Besselian elements

The Bessel elements are tabulated sizes that are in mutual coverage of two celestial bodies of the calculation and prediction of local situations one observation site on the earth. The principle is used primarily for solar eclipses, but even with a star or planet occultations by the Moon and the transits of Venus and Mercury from the sun it is used. In lunar eclipses is a similar method of application, which does not fall in this case, the shadow on the earth but on the moon.

For solar eclipses, for example, based on the Bessel elements, the cover duration are determined in a particular place, or is it the path can be determined, where the umbra passes over the surface of the earth during a total solar eclipse. This calculation method was developed in the 1820s by Friedrich Wilhelm Bessel and later refined by William Chauvenet.

The basic idea is that the Bessel elements of the movement caused by the covering celestial bodies shadow - during solar eclipses, this is the Moon's shadow - reflect on the fundamental plane suitably chosen, so-called. It is the geocentric normal plane of the shadow axis. By this is meant that this layer goes through the center of the earth and is perpendicular to the axis of the shadow cone, that is, the straight line passing through the centers of the clad and the covering celestial body.

To describe the motion of the shadow in this level, indicating comparatively less quantities is sufficient - with sufficient accuracy. This is not least because the shade during the entire course of darkness in this plane is always circular and no perspective distortion subject. Assuming the conditions are calculated at the surface until the next step by projection where only the approximate spherical shape of the Earth, the Earth's rotation and the position and height of the observing site must be considered.

• 3.1 occultations of stars by the moon
• 3.2 coverage of the planet by the moon

History

This method, star and planet occultations and eclipses to describe, was drafted by the German scientist Friedrich Wilhelm Bessel in the 1820s. The first work on the subject Bessels occultations can be found in the Astronomische Nachrichten No. 50 of 1824, in which he was doing some calculations based on previously observed stellar occultations. In 1829 he published a work generalizing Concerning the prediction of occultations of stars in the Astronomische Nachrichten No. 145 In the same year, he developed the idea further by generalizing the procedure with the aim of applying for planetary occultations and eclipses.

Up to this point two independent methods were used at different locations for the calculation. The first method was used to determine the circumstances as they presented themselves to an observer at a particular location. The method used here was already back to Johannes Kepler and was later further developed by Jérôme Lalande and Johann Gottlieb Friedrich von Berger beans. The second method, which is due to Joseph -Louis Lagrange, was used to calculate the timing of the conjunction. Since this procedure was referring to the Earth's center and could not make any statement about local conditions on the Earth's surface, it has been less frequently applied for the calculation of eclipses than the first. However, it simplified many other astronomical calculations. Bessel's approach was now to Lagrange's method further so that therefore the calculation of local conditions was possible, so he reached a combination of both methods.

In the second volume of his astronomical investigations Bessel published in 1842 four sections comprehensive treatise entitled Analysis of eclipses. In it, he summarized his previously published work together on this issue and rounded it off with some additions. This publication served as the basis for many astronomers who later grappled with this issue. Peter Andreas Hansen used in his 1858 published work theory of eclipses and related phenomena deviating from the Bessel line of intersection of the ecliptic with the fundamental plane as the axis. Bessels variant, the use of the equatorial plane instead of the ecliptic, but had some advantages, such as 1863 William Chauvenet emphasized. Chauvenet, an American astronomer who followed in his Manual of Spherical and Practical Astronomy for the most part the process Bessels, but developed for some subproblems own solutions. Chauvenets representation was then the basis for many further developments in this area.

Even in today's time when not done the calculations of eclipses manually, but electronically, the Bessel elements have not lost their importance. On the contrary, they represent the link between the calculations of the time of occurrence of an eclipse, and the calculations of the local conditions dar. Many computer programs are dedicated to one of the two calculations, the Bessel elements act as a kind of interface.

Eclipses

The position of the distant covered and the covering surrounding and the location of the observer on the earth must be known to describe the mutual occultation of stars. In panel works the path data of stars (ephemeris ) by declination and right ascension are given. These angles refer to as geocentric coordinates on the Earth's center, so that from them the constellation coverage is not to be found, which can be observed at a given point of the earth's surface.

To describe a coverage at a point on the earth's surface consisting of panels removed or otherwise known orbital data of the two celestial bodies must be converted. The Bessel elements used to describe the experience as well as the size of the core and penumbra in the fundamental plane. It is not difficult on the one hand, starting to describe the course of the shadow in this level of the path data of the heavenly bodies, on the other hand quite a simple conversion to an observation point is possible. For the latter, the conversion Bessel elements also contain information such as the fundamental plane is rotated relative to the zero meridian and the equatorial plane.

The occultation of the sun by the moon presents with regard to the description of the situation on the ground is the most complicated Okkultationstyp, since both the covered body - the sun - as well as covering the body - the moon - have non-negligible visual angle. In addition, the apparent movement of the sun must be taken into account during the coverage.

Definition of the Bessel elements

First, a rectangular coordinate system is introduced, which is referred to as fundamental or Bessel cal coordinate system. This is based on the shadow axis, the straight line connecting the center of the sun and moon. The parallel of the shadow axis passing through the center of the Earth, the axis of the Bessel fundamental coordinate system represents and constantly follows the shadow, the coordinate system that is rotating with the direction of the shadow axis. The fundamental plane is at the center of the earth perpendicular to this axis. In the fundamental plane, the position and size of the core and penumbra by means of the will - and coordinate described. The axis is the line of intersection of the fundamental plane with the equatorial plane and points to the east, the axis points to the north.

The first two values ​​of the Bessel elements are the coordinates of the intersection point and the axis of the shadow at the fundamental level. The direction of the shadow axis - corresponding to the direction of the axis - is given by the declination and the Ephemeridenstundenwinkel. The radius of the penumbra cone in the fundamental plane is described by which of the umbra cone through. is negative for a total eclipse, positive for a ring. The values ​​, and are typically reported in units of the equatorial radius of the earth.

In addition to these six variables that change over the course of the darkness, there are two other variables that can be considered constant: The sizes and define the half opening angle of the half-or umbra cone.

Calculation of Bessel's elements

Solar eclipses are calculated from the time course of the geocentric positions of the Sun and Moon, which are available on their ephemeris. [A 1] is a way of calculating the occurrence of eclipses to convert the positions of the sun and moon immediately to the fundamental coordinate system. Then can be pretty easy to determine if and when the shadow axis, the fundamental plane pierces within the terrestrial globe - which means that there have a central, ie total or annular eclipse.

There are other ways of calculating the occurrence of eclipses, for example through the darkness limits. But even in this case, the positions of the sun and moon for the darkness in the course of the fundamental coordinate system must be converted in order to calculate based on the Bessel elements local conditions at every location on earth can.

Based on the geocentric coordinates and the distances of the sun and moon, the Bessel elements for a given time can be calculated. From declination and right ascension and the distance initially can be the position vectors of the Sun and Moon determined as follows:

As the unit of distance is usually of the equatorial radius of the earth. In the literature, the removal is often expressed by the parallax can be taken from the Ephemeridentafeln. Since the parallax refers to the radius of the Earth as a base, the distance in units of the equatorial radius can be calculated by.

The following are calculated as the first Bessel's elements and the declination of the Ephemeridenstundenwinkel, so the equatorial coordinates of the direction of the shadow axis. Instead of the hour angle in this case the right ascension is first calculated from this can be determined by the formula of the hour angle, with the related Greenwich sidereal time equivalent.

For conversion to the fundamental coordinate system unit vectors, and which point in the direction of the coordinate axes of this coordinate system, expressed by the two variables, and:

Since the direction of the axis corresponds to the difference of the position vectors from the Earth to the Sun and Moon, the unit vector in the direction of the axis can also be expressed as follows:

Equating the two representations of it is now possible and thus determine all unit vectors of the fundamental coordinate system.

Using these unit vectors the coordinates of the Sun and Moon in this coordinate system can now be determined. And coordinates of the Sun and Moon identical - Due to the definition of the fundamental plane are. These represent simultaneously the intersection of the shadow axis with the fundamental plane and represents the next determined Bessel elements. Furthermore, the coordinate of the moon is determined, as it is required for the calculation of the shadow radii.

The angle between the shadow axis and the tangent to the sun and moon, form the coats of the cone half-and umbra, can be determined by means of an auxiliary triangle. The tangents are shifted in parallel, so they go through the center of the Moon (pictured at right ). Hypotenuse of both triangles is the line connecting the Sun and the Moon midpoint, the Gegenkatheten the unknown angle form the parallel shifted to the tangents at right angles routes through the center of the sun. In these right triangles each the length of two sides is known, on the one hand, the distance between the sun and the moon, on the other hand the length of the opposite side, which corresponds to the penumbra of the sum of solar and lunar radius, when the umbra of the difference of these two quantities. Thus, the following applies:

To calculate the last two remaining Bessel's elements and the radii of semi - and umbra in the fundamental plane, the distance between the points of intersection of the tangents is required with the shadow axis from the fundamental plane. For the penumbra with designated point of this is on the shadow axis between the sun and moon and represents the pinnacle of the penumbra cone dar. The intersection is also at the shadow axis and the top - that is, the end point - the umbra. The following applies:

By means of these distances of the points from the fundamental plane can be the radii of the cone of shadow in this plane be determined as follows:

If the cone tip of the umbra seen from the moon behind the fundamental plane falls, so there is a total solar eclipse, is negative, otherwise positive, which is the case with an annular solar eclipse. According to the convention, the sign of the core shadow radius is selected so that it is given negative in the case of total visibility with annular visibility however positive. The sizes and are always positive.

For calculating a moon radius is selected, representing an averaging of the irregularities of the moon edge (). However, since the totality of an eclipse is not present, as long as shining through the deepest moon valley sun's rays reach the observation point yet, also a second, smaller value () is used to calculate the zone of totality and duration.

Indication of the Bessel elements

The Bessel elements are time dependent. To describe a coverage, they must therefore be specified for a period covering, for example, for a complete description of an eclipse of the sun for several hours.

Another variant is to specify the Bessel elements for a reference time (), for example, the maximum nearest hour in Terrestrial Dynamic Time ( TDT ), and in addition the hourly changes for all not to be regarded as constant elements. This allows the calculation of the values ​​for other time points of the darkness curve as a linear function of time.

The statement in polynomial form provides a slightly more accurate approximation compared with the linear interpolation. In this case be specified for the variable quantities, in addition to the value at the time of up to three polynomial. Calculating the value at a particular time is then of the form:

This corresponds to the varying sizes of the difference to the time in hours.

In practice it is often resorted elements on the Bessel published by the Goddard Space Flight Center of NASA in polynomial form. Information published in the Astronomical Almanac Bessel elements of value for practical reasons already applying the candidate trigonometric functions (sine and cosine ) is given, also the sizes, which for the entire darkness course can be regarded approximately as a constant hourly changes in the sizes and.

For example the application of the Bessel elements

In the following example, the first Bessel elements for a given time are calculated, with which the position and size of the core and penumbra cone at the fundamental level at this time are known. Then, for practical applications must be examined, such as points on the surface are relative to this shadow cones. All sizes required for this are determined by the geometry of the earth. In the example, it is investigated whether a given location is within the umbra cone.

Determination of the Bessel elements for a specific time

The adjacent table shows the Bessel elements of the solar eclipse of August 11, 1999 in polynomial form. The aim is now, for 12:34:03 CEST ( 10:34:03 UT corresponds to ) the position of the umbra in the fundamental plane to calculate.

First, the difference to the reference time (11:00:00 TDT ) is to be determined. This is still the difference between TDT and Universal Time (UT ) taken into account, which amounted to 63.7 seconds at the time of darkness:

The coordinates of the intersection point of the shadow axis with the fundamental plane for the desired time can be calculated as follows:

Analogously, calculate declination and hour angle ( the missing values ​​in the table for or are to be set to 0 ):

Likewise, it is now possible, the radius of the umbra in the fundamental plane for this time calculate:

The penumbra radius can be calculated in the same manner, it is, however, not required for the following calculation.

Test whether a given point at this time is in the zone of totality

In the first step, the Bessel elements of darkness of 11 August 1999 has been calculated for 12:34:03 CEST. Now you want to check whether the Schlossplatz in Stuttgart (48 ° 46 ' 43 "N, 9 ° 10' 48" O48.778559.1799111111111 ) was at this time in the zone of totality. For this purpose, the coordinates of the Schlossplatz be converted into the fundamental coordinate system. Are these coordinates determined, can be easily determined whether this point lies within the shadow cone, since the shadow axis is perpendicular yes, by definition, on the fundamental plane.

First of all this are the given geodesic coordinates of the castle square ( = 48.77855 ° and = 9.17991 ° ), including the ellipsoidal height ( = 295 m [A 2] ) in geocentric spherical coordinates ( and ) to be converted, the length remains unchanged. For this purpose, the numerical eccentricity of the ellipsoid of revolution of the Earth and two other derived, broad -dependent auxiliary variables are used:

With the equatorial radius = 6,378,137 m can be the geocentric coordinates calculated as follows:

It expresses the distance of the castle square from the Earth's center in units of equatorial radius of, the angle between the equatorial plane and the Palace Square facing the geocenter position vector.

As an auxiliary variable of the hour angle of the observation location is now determined with respect to the axis of the fundamental coordinate system. It should be noted that in the Bessel elements of the hour angle, assuming a Ephemeridentags (corresponding to the dynamic time formerly Ephemeris Time ) is calculated. However, since the actual rotation of the earth is not quite regular, it must first be corrected for the time difference between Dynamic Time and Universal Time, which corresponds to. To calculate the corresponding angular correction, the sidereal day length is decisive, the difference with the synodic day length ( solar day ) is taken into account by a factor of 1.002738. [A 3] Actually need the longitude of the observer () of peeled, since both angles but in are measured opposite direction, it is an addition.

This allows the Cartesian coordinates, and the palace square in the fundamental coordinate system determined as follows, where the slope of the fundamental plane with respect to the geodetic coordinate system is taken into account by the declination:

Setting the radius of the cut surface of the umbra cone in through the palace square, the fundamental plane parallel plane is closer to the Sun and Moon and is therefore slightly larger than the radius of the umbra in the fundamental plane. It can be based on the specified elements in the Bessel cone angle of the shadow cone ( ) and the distance of the castle square of the fundamental plane () calculate. It should be noted that the radius of the umbra is negatively specified during a total eclipse by definition.

The distance of the castle square of the shadow axis in the same plane can be determined as follows:

Since the distance of the Schlossplatz in this plane is smaller than the radius of the shadow cone, the Palace Square was so at the appropriate time within the umbra. Because it was raining at that time in Stuttgart, but no observation of the eclipsed sun was possible on the Palace Square.

By iteratively performing these calculations for a period can, in principle, the contact times at a certain place determined. There are also direct methods to calculate the contact time.

More star occultations by the Moon

Occultations of stars by the moon

In occultations calculating the Bessel elements to solar eclipses can be greatly simplified because it is sufficiently accurate to regard the overcast sky body as infinitely far away. This assumption makes it possible to saw the light rays of the remote object that reach the Earth-Moon system as a parallel. Thus, it appears that the direction of the shadow axis, ie the axis of the Bessel fundamental coordinate system throughout the course of the coverage exactly pointing in the direction of the star and is thus given by the equatorial coordinates of the star from the outset.

A further simplification compared to a solar eclipse is that no core and penumbra cone has to be written, but that it is sufficient to be understood "shadow" as is perpendicular to the fundamental plane cylinder. The radius of this cylinder corresponding to the moon radius corresponding to 0.2725 of the equatorial radius of the earth. Specifying variable shade radii and opening angles is therefore unnecessary.

The fundamental plane is chosen in analogy to the solar eclipses, ie the line passing through the center of the earth plane normal to this axis shadow. The line of intersection of the fundamental level of the equatorial plane is the axis, and is facing East, perpendicular to the Earth's center is in the axis and pointing north. As with solar eclipses, all figures in this coordinate system in units of the equatorial radius.

Unlike the eclipse is not a full hour, but the time of conjunction is chosen in right ascension as the reference date for the Bessel's elements, ie, the time to have the star and moon the same right ascension. At this time, the coordinate of the cylinder axis is equal to 0, so that in the tables, only the coordinate of the axis of the cylinder is indicated in the fundamental plane. The Bessel elements of an occultation are defined as follows:

For predictive calculations, it is sufficient, and to be regarded as constant during the entire course of the cover.

Coverage of the planet by the moon

The method of Bessel's elements can be applied to any star occultations, when the luminaries are sufficiently accurate spherical. It is only the position and size of the sun at replaced by the respective planet. Exceptions were Bessel in 1842, only the planets Jupiter and Saturn, since the deviation from the spherical shape was measurable at that time. To predict the visibility area for occultations of planets by the moon, the same simplified method can be applied as stellar occultations ( see above).

However, if the contact time can be accurately determined, an optional deviation from the spherical shape of the planet is taken into account and which part of the planetary disk is irradiated at the time of the cover of the sun. This procedure was described in 1865 by Chauvenet, since Bessel's method for the meantime become more precise methods of observation was not accurate enough. Here, the sunlit and visible from the Earth part of the planet is viewed directly and not shown in a fundamental plane.

Transit of the inferior planets

At transit of the inferior planets Venus and Mercury from the sun is the celestial bodies covering the planet. This may the sun never completely cover, because the umbra is much too short to fall to the ground. Also for these astronomical events Bessel elements to calculate the local conditions are used. It can be used exactly the same method of calculation as during solar eclipses, the planet takes on the role of the moon.

Since the removal of the lower planet Earth is much greater than that of the moon, there is the possibility of a simplified calculation of the time points of entry and exit of the planet wheel from the sun in transits. This method does not require the conversion of the ephemeris into the Bessel fundamental coordinate system. It makes use of advantage that the squared or raised to a higher power parallax of the planet is so small that it can be neglected. Starting from the center of the earth related to the contact times the corresponding time points can be calculated at any point of the earth in this way. The principle of this simplified calculation goes back to Lagrange and was improved by William Chauvenet by he was aware of the Earth flattening.

Lunar Eclipses

In a lunar eclipse, a terrestrial observer is on the celestial body that casts the shadow. Thus you can see from all the places on earth exactly the same course of darkness, provided the moon is visible. Bessel's method is also suitable for the calculation of lunar eclipses, although the calculation of the local conditions on the ground in this case is not required.

As with solar eclipses, the fundamental coordinate system refers to the shadow axis always passes through the center of the Earth at lunar eclipses. The calculation is similar to that during solar eclipses. Right ascension and declination of the shadow axis are obtained in this case directly from the corresponding values ​​of the sun, the axis but shows away from the sun. Thus, the following applies:

In the same way as during solar eclipses the geocentric position vector of the Moon can be converted into the fundamental system. About the - and - component the position of the Moon's center of gravity with respect to the shadow axis can be determined. Since all use angle have their apex at the center of the earth, no length information is required, in contrast to solar eclipses for the conversion. The coordinates and refer to the unit sphere. The derived angles are given in seconds of arc. The formulas used in this case correspond to the lack of unit conversion was used during the eclipse.

From this, the angular distance of the moon midpoint of the shadow axis directions:

The size of the radii of semi - and umbra are also given as geocentric angle of view. The sizes and describe here the visual angle of the shadow radii in lunar orbit. In the adjacent figure, the dashed line indicates the moon's orbit. The angle is the angle of vision of Earth's radius seen from the moon, and thus corresponds to the parallax of the moon. Since this angle is an exterior angle of the triangle, applies to the visual angle of the umbra in lunar orbit

With the half opening angle of the umbra cone. Analog can be derived on the triangle another angle relationship: the exterior angle equal to the angle of vision of the solar radius of the earth, the angle of the geocentric parallax of the sun. Thus, the following applies:

From both angular relationships now the sought angle can be determined by elimination of the cone angle:

Analogously, also the geocentric angle of view of the penumbra radius in lunar orbit can be determined. For this results in the following relationship:

To assist in the determination of the contact times of darkness, three more auxiliary variables are derived from the sizes of the core and penumbra and the moon radius. These are the visual angle for the distance of the moon midpoint of the shadow axis during a particular contact, which are calculated from the visual angles of the shadow radius and the angle of view of the lunar radius:

The calculations presented so far used only angle to the shadow axis and arrived with no definition of the fundamental level. If you want to calculate when certain lunar craters - so distinctive points of the lunar surface - or a - emerge into the umbra, this is possible if one so chooses, the fundamental plane, that it goes through the center of the Moon - in a similar way as for points of the earth's surface at solar eclipses.

When checking the calculated contact times and in particular the umbra - entry and exit points in time certain lunar craters, the calculated data thus show no useful correspondence with reality. This is due to the fact that the Earth does not cast sufficient circular shadow due to their flattening. Second, it is up to the Earth's atmosphere, zoom through which the shadow cone. To compensate for these effects, it is customary to introduce the formulas for calculating the size of the half-and umbra cone two correction factors:

But even so calculated data show no particularly precise correspondence with reality. This is mainly attributed to the fact that the flattening of the atmosphere is not significantly larger than the ground level. An attempt is made on the basis of observational data of various lunar eclipses to develop a more accurate correction procedures.