equilibrium radiating star on the "spectrum-luminosity" diagram. This parameter is the initial chemical composition of the star. If the relative abundance of heavy elements decreases, the star will "fall" in the diagram below. It is this circumstance that explains the presence of a sequence of subdwarfs.
As mentioned above, the relative abundance of heavy elements in these stars is ten times less than in main sequence stars.
The residence time of a star on the main sequence is determined by its initial mass. If the mass is large, the radiation of the star has a huge power, and it quickly consumes its hydrogen "fuel" reserves. For example, main-sequence stars with a mass several tens of times greater than the solar mass (these are hot blue giants of the spectral type O) can radiate steadily while being on this sequence for only a few million years, while stars with a mass close to solar, are on the main sequence 10-15 billion years.
The "burning out" of hydrogen (i.e., its transformation into helium in thermonuclear reactions) occurs only in the central regions of the star. This is explained by the fact that the stellar matter is mixed only in the central regions of the star, where nuclear reactions take place, while the outer layers keep the relative content of hydrogen unchanged. Since the amount of hydrogen in the central regions of the star is limited, sooner or later (depending on the mass of the star) almost all of it will "burn out" there.
Calculations show that the mass and radius of its central region, in which nuclear reactions take place, gradually decrease, while the star slowly moves to the right in the "spectrum - luminosity" diagram. This process occurs much faster in relatively massive stars. If we imagine a group of simultaneously formed evolving stars, then over time the main sequence on the "spectrum - luminosity" diagram, constructed for this group, will, as it were, bend to the right.
What will happen to a star when all (or almost all) hydrogen in its core "burns out"? Since the release of energy in the central regions of the star ceases, the temperature and pressure there cannot be maintained at the level necessary to counteract the gravitational force that compresses the star. The core of the star will begin to shrink, and its temperature will rise. A very dense hot region is formed, consisting of helium (to which hydrogen has turned) with a small admixture of heavier elements. A gas in this state is called "degenerate". It has a number of interesting properties, which we cannot dwell on here. In this dense hot region, nuclear reactions will not occur, but they will proceed quite intensively on the periphery of the nucleus, in a relatively thin layer. Calculations show that the luminosity of the star and its size will begin to grow. The star, as it were, "swells" and begins to "descend" from the main sequence, moving into the region of red giants. Further, it turns out that giant stars with a lower content of heavy elements will have a higher luminosity for the same size. When a star passes into the stage of a red giant, the rate of its evolution increases significantly.
The next question is what will happen to the star when the helium-carbon reaction in the central regions has exhausted itself, as well as the hydrogen reaction in the thin layer surrounding the hot dense core? What stage of evolution will come after the stage of the red giant? The totality of observational data, as well as a number of theoretical considerations, indicate that at this stage of the evolution of stars, the mass of which is less than 1.2 solar masses, a significant part of their mass, which forms their outer shell, "drops."
So, due to the specific instability described above, large-scale gas motions occur in the convective layers of stars. The hotter masses of gas rise from the bottom up, while the colder masses sink. There is an intensive process of mixing of the substance. Calculations show, however, that the difference in the temperature of the moving elements of the gas and the environment is completely negligible, only about 1 K - and this is at a temperature of the substance of the bowels of the order of ten million kelvins! This is explained by the fact that convection itself tends to equalize the temperature of the layers. The average speed of the rising and falling gaseous masses is also insignificant - only about a few tens of meters per second. It is useful to compare this velocity with the thermal velocities of ionized hydrogen atoms in the interiors of stars, which are on the order of several hundred kilometers per second. Since the speed of movement of gases involved in convection is tens of thousands of times less than the thermal velocities of particles of stellar matter, the pressure caused by convective flows is almost a billion times less than ordinary gas pressure. This means that convection has absolutely no effect on the hydrostatic equilibrium of the stellar interior matter, which is determined by the equality of the forces of gas pressure and gravity.
One should not think of convection as some kind of ordered process, where areas of gas rise regularly alternate with areas of its lowering. The nature of the convective motion is not "laminar", but "turbulent"; i.e., it is extremely chaotic, randomly changing in time and space. The chaotic nature of the movement of gas masses leads to complete mixing of matter. This means that the chemical composition of the region of the star covered by convective motions must be uniform. The latter circumstance is of great importance for many problems of stellar evolution. For example, if as a result of nuclear reactions in the hottest (central) part of the convective zone, the chemical composition has changed (for example, there is less hydrogen, part of which has turned into helium), then in a short time this change will spread to the entire convective zone. Thus, “fresh” nuclear hot can continuously enter the “nuclear reaction zone” - the central region of the star, which, of course, is of decisive importance for the evolution of the star. At the same time, there may well be situations where there is no convection in the central, hottest regions of the star, which leads in the course of evolution to a radical change in the chemical composition of these regions. This will be discussed in more detail in Section 12.
In § 3, we have already said that thermonuclear reactions are the sources of energy for the Sun and stars, which ensure their luminosity during gigantic "cosmogonic" periods of time, calculated for stars of not too large mass in billions of years. Now we will dwell on this important issue in more detail.
The foundations of the theory of the internal structure of stars were laid by Eddington even when the sources of their energy were not known. We already know that a number of important results concerning the condition of equilibrium of stars, temperature and pressure in their interiors, and the dependence of luminosity on mass, chemical composition (which determines the average molecular weight), and opacity of matter, could be obtained even without knowing the nature of stellar energy sources. Nevertheless, an understanding of the essence of energy sources is absolutely necessary to explain the duration of the existence of stars in an almost unchanged state. Even more important is the importance of the nature of stellar energy sources for the problem of the evolution of stars, that is, the regular change in their main characteristics (luminosity, radius) over time. Only after the nature of the sources of stellar energy became clear did it become possible to understand the Hertzsprung-Russell diagram, the basic regularity of stellar astronomy.
The question of the sources of stellar energy was raised almost immediately after the discovery of the law of conservation of energy, when it became clear that the radiation of stars is due to some kind of energy transformations and cannot take place forever. It is no coincidence that the first hypothesis about the sources of stellar energy belongs to Mayer, the man who discovered the law of conservation of energy. He believed that the source of radiation from the Sun is the continuous fallout of meteoroids onto its surface. Calculations, however, have shown that this source is clearly insufficient to ensure the observed luminosity of the Sun. Helmholtz and Kelvin tried to explain the prolonged radiation of the Sun by its slow contraction, accompanied by the release of gravitational energy. This hypothesis, which is very important even (and especially!) for modern astronomy, turned out to be untenable for explaining the radiation of the Sun over billions of years. We also note that at the time of Helmholtz and Kelvin, there were no reasonable ideas about the age of the Sun. Only recently it became clear that the age of the Sun and the entire planetary system is about 5 billion years.
At the turn of the XIX and XX centuries. one of the greatest discoveries in human history was made - the discovery of radioactivity. Thus, a completely new world of atomic nuclei opened up. It took, however, more than one decade for the physics of the atomic nucleus to become on a solid scientific basis. Already by the 1920s it became clear that the source of the energy of the Sun and stars should be sought in nuclear transformations. Eddington himself also thought so, but it was not yet possible to indicate specific nuclear processes occurring in real stellar interiors and accompanied by the release of the required amount of energy. How imperfect the knowledge of the nature of stellar energy sources was at that time can be seen, if only from the fact that Jeans, the greatest English physicist and astronomer of the beginning of our century, believed that such a source could be ... radioactivity. This, of course, is also a nuclear process, but it is easy to show that it is completely unsuitable for explaining the radiation of the Sun and stars. This can be seen at least from the fact that such an energy source is completely independent of external conditions - after all, radioactivity, as is well known, is a process spontaneous. For this reason, such a source could in no way "adjust" to the changing structure of the star. In other words, there would be no "adjustment" of the radiation of the star. The whole picture of stellar radiation would sharply contradict observations. The first to understand this was the remarkable Estonian astronomer E. Epik, who, shortly before the Second World War, came to the conclusion that only thermonuclear fusion reactions can be the source of energy for the Sun and stars.
Only in 1939 did the famous American physicist Bethe give a quantitative theory of nuclear sources of stellar energy. What are these reactions? In § 7 we already mentioned that in the depths of stars there should be thermonuclear reactions. Let's dwell on this in a little more detail. As is known, nuclear reactions, accompanied by the transformation of nuclei and the release of energy, occur when particles collide. Such particles can be, first of all, the nuclei themselves. In addition, nuclear reactions can also occur when nuclei collide with neutrons. However, free (ie, not bound in nuclei) neutrons are unstable particles. Therefore, their number in the interiors of stars should be negligible. On the other hand, since hydrogen is the most abundant element in stellar interiors and is completely ionized, collisions of nuclei with protons will occur especially often.
In order for the proton to be able to penetrate into the nucleus with which it collides during such a collision, it must approach the latter at a distance of about 10 -13 cm. , colliding proton. But in order to approach the nucleus at such a small distance, the proton must overcome a very significant force of electrostatic repulsion ("Coulomb barrier"). After all, the nucleus is also positively charged! It is easy to calculate that in order to overcome this electrostatic force, the proton needs to have a kinetic energy that exceeds the potential energy of the electrostatic interaction
The third problem is the low level of radiation of the star in the visible range. On fig. Figure 8.7 shows the spectra of the Sun and an M6-class dwarf with the same chemical composition. For convenience of comparison, the height of the maxima in these spectra is assumed to be the same. A sharp drop in the spectrum of an M-dwarf in the region of wavelengths shorter than 0.7 μm would deprive terrestrial organisms of most of the radiation they use for photosynthesis (Sec. 2.5.2).
Of course, even the lack of conditions for photosynthesis on the planets of an M-dwarf is not a fundamental obstacle to the development of life, since on Earth, for example, there are microorganisms whose life is not associated with photosynthesis (Sec. 2.5.2). Moreover, some terrestrial bacteria use radiation with a wavelength of more than 0.7 microns for photosynthesis. So the weakness of the visible radiation of M-dwarfs cannot be considered an insurmountable problem.
Radiation variability of M-dwarfs
This last problem doesn't look fatal either. All stars flare, including the Sun. A flare is a sharp increase in the emission of electromagnetic radiation and charged particles from a compact region of the photosphere, often associated with star spots [Referring to dark spots on the surface of a star, similar to sun spots. They are characterized by a high energy density of the magnetic field. - Note. ed.]. The flash can last several minutes, although it usually fits in a few tens of seconds; but even a long flash has a short powerful peak that begins with a slow rise and ends with a slow fall. Flashes especially intensify X-ray and ultraviolet (UV) radiation, which poses the greatest danger to living organisms. X-rays are less of a threat because they do not penetrate the planet's atmosphere, but UV radiation is a real danger, especially since its intensity at the time of the outbreak increases by about 100 times. Fortunately, the UV radiation of M-dwarfs in the undisturbed state is so weak (Fig. 8.7) that even with a hundredfold increase, its level at the surface of the planet (having an Earth-like atmosphere) will only be several times higher than the flux at the Earth's surface coming from the quiet Sun.
Although the flare power is low, young M-dwarfs flare much more frequently than the Sun, sometimes several times a day. Fortunately, the frequency of flares decreases with the age of the star: it decreases significantly after about 1 billion years. So the frequent outbursts of a star can only detain the emergence of life on the surface of the planet. And they cannot affect life in the planet's crust or in the depths of its oceans at all.
Another type of variability is due to a change in the luminosity of a star when dark spots appear on its surface. Stars of spectral type M can have spots much larger than those of the Sun; therefore, the luminosity of such stars can decrease by tens of percent, and this can last up to several months. However, calculations show that on planets with an atmosphere, a decrease in temperature will not be catastrophic even for surface inhabitants.
Thus, there is no good reason to exclude the ubiquitous M-dwarfs from the list of stars capable of hosting planets suitable for life, the manifestations of which we could detect from afar.
Galactic Life Zone
Not only the star has a life zone, but also the Galaxy. On fig. 8.8 schematically shows our Galaxy when viewed edge-on; its main components are distinguished: a thin disk, a thick disk, a central thickening (bulge), and a halo (Sec. 1.3.2). Note that the thick disk includes the thin disk, but differs from it in the type of stellar population. The number of stars contained in the thin disk, thick disk, bulge and halo is roughly 100:20:10:1, so that the thin disk contains about 3/4 of all the stars in the Galaxy.
The Galactic life zone can be determined by estimating the probability of the existence of habitable planets in each of the components of the Galaxy.
As noted in Section 8.2.2, the main factor determining the possibility of the emergence of life is the metallicity of the substance from which a star and its planetary system are formed: for the birth of habitable planets, the metallicity of a star must, apparently, be at least half that of the Sun. The history of star formation in the thin disk is the longest; the metallicity of its interstellar medium began to increase at the dawn of the history of the Galaxy and continues to increase to this day. That is why
the thin disk is the most promising for the search for habitable worlds. True, its outer regions contain fewer heavy elements, so there should be fewer suitable planets there. The thick disk is inhabited by significantly older and less metallic stars, so it is unlikely to find habitable planets there. Even older stars inhabit the galactic halo, which means that habitable planets should be even rarer there. About 1% of the halo stars are concentrated in globular star clusters (Fig. 1.14), which are also present in the bulge of the Galaxy, where the era of rapid star formation has already ended, but the formation of stars continues little by little. In this region, apparently, habitable planets can also exist, although heavy elements are represented there in a different proportion than in a thin disk, and it is difficult to say what this could lead to.
In addition to metallicity, there are two more factors that affect the habitability of planets - this is a sharp increase in penetrating radiation and gravitational perturbations of orbits. In chapter 7 it was said that many planets could be sterilized by powerful radiation flows, for example, in supernova explosions; and some planetary systems could be destroyed by the gravitational influence of nearby stars. Supernova outbursts occur throughout the disk, but relatively less frequently in its outer low-density regions. In the inner regions of the disk and in the central bulge, they pose a serious threat to life. The situation is the same in globular clusters, where the evolution of massive stars long ago ended with supernova explosions that filled the star cluster with deadly radiation.
The gravitational perturbations of planetary orbits are also particularly strong in
the bulge and globular clusters, since the stars are much more closely packed there.
Thus, the largest number of stars with habitable planets should be expected in a thin disk, especially in its middle annular region enclosed between a dense central part and a rarefied periphery. It is in this ring that our Sun is located! Since the thin disk contains about three-quarters of the stars in the Galaxy, we must exclude more than a quarter of all stars from consideration. In addition, some of the remaining stars, for the above reasons, do not have planets, the presence of life on which could be registered from afar.
So, if we do not discard M-dwarfs (with the exception of 5-10% of the youngest ones), then we can say that approximately half of the stars in the Galaxy have planets on which life can be detected from afar. We emphasize that this estimate is very This is a rough estimate and represents an upper limit that will be lowered in later sections of the book as additional constraints are considered, both in terms of planet formation and survival.
findings
* The external characteristics of stars and their evolution are clearly described by the Hertzsprung-Russell diagram, which demonstrates the luminosity of a star and its effective temperature or other parameters related to them, for example, instead of effective temperature, the spectral class (O, B, A, F, G, K and M ).
* The evolution of a star is mainly determined by its mass, with which it enters the main sequence. Stars with a mass of up to approximately 8 M¤ become giants in the course of evolution and throw off their shells in the form of planetary nebulae, and their remnants turn into white dwarfs. More massive stars turn into supergiants and then explode as supernovae, and their remnants turn into neutron stars or black holes.
* The duration of the evolution of a star on the main sequence decreases sharply with an increase in its initial mass, so different stars have very different life expectancy - from the moment of birth of a star to the ejection of a planetary nebula or a supernova explosion.
* The abundance of stars of different spectral types decreases from M to O, so that M dwarfs are the most common.
* Earth-like planets seem to be the most convenient for the development of life on the surface. In order for the manifestations of life in terms of its impact on the atmosphere and surface of the planet to become noticeable from a great distance, the planet must spend at least 2 billion years in the life zone.
* Planets, on which manifestations of life can be registered from a great distance, most likely, can be possessed by main sequence stars of spectral classes F, G, K and M (i.e. with masses less than about 2M ¤), which have high metallicity. Their lifetime on the main sequence should exceed 2 billion years, and they should be older than 2 billion years. From these, we must exclude close binary stars, as well as systems sterilized by supernova explosions, and systems experiencing a strong gravitational influence of neighbors. But there is no good reason to exclude M-dwarfs from consideration.
* Most of the stars with habitable planets, apparently, should be concentrated in the thin disk of the Galaxy, far from its inner and outer edges.
* As a rough upper estimate, we can assume that half of the stars in the Galaxy have planets on which life can be detected by observations from a large distance. These stars include M dwarfs, except for 5–10% of the youngest ones. Reduced score very rude; it will be reduced in later sections of the book as additional constraints are considered, both in terms of planet formation and their survival.
Questions
The answers are given at the end of the book.
Question 8.1.
Indicate, justifying your choice, which of the stars listed below should be excluded from the list that can have planets on which life can be detected from afar (recall that the number V indicates the stars of the main sequence).
(1) Star of spectral type A3V.
(2) A binary system containing a solar-mass star and an M dwarf separated by 3 AU.
(3) A star with the mass of the Sun that belongs to a globular cluster.
(4) A G2V star with an age of 1 Gyr.
(5) A star of spectral type M0V with an age of 5 billion years, located in the thick disk of the Galaxy approximately in the middle of its radius.
Question 8.2.
Some of the stars with giant planets have a metallicity of less than 1%. Explain why this does not contradict the statement that such stars are unlikely to have planets with life on the surface (Section 8.2.2).
Figure captions
Fig.8.1.
The Hertzsprung–Russell diagram shows where the most common types of stars cluster. The slanted straight lines correspond to constant stellar radii (in units of the solar radius), and the numbers on the main sequence indicate the stellar masses (in units of the solar mass).
Rice. 8.2.
Radiation spectra of a black body at temperatures of 8000, 6000 and 4000 K.
Rice. 8.3.
Evolutionary tracks on the Hertzsprung–Russell diagram for main sequence stars whose mass (in solar masses) is shown in the figure. The tracks end at those points where catastrophic changes begin in the star.
Fig.8.4.
The line shows the initial mass function for the stars of the Galactic disk (the scale along the y-axis is arbitrary). The dots indicate the number of stars in the vicinity of the Sun
in a unit interval of masses.
Rice. 8.5.
The boundaries of life zones around dwarf stars: spectral class M0 with a mass of 0.5 M ¤ and class G2 with a mass of 1.0 M ¤ (solar metallicity).
Rice. 8.6.
Gravitational (tidal) deformation of the planet. The axis of extension deviates from the direction to the star due to the rapid rotation of the planet (until the moment when the daily rotation begins to occur synchronously with the orbital).
Rice. 8.7. Spectra of the Sun and an M6 dwarf with the same chemical composition. To equalize the spectral maxima, the vertical scales are chosen different.
Rice. 8.8. Scheme of the structure of the Galaxy (edge-on view). The main structural elements are highlighted, the boundaries of which are actually not as sharp as in the figure.
Captions on the drawings
Fig.8.1.
3 - Supergiants
4 - Giants
5 - Main sequence
6 - White dwarfs
Rice. 8.2.
1 – Wavelength, µm
2 - Radiation power, 10 6 W m -2 μm -1
Rice. 8.3.
1 - Effective temperature, K
2 - Luminosity (in units of solar luminosity)
3 - Initial Main Sequence
4 - Final Main Sequence
Fig.8.4.
1 – Mass, 1 M ¤
2 – Relative number of stars in the mass interval 1 M ¤
Rice. 8.5.
1 - Age of the star (billion years)
2 - Distance from the star (AU)
3 - 1.0 solar masses
4 - 0.5 solar masses
Rice. 8.6.
1 - Rotation
2 - To the star
Rice. 8.7.
1 – Wavelength, µm
2 - Radiation power (relative units)
3 - Sun
4 - Dwarf M6
Rice. 8.8.
1 - 100,000 light years
3 - Thick disk (about 4000 light years thick)
5 - Thin disk (about 1200 light years thick)
Stars: their birth, life and death [Third edition, revised] Shklovsky Iosif Samuilovich
Chapter 7 How do stars radiate?
Chapter 7 How do stars radiate?
At a temperature of the order of ten million kelvins and a sufficiently high density of matter, the interior of the star should be "filled" with a huge amount of radiation. The quanta of this radiation continuously interact with matter, being absorbed and re-emitted by it. As a result of such processes, the radiation field acquires equilibrium character (strictly speaking, almost equilibrium character - see below), i.e. it is described by the well-known Planck formula with the parameter T, equal to the temperature of the environment. For example, the radiation density at a frequency
in a unit frequency interval is equal to
An important characteristic of the radiation field is its intensity, usually denoted by the symbol I
The latter is defined as the amount of energy flowing through an area of one square centimeter in a unit frequency interval in one second within a solid angle of one steradian in some given direction, and the area is perpendicular to this direction. If the intensity is the same for all directions, then it is related to the radiation density by a simple relation
Finally, of particular importance for the problem of the internal structure of stars is radiation flux, denoted by the letter H. We can define this important quantity in terms of the total amount of energy flowing outward through some imaginary sphere surrounding the center of the star:
| (7.5) |
If energy is "produced" only in the innermost regions of the star, then the quantity L remains constant, i.e. does not depend on an arbitrarily chosen radius r. Assuming r = R, i.e. the radius of the star, we will find the meaning L: obviously it's simple luminosity stars. As for the amount of flow H, then it changes with depth as r -2 .
If the radiation intensity in all directions were strictly the same(i.e., as they say, the radiation field would be isotropic), then the flow H would be equal to zero[18]. This is easy to understand if we imagine that in an isotropic field the amount of radiation flowing through a sphere of arbitrary radius outside, equal to the number inflowing inside this imaginary sphere of energy. Under conditions of stellar interiors, the radiation field almost isotropically. This means that the value I overwhelmingly superior H. We can verify this directly. According to (7.2) and (7.4) for T= 10 7 K I\u003d 10 23 erg / cm 2
erased, and the amount of radiation flowing in any one direction (“up” or “down”) will be somewhat larger: F = I = 3
10 23 erg / cm 2
with. Meanwhile, the magnitude of the solar radiation flux in its central part,. somewhere in the distance
100 000 km from its center (this is seven times less than the solar radius), will be equal to H = L/ 4r 2 = 4
10 33 / 10 21 = 4
10 12 erg / cm 2
s, i.e. a thousand billion times less. This is explained by the fact that in the interior of the sun, the radiation flux outward ("up") is almost exactly equal to the flux inward ("down"). It's all about "almost". The negligible difference in the intensity of the radiation field determines the entire picture of the star's radiation. It is for this reason that we made the reservation above that the radiation field is almost in equilibrium. With a strictly equilibrium radiation field, there should not be any radiation flux! We emphasize once again that the deviations of the real radiation field in the interiors of stars from the Planck field are completely negligible, as can be seen from the smallness of the ratio H/F
At T
10 7 K, the maximum energy in the Planck spectrum is in the X-ray range. This follows from Wien's law, well known from the elementary theory of radiation:
| (7.6) |
10 -8 cm or 3? - typical x-ray range. The amount of radiant energy contained in the interior of the Sun (or some other star) depends strongly on the distribution of temperature with depth, since u T 4 . The exact theory of stellar interiors makes it possible to obtain such a dependence, from which it follows that our luminary has a radiant energy reserve of about 10 45 erg. If nothing had restrained the quanta of this hard radiation, they would have left the Sun in a couple of seconds and this monstrous flash would undoubtedly have burned all life on the surface of the Earth. This does not happen because the radiation is literally "locked" inside the Sun. The huge thickness of the Sun's matter serves as a reliable "buffer". Radiation quanta, continuously and very often absorbed by atoms, ions and electrons of the plasma of the solar substance, only extremely slowly "leak" out. In the process of such "diffusion" they significantly change their main quality - energy. If in the interiors of stars, as we have seen, their energy corresponds to the X-ray range, then from the surface of the star the quanta come out already very "lean" - their energy already corresponds mainly to the optical range.
The main question arises: what determines the luminosity of a star, i.e., the power of its radiation? Why does a star, which has huge energy resources, spend them so “economically”, losing only a small, albeit quite definite, part of this “reserve” for radiation? Above, we estimated the radiant energy reserve in the interiors of stars. It should be borne in mind that this energy, interacting with matter, is continuously absorbed and renewed in the same amount. The “reservoir” for the “available” radiant energy in the interior of stars is thermal the energy of the particles of matter. It is not difficult to estimate the value thermal energy stored in a star. For definiteness, consider the Sun. Assuming, for simplicity, that it consists only of hydrogen, and knowing its mass, it is easy to find that there are approximately 2
10 57 particles - protons and electrons. At a temperature T
10 7 K the average energy per particle will be equal to kT = 2
10 -9 erg, whence it follows that the supply of thermal energy of the Sun W T constitutes a very significant
10 48 erg. At the observed power of solar radiation L
10 33 erg/s this reserve is enough for 10 15 seconds or
30 million years. The question is, why does the Sun have exactly the luminosity that we observe? Or, in other words, why does a ball of gas with a mass equal to the mass of the Sun, which is in a state of hydrostatic equilibrium, have a completely defined radius and a completely defined temperature of the surface from which the radiation comes out? For the luminosity of any star, including the Sun, can be represented by a simple expression
|
(7.7) |
where T e- temperature of the solar surface [ 19 ]. After all, in principle, the Sun with the same mass and radius could have a temperature of, say, 20,000 K, and then its luminosity would be hundreds of times greater. However, this is not the case, which, of course, is not an accident.
Above, we talked about the store of thermal energy in a star. Along with thermal energy, the star also has a solid supply of other types of energy. First of all, consider gravitational energy. The latter is defined as the energy of the gravitational attraction of all the particles of the star to each other. She is, of course, potential star energy and has a minus sign. Numerically, it is equal to the work that must be expended in order to “pull” all parts of the star to an infinitely large distance from its center, overcoming the force of gravity. An estimate of the magnitude of this energy can be made if we find the energy of the gravitational interaction of the star with itself:
Let us now consider a star not in an equilibrium, stationary state, but in a stage of slow contraction (as is the case for a protostar; see § 5). In the process of contraction, the gravitational energy of the star slowly decreases(remember that it is negative). However, as can be seen from formula (7.9), only half The released gravitational energy will turn into heat, i.e., it will be spent on heating the substance. The other half of the released energy must leave star in the form of radiation. It follows from this that if the star's radiative energy source is its compression, then the amount of energy radiated during its evolution is equal to its thermal energy reserve.
Leaving aside for now the very important question of why a star has very definite luminosity, we immediately emphasize that if we consider the release of its gravitational energy in the process of compression as the source of energy of a star (as was believed at the end of the 19th century), then we will encounter very serious difficulties. The point is not that to ensure the observed luminosity, the radius of the Sun must decrease by about 20 meters annually - such an insignificant change in the size of the Sun is not able to be detected by modern observational astronomy. The difficulty is that the reserve of the gravitational energy of the Sun would be enough only for 30 million years of radiation of our luminary, provided, of course, that it radiated in the past about the same as now. If in the 19th century, when the famous English physicist Thompson (Lord Kelvin) put forward this "gravitational" hypothesis of maintaining solar radiation, knowledge about the age of the Earth and the Sun was very vague, but now this is no longer the case. Geological data with great reliability allow us to assert that the age of the Sun is calculated at least several billion years, which is a hundred times more than the "Kelvin scale" for its life.
From this follows a very important conclusion that neither thermal nor gravitational energy can provide such a long-term radiation of the Sun, as well as the vast majority of other stars. Our age has long pointed to a third source of energy from the radiation of the sun and stars, which is of decisive importance for our whole problem. This is about nuclear energy(see § 3). In § 8 we will speak in more detail and specifically about those nuclear reactions that take place in the stellar interior.
The amount of stock of nuclear energy W i = 0 , 008Xc 2 M
10 52 erg exceeds the sum of the gravitational and thermal energy of the Sun by more than 1000 times. The same applies to the vast majority of other stars. This reserve is enough to maintain the radiation of the Sun for a hundred billion years! Of course, it does not follow from here that the Sun will radiate for such a huge period of time at the present level. But in any case, it is clear that the Sun and stars have more than enough reserves of nuclear fuel.
It is important to emphasize that nuclear reactions occurring in the interior of the Sun and stars are thermonuclear. This means that although fast (and therefore quite energetic) charged particles react, they still thermal. The fact is that the particles of a gas heated to a certain temperature have Maxwellian velocity distribution. At a temperature
10 7 K, the average energy of thermal motions of particles is close to 1000 eV. This energy is too small to overcome the Coulomb repulsive forces during the collision of two nuclei and get into another nucleus and thereby cause a nuclear transformation. The required energy must be at least ten times greater. It is essential, however, that in the case of a Maxwellian distribution of velocities, there will always be particles whose energy will significantly exceed the average. True, there will be few of them, but only they, colliding with other nuclei, cause nuclear transformations and, consequently, the release of energy. The number of such abnormally fast, but still "thermal" nuclei very sensitively depends on the temperature of the substance. It would seem that in such a situation, nuclear reactions, accompanied by the release of energy, can quickly increase the temperature of matter, which, in turn, increases their speed sharply, and the star could use up its supply of nuclear fuel in a relatively short time by increasing its luminosity. After all, energy cannot accumulate in a star - this would lead to a sharp increase in gas pressure and the star would simply explode like an overheated steam boiler. Therefore, all the nuclear energy released in the interior of stars must leave the star; this process determines the luminosity of the star. But the fact of the matter is that whatever thermonuclear reactions are, they cannot go on in a star at an arbitrary speed. As soon as, at least to an insignificant degree, local (i.e., local) heating of the stellar matter occurs, the latter, due to increased pressure will expand, why, according to the Clapeyron formula, will happen cooling. In this case, the rate of nuclear reactions will immediately fall and the substance will thus return to its original state. This process of restoring hydrostatic equilibrium disturbed due to local heating, as we saw earlier, proceeds very quickly.
Thus, the rate of nuclear reactions, as it were, "adjusts" to the temperature distribution inside the star. As paradoxical as it may sound, the luminosity of a star does not depend from nuclear reactions occurring in its bowels! The significance of nuclear reactions lies in the fact that they are, as it were, support a steady temperature regime at a level determined by the structure of the star, ensuring the luminosity of stars during "cosmogonic" time intervals. Thus, a "normal" star (for example, the Sun) is a superbly adjusted machine that can operate in a stable mode for a huge time.
Now we must approach the answer to the main question that was posed at the beginning of this section: if the luminosity of a star does not depend on the sources of energy in it, then what determines it? To answer this question, one must first of all understand how energy is transported (transferred) from the central parts to the periphery in the interiors of stars. Three main methods of energy transfer are known: a) thermal conductivity, b) convection, c) radiation. In most stars, including the Sun, the mechanism of energy transfer by heat conduction is completely inefficient compared to other mechanisms. The exception is the subsoil white dwarfs, which will be discussed in § 10. Convection occurs when thermal energy is transferred along with matter. For example, a heated gas in contact with a hot surface expands, hence its density decreases and it moves away from the heating body - it just "pops up". In its place, a cold gas descends, which again heats up and rises, etc. Such a process can, under certain conditions, proceed quite rapidly. Its role in the most central regions of relatively massive stars, as well as in their outer, "subphotospheric" layers, can be very significant, as will be discussed below. The main process of energy transfer in stellar interiors is still radiation.
We have already said above that the radiation field in the stellar interior almost isotropically. If we imagine a small volume of stellar matter somewhere in the interior of a star, then the intensity of the radiation coming "from below", i.e., in the direction from the center of the star, will be slightly greater than from the opposite direction. It is for this reason that inside the star there is flow radiation. What determines the difference between the intensities of the radiation coming "from above" and "from below", i.e., the radiation flux? Imagine for a moment that the substance of the stellar interior is almost transparent. Then through our volume "from below" the radiation that originated far from it, somewhere in the very central region of the star, will pass. Since the temperature there is high, the intensity will be very significant. On the contrary, the intensity coming "from above" will correspond to the relatively low temperature of the outer layers of the star. In this imaginary case, the difference between the radiation intensities "from below" and "from above" will be very large and will correspond to a huge flow radiation.
Now imagine the other extreme: the matter of the star is very opaque. Then from the given volume it is possible to "see" only at a distance of the order l/
Absorption coefficient calculated per unit mass [20]. In the bowels of the Sun, the value l/
Close to one millimeter. It is even strange at first glance that a gas can be so opaque. After all, we, being in the earth's atmosphere, see objects that are tens of kilometers away! Such a huge opacity of the gaseous substance of the stellar interior is explained by its high density, and most importantly, by its high temperature, which makes the gas ionized. It is clear that the difference in temperature over one millimeter must be absolutely negligible. It can be roughly estimated by assuming that the temperature difference from the center of the Sun to its surface is uniform. Then it turns out that the temperature difference at a distance of 1 mm is close to one hundred thousandth of a degree. Accordingly, the difference between the intensity of the radiation coming "from above" and "from below" will also be negligible. Consequently, the radiation flux will be negligibly small compared to the intensity, as discussed above.
Thus, we come to the important conclusion that the opacity of stellar matter determines the energy passing through it. flow radiation, and hence the luminosity of the star. The greater the opacity of the stellar matter, the lower the radiation flux. In addition, the radiation flux must, of course, still depend on how quickly the temperature of the star changes with depth. Let us imagine a heated ball of gas, the temperature of which is strictly constant. It is quite obvious that in this case the radiation flux would be equal to zero, regardless of whether the radiation absorption is large or small. After all, for any
the intensity of radiation "from above" will be equal to the intensity of radiation "from below", since the temperatures are strictly equal.
Now we can fully understand the meaning of the exact formula that relates the luminosity of a star to its main characteristics:
|
(7.10) |
where symbol
means the change in temperature when moving one centimeter from the center of the star. If the temperature were strictly constant, then
would be zero. Formula (7.10) expresses what has already been discussed above. The radiation flux from a star (and hence its luminosity) is the greater, the lower the opacity of the stellar matter and the greater the temperature drop in the stellar interior.
Formula (7.10) makes it possible, first of all, to obtain the luminosity of a star if its main characteristics are known. But before moving on to numerical estimates, we will transform this formula. Express T through M, using formula (6.2), and accept that
3M/ 4R 3 .
Then, assuming
Will have
|
(7.11) |
A characteristic feature of the obtained formula is that the dependence of the luminosity on the radius of the star has dropped out of it. Although the dependence on the average molecular weight of the substance of the stellar interior is quite strong, the value itself
For most stars, it varies within insignificant limits. Opacity of stellar matter
depends primarily on the presence of heavy elements in it. The fact is that hydrogen and helium in the conditions of stellar interiors fully are ionized and in this state almost cannot absorb radiation. Indeed, in order for a radiation quantum to be absorbed, it is necessary that its energy be completely spent on the detachment of an electron from the nucleus, i.e., on ionization. If the atoms of hydrogen and helium are completely ionized, then, to put it simply, there is nothing to tear off [21]. Another thing is heavy elements. They, as we have seen above, retain some more of their electrons in their innermost shells and can therefore absorb radiation fairly effectively. Hence it follows that although the relative abundance of heavy elements in stellar interiors is low, their role is disproportionately large, since it is they that mainly determine the opacity of stellar matter.
The theory leads to a simple dependence of the absorption coefficient on the characteristics of the substance (Kramers formula):
| (7.12) |
Note, however, that this formula is rather approximate. Nevertheless, it follows from it that we will not make a very big mistake if we set the quantity
not very much varying from star to star. Exact calculations show that for hot massive stars
1, while for red dwarfs the value
10 times more. Thus, it follows from formula (7.11) that the luminosity of a "normal" (ie, in equilibrium on the main sequence) star primarily depends on its mass. If we substitute the numerical value of all the coefficients included in the formula, then it can be rewritten in the form
|
(7.13) |
This formula makes it possible to determine absolute the luminosity of a star if its mass is known. For example, for the Sun, we can assume that the absorption coefficient
20, and the average molecular weight
0, 6 (see above). Then L/L
5, 6. We should not be embarrassed by the fact that L/L
It did not turn out to be equal to one. This is due to the extreme roughness of our model. Exact calculations, taking into account the distribution of the temperature of the Sun with depth, give the value L/L
close to unity.
The main meaning of formula (7.13) is that it gives the dependence of the luminosity of a main sequence star on its masses. Therefore formula (7.13) is usually called "mass - luminosity dependence". Let us once again pay attention to the fact that such an important characteristic of a star as its radius, is not included in this formula. There is no hint of the dependence of the luminosity of a star on the power of energy sources in its depths. The last circumstance is of fundamental importance. As we have already emphasized above, a star of a given mass, as it were, regulates the power of energy sources, which "adjust" to its structure and "opacity".
The "mass - luminosity" relationship was first derived by the outstanding English astronomer Eddington, the founder of modern theories of the internal structure of stars. This dependence was found by him theoretically and only subsequently was confirmed on extensive observational material. The agreement of this formula, obtained, as we have seen above, from the simplest assumptions, with the results of observations is generally good. Some discrepancies take place for very large and very small stellar masses (i.e., for blue giants and red dwarfs). However, further improvement of the theory allowed these discrepancies to be eliminated ...
Above, we presented the relationship between the radiation flux and the temperature difference, based on the assumption that energy is transferred from the interior of the star to the outside only by radiation (see formula (7.10)). In the interiors of stars, the condition radiant equilibrium. This means that each element of the star's volume absorbs exactly as much energy as it radiates. However, this balance is not always sustainable. Let's explain this with a simple example. Let's single out a small volume element inside the star and mentally move it up (ie, closer to the surface) a short distance. Since as we move away from the center of the star, both the temperature and pressure of the gas that forms it will decrease, our volume should expand with such a movement. We can assume that in the process of such a movement between our volume and the environment there is no energy exchange. In other words, the expansion of the volume as it moves up can be considered adiabatic. This expansion will take place in such a way that its internal pressure will always be equal to the external pressure of the environment. If we, after moving, imagine our volume of gas "to itself", then it will either return back to its original position, or will continue to move up. What determines the direction of volume movement?
and P denote density and pressure. After the volume has moved upwards (or, in other words, "underwent a perturbation"), and its internal pressure is balanced by the pressure of the environment, its density must differ from the density of the indicated medium. This is explained by the fact that in the process of lifting and expanding our volume, its density changed according to a special, so-called "adiabatic" law. In this case we will have
|
(7.15) |
In the interiors of stars, as a rule, the transfer of energy is carried out by means of radiation. This is explained sustainability medium in relation to perturbations of its “immobility” (see above). But there are such layers in the interiors of a number of stars, and even entire large regions, where the stability condition, which was obtained above, is not satisfied. In these cases, the bulk of the energy is transferred by convection. This usually happens when the transfer of energy by radiation for some reason is limited. This can happen, for example, with too much opacity.
Above, the basic relation "mass - luminosity" was obtained from the assumption that the transfer of energy in stars is carried out only by radiation. The question arises: if energy transfer by convection also takes place in a star, will this dependence not be violated? It turns out not! The fact is that "completely convective stars", i.e., such stars, in which everywhere, from the center to the surface, the energy transfer would be carried out only by convection, does not exist in nature. Real stars have either only more or less thin layers, or large regions in the center where convection plays a dominant role. But it is enough to have at least one layer inside the star, where the energy transfer would be carried out by radiation, so that its opacity would most radically affect the "throughput" of the star in relation to the energy released in its depths. However, the presence of convective regions in the interiors of stars will, of course, change the numerical value of the coefficients in formula (7.13). This circumstance, in particular, is one of the reasons why the solar luminosity calculated by us using this formula is almost five times higher than the observed one.
So, due to the specific instability described above, large-scale gas motions occur in the convective layers of stars. The hotter masses of gas rise from the bottom up, while the colder masses sink. There is an intensive process of mixing of the substance. Calculations show, however, that the difference in the temperature of the moving elements of the gas and the environment is completely negligible, only about 1 K - and this is at a temperature of the substance of the bowels of the order of ten million kelvins! This is explained by the fact that convection itself tends to equalize the temperature of the layers. The average speed of the rising and falling gaseous masses is also insignificant - only about a few tens of meters per second. It is useful to compare this velocity with the thermal velocities of ionized hydrogen atoms in the interiors of stars, which are on the order of several hundred kilometers per second. Since the speed of movement of gases involved in convection is tens of thousands of times less than the thermal velocities of particles of stellar matter, the pressure caused by convective flows is almost a billion times less than ordinary gas pressure. This means that convection has absolutely no effect on the hydrostatic equilibrium of the stellar interior matter, which is determined by the equality of the forces of gas pressure and gravity.
One should not think of convection as some kind of ordered process, where areas of gas rise regularly alternate with areas of its lowering. The nature of the convective motion is not "laminar", but "turbulent"; i.e., it is extremely chaotic, randomly changing in time and space. The chaotic nature of the movement of gas masses leads to complete mixing of matter. This means that the chemical composition of the region of the star covered by convective motions must be uniform. The latter circumstance is of great importance for many problems of stellar evolution. For example, if as a result of nuclear reactions in the hottest (central) part of the convective zone, the chemical composition has changed (for example, there is less hydrogen, part of which has turned into helium), then in a short time this change will spread to the entire convective zone. Thus, the “nuclear reaction zone” - the central region of the star - can continuously receive “fresh” nuclear hot, which, of course, is of decisive importance for the evolution of the star [22]. At the same time, there may well be situations where there is no convection in the central, hottest regions of the star, which leads in the course of evolution to a radical change in the chemical composition of these regions. This will be discussed in more detail in Section 12.
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