Mars Daylight Math: Roffman vs. NASA (REMS Team & Ashima)

 The first step is to specify the latitude, φ. For purposes of my calculation, the latitude used was the actual latitude of the MSL on Mars, namely 4.59° South (written as -4.59). Next the declination (δ) must be calculated: δ = arcsin((sin(25.19)*sin(λsun)).  The required input is the Ls, which is λsun. The hour angle is then H = arccos((SIN(-.17) - SIN(φ)*SIN(δ))/(COS(φ)*COS(δ))).  This output must be in degrees.  The text speaks of calculating the solar transit (local noon).  However, it also notes that the sunrise and sunset times are the local noon time -/+ 1.027491*H/360.  By taking the equations for sunrise and sunset times (the actual formulae for the Julian dates are irrelevant as they share a common term that will be canceled) and subtracting the sunrise from the sunset one, a number is obtained.  This is the time interval between the sunset and sunrise times, and is hence the length of day.  The length of day is: 2*1.027491*H/360.  Multiply by 24 to get hours.

How Consistent are REMS Team and Ashima Research Mars Weather Reports? Given that Ashima gets its information from REMS we would expect them to provide identical data for the same sols. Usually they do, but not always. Close examination of Figure 2 for Ls 90 (the first day of winter at MSL Curiosity) shows matching reports in all aspects expect the terrestrial date for Sol 544, which Ashima gives as February 15, 2014 while REMS lists February 16, 2014. In comparing the length of day with the David Roffman calculation, taking the difference between sunrise and sunset the Government teams indicate 11 hours 43 minutes. They are rounding off both sunrise and sunset times to the nearest minute. David’s calculation is only for day length, rounding off to the nearest tenth of a minute (6 seconds). He indicates 11 hours 43.8 minutes.

     For Ls 270, the first day of summer at MSL Curiosity is more typical of reporting problems seen by both REMS and Ashima. Normally it takes about 2 sols to go through each Ls number change. An on-line Mars calendar shows that February 23, 2013 was Ls 269.7 while February 24, 2013 was Ls 270.3. So the solstice occurred between the times used for these two dates. But here at Ls 270 Ashima lists the sol as 191, while REMS indicates 197! Ashima shows a terrestrial date of February 23, 2013 while REMS shows February 24, 2013. A six sol difference is NOT possible for Mars to go through one sol number! We assume that the mistake is due to sloppy record keeping at Ashima, because (as of April 11, 2014) they also indicate that February 17, 2013 was another Sol 191. That was on February 17, 2013 (6 days before the more correct date on February 23, 2013).

       Ashima first claims that Sol 191 is February 23, 2013. But then they claim that February 23, 2013 is also sol 198! Now remember, these rocket scientists work for NASA/JPL! Why is unclear unless there is some affirmative action program which encourages the hiring of mathematically challenged incompetents.

      Both Ashima and REMS agree that the minimum air temperature was -68 °C for Ls 270, but Ashima shows a high of 4°C (39.2°F), while REMS indicatives only -3°C (26.6°F), a difference of 7°C (12.6°F). They do not agree about sunrise or sunset times. Ashima indicates sunrise at 06:00, REMS shows 06:04, For sunset Ashima shows 18:17, REMS shows 18:22. Thus for time that the sun is up Ashima shows 12 hours 17 minutes, and REMS shows 12 hours 18 minutes, while David Roffman’s calculation reveals 18 hours 19.3 minutes. Trust David.

In addition to posting constantly wrong and never changing sunrise and sunset times for so long, Ashima Research and the REMS Team often mislabel the dates and associated MSL Sols for the weather data supposedly coming back from Mars. Table 2 below details the history of these errors between August 22, 2012 (Sol 15) and January 2, 2013 (Sol 146). This information may be useful for data set references purposes when evaluating data from the MSL.

Understanding where Mars is in its orbit. The solar longitude Ls is the Mars-Sun angle, measured from the Northern Hemisphere spring equinox where Ls=0. Ls=90 thus corresponds to summer solstice, just as Ls=180 marks the autumn equinox and Ls=270 the winter solstice (all relative to the northern hemisphere). In trying to understand where Mars is in its orbit, it’s important to know that while all Martian months involve a change of 30 degrees, there is a large variation in number of days in each of the 12 months there, and also a large variation in angular speed around the sun (see Figure 3 and Table 3 below). In working up estimates for the length of any given day I used Table 2. This allowed me to project within a day or 2 over the Martian year where the planet would be, so that I could compare the length of day on Mars with a similar position in Earth’s orbit.  Small errors were introduced where the daily rate of change in Ls was not quite the monthly average shown in the last column of Table 3.

While the slightly greater tilt of the Martian axis was not much of a factor around the equinoxes (Month 1, Sol 1; and Month 7, Sol 373), the tilt does make a significant difference at the solstices (Month 3, Sol 193; and Month 10, Sol 515). Note that aphelion (when Mars is furthest from the sun) and perihelion (when Mars is closest to the sun) does not correspond to the start of any Martian month. The next perihelion for Mars will be on December 12, 2014 which on the Martian calendar is Month 9, Sol 485, Ls 250.7. It will then be 1.38 AU from the sun. The next aphelion for Mars will be on January 2, 2014 which on the Martian calendar is Month 3, Sol 151, Ls 70.6. It will then be 1.67 AU from the sun.

        For the sake of quick reference, Table 4 contains a few useful facts for comparing Earth and Mars orbital parameters and orbital elements.

Table 5 below can be used  to relate Martian Ls to hours and minutes of daylight at the latitude (4.59 South) of the Mars Science Laboratory rover Curiosity.