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

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How can we trust scientists who publish faulty data for so long even after we have pointed out obvious errors? (Updated 8/3/2016)

With the exception of MSL Sol 56, for nearly nine months Ashima Research published the daily MSL sunrise time as 6 am and the daily sunset time as 5 pm (11 hours of daylight, 13 hours of darkness). I, Barry Roffman, deemed this position 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. I took the first shot at estimating how the daylength would vary using common sense, whereas my son, David, followed up my results with on-line formulas and a good bit of trigonometry. The results of both methods strongly agreed with each other, and proved that the Ashima figures published had no resemblance to the truth. In fact, Ashima published a concession about their numerous errors (some whch continue to this on April 11, 2014). David's calculations were essential accepted by Ashima Research and by the REMS Team working for NASA/JPL. This article will first focus on his method of correcting the daylight data, and the highlight some of the problems that still remain with Government data about Mars.

Figure 1 above - From August 2012 to May 2013 sunrise/sunset times publsihed by Ashima Research were ludicrous. They never changed and always showed an impossible 11 hours of daylight and 13 hours of night. This included their report for the equinox at Ls 270 when spring started for MSL Curiosity. Figure 1 shows their initial error and their tardy correction in May, 2013. .

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.

TABLE 1 - SUMMARY OF DAVID A. ROFFMAN'S MARS DAYLIGHT CALCULATIONS.

Figure 2 - Ashima and the REMS team essentially came to agree with each other and the earlier David Roffman calculation for the arrival of summer at the Curiosity. There were several differences between Ashima and the REMS Team about the arrival of winter, but both again essentially agreed with the Roffman calculation about daylight 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.

  

TABLE 2 – ASHIMA DATA ERRORS UP THROUGH JANUARY 2, 2013

NOTE: Most of these mistakes which were posted by us on January 2, 2013 were repaired by May, 2013, but as we indicate above in conjunction with Ls 90, such errors in reporting do continue into 2014.

DATE

SOL

DATA BELOW TAKEN FROM THE ASHIMA SITE AT http://marsweather.com/data

 
   

FIRST COMMENT

SECOND COMMENT

23-Aug

16

   

25-Aug

19

2 days forward, 3 sols forward.

 

27-Aug

21

   

29-Aug

23

   

28-Aug

22

right info, wrong order

 

1-Sep

25

Data days Aug 28, 29, (missing 30 and 31) and Sep 1 (5 days), Sols over period 22, 23, 24 (missing) and 25 (4 Sols)

REMS reports 742 hPa, Ashima later corrects their copied figure

1-Sep

25

Same date and Sol, but different pressure & temperature. Ls changes from 165 to 162 (too much and wrong way).

 

2-Sep

26

Ls stays at 165.

 

3-Sep

27

Ls stays at 165.

 

4-Sep

28

Ls now 166

 

5-Sep

29

Last day REMS reports over 742 hPa (747 today)

 

6-Sep

30

   

7-Sep

31

   

8-Sep

32

   

10-Sep

34

Missing Sol 33, but spacing is correct.

 

11-Sep

35

Pressure jumps off curve from 7.48 to 7.99 hPa

 

12-Sep

35

Same Sol number as 11 Sep

 

13-Sep

37

Sol number consistent with 11 Sep.

 

14-Sep

38

   

15-Sep

39

   

16-Sep

40

Pressure jumps off curve from 7.57 hPa to 8.04 hPa

 

17-Sep

41

Pressure back to 7.52 hPa

 

18-Sep

42

   

19-Sep

43

   

20-Sep

44

   

21-Sep

45

   

22-Sep

46

   

23-Sep

47

   

24-Sep

48

   

25-Sep

49

   

26-Sep

50

   

27-Sep

51

   

28-Sep

52

   

29-Sep

53

   

30-Sep

54

   

1-Oct

55

   

2-Oct

56

   

3-Oct

57

   

4-Oct

58

   

5-Oct

59

   

6-Oct

60

   

7-Oct

61

   

8-Oct

62

   

9-Oct

63

   

11-Oct

65

Low -72, Ls 186

 

11-Oct

66

Low -73, Ls 187, 2 days in a row labeled 11 Oct

 

13-Oct

67

Ls 187. This date and Ls is consistent with the 1st 11 oct.

 

14-Oct

68

No pressure available

 

16-Oct

70

15 Oct and Ls 69 missing

 

17-Oct

71

   

18-Oct

72

   

19-Oct

73

   

20-Oct

74

   

22-Oct

75

Date changes by 2, Sol by 1.

 

23-Oct

76

   

24-Oct

77

   

3-Nov

87

   

5-Nov

89

4 Nov and sol 88 missing. Ls 202.

 

6-Nov

90

   

7-Nov

91

   

8-Nov

92

   

10-Nov

89

2 days forward, 3 sols back? Ls 205, Pressure 8.22 hPa

 

11-Nov

89

Ls 205 (third Sol 89). Pressure 8.16 hPa

 

12-Nov

96

1 day forward, 7 sols forward. Ls 205, Pressure 8.27 hPa

 

13-Nov

97

Ls 205

 

14-Nov

98

Ls 206. An identical sol 98 is on the Ashima display.

 

14-Nov

98

   

15-Nov

99

   

18-Nov

102

Nov 16 & 17, Sols 100 and 101 missing.

 

20-Nov

104

Nov 19, Sol  103 missing.

 

25-Nov

108

Nov 21 (Sol  105), 22 (Sol 106), 23 (Sol 107) and 24 (Sol 108) Missing. Nov 25, shown as Sol 108 should be Sol 109.   Both Ashima and REMS report Sol 108 as November 25. Sol 109 and 110 missing.

Ashima reports this Sol at Ls 221. In reality November 24 is Sol 108 at Ls 212.4 and November 25 is Sol 109 at Ls 213.1. There is no pressure or other data reported out for the previous 4 days when a regional dust storm was southwest of MSL.

28 Nov

111

Missing.

 

Nov 29

111

 

 

30 Nov

112

 

 

1 Dec

113

LATER SOL 144 WAS ALSO LABELED AS DEC 1

 

2 Dec

114

 

 

2 Dec

115

 

 

3 Dec

116

 

 

8 Dec

121

 

 

9 Dec

122

No report

 

10 Dec

123

 

 

11 Dec

124

 

 

22 Dec

135

 

 

23 Dec  to 39 Dec

136 to 143

No reports – Vacation?

 

31 Dec

144

 A SECOND REPORT LABELED SOL 144 WAS DATED DEC 1, BUT PRESSURE

DROPPED ON IT TO 851 FROM 907 HPA.

 

1 Jan

145

 

 

2 Jan

146

Date labeled as ene (Spanish for January)

 

 

 

Figure 3 - We found it necessary to correct the REMS Team and JPL about which month it was on Mars back in August 2012. They made the correction, but the level of mistakes made has been high enough to raise questions about what is really going on with the REMS Team. Are they being prevented from revealing the truth about Martian air pressure, or were they simply incapable of distinguishing between hPa and Pa pressure units?


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.

 

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

  
 

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.

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
69 -4.59 23.4127291 88.19380533 0.503435229 12.0824455 12.329895 -0.24745 -14.8469709 11 hr 45.15303 min
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
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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
79 -4.59 24.69586459 88.07206189 0.502740283 12.06576679 12.329895 -0.264128 -15.8476932 11 hr 44.15231 min
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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92 -4.59 25.17358442 88.02612142 0.502478042 12.059473 12.329895 -0.270422 -16.2253206 11 hr 43.77468 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
100 -4.59 24.78126436 88.06387452 0.502693547 12.06464513 12.329895 -0.26525 -15.9149928 11 hr 44.08501 min
101 -4.59 24.69586459 88.07206189 0.502740283 12.06576679 12.329895 -0.264128 -15.8476932 11 hr 44.15231 min
102 -4.59 24.60250317 88.08100016 0.502791305 12.06699132 12.329895 -0.262904 -15.7742213 11 hr 44.22578 min
103 -4.59 24.50122608 88.09068169 0.50284657 12.06831768 12.329895 -0.261577 -15.6946397 11 hr 44.30536 min
104 -4.59 24.39208299 88.10109833 0.502906031 12.06974475 12.329895 -0.26015 -15.6090158 11 hr 44.39098 min
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107 -4.59 24.01800702 88.1366692 0.50310908 12.07461792 12.329895 -0.255277 -15.3166257 11 hr 44.68337 min
108 -4.59 23.8779663 88.14993419 0.5031848 12.0764352 12.329895 -0.25346 -15.2075885 11 hr 44.79241 min
109 -4.59 23.73035947 88.16388591 0.503264441 12.07834657 12.329895 -0.251548 -15.0929064 11 hr 44.90709 min
110 -4.59 23.57525619 88.1785134 0.503347938 12.08035052 12.329895 -0.249544 -14.9726694 11 hr 45.02733 min
111 -4.59 23.4127291 88.19380533 0.503435229 12.0824455 12.329895 -0.24745 -14.8469709 11 hr 45.15303 min
112 -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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122 -4.59 21.15838967 88.40234802 0.50462565 12.1110156 12.329895 -0.218879 -13.132765 11 hr 46.86723 min
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
129 -4.59 19.31551336 88.56832925 0.505573118 12.13375483 12.329895 -0.19614 -11.7684112 11 hr 48.23159 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
134 -4.59 17.82835026 88.69967872 0.506322898 12.15174955 12.329895 -0.178145 -10.688728 11 hr 49.31127 min
135 -4.59 17.51525716 88.72706138 0.506479206 12.15550094 12.329895 -0.174394 -10.4636445 11 hr 49.53636 min
136 -4.59 17.19720459 88.75478643 0.506637468 12.15929924 12.329895 -0.170596 -10.2357466 11 hr 49.76425 min
137 -4.59 16.87430512 88.78284203 0.506797617 12.16314282 12.329895 -0.166752 -10.0051316 11 hr 49.99487 min
138 -4.59 16.54667169 88.81121657 0.506959587 12.1670301 12.329895 -0.162865 -9.77189496 11 hr 50.22811 min
139 -4.59 16.21441747 88.83989864 0.507123313 12.17095951 12.329895 -0.158936 -9.5361304 11 hr 50.46387 min
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142 -4.59 15.19106443 88.92767948 0.507624391 12.18298538 12.329895 -0.14691 -8.81457822 11 hr 51.18542 min
143 -4.59 14.84146195 88.95748228 0.507794513 12.18706832 12.329895 -0.142827 -8.56960138 11 hr 51.4304 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
147 -4.59 13.40365796 89.07913482 0.508488941 12.20373458 12.329895 -0.12616 -7.56962624 11 hr 52.43037 min
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149 -4.59 12.66270765 89.14129193 0.508843751 12.21225002 12.329895 -0.117645 -7.05869928 11 hr 52.9413 min
150 -4.59 12.28711642 89.17267137 0.509022874 12.21654897 12.329895 -0.113346 -6.80076253 11 hr 53.19924 min
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152 -4.59 11.52626529 89.2359892 0.50938431 12.22522344 12.329895 -0.104672 -6.2802945 11 hr 53.71971 min
153 -4.59 11.14123007 89.26791115 0.509566529 12.22959671 12.329895 -0.100298 -6.01789839 11 hr 53.9821 min
154 -4.59 10.75327109 89.29999786 0.509749689 12.23399255 12.329895 -0.095902 -5.75414796 11 hr 54.24585 min
155 -4.59 10.3625 89.33224165 0.509933746 12.23840991 12.329895 -0.091485 -5.48910631 11 hr 54.51089 min
156 -4.59 9.969028179 89.36463506 0.510118657 12.24284777 12.329895 -0.087047 -5.22283479 11 hr 54.77717 min
157 -4.59 9.572966702 89.39717083 0.51030438 12.24730513 12.329895 -0.08259 -4.95539311 11 hr 55.04461 min
158 -4.59 9.17442634 89.42984189 0.510490876 12.25178102 12.329895 -0.078114 -4.68683933 11 hr 55.31316 min
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162 -4.59 7.557680868 89.56174433 0.511243812 12.2698515 12.329895 -0.060044 -3.60261083 11 hr 56.39739 min
163 -4.59 7.148397059 89.59499233 0.511433601 12.27440643 12.329895 -0.055489 -3.32931467 11 hr 56.67069 min
164 -4.59 6.737292666 89.62833699 0.511623942 12.27897461 12.329895 -0.05092 -3.05522391 11 hr 56.94478 min
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167 -4.59 5.494141598 89.72889286 0.512197944 12.29275065 12.329895 -0.037144 -2.22866191 11 hr 57.77134 min
168 -4.59 5.076839171 89.76256672 0.512390164 12.29736393 12.329895 -0.032531 -1.95186522 11 hr 58.04813 min
169 -4.59 4.658257039 89.79630913 0.512582775 12.3019866 12.329895 -0.027908 -1.67450502 11 hr 58.32549 min
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175 -4.59 2.125894518 89.99991384 0.513745008 12.3298802 12.329895 -1.48E-05 -0.00088898 11 hr 59.99911 min
176 -4.59 1.701352853 90.03399304 0.513939542 12.33454901 12.329895 0.004654 0.27923957 12 hr 0.27924 min
177 -4.59 1.276386306 90.06810071 0.514134238 12.33922172 12.329895 0.009327 0.55960221 12 hr 0.559602 min
178 -4.59 0.851101167 90.10223209 0.51432907 12.34389767 12.329895 0.014003 0.8401597 12 hr 0.84016 min
179 -4.59 0.425603669 90.13638243 0.51452401 12.34857623 12.329895 0.018681 1.12087298 12 hr 1.120873 min
180 -4.59 2.98768E-15 90.17054697 0.51471903 12.35325673 12.329895 0.023362 1.40170306 12 hr 1.401703 min
181 -4.59 -0.425603669 90.20472098 0.514914105 12.35793853 12.329895 0.028044 1.68261099 12 hr 1.682611 min
182 -4.59 -0.851101167 90.23889972 0.515109207 12.36262098 12.329895 0.032726 1.96355777 12 hr 1.963558 min
183 -4.59 -1.276386306 90.27307843 0.515304309 12.36730342 12.329895 0.037408 2.24450426 12 hr 2.244504 min
184 -4.59 -1.701352853 90.30725231 0.515499383 12.3719852 12.329895 0.04209 2.52541109 12 hr 2.525411 min
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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