Timekeeping on Mars

Martian season lengths and time as compared to seasons on the Earth.

Various schemes have been used or proposed for timekeeping on the planet Mars independently of Earth time and calendars.

Mars has an axial tilt and a rotation period similar to those of Earth. Thus it experiences seasons of spring, summer, autumn and winter much like Earth, and its day is about the same length. Its year is almost twice as long as Earth's, and its orbital eccentricity is considerably larger, which means among other things that the lengths of various Martian seasons differ considerably, and sundial time can diverge from clock time more than on Earth.

Time of day

The average length of a Martian sidereal day is 24h 37m 22.663s (88,642.66300 seconds based on SI units), and the length of its solar day (often called a sol) is 24h 39m 35.244147s (88,775.244147 seconds). The corresponding values for Earth are 23h 56m 4.0916s and 24h 00m 00.002s, respectively. This yields a conversion factor of 1.02749125170 days/sol. Thus Mars' solar day is only about 2.7% longer than Earth's.

A convention used by spacecraft lander projects to date has been to keep track of local solar time using a 24-hour "Mars clock" on which the hours, minutes and seconds are 2.7% longer than their standard (Earth) durations. For the Mars Pathfinder, Mars Exploration Rover, Phoenix, and Mars Science Laboratory missions, the operations team has worked on "Mars time", with a work schedule synchronized to the local time at the landing site on Mars, rather than the Earth day. This results in the crew's schedule sliding approximately 40 minutes later in Earth time each day. Wristwatches calibrated in Martian time, rather than Earth time, were used by many of the MER team members.[1][2]

Local solar time has a significant impact on planning the daily activities of Mars landers. Daylight is needed for the solar panels of landed spacecraft. Its temperature rises and falls rapidly at sunrise and sunset because Mars does not have the Earth's thick atmosphere and oceans that buffer such fluctuations.

Alternative clocks for Mars have been proposed, but no mission has chosen to use such. These include a metric time schema, with "millidays" and "centidays", and an extended day which uses standard units but which counts to 24h 39m 35s before ticking over to the next day.

The analemma for Mars

As on Earth, on Mars there is also an equation of time that represents the difference between sundial time and uniform (clock) time. The equation of time is illustrated by an analemma. Because of orbital eccentricity, the length of the solar day is not quite constant. Because its orbital eccentricity is greater than that of Earth, the length of day varies from the average by a greater amount than that of Earth, and hence its equation of time shows greater variation than that of Earth: on Mars, the Sun can run 50 minutes slower or 40 minutes faster than a Martian clock (on Earth, the corresponding figures are 14m 22s slower and 16m 23s faster).

Mars has a prime meridian, defined as passing through the small crater Airy-0. However, Mars does not have time zones defined at regular intervals from the prime meridian, as on Earth. Each lander so far has used an approximation of local solar time as its frame of reference, as cities did on Earth before the introduction of standard time in the 19th century. (The two Mars Exploration Rovers happen to be approximately 12 hours and one minute apart.)

Note that the modern standard for measuring longitude on Mars is "planetocentric longitude", which is measured from 0°–360° East and measures angles from the center of Mars. The older "planetographic longitude" was measured from 0°–360° West and used coordinates mapped onto the surface.[3]

Coordinated Mars Time (MTC)

MTC is a proposed Mars analog to Universal Time (UT1) on Earth. It is defined as the mean solar time at Mars's prime meridian. The prime meridian was first proposed by German astronomers Wilhelm Beer and Johann Heinrich Mädler in 1830 as marked by the fork in the albedo feature later named Sinus Meridiani by Italian astronomer Giovanni Schiaparelli. This convention was readily adopted by the astronomical community, the result being that Mars had a universally accepted prime meridian half a century before the International Meridian Conference of 1884 established one for Earth. The definition of the Martian prime meridian has since been refined on the basis of spacecraft imagery as the center of the crater Airy-0 in Terra Meridiani. The name "MTC" is intended to parallel the Terran Coordinated Universal Time (UTC), but this is somewhat misleading: what distinguishes UTC from other forms of UT is its leap seconds, but MTC does not use any such scheme. MTC is more closely analogous to UT1.

Use of the term "MTC" as the name of a planetary standard time for Mars first appeared in the Mars24[4] sunclock coded by the NASA Goddard Institute for Space Studies. It replaced Mars24's previous use of the term "Airy Mean Time" (AMT), which was a direct parallel of Greenwich Mean Time (GMT). In an astronomical context, "GMT" is a deprecated name for Universal Time, or sometimes more specifically for UT1.

AMT has not yet been employed in official mission timekeeping. This is partially attributable to uncertainty regarding the position of Airy-0 (relative to other longitudes), which meant that AMT couldn't be realized as accurately as local time at points being studied. At the start of the Mars Exploration Rover missions, the positional uncertainty of Airy-0 corresponded to roughly a 20-second uncertainty in realizing AMT.

Time zones

Each lander mission so far has used its own time zone, corresponding to average local solar time at the landing location. Of the six successful Mars landers to date, five employed offsets from local mean solar time (LMST) for the lander site while the sixth (Mars Pathfinder) used local true solar time (LTST).[5][6]

Pathfinder

Mars Pathfinder used the local apparent solar time at its location of landing. Its time zone was AAT-02:13:01, where "AAT" is Airy Apparent Time, meaning apparent solar time at Airy-0.

Spirit and Opportunity

The two Mars Exploration Rovers do not use precisely the LMST of the landing points. For mission operations purposes, they defined a time scale that would match the clock used for the mission to the apparent solar time about halfway through the nominal 90-sol prime mission. This is referred to in mission planning as "Hybrid Local Solar Time" (HLST). The time scales are uniform in the sense of mean solar time (they are actually mean time of some longitude), and are not adjusted as the rovers travel. (The rovers have traveled distances that make a few seconds difference to local solar time.) The HLST of Spirit is AMT+11:00:04 whereas the LMST at its landing site is AMT+11:41:55. The HLST of Opportunity is AMT-01:01:06 whereas the LMST at its landing site is AMT-00:22:06. Neither rover is likely to ever reach the longitude at which its mission time scale matches local mean time. For science purposes, Local True Solar Time is used.

Curiosity

The Curiosity Rover opts for the use of Local Mean Solar Time at its landing location. The local time of the Curiosity Rover is AMT+09:09:46.

Other

With the location of Airy-0 now known much more precisely than when these missions landed, it is technically feasible for future missions to use a convenient offset from Airy Mean Time, rather than completely non-standard time zones. Although there are currently no official timezone system for Mars other than AMT/MTC, Ares Astronautics has also put forward a solution to this problem based upon the USGS Martian Quadrangles officially known as the Arean Time Zone System.[7]

Sols

The term sol is used by planetary astronomers to refer to the duration of a solar day on Mars.[8] A mean Martian solar day, or "sol", is 24 hours, 39 minutes, and 35.244 seconds.[6]

When a spacecraft lander begins operations on Mars, the passing Martian days (sols) are tracked using a simple numerical count. The two Viking missions, Mars Phoenix and the Mars Science Laboratory rover Curiosity count the sol on which each lander touched down as "Sol 0"; Mars Pathfinder and the two Mars Exploration Rovers instead defined touchdown as "Sol 1".[9]

Although lander missions have twice occurred in pairs, no effort was made to synchronize the sol counts of the two landers within each pair. Thus, for example, although Spirit and Opportunity were sent to operate simultaneously on Mars, each counted its landing date as "Sol 1", putting their calendars approximately 21 sols out of sync. Spirit and Opportunity differ in longitude by 179 degrees, so when it is daylight for one it is night for the other, and they carried out activities independently while both were operational.

On Earth, astronomers often use Julian Dates – a simple sequential count of days – for timekeeping purposes. A proposed counterpart on Mars is the Mars Sol Date (MSD), which is a running count of sols since December 29, 1873 (coincidentally the birth date of astronomer Carl Otto Lampland). Another proposal suggests a start date (or epoch) in the year 1608 (invention of the telescope). Either choice is intended to ensure that all historically recorded events related to Mars occur after it. The Mars Sol Date is defined mathematically as MSD = (Julian Date using International Atomic Time - 2451549.5 + k)/1.02749125 + 44796.0, where k is a small correction of approximately 0.00014 d (or 12 s) due to uncertainty in the exact geographical position of the prime meridian at Airy-0 crater.

The word "yestersol" was coined by the NASA Mars operations team early during the MER mission to refer to the previous sol (the Mars version of "yesterday"), and came into fairly wide use within that organization during the Mars Exploration Rover Mission of 2003.[10] It was even picked up and used by the press. Other neologisms include "tosol" (for "today" on Mars), as well as one of three Mars versions of "tomorrow": "nextersol", "morrowsol", or "solmorrow".[11] NASA planners coined the term "soliday" at least as far back as 2012 to refer to days off due to time phasing or the synching of planetary schedules.[12]

Martian year

The length of time for Mars to complete one orbit around the Sun is its sidereal year, and is about 686.98 Earth solar days, or 668.5991 sols. Because of the eccentricity of Mars' orbit, the seasons are not of equal length. Assuming that seasons run from equinox to solstice or vice versa, the season Ls 0 to Ls 90 (northern-hemisphere spring / southern-hemisphere autumn) is the longest season lasting 194 Martian sols, and Ls 180 to Ls 270 (northern hemisphere autumn / southern-hemisphere spring) is the shortest season, lasting only 142 Martian sols.[13] One commonly used system in the scientific literature denotes year number relative to Mars Year 1 (MY1) beginning with the northern Spring equinox of April 11, 1955.[14]

As on Earth, the sidereal year is not the quantity that is needed for calendar purposes. Rather, the tropical year would be likely to be used because it gives the best match to the progression of the seasons. It is slightly shorter than the sidereal year due to the precession of Mars' rotational axis. The precession cycle is 93,000 Martian years (175,000 Earth years), much longer than on Earth. Its length in tropical years can be computed by dividing the difference between the sidereal year and tropical year by the length of the tropical year.

Tropical year length depends on the starting point of measurement, due to the effects of Kepler's second law of planetary motion. It can be measured in relation to an equinox or solstice, or can be the mean of various possible years including the March (northward) equinox year, June (northern) solstice year, the September (southward) equinox year, the December (southern) solstice year, and other such years. The Gregorian calendar uses the March equinox year.

On Earth, the variation in the lengths of the tropical years is small, but on Mars it is much larger. The northward equinox year is 668.5907 sols, the northern solstice year is 668.5880 sols, the southward equinox year is 668.5940 sols, and the southern solstice year is 668.5958 sols. Averaging over an entire orbital period gives a tropical year of 668.5921 sols. (Since, like Earth, the northern and southern hemispheres of Mars have opposite seasons, equinoxes and solstices must be labelled by hemisphere to remove ambiguity.)

Martian calendars in science

Long before mission control teams on Earth began scheduling work shifts according to the Martian sol while operating spacecraft on the surface of Mars, it was recognized that humans probably could adapt to this slightly longer diurnal period. This suggested that a calendar based on the sol and the Martian year might be a useful timekeeping system for astronomers in the short term and for explorers in the future. For most day-to-day activities on Earth, people do not use Julian days, as astronomers do, but the Gregorian calendar, which despite its various complications is quite useful. It allows for easy determination of whether one date is an anniversary of another, whether a date is in winter or spring, and what is the number of years between two dates. This is much less practical with Julian days count. For similar reasons, if it is ever necessary to schedule and co-ordinate activities on a large scale across the surface of Mars it would be necessary to agree on a calendar.

American astronomer Percival Lowell expressed the time of year on Mars in terms of Mars dates that were analogous to Gregorian dates, with 20 March, 21 June, 22 September, and 21 December marking the southward equinox, southern solstice, northward equinox, and northern solstice, respectively; Lowell's focus was on the southern hemisphere of Mars because it is the hemisphere that is more easily observed from Earth during favorable oppositions. Lowell's system was not a true calendar, since a Mars date could span nearly two entire sols; rather it was a convenient device for expressing the time of year in the southern hemisphere in lieu of heliocentric longitude, which would have been less comprehensible to a general readership.[15]

Italian astronomer Mentore Maggini's 1939 book describes a calendar developed years earlier by American astronomers Andrew Ellicott Douglass and William H. Pickering, in which the first nine months contain 56 sols and the last three months contain 55 sols. Their calendar year begins with the northward equinox on 1 March, thus imitating the original Roman calendar. Other dates of astronomical significance are: northern solstice, 27 June; southward equinox, 36 September; southern solstice, 12 December; perihelion, 31 November; and aphelion, 31 May. Pickering's inclusion of Mars dates in a 1916 report of his observations may have been the first use of a Martian calendar in an astronomical publication.[16] Maggini states: "These dates of the Martian calendar are frequently used by observatories...."[17] Despite his claim, this system eventually fell into disuse, and in its place new systems were proposed periodically which likewise did not gain sufficient acceptance to take permanent hold.

In 1936, when the calendar reform movement was at its height, American astronomer Robert G. Aitken published an article outlining a Martian calendar. In each quarter there are three months of 42 sols and a fourth month of 41 sols. The pattern of seven-day weeks repeats over a two-year cycle, i.e., the calendar year always begins on a Sunday in odd-numbered years, thus effecting a perpetual calendar for Mars.[18]

Whereas previous proposals for a Martian calendar had not included an epoch, American astronomer I. M. Levitt developed a more complete system in 1954. In fact, Ralph Mentzer, an acquaintance of Levitt's who was a watchmaker for the Hamilton Watch Company, built several clocks designed by Levitt to keep time on both Earth and Mars. They could also be set to display the date on both planets according to Levitt's calendar and epoch (the Julian day epoch of 4713 BCE).[19][20]

Charles F. Capen included references to Mars dates in a 1966 Jet Propulsion Laboratory technical report associated with the Mariner 4 flyby of Mars. This system stretches the Gregorian calendar to fit the longer Martian year, much as Lowell had done in 1895, the difference being that 20 March, 21 June, 22 September, and 21 December marks the northward equinox, northern solstice, southward equinox, southern solstice, respectively.[21] Similarly, Conway B. Leovy et al. also expressed time in terms of Mars dates in a 1973 paper describing results from the Mariner 9 Mars orbiter.[22]

British astronomer Sir Patrick Moore described a Martian calendar of his own design in 1977. His idea was to divide up a Martian year into 18 months. Months 6, 12 and 18, have 38 sols, while the rest of the months contain 37 sols.[23]

American aerospace engineer and political scientist Thomas Gangale first published regarding the Darian calendar in 1986, with additional details published in 1998 and 2006. It has 24 months to accommodate the longer Martian year while keeping the notion of a "month" that is reasonably similar to the length of an Earth month. On Mars, a "month" would have no relation to the orbital period of any moon of Mars, since Phobos and Deimos orbit in about 7 hours and 30 hours respectively. However, Earth and Moon would generally be visible to the naked eye when they were above the horizon at night, and 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.[24][25][26]

Czech astronomer Josef Šurán offered a Martian calendar design in 1997, in which a common year has 672 Martian days distributed into 24 months of 28 days (or 4 weeks of 7 days each); in skip years an entire week at the end of the twelfth month is omitted.[27]

Martian time in fiction

Allusions to timekeeping on Mars in fiction tend to be cursory, meant only as part as the exotic ambience of another world.

The first known reference to time on Mars appears in Percy Greg's 1880 novel Across the Zodiac. The primary, secondary, tertiary, and quaternary divisions of the sol are based on the number 12. Sols are numbered 0 through the end of the year, with no additional structure to the calendar. The epoch is "the union of all races and nations in a single State, a union which was formally established 13,218 years ago".[28]

Edgar Rice Burroughs described the divisions of the sol into zodes, xats, and tals;[29] he may have been the first to make the mistake of describing the Martian year as lasting 687 Martian days, but he was far from the last.[30]

In Robert A. Heinlein's 1949 novel Red Planet, humans living on Mars use a 24-month calendar, alternating between familiar Earth months and newly created months such as Ceres and Zeus. For example, Ceres comes after March and before April, while Zeus comes after October and before November.[31]

Arthur C. Clarke's 1951 novel The Sands of Mars mentions in passing that "Monday followed Sunday in the usual way" and "the months also had the same names, but were fifty to sixty days in length".[32]

In H. Beam Piper's 1957 short story "Omnilingual", the Martian calendar and the periodic table are the keys to archaeologists' deciphering of the records left by the long dead Martian civilization.[33]

Kurt Vonnegut's 1959 novel The Sirens of Titan describes a Martian calendar divided into twenty-one months: "twelve with thirty sols, and nine with thirty-one sols", for a total of only 639 sols.[34]

D. G. Compton states in his 1966 novel Farewell, Earth's Bliss, during the prison ship's journey to Mars, "Nobody on board had any real idea how the people in the settlement would have organised their six-hundred-and-eighty-seven-day year".[35]

In Kim Stanley Robinson's Mars Trilogy, clocks retain Earth-standard seconds, minutes, and hours, but freeze at midnight for 39.5 minutes. As the fictional colonization of Mars progresses, this "timeslip" becomes a sort of witching hour, a time when inhibitions can be shed, and the emerging identity of Mars as a separate entity from Earth is celebrated. (It is not said explicitly whether this occurs simultaneously all over Mars, or at local midnight in each longitude.) Also in the Mars Trilogy, the calendar year is divided into twenty-four months. The names of the months are the same as the Gregorian calendar, except for a "1" or "2" in front to indicate the first or second occurrence of that month (for example, 1 January, 2 January, 1 February, 2 February).

In the manga and anime series Aria by Kozue Amano, set on a terraformed Mars, the calendar year is also divided into twenty-four months. Following the modern Japanese calendar, the months are not named but numbered sequentially, running from 1st Month to 24th Month.[36]

The Darian calendar is mentioned in several works of fiction set on Mars:

In both The Martian (Weir novel), a 2011 science fiction novel by Andy Weir, and The Martian (film), a 2015 American film directed by Ridley Scott, based on the Weir novel, sols are counted and referenced frequently.

In Ian McDonald's Desolation Road, set on a terraformed Mars (referred to by the book's characters as "Ares"), characters follow an implied 24-month calendar whose months are portmanteaus of Gregorian months, such as "Julaugust," "Augtember" and "Novodecember."

Formulas to compute MSD and MTC

MSD can be computed from the Julian date referred to Terrestrial Time (TT), as[37]

MSD = (JDTT − 2405522.0028779) / 1.0274912517

Terrestrial time, however, is not as easily available as Coordinated Universal Time (UTC). TT can be computed from UTC by first adding the difference TAI−UTC, which is a positive integer number of seconds occasionally updated by the introduction of leap seconds (see current number of leap seconds), then adding the constant difference TT−TAI = 32.184 s. This leads to the following formula giving MSD from the UTC-referred Julian date:

MSD = (JDUTC + (TAI−UTC)/86400 − 2405522.0025054) / 1.0274912517

where the difference TAI−UTC is in seconds. JDUTC can in turn be computed from any epoch-based time stamp, by adding the Julian date of the epoch to the time stamp in days. For example, if t is a Unix timestamp in seconds, then

JDUTC = t / 86400 + 2440587.5

It follows, by a simple substitution:

MSD = (t + (TAI−UTC)) / 88775.244147 + 34127.2954262

MTC is simply the fractional part of MSD, in hours, minutes and seconds:[6]

MTC = (MSD mod 1) × 24 h

For example, at the time this page was last generated (23 Aug 2017, 03:04:12 UTC):

See also

References

  1. ↑ "Watchmaker With Time to Lose". JPL Mars Exploration Rovers. 2014. Retrieved 22 January 2015.
  2. ↑ Redd, Nola Taylor (18 March 2013). "After Finding Mars Was Habitable, Curiosity Keeps Roving". space.com. Retrieved 22 January 2015.
  3. ↑ "Mars Express - Where is zero degrees longitude on Mars?". European Space Agency. 19 August 2004. Retrieved 13 July 2012.
  4. ↑ "Mars24 Sunclock - Time on Mars". NASA Goddard Institute for Space Studies. 5 August 2008. Retrieved 13 July 2012.
  5. ↑ Allison, M.; McEwen, M. (2000). "A post-Pathfinder evaluation of areocentric 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..
  6. 1 2 3 Allison, Michael (5 August 2008). "Technical Notes on Mars Solar Time". NASA Goddard Institute for Space Studies. Retrieved 13 July 2012.
  7. ↑ Little, Dean. "Arean Time Zone System". Ares Astronautics.
  8. ↑ "Opportunity's View, Sol 959 (Vertical)". NASA. Retrieved 13 July 2012.
  9. ↑ "Phoenix Mars Mission - Mission - Mission Phases - On Mars". Phoenix.lpl.arizona.edu. 29 February 2008. Retrieved 13 July 2012.
  10. ↑ Rusch, Elizabeth (2012). The Mighty Mars Rovers: The Incredible Adventures of Spirit and Opportunity. ISBN 9780547822808.
  11. ↑ Martínez-Frías, Jesús (28 September 2002). "Marte: 'yestersol', 'tosol' y 'solmorrow'" [Mars: 'yestersol', 'tosol', and 'solmorrow']. El Mundo (in Spanish). Madrid, Spain: Unidad Editorial S.A. Retrieved 23 April 2014.
  12. ↑ "MSL abbreviations and acronyms". an.rsl.wustl.edu. 31 October 2012. Retrieved 19 December 2015.
  13. ↑ J. Appelbaum and G. A. Landis, Solar Radiation on Mars-- Update 1991, NASA Technical Memorandum TM-105216, September 1991 (also published in Solar Energy, Vol. 50 No. 1 (1993)).
  14. ↑ Clancy, R. T.; Sandor, B. J.; Wolff, M. J.; Christensen, P. R.; Smith, M. D.; Pearl, J. C.; Conrath, B. J.; Wilson, R. J., (2000) An intercomparison of ground-based millimeter, MGS TES, and Viking atmospheric temperature measurements: Seasonal and interannual variability of temperatures and dust loading in the global Mars atmosphere, Journal of Geophysical Research, 105 (E4).
  15. ↑ Lowell, Percival. (1895-01-01). Mars. Houghton, Mifflin.
  16. ↑ Pickering, William H. (1916-01-01). "Report on Mars, No. 17." Popular Astronomy, Vol. 24, p.639.
  17. ↑ Maggini, Mentore. (1939-01-01). Il pianeta Marte. Scuola Tip. Figli Della Provvidenza.
  18. ↑ Aitken, Robert G. (1936-12-01). "Time Measures on Mars." Astronomical Society of the Pacific Leaflets, No. 95.
  19. ↑ Levitt, I. M. (1954-05-01). "Mars Clock and Calendar." Sky and Telescope, May, 1954, pp. 216-217.
  20. ↑ Levitt, I. M. (1956-01-01). A Space Traveller's Guide to Mars. Henry Holt.
  21. ↑ Capen, Charles F. (1966-01-01). "The Mars 1964-1965 Apparition." Technical Report 32-990. Jet Propulsion Laboratory, California Institute of Technology..
  22. ↑ Leovy, C. B., G. A. Briggs, B. A. Smith. (1973-07-10). "Mars atmosphere during the Mariner 9 extended mission: Television results." Journal of Geophysical Research, Volume 78, Issue 20.
  23. ↑ Moore, Patrick. (1977-01-01). Guide to Mars. Lutterworth Press.
  24. ↑ Gangale, Thomas. (1986-06-01). "Martian Standard Time". Journal of the British Interplanetary Society. Vol. 39, No. 6, p. 282-288.
  25. ↑ Gangale, Thomas. (1998-08-01). "The Darian Calendar". Mars Society. MAR 98-095. Proceedings of the Founding Convention of the Mars Society. Volume III. Ed. Robert M. Zubrin, Maggie Zubrin. San Diego, California. Univelt, Incorporated. 13-Aug-1998.
  26. ↑ Gangale, Thomas. (2006-07-01). "The Architecture of Time, Part 2: The Darian System for Mars." Society of Automotive Engineers. SAE 2006-01-2249.
  27. ↑ Šurán, Josef. (1997-06-01). "A Calendar for Mars." Planetary and Space Science, Vol. 45, No. 6. pp. 705-708.
  28. ↑ Greg, Percy. (1880-01-01). Across the Zodiac: The Story of a Wrecked Record. Trübner.
  29. ↑ Burroughs, Edgar Rice. (1913-01-01). The Gods of Mars. All-Story. January–May.
  30. ↑ Burroughs, Edgar Rice. (1913-12-01). The Warlord of Mars. All-Story Magazine, December, 1913-March, 1914.
  31. ↑ "Heinlein Concordance "Red Planet"". Heinlein Society. 2013. Retrieved 22 January 2015.
  32. ↑ Clarke, Arthur C. (1951-01-01). The Sands of Mars. Sidgwick & Jackson.
  33. ↑ Piper, H. Beam. (1957-02-01). "Omnilingual." Astounding Science Fiction, February.
  34. ↑ Vonnegut, Kurt. (1959-01-01). The Sirens of Titan. Delacorte.
  35. ↑ Compton. D. G. (1966-01-01). Farewell, Earth's Bliss. Hodder & Stoughton .
  36. ↑ Amano, Kozue (February 2008). "Navigation 06: My First Customer". Aqua volume 2. Tokyopop. p. 7. ISBN 978-1-4278-0313-9.
  37. ↑ This is a trivial simplification of the formula (JDTT − 2451549.5) / 1.0274912517 + 44796.0 − 0.0009626 given in Mars24 Algorithm and Worked Examples.
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