National Aeronautics and Space Administration

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, the standard Java date/time classes do not keep track of leap seconds, i.e., they keep time in UT rather than UTC. Consequently, when Mars24 calls System.currentTimeMillis(), it expects 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.

Although there's plenty of sample code available on-line that 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.

JD_UT = 2440587.5 + (millis / 8.64×10⁷ ms/day)

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 = (JD_UT - 2451545.0) / 36525.

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, 2017, the TAI-UTC value is 37.0 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 37.0s + 32.184s = 69.184s.

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 accounts for the leap second added January 1, 2017. It does not allow for any leap seconds that may have been subsequently added, which as of March 1, 2023, there have been none. Bulletin C 33 from the IERS Earth Orientation Centre indicates that no leap second addition will occur any earlier than January 1, 2024., if then.

JD_TT = JD_UT + [(TT - UTC) / 86400 s·day^-1]

Δt_J2000 = JD_TT - 2451545.0

M = 19.3871° + 0.52402073° Δt_J2000

α_FMS = 270.3871° + 0.524038496° Δt_J2000

PBS = Σ_(i=1,7) A_i cos [ (0.985626° Δt_J2000 / τ_i) + φ_i]

where 0.985626° = 360° / 365.25, and

The equation of center is the true anomaly minus mean anomaly.

ν - M = (10.691° + 3.0° × 10^-7 Δt_J2000) sin M + 0.623° sin 2M + 0.050° sin 3M + 0.005° sin 4M + 0.0005° sin 5M + PBS

L_s = α_FMS + (ν - M)

EOT = 2.861° sin 2L_s - 0.071° sin 4L_s + 0.002° sin 6L_s - (ν - M)

The above result for EOT is in degrees. Multiply by (24 h / 360°) = (1 h / 15°) to obtain the result in hours.

This is the mean solar time at Mars's prime meridian.

MST = mod₂₄ { 24 h × ( [(JD_TT - 2451549.5) / 1.0274912517] + 44796.0 - 0.0009626 ) }

The function mod_X 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.

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 = mod₂₄ { MST - Λ (24 h / 360°) } = mod₂₄ { MST - Λ (1 h / 15°) }

LTST = LMST + EOT (24 h / 360°) = LMST + EOT (1 h / 15°)

Λ_s = MST (360° / 24 h) + EOT + 180° = MST (15° / h) + EOT + 180°

δ_s = arcsin {0.42565 sin L_s)} + 0.25° sin L_s

R_M = 1.52367934 × (1.00436 - 0.09309 cos M - 0.004336 cos 2M - 0.00031 cos 3M - 0.00003 cos 4M)

l_M = L_s + 85.061° - 0.015° sin (71° + 2L_s) - 5.5°×10^-6 Δt_J2000

b_M = -(1.8497° - 2.23°×10^-5 Δt_J2000) sin (L_s - 144.50° + 2.57°×10^-6 Δt_J2000)

For any given point on Mars's surface, we want to determine the angle of the sun. The zenith angle is:

Z = arccos (sin δ_s sin φ + cos δ_s cos φ cos H)

where φ is the planetographic latitude, Λ is the planetographic longitude, and H the hour angle, Λ - Λ_s.

The solar elevation is simply 90° - Z.

The second element of the sun's location as seen from a point on Mars's surface is its azimuth, i.e., compass angle relative to due north.

A = arctan (sin H / (cos φ tan δ_s - 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.)