Astronomy answer book for grade 11 for lesson number 2 (workbook) - Heavenly sphere

1. Complete the sentence.

A constellation is a section of the starry sky with a characteristic observed group of stars.

2. Using the map of the starry sky, enter the constellation schemes with bright stars in the corresponding columns of the table. In each constellation, select the brightest star and indicate its name.

3. Complete the sentence.

The position of the planets is not indicated on star charts, since the charts are intended to describe the stars and constellations.

4. Place the following stars in decreasing order of magnitude:

1) Betelgeuse; 2) Spica; 3) Aldebaran; 4) Sirius; 5) Arcturus; 6) Capella; 7) Procyon; 8) Vega; 9) Altair; 10) Pollux.

5. Complete the sentence.

Stars of the 1st magnitude are 100 times brighter than stars of the 6th magnitude.

The ecliptic is the apparent annual path of the Sun among the stars.

6. What is called the celestial sphere?

An imaginary sphere of arbitrary radius.

7. Indicate the names of points and lines of the celestial sphere, indicated by numbers 1-14 in Figure 2.1.

1. North pole of the world
2. zenith; zenith point
3. vertical line
4. celestial equator
5. west; west point
6. center of the celestial sphere
7. midday line
8. south; point south
9. skyline
10. East; point east
11. south pole of the world
12. nadir; toka nadir
13. north point
14. line of the celestial meridian

8. Using Figure 2.1, answer the questions.

How is the axis of the world located relative to the earth's axis?

Parallel.

How is the axis of the world located relative to the plane of the celestial meridian?

Lies on a plane.

At what points does the celestial equator intersect with the horizon line?

At the points of the east and west.

At what points does the celestial meridian intersect with the horizon line?

At points north and south.

9. What observations convince us of the daily rotation of the celestial sphere?

If you watch the stars for a long time, the stars appear to be a single sphere.

10. Using a moving star chart, write in the table two or three constellations visible at latitude 55 ° in the Northern Hemisphere.

The solution to the 10th task corresponds to the reality of the events of 2015, however, not all teachers check the solution of the task of each student on the star map for compliance with reality

Auxiliary celestial sphere

Coordinate systems used in geodetic astronomy

Geographic latitudes and longitudes of points on the earth's surface and azimuths of directions are determined from observations of celestial bodies - the Sun and stars. For this, it is necessary to know the position of the luminaries both relative to the Earth and relative to each other. The positions of the luminaries can be specified in expediently chosen coordinate systems. As is known from analytical geometry, to determine the position of the luminary s, you can use a rectangular Cartesian coordinate system XYZ or polar a, b, R (Fig. 1).

In a rectangular coordinate system, the position of the luminary s is determined by three linear coordinates X, Y, Z. In the polar coordinate system, the position of the luminary s is set by one linear coordinate, the radius vector R = Os and two angular ones: the angle a between the X axis and the projection of the radius vector onto the XOY coordinate plane, and the angle b between the XOY coordinate plane and the radius vector R. The relationship between rectangular and polar coordinates is described by the formulas

X = R cos b cos a,

Y = R cos b sin a,

Z = R sin b,

These systems are used in cases where the linear distances R = Os to celestial bodies are known (for example, for the Sun, Moon, planets, artificial satellites of the Earth). However, for many luminaries observed outside the solar system, these distances are either extremely large compared to the radius of the Earth, or are unknown. To simplify the solution of astronomical problems and to do without distances to the stars, it is assumed that all the stars are at an arbitrary, but the same distance from the observer. Usually this distance is taken equal to one, as a result of which the position of the luminaries in space can be determined not by three, but by two angular coordinates a and b of the polar system. It is known that the locus of points equidistant from a given point "O" is a sphere centered at this point.

Auxiliary celestial sphere - an imaginary sphere of arbitrary or unit radius, onto which images of celestial bodies are projected (Fig. 2). The position of any star s on the celestial sphere is determined using two spherical coordinates, a and b:

x = cos b cos a,

y = cos b sin a,

z = sin b.

Depending on where the center of the celestial sphere O is located, there are:

1)topocentric the celestial sphere - the center is on the surface of the Earth;

2)geocentric the celestial sphere - the center coincides with the center of mass of the Earth;

3)heliocentric the celestial sphere - the center is aligned with the center of the Sun;

4) barycentric celestial sphere - the center is at the center of gravity of the solar system.

The main circles, points and lines of the celestial sphere are shown in Fig. 3.

One of the main directions relative to the Earth's surface is the direction plumb line, or gravity at the observation point. This direction crosses the celestial sphere at two diametrically opposite points - Z and Z. "Point Z is located above the center and is called zenith, Z "- under the center and is called nadir.

Draw through the center a plane perpendicular to the plumb line ZZ. "The large NESW circle formed by this plane is called celestial (true) or astronomical horizon... This is the main plane of the topocentric coordinate system. It has four points S, W, N, E, where S - point south, N - North point, W - point west, E - point east... Direct NS is called midday line.

The straight line P N P S, drawn through the center of the celestial sphere parallel to the axis of rotation of the Earth, is called axis of the world... Points P N - north pole of the world; P S - south pole of the world... Around the axis of the World there is a visible daily movement of the celestial sphere.

Draw through the center a plane perpendicular to the axis of the world P N P S. The large circle QWQ "E, formed as a result of the intersection of this plane with the celestial sphere, is called celestial (astronomical) equator... Here Q - the highest point of the equator(above the horizon), Q "- the lowest point of the equator(under the horizon). The celestial equator and the celestial horizon intersect at points W and E.

The plane P N ZQSP S Z "Q" N, containing the plumb line and the axis of the World, is called true (celestial) or astronomical meridian. This plane is parallel to the plane of the earth's meridian and perpendicular to the plane of the horizon and equator. This is called the origin plane.

Draw through ZZ "a vertical plane perpendicular to the celestial meridian. The resulting circle ZWZ" E is called first vertical.

The great circle ZsZ "along which the vertical plane passing through the star s intersects the celestial sphere is called vertical or circle of heights of the sun.

The large circle P N sP S passing through the star perpendicular to the celestial equator is called around the declination of the luminary.

The small circle nsn "passing through the star parallel to the celestial equator is called diurnal parallel. The apparent diurnal movement of the luminaries occurs along diurnal parallels.

The small circle asa "passing through the star parallel to the celestial horizon is called circle of equal heights, or almucantara.

In a first approximation, the Earth's orbit can be taken as a flat curve - an ellipse, in one of the focuses of which is the Sun. The plane of the ellipse taken as the Earth's orbit , called a plane ecliptic.

In spherical astronomy, it is customary to talk about the apparent annual motion of the Sun. The large circle ЕgЕ "d, along which the apparent movement of the Sun occurs during the year, is called ecliptic... The plane of the ecliptic is inclined to the plane of the celestial equator at an angle approximately equal to 23.5 0. In fig. 4 shows:

g - vernal equinox point;

d - the point of the autumnal equinox;

E - the point of the summer solstice; E "- the point of the winter solstice; R N R S - axis of the ecliptic; R N - north pole of the ecliptic; R S - south pole of the ecliptic; e - inclination of the ecliptic to the equator.

The celestial sphere is an imaginary sphere of arbitrary radius used in astronomy to describe the relative positions of the stars in the sky. For simplicity of calculations, its radius is taken equal to one; the center of the celestial sphere, depending on the problem being solved, is combined with the pupil of the observer, with the center of the Earth, Moon, Sun, or generally with an arbitrary point in space.

The concept of the celestial sphere originated in ancient times. It was based on the visual impression of the existence of a crystal dome of the sky, on which the stars are supposedly fixed. The celestial sphere in the minds of the ancient peoples was the most important element of the Universe. With the development of astronomy, this view of the celestial sphere disappeared. However, the geometry of the celestial sphere, laid down in antiquity, as a result of development and improvement, has received a modern form, in which, for the convenience of various calculations, it is used in astrometry.

Consider the celestial sphere as it appears to the Observer at mid-latitudes from the Earth's surface (Fig. 1).

Two straight lines, the position of which can be established experimentally with the help of physical and astronomical instruments, play an important role in defining the concepts associated with the celestial sphere. The first is a plumb line; this is a straight line that coincides at a given point with the direction of the action of gravity. This line, drawn through the center of the celestial sphere, crosses it at two diametrically opposite points: the upper one is called the zenith, the lower one - the nadir. The plane passing through the center of the celestial sphere perpendicular to the plumb line is called the plane of the mathematical (or true) horizon. The line of intersection of this plane with the celestial sphere is called the horizon.

The second straight line is the axis of the world - a straight line passing through the center of the celestial sphere parallel to the axis of rotation of the Earth; around the axis of the world there is a visible daily rotation of the entire firmament. The points of intersection of the axis of the world with the celestial sphere are called the North and South poles of the world. The most conspicuous of the stars near the North Pole of the world is the Pole Star. There are no bright stars near the South Pole of the world.

The plane passing through the center of the celestial sphere perpendicular to the axis of the world is called the plane of the celestial equator. The line of intersection of this plane with the celestial sphere is called the celestial equator.

Recall that the circle that is obtained when the celestial sphere intersects with a plane passing through its center is called a large circle in mathematics, and if the plane does not pass through the center, then a small circle is obtained. The horizon and celestial equator are large circles of the celestial sphere and divide it into two equal hemispheres. The horizon divides the celestial sphere into visible and invisible hemispheres. The celestial equator divides it into the Northern and Southern Hemispheres, respectively.

With the diurnal rotation of the firmament, the luminaries rotate around the axis of the world, describing small circles on the celestial sphere, called diurnal parallels; luminaries, remote from the poles of the world by 90 °, move along the great circle of the celestial sphere - the celestial equator.

Having determined the plumb line and the axis of the world, it is not difficult to define all the other planes and circles of the celestial sphere.

The plane passing through the center of the celestial sphere, in which both the plumb line and the axis of the world lie, is called the plane of the celestial meridian. The great circle from the intersection of this plane with the celestial sphere is called the celestial meridian. That of the points of intersection of the celestial meridian with the horizon, which is closer to the North Pole of the world, is called the point of the north; diametrically opposite - the point of the south. The straight line passing through these points is the midday line.

Horizon points that are 90 ° from north and south are called east and west points. These four points are called the main points of the horizon.

The planes passing through the plumb line cross the celestial sphere in large circles and are called verticals. The celestial meridian is one of the verticals. The vertical, perpendicular to the meridian and passing through the points of east and west, is called the first vertical.

By definition, the three main planes - the mathematical horizon, the celestial meridian and the first vertical - are mutually perpendicular. The plane of the celestial equator is perpendicular only to the plane of the celestial meridian, forming a dihedral angle with the plane of the horizon. At the geographic poles of the Earth, the plane of the celestial equator coincides with the plane of the horizon, and at the equator of the Earth it becomes perpendicular to it. In the first case, at the geographic poles of the Earth, the axis of the world coincides with the plumb line and any of the verticals can be taken as the celestial meridian, depending on the conditions of the task at hand. In the second case, at the equator, the axis of the world lies in the plane of the horizon and coincides with the midday line; In this case, the North Pole of the world coincides with the point of the north, and the South Pole of the world - with the point of the south (see Fig.).

When using the celestial sphere, the center of which is aligned with the center of the Earth or some other point in space, a number of features also arise, however, the principle of introducing the basic concepts - horizon, celestial meridian, first vertical, celestial equator, etc. - remains the same.

The main planes and circles of the celestial sphere are used when introducing horizontal, equatorial and ecliptic celestial coordinates, as well as when describing the features of the apparent diurnal rotation of luminaries.

The large circle formed when the celestial sphere intersects with a plane passing through its center and parallel to the plane of the earth's orbit is called the ecliptic. The apparent annual movement of the Sun takes place along the ecliptic. The point of intersection of the ecliptic with the celestial equator, at which the sun passes from the southern hemisphere of the celestial sphere to the northern, is called the vernal equinox. The opposite point of the celestial sphere is called the autumnal equinox. A straight line passing through the center of the celestial sphere perpendicular to the plane of the ecliptic intersects the sphere at two poles of the ecliptic: the North Pole - in the Northern Hemisphere and the South - in the Southern Hemisphere.

3. Heavenly sphere. Basic planes, lines and points of the celestial sphere.

Under celestial sphere it is customary to understand a sphere of arbitrary radius, the center of which is at the point of observation, and all the celestial bodies or luminaries surrounding us are projected onto the surface of this sphere

The rotation of the celestial sphere for an observer on the surface of the Earth reproduces daily movement shone in the sky

ZOZ"- plumb (vertical) line,

SWNE- true (mathematical) horizon,

aMa"- almucantarat,

ZMZ"- circle of height (vertical circle), or vertical

P OP"- the axis of rotation of the celestial sphere (axis of the world),

P- the north pole of the world,

P" - the south pole of the world,

Ð PON= j (latitude of the place of observation),

QWQ" E- celestial equator,

bMb"- diurnal parallel,

PMP"- declination circle,

PZQSP" Z" Q" N- celestial meridian,

NOS- midday line

4. Systems of celestial coordinates (horizontal, first and second equatorial, ecliptic).

Since the radius of the celestial sphere is arbitrary, the position of the star on the celestial sphere is uniquely determined by two angular coordinates, if the main plane and the origin are set.

In spherical astronomy, the following celestial coordinate systems are used:

Horizontal, 1st equatorial, 2nd equatorial, Ecliptic

Horizontal coordinate system

Main plane - the plane of the mathematical horizon

1)Ð mOM = h (height)

0 £ h£ 90 0

–90 £ 0 h £ 0

or Ð ZOM = z (zenith distance)

0 £ z£ 180 0

z + h = 90 0

2) Р SOm = A(azimuth)

0 £ A£ 360 0

1st equatorial coordinate system

The main plane is the plane of the celestial equator

1) Р mOM= d (declination)

0 £ d £ 90 0

–90 0 £ d £ 0

or Ð POM = p (pole distance)

0 £ p£ 180 0

p+ d = 90 0

2) Р QOm = t (hour angle)

0 £ t£ 360 0

or 0 h £ t£ 24 h

All horizontal coordinates ( h, z, A) and hour angle t the first equatorial SC are continuously changing during the diurnal rotation of the celestial sphere.

The declination d does not change.

Must be entered instead of t such an equatorial coordinate, which would be measured from a point fixed on the celestial sphere.

2nd equatorial coordinate system

O main plane - the plane of the celestial equator

1) Р mOM= d (declination)

0 £ d £ 90 0

–90 0 £ d £ 0

or Ð POM = p (pole distance)

0£ p£ 180 0

p+ d = 90 0

2) Ð ¡ Om= a (right ascension)

or 0 h £ a £ 24 h

The horizontal SC is used to determine the direction to the star relative to terrestrial objects.

The 1st equatorial SC is used mainly for determining the exact time.

2-th equatorial SC is generally accepted in astrometry.

Ecliptic SC

The main plane is the plane of the ecliptic E¡E "d

The plane of the ecliptic is inclined to the plane of the celestial meridian at an angle ε = 23 0 26 "

PP "- axis of the ecliptic

E - the point of the summer solstice

E "- the point of the winter solstice

1) m = λ (ecliptic longitude)

2) mM= b (ecliptic latitude)

5. Daily rotation of the celestial sphere at different latitudes and phenomena associated with it. The daily movement of the sun. Change of seasons and thermal belts.

Measurements of the Sun's altitude at noon (i.e., at the moment of its upper culmination) at the same geographical latitude showed that the declination of the Sun d Ÿ during the year varies from +23 0 36 "to -23 0 36", two times passing through zero.

Right ascension of the Sun a Ÿ throughout the year also constantly changes from 0 to 360 0 or from 0 to 24 h.

Considering the continuous change in both coordinates of the Sun, it can be established that it moves among the stars from west to east along a large circle of the celestial sphere, which is called ecliptic.

March 20-21, the Sun is at point ¡, its declination δ = 0 and right ascension a Ÿ = 0. On this day (vernal equinox) the Sun rises exactly at the point E and goes to the point W... The maximum height of the center of the Sun above the horizon at noon this day (upper climax): hŸ = 90 0 - φ + δ Ÿ = 90 0 - φ

Then the Sun will move along the ecliptic closer to point E, i.e. δ Ÿ> 0 and a Ÿ> 0.

On June 21-22, the Sun is at point E, its declination is maximum δ Ÿ = 23 0 26 ", and right ascension is a Ÿ = 6 h. At noon of this day (summer solstice), the Sun rises to its maximum height above the horizon: hŸ = 90 0 - φ + 23 0 26 "

Thus, in mid-latitudes, the Sun NEVER is at its zenith

Latitude of Minsk φ = 53 0 55 "

Then the Sun will move along the ecliptic closer to point d, i.e. δ Ÿ will start to decrease

Around September 23, the Sun will come to point d, its declination δ Ÿ = 0, right ascension a Ÿ = 12 h. This day (the beginning of the astronomical autumn) is called the day of the autumnal equinox.

On December 22-23 the Sun will be at point E ", its declination is minimal δ Ÿ = - 23 0 26", and right ascension a Ÿ = 18 h.

Maximum height above the horizon: hŸ = 90 0 - φ - 23 0 26 "

The change in the equatorial coordinates of the Sun is uneven throughout the year.

Declination changes fastest when the Sun moves near the equinox points, and slowest near the solstice points.

Right ascension, on the contrary, changes more slowly near the equinox points, and faster - near the solstice points.

The apparent motion of the Sun along the ecliptic is associated with the actual motion of the Earth in its orbit around the Sun, as well as with the fact that the Earth's axis of rotation is not perpendicular to the plane of its orbit, but makes an angle ε = 23 0 26 ".

If ε = 0, then at any latitude on any day of the year the day would be equal to the night (excluding refraction and the size of the Sun).

Polar days lasting from 24 h to six months and the corresponding nights are observed in the polar circles, the latitudes of which are determined by the conditions:

φ = ± (90 0 - ε) = ± 66 0 34 "

The position of the axis of the world and, consequently, the plane of the celestial equator, as well as points ¡and d, is not constant, but periodically changes.

Due to the precession of the earth's axis, the axis of the world describes a cone around the axis of the ecliptic with an opening angle of ~ 23.5 0 in 26,000 years.

Due to the disturbing action of the planets, the curves described by the poles of the world do not close, but contract into a spiral.

T

.To. both the plane of the celestial equator and the plane of the ecliptic slowly change their position in space, then the points of their intersection (¡and d) slowly move to the west.

Travel speed (total annual precession in the ecliptic) per year: l = 360 0 /26 000 = 50,26"".

Total annual precession at the equator: m = l cos ε = 46.11 "".

At the beginning of our era, the vernal equinox was in the constellation Aries, from which it received its designation (¡), and the autumnal equinox was in the constellation Libra (d). Since then, point ¡has moved to the constellation Pisces, and point d to the constellation Virgo, but their designations have remained the same.

One of the most important astronomical problems, without which it is impossible to solve all other problems of astronomy, is to determine the position of the celestial body on the celestial sphere.

The celestial sphere is an imaginary sphere of an arbitrary radius, described from the eye of the observer, as from the center. On this sphere, we project the position of all celestial bodies. Distances on the celestial sphere can only be measured in angular units, degrees, minutes, seconds, or radians. For example, the angular diameters of the Moon and the Sun are approximately 30 minutes.

One of the main directions, relative to which the position of the observed celestial body is determined, is the plumb line. A plumb line anywhere in the world is directed towards the center of gravity of the Earth. The angle between the plumb line and the plane of the earth's equator is called astronomical latitude.

Rice. 1. Position in space of the celestial sphere for an observer at latitude relative to the Earth

A plane perpendicular to the plumb line is called the horizontal plane.

At every point on the Earth, the observer sees half a sphere, smoothly rotating from east to west, together with stars that seem to be attached to it. This apparent rotation of the celestial sphere is explained by the uniform rotation of the Earth around its axis from west to east.

The plumb line crosses the celestial sphere at the zenith point, Z, and at the nadir point, Z ".

Rice. 2. Heavenly sphere

The great circle of the celestial sphere, along which the horizontal plane passing through the eye of the observer (point C in Fig. 2) intersects with the celestial sphere, is called the true horizon. Recall that the great circle of the celestial sphere is the circle passing through the center of the celestial sphere. The circles formed by the intersection of the celestial sphere with planes that do not pass through its center are called small circles.

A line parallel to the earth's axis and passing through the center of the celestial sphere is called the axis of the world. It crosses the celestial sphere at the north pole of the world, P, and at the south pole of the world, P ".

From fig. 1 shows that the axis of the world is tilted to the plane of the true horizon at an angle. The apparent rotation of the celestial sphere occurs around the axis of the world from east to west, in the direction opposite to the true rotation of the Earth, which rotates from west to east.

The great circle of the celestial sphere, the plane of which is perpendicular to the axis of the world, is called the celestial equator. The celestial equator divides the celestial sphere into two parts: north and south. The celestial equator is parallel to the equator of the Earth.

The plane passing through the plumb line and the axis of the world crosses the celestial sphere along the line of the celestial meridian. The celestial meridian intersects with the true horizon at the points north, N, and south, S. And the planes of these circles intersect along the noon line. The celestial meridian is a projection onto the celestial sphere of the earth meridian, on which the observer is located. Therefore, there is only one meridian on the celestial sphere, because the observer cannot be on two meridians at the same time!

The celestial equator intersects the true horizon at points east, E, and west, W. The EW line is perpendicular to noon. Point Q is the top of the equator and Q "is the bottom of the equator.

Large circles with planes passing through a plumb line are called verticals. The vertical passing through points W and E is called the first vertical.

Large circles, the planes of which pass through the axis of the world, are called declination circles or hour circles.

Small circles of the celestial sphere, the planes of which are parallel to the celestial equator, are called celestial or diurnal parallels. They are called daily because the daily movement of heavenly bodies occurs along them. The equator is also a diurnal parallel.

The small circle of the celestial sphere, the plane of which is parallel to the plane of the horizon, is called almucantarat

Tasks

Name	Formula	Explanations	Notes (edit)
The height of the luminary at the upper culmination (between the equator and the zenith)	h = 90 ° - φ + δ z = 90 ° - h	d - declination of the star, j- latitude of the place of observation, h- the height of the luminary above the horizon z- zenith distance of the luminary
The height of the luminary to the top. climax (between the zenith and the pole of the world)	h= 90 ° + φ – δ
The height of the luminary at the bottom. climax (non-setting star)	h = φ + δ - 90 °
Latitude of a non-setting star, both culminating north of zenith	φ = (h in + h n) / 2	h in- the height of the luminary above the horizon at the upper climax h n- the height of the star above the horizon at the bottom culmination	If not north of the zenith, then δ =(h in + h n) / 2
Orbital eccentricity (the degree of elongation of the ellipse)	e = 1 - r p / a or e = r a / a - 1 or e = (1 - in 2 /a 2 ) ½	e - eccentricity of an ellipse (elliptical orbit) - the ratio of the distance from the center to the focus to the distance from the center to the edge of the ellipse (half of the major axis); r p - orbital perigee distance r a - apogee orbital distance a - semi-major axis of the ellipse; b - semi-minor axis of the ellipse;	An ellipse is a curve in which the sum of the distances from any point to its foci is a constant value equal to the major axis of the ellipse
Semi-major axis of the orbit	r p + r a = 2a
The smallest value of the radius vector at the periapsis	r p = a ∙ (1-e)
The largest value of the radius vector at the apocenter (aphelion)	r a = a ∙ (1 + e)
Ellipse flattening	e = (a - b) / a = 1 - b / a = 1 - (1 - e 2 ) 1/2	e - squeeze ellipse
Semi-minor axis of an ellipse	b = a ∙ (1 - e 2 ) ½
Area constant