Bases of celestial mechanics

3. Basics of celestial mechanics

The movement of the stars

3. Bases of celestial mechanics

The movement of the stars

At first the beginner is disconcerted by the apparent movement of the stars. The stars maintain their apparent distance from each other, but they appear each night in a somewhat different position and then move on further. Some stars and constellations are visible all year round in the night sky, but others disappear. After a few hours in the western horizon, new stars and constellations then appear in their position. “The movement” of the stars is very slow and is hardly detectable by observers.

Fig 49: The earth rotates once every 24 hours on its own axis. The Earth’s axis is not vertical, on the other hand, but is inclined by 23.27 º from the plane of the orbit, in the direction of the sun

If however a telescope with higher magnification is pointed towards a star, then the star will disappear after some minutes from the visual field of the telescope and one must “adjust” the telescope onto the new star position.

An experiment can show very easily that the position of the stars change (evidence that the earth rotates):

Look for a bright star or constellation, which appears over a prominent point on earth, such as a house, a tree or a mast. Note the time and observe the position of the star or the constellation one hour later. What do you determine?

You will find that the stars moved westward with reference to the prominent point. They did not change their position to each other.

If you observe these star on the following nights at the same time, you will determine that they are positioned over the point approximately four minutes earlier each night. Does the earth rotate about its own axis more slowly than once in 24 hours?

Yes! It takes exactly 23 hours, 56 minutes and 26 seconds. This difference becomes balanced by the intercalary days.

Circumpolar stars and constellations

If we are on the fiftieth northern degree of latitude above the equator, then the celestial north pole is exactly 50 degree over the northern horizon. All stars, which are less than 50 degrees of arc from the polar star, never set below our horizon. We call these stars “ Circumpolar”. The more further south we are, the lower the Pole star is in the sky then the area covered by the circumpolar stars is decreased. At the equator there are therefore no circumpolar stars. Exactly at the North and South Poles however the stars neither rise nor set, but circle the horizon at a constant altitude.

Fig 50: If at 4:15 (left) the Pleiades and the constellation are positioned above a prominent point, then it will be determined one hour later that they have moved westwards. However they have maintained their position relative to each other.

Apart from the circumpolar constellations, the selection of available celestial objects depends on the season. By means of a rotating star map, one can determine the visibility of the constellations for the reSpotting scope observation places at each season. Previously mentioned yearbooks and technical periodicals offer further orientation assistance. After these fundamentals we would now like to present some objects which are worth seeing. We are limited here to easy and moderately difficult objects.

Circumpolar constellations: The constellations of the Great and Little Bear, Lynx, Cassiopeia, Cepheus, Camelopardalis and Lizard never go down in our latitudes. One can observe them during each season. The observation conditions depend also on the observation date, because circumpolar constellations are positioned either low or high in the sky.

The Pole star is clearly visible at all times. It is very close to the celestial pole and is a double star, which many people do not do not realise. About 18 arc seconds from “Polaris “ we can make out a small faint star. The Great Bear contains the most famous pair of double stars in the sky. Mizar and Alkor, which we already described in the introduction. The two can be readily identified and be checked with the naked eye, and have been used since long ago as eye testers. In the telescope we find a further companion beside Alkor, which is only 14 arc seconds distant and is a physical double star. Mizar and Alkor however are only spatially close together.

A deep red star can be found in the constellation of Cepheus. Because of its colour, ?(micro)-Cephei is called the garnet star. b-Cephei is a beautiful double star. Two stars of differing brightness stand at a distance of 13 arc seconds apart.

The five brightest stars in Cassiopeia form the remarkable “W” in the sky. With binoculars we can make out the open star clusters M103 and M52, which are members of our Milky Way. h Cassiopeia is a double star. A yellowish and a reddish star circle each other at a distance of 13 arc seconds.

Circum polar stars and polar star photographed

Circum polar stars can be make visible photographically. The best time is at the beginning of the year. In the summer the night is too bright for such photography.

Telescopes with equatorial mountings and tracking motors or computer control are suitable for astrophotography.

You will need a camera with a cable release, a sensitive film (400 ASP/27 DIN or less is sufficient) and a stable stand. It is important that the shutter of the camera has a control for selecting the exposure time “B“(arbitrary). This way we can leave the shutter of the camera open for any length of time and expose the film over a long period.

Insert the film into the camera, set the sensitivity of film and turn the wheel for the exposure time to step “B” .The camera is now attached to the stand and aligned onto some bright stars. Screw the cable release into the trip button. Set the focus to infinity. The diaphragm is completely opened. Open camera shutter for at least 30 minutes by pressing and tightening the camera release. Depending upon the sensitivity of the chosen film, you can take such a picture with up to, or over two hours exposure time. Lock the cable release after pressing, with the locking screw. When the time has elapsed, simply loosen the locking screw again and the shutter closes again.

A useful trick before you operate the cable release, likewise before completion of the exposure, is to cover the camera objective with a dark cardboard box. In this way you will not blur the picture and the lines and/or star arc created and will not show serrations at the beginning and at the end of the exposure. During the exposure time the view finder of the camera is not available.

Fig 51: A SLR-camera with cable release (A). The exposure time is set to “B”(Bulb)

If you bring the film to your dealer for developing, be certain to point out that these are astronomical photographs, otherwise the pictures will not be processed using automatic development. Try different exposure . Experiment!

Fig 52: During the exposure of the film the stars continue to move in the night sky. In this photo by M. Stoelker – which was taken in the spring, it can be seen which stars disappear beneath the horizon in a short period of time, that is “sink”. 2 hour exposure, taken with a 400 ASA Film.

On the photographs it becomes visible that the stars turn along different paths apparently around a central point. This central point is the pole star.

With stars, which are visible as circular arcs in the photo are the previously described circumpolar stars, i.e. these stars are always to be seen pole area of the sky.

In which part of the sky do we find the “circumpolar” stars?

If we turn northwards, we will find the constellation of the Great Bear. This constellation is “circumpolar” i.e. we can see it each night at all times in the sky.

Fig 53: The same picture as the one on the right, showing here the stars of the circumpolar regions, which never sink below the horizon.

Depending upon the season, the Great Bear is sometimes close to the horizon and sometimes can be seen almost vertically above us. Whatever the position might be, the two stars “at the front of the plough” always point in the direction of the pole star.

If we imagine a line, which extends from the Pole star perpendicularly to the horizon, it will meet the horizon at the so-called North point. All stars, which lie between the Pole star and the North point, will never dip below the horizon. They are visible throughout the year, these are circumpolar stars.

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Fig. 54: The graphic shows the area of circumpolar stars between the Pole star and the North point.

3.1 Why does the sky change throughout the year?

If you imagine, your telescope is firmly set up and points at a certain time to Sirius, which is the brightest star in the northern sky, then you will see Sirius again after one complete earth revolution again in the eyepiece and in you will see that it is moving. The earth actually rotates once around its own axis in 23 hours, 56 minutes and 26 seconds. If we looked in the eyepiece 24 hour later, then we would have missed the passage of Sirius in the eyepiece by exactly 3 minutes and 34 seconds. For this reason a bright object such as Sirius rises daily exactly 3 minutes and 34 seconds earlier. In ten days that is approximately 35 minutes. The same applies for the other non circumpolar stars. The same also naturally applies to constellations, which rise daily about 4 minutes earlier.

Observe Sirius, the main star in the Canis Major Constellation above a prominent point on your visible horizon and make a note daily over a period of around ten days at which time it is in this position. After ten days Sirius would be in this position about 35 minutes earlier. The duration of one revolution of the earth is called an astronomical Earth day or also a sidereal day.

For the sake of simplicity we have divided the day into 24 hours and we therefore accept that the constellations in the course of a year will move day by day and so will the typical spring summer autumn and winter constellations seen during the evenings.

3.1.1. Why are there intercalary days and leap years?

Our sky, in the astronomical sense, is very varied, because in the course of orbiting the sun the Earth describes one plane around the sun and thereby moves in a circular path around the central object of our solar system.

During this orbit the earth rotates 365 times about its polar axis, therefore 365 sunrises and sunsets and somewhat less than 6 hours will go by. It was agreed many years ago that as far as the calendar is concerned, there would be 365 days in a year. Nature however requires a few more hours.

Every four years we acknowledge this time deficit in the duration from 365 days and every four years an additional day is added onto our calendar.

This way we prevent the seasons from being pushed back, in terms of the calendar, one day every four years. Your birthday remains, for example on 27th August, nothing changes . The weather however changes. In the spring there is a date, that is to say a calendar day when the sun is visible above the horizon for twelve hours and is below the horizon for twelve hours. Spring begins on the 21st of March each year. In relation to annual weather this means a constant shift of the weather periods for all calendar months, once in, for instance, 365x~4 years. A certain birthday, for example in the summer on the 5th July would move into spring. There are a great many customs and traditions, and cultural rites everywhere in the world, which depend to a large extent on the weather. By adding an additional day, the intercalary day, these celebrations and events remain on the prescribed calendar date and places the beginning of spring punctually every year on March 21st.

This point in time is referred to both as the date of the first day of spring and also the first night of spring. The sun remains central during this period always at a certain point in the sky, for the first day of spring. If one did not add the intercalary day every four years, the sun after four years would reach the point for the first day of spring on the 22nd March, thus one day later, therefore the beginning of spring would move every four years by one day. Please do not confuse this with the fact that the rotation of the Earth takes a little less than 24 hours.

The rule that a day has 24 hours and that the year represents exactly 365 days is just a practical simplification for mankind. One Earth year (astronomical) and one terrestrial year (calendar) are therefore different.


The spring:

The dominant constellation in the spring sky is Leo (the lion). Leo is easy to recognise, it has a very distinctive appearance. In the constellation of Leo are to be found several galaxies, which are not very easy to find because their brightness is not very great. It involves M65,M66 and also M96, all of which are spiral galaxies.




Somewhat to the west of the constellation of Leo, the constellation of Cancer can be found. Cancer is a rather inconspicuous constellation, in which there are two beautiful open star clusters. The splendid Manger or Beehive, as the star cluster M 44 is popularly called, is reduced in the binoculars into a beautiful single star. It is possible to see at least 40 stars approximately 500 light-years away.Alittle further south the open star cluster M 67 can be found, which is substantially smaller, but is nevertheless impressive because of its high concentration of stars. The star cluster is about 2,700 light-years distant.

East of Leo can be found the constellation of Berenice’s Hair and to the south lies Virgo. The attraction of these constellations is the Virgo Cluster. If the telescope is pointed towards the Virgo Cluster and if the area is carefully examined, some small blurred “stars” can be seen. These are a distant galaxy, which can often only be recognised as a galaxy after very careful observation. The distance of this galactic cluster is also over 40 million lightyears away.

This is obviously only a small part of the visible sky. A view of a detailed star map reveals an abundance of further objects. The still comparatively dark nights in the spring and often surprisingly good weather can often make these nights very entertaining. Somewhat to the west of the constellation of Leo, the constellation of Cancer can be found. Cancer is a rather inconspicuous constellation, in which there are two beautiful open star clusters. The splendid Manger or Beehive, as the star cluster M 44 is popularly called, is reduced in the binoculars into a beautiful single star. It is possible to see at least 40 stars approximately 500 light-years away. A little further south the open star cluster M 67 can be found, which is substantially smaller, but is nevertheless impressive because of its high concentration of stars. The star cluster is about 2,700 light-years distant.

East of Leo can be found the constellation of Berenice’s Hair and to the south lies Virgo. The attraction of these constellations is the Virgo Cluster. If the telescope is pointed towards the Virgo Cluster and if the area is carefully examined, some small blurred “stars” can be seen. These are a distant galaxy, which can often only be recognised as a galaxy after very careful observation. The distance of this galactic cluster is also over 40 million light-years away.

This is obviously only a small part of the visible sky. A view of a detailed star map reveals an abundance of further objects. The still comparatively dark nights in the spring and often surprisingly good weather can often make these nights very entertaining.


The summer:

In the summer it becomes dark either late or never absolutely dark. This is not advantageous for astronomical observations. Clear weather and pleasant temperatures make observing fun. Moonless nights are still dark enough in the summer to admire the Milky Way. Even with binoculars one seems to drown in the sea of stars. Relax! With its many open star clusters and gas nebulae, the Milky Way provides much entertainment. The three main constellations, whose main stars are referred to as the summer triangle, are the Swan, the Lyre and the Eagle with their bright stars Deneb, Vega, and Altair. The constellation Swan, which lies within the Milky Way band, has one of the most beautiful double stars of all. The pair of stars is called Albireo and represents the head of the swan. At a distance of 34 arc seconds there are a yellowish star and a sapphire-blue star. They are easily recognised by their different colours.




In the constellation in the Lyre there is another beautiful double star to be found, called ?(epsilon)-Lyrae. ?-Lyrae is close to Vega. The two components stand apart, that is nearly 1/10 of the moon’s diameter. With high magnification and good seeing one can separate the two stars into two closely neighbouring stars, which are distant from each other by about two and a half arc seconds. Here we have a genuine four-fold system, that is to say stars which form a gravitational system similar to the Earth - Moon system.

Probably the most well-known object in the Lyre is the Lyre Ring Nebula or M57. In order to find this jewel, we must point our telescope at ?-Lyrae and then slowly move in the direction of ?-Lyrae. With low magnification, a faint smoke ring can be seen half way. With higher magnification the structure of the ring becomes clearer. This object is a planetary nebula, which does not have anything to do with planets, however, despite the name. One sees the dust and gas of an imploded star, which became a white dwarf which shines due to the hot remains of the star.

West of the Lyre is the constellation of Hercules, which also contains two objects from the Messier Catalogue. One is the globular star cluster M 92 and the other is the globular star cluster M 13, which qualifies as the most beautiful globular star cluster of the northern skies. M 13 can be identified with the binoculars as a small, blurred “star”, but in the telescope reveals its true beauty in the sky. In a small telescope the edges of the single stars can be seen.

Even with binoculars we can penetrate deeply into the band of the Milky Way. If we move the binoculars to the south towards the Sagittarius constellation, then we can discover gas nebulae and star clusters with a good view of the horizon. Amongst them, for example, the Wild Duck Cluster is in the constellation Scutum, which qualifies, with many amateur astronomers, as their favourite object. In addition the Omega Nebula and the Eagle Nebula also qualify. They consist of enormous hydrogen clouds and are the birth place of the stars.


The autumn:

In the autumn, slowly the summer constellations say good-bye and after midnight we already risk the possibility of a view of the forthcoming winter sky. The appreciably longer nights allow astronomical observations to be started in the evening. The most remarkable constellation in the autumn is Pegasus. Pegasus has galaxies to offer, which shine very weakly however. The globular star cluster M 15, is well worthwhile. This is 31,000 light-years distant. M 15 is not as impressive as M 13, but can in addition, be separated into individual stars.




East of the constellation of Pegasus is the constellation Andromeda. In this constellation is one of the most famous galaxies of all, the Andromeda Nebula or M31. These are 2.2 million light-years distant and the spiral nebula is already recognisable on dark nights as a blurred “star”. However, with the telescope the bright core of our neighbouring galaxy can be seen. Because of the size of the object in the sky, one can only bring part of the galaxy into the visual field of the telescope. If one looks in the region of the core, you will see more detail of the spiral arms. The Andromeda Galaxy has two companion galaxies, which can be easily recognised. One is the galaxy M 32 and the other the galaxy NGC 205, both are elliptical galaxies.

The constellations of Cassiopeia and Perseus in the autumn are very high in the sky. The two constellations are still located in the Milky Way and offer some beautiful open star clusters. The most beautiful star cluster, perhaps even the most beautiful of all, is in the constellation of Perseus. It is the double star cluster of h and à Persei (NGC 884/NGC 889). These two are located only 50 arc minutes apart and can be recognised in the binoculars as a pretty couple. You can view this object, which is very beautiful to see, in your telescope with less than times 50 magnification. We can then see a double star cluster, 8000 light-years away, with approximately 400 stars.


The Winter:

Because the nights begin early in the winter, work can commence early in the evening. A great deal of love for the hobby is re q u i red at temperatures below zero degree. The correct clothes, warm drinks and a place for warming up, are good preconditions for a pleasant night of observation.




Fig 30: The Pleiades open star cluster M45 from C.Kimball

Fig 46: Ring nebula M57, taken by M. Moilanen and A. Oksanen

Fig 55: The Cancer Nebula, M1, taken by J.Newton

Fig 56: The Orion Nebula, M42, taken by C.Kimball

The observation in cold winter is worthwhile, because the winter sky has to offer some spectacular views. The constellations Charioteer, Taurus, Gemini and Orion dominate the winter sky with their bright stars. These constellations have still more to offer however. Have a look at the constellation of Taurus, which has two bright open star clusters. On the one hand Taurus presents the Pleiades, which are also known as the Seven Sisters, on the other the Hyades, in whose centre the bright star Aldebaran is located. The Pleiades consist of at least 500 young stars, which were formed over 100 million yeas ago. With the naked eye one can see at least six stars, with good conditions up to nine. The Pleiades (M 45) are close to the plane of the Earth's orbit and therefore now and then get a visit from the moon, which leads to interesting star cover. The Hyades also represents an open star cluster, which is close to the level of the ecliptic, This is how we refer to the path, which is described by the annual orbit of the earth around the sun. The moon also regularly passes through. The star Aldebaran is not Hyades star, it stands spatially in front of the Hyades.

In the Taurus constellation the object M 1 is located. This is the first entry in the Messier Catalogue. M 1 is the remnants of a supernova, which occurred in the year 1054 AD and was recorded in writing in China. Because of its appearance, M 1 is also known as the Cancer Nebula. In the centre of the Cancer Nebula is a fast rotating Pulsar, which energizes the surrounding materials and causes them to shine.

The constellation Charioteer (Auriga) lies in the Milky Way and offers several open star clusters close to the bright star Capella. These are not as bright as the Hyades and Pleiades, but are however worthwhile objects because of the wealth of stars. These are the star clusters M 36, M 37 and M 38 from the Messier Catalogue, which look like nebulae in the binoculars.

One of the most well-known winter constellations is the Constellation Orion, which reminds us of the sky hunter Orion in Greek mythology. The three belt stars, which is known as also Jacob’s staff are prominent. The Orion Nebula (M 42) is a most remarkable object which symbolises the sword of Orion, the mythical sky hunter. The nebula is the brightest gas nebula in our sky. An enormous hydrogen cloud becomes illuminated by young, hot stars. In the middle in the Orion nebula can be seen a constellation of four stars, which are called trapezoidal stars. In larger telescopes two further stars are visible. The Orion nebula is 1,600 lightyears distant and has a diameter of over 66 arc minutes. It is four times larger in the sky than the disk of the full moon. In the telescope,however only the bright centre can be seen.

Southeast the constellation of Orion, is the constellation of the Great Dog (Canis Major). In it is the brightest star in the sky. The Dog Star, also called Sirius, flickers in a whole range of colours due to its proximity to the horizon.

North of the Constellation of Orion is the constellation of the Twins. The star Castor ranks as the brightest stars of the Twins. In the telescope Castor can be seen as double star. The two stars are only 3 arc seconds in the sky from each other. A beautiful object in the Twins is the open star cluster M 35, which can be seen in the binoculars as small nebula spots.

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3.2 Use of the rotating star map

In order to be able to plan observation nights better, there are practical rotating star maps, manufactured from plastic or cardboard, in addition to star map software for computers. The observation date and the desired observation time are set on a scale along the edge of the circular star map. A circular template cut-out window indicates the section of the sky, which will be visible on the observation day at the desired time. Here we will describe briefly, how to deal practically with such maps. It is important to mention here that diff e rent maps for the Northern and for the Southern Hemispheres of the Earth must be purchased. This must be borne in mind when obtaining a map.

Local time on swivelling star maps is set for the northern hemisphere for Central Europe. For the star map Central European Time (CET) applies. During the summer time, please subtract one hour from the local time (this is to convert from winter time to summer time).


When it must be very precise:

Determine the degree of longitude of your location (e.g. 10° East for Hamburg) and the difference between the reference meridian for the CET (15° East). Thus 15° – 10° = 5°, you then multiply this difference by 4. The result is the number of minutes, which you must subtract from the CET. You have now specified the so-called, true local time at your observation site. You can now set this onto the star map.

This data can also be obtained from the internet. The web page is such an efficient data base.


What can I see at the moment?

Turn the top of the star map in such a way that the CET or the true local time coincides with the current date. Now turn the entire map so that the appropriate horizon (north, south, east, west) agrees with your own line of sight at your geographical location. Now the cut-out on the map will show the actual sky to be seen at that instant.


Where is the position of the sun?

Turn the pointer so that it coincides with the current date. Where the pointer now cuts the line of the ecliptic (the apparent path of the sun in the sky), is the present position of the sun, seen from your location.

When will it become light or dark? Locate the position of the sun, as described above. Now turn the top of the map so that the place of the sun coincides with a dawn line. The rotating star map is now set accordingly.
The following apply:

• civil dawn - brightest stars recognisable

• nautical dawn - constellations recognisable

•astronomical dawn - beginning/end of darkness


Where is the moon or planets positioned?

Look up in an astronomical yearbook the coordinates of the desired planet. Now turn the pointer, until the right ascension (hour value) of the planet is set on the hour circle. The declination, that is the angular height above the celestial equator (expressed in degrees) is read off from the pointer scale. Note: The moon and planets are always on, or close to the ecliptic. That has to do with the history of our solar system.


To determine the present sidereal time

Sidereal time (ST) is needed to point the telescope on celestial objects using coordinates. Turn the top of the star map so that the date and local time coincide with each other. Turn the pointer so that it points exactly to the south point on the map. Now you can read off the sidereal time (hour angle of the point of spring) from the hour scale.



Fig 57: The rotatable star map is a practical aid for planning an observation session.

3.3 Why can we only see part of the sky?

The answer is very simple. The earth is a sphere and lying in the meadow looking into the sky, we cannot see laterally towards the horizon around the Earth's curvature. We are presented with a sky, which can be described spatially as a very large transparent hemisphere. The Earth's curvature can be seen in coastal regions, on the beach, with binoculars or telescope, We can see sailing boats rising above or disappearing below the horizon, without damage, as a consequence of the Earth's curvature. Because the earth is of spherical shape, only half of it is illuminated by the sun at any one time. The opposite side is in shadow and forms the night side, so we only able to see half of the sky from the earth.

3.3.1 The eyes’ field of view

On the other hand, with our eyes we can only see at the most a field of vision of an angle of 110°. Of this only 5° can be seen sharply with healthy eyes, which are however controlled, more or less unconsciously, in such a way that whatever is of interest to us is centred automatically into the 5° range. As the visual field of an eyepiece is much smaller now than the total visual field of the eye, then one speaks of so-called tunnel vision: one sees only a small area, surrounded by blackness. Good, standard eyepieces have a visual field of approx. 50°, which can be comfortably observed. In addition, t h e re are also wide angle eyepieces with a visual field increasing to over 80° - giving the impression when observing of not looking through the telescope but almost floating in space past the object, as the eyepiece illuminates nearly the entire visual field of the eye.

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