
This is my search for Chodesh: The New Moon worship day; just what and where is it? Further, whether anyone wants to keep the day or not, this investigation brings out a lot of fascinating points that anyone studying the Bible will find extremely interesting:
I started on this search with the only understanding of chodesh or the Bible new moon day basically just the new moon mentioned on my wall calendar. Anyone reading this should consider this as a reasonable starting point in their mind. The conclusions that I have drawn and the day I have constructed are not the same as the traditional Jewish new moon day. The Bible reasons for this are given. I may be wrong but this works for me. The working solution is very simple and based on the fact that the phases of the moon appear the same everywhere on the earth because of the distance of the moon from the earth. Though not the intent, the numerical solution could also be worked just as easily for the Jewish calendar though not really necessary since these calendars are readily available. Note this is nothing fancy—it's just a number to subtract from the astronomical date to get the correct day—approximately. Very Important Note This study investigates the general new moon worship day of the Bible.
It became clear that this was not as simple as it looked and it was possible that a few different approaches could be taken. One
alternative view was constructed and a numerical approximation for this was used from late 2011 through 2014. However, late 2014 an historical
search for the crucifixion date was undertaken which just might show which new moon
worship day approach was indeed most likely to be correct. This search was successful and one of the new moon
approaches did indeed supply a reasonable date for the crucifixion. And this result showed just which new moon worship day was most likely to
be the correct new moon worship day of the Bible. More information about this is given in the Important Historical Update at the end of this study. The numerical solution obtained was tested with the Jewish calendar for 2011 and in every case succeeded in landing on a Rosh Chodesh date. Some months the Jewish calendar listed 2 Rosh Chodesh days. Every time this happened the numerical solution was successful in landing on one of them. Further, using the Jewish calendar it was possible to make a small correction in the numerical solution if one wanted to use it to obtain Jewish new moon days. Update for 2013: As will be seen, the Jewish
correction proposed worked OK for 2011, and also
for 2012. However it did not do well for 2013. This is commented on in
the PostScript at the end of this study. Stephen.
So toss away any preconceived ideas you may have of this day and come on a curious walk as we wander about trying to figure out just what and where chodesh is. Note: chodesh
is the Hebrew word
that is translated mostly as month, but sometimes as new moon depending
on the context in the verse. The search for chodesh Very early in my Christian walk I came across Isaiah 66:22,23: For
as the new heavens and the new earth, which I will make, shall remain
before me, saith the LORD, so shall your seed and your name remain. This was certainly not a
problem. I believed Isaiah was here clearly stating that there would be
a weekly worship day of rest, a sabbath, which will be kept by God's
people for eternity. There was also mention of another worship day, the
newmoon, whatever that entailed. It was not very clear and I have not
heard much about it if anything at church. For all the years I have
been a Christian I have never heard a sermon about this extra day of
worship.
The years rolled by. As a
Christian I referred to these verses many times but as usual left the
newmoon part in the toohardbasket. Probably like most other
Christians who were travelling this road.
Late in 2010 this all changed. I was studying some verses in the Old Testament and realised that they definitely had a worship connection with the new moon. The new moon itself was not mentioned in the verses but to me it was very clear that the worship being outlined clearly was linked to the new moon worship day. And the worship I was looking at had absolutely nothing to do with the Law of Moses containing sacrifices and ordinances. By new moon here, I mean specifically a day of worship as outlined in Isaiah: “one new moon to another, ... shall all flesh come to worship...” From this point onwards I decided that this new moon worship day needed another look. So I decided to search for this worship day. Where would you start? Well the most obvious place to start looking for a newmoon worship day is of course, the newmoon. The first place to look was on the internet for sites that might list moon phase dates. This was successful and the first chart I obtained was as follows:
These dates were for UTC which I assumed was a certain time or location so checked where I could adjust this on the page. Setting the location for my approximate time location I then obtained the chart:
From
here,
it begins to get very interesting. One site gave the following information: So,
how long does each phase of the Moon last? Well, the phases are just
names we give to certain points along the Moon's smooth path around
Earth. Technically, each phase, just like the one called a New Moon,
when the Moon is exactly between Earth and the Sun, lasts only a
brief instant. But to our eyes, a New Moon can last for a few days,
representing the time that the Moon appears in the sky too near the
Sun's position for us to see it at all. The time it takes the Moon to
go through all its phases is about a month, and that was so important
to our ancestors that they created the period of time we call a
month. Maybe you've even noticed that the word month is like the word
moon. So
what we
call the new moon could last a few days. Is this the period of
worship though?
Thus
saith the Lord GOD; The gate of the inner court that looketh toward
the east shall be shut the six working days; but on the sabbath it
shall be opened, and in the day of the new moon it shall be opened. We are obviously not interested in any sacrifices mentioned with the newmoon but nevertheless these verses give us the length of time: the day of the new moon. So the new moon worship period is one day in length, basically as expected but this was now conclusive.
The next step would be to get some sort of confirmation that I was on the right track. The obvious spot to check was Jewish sites that might mention this. I very quickly ran into the term “Rosh Chodesh” which appeared to be the Jewish worship day for the new moon or beginning of the month. At this time I also found out that the Jewish month starts with the new moon. I was also going to find out that this was a very loose statement. What is Rosh Chodesh? Rosh
Chodesh is the monthly celebration of the New Moon, according to the
Jewish calendar. The Jewish calendar follows lunar months, each with
29 or 30 days, although the year is solar. Some scholars believe that
lunar months derive from ancient nomadic calendars and solar years
are the invention of agricultural societies; the Jewish calendar
combines the two. Many Jewish festivals are tied to the lunar cycle;
for example Sukkot and Passover begin on the full moon, in the middle
of the month. Since 12 lunar months do not add up to one complete
solar year, additional “leap months” are intercalated into the
calendar in seven years out of a 19year cycle. This looked very promising and straightforward. The exact astronomical time of the new moon would give the day for worship and the first day of the Jewish month. But once I started looking at Jewish calendars on the internet I found this was anything but the truth. The first calendar I pulled up was:
The first thing I
noticed
there were TWO days listed as Rosh Chodesh for March 2011: Sunday 6 AND
Monday 7! 5. The Hebrew New Month, as it goes
according to this
calculation
is either on the 30th or 31st day each month, as a month is 29.5
days, we have alternating New Months, being the 30th (and) or 31st
day. The following quotation [found at a later time and added here for continuity] also helps explain this: Sometimes
Rosh Chodesh is one day, but sometimes it is celebrated for two days.
Months are based on the lunar cycle, of course. The amount of time
required for the moon to make one complete revolution around the
earth is determined by the conjunction of the earth, moon, and sun
(i.e. they lie along a line). When this happens, it is the new moon.
A complete revolution takes 29 days, 12 hours, 44 minutes, 31/2
seconds. Days in our calendar must begin at sundown, regardless of
when the conjunction actually takes place. Hence it is necessary to
either add or subtract a half a day from each calendar month. For
this reason, Hebrew months alternate between 29 and 30 days in
length, with the actually moment of the new moon falling in between.
The second thing I noticed is that the day for the new moon did not sit well with my other chart obtained from the internet. For Sydney times I had the new moon in March on Saturday the 5^{th} at 7:46am. This would also be daylight saving time for where I was in Australia. The site I obtained the chart from also had UTC times. This gave the new moon on March 4 at 8:46pm. This was even worse and further away from the Jewish dates. I was not sure on the UTC times so decided to look further. And I couldn't have done better. I found a page from NASA which gave moon phases from 2001 to 2100. This of course was a long page and the part of interest with top of page is as follows: This did help a lot. The NASA site also explained that “Universal Time is equivalent to Greenwich Mean Time or GMT.” So I could use the chart to determine any new moon times on the planet anywhere. Since the NASA chart gave the same times as my original UTC chart I could then use either one. For Sydney Australia it was simply a matter of adding 10 hours plus one for daylight saving. But it didn't help the Jewish calendar Rosh Chodesh dates I was finding. I wrote this time to a couple of different Jewish groups in Sydney about this: I
have had a reply from another Jewish group about the New Moon but
there is still something wrong with the dates. I have had no reply. Anyway about this
time I
was also looking on the internet to check the lunar month time and
had also worked out a rough approximation using the first date chart
I obtained. For February to October I worked out 10 periods and took
the average and got something round 29.53 days ignoring the minutes.
I think with the minutes this went to about 29.54. These figures are
reasonable and are in the range given by NASA top of page previous.
A little more information about this... In one of the replies was a comment: We have a New Moon every 29 days, 12 hours and 44 1/18 minutes, so once you have the new moon for any given month you can figure every future one by adding the 29.12.44.1/8. Now this is not
exactly
correct. Apparently the period changes through the yearly cycle but
over a reasonable period of time the average would give the value
quoted. I decided to plot and average the values for 2011 with
results as follows.
Through 2011 the
synodic
month length ranged from 29.34 days up to about 29.76 days. The other thing that happened here I became aware of two different month periods. There are actually a few but probably the two most well known ones are the sidereal and synodic months. A
synodic
month is
29.53059 days (29 days, 12 hours,
44 minutes, 2.8 seconds) and is measured from New Moon to New Moon. So the
synodic month is
actually what we are interested in here and should be the same in
theory as the Jewish month. The Moon's
orbital period in a nonrotating frame of reference (which on average
is equal to its rotation period) is about 27.32166 days (27 days, 7
hours, 43 minutes, 11.6 seconds). This is known as a
sidereal month and
is measured by observing how long it takes the Moon to pass a fixed star
on the celestial sphere.
Simply put, the sidereal month is the time it takes the moon to complete one orbit around the earth. Wait
a
minute! Isn't one orbit of the moon around the earth meant to bring
the moon from one new moon to the next new moon? So shouldn't these
two periods be the same in length? Why are they different? A
synodic month is longer than a sidereal month because the EarthMoon
system is orbiting the Sun in the same direction as the Moon is
orbiting the Earth. Therefore, the Sun appears to move with respect
to the stars, and it takes about 2.2 days longer for the Moon to
return to the apparent position of the Sun. The synodic month is the
most common way of expressing the lunar cycle. To my shame I have to admit I have not really run across this before or even given it much thought—until now, after having read about it in the investigation of this matter. The best news is that it is actually very easy to show using a diagram. The mathematics of it is really very simple. Thankfully. This sort of thing as you could imagine could get very messy very quickly. In this case it doesn't! It can be done very simply by approximating the orbit of the earth as circular. Now someone is going to say you can't do that—the Earth's orbit is elliptic. True, but the eccentricity of the orbit makes it nearly circular: The
eccentricity of the Earth's orbit is currently about 0.0167, meaning
that the Earth's orbit is nearly circular, the semiminor axis is
98.6% of the semimajor axis. The
eccentricity of an ellipse, usually denoted by
or e, is the ratio of the distance between the two foci, to the
length of the major axis or e = 2f/2a = f/a. For an ellipse the
eccentricity is between 0 and 1 (0<e<1). When the eccentricity
is 0 the foci coincide with the center point and the figure is a
circle. Actually this approximation works pretty well and it can be used to evaluate the Moon's orbital sidereal month given the length of the synodic month! Here's a diagram I have constructed to show this [nothing to scale of course]:
The diagram shows the moon positions for one synodic and sidereal month. The Earth position is for one synodic month but for a simple comparison the moon position for the sidereal month is shown at the same time. As you can see from the diagram the moon has to travel some extra distance beyond a single orbit of the Earth to get to the next new moon. We can work out the extra angle and time as follows. To work out the angle of travel of the Earth we use the length of the synodic month of approximately 29.53 days. The angle of travel is simply what this gives given the year of 365.25 days marking out 360 degrees for the orbital circle [approximating the ellipse]. This calculation is simply:
So
as the
moon travels from one new moon to the next the Earth travels
approximately 29.1 degrees in its orbit around the Sun. Further,
it
is possible to use this result to find an approximate calculation for
the sidereal month given the synodic month is 29.53 days.
This
approximate calculation agrees with the true sidereal month value to
5 significant digits. That's a pretty good approximation!
or approximately 2.2 days as stated in the previous quotation. Before continuing with the new moon investigation there is a very important question to ask here. What is the obvious question to ask from looking at the diagram? Have a quick look at it again. Note especially the new moon positionings. Just what else do you think of when the moon is directly between the Earth and the Sun? Answer: when the moon is directly between the Earth and the Sun would not that give us a solar eclipse? If we have a new moon every 29.53 days why don't we have a solar eclipse then too? It's a good question and struck me when I was reviewing this study. I looked for the answer and found the following information: The Moon is a cold, rocky body about 2,160 miles (3,476 km) in diameter. It has no light of its own but shines by sunlight reflected from its surface. The Moon orbits Earth about once every 29 and a half days. As it circles our planet, the changing position of the Moon with respect to the Sun causes our natural satellite to cycle through a series of phases:
The
phase known as New Moon can not actually be seen because the
illuminated side of the Moon is then pointed away from Earth. The
rest of the phases are familiar to all of us as the Moon cycles
through them month after month. Did you realize that the word month
is derived from the Moon's 29.5 day period? Admittedly we only needed the last paragraph but there was a wealth of information here that just could not be missed; will mention some of this again later. Curiously though this site has the orbit of the moon as the synodic length, not the sidereal. A simple mistake. Now back to the investigation... I
was still
investigating the Jewish sites on the internet trying to make sense
of all this why their dates for Rosh Chodesh did not seem to fit with
the astronomical information I was finding.
H2320 חדשׁ chôdesh 2320 is the Strong's Concordance number for this word. The H before the number above just signifies this is a Hebrew word since there are also Strong's numbers for Greek words from the New Testament. You can see the Hebrew word for this below the number above and it is read from right to left and would be pronounced, of course, as “chodesh.” So
why do
all the Jewish sites refer to Rosh Chodesh instead of just Chodesh? I
kept searching and found the answer. Opening
Ceremony We
are gathered to celebrate Rosh Chodesh, the beginning of the new
Jewish month. When introducing the commandment of Rosh Chodesh, the
Torah says: "This month shall mark for you the beginning of the
months; it shall be the first of the months for you" (Exodus
12:2). Commenting on the fact that the verse says "This month
shall mark for you," Rabbi Samson Raphael Hirsch writes: "This
renewal of the moon shall be a beginning of renewals for you.
Noticing, realizing the fresh birth of the moon shall induce you to
achieve a similar rejuvenation. You are to fix your moons, your
periods of time by taking note of this ever fresh recurring
rejuvenation." Rosh Chodesh is a time for introspection and
reflection on the previous month that was, and on the new month that
will be. This does appear to be where the term came from. Looking at Exodus 12:2: This^{H2088} month^{H2320} shall be unto you the beginning^{H7218} of months:^{H2320} it^{H1931} shall be the first^{H7223} month^{H2320} of the year^{H8141} to you. You can see the Strong's number H2320 translated here as months. That's Chodesh. The Hebrew word before it, H7218 is:
H7218 ראשׁ rô'sh So
together
these two words give Rosh Chodesh and mean the beginning of the
month. So far my investigation was aimed at looking at the day of the new moon, and when was it. To my satisfaction I had been completely successful in this attempt. But now things were about to get a whole lot worse before they got better. Get better they did and I did eventually completely resolve this muddle, to my satisfaction, at least. But what went wrong? You have a good idea so far. None of the astronomical new moon dates I was finding seemed to fit any of the Jewish calendars. Then I started to find pictures like the following on some of the Jewish sites:
Look at the picture. Note the title on the right hand side says: ROSH CHODESH: A New Moon. Well maybe this is what the new moon looks like to the Jewish mind but to me I thought it looked more like the phase following the new moon. So the waters may be muddying a bit here. Digging a bit deeper I found the following: According
to the Mishnah and Tosefta, in the Maccabean, Herodian, and Mishnaic
periods, new months were determined by the sighting of a new
crescent, with two eye witnesses required to testify to the Sanhedrin
to having seen the new lunar crescent at sunset. The practice in
the time of Gamaliel II (c. 100 CE) was for witnesses to select the
appearance of the moon from a collection of drawings that depicted
the crescent in a variety of orientations, only a few of which could
be valid in any given month. So here we are told the new month was determined by the sighting of the new lunar crescent at sunset. The next quote gives some information about the judicial process for these sightings: An
essential element of the Jewish calendar is the determination of Rosh
Chodesh, the day when each month begins. Since our months, by
biblical law, begin with the sighting of the new moon (molad),
which can occur on either the thirtieth or thirtyfirst night of a
given month, we need to determine which of these is Rosh Chodesh. In
ancient times, this was done through eyewitnesses, who would testify
before the High Court in Jerusalem that they had seen the new moon
the previous evening; once they had been crossexamined and judged to
be trustworthy, the court would declare the day to be sanctified
(Mekudash) as Rosh Chodesh. This process is known as Kiddush
Hachodesh (sanctification of the month).
Nowadays, we have a fixed calendar, and we observe either one day of
Rosh Chodesh (if the thirtieth day becomes the first of the new
month) or two (the thirtieth and the thirtyfirst, the latter
becoming the first day of the new month). Nevertheless, on the
Shabbat preceding Rosh Chodesh (for all months but Tishrei), we make
a formal announcement in the synagogue about it, amid prayers for
peace and prosperity in the coming month, which we call Birkat
Hachodesh (blessing of the month). What is the
relationship between them? The Magen Avraham writes (OC 417:1): Rav
Akiva Eger raises an objection (Hagahot, ad loc.):
I am ignorant and do not know where we find that Kiddush Hachodesh was performed standing. On the contrary, [what is stated] at the beginning of Rosh Hashana implies that it was performed seated. Rav
Akiva Eger appears to be referring to the first mishna in the
third chapter of Tractate Rosh Hashana: Even
without this proof from Rosh Hashana, it seems to be clear
that the court must be seated when it sanctifies the month, for all
judicial actions require that the judges be seated, and Kiddush
Hachodesh is a judicial action!
For any nonJews reading this the last quote may even seem quite humorous. Nevertheless, the importance of this last quote cannot be underestimated. It introduces something completely new. The molad. What is the molad? The
Jewish calendar is lunarbased, with each month representing one
lunar cycle  the time it takes for the moon to complete one orbit
around the earth. Well, this is interesting. The lunar month for the moon to complete one orbit around the Earth is the sidereal month of approximately 27.32 days, not the synodic month which is wanted here. The synodic month would give the correct time from first thin crescent to the next first thin crescent. And we know this is the same as from new moon to new moon. It's an easy mistake to make and we would not condemn the writer of the above article for making it. This last quotation is telling us that Chodesh is determined as the day the thin crescent appears after the new moon. Another important point that we now have is that Rosh Chodesh should be determined for the Jerusalem time zone. I may disagree with this later but for continuity and for trying to align my results so far with the Jewish calendar we will take this approach here. Later found another site that confirms above more clearly; that is, the crescent approach, and Jerusalem as the base position for determining Chodesh for the Jewish calendar:
Note the text under the picture: “Rosh Chodesh, the new month, is marked by the appearance of the first sliver of the new moon in the sky above Jerusalem in Israel.” This says it all in one sentence! The time correction for Jerusalem can be found from any map showing time zones. I found one as follows:
From http://aa.usno.navy.mil/faq/docs/world_tzones.php Jerusalem would be
in
zone B. Sydney Australia is in zone K and the above chart gives
Universal Time  (10) = GMT + 10 which sounds correct. So the
correction for Jerusalem would then be GMT + 2. This sounds right. Some more on the crescent: In
parashat hashavua (weekly Torah reading) Bo, we read: “Then Adoshem
said to Moses and to Aaron in the land of Egypt, saying: this renewal
of the moon shall be for you the beginning of new moons; it shall be
to you the first of the months of the year.” (Exodus 12:12) And we
note that Akeidat Yitzchak (Rabbi Yitzchak ben Moshe Arama,
14201494) teaches that this was the first commandment we received as
a nation, while we were still in Egypt. By the time I got to the end of all this I was really puzzled. Who was right? Did the month start with the new moon or this first thin crescent? Could I go by the Bible references to the new moon? After all, the prophet Isaiah conclusively states from one new moon to another. That should be the end of the matter. Or is it? The problem may be the translation. The Hebrew word translated new moon is Chodesh. But these Jewish quotations are all saying they go looking for chodesh by searching for this first thin crescent. In other words, it's their spin on it and how they have translated the instructions the Lord gave Moses in Exodus chapter 12. I was struggling here with how the Jews were interpreting the instructions given to them about thirty five hundred years ago. Simply put, our current understanding of these matters, new moon terminology etc just may not be sufficient to sort this out. The only way to resolve this problem was to go back to the early part of the Bible and see if it was possible to determine just how the month was supposed to be constructed. Taking this approach I was actually able to find the answer. We proceed as follows... The moon was given for seasons: And
God said, Let there be lights in the firmament of the heaven to
divide the day from the night; and let them be for signs, and for
seasons, and for days, and years: The word translated as season is:
H4150 מועדה מעד מועד mô‛êd mô‛êd mô‛âdâh We would read this as a fixed time, a period of time. Note it states conventionally a year. But years is already mentioned in the verse so this period of time is not a year. But the word seems to be more saying a set time instead of a period. That is, the moon is marking a set time. Where else is this thought conveyed? Psalm 104:19 gives: He
appointed the moon for seasons: the sun knoweth his going down. and the Good News Bible gives this as: You created the moon to mark the months; the sun knows the time to set. Now lets have a
look
again at the main verse the Jewish sites refer to for the beginning
of months. This
month shall
be unto
you the beginning of months: it shall
be
the
first month of the year to you. The three words in this verse translated month are all chodesh. This is clearly not where months began for they are mentioned well before this. The statement in Exodus chapter 12 is just laying out where the Jewish year started. This month shall be the first month of the year. But it could also be translated as new moons for the year. This is the new moon, it shall be the beginning of the month [Rosh Chodesh], ... Another interesting point about month: A month is a unit of time,
used with calendars,
which was first used and invented in Mesopotamia, as a natural period
related to the motion of the Moon;
month
and Moon
are cognates.
The traditional concept arose with the cycle of moon
phases; such months
(lunations) are synodic months
and last approximately 29.53 days. So here is another point usually missed. Moon and month are cognates: In linguistics, cognates
are words that have a common etymological
origin. The word derives from the Latin cognatus
(blood relative). So the term new moon could also mean new month. That's pretty clear. From the verse it
does
sound like the new moon is the beginning of the month. Is there any
other way to determine this? Verse
5: And the evening and the morning were the first day. As
each day came from the hand of the Creator He stated that the day is
classified as the evening followed by the morning. Of course by
morning we don't literally mean just the morning say till about 12
mid day. The following two verses make this point very clear:
It shall
be unto
you a sabbath of rest, and ye shall afflict your souls: in the ninth day
of the month at even, from even unto even, shall ye celebrate your
sabbath. And
at even, when the sun did set, … The
sabbath was the seventh day of the week and we could similarly apply
the previous day length verses to give: And
the evening and the morning were the seventh day. Of
course the Bible doesn't say it in these words. The comment has been
made for the previous six days of the week. And when we come to the
second chapter of Genesis, the seventhday is clearly mentioned. Verse 2: And on the seventh day God ended his work which he had made; and he rested on the seventh day from all his work which he had made. From my computer
dictionary the meaning for sabbath is given as: [Sabbath]
A day of rest and worship: Sunday for most Christians; Saturday for
the Jews and a few Christians; Friday for Muslims. Anyway you look at this the sabbath is a day. And it is the seventh day of the week mentioned in Genesis chapter two. The day construct from Genesis chapter one gives the evening followed by the morning. The two hebrew words translated evening and morning are ereb and boquer. The Brown, Driver, and Briggs definitions for these days are: H6153 ערב ‛ereb and H1242 בּקר bôqer So what the verses
are
saying is that the sunset comes first giving the start of the day.
This is reflected in the verses previously quoted; Lev 23:32 and Mark
1:32: from
even unto even, shall ye celebrate your sabbath. So the full day period is defined as being from sunset to sunset. The following diagram shows this:
The important part to glean from this is that the dark part of the day comes first followed by the light part:
This is how the
Creator
has defined the day in Genesis.
Well, it's pretty hard to get this wrong. The dark part must come first. According to sunset for the beginning of the dark part of the day the answer must simply be the start of the new moon. This would simply be determined as the day of the new moon. So the Bible month would be as follows:
And it would
appear to
be
the mind of God on this matter. Just about everywhere you look the
month is defined as being from new moon to new moon; all the way from
NASA down to the most general sites on astronomy: the
mean synodic month (New Moon to New Moon) The New Moon phase is
uniquely recognized as the beginning
of
each calendar month A synodic
month is 29.53059 days (29 days, 12 hours,
44 minutes, 2.8 seconds) and is measured from New Moon to New Moon. Lunar
Month: The period between successive new moons (29.531 days). lunar
month. the period of a complete revolution of the moon around the
earth, as the period between successive new moons (synodic month),
equal to 29.531 days The
month was originally defined as the time between one new
moon to the next, a period now called the synodic
month. Sometimes
other phases of the moon are mentioned but the most common is the new
moon. Before continuing I have just been made aware of something very important that sheds more light on this and actually backs this premise up and reinforces it. It strengthens the argument here for the Lord's construction of the day. We have stated correctly that the construction for the day is sunset to sunset. That is how it is stated in the Bible. It is how the Lord defines a day. From sunset to sunset. I have applied this approach to the moon for the construction of the month. This is far more apt than originally thought. Here's the quote: Many
students get confused about the geometry of the orbit of the Moon
because they have heard this term "the dark side" of the
Moon (besides being a famousat least to us old timersalbum by
Pink Floyd). If you examine the figures above about how the phases of
the Moon change, you see that for at least one half of the lunar
orbit, each spot on the Moon has daylight. That is there are 14.75
days when the Sun is visible each orbit, and 14.75 days each orbit
when the Sun is not visible. At any one time (just like on the
Earth!!!), there is one half of the Moon which is in darkness, and
one half which is in the sunlight. It is true that the Moon spins
much more slowly than the Earth, but the Moon has "days"
and "nights". It is just that these days and nights are
14.75 Earth days long! So the farside of the Moon which we cannot
see is not the "dark side", it is the farside. At full
Moon, the farside is completely dark. But at New Moon, the farside
is fully illuminated (as seen from the Sun!). What is the point of this quote? We only see one side of the moon. Why? Because the moon is also rotating on its axis. The time the moon has to do “one orbit” about its axis takes the same time as one synodic month. Now have a real good look at the following picture. You should recognize it from the previous—just adjusted a little to help understand what you are about to read.
The 1 Earth day at the beginning is included in the 29.53 days. I have put a little yellow spot on the moon. Now as the moon orbits the Earth after one synodic month the little yellow spot moves too. Note the extra moon added for the 3^{rd} Earth position for the synodic month. This is just showing the moon at the opposite side of the Earth. This of course would be half way through the synodic month but for comparison showing it here in the 3^{rd} Earth position. The importance of this is that you can see that the Moon IS actually rotating about its own axis as it also rotates about the Earth. Now consider the yellow moon spots on the New Moons. If you were standing on these yellow spots on the moon it would be midday. In other words the New Moon to New Moon is midday to midday for the yellow spot position. What this means is that the New Moon to New Moon has covered EXACTLY one Moon day. What we call the synodic month is no more than the day for the Moon. The synodic month IS one Moon Day. Now the next bit is a little tricky. When I first thought this out I had to admit I didn't see this one coming. Here it is. Look carefully at the Moon positions for the New Moon with the yellow spot marking midday to midday. That's one Moon Day. It's equivalent to 29.53 [approximately] Earth days. For the Moon, it has had to rotate an extra 29.1 degrees to get there. For one Moon Day the Moon has to rotate 360 + 29.1 = 389.1 degrees on its own axis. Now someone's going to say, No way! That's crazy! One Earth Day is one rotation of the Earth about it's axis. So a day for the Moon has to be the same. Guess what. That's wrong too. Have a real close look at that picture again. See the yellow spot I put on the Earth. For midday to midday on the Earth. For one Earth day the Earth has moved approximately 0.986 degrees in its orbit around the Sun. For the yellow spot on the Earth to get to the next midday the Earth has to rotate that extra 0.986 degrees. This means that for one Earth Day the Earth has to rotate approximately 360.986 degrees!! Think about this for minute or two. It takes a bit of getting used to! The angle value was found as calculated before, but for 1 day we obtain 1/365.25 * 360 degrees and gives approximately 0.985626... or approx 0.986 degrees. So what's the point of all this?
Before continuing, more recently I have found further support from the Bible for the month starting at the beginning of the dark phase of the moon. In Psalms chapter 81 we read:
Blow the trumpet at the new moon, at the full moon, on our feast day. There
is mention of the new moon and the full moon in this verse. Though
this may be hotly debated I believe there is little doubt that this
verse shows some form of Bible merism. This is a figure of speech
that is commonly used in the Bible. Some definitions follow: 13. Merism:
the use of two opposite
statements to signify the whole; e.g., day and night, springtime and
harvest, hell and high water (Bullinger, p. 435). Note that
Bullinger lists these passages under synecdoche, for merism is a kind
of synecdoche. But we shall use a separate category. (literature,
rhetoric)
Referring to something by its polar extremes. containing
a pair of oppositional terms commonly thought of as binaries. It is
an expression known as a "merism," wherein oppositional
terms are jointly employed to form a totality.
Now some people disagree with the full moon translation part in this verse. But it is curious that all the Jewish versions quoted above do mention
the full moon. If we do accept the full moon part then this means that we have the two extremes in moon phase mentioned
in this verse: the new moon and the full moon. And they are polar
extremes or opposites. If we take this view then it is unlikely that the first sliver or waxing crescent could be understood to be the Biblical new moon in this verse. The more general accepted
approach stated for the new moon worship day is the waxing crescent. Well this is the approach the Jews claim to take though their calendar is
constructed a little differently. Anyway, an alternative approach we are taking here is to work this from the beginning of the dark
phase instead using the Bible construct for a day. 2014 Sep note: As novel as this approach is it doesn't have historical support and the other method of the waxing crescent appears to be more correct. See the Important Historical Update at the end of this study. And the two main flaws in the previous arguments [I was always aware of...] are 1. we are not located on the moon so sunset on the moon as the beginning of the moon day or month is not completely clear; and 2. Bible merism: could mean signifying the whole and not just being extreme oppositeswhich would actually support the astronomical new moon an interesting thought. However the following numerical method outlined is sound and can be applied to either approach. So I am not going to disturb any of the following working. The rest of this discussion will now focus on determining just where we start the beginning of the dark moon phase to place our location for chodesh according to the Bible construct. Simply put, this will give us the correct day [as best as we can find it] for the new moon worship day according to the Bible. Some Fun Stuff Now this is where it gets really interesting. According to the Bible model we need to determine the start of the new moon phase. There is not much information on the internet about when the actual phases begin or end. The simplest approach mentioned is to just divide the lunar month into 8 sections. We will look at this a little later. But first some interesting points to ponder. Before proceeding we need to consider a few points:
There is a minor correction to the second point I have found: the
moon does not orbit the exact center of the Earth, but a point on a
line between the center of the Earth and the Moon, approximately
1,710 km below the surface of the Earth, where their respective
masses balance. This is the point about which the Earth and Moon
orbit as they travel around the Sun. It's still a
wobbly
curve. If we are going to look at when the phases may be changing we need to be aware of the following. The Moon is
approximately
3500 km wide.
The sizes here and the distance are reasonably to scale. This is it. It's hard to believe from all the nice sketches that are usually drawn. Of course they have to be drawn that way otherwise you just wouldn't see the detail being shown. The shadow speed
on the
moon. There are two: on the surface itself, and on the face [we see
from Earth]. Moon circumference
= 2πr
= π ×
diameter = π ×
3500 km = 10995.6 km. I used a rough
approximation for the diameter so will go with the Wikipedia value. For 2π
radians, 1 rotation = 27.322 days [one siderial month] Radial speed: we
have 2π
rad = circumference
We can check this:
So the moon
equatorial
surface speed = 1738.14 ×
0.009582 km/hr = 16.65 km/hr as above. The problem we are
now
faced with is, how long is a piece of string? Technically,
each phase, just like the one called a New Moon, when the Moon is
exactly between Earth and the Sun, lasts only a brief instant. But to
our eyes, a New Moon can last for a few days, representing the time
that the Moon appears in the sky too near the Sun's position for us
to see it at all. How many days is a few days? Well, the following quote is from the U.S. Naval Observatory about the “Crescent Moon Visibility and the Islamic Calendar.” So why are we now interested in the crescent moon? Well when the crescent moon becomes visible this basically marks the visible end of the new moon phase [correctly or incorrectly]. That will help us determine the length of the new moon phase. Here's the quote and it reads pretty dismally: Crescent
Visibility: 15.5 hrs that's hardly a day. Doubling this gives us 31 hours and not much more than a day. The general internet sites showing pics of the moon on various days also add to this muddle. As follows it looks like: http://moonphases.willyweather.com.au/nsw/sydney/sydney.html
http://paulcarlisle.net/mooncalendar/
Well looking at all these site pics showing phases of the moon about the astronomical new moon date the length of the new moon phase could be anywhere from 1 to 6 days. 2 of them show 1
day for
the phase. Of course, all this is a little loose. As the pics were each generated for just one time of the day, noon or midnight or whatever. Saying that, there is clearly some leeway as the new moon could actually extend into the next day up to about noon [or midnight etc]. So these figures could all be extended by one day! That would give anywhere from 2 to 7 days for the length of the new moon phase! Is there any way to sort this out?? A numerical division may not give the exact answer but it could be a good guide just what direction to move in. Simply put, lets take the time period for the month and divide by the number of phases. We have 8 phases generally accepted: New Moon, Waxing Crescent, First Quarter, Waxing Gibbous, Full Moon, Waning Gibbous, Third or Last Quarter, Waning Crescent, and then a New Moon again. This would give
Halving this amount would give 1.85 days either side of the astronomical New Moon time. It's probably reasonable but this result was obtained using a linear approximation to divide the phases. The problem is, as we see them on the face of the Moon, the phases are not linear. Here's why. We worked out before that the radial speed of the Moon about its axis is about 0.009582 rad/hr. This translates to the equatorial surface speed approximately at 16.65 km/hr. Is this the same as the shadow speed across the Moon's surface? Apparently not. The reason for this is that all the phases occur in the synodic month of 29.53 days. The equatorial surface speed was calculated using the sidereal month of 27.3216 days. Why the difference then? Basically these are different because the Moon is also orbiting the Earth and it's just not a simple rotation about its axis. Let's consider a diagram of the Moon with the shadow line rotating. The Moon is also rotating but for this diagram we will ignore that. Also given that we will determine an equatorial speed in km/hr we still won't really see this speed correctly because of the curvature of the Moon. When we look at the Moon it is so far away it is like looking at a flat disc. The following diagram shows this better:
This is the view from the Southern Hemisphere. The Moon is rotating clockwise and the line of shadow/light appears to rotate anticlockwise. From the Southern Hemisphere the line dividing the dark and light parts of the face of the Moon travels from left to right. As θ travels from zero to 2π we see that the full cycle of phases is taken: θ = 0 New Moon Note that during this same period of time the Moon has rotated 389.1 degrees about its own axis which is more than 2π radians. As the shadow line rotates we see the shadow/light position on the face denoted as x_{1} and x_{2} [depending which division line we see] move forwards across the face of the Moon from left to right. In the Northern Hemisphere the direction of travel will be from right to left. Note: x_{2} = −x_{1} but for simplicity in the diagram above labelled it as x_{2}. We can safely assume that the angle theta is increasing approximately at a constant rate. But as it passes through π and zero again, the point x_{1} [or x_{2}] will appear to slow in movement. Note that as θ passes through zero the shadow line will move out of view behind the face of the Moon. As this happens the opposite end of the diameter shadow line will appear at the other extreme. So as point x_{1} disappears from view x_{2} appears at the other extreme and moves across the face of the Moon. And the process repeats. You see this as the Moon moves through the New Moon phase. The shadow line shows a waning crescent at one side and then disappears. And then moving through the New Moon phase the next thing you see is a waxing crescent appear at the OPPOSITE side of the face. As the shadow line gets to the extreme edges the speed will actually drop to zero instantaneously and then increase again with maximum speeds attained at π/2 and 3π/2. These are the radial positions for the First and Last Quarters. The slowest speeds [of zero km/hr] will be attained at the instantaneous points of the New and Full Moons. The mathematics of this gives us an amazing result. The longest phases are the New and Full Moons. All the other phases are shorter! We can determine the formula for x_{1} in terms of θ mathematically as
where r is the radius of the Moon. A corresponding velocity formula for x_{1} as it moves across the face of the Moon can be found as:
The radial rate of change here is different to the radial velocity of the Moon. This is because this calculation is independent of how fast the Moon is rotating itself about its axis. We simply have the phases during the synodic month of 29.53 days. That's it. Having said that, we can work out the value in km/hrs as follows: θ travels 2π radians in 29.53 days so that gives
Using l = rθ then the equatorial shadow speed can be found as:
This is on the surface of the Moon. To adjust this for the face of the Moon we require the previous formula:
The equatorial shadow speed we worked out was 15.41 km/hr. The last formula will obtain this value in magnitude for θ = π/2 and 3π/2 at the First and Third Quarter Moons when the shadow is closest to us and moving the fastest. And at the extremes corresponding to the astronomical New Moon and Full Moon the speed drops instantaneously to zero as mentioned before. This equatorial shadow speed formula was important as it tells us that the shadow is moving the fastest at the First and Last Quarter Moons and slowest at the New and Full Moon phases. This means that the New and Full Moon phases are longer than the other phases. The lengths of the phases is not constant. Perhaps we didn't really need this shadow speed formula but it was important to know how the position x_{1} moves as the shadow line moves with a constant radial velocity. It's not linear so 3.7 days for each phase won't work. What will? As stated before we see the face of the Moon more as a flat disc from the Earth. Dividing up the phases may be simply a linear division along the horizontal diameter. We will show this approach is actually reasonable but first let's try the calculation. The formula we have is:
but we can simply work this for a unit circle with r = 1 and obtain identical results. In one run of x_{1} for θ from π to 2π this translates to x_{1} from −1 to 1. We have Full Moon, Waning Gibbous, Last Quarter, Waning Crescent, and finally New Moon. The Full and New Moon phases are at the extremes so we only have half of these. This gives: 0.5 Full Moon + Waning Gibbous + Last Quarter + Waning Crescent + 0.5 New Moon = 4 full phases. The unit circle diameter length is represented by 2 units. So 2/4 = 0.5. Each phase is represented by 0.5 units. The positions of interest are as follows. The first position is the end of the Full Moon phase at: −1 + 0.25 = −0.75 then the end of the Waning Gibbous phase should be −0.75 + 0.5 = −0.25 the end of the Last Quarter Moon phase should be −0.25 + 0.5 = 0.25 and finally the end of the Waning Crescent phase should be 0.25 + 0.5 = 0.75 Working out the first angle for the half Full Moon position gives cos^{−1}(−0.75) = 3.8643 rad [in the third quadrant] The angle then contained in the half of the Full Moon period is 3.8643 − π = 0.723 rad or approximately 3.4 days and should be the same for the New Moon phase. We don't have to work out any other values as this gives the New Moon phase length to be approximately 6.8 days. This is probably far too high and the Waning and Waxing Crescents will most likely be seen too far into this phase for this number to be of any use. It is, however, interesting to note that one of the values determined from the web sites showing Moon phases did allow for a length of 7 days. In general though this does sound a little high. The next approach was to determine, if possible, a mathematical formula for the percentage illumination of the face of the Moon. This may be a little more helpful than the last approach with the angles though the two approaches are actually related. It turned out that this formula was extremely helpful in determining a upper bound [more about this later]. So what are we doing? You look at the face of the moon and there is some part in light and the rest dark. What is the percentage of the moon face that is illuminated? Basically we have something that looks like:
It may actually be difficult to determine the correct shape of the curve to get this exactly so some simple model approximations were considered as follows [though this was resolved using spherical coordinate geometry]. The models I investigated were rectangular, trapezoidal, parabolic, and exact using spherical coordinate geometry. They all yielded the same formula solution. To simplify the mathematics we will consider the model dimensions about 1 or similar for calculating ratios. Further only the simplest model is covered here. It should be pretty easy for anyone to follow. If anyone is keen to see the other working time permitting I will place them on a separate page.
Rectangular Model:
Remember we are after a percentage of illumination. This is like a ratio so the actual correct area is not needed. A simple approximation should suffice. The light rectangular area is (1 − x_{1})h ; the total rectangular area is 2h. Then the percentage illumination is approximately:
For our purposes this formula works pretty well. The other mathematical models all gave the same formula. This formula will be used shortly and we will find it very helpful. Now the question. Can we make any sense of all of this? A working solution First the visual sighting approach. I have tried looking at a few sunset times to see if I could notice the new moon—either view; first sliver of crescent or dark moon and have to admit not being successful with either. The crescent [waxing or waning] was more noticeable the further from the astronomical new moon time I looked, as expected. But what about the location of the moon to the sun? The astronomical time is dependent on the moon sun positioning, not sunset times. Simply put, some new moon times [astronomical] are given for during the night—depending on your location. But you can't see the new moon during the night as the moon sets and rises with the sun. Similarly the event could occur at midday and again a sunset reading would be hours away. A simple solution to this problem would be to determine some approximate numerical value that could be subtracted, or added to the astronomical new moon time according to whichever “new moon” you were using. The ancient Israelites were far from stupid and it would not have taken them too many years to work out the approximate value for the synodic month [if they already didn't know it] and the fact that the periods follow a sinusoidal pattern through the year. This information could have been easily used to determine the approximate date of the next new moon at any time of the year. Of course where we are now we readily have available astronomical new moon times so don't need a sinusoidal chart, just a numerical value to find approximate new moon dates. By new moon dates here we mean either the Jewish new moon or the Bible model. Also since the
sighting
approach is all approximate this is not really an exact science so as
an alternative we don't really need to have a number to 10 decimal
places of precision. A rough approximation should suffice. With the
information we have at hand it is actually possible to determine a
rough approximate value. Further the value we
obtain appears to hold reasonably well with the Jewish calendar
dates. Also using the Jewish calendar it is possible to make a
correction in the value, if we want to use it. We can easily obtain a lower value by looking at the data from the internet sites for [astronomical] new moon periods. From before we obtained: 2 of them show 1
day for
the phase. But allowing for the reading extending half a day in either direction extends these periods by one day giving: 2, 3, 4, 6, and 7
days
displaying as possible periods.
I rotated this again and took some measurements as follows: By drawing a line and measuring this approximately I obtained a value of 0.826 on the right hand side where the shadow changes to light. This is probably a bit rough but should be reasonable for what we are doing here. The percentage illumination can be found using the formula we obtained to give Illumination = (1 – 0.826) × 50% ≈ 8.7% and using cosθ = x_{1} =0.826 we find θ ≈ 0.5988 rad and from this we get approximately 2.8 days. For 8.7% illumination the light part looks pretty obvious so I would suggest dropping this. Trying for simple numbers I would suggest dropping 2.8 down to 2 days, as 2.5 probably would still be showing a fair section illuminated which would be visible. We could also look at the pic obtained from the other Jewish site mentioning the new moon. Again a rotated version gives:
Doing an approximate measurement on this we obtain:
The value obtained
here
is approximately 0.906. Illumination = (1 – 0.906) × 50% ≈ 4.7% and
using cosθ
= x_{1}
=0.906 we find θ ≈ 0.437 rad and from
this we get approximately 2.05 days. As before, we could definitely set 2.0 days as a top bound for an approximate numerical value. So what do we have? We have a lower value of 1.0 days. We have an upper value of 2.0 days. It should be pretty obvious just where this is going. The answer is clearly between these two numbers. Somewhere. What's the simplest value we could try that would be very easy to use. It's not really hard to guess this. Mathematically there is a numerical method called the bisection method that is used when you have two initial guesses as the solution to a problem. The next guess is simply the average point. In this case we would obtain 1.5 days and this seems like a good point to start. It may even do completely. Let's just see how a value of 1.5 days works. Can we test this? Yes we can by using the value forwards and see how it compares with the Jewish new moon dates.
We will use this value for determining Jewish Rosh Chodesh dates for 2011. The first astronomical new moon date is 4 Jan 11.03am. We are using here, of course, Jerusalem times. Moving forwards 1.5 days is very easy: just add 1 to the day and add 12 hours—simple by changing am/pm to pm/am. If the original time is am it just moves to pm. But if the original time is pm you move to am but for the next day so add 1 to the date. For 4 Jan 11.03am 1 day takes us to 5 Jan 11.03am. Then adding 12 hrs we get to 5 Jan 11.03pm. The final step in this process is to compare this time with the sunset time. Obviously this really only needs to be done if you are likely to be anywhere near it. At 11.03pm at night this is not likely to be the case. However, it is late at night and obviously AFTER sunset of 5 January. The Bible reckoning for a day is from sunset to sunset. This is how the Jews keep the day and why they keep sabbath from Friday sunset to Saturday sunset. Since the time we have arrived at is AFTER sunset we are really in the next Jewish day. Then we take chodesh as the NEXT day AFTER January 5. We then arrive at 6 January for chodesh. The Jewish calendar for January 2011 gives Rosh Chodesh on 6 January. This process was continued with all the astronomical new moon dates in 2011 and every one of them landed on a Jewish new moon or Rosh Chodesh day using the numerical value of 1.5 days. Sometimes the Jewish calendar had 2 Rosh Chodesh dates in the month. Each time this happened the 1.5 day calculation landed on one of them. For 2011 in EVERY case the method was completely successful in landing on a Jewish Rosh Chodesh date.
The next question to ask was, just how well did the numerical value of 1.5 days fit the Jewish calendar? There appeared to be a reasonable amount of flexibility especially when the Jewish calendar gave 2 Rosh Chodesh dates. The main points of interest then fell on the single day calculations. For example, April gave a range of 10.32 hours to +13.29 hours flexibility in the 1.5 day numerical value. This is explained in a bit more detail as follows. The Jerusalem astronomical new moon time for April is given as 3 Apr 5.32pm. Adding 1 day takes us to 4 Apr 5.32pm. Now adding 12 hrs we get to 5 Apr 5.32am. Note we moved to am but the next morning so the date moves to 5 April.
With the numerical value of 1.5 days we have landed at 5.32am. We take chodesh as 5 April. The question we are asking is how much variability from 1.5 could we take and still stay in 5 April as chodesh? To answer this properly requires the sunset times at both ends of the Jewish day. So we need the previous sunset and the sunset for 5 April. For Jerusalem these were found as 7.00pm and 7.01pm. From 7.00pm the previous night we have 5 hours to midnight and to 5.32am we have 5 + 5.32 or 10.32 hours or 10 hours 32 minutes. So in theory the value of 1.5 days could be reduced nearly 10 hours 32 minutes and still give a Rosh Chodesh date of 5 April. Similarly at the other end of the day we obtain 7.01pm or 1901 – 0532 giving 1329 or 13 hours 29 minutes we could extend the 1.5 numerical value and still obtain a Rosh Chodesh date of 5 April. The date for April shows that the 1.5 numerical value is nearly in the middle or centrally located for these extremes. However, trying some other dates we may be able to tighten the boundaries a bit. This is indeed the case and an interesting adjustment can be obtained using February and November.
The Jerusalem astronomical new moon time for February is 3 Feb 4.31am. Adding 1.5 days takes us to 5 Feb 4.31pm. Sunset is 5.17pm so we take chodesh as 5 Feb. This is one of the months the Jewish calendar gives 2 Rosh Chodesh dates: 4 and 5 Feb. However we only have leeway of 46 minutes to sunset. So we have a right hand bound on the numerical value of 0.46. Similarly for November we start at 25 Nov 8.10am. Adding 1.5 days takes us to 26 Nov 8.10pm. The Jerusalem sunset time for 26 Nov is 4.36pm. Since we have landed at 8.10pm this is well after sunset and we choose the next day 27 November for chodesh. Further this only gives leeway of 8.10 – 4.36 = 3.24 or 3 hours and 24 mins. So to fit the Jewish calendar the numerical value of 1.5 days is squeezed on the left by 3 hrs 24mins and on the right by 46 mins. A simple adjustment for the Jewish calendar would be to have our numerical estimate for chodesh centrally located between these two extremes. This adjustment can be easily determined by taking the average of the values. – 3.34 + 0.46 gives – 2.48 and then divide by two to obtain –1.24 or approximately minus 1 and a half hours. [the decimal parts in these numbers are minutes so 0.60 would become 1.00, that is 60 mins = 1 hr]. The adjustment in our numerical approximation would be to subtract 1 hr 30 mins. That's it. It is not really much at all. The value of 1.5 days was a very good estimate. We could nearly leave it as it is. But an adjustment of 1 hr 30 mins is not hard to do. For an approximate value for finding the Jewish chodesh dates the adjustment would be to subtract 1 hr 30 mins from the numerical estimate of 1.5 days giving 1 day 10 hrs and 30 mins. Update for 2013: As will be seen, the Jewish
Model proposed of 1 day 10 and a half hours worked OK for 2011, and also
for 2012. However it did not do well for 2013. This is commented on in
the PostScript at the end of this study. For historical interest, and
of course, for this investigation, the numerical value for the Jewish
Model will be left where it is for the rest of this study. Stephen.
What about the Bible model we have proposed? Do we subtract or add this adjustment? The numerical value in the Bible model is for the reverse direction from the astronomical new moon time. But basically as we travel through the phases of the moon we are moving in a positive direction from the waning crescent to the new moon and then the waxing crescent. Approximate numerical values in time are suggested for these two points that would be in reasonable accord with visual sightings. An adjustment has been proposed which would have us move the numerical value for the waxing crescent phase back one and a half hours. This suggests that the previous phase approximation could also be out and should be moved back the same amount of time.
However, some may argue by symmetry it should be subtracted. Unfortunately this is a pretty good point. But from the sample of new moon pictures obtained from internet sites giving possible ranges of the phase from 2 days to 7 days I would be very hesitant to subtract any value from our numerical approximation. It is an approximation. It does actually work well for the Jewish dates even without the adjustment. That the Jewish calendar dates suggest an adjustment of only 1 and a half hours is pretty good evidence that the approximation is pretty close to where it should be. As such for the simplicity of the calculations I propose that the numerical value of 1 and a half days be left where it is for the Bible model. Simply put, it's close enough. This brings us to a final question. One Rosh Chodesh day or two? This may seem a silly question but it must be noted that when you look at the Jewish calendar just about every couple of months they have 2 Rosh Chodesh days listed next to each other. Some of the previous Jewish quotes we looked at mentioned this: 5. The Hebrew New Month, as it goes
according to this calculation
is either on the 30th or 31st day each month, as a month is 29.5
days, we have alternating New Months, being the 30th (and) or 31st
day. Sometimes
Rosh Chodesh is one day, but sometimes it is celebrated for two days.
Months are based on the lunar cycle, of course. The amount of time
required for the moon to make one complete revolution around the
earth is determined by the conjunction of the earth, moon, and sun
(i.e. they lie along a line). When this happens, it is the new moon.
A complete revolution takes 29 days, 12 hours, 44 minutes, 31/2
seconds. Days in our calendar must begin at sundown, regardless of
when the conjunction actually takes place. Hence it is necessary to
either add or subtract a half a day from each calendar month. For
this reason, Hebrew months alternate between 29 and 30 days in
length, with the actually moment of the new moon falling in between.
As noted before there is a simple error in this quote—the period of time for the moon to make one revolution about the earth is the siderial month of 27.32 days, not the synodic month which is actually wanted here. A simple mistake. But we have the reasons given for the 2 days. The synodic month is of length 29.5 days approximately. So one month it is celebrated on the 30^{th} day and the next on the 31^{st}. But to be sure they don't get it wrong for the second one they do both: the 30^{th} and 31^{st} days. An interesting solution. Should we be interested in doing this too? The best way to investigate this is to try our numerical solution for the Jewish calendar for 2011 and see just how the Rosh Chodesh days fall with the numerical solution. We of course will use the Jewish calendar correction for 2011: 1 day 10 hours and 30 mins. And for the Jewish calendar dates we will use Jerusalem times. The following table will list the astronomical new moon dates and times, the numerical solution with the adjustment for the Jewish calendar, the sunset time for Jerusalem, the selected date for chodesh, and the actual Rosh Chodesh dates given in the Jewish calendar. We then obtain:
This table raises more problems than it solves. I was expecting that the numerical times for the 2 Rosh Chodesh dates would be near the sunset times. This actually does appear to be correct for a majority of the months: February, May, July, and August. The last three fit well with the numerical value near the sunset time in the middle. But February is near the edge of the second day. The adjusted numerical value as expected appears to be centrally located. The extremes of February and November give numerical times approximately 2 hours near sunset at both ends. For February another few hours will take us to another day for Chodesh. Similary for November a few hours back will force us to the previous day for Chodesh. There appears to be some issues with the workings for the months with 2 Rosh Chodesh dates. According to the Jewish quotes this should happen every couple of months as they adjust for the half a day in the lunar month. But this calendar shows 2 days for the months of February and March consecutively and then again three months in a row later in the year: August, September, and October. The answer to this puzzle is that perhaps the Jewish calendar is using exactly 29.53 days to go from one lunar month to the next. The value of 29.53 days is an approximate average value. The periods for the actual synodic months vary through 2011 with a maximum value of about 29.76 attained in the first half of the year and a minimum value of 29.34 through the second half. This is reasonably obvious from the table constructed above. For the first half of the year the approximation of 29.53 days is lagging behind the actual synodic month so the placement of the two days in February and May are showing an earlier position. The numerical value lands on the second of the dates. This doesn't appear to be the case for March but I believe that that is most likely because March shouldn't actually have 2 days anyway. Similarly for all the 2date months in the second half of the year, EVERY one has the numerical solution landing early on the first day as the true synodic month is being followed and lagging with the Jewish calendar. The numerical
solution
was found so as to follow the actual time of the astronomical new
moon. As such, it should follow the actual synodic month more closely
as the period lengths change through the year. It's not perfect and
perhaps even needs adjustment as the lengths shorten and get longer.
But even as it stands, for 2011 at least, it should follow the actual
synodic months closer than just using 29.53 days continually between
each month. Having said that, I have no hesitation whatsoever saying
that some of the Rosh Chodesh dates given in the Jewish calendar for
2011 are questionable. Mainly Feb 4, Mar 7, May 4, Jul 2, Sep 30, Oct
29, and Dec 27. OK this is now official. I have been able to obtain actual Molad times for Jerusalem. From a table I have extracted 4 Molad dates: 3 Feb 11.16am From some correspondence I received earlier [not fully quoted] the comment was given: We
have a New Moon every 29 days, 12 hours and 44 1/18 minutes, so once
you have the new moon for any given month you can figure every future
one by adding the 29.12.44.1/8. The correspondence date was Thursday March 10. The last Friday referred to here would then be 5 March as given above. The Molad table gives the time as 12.00am so is the same time in the correspondence of 12:00.07 just past midnight. Note the quote calls this the New Moon. This is supposedly the exact “astronomical” time of the Jewish new moon according to this quote. We now call this into serious question. Note the 4 Molad date and times I have listed above. From the February to March date time we obtain doing this as a subtraction 29 days 12 hours and 44 minutes. Similarly for the August to September date and time we would also obtain by subtraction a value of 29 days 12 hours and 44 minutes. That is, the the next molad date and time has been obtained from the previous month by adding exactly 29 days 12 hours and 44 minutes. So what's wrong with that? I picked these 4 months specifically for a reason. The first two months of February to March give the longest astronomical length in the synodic month for 2011. Approximately 29.76 days or 29 days 18 hours and 15 minutes. Similarly for the last two months of August and September, these actually give the shortest astronomical length for the synodic month in 2011: Approximately 29.34 days or 29 days 8 hours and 5 minutes. So the Jewish calendar is clearly not following the lunar synodic months correctly through the year. The molad times as given in these tables sadly would most likely be just about all wrong. Update for 2013: The Jews are actually aware of these discrepancies and readily admit them:
JUDAISM 101 Considering this, the next few paragraphs could most likely be jumped, but I will leave them in for historical interest. Stephen. This actually does sound pretty bad and any Jews reading this are most likely to be disheartened. But on the good side most of the Rosh Chodesh dates given in the 2011 calendar do appear to be reasonable. This is good news. I would propose a simple solution to this problem. Pretty much what I have worked out in this search. Adding some numerical value to the astronomical new moon time to arrive at the correct date for Chodesh. It certainly couldn't do worse than using a table of values that are mostly incorrrect as the length of the synodic month changes through the year. And having a numerical value linked to the astronomical new moon time will ensure that as the month length changes the dates for Rosh Chodesh will follow accordingly. Will
this work for the
months with 2 Rosh Chodesh dates? Yes. I have given this some thought
and believe that the Jewish approach of 2 Rosh Chodesh dates does have
some merit. Specifically when the calculation falls just before and
near a sunset time. Both days could be called.
I have also given this some further thought and finally decided that the Jewish approach may be safer when the time falls about half way or close to the sunset time either before or after. This will add a few extra days to the original proposed chart: 4 May, 2 Jul, and 26 Nov. Taking this
approach a
proposed table for 2011 would then be as follows:
Note that for the middle of the year we find the numerical model gives the days exactly the same as the Jewish calendar. But as we move away from the middle of the year the synodic month lengths change and some discrepancies appear in the dates chosen. Closing thoughts This approach is pretty simple but some care may be needed. It could be argued that the numerical value is for a standard month which may be the 29.53 days. This could be investigated just which length this was optimal for. Having determined the correct length [which hopefully should be for 29.53 days] it could be further investigated whether the number needed any adjustment as the month lengths changed since it is tied to a fixed length. The adjustment most likely should be minimal and may not be necessary. However, note the following: Because
of perturbations in the orbits of the Earth and Moon, the actual time
between lunations may range from about 29.18 to about 29.93 days. The
longterm average duration is 29.530589 days (29 d 12 h 44 min 2.9
s). This gives a worst possible range of 0.75 days or 18 hours. This may be extreme and for 2011 only about a 10 hour range occurred. Still this may mean that a close watch on the synodic month lengths should be given each year. End of the walk ... nearly! I have finally got
to the
end of this walk. As I put pen to paper and keyboard to screen it is
early November 2011. It's not hard to figure out when this
started—just look at the early new moon dates investigated in this
study. The first one mentioned was March 6. So here we are 8 months
later. It has been a very interesting walk. An awful lot of ground
has been covered in these months. Maybe you do not agree with my
conclusions but I hope you have enjoyed the walk. Important Historical Update! [Sep 2014] Finally
to finish off we add a puzzle here for our athiest friends. A quote
from the Bible and a wonderful bit of information about the moon from a
site that probably doesn't have any Christian leaning. Enjoy! Gen 1:16 And God made the two great lights, the greater light to rule the day, and the lesser light to rule the night; he made the stars also. The
Moon orbits the Earth  or more accurately the Earth and Moon revolve
around a common barycentre  at a centretocentre distance of
384,000 km. The Sun, some 150,000,000 km away, is 1,400,000 km
across. It is an almost unbelievable coincidence that from the
surface of the Earth, the Sun and the Moon appear more or less
exactly the same size  around 32 arc minutes. It is this fact which
allows us to see the Sun’s corona during a solar eclipse. No other
combination of bodies anywhere in the Solar System give rise to this
phenomenon. Stephen Buckley POSTSCRIPT The above study basically ended
in November of 2011. The original plan was to show here how the Jewish model shaped up with each
current year. However, it did not do well for 2013 so this approach will not be followed. This is further commented on below. First, how did it do for 2012?
As you can see for 2012 the
Jewish Model returned very reasonable dates. But notice what happens
when we try the Jewish Model for 2013...
Only 3 of the proposed dates
were in agreement with the Jewish calendar for 2013! At first I thought
that the Jewish Model had completely bungled it. But the problems here
were actually far beyond any numerically proposed value for the Jewish
Model. It appears that the Jewish Calendar for 2013 shows just what
sort of problems you can have by rigidly adhering to a fixed value for
the length of the lunar synodic month as the moon wanders through the year.
Have a real close look at the Jewish Rosh Chodesh dates given in the
last column and note the superscripts.
It appears that for 2013 the Jewish Calendar actually drifted to the point where half of the Jewish Rosh Chodesh dates given actually fell on the same day as the Astronomical New Moon! Further than this, the Jewish Calendar had actually drifted so far that 2 of the Jewish Rosh Chodesh dates fell BEFORE the Astronomical New Moon and one of them actually fell on the same day that the NT Bible Model proposed and these were for the beginning of the Dark Moon phase!!! Anyway this has already been commented upon in the previous 2013 insert. The Jews readily acknowledge these discrepancies. What do these results mean for
the numerical value for the Jewish Model proposed in this study? Since
2 of the Rosh Chodesh dates given in the Jewish calendar for 2013 actually fell before the Astronomical New Moon, NO
numerical value will work properly with the Jewish calendar.
And what do these results mean
for the NT Bible Model proposed in this study? If nothing else,
strengthen it! The NT Bible Model does
follow the moon correctly through the year.
There is nothing wrong with using a fixed value for the months as in the Jewish calendar. It makes more sense to have months a fixed length instead of continually wandering each year with different values. At least for calendar construction, this side of time. However, as I have found in this study, anyone wanting to keep the Biblical New Moon worship day as mentioned in Isaiah and Revelation for the coming ages of eternity should find the NT Bible Model approach here constructed to be of some use. Note: it is possible to use the NT Bible Model approach to work approximations for the crescent version also. And according to the historical update given above this is the direction the NT Bible Model is going to take. Any extra notes, references etc
Though this study ended in November of 2011 there have been some minor amendments. Stephen Buckley Email: stephen@chodesh.info Last revised: 8 Oct 2014.
