Martian Heliochronometer Sundial

Summary

With a simple modification to the analemma plate, you can convert your Earth-based heliochronometer into an accurate Martian sundial using one of these printable models:

1. Planetary Geared Sundial/Clock Hybrid - The Best of both Worlds with Reading Accuracy Optimization by yba2cuo3 | Download free STL model | Printables.com
2. Gard Heliochronometer Sundial Derivative by yba2cuo3 | Download free STL model | Printables.com
3. Heliochronometer - World's Most Accurate Sundial by yba2cuo3 | Download free STL model | Printables.com
4. Heliochronometer (version 2) - World's Most Accurate Sundial by yba2cuo3 | Download free STL model | Printables.com

Understanding and displaying the analemma is crucial for constructing an accurate sundial or heliochronometer on Mars. By incorporating Mars's specific analemma into these reference designs, we can account for the planet's unique equation of time, enhancing the precision of timekeeping devices for future explorers and colonists on the Red Planet.

While you won't be able to test this sundial unless you're on Mars, it serves as a captivating conversation piece and an excellent educational tool. This project can ignite curiosity about planetary timekeeping, astronomy, and the future of space exploration. Who knows—some of you might be fortunate enough to set foot on the Red Planet in your lifetime!

For useful background information on heliochronometers; i.e. history, theory of operation, design, how to align and use, etc.  refer to this useful printable:  Heliochronometer - World's Most Accurate Sundial by yba2cuo3 | Download free STL model | Printables.com

                                            Figure 1:  Mars Analemma Plate

Main Features

• Accurate to within 1 minute throughout the entire Martian solar year from any location in the northern hemisphere;
• 5 minute reading resolution on the main dial plate;
• 0 to 5 minutes reading resolution on a large, easy to read, secondary Vernier scale, in increments of 30 seconds;
• 1 degree latitude adjustment resolution;
• 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 offset adjustments;
• 4 base options are available; 3 with integrated magnetic compass and levels, and 1 without.  For compass/level options, refer to these printable links:
• Tri-leg:  Improved Heliochronometer Sundial Base Design with Integrated Bubble Levels and Magnetic Compass by yba2cuo3 | Download free STL model | Printables.com
• 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).
• Added deeper cut mark versions for text, dial tick marks, etc. Look for the files with (deeper) in the name.

Print Settings

• Printer brand:  Prusa
• Model:  i3 MK2S
• Supports:  Yes
• Resolution:  0.10mm DETAIL or 0.05mm ULTRADETAIL
• Infill:  20%
• Brim: Yes - 10 to 20mm
• Filament brand:  Doesn't matter
• Filament material:  ABS
• 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 75% on top layers will improve tick mark production.

Construction

Just replace your original analemma vertical arm/plate with the model provided in this printable.  

Analemma Plate Design

This design is constructed out of ABS plastic filament.  Check the Technical Details section & How was the Analemma Curve Designed into this Heliochronometer from this other printable:  Heliochronometer - World's Most Accurate Sundial by yba2cuo3 | Download free STL model | Printables.com  

Just like on Earth, Mars has its own analemma—a figure that represents the Sun's position in the sky at the same  mean solar time over the course of a Martian year. However, the shape of Mars's analemma is significantly different from Earth's. While Earth's analemma forms a figure-eight due to the combination of its axial tilt and orbital eccentricity, Mars's analemma resembles a teardrop shape (see images below).

                Figure 2:  Sky View of Earth Analemma (left) vs. Mars Analemma (right)

The unique teardrop shape of Mars's analemma arises from its orbital characteristics. Mars has a more eccentric orbit compared to Earth, meaning it deviates more from a perfect circle. This higher eccentricity, coupled with its axial tilt, results in the asymmetrical, teardrop-shaped analemma. This difference is crucial for accurate timekeeping on Mars, as it affects the equation of time—the discrepancy between apparent solar time and mean solar time.

To generate the data for Mars's analemma, I wrote a Python script (attached) that calculates the Sun's apparent position at consistent intervals throughout the Martian year. The resulting Equation of Time data was then imported into Microsoft Excel for further computations and eventual plotting of the analemma curve.  See attached pdf document on how this was accomplished. This process allowed for precise modeling of the Sun's analemma on Mars, which is essential for designing an accurate Martian heliochronometer. 

                      Figure 3:  Equation of Time (EoT) over the course of a Martian Year (668 sols)

                         Figure 4:  Plotted Analemma for Mars

Figure 5:  Scaled Analemma for Heliochronometer Plate with 1st Day of Martian Month and Astronomical Markers (solstices, equinoxes, etc)

Background Information

Using a standard sundial on Mars would result in significant inaccuracies; with readings varying approximately 41 minutes slow to 53 minutes fast over the Martian solar year. In contrast, a standard sundial on Earth deviates by only about 16 minutes fast to 14 minutes slow throughout the year if not corrected. This substantial discrepancy highlights the importance of incorporating a Martian heliochronometer equipped with its own equation of time corrections. By compensating for the differences between true solar time and mean solar time—and accounting for solar declination as functions of areocentric longitude—a Martian heliochronometer provides much more accurate timekeeping on the Red Planet.

A Martian solar day, known as a "sol," is approximately 1.02749125 Earth days long. Mars completes one orbit around the Sun relative to the stars (its sidereal year) in about 668.5921 sols, or 686.98 Earth days—approximately 1.88 Earth years. When expressing Martian time consistently in Martian units, the correct figure is 668.59 sols per vernal equinox year.

Interestingly, Martian clocks use the same time units as we do on Earth: 60 seconds per minute, 60 minutes per hour, and 24 hours per sol. However, each of these Martian units is slightly longer—about 2.7%—than their terrestrial counterparts. This adjustment methodology is considered to be the most practical approach and one which is the most accepted in Martian clock applications.

A more significant difference between Earth and Mars is the length of their years. Because Mars is nearly 80 million kilometers farther from the Sun, it takes almost twice as long to complete one orbit. This results in a Martian year lasting about 687 Earth days.

The Martian calendar begins with Sol 1, defined as the time when Mars is at 0 degrees areocentric longitude, marking the vernal equinox. Assuming that seasons run from equinox to solstice or vice versa, the northern hemisphere's spring (southern hemisphere's autumn) is the longest season on Mars, lasting 194 sols. Conversely, the northern hemisphere's autumn (southern hemisphere's spring) is the shortest season, lasting only 142 sols.

The Darian Calendar and Martian Timekeeping

American aerospace engineer and political scientist Thomas Gangale introduced the Darian calendar in 1986, with further details published in 1998 and 2006. Designed to accommodate the longer Martian year, the calendar divides it into 24 months, maintaining a notion of a "month" that is reasonably similar in length to an Earth month.

On Mars, a "month" does not correlate with the orbital periods of its moons, Phobos and Deimos, which orbit the planet in approximately 7 hours and 30 hours, respectively. However, Earth and its Moon would generally be visible to the naked eye from Mars when above the horizon at night. Interestingly, the time it takes for the Moon to move from maximum separation in one direction to the other and back, as seen from Mars, is close to a lunar month.

Given that a Martian year is nearly twice as long as an Earth year, dividing it into twice as many months—24 instead of 12—is a logical approach. This method offers several advantages. Humans are already familiar with the mean Earth month of approximately 30.44 days. By dividing the Martian year (668.59 sols) by 24, we get a mean Martian month of about 27.86 sols, or 28.62 Earth days. This results in only a 6% difference between the mean Martian month and the mean Earth month, making it easier for humans to adjust to a slightly shorter month rather than one nearly twice as long.

Moreover, although a 28-sol month has no astronomical basis on Mars, it aligns with human biological cycles, which average around 28 days. This synchronization enhances the calendar's relevance to human experience. After all, the primary purpose of a calendar is to mark the passage of time in human terms.

History

In early Roman religion, Mars was the god of vegetation and fertility, and his festivals signified the return of life to the land. This was before Rome became an imperial power and Mars, the farm guardian, was transformed into a god of war. During this pastoral era, Romulus chose to begin his calendar with the vernal equinox, naming the first month of the year after Mars—the provider and protector of the Roman people.

Similarly, the Darian calendar is intended to symbolize the beginning of life on the planet named for Mars. If ancient life once flourished there, it represents the return of life to Mars. Therefore, the vernal equinox is chosen as the starting point of the Martian year. On Earth, the vernal equinox is a standard astronomical reference that marks the beginning of the astronomical year, and it seems reasonable to extend this concept to Mars.

The current position of the Martian vernal equinox lies on the western edge of the constellation Sagittarius. As a result, the first month of the Darian calendar is named Sagittarius, with the subsequent months following in their appropriate order.

In the Darian calendar, common years of 668 sols consist of 20 months of 28 sols and four months of 27 sols. The 27-sol months occur at the end of each quarter. In leap years of 669 sols, the last month of the year—which also concludes the fourth quarter—instead of having 27 sols, contains the standard 28 sols. The leap sol is therefore the last sol of the year, rather than being inserted somewhere in the middle as it is in Earth's Gregorian calendar.

Table 1:  The Martian Seasons

Table 2:  Annual Martian Astronomical Events (Nominal Dates)

Table 3:  First Day of Martian Months, Names and Durations                             

Total Martian Days:   668 days

What would the Analemma look like on other planets in our Solar System?

Analemmas vary from planet to planet. Each one has its own unique look.  Here's a simple breakdown:

• Mercury: dot
• Venus and Jupiter: elliptical shape
• Saturn:  teardrop with a small loop
• Neptune and Uranus? figure-eight pattern

Figure 6:  Analemma as seen from the planets of the Solar System. © Vito Technology, Inc.

Please hit the like button if you think this is a cool design.  I hope you enjoy building this & thanks for your support!

References:

1. NASA GISS: Mars24 Sunclock — Algorithm and Worked Examples
2. Martian Daylight Time (ops-alaska.com)
3. Sundials - Their Construction & Use, R. Newton Mayall & Margaret Mayall, Dover Publications Inc., 1994