Хэрэглэгч:Timur/Ноорог/Тэнгэрийн координатын системүүдийн харилцан хамаарал

Дэлхийн хоногийн эргэлт

Diurnal motion is an astronomical term referring to the apparent daily motion of stars around the Earth, or more precisely around the two celestial poles. It is caused by the Earth's rotation on its axis, so every star apparently moves on a circle, that is called the diurnal circle. The time for one complete rotation is 23 hours, 56 minutes and 4.09 seconds (1 sidereal day).

Direction of the motion in the Northern hemisphere:

• looking to the north, below the North Star: left-right, west-east
• looking to the north, above the North Star: right-left, east-west
• looking to the south: left-right, east-west

Thus northern circumpolar stars move anti-clockwise around the North Star.

At the North Pole, north, east and west are not applicable, the motion is simply left-right, or looking vertically upward, anti-clockwise around the zenith.

For the southern hemisphere, interchange north/south and left/right, and replace North Star by southern celestial pole. The circumpolar stars move clockwise around it. East/west are not interchanged.

At the equator both celestial poles are at the horizon and motion is anti-clockwise (i.e. to the left) around the North Star and clockwise (i.e to the right) around the southern celestial pole. All motion is from east to west, except for the two stationary points.

The daily path of an object on the celestial sphere, including the possible part below the horizon, has a length proportional to the cosine of the declination. Thus the speed of the diurnal motion of a celestial object is this cosine times 15 °/hr = 15'/min = 15"/s, i.e. (compare angular diameter):

• up to a Sun or Moon diameter every two minutes
• ca. four seconds for the largest planet
• 2000 diameters of the largest stars per second

Diurnal motion can be seen in time-exposure photography. Circumpolar stars close to the celestial pole move only slowly. Conversely, following the diurnal motion with the camera, to eliminate it on the photograph, can best be done with an equatorial mount, which requires adjusting the right ascension only; a telescope may have a motor to do that automatically (sidereal drive).

Высота h, зенитное расстояние z, азимут A и часовой угол t светил постоянно изменяются вследствие вращения небесной сферы, так как отсчитываются от точек, не связанных с этим вращением. Склонение δ, полярное расстояние p и прямое восхождение α светил при вращении небесной сферы не изменяются, но они могут меняться из-за движений светил, не связанных с суточным вращением.

Дэлхийн эргэлтийн тэнхлэгийн прецесслэх хөдөлгөөн

Precessional movement as seen from 'outside' the celestial sphere

The precession of Earth's axis of rotation with respect to inertial space is also called the precession of the equinoxes. Like a wobbling top, the direction of the Earth's axis is changing. While today, the Earth's North Pole points roughly to Polaris, over time, that will change. Also, because of this wobble, the position of the earth in its orbit around the sun at the moment of the equinox (the start of spring or fall) will also change.

The term precession typically refers only to the largest periodic motion. Other changes of Earth's axis are nutation and polar motion; their magnitude is very much smaller.

Currently, this annual motion is about 50.3 seconds of arc per year or 1 degree every 71.6 years. The process is slow, but cumulative. A complete precession cycle covers a period of approximately 25,765 years, the so called Platonic year, during which time the equinox regresses a full 360° through all twelve constellations of the zodiac. Precessional movement is also the determining factor in the length of an Astrological Age.

In ancient times the precession of the equinox referred to the motion of the equinox relative to the background stars of the twelve constellations of the zodiac. It acted as a method of keeping time in the Great Year.

Hipparchus is credited with discovering that the positions of the equinoxes move westward along the ecliptic compared to the fixed stars on the celestial sphere. The exact dates of his life are not known, but astronomical observations attributed to him date from 147 BC to 127 BC and were described in his writings, none of which survive to date.

Values

Simon Newcomb's calculation at the end of the nineteenth century for general precession (known as p) in longitude gave a value of 5,025.64 arcseconds per tropical century, and was the generally accepted value until artificial satellites delivered more accurate observations and electronic computers allowed more elaborate models to be calculated. Lieske developed an updated theory in 1976, where p equals 5,029.0966 arcseconds per Julian century. Modern techniques such as VLBI and LLR allowed further refinements, and the International Astronomical Union adopted a new constant value in 2000, and new computation methods and polynomial expressions in 2003 and 2006; the accumulated precession is:

p_A = 5,028.796195×T + 1.1054348×T² + higher order terms,

in arcseconds per Julian century, with T, the time in Julian centuries (that is, 36,525 days) since the epoch of 2000.

The rate of precession is the derivative of that:

p = 5,028.796195 + 2.2108696×T + higher order terms

The constant term of this speed corresponds to one full precession circle in 25,772 years.

The precession rate is not a constant, but (at the moment) slowly increasing over time, as indicated by the linear (and higher order) terms in T. In any case it must be stressed that this formula is only valid over a limited time period. It is clear that if T gets large enough (far in the future or far in the past), the T² term will dominate and p will go to very large values. In reality, more elaborate calculations on the numerical model of solar system show that the precessional constants have a period of about 41,000 years, the same as the obliquity of the ecliptic. Note that the constants mentioned here are the linear and all higher terms of the formula above, not the precession itself. That is, p = A + BT + CT² + … is an approximation of p = A + Bsin (2πT/P), where P is the 410-century period.

Theoretical models may calculate the proper constants (coefficients) corresponding to the higher powers of T, but since it is impossible for a polynomial to match a periodic function over all numbers, the error in all such approximations will grow without bound as T increases. In that respect, the International Astronomical Union chose the best developed available theory. For up to a few centuries in the past and the future, all formulas do not diverge very much. For up to a few thousand years in the past and the future, most agree to some accuracy. For eras farther out, discrepancies become too large - the exact rate and period of precession may not be computed, even for a single whole precession period.

The precession of Earth's axis is a very slow effect, but at the level of accuracy at which astronomers work, it does need to be taken into account on a daily basis. Note that although the precession and the tilt of Earth's axis (the obliquity of the ecliptic) are calculated from the same theory and thus, are related to each other, the two movements act independently of each other, moving in mutually perpendicular directions.

Over longer time periods, that is, millions of years, it appears that precession is quasiperiodic at around 25,700 years, however, it will not remain so. According to Ward, when the distance of the Moon, which is continuously increasing from tidal effects, will have gone from the current 60.3 to approximately 66.5 Earth radii in about 1,500 million years, resonances from planetary effects will push precession to 49,000 years at first, and then, when the Moon reaches 68 Earth radii in about 2,000 million years, to 69,000 years. This will be associated with wild swings in the obliquity of the ecliptic as well. Ward, however, used the abnormally large modern value for tidal dissipation. Using the 620-million year average provided by tidal rhythmites of about half the modern value, these resonances will not be reached until about 3,000 and 4,000 million years, respectively. Long before that time (about 2,100 million years from now), due to the increasing luminosity of the Sun, however, the oceans of the Earth will have boiled away, which will alter tidal effects significantly.

Дэлхийн эргэлтийн тэнхлэгийн прецесс дэх нутаци

Rotation (green), Precession (blue) and Nutation in obliquity (red) of the Earth

Nutation is a slight irregular motion (etymologically a "nodding") in the axis of rotation of a largely axially symmetric object, such as a gyroscope or a planet.

The nutation of a planet is due to the fact that the tidal forces which cause the precession of the equinoxes vary over time so that the speed of precession is not constant. It was discovered in 1728 by the English astronomer James Bradley, but was not explained until 20 years later.

Because the dynamics of the planets are so well known, nutation can be calculated within seconds of arc over periods of many decades. There is another disturbance of the Earth's rotation called polar motion that can be estimated only a few months ahead, because it is influenced by rapidly and unpredictably varying things such as ocean currents, wind systems, and motions in the Earth's core.

Values of nutation are usually divided into components parallel and perpendicular to the ecliptic. The component which works along the ecliptic is known as the nutation in longitude. The component perpendicular to the ecliptic is known as the nutation in obliquity. Celestial coordinate systems are based on an "equator" and "equinox," which means a great circle in the sky that is the projection of the Earth's equator outwards, and a line, the Vernal equinox intersecting that circle, which determines the starting point for measurement of right ascension. These items are affected both by precession of the equinoxes and nutation, and thus depend on the theories applied to precession and nutation, and on the date used as a reference date for the coordinate system. In simpler terms, nutation (and precession) values are important in observation from Earth for calculating the apparent positions of astronomical objects.

Nutation of Earth

In the case of Earth, the principal sources of tidal force are the Sun and Moon, which continuously change location relative to each other and thus cause nutation in Earth's axis. The largest component of Earth's nutation has a period of 18.6 years, the same as that of the precession of the Moon's orbital nodes. However, there are other significant periodical terms which must be calculated depending on the desired accuracy of the result. A mathematical description (set of equations) that represents nutation is called a "theory of nutation" (see, e.g. [1]). Generally, the theory is really a theory, in the sense that it applies physical laws and astronomical measurements; however, there may be parameters which are adjusted in a more or less ad hoc way to obtain the best fit to data. As can be seen from the IERS publication just cited, nowadays simple rigid-body mechanics do not give the best theory; one has to account for deformations of the solid Earth.

Values

The principal term of nutation is due to the regression of the moon's nodal line and has the same period of 6798 days (18.6 years). It reaches 17" in longitude and 9" in obliquity. All other terms are much smaller. The next largest, with a period of 183 days (0.5 year) has amplitudes 1.3" and 0.6" respectively. Interestingly the periods of all terms larger than 0.0001" (about as accurate as one can measure), which lie between 5.5 and 6798 days seem to avoid the range from 34.8 to 91 days. It is therefore customary to split the nutation into long-period and short-period terms. The long-period terms are calculated and mentioned in the almanacs, while the additional correction due to the short-period terms is usually taken from a table.

Гэрлийн аберраци

The aberration of light (also referred to as astronomical aberration or stellar aberration) is an astronomical phenomenon which produces an apparent motion of celestial objects. It was discovered and later explained by the third Astronomer Royal, James Bradley, in 1725, who attributed it to the finite speed of light and the motion of Earth in its orbit around the Sun.

At the instant of any observation of an object, the apparent position of the object is displaced from its true position by an amount which depends upon the transverse component of the velocity of the observer, with respect to the vector of the incoming beam of light (i.e., the line actually taken by the light on its path to the observer). In the case of an observer on Earth, the direction of its velocity varies during the year as Earth revolves around the Sun (or strictly speaking, the barycenter of the solar system), and this in turn causes the apparent position of the object to vary. This particular effect is known as annual aberration or stellar aberration, because it causes the apparent position of a star to vary periodically over the course of a year. The maximum amount of the aberrational displacement of a star is approximately 20 arcseconds in right ascension or declination. Although this is a relatively small value, it was well within the observational capability of the instruments available in the early eighteenth century.

Aberration should not be confused with stellar parallax, although it was an initially fruitless search for parallax that first led to its discovery. Parallax is caused by a change in the position of the observer looking at a relatively nearby object, as measured against more distant objects, and is therefore dependent upon the distance between the observer and the object.

In contrast, stellar aberration is independent of the distance of a celestial object from the observer, and depends only on the observer's instantaneous transverse velocity with respect to the incoming light beam, at the moment of observation. The light beam from a distant object cannot itself have any transverse velocity component, or it could not (by definition) be seen by the observer, since it would miss the observer. Thus, any transverse velocity of the emitting source plays no part in aberration. Another way to state this is that the emitting object may have a transverse velocity with respect to the observer, but any light beam emitted from it which reaches the observer, cannot, for it must have been previously emitted in such a direction that its transverse component has been "corrected" for. Such a beam must come "straight" to the observer along a line which connects the observer with the position of the object when it emitted the light.

Aberration should also be distinguished from light-time correction, which is due to the motion of the observed object, like a planet, through space during the time taken by its light to reach an observer on Earth. Light-time correction depends upon the velocity and distance of the emitting object during the time it takes for its light to travel to Earth. Light-time correction does not depend on the motion of the Earth—it only depends on Earth's position at the instant when the light is observed. Aberration is usually larger than a planet's light-time correction except when the planet is near quadrature (90° from the Sun), where aberration drops to zero because then the Earth is directly approaching or receding from the planet. At opposition to or conjunction with the Sun, aberration is 20.5" while light-time correction varies from 4" for Mercury to 0.37" for Neptune (the Sun's light-time correction is less than 0.03").

Explanation

It has been stated above that aberration causes a displacement of the apparent position of an object from its true position. However, it is important to understand the precise technical definition of these terms.

Apparent and true positions

Figure 1. Diagram illustrating stellar aberration

The apparent position of a star or other very distant object is the direction in which it is seen by an observer on the moving Earth. The true position (or geometric position) is the direction of the straight line between the observer and star at the instant of observation. The difference between these two positions is caused mostly by aberration.

Aberration occurs when the observer's velocity has a component that is perpendicular to the line traveled by light between the star and observer. In Figure 1 to the right, S represents the spot where the star light enters the telescope, and E the position of the eye piece. If the telescope does not move, the true direction of the star relative to the observer can be found by following the line ES. However, if Earth, and therefore the eye piece of the telescope, moves from E to E’ during the time it takes light to travel from S to E, the star will no longer appear in the center of the eye piece. The telescope must therefore be adjusted so that the star light enters the telescope at spot S’. Now the star light will travel along the line S’E’ and reach E’ exactly when the moving eye piece also reaches E’. Since the telescope has been adjusted by the angle SES’, the star's apparent position is hence displaced by the same angle.

Moving in the rain

Many find aberration to be counter-intuitive, and a simple thought experiment based on everyday experience can help in its understanding. Imagine you are standing in the rain. There is no wind, so the rain is falling vertically. To protect yourself from the rain you hold an umbrella directly above you.

Now imagine that you start to walk. Although the rain is still falling vertically (relative to a stationary observer), you find that you have to hold the umbrella slightly in front of you to keep off the rain. Because of your forward motion relative to the falling rain, the rain now appears to be falling not from directly above you, but from a point in the sky somewhat in front of you.

The deflection of the falling rain is greatly increased at higher speeds. When you drive a car at night through falling rain, the rain drops illuminated by your car's headlights appear to fall from a position in the sky well in front of your car.

Types of aberration

There are a number of types of aberration, caused by the differing components of the Earth's motion:

• Annual aberration is due to the revolution of the Earth around the Sun.
• Planetary aberration is the combination of aberration and light-time correction.
• Diurnal aberration is due to the rotation of the Earth about its own axis.
• Secular aberration is due to the motion of the Sun and solar system relative to other stars in the galaxy.

Annual aberration

As the Earth revolves around the Sun, it is moving at a velocity of approximately 30 km/s. The speed of light is approximately 300,000 km/s. In the special case where the Earth is moving perpendicularly to the direction of the star (i.e. if SEE’ in the diagram is 90 degrees), the angle of displacement, SES’, would therefore be (in radians) the ratio of the two velocities, i.e. 1/10000 or about 20.5 arcseconds.

This quantity is known as the constant of aberration, and is conventionally represented by κ. Its precise accepted value is 20".49552 (at J2000).

Figure 2. Diagram illustrating the effect of annual aberration on the apparent position of three stars at ecliptic longitude 270 degrees, and ecliptic latitude 90, 45 and 0 degrees, respectively

The plane of the Earth's orbit is known as the ecliptic. Annual aberration causes stars exactly on the ecliptic to appear to move back and forth along a straight line, varying by κ on either side of their true position. A star that is precisely at one of the ecliptic's poles will appear to move in a circle of radius κ about its true position, and stars at intermediate ecliptic latitudes will appear to move along a small ellipse (see figure 2).

A special case of annual aberration is the nearly constant deflection of the Sun from its true position by κ towards the west (as viewed from Earth), opposite to the apparent motion of the Sun along the ecliptic. This constant deflection is often erroneously explained as due to the motion of the Earth during the 8.3 minutes that it takes light to travel from the Sun to Earth. The latter is a type of parallax, and actually causes the apparent motion of the Sun along the ecliptic towards the east relative to the fixed stars. (8.316746 minutes divided by one sidereal year (365.25636 days) is 20.49265", very close to κ, but of opposite sign, east vs. west.) Nor is this the Sun's light-time correction because the Sun is almost motionless, moving around the barycenter (center of mass) of the solar system by usually much less than 0".03 (as viewed from Earth) during 8.3 minutes.

Aberration can be resolved into east-west and north-south components on the celestial sphere, which therefore produce an apparent displacement of a star's right ascension and declination, respectively. The former is larger (except at the ecliptic poles), but the latter was the first to be detected. This is because very accurate clocks are needed to measure such a small variation in right ascension, but a transit telescope calibrated with a plumb line can detect very small changes in declination.

Figure 3. Diagram illustrating aberration of a star at the north ecliptic pole

Figure 3, above, shows how aberration affects the apparent declination of a star at the north ecliptic pole, as seen by an imaginary observer who sees the star transit at the zenith (this observer would have to be positioned at latitude 66.6 degrees north – i.e. on the arctic circle). At the time of the March equinox, the Earth's orbital velocity is carrying the observer directly south as he or she observes the star at the zenith. The star's apparent declination is therefore displaced to the south by a value equal to κ. Conversely, at the September equinox, the Earth's orbital velocity is carrying the observer northwards, and the star's position is displaced to the north by an equal and opposite amount. At the June and December solstices, the displacement is zero.

Note that the effect of aberration is out of phase with any displacement due to parallax. If the latter effect were present, the maximum displacement to the south would occur in December, and the maximum displacement to the north in June. It is this apparently anomalous motion that so mystified Bradley and his contemporaries.

Planetary aberration

Planetary aberration is the combination of the aberration of light (due to Earth's velocity) and light-time correction (due to the object's motion and distance). Both are determined at the instant when the moving object's light reaches the moving observer on Earth. It is so called because it is usually applied to planets and other objects in the solar system whose motion and distance are accurately known.

Diurnal aberration

Diurnal aberration is caused by the velocity of the observer on the surface of the rotating Earth. It is therefore dependent not only on the time of the observation, but also the latitude and longitude of the observer. Its effect is much smaller than that of annual aberration, and is only 0".32 in the case of an observer at the equator, where the rotational velocity is greatest.

Secular aberration

The Sun and solar system are revolving around the center of the Galaxy, as are other nearby stars. It is therefore possible to conceive of an aberrational effect on the apparent positions of other stars and on extragalactic objects. However, the change in the solar system's velocity relative to the center of the Galaxy varies over a very long timescale, and the consequent change in aberration would be extremely difficult to observe. Therefore, this so-called secular aberration is usually ignored when considering the positions of stars.

However, it is possible to estimate the displacement between the apparent and true position of a nearby star whose distance and motion are known. Newcomb gives the example of Groombridge 1830, where he estimates that the true position is displaced by approximately 3 arcminutes from the direction in which we observe it. This calculation also includes an allowance for light-time correction, and is therefore analogous to the concept of planetary aberration.

Агаар мандал дахь гэрлийн хугарал

Atmospheric refraction is the deviation of light or other electromagnetic wave from a straight line as it passes through the atmosphere due to the variation in air density as a function of altitude. Atmospheric refraction near the ground produces mirages and can make distant objects appear to shimmer or ripple.

Atmospheric refraction causes astronomical objects to appear higher in the sky than they are in reality. It affects not only lightrays but all electromagnetic radiation, although in varying degrees (see dispersion (optics)). For example in visible light, blue is more affected than red. This may cause astronomical objects to be spread out into a spectrum in high-resolution images.

Whenever possible astronomers will always schedule their observations around the time of culmination of an object when it is highest in the sky. Likewise sailors will never shoot a star which is not at least 20° or more above the horizon. If observations close to the horizon cannot be avoided, it is possible to equip a telescope with control systems to compensate for the shift caused by the refraction. If the dispersion is a problem too, (in case of broadband high-resolution observations) atmospheric refraction correctors can be employed as well (made from pairs of rotating glass prisms). But as the amount of atmospheric refraction is function of temperature and pressure as well as humidity (the amount of water vapour especially important at mid-infrared wavelengths) the amount of effort needed for a successful compensation can be prohibitive.

It gets even worse when the atmospheric refraction is not homogenous, when there is turbulence in the air for example. This is the cause of twinkling of the stars and deformation of the shape of the sun at sunset and sunrise.

• Values

The atmospheric refraction is zero in the zenith, is less than 1' (one arcminute) at 45° altitude, still only 5' at 10° altitude, but then quickly increases when the horizon is approached. On the horizon itself it is about 34' (according to FW Bessel), just a little bit larger than the apparent size of the sun. Therefore if it appears that the setting sun is just above the horizon, in reality it has already set. Formulae to calculate the times of sunrise and sunset do not calculate the moment that the sun reaches altitude zero, but when its altitude is -50': 16' for the radius of the sun (solar positions are for the centre of the sun-disc, but sunrise and sunset usually refer to the appearance and disappearance of the upperlimb) plus 34' for the refraction. In the case of the Moon one should apply additional corrections for the horizontal parallax of the moon, its apparent diameter and its phase, although the latter is seldom done.

The refraction is also a function of temperature and pressure. The values given above are for 10 °C and 1003 mbar. Add 1% to the refraction for every 3° C colder, subtract if hotter (hot air is less dense, and will therefore have less refraction). Add 1% for every 9 mbar higher pressure, subtract if lower. Evidently day to day variations in the weather will affect the exact times of sunrise and sunset as well as moonrise and moonset, and for that reason are never given more accurately than to the nearest whole minute in the almanacs.

Finally as the atmospheric refraction is 34' on the horizon, but only 29' half a degree above it, the setting or rising sun seems to be flattened by about 5' or 1/6 of its apparent diameter.

• Random refraction effects

Turbulence in the atmosphere magnifies and de-magnifies star images, making them appear brighter and fainter on a time-scale of milliseconds. The slowest components of these fluctuations are visible to the eye as twinkling (also called “scintillation”).

Turbulence also causes small random motions of the star image, and produces rapid changes in its structure. These effects are not visible to the naked eye, but are easily seen even in small telescopes. They are called “seeing” by astronomers.