Determining the distance to solar system bodies. a) Determination of the radius of the Earth

And leaves the battlefield,
And Apollo retreats.
Other knights start
To the networks of Saturn's rings,
To where Io's breath burns
And it feels like the end
That Amazing System
Domains of the Royal Star,
Which we are all natives of.
I. Galkin

Lesson 5/11

Subject: Determination of distances to SS bodies and sizes of these celestial bodies.

Target: Consider various ways determining the distance to SS bodies. Give the concept of horizontal parallax and establish a method for finding the distance and size of bodies through horizontal parallax.

Tasks :
1. Educational: Introduce the concepts of geometric (parallactic), “radar” and “laser” methods for determining distances to solar system bodies. Derive a formula for determining the radius of the celestial bodies of the Solar System (concepts: linear radius, angular radius). Use problem solving to continue building calculation skills.
2. Educating: revealing the topic of the lesson what modern science has various methods determining distances to celestial bodies and their sizes in order to obtain reliable information about the scale of the solar system and the sizes of the celestial bodies included in it, to contribute to the formation of an ideological idea about the knowability of the world.
3. Developmental: show that the seemingly insoluble problem of determining the distances to celestial bodies and the radii of celestial bodies is currently being solved by various methods.

Know:
Level I (standard)- methods for determining distances to SS bodies, the concept of basis and parallax, a method for determining the size of the Earth and any celestial body.
II level- methods for determining distances to SS bodies, the concept of basis and parallax, a method for determining the size of the Earth and any celestial body. That the diameter of the Moon is as many times smaller than the diameter of the Sun as the distance from the Moon to the Earth is smaller than the distance from the Earth to the Sun.

Be able to:
Level I (standard)
II level- determine distances to SS bodies using parallax and radar data, determine the size of celestial bodies.

Equipment: Tables: “Solar System”, theodolite, film “Radar”, slides, filmstrip “Determination of distances to celestial bodies”. CD - "Red Shift 5.1". SHAK.

Intersubject communication: Degree and radian measures of angle, adjacent and vertical angles. Ball and sphere (mathematics, 5, 7, 10, 11 classes). Distance from the Earth to the Moon and the Sun. Comparative sizes Sun and Earth, Earth and Moon (natural history, 5th grade). Spread speed electromagnetic waves. Radar method (physics, 11 classes).

Lesson progress:

I. Survey of students (5-7 minutes). Dictation.

II New material

1) Determination of distances to celestial bodies.
There is no one thing in astronomy universal method determining 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 crudest method or by the accuracy of the measurement of the astronomical unit of length (AU).
1st method: (known) According to Kepler's third law, it is possible to determine the distance to the SS bodies, knowing the periods of revolutions and one of the distances.

Approximate method.

2nd method: Determination of distances to Mercury and Venus at moments of elongation (from right triangle according to the elongation angle).
3rd method: Geometric (parallactic).
Example: Find the unknown distance AC.

[AB] - Basis is the main known distance, since the angles CAB and CBA are known, then using the formulas of trigonometry (theorem of sines) it is possible to? find the unknown side, i.e. Parallax displacement is the change in direction of an object when the observer moves.
Parallax is the angle at which the basis is visible from an inaccessible place(AB is a known segment). Within the SS, the equatorial radius of the Earth R = 6378 km is taken as the basis.

Let K be the location of the observer from which the luminary is visible on the horizon. From the figure it can be seen that from a right triangle the hypotenuse, the distance D equals: , since with a small value of the angle, if we express the value of the angle in radians and take into account that the angle is expressed in seconds of arc, and 1rad =57.3 0 =3438"=206265" , then the second formula is obtained.

The angle (ρ) at which the equatorial radius of the Earth would be visible from a luminary located on the horizon (? R - perpendicular to the line of sight) is called the horizontal equatorial parallax of the luminary.
Because no one will observe from the luminary due to objective reasons, then the horizontal parallax is determined as follows:

1. we measure the height of the luminary at the moment of the upper culmination from two points earth's surface located on the same geographic meridian and having known geographic latitudes.
2. All angles (including parallax) are calculated from the resulting quadrilateral.

From history: The first parallax measurement (parallax of the Moon) is made at 129g to NE Hipparchus(180-125, Ancient Greece).
For the first time, distances to celestial bodies (Moon, Sun, planets) are estimated Aristotle(384-322, Ancient Greece) in 360 BC in the book “On the Sky” →too imprecise, for example, the radius of the Earth is 10,000 km.
In 265g to NE Aristarchus of Samos(310-230, Ancient Greece) in the work “On the magnitude and distance of the Sun and Moon” determines the distance through lunar phases. So his distances to the Sun (according to the phase of the Moon in 1 quarter of a right triangle, i.e. for the first time he uses the basic method: ZS=ZL/cos 87º≈19*ZL). The radius of the Moon was determined to be 7/19 of the radius of the Earth, and of the Sun 6.3 of the radius of the Earth (actually 109 times). In fact, the angle is not 87º but 89º52" and therefore the Sun is 400 times farther than the Moon. The proposed distances have been used by astronomers for many centuries.
In 240g to NE ERATOSTHENES(276-194, Egypt) having made measurements on June 22 in Alexandria of the angle between the vertical and the direction of the Sun at noon (he believed that since the Sun is very far away, the rays are parallel) and using recordings of observations on the same day of the fall of light rays into a deep well in Siene (Aswan) (in 5000 stadia = 1/50 of the earth's circumference (about 800 km), i.e. the Sun was at its zenith) receives an angle difference of 7º12" and determines the size globe, obtaining a circumference of the ball of 39690 km (radius = 6311 km). This is how the problem of determining the size of the Earth was solved using the astrogeodetic method. The result was not produced until the 17th century, only astronomers at the Baghdad Observatory in 827 slightly corrected his error.
In 125g to NE Hipparchus quite accurately determines (in Earth radii) the radius of the Moon (3/11 R ⊕ ) and the distance to the Moon (59 R ⊕ ).
Accurately determined the distance to the planets, taking the distance from the Earth to the Sun as 1 AU, N. Copernicus.
The closest body to the Earth, the Moon, has the greatest horizontal parallax. R? =57"02" ; and for the Sun Р ¤ =8.794 "
Problem 1 : textbook Example No. 6 - Find the distance from the Earth to the Moon, knowing the parallax of the Moon and the radius of the Earth.
Problem 2 : (on one's own). At what distance from the Earth is Saturn if its parallax is 0.9".
4th method Radar: impulse→object →reflected signal→time. Proposed by Soviet physicists L.I. Mandelstam And N.D. Papaleksi. The rapid development of radio technology has given astronomers the opportunity to determine distances to solar system bodies using radar methods. In 1946, the first radar of the Moon was carried out by Bai in Hungary and the USA, and in 1957-1963 - radar of the Sun (studies of the solar corona have been carried out since 1959), Mercury (since 1962 at ll = 3.8, 12, 43 and 70 cm), Venus, Mars and Jupiter (in 1964 at waves l = 12 and 70 cm), Saturn (in 1973 at wave l = 12.5 cm) in the UK, USSR and USA. The first echo signals from the solar corona were received in 1959 (USA), and from Venus in 1961 (USSR, USA, Great Britain). According to the speed of propagation of radio waves With= 3 × 10 5 km/sec and over time t(sec) the passage of a radio signal from the Earth to a celestial body and back, it is easy to calculate the distance to the celestial body.
V EMW =С=299792458m/s≈3*10 8 m/s.

The main difficulty in studying celestial bodies using radar methods is due to the fact that the intensity of radio waves during radar is attenuated in inverse proportion to the fourth power of the distance to the object being studied. Therefore, radars used to study celestial bodies have antennas large sizes and powerful transmitters. For example, the radar installation of the deep space communications center in Crimea has an antenna with a main mirror diameter of 70 m and is equipped with a transmitter with a power of several hundred kW at a wave of 39 cm. The energy directed to the target is concentrated in a beam with an opening angle of 25".
From the radar of Venus, the value of the astronomical unit has been clarified: 1 a. e. = 149 597 870 691 ± 6 m ≈149.6 million km, which corresponds to Р ¤ = 8.7940". This is how the processing of data from radar measurements of the distance to Venus carried out in the Soviet Union in 1962-75 (one of the first successful experiments on the radar of Venus was carried out by employees of the Institute of Radio Engineering and Electronics of the USSR Academy of Sciences in April 1961 with a long-distance space communications antenna in Crimea, l = 39 cm) gave a value of 1 AU = 149597867.9 ± 0.9 km XVI. General Assembly The International Astronomical Union adopted in 1976 the value 1 au = 149597870 ± 2 km. By means of radar from spacecraft, the surface relief of planets and their satellites is determined, and their maps are compiled.
The main antennas used for planetary radar are:
= Evpatoria, Crimea, diameter 70 m, l = 39 cm;
= Arecibo, Puerto Rico, diameter 305 m, l = 12.6 cm;
= Goldstone, California, diameter 64 m, l = 3.5 and 12.6 cm, in bistatic mode reception is carried out on the VLA aperture synthesis system.

With the invention of Quantum Generators ( laser) in 1969, the first laser location of the Moon was carried out (a mirror for reflecting a laser beam on the Moon was installed by US astronauts "Apollo - 11" on July 20, 1969), the measurement accuracy was ±30 cm. The figure shows the location of laser corner reflectors on the Moon installed during the spacecraft flight "Luna-17, 21" and "Apollo - 11, 14, 15". All, with the exception of the Lunokhod-1 (L1) reflector, are still working.
Laser (optical) location is needed for:
-solving space research problems.
-solving space geodesy problems.
-clarification of the issue of the movement of the earth’s continents, etc.

2) Determination of the sizes of celestial bodies.

a) Determination of the radius of the Earth.

b) Determination of the size of celestial bodies.

III. Fixing the material

1. Example 7(page 51).
2. CD- "Red Shift 5.1" - Determine on at the moment distance of the lower (terrestrial planets, upper planets, giant planets) from the Earth and the Sun in a.u.
3. The angular radius of Mars is 9.6" and the horizontal parallax is 18". What is the linear radius of Mars? [From formula 22 we get 3401.6 km. (actually 3396 km)].
4. What is the distance between the laser reflector on the Moon and the telescope on Earth if the pulse returns after 2.43545 s? [ from the formula R=(c . t)/2 R=3 . 10 8. 2.43545/2≈365317500.92m≈365317.5km]
5. The distance from the Earth to the Moon at perigee is 363,000 km, and at apogee 405,000 km. Determine the horizontal parallax of the Moon at these positions. [ from the formula D=(206265"/p)*R ⊕ hence p=(206265"/D)*R ⊕ ; p A = (206265"/405000)*6378≈3248.3"≈54.1", p P = (206265"/363000)*6378≈3624.1"≈60.4"].
6. with pictures for chapter 2.
7. Additionally, for those who did it - a crossword puzzle.

Result:
1) What is parallax?
2) In what ways can you determine the distance to SS bodies?
3) What is a basis? What is taken as the basis for determining the distance to SS bodies?
4) How does parallax depend on the distance of the celestial body?
5) How does the size of a body depend on the angle?
6) Ratings

Homework: §11; questions and tasks p. 52, pp. 52-53 know and be able to. Repeat the second chapter in its entirety. , .
You can ask for this section to prepare a crossword puzzle, a survey, an essay about one of the astronomers or the history of astronomy (one of the questions or directions).
Can you suggest practical work "Determination of the size of the Moon."
During the full moon, using two rulers connected at right angles, the apparent dimensions of the lunar disk are determined: since triangles KCD and KAB are similar, it follows from the triangle similarity theorem that: AB/CD = KB/KD. Moon diameter AB = (CD . KB)/KD. You take the distance from the Earth to the Moon from reference tables (but it’s better if you can calculate it yourself).

I designed the lesson members of the Internet Technologies circle - Leonenko Katya(11kl)

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According to the theory of universal gravitation, any massive, isolated body rotating around an axis at a certain speed (not very fast) should take on a shape close to a ball. Indeed, all observed massive celestial bodies (Sun, Moon, planets) have shapes that differ little from regular spheres. The spherical shape of the Earth is clearly visible in its photographs taken from space (1967-1969).

The sphericity of the Earth makes it possible to determine its size in a way that was first used by Eratosthenes in the 3rd century. BC e. The idea of ​​this method is simple. Let's take two points on the globe O 1 and O 2, lying on the same geographical meridian (Fig. 38). Let us denote the length of the meridian arc O 1 O 2 (for example, in kilometers) by , and its angular value (for example, in degrees) - through °. Then the arc length is 1° meridian will be equal, and the length of the entire circumference of the meridian where R is the radius of the globe. From here

Angular value of arc ° equals the difference geographical latitudes points O 1 and O 2, i.e. ° = -.

It is much more difficult to determine linear distance between points O 1 and O 2. Arc length determined by calculations using a special method that requires direct measurement of only a relatively small distance - the basis and a number of angles. This method was developed in geodesy and is called triangulation.

The essence of the triangulation method is as follows. On both sides of the arc O 1 O 2 (Fig. 39), the length of which must be determined, several points A, B, C, ... are selected at distances of 30-40 km from one another. Points are selected so that at least two other points are visible from each. At all points, geodetic signals are installed - towers in the shape of pyramids - several tens of meters high. At the top of the signal there is a platform for the observer and the instrument. The distance between any two points, for example O 1 A, is selected on a completely flat surface and taken as the basis. The length of the base is very carefully measured directly using special measuring tapes. The most accurate modern measurements of a 10 km long baseline are made with an error of ±2 mm. Then install a goniometer tool (theodolite)

sequentially at points O 1, A, B, C, ..., O 2 and measure all angles of triangles O 1 AB, ABC, BCD, ... Knowing in triangle O 1 AB all angles and side O 1 A (base) , we can calculate its other two sides O 1 B and AB. These calculations take into account that the triangles are not flat, but spherical. Next, having determined the azimuth of the direction of the side O 1 B (or O 1 A) from point O 1, you can project the broken line O 1 BDO 2 (or O 1 ACEO 2) onto the meridian O 1 O 2, i.e. obtain the length of the arc O 1 O 2 in linear measures.

6.2. Determining distances to celestial bodies

Knowing the horizontal equatorial parallax p 0 of the luminary, it is easy to determine its distance from the center of the Earth (see Fig. 20). Indeed, if TO = R 0 is the equatorial radius of the Earth, TM = is the distance from the center of the Earth to the luminary M, and the angle p is the horizontal equatorial parallax of the luminary p 0, then from the right triangle TOM we have

For all luminaries except the Moon, the parallaxes are very small. Therefore, formula (3.1) can be written differently, putting

namely,

(3.2)

Distance is obtained in the same units in which the Earth's radius R 0 is expressed. Using formula (3.2), the distances to the bodies of the Solar System are determined. The rapid development of radio technology has given astronomers the opportunity to determine distances to solar system bodies using radar methods. In 1946, radar of the Moon was carried out, and in 1957-1963, radar of the Sun, Mercury, Venus, Mars and Jupiter was carried out. From the speed of propagation of radio waves c = 3 × 105 km/sec and from the time interval t (sec) of the passage of a radio signal from the Earth to the celestial body and back, it is easy to calculate the distance to the celestial body

DETERMINATION OF DISTANCES AND SIZES OF BODIES IN THE SOLAR SYSTEM

Razumov Viktor Nikolaevich,

teacher at Municipal Educational Institution "Bolsheelkhovskaya Secondary School"

Lyambirsky municipal district of the Republic of Mordovia

10-11 grade

UMK B.A.Vorontsov-Velyaminov

Shape and size of the Earth

Eratosthenes

(276 -194 BC)

Eratosthenes method:

• measure the arc length of the earth's meridian in linear units and determine which part full circle this arc constitutes;
• Having received this data, calculate the length of an arc of 1°, and then the length of the circle and the value of its radius, i.e., the radius of the globe.

Meridian arc length in degree measure equal to the difference in geographical latitudes of two points: φB – φA.

The Greek scientist Eratosthenes, who lived in Egypt, made the first fairly accurate determination of the size of the Earth.

Eratosthenes

(276 -194 BC)

To determine the difference in geographical latitudes, Eratosthenes compared midday altitude Suns on the same day in two cities located on the same meridian.

At noon on June 22 in Alexandria, the Sun is 7.2° from the zenith. On this day at noon in the city of Siena (now Aswan), the Sun illuminates the bottom of the deepest wells, i.e. it is at its zenith. Therefore, the arc length is 7.2°. The distance between Syene and Alexandria (800 km) according to Eratosthenes is 5000 Greek stadia, i.e. 1st stage = 160 m.

= , L=250,000 stadia or 40,000 km, which corresponds to modern measurements of the circumference of the globe.

The calculated radius of the Earth according to Eratosthenes was 6,287 km.

Modern measurements give a value of 6,371 km for the average radius of the Earth.

Basis

A method based on the phenomenon of parallactic displacement and involving the calculation of distance based on measurements of the length of one of the sides (base - AB) and two angles A and B in the triangle ACB, is used if it is impossible to directly measure the shortest distance between points.

Parallax displacement is a change in direction of an object

when the observer moves.

To determine the length of the arc, a system of triangles is used - a triangulation method that was first used back in 1615.

Points at the vertices of these triangles are selected on both sides of the arc at a distance of 30-40 km from each other so that at least two others are visible from each point.

The measurement accuracy of a 10 km long baseline is about 1 mm.

By measuring the angles in a triangle, one of the sides of which is the basis, using a goniometer instrument (theodolite), surveyors are able to calculate the length of its other two sides.

Basis

Triangulation, 16th century drawing

Triangulation execution scheme

To what extent the shape of the Earth differs from a sphere became clear at the end of the 18th century.

To clarify the shape of the Earth, the French Academy of Sciences equipped two expeditions: to equatorial latitudes South America in Peru and in Finland and Sweden near the Arctic Circle.

Measurements have shown that the length of one degree of meridian arc in the north is greater than near the equator.

This meant that the shape of the Earth is not a perfect sphere: it is flattened at the poles. Its polar radius is 21 km shorter than the equatorial one.

For a school globe at a scale of 1:50,000,000, the difference between these radii will be only 0.4 mm, i.e. completely unnoticeable.

The ratio of the difference between the equatorial and polar radii of the Earth to the equatorial radius is called compression. According to modern data, it is 1/298, or 0.0034, i.e. the cross section of the Earth along the meridian will be ellipse.

Currently, the shape of the Earth is usually characterized by the following quantities:

ellipsoid compression –1: 298.25;

average radius – 6371.032 km;

the circumference of the equator is 40075.696 km.

In the 20th century Thanks to measurements, the accuracy of which was 15 m, it turned out that the earth's equator also cannot be considered a circle.

The oblateness of the equator is only 1/30,000 (100 times less than the oblateness of the meridian).

More accurately, the shape of our planet is conveyed by a figure called ellipsoid, in which any section by a plane passing through the center of the Earth is not a circle.

Determination of distances in the solar system. Horizontal parallax

Horizontal parallax of the luminary

It was possible to measure the distance from the Earth to the Sun only in the second half of the 18th century, when the horizontal parallax of the Sun was first determined.

Horizontal parallax ( p) is the angle at which the radius of the Earth is visible from the luminary, perpendicular to the line of sight.

A solar parallax value of 8.8” corresponds to a distance of 150 million km. One astronomical unit (1 AU) is equal to 150 million km.

For small angles expressed in radians, sin p ≈ p.

The parallax of the Moon is of greatest importance, averaging 57".

In the second half of the 20th century. the development of radio technology has made it possible to determine distances

to the bodies of the Solar System via radar.

The first object among them was the Moon. Based on radar observations of Venus, the value of the astronomical unit was determined with an accuracy of the order of a kilometer.

Currently, thanks to the use of lasers, it has become possible to carry out optical location of the Moon.

In this case, distances to the lunar surface are measured with an accuracy of centimeters.

Example of problem solution

How far is Saturn from Earth when its horizontal parallax is 0.9"?

Given:

p1=0.9“

D= 1 a.u.

p  = 8.8“

D1 = R,

D  = R,

Solution:

D1 = = = 9.8 a.u.

Answer: D1 = 9.8 AU

Determining the size of the luminaries

Knowing the distance to the star, you can determine its linear dimensions by measuring its angular radius r. The formula connecting these quantities is similar to the formula for determining parallax:

Example of problem solution

What is the linear diameter of the Moon if it is visible from a distance of 400,000 km at an angle of approximately 30"?

Given:

D= 400000 km

ρ = 30’

Solution:

If ρ is expressed in radians, then r = D ρ

d = = 3490 km.

Answer: d= 3490 km.

Considering that the angular diameters of even the Sun and Moon are approximately 30", and all planets are visible to the naked eye as points, we can use the relationship: sin р ≈ р.

Hence,

If the distance D it is known then r = Dρ, where the value ρ expressed in radians.

Questions (p. 71)

1. What measurements made on the Earth indicate its compression?

2. Does the horizontal parallax of the Sun change throughout the year and for what reason?

3. What method is used to determine the distance to the nearest planets at the present time?

Homework

2) Exercise 11 (p.71)

1. What is the horizontal parallax of Jupiter observed from the Earth at opposition, if Jupiter is 5 times farther from the Sun than the Earth?

2. The distance of the Moon from the Earth at the point of its orbit closest to the Earth (perigee) is 363,000 km, and at the most distant point (apogee) - 405,000 km. Determine the horizontal parallax of the Moon at these positions.

3. How many times is the Sun larger than the Moon if their angular diameters are the same and their horizontal parallaxes are 8.8" and 57" respectively?

4. What is the angular diameter of the Sun as seen from Neptune?

• Vorontsov-Velyaminov B.A. Astronomy. Basic level. 11th grade : textbook/ B.A. Vorontsov-Velyaminov, E.K.Strout. - M.: Bustard, 2013. – 238 p.
• CD-ROM “Library of electronic visual aids"Astronomy, grades 9-10." Physicon LLC. 2003
• http://static.webshopapp.com/shops/021980/files/053607438/fotobehang-planeten-232cm-x-315cm.jpg
• http://images.1743.ru/images/1743/2017/06_june/image_18062017102234_14977633549594.jpg
• http://www.creationmoments.com/sites/creationmoments.com/files/images/What%27s%20the%20Right%20Answer.jpg
• https://videouroki.net/videouroki/conspekty/geom9/26-izmieritiel-nyie-raboty.files/image021.jpg
• http://www.muuseum.ut.ee/vvekniga/pages/data/geodeesia/1-CD006-Triangulation_16th_century.jpg
• http://elima.ru/i/12/000054e.jpg
• http://otvet.imgsmail.ru/download/182729882_1ef2e5f39d37858546ff499b3558a78a_800.png
• http://www.radartutorial.eu/01.basics/pic/radarprinzip.bigger.jpg

Determining distances to solar system bodies is based on measuring their horizontal parallaxes.

The angle between the directions in which the light was shining M" would be visible from the center of the Earth and from some point on its surface, is called daily parallax luminaries (Fig. 2.3). In other words, the daily parallax is the angle p", under which the radius of the Earth at the observation site would be visible from the luminary.

Rice. 2.3. Daily parallax.

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 r.

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 p.

Within the Solar System, distances to celestial bodies are defined as geocentric, i.e. from the center of the Earth to the center of the celestial body. In Fig. 2.3 distance r to the luminary M There is TM.

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. This radius is taken to be the equatorial radius of the Earth RÅ = 6378 km, and the horizontal parallaxes calculated for it are called horizontal equatorial parallaxes. It is these parallaxes of the bodies of the Solar System that are given in all reference books.

Knowing the horizontal parallax r luminary, it is easy to determine its geocentric distance. Indeed, if THAT = RÅ is the equatorial radius of the Earth, TM = r- distance from the center of the Earth to the star M, and the angle r - horizontal parallax of the luminary , then from a right triangle VOLUME we have

Where p²- horizontal parallax in arcseconds. Distance r is obtained in the same units in which the radius of the Earth is expressed R Å .

The horizontal parallax of a luminary can be determined by daily parallactic displacement this luminary in the sky, which is obtained as a result of a change in the position of the observer as a result of his movement along the surface of the Earth.

Horizontal parallax of the Sun r ¤= 8",79 corresponds to the average distance of the Earth from the Sun, equal to approximately 149.6 × 10 6 km. This distance in astronomy is taken as one astronomical unit (1 a.e.), i.e. 1 a.e.= 149.6 × 10 6 km. The distance to the bodies of the Solar System is usually expressed in astronomical units. For example, Mercury is at a distance of 0.387 AU from the Sun, and Pluto is at a distance of 39.4 AU.

If the semimajor axes of planetary orbits are expressed in astronomical units, and the orbital periods of planets are expressed in years, then for the Earth a = 1 a.e., T = 1 year and the period of revolution around the Sun of any planet, taking into account formula (2.7), is determined as

(a more accurate formula is obtained in the general theory of relativity).