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(Updated May 20, 2008)
The calculations made by Mars24 to determine the time for a given location on Mars are primarily based on Allison and McEwen (2000) (henceforth AM2000). We also refer to Allison (1997) (henceforth A1997), of which AM2000 was a thorough update. However, some typographical errors appeared in the published version of AM2000, and some calculations have been revised since publication of that paper because of the availability of new data. (These updates may appear in a paper in preparation by Allison and Ferrier.)
Consequently, we provide here step-by-step documentation of the equations employed by Mars24 for users who wish to implement their own Martian timekeeping applications. At the end of this presentation, we also provide worked examples for verification of intermediate results.
Our Mars time calculations will use the parameter ΔtJ2000, the elapsed time in days since the J2000 epoch, i.e., 12:00 on Jan. 1, 2000 (TT). The following describes how we get there from a Java call which returns the system time. If one has an alternative scheme for obtaining ΔtJ2000, then these steps can be skipped.
A-1. Get a starting Earth time.
Mars24 is written in Java, so we use the
System.currentTimeMillis() method to find out the number of
milliseconds, millis, that have elapsed since 00:00:00 on Jan. 1, 1970
(i.e., the Unix epoch).
Unfortunately, most if not all Java implementations
(or the operating systems on which they run) do not keep track of
leap seconds, i.e.,
they keep time in UT rather than UTC. Consequently, when Mars24 calls
System.currentTimeMillis(), it assumes that
the value returned uses UT rather than UTC.
Only A-2 in the following steps explicitly uses millis. However, there are some displays in Mars24 which also use other readings based on the value of millis, so there is a possibility that if Mars24 is used on a computer and Java implementation which do keep track of leap seconds, display errors could result.
A-2. Convert millis to Julian Date (UT).
Although there's plenty of sample code available on-line which demonstrates how to convert a Gregorian calendar date to a Julian Date, we simply use the offset from a known, recent Julian Date. Again, we use the Unix epoch, 00:00:00 on Jan. 1, 1970.
JDUT = 2440587.5 + (millis / 8.64×107 ms/day)
A-3. Determine time offset from J2000 epoch (UT).
This step is optional; we only need to make this calculation if the date is before Jan. 1, 1972. Determine the elapsed time in Julian centuries since 12:00 on Jan. 1, 2000 (UT).
T = (JDUT - 2451545.0) / 36525.
A-4. Determine UTC to TT conversion. (Replaces AM2000, eq. 27)
Terrestrial Time (TT) advances at constant rate, as does UTC, but no leap seconds are inserted into it and so it gradually gets further ahead of UTC. The best way to determine the difference between TT and UTC is to consult a table of leap seconds. Alternatively, one could try to use an empirical formula.
In Mars24 we, oddly enough, use both methods. We use the USNO table for dates after Jan. 1, 1972, and a formula for dates prior to then. In consulting the USNO table, however, it is important to note that the table provides values for the TAI-UTC difference, where TAI is International Atomic Time. To obtain the TT-UTC difference, add 32.184 seconds to the value of TAI-UTC. For example, the USNO table indicates that on Jan. 1, 2006, the TAI-UTC value is 33.0 seconds, and thus, the value for TT-UTC on that date (and until the next date on which another leap second is added to the clock) would be 33.0s + 32.184s = 35.184s.
The formula applied for dates prior to Jan. 1, 1972, is similar to AM2000, eq. 27, but has been revised and includes additional terms:
TT - UTC = 64.184s + 59 s × T - 51.2 s × T2 - 67.1 s × T3 - 16.4 s × T4
(Note: Mars24 uses the USNO table which includes the leap second added Jan. 1, 2006. Obviously, then, it does not allow for any leap seconds which might be subsequently added. Bulletin C 33 from the IERS Earth Orientation Centre indicates this will not occur any earlier than Jan. 1, 2009.)
A-5. Determine Julian Date (TT).
JDTT = JDUT + [(TT - UTC) / 86400 s·day-1]
A-6. Determine time offset from J2000 epoch (TT). (AM2000, eq. 15)
ΔtJ2000 = JDTT - 2451545.0
Now we turn our attention to Mars, first determining some orbital paramaters.
B-1. Determine Mars mean anomaly. (AM2000, eq. 16)
M = 19.3870° + 0.52402075° ΔtJ2000
B-2. Determine angle of Fiction Mean Sun. (AM2000, eq. 17)
αFMS = 270.3863° + 0.52403840° ΔtJ2000
B-3. Determine perturbers. (AM2000, eq. 18)
PBS = Σ(i=1,7) Ai cos [ (0.985626° ΔtJ2000 / τi) + φi]
where 0.985626° = 360° / 365.25, and
| i | Ai
| τi |
φi |
|
|---|---|---|---|
| 1 | 0.0071 | 2.2353 | 49.409 |
| 2 | 0.0057 | 2.7543 | 168.173 |
| 3 | 0.0039 | 1.1177 | 191.837 |
| 4 | 0.0037 | 15.7866 | 21.736 |
| 5 | 0.0021 | 2.1354 | 15.704 |
| 6 | 0.0020 | 2.4694 | 95.528 |
| 7 | 0.0018 | 32.8493 | 49.095 |
B-4. Determine Equation of Center. (Bracketed term in AM2000, eqs. 19 and 20)
The equation of center is the true anomaly minus mean anomaly.
ν - M = (10.691° + 3.0° × 10-7 ΔtJ2000) sin M + 0.623° sin 2M + 0.050° sin 3M + 0.005° sin 4M + 0.0005° sin 5M + PBS
B-5. Determine areocentric solar longitude. (AM2000, eq. 19)
Ls = αFMS + (ν - M)
C-1. Determine Equation of Time. (AM2000, eq. 20)
EOT = 2.861° sin 2Ls - 0.071° sin 4Ls + 0.002° sin 6Ls - (ν - M)
The above result for EOT is in degrees. Multiply by (24 h / 360°) = (1 h / 15°) to obtain the result in hours.
C-2. Determine Coordinated Mars Time. (AM2000, eq. 22, modified)
This is the mean solar time at Mars' prime meridian.
MTC = mod24 { 24 h × ( [(JDTT - 2451549.5) / 1.027491252] + 44796.0 - 0.00096 ) }
The function modX indicates a re-setting of the function parameter, a cyclical value, to a value between 0 and X. In this case, we apply mod24 to indicate that values outside the range 0-24 should be re-set to be within that range, e.g., mod24 (30) = 6.
C-3. Determine Local Mean Solar Time.
The Local Mean Solar Time for a given planetographic longitude, Λ, in degrees west, is easily determined by offsetting from the mean solar time on the prime meridian.
LMST = MTC - Λ (24 h / 360°) = MTC - Λ (1 h / 15°)
C-4. Determine Local True Solar Time. (AM2000, eq. 23)
LTST = LMST + EOT (24 h / 360°) = LMST + EOT (1 h / 15°)
C-5. Determine subsolar longitude
Λs = (MTC + EOT) (360° / 24 h) + 180° = (MTC + EOT) (15°/h) + 180°
D-1. Determine solar declination (planetographic). (A1997, eq. 5)
δs = arcsin {0.42565 sin Ls)} + 0.25° sin Ls
D-2. Determine heliocentric distance. (AM2000, eq. 25, corrected)
RM = 1.523679 × (1.00436 - 0.09309 cos M - 0.004336 cos 2M - 0.00031 cos 3M - 0.00003 cos 4M)
D-3. Determine heliocentric longitude. (AM2000, eq. 26)
lM = Ls + 85.061° - 0.015° sin (71° + 2Ls) - 5.5°×10-6 ΔtJ2000
D-4. Determine heliocentric latitude.
bM = - (1.8497° - 2.23°×10-5 ΔtJ2000) sin (Ls - 144.50° + 2.57°×10-6 ΔtJ2000)
D-5. Determine local solar elevation
For any given point on Mars's surface, we want to determine the angle of the sun. The zenith angle is:
Z = arccos (sin δs sin φ + cos δs cos φ cos H)
where φ is the planetographic latitude, Λ is the planetographic longitude, and H the hour angle, Λ - Λs.
The solar elevation is simply 90° - Z.
D-6. Determine local solar azimuth
The second element of the sun's location as seen from a point on Mars' surface is its azimuth, i.e., compass angle relative to due north.
A = arctan (sin H / (cos φ tan δs - sin φ cos H))
(Note: When applying this equation in your computer code or spreadsheet, use the atan2 function so that the correct quadrant is obtained.)
An easy-to-remember benchmark for calibrating clocks is a date and time at which the "standard time" for both Mars and Earth was almost the same. When the time on Earth was 00:00:00 on Jan. 6, 2000 (UTC), it was just 21 Mars-seconds away from also being mean midnight at the Mars prime meridian. Let's run through the calculations to see if we obtain MTC = 23:59:39. For longitude and latitude, we specify 0°W 0°N.
| Eq. | Parameter | Value |
|---|---|---|
| A-1 | millis | 947116800000 ms |
| A-2 | JDUT | 2451549.5 |
| A-3 | T | — |
| A-4 | TT - UTC | 64.184 s |
| A-5 | JDTT | 2451549.50074 |
| A-6 | ΔtJ2000 | 4.50074 |
| B-1 | M | 21.74548° |
| B-2 | αFMS | 272.74486° |
| B-3 | PBS | 0.00142° |
| B-4 | ν - M | 4.44191° |
| B-5 | Ls | 277.18677° |
| C-1 | EOT | -5.18764° = -0.34584 h = -00:20:45 |
| C-2 | MTC | mod24 (23.99431 h) = 23.99431 h = 23:59:39 |
| C-3 | LMST | 23.99431 h = 23:59:39 |
| C-4 | LTST | 23.64847 h = 23:38:54 |
| C-5 | Λs | 174.72703° |
| D-1 | δs | -25.22838° |
| D-2 | RM | 1.39358 AU |
| D-3 | lM | 2.26270° |
| D-4 | bM | -1.35959° |
| D-5 | Z | 154.26182° |
| D-6 | A | 191.03687° |
The Mars Exploration Rover A, Spirit, landed on Jan. 4, 2004 (UTC; in U.S. timezones it was late Jan. 3). For the planned landing target, local true midnight prior to touchdown was expected at 13:46:31 UTC on Jan. 3. We specify longitude and latitude of 184.702°W -14.640°N. We expect to obtain that the Mars local time at that location was LTST = 00:00:00 and the solar azimuth was 180°.
| Eq. | Parameter | Value |
|---|---|---|
| A-1 | millis | 1073137591000 ms |
| A-2 | JDUT | 2453008.07397 |
| A-3 | T | — |
| A-4 | TT - UTC | 64.184 s |
| A-5 | JDTT | 2453008.07471 |
| A-6 | ΔtJ2000 | 1463.07471 |
| B-1 | M | 786.06851° → 66.06851° |
| B-2 | αFMS | 1037.09363° → 317.09363° |
| B-3 | PBS | 0.01614° |
| B-4 | ν - M | 10.22959° |
| B-5 | Ls | 1047.32322° → 327.32322° |
| C-1 | EOT | -12.77557° = -0.85171 h = -00:51:06 |
| C-2 | MTC | mod24 (1109173.16542 h) = 13.16542 h = 13:09:55 |
| C-3 | LMST | 0.85196 h = 00:51:07 |
| C-4 | LTST | 0.00025 h = 00:00:00 |
| C-5 | Λs | 4.70575° |
| D-1 | δs | -13.42105° |
| D-2 | RM | 1.47767 AU |
| D-3 | lM | 52.37469° |
| D-4 | bM | 0.08962° |
| D-5 | Z | 151.93895° |
| D-6 | A | 179.99225° |
Discussion of the data and formulae used by Mars24 may be found in: