A Third DIY Garden Heliochronometer

My wife and I enjoy visiting gardens, in part to gain inspiration for our own. In the summer of 2016, such outings led me to consider installing a sundial in our garden.

Knowing little about sundials, I inspected those we encountered and was dismayed to discover that none agreed with my watch. Even those that claimed to reflect clock time required additional consultation of a table or graph.

‘What we need,’ I said to my wife, ‘is a sundial from which clock time can be easily read by anybody.’

My subsequent journey into the fascinating world of sundials is detailed on the above web pages. Those pages also include a description of the underlying physics, although anyone interested in how sundials work should also visit the website of the British Sundial Society at: http://sundialsoc.org.uk/discussions/how-do-sundials-work/.

I had never intended to build two sundials, let alone three. Experience of operating the first two, however, and contact with the British Sundial Society, led me to understand how improvements could be made in the design and construction of my DIY garden heliochronometers. This endeavour, therefore, turned into an ongoing quest to construct my ideal garden sundial.

The following notes describe the operation of my third dial, which I have called Mark III.

Mark III differs from my previous sundials in that it includes latitude and longitude correction mechanisms such that it can be used anywhere in the British Isles. It also includes a date scale.

Dimensioned photographs and relevant graphics files are also included below, should anyone wish to construct something similar.

I have not explained in detail on this web page the physics involved, and thus I realise that some aspects will not immediately be apparent to those with no knowledge of equatorial heliochronometers. Please see the web pages related to Mark I and Mark II, and the British Sundial Society's website, for such technical background.

A video in which I describe the operation of my Mark III sundial can be viewed here:

An equatorial sundial has the plane of its dial plate parallel to that of the Earth’s equatorial plane, and its gnomon orientation parallel to that of the Earth’s axis.

A heliochronometer is a sundial that aims to display a time that corresponds, as closely as possible, to that displayed by a properly adjusted clock.

Fig. 2. Latitude scale.

My Mark I and Mark II heliochronometers were designed specifically for the latitude of my garden, which is 50.94° north – 51° for practical purposes. Mark III is designed to be adjustable for any latitude within the British Isles.

Fig. 1 shows the latitudes and longitudes of the furthest points North, South, East and West within the British Isles.

Furthest point in the British Isles to the:	Location	Latitude	Longitude
North	Out Stack, Shetland Islands, Scotland	60.85° N	0.87° W
East	Lowestoft Ness, Suffolk, England	52.48° N	1.77° E
South	Les Minquiers Reef, Jersey, Channel Islands	48.95° N	2.13° W
West	Rockall, Harris, Scotland	57.58° N	13.68° W

Fig. 1. Extremities of the British Isles.

It can be seen that the furthest point North in the British Isles is Out Stack in the Shetland Islands, Scotland, at a latitude of 60.85° N. The furthest point South is Les Minquiers Reef in Jersey, Channel Islands, at a latitude of 48.95° N.

The base plate of Mark III is connected to the dial plate with hinges such that the above latitudes can be accommodated. The angle itself is controlled with a brass screw jack, designed for a skylight window.

The latitude scale can be seen in Fig. 2.
The screw jack can be seen in Fig. 3 and Fig. 4.

The angle between the base plate and the dial plate, with the axis of the hinge pivots as the origin, is the co-latitude, not the latitude. The scale, however, is calibrated to read the associated latitude. This is why the angle shown by the latitude scale increases as the actual angle that it measures decreases.

There is no mechanism built into the heliochronometer to set the north-south orientation.

An approximate north-south orientation can be achieved by the use of a compass or any other appropriate method.

Due to the accuracy of modern clocks, it is possible to align the sundial accurately by making all other relevant adjustments and then orientating the instrument such that it reads the expected time.

I will describe the mechanism for the EoT correction prior to addressing the question of longitude, as the EoT correction scales play a part in the longitude correction process.

The EoT correction for Mark III is effected in exactly the same way as for Mark II.

The angular rotation of the timescale arc to correct for the EoT on a given day is achieved by the alignment of a pair of lines that are specific to that day. One line of each pair is located on the outer EoT scale and the other on the inner EoT scale which is attached to the top of the rotating timescale arc (Fig. 5).

In practice, the EoT correction adjustment requires a small, but very accurate, movement of the timescale arc each day. This is effected by a centrally placed knob, connected to a drive belt that moves the timescale arc. The knob and the drive belt are shown in Fig. 6 and Fig. 7.

The above EoT correction mechanism requires alignment of two lines for each day which are set at the correct angular displacement to compensate for the day's EoT correction.

The location of one line-pair in relation to any other line-pair, however, is irrelevant as long as the final configuration does not make the scale difficult to read.

Thus I was able to group all GMT dates on the east side of the sundial’s longitudinal axis, and all BST dates are on the west side (Fig. 8).

The sequence of GMT dates runs clockwise and the sequence of BST dates runs anticlockwise, purely for aesthetic reasons.

Each pair of EoT correction lines for GMT dates are displaced clockwise (looking from the top of the dial plate) by 15°, or one hour, in relation to those for the BST dates, to reflect the one hour difference between GMT and BST.

GMT changes to BST on the last Sunday in March, and BST changes to GMT on the last Sunday in October. This means that dates between 25th March and 30th March inclusive, and also dates between 25th and 30th October inclusive, might, depending on the year, fall within either the period of BST or the period of GMT. These dates therefore appear both on the BST and on the GMT sections of the scale.

There is no line-pair for 31st October on the BST section of the scale as 31st October must always fall within the period of GMT. Similarly, there is no line-pair for 31st March on the GMT section of the scale as the 31st March must always fall within the period of BST.

The angular spacing of timescale graduations are equal on an equatorial sundial. Fifteen degrees always represents one hour. For this reason, a longitude correction can be effected by simply rotating the time scale arc by the longitude difference between where the sundial is sited and the relevant time zone meridian..

My garden is at a longitude of 1.178° west of the Greenwich Meridian. This means that when it is solar noon on the Greenwich Meridian, it is 4.7 minutes before solar noon in my garden.

Correcting my sundial to correspond with a simultaneous reading at Greenwich requires the timescale arc to be rotated 1.178° anticlockwise when viewed from the top of the dial plate. This advances the time shown by the gnomon shadow on my timescale by 4.7 minutes. Thus when it is solar noon in my garden, the timescale indicates 4.7 minutes past noon. This is the time that would be read, at the same moment, from a similar dial located at Greenwich, and adjusted for the longitude of the Prime Meridian.

The same principle applies for any other longitude. Locations west of the Greenwich Meridian require an anti-clockwise correction of the timescale arc (looking from the top of the dial plate) by the angle of longitude west of Greenwich. Locations east of the Greenwich Meridian require a clockwise correction of the timescale arc (looking from the top of the dial plate) by the angle of longitude east of Greenwich.

The outer EoT scale is rotated by the angle required to compensate for longitude. Thus, when the sundial is correctly adjusted for longitude, all EoT corrections will include the longitude correction.

The outer EoT correction scale would usually require very infrequent adjustment for longitude - only when the whole sundial was relocated. This scale is therefore held in place by two wing nuts, one of which is visible in Fig. 7, above.

The adjustment can be easily made, although is not quite as convenient to employ as the mechanisms to effect the EoT and date scale adjustments described elsewhere in these notes.

Fig. 13. The date scale graphic.

The sun’s declination changes during the year. It forms an angle of 0° to the equatorial plane at the equinoxes, and it forms an angle of 23.45° above or below the equatorial plane at the summer and winter solstices respectively.

As the plane dial plate is parallel with that of the Earth's equatorial plane, these are also the angles that the sun's declination forms with relation to the dial plate of the sundial.

A protrusion, called a nodus, is connected to the gnomon (Fig.10 and Fig. 11).

Because of the changing solar declination (its apparent height in the sky), a shadow cast by a nodus onto an appropriately calibrated scale can indicate the date.

This nodus shadow moves horizontally during the day as the sun appears to move across the sky, and so a static date scale would either need to be wide, or only be useable for a portion of the day.

Also the movement of the nodus shadow during the course of the year, from its highest point to its lowest point, and then back again, means the calibrations on a date scale can become very confusing to read.

My date scale graphic is shown in Fig. 13. It attempts to minimise both of the above problems by being able to move in an arc across the dial plate. The scale can thus be positioned so that the nodus shadow falls on any point within the width of the scale.

This has allowed the scale to be fairly narrow, and has also meant that dates from June to December can be separated from the December to June dates in order to improve readability.

In practical use, the date scale must be moved frequently and quickly. It does not, however, require to be positioned with a high degree of accuracy. It is therefore moved by hand, by directly pulling it around its track with the knob shown in Fig. 12.

The date scale is positioned 330 mm from the gnomon, and the scale is curved such that all points on its surface are at that distance.

The blue equinox line on the scale, together with the nodus, lie on a plane that is parallel to the equatorial plane.

Due to the variation in the time of the vernal, and related autumnal, equinoxes, no date scale can ever be totally accurate from year to year.

I am indebted to Dr. Frank King, Chair of the British Sundial Society, for explaining to me an approach to provide a 'best fit' for a date scale that is designed for use over many years.

This is the approach that he used in his design of the sundial at the London Stock Exchange (Fig 14a and Fig. 14b).
This is also the approach that I have adopted for my sundial.

The information required from the method is simply the declination to be used for each date line on the scale. Thereafter, the positioning of each date line is a matter of simple trigonometry.

Method for calculating declinations for the date scale:
1 - Decide upon a notional design life for the scale that spans a multiple of 4 years.
2 - Select a period of that duration which begins on a leap year.
I have considered 36 years from 2020 to 2055.
3 - Select a year in the middle of the range, and identify the date and time of the vernal equinox for that year.
My chosen year was 2038, and the vernal equinox in that year occurs at 12:40 GMT on 20th March.
4 - If the chosen year is not a leap year, identify the next leap year on which the vernal equinox falls at around the same date and time.
2038 is not a leap year. The next appropriate leap year is 2104 when the vernal equinox occurs at 12:15 GMT on 20th March.
5 - Use, for the scale, the declinations at 12:00 from 1st March on that leap year to 28th February on the following year.
The declinations I used were all for 12:00, and used the dates from 1st March 2104 to 28th February 2105.

All declinations for the required dates and time were derived from the HORIZONS Web-Interface provided by NASA's Jet Propulsion Laboratory:
https://ssd.jpl.nasa.gov/horizons.cgi.

Fig. 14 shows the nodus shadow on the date scale for the 2nd of October.

The sundial stands on three adjustable feet, designed for use with furniture (Fig. 15). Having three points of support allows any object to stand on uneven surfaces. It also simplifies the levelling process.

The horizontal attitude of the base plate is checked by a spirit level, designed for record player turntables. The spirit level is fixed to the base plate of the sundial (Fig. 16).

The plinth for the sundial is shown in Fig. 17. It includes a hole in the top plate to allow the operation of the screw jack that effects the latitude adjustment of the sundial.

The plinth has adjustable feet of the same type as the sundial. These are not primarily for levelling the sundial. They are simply included because the plinth has four legs, and adjustment is thus needed to allow it to stand firmly on uneven surfaces.

The scales were drawn with the use of two free programs that can be downloaded from the Internet.
These were:
Inkscape: https://inkscape.org/en/
Graph: https://www.padowan.dk/download/
Calculations for the scales were made on an Excel spreadsheet.

The adhesive on the vinyl scales was strong enough to hold the scales in place. When attached to previously varnished surfaces, however, the scales could be peeled away to allow final positioning.

The scales were finally varnished with four coats of Ronseal Crystal Clear Outdoor Varnish, primarily to seal the edges to the wood so that they would be more resistant to peeling away. Varnishing is probably not necessary to protect the vinyl.

A .pdf file containing all the graphics can be downloaded here.
A .svg file containing all the graphics can be downloaded here.

The design of the gnomon supports is derived from that of microphone booms. These allow as much adjustment of the gnomon position as is required.

The correct positioning and alignment of the gnomon is achieved with a gnomon alignment tool that attaches to mounts which are part of the sundial’s structure.

When the gnomon alignment tool is mounted, the gnomon is positioned by adjusting the gnomon support booms such that the gnomon string is in contact with the line on the face of the alignment tool (Fig. 21). A rebate in the tool allows positioning of the nodus, and also allows the gnomon string to lay flat on the alignment line.

The boom construction of the gnomon supports allows them to be easily mounted at any point on the sundial’s structure. This has freed the longitudinal axis of the sundial for location of the EoT scale adjustment knob.

The boom construction, however, cannot sustain the tension of a wire gnomon, as was used in my previous two sundials. The gnomon is therefore made from 0.8mm diameter Kevlar string, as used for stunt kites. The string also allows easy fitting and adjustment of the drilled bead that is the date scale nodus.

The design of the sundial is such that the gnomon supports are attached to the top and bottom of the dial plate on either side of the longitudinal, north-south, axes of the instrument.

The situation between the autumnal and the vernal equinoxes is straightforward: The sun rise is due east and the sunset is due west at the equinoxes. Between the autumnal and vernal equinoxes, when the sun shines from below the plane of the dial plate in the northern hemisphere, the sun neither rises north of due east nor sets north of due west. Structures that protrude below the timescale must simply lie to the north of a line running east-west, in the plane of the dial plate, which passes through the gnomon.

The situation in the northern hemisphere in summer is more complicated because the sun rises north of due east and sets north of due west, with the maximum angles, north of east and north of west, occurring at the summer solstice on 21st of June.

The timescale of the sundial runs from 06:00 to 21:00, and so it is important to know, for the British Isles, the maximum angle north of east that the sun appears at 06:00 at the summer solstice, and the maximum angle north of west that the sun appears at 21:00 at the summer solstice.

Any structure that protrudes from the top of the dial place must be placed at an angle to a line running east-west, in the plane of the dial plate, with the gnomon as the origin, that is greater than these maximums.

Because the whole of the British Isles is in the same time zone, solar noon becomes later when compared to clock time as one moves west.

The sun, on a given day and a given time, will appear to have moved further across the sky when viewed from eastern locations than from western locations. Similarly, it will appear to have moved less far across the sky when viewed from western locations than from eastern locations. .

This means that at 06:00 BST on 21st June, the maximum angle of the sun, north of east, would be measured from the western extremity of the British Isles because the sun would not appear to have moved as far across the sky by 06:00 BST, viewed from the west, as it would appear to have moved if viewed from the east at the same time.

Conversely, at 21:00 BST on 21st June, the maximum angle of the sun, north of west, would be measured from the eastern extremity of the British Isles because the sun would appear to have moved further across the sky by 21:00 BST, viewed from the east, than it would appear to have moved if viewed from the west at the same time.

Maximum angle of the sun north of east at 06:00 on 21st June 2018 = 36.36° at the furthest western point of the British Isles - Rockall, Harris, Scotland
Maximum angle of the sun north of west at 21:00 on 21st June 2018 = 38.35° at the furthest eastern point of the British Isles - Lowestoft Ness, Suffolk, England

Note that these are not the locations of the earliest sunrise or latest sunset. This is because sunrise and sunset relate to the orientation of the terminaor. At any given moment, the azimuth of the sun, and hence its angle north of east, or north of west, relates only to longitude. Shining a torch on a globe, to simulate sunlight falling on th Earth, makes this clearer than any explanation.

The above angles were derived from https://www.suncalc.org. These angles are similar for future years.

Following from the above, structures can be placed on the top of the dial plate as long as they are located at an angle to a line running east-west, with the gnomon as the origin, that is greater than 40° north of east or 40° north of west (see Fig. 23).

In addition to the above, no part of the gnomon support structure that lies less than 40° north of due east, or 40° north of due west, must cast a shadow on the timescale .

The gnomon supports must, therefore, be set at a position in relation to the plane of the dial plate (equator) that takes account of the sun's maximum declination of 23.45° – see Fig. 24 and Fig 25.

No part of the gnomon support structure must be positioned to lie between the plane of the dial plate and the lines that mark the sun's maximum declination.

It can be seen from photographs on this page that the timescale is positioned at right angles to the plane of the dial plate. This allows the timescale to be easily read, from above, throughout the year.

In addition, the timescale annulus is absent for the section that would represent the period from 21:00 to 06:00. This allows the timescale to be read from 9:00 to 18:00 on days close to an equinox, when the sun is near alignment with the plane of the Earth’s equator – and thus near alignment with the plane of the instrument’s dial plate. These are periods during which a timescale that was a complete annulus would be in its own shadow.

Looking from the Earth, the Sun has an angular diameter of 0.5° – an angle equivalent to two minutes on the time scale of an equatorial sundial..

My sundial uses 0.8 mm diameter Kevlar string for the gnomon, and one minute on the time scale is 0.85 mm in width.

The width of the shadow cast by the gnomon onto the time scale, therefore, is around the equivalent of the minimum possible two minutes.

An estimate of the centre of the shadow can be made to improve accuracy to within one minute of clock time.

A dry, varnished surface sticks to another dry, varnished surface if they are in contact for any period of time. This adhesion is not great, but it makes for a stiff or jerky beginning to any adjustment. All surfaces that move against other surfaces on this sundial are therefore covered with woven, self-adhesive ptfe tape.

It may also be noted in Fig 27 that the rear face of the date scale channel is lined with 3mm draft excluder. This provides a slightly flexible surface to both support and facilitate ease of movement of the date scale in its channel.

Circles for my previous two sundials were cut using a jigsaw, mounted in a homemade jig. Such an arrangement was very prone to inaccuracy due to the jigsaw blade wandering.

All circles and slots for Mark III were cut with a router (Fig. 29) which allowed near perfect circles.

Fig. 31, Fig. 32, Fig.33 and Fig. 34 show the basic dimensions of the structure.

Fig. 35 shows the inside of the time scale arc. The woven ptfe strips are visible and also the pillars that provide attachment points for the melamine surface veneer, the drive belt and the guide spacer.

The drive belt is attached to the timescale arc below the bearing surface on which it slides and rotates. This creates a slight downward pull on the time scale arc, leading to small vertical rotation that tends to lift the edge of the time scale arc at the curve on which the inner and outer EoT scales meet.

The black, rectangular guide spacer, visible in Fig. 35, Fig. 36 and Fig. 37, gently touches the underside of the bearing surface on which the timescale arc slides and rotates. It thus prevents the unwanted lifting of the time scale arc.

The guide spacer is made from a section of drive belt because the rubber-like surface suits the task.

The holes in the melamine veneer, visible in Fig. 36 and Fig. 37, are for rainwater drainage.


Fig. 31. Basic dimensions.	Fig. 32. Basic dimensions.

Fig. 33. Basic dimensions.	Fig. 34. Basic dimensions.

Fig. 35. Time scale arc.

Fig. 36. Time scale arc.	Fig. 37. Time scale arc.


Fig. 3. The screw jack that controls the angle of the dial plate and hence the latitude setting.	Fig. 4. The screw jack as viewed from the underside of the base plate.


Fig. 6. The centrally placed knob that allows adjustment of the timescale arc and hence the inner EoT correction scale.	Fig. 7. The pulley and drive belt mechanism that moves the timescale arc and hence the inner EoT correction scale.


Fig. 10. The date scale showing the track in which it runs and the nodus on the gnomon.	Fig. 11. The nodus.	Fig. 12. The rear of the date scale showing the knob used to move it.


Fig. 14a and Fig. 14b. The dial on the London Stock Exchange - designed by Dr Frank King.


Fig. 15. The underside of the base plate showing the three adjustable feet.	Fig. 16. The fixed spirit level.


Fig. 18. The upper gnomon support.	Fig. 19. The lower gnomon support.


Fig. 20. The gnomon alignment tool.	Fig. 21. The gnomon alignment tool.


Fig. 22. In summer, the sun rises north of due east and sets north of due west.	Fig. 23. Maximum declination of the sun (23.45°) at the summer solstice.	Fig. 24. Maximum declination of the sun (23.45°) at the winter solstice.


Fig. 25. The dial plate.	Fig. 26. The shadow of the gnomon on the timescale.