Equations and Worked Examples

The calculations made by Mar24 to determine the time for a given location on Mars is 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. Consequently, we provide here documentation of the equations used by Mars24 for users who wish to implement their Martian timekeeping applications. At the end of this presentation, we also provide worked examples for verification of results.

Equations

Determine Julian Day

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 on Jan. 1, 1970 (aka, the Unix epoch).

A-2. Convert millis to Julian Day (UT).

Although there's sample code available on-line which demonstrates how to convert a Gregorian calendar date to a "Julian Day", we simply use the offset from a known, recent Julian Day, with 00:00 on Jan. 1, 1970 being the obvious choice.

JDUT = JDUnix + (millis / 8.64×107 ms·day-1) = 2440587.5 + (millis / 8.64×107 ms·day-1)

A-3. Determine time offset from J2000 epoch (UT).

Most of our calculations are in terms of days since 12:00 on Jan. 1, 2000 (TT). First, determine the number of days since that epoch (UT).

ΔtJ2000,UT = JDUT - JD2000 = JDUT - 2451545.0

A-4. Determine UTC to TT conversion. (Replaces AM2000, eq. 27)

Coordinated Universal Time (UTC) and Terrestrial Time (TT) are not the same. As of late 2003, there is a difference of 64.184 seconds between the systems. Perhaps the best way to determine the number of seconds to add to the UTC time to obtain the TT time is to consult a table indicating when leap seconds were added to the UTC time. Alternatively, one could try to use an empirical formula. In Mars24, we currently do the latter, with a caveat.

If ΔtJ2000,UT < 0:

T = ΔtJ2000,UT / 36525
TT - UTC = 64.184s + 59 s × T - 51.2 s × T2 - 67.1 s × T3 - 16.4 s × T4

If ΔtJ2000,UT ≥ 0:

TT - UTC = 64.184s

However, this latter condition will only be correct until the next occasion in which a leap second is added to the clock. As of the end of 2003, no leap seconds have been added to the clock in five years. The next opportunity when one might be added is June 30, 2004. When that happens, we will have to modify this step of the process.

A-5. Determine Julian Day (TT).

JDTT = JDUTC + [(TT - UTC) / 86400 s·day-1]

A-6. Determine time offset from J2000 epoch (TT). (AM2000, eq. 15)

ΔtJ2000 = JDTT - JD2000
= JDTT - 2451545.0

Determine Mars Parameters of Date

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:

Table 1
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 bracket term. (AM2000, eqs. 19 and 20)

Ls - αFMS = (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 + (Ls - αFMS)

Determine Mars Time

C-1. Determine Equation of Time. (AM2000, eq. 20)

EOT = 2.861° sin 2Ls - 0.071° sin 4Ls + 0.002° sin 6Ls - (Ls - αFMS)

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 = 24 h × mod24 {[(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 solar time for a given longitude is easily determined by offsetting from the mean solar time on the prime meridian.

LMST = MTC - ΛW (24 h / 360°)

C-4. Determine Local True Solar Time. (AM2000, eq. 23)

LTST = LMST + EOT (24 h / 360°)

Additional Calculations

D-1. Determine solar declination. (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 angle. (AM2000, eq. 26)

λM = Ls + 85.061° - 0.015° sin (71° + 2Ls) - 5.5°×10-6 ΔtJ2000

D-4. Determine heliocentric latitude.

ΨM = - (1.8497° - 2.23°×10-5 ΔtJ2000) sin (Ls - 144.50° + 2.57°×10-6 ΔtJ2000)

Worked Examples

MER-A Landing

The Mars Exploration Rover A clock starts at UTC 2004-01-03 13:46:31. The lander is expected to touch down at longitude 184.702°W. The results follow in Table 2.

Table 2
Eq.ParameterValue
A-1 millis 1073137591000 ms
A-2 JDUT 2453008.07397
A-3 ΔtJ2000,UT 1463.07397
A-4 TT - UTC 64.184 s
A-5 JDTT 2453008.07471
A-6 ΔtJ2000 1463.07471
B-1 M 786.06850° → 66.06850°
B-2 αFMS 1037.09363° → 317.09363°
B-3 PBS 0.01614°
B-4 Ls - αFMS 10.22958°
B-5 Ls 1047.32322° → 327.32322°
C-1 EOT -12.77557° = -0.85170 h = -00:51:06
C-2 MTC mod24 (1109173.16542 h) = 13.16542 h = 13:09:55
C-3 LMST 0.85195 h = 00:51:07
C-4 LTST 0.00025 h = 00:00:00
D-1 δs -13.42075°
D-2 RM 1.47767 AU
D-3 λM 52.37469°
D-4 ΨM 0.08962°

Mars Pathfinder Landing

Mars Pathfinder landed at UTC 1997-07-04 16:07:25 at longitude 33.55°W.

Table 2
Eq.ParameterValue
A-1 millis 868035415000
A-2 JDUT 2450634.20619
A-3 ΔtJ2000,UT -910.79381
A-4 TT - UTC 62.68196 s
A-5 JDTT 2450634.20692
A-6 ΔtJ2000 -910.79308
B-1 M -7.99164° → 352.00836°
B-2 αFMS -206.90425° → 153.09575°
B-3 PBS 0.00663°
B-4 Ls - αFMS -10.37079°
B-5 Ls -217.27504° → 142.72496
C-1 EOT 7.65102° = 0.51007 h = 00:30:36
C-2 MTC mod24 (1053724.68645 h) = 4.68645 h = 04:41:11
C-3 LMST 2.44978 h = 02:26:59
C-4 LTST 2.95985 h = 02:57:35
D-1 δs 15.09047°
D-2 RM 1.55592 AU
D-3 λM 227.79190°
D-4 ΨM 0.05800°

Reference

Discussion of the data and formulae used by Mars24 may be found in: