Stars are very different: small and large, bright and not very bright, old and young, hot and cold, white, blue, yellow, red, etc.

The Hertzsprung-Russell diagram allows you to understand the classification of stars.

It shows the relationship between absolute magnitude, luminosity, spectral type, and surface temperature of a star. The stars in this diagram are not arranged randomly, but form well-defined areas.

Most of the stars are located on the so-called main sequence. The existence of the main sequence is due to the fact that the stage of hydrogen burning is ~90% of the evolutionary time of most stars: the burning of hydrogen in the central regions of the star leads to the formation of an isothermal helium core, the transition to the red giant stage, and the departure of the star from the main sequence. The relatively brief evolution of red giants leads, depending on their mass, to the formation of white dwarfs, neutron stars, or black holes.

Being at different stages of their evolutionary development, stars are divided into normal stars, dwarf stars, giant stars.

Normal stars are the main sequence stars. Our sun is one of them. Sometimes such normal stars as the Sun are called yellow dwarfs.

yellow dwarf

A yellow dwarf is a type of small main sequence star with a mass between 0.8 and 1.2 solar masses and a surface temperature of 5000–6000 K.

The lifetime of a yellow dwarf is on average 10 billion years.

After the entire supply of hydrogen burns out, the star increases many times in size and turns into a red giant. An example of this type of star is Aldebaran.

The red giant ejects its outer layers of gas, forming planetary nebulae, and the core collapses into a small, dense white dwarf.

A red giant is a large reddish or orange star. The formation of such stars is possible both at the stage of star formation and at the later stages of their existence.

At an early stage, the star radiates due to the gravitational energy released during compression, until the compression is stopped by the onset of a thermonuclear reaction.

At the later stages of the evolution of stars, after the hydrogen burns out in their interiors, the stars descend from the main sequence and move to the region of red giants and supergiants of the Hertzsprung-Russell diagram: this stage lasts about 10% of the time of the “active” life of stars, that is, the stages of their evolution , during which nucleosynthesis reactions take place in the stellar interior.

The giant star has a relatively low surface temperature, about 5000 degrees. A huge radius, reaching 800 solar and due to such large sizes, a huge luminosity. The maximum radiation falls on the red and infrared regions of the spectrum, which is why they are called red giants.

The largest of the giants turn into red supergiants. A star called Betelgeuse in the constellation Orion is the most striking example of a red supergiant.

Dwarf stars are the opposite of giants and can be as follows.

A white dwarf is what remains of an ordinary star with a mass not exceeding 1.4 solar masses after it passes through the red giant stage.

Due to the absence of hydrogen, a thermonuclear reaction does not occur in the core of such stars.

White dwarfs are very dense. They are no larger than the Earth in size, but their mass can be compared with the mass of the Sun.

These are incredibly hot stars, reaching temperatures of 100,000 degrees or more. They shine on their remaining energy, but over time, it runs out, and the core cools down, turning into a black dwarf.

Red dwarfs are the most common stellar-type objects in the universe. Estimates of their abundance range from 70 to 90% of the number of all stars in the galaxy. They are quite different from other stars.

The mass of red dwarfs does not exceed a third of the solar mass (the lower mass limit is 0.08 solar, followed by brown dwarfs), the surface temperature reaches 3500 K. Red dwarfs have a spectral type M or late K. Stars of this type emit very little light, sometimes in 10,000 times smaller than the Sun.

Given their low radiation, none of the red dwarfs are visible from Earth to the naked eye. Even the closest red dwarf to the Sun, Proxima Centauri (the closest star in the triple system to the Sun) and the closest single red dwarf, Barnard's Star, have an apparent magnitude of 11.09 and 9.53, respectively. At the same time, a star with a magnitude of up to 7.72 can be observed with the naked eye.

Due to the low rate of hydrogen combustion, red dwarfs have a very long lifespan - from tens of billions to tens of trillions of years (a red dwarf with a mass of 0.1 solar masses will burn for 10 trillion years).

In red dwarfs, thermonuclear reactions involving helium are impossible, so they cannot turn into red giants. Over time, they gradually shrink and heat up more and more until they use up the entire supply of hydrogen fuel.

Gradually, according to theoretical concepts, they turn into blue dwarfs - a hypothetical class of stars, while none of the red dwarfs has yet managed to turn into a blue dwarf, and then into white dwarfs with a helium core.

Brown dwarfs are substellar objects (with masses in the range of about 0.01 to 0.08 solar masses, or, respectively, from 12.57 to 80.35 Jupiter masses and a diameter approximately equal to that of Jupiter), in the depths of which, in contrast from main sequence stars, there is no thermonuclear fusion reaction with the conversion of hydrogen into helium.

The minimum temperature of main sequence stars is about 4000 K, the temperature of brown dwarfs lies in the range from 300 to 3000 K. Brown dwarfs constantly cool down throughout their lives, while the larger the dwarf, the slower it cools.

subbrown dwarfs

Subbrown dwarfs or brown subdwarfs are cold formations that lie below the brown dwarf limit in mass. Their mass is less than about one hundredth of the mass of the Sun or, respectively, 12.57 masses of Jupiter, the lower limit is not defined. They are more commonly considered planets, although the scientific community has not yet come to a final conclusion about what is considered a planet and what is a subbrown dwarf.

black dwarf

Black dwarfs are white dwarfs that have cooled down and therefore do not radiate in the visible range. Represents the final stage in the evolution of white dwarfs. The masses of black dwarfs, like the masses of white dwarfs, are limited from above by 1.4 solar masses.

A binary star is two gravitationally bound stars revolving around a common center of mass.

Sometimes there are systems of three or more stars, in such a general case the system is called a multiple star.

In cases where such a star system is not too far removed from the Earth, individual stars can be distinguished through a telescope. If the distance is significant, then it is possible to understand that a double star is possible before astronomers only by indirect signs - fluctuations in brightness caused by periodic eclipses of one star by another and some others.

New star

Stars that suddenly increase in luminosity by a factor of 10,000. A nova is a binary system consisting of a white dwarf and a main sequence companion star. In such systems, gas from the star gradually flows into the white dwarf and periodically explodes there, causing a burst of luminosity.

Supernova

A supernova is a star that ends its evolution in a catastrophic explosive process. The flare in this case can be several orders of magnitude greater than in the case of a new star. Such a powerful explosion is a consequence of the processes taking place in the star at the last stage of evolution.

neutron star

Neutron stars (NS) are stellar formations with masses of the order of 1.5 solar masses and sizes noticeably smaller than white dwarfs, the typical radius of a neutron star is, presumably, of the order of 10-20 kilometers.

They consist mainly of neutral subatomic particles - neutrons, tightly compressed by gravitational forces. The density of such stars is extremely high, it is commensurate with, and according to some estimates, may be several times higher than the average density of the atomic nucleus. One cubic centimeter of NZ matter would weigh hundreds of millions of tons. The force of gravity on the surface of a neutron star is about 100 billion times greater than on Earth.

In our Galaxy, according to scientists, there can be from 100 million to 1 billion neutron stars, that is, somewhere around one in a thousand ordinary stars.

Pulsars

Pulsars are cosmic sources of electromagnetic radiation coming to Earth in the form of periodic bursts (pulses).

According to the dominant astrophysical model, pulsars are rotating neutron stars with a magnetic field that is tilted to the axis of rotation. When the Earth falls into the cone formed by this radiation, it is possible to record a radiation pulse that repeats at intervals equal to the period of revolution of the star. Some neutron stars make up to 600 revolutions per second.

cepheid

Cepheids are a class of pulsating variable stars with a fairly accurate period-luminosity relationship, named after the star Delta Cephei. One of the most famous Cepheids is the North Star.

The above list of the main types (types) of stars with their brief characteristics, of course, does not exhaust the entire possible variety of stars in the Universe.

The Hertzsprung-Russell Diagram (HR Diagram)

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Hertzsprung-Russell diagram

The most important physical characteristics of a star are temperature and absolute magnitude. Temperature indicators are closely related to the color of the star, and the absolute stellar magnitude - with the spectral type. Recall that according to the currently used classification, stars, in accordance with their spectra, as already mentioned in the "Spectral classes" section of the site, are divided into seven main spectral classes. They are designated in Latin letters O, B, A, F, G, K, M. It is in this sequence that the temperature of stars decreases from several tens of thousands of degrees for class O (very hot stars) to 2000-3000 degrees for class M stars..

Those. a measure of brilliance, expressed as the amount of energy emitted by a star. It can be calculated theoretically, knowing the distance to the star.

In 1913, the Danish astronomer Einar Hertzsprung and the American Henry Norris Ressel independently came up with the same idea to build a theoretical graph linking two main stellar parameters - temperature and absolute stellar magnitude. The result was a diagram that was given the names of two astronomers - the Hertzsprung-Russell diagram (abbr. HRD), or, more simply, the G-R diagram. As we will see later, the Hertzsprung-Russell diagram helps to understand the evolution of stars. In addition, it is widely used to determine the distances to star clusters.

Each point on this diagram corresponds to a star. The luminosity of the star is plotted along the y-axis (vertical axis), and the temperature of its surface is plotted along the abscissa (horizontal axis). If we determine its temperature by the color of a star, then we will have at our disposal one of the values ​​\u200b\u200bnecessary for constructing a G-R diagram. If the distance to the star is known, then its apparent brightness in the sky can be used to determine the luminosity. Then we will have at our disposal both quantities necessary for constructing the G-R diagram, and we will be able to put a point on this diagram that corresponds to our star.

The sun is placed in the diagram opposite luminosity 1, and since the surface temperature of the sun is 5800 degrees, it is almost in the middle of the H-R diagram.

Stars with a luminosity greater than the sun are located in the diagram above. For example, the number 1000 means that stars are located at this level, the luminosity of which is 1000 times greater than the luminosity of the Sun.

Stars with less luminosity, such as Sirius B - a white dwarf from the Sirius system - lie below. Stars that are hotter than the Sun, such as Sirius A and Zeta Aurigae B, a hot star from the Zeta Aurigae and Spica system in the constellation Virgo, lie to the left of the Sun. Cooler stars, like Betelgeuse and the red supergiant from the Zeta Aurigae system, lie to the right.

Since cool stars emit red light and hot stars emit white or blue light, the diagram shows red stars on the right and white or blue stars on the left. At the top of the diagram are stars with high luminosity, and at the bottom - with low luminosity.

Main sequence

Most of the stars in the H-R diagram are located within a diagonal stripe running from the upper left corner to the lower right. This band is called "main sequence" . The stars on it are called "main sequence stars". Our Sun belongs to the main sequence stars and is located in that part of it that corresponds to yellow stars. At the top of the main sequence are the brightest and hottest stars, and at the bottom right are the dimmest and, as a result, long-lived.

The stars of the main sequence are in the most "calm" and stable phase of their existence, or, as they say, the phase of life.

The source of their energy is. According to modern estimates of the theory of stellar evolution, this phase makes up about 90% of the life of any star. That is why most stars belong to the main sequence.

According to the theory of stellar evolution, when the supply of hydrogen in the interior of a star runs out, it leaves the main sequence, deviating to the right. In this case, the temperature of the star always falls, and the size increases rapidly. A complex, increasingly accelerating movement of the star along the diagram begins.

Red giants and white dwarfs

Separately - to the right and above the main sequence, there is a group of stars with a very high luminosity, and the temperature of such stars is relatively low - these are the so-called red giant stars and supergiants . These are cold stars (approximately 3000°C), which, however, are much brighter than stars with the same temperature in the main sequence. One square centimeter of the surface of a cold star radiates a relatively small amount of energy per second. The large total luminosity of a star is explained by the fact that its surface area is large: the star must be very large. Stars are called giants, the diameter of which is 200 times the diameter of the Sun.

In the same way, we can consider the lower left part of the diagram. There are hot stars with low luminosity. Since a square centimeter of the surface of a hot body radiates a lot of energy per second, and the stars from the lower left corner of the diagram have a low luminosity, we must conclude that they are small in size. Bottom left, thus, are located white dwarfs , very dense and compact stars, on average 100 times smaller than the Sun, with a diameter commensurate with the diameter of our planet. One such star, for example, is a satellite of Sirius called Sirius B.

Star sequences of the Hertzsprung-Russell diagram in the accepted conditional numbering

On the Hertzsprung-Russell diagram, in addition to the sequences we have considered above, astronomers actually distinguish several more sequences, and the main sequence has a conditional number V . Let's list them:

Ia - a sequence of bright supergiants,
Ib is a sequence of weak supergiants,
II- a sequence of bright giants,
III- sequence of weak giants,
IV is the sequence of subgiants,
V - main sequence,
VI - sequence of subdwarfs,
VII is a sequence of white dwarfs.

In accordance with this classification, our Sun with its spectral type G2 is designated as G2V .

Thus, already from general considerations, knowing the luminosity and surface temperature, it is possible to estimate the size of the star. Temperature tells us how much energy one square centimeter of a surface radiates. The luminosity, equal to the energy that the star emits per unit time, allows you to find out the size of the radiating surface, and hence the radius of the star.

It is also necessary to make a reservation that it is not so easy to measure the intensity of light coming to us from stars. The Earth's atmosphere transmits not all radiation. Short-wavelength light, for example, in the ultraviolet region of the spectrum, does not reach us. It should also be noted that the apparent stellar magnitudes of distant objects are weakened not only due to absorption by the Earth's atmosphere, but also due to the absorption of light by dust particles present in interstellar space. It is clear that even a space telescope that operates outside the Earth's atmosphere cannot be rid of this interfering factor.

But the intensity of light passing through the atmosphere can be measured in different ways. The human eye perceives only a fraction of the light emitted by the sun and stars. Light rays of different lengths, having different colors, do not equally intensely affect the retina, photographic plate or electronic photometer. When determining the luminosity of stars, only light that is perceived by the human eye is taken into account. Therefore, for measurements, it is necessary to use instruments that, with the help of color filters, simulate the color sensitivity of the human eye. Therefore, on the G-R diagrams, instead of the true luminosity, the luminosity in the visible region of the spectrum, perceived by the eye, is often indicated. It is also called visual luminosity. The values ​​of true (bolometric) and visual luminosity can differ quite strongly. So, for example, a star with a mass 10 times greater than the sun radiates about 10 thousand times more energy than the Sun, while in the visible spectrum it is only 1000 times brighter than the Sun. For this reason, the spectral type of a star is often replaced today with another equivalent parameter called "color index"; or "color index" displayed on the horizontal axis of the chart. In modern astrophysics, the color index is, in fact, the difference between the stellar magnitudes of a star in different ranges of the spectrum (it is customary to measure the difference between stellar magnitudes in the blue and visible parts of the spectrum, called B-V or B minus V from English Blue and Visible). This parameter shows the quantitative distribution of energy that a star radiates at different wavelengths, and this is directly related to the surface temperature of the star.

The G-R diagram is usually given in the following coordinates:
1. Luminosity - effective temperature.
2. Absolute magnitude - color index.
3. Absolute magnitude - spectral class.

The physical meaning of the G-R diagram

The physical meaning of the G-R diagram is that after plotting the maximum number of experimentally observed stars on it, it is possible to determine the patterns of their distribution by the ratio of the spectrum and luminosity by their location. If there were no dependence between the luminosities and their temperatures, then all the stars would be distributed evenly on such a diagram. But the diagram reveals several regularly distributed groupings of stars that we have just considered, called sequences.

The Hertzsprung-Russell diagram is of great help in studying the evolution of stars throughout their existence. If it were possible to trace the evolution of a star throughout its life, i.e. over several hundred million or even several billion years, we would see it slowly shifting along the G-R diagram in accordance with the change in physical characteristics. The movements of stars along the diagram depending on their age are called evolutionary tracks.

In other words, the G-R diagram helps to understand how stars evolve throughout their existence. Reverse calculation using this diagram, you can calculate the distances to the stars.

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The picture above has nothing to do with the Chelyabinsk car; this picture is called the Hertzsprung-Russell diagram, and it shows patterns in the distribution of stars by luminosity and color (spectral class). Probably everyone who read at least some popular science book on astronomy saw this picture and remembered that the vast majority of stars in the Universe are on the "main sequence", that is, they are located near the curve that goes from the upper left to the right lower corner of the Hertzsprung-Russell diagram. Stars on the main sequence are stable, and can move very slowly along it for many billions of years, slowly converting hydrogen into helium; when the nuclear fuel runs out, an ordinary star leaves the main sequence, becoming a red giant for a short time, and then collapsing forever into a white dwarf, which gradually fades.

So, the metaphor is that you can draw a similar picture about startups, and it will also turn out that there is a narrow zone of stability - the "main sequence" - and there are unstable states beyond it. The axes can be cash burn (rate of spending investments) and the growth rate of key metrics (each project has its own, of course; in the most typical case, this is the number of users).

On the main sequence - projects that are able to balance one with the other. The ideal situation is a neat, smooth movement along it: expenses gradually increase, and the growth rate increases proportionally (namely, the growth rate, not the metrics themselves!). In other words, the money invested gives explosive growth - the startup "takes off".
A huge graveyard of dwarfs is under the main sequence. These projects are frozen, they do not eat up money, or they use up a very small, unchanged amount of them (roughly speaking, hosting costs) - but the metrics are stable, do not grow or practically do not grow. Maybe someone comes in, registers, even starts using it - but this will not lead to a new round of growth. (From personal experience, this is, of course, 9facts).
Above the main sequence are artificially inflated giants. Money burns out very quickly (like helium!), but this happens in the wrong place, or simply too early - the market is not yet ready to respond with a corresponding increase in metrics. On the spectrogram of such a startup, characteristic features are very clearly visible: bloated staff, lack of organic growth of users (growth only through the purchase of traffic), throwing from side to side. In the anamnesis, as a rule, a "wild investor" - someone who strongly believed in the idea, but at the same time is not professionally involved in the development of startups, cannot assess the needs of the project at the next stage, and gives too much money. (And this was also all we had with 9facts, by the way).
Very often you can observe how a project goes exactly the same way as a star in the process of its evolution: from the main sequence to giants (they mistakenly decided that they grabbed the model that would provide explosive growth and started pumping money), and then to dwarfs ( money is gone). Well, a few more amusing analogies can be seen within this rich metaphor.

And the productivity of this metaphor is this.
1) The main sequence is very narrow. This is a thin path, it is impossible to walk along it without a very clear understanding of how the venture industry works in general (I will take this opportunity to once again advertise , and ), without a very clear concentration on the essence of your product, without identifying and controlling your own key metrics. without experienced pilots, without involvement, diligence, even fanaticism. Step to the left, step to the right - and it will be difficult, almost impossible to return. If, nevertheless, a gathering has occurred, you must drop everything and try to return. This is the usefulness of my metaphor for a startup.
2) If the project is obviously outside the main sequence - it makes no sense to invest in it, it makes no sense to consider it. There is no chance. In particular, it makes no sense to consider a project that has not even begun yet, but whose main parameters from the very beginning suggest a deviation from the main sequence (“we will immediately hire 30 people”). This is the benefit of my metaphor for the investor, it helps a lot to save time.
3) And of course, we must not forget that generalizations and dogmas are useful only when you remember their rationale, and you can understand for yourself why in this particular situation the generalization will not work, and the dogma can be violated.

And finally, a few words about what the main sequence looks like for startups. (Of course, this can only be said in a very generalized way, markets, countries, etc. are very different).
It all starts in that part of the schedule where there are no users yet - and at this stage the team cannot have more than 2-3 people, and it cannot burn hundreds of thousands of rubles a month, but it would be better not to burn anything at all. The prototype is ready, the main hypothesis is formulated, promotion attempts have been started, seed funding has been raised - a team can have 5-6 people, it can spend a couple of hundred thousand a month, but there must be customers, even if in beta testing mode, and a significant part of the money should be directed not to development. The product has been created, customers are using it and have started paying the first money, we have managed to attract serious funding from business angels - the main thing at this stage is to stop the growth of development costs at some point, focusing on business development and obtaining sustainable metrics; You can't spend millions. Stable growth has been achieved, the first venture round of financing has been attracted - this is not a reason for uncontrolled inflation of staff and careless handling of money, successful projects here grow to 10-20 people, and keep their costs within 50-100 thousand dollars a month. Etc.

In short, everything is like in space, with only one difference.
There - 90% of the stars are on the main sequence, and it will not be a big exaggeration for us to say that 90% of startups are trying to find themselves outside it.
From interviews and pitches just this week:
- startup A has already spent $1.5M over two years on product development, the demand for the solution has not been proven, the user base is not growing, they are trying to attract another $2M - mainly to continue development (and who will give them? and, most importantly, by what estimate?) ,
- startup B has run out of all the money raised at the seed stage, and the founders continue to tinker with it in parallel with the main work, while competitors have gone ahead at a good pace; at one time, the founders did not take decent investments at a good estimate, trying not to blur and relying on their own strengths, and now they already agree to a much lower estimate, but ...,
- startup B is trying to raise several tens of millions of rubles at the idea stage, planning to assemble a team of about 20 people to create a prototype and test the hypothesis,
... etc.

Posted on Feb. 17th, 2013 at 02:10 pm |

The section is very easy to use. In the proposed field, just enter the desired word, and we will give you a list of its meanings. I would like to note that our site provides data from various sources - encyclopedic, explanatory, word-building dictionaries. Here you can also get acquainted with examples of the use of the word you entered.

To find

What does "main sequence" mean?

Encyclopedic Dictionary, 1998

main sequence

The MAIN SEQUENCE of the Hertzsprung-Russell diagram is a narrow band on this diagram, within which the vast majority of stars are located. Crosses the diagram diagonally (from high to low luminosities and temperatures). Main sequence stars (in particular, the Sun) have the same source of energy - thermonuclear reactions of the hydrogen cycle. Stars are on the main sequence for about 90% of the time of stellar evolution. This explains the predominant concentration of stars in the main sequence region.

Wikipedia

Main sequence

Main sequence- an area on the Hertzsprung-Russell diagram containing stars, the energy source of which is the thermonuclear reaction of helium fusion from hydrogen.

The main sequence is located in the vicinity of the diagonal of the Hertzsprung-Russell diagram and runs from the upper left corner (high luminosities, early spectral types) to the lower right corner of the diagram. Main sequence stars have the same source of energy (the “burning” of hydrogen, primarily the CNO cycle), and therefore their luminosity and temperature are determined by their mass:

L=M,

where is the luminosity L and mass M measured in units of solar luminosity and mass, respectively. Therefore, the beginning of the left part of the main sequence is represented by blue stars with masses of ~50 solar masses, and the end of the right part is represented by red dwarfs with masses of ~0.0767 solar masses.

The existence of the main sequence is due to the fact that the stage of hydrogen burning is ~90% of the time of evolution of most stars: the burning of hydrogen in the central regions of the star leads to the formation of an isothermal helium core, the transition to the red giant stage, and the departure of the star from the main sequence. The relatively short evolution of red giants leads, depending on their mass, to the formation of white dwarfs, neutron stars or black holes.

The section of the main sequence of star clusters is an indicator of their age: since the rate of evolution of stars is proportional to their mass, for clusters there is a "left" break point of the main sequence in the region of high luminosities and early spectral classes, which depends on the age of the cluster, since stars with a mass exceeding a certain limit, set by the age of the cluster, left the main sequence. Lifetime of a star on the main sequence $\tau_(\rm MS)$ depending on the initial mass of the star M with respect to the modern solar mass $\begin(smallmatrix)M_(\bigodot)\end(smallmatrix)$ can be estimated by the empirical formula:

$$\begin(smallmatrix) \tau_(\rm MS)\ \approx \ 6\cdot\ 10^(9) \text(years) \cdot \left[ \frac(M_(\bigodot))(M) + \ 0.14 \right]^(4) \end(smallmatrix)$$

In the Stellar Equilibrium problem, it was discussed that on the Hertzsprung-Russell diagram (connecting the color and luminosity of stars), most of the stars fall into the "band", which is commonly called the main sequence. Stars spend most of their lives there. A characteristic feature of main sequence stars is that their main energy release is due to the “burning” of hydrogen in the core, in contrast to T Tauri stars or, for example, giants, which will be discussed in the afterword.

It has also been discussed that different colors (the "temperature" of the surface) and luminosities (energy emitted per unit time) correspond to different masses of main sequence stars. The mass range starts from tenths of the mass of the Sun (for dwarf stars) and extends to hundreds of solar masses (for giants). But massiveness comes at the price of a very short life on the main sequence: giants spend only millions of years (and even less) on it, while dwarfs can live on the main sequence for up to ten trillion years.

In this problem, we will “from first principles”, using the results of previous problems (Stellar Equilibrium and Photon Wandering), understand why the main sequence is almost a straight line on the diagram, and how the luminosity and mass of stars are related on it.

Let be u is the energy of photons per unit volume (energy density). By definition, luminosity L is the energy radiated from the surface of a star per unit of time. In order of magnitude \(L\sim \frac(V u)(\tau) \), where V- the volume of the star, τ - a certain characteristic time for the transfer of this energy to the outside (the same time for which the photon leaves the interior of the star). As volume, again in order of magnitude, we can take R 3 , where R is the radius of the star. The energy transfer time can be estimated as R 2 /lc, where l is the mean free path, which can be estimated as 1/ρκ (ρ is the star matter density, κ is the opacity coefficient).

In equilibrium, the photon energy density is expressed according to the Stefan-Boltzmann law: u = aT 4 , where a is some constant, and T is the characteristic temperature.

Thus, omitting all constants, we obtain that the luminosity L is proportional to \(\frac(T^4 R)(\rho\kappa).\)

We also have that the pressure P must be balanced by gravity: \(P\sim \frac(M\rho)(r).\)

The compression of stars during their formation stops when an intense burning of hydrogen begins in the very center, which produces sufficient pressure. It happens at a certain temperature T, which does not depend on anything. Therefore, by and large, the characteristic temperature (in fact, this is the temperature in the center of the star, not to be confused with the surface temperature!) is the same for main sequence stars.

Task

1) For medium-mass stars (0.5< M/M ☉ < 10) давление обусловлено давлением газа P = ν RT ~ ρ T, and the opacity (for photons) is caused by Thomson scattering on free electrons, due to which the opacity coefficient is constant: κ = const. Find dependence of the luminosity of such stars on their mass. Rate the luminosity of a star that is 10 times as massive as the Sun (relative to the luminosity of the Sun).

2) For low-mass stars, the pressure is still determined by the gas pressure, and the opacity coefficient is determined mainly by other scatterings and is given by the Kramers approximation: κ ~ ρ/ T 7/2 . Decide the same problem for low-mass stars by estimating the luminosity of a star that is 10 times lighter than the Sun.

3) For massive stars with masses greater than several tens of solar masses, the opacity coefficient is due only to Thomson scattering (κ = const), while the pressure is due to the pressure of photons, not gas ( P ~ T 4). Find the dependence of luminosity on mass for such stars, and rate the luminosity of a star that is 100 times more massive than the Sun (be careful, you cannot compare with the Sun here, you need to take an intermediate step).

Hint 1

Accepting that M ~ ρ R 3, use approximate expressions for luminosity and pressure, as well as an expression for density and opacity to get rid of ρ. Characteristic temperature T is the same everywhere, as noted above, so it can also be omitted everywhere.

Hint 2

In the last paragraph, there is one dependence for solar-mass stars, and another for heavy ones, so it is impossible to immediately compare with the Sun. Instead, first calculate the luminosity for some intermediate mass (for example, 10 solar masses) using the formula for medium-mass stars, then use the formula for massive stars to find the luminosity of a star 100 times heavier than the Sun.

Decision

For stars in which the pressure that opposes gravity is provided by the pressure of an ideal gas P ~ ρ T, you can write P ~ Mρ/ R~ ρ (assuming T for a constant). Thus, for such stars we get that M ~ R which we will use below.

Note that this expression says that a star that is 10 times as massive as the Sun has about 10 times the radius.

1) Taking κ and T for constants, as well as setting ρ ~ M/R 3 and using the relation obtained above, we obtain for medium-mass stars L ~ M 3 . This means that a star 10 times more massive than the Sun will radiate 1000 times more energy per unit time (with a radius exceeding the sun's only 10 times).

2) On the other hand, for low-mass stars, assuming κ ~ ρ/ T 7/2 (T- still a constant), we have L ~ M 5 . That is, a star that is 10 times less massive than the Sun has a luminosity 100,000 times less than the sun (again, with a radius less than 10 times).

3) For the most massive stars, the ratio M ~ R no longer works. Since the pressure is provided by the photon pressure, P ~ Mρ/ r ~ T 4 ~ const. Thus, M ~ R 2 , and L ~ M. It is impossible to immediately compare with the Sun, since for stars of solar masses there is a different dependence. But we have already found out that a star 10 times more massive than the Sun has a luminosity 1000 times greater. You can compare with such a star, it gives that the star is 100 times more massive than the Sun, it radiates about 10,000 times more energy per unit of time. All this determines the shape of the main sequence curve on the Hertzsprung-Russell diagram (Fig. 1).

Afterword

As an exercise, let's also evaluate the slope of the main sequence curve in the Hertzsprung-Russell diagram. For simplicity, consider the case L ~ M 4 - the middle option between the two considered in the solution.

By definition, the effective temperature (the "temperature" of the surface) is

\[ \sigma T_(\mathrm eff)^4=\frac(L)(4\pi R^2), \]

where σ is some constant. Given that M ~ R(as we found above), we have (on average) \(L\sim T_(\rm eff)^8 \) for main sequence stars. That is, the temperature of the surface of a star that is 10 times more massive than the Sun (and shines 1000 times more intensely) will be 15,000 K, and for a star with a mass 10 times less than the sun (which shines 100,000 times less intensely) - about 1500 K .

Summarize. In the interiors of main sequence stars, “heating” takes place with the help of thermonuclear burning of hydrogen. Such combustion is a source of energy that is enough for trillions of years for the lightest stars, for billions of years for solar-mass stars, and for millions of years for the heaviest.

This energy is transformed into the kinetic energy of the gas and the energy of photons, which, interacting with each other, transfer this energy to the surface, and also provide enough pressure to counteract the gravitational contraction of the star. (But the lightest stars ( M < 0,5M☉) and heavy ( M > 3M☉) transfer also occurs with the help of convection.)

On each of the diagrams in Fig. 3 shows stars from the same cluster, because the stars from the same cluster were presumably formed at the same time. The middle diagram shows the stars in the Pleiades cluster. As can be seen, the cluster is still very young (its age is estimated at 75–150 million ns), and most of the stars are on the main sequence.

The left diagram shows a cluster that has just formed (up to 5 million years old), in which most of the stars have not even been “born” yet (if the entry onto the main sequence is considered a birth). These stars are very bright, since most of their energy is due not to thermonuclear reactions, but to gravitational contraction. In fact, they are still contracting, moving gradually down the Hertzsprung-Russell diagram (as shown by the arrow) until the temperature at the center rises enough to start effective thermonuclear reactions. Then the star will be on the main sequence (black line in the diagram) and will be there for some time. It is also worth noting that the heaviest stars ( M > 6M☉) are born already on the main sequence, that is, when they form, the temperature in the center is already high enough to initiate the thermonuclear combustion of hydrogen. Because of this, we do not see heavy protostars (on the left) in the diagram.

The right diagram shows an old cluster (12.7 billion years old). It can be seen that most of the stars have already left the main sequence, moving "up" in the diagram and becoming red giants. We will talk about this in more detail, as well as the horizontal branch, another time. However, it is worth noting here that the heaviest stars leave the main sequence before anyone else (we have already noted that you have to pay for high luminosity with a short life), while the lightest stars (to the right of the main sequence) continue to be on it. Thus, if the "inflection point" is known for the cluster - the place where the main sequence breaks off and the giant branch begins, one can fairly accurately estimate how many years ago the stars formed, that is, find the age of the cluster. Therefore, the Hertzsprung-Russell diagram is also useful for identifying very young and very old star clusters.