Submitted by mossy2100 on Tue, 04/25/2017 - 00:33

In the Utopian Calendar, rather than using an era label like BC/BCE or AD/CE as in the Gregorian Calendar, mirs are simply numbered using integers, i.e. … –3, –2, –1, 0, 1, 2, 3, etc. Defining an epoch for the calendar means specifying which mir is numbered 0. As we have already decided that the mir begins with the northern vernal equinox (NVE), the next question is which one will define the beginning of mir 0.

The Martian NVE that occurred at approximately 12 March 1609, 19:06:20 UTC is used. This particular NVE makes a good starting point for the calendar because this is approximately when telescopes started being used to study Mars. This epoch is called the Telescopic Epoch.

The year 1609 represented the beginning of a new relationship between humans and Mars. It was the year that Johannes Kepler published Astronomia Nova, in which his first two laws of planetary motion were explained. He arrived at these laws by studying Tycho Brahe’s observations of Mars, which enabled him to determine that Mars' orbit was elliptical rather than circular. In September of the following year, Galileo became the first person known to have observed Mars through a telescope.

Before that, Mars had only ever been viewed with the naked eye. Selecting the Telescopic Epoch for mir numbering means that the dates of all telescopic observations of Mars, and therefore all maps, missions, and everything that has happened on Mars since, and that will happen in the future, can be expressed with non-negative mir numbers.

The use of this epoch for Martian timekeeping was first suggested in 1999 by Peter Kokh, who was the President of the Moon Society for many years.

MTC (Coordinated Mars Time) at the NVE was approximately M:991.372 (in Utopian notation), i.e. almost midnight. The closest midnight is about 13 minutes later, at approximately 12 March 1609, 19:19:06 UTC. This is used as the actual starting time of the first sol in mir 0 (referred to as sol 0 from hereon), and thus of the epoch.

How this epoch was computed

The astronomer Jean Meeus calculated the equinoxes and solstices of Mars from 1874–2126 and published these in Astronomical Tables of the Sun, Moon and Planets. They are also available here, on the Martian Time website. Ideally I would have directly computed the Julian Date for the NVE for 1609 using the same astronomical formulae that Meeus used, but this would have required computational resources I do not presently have access to; therefore, for practical purposes, I have taken a different approach.

Using these values, I calculated a line of best fit through the data points using the method of least squares, to derive a formula relating mir numbers to Julian Dates of Mars’ northern venal equinoxes. Specifying the NVE of 1874 as mir 141 causes the NVE of 1609 to be mir 0. The resulting formula is:

JDUTC = 686.97103295817 * mir + 2308806.29606

For mir = 0, the Julian Date for the NVE is therefore 2308806.29606.

Using formulas adapted from Numerical Recipes in C: The Art of Scientific Computing, the Gregorian Calendar datetime corresponding to this Julian Date is found to be 1609-03-12 19:06:20 UTC.

The data points provided by Meeus fit this line very closely, with a maximum error of only about ±0.03 days, or 42 minutes and 12 seconds. This may seem a large inaccuracy, but fortunately this is not the precise start datetime for the epoch that is required. The task now is to determine the closest midnight, which will be the beginning of sol 0.

To achieve this, we first need to find the local time at 0°, at the moment of the NVE.

This requires knowing the Julian Date in terms of Terrestrial Time (TT), rather than UTC, as these are slightly different due to variations in the length of Earth’s day. The difference depends how far in the past or future we go.

To calculate ∆T, which is the difference in seconds between TT and UTC, formulae were obtained from Morrison & Stephenson (2004) – Historical Values of the Earth’s Clock Error ΔT and the Calculation of Eclipses.

Step 1. Calculate the number of Julian centuries before the start of the year 1820.

cy = (1820 - year) / 100
  = (1820 - 1609) / 100
  = 2.11

Step 2. Calculate the difference in seconds between TT (Terrestrial Time) and UTC (Universal Coordinated Time).

For dates before 1770 the following formula is used:

∆T = TT - UTC
  = -20 + 32 cy2
  = -20 + (32 * 2.112)
  = 122 seconds

According to Morrison & Stephenson, the error in this calculation is about 20 seconds.

From this information we can calculate MTC, i.e. the local time at 0° longitude. The algorithm for this calculation come from Allison & McEwen (2000) A post-Pathfinder evaluation of aerocentric solar coordinates with improved timing recipes for Mars seasonal/diurnal climate studies, also shown on this webpage at the NASA Goddard website: Mars24 Sunclock — Time on Mars — Algorithm and Worked Examples. This webpage also provides formulae for calculating ∆T, but these are only useful for dates from 1770 onwards.

Step 3. Calculate the Julian Date in Terrestrial Time.

JDTT = JDUTC + (∆T / 86400)
  = 2308806.29606 + (122 / 86400)
  = 2308806.29747

Step 4. Calculate the Mars Sol Date (MSD).

The Mars Sol Date is a proposed Martian equivalent of the Julian Date. It appears in the Allison & McEwen paper, and was presumably invented by them.

The calculation is done by subtracting an offset, then dividing by the number of days per sol (1.0274912517).

MSD = (JDTT - 2405522.00288) / 1.0274912517
  = (2308806.29747 - 2405522.00288) / 1.0274912517
  = -94128.008628

Step 5. Calculate MTC (Coordinated Mars Time).

MTC = MSD - floor(MSD)
  = -94128.008628 - (-94129)
  = 0.991372 sols

(The function floor() rounds a value down to the nearest integer.)

This gives the time of day as a fraction of a sol. Serendipitously, the NVE occurred just before local midnight at 0° longitude.

As this time is just before midnight, it makes more sense to start the calendar at the midnight following this point in time, rather than the one preceding it.

To find the Julian Date of this midnight, we must add the time remaining in the sol to our original Julian Date for the NVE. Since Julian Dates are in days, the required value must be in days.

∆D = (1 - 0.991372) * 1.0274912517
  = 0.00886519 days

Adding this value gives the Julian Date of midnight, MTC, which is the start of sol 0.

JDepoch = JDUTC + ∆D
  = 2308806.29606 + 0.00886519
  = 2308806.30493

This Julian Date equals the Gregorian datetime of 1609-03-12 19:19:06 UTC.

Although this timestamp is shown accurate to the second, the calculation is not actually this accurate. For example, the calculation of ∆T for 17th-century dates is only an approximation. Also, since the length of both the day and the sol change over time, the number of days per sol also varies, whereas here a constant value of 1.0274912517 is used. The calculation of the precise moment of the epoch beginning will no doubt be refined in the future using more advanced methods.