Heliochronometer - World's Most Accurate Sundial

Summary

The Heliochronometer is just what its name implies—a solar chronometer.  It is not a toy, or ordinary garden ornament, but an astronomical instrument for precision timekeeping as well as education.

Main Features

• Accurate to within 5 minutes (ver. 1), or 1 minute (ver. 2), throughout the entire solar year from any location in the northern (or southern) hemisphere. In practical terms, however, the accuracy will be determined by the quality of it’s construction, assembly and precision of it's final alignment;
• Displays Local Mean solar time directly by computing corrections to the true, or apparent solar time.  It achieves this by utilizing a visual-mechanical computer, also known as an analemma plate;
• Converts Local Mean Time to Standard Local Time via meridian dial adjustments;
• Compensates for Daylight Savings Time;
• Latitude adjustable;
• 4 base options are available; 3 with integrated magnetic compass and levels, and 1 without.  For compass/level options, refer to these printable links:

1. Tri-leg:  Improved Heliochronometer Sundial Base Design with Integrated Bubble Levels and Magnetic Compass by yba2cuo3 | Download free STL model | Printables.com
2. Leveling base & tripod head:  Heliochronometer Sundial Mount for Head Leveling Base or Tripod Ball by yba2cuo3 | Download free STL model | Printables.com

• Assembly measures 180x200x200mm (W x L x H).

List of Upgrades in Ver. 2:

• Higher reading resolution dial plate; i.e. 5 minutes vs. original 15;
• Introduction of a secondary Vernier scale to the Alidade which allow time readings to within ½ minute. As in ver. 1, the accuracy will be determined by the quality of it’s construction, assembly and precision of it's final alignment;
• Improved Analemma curve resolution;
• Replaceable Nodus cone with smaller circular sight to improve reading resolution (or adopt your own custom design);
• Improved dial holder ends with better meridian marker and adaptable pedestal for tripod base; 
• As in ver. 1, 4 base options are available; 3 with integrated magnetic compass and levels, and 1 without.  See links above to download the base designs.

This heliochronometer was constructed out of ABS plastic filament.  Check the Technical Details section below for the various parts that make up this heliochronometer. The analemma curve was calculated & plotted using MS Excel and then scaled to match the size (diameter) of the heliochronometer. Check the description below on How was the Analemma Curve Designed into this Heliochronometer.  

The dial design also accommodates Daylight Savings Time (DST) by allowing the rotation of the dial plate. The design presented here is for use in the northern hemisphere, however, a southern hemisphere dial plate can be easily generated using an online calculator like this one; https://www.blocklayer.com/sundial-equatorial, or use the attached file. Conversely, ver. 2 dial design was generated via a python script I wrote which outputted an SVG file, edited in Inkscape, then imported into Blender for STL generation.

Why build a heliochronometer?

Why would a person who owns a timepiece go to the trouble of building a sundial?  Because they are motivated in part by the intellectual charm of a device which, without moving parts, can convert the sun's changing position directly into time.  In the course of developing and constructing this sundial, one is exposed to a fascinating and well-defined mixture of mathematics, geometry, geography and astronomy.  Building an accurate sundial is a challenge to anyone's creative talents, and its construction will put a person craftsmanship to an exacting test.

Many sundials, both portable and stationary, were made in the 18th and 19th centuries which incorporated the equation of time in their construction. This enabled one to read local mean time directly from the dial and the analemma was the device that made this direct reading possible. Among them, the heliochronometer stood out as one of the most elegant mechanical and optical sundial instruments of its day. They were heavily relied upon before accurate clocks became readily available to the general public. This was the type of dial used by the railroads in France for setting watches, as late as 1900. Embedded within its design is the celestial alignment and tracking of the motion of our planet around the sun. Once its sights are aligned to the sun’s rays, it actually measures the earth’s location in our solar system.  

Figure 1:  Shadows from the Nodus sight moves along the Analemma curve during the course of a solar year.  

What exactly is a Analemma?  It is the figure 8 pattern found on the back projection plate of a heliochronometer.  One can actually visualize this by measuring the position of the sun in the sky at a fixed time throughout a calendar year.  The pattern which is projected onto the sky would be in the form of a figure 8.  In the case of the Analemma plate, it is a computed curve derived from the Equation of Time (see explanation below), and is used to convert the sun’s true, or apparent solar time, to mean solar time. The Analemma’s vertical axis is the sun’s negated declination and its horizontal axis is the correction from true solar time to mean solar time; (derived from the Equation of Time).  An MS Excel spreadsheet was used to generate the plot and the resulting analemma curve was sized accordingly so to be compatible with diameter of the dial plate and height of the nodus sight. See attached image in the file section for more details.

Figure 2:   The Analemma plate.  The diamond shapes indicate the 2 Solstices (winter/summer) and the 2 Equinoxes (spring/autumn). The circular marks along the curve indicates the 1st day of a specific month.  Letters next to the analemma curve indicates the month.  Note that the shadow casted by the Nodus needs to be manually aligned to the analemma curve for any specific month to correctly read the time.

Figure 3:  Apollo's Analemma, Image Credit & Copyright: Anthony Ayiomamitis (TWAN)

Print Settings

• Printer brand:  Prusa
• Model:  i3 MK2S
• Supports:  Yes
• Resolution:  0.15mm OPTIMAL
• Infill:  50%
• Brim: Yes - 10mm
• Filament brand:  Doesn't matter
• Filament material:  ASA or ABS (easier for printing)
• Filament color:  Doesn't matter
• Special Notes:  
  • Print in an enclosure for best results. 
  • Use a darker color filament at a specific layer height to highlight the text if you have a single extruder. 
  • Slow printer speed to 80% on top layers will improve tick mark production.

Construction

The construction of this sundial is relatively simple, making use of M4 hardware.  A list of assembly material is provided below, along with where it's used.  Also check the description associated with each file for more assembly details.  All parts can be easily disassembled and reassembled to facilitate transportation.

List of Required Assembly Hardware

All HW is Stainless Steel Button Head Hex Socket Head Cap Screws and Nuts, unless specified otherwise.

Post Processing Tools

1. Deburring tool for removing excess plastic from printed parts
2. Hand Drill or Drill Press
3. 3.3mm or 1/8" drill bit for enlarging holes for M4 tap
4. M4 tap for making threads

For Sundial Alignment

1. Magnetic Compass
2. Circular Bubble Level

Alternatives:   Smart phone with:  1) Compass or GPS app, 2) level app.

TECHNICAL DETAILS

What makes up a heliochronometer?

The heliochronometer consists of four basic parts: 1) Base, 2) Equatorial Dial Plate, 3) Alidade, or sighting instrument, and 4) Analemma.

1. The base is physically fixed in place.  It secures the equatorial dial plate to a solid, non movable surface and allows for its adjustment in declination to accommodate for different latitudes;
2. The equatorial dial plate is attached to the base so that it may be rotated about its center.  Its axis is aligned to the celestial north pole so that its surface is parallel to the equator;
3. The alidade is attached to the dial plate so that it can be rotated about its center, which is coincident with the center of the dial plate. Consisting of a flat plate, the alidade has two fixed upright arms perpendicular to the dial plate. One arm contains the front sight or nodus, the other the analemma;
4. The analemma consists of an imprint of the analemma curve on a flat (or sometimes curved) surface and is based on the equation of time.  Most analemma plates contain basic curves.  Alternative design suggestions:  One can make the design more interesting by adding the months of the year, marking every five days on the analemma and indicating the degrees of declination on one side and the degrees of altitude on the other side of the analemma. With this data incorporated, a dial becomes very useful. It can be used to show the declination and altitude of the sun, the day of the year, apparent time, mean time, standard time, and the equation of time.

A further design feature was incorporated in the equatorial dial plate’s construction to allow for the direct reading of Standard Time from Local Mean Time.  This was achieved in this design by allowing the dial plate to be rotated on the base such that the longitudinal offset from prime meridian can be dialed in and locked into place with a screw. 

Figure 4:  Parts of a heliochronometer

How to Align:

Can't stress enough the importance of having a well aligned heliochronometer in order to make accurate and consistent readings.  Please follow these steps:

1. Place the heliochronometer on a flat, relatively level surface;
2. Adjust the latitude (elevation) of the dial plate for 90 degrees; i.e. horizontal;
3. With the longitude (prime meridian) correction on the dial plate set to zero, move the alidade so that it is pointing at 12 noon. Keep the alidade position fixed throughout the remainder of the adjustments;
4. Download a GPS compass app on your smartphone, or use a magnetic compass. You will need to determine true north and not magnetic north.  If you use a compass, you will need to adjust/compensate for the magnetic declination offset in your region;
5. Place your phone (or compass) flat on the dial plate and keep its side against the alidade. While monitoring your smartphone app (or compass dial), rotate the sundial base until the alidade is pointing true north (or south). Secure the base once this is done so that it can no longer rotate;
6. Using a bubble level; (circular bubble works best), or smartphone leveling app, place the level on top of the alidade and adjust all three leveling screws on the sundial base to achieve a perfect level;
7. Once the alidade is level & true north is set in place, adjust for your latitude by tilting the dial plate in elevation to match your latitude;
8. Adjust your dial plate for the difference in longitude between your location and your standard time prime meridian.  If you are on daylight savings, adjust for that also; (see above descriptions for procedures);
9. Tighten all screws.

Alternative Method for Sundial Alignment (simplest & most accurate, but less convenient): 

Each sundial has some inaccuracies in it’s build quality and assembly. This procedure describes a way to somewhat compensate for these inconsistencies & doesn't rely on a smartphone or compass for installation, but it does rely on it to be completely level.  Note that this alignment procedure is only possible during specific times of the solar year; i.e. 1st of the month (circles), or summer/winter solstices, spring and autumn equinoxes (diamonds), as indicated on the analemma curve plate.

1. Make sure your sundial is level following the procedures outlined above;
2. Adjust your dial plate for the difference in longitude between your location and the standard time meridian & if you are on daylight savings; (see above descriptions for procedures);
3. Set your latitude; (dial plate elevation), at the protractor;
4. Pick a day which has an identifying marker on the analemma, like the 1st day of the month, or the equinoxes, or solstices. Choose a time also which has a clear mark indication on the dial plate.  A good choice would be on-the-hour. Noon works best;
5. Move the alidade so that it points to the actual local standard time in your time zone;
6. Now, without moving the alidade, rotate the entire sundial at its base.  Do not rotate the dial plate!  Rotate the sundial base in azimuth so that the sunlight passing through the alidade sighting hole; i.e. nodus, is directly on the marker you chose in step 4 of analemma curve. This will align the sundial for true north;
7. If the light spot has moved from the date marker you selected in step 4, you will need to re-adjust the sundial so that everything registers correctly; i.e. 1) alidade is pointing to the correct time, 2) light spot is on the analemma curve (azimuth adjustment), light spot is on the correct date marker of the analemma curve (elevation adjustment). If the time seems to be off over the course of the day, check the levelness of the base;
8. Tighten all screws.

How to Use:

1. Make sure your sundial has been aligned first!
2. If you haven't done so already, you must first adjust the dial plate for the difference in longitude between your location and your standard time meridian. See example below on how to calculate the offset for Local Mean Time (LMT) to Standard Time (ST). Unloosen the 4 dial lock screws and turn the dial plate until the prime meridian line on the dial; i.e. line extending to the outer edge of dial plate, opposite 12 o'clock, is at the time correction you calculated; i.e. difference between your standard time meridian and your longitude. Since this always remains the same, the dial plate may be locked in this position once this step is done. The alidade point will now indicate standard time.
3. To obtain the local mean time at any hour of any day, just turn the alidade until the sunlight passing through the sighting hole is centered on that portion of the analemma curve corresponding to the current month of the year. In this position, the pointer on the alidade will indicate local mean time and if corrected, local standard time.

Here is an example using the more complex and accurate ver. 2 heliochronometer design with a secondary Vernier scale:

• The sun spot cast by the Nodus sight is aligned onto the Analemma curve for the current month. In this case it is September.  It also happens to be the 25th, so the sun spot is just above the Autumnal Equinox indicated by the diamond shape; (see below)

• The secondary Vernier scale located on the analemma side of the alidade is from 0 to 5 mins, in increments of 30 secs; (see below)

• Therefore, for this example, the time is read as follows;

1. The zero mark on the Vernier indicates 10 mins past 3 o’clock on the dial plate; i.e. 2x 5 mins minor tick marks past the main 3 hour mark;
2. Now look for the next aligned marks on the Vernier scale which meets up with those on the dial plate.  This happens to be at 3mins (± 30secs).  Therefore, the time would read 3: 10+3mins = 3:13 PM PST.

Can the Dial be adjusted to show Daylight Savings Time? Yes! In some locations, clocks are adjusted forward one hour for Daylight Savings Time (DST). To adjust for DST the dial plate is rotated ahead (counter-clockwise) by 15 degrees, or 1 hour. The longitudinal offset is then based off daylight savings.  

See description below on how to set your sundial to read ST or DST directly.

Background: What are the different types of time?

Before learning how a heliochronometer really works, we need to understand the different types of times: i.e.

• A sundial shows True or Apparent Solar Time. Because the Earth's rotation is not constant, solar days vary slightly in length as it follows the ecliptic. This means that the speed of true solar time is not constant. It must be remembered that a sundial measures the hour angle of the true Sun as observed in the sky. In reality, the Sun’s true hour angle is due to two motions: diurnal motion, i.e. the motion of the Earth as it turns on its axis; and annual motion, i.e. the apparent eastward displacement of the Sun along the ecliptic. More about this later; 

• Mean Solar Time is based on the length of a mean or average solar day, which is 24 hours long. It moves at a constant speed along the celestial equator. All hours have the same length regardless of the season. Mean solar time might be faster or solar than the true solar time depending on the time of year; 

• Local Mean Time (LMT) is the Mean Solar Time for a specific location on Earth. It is the same for all locations that share the same longitude; 

• Standard Time (ST) is also referred to as the official time of a region ascertained by the distance from the Prime Meridian of the meridian running through the area. For example; Pacific Standard Time (PST) →  (UTC−08:00), has a prime meridian at 120 degrees longitude. LMT can easily be derived from ST by adding or subtracting 4 minutes for each degree away from the prime meridian, or 360°/24h = 15° for every time zone hour.  It follows the simple relationship of 1 hour or 60 minutes for every 15° of longitude; or 4m per degree.

Converting time & setting your dial plate to convert to ST : 

True Solar Time + EOT → Local Mean Time + Longitudinal Time Correction → Standard Time

Here is simple example of how to convert Local Mean Time to Standard Time (or vise versa) for a sundial situated in Vancouver, BC, Canada.  Note that your dial plate only needs to be set once for your location to adjust for Standard Time.

Vancouver BC:  Longitude -123° 07m; which is 3° 7m West of the Pacific Standard Time (PST) zone prime meridian at 120°.  Therefore, the time correction (TC) would be: 

TC = (3 + 7/60) x 4m/deg = 12.467m or 12m 28s.

Since Vancouver is West of the PST time zone prime meridian:  LMT = PST - TC

 or conversely: PST = LMT + TC 

In this example, sundials in Vancouver would be running behind sundials situated on the pacific standard prime meridian.  Therefore, the dial plate would need to be rotated counter-clockwise; i.e. ahead from it’s prime meridian mark by 12-1/2 minutes to properly read PST. Note that every minor tick mark on the main dial is 5 minutes, so the rotation would be 2-½ minor ticks ahead on the dial.  The adjustments are highlighted in the figures below:

Figure 5:  Dial on Prime Meridian (left).  Dial adjusted for PST in Vancouver; i.e. LMT+ 12-1/2 mins (right)

For adjustment to daylight savings time, the dial needs to be further rotated forward in time by an additional 1 hour; (see below).

Figure 6:  Dial rotated an additional 1 hour ahead to display daylight savings directly.

What makes it work (more details)

How does the heliochronometer actually tell time?

To compute standard time the heliochronometer makes use of several pieces of information: the observer’s latitude and longitude, the hour angle of the sun, the direction of true north, the month of the year, the declination of the sun and an astronomical formula called the Equation of Time (EOT).

What is The Equation of Time? The Equation of Time (EOT) is the difference between true or apparent solar time and mean solar time over the course of a year. The difference in time is due to two effects: the eccentricity of the Earth’s orbit and the obliquity or tilt of the Earth’s rotational axis. 

• Eccentricity of Earth’s Orbit: The Earth’s orbit around the sun is not a perfect circle but an ellipse. This means the Earth’s orbital speed varies throughout the year, moving faster when it is closer to the Sun (perihelion) and slower when it is farther (aphelion). Due to this variation in speed, the apparent movement of the Sun across the sky does not occur at a constant rate when measured against a uniformly ticking clock. When the Earth is moving faster in its orbit, the solar day (the time from one solar noon to the next) is slightly longer than the average, and when the Earth is moving slower, the solar day is slightly shorter. 

• Axial Tilt of the Earth: The Earth’s axis is tilted at an angle of about 23.5 degrees relative to the plane of its orbit around the Sun. This tilt causes the Sun’s apparent path in the sky (the ecliptic) to be inclined relative to the celestial equator. As a result, the rate at which the Sun appears to move along the celestial equator varies throughout the year. When the Sun’s path makes a steep angle with the equator (as during the solstices), its apparent eastward motion along the equator is slower than average. Conversely, when this path is more parallel to the equator (as during the equinoxes), its apparent motion is faster. As mentioned, these two combined effects result in a discrepancy between true or apparent solar time and mean solar time. The EOT is a way of quantifying this discrepancy. It varies throughout the year, typically ranging between about -14 and +16 minutes. It reaches its maximum and minimum values around the times of the summer and winter solstices and is zero near the spring and autumn equinoxes. This discrepancy is why a sundial will sometimes run ahead of a clock and at other times fall behind it, and why the length of a true solar day is not exactly 24 hours all year round. The EOT corrects for these variations, allowing us to reconcile solar time with the 24-hour clock used in daily life.

How was the Analemma Curve Designed into this Heliochronometer

Simplifying the math, the relationship of the distance between the nodus and analemma vertical arms is as follows:

• Dimensions of fhe Analemma curve in Y-axis (mm):  B x (867.17 x 10E-3);
• Dimensions of the Analemma curve in the X-axis (mm):  B x (133.849 x 10E-3);

where, B is the horizontal (internal) spacing between the nodus and analemma plates mounted on the horizontal alidade.  The distance B/2 is at the center of the dial plate, or the pivot point of the alidade.  

Design Example:

A designer decides on a dial plate diameter (D) of 160mm and a alidade (horizontal arm) length of 130mm.  Based on these dimensions, the user chooses a distance B of 80mm.  Again, this corresponds to the spacing between the vertical nodus arm and analemma arm (or plates).  

Using the attached analemma SVG plot in the file section, one would scale the total height and width of the desired curve to be:

• Y-axis analemma height:  80 x (867.17 x 10E-3) = 69.3736 mm
• X-axis analemma width:  80 x (133.849 x 10E-3) = 10.7079 mm

This newly scaled curve now requires to be superimposed on a vertical plate and be properly aligned to the nodus hole.  How is this achieved?  Answer: there is an isolated dot (alignment mark) on the analemma plot, located between the spring and autumn (diamond shaped) equinoxes.  This dot needs to be perfectly aligned with the nodus hole location in both the X and Y directions.  In this particular example, the nodus height above the dial plate (C) is 50mm.  Note that the nodus hole also needs to be centered along the center line (X-axis) of the dial plate; i.e. D/2 mm.  Therefore, the alignment dot on the scaled analemma curve would need to be exactly 50mm from the surface of the dial plate (same height as the nodus hole), and also be aligned to the center line of the dial plate (D/2).

In most cases, the diameter of the dial plate would be larger than B to accommodate for extra space for the time markings and numerals, etc. which are normally located at the edge of the dial.

Suggestions for any improvements are always welcomed!

Category: Outdoor & Garden