A Cautionary Note.
The Grey Hill Stone Circle and Adjacent Stones (Site Description)
The Alignment of the Stone-Row and of the Notch.
Solar Declination δs
The Effect of Declination on Azimuth of Rising and Setting.
The Effect of Latitude on Azimuth of Rising and Setting.
The Effect of Altitude of the Horizon on Azimuth of Rising and Setting.
Calculating the Declinations of the Grey Hill Azimuths.
Discussion of Results.
Azimuth of sunrise on days around the winter solstice.
Possible Alignment to the South.
Appendix I. Obliquity of the Ecliptic (ε).
Appendix II. Precession of the Equinoxes.
Appendix III. Atmospheric Refraction.
Appendix IV. Survey Technique - Data Gathering and Processing.
Some Grey Hill Websites.
Llanfair Iscoed Stone & Castl
A technique for finding AZP using a theodolite and clock to 'shoot the sun'
A very good introduction to archaeoastronomy can be found on this Wikipedia page, and for a full account of the topic the reader should go there. Archaeoastronomy is more a branch of archaeology than of astronomy, so it includes all of the subtleties and richness that archaeologists bring to their studies. It is therefore simplistic to define the topic merely as an attempt to find in archaeological sites evidence of astronomical enthusiasms on the part of their creators. Archaeologists will naturally seek a more integrated approach which will interpret these artifacts within the context of an emerging understanding of ancient cultures as a whole, of which interest in the observation of the heavens is just a part. Respected professional astronomers (for example Gerald Hawkins - author of Stonehenge Decoded) who have ventured into this area have exposed themselves to criticism for adopting too narrow an approach, and their findings have often been rejected on the grounds that they do not integrate well into a broader understanding of the cultures to which they refer. Conversely, when the astronomers produce hard evidence, the archaeologists do well to find a place for their findings in their wider view, and this has often been challenging for them. Mainstream archaeologists have been restrained in their acceptance of astronomical interpretations of stone circles, rows, avenues, etc, but the general idea that ancient cultures can be expected to represent in their artifacts interest in the seasonal movements of the sun, and sometimes of the moon and stars, has firmly taken root. In many cases alignments to solsticial sunrise or sunset, or particular lunar events have become accepted as legitimate interpretations of megalithic and other monuments. (Burl (1995) p. 18). Convincing alignments to stars are rarer.
I became interested in this topic when I expressed to a friend my sense of outrage that the explanatory board which has been erected at the site of Harold's Stones, Trellech, essentially said that no-one had the least idea what the stones are about. He said I should look into it myself, so I did. These stones are three in number, are impressively large – up to 4m in height, and stand in a compact row about 7m long. They are suggestively almost in a line, but not quite so. They are supposed to date from the Bronze Age. One of them leans dramatically, suggesting it might have moved over the millennia, and opening the possibility that they might once have been precisely in line. I chose to see in the stones a line that points somewhere, but others have seen in them a fragment of a very large circle that goes nowhere. We are immediately in the realm of subjective response; another commenter on the internet even saw the leaning stone as a giant phallus.
|But already the seeds of what was to come
had been sown; I
had read Thom.
Thom was a professor of engineering from the University of
Oxford, who took up archaeoastronomy in his retirement.
He surveyed hundreds of megalithic sites with
meticulous care, and his surveys have won him enormous respect, though
interpretations of them have often been contentious.
He identified several sites which he regarded
as being solar or lunar observatories on the grounds that they embodied
astronomical alignments. The
solar site is at Ballochroy on the western side of the Mull of Kintyre,
while an even more
astonishing lunar site is at Callanish
on Lewis. In order to understand my Grey Hill case-study, it
is necessary first to look at Thom's interpretation of Ballochroy.
Fig 2 is a famous illustration from Thom's Megalithic Sites in Britain (p 142). The bottom left inset shows an approximate alignment of three stones, similar to Harold's Stones. Beyond the stones lies a kist (a slab-sided burial pit) and a view over the sea to a small island - Cara - in the Sound of Gigha, 13km distant. The plan on the right shows the alignment of the stones and the kist, while the middle left inset shows the trajectory of the sun as it sets on the winter solstice, as seen from the stones. It is Thom's contention that the alignment of the stones was designed to point to the midwinter sunset, and that the position of the site was chosen to achieve this particular pattern of the setting midwinter sun in relation to the undulations of the horizon created by Cara. Note that, as with Harold's stones, the alignment created by the stones is indefinite - the stones are not perfectly in line, and their width allows a wide range of possible azimuths. They do not even point exactly to Cara, but to the left of the island. The stones are merely a pointer or gesture to the real alignment, which is from the last stone to the SW (the 12 ft stone) to Cara, and this is a very precise alignment, designed to produce a definite visual effect.
The middle stone at Ballochroy is a slab, erected with its wide faces broadside-on to the alignment to Cara (see site plan, Fig.2), an arrangement which seems unintuitive since it appears to block movement in the direction of the alignment.It invites a view along the faces of the slab, at right angles to the alignment with the other stones. Thom found that this is the direction of the midsummer sunset (Fig.3), and again, the path of the sun as it sets on the horizon is pleasing; it first collides head-on with Ben Corra - one of the Paps of Jura - and then 'rolls' behind its profile, leaving just a glimpse of visible surface (see top-left inset, Fig. 2).
|Ballochroy thus appears
to be a
splendidly unusual site, combining two important solsticial events in
one monument. To have found a site that makes this possible
extraordinary good luck on the part of its makers (or the result of an
extremely diligent search), since the distances to the markers on the
horizon (13km and 30.6km) mean that the position of the monument must
be shifted over large distances to achieve only small adjustments in
the effects produced at the horizon. In the case of
Ben Corra, to
adjust the position of the setting sun by 1º, the central stone must be
shifted by 0.53km, and such a shift (if needed) might well relocate the
monument to some completely unsuitable locus where the local
terrain is unconducive, or sightlines are obscured by local
impediments. Such a difficulty will be discussed later in
to Grey Hill.
Note that in the second alignment at Ballochroy there is no other rearsight than the unexpectedly rotated slab, and none is in fact needed. The faces of the rearsight slab merely gesture in the direction of the distant foresite of Ben Corra and the focus of the alignment is actually improved by eliminating any imprecise local alignment with other nearby stones. Aubrey Burl (p177) was unconvinced by this interpretation of the middle stone. He comments that the alignment to Ben Corra is probably mere coincidence, giving his reason that only the central stone is properly orientated. He seems to expect that all three stones should point accurately to Ben Corra if this significance is really intended. Other critics of Thom's interpretation of this site have commented that the kist was probably originally covered by a cairn, which would have obstructed the view to the midwinter sunset over Cara. Unfortunately no-one knows exactly how high this cairn might have been, and the sight-line is now obstructed by trees, so little can be done to test this idea.
|In Megalithic Lunar Observatories,
Thom identifies many such alignments of risings and settings over
horizon features (in these
cases mainly of the moon), and seems to have
established the principle of alignments of this kind convincingly.
Hard-nosed critics will remark that, as in the case of
ley-lines a fully statistical analysis is required, since otherwise if
a horizon contains many lumpy features, the probability of one of them
matching a solsticial or lunar event by chance becomes unacceptably
large. For example, Jura has many paps (see Fig. 3), so one
is bound to fit; perhaps
this is what was in Burl's mind. Similarly, Cora has three peaks,
so you have three chances of finding something suitable (Fig.
However, if a horizon profile is mainly flat, and a single feature
on it aligns well, the case is more convincing. It was with
freshly in my mind that I embarked on my expedition to investigate
It turned out that these stones are aligned very roughly with the midwinter sunset, with an azimuth of approximately 46.8º, or its reciprocal, 226.8º . The stones are broad, closely spaced, and out of line, so it is impossible to establish an alignment with a precision of better than +/-5º. Furthermore, the site is in a low position without distant views or low horizons, so it is not a candidate for Thom's interpretative approach.
However, on the same trip, I included a visit to Grey Hill, and this is entirely a different matter. The location is on an open hillside which slopes to the SE, with spectacular and panoramic views of the Bristol Channel and the escarpment of the Cotswolds, North of Bath, beyond.
circle is at ST 438 944 and can be approached from the track
which leads upwards from the Wentwood Reservoir car park at ST 428 939,
or from the track from Llanfair Iscoed. The former route is
marked as a footpath on the OS map and includes a
assault on the 275m summit. From here there are exhilarating
panoramic views of the Bristol Channel to the South, and inland to
forests and distant mountains; the magic of the location is immediately
striking. There is reputed to be a cairn on this summit, but
have not found it.
The footpath which ascends to the summit continues along the ridge of the hill and passes disturbed ground which has been interpreted as recent small-scale stone quarrying. After about 100m there is a lesser track to the right which descends the hill, passing first the standing stone marked on the OS map (called here "Standing Megalith 1" - SM1), and then leading to the circle with its attendant second megalith (SM2) at a distance of 57.3m from SM1. (See Fig. 5). Immediately to the left, and a little lower than SM2 is a smaller stone (SS1 - see Fig.14). Between SM1 and SM2, 35.7m from SM1, there is a scatter of other, recumbent, stones sketched in Fig. 5 as RS1 - RS6. The stones designated SM1, RS1 - RS3 and SM2 form a definite straight row which is approximately tangential to the circle, SM2 being about 1.5 to 2m from the eastern edge of the circle. The track continues to descend below the circle, and after about a further 100m, a second small stone (SS3) will be seen on the left at ST 44088 93282. There is said to be a "cairn cemetery" near this stone at ST 44086 93273, but I have not found this. It is possible that SS3 lies on the SM1-SM2 alignment, but this cannot be verified since SS3 cannot be seen from the other stones.
RS4 - RS6 are to the right of RS1 - RS3 as seen when descending the hill, and seem to be aligned directly to the centre of the circle. All of the stones mentioned so far are of the conglomerate type (see below - Geology).
There are reports of a third smaller stone (SS2) above SM1, but I have not been able to find this.
The circle can also be approached by following a track which descends immediately from the summit. This leads diagonally across the slope of the hill, and emerges beneath the circle (Fig. 5). On this path the first sign of the monument is a distant view of SM2.
|The terrain near and above the
monument is mainly open, with a thick cover of bracken and
brambles, and a
thin scatter of low shrub. The bracken is chest-high in summer,
but even when the bracken is not in growth, thorny vegetable debris at
least 30cm deep covers the ground, making exploration beyond the
trodden areas very difficult; it is possible that much could
missed and that other stones remain to be found. The open terrain
allows panoramic views of the Bristol Channel, and the hills beyond.
A little below the circle, an area of open birch forest is
entered, again with bracken and thorny undergrowth. The trees
obscure the views and also prevents SS3 from
being viewed from the region of the circle.
SM1 rewards careful examination. It is rectangular in cross-section, with its wider faces (which are bedding planes) parallel to the axis of the stone-row. There appear to be two cup-marks, one on the edge of the stone facing the circle and the horizon notch, the other on the face directed towards the south-west. The former is somewhat oval with its long axis parallel to the long axis of the stone; it is located about 1.3m from the ground and 18cm from the left-hand edge of the stone (Fig. 9). I am by no means an expert on cup-marks; the only one I have previously seen is that said to be on the middle stone at Trellech - Fig. 1. The mark illustrated in Fig 9 might be questionable on the grounds of its oval shape and the generally much-eroded surface of the stone, which is normal to the bedding planes and therefore quite rough. As can be seen from Fig. 9, this mark is quite deep, and is unlike the erosion features on the stone; this argues in its favour. It has been suggested that this hollow marks the position of a pebble which has dropped out, but the visible pebbles are so much smaller that this interpretation is not credible.
SM2 also bears a mark which is possibly a cup-mark, but this is less convincing than those on SM1. Like SM1, SM2 is also rectangular in section, with its longer faces (which are bedding planes) parallel to the alignment. The putative cup-mark is on the NE face 45cm from the ground, and 20cm from the left-hand edge. All of the supposed cup-marks that I have so far seen (including that at Trellech) seem to me to be indistinct and in need of expert authentification. An example of a heavily cup-marked stone can be found here.
Fig. 9. Oval Cup-mark on SE edge of SM1, facing the horizon notch. Note erosion along bedding planes and pebbly texture of the conglomerate.
The second cup-mark is more convincing; it is circular, and is on a much smoother surface (the face which is directed towards the south-west). It is towards the bottom of SS1, and right of centre. A channel, which is probably an erosion feature, leads to it from above. This mark is not illustrated; its possible significance is discussed below.
The stone circle itself is about 10m in diameter and is formed from 13 or so small stones, mostly less than 70cm in maximum extent, some of which are conglomerate, others sandstone (Fig. 7). The stones are all prostrate and barely break the surface of the vegetation except where this is trodden down. The circle seems to be untypical in that several of the stones (those towards the southeast) are actually touching each other in a manner that suggests that this might have once been a curbed cairn or barrow similar in some respects to that at Balnuaran.
Inside the circle are several other stones, at least two of which (FM1, FM2) could be large fallen megaliths (Figs. 7, 8, 11). Burl (1997) states that the two large stones were standing in the nineteenth century, and speculates that they might have been part of a large cist or even a chambered tomb. For a small circle to contain within itself two largish (presumed) standing stones again seems untypical. One of the smaller prostrate stone inside the circle can be seen in Fig.8. These stones, with SM1 and SM2 are all conglomerate, and all appear to have a characteristically stepped or shouldered top. Whether this shape is a result of natural fracturing, or deliberate shaping is impossible to say. In most cases it is clear that both 'cuts' required to make the step are normal to the bedding planes, so this shape would not be formed by natural flaking parallel to the planes. Despite this, and the very artificial appearance of FM2 (Fig. 11) I feel that, pending further investigation, the null hypothesis should be that it is a natural fracturing pattern. The presumed megalith at nearby Llanfair Iscoed has a similar profile.
Geology. Many of the stones of the circle and stone-row are small, recumbent, and barely rise above the vegetation cover (Fig. 13). Sometimes it is difficult to say whether they are part of a man-made structure, or merely part of the natural rocky surface. However, the stones are of two distinct types - reddish-brown sandstone, and greyish quartzite-conglomerate, and this usually allows the ambiguity to be resolved. The natural outcrops near the circle are of sandstone only, while the conglomerate seems to have been brought to the site from further up the hill. The rock-types reflect the nature of the local geology. Grey Hill is mainly composed of rocks of the Brownstone Formation of the Old Red Sandstone (Devonian Period). However, the summit of the hill above the stone circle is capped by an outcrop of Devonian pebbly conglomerate of the Quartz Conglomerate Formation. The two types of rock are usually easily distinguishable by eye, the former being a brownish-red, uniformly fine-grained sandstone, the latter being a grey matrix with small quartzite pebbles ranging in size up to 2cm embedded in it. Stones of the conglomerate type have faces which can be clearly identified on the basis of weathering features as being either parallel to the bedding planes, or normal to them, while stones of the sandstone type do not have obvious bedding planes. The density of pebbles in the conglomerate varies; it is usually extremely dense, but examples can be found where there are only one or two in a 10cm square of surface. Since the colour of the sandstone can merge into grey, while some samples of conglomerate have few inclusions, it is sometimes difficult to tell them apart. However, the sandstone is absolutely without inclusions, so the presence of only a single quartzite pebble can be taken as indicating that a particular stone is of the conglomerate type, and has therefore been brought to the site. Naturally placed stones exposed on the footpath near the circle are of the sandstone type, but the two standing stones, and those stones which are clearly part of the circle are of conglomerate. Thus any stone found near the circle that has even a few quartzite pebbles in it is an example of conglomerate that has been brought to the location from the beds higher up the hill, and the stone in question is part of the monument.
Fig. 11. FM2. Note shouldered profile.
Conversely sandstone stones should be treated with suspicion. It is possible that only the conglomerate yields large stones suitable as megaliths, but it is also possible that the occasional largish piece of sandstone was found locally and was included in the circle. So care must be taken when including a particular stone as part of the man-made arrangement; both its geological type and its size could be useful indicators. This point is particularly relevent to the interpretation of the recumbent stones of the stone-row; they all seem to me to be conglomerate, and therefore deliberately placed (see Fig. 13).
Fig 13. One of the Recumbent Stones. Small and barely breaking the surface of the turf, but clearly conglomerate. But for its distinctive and alien lithography this could be taken for a naturally outcropping stone (see Fig.14).
Fig. 6. SM 1. Note shouldered profile. Cup-mark visible above tripod leg; c.f. Fig 9.
Fig. 8. FM1 with SM2 in background. Note shouldered profile of both stones.
Fig10. SM2 with SM1 in background. Note shouldered profile of both stones.
Fig. 12. SM2 and lower part of circle, with Bristol Channel & Cotswold Escarpment
Fig. 14. SM2 and SS1 with Path Between. Note small native sandstone block in centre foreground. Stones like this - some larger - abound on the site.
The technique used for determining azimuths is described below; it is based on the technique used by Thom (1971) using a theodolite, a nautical almanac, and a sighting of the sun's azimuth at a know time. An altitude-azimuth profile of the Grey Hill horizon notch, determined using Thom's technique, and viewed from a position immediately in front of SM1 is illustrated left. Note that the notch is about 1/4º wide at its base, and widens to about half a degree towards its top. It could therefore accommodate the rising solar disc (which is about half a degree wide) very neatly. The average of three determinations of the azimuth for the centre of the notch was found to be 127.74º.
The azimuth of the alignment of the stone row could not be determined so precisely, since SM1 and SM2 are relatively broad, and are not very widely spaced. Viewed from a position immediately in front of SM1, the left edge of SM2 had an azimuth of 129.26º, and its right hand edge 130.10º, the average of these, which could be taken as representative of the alignment of the stones is 129.68º.
Fig. 15. Altitude-Azimuth plot of the Horizon Notch at Grey Hill.
The notch is the declivity between Hanging Hill and Frozen Hill on the Cotswold Escarpment North of Bath (~ST 717 708). Click here to see a view of this notch as seen from Horfield Common, Bristol BS9. The region between these two hills was the site of the Civil War battle of Lansdown, 5th July 1643
and h are all characteristics
of the particular site and alignment in question, while
δ is a characteristic of a particular celestial
object. It is
therefore convenient to group A, λ
h together, and calculate from them the declination of the body which
azimuth A, using values of λ
h for a particular site.
The declination calculated in this way is often loosely referred to as the declination of a site or of an alignment, but in reality it is the declination of the celestial body which would rise or set on the alignment in question. The value of characterising an alignment by δ is that the celestial body to which the alignment points can be identified from a table of declinations of astronomical objects, suitable adjustment being made for changes which occur over time. Alignments to the same celestial body (same δ) will have different azimuths at different locations, so A cannot be used directly to compare different sites, or to identify a particular celestial body, except in a very approximate way.
The relationship between A, δ, λ, and h is this, taken from Thom, (1971) p17:-
sin(δ) = sin(λ)*sin(h) + cos(λ)*cos(h)*cos(A) ... (1)
Each of these variables will now be discussed, but if you would like to skip directly to the processing of the Grey Hill data, click here.
In southern England ( λ = 51º, say) stars with declination down to -39º will be above the horizon at some time in the year. All stars in the northern celestial hemisphere will be visible at some time of the year, and some (those with a declination greater than 90 - λ) will never set. This is illustrated in Fig. 17 and discussed further below.
Solar Declination δs. The declinations of stars are fixed in the short term; minor variations such as those due to parallax, proper motion, and precession of the equinoxes are only noticeable if very precise observations are made. But the declination of the sun varies greatly throughout the year; it rises higher in the sky in summer, and lower in the winter. In northern latitude summer the sun has a positive declination, which increases to a maximum of 23.44º at the time of the summer solstice, while in winter its declination is negative, and falls to a minimum of -23.44º at the time of the winter solstice. At the equinoxes δs is zero. The apparent path of the sun on the celestial sphere as it moves above and below the equator at different times during the year is marked by a line called the ecliptic (Fig. 16), which is the projection of the plane of the earth's orbit on to the sphere. This plane is inclined to the equatorial plane by an angle called the obliquity of the ecliptic (ε), which at present is 23.44º. This quantity sets the maximum and minimum values of the solar declination as +/-ε. ε can also be defined as the angle between the earth's polar axis and the normal to the plane of its orbit. Historic variations in ε and thus of the declination of the sun at the solstices are discussed below. Since δ is fixed for the stars, but variable for the sun, it follows that the azimuth of a rising star at a particular location is constant through the year, whereas for the sun, the azimuths of its rising and setting will vary greatly. During the summer the sun rises in the north-east and sets in the north west, describing a wide, high arc over the sky during a day, but in winter it rises in the south-east, and sets in the south-west, describing a much narrower and lower arc as it moves across the sky.
The Effect of Declination on Azimuth of Rising and Setting. The effect of the seasonal changes in declination of the sun on its azimuths of rising and setting is mentioned above, and the same effect of declination on azimuth is true for all celestial bodies, except that in the cases of stars, δ is constant throughout the year. The general effect is illustrated below. Note that the vertical axis is actually altitude (angular height above the horizon); the declination figures refer to a parameter which is appropriate to the various curved lines.
In Fig. 17a, the celestial equator is the heavy line with a declination of zero which passes just below Aldeboran . This, and the other lines show the paths of points on the celestial sphere as it rotates once each day towards the West. Stars above the equator are in the Northern celestial hemisphere, while those below it are in the southern hemisphere. Stars near the N pole are circumpolar - they never set (Fig. 17b). This is true for stars in Fig 17a whose declination exceeds 35º (i.e. 90 - 55). The W shaped constellation of Cassiopia is close to this limit, as are some of the stars of the constellation Auriga (which includes Capella), and these stars will just graze the horizon as they rotate around the pole daily. All of the stars to the left of the diagram rise and set normally, but clearly the azimuths of setting decrease as the declinations of the stars are less. The azimuths of rising will increase in the same way, narrowing the arc which the stars describe in the sky. If the altitude of the horizon is zero, azimuths of rising and setting will be symmetrical about the North-South meridian. Stars whose declinations are less than -35º (i.e. 90 - 55) will never rise, and will never be seen from this latitude (+55º), while stars whose declination slightly exceed this limit will just break the horizon briefly, and will appear low on the horizon at azimuths which are close to due south. Thus in general terms, a larger azimuth of setting, or a smaller azimuth of rising indicates a higher declination.
The Effect of Latitude on Azimuth of Rising and Setting. Fig 17 refers to a latitude of 55º. If the latitude is increased, the altitude of the celestial equator will fall, pushing some southern hemisphere stars (for example Sirius, Fig 17) below the horizon, while the altitude of the pole will rise, lifting some northern stars (e.g. Capella) into circumpolar paths. The number of stars which rise and set will be diminished, and their points of setting will be spread out along the S-NW horizon; some azimuths (such as that of Capella) will increase while others (e.g. Sirius) will decrease. At the limit, when λ = 90º (at the N pole), all visible stars (those of the northern celestial hemisphere) will be circumpolar, and will neither rise nor set. At the equator (λ = 0º) all stars, whether northern or southern, will rise and set, and their azimuths of rising and setting will be distributed at all angles around the horizon.
The Effect of Altitude of the Horizon on Azimuth of Rising and Setting. In Fig 17, look at the trajectory of Aldeboran as it sets, following the line of the celestial equator. Is skirts the western edge of a distant hill, and sets almost due West. If the hill had been a little higher or nearer (i.e. if the altitude of the horizon in the region of this azimuth had been a little higher) the star might have set on the summit of the hill, at a noticeably smaller azimuth. Thus when the horizon is higher, azimuths of setting will be reduced, and correspondingly those of rising will be increased. This effect will be more pronounced at higher latitudes, where the trajectories of the stars, as they rise from, and set into the horizon, are at lower angles.
The Effect of Atmospheric Refraction. This phenomenon is discussed in greater detail below. At present it is sufficient to say that atmospheric refraction has the effect of increasing by about half a degree the apparent altitude of a celestial body when it is near the horizon. This can be expressed by saying that the altitude of a body seen on the horizon is about half a degree less than its observed value. Thus if the horizon has a measured altitude of zero, a star with a true altitude (in geometric and astronomical terms) of -0.5º will appear to be on the horizon.
Azimuth A = 127.737º
Latitude λ = 51.638º
The altitude (h) of the notch requires special consideration. The measured altitude of the deepest part of the notch is -0.27º. If we wish the sun to rise in the notch in such a way that the first flash of its disc appears at the bottom of the notch, we must subtract 0.25º as the semi-diameter of the sun's disc, giving -0.52º as the apparent altitude of the centre of the sun. Then we must apply a correction for atmospheric refraction. Equation (5) gives the refraction as 0.707º at 10ºC or 0.733º at 0ºC, a more realistic figure for a late December morning. The true altitude (ht) of the centre of the rising sun's disc is then -1.253º
II. Precession of the Equinoxes.
phenomenon is not of any importance when solar alignments are being
However, precession causes an important long-term variation
of the declinations of stars, and it is therefore possible that some
confusion could arise. The topic is discussed here in order
carify this difference between the sun and other stars in this
The sun's gravitational attraction acts on the earth's equatorial bulge, and tends to draw it into the plane of the ecliptic (i.e. the plane of the earth's orbit). This torque, acting on the rotating earth causes it to precess in a manner which is analogous to a top whose axis of rotation is inclined. The axis of rotation of the earth describes a pair of cones - see Fig. 24 (only the northern cone is shown). The period of the precessional motion is about 26,000 years, and over this period all stars will exhibit changes of declination amounting to 2*ε, a figure in the region of 45 to 48 degrees. Over half this period (i.e. 13,000 years) some stars could change their declination by the same amount. For example, in Fig.23, the present direction of the earth's axis is aligned approximately to Polaris, but when precession takes the axis to an orientation at the diametrically opposite position on the base of the cone, Polaris will be about 47º away from the celestial north pole, and its declination will have changed accordingly over a period of 13,000 years.
However, precession does not change the magnitude of the obliquity (i.e. the half-angle of the cone). Since it is the magnitude of ε which fixes the limits of the annual variation of δs, precession does not affect this, and we need not take precession into account when considering the historic changes in δs. As illustrated above in the case of Polaris, changes in declinations of stars are very different. Whereas ε changes by only about 0.5º over a period of 5,000 years precession can cause very much greater changes in the declinations of stars. Consequently archaeo-astronomical calculations of alignments to stars are more complicated, and the differences from modern alignments to the same stars are much greater than is the case with the sun.
Fig. 24. Precession of the Eath's Polar Axis (Precession of the Equinoxes).
III. Atmospheric Refraction.
a) Astronomical Refraction. When a ray of light from a celestial body enters the earth's atmosphere from the vacuum of space, it encounters an optical medium of increasing density and therefore of increasing refractive index. Atmospheric density increases as the surface of the earth is approached, and this causes the ray to follow a curved path which is concave towards the ground; this is astronomical refraction (Fig. 25). Refraction is greatest when the object appears to be at the horizon, since under these conditions the extent of air through which the ray passes is at its greatest.
Fig. 25. Atmospheric Refraction (Astronomical Refraction).
Table 3. Thom's Refraction Figures.
ha = 0.03*ht^2 + 0.8494*ht + 0.483 ... (3)
ht = -0.0472*ha^2 + 1.2258*ha - 0.5831 ... (4)
Fig 26. Thom's Refraction Figures and Polynomial fit.
gives this equation for refraction (R, measured in arcmin) as a
function of ha:
R = cot(ha + 7.31/(ha + 4.4)) ... (5)
while Saemundsson gives the following for R in terms of ht:
R = 1.02*cot(ht + 10.3/(ht + 5.11)) ... (6)
In both cases, the equations assume an atmospheric pressure (P) of 101.0 kPa, and a temperature (T) of 10ºC. Adjustment for other temperatures and pressures can be made by multiplying the figures for R by (P/101)*(283/(273 + T))
Comparison with Thom's figures is given in Table 4. Again there are minor but unimportant differences, the maximum difference being 13" if the value for negative apparent altitude is excluded.
Table 4. Comparison of Refraction Figures
1) A Suitable Theodolite and Tripod. I used the Carl-Zeiss Jena Theo-120. This is particularly well suited to this task since it has a circular sun-graticule in its telescope and is equipped with an eyepiece diagonal and sun filter as standard equipment. In these respects it is quite unlike other more conventional theodolites. The Theo-120 is described as a Kleintheodolit, and is smaller and lighter than many others. It is a transit type (i.e. the vertical circle is graduated over the full 360º, allowing centering and collimation errors to be eliminated by reversing the telescope about the vertical and horizontal axes simultaneously and averaging the results). It can also be used with the repetition technique (the bottom plate can be released and rotated, allowing the horizontal circle to be read in several different positions, and the results averaged to reduce systematic errors).
The circles are marked in grad (400 grad = 360º) and decimals. The least-count is 0.1 grad (0.09 deg or 5.4'). This makes the instrument considerably less precise than the industry standard of 20" or 0.33', but the circles and their indices are very finely engraved, and interpolation to 0.02 grad (1.8') or even 0.01grad (54") can be made with confidence. It is particularly convenient that the scale-reading microscope swivels independently of the telescope, so that only one eyepiece diagonal is required. For the present task the instrument is an ideal compromise between portability, convenience and precision, and seems to be made for the job; I feel I was very lucky to find it.
2) A Nautical Almanac or Ephemeris. This was found on-line as a downloadable .xls file.
3) An accurate Clock. I used the quartz clock in a small laptop, set to the nearest second using the MSF time and frequency standard maintained by the National Physical Laboratory.
4) A Computational Aid. I used a small laptop computer running home-brewed VB4 software.
5) A Way of determining the Latitude and Longitude of the Site. A GPS device, a suitable mobile phone app, or Wikki maps are all suitable. I used a Garmin Etrex and an app called GPS Grid Reference and also Wikki Maps. They all agree well.
Fig. 27. Theo-120 Kleintheodolit. With eyepiece diagonal and sun filter fitted.
screenshot of a Visual Basic
program to perform these calculations is shown in Fig
28. It includes pre-set data
corresponding to the values given in Thom's worked example
(illustrated), which was included as a check on correct operation of
the program. The software is tailored to the angle format of
Theo-120, and includes presets for lat & long of my own home,
of the Grey Hill site. Other locations can be entered
It allows up to five determinations of APZ to be averaged,
calculates their standard deviation. Time is determined
automatically from the PC's internal clock when a mouse button is
pressed. The software has been designed with care to simplify
operation in the field, the keyboard cursor (focus) moving
automatically to the
next data box once data has been entered. Selectable angular
offsets have been included to simplify operation when the theodolite is
used in transit mode according to plate-left or plate-right
If anyone would like to use this software I would be happy to edit it to allow theodolite data to be input in other formats. I will email it to anyone who asks, but the installation package is over 11MB, so be warned. Use the email link on my homepage, or consult qrz.com.
I have also developed software for calculating declination from azimuth, and vice-versa, and this application can also be used to calculate day of year from altitude and azimuth (Fig29); this is also available to anyone who wants it.
Fig. 28. Program for Calculating APZ of Theodolite and True Azimuths of Targets.