method of finding paschal visable new moon by calendar review

Review Paschal Moon Age by Lunation Number Method OP Armstrong 4/15

Page 1

This review looked at use of Lunation Number to find New Moon date and subsequent moon age for sighted moon. The lunation number method is claimed to represent moon age +/- 0.25 days. This appears roughly correct for the values checked in Table 1. These dates ranged between CE2024 and 3975BC, nearly a 6000 year time

span. The lunation number method was checked against a routine published online, Torah-Times and by a second method from Fourmilabs. The error seems to be between 0.25 and 0.50 days. This is due to perturbations in rotation cycle of earth sun and moon. Table-1 shows the results comparison for these three methods. The average difference between the Lunation Number and TT method was -0.02 for 38 points over a span of 6,000 years. A larger error was seen when LN method was compared to Walkers New Moon Age. This begged the question of who has a more correct New Moon Date? Solar Eclipses happen on or during a new moon. Looking at eclipse of Wednesday, -2470 on Mar12 in Table-2 shows TT method of new moon is only 0.01 day off from the Eclipse date, whilst Walkers Fourmilab new moon age differs by 0.60 days and the LN method varies by 0.22 days. This is confirmed by review of data in Table 3. It shows little difference between Walkers new moon and TT at current dates but as age increases, so also does the variance between Walkers new moon and the TT new moon. The graph depicts this more aptly. As time increases to the left, so also does the variation between Walkers method and TTs method. Even with time correction one can see less but substantial difference. This anomaly is also confirmed by use of solar and lunar eclipse data. A copy of the spread sheet for calculation of the various items is available upon request. JD#-> Julian Day Number, JD.LN->age of new visible new moon by Lunation Number method, T.time.age->TorahTimes web page age of visible new moon, 4milLab-> new moon by Walkers web page, .corr-> corrected age by Bromberg Method: click links for details.

Table 1 summary of T3 Item/Method 4mi Lu# TT

Avg Error -0.2 -0.02

Std Dev 0.31 0.29

Max Age 1.84 2.2 1.92

Min Age 0.68 0.88 0.8

Table 2

Solar Eclipse Spot Check

JD New Moon Method

818980.79 Eclipse Sar208

818981.39 Walker New Moon

818980.97 JD of Lunation#

818980.78 Torah Times W-2470Mar12 date


Page 2

Table 3 Method of Finding Paschal Visible New Moon by Calendar review GoPg1

a b c d e Walker JD Moon Days+ i moon diff L m

DOW JD# Tcal YR Mo day NewMoon walker This err.w Tcal.d tc-me 4.ch cal'dr

mon 2460381.156 2024 Mar 11 2460379.88 1.28 1.31 -0.03 1.28 -0.03 2Vdr 1Ad.s

wed 2457842.165 2017 Mar 29 2457840.62 1.54 1.94 -0.40 1.54 -0.40 2Nis 2Nis

thur 2457458.156 2016 Mar 10 2457456.58 1.58 1.82 -0.24 1.58 -0.24 1Vdr 30Ad.r

sat 2457103.160 2015 Mar 21 2457101.90 1.26 1.20 0.06 1.26 0.06 2Nis 1Nis

sun 2439562.156 1967 Mar 12 2439560.69 1.47 1.37 0.10 1.47 0.10 1Vdr 30Ad.r

sun 2432268.161 1947 Mar 23 2432267.19 0.97 1.43 -0.46 0.97 -0.46 3Nis 2Nis

sat 2421312.162 1917 Mar 24 2421310.67 1.49 1.27 0.22 1.49 0.22 2Nis 1Nis

wed 2420574.158 1915 Mar 17 2420572.32 1.84 1.54 0.30 1.84 0.30 3Nis 2Nis

sat 2414018.167 1897 Apl 3 2414016.68 1.48 1.33 0.15 1.48 0.15 2Nis 2Nis

sun 2384045.156 1815 Mar 12 2384043.65 1.51 1.88 -0.37 1.51 -0.37 1Vdr 30Ad.r

mon 2311015.165 1615 Mar 30 2311013.80 1.37 1.02 0.35 1.37 0.35 1Nis 29Ad.s

tues 2237957.160 1415 Mar 21 2237955.83 1.34 1.70 -0.36 1.34 -0.36 2Nis 1Nis

tues 2164898.156 1215 Mar 10 2164896.69 1.47 1.38 0.09 1.47 0.09 2Nis 1Nis

thur 2091869.164 1015 Mar 30 2091867.50 1.67 1.52 0.15 1.68 0.16 2Nis 1Nis

fri 2018811.160 815 Mar 20 2018809.42 1.74 2.20 -0.46 1.77 -0.43 3Nis 2Nis

thur 1945751.155 615 Mar 9 1945749.97 1.18 0.88 0.30 1.21 0.33 2Nis 1Nis

sat 1872722.164 415 Mar 28 1872721.43 0.73 1.02 -0.29 0.80 -0.22 2Nis 1Nis

sun 1799664.160 215 Mar 19 1799662.54 1.62 1.70 -0.08 1.70 0.00 3Nis 2Nis

thur 1763135.158 115 Mar 14 1763133.61 1.55 2.04 -0.49 1.64 -0.40 2Nis 1Nis

sat 1746715.164 70 Mar 29 1746714.41 0.76 1.05 -0.29 0.86 -0.19 2Nis 1Nis

fri 1733190.160 33 Mar 18 1733189.05 1.11 1.06 0.05 1.21 0.15 1Nis 29Adr

fri 1732098.161 30 Mar 22 1732096.35 1.81 1.69 0.12 1.92 0.23 3Nis 2Nis

thur 1731005.163 27 Mar 25 1731004.35 0.81 1.32 -0.51 0.91 -0.41 2Nis 1Nis

fri 1730621.154 26 Mar 6 1730620.41 0.74 1.22 -0.48 0.85 -0.37 1Nis 29Adr

mon 1730267.159 25 Mar 17 1730265.76 1.39 1.58 -0.19 1.50 -0.08 1Nis 29Adr

sun 1726605.162 15 Mar 8 1726603.72 1.44 1.38 0.06 1.54 0.16 3Nis 2Nis

thur 1721142.162 0 Mar 23 1721141.10 1.06 1.54 -0.48 1.17 -0.37 2Nis 1Nis

sun 1719311.160 -5 Mar 19 1719309.50 1.66 1.43 0.23 1.76 0.33 3Nis 2Nis

sat 1715649.160 -15 Mar 9 1715647.96 1.20 1.23 -0.03 1.30 0.07 2Nis 1Nis

tues 1554589.164 -456 Mar 21 1554587.89 1.28 1.06 0.22 1.44 0.38 2Nis 1Nis

wed 1365387.158 -974 Mar 15 1365386.01 1.14 1.54 -0.40 1.40 -0.14 3Nis 2Nis

thur 1187140.156 -1462 Mar 6 1187139.12 1.04 1.17 -0.13 1.39 0.22 2Nis 1Nis

sat 1030097.158 -1892 Mar 17 1030095.89 1.27 1.85 -0.58 1.71 -0.14 2Nis 1Nis

tues 864991.152 -2344 Feb 29 864990.47 0.68 1.37 -0.69 1.25 -0.12 2Nis 1Nis

fri 634918.162 -2974 Mar 2 634917.16 1.00 1.72 -0.72 1.76 0.04 2Nis 1Nis

Sun 342712.157 -3774 Mar 19 342711.95 0.21 0.52 -0.31 1.24 0.72 30Nis 30Nis

thur 306183.155 -3874 Mar 14 306183.24 0.08 0.86 -0.78 0.99 0.13 2Nis 1Nis

Sun 269653.153 -3974 Mar 8 269653.47 -0.32 0.19 -0.51 0.79 0.60 Vdr29 Vdr28


Page 3

Date.Lunation.Calculator2.xlsx

The basic formulation for lunar age by Lunation number follows Bradley Schaefers widely accepted method with slight modification. Gopg1 Determine 1st day of Nissan by use of WHOLE Julian Day number nearest Gregorian March 20 Equinox so moon age falls between: 0.85< (read: 0.85 is less than expression) { 29.53 * ABS ((JD#w - 2451550 - 0.048) / 29.530588853 INT ((JD#w -2451550 - 0.048)/29.530588853))} < 2.2 (read: expression is less than 2.2) recheck date if ok ->end where JD#w = INT(JD#) INT is function for getting the integer part of number, remove spaces to paste formula in Excel.

Tables and graphs on page 4 show eclipse dates for New and full

moons. This data indicates that Walkers deviation from zero error increases in older times. The Lunation number method has a random error of about day. A full moon gives a lunar eclipse and a solar eclipse happens only in the new moon. Thus the use of eclipse dates provides a means to cross check accuracy of lunar phase predictions. In conclusion, the Calculated Hebrew Calendar has enough error to warn against using it to accurately state time of Paschal moon in Jerusalem.

Table 3 Column Schedule

a Day Of Week

b JD# of Visible Moon by Tcal

c Year, AY notation

d Month of Visible Gregorian

e Gregorian month date visible

f JD# of New Moon by Walker 4mi

g Moon Age, days, Walker-Tcal

h Moon Age, days, by Lunation#

i Difference of (g)-(h)

j Visible moon age days by Tcal

k Difference of (Tcal)-(Lun#)

L CHC Date by walker

m CHC Date: Calendrica ME2002

excel exemplar for date conversion and moon data by JD#, getZcalc

description of excel keys


Page 4

Verify Full Moon Date by Lunar Eclipse Date

Date-G eclps JD# wlkr.FM.JD FM.by JD#

-1997Oct14 991956.79 991956.728 991956.383

-1693Jun20 1102873.67 1102873.610 1102873.275

-1393May7 1212402.77 1212402.708 1212402.229

-1002May6 1355212.77 1355212.740 1355212.157

-749Sept25 1447760.85 1447760.824 1447761.022 Sat-

503Jly10 1537533.85 1537533.825 1537534.012

-300Feb20 1611537.69 1611537.668 1611537.668

-4Mar21 1719679.39 1719679.372 1719678.684

Sat25Mar1 1730250.76 1730250.747 1730250.635

Tue29Jun12 1731815.44 1731815.428 1731815.756

Fri329Oct25 1841522.21 1841522.199 1841521.894

Fri803Nov7 2014659.51 2014659.503 2014659.736

F1200Dec29 2159713.75 2159713.747 2159713.989

1493Apl11 2266467.56 2266467.557 2266467.067 1917Jly4 2421414.40 2421414.404 2421414.067

2050Oct30 2470109.64 2470109.636 2470110.008

Verify New Moon Dates by Solar Eclipse Dates

SAR Date NuMoon JD# Walker CF day JD# eclip

203 W-2470Mar12 818980.97 818981.39 0.71 818980.79

5 Sa-2468Aug13 819866.89 819866.71 0.79 819866.11

220 Mo-2429Jly5 834071.10 834071.54 0.39 834070.95

222 W-2353Jun5 861800.33 861799.87 -0.06 861799.97

9 M-2352May25 862154.69 862155.22 0.47 862154.64

213 W-2343May16 865432.59 865433.18 -0.06 865432.62

4 W-1807Jly31 1061279.46 1061279.21 0.57 1061279.27

43 W-1206Oct19 1280868.91 1280869.35 0.14 1280869.39

34 W-906Sept5 1390397.87 1390398.14 0.07 1390398.17

63 W-356Feb24 1591087.75 1591088.08 0.47 1591088.10

62 Sat83Dec25 1751734.15 1751734.09 -0.35 1751734.11

109 Su993Aug25 2083982.809 2083982.85 0.21 2083982.86

125 Sa1655Feb6 2325572.556 2325573.04 0.03 2325573.04

127 M1933Aug21 2427305.435 2427305.74 -0.12 2427305.74

NB: Listed correction factors, (CF) are for full moon & =CF Synodic + CF TT, Eclipse-data, goPg1

-0.40

-0.20

0.00

0.20

0.40

0.60

950000 1450000 1950000 2450000

Erro

r J.

D.

Julian Day Nr., JD#

Full Moon (FM 14.75d) by Lunar Eclipse (LE)

LEFM-Walker FM

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

7.8E+05 1.3E+06 1.8E+06 2.3E+06

err

or,

in

day

s

Date by Julian Day Number, range 2500BC to 1933

New Moon by Solar Eclipse

#e-#JDn

#e-3wnu+c

#e-3wnu


Page 5

These two charts show that use of Uncorrected Mean New Moon has substantial error. Possibly the Hebrew calendar uses a tabular correction term to reduce not eliminate error. But even so one can see about plus-minus 10 hours difference in the actual and the mean new moon by Lunation Number.

The above chart show even greater error of plus minus 0.85 days. A simple correction by Fourier series reduces the average error by a significant amount +/- 0.20 days. This is still substantial compared to a sighted New Moon The mean new moon by Lunation number is determined by Excel notation: =29.53 * ABS((JD# - 2457101.907) / 29.530588853-INT((JD# -2457101.907)/29.530588853)) To which one may or may not add a correction factor.

NB: these graphics are exemplars: to see clearly click referenced sites. Notes on Synodic & Lunation time correction factors from Various Sources. It appears the New Moon Time adjustment and the Synodic correction factors should both be added to the mean new moon date. This is based on a review of the Solar eclipse data (above). The Synodic Correction Factor for full and new moons are different. This calculation is concerned only with new moon. Bromberg Discussion graphics are reproduced as exemplar only. Reader advised to link click for their full text. There for calculating

but for visible new moon (not conjunction) consult, Torah-Times or Keralite Sites that are keen on finding visible crescent. For visible crescent kindly note that altitude, background light, and atmospheric conditions are important.


Page 6

One way to represent the mean moon error is by use of Fourier Series. The Fourier theorem states that any periodic function can be represented by summing a series of Sine and Cosine functions, using system frequencies. Various terms were tried. The result is given by the equations in Table to left. For 150 new moon points, the error span was reduced from +/- 0.8 days to an average of zero error and a S.D. of 0.08days. When coding

Excel for the formula, replace 1 by 2 in the above formula, as formula = F/2. Long term cycles add for the delta T factor by Fourier regression. The correction is added to JD# to estimate conjunction at moon age of 28.5 days. The more accurate celestial mechanic calculations use over 28 correction terms plus a delta T. The delta T factor is only reliable to around 2000BC. Some routines extrapolate T for older dates. When

extrapolation of T is used, know that results have substantial uncertainty. The T error may exceed the error

of the Lunation Number Conjunction. Discussion of Synodic CF from Wiki and others:

These correction factors and Dr. Bromberg discussion of the Meridian Location for Hillels Calculated Hebrew Calendar (CHC) show that computation of the CHC Molad does not correspond to actual new moon (Ps81.3) time in Jerusalem. The CHC, like other calculated calendars, only provide an estimate of when a conjunction, not a visible new moon, should happen. The Law and Prophets do not say preserve the molad. They say Sanctify the new moon. It seems these things are currently a historical item. It is likely, that during the time of Elijah and the Millennium, the Mosaic calendar will be instituted from Jerusalem, based upon a visible new moon.

Acts 15 tells that many things of the Law were so hard that even the Fathers had trouble keeping every Jot and Tittle. Colossians 2:16 warns,

so let none judge you in: food, drink, a festival, a new moon, or Sabbaths, which are a shadow of things to come. Also in Romans we are

warned not to judge others if they so desire to keep such customs, for all will give account of their actions in the judgment.

17

Considerations on Grattan Guinness Cycle, of 391Yrs by OP Armstrong 7.20/15Find mean new moon of equinox by Julian day number (JD#) Epact

Epact is age of moon (in days) upon January 1, for any year, Y,

expressed in proleptic Gregorian reform of Julian calendar system.

Linear method to find Epact is a counting method to determine moon age

at 1January using average lunation time. The Metonic counting method

was one of 1st ways to determine lunar age. The JD# of January 01 for

any Gregorian year from -4010 to 3055 is calculated with high precision

by Formula 3 of Table. Given JD# of 1January in a particular year, the

moon age is the Epact, formula 1 or 2 of Table.

Sundry Epact systems have been proposed, this proposal uses the

simplicity of the Julian Day number system. The counting of days by

JD# for astronomical events is the natural order. The calendar

conversion, then orders the date in any calendar of choice and thusly

reckons the effect of leap years. Leap days are calendar events that are

not associated with astronomical events.

A mean lunation is about 29.51 day. Thus, days unto end of Epact

Moon is 29.51 less Epact. This means the 1st moon in January is also

(29.51-E) days from 1January. There are then 2 more moons from this

time unto the first moon in March, this being an additional 59 days. This

is by Table formula 5, Marchs first moon.For any proleptic Gregorian year, Y, the JD# of astronomical

Vernal/Spring equinox is by Table formula 4, JD#.Equinox

The Catholic system calls the 1st Sunday after the 1st full moon after

spring equinox to be Easter Sunday. The full moon is about 14.75 days

after new moon. Thus 14.75 added to JD# of first March moon and it is

just simple counting to find Easter Sunday JD#. If the first March moon

JD# plus 14.75 is less than VE, then increment by (28.5 +14.75) days to

1st March moon and find next Sunday. All this is calculated by JD# and

the final JD# is converted to Gregorian calendar date, if needed.

Because the Catholic VE is artificially set to 21 March, then to get the

Gregorian Easter date, simply substitute JD# of 21March for year, Y.

This JD# is by Table Formula 7. The counting of week days from a JD#

is by Table formula 6.

The original 19 year Metonic cycle was eventually found to be deficient by

one day in 228 years. This secondary correction is still linear.

However for holiday dates, the Epact method is deemed

suitable. Typical Epact tables or Easter formula are given for

years after 1900 and valid for about 300 years forward. If

a larger range of dates are desired then another method

such as this is needed. The below graph compares the

methods over 7000 year range. From this graph, is seems

the Lunation Number Epact is better than a Metonic Cycle.

Name (Nu) Excel Name Formula-Yr-year, JD-JulianDay.astro

Epact.Cassidy.129.09-MOD(MOD(Yr,19)*11-INT((Yr-1502.57-

12*MOD(Yr,19))/228),29.983)

Epact.Lunation# (2)

(1+MOD((365.242454*(-4006-Yr)),29.5306)) &if>=30, then subtract 30

JD# Jan1-(3) 257898.52-365.242454*(-4006-Yr)

JD# Equinox-(4)

(2457102.448+(Yr-2015)*365.2422)+((-0.0005947871)*((Yr-2015)/1000)^4+(-

0.00392591)*((Yr-2015)/1000)^3+(0.013808751)*((Yr-2015)/1000)^2+(0.1590901)*((Yr-2015)/1000))

March 1st Moon JD#.Jan1 + Epact + 59

Day of Week-(6)(7-INT(MOD((1.5+JD#),7))) one is Sunday and 7 is

Saturday, etc

JD#21March-(7) 257978.00-365.242454*(-4006-Yr)

method of finding paschal visable new moon by calendar review

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