Grey Hill (Gwent): a Case Study in Archaeo-Astronomy

G4OEP Homepage

A Cautionary Note.

grey hill stone circle

Archaeo-astronomy
The Grey Hill Stone Circle and Adjacent Stones (Site Description) 
The Alignment of the Stone-Row and of the Notch.  
Declination δ.
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.
A Possible Alignment to the South.
Cup Marks
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.
Software
References.
Some Grey Hill Websites.
Llanfair Iscoed Stone & Castl
A technique for finding AZP using a theodolite and clock to 'shoot the sun'
Synopsis.  The megalithic site at Grey Hill (Gwent) includes two large standing stones which seem to point to a solitary notch on a distant horizon.  In this respect the site seems to have similarities with those described by Alexander Thom (1967, 1971).  The azimuth of this long-range alignment is investigated as a possible marker for the winter solstice.
Archaeo-astronomy.  
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.
Harolds Stones Standing Stones, Stone Row

Fig 1.  Harold's Stones, Trellech, From South East.  
Note the poor alignment of the row.  Note also the large cup-mark near left edge of central stone, facing the alignment to the south-west.

My initial researches were frustrated by inaccurate information. A source derived from the Ordnance Survey said that the supposed alignment has an azimuth of 72º , so I ransacked the literature to try to give this a meaning.  The mistake was serendipitous; 72º is an obscure figure in this context, and in my search for relevance, I  read much more widely than I would have done if the correct figure had been given at the outset.  I eventually found in Thom (1967) pp136-141 six references to the declination corresponding to this angle, all supposed to be to the rising of Antares in an appropriate historical epoch.  But  I was not happy about this; Antares is not a very bright star, and I could see no reason why anyone should take an interest in its rising to such a degree that they would manhandle three enormous monoliths into position in its honour.  A solsticial or equinoctial alignment I could accept, since the importance of establishing a simple marker for the turning of the seasons and as an index for the counting of days has real practical significance for communities who live outdoors and depend on pastoralism or agriculture for their livelihood.  That was the kind of thing that I wanted to find, so I decided that the OS was wrong and that the only way to progress was to measure the alignment myself.

But already the seeds of what was to come had been sown; I had read Thom.  Alexander 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 his 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 strikingly precise astronomical alignments.  The classic 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.  
Ballochroy stone row. alignment. archaeology

Fig 2.  The Alignments at Ballochroy from Thom Megalithic Sites in Britain. p 142

However, to create this effect it is not necessary for the stones to be precisely positioned, which might be difficult given the primitive means available for moving them.  Even if the stones were displaced by as much as 25m laterally, the error in the alignment would be only arctan(25/13000) = 0.11º - a just-noticeable inaccuracy, given that the diameter of the sun is about 0.5º.  Note also that the procedure for determining the position of  the index stone (the rearsight) is simplicity itself: having previously determined the approximate location, you simply observe the sunset, walk left or right until the desired visual effect is achieved, and then plant a stake. This would be repeated for several days in succession before the solstice until the southward (leftward) migration of the setting point was seen to stop or reverse, at which point the final position of the stake would indicate the correct position for the stone which is to act as rearsight.   The rest of the alignment can then be laid out at one's leisure.  

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 is 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 relation 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.
Ballochroy stone row alignment. Grey Hill

 Fig. 3. Midsummer Sunset at Ballochroy
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 supposed 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. 2). However, if a horizon profile is mainly flat, and a single feature on it aligns well, the case is more convincing.  It was with these ideas freshly in my mind that I embarked on my expedition to investigate Harold's Stones.

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.


Ballochroy stone row. alignment. archaeology. Grey Hill
Fig. 4.  Other Examples of Alignments involving Horizon Features.  Thom (1971), p 72
I had read up the site superficially on the internet, and located it on the Landranger 172 OS map, which indicates a stone circle and a single isolated standing stone.  When I arrived at the site I was surprised to find two large standing stones, about 50m apart, with other recumbent stones between them, the row being tangential to the circle, and on its NE side. (see detailed description, and plan below). Viewed from the upper stone, the lower one is well below the horizon (see picture above), but looking up I saw to my astonishment a sharp notch in an otherwise featureless horizon.  I had a small sighting compass with me, and found the azimuth of the alignment to the notch to be about 130º.  I very nearly fell off my perch - this is the direction of the midwinter sunrise.  Here, it seemed, was a classic Thom alignment - a Welsh Ballochroy !  I decided to return to the site equipped to make a proper survey - to determine accurately the azimuths of the stone row, and of the alignment from the site to the notch; it was already clear that they were not quite the same, the notch being a little to the left (i.e. the east) of the lower stone when viewed from the upper stone.

A detailed description of the site is given next, but if you want to go directly to a discussion of the alignments, click here.



The Grey Hill Stone Circle and Adjacent Stones.

Location.
 
ST 438 944 (Landranger 172)  Map. Near Llanfair Iscoed, Gwent. 7 km NW of Caldicot, 10km West of Chepstow, 14 km East of Newport.  Accessible from M48, A48, M4.

Site Description.
Websites. Grey Hill has been much studied, and is regarded as an important historical site with features dating from all periods of human activity ranging from the early Bronze Age, through the Mediaeval period to more recent times. But notwithstanding this, frustratingly little material is readily available on the internet. 
The OS map indicates only the stone circle and a single standing stone as features of interest, but a careful investigation of the area around the circle reveals much more.

The 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 steep frontal assault on the 275m summit.  From here there are exhilarating and 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 I 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.
Grey Hill. Stone row alignment
Fig. 5.  Schematic Map of the Grey Hill Megalithic Site.  Not accurately to scale.  The footpaths are schematic; SS1 is actually on the other side of the join in the paths.  The diagram of the circle and FM1, FM2 does not accurately represent the positions of the stones or their number.
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 be 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.

Grey Hill. Stone row alignment. megalithic Circle
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.

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.

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.

Grey Hill. Stone row alignment. megalithic Circle

Fig. 11.  FM2.  Note shouldered 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.

 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).


Grey Hill. Stone row alignment. megalithic Circle
  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.



Grey Hill. Stone row alignment. megalithic Circle
Fig. 7.  Lower part of circle, with SM2 in Background and FM1 between.

Grey Hill
Fig. 8.  FM1 with SM2 in background.  Note shouldered profile of both stones.


Grey Hill. Stone row alignment. megalithic Circle
Fig10.  SM2 with SM1 in background.  Note shouldered profile of both stones.


Grey Hill. Stone row alignment. megalithic Circle
Fig. 12. SM2 and lower part of circle, with Bristol Channel & Cotswold Escarpment
in Background.


 
Grey Hill. Stone row alignment. megalithic Circle

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 Alignment of the Stone-Row and of the Notch.  

The most obvious way of characterising an alignment is to give its azimuth - i.e. the angle, increasing clockwise, which the alignment makes with respect to due North.  But if the intention is to link an alignment to an astronomical event  this measure is unsatisfactory, since the same event observed from different locations will have different azimuths; the reasons for this will be explained.  However, the first step in the interpretation of an alignment is to measure its azimuth.

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.

Grey Hill. Stone row alignment. megalithic Circle
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
http://www.battlefieldstrust.com/media/189.pdf
In order to find out whether the sun would rise on either of these azimuths at the winter solstice, a fairly elaborate calculation is required.  The azimuth (A) of a rising celestial body (e.g. the sun) is related to the declination of the body (δ), the latitude of the observer (λ), and the altitude (h) of the horizon over which the body rises.  A correction to h for atmospheric refraction is needed, taking into consideration atmospheric temperature and pressure, and the value of δ must be appropriate to the historical epoch which is assumed.

A, λ 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 and h together, and calculate from them the declination of the body which rises at azimuth A, using values of λ and 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.

Declination δ.  This, with right ascension,  is one of a pair of polar coordinates which describes the position of an astronomical body on the celestial sphere.  For the purposes of observational astronomy, the sky is imagined to be a hollow sphere, approximately half of which can be seen by an observer at a particular location and time of day.  The stars, sun, moon, etc are imagined to be located on the inner surface of the sphere.  The plane of the earth's equator and the points immediately above the north and south poles are projected on to the sphere, and other lines similar to terrestrial latitude and longitude are marked on the sphere.  These form the basis of a coordinate system by which the locations of the stars, etc can be specified. The earth's axial rotation causes the sphere to appear to rotate once each day, but in fact it rotates slightly more than this. The additional increment of rotation is caused by the earth's orbital motion around the sun, and over a year this adds up to one extra turn. This difference between a day and the period of rotation of the celestial sphere is the difference between solar time and sidereal time.  Because of its rotation, lines on the celestial sphere which look like lines of longitude do not in fact remain fixed relative to projections of terrestrial longitude, so an alternative name and definition is required; these are lines of right ascension (Fig.16).
  
However, the lines which look like lines of latitude do remain fixed relative to projections of terrestrial latitude.  These lines on the celestial sphere are termed lines of declination; declination is thus the celestial equivalent of latitude.  A star with a particular declination will pass directly overhead once per sidereal day as seen by an observer whose terrestrial latitude λ has the same value as δ.  Like latitude, declination is signed; a positive value signifies a position north of the celestial equator, while a negative value indicates a southern position.  Viewed from intermediate northern latitudes, the celestial equator arches from east to west across the sky so that there are always some stars in the southern celestial hemisphere which are visible below it.   At lower latitudes the arch is higher, and more southern stars become visible. Provided that a star has a declination greater than 90 - λ it will rise above the horizon of a location of latitude λ at some time in the year (give or take the height of the horizon).   

Fig 16.  Celestial Sphere and stellar co-ordinates.  Note that the angle marked 'inclination' is in fact the obliquity of the ecliptic ε.  Declination δ is the angle between the celestial equator and a celestial object,  measured along a celestial meridian.  Note that as the sun follows its annual path around the ecliptic, its declination varies between +/-ε, and is zero at the equinoxes

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.



Fig 17a.  Altitude-Azimuth plot of the trajectories of stars with declination as a parameter. The stars follow paths of constant declination as they set towards the West.
grey hill stone row

Fig 17b.  Circumpolar and other stars. Time exposure of star trails.

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.  

Calculating the Declinations of the Grey Hill Azimuths.

1). The Horizon Notch. For this alignment the following data applies:-

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
 
The Horizon Notch Alignment.

Altitude ht =  -1.253º
Azimuth A = 127.737º
Latitude λ = 51.638º
We now use Equation 1 to get:
 
Declination of Notch:  δn = -23.39º
 
In the case of the alignment from SM1 to SM2, two azimuths were determined from a point immediately in front of the centre of SM1; one (a) to the left hand edge of SM2, the other (b) to its right-hand edge.  The altitude of the horizon was -0.160º in both cases.

The Stone-Row Alignment.

Measured altitude ha =  -0.160º
Corrected altitude ht  = -1.10º
Latitude λ = 51.638º

a) To left hand edge:
Azimuth A = 127.55º
Declination :  δlhs = -23.155º

 a) To left hand edge:
Measured azimuth A = 128.31º    
Declination :  δrhs = -23.561º

Mean Azimuth 127.93
Mean declination for SM1-SM2: -23.36º


Comparison with the figures in Table 2 clearly indicates that the declinations of these alignments are too small to agree accurately with historical values of the minimuum declination of the sun (i.e. the declination of the sun at the winter solstice: -ε).  A more meaningful comparison might be to compare the measured azimuths with calculated azimuths using historic values for 
-ε to give an 'error' in alignment. Table 1 shows these errors, a value of -1.1º being used for ht.  A likely estimated date for the monument is c1800BC, so the 'error' for the notch is about 1.22º and that for the mid-point stone alignment is slightly less at about 1.0º.  The absolute values of the azimuths and declinations are too small, indicating that the sun could rise on the alignments of the monument, not exactly on the winter solstice, but a few days before, and a few days after.  This is illustrated in Fig. 18,which shows that the sun will rise on the stone alignment +/-12 days relative to the solstice, and on the notch alignment +/-13days. This plot was derived using equations given in Thom (1967) p24 with appropriate historic values for the variables (p109).

Date  BC                                                                     Calculated Azimuth for Midwinter Sunrise (º) Error of Notch Alignment (º) Error of Middle SM1-SM2 Alignment (º)
1000 128.79 1.053 0.86
1500 128.90 1.161 0.97
1800 128.96 1.223 1.03
2000 129.00 1.263 1.07

Table 1.  Errors in the Alignments.
Grey Hill. Stone row alignment. megalithic Circle
Fig. 18.  Azimuth of sunrise on days around the winter solstice.

Discussion of Results.  I was greatly disappointed that the notch alignment at Grey Hill did not turn out to be a classic of the type described by Thom.  Unlike some of the the horizons described by him (e.g. Figs 2 & 3), that at Grey Hill is ideal in the sense that it is mainly featureless, but has a single distinct notch which allows no ambiguity as to where the supposed alignment is intended to be.  As noted above, a lumpy horizon with several possible target features roughly in the right direction (e.g. Fig. 3) invites the criticism that, if one of them gives the correct alignment for an astronomical event, it does so by accident.  The essence of the idea of a chosen alignment is that the site has precisely lcated so that a single alignment is accurately indicated by a single feature on the horizon which acts as a foresight, a stone on the site itself acting as a rearsight.  Grey Hill seems to offer this possibility in an ideal way, but the alignment, as constructed, is incorrect by about 1.2º, which must be explained.  This error can be interpreted in perhaps two ways - firstly that the supposed intention to mark the midwinter sunrise is simply a fantasy of my own mind, and in no way reflects the ideas of the builders of the monument;  the approximate alignment is thus merely accidental, and is of no particular significance.  If the alignment has any meaning at all, it is something else entirely.  This possibility must be admitted, but I do not find it attractive or convincing - at least not until I stumble across an alternative convincing explanation.  That the row has no meaning at all, I find difficult to accept.  In this context it must be mentioned that Burl (1993), discussing the stone-rows of Dartmoor, comments that they frequently link a circle at the lower end with a cairn higher up a slope, and have no other identifiable function.  It is possible that Grey Hill could fit this pattern, should a suitably aligned cairn be found.  

Secondly, we might suppose that the evidence provided by Thom and the other archaeo-astronomers indicates convincingly that the builders of stone circles and stone rows had a strong interest in solsticial alignments, and in many cases used features on the horizon as foresights for the construction of accurate solsticial alignments relative to their monuments as rear sights.
Burl, a more conservative commentator writes, (1995, p 18) ' More and more firm evidence has become available to prove that prehistoric societies built astronomical alignments into some of their ritual monuments.  The lines were imprecise, but they exist and can be recovered'.  From this point of view the small error might seem typical and not in need of explanation.  The evidence suggests that in order to create the alignments the builders selected sites with two (or more) criteria in mind - (a) the aesthetic of the location (elevated airy terrain with low horizons and distant views - ideally to a single foresight), and (b) (according to Thom) the creation of a precise alignment: the alignment can be adjusted by moving the rearsight (i.e. the monument) laterally until the solsticial sunrise occurs at the required point on the horizon (i.e. the notch or peak).  Why then is the Grey Hill notch azimuth 1.2º too small ?  A simple answer to this is that in order to correct the alignment, the site of the monument must be moved off Grey Hill entirely.  The sad truth about Grey Hill and the notch between Hanging Hill and Freezing Hill is that there is no location on Grey Hill from which the notch appears precisely in line with the midwinter sunrise.  The required lateral displacement is 36*tan(1.2) = 0.75km (36km being the distance to the notch), and this would take the monument off the hill.  

My interpretation of this situation is therefore that the Bronze Age people who erected these stones recognised in Grey Hill an apparently ideal location for a monument aligned to the winter solstice sunrise.  The site is strikingly beautiful and has an enticing horizon feature in the required direction.  But the notch, though very nearly correct, is frustratingly impossible to align.  They therefore made a compromise; they constructed their monument on Grey Hill in recognition of its suitability according to criterion (a), but accepted that they could not fulfil criterion (b).  Notwithstanding this, the stone-row can still be used as a midwinter marker, it is just that the sun rises in the notch 12 days too soon, and again 12 days too late.  If the intention was to identify the precise day of the solstice, this alignment would still be useful;  in fact it would be more accurate than a correctly aligned marker.  Look at the peak of the curve of Fig. 18: if the notch were perfectly aligned, the sun would rise in it for up to six consecutive days with almost no perceptible difference, but with it aligned as it is, the difference between the azimuths 13 days before and 12 days before is immediately apparent - about 1/6 degree, or a third of the sun's diameter. Since the slope of the curve is so much greater at the +/-12 day points, interpolation between them actually gives a better estimate of the precise day of the solstice.  Here we have a compromise between the aesthetic of precision (the sun rising exactly in the notch on the solstice) and practical usefulness.  I am not suggesting that the lack of true alignment is deliberate, but it is certainly serendipitous from a practical point of view, and this might have made it acceptable to those who constructed the stone row.

I am fully aware that there is an element of casuistry in this; the interpretation of the row as an instrument for determining midwinter would be much more convincing if the alignment were precise.  But there is also a certain rigour in that the stone-row is forced to comply with the realities of the site, which is intractable.  If the profile of the Cotswolds escarpment were as irregular as that of the island of Jura (Fig. 3) there would have been no difficulty in finding a precise alignment to one of the many peaks and hollows, but conversely the convincing power of the alignment would be reduced, as suggested by Burl.  The people of Grey Hill faced a more rigorous problem than did those of Ballochroy, and found only an imperfect solution.  

Comparing these two sites, I feel that there is a defect in the way that Thom presents his results in that he does not give a wider perspective of the horizons he discusses.  Thus his profile of Ben Corra given in Fig.2 invites an impression which is belied by the photograph of Fig. 3.  This tends to make the proposed alignment seem facile, whereas in reality finding a site for a precisely aligned and unambiguous solsticial marker is extremely difficult.

A Possible Alignment to the South.  If one stands in front of the upper standing monolith (SM1) and allows one's eye to roam over the distant horizon another broader declivity which is almost exactly due south will be noticed.  This is extremely distant, beyond a nearer range of hills, and will only be visible when the air is clear.  A profile is given as Fig. 19.  This horizon is probably near Warren Hill or Castle Hill, west of Crewkerne, almost on the south coast of Devon, and 87km distant.  It seems very unlikely that the site of SM1 was chosen in relation to this feature, but it is possible that the makers of the row noticed it as I have done, and marked it with the cup-mark on the south-western face of SM1 mentioned above. It might be fruitful to explore the terrain south of SM1 in the hope of finding another fallen monolith which marks this direction.

If this horizon feature has indeed been indicated by a cup-mark or a now-fallen monolith, we could infer that the makers of this monument recognised the significance of a southerly direction, but what significance did they attached to such an alignment ?  Burl (1993) (p155) comments that meridional alignments are seldom commented on, but are too frequent (in Brittany) to be accidental; he also remarks on the N-S alignment at Callanish (Lewis).  He asks three questions:  (1) how were the alignments achieved when there was no pole star (see comments on precession) ?  (2) why are the alignments often 5º or more adrift from true north-south ? and (3) why should the row-builders have an interest in a direction (N or S) where neither sun nor moon can ever rise or set ?    He answers first two questions by postulating a simple technique for finding north or south: the directions of sunrise and sunset on the same day near one of the solstices are marked in relation to a stone (A) by stakes B & C, equidistant from A (as determined by a rope of fixed length, or perhaps merely by pacing).  The midpoint (D) of line BC is then found by stretching the rope between B & C and folding it in half.  AD is then the north-south meridian, or at least approximately so.


Fig. 19.  Horizon Dip to South.




Fig. 20.  Displacement of  'North' due to Irregular Horizon. 
 Burl (1993) (p155)
When this technique is used, characteristic errors will arise from undulations of the horizon.  This is shown in Fig 20 which illustrates sunrise and sunset in the northern summer.  When the horizon is of uniform height (case 1), the azimuths of rising and setting will be equally disposed about true North, but if the horizon is higher at the point of sunset (case 2), the azimuth of sunset will decrease, and the midpoint (Bronze Age "North") will be displaced a few degrees towards the west.  If the sunrise horizon is higher, the azimuth of sunrise will be increased, and the bisection will swing towards the east.  If the same technique is used in the winter a southerly direction will be found, but a higher horizon at sunrise will slew the supposed position of 'south' towards the west.

Burl answers his third question by postulating that the designers of the meridional alignments were interested in the midpoint between sunrise and sunset; we could add that they were doubtless aware that this is the direction in which the sun is highest in the sky. Burl comments that if the stone-men used the former definition of north (or, when appropriate, south), then an alignment such as Fig. 20, case 2, which we would describe as incorrect by 2
º, would be perfectly correct to them. Sadly this hypothesis is essentially untestable, since the 'errors' depend on the altitude of the horizon at the rising and setting points used for the bisection.  These points change day by day, and there is no way of determining which ones were used.  However, if 'errors' were to be found to be consistently related to mid-points between solsticial sunrises and sunsets, this would be significant.

Cup Marks.  I have suggested above that the Grey Hill stones bear three cup-marks, two on SM1, and one on SM2 and I have commented on the mark on one of Harold's Stones, which faces in the direction of the supposed alignment of these stones.  I have postulated that the two marks on SM1 might also be intended to indicate particular aligments - to the midwinter sunrise, and meridionally to the south.  This leaves the supposed mark on SM2 to be interpreted, and suggests that a fourth alignment might be found in a generally eastern direction from SM2.  I have not yet found references to cup marks being related to alignments except some comments here in relation to a stone on the site of the Crick round barrow, which is within 10km of Grey Hill, and about 20km from Trellech. Burl (1995) p 19 also mentions "a cup-marked pillar" as an example of an indication that a stone might be intended as a local foresight.


Appendix I Obliquity of the Ecliptic (ε).  This quantity sets the maximum and minimum values of the solar declination δs, and varies over the millennia.  If we wish to test an alignment which seems to be so a solstitial sunrise or sunset we must take this variation into consideration. 

The  obliquity of the ecliptic is the axial tilt of the earth's axis of rotation; i.e. it is the angle between the polar axis of the earth and the normal to the plane of its orbit around the sun (Fig. 21).  It is also the angle on the celestial sphere between the North Celestial Pole (NCP) and the North Ecliptic Pole (NEP).  This angle is important because it sets the upper and lower limits of the declination of the sun; the maximum and minimum values of δs are +/-ε at the northern summer and winter solstices respectively. Historically, over the last few millennia ε has been falling at a low but varying rate as a consequence of planetary perturbations of the earth's orbit. At present it is falling at 47" per century. A 10th order polynomial for the long-term variation of ε accurate to within a few arcseconds over a period of 10,000 years is given in Laskar:

Here t is in units of 10,000 Julian years measured from J2000 (2000 AD), and ε is expressed in arcsecs.

Thom (1967) p 20 refers to this equation (below), which he attributes to De Sitter.  Here t is measured in centuries from 1900 AD and 
ε is in degrees.

 ε = 23.4523 - 0.013078*t - 1.63889*10^-6 *t^2 -0.516667*10^-6*t^3 ... (2)

As shown in Fig.22, there is a slight disagreement between these figures which increases at earlier dates, and amounts to 22 arcsec at 2500 BC. For our present purposes this is not important, and either equation can be used.  A table of calculated values is given in Table 2.  At present, ε is 23.439º.


Fig. 22.  Historical Obliquity of the Ecliptic.



Fig. 21. Obliquity of the Ecliptic.



   Year  BC Laskar De Sitter
2000 AD 23.43929 23.43922
1000 23.81438 23.81758
1250 23.84285 23.84648
1500 23.87065 23.87474
1600 23.88157 23.88586
1700 23.89238 23.89687
1750 23.89774 23.90233
1800 23.90307 23.90776
1900 23.91364 23.91854
2000 23.92409 23.92919
2250 23.94963 23.95528
2500 23.97432 23.98054

  Table 2. Historical Obliquity of the Ecliptic ε (degrees).  Solstitial values for the declination of the sun are +/-ε.


Fig. 23.  Historical Obliquity of the Ecliptic. (Laskar)

Appendix II.  Precession of the Equinoxes.  This phenomenon is not of any importance when solar alignments are being considered.  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 to carify this difference between the sun and other stars in this respect.   

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).
Appendix 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).

This, and the usual considerations of Snell's Law, causes refraction to decrease with altitude; it is zero when the ray approaches from the zenith.  Viewed from the ground, the apparent altitude of the source is along a tangent to this curved path, but the true angular position of the source is lower.  Thus, when an object is seen just above a low horizon, the apparent altitude (ha) is near zero, but the true altitude (ht) is less than this - i.e. it is negative. The observer is thus able to see a celestial object when it is actually below the horizon.  Apparent altitude is greater than true altitude by an amount (R) which depends on atmospheric pressure and temperature, but the difference is usually close to 0.5º when the object is close to the horizon.  When the apparent altitude of a celestial object is calculated, its value must be increased by an appropriate correction to give the apparant altitude as seen by an observer; simply adding 0.5º will give a rough estimate, but a more refined approach is possible. Conversely, when the apparent altitude is measured, the true astronomical altitude must be estimated by subtracting an appropriate correction (R).

Thom (1971) gives this table of values (Table 3) for astronomical refraction at 101.25 kPa (29.9"Hg) and 7.222ºC (45ºF), from which the plot, and the polynomials (3) & (4) are derived:-


Apparent Altitude
(Aa º)
Refraction (R º) True Altitude (At º)
-0.3333 0.66667 -0.99997
0 0.5800 -0.5800
0.3333 0.5100 -0.1770
0.6667 0.453333 0.213367
1 0.408333 0.591667
1.3333 0.368333 0.964667
1.6667 0.3350 1.332

Table 3. Thom's Refraction Figures.

h
a = 0.03*ht^2 + 0.8494*ht + 0.483 ... (3)

h
t = -0.0472*ha^2 + 1.2258*ha - 0.5831 ... (4)

Fig 26.  Thom's Refraction Figures and Polynomial fit.
Bennett gives this equation for refraction (R, measured in arcmin) as a function of ha:

R = cot(h
a + 7.31/(ha + 4.4)) ... (5)

while Saemundsson gives the following for R in terms of ht:

R = 1.02*cot(h
t + 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.


Apparent altitude (º) Refraction Thom (º) Refraction Bennett (º)
-0.3333 0.666667 0.658495
0 0.5800 0.580322
0.333 0.5100 0.513483
0.6667 0.453333 0.456972
1 0.408333 0.409505
1.333 0.368333 0.369518
1.667 0.3350 0.335525

Table 4. Comparison of Refraction Figures


b) Terrestrial Refraction. The refraction experienced by an approximately horizontal ray between two terrestrial objects is termed terrestrial refraction. This is relevant when the profile of a distant horizon is plotted from height and range found from an OS map, or when it is plotted from surveyed altitudes and azimuths.  Thom (1971) p30 gives an equation which relates refraction angle (T - seconds of arc) to a refraction constant (K, found by measurement to lie between 5 and 13, depending on conditions), length of ray (L - feet), temperature (t - ºR), and atmospheric pressure (P - inches of mercury):-

T = K*L*P/t^2

With L = 36km (appropriate at the Grey Hill site), T= 16C, P = 101.4kPa, and appropriate unit conversion, this gives T=13*K seconds of arc or 1' to 2.8'.  

Thom comments that the lowest values of K were obtained during early afternoon, and the highest after sunset, so a low value can be assumed as normal. At the level of ~1', this effect can often be ignored (though not in accurate surveying work, where angles are routinely measured with a precision greater than 20"); it is always much less than astronomical refraction.

Appendix IV.  Survey Technique - Data Gathering and Processing.
As suggested above, the intention of the site survey was to determine accurately the azimuth (A) of the alignment defined by SM1 and the Hanging Hill - Freezing Hill notch, and its declination δ.  If a theodolite is located at point A, and B and C are two other remote points, the angle BAC can be found by taking two readings of the theodolite's horizontal circle, and subtracting one from the other.  But if the azimuth of B or C is required, the angle-readings for B or C must be referenced to true north, and a technique must be used to find the azimuth of the plate zero (APZ) - that is the true azimuth of the telescope's sight-line when the horizontal circle reading is zero. When the theodolite is screwed down on to a tripod, this quantity is randomised.  But if APZ is found, it can be added to the indicated horizontal angle for a target to give its true azimuth. Three techniques for finding APZ are:

1) Using a magnetic compass to set APZ to zero.  The theodolite is set to read a horizontal angle of 0, and, with the lower plate released, the instrument is rotated to point North as indicated by a compass held against one of the trunnion supports.  The lower plate is then clamped, and the upper part of the theodolite is rotated to centre the target on the graticule. The azimuth of the target is then the horizontal plate reading plus or minus the magnetic declination.  This technique will yield accuracy no better than that of the compass alone, and in terms of convenience, a good prismatic compass fitted to a light tripod will be preferred.

2) Using an OS map and a protractor.  This will give slightly better accuracy.  The theodolite is directed towards a distant object identifiable on an OS map, and the indicated bearing is (b) read from the horizontal circle.  The azimuth of the reference object (r) is then found from the OS map using a ruler and protractor, correction being made for deviation of Grid North from True North (d). APZ is then r - b.  Precision depends on that of the protractor (0.5º for a large protractor) while accuracy depends on that of d, which varies across a map sheet, and can rarely be found more accurately than +/-10'.  

 3) A technique for finding AZP using a theodolite and clock to 'shoot the sun' is described in Thom (1971), pp119 ff.  This is a precise technique, which if used carefully is limited only by the characteristics of the theodolite.  Thom gives a worked example as Table B1, p121, but his description of the technique is cryptic to say the least.  I found that deciphering this, and working out an algorithm suitable for programming on a PC or laptop was a major intellectual achievement.  Here is a brief resumé of Thom's technique, as adapted for modern use:

Materials Required:

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 .

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.

The essence of the technique that Thom describes is to use a Nautical Almanac to find the local hour angle (H) at the GMT time (t) that an observation of the indicated azimuth (or plate bearing - PB) of the sun is measured.  The sun's true azimuth (A) is then found using this equation, with δ as the sun's current declination, also taken from the Almanac, and λ the latitude of the point at which the observation is made:-

cot(A) = sin(λ)*tan(H) - cos(λ)*tan(δ)/sin(H)  ... Thom (1971), p120

When A is known, APZ is found as APZ  = A - PB

Thereafter, the azimuth of any target having a plate bearing B is APZ + B.

In an ideal world, H could be found directly from t simply by multiplying the difference between t and local noon by 15º/hr (i.e 360º/day).  But in reality we must tangle with the equation of time (i.e. the amount by which clock time differs from solar time at different moments in the year).  It is this complication which requires that we obtain the Greenwich Hour Angle (GHA) from tabulated data, and establish H from that.  We now shift our point of reference to Greenwich, for which data is tabulated.  The almanac gives GHA at hourly intervals throughout the year, so we read the tabulated value for the preceding hour, GHAp, then calculate H as:

 H = GHAp + (t - w)*15 + φ  

where (t - w) is the fraction of an hour since the last whole hour (w), and φ is the longitude. (t - w)*15 is a correction for the fraction of an hour which has elapsed since the time for the tabulated value of GHA, while φ is a correction for the fact that our point of observation will not generally be on the Greenwich meridian.

Overall, I have found this technique a difficult one to use.  I came to it having previously never handled a theodolite at all, and mastering the various manipulations so that the required procedure could be executed slickly and reliably in the field did not come easily.  But with practice excellent results can be obtained with repeatability comparable with that indicated by Thom.  At Grey Hill, three determinations ab initio of the low point of the notch were all within 1' of each other.

Software. A 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 the Theo-120, and includes presets for lat & long of my own home, and of the Grey Hill site.  Other locations can be entered manually.  It allows up to five determinations of APZ to be averaged, and 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 configuration.

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.



Fig. 29.  Calculator for Azimuths, Declinations, and Dates.


A Cautionary Note.  Prior to my second visit to Grey Hill (i.e the visit during which I made a precise measurement of the azimuth of the horizon notch), I spent some time on Horfield Common (Bristol) practicing using the theodolite, and in particular perfecting the technique for determining azimuth from sun-shots. During these activities in April 2015, I realised that Horfield Common was a useful place for actually re-enacting the construction of a stone-row alignment directed to the winter solstice sun-rise.  I predicted the position at which a rear-sight should be located in order that a prominent local feature should act as a foresight for the sunrise.  In December 2015 I returned to the Common to test this prediction, and found that it was gratifyingly accurate.  However, I was very much surprised to find not only confirmation that my rearsight corresponded to an existing feature on the Common, but also that this solsticial alignment also coincided very precisely with a topological alignment with three other prominent features in the local urban landscape of Bristol.  This 5-point alignment is reminiscent in a general way of the "ley line" alignments which enthusuasts have been inspired to find, following Alfred Watkins' Old Straight Track, and is an uncommonly precise example of such a thing.  The Horfield Common alignment (HFA) cannot be anything other than a striking coincidence, but it involves more points, and is more accurate than the Grey Hill alignment about which I had become so enthusiastic.  The discovery of the HFA came as a serious shock to me, and very much pricked the bubble that I had inflated around Grey Hill.  A description of the HFA can be found here, together with a self-deprecatory and ironic interpretation.  I cannot help feeling that the discovery of the HCA undermines what I have written here about Grey Hill, and serves as a reminder of the extreme caution which must be exercised when interpreting sites such as Grey Hill.  I do not doubt that true archaeological alignments can be found, but I am no longer certain as to how they can be properly identified, given that such a striking but utterly spurious alignment as the HCA was so easily found.

References.

1. Hawkins G S, Stonehenge Decoded, Barnes & Noble Books, New York, (1965).

2. Thom A, Megalithic Sites in Britain, Oxford University Press, (1967)

3.  Thom A, Megalithic Lunar Observatories, Oxford University Press, (1971).

4.  Burl A, From Carnac to Callanish, Yale University Press, (1993)

5. Laskar J, New Formulas for the Precession, Valid over 10,000 Years, Astronomy and Physics, Vol 157, p68 (1968)

6. Sitter W De, On the System of Astronomical Constants,  Bull. astr. Insts Neth. 8, 213 (1938)

7. Bennett G G, "The Calculation of Astronomical Refraction in Marine Navigation". Journal of Navigation 35: 255–259 (1982).

8. Sæmundsson, Þorsteinn,  Astronomical Refraction. Sky and Telescope 72: 70 (1986).


9. Burl A, A guide to the Stone Circles of Britain, Ireland and Brittany,  Yale University Press, New Haven & London (1995).

Some Grey Hill Websites

http://www.britainexpress.com/attractions.htm?attraction=393
http://www.megalithic.co.uk/article.php?sid=962

http://thejournalofantiquities.com/2014/05/14/gray-hill-stone-circle-llanfair-discoed-monmouthshire-sir-fynwy/
http://www.researchgate.net/profile/Helen_Lewis4/publication/256895168_Relating_land_use_to_the_development_of_coaxial_field_systems_preliminary_geoarchaeological_investigations_at_Gray_Hill_Caerwent_Wales_and_Shovel_Down_Dartmoor_England/links/00b4952406fc22d1d3000000.pdf

Llanfair Iscoed. The Castle and a standing stone in a nearby field are worth visiting
http://www.castlewales.com/llanvair.html

Astonishingly, this stone also has a shouldered profile.  It is not clear whether it is from the prehistoric period, but its location seems to me to be quite evocative; the little valley in which it stands is definitely atmospheric, and focusses the gaze towards an attractive view.  I might have felt motivated to place a stone there myself.  But it is interesting that other visitors to the site turn their back on this view, and see no significance in it, which makes me think that my personal sense of place might be idiosynchratic: this link is to a picture on the site whose URL is above.

To get there climb the hill out of the village towards Wentwood, and look for a gate into a field on the left once you pass a modern bungalow near the top.  The castle is concealed behind the trees on the left of this picture.  It can also be seen from the churchyard on the left as you climb out of the village.