## Technical Notes on Mars Solar Time as Adopted by the Mars24 Sunclock

By Michael Allison and Robert Schmunk
NASA Goddard Institute for Space Studies

(Updated Dec. 13, 2005)

These notes provide a technical brief on the definitions of numeric readouts in Mars24. Further details can be found in the appended references. A less technical account of solar time on Mars is provided in the 1998 article "Telling Time on Mars" on the NASA GISS website. Information about the specific controls and displays in Mars24 is provided in the accompanying User's Guide.

### Mars Solar Days and 24-hr Clock Convention

Following the long-standing practice originally adopted in 1976 by the Viking Lander missions, the daily variation of Mars solar time is reckoned in terms of a "24-hour" clock, representing a 24-part division of the planet's solar day, along with the traditional sexagesimal subdivisions of 60 minutes and 60 seconds. A Mars solar day has a mean period of 24 hours 39 minutes 35.244 seconds, and is customarily referred to as a "sol" in order to distinguish this from the roughly 3% shorter solar day on Earth. The Mars sidereal day, as measured with respect to the fixed stars, is 24h 37m 22.663s, as compared with 23h 56m 04.0905s for Earth.

### Mars Solar Seasons

The apparent seasonal advance of the Sun at Mars is commonly measured in terms of the areocentric longitude Ls, as referred to the planet's vernal equinox (the ascending node of the apparent seasonal motion of the Sun on the planet's equator). As defined, Ls = 0°, 90°, 180°, and 270° indicate the Mars northern hemisphere vernal equinox, summer solstice, autumnal equinox, and winter solstice, respectively.

In terms of Ls, the seasonally variable, planet-centered solar declination d equals arcsin [(sin e)(sin Ls)], where the obliquity e is the inclination of the planet's spin axis with respect to the plane of its orbit. For an accurate account of the solar illumination relative to the plane of a locally flat surface, the solar declination can be corrected for the small difference appropriate to the so-called planetographic measure of latitude on an oblate sphere, as it is in the Mars24 sunclock.

As a result of the Mars's orbital eccentricity, Ls advances somewhat unevenly with time, but can be efficiently evaluated as a trigonometric power series for the orbital eccentricity and the orbital mean anomaly measured with respect to the perihelion. The areocentric longitude at perihelion, Ls,p = 251°.000 + 0°.00645×(yr - 2000), indicates a near alignment of the planet's closest approach to the Sun in its orbit with its winter solstice season, as related to the occasional onset of global dust storms within the advance of this season.

### Mars Orbit Periods

The period for the repetition of the planet-centered measure of mean Solar longitude is referred to as the tropical year. (This period is linked to the rate of advance of the "Fictitious Mean Sun," as discussed below.) The Mars tropical year is 686.9726 day or 668.5921 sol. For comparison, the Mars sidereal year, as measured with respect to the fixed stars, is 668.5991 sol. The difference between these values results from the precession of the planet's spin axis.

The mean interval between the repetition of the planet's perihelion passage, or anomalistic year, is 668.6146 sol, and corresponds to the rate of advance of the planet's orbital mean anomaly. The mean repetition period for a particular solar season varies with the Ls. The mean repetition intervals for the vernal equinox, summer solstice, autumnal equinox, and winter solstice on Mars are 668.5906 sol, 668.5880 sol, 668.5940 sol, and 668.5958 sol, respectively, and the average of these is just the tropical year.

### Mean and True Solar Time

Also as a result of a planet's orbital eccentricity, as well as its obliquity, there is a seasonally variable discrepancy between the even advance of an artificially defined Mean Solar Time and of the True Solar Time corresponding to the actual planet-centered position of the Sun in its sky. Following the conventional usage of terrestrial timekeeping, Mean Solar Time on Mars has been defined in reference to the so-called Right Ascension of the Fictitious Mean Sun (FMS). As defined, the FMS is the angle between the planet's vernal equinox, measured along the plane of its equator, and an artificially defined "dynamical mean Sun" advancing at a rate corresponding to the planet's solar tropical year (i.e., the Mars FMS advances at a rate of 360°/686.9726 day or 0.5240384°/day). Its numerical value (to within an arbitrary multiple of 360°) is just the sum of the orbital mean anomaly, M, and the areocentric longitude at perihelion, Ls,p. The FMS at Mars was evaluated by Allison and McEwen (2000) (hereafter "AM2000") as a mean fit to an accurate calculation of the areocentric longitude over 134 Mars orbits (for the years 1874-2127), adjusted in its angular placement by the (~0°.0046) solar aberration. This evaluation has been adopted by the Mars Exploration Rover project for its definition of Mars Mean Solar Time (cf. Roncoli et al., 2002).

The difference between the True Solar Time (TST) and the Mean Solar Time (MST), equivalent in the corresponding angular measure to the difference between the right ascensions of the FMS and the true Sun, is referred to as the Equation of Time (EOT). For Earth, the EOT varies between -14.2min and +16.3min. Mars, with its more than five times larger orbital eccentricity, has an EOT varying between -51.1min and +39.9min. The parametric plot of the EOT vs. the solar declination is called the solar analemma. For Earth, this takes the form of the figure-8 pattern marked on some sundials and globes (for the latter typically in the empty space of the South Pacific). For Mars, the analemma assumes the shape of a raindrop or mis-shapen pear.

### Local and "Zonal" Times

In the mid-1800s the use of locally measured and defined time on Earth was gradually supplanted by the use of time zones in order to facilitate standardization of railroad schedules and, to a lesser extent, of recording scientific observations. This process culminated in 1884 in an international conference which created the global system of time zones and specified the longitude of Greenwich as the prime meridian. Each zone is approximately 15° wide, the exact width and shape being subject to political boundaries and significant geographic features, and within each clocks are referenced to the same hour.

The prime meridian of Mars is defined by the location of the crater Airy-0 (De Vaucoulers et al., 1973), named in honor of the British astronomer George Biddel Airy, who built the telescope at Greenwich whose location defines the prime meridian on Earth. Although it might be tempting to refer to "standard" time on the Mars prime meridian as "Airy Mean Time" (AMT) in analogy to Earth's "Greenwich Mean Time" (GMT), the latter term has been supplanted by Coordinated Universal Time (UTC) in international timekeeping services. Mars24 therefore refers to the Mean Solar Time on the Mars prime meridian as Coordinated Mars Time, or MTC, by analogy to the terrestrial UTC.

Mars24 includes the option to display the local time at the selected location in terms of similarly constructed "Mars time zones". We have defined these Mars time zones to be exactly 15° wide and centered on successive 15° multiples of longitude, at 0°, 15°, 30°, etc. We have not attempted to name these time zones, as for example "Olympus Standard Time", but do identify their read-out with a suffix indicating the timezone offset. Thus, for the the case of Olympus Mons the timezone identifier is "MTC-9", or nine hours behind MTC.

### Mission Times

Each landed mission on Mars has adopted a different reference for its solar time keeping.

The Viking Lander "Sol Numbers" were reckoned from a zero starting point at each of the local true solar midnights immediately prior to their touchdowns. Time tags for the Mars Pathfinder Lander were referenced with respect to the local true solar time, with elapsed sols reckoned from the local true solar midnight preceding its landing, but designated with a starting number "1" rather than zero. Conversion formulae are provided by AM2000.

The derivation of mission time adopted by the Mars Exploration Rover Project by mission controllers at the NASA Jet Propulsion Laboratory is an offset modification of an evenly advancing mean solar time, based on its definition by Roncoli et al. (2002). For MER-A Spirit, the difference between lander mission time and local mean solar time is more than 41 minutes; for MER-B Opportunity the difference is more than 37 minutes. At approximately the middle of each of the MER A and B nominal missions (i.e., the 45th sol after landing), lander mission time should align with local true solar time to within 30 seconds. As with Pathfinder, "Sol 1" for each MER lander denotes the solar day on which it touched down on the Martian surface.

### Mars Sol Date (MSD)

Although numerous month-year calendars have been proposed for Mars, we have not attempted to support any of these as a feature of Mars24. We have, however, included in the Mars24 displays the "Mars Sol Date" (MSD) defined by AM2000. This represents a sequential count of Mars solar days elapsed since 1873 December 29 at approximately Greenwich noon (Julian Day 2405522.0). This epoch was prior to the great 1877 perihelic opposition of Mars and precedes nearly all detailed observations of temporal changes on the planet. It corresponds to a Mars Ls of 277°, approximately the same planetocentric solar longitude as that for the Earth on the same date. MSD 44796.0 is approximately coincident with 2000 January 6.0, at a near-coincidence of prime meridian midnights on the two planets and a repetition of Mars Ls = 277°. The period 44796 sols also represents a near commensurability of 126 Julian years and 67 Mars tropical revolutions. In principle, the MSD could be used as a coherent sol-date reference for a variety of Mars missions.

### Mars24 Accuracy

Mars24 uses the short-series representation of the seven-largest short-period planetary perturbations of the Mars orbital longitude specified by AM2000, as adapted from Simon et al. (1994). Detailed comparisons with an accurate ephemeris suggest that the maximum error in the calculated Ls by the adopted algorithm is 0°.008 over ±100 years of J2000. According to the implicit dependence of the calculated EOT on the Ls, the resulting True Solar Time (TST) can be estimated to be in error by as much as 3sec. The error in the currently coded conversion between TT and UTC is itself in error for some periods in the post-1975 era by as much as 3sec. There is also as yet a small (~0°.004) uncertainty in the inertial location of the Mars prime meridian as defined by the crater Airy-0 (Duxbury et al., 2001).

Of course, the calculation of Local (True or Mean) solar time cannot be any more accurate than the longitudinal placement of the local point of interest! Predicted solar times for a given lander location may therefore need to be revised as improved knowledge of their locations becomes available. Although the precide coordinates of the two MER landers were obtained quickly after their landing, disputes as to the precise locations of the two Viking landers continued for more than 20 years.

Mars24's estimate of times of local sunrise and sunset, as well as times of Earthrise and Earthset, requires correct input of planetographic latitude. Even so, the results may be in error due to local topography and atmospheric refraction. Comparison to known times for a limited number of solar and Earth rise and set events at Mars lander sites suggest that absent topographic and atmospheric effects, the error in calculating these times is less than 30 seconds.

### Mars24 Update Summary

The current version of Mars24 is posted in the Software Tools section of the NASA GISS website. Interested users should revisit the website periodically to check for updates. Alternatively, you may e-mail the author and request to be added to the update announcements mailing list. (The author also welcomes e-mail from users willing to beta test forthcoming versions.)

A brief history of significant timekeeping modifications to the code is summarized as follows:

- April 1998: Version 1.0 applet released, using timing formulae published in Allison (1997). An outgrowth of an earlier Mars Pathfinder clock applet, Version 1.0 allowed lookup of the time anywhere on the planet and displayed a simple equirectangular sunclock.
- January 2000: Version 2.0 released, using formulae subsequently published in AM2000.
- September 2000: Version 3.0 released.
- January 2002: Version 4.0 released, using a revised specification of the inertial location of the Mars prime meridian, replacing that given in Table 4 of AM2000, consistent with the recommendations of Seidelmann et al. (2000, 2002).
- July 2003: Version 5.0 released. First stand-alone application version.
- August 2003: The conversion between the terrestrial time TT and Universal Coordinated Time UTC as given by Equation 27 of AM2000 was modified for greater accuracy with a fourth-order polynomial representation, accurate to within 3 s for 1975-2005.
- October 2003: Calculation of Mars' heliocentric longitude and latitude refined, which in turn allowed for calculation of the light-distance between Earth and Mars to within one second.
- December 2003: Calculation of local sunrise and sunset times added.
- January 2004: Calculation of sub-Earth point added.
- February 2004: Calculation of local earthrise and earthset times added.
- November 2005: Version 5.5 released. Analemma plot added.

### Contacts

The NASA GISS Mars24 application was written by Dr. Robert B. Schmunk. If you wish to be notified when new versions of Mars24 are released or if you would like to report a bug, please contact:

> Robert B. Schmunk
> NASA Goddard Institute for Space Studies
> 2880 Broadway
> New York, NY 10025 USA
> rschmunk@giss.nasa.gov

Technical questions related to the definition of Mars Solar Time and the mathematical algorithms adopted for its calculation by Mars24 may be directed to:

> Michael D. Allison
> NASA Goddard Institute for Space Studies
> 2880 Broadway
> New York, NY 10025 USA
> mallison@giss.nasa.gov

### Acknowledgments

We thank members of the Mars Climate Orbiter and Mars Exploration Rover Projects at NASA's Jet Propulsion Laboratory for their encouragement over many years on behalf of the development of our solar timing algorithms and Mars24 sunclock.

The base map image used by Mars24 in its sunclock is derived from an image created by the Mars Global Surveyor MOLA and MOC teams for the National Geographic Society. See the MOLA science webpage at NASA Goddard Space Flight Center.

Mars24 includes software developed by the JDOM Project and which is copyright 2000-2003 Jason Hunter and Brett McLaughlin. For more information about JDOM, including source code and conditions of use, please see the JDOM homepage.

Mars24 includes software developed as part of the GNU JAXP project, including the AElfred2 parser. For more information about GNU JAXP, including source code and conditions of use, please see the GNU JAXP homepage.

Mars24 uses QuantizeFilter by Jerry Huxtable. For more information, including source code, please see the JH Labs website.

### References

- Allison, M. 1997. Accurate analytic representations of solar time and seasons on Mars with applications to the Pathfinder/Surveyor missions. Geophys. Res. Lett. 24, 1967-1970, doi:10.1029/97GL01950.
- Allison, M., and M. McEwen 2000. A post-Pathfinder evaluation of aerocentric solar coordinates with improved timing recipes for Mars seasonal/diurnal climate studies. Planet. Space Sci. 48, 215-235, doi:10.1016/S0032-0633(99)00092-6.
- De Vaucouleurs, G., M.E. Davies, and F.M. Sturms 1973. Mariner 9 areographic coordinate system. J. Geophys. Res. 78, 4395-4404.
- Duxbury, T., R.L. Kirk, B.A. Archinal 2001. Mars Geodesy/Cartography Working Group recommendations on Mars cartographic constants and coordinate systems. International Society for Photogrammatry and Remote Sensing (ISPRS) Working Group Meeting IV/9.
- Folkner, W.M., C.F. Yoder, D.N. Yuan, E.M. Standish, and R.A. Preston 1997. Interior structure and seasonal mass redistribution of Mars from radio tracking of Mars Pathfinder. Science 278, 1749-1751, doi:10.1126/science.278.5344.1749.
- Roncoli, R., B. Strauss, and D. Highsmith 2002. Mars Exploration Rover Project planetary constants and models - Version 2. Interoffice Memorandum IOM 312.F-02-003. NASA Jet Propulsion Laboratory
- Seidelmann, P.K., V K. Abalaki, M. Burs, M E. Davies, C. de Bergh, J.H. Lieske, J. Oberst, J.L. Simon, E.M. Standish, P. Stooke, and P.C. Thomas 2002. Report of the IAU/IAG Working Group on cartographic coordinates and rotational elements of the planets and satellites: 2000. Celest. Mech. Dynam. Astron. 82, 83-110, doi:10.1023/A:1013939327465. (Also cf., Corrigendum, 2002. Celest. Mech. Dynam. Astron. 84, 429, doi:10.1023/A:1021101631766.)
- Simon, J.L., P. Bretagnon, J. Chapront, M. Chapront-Touzé, G. Francour, and J. Lasker 1994. Numerical expressions for precession formulae and mean elements for the Moon and planets. Astron. Astrophys. 282, 663-683.