Astronomical Easter Comparison & Calculation by OP Armstrong 10.25/15

Table-A Calculation Formula for Astronomical Easter Calculation, Gregorian Year JD# MethodA B C D E F G H I J

YR pEpact JD#1Jan JD#JanNM JD#PFM1 JD#Eqnx JD#PFM WkDy JD# Easter D#ck

-4000 24.3 260090.45 260113.8 260187.8 260170.2 260187.8 7 260188.8 1

2019 6.0 2458484.8 2458489.8 2458563.8 2458563.4 2458563.8 5 2458566.8 1

Yr pEPCT JD#Jan1.3 (B-1)+C 74.02+D F.3 or F.7 IF(E>=F,E,E+29.5) f.6(G)IF(H=1,G+7,8-

INT(H)+G)f.6(i)

The above Table-A is a tabulation of calculation steps for finding Astronomical date of Easter. The

example calculations use formula-2 of Table-B to determine moment of January New Moon for

selected year, expressed as astronomical Julian Day number, JD#. Alternatively one could use

any number of resources to arrive at a value for column B. Just be sure to express the moment as

an Astronomical formatted Julian Day Number, JD#. Julian calendar or Gregorian calendar dates

may be verified by the day of week. The numbers in column F, JD#.Equinox, are for equinox

moment by formula 4 of Table B. The month number and day-of-month may be determined in a

spreadsheet by adding two more columns and using formula 8 and 9,Table B, given that Year, Yr,

is stated as input in column-A. Alternatively, the Easter JD# may be converted back to Gregorian

Date by several free programs. This routine was compared by 70 dates. If using JMT in place of

GMT (UT), then add an offset of 0.098 days to step “I” and “J”.

A check was made against the WCC Easter dates table for years 2025 to 2001. Against that Table

this method using formula 2 and 4 reproduced their result. Other checked instances returned

results that matched either the Catholic Easter dates or alternative astronomical calculation

results. However the data of Ovidiu Vaduvescu did not confirm the astronomical values set forth in

the WCC document nor results of this calculation. If Equinox and PFM dates are closer than one

(1) day, verification by a more precise routine is advised on slide-9. NEXT N3 1

Astronomical Easter Comparison & Calculation by OP Armstrong 9.5/15

Name (Nu) TABLE-B Excel Astronomical Name Formula: Yr-year; JD#-Julian Day

pEpact.Cassidy.f1 29.09-MOD(MOD(Yr,19)*11-INT((Yr-1502.57-12*MOD(Yr,19))/228),29.983)

pEpact.Cassidy.0.f1b 29.5-MOD(MOD(Yr,19)*11-INT((Yr-1584-12*MOD(Yr,19))/228),30)

pEpact.Lunation#.f2IF((1+MOD((365.242454*(-4006-Yr)),29.5306))>=30,((1+MOD((365.242454*(-4006-

Yr)),29.5306))-30),(1+MOD((365.242454*(-4006-Yr)),29.5306)))

JD# Jan1.f3 257898.52-365.242454*(-4006-Yr)

JD# Equinox.f41st Page

(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.f5 JD#.Jan1 + pEpact + 59

Day of Week.f6 (1+INT(MOD((1.5+JD#),7))) one is Sunday and 7 is Saturday, etc

JD#21March.f7 257978.00-365.242454*(-4006-Yr)

Day of Month.f81+INT(MOD(((INT(MOD(((INT(JD#+0.5)+(-

37+INT(0.5+0.75*INT((INT(JD#+0.5)-4479.5)/36524.25))))-59.25),365.25)))+0.5),30.6))

Month Number.f9 , 3=March, 4=April

1+MOD((2+INT(((INT(MOD(((INT(JD#+0.5)+(-37+INT(0.5+0.75*INT((INT(JD#+0.5)-4479.5)/36524.25))))-

59.25),365.25)))+0.5)/30.6)),12)

2Next page

Look at 135 different years between 4000BC and

2038AD. The average lunar conjunction age at

4pm Easter Sunday was about 19. days, with a

minimum of 15.5 days and a max of 22.2 days.

Without the postponement rule, then some Easter

Sundays would land before a lunar age of 13.5 day

This gives a very early Lunar Good Friday of just

12.5 days. Application of the rule "Sunday

following the Paschal Moon" keeps Good Friday in

better alignment with the 14th lunar day. For at the

minimum found above, Good Friday falls more

closely on the 14th or 15th lunar day.

Recapping the "raised up on 3rd day", Good Friday

was day one by inclusive counting as Jesus was

laid in grave before sunset on the Sabbath. Then

day 2 was Friday after sunset to sunset Saturday,

and day 3 inclusive count was Saturday after

sunset unto about sunrise Sunday. Thus the

reason for sunrise Easter service is to memorialize

"Christ our Passover", "a feast by an ordinance for

ever." To the end that others may wish to abolish

this ".- feast by an ordinance for ever", they neglect

the blessings of the Almighty and sadly invite the

fires of judgment to blot their memory from off the

earth!

UTC to Jerusalem Meridian Time

Adjustment to Jerusalem time can be

accomplished by adding 2.33 hours or 0.098 days

to the “if” statements and day of week calculations.

Astronomical Easter Comparison & Calculation by OP Armstrong 9.5/15

Why the Sunday after the Paschal Full Moon?

Take a look at either the proleptic Catholic or

Astronomical Easters for years 30 and 33AD. For these

dates, the proleptic Gregorian Easter dates are April 7

and April 3, respectively. Next determine the Lunar

Conjunction (astronomical new moon) age of 17.9 and

17.2 days, respectively, at 4pm Sunday. Given that Jesus

died before Friday sunset and was buried before sunset,

so 1 day back is, Saturday 4pm and another day back is

Friday 4pm. Thus at time of Jesus death, the Lunar days

are 15.9 and 15.2 respectively. Given that the visible new

crescent or sighted moon is some where about 1 day

after conjunction. So on Good Friday the lunar sighted

moon of these years is 14 days in age. Thus Good Friday

of years 30AD and 33AD would either be on Nissan 14 or

15. However, if this skip week, “after” were not used, then

sometimes the Easter would come early to the lunar

calendar.

Why the Postponement Rule?

From this it is seen that reckoning Easter by these

methods speaks of "Christ our Passover", Jn18:39, 19:14,

1Cor5:7, Heb11:28, Num9:13, Ex12:14.

Since the Catholic Easter concurs the Astronomical

Easter in about than 90% of the dates, keeping Good

Friday and Easter are keeping "Christ our Passover" as

"a memorial; and .... a feast to the LORD throughout your

generations; ... feast by an ordinance for ever." Ex12:14.

The postponement rule importance is illustrated by

calculation. NEXT3

Astronomical Easter Comparison & Calculation by OP Armstrong 10.25/15

"And ye shall count unto you from the morrow after theSabbath, from the day that ye brought the sheaf of the waveoffering; seven Sabbaths shall be complete: Even unto themorrow after the seventh Sabbath shall ye number fiftydays; and ye shall offer a new meat offering unto the LORD."Lev 23: 15/16. The Fifty days were numbered from aSaturday, so the 50th day falls always on a Sunday, 49days or7 weeks after Easter SundayFor Christians, the Easter Sunday sets the precedent to findthe Sunday in the 7th month. The earliest Easter being 22March. By this the 7th lunar month starts around the lastweek of September. The latest Easter falls on 25 April, bywhich the 7th lunar month looks to fall on last week ofOctober. The 1st Sunday after and a 3rd Sunday would be alogical memorial for the Autumn Feasts.Effect of using Jerusalem as Principle MeridianExercise Caution when the Universal Time new moonmoment is on a Saturday night after 21:30 hours. As in 1998,when the Full Moon moment was Saturday at 22:23. The+2h21m offset between GMT and Jerusalem, gives a Paschalfull moon on 00:44 Sunday 12 Apr. This astronomical Easteris delayed unto the following Sunday. Compared to aCatholic and/or Universal Time Easter of 12April. This showsthe principle difference in the two methods. The Catholiccalculation is stepped in days and weeks. The Astronomicalmethod is a moment defined calculation. Thus it dependsupon the details of complex astronomical calculations.NEXT

Frequency of VariationFor 24 years, from 2001 to 2025, only once did theAstronomical Easter differ from the Catholic Easter. Thatbeing 2019, with Astro Easter on 24March, and CatholicEaster on 21April. Many of the tabulated dates of thetable were evaluated for known difficult dates likely tooffer discrepancy between the two methods. Even so, only11 of 60 years show variance between the methods. Somost likely, 90% of years will show the Catholic Easter andLunar Easter are same.

Other Feasts: Pentecost, Tabernacles, and The Lord's Supper

The other two of the three main feasts were Pentecostand Tabernacles. For Christians, Pentecost (7 weeks afterEaster) celebrates the Holy Spirit. The feast of Tabernaclescelebrates being freed from bondages of the flesh, theworld, and eternal death, thru election by God the Father.By celebrating these 3 feasts, Passover (hope of a morebetter resurrection thru Jesus Christ), Pentecost (hope ofcontinual renewal by Holy Spirit), and Tabernacles (hopeof election and freedom from bondage thru grace ofFather God), the Holy Trinity can be honored by Christians.Because Jesus initiated the Holy Communion, thencommunion on these days should be held in high esteem.The 1st of the first and seventh lunar months are seen asspecial days or Sabbath's. Also, the middle of thesemonths are feasts of remembrance.Pentecost is by the definition, always on a Sunday byvirtue of the week count, 7th Sunday after Easter.

4

Astronomical Easter Comparison & Calculation by OP Armstrong 10.25/15

Comparison of Three Major Hebrew FeastsThe Table at right compares the three feasts set forth byMoses. Two of these feast weeks are routinely celebratedin most churches. The first is Easter Holy week. That startswith Palm Sunday, followed by Good Friday, and lastly,Resurrection Sunday or Easter Sunday. The first of thesetwo Holy Days of Palm Sunday and Good Friday, correspondapproximately to the 10th and 15th days of a lunar month.This provisioned that the 1st day of the lunar month fallsupon a Friday, the 8th day being a Friday also, the 10th day isthen a Sunday, and the 15th day being a Friday. It isprovisioned that Jesus died on a Friday and His emptytomb found early Sunday; thus the term Holy Week.The feast of Pentecost is celebrated in most churches 7weeks after Easter. These first two Mosaic Feasts beingcelebrated, begs the question? If two are celebrated, thenwhy not the 3rd and if so, then how so? If one remembersthe words of the blessed Savior, “When you Fast” theywere not in the permissive sense but in the imperativeinstance. The 40 days of Lent could be taken as somethingdealing with fasting. However the Master was noted tohave kept the Feast of Tabernacles. So then does it notseem proper for churches to also remember this 3rd Feast?The Sunday following Passover is typically Catholic EasterSunday. As Easter is the Sunday following the first full moonafter Spring Equinox, then the start of the lunar month isabout 2 weeks prior.A lunar month is about 28.5 days in length. Six lunarmonths are then 171 days or 24.4 weeks.

The 3 Mosaic Feasts of 1st

month, 7th

Month Ex23:14, and Pentecost

Reference Nissan, Lev23, Nu28:16Pentecost,

Du16:16-17Trumpets -Tishari

MonthDay/

date+\-March/April May/June Ex23:14 September/Oct.

1st Day-A Sabbath

Sabbath, 1st Day of 1st

Month15th Nissan + 49D, Easter + 49 days

Sabbath & blow Trumpets 1st of 7th,

New Moon & sin

offering

7th DayA Sabbath Lev23:8,

Ex12:6A Sabbath

10th pick passover Lamb, Psalm Sunday

9th evening to 10th

eveningday of atonement,

fasting

14thPassover: sunset

Deut16:6, kill lamb, Ex12:6

15thunleven bread 7 days, 15th is Good Friday &

17th Easter

Tabernacles for 7 days, start with a

Sabbath

21st evening end unleven

22nd offering for 7 day

23rd Assembly

Taking out the two and a fraction of weeks elapsed untoEaster, then 22 weeks after Easter corresponds to nearly thestart of the seventh lunar month. A comparison of dates 22weeks after Easter to 1 Tishrei shows the two dates agreewithin a few days. The odd exceptions being when Hebrewcalendar postponement rules are applied. One possiblechurch memorial to this time would be Eucharist 22 and 24weeks after Easter. BACK NEXT

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Astronomical Easter Comparison & Calculation by OP Armstrong 10.25/15

First. Page

Easter

by UTC UTC

YR Astro Cath this astro Ful l Moon Equinox

2038 28Mr 25Ap 28-Mar 2465503.59 2465503.03

2019 24Mr 21Ap 24-Mar 2458563.57 2458563.42

1998 19Ap 12Ap 19-Apr 2450915.43 2450893.33

1967 02Ap 26Mr 2-Apr 2439575.64 2439570.82

1962 25Mr 22Ap 25-Mar 2437744.83 2437744.60

1954 25Ap 18Ap 25-Apr

1943 28Mr 25Ap 28-Mar 2430805.42 2430805.00

1927 24Ap 17Ap 24-Apr

1876 16Ap 09Ap 9-Apr 2406353.32 2406333.76

1845 23Mr 30Mr 30-Mar 2395014.35 2395011.24

1829 19Ap 26Ap 26-Apr 2389167.08 2389167.36

1825 03Ap 10Ap 10-Apr 2387719.77 2387706.39

1818 22Mr 29Mr 29-Mar 2385151.09 2385149.70

1805 14Ap 21Ap 21-Apr 2380425.49 2380401.54

1802 18Ap 25Ap 25-Apr 2379333.61 2379305.83

1744 29Mr 05Ap 29-Mar

1724 09Ap 16Ap 9-Apr

1700 04Ap 11Ap 4-Apr

550 26Ap 19Ap 26-Apr 1922022.20 1922022.43

-61 me26Mr NoCalc 26-Mar 1698859.401 1698859.395

97 YR's Astro v Catholic Easter Variants

2038AD-1AD

Only one application offers astronomical or‘uniform’ Easter calculation. Kalendis has notbeen ported to mobile devices. This Excelspreadsheet will find the Astronomical Easterfor years between 4007BC and 3027AD. In Tableto right, are differences between this calc andothers Astro-Easter. For these, exact values offull Moon and Equinox are input. The evaluatedUniform Easter dates by this excel sheet werethe same day as those from Kalendis. The belowchart shows variations of my Full Moon Date toKalendis. The normal range is +/- 1 hour forabout 7000 years. Beware, not allimplementations of mobile Excel have theneeded numerical accuracy to evaluate thesecomplex expressions. Only IOS-Numbers and MS-Excel routines could accurately execute thesecomplex calculations.

Astronomical Easter Comparison & Calculation by OP Armstrong 10.25/15

Excel Formula for Catholic Easter by Julian Day Number for Calendar Converter or direct Excel Cell

C var !. Single Input of Gregorian Year, if BC then 1-Yr.BC & find Easter in 7 steps, i.e. 30AD=> 30

.f10 P' =MOD(-8-11*MOD(Yr,19)+INT((Yr-1600)/100)-INT((Yr-1600)/400)-INT((8*INT((Yr-1400)/100))/25),30) 14

.f11 P =P'-IF(P'=29,1,IF((1+MOD(Yr,19))>11,IF(P'=28,1,0),0)) 14

.f12 D.1 =118+INT(365.25*(Yr+4712))-INT(0.75*INT(((Yr)/100)+49)) 1732097

.f13 D.2 =D.1+P 1732111

.f6 D.3 =1+INT(MOD((1.5+D.2),7)) 5

.f14 D.4 =IF(D.3=1,D.2+7,8-D.3+D.2)&"Catholic Easter as Julian Day Number" 1732114

end Easter use D.4 to get Day of Month, f.9, &Month#, f.8, &day-of-Week,f.6 as =D&f.n&… D7M4wkd1

Above is Catholic Easter formulation using Gregorian Year asinput to find Easter Date as a Julian Day Number. Thissimplifies several steps as compared to other methods. Theresultant JD# may be used to rapidly find Pentecost Sundayor any number of Easter dependant days. The JD# can berapidly changed to calendar dates via formula f.6, f.8, & f.9for most, if not all, dates. Beware excel dates do not displayfor years prior to 1900. Thus the above work around. TheEaster JD# can be ported to excel date system for years 1900forward. Simply determine the offset to Excel date numberand JD#, then subtract offset to other dates. Apply thismethod when a wide range of dates are to be reviewed.

Most Easter routines, but not this one, are valid for a fewhundred years after 1901. This is substantial since Christour Passover was slain from the foundation of the world.Thus there has been a perpetual Easter since the day thisworld was founded. That date was when Adam sinned,4000 years prior to Jesus Baptism at river Jordan by SaintJohn. The day of resurrection was hidden by God andrevealed by Christ earthly Easter day in 30AD.

The following list dates when Astronomical Easter is notsame date as the Catholic Easter, about 1 in 12 years.

Next page

257973-365.242454*(-4006-Y)+MOD((365.242454*(-4006-Y)),29.5306)+CFCF= -0.40614*SIN(l')+ 0.01614*SIN(2l’) + 0.17302*SIN(l) - 0.17+ CFtCFt = - C^3/999999.45 - C^2/4028.335 - C/64.259 + 1/ 547.41

Another estímate of astronomical PFM is

given at left. This uses Delaunay arguments

for lunar and sun anomalies, expressed in

Julian century, C of J2000. 7

Astronomical Easter Comparison & Calculation by OP Armstrong 10.25/15

This tabulation covers from years 2050 back to 60BC. Aspan of about 2100 years. The above Tabulation showsthat some centuries have less discrepancy than others.The cycle of 391-19 or 372 years can be seen in someskip sequences. The average is about 11 of 12 yearsagree and 1 in 12 years are not in agreement betweenthe two Easter Methods. However the pattern is notuniform. Some centuries have fewer concurrences thanothers. Next page

Given the primitive nature of physics and astronomy atthe time of the original formulation of the EasterCalendar, the Catholic method mostly agrees with theAstronomical formulation. That being stated, thePaschal Moon of the Catholic Easter is not a truemoon. In fact there is similarity to ‘Molad’ of thecalculated Hebrew Calendar. The Molad event is timedto start around the autumn equinox vs. around orabout the Spring Equinox for Catholic Easter.

8

Variant Astro-Easter Years to Catholic Variant Uniform Easter Years to Catholic

2049 1700 1457 1206 915 543 254 1903 1558 1351 1074 716 384 73

2045 1693 1453 1199 895 536 242 1900 1552 1332 1061 685 367 66

2038 1685 1446 1182 881 519 235 1876 1551 1331 1030 634 343 42

2019 1666 1429 1162 861 516 218 1873 1527 1328 1003 614 323 22

1981 1629 1427 1155 854 509 216 1845 1514 1313 998 607 320 15

1974 1622 1419 1142 846 496 191 1829 1507 1311 979 590 313 -5

1967 1609 1409 1135 837 489 188 1825 1503 1308 960 588 303 -29

1962 1598 1408 1128 827 482 171 1818 1487 1294 959 587 292 -49

1954 1590 1402 1111 817 474 168 1805 1484 1284 941 570 289 -53

1943 1582 1389 1108 810 469 144 1802 1483 1277 935 569 273 -56

1927 1578 1375 1101 783 438 137 1778 1473 1237 932 563 269 1998

1924 1571 1370 1088 736 411 124 1744 1465 1226 922 550 262

1923 1362 1081 729 1724 1463 1218

Astronomical Easter Comparison & Calculation by OP Armstrong 10.25/15

Uniform Easter Spreadsheet Method: Find full moonusing longitude routine and: Moon age = (Solar lessMoon Longitudes)*29.5306/360Since Full moon is 180 degree, then full moon age ishalf of 29.530… Three approximations (February,March, April) are used to find Paschal full moon date.When solved yield average days between spring fullmoons for the year. The first guess when corrected,may not meet Paschal Criteria after correction. Thus2 checks are used. Given the Equinox for a year, asEQ.jd, then the logics are: 1) IF(guess < EQ.jd, guess,

then new is (old guess+29.5), Guess-1 is 257972.7-

365.242454*(-4006-Yr)+MOD((365.242454*(-4006-Yr)),29.5306)

from which guess, an age is determied. This AGE is

converted to full moon correction, CF, for an improved guess

as follows: Cfi =29.5306/2 - IF(AGE>0, AGE, 29.5306+AGE)

Guess2 = Guess1 + CF1, other iterations are madeas Guess3 = Guess2 +CF2, the 4th and final value,Guess4 = Guess3 + CF3 is taken as the Full Moondate. A final check is used to ensure the Full Moondate falls after the Spring Equinox. Macro loops areavoided by using successive Full Moon dates:estimate, estimate +/- 29.5 days and logically selectPFM. The day, 257973, is March FM of Gregorianyear -4006. The file is embedded on page 6 of theMS document file. This file may be used to bettersee the method. A simplified method screened theprior table for validation by 70 term method. FM JD#

Constants: April 258002.07, March257972.70, February 257943.37

The Paschal Full Moon is defined as first moon, after

the Spring or Vernal Equinox. This spreadsheet

method of longitude was corrected and adapted after

T. Alonso Albi’s adaptation of P.Duffet but with

additional terms. T. Albi’s original lambda terms were

only about 24. Accuracy was improved by using more

Lambda terms. The total Sine terms comes to about

70. Evaluation of Delaunay’s arguments from Chapront

& Chapront ELP-S2001 was used to improve

accuracy. The ELP-S2001 4th order terms are reported

to be within 10 arc seconds down to 1500BC, if using

the full series of 100’s of adjustments. Here only about

70 terms are employed. The original method does not

account for Tidal Braking. Thus, an Equation of Time is

used to extend the solution validity. The extended

solution calculates Paschal Full moon dates to within

an hour of NASA or Kalendis for years between

4000BC to 3000AD. Typically less than 60 minutes

difference is found. The graph on page 6 shows the

variation for 110 points over a 7100 year span.

It is incorrect to apply angle reduction, Mod360, for

values less than zero. The spreadsheet solution is as

follows: reduce angle by A/360=rA, then find

IF(ABS(rA)>1, 360*(ABS(rA)-INT(ABS(rA)))*SIGN(rA), 360*rA).

This yields a correctly reduced angle for either positiveor negative input values. It is possible to improveaccuracy by limiting the year range, to just a few century.

Next page 9

Astronomical Easter Comparison & Calculation by OP Armstrong Nov7/15

New Moon Moment by SpreadsheetThe prior calculations were for finding a Full Moon momentby longitudes. Here a new moon moment is also calculatedby longitudes. This improved the accuracy of “moon age”cell in spreadsheet ‘Gregorian date to/from Julian Day’. Theprior calc used lunation number. The Solar to Lunarlongitudes method improves accuracy when used withadaption. The adaptation was to tailor an EOT specific forthis data set. In the first instance, Broomberg’s 4th orderlunation number correction was adjusted to minimumerror around J2000’. The results of that fit are given ingraph of this page. This gave +/- one hour error around themean between 4000BCE and 3000AD. Next a 3rd ordersupplemental correction was fitted in terms of JulianMillennia or c’/10. Where c’ is (JD#- 2451550.098)/36525.

The new moon moment was found by setting the longitudedifference to zero. The EOT is required because Longitudemethods do not account for tidal braking of earth rotation.This method is known as a truncated series of the VSOPanalytical series, as compared to the gold standard directintegration of DE200, as discussed in USNO write ups. Takeas example March 30 AD: guess M15 and JD# is 1732091,and EOT date is 1732095.950, and is used for second guess,the third and final calc gives a JD# of 1732096.244 for theEOT corrected date when moon age is zero, this is Mar2017:51HR vs. NASA table of 17:47 or 4 minute later. Likewisefor April 457BC , start midmonth at 1554614.00 to arrive atEOT JD# 1554617.311 or 19:28HR vs.19:25 NASA.BACK NEXT

10

For 128 points, the Standarddeviation was about 3 minutes ata mean of about zero. The RMSmean was 2 minutes. Themaximum/minimum differenceswere 10 and -7 minutes. Thisnearly corresponds to the 12 mingap of 2 times SD. This methodexplains 98% of determinedvariance.

Because the original equation is in hours, divide by 24 togive days. This correction is then added to the Broombergterm. The sum is then subtracted to yield a new moontime, +/- 6minute with 97.5% confidence.