How to find the distance from the earth to a star. How are distances to stars measured? Determining distance by relative speeds

Scientists have long assumed that stars have the same physical nature as the Sun. Due to the colossal distances, the disks of stars are not visible even with powerful telescopes. To compare stars with each other and with the Sun, it is necessary to find methods for determining the distances to them. The main method is the method of parallactic displacement of stars, which we discussed earlier. Since the radius of the Earth is too small compared to the distance to the stars, it is necessary to choose a larger basis for measuring the parallactic displacement of the stars. Even N. Copernicus understood that, according to his heliocentric system, close stars against the background of distant stars should describe ellipses as a result of the annual movement of the Earth around the Sun.

The apparent movement of a closer star against the background of very distant stars occurs along an ellipse with a period of 1 year and reflects the movement of the observer along with the Earth around the Sun. The position of the Earth in orbit and the position of the star in the sky visible from Earth in this figure are indicated by the same numbers. The small ellipse described by a star is called a parallactic ellipse. In angular measure, the semi-major axis of this ellipse is equal to the angle at which the semi-major axis of the earth's orbit is visible from the star, perpendicular to the direction to the star. This angle is called annual parallax(\(\pi\)). The parallactic displacements of stars serve as irrefutable evidence of the Earth's revolution around the Sun.

Distances to stars are determined by their annual parallactic displacement, which is determined by the movement of the observer (along with the Earth) along the Earth's orbit.

If \(CT = a\) is the average radius of the Earth's orbit, \(SC = r\) is the distance to the star \(S\) from the Sun \(C\), and the angle \(\pi \) is the annual parallax of the star , That \

Since the annual parallaxes of stars are estimated in decimal fractions of a second, and 1 radian is equal to \((206\:265)""\), the distance to the star can be determined from the relation \

When measuring distances to stars, the astronomical unit is too small. Therefore, for the convenience of determining distances to stars in astronomy, a special unit of length is used - parsec (pc), the name of which comes from the words “parallax” and “second”. Parsec- this is the distance from which the radius of the earth's orbit would be visible at an angle of \(1""\).

According to the formula \(r = \frac((206\:265)"")((\pi)"")\), \(1\:pk = 206\:265\:a.\:e. = 3.086 \cdot 10^(13)\:km\). Thus, the distance to stars in parsecs will be determined by the expression \

Distances to solar system bodies are usually expressed in astronomical units. Distances to celestial bodies outside the Solar System are usually expressed in parsecs, kiloparsecs (\(1\:kpc = 10^(3)\:pc\)) and megaparsecs (\(1\:Mpc = 10^(6 )\:pk\)), as well as in light years (\(1\:st.\:g. = 9.46 \cdot 10^(12)\:km = 63\:240\:a.\: e. = 0.3067\:pc\) or \(1\:pc = 3.26\:sv.\:g.\)). Light year- the distance that electromagnetic radiation (in vacuum) travels in 1 year.

The lower limit of parallax measurements does not exceed \((0.005)""\), which makes it possible to determine distances of no more than 200 pc. Distances to even more distant objects are determined less accurately and using other methods.

Introduction................................................. ............................ 3

Determination of distances to space objects. 3

Determining distances to planets.................................................... .............. 4

Determining distances to the nearest stars.................................................... 4

Parallax method............................................................................................. 4

Determination of distance by relative speeds.........................

Cepheids.............................................................................................................. 8

Bibliography................................................ ........... 9

Introduction.

Our knowledge of the Universe is closely related to man's ability to determine distances in space. Since time immemorial, the question “how far?” played a primary role for the astronomer in his attempts to understand the properties of the Universe in which he lives. But no matter how great was man’s desire for knowledge, it could not be realized until highly sensitive and perfect instruments were at people’s disposal. Thus, although ideas about the physical world continually evolved over the centuries, the veils that hid the milestones of space remained intact. In all centuries, philosophers and astronomers have thought about cosmic distances and diligently searched for ways to measure them. But everything was in vain, since the tools necessary for this could not be made. And finally, after telescopes had been used for many years by astronomers and the first geniuses had devoted their talent to studying the riches obtained by these telescopes, the time had come for the union of precise mechanics and perfect optics, which made it possible to create an instrument capable of solving the problem of distances. Barriers were removed, and many astronomers combined their knowledge, skill and intuition to determine the colossal distances that separate the stellar worlds from us.

In 1838, three astronomers (in different parts of the world) successfully measured the distances to some stars. Friedrich Wilhelm Bessel in Germany determined the distance to the star Cygnus 61. The outstanding Russian astronomer Vasily Struve determined the distance to the star Vega. At the Cape of Good Hope in South Africa, Thomas Henderson measured the distance to the closest star to the Sun - Alpha Centauri. In all these cases, astronomers measured incredibly small angular distances to determine the so-called parallax. Their success was due to the fact that the stars to which they measured distances were relatively close to Earth.

Determination of distances to space objects.

In astronomy, there is no single universal way to determine distances. As we move from close celestial bodies to more distant ones, some methods for determining distances are replaced by others, which, as a rule, serve as the basis for subsequent ones. The accuracy of distance estimation is limited either by the accuracy of the roughest method or by the accuracy of measurement of the astronomical unit of length (AU), the value of which is known from radar measurements with a root-mean-square error of 0.9 km. and is equal to 149597867.9 ± 0.9 km. Taking into account various changes a. e. The International Astronomical Union adopted the value 1 a in 1976. e. = 149597870 ± 2 km.

Determination of distances to planets.

Average distance r planets from the Sun (in fractions of AU) are found by their period of revolution T :

Where r expressed in a. e., a T - in earth years. Mass of the planet m compared to the mass of the sun m c can be neglected. The formula follows from Kepler's third law (the squares of the periods of revolution of the planets around the Sun are proportional to the cubes of their average distances from the Sun).

Distances to the Moon and planets are also determined with high accuracy using planetary radar methods.

Determining the distances to the nearest stars.

Parallax method.

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 maximum angle at which 1 AU is visible from the star. e. (semimajor axis of the earth's orbit), perpendicular to the direction of the star. This angle (p), called the annual or trigonometric parallax of the star, equal to half of its apparent displacement per year, serves to measure the distance to it based on the trigonometric relationships between the sides and angles of the triangle ZSA, in which the angle p and the basis is the semimajor axis of the earth's orbit. (see Fig. 1).

Distance r to the star, determined by the value of its trigonometric parallax p, is equal to:

r = 206265""/p (a.u.),

where parallax p is expressed in arcseconds.

For the convenience of determining distances to stars using parallaxes, astronomy uses a special unit of length - the parsec (ps). A star located at a distance of 1 pc has a parallax of 1 "". According to the above formula, 1 ps = 206265 a. e. = 3.086 10 18 cm.

Along with the parsec, another special unit of distance is used - the light year (i.e., the distance that light travels in 1 year), it is equal to 0.307 ps, or 9.46 10 17 cm.

The closest star to the Solar System, the 12th magnitude red dwarf Proxima Centauri, has a parallax of 0.762, i.e. the distance to it is 1.31 ps (4.3 light years).

The lower limit for measuring trigonometric parallaxes is ~0.01"", so they can be used to measure distances not exceeding 100 ps with a relative error of 50%. (At distances up to 20 ps, the relative error does not exceed 10%.) Distances to about 6000 stars have been determined so far by this method. Distances to more distant stars in astronomy are determined mainly by the photometric method.

Table 1. Twenty nearest stars.

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 apparent brightness 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 distance to them. Expression of illuminances in magnitudes ( m – apparent magnitude, M – absolute magnitude) leads to the following basic formula for photometric distances r f (ps).

The distance between the Earth and the Moon is enormous, but it seems tiny compared to the scale of space.

Space, as we know, is quite large, and therefore astronomers do not use the metric system, which is familiar to us, to measure them. In the case of distances up to (384,000 km), kilometers may still be applicable, but if we express the distance to Pluto in these units, we get 4,250,000,000 km, which is less convenient for recording and calculations. For this reason, astronomers use other units of distance measurement, which you will read about below.

The smallest of these units is (a.u.). Historically, one astronomical unit is equal to the radius of the Earth’s orbit around the Sun, otherwise it is the average distance from the surface of our planet to the Sun. This measurement method was most suitable for studying the structure of the Solar system in the 17th century. Its exact value is 149,597,870,700 meters. Today, the astronomical unit is used in calculations with relatively small lengths. That is, when studying distances within the Solar System or planetary systems.

Light year

A slightly larger unit of length in astronomy is . It is equal to the distance that light travels in a vacuum in one earthly, Julian year. It also implies zero influence of gravitational forces on its trajectory. One light year is about 9,460,730,472,580 km or 63,241 AU. This unit of measurement of length is used only in popular science literature for the reason that the light year allows the reader to get a rough idea of distances on a galactic scale. However, due to its inaccuracy and inconvenience, the light year is practically not used in scientific work.

Parsec

The most practical and convenient unit for astronomical calculations is the unit of measurement of distance. To understand its physical meaning, one should consider the phenomenon of parallax. Its essence is that when the observer moves relative to two bodies distant from each other, the apparent distance between these bodies also changes. In the case of stars, the following happens. As the Earth moves in its orbit around the Sun, the visual position of stars close to us changes somewhat, while distant stars, acting as a background, remain in the same places. The change in the position of a star when the Earth moves by one radius of its orbit is called annual parallax, which is measured in arcseconds.

Then one parsec is equal to the distance to a star whose annual parallax is equal to one arcsecond - the unit of measurement of angle in astronomy. Hence the name “parsec”, a combination of two words: “parallax” and “second”. The exact value of a parsec is 3.0856776 10 16 meters or 3.2616 light years. 1 parsec is equal to approximately 206,264.8 AU. e.

Laser ranging and radar method

These two modern methods are used to determine the exact distance to an object within the Solar System. It is done as follows. Using a powerful radio transmitter, a directed radio signal is sent towards the object of observation. After which the body repels the received signal and returns it to Earth. The time spent by the signal to cover the path determines the distance to the object. The radar accuracy is only a few kilometers. In the case of laser ranging, instead of a radio signal, the laser sends a light beam, which allows similar calculations to determine the distance to the object. Laser location accuracy is achieved down to fractions of a centimeter.

Trigonometric parallax method

The simplest method for measuring the distance to distant space objects is the trigonometric parallax method. It is based on school geometry and consists of the following. Let's draw a segment (basis) between two points on the earth's surface. Let's select an object in the sky, the distance to which we intend to measure, and define it as the vertex of the resulting triangle. Next, we measure the angles between the basis and straight lines drawn from the selected points to the body in the sky. And knowing the side and two adjacent angles of a triangle, you can find all its other elements.

The value of the selected basis determines the accuracy of the measurement. After all, if the star is located at a very large distance from us, then the measured angles will be almost perpendicular to the basis and the error in their measurement can significantly affect the accuracy of the calculated distance to the object. Therefore, you should choose the most distant points on as a basis. Initially, the radius of the Earth acted as a basis. That is, observers were located at different points on the globe and measured the mentioned angles, and the angle located opposite the base was called horizontal parallax. However, later they began to take a larger distance as a basis - the average radius of the Earth's orbit (astronomical unit), which made it possible to measure the distance to more distant objects. In this case, the angle lying opposite the basis is called the annual parallax.

This method is not very practical for research from the Earth for the reason that, due to interference from the Earth’s atmosphere, it is not possible to determine the annual parallax of objects located more than 100 parsecs away.

However, in 1989, the European Space Agency launched the Hipparcos space telescope, which made it possible to identify stars at distances of up to 1000 parsecs. As a result of the data obtained, scientists were able to create a three-dimensional map of the distribution of these stars around the Sun. In 2013, ESA launched a follow-up satellite, Gaia, which has 100 times better measurement accuracy, making it possible to observe all stars. If human eyes had the precision of the Gaia telescope, we would be able to see the diameter of a human hair from a distance of 2,000 km.

Standard candle method

To determine the distances to stars in other galaxies and the distances to these galaxies themselves, the standard candle method is used. As you know, the further the light source is located from the observer, the dimmer it appears to the observer. Those. the illumination of a light bulb at a distance of 2 m will be 4 times less than at a distance of 1 meter. This is the principle by which the distance to objects is measured using the standard candle method. Thus, by drawing an analogy between a light bulb and a star, one can compare the distances to light sources with known powers.

Standard candles in astronomy are objects whose source power is known. It can be any kind of star. To determine its luminosity, astronomers measure the surface temperature based on the frequency of its electromagnetic radiation. After which, knowing the temperature that allows one to determine the spectral class of the star, its luminosity is determined using. Then, having the luminosity values and measuring the brightness (apparent magnitude) of the star, you can calculate the distance to it. This standard candle allows you to get a general idea of the distance to the galaxy in which it is located.

However, this method is quite labor-intensive and is not highly accurate. Therefore, it is more convenient for astronomers to use cosmic bodies with unique features for which the luminosity is initially known as standard candles.

Unique standard candles

The most commonly used standard candles are variable pulsating stars. Having studied the physical characteristics of these objects, astronomers learned that Cepheids have an additional characteristic - a pulsation period, which can be easily measured and which corresponds to a certain luminosity.

As a result of observations, scientists are able to measure the brightness and pulsation period of such variable stars, and therefore their luminosity, which allows them to calculate the distance to them. Finding a Cepheid in another galaxy makes it possible to relatively accurately and simply determine the distance to the galaxy itself. Therefore, this type of star is often called the “beacons of the Universe.”

Although the Cepheid method is most accurate at distances up to 10,000,000 pc, its error can reach 30%. To increase accuracy, you will need as many Cepheids as possible in one galaxy, but even in this case the error is reduced to no less than 10%. The reason for this is the inaccuracy of the period-luminosity relationship.

Cepheids are “beacons of the Universe.”

In addition to Cepheids, other variable stars with known period-luminosity relationships can be used as standard candles, as well as supernovae with known luminosity for the largest distances. Close in accuracy to the Cepheid method is the method with red giants as standard candles. As it turned out, the brightest red giants have an absolute magnitude in a fairly narrow range, which makes it possible to calculate the luminosity.

Distances in numbers

Distances in the Solar System:

1 a.u. from Earth to = 500 St. seconds or 8.3 light. minutes
30 a. e. from the Sun to = 4.15 light hours
132 a.u. from the Sun - this is the distance to the spacecraft "", was noted on July 28, 2015. This object is the most distant of those that have been constructed by man.

Distances in the Milky Way and beyond:

1.3 parsecs (268144 AU or 4.24 light years) from the Sun to the closest star to us
8,000 parsecs (26 thousand light years) - the distance from the Sun to the Milky Way
30,000 parsecs (97 thousand light years) - the approximate diameter of the Milky Way
770,000 parsecs (2.5 million light years) - distance to the nearest large galaxy -
300,000,000 pc - the scale at which it is almost uniform
4,000,000,000 pc (4 gigaparsecs) is the edge of the observable Universe. This is the distance traveled by the light recorded on Earth. Today, the objects that emitted it, taking into account , are located at a distance of 14 gigaparsecs (45.6 billion light years).

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 semi-major 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 detect 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 the 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.

Introduction................................................. ............................ 3

Determination of distances to space objects. 3

Determining distances to planets.................................................... .............. 4

Determining distances to the nearest stars.................................................... 4

Parallax method. ............................................................................................ 4

Photometric method for determining distances. ................................. 6

........................

Cepheids. ............................................................................................................. 8

Bibliography................................................ ........... 9

Introduction.

Determination of distances to space objects.

Determination of distances to planets.

Average distance r planets from the Sun (in fractions of AU) are found by their period of revolution T :

Distances to the Moon and planets are also determined with high accuracy using planetary radar methods.

Determining the distances to the nearest stars.

Parallax method.

Distance r to the star, determined by the value of its trigonometric parallax p, is equal to:

r = 206265""/p (a.u.),

where parallax p is expressed in arcseconds.

Determination of distance by relative speeds.

An indirect indicator of the distance to stars is their relative speeds: as a rule, the closer the star, the more it moves along the celestial sphere. Of course, it is impossible to determine the distance in this way, but this method makes it possible to “catch” nearby stars.

There is also another method for determining distances from velocities that is applicable to star clusters. It is based on the fact that all stars belonging to the same cluster move in the same direction along parallel trajectories. By measuring the radial velocity of stars using the Doppler effect, as well as the speed with which these stars move relative to very distant, that is, conventionally fixed stars, we can determine the distance to the cluster of interest to us.

Cepheids.

An important method for determining photometric distances in the Galaxy and to neighboring stellar systems - galaxies - is based on the characteristic property of variable stars - Cepheids.

The first Cepheid discovered was d Cephei, which changed its brightness with an amplitude of 1, temperature (by 800K), size and spectral type. Cepheids are unstable stars of spectral classes from F6 to G8, which pulsate as a result of an imbalance between gravity and internal pressure, and the curve of changes in their parameters resembles a harmonic law. Over time, the vibrations weaken and die out; to date, a gradual cessation of variability has been discovered in the star RU Giraffi, discovered in 1899. By 1966, its variability had completely ceased. The periods of various Cepheids range from 1.5 hours to 45 days. All Cepheids are giants of great luminosity, and the luminosity strictly depends on the period according to the formula:

M= – 0.35 – 2.08 lg T .

Since, in contrast to the above Hertzsprung–Russell diagram (see Fig. 2), the dependence is clear, the distances can be determined more accurately. 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 fainter the Cepheid in absolute value. Knowing from observations the period T , you can find the absolute magnitude M , and, knowing the absolute magnitude and finding the apparent magnitude from observations m , you can find the distance. This method of finding distances is used not only to determine the distance to the Cepheids themselves, but also to determine the distances to distant galaxies in which Cepheids were discovered (this is not very difficult to do, since Cepheids have a fairly high luminosity).