Martian Sunrise & Sunset Calculations by David and Barry Roffman

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Over eight months of daily mistakes by Ashima Research until May, 2013 (Updated on 6/19/2017)

With the exception of Sol 56, for the nine months of MSL operations on Mars, Ashima Research (working for JPL) published the daily MSL sunrise time as 6 am and the daily sunset time as 5 pm (11 hours of daylight, 13 of darkness). My father, Barry Roffman, deemed this data as instinctually wrong given that (1) the tilt of Earth’s axis = 23.44°, and (2) The tilt of the Martian axis = 25.19°. The difference is 1.75°. It seemed that the length of the Martian day should vary in a matter close to that seen on Earth. He took the first shot at estimating how the daylength would vary using common sense, I followed up his results with on-line formulas and a good bit of trigonometry. The results of both methods strongly agreed with each other, and prove that the Ashima figures published had no resemblance to the truth. In this article for  the sake of brevity for those only concerned about how to do the math I will first explain how I executed the most accurate calculation. Then I will show how my father (via what he calls common sense) reached a close estimation to what I found. On May 11, 2013 Ashima Research revised its figures to agree (within a minute or two) with my daylight calculations. They and JPL also agreed with my father's demand (to JPL Public Relations specialist - Guy Webster) to list wind as N/A rather than the never changing two meters per second from the east that the REMS Team had on its reports and that Ashima repeated on its site. 

THE CALCULATION BY DAVID A. ROFFMAN.

 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. The spreadsheet is shown as Table 1 below. The more extensive Table 5 showing daylength for every Ls is shown at the end of this article.

Table 1 above: SUMMARY OF DAVID A. ROFFMAN'S MARS DAYLIGHT CALCULATIONS WITH A COMPARISON TO THE FIGURES ESTIMATED PREVIOUSLY BY BARRY S. ROFFMAN.

Table 2 above: Sunrise and sunset times for Mars Science Laboratory calculated by Barry S. Roffman.
 

THE CALCULATION AND STATEMENT BY BARRY S. ROFFMAN

       My initial estimate was for local MSL times that varied from 6:07:30 am for sunrise and sunset at 17:52:30 (11 hours 45 minutes of daylight and 12 hour 15 minutes of darkness (Ls 270) to 12 hours 29 minutes of daylight and 11 hours 31 minutes of darkness around Ls 270 with sunrise at about 5:45:30 and sunset at 18:14:30. If you look at Ashima's reports on Figure 3 you will see that the length of day figures constitute 25% of their daily reports. They were contacted by me on numerous occasions about this repeated error, yet they and JPL persisted in publishing erroneous information until May 11, 2013. See Table 2 for a better approximation of MSL sunrise and sunset times. Before going on let me say this to Ashima. I have noticed that you got out of the business of publishing false Mars weather data with your last public report issued for MSL Sol 737 for September 1, 2014. Without consent by my son, David, I have been tough on your group and I hope that it hasn't hurt any careers there. I wouldn't have been as hard if you had answered my initial concerns. While mistakes about day length may seem like something not worth picking a fight about, the pattern of false weather data published by JPL is another matter. I've raised my son to be a scientist, but my efforts have been complicated by impressive names that stand behind easy to refute data. For over six years I was troubled by the question of whether mistakes uncovered were due to human error or poltically motivated disinformation. As of November 15, 2015 (with an exception of day light math) it's my belief that the answer often lies with disinformation.  This of course requires a purpose which is beyond the scope of this article on my son's site. On this site we merely focus on proving that erroneous data has been published and we attempt to demonstrate what a closer approximation of the truth is. Now, back to the issue at hand - how I knew that the original never changing daylength data was wrong.

       As my son mentioned above, a reasonable approximation of actual sunrise and sunset times at the Mars Science Laboratory (MSL) was based on the following initial assumptions: (1) Tilt of Earth’s axis = 23.44°. (2) Tilt of the Martian axis = 25.19°. The difference is 1.75°. (3) The latitude of MSL is 4.59° S. (4) For purposes of calculating how the length of Martian day light varies with its position in its orbit around the sun, because the Martian axis is tilted an extra 1.75°, the effective latitude of MSL for purposes of comparison with Earth will be 4.59° + 1.75° = 6.34° S. (5) Local time at MSL is based on a longitude of 0°. While Table 2 shows the local sunrise times of 4.59° S, the times offered for 6.34° S are a better approximation of what is seen at MSL. Earth’s daylight was derived from a calculator at http://www.satellite-calculations.com/Satellite/Sunrise_and_sunset.htm.

The entering arguments were for both 4.59° S and 6.38° S, with length of daylight displayed for both latitudes and emphasis on 6.38° S. It is further assumed that local apparent noon is half way through the day length, and that therefore local sunrise occurs at a time that corresponds to local apparent noon minus half the daylight while local sunset occurs at a time corresponding to local apparent noon plus half the daylight hours. From these assumptions it can quickly be seen that in contrast to errors published by Ashima Research in conjunction with the Remote Environmental Monitoring Station (REMS) Team employed by NASA/JPL for publishing all weather data gathered by MSL; the length of the day does indeed vary from their constant 13 hours of night time and 11 hours of day light with a continuing sunrise at 6 am and sunset at 5 PM. In fact, the constant day length of 11 hours at MSL will never occur at MSL. I calculated that the shortest daylight will be about 11 hours 45 minutes. My son's more accurate calculation indicated 11 hours 43.8 minutes. This occurred for several days on either side of Ls 90 around February 7, 2014. The longest daylight by my crude math was about 12 hours 29 minutes, but David found it to be 12 hours 19.3 minutes. This occurs for several days around Ls 270 which occurred on February 26, 2013. While there are many days when sunrise will occur near 6 am as Ashima shows from August 22, 2012 to December 31, 2012 (it was at 6 am August 26 to 27, 2012), sunset never occurs as early as 5 pm. By my math the earliest sunset will be about 5:52:30 pm around Ls 90.

       That Ashima/REMS Team and JPL were so wrong in publishing basic information was merely the entering argument in our case that the NASA/JPL Emperor has no clothes, and that much of the Mars weather data published by NASA/JPL since the Vikings have landed is false. Our full case for this is made in our report, MARS CORRECT: CRITIQUE OF ALL NASA MARS WEATHER DATA and throughout this web site. The primary purpose of my research efforts is to spur serious funding for manned missions to Mars and to ensure that any evidence uncovered having a bearing on a history of advanced life forms there will published in an honest and unclassified manner.

Figure 2 - Erroneous sunrise and sunset times by Ashima that were later corrected to match our math.

Issues considered by Barry Roffman in refining the sunrise sunset calculations. If you look at the part of Figure 3 that shows the sole exception to Ashima’s original reports about 13 hour nights and 11 hour days (Sol 56 on October 2, 2012) you can get an idea of how difficult that it was for Ashima to nail down sunrise and sunset times at MSL. Was their estimate that sunrise would occur at 5:31 am and sunset at 5:09 pm correct for that day? No. That amounted to 11 hours 38 minutes of daylight by my math. In accordance with my son's calculation, that would equate to a sol with about 12 hours 1 minute 41 seconds of daylight when the planet was at Ls 181.2. This would be only three days after the southern hemisphere had just left the winter and entered the spring. At the equinox we expect daylight and hours after daylight to be close to 12 hour each. My cruder method indicated 12 hour 7 minutes of daylight for Sol 56, so there was a 5 minute 19 second difference between our methods (amounting to a little less than a 3 minute disagreement for sunrise and sunset), but a 23 minute error for daylight by Ashima, whose error grew to over 1 hour 19 minutes of daylight at Ls 270.

 

       As an Orthodox Jew, I watch sunset times very closely throughout the year because it affects when we begin the Sabbath. I know from experience that the days are not equal where I live at the “equinox,” nor does the earliest Sabbath come at the winter solstice, nor does the latest Sabbath come at the summer solstice. For example, in Washington, D.C. in 2013 the vernal equinox occurs on March 20 but the length of day then is actually 12 hours 9 minutes. It will be 12 hours 1 minute long on March 17 (and 11 hours 59 minutes long on March 16). The autumnal equinox occurred on September 22, 2012, but the length of day then was actually 12 hours 8 minutes. It was 12 hours long on September 25, 2012. When we look at the equinoxes on Mars, we must be careful to not confuse them with 12 hours of daylight and 12 hours of the sun down. However, we would expect them to be close to 12 hours each. The problem with the equinox in Washington is discussed at http://www.wjla.com/blogs/weather/2012/09/autumnal-equinox-equal-hours-of-daylight-darkness-or-not--16735.html. Of particular note is the following:

Sunrise: 6:56 a.m.  Sunset: 7:04 p.m..  So if there isn't equal parts of daylight and darkness on Saturday, why is it the equinox?  The day we have 12 hours of daylight and 12 hours of darkness actually falls on Tuesday, Sept. 25.  Sunrise: 6:59 a.m.  Sunset: 7:00 p.m.   Shouldn't Tuesday be the autumnal equinox?

Well, we have the atmosphere to thank for this oddity.  Also, the definition of sunrise and sunset.  Sunrise occurs the moment the tip of the sun can be seen on the horizon and sunset is the last minute the sun can be seen before it dips below the horizon.  Also, keep in mind our atmosphere refracts, or bends, light, which makes it appear as if the sun is rising or setting earlier. 

The true equinox occurs when the center of the sun's disk crosses the celestial equator and this occurs at 10:59 a.m. EDT on September 22.  At the same time the equinox occurs in D.C., it occurs across the globe.

       So we have two slightly complicating factors, although the sun subtends less of an angle from Mars than it does from Earth. However, Mars also turns about 37 minutes slower than Earth does each day. This leads to a question of how to measure time on Mars. When we performed an extensive audit of Viking lander pressures and temperatures for our Report, we used the Viking Project time scheme of 25 time bins per Martian sol, each with about 59 minutes in them. However, for purposes of times quoted in my article, I assume 24 hour on Mars, with each hour having 60 minutes but with each minute having a greater duration than on Earth (each Mars second will be about 1.02583333 Earth seconds). The goal here is for a sense of proportionality. So 12 hours and 10 minutes of daylight in this scheme will actually be equal to about 5.5 seconds longer than on Earth. David's formula uses a conversion figure of 1.027491 rather than the 1.02583333 figure that I employed because he (correctly) applies the correction for a solar day rather than one for a sidereal day.

       The bottom line on the issue of Sol 56 on Mars is this. MSL landed close to the equator (4.59°). Each degree of latitude change in Mars is only about 59 km. So that's just 270.81 km (169.25 miles) from the equator. Although the tilt of Mars' axis is 1.79° more than Earth's, this will not be a significant factor around the solstices so close to the equator. The length of the Martian day this close to the solstice at Ls 181.2 will be close to 12 hours. Obviously, this is also also true for the day before and after Sol 56, when Ashima reported 13 hours of night for each sol as it did for all other sols until at least Sol 177 on February 5, 2013. See Figure 3.

Figure 3: A few of the many serious mistakes made by the REMS Team and Ashima Research. While those pressures listed about 740 hPa are labeled as having wrong units because they are 100 times greater than pressures generally accepted, the actUal weather on Mars seems to suggest that such high pressures are indeed possible. Note the only exception at sol 56 for 13 hours of night, sandwiched between sols 55 and 57 when Ashima reported 13 hour of night.

 

Figure 4 - REMS Team, Ashima, and Roffman calculations for day length at winter and summer solstices at Gale Crater, Mars.

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 2 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.

 

TABLE 3 – THE MARTIAN MONTHLY CALENDAR

MARS

MONTH

END SOL

START SOL

DAYS

Degrees per month

ΔLs Degrees per Day

1

61

0

61

30

0.491803

2

127

61

66

30

0.454545

3

193

127

66

30

0.454545

4

258

193

65

30

0.461538

5

318

258

60

30

0.5

6

372

318

54

30

0.555556

7

422

372

50

30

0.6

8

468

422

46

30

0.652174

9

515

468

47

30

0.638298

10

562

515

47

30

0.638298

11

613

562

51

30

0.588235

12

669

613

56

30

0.535714

 

      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 4 - ORBITAL PARAMETERS

SOURCE:

 

 

MARS

EARTH

RATIO

Semimajor axis (106 km)

227.92

149.60

1.524

Sidereal orbit period (days)

686.980

365.256

1.881

Perihelion (106 km)

206.62

147.09

1.405

Aphelion (106 km)

249.23

152.10

1.639

Mean orbital velocity (km/s)

24.13

29.78

0.810

Max. orbital velocity (km/s)

26.50

30.29

0.875

Min. orbital velocity (km/s)

21.97

29.29

0.750

Orbital Inclination (deg)

1.850

0.000

--

Orbit eccentricity

0.0935

0.0167

5.599

Sidereal rotation period (hrs)

24.6229

23.9345

1.029

Length of day (hrs)

24.6597

24.0000

1.027

Obliquity to orbit/axis tilt (deg)

25.19

23.44

1.075

NOTE: THE CALCULATIONS ON TABLE 5 BELOW ARE FOR THE 2012 LANDING LATTITUDE OF 4.59 DEGREES SOUTH. THE ROVER IS NOW, IN 2017 ABOUT 10 MILES SOUTH (ABOUT 4.7 DEGREES SOUTH). WE WILL PUBLISH AN UPDATED TABLE FOR THIS LATITUDE BY JUNE 21, 2017. THE MAXIMIM DIFFERENCE IN DAY LENGTH  BETWEEN THE TWO LATITUDES IS ABOUT 26.37 SECONDS AT LS 270 (START OF SUMMER), OR 25.69 SECONMDS AT LS 90 (START OF WINTER).

TABLE 5: The following chart 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.

λsun Latitude     Day Length = Daylight Half Sol Difference Minutes Hours plus  
 (0 for spring  (phi) δdegrees =  H = arccos((SIN(-.17) - SIN(lw)*SIN(δ))/(COS(lw)*COS(δ))) 2*1.027491*H/360 In Hours in Hours Half day - above or minutes  
in northern     arcsin((sin(25.19)*sin(λsun))     David's   Daylight below 12 of    
hemisphere)         Calculation   (G-F) hours for daylight  
This is Ls         (=E value * 24)     daylight      
0 -4.59 0 90.17054697 0.51471903 12.35325673 12.329895 0.023362 1.40170306 12 hr 1.401703 min
1 -4.59 0.425603669 90.13638243 0.51452401 12.34857623 12.329895 0.018681 1.12087298 12 hr 1.120873 min
2 -4.59 0.851101167 90.10223209 0.51432907 12.34389767 12.329895 0.014003 0.8401597 12 hr 0.84016 min
3 -4.59 1.276386306 90.06810071 0.514134238 12.33922172 12.329895 0.009327 0.55960221 12 hr 0.559602 min
4 -4.59 1.701352853 90.03399304 0.513939542 12.33454901 12.329895 0.004654 0.27923957 12 hr 0.27924 min
5 -4.59 2.125894518 89.99991384 0.513745008 12.3298802 12.329895 -1.48E-05 -0.00088898 11 hr 59.99911 min
6 -4.59 2.549904928 89.96586793 0.513550664 12.32521595 12.329895 -0.004679 -0.28074393 11 hr 59.71926 min
7 -4.59 2.973277609 89.93186014 0.513356538 12.32055692 12.329895 -0.009338 -0.56028545 11 hr 59.43971 min
8 -4.59 3.395905971 89.89789539 0.513162658 12.31590379 12.329895 -0.013991 -0.83947332 11 hr 59.16053 min
9 -4.59 3.817683284 89.86397861 0.512969051 12.31125723 12.329895 -0.018638 -1.11826682 11 hr 58.88173 min
10 -4.59 4.23850266 89.83011482 0.512775747 12.30661793 12.329895 -0.023277 -1.39662469 11 hr 58.60338 min
11 -4.59 4.658257039 89.79630913 0.512582775 12.3019866 12.329895 -0.027908 -1.67450502 11 hr 58.32549 min
12 -4.59 5.076839171 89.76256672 0.512390164 12.29736393 12.329895 -0.032531 -1.95186522 11 hr 58.04813 min
13 -4.59 5.494141598 89.72889286 0.512197944 12.29275065 12.329895 -0.037144 -2.22866191 11 hr 57.77134 min
14 -4.59 5.910056641 89.69529294 0.512006146 12.2881475 12.329895 -0.041748 -2.50485086 11 hr 57.49515 min
15 -4.59 6.324476385 89.66177244 0.511814801 12.28355523 12.329895 -0.04634 -2.78038691 11 hr 57.21961 min
16 -4.59 6.737292666 89.62833699 0.511623942 12.27897461 12.329895 -0.05092 -3.05522391 11 hr 56.94478 min
17 -4.59 7.148397059 89.59499233 0.511433601 12.27440643 12.329895 -0.055489 -3.32931467 11 hr 56.67069 min
18 -4.59 7.557680868 89.56174433 0.511243812 12.2698515 12.329895 -0.060044 -3.60261083 11 hr 56.39739 min
19 -4.59 7.965035114 89.52859902 0.51105461 12.26531063 12.329895 -0.064584 -3.87506286 11 hr 56.12494 min
20 -4.59 8.370350533 89.49556259 0.510866028 12.26078468 12.329895 -0.06911 -4.14661995 11 hr 55.85338 min
21 -4.59 8.773517562 89.46264138 0.510678105 12.25627451 12.329895 -0.07362 -4.41722994 11 hr 55.58277 min
22 -4.59 9.17442634 89.42984189 0.510490876 12.25178102 12.329895 -0.078114 -4.68683933 11 hr 55.31316 min
23 -4.59 9.572966702 89.39717083 0.51030438 12.24730513 12.329895 -0.08259 -4.95539311 11 hr 55.04461 min
24 -4.59 9.969028179 89.36463506 0.510118657 12.24284777 12.329895 -0.087047 -5.22283479 11 hr 54.77717 min
25 -4.59 10.3625 89.33224165 0.509933746 12.23840991 12.329895 -0.091485 -5.48910631 11 hr 54.51089 min
26 -4.59 10.75327109 89.29999786 0.509749689 12.23399255 12.329895 -0.095902 -5.75414796 11 hr 54.24585 min
27 -4.59 11.14123007 89.26791115 0.509566529 12.22959671 12.329895 -0.100298 -6.01789839 11 hr 53.9821 min
28 -4.59 11.52626529 89.2359892 0.50938431 12.22522344 12.329895 -0.104672 -6.2802945 11 hr 53.71971 min
29 -4.59 11.90826481 89.20423991 0.509203076 12.22087382 12.329895 -0.109021 -6.54127143 11 hr 53.45873 min
30 -4.59 12.28711642 89.17267137 0.509022874 12.21654897 12.329895 -0.113346 -6.80076253 11 hr 53.19924 min
31 -4.59 12.66270765 89.14129193 0.508843751 12.21225002 12.329895 -0.117645 -7.05869928 11 hr 52.9413 min
32 -4.59 13.0349258 89.11011015 0.508665757 12.20797816 12.329895 -0.121917 -7.31501128 11 hr 52.68499 min
33 -4.59 13.40365796 89.07913482 0.508488941 12.20373458 12.329895 -0.12616 -7.56962624 11 hr 52.43037 min
34 -4.59 13.76879101 89.04837498 0.508313355 12.19952051 12.329895 -0.130374 -7.82246991 11 hr 52.17753 min
35 -4.59 14.13021166 89.0178399 0.508139052 12.19533724 12.329895 -0.134558 -8.07346609 11 hr 51.92653 min
36 -4.59 14.48780649 88.98753908 0.507966086 12.19118607 12.329895 -0.138709 -8.32253663 11 hr 51.67746 min
37 -4.59 14.84146195 88.95748228 0.507794513 12.18706832 12.329895 -0.142827 -8.56960138 11 hr 51.4304 min
38 -4.59 15.19106443 88.92767948 0.507624391 12.18298538 12.329895 -0.14691 -8.81457822 11 hr 51.18542 min
39 -4.59 15.53650028 88.89814092 0.507455776 12.17893863 12.329895 -0.150956 -9.05738304 11 hr 50.94262 min
40 -4.59 15.87765582 88.86887707 0.50728873 12.17492952 12.329895 -0.154965 -9.29792978 11 hr 50.70207 min
41 -4.59 16.21441747 88.83989864 0.507123313 12.17095951 12.329895 -0.158936 -9.5361304 11 hr 50.46387 min
42 -4.59 16.54667169 88.81121657 0.506959587 12.1670301 12.329895 -0.162865 -9.77189496 11 hr 50.22811 min
43 -4.59 16.87430512 88.78284203 0.506797617 12.16314282 12.329895 -0.166752 -10.0051316 11 hr 49.99487 min
44 -4.59 17.19720459 88.75478643 0.506637468 12.15929924 12.329895 -0.170596 -10.2357466 11 hr 49.76425 min
45 -4.59 17.51525716 88.72706138 0.506479206 12.15550094 12.329895 -0.174394 -10.4636445 11 hr 49.53636 min
46 -4.59 17.82835026 88.69967872 0.506322898 12.15174955 12.329895 -0.178145 -10.688728 11 hr 49.31127 min
47 -4.59 18.13637166 88.67265049 0.506168613 12.14804671 12.329895 -0.181848 -10.9108982 11 hr 49.0891 min
48 -4.59 18.43920959 88.64598892 0.506016421 12.14439411 12.329895 -0.185501 -11.1300544 11 hr 48.86995 min
49 -4.59 18.73675284 88.61970643 0.505866393 12.14079344 12.329895 -0.189102 -11.3460945 11 hr 48.65391 min
50 -4.59 19.02889074 88.59381562 0.505718601 12.13724643 12.329895 -0.192649 -11.5589151 11 hr 48.44108 min
51 -4.59 19.31551336 88.56832925 0.505573118 12.13375483 12.329895 -0.19614 -11.7684112 11 hr 48.23159 min
52 -4.59 19.59651149 88.54326023 0.505430017 12.1303204 12.329895 -0.199575 -11.9744768 11 hr 48.02552 min
53 -4.59 19.87177679 88.5186216 0.505289372 12.12694494 12.329895 -0.20295 -12.1770045 11 hr 47.823 min
54 -4.59 20.14120185 88.49442652 0.50515126 12.12363024 12.329895 -0.206265 -12.3758864 11 hr 47.62411 min
55 -4.59 20.4046803 88.47068824 0.505015755 12.12037812 12.329895 -0.209517 -12.5710133 11 hr 47.42899 min
56 -4.59 20.66210688 88.44742013 0.504882934 12.11719042 12.329895 -0.212705 -12.7622755 11 hr 47.23772 min
57 -4.59 20.91337758 88.42463557 0.504752873 12.11406896 12.329895 -0.215826 -12.949563 11 hr 47.05044 min
58 -4.59 21.15838967 88.40234802 0.50462565 12.1110156 12.329895 -0.218879 -13.132765 11 hr 46.86723 min
59 -4.59 21.3970419 88.38057095 0.50450134 12.10803216 12.329895 -0.221863 -13.311771 11 hr 46.68823 min
60 -4.59 21.62923453 88.35931782 0.504380021 12.10512051 12.329895 -0.224775 -13.4864701 11 hr 46.51353 min
61 -4.59 21.85486944 88.33860208 0.50426177 12.10228248 12.329895 -0.227613 -13.656752 11 hr 46.34325 min
62 -4.59 22.07385031 88.31843711 0.504146663 12.0995199 12.329895 -0.230375 -13.8225067 11 hr 46.17749 min
63 -4.59 22.28608263 88.29883622 0.504034775 12.0968346 12.329895 -0.23306 -13.9836246 11 hr 46.01638 min
64 -4.59 22.4914739 88.27981262 0.503926183 12.09422839 12.329895 -0.235667 -14.1399972 11 hr 45.86 min
65 -4.59 22.6899337 88.26137938 0.503820961 12.09170306 12.329895 -0.238192 -14.2915171 11 hr 45.70848 min
66 -4.59 22.8813738 88.24354941 0.503719182 12.08926038 12.329895 -0.240635 -14.4380781 11 hr 45.56192 min
67 -4.59 23.0657083 88.22633545 0.50362092 12.08690209 12.329895 -0.242993 -14.5795756 11 hr 45.42042 min
68 -4.59 23.24285371 88.20975 0.503526246 12.0846299 12.329895 -0.245265 -14.7159068 11 hr 45.28409 min
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70 -4.59 23.57525619 88.1785134 0.503347938 12.08035052 12.329895 -0.249544 -14.9726694 11 hr 45.02733 min
71 -4.59 23.73035947 88.16388591 0.503264441 12.07834657 12.329895 -0.251548 -15.0929064 11 hr 44.90709 min
72 -4.59 23.8779663 88.14993419 0.5031848 12.0764352 12.329895 -0.25346 -15.2075885 11 hr 44.79241 min
73 -4.59 24.01800702 88.1366692 0.50310908 12.07461792 12.329895 -0.255277 -15.3166257 11 hr 44.68337 min
74 -4.59 24.15041507 88.12410154 0.50303734 12.07289616 12.329895 -0.256999 -15.419931 11 hr 44.58007 min
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77 -4.59 24.50122608 88.09068169 0.50284657 12.06831768 12.329895 -0.261577 -15.6946397 11 hr 44.30536 min
78 -4.59 24.60250317 88.08100016 0.502791305 12.06699132 12.329895 -0.262904 -15.7742213 11 hr 44.22578 min
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80 -4.59 24.78126436 88.06387452 0.502693547 12.06464513 12.329895 -0.26525 -15.9149928 11 hr 44.08501 min
81 -4.59 24.85866022 88.05644508 0.502651138 12.06362731 12.329895 -0.266268 -15.9760623 11 hr 44.02394 min
82 -4.59 24.92801369 88.04978001 0.502613092 12.0627142 12.329895 -0.267181 -16.0308487 11 hr 43.96915 min
83 -4.59 24.98929018 88.04388513 0.502579442 12.06190661 12.329895 -0.267988 -16.0793042 11 hr 43.9207 min
84 -4.59 25.04245901 88.03876561 0.502550218 12.06120524 12.329895 -0.26869 -16.1213862 11 hr 43.87861 min
85 -4.59 25.08749348 88.03442597 0.502525447 12.06061072 12.329895 -0.269284 -16.1570578 11 hr 43.84294 min
86 -4.59 25.12437093 88.03087006 0.502505148 12.06012356 12.329895 -0.269771 -16.1862871 11 hr 43.81371 min
87 -4.59 25.15307274 88.02810106 0.502489342 12.05974421 12.329895 -0.270151 -16.2090481 11 hr 43.79095 min
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90 -4.59 25.19 88.02453664 0.502468995 12.05925589 12.329895 -0.270639 -16.2383473 11 hr 43.76165 min
91 -4.59 25.18589559 88.02493293 0.502471258 12.05931018 12.329895 -0.270585 -16.2350899 11 hr 43.76491 min
92 -4.59 25.17358442 88.02612142 0.502478042 12.059473 12.329895 -0.270422 -16.2253206 11 hr 43.77468 min
93 -4.59 25.15307274 88.02810106 0.502489342 12.05974421 12.329895 -0.270151 -16.2090481 11 hr 43.79095 min
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99 -4.59 24.85866022 88.05644508 0.502651138 12.06362731 12.329895 -0.266268 -15.9760623 11 hr 44.02394 min
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102 -4.59 24.60250317 88.08100016 0.502791305 12.06699132 12.329895 -0.262904 -15.7742213 11 hr 44.22578 min
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108 -4.59 23.8779663 88.14993419 0.5031848 12.0764352 12.329895 -0.25346 -15.2075885 11 hr 44.79241 min
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123 -4.59 20.91337758 88.42463557 0.504752873 12.11406896 12.329895 -0.215826 -12.949563 11 hr 47.05044 min
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127 -4.59 19.87177679 88.5186216 0.505289372 12.12694494 12.329895 -0.20295 -12.1770045 11 hr 47.823 min
128 -4.59 19.59651149 88.54326023 0.505430017 12.1303204 12.329895 -0.199575 -11.9744768 11 hr 48.02552 min
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131 -4.59 18.73675284 88.61970643 0.505866393 12.14079344 12.329895 -0.189102 -11.3460945 11 hr 48.65391 min
132 -4.59 18.43920959 88.64598892 0.506016421 12.14439411 12.329895 -0.185501 -11.1300544 11 hr 48.86995 min
133 -4.59 18.13637166 88.67265049 0.506168613 12.14804671 12.329895 -0.181848 -10.9108982 11 hr 49.0891 min
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137 -4.59 16.87430512 88.78284203 0.506797617 12.16314282 12.329895 -0.166752 -10.0051316 11 hr 49.99487 min
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146 -4.59 13.76879101 89.04837498 0.508313355 12.19952051 12.329895 -0.130374 -7.82246991 11 hr 52.17753 min
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157 -4.59 9.572966702 89.39717083 0.51030438 12.24730513 12.329895 -0.08259 -4.95539311 11 hr 55.04461 min
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181 -4.59 -0.425603669 90.20472098 0.514914105 12.35793853 12.329895 0.028044 1.68261099 12 hr 1.682611 min
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192 -4.59 -5.076839171 90.57987934 0.517055616 12.40933477 12.329895 0.07944 4.7663857 12 hr 4.766386 min
193 -4.59 -5.494141598 90.61378552 0.517249162 12.41397988 12.329895 0.084085 5.04509203 12 hr 5.045092 min
194 -4.59 -5.910056641 90.64763553 0.517442387 12.41861729 12.329895 0.088722 5.3233367 12 hr 5.323337 min
195 -4.59 -6.324476385 90.6814237 0.51763526 12.42324623 12.329895 0.093351 5.60107301 12 hr 5.601073 min
196 -4.59 -6.737292666 90.7151442 0.517827746 12.4278659 12.329895 0.097971 5.87825312 12 hr 5.878253 min
197 -4.59 -7.148397059 90.74879108 0.518019812 12.43247548 12.329895 0.10258 6.15482801 12 hr 6.154828 min
198 -4.59 -7.557680868 90.78235821 0.518211422 12.43707414 12.329895 0.107179 6.43074738 12 hr 6.430747 min
199 -4.59 -7.965035114 90.81583931 0.518402542 12.44166101 12.329895 0.111766 6.70595964 12 hr 6.70596 min
200 -4.59 -8.370350533 90.84922794 0.518593134 12.44623521 12.329895 0.11634 6.98041177 12 hr 6.980412 min
201 -4.59 -8.773517562 90.88251747 0.51878316 12.45079583 12.329895 0.120901 7.2540493 12 hr 7.254049 min
202 -4.59 -9.17442634 90.91570108 0.518972581 12.45534195 12.329895 0.125447 7.52681622 12 hr 7.526816 min
203 -4.59 -9.572966702 90.94877178 0.519161358 12.45987259 12.329895 0.129978 7.79865495 12 hr 7.798655 min
204 -4.59 -9.969028179 90.98172234 0.519349449 12.46438678 12.329895 0.134492 8.06950623 12 hr 8.069506 min
205 -4.59 -10.3625 91.01454537 0.519536812 12.4688835 12.329895 0.138988 8.33930913 12 hr 8.339309 min
206 -4.59 -10.75327109 91.04723322 0.519723404 12.47336169 12.329895 0.143467 8.60800091 12 hr 8.608001 min
207 -4.59 -11.14123007 91.07977805 0.519909179 12.4778203 12.329895 0.147925 8.87551704 12 hr 8.875517 min
208 -4.59 -11.52626529 91.11217177 0.520094092 12.4822582 12.329895 0.152363 9.14179113 12 hr 9.141791 min
209 -4.59 -11.90826481 91.14440608 0.520278094 12.48667426 12.329895 0.156779 9.40675484 12 hr 9.406755 min
210 -4.59 -12.28711642 91.17647243 0.520461138 12.49106731 12.329895 0.161172 9.67033791 12 hr 9.670338 min
211 -4.59 -12.66270765 91.20836202 0.520643173 12.49543615 12.329895 0.165541 9.93246807 12 hr 9.932468 min
212 -4.59 -13.0349258 91.24006582 0.520824147 12.49977953 12.329895 0.169885 10.193071 12 hr 10.19307 min
213 -4.59 -13.40365796 91.27157453 0.521004008 12.50409619 12.329895 0.174201 10.4520704 12 hr 10.45207 min
214 -4.59 -13.76879101 91.30287862 0.5211827 12.50838481 12.329895 0.17849 10.7093877 12 hr 10.70939 min
215 -4.59 -14.13021166 91.33396827 0.521360169 12.51264405 12.329895 0.182749 10.9649424 12 hr 10.96494 min
216 -4.59 -14.48780649 91.36483344 0.521536356 12.51687254 12.329895 0.186978 11.2186518 12 hr 11.21865 min
217 -4.59 -14.84146195 91.3954638 0.521711203 12.52106887 12.329895 0.191174 11.4704312 12 hr 11.47043 min
218 -4.59 -15.19106443 91.42584877 0.521884649 12.52523157 12.329895 0.195337 11.7201935 12 hr 11.72019 min
219 -4.59 -15.53650028 91.45597753 0.522056632 12.52935917 12.329895 0.199464 11.9678497 12 hr 11.96785 min
220 -4.59 -15.87765582 91.48583897 0.52222709 12.53345016 12.329895 0.203555 12.2133086 12 hr 12.21331 min
221 -4.59 -16.21441747 91.51542175 0.522395957 12.53750296 12.329895 0.207608 12.456477 12 hr 12.45648 min
222 -4.59 -16.54667169 91.54471427 0.522563167 12.541516 12.329895 0.211621 12.6972593 12 hr 12.69726 min
223 -4.59 -16.87430512 91.57370467 0.522728652 12.54548765 12.329895 0.215593 12.9355583 12 hr 12.93556 min
224 -4.59 -17.19720459 91.60238085 0.522892344 12.54941625 12.329895 0.219521 13.1712745 12 hr 13.17127 min
225 -4.59 -17.51525716 91.6307305 0.523054172 12.55330012 12.329895 0.223405 13.4043066 12 hr 13.40431 min
226 -4.59 -17.82835026 91.65874105 0.523214064 12.55713753 12.329895 0.227243 13.6345513 12 hr 13.63455 min
227 -4.59 -18.13637166 91.68639972 0.523371947 12.56092674 12.329895 0.231032 13.8619035 12 hr 13.8619 min
228 -4.59 -18.43920959 91.71369352 0.523527748 12.56466596 12.329895 0.234771 14.0862566 12 hr 14.08626 min
229 -4.59 -18.73675284 91.74060925 0.523681391 12.56835338 12.329895 0.238458 14.3075019 12 hr 14.3075 min
230 -4.59 -19.02889074 91.76713353 0.523832799 12.57198717 12.329895 0.242092 14.5255296 12 hr 14.52553 min
231 -4.59 -19.31551336 91.79325281 0.523981895 12.57556548 12.329895 0.24567 14.7402282 12 hr 14.74023 min
232 -4.59 -19.59651149 91.81895336 0.524128601 12.57908643 12.329895 0.249191 14.9514849 12 hr 14.95148 min
233 -4.59 -19.87177679 91.84422133 0.524272838 12.58254811 12.329895 0.252653 15.1591858 12 hr 15.15919 min
234 -4.59 -20.14120185 91.86904273 0.524414525 12.58594861 12.329895 0.256054 15.3632159 12 hr 15.36322 min
235 -4.59 -20.4046803 91.89340345 0.524553583 12.589286 12.329895 0.259391 15.5634593 12 hr 15.56346 min
236 -4.59 -20.66210688 91.91728931 0.524689931 12.59255833 12.329895 0.262663 15.7597993 12 hr 15.7598 min
237 -4.59 -20.91337758 91.94068604 0.524823486 12.59576366 12.329895 0.265869 15.9521188 12 hr 15.95212 min
238 -4.59 -21.15838967 91.96357934 0.524954167 12.59890001 12.329895 0.269005 16.1403001 12 hr 16.1403 min
239 -4.59 -21.3970419 91.98595489 0.525081893 12.60196544 12.329895 0.27207 16.3242254 12 hr 16.32423 min
240 -4.59 -21.62923453 92.00779835 0.525206582 12.60495796 12.329895 0.275063 16.5037771 12 hr 16.50378 min
241 -4.59 -21.85486944 92.02909542 0.525328152 12.60787564 12.329895 0.277981 16.6788375 12 hr 16.67884 min
242 -4.59 -22.07385031 92.04983186 0.525446521 12.6107165 12.329895 0.280821 16.8492895 12 hr 16.84929 min
243 -4.59 -22.28608263 92.06999349 0.525561609 12.61347862 12.329895 0.283584 17.0150167 12 hr 17.01502 min
244 -4.59 -22.4914739 92.08956627 0.525673336 12.61616007 12.329895 0.286265 17.1759035 12 hr 17.1759 min
245 -4.59 -22.6899337 92.10853627 0.525781622 12.61875894 12.329895 0.288864 17.3318356 12 hr 17.33184 min
246 -4.59 -22.8813738 92.12688974 0.525886389 12.62127334 12.329895 0.291378 17.4826998 12 hr 17.4827 min
247 -4.59 -23.0657083 92.14461314 0.525987559 12.62370143 12.329895 0.293806 17.6283848 12 hr 17.62838 min
248 -4.59 -23.24285371 92.16169313 0.526085057 12.62604137 12.329895 0.296146 17.7687812 12 hr 17.76878 min
249 -4.59 -23.4127291 92.17811668 0.526178807 12.62829137 12.329895 0.298396 17.9037815 12 hr 17.90378 min
250 -4.59 -23.57525619 92.19387101 0.526268737 12.6304497 12.329895 0.300555 18.033281 12 hr 18.03328 min
251 -4.59 -23.73035947 92.20894369 0.526354776 12.63251463 12.329895 0.30262 18.1571773 12 hr 18.15718 min
252 -4.59 -23.8779663 92.22332264 0.526436856 12.63448453 12.329895 0.30459 18.2753712 12 hr 18.27537 min
253 -4.59 -24.01800702 92.23699616 0.526514908 12.63635779 12.329895 0.306463 18.3877667 12 hr 18.38777 min
254 -4.59 -24.15041507 92.24995301 0.526588869 12.63813286 12.329895 0.308238 18.494271 12 hr 18.49427 min
255 -4.59 -24.27512709 92.26218235 0.526658678 12.63980827 12.329895 0.309913 18.5947953 12 hr 18.5948 min
256 -4.59 -24.39208299 92.27367385 0.526724275 12.64138259 12.329895 0.311488 18.6892546 12 hr 18.68925 min
257 -4.59 -24.50122608 92.2844177 0.526785603 12.64285448 12.329895 0.312959 18.7775683 12 hr 18.77757 min
258 -4.59 -24.60250317 92.29440462 0.526842612 12.64422268 12.329895 0.314328 18.85966 12 hr 18.85966 min
259 -4.59 -24.69586459 92.30362588 0.526895249 12.64548598 12.329895 0.315591 18.9354581 12 hr 18.93546 min
260 -4.59 -24.78126436 92.31207337 0.52694347 12.64664328 12.329895 0.316748 19.0048959 12 hr 19.0049 min
261 -4.59 -24.85866022 92.3197396 0.526987231 12.64769354 12.329895 0.317799 19.0679117 12 hr 19.06791 min
262 -4.59 -24.92801369 92.32661769 0.527026493 12.64863583 12.329895 0.318741 19.1244491 12 hr 19.12445 min
263 -4.59 -24.98929018 92.33270145 0.527061221 12.6494693 12.329895 0.319574 19.1744572 12 hr 19.17446 min
264 -4.59 -25.04245901 92.33798536 0.527091383 12.65019319 12.329895 0.320298 19.2178906 12 hr 19.21789 min
265 -4.59 -25.08749348 92.34246461 0.527116952 12.65080684 12.329895 0.320912 19.2547097 12 hr 19.25471 min
266 -4.59 -25.12437093 92.34613509 0.527137904 12.65130969 12.329895 0.321415 19.2848808 12 hr 19.28488 min
267 -4.59 -25.15307274 92.34899343 0.52715422 12.65170128 12.329895 0.321806 19.3083761 12 hr 19.30838 min
268 -4.59 -25.17358442 92.35103699 0.527165885 12.65198125 12.329895 0.322086 19.325174 12 hr 19.32517 min
269 -4.59 -25.18589559 92.35226389 0.527172889 12.65214933 12.329895 0.322254 19.335259 12 hr 19.33526 min
270 -4.59 -25.19 92.35267298 0.527175224 12.65220537 12.329895 0.32231 19.3386217 12 hr 19.33862 min
271 -4.59 -25.18589559 92.35226389 0.527172889 12.65214933 12.329895 0.322254 19.335259 12 hr 19.33526 min
272 -4.59 -25.17358442 92.35103699 0.527165885 12.65198125 12.329895 0.322086 19.325174 12 hr 19.32517 min
273 -4.59 -25.15307274 92.34899343 0.52715422 12.65170128 12.329895 0.321806 19.3083761 12 hr 19.30838 min
274 -4.59 -25.12437093 92.34613509 0.527137904 12.65130969 12.329895 0.321415 19.2848808 12 hr 19.28488 min
275 -4.59 -25.08749348 92.34246461 0.527116952 12.65080684 12.329895 0.320912 19.2547097 12 hr 19.25471 min
276 -4.59 -25.04245901 92.33798536 0.527091383 12.65019319 12.329895 0.320298 19.2178906 12 hr 19.21789 min
277 -4.59 -24.98929018 92.33270145 0.527061221 12.6494693 12.329895 0.319574 19.1744572 12 hr 19.17446 min
278 -4.59 -24.92801369 92.32661769 0.527026493 12.64863583 12.329895 0.318741 19.1244491 12 hr 19.12445 min
279 -4.59 -24.85866022 92.3197396 0.526987231 12.64769354 12.329895 0.317799 19.0679117 12 hr 19.06791 min
280 -4.59 -24.78126436 92.31207337 0.52694347 12.64664328 12.329895 0.316748 19.0048959 12 hr 19.0049 min
281 -4.59 -24.69586459 92.30362588 0.526895249 12.64548598 12.329895 0.315591 18.9354581 12 hr 18.93546 min
282 -4.59 -24.60250317 92.29440462 0.526842612 12.64422268 12.329895 0.314328 18.85966 12 hr 18.85966 min
283 -4.59 -24.50122608 92.2844177 0.526785603 12.64285448 12.329895 0.312959 18.7775683 12 hr 18.77757 min
284 -4.59 -24.39208299 92.27367385 0.526724275 12.64138259 12.329895 0.311488 18.6892546 12 hr 18.68925 min
285 -4.59 -24.27512709 92.26218235 0.526658678 12.63980827 12.329895 0.309913 18.5947953 12 hr 18.5948 min
286 -4.59 -24.15041507 92.24995301 0.526588869 12.63813286 12.329895 0.308238 18.494271 12 hr 18.49427 min
287 -4.59 -24.01800702 92.23699616 0.526514908 12.63635779 12.329895 0.306463 18.3877667 12 hr 18.38777 min
288 -4.59 -23.8779663 92.22332264 0.526436856 12.63448453 12.329895 0.30459 18.2753712 12 hr 18.27537 min
289 -4.59 -23.73035947 92.20894369 0.526354776 12.63251463 12.329895 0.30262 18.1571773 12 hr 18.15718 min
290 -4.59 -23.57525619 92.19387101 0.526268737 12.6304497 12.329895 0.300555 18.033281 12 hr 18.03328 min
291 -4.59 -23.4127291 92.17811668 0.526178807 12.62829137 12.329895 0.298396 17.9037815 12 hr 17.90378 min
292 -4.59 -23.24285371 92.16169313 0.526085057 12.62604137 12.329895 0.296146 17.7687812 12 hr 17.76878 min
293 -4.59 -23.0657083 92.14461314 0.525987559 12.62370143 12.329895 0.293806 17.6283848 12 hr 17.62838 min
294 -4.59 -22.8813738 92.12688974 0.525886389 12.62127334 12.329895 0.291378 17.4826998 12 hr 17.4827 min
295 -4.59 -22.6899337 92.10853627 0.525781622 12.61875894 12.329895 0.288864 17.3318356 12 hr 17.33184 min
296 -4.59 -22.4914739 92.08956627 0.525673336 12.61616007 12.329895 0.286265 17.1759035 12 hr 17.1759 min
297 -4.59 -22.28608263 92.06999349 0.525561609 12.61347862 12.329895 0.283584 17.0150167 12 hr 17.01502 min
298 -4.59 -22.07385031 92.04983186 0.525446521 12.6107165 12.329895 0.280821 16.8492895 12 hr 16.84929 min
299 -4.59 -21.85486944 92.02909542 0.525328152 12.60787564 12.329895 0.277981 16.6788375 12 hr 16.67884 min
300 -4.59 -21.62923453 92.00779835 0.525206582 12.60495796 12.329895 0.275063 16.5037771 12 hr 16.50378 min
301 -4.59 -21.3970419 91.98595489 0.525081893 12.60196544 12.329895 0.27207 16.3242254 12 hr 16.32423 min
302 -4.59 -21.15838967 91.96357934 0.524954167 12.59890001 12.329895 0.269005 16.1403001 12 hr 16.1403 min
303 -4.59 -20.91337758 91.94068604 0.524823486 12.59576366 12.329895 0.265869 15.9521188 12 hr 15.95212 min
304 -4.59 -20.66210688 91.91728931 0.524689931 12.59255833 12.329895 0.262663 15.7597993 12 hr 15.7598 min
305 -4.59 -20.4046803 91.89340345 0.524553583 12.589286 12.329895 0.259391 15.5634593 12 hr 15.56346 min
306 -4.59 -20.14120185 91.86904273 0.524414525 12.58594861 12.329895 0.256054 15.3632159 12 hr 15.36322 min
307 -4.59 -19.87177679 91.84422133 0.524272838 12.58254811 12.329895 0.252653 15.1591858 12 hr 15.15919 min
308 -4.59 -19.59651149 91.81895336 0.524128601 12.57908643 12.329895 0.249191 14.9514849 12 hr 14.95148 min
309 -4.59 -19.31551336 91.79325281 0.523981895 12.57556548 12.329895 0.24567 14.7402282 12 hr