Distances to space objects (methods of determination)

Determination of distances to planets.

Wed. The planet's distance r from the Sun (in fractions of an AU) is found from its orbital period T:
, (1)
where r is expressed in AU and T in Earth years. The mass of the planet compared to the mass of the Sun can be neglected. Formula (1) follows from 3. Distances to the Moon and planets are determined with high accuracy by radar methods (see).

Determining the distances to the nearest stars.

Due to the annual motion of the Earth in its orbit, nearby stars move slightly relative to distant “fixed” stars. Over the course of a year, such a star describes a small ellipse on the celestial sphere, the dimensions of which become smaller the further away the star is. In angular measure, the semimajor axis of this ellipse is approximately equal to the value of max. the angle at which 1 AU is visible from the star. (semimajor axis of the earth's orbit), perpendicular to the direction of the star. This angle (), called annual or trigonometric. parallax of a star, serves to measure the distance to it based on trigonometry. relationships between the sides and angles of the triangle ZSA, in which the angle and basis are known - the semi-major axis of the earth's orbit (Fig. 1).

The distance r to the star, determined by its trigonometric value. parallax is equal to:
(a.u.), (2)
where parallax is expressed in arcseconds.

The closest star to the Solar System, the red dwarf 12th Proxima Centauri, has a parallax of 0.762, i.e. the distance to it is 1.32 pc (4.3 light years).

The lower limit of measurements is trigonometric. parallaxes ~ 0.01", so they can be used to measure distances not exceeding 100 pc (with a relative error of 50%). At distances up to 20 pc, the relative error does not exceed 10%. Distances to more distant stars in astronomy are determined in mainly by photometric method (see below).

In addition to parallactic displacements of nearby stars, only two cases can be noted when visible displacements of cosmic details. objects across the sky can also be used to accurately determine distances to them. This is several. moving nearby star clusters and rapidly moving gaseous shells or clumps. An example of the phenomenon. novae and supernovae, for the expanding shells of which, along with the apparent expansion rate in arcseconds, a spectrum can be determined. way radial expansion rate.

Photometric method for determining distances.

The illuminance created by light sources of equal power is inversely proportional to the squares of the distances to them. Consequently, the visible brilliance of identical luminaries (i.e., the illumination created near the Earth on a single area perpendicular to the rays of light) can serve as a measure of the distances to them. Expression of illuminances in magnitudes ( m- visible, M- absolute magnitude) leads to the following basic. f-le photometric. distances r f (pc):
. (3)

For luminaries whose trigonometric formulas are known. parallaxes can be determined by M using the same formula, compare physical Saints with abs. stellar magnitudes. This comparison showed that abs. the magnitudes of many classes of luminaries (stars, galaxies, etc.) can be estimated by a number of their physical properties. St.

Basic method of estimating abs. magnitudes of stars spectral: in the spectra of stars of the same spectral class, features are found that indicate their abs. magnitude (most often this is an increase in the lines of ionized atoms with increasing luminosity of stars). Based on these characteristics, stars are divided into luminosity classes (see). By classes and smaller subclasses of luminosity, estimated from the spectra of stars, one can find the abs. values ​​with an error of up to 0.5 m. This error corresponds to a relative error of 30% when determining r f using f-le (3).

There is a special tool for determining distances to star clusters. a method using the apparent magnitude-show colors diagram of the stars in the cluster. It is compared with the “absolute magnitude-show colors” diagram, which is compiled from stars of the same type in clusters close to us (Fig. 2). The vertical shift between the compared diagrams is equal to the distance modulus ( m-M), according to Krom using formula (3) and find the so-called. photometric distance r f of the star cluster (with a relative error of 20%).

For long-period Cepheids (oscillation periods from 1 to 146 days), belonging to the stellar population of type I (the flat component of the Galaxy), an important period-luminosity relationship has been established, according to which the shorter the period of brightness oscillations, the weaker the Cepheid in absolute terms. size. Using this dependence, you can determine the abs. the magnitude of Cepheids by the duration of their periods of brightness fluctuations and, therefore, photometric. distances to Cepheids and star clusters, spiral arms and star systems where they are observed (see). The error in determining distances from Cepheids is on average 40% for star clusters (in some cases less).

Determination of extragalactic distances.

Distances to nearby galaxies were determined by estimates of the apparent magnitudes of Cepheids and the brightest stars in these star systems. More than a thousand Cepheids were found in, several. hundreds - in the Andromeda Nebula. Cepheids have also been detected in seven irregular and spiral galaxies located within a radius of approx. 3 Mpc around our Galaxy.

In systems where it is not possible to detect Cepheids, they look for the brightest supergiant stars and giants of the highest luminosity classes. The brightest supergiants have been discovered in several hundreds of spiral and irregular galaxies within a radius of up to 10 Mpc (their absolute magnitudes range from -9 to -10 m). In elliptical In galaxies, the population of type I (long-period Cepheids, supergiants and hot gas nebulae) is absent. Odanko small elliptical. galaxies of our Local Group (see) in the photographs stars are breaking up, the brightest of which turned out to be red giants, similar to the giants in the globular star clusters of our Galaxy (the absolute magnitudes of these giants reach -2 m, detection radius - approx. 1 Mpc). From red giants it is possible to estimate photometrics. distances to elliptical galaxies within the Local Group of galaxies with an error of 20%.

and are also used as distance indicators.

Bright gas nebulae are observed in some galaxies. It turned out that the linear sizes of the largest nebulae in galaxies are almost the same. Therefore, by measuring the angular dimensions d" of the brightest nebula in a cosmic galaxy, it is possible to determine the distance r to this galaxy. This method is applicable to spiral and irregular galaxies up to distances of 15 Mpc. The error of this method is at least 10%.

To other galaxies photometric. distances can be determined in a more crude way by estimating the integral magnitude of the galaxy. According to the features of the external The type of spiral galaxies (thickness, length of spiral arms, surface brightness, etc.) can often roughly estimate the luminosity of the galaxy or, at least, establish that the galaxy is not a dwarf galaxy. In the latter case, its abs. the integral value can be conventionally taken equal to -20 m(cf. value for giant galaxies) and roughly estimate the distance from the apparent magnitude.

At large distances (> 1000 Mpc) the visible brightness of galaxies and other cosmic. objects is weakened not only due to the photometric law of the square of the distance, but also, in addition to the absorption of light, due to the “reddening” of distant radiation sources, reflecting the expansion of the Universe, which must be taken into account when determining the photometric. distances

Determining distances by redshift

Comparison of photometric distances to galaxies with a shift value z of their spectrum. lines to the red end of the spectrum showed that the value is proportional to the distance r (): z=Hr/c, where H is the Hubble constant. From this we obtain the formula for determining distances to distant galaxies, radio galaxies and quasars:
r=cz/H (Mpc). (4)

Up to 500 Mpc, the system is extragalactic. distances (photometric and Hubble) was verified by direct determinations of distances to supernovae from measurements of their surface temperatures and shell expansion rates. There are no reliable estimates of significantly larger distances yet.
Publications with words: distance to galactic star clusters - distance

Rice. 1. Daily parallax

In other words, the daily parallax is the angle R", under which the radius of the Earth drawn to the observation point would be visible (Fig. 1).

For a star located at the zenith at the time of observation, the daily parallax is zero. If it was shining M is observed on the horizon, then its daily parallax takes on a maximum value and is called horizontal parallax of the river.

From the relationship between the sides and angles of triangles VOLUME" And VOLUME(Fig.1) we have

R / Δ = sin p / / sin z / and R / Δ = sin p (1)

From here we get

sin p / =sin p sin z / . (2)

The horizontal parallax for all bodies of the Solar System is small (for the Moon on average R - 57", near the Sun p = 8.79", the planets are less than 1").

Therefore, the sines of the angles R and p" in the last formula can be replaced by the angles themselves and written

p" = p sin z". (3)

Due to daily parallax, the star appears to us lower above the horizon than it would be if the observation was carried out from the center of the Earth; in this case, the influence of parallax on the height of the luminary is proportional to the sine of the zenith distance, and its maximum value is equal to the horizontal parallax R.

Since the Earth has the shape of a spheroid, in order to avoid disagreements in determining horizontal parallaxes, it is necessary to calculate their values ​​for a certain radius of the Earth. The equatorial radius of the Earth Ro = 6,378 km is taken as such a radius, and the horizontal parallaxes calculated for it are called horizontal equatorial parallaxes p O . It is these parallaxes of the bodies of the Solar System that are given in all reference books.

1. Determination of distances to planets.

Another method of determination is associated with the use of Kepler's third (refined) law. In this case, the average distance r of the planet from the Sun (in fractions of an AU) is found from its orbital period T:

where r is expressed in AU and T in Earth years. The mass of the planet compared to the mass of the Sun can be neglected. Formula (4) follows from Kepler's 3rd law. Distances to the Moon and planets are determined with high accuracy using radar methods.

How can you measure the distance to stars?

Horizontal parallax method

The globe, staying at a distance of 149.6 million kilometers from the Sun, “winds” quite a distance in orbit over the course of a year.

However, truly gigantic distances begin outside. Only at the beginning of the 20th century did scientists manage to make fairly accurate measurements and for the first time establish the distance to some stars.

The method of determining the distance to the stars consists of accurately determining the direction towards them (that is, determining their position on) from both ends of the diameter of the earth's orbit and is called "Horizontal parallax method". To do this, you just need to determine the direction to the star at moments separated by six months, since during this time the Earth itself carries the observer with it from one side of its orbit to the other.

The displacement of the star (of course, apparent), caused by a change in the observer’s position in space, is extremely small, barely perceptible. But, it was measured with an accuracy of 0″.01. Is it a lot or a little? Judge for yourself - it’s the same as examining the edge of a coin from Ryazan thrown by a passer-by in Moscow on Red Square.

It is clear that at such distances and distances, the meters and kilometers we are accustomed to are no longer suitable. Truly large distances, that is, cosmic distances, are more conveniently expressed not in kilometers, but in light years, that is, in those distances that light, propagating at a speed of 300,000 km/sec, travels in a year.

Using the described method, it is possible to determine distances to stars much further away than three hundred light years. The light from the stars of some distant star systems reaches us hundreds of millions of light years away.

This does not mean at all, as is often thought, that we are observing stars that may no longer exist in reality. There is no need to say that “we see in the sky something that in reality no longer exists.” In fact, the vast majority of stars change so slowly that millions of years ago they were the same as they are now, and even their visible places in the sky change extremely slowly, although stars move quickly in space. Thus, the stars as we see them are, in general, the same at the present time.

Distances to distant celestial objects, such as stars, cannot be measured directly. They are calculated based on the measured parameters of these objects, such as the brightness of the star or the periodic change in its coordinates. Currently, several methods have been developed for calculating stellar distances, and each of them has its own limits of applicability. Let's take a closer look at how scientists determine the distance to stars.

Using parallax

Parallax is the displacement of the observed object relative to the distant background when the position of the observer changes. Knowing the distance between the observation points (parallax basis) and the magnitude of the angular displacement of the object, it is easy to calculate the distance to it. The smaller the offset value, the further away the object is. Interstellar distances are enormous, and in order to increase the angle, they use the largest possible basis - for this they measure the position of the star at opposite points of the earth's orbit. This method is called stellar annual parallax.

Now it’s easy to understand how to reach the stars using the annual parallax method. It is calculated as one of the sides of the triangle formed by the observer, the Sun and the distant star, and is equal to r = a/sin p, where: r is the distance to the star, a is the distance from the Earth to the Sun and p is the annual parallax of the star. Since the parallaxes of all stars are less than 1 arcsecond (1’’), the sine of the small angle can be replaced by the value of the angle itself in radian measure: sin p ≈ p’’/206265. Then we get: r = a∙206265/p’’, or, in astronomical units, r = 206265/p’’.

Units of interstellar distances

It is clear that the resulting formula is inconvenient, as is the expression of colossal distances in kilometers or astronomical units. Therefore, the parsec (“parallax second”; abbreviated as pc) is adopted as a generally accepted unit in stellar astronomy. This is the distance to a star whose annual parallax is 1 second. In this case, the formula takes a simple and convenient form: r = 1/p pc.

One parsec is equal to 206,265 astronomical units or approximately 30.8 trillion kilometers. In popular literature and articles, a unit such as the light year is often used - the distance that electromagnetic waves travel in a vacuum in a year without being influenced by gravitational fields. One light year is equal to about 9.5 trillion kilometers, or 0.3 parsecs. Accordingly, one parsec is approximately 3.26 light years.

Parallax Accuracy

The accuracy of parallax measurements under ground conditions currently allows the determination of distances to stars of no more than 200 parsecs. Further improvements in accuracy are achieved through observations using space telescopes.

Thus, the European satellite Hipparchos (HIPPARCOS, launched in 1989) made it possible, firstly, to increase this distance to 1000 pc, and secondly, to significantly clarify the already known stellar distances. The European satellite Gaia, or Gaia, launched in 2013, has increased the accuracy of measurements by another two orders of magnitude. Using Gaia data, astronomers both determine the distance to stars within a radius of 40 kiloparsecs and hope to discover new exoplanets. Space Telescope named after. Hubble achieves accuracy comparable to Gaia. It is probably close to the limit for optical measurements.

Despite this limitation, trigonometric annual parallax serves as a calibration basis for other methods for determining stellar distances.

Photometry. The concept of magnitude

Photometry in astronomy deals with measuring the intensity of electromagnetic radiation emitted by a celestial object, including in the optical range. Based on photometric parameters, various methods are used to determine the distance to both stars and other distant objects, for example, galaxies. One of the basic concepts used in photometric methods is magnitude, or brightness (denoted by the index m).

Visible, or relative (for the optical range - visual) stellar magnitude is measured directly by the brightness of the star and has a scale in which an increase in magnitude characterizes a decrease in brightness (this has happened historically). For example, the Sun has an apparent magnitude of -26.7 m, Sirius has a magnitude of -1.46 m, and the closest star to the Sun, Proxima Centauri, has a magnitude of +11.05 m.

Absolute magnitude is a calculated parameter. It corresponds to a visible star if this star were at a distance of 10 pc. This parameter relates the brightness of an object to its distance. For the stars given as an example, the absolute magnitude is: for the Sun +4.8 m, for Sirius +1.4 m, for Proxima +15.5 m. The distances of these stars are 0.000005, 2.64 and 1.30 parsecs, respectively. They differ in a very important astrophysical parameter - luminosity.

Spectra and luminosity of stars

Astronomers call luminosity L the total energy emitted by a star (or other object) per unit time, that is, the power of the star. Luminosity can be expressed in terms of absolute magnitude, however, unlike it, it does not depend on distance.

Based on their emission spectrum, which primarily reflects temperature (color depends on it), stars are divided into several spectral classes. Stars of the same spectral class are characterized, as a rule, by the same luminosity (there are exceptions here, but they are identified by the features of the spectrum). The “spectrum - luminosity” (or “color - magnitude”) relationship is displayed on the so-called Hertzsprung-Russell Diagram.

This diagram makes it possible to estimate their absolute magnitudes based on the spectral types of stars. And since the absolute value is connected by a simple relationship with the distance and with the visible, observable value, then it is already clear to us how the distance to the stars is determined. The formula is as follows: log r = 0.2(m - M)+1. Here r is the distance, m is the apparent magnitude and M is the absolute magnitude. The accuracy of this method is low, but it allows you to estimate the distance.

Standard candles in astronomy

There are stars whose luminosity is characterized by unambiguous correspondence to a certain physical parameter. Thanks to this, astronomers, with good accuracy, use the inverse square law to determine the distance to stars as a function of the brightness decline. The smaller the apparent magnitude of such a star, the further away the star itself is located. Such objects include, for example, Cepheids and type Ia supernovae.

Cepheids - the variables of which are strictly related to the pulsation period. By measuring the brightness and period of such a star, it is easy to calculate the distance to it. Cepheids are very bright stars. Modern telescopes are capable of resolving Cepheids in other galaxies and thus establishing the distance to the galaxy.

Type Ia supernovae are explosions of a specific type of star in close binary systems. The explosion occurs when the star reaches a certain critical mass value and always has the same luminosity and pattern of brightness decay, which also makes it possible to calculate the distance. The brightness of supernovae can be comparable to the brightness of an entire galaxy, so with their help astronomers can estimate distances on very large, cosmological scales - on the order of billions of parsecs.

Farthest

Many people know about the star closest to us - Proxima Centauri. But which of the currently known stars is located the farthest?

Belonging to our Galaxy, it was discovered not so long ago. It lies outside the Milky Way's spiral disk, on the outer edge of the galactic halo, at a distance of about 122,700 pc, or 400,000 light-years, in the constellation Libra. It is a red giant of 18 magnitude. Of course, more distant stars are also known, but it is difficult to determine exactly whether they belong to our Galaxy.

Well, which star of all known in the Universe is the most distant from us? It has the romantic name MACS J1149+2223 Lensed Star-1, or simply LS1, and is located 9 billion light years away. Its discovery is an astronomical success, since seeing a star at such a distance was possible only due to the event of gravitational microlensing in a distant galaxy, which in turn was lensed by a closer one. In this case, a different method was used to calculate the distance - by cosmological redshift. This method is used to determine distances to the most distant objects in the Universe, which cannot be resolved into individual stars. And LS1 is one of the most amazing and beautiful examples of how astronomers determine distances to stars.

Stars are the most common type of celestial body in the Universe. There are about 6000 stars up to the 6th magnitude, about a million up to the 11th magnitude, and about 2 billion of them in the entire sky up to the 21st magnitude.

All of them, like the Sun, are hot, self-luminous balls of gas, in the depths of which enormous energy is released. However, even in the most powerful telescopes, stars are visible as luminous points, since they are very far from us.

1. Annual parallax and distances to stars

The radius of the Earth turns out to be too small to serve as a basis for measuring the parallactic displacement of stars and for determining the distances to them. Even in the time of Copernicus, it was clear that if the Earth really revolves around the Sun, then the apparent positions of the stars in the sky should change. In six months, the Earth moves by the diameter of its orbit. The directions to the star from opposite points of this orbit should be different. In other words, the stars should have a noticeable annual parallax (Fig. 72).

The annual parallax of a star ρ is the angle at which the semimajor axis of the Earth's orbit (equal to 1 AU) could be seen from the star if it is perpendicular to the line of sight.

The greater the distance D to the star, the less its parallax. The parallactic shift in the position of a star in the sky throughout the year occurs in a small ellipse or circle if the star is at the pole of the ecliptic (see Fig. 72).

Copernicus tried but failed to detect the parallax of stars. He correctly argued that the stars were too far from the Earth for the instruments that existed at that time to notice their parallactic displacement.

For the first time, a reliable measurement of the annual parallax of the star Vega was carried out in 1837 by the Russian academician V. Ya. Struve. Almost simultaneously with him, in other countries the parallaxes of two more stars were determined, one of which was α Centauri. This star, which is not visible in the USSR, turned out to be the closest to us, its annual parallax is ρ = 0.75". At this angle, a wire 1 mm thick is visible to the naked eye from a distance of 280 m. It is not surprising that for so long they could not notice such stars in stars small angular displacements.

Distance to star where a is the semimajor axis of the earth's orbit. At small angles if p is expressed in arcseconds. Then, taking a = 1 a. That is, we get:

Distance to the nearest star α Centauri D=206,265": 0.75" = 270,000 AU. e. Light travels this distance in 4 years, while from the Sun to the Earth it travels only 8 minutes, and from the Moon about 1 s.

The distance that light travels in a year is called a light year. This unit is used to measure distance along with parsec (pc).

Parsec is the distance from which the semimajor axis of the earth's orbit, perpendicular to the line of sight, is visible at an angle of 1".

The distance in parsecs is equal to the reciprocal of the annual parallax expressed in arcseconds. For example, the distance to the star α Centauri is 0.75" (3/4"), or 4/3 pc.

1 parsec = 3.26 light years = 206,265 AU. e. = 3*10 13 km.

Currently, measuring annual parallax is the main method for determining distances to stars. Parallaxes have already been measured for many stars.

By measuring the annual parallax, the distance to stars located no further than 100 pc, or 300 light years, can be reliably determined.

Why is it not possible to accurately measure the annual parallax of more distant stars?

The distance to more distant stars is currently determined by other methods (see §25.1).

2. Apparent and absolute magnitude

Luminosity of stars. After astronomers were able to determine the distances to stars, it was found that stars differ in apparent brightness not only because of the difference in distance to them, but also because of the difference in their luminosity.

The luminosity of a star L is the power of light energy emitted compared to the power of light emitted by the Sun.

If two stars have the same luminosity, then the star that is farther away from us has lower apparent brightness. You can compare stars by luminosity only if you calculate their apparent brightness (stellar magnitude) for the same standard distance. This distance in astronomy is considered to be 10 pc.

The apparent magnitude that a star would have if it were at a standard distance from us D 0 = 10 pc is called the absolute magnitude M.

Let us consider the quantitative relationship between the apparent and absolute magnitudes of a star at a known distance D to it (or its parallax p). Let us first remember that a difference of 5 magnitudes corresponds to a difference in brightness of exactly 100 times. Consequently, the difference in the apparent magnitudes of two sources is equal to unity when one of them is exactly one factor brighter than the other (this value is approximately equal to 2.512). The brighter the source, the smaller its apparent magnitude is considered. In the general case, the ratio of the apparent brightness of any two stars I 1:I 2 is related to the difference in their apparent magnitudes m 1 and m 2 by a simple ratio:

Let m be the apparent magnitude of a star located at a distance D. If it were observed from a distance D 0 = 10 pc, its apparent magnitude m 0 would, by definition, be equal to the absolute magnitude M. Then its apparent brightness would change by

At the same time, it is known that the apparent brightness of a star varies inversely with the square of the distance to it. That's why

(2)

Hence,

(3)

Taking logarithm of this expression, we find:

(4)

where p is expressed in arcseconds.

These formulas give the absolute magnitude of M according to the known apparent magnitude m at a real distance to the star D. Our Sun from a distance of 10 pc would look approximately like a star of the 5th visible magnitude, i.e. for the Sun M ≈5.

Knowing the absolute magnitude M of any star, it is easy to calculate its luminosity L. Taking the luminosity of the Sun L = 1, by definition of luminosity we can write that

The values ​​of M and L in different units express the power of the star's radiation.

A study of stars shows that their luminosity can differ by tens of billions of times. In stellar magnitude, this difference reaches 26 units.

Absolute values stars of very high luminosity are negative and reach M = -9. Such stars are called giants and supergiants. The radiation of the star S Dorado is 500,000 times more powerful than the radiation of our Sun, its luminosity is L=500,000, dwarfs with M=+17 (L=0.000013) have the lowest radiation power.

To understand the reasons for significant differences in the luminosity of stars, it is necessary to consider their other characteristics, which can be determined based on radiation analysis.

3. Color, spectra and temperature of stars

During your observations, you noticed that the stars have different colors, clearly visible in the brightest of them. The color of a heated body, including a star, depends on its temperature. This makes it possible to determine the temperature of stars by the energy distribution in their continuous spectrum.

The color and spectrum of stars are related to their temperature. In relatively cool stars, radiation in the red region of the spectrum predominates, which is why they have a reddish color. The temperature of red stars is low. It grows sequentially as it moves from red stars to orange, then to yellow, yellowish, white and bluish. The spectra of stars are extremely diverse. They are divided into classes, designated by Latin letters and numbers (see back flyleaf). In the spectra of cool red class M stars with a temperature of about 3000 K, absorption bands of the simplest diatomic molecules, most often titanium oxide, are visible. The spectra of other red stars are dominated by carbon or zirconium oxides. Red stars of the first magnitude class M - Antares, Betelgeuse.

In the spectra of yellow class G stars, which includes the Sun (with a temperature of 6000 K on the surface), thin lines of metals predominate: iron, calcium, sodium, etc. A star like the Sun in spectrum, color and temperature is the bright Capella in the constellation Auriga.

In the spectra of class A white stars, like Sirius, Vega and Deneb, the hydrogen lines are the strongest. There are many weak lines of ionized metals. The temperature of such stars is about 10,000 K.

In the spectra of the hottest, bluish stars with a temperature of about 30,000 K, lines of neutral and ionized helium are visible.

The temperatures of most stars range from 3000 to 30,000 K. A few stars have temperatures around 100,000 K.

Thus, the spectra of stars are very different from each other and from them one can determine the chemical composition and temperature of the atmospheres of stars. A study of the spectra showed that hydrogen and helium are predominant in the atmospheres of all stars.

Differences in stellar spectra are explained not so much by the diversity of their chemical composition as by differences in temperature and other physical conditions in stellar atmospheres. At high temperatures, molecules break down into atoms. At an even higher temperature, less strong atoms are destroyed, they turn into ions, losing electrons. Ionized atoms of many chemical elements, like neutral atoms, emit and absorb energy at certain wavelengths. By comparing the intensity of absorption lines of atoms and ions of the same chemical element, their relative amount is theoretically determined. It is a function of temperature. Thus, the temperature of their atmospheres can be determined from the dark lines in the spectra of stars.

Stars of the same temperature and color, but different luminosities, have generally the same spectra, but differences in the relative intensities of some lines can be seen. This occurs because at the same temperature the pressure in their atmospheres is different. For example, in the atmospheres of giant stars there is less pressure and they are more rarefied. If we express this dependence graphically, then from the intensity of the lines we can find the absolute magnitude of the star, and then using formula (4) we can determine the distance to it.

Example of problem solution

Task. What is the luminosity of the star ζ Scorpii if its apparent magnitude is 3 and the distance to it is 7500 ly. years?

Exercise 20

1. How many times is Sirius brighter than Aldebaran? Is the sun brighter than Sirius?

2. One star is 16 times brighter than the other. What is the difference in their magnitudes?

3. Vega's parallax is 0.11". How long does the light from it take to reach the Earth?

4. How many years would it take to fly towards the constellation Lyra at a speed of 30 km/s for Vega to become twice as close?

5. How many times is a star of magnitude 3.4 fainter than Sirius, which has an apparent magnitude of -1.6? What are the absolute magnitudes of these stars if the distance to both is 3 pc?

6. Name the color of each of the stars in Appendix IV according to their spectral type.