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5.  Lunar Motion and Data:  An Illustrated Primer



 
 
 I
N
D
E
X
How Does the Moon Work?
Four Basic Moonphases
– New Moon
– First Quarter
– Full Moon
– Last Quarter
Lunar Orbit vs Lunar Month
Fluctuation in Length of Synodic Month
Causes for Synodic Month Fluctuation
– Overview of Geometry and Motions Involved
– Cause 1:  Variable Speed of Orbiting Moon
– Cause 2:  Same Angle, Different Distance
Range of Variation from Mean Synodic Month
Was There a Full Moon on 2-20-11013 BC?
Comparison With 37 Modern Full Moons
Was There a New Moon on 2-1-4990 BC?
Comparison With 37 Modern New Moons
Lunar topics in Other Chapters
Calendars based on the moon  /  Chapter 1
Was the moon created full?  --Discussion  /  Chapter 2
The Full-Moon-Cycle  /  Chapter 2
Creation lag-time fits Full-Moon-Cycle  /  Chapter 2
Seventeen, 7000, and the Full-Moon-Cycle  /  Chapter 2
Precession of Moon's elliptical orbit  /  Chapter 2
1472 Precessions, Creation-to-Destruction  /  Chapter 2



How does the Moon work?

The following is a beginner's look at the Moon, allowing newcomers to astronomy the chance to understand and trust this study's lunar evidence.

These original illustrations were crafted to present the Bible student with a simplified minimum of information necessary to understand God's great Celestial Clock.



There are three celestial bodies involved with moon-phases:


The Sun shines light on the Moon which we see here on Earth.




It takes motion among these bodies to create the different moonphases.  Here is an overhead look at how they move:

The Earth rotates counter-clockwise.

The Moon orbits the Earth counter-clockwise.

The Sun remains fixed.

The Earth/Moon system orbits the Sun counter-clockwise.




Only one of these motions is involved with moon-phases-- the Moon's orbit around the Earth.

The daily rotation of the Earth is not connected with moon-phases.  It merely allows everyone on Earth to see the Moon each day.

The Moon is a sphere (ball).  One half is always lit-- the side facing the Sun.   As the Moon slowly progresses around the earth each month, we get a glimpse of its surface from all angles, seeing both lit areas and dark.

The process of lunar motion and observable phases is probably not understood by most.  Perhaps the greatest factor confusing casual observers is the speed at which the Earth rotates.  The Moon seems to be moving a lot, when actually most of its apparent "motion" is an illusion caused by our vantage point from the rotating Earth.

The Moon's motion, per day, is in reality not that great.




Four Basic Moonphases

The location of the Sun, Moon, and Earth determine how the Moon will appear to anyone standing on the Earth.  Let us step off the Earth and see where our three key players are during the lunar cycle.

Note that in each image, the Earth and Moon are lined up to provide an "Earth's-eye-view" of the moon-phase.


The New Moon starts the lunar cycle.

It rises and sets with the sun.  Hence, a perfectly "new Moon" is impossible to observe until enough separation occurs between the two bodies.  

It will first be visible as a thin crescent Moon at one of the next sunsets, after the Sun has set.  At that time, the (recently) new Moon will be visible just above the horizon.  At this point it "re-appears" after having been lost in the bright Sun for the previous couple of days.

A new "lunar month" may be decreed on the day of this first sighting, or back on the day the astronomical new Moon is calculated to have occurred, depending upon the convention of those observing such a "lunar" calendar.


First quarter (one week later)

The Moon is 90 degrees east from the Sun, and rises six hours after it.



Full Moon (one week later)

The Moon is directly opposite the Sun, rising twelve hours after it.


Last quarter (one week later)

The Moon is 90 degrees west of the Sun, and rises six hours before it.




Lunar Orbit  vs  Lunar Month

As the Moon travels around the Earth, there are two competing reference points to determine when a complete trip has been accomplished:

    1.  The stars

  • Moon arrives at the same spot relative to the starfield (distant, unmoving)
  • Moon describes a 360-degree arc
  • Period: 27.32216 days (fixed)
  • Moon completes a "lunar orbit"

    2.  The Sun

  • Moon arrives at the same spot relative to the Sun (i.e., same moon-phase)
  • Moon describes a 360 + 12.37 = 372.37 -degree arc
  • Period:  29.530588853 days (variable; to within 1/2 day)
  • Moon completes a "synodic month" (lunar month)

Note the Period of (2.) above: The 29.5 days includes 27.322 days (to orbit 360 degrees) plus an extra two days the Moon must travel to "catch up" with the Earth's angular change from the Sun.  Remember, throughout the last month, the Earth-Moon system has moved in its orbit around the Sun.

How much of an angle?  A fraction of the entire year's orbit, equal to (in degrees):

Angle  =  (days in Earth's orbit)  /  (days in synodic month)

          =  365.2422  /  29.530588853

          =  12.36827 degrees

This is the change in the Earth's angle from the Sun, accrued over one month of its incessant counter-clockwise orbit.  The Moon must therefore add this same angle to its counter-clockwise motion around the Earth, to resume its starting position relative to the Sun.

Angular motion of Earth around Sun (yellow angle)
= same angle Moon must add to its orbit (red angle)
to attain same phase.




Fluctuation in Length of Synodic Month  

From "Hebrew Calendar" / Wikipedia, the free encyclopedia:
(
http://en.wikipedia.org/wiki/Hebrew_calendar )

"A synodic month is the period between two lunar conjunctions, such as between two new moons."  
A C T U A L   ( Astronomical, or True )    N e w   M o o n
synodic month
( lunar month
or lunation )
> period between two lunar conjunctions, such as between two new moons
> duration varies within 1/2 day of average
starting point:

> actual lunar conjunction

"Since the actual length of a synodic month varies by several hours from month to month, the [Hebrew] calendar is based on a long-term average length called the mean synodic month.  The virtual [calculated] lunar conjunctions at the start of each mean synodic month [long-term-average length month] are called molads"


C A L C U L A T E D   ( Virtual, or Cyclic )   N e w   M o o n
mean synodic month

> long-term average length synodic month,
defined as 29.530588853 days
starting point:


> "cyclic new moon" or "molad" (a Hebrew word meaning "birth"):  the virtual (calculated) lunar conjunction at start of mean synodic month

"The mean synodic month used in the Hebrew calendar is exactly 765433/25920 days, or 29 days, 12 hours, and 793 parts (44+1/18 minutes) (ie 29.5306 days).  This interval exactly matches the mean synodic month determined by the Babylonians before 250 BCE and as adopted by the Greek astronomer Hipparchus and the Alexandrian astronomer Ptolemy.  Its remarkable accuracy (less than one second from the true value) is thought to have been achieved using records of lunar eclipses from the eighth to fifth centuries BCE."  

"...[D]ue to the eccentricity of Earth's orbit, series of shorter lunations alternate with series of longer lunations.  Consequently the actual lunar conjunction moments can range from 12 hours earlier than to 16 hours later than the molad moment [virtual, or calculated conjunction]..."

From Ortelius.de / Glossary page:
( http://www.ortelius.de/kalender/gloss_en.php )

"The cyclic calculation [of the Moon] assumes the lunar month [mean synodic month] to have a constant and invariable (mean) length whereas the true lunar months vary in length considerably due to the irregularities of the moon's orbit.  The calculated new and full moons are called cyclic new/full moons as against the true new/full moons determined by observation."



Causes for Synodic Month Fluctuation

The causes of fluctuation in the actual, precise length of the moon-phase cycle (synodic month) are at least two-fold-- each a consequence of the elliptical nature common to planetary orbits.

Both the Earth and the Moon have slightly elliptical orbits.  This is in accord with Keppler's First Law of planetary motion:

"The orbit of every planet is an ellipse with the Sun at a focus."
Actually, Keppler's laws apply anywhere a relatively small object orbits a relatively large object.  So in terms of the Earth-Moon system, the law would read:
"The orbit of  the Moon  is an ellipse with  the Earth  at a focus."
Many may be quick to assume that orbits are perfectly circular.  Indeed a circular orbit is possible... as a circle does qualify geometrically as an ellipse:  that is, one with foci zero distance apart.  Yet for whatever divine reason, most orbits were created with slight "imperfections":  tilted, and/or elliptical.

Yet these complications add richness and complexity to the behavior of, and interraction between, celestial bodies.

As we learned the above section,
Fluctuation in Length of Synodic Month, the elliptical nature of both the Earth's orbit, and the Moon's, plays a role in the cyclical variations of synodic month duration (i.e., new-moon to new-moon).

To gain clarity on why an elliptical orbit affects the timing of things, we will focus on the eccentricity of the Moon's orbit alone.  In the following illustrations, the slightly elliptical shape of the Moon's orbit is exaggerated to emphasize the principles involved.


Below we see the Earth-Moon system in motion around the Sun.  Notice the left-to-right orientation of the Moon's orbit in all four locations.  From this we see that, although its orientation relative to distant stars stays the same, its orientation with respect to the Sun-Moon-Earth alignment (dotted blue line) undergoes much change.


Elliptical Orbit of Moon, Over Several Months
Dotted blue line:  _ _ _ _ _ _ _ _ _ _ _
> Points from the Earth to the Moon... therefore it sweeps counter-clockwise as the Moon circles the Earth (red arrows).
> In diagram, it is "photographed" when the Sun becomes the third body in this alignment, at conjunction-- the "new moon".

Solid blue line:   ____________
> Points along axis of the Moon's elliptical orbit, providing a line of reference.

Note the angular difference, in the position required for conjunction, that accumulates with each lunar month.

Note:  Although the Moon's orbit is shown pointing left-to-right throughout the year, in reality the orientation of this orbit does shift slowly, counter-clockwise.  Since the rate of this shift is gradual (8.85 years, ~45 deg/year), and independent of the other motions displayed, it is safe to exclude it from this illustration.



To explain the first, we consult Keppler's second law of planetary motion:

"Arc sections of equal time sweep out areas of equal size."
The key implication of this proportionality of motion-to-area, is that orbital velocity at any given time has differing values, relative to the changing distance from the focus of the orbit.  To sweep out an equal "pie slice" on the distant portion of an orbit, with its longer straight-line distances from focus to satellite, there need be a shorter third "side" to this geometric shape-- the arced area along the orbital path.  Therefore objects in orbit must move slower when more distant from, and faster when nearer to, the orbited body.

Consider the following examples, extracted from the overview-graphic above.  In it, consider each "dot" as the location of the Moon from day to day.  The uneven spacing has been exaggerated to make the point.

November
May   ( inverted )
The slower-moving distant moon must travel 6 "days" to cover the same arc angle. The faster-moving near moon covers the same arc angle more quickly (in 1 "day").



The second cause of fluctuation is purely geometric, having nothing to do with orbital speed.

Due to different distances from the orbital focus, different amounts of the ellipse are intercepted:

November
May   ( inverted )
The same arc angle, when measured from the focus location on the left, encompasses a longer portion of the orbital track (in yellow), requiring more time to cover. The same arc angle, when measured to the near portion of the orbit, contains a shorter (yellow) portion of the orbital path, which can be covered in less time.



Range of Variation from Mean Synodic Month

The moon cycle operates with great regularity over long periods of time, yet due to explainable, intricate factors (i.e., eccentricities of orbits, gravitational interaction), fluctuation in the duration of a given cycle always exists.  Such variance is contained within a narrow window of a few hours from the calculated timespan.  

Yet for the Bible student, unacquainted with astronomy, to gain confidence in this claimed cycle-length regularity, we ask:  How many hours range is there, between the actual or astronomical new moon (at the start of the synodic month), and the calculated or virtual new moon (at the start of the mean synodic month)?

We will see that for 2005, 2006 and 2007 the lunations stay within a range of roughly six hours, either side of the long-term average, maintaining a steady, if slightly erratic, oscillation.

========================================================
Intervals between full moons (synodic months) are 
derived in this chart for the years 2005, 2006, and 2007

Column on right has synodic month deviation from average
full moon:  lateness (+) or earliness (-) 

Values range from [+6h:12m] to [-5h:58m]

---------------------------------------------------------
FULL MOON.....  SEPARATION.......   SYNODIC MO.    DEVIAT.

                                    ...avg = 29d+12:44...
Month-Day-Time  Part  Whole  Part                
     dd hh mm   day   days   day    dd + hh:mm      hh:mm
2005 ---------  ----  -----  ----   ----------    -------
JAN. 25 10:32\_
             / 13:28 + 29 +  4:54 = 29 + 18:22    +  5:38
FEB. 24  4:54\_
             / 19:06 + 28 + 20:58 = 29 + 16:04    +  3:20
MAR. 25 20:58\_
             /  3:02 + 29 + 10:06 = 29 + 15:08    +  2:24
APR. 24 10:06\_ 
             / 13:54 + 28 + 20:18 = 29 + 10:12    -  2:32
MAY  23 20:18\_   
             /  3:42 + 29 +  4:14 = 29 +  7:56    -  4:48
JUNE 22  4:14\_   
             / 19:46 + 28 + 11:00 = 29 +  6:46    -  5:58
JULY 21 11:00\_   
             / 13:00 + 28 + 17:53 = 29 +  6:53    -  5:51
AUG. 19 17:53\_   
             /  6:07 + 29 +  2:01 = 29 +  8:08    -  4:36
SEPT.18  2:01\_   
             / 21:59 + 28 + 12:14 = 29 + 10:13    -  2:31
OCT. 17 12:14\_   
             / 11:46 + 29 +  0:57 = 29 + 12:43    -  0:01
NOV. 16  0:57\_   
             / 23:03 + 28 + 16:15 = 29 + 15:18    +  2:34
DEC. 15 16:15\_   
2006 --------/  7:45 + 29 +  9:48 = 29 + 17:33    +  4:49
JAN. 14  9:48\_      
             / 14:12 + 29 +  4:44 = 29 + 18:56    +  6:12
FEB. 13  4:44\_   
             / 19:16 + 28 + 23:35 = 29 + 18:51    +  6:07
MAR. 14 23:35\_   
             /  0:25 + 29 + 16:40 = 29 + 17:05    +  4:21
APR. 13 16:40\_   
             /  7:20 + 29 +  6:51 = 29 + 14:11    +  1:27
MAY  13  6:51\_    
             / 17:09 + 28 + 18:03 = 29 + 11:12    -  1:32
JUNE 11 18:03\_   
             /  5:57 + 29 +  3:02 = 29 +  8:59    -  3:45
JULY 11  3:02\_ 
             / 20:58 + 28 + 10:54 = 29 +  7:52    -  4:52
AUG.  9 10:54\_   
             / 13:06 + 28 + 18:42 = 29 +  7:48    -  4:56
SEPT. 7 18:42\_  
             /  5:18 + 29 +  3:13 = 29 +  8:31    -  4:13
OCT.  7  3:13\_    
             / 20:47 + 28 + 12:58 = 29 +  9:45    -  2:59
NOV.  5 12:58\_    
             / 11:02 + 29 +  0:25 = 29 + 11:27    -  1:17
DEC.  5  0:25\_    
2007 --------/ 23:35 + 28 + 13:57 = 29 + 13:22    +  0:38
JAN.  3 13:57\_
             / 10:03 + 29 +  5:45 = 29 + 15:48    +  3:04
FEB.  2  5:45\_
             / 18:15 + 28 + 23:17 = 29 + 17:32    +  4:48
MAR.  3 23:17\_
             /  0:43 + 29 + 17:15 = 29 + 17:58    +  5:14
APR.  2 17:15\_
             /  6:45 + 29 + 10:09 = 29 + 16:54    +  4:10
MAY   2 10:09\_
             / 13:51 + 29 +  1:04 = 29 + 14:55    +  2:11
JUNE  1  1:04\_ 
             / 22:56 + 28 + 13:49 = 29 + 12:45    +  0:01
JUNE 30 13:49\_
             / 10:11 + 29 +  0:48 = 29 + 10:59    -  1:45
JULY 30  0:48\_
             / 23:12 + 28 + 10:35 = 29 +  9:47    -  2:57
AUG. 28 10:35\_
             / 13:25 + 28 + 19:45 = 29 +  9:10    -  3:34
SEPT.26 19:45\_
             /  4:15 + 29 +  4:52 = 29 +  9:07    -  3:37
OCT. 26  4:52\_
             / 19:08 + 28 + 14:30 = 29 +  9:38    -  3:06
NOV. 24 14:30\_
             /  9:30 + 29 +  1:16 = 29 + 10:46    -  1:58
DEC. 24  1:16

Full moon dates obtained from:
http://aa.usno.navy.mil/data/docs/MoonPhase.php





Was There a Full Moon on 2-20-11013 BC?


Comparison With 37 Modern Full Moons

 
========================================================
2-20-11013 BC and 2-19-11013 BC:  Which fits an even 
lunation count between then and now?  Compared to full
moons of 2005, 2006, and 2007.

1 Day = 0.0338632 mean synodic mo.s, therefore closest 
day (2-20 vs 2-19) to full moon must have # of lunations
(LUN) with remainder (R) within [0.098307 < R > 0.01693]

2-19 "wins" are indented 1 space

-------------------------- 2005 ------------------------
POSS. FULL MOON   FULL MOON   DAYSPAN       LUN   .  R 

2-20-11013 BC --  1-25-2005 = 4754336 dys = 160996.99276
2-20-11013 BC --  2-24-2005 = 4754366 dys = 160998.00866
2-20-11013 BC --  3-25-2005 = 4754395 dys = 160998.99069
2-20-11013 BC --  4-24-2005 = 4754425 dys = 161000.00659
2-20-11013 BC --  5-23-2005 = 4754454 dys = 161000.98862
2-20-11013 BC --  6-22-2005 = 4754484 dys = 161002.00452
2-20-11013 BC --  7-21-2005 = 4754513 dys = 161002.98655
 2-19-11013 BC -  8-19-2005 = 4754543 dys = 161004.00244
2-20-11013 BC --  9-18-2005 = 4754572 dys = 161004.98448
 2-19-11013 BC - 10-17-2005 = 4754602 dys = 161006.00037
 2-19-11013 BC - 11-16-2005 = 4754632 dys = 161007.01627
 2-19-11013 BC - 12-15-2005 = 4754661 dys = 161007.99830

-------------------------- 2006 ------------------------
 2-19-11013 BC -  1-14-2006 = 4754691 dys = 161009.01420
2-20-11013 BC --  2-13-2006 = 4754720 dys = 161009.99623
 2-19-11013 BC -  3-14-2006 = 4754750 dys = 161011.01212
2-20-11013 BC --  4-13-2006 = 4754779 dys = 161011.99416
2-20-11013 BC --  5-13-2006 = 4754809 dys = 161013.01005
2-20-11013 BC --  6-11-2006 = 4754838 dys = 161013.99209
2-20-11013 BC --  7-11-2006 = 4754868 dys = 161015.00798
2-20-11013 BC --  8- 9-2006 = 4754897 dys = 161015.99001
 2-19-11013 BC -  9- 7-2006 = 4754927 dys = 161017.00591
2-20-11013 BC -- 10- 7-2006 = 4754956 dys = 161017.98794
 2-19-11013 BC - 11- 5-2006 = 4754986 dys = 161019.00384
2-20-11013 BC -- 12- 5-2006 = 4755015 dys = 161019.98587

-------------------------- 2007 ------------------------
2-20-11013 BC --  1- 3-2007 = 4755044 dys = 161020.96790
2-20-11013 BC --  2- 2-2007 = 4755074 dys = 161021.98380
 2-19-11013 BC -  3- 3-2007 = 4755104 dys = 161022.99969
 2-19-11013 BC -  4- 2-2007 = 4755134 dys = 161024.01559
2-20-11013 BC --  5- 2-2007 = 4755163 dys = 161024.99762
2-20-11013 BC --  6- 1-2007 = 4755193 dys = 161026.01352
2-20-11013 BC --  6-30-2007 = 4755222 dys = 161026.99555
2-20-11013 BC --  7-30-2007 = 4755252 dys = 161028.01145
2-20-11013 BC --  8-28-2007 = 4755281 dys = 161028.99348
 2-19-11013 BC -  9-26-2007 = 4755311 dys = 161030.00938
2-20-11013 BC -- 10-26-2007 = 4755340 dys = 161030.99141
 2-19-11013 BC - 11-24-2007 = 4755370 dys = 161032.00730
2-20-11013 BC -- 12-24-2007 = 4755399 dys = 161032.98934
--------------------------------------------------------
SUMMARY AND COMMENT ON RESULTS:

2-19 Prevailed 12 of 37 lunations
2-20 Prevailed 25 of 37 lunations

  32% 2-19  :  68% 2-20

As "Biblical" 2-20-11013 BC began at evening 2-19-11013, 
and ended at evening 2-20-11013, 2-20-11013 consisted of: 

  25% 2-19  :  75% 2-20

The above table was tabulated using Bible-calculator, with full moon dates from:
http://aa.usno.navy.mil/data/docs/MoonPhase.php





Was There a New Moon on 2-1-4990 BC?


Comparison With 37 Modern New Moons

=====================================================
2-1-4990 BC and neighboring dates:  Which fits an even 
lunation count between then and now?  Compared to new
moons of 2005, 2006, and 2007.

1 Day = 0.0338632 mean synodic mo.s, therefore closest day 
(2-1, 2-2 or 1-31) to new moon must have # of lunations
(LUN) with remainder (R) within [0.098307 < R > 0.01693]

2-2 and 1-31 "wins" are indented 1 space  

------------------------ 2005 -----------------------
POSS. NEW MOON   NEW MOON   DAYSPAN       LUN   .  R 

2-1-4990 BC --  1-10-2005 = 2554485 dys = 86503.01599
2-1-4990 BC --  2- 8-2005 = 2554514 dys = 86503.99803
2-1-4990 BC --  3-10-2005 = 2554544 dys = 86505.01392
2-1-4990 BC --  4- 8-2005 = 2554573 dys = 86505.99595
2-1-4990 BC --  5- 8-2005 = 2554603 dys = 86507.01185
2-1-4990 BC --  6- 6-2005 = 2554632 dys = 86507.99388
2-1-4990 BC --  7- 6-2005 = 2554662 dys = 86509.00978
 2-2-4990 BC -  8- 5-2005 = 2554691 dys = 86509.99181
2-1-4990 BC --  9- 3-2005 = 2554721 dys = 86511.00771
 2-2-4990 BC - 10- 3-2005 = 2554750 dys = 86511.98974
 2-2-4990 BC - 11- 2-2005 = 2554780 dys = 86513.00564
 2-2-4990 BC - 12- 1-2005 = 2554809 dys = 86513.98767
 2-2-4990 BC - 12-31-2005 = 2554839 dys = 86515.00356

------------------------ 2006 -----------------------
 2-2-4990 BC -  1-29-2006 = 2554868 dys = 86515.98560
 2-2-4990 BC -  2-28-2006 = 2554898 dys = 86517.00149
 2-2-4990 BC -  3-29-2006 = 2554927 dys = 86517.98352
2-1-4990 BC --  4-27-2006 = 2554957 dys = 86518.99942
2-1-4990 BC --  5-27-2006 = 2554987 dys = 86520.01532
2-1-4990 BC --  6-25-2006 = 2555016 dys = 86520.99735
2-1-4990 BC --  7-25-2006 = 2555046 dys = 86522.01324
2-1-4990 BC --  8-23-2006 = 2555075 dys = 86522.99528
2-1-4990 BC --  9-22-2006 = 2555105 dys = 86524.01117
 2-2-4990 BC - 10-22-2006 = 2555134 dys = 86524.99321
2-1-4990 BC -- 11-20-2006 = 2555164 dys = 86526.00910
 2-2-4990 BC - 12-20-2006 = 2555193 dys = 86526.99113

------------------------ 2007 -----------------------
 2-2-4990 BC -  1-19-2007 = 2555223 dys = 86528.00703
 2-2-4990 BC -  2-17-2007 = 2555252 dys = 86528.98906
 2-2-4990 BC -  3-19-2007 = 2555282 dys = 86530.00496
 2-2-4990 BC -  4-17-2007 = 2555311 dys = 86530.98699
2-1-4990 BC --  5-16-2007 = 2555341 dys = 86532.00289
 2-2-4990 BC -  6-15-2007 = 2555370 dys = 86532.98492
2-1-4990 BC --  7-14-2007 = 2555400 dys = 86534.00081
 1-31-4990 BC - 8-12-2007 = 2555430 dys = 86535.01671
2-1-4990 BC --  9-11-2007 = 2555459 dys = 86535.99874
2-1-4990 BC -- 10-11-2007 = 2555489 dys = 86537.01464
2-1-4990 BC -- 11- 9-2007 = 2555518 dys = 86537.99667
2-1-4990 BC -- 12- 9-2007 = 2555548 dys = 86539.01257
-----------------------------------------------------
SUMMARY AND COMMENT ON RESULTS:

2- 1 Prevailed 21 of 37 lunations
2- 2 Prevailed 15 of 37 lunations
1-31 Prevailed  1 of 37 lunations

  3% 1-31  :  57% 2-1  :  41% 2-2

Subtracting 3% from both sides of 2-1-4990 yields:

              57% 2-1  :  38% 2-2

This is a ratio of:

                    3  :  2
                 (2-1)   (2-2)

Elligible dates for New Moon closest to February 1, 4990 BC are:

  (1-31), (2-1), (2-2)

The median and mode of these dates is:

  (2-1)

The above table was tabulated using Bible-calculator, with new moon dates from:
http://aa.usno.navy.mil/data/docs/MoonPhase.php

John O'Leary / Biblecalculator.com