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New Moon arithmetic

Irv Bromberg
Dear Calendar People:

Re:  <http://en.wikipedia.org/wiki/New_moon>

I am trying to understand the arithmetic for the constants and
coefficients given on the above web page for determining the moment of
the mean lunar conjunction.

Two are defying me:

The quadratic coefficient on the approximate formula for D, that is
102.026E-12, which somehow has a relation to Chapront's tidal
acceleration figure of -25.858 arcseconds/century/century.

The Delta T approximation quadratic coefficient of -235E-12, which is
somehow related to the long-term mean Delta T increase of +31
seconds/century/century.

Help, please!


-- Irv Bromberg, Toronto, Canada

<http://www.sym454.org/>
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Re: New Moon arithmetic

Tom Peters-2
> Dear Calendar People:
>
> Re:  <http://en.wikipedia.org/wiki/New_moon>
>
> I am trying to understand the arithmetic for the constants and
> coefficients given on the above web page for determining the moment of
> the mean lunar conjunction.

I wrote that page (or at least all those numerical details)

> Two are defying me:
>
> The quadratic coefficient on the approximate formula for D, that is
> 102.026E-12, which somehow has a relation to Chapront's tidal
> acceleration figure of -25.858 arcseconds/century/century.

The tidal acceleration is just one component of the net coefficient of the
T-squared term in D.
The coefficient of T-squared in D is measured in angular measure per
Julian century (36525 days) squared.  The expression for the conjunction
that has the coefficient 102.026E-12 is measured in days per lunation
squared.  It involves a proper mix of factors 36525, 29.5305888610, 86400,
60*60*360, which I cannot reproduce from the top of my head.  Hope that
you can figure it out now.

> The Delta T approximation quadratic coefficient of -235E-12, which is
> somehow related to the long-term mean Delta T increase of +31
> seconds/century/century.

It involves a change in time units.
31 / ((36525/29.53059)^2) / 86400 = 235E-12 days per lunation^2

HTH,
    Tom Peters
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Re: New Moon arithmetic

Irv Bromberg
> Dear Calendar People:
>
> Re:  <http://en.wikipedia.org/wiki/New_moon>
>
> I am trying to understand the arithmetic...

On Mar 13, 2006, at 04:34, Tom Peters wrote:
> I wrote that page (or at least all those numerical details)
>
> The tidal acceleration is just one component of the net coefficient of
> the T-squared term in D.

> The coefficient of T-squared in D is measured in angular measure per
> Julian century (36525 days) squared.  The expression for the
> conjunction
> that has the coefficient 102.026E-12 is measured in days per lunation
> squared.  It involves a proper mix of factors 36525, 29.5305888610,
> 86400,
> 60*60*360, which I cannot reproduce from the top of my head.  Hope that
> you can figure it out now.

Bromberg replies:

OK, by "properly mixing up the factors", I obtained the following which
agrees very closely, but is it the correct arithmetic?


Tidal_Acceleration = -25.858 arcseconds/cy/cy

Half_Tidal_Acceleration/3600 = -25.858 / 2 / 3600 = -0.003591389
degrees/cy/cy

Mean_Lunation_Fraction = -0.003591389 / 360 = -9.97608E-06 (since a
full mean lunation is 360° of mean lunar elongation with respect to the
mean solar longitude)

Quadratic_Coefficient = above squared = 9.95222E-11


This is almost exactly the same as the Wiki quadratic coefficient which
is 102.026E-12 = 1.02026E-10.

So, is this the "proper mix"?


-- Irv Bromberg, Toronto, Canada

<http://www.sym454.org/>
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Re: New Moon arithmetic

Tom Peters-2
>> Dear Calendar People:
>>
>> Re:  <http://en.wikipedia.org/wiki/New_moon>
>>
>> I am trying to understand the arithmetic...
>
> On Mar 13, 2006, at 04:34, Tom Peters wrote:

>> The tidal acceleration is just one component of the net coefficient of
>> the T-squared term in D.

> Bromberg replies:
>
> OK, by "properly mixing up the factors", I obtained the following which
> agrees very closely, but is it the correct arithmetic?
>
> Tidal_Acceleration = -25.858 arcseconds/cy/cy

Like I wrote, the coefficient of T-squared in the expression "D" for the
day of New Moon on the Wikipedia page, is derived from the coefficient of
T-squared in the expression for the difference in mean longitude of Moon
and Sun (Delauney parameter D).  It is not just the tidal acceleration,
there are more contributions to the T-squared term.  Go back to Chapront's
expression for D to find the proper value.  From the explanation for the
quadratic term on the Wikipedia page you can also find: -5.8681 -0.9817 =
-6.8498 "/cy/cy.

Suppose it is C "/cy/cy
Suppose the linear term is B "/cy (should be about
36525*360*60*60/29.5305888610 = 1602961601 "/cy)
Then the mean motion of the Moon is B/36525 "/day (at J2000)
And the lunation length L is 36525*360*60*60/B days (at J2000)
So the effect of C on the lunation length is -C*36525/B day/cy/cy ("-"
because smaller increase in longitude means longer duration of lunation).
Since 1 century is (36525/L) lunations, the effect per kunation is:
(-C/B)*(36525/((36525/L)^2))
= +6.8498/1602961601 * 0.02387558... = +102.026E-12 day/lunation/lunation .

Like it says on the Wikipedia page.

--
  Tom Peters
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Re: New Moon arithmetic

Irv Bromberg
On Mar 13, 2006, at 12:46, Tom Peters wrote:

>>> The tidal acceleration is just one component of the net coefficient
>>> of the T-squared term in D.
>
>> Bromberg replies:
>>
>> OK, by "properly mixing up the factors", I obtained the following
>> which
>> agrees very closely, but is it the correct arithmetic?
>>
>> Tidal_Acceleration = -25.858 arcseconds/cy/cy
>
> Like I wrote, the coefficient of T-squared in the expression "D" for
> the
> day of New Moon on the Wikipedia page, is derived from the coefficient
> of
> T-squared in the expression for the difference in mean longitude of
> Moon
> and Sun (Delauney parameter D).  It is not just the tidal acceleration,
> there are more contributions to the T-squared term.  Go back to
> Chapront's
> expression for D to find the proper value.  From the explanation for
> the
> quadratic term on the Wikipedia page you can also find: -5.8681
> -0.9817 =
> -6.8498 "/cy/cy.

Bromberg replies:

As per previous correspondence, what I'm after is the mean lunar
conjunction, ignoring all periodic effects.

This should include the starting point, which is the first mean lunar
conjunction after J2000 (actually the Wiki page uses J2000 minus 1/2
day), the linear term which is the mean synodic month at J2000, as you
calculated below, and that portion of the quadratic term which is due
to the secular acceleration of the mean duration of the lunation,
finally adjusted for Delta T.

As stated by Chapront, and by you, there are long-period contributions
too, but I feel that they should only be included if they are
physically plausible and it is understood what is causing them, and if
it is valid to include them outside of the intended use of the
polynomial, that is as the base for the summation of a series of
periodic terms.

Are any of the long-period "quasi" secular changes in the mean duration
of the lunation physically explained?

Ignoring a non-negligible long-period periodic effect could risk
drifting out of sync for many centuries or even millennia.
Ideally such an effect, if truly periodic, should be taken into account
using an appropriate periodic term, as per previous correspondence.

This doesn't have to work "forever".  A calculation that is
"reasonably" valid (say, within one hour, not counting the periodic
variations) for the mean lunar conjunction for a "mere" 10 millennia
into our future would be nice.  A step above using a constant duration
of the lunation = traditional molad interval (even if trimmed by 3/5
second to better match the modern mean synodic month), which in only
about 5 millennia would be late by about a day.

> Suppose it is C "/cy/cy
> Suppose the linear term is B "/cy (should be about
> 36525*360*60*60/29.5305888610 = 1602961601 "/cy)
> Then the mean motion of the Moon is B/36525 "/day (at J2000)
> And the lunation length L is 36525*360*60*60/B days (at J2000)
> So the effect of C on the lunation length is -C*36525/B day/cy/cy ("-"
> because smaller increase in longitude means longer duration of
> lunation).
> Since 1 century is (36525/L) lunations,

Bromberg says:

But L is the mean synodic month (in Julian days) at J2000, and is not
constant, so doesn't the estimate of lunations per century need to take
its variation into account?

-- Irv Bromberg, Toronto, Canada

<http://www.sym454.org/>
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Re: New Moon arithmetic

Irv Bromberg
In reply to this post by Tom Peters-2
Dear Calendar People:

The reason why I'm so persistent about this new moon arithmetic issue:

Using my implementation of Jean Meeus' "Astronomical Algorithms" 2nd
edition lunar and solar algorithms, I evaluated the relationship
between the traditional molad of the Hebrew calendar and the actual
lunation moments, referred to the meridian of Jerusalem, produced by
those algorithms.  This generated a curve of scattered points, and
least squares regression fitted a quadratic that has its minimum = +26
minutes in the era of Hillel II (Hebrew year 4111, Julian year 350 AD.  
That is the closest that the molad ever agrees with Jerusalem mean
solar time for the mean lunar conjunction.

That +26 minute offset implies that the original molad reference
meridian was 26 minutes of time to the east of Jerusalem, converted to
longitude that is 26/1440 * 360 = 6 1/2 degrees east of Jerusalem,
which was about mid-way between Israel and Babylonia.  That is an
interesting finding, but it is odd to suggest that the sages decided to
compromise by "splitting the difference" between the two major Jewish
centers at that time.  Since the arithmetic for the molad was probably
developed in Babylonia by Shmuel the Astronomer of Nehardea (he lived
one century before Hillel II), and since Shmuel had close ties with
Babylonian astronomers, I would have expected the molad reference
meridian to have been in Babylonia itself.  I considered it unlikely
that Shmuel or other sages would have had the knowledge necessary to
shift the reference meridian mid-way towards Israel, and, if they were
going to shift it at all, why wouldn't they shift it all of the way to
Jerusalem?

As it stands, the Wiki Mean New Moon approximation, which hopefully is
more accurate because it is based on >3 decades of Laser Lunar Ranging,
and which arguably employs a better estimate of the long-term Delta T
rate, had a minimum of +44.5 minutes, corresponding to 11+1/8 degrees
east of Jerusalem, which was essentially the meridian of Babylon.  This
makes sense to me, but it going to seriously bother a lot of people who
fervently believe that the molad moment refers to Jerusalem time.

Therefore I want to know exactly how the arithmetic is derived, and to
ensure to the degree that is possible, that the mean lunar conjunction
arithmetic is as accurate as reasonable.

-- Irv Bromberg, Toronto, Canada

<http://www.sym454.org/>
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Re: New Moon arithmetic

Tom Peters-2
In reply to this post by Irv Bromberg
> On Mar 13, 2006, at 12:46, Tom Peters wrote:
>>>> The tidal acceleration is just one component of the net coefficient
>>>> of the T-squared term in D.

> Bromberg replies:
>
> As per previous correspondence, what I'm after is the mean lunar
> conjunction, ignoring all periodic effects.
>
> This should include the starting point, ...,
> the linear term which is the mean synodic month at J2000, ...,
> and that portion of the quadratic term which is due
> to the secular acceleration of the mean duration of the lunation,

The quadratic and higher-order terms contain some other real secular
effects, e.g. those due to the flattening of the Earth.  Please consult
Chapront's original ELP paper from 1982 for a break-down of the components
that make up the quadratic term.

> As stated by Chapront, and by you, there are long-period contributions
> too, but I feel that they should only be included if they are
> physically plausible and it is understood what is causing them,

Irv, all this is physically plausible.  The ELP describes the motion of
the Moon as a function of its orbital parameters, has expressions for
mutual derivatives of these, and has built series expansions of Fourier
terms and polynomials.  The constants of this model have been fitted to
the numerical integrations DExxx from the JPL, which are a state-of-the
art physical model of gravitational and non-gravitational forces that run
the Solar System, and have been fitted to all suitable observations.

If you are trying to understand e.g. the motion of the perigee by some
periodic pull from the Sun, you isolate a particular abstract concept
(perigee) and describe it in some particular reference frame (Earth-Sun
fixed) and look only to the 3-body problem.  This is a valid but limited
model, and what you get is an approximation.
  For example, an elliptic orbit is a valid model to describe a planet's
motion: but it it exact only in a two-body system, and in the case of
the Moon you need to take account of at least 3 (EMS) to be even
approximately accurate.  But still you can use the model of an alliptic
orbit.  At any moment in time the Moon has a position and a velocity,
and you can derive an instantaneous elliptic orbit for it (osculating
elements).  Then you need to derive expressions that describe the
evolution of these osculating elements under influence of all other
bodies, which also disturb each other: this leads to many complex
higher-order terms of all kinds of periods.  That can be worked out as
series expansions of periodic terms, and polynomials.
  Similarly this is true for your quest for the evolution of the "true"
mean length of the synodic month.

The differential equations of motions are known, but they can not be
solved to give an exact mathematical expression.  There are 2 approaches,
both of which can give results to any desired level of accuracy.
  One is numerical integration, which is limited to the period of time
that you compute it for, and become less accurate as you extrapolate
further into the past or future.
  The other is series expansion, be it Fourier, Chebychev, or Taylor
series: these have to be truncated at some level, so have a limited
accuracy that becomes apparent as you move further away from the
present.
  For example, you can compute the position along an elliptic orbit using
Kepler's equations; or you can use a series expansion (Equation of
Centre) that has terms (2e-1/4*e^3+...)*sin(M) + (5/4*e^2+...)*sin(2M) +
(13/12*e^3-...)*sin(3M) etc.  You have to truncate both the powers of e
and the number of terms at some level.

> Are any of the long-period "quasi" secular changes in the mean duration
> of the lunation physically explained?

Of course.  The various models are strictly based on physics.  Their
constants are matched to observations.  Any residuals (observed minus
calculated) are measurement errors and unmodelled (i.e. not
well-understood) physics.  The "quasi" secular changes are a choice of
representation in the model, not a hack to have some heuristic expression
match some non-understood observations!

> This doesn't have to work "forever".  A calculation that is
> "reasonably" valid (say, within one hour, not counting the periodic
> variations) for the mean lunar conjunction for a "mere" 10 millennia
> into our future would be nice.

The ELP is supposedly valid between -4000 and +6000, with some defined
level of inaccuracy: it gets worse near the ends.  But you have to take
account of all secular terms, both in periodic and in polynomial terms.


> Bromberg says:
>
> But L is the mean synodic month (in Julian days) at J2000, and is not
> constant, so doesn't the estimate of lunations per century need to take
> its variation into account?

Yes, that is why there is a quadratic term!
  All this is derived from the expression for the mean elongation of the
Moon from Sun, as given in the ELP2000; this was first published in
1982, and updated in 1985|1988, 1997, and 2002.  Chapront gives a 4-th
order polynomial centered at J2000.
  If you want to know the mean angular velocity of the Moon w.r.t. the
Sun, you differentiate the expression for the mean elongation, and this
gives you a 3rd-order polynomial for the angular velocity.  Fill in any
value for T and you get the actual mean angular velocity for that time.
Divide 360 deg. by this instantaneous mean velocity, and you get the
instantaneous length of the synodic month.
  I divided 360 deg. by the polynomial itself: you get a new polynomial
expression using the Taylor expansion:
    1/(1+x) = 1-x +x^2 - ...
  I truncated the series at the quadratic term, and also scaled from
centuries to lunations as unit of time.  Yes, the lunation length itself
is variable, so strictly speaking the time unit is that of J2000:
variation of the lunation length leads to higher-order terms that are so
small that they can be ignored for centuries, or even millennia unless
you want highest accuracy.  Since the uncertainty in DeltaT is the major
error, those higher-order terms are relatively unimportant.

  HTH,
--
    Tom Peters
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Re: New Moon arithmetic

Tom Peters-2
In reply to this post by Irv Bromberg
> Dear Calendar People:
>
> The reason why I'm so persistent about this new moon arithmetic issue:
>
> Using my implementation of Jean Meeus' "Astronomical Algorithms" 2nd
> edition lunar and solar algorithms, I evaluated the relationship
> between the traditional molad of the Hebrew calendar and the actual
> lunation moments, referred to the meridian of Jerusalem, produced by
> those algorithms.  This generated a curve of scattered points, and
> least squares regression fitted a quadratic

While this is a valid method to model scattered data, you should keep 2
things in mind:
1) A quadratic polynomial is not necessarily the proper model.  A
third-(or higher) order polynomial wil always give a better fit, and the
quadratic term in that one will be different from your original one.
2) The parameters of the polynomial depend on the data that you fit, in
particular the interval of time that you compute it for: if you use a
longer or shorter interval, even if centered on the same epoch, then the
parabola will be different.

So there is always some arbitrariness in your results.

But more fundamentally: you compute accurate positions based on a
computational method that exists of polynomials + periodic terms; and then
you fit a polynomial to the results.  That might make sense if you want to
have a compact representation of the full complex behaviour: but then the
way to go is using cubic splines (i.e. contiguous 4th order polynomials)
or Chebychev polynomials (i.e. polynomials of sines), and these are
typically applied to short intervals of time.
  If you want a polynomial representation spanning millennia, you should
just use the ones from Meeus (essentially from Chapront's ELP) and leave
out all the periodic terms.  Yes, Chapront's polynomials may contain
contributions of very-long-periodic perturbations that have been
modelled by a polynomial rather than a sine: but Meeus's full
expressions do not contain them either.  Moreover, by fitting a
polynomial over some limted period of time (even if it covers millennia)
you are doing exactly the same: approximating sine curves.  For
instance, the great inequalty of Jupiter and Saturn has a period of 9
centuries or so, and the great Venus term 18 centuries, so if you cover
just 2 millennia you will not yet neatly average them out.  Luckily all
planetary perturbations to the Moon's motion are small.


> As it stands, the Wiki Mean New Moon approximation, which hopefully is
> more accurate because it is based on >3 decades of Laser Lunar Ranging,
> and which arguably employs a better estimate of the long-term Delta T
> rate, had a minimum of +44.5 minutes

The difference with 26 minutes that you found from the full theory is
unexpected.  The Wikipedia expression is essentially the same as Meeus's.
My Wikipedia expression has a better tidal term; the difference
contributes (0.9817")*(17*17)/(0.5"/s) = 567s = 9.5 min (assuming Hillel
ca. 300 AD, Moon moving about 0.5"/s).  Also I truncated the New Moon
expression to quadratic terms, but I do not believe the 3rd-order and
higher terms reach as much as the 555" that are needed to explain the
difference of 18.5 min.
Are you doing all this in UT instead of ET?  There may be a difference in
the estimates of DeltaT.

What you can do is just use the polynomial expression for D (the mean
elongation of the Moon from the Sun) from Meeus or directly from Chapront,
and then find times of conjunction (D=0 modulo 360) around 300 AD by
iteration: the differences between the times of the syzygies (which
incidentally are in ET) give you the mean lunation lengths at that epoch
(in SI seconds).
  But the challenge will be to find the correct value of DeltaT.  You
shouldn't use any polynomial approximation, but rather interpolate the
observed values provided by Morrison and Stephenson for the period (they
contain the accumulated effect of all kinds of non-tidal forces that
change the rotation of the Earth but not the motion of the Moon).

  Anyway the uncertainty in DeltaT in the first few centuries AD is easily
10 minutes or so or more, so you may not be able to conclusively
distinguish between the meridians of Babylon, Jerusalem, or somewhere in
between.

--
    Tom Peters
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Re: New Moon arithmetic

VictorEngel
In reply to this post by Irv Bromberg
Dear Tom and Calendar People,

>   Anyway the uncertainty in DeltaT in the first few centuries
> AD is easily
> 10 minutes or so or more, so you may not be able to conclusively
> distinguish between the meridians of Babylon, Jerusalem, or
> somewhere in
> between.


What about the precision and accuracy of the observations made at the time
-- especially if the observations are first new moon visibility, as opposed
to, say eclipse observations, which, it seems to me would be much more
accurate.

Victor
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Re: New Moon arithmetic

Amos Shapir
In reply to this post by Irv Bromberg
At that time, people were not even aware of the difference due to longitude;
it was natural for them to assume that astronomical phenomena (including
sunrise and sunset) occur all over the world at the same time.  So if they
had access to calculations or tables created in Babylon, they had used
these, and assumed that the same numbers applied everywhere.  I don't think
longitudes were considered until at least the 14th century AD.


>From: Irv Bromberg <[hidden email]>
>Date: Tue, 14 Mar 2006 01:08:44 -0500
>
>That +26 minute offset implies that the original molad reference meridian
>was 26 minutes of time to the east of Jerusalem, converted to longitude
>that is 26/1440 * 360 = 6 1/2 degrees east of Jerusalem, which was about
>mid-way between Israel and Babylonia.  That is an interesting finding, but
>it is odd to suggest that the sages decided to compromise by "splitting the
>difference" between the two major Jewish centers at that time.  Since the
>arithmetic for the molad was probably developed in Babylonia by Shmuel the
>Astronomer of Nehardea (he lived one century before Hillel II), and since
>Shmuel had close ties with Babylonian astronomers, I would have expected
>the molad reference meridian to have been in Babylonia itself.  I
>considered it unlikely that Shmuel or other sages would have had the
>knowledge necessary to shift the reference meridian mid-way towards Israel,
>and, if they were going to shift it at all, why wouldn't they shift it all
>of the way to Jerusalem?
>

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Re: New Moon arithmetic

Tom Peters-2
> At that time, people were not even aware of the difference due to
> longitude;
> it was natural for them to assume that astronomical phenomena (including
> sunrise and sunset) occur all over the world at the same time.  So if they
> had access to calculations or tables created in Babylon, they had used
> these, and assumed that the same numbers applied everywhere.  I don't
> think
> longitudes were considered until at least the 14th century AD.

That is not true.  People have known that the Earth is round at least
since Aristotle (4th cy BC), and longitudes have been used at least since
Eratosthenes (3rd cy BC).  At least since Hipparchos (2nd cy BC) people
have known that astronomical phenomena occur at different local times at
different places, and H. (or one of his predecessors) proposed to
determine local longitude from timing e.g. a lunar eclipse.  All this is
in the Almagest, and any competent astronomer (which I suppose includes
Hillel) should have known about the longitude effect.  Incidentally, the
Almagest uses the meridian of Alexandria.

--
    Tom Peters
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Re: New Moon arithmetic

Lance Latham
In reply to this post by Irv Bromberg
RE:

> Therefore I want to know exactly how the arithmetic
> is derived, and to
> ensure to the degree that is possible, that the mean
> lunar conjunction
> arithmetic is as accurate as reasonable.

Lance replies:
Not a solution, but 2 thoughts here.
(1) Before taking these displacements to the bank, and
inferring positions, one should consider the role of
error. Was Babylonian or Jewish astronomy accurate
enough to justify the concusions?

(2) Babylonia as a source of Jewish astronomy makes
more sense to me. Jewish astronomers could have taken
away a corpus of knowledge that had the location of
Babylon occulted as a kind of 'given', without having
the understanding of how to correct that factor for
another location.

-Lance


Lance Latham
[hidden email]
Phone:    (518) 274-0570
Address: 78 Hudson Avenue/1st Floor, Green Island, NY 12183
 




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Re: New Moon arithmetic

Irv Bromberg
In reply to this post by Tom Peters-2
Dear Calendar People:

New chart posted today:  "The Duration of the Lunation", as the number
of hours and minutes in excess of 29 days versus the cumulative
percentage, for past, present, and future:

<http://individual.utoronto.ca/kalendis/hebrew/rect.htm#pct>

-- Irv Bromberg, Toronto, Canada

<http://www.sym454.org/>
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Re: New Moon arithmetic

VictorEngel
In reply to this post by Irv Bromberg
What is the date range for present?

> -----Original Message-----
> From: East Carolina University Calendar discussion List
> [mailto:[hidden email]]On Behalf Of Irv Bromberg
> Sent: Thursday, March 16, 2006 2:39 PM
> To: [hidden email]
> Subject: Re: New Moon arithmetic
>
>
> Dear Calendar People:
>
> New chart posted today:  "The Duration of the Lunation", as
> the number
> of hours and minutes in excess of 29 days versus the cumulative
> percentage, for past, present, and future:
>
> <http://individual.utoronto.ca/kalendis/hebrew/rect.htm#pct>
>
> -- Irv Bromberg, Toronto, Canada
>
> <http://www.sym454.org/>
>
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Re: New Moon arithmetic

Irv Bromberg
>> -----Original Message-----
>> From: East Carolina University Calendar discussion List
>> [mailto:[hidden email]]On Behalf Of Irv Bromberg
>> Sent: Thursday, March 16, 2006 2:39 PM
>> To: [hidden email]
>> Subject: Re: New Moon arithmetic
>>
>> New chart posted today:  "The Duration of the Lunation", as the number
>> of hours and minutes in excess of 29 days versus the cumulative
>> percentage, for past, present, and future:
>>
>> <http://individual.utoronto.ca/kalendis/hebrew/rect.htm#pct>

On Mar 16, 2006, at 17:14, Engel,Victor wrote:

> What is the date range for "present era" ?

Bromberg replies:

As stated on the chart, for each era it is plus or minus 1000 lunar
months.
The "present era" is this month, so from 1000 months ago until 1000
months into the future is called "Present Era" on that chart.

I suppose that I could state explicitly that March 2006 was the
"present" used for that chart, but I didn't want to get into cluttering
it with calendar dates (Gregorian, Hebrew etc.), nor cluttering it with
start/end dates for each era.

I tried to use present -2000 years for the past and +2000 for the
future, but their lines were too close to discern from the present era
line, even when stretched on larger paper layout.  I had to go to -5000
years ago for the past, and +5000 years into the future in order to get
chart lines that were sufficiently distinct to actually see the trend.  
This is beyond the best accuracy range of my Meeus lunar algorithms, of
course, but on the scale of that chart, more accurate algorithms would
not make any visually apparent difference.

Note that Delta T is not a factor in the accuracy here, even though
included, because the duration of each lunation is from one lunar
conjunction to the next, over which time Delta T would hardly change.


-- Irv Bromberg, Toronto, Canada

<http://www.sym454.org/>
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Re: New Moon arithmetic

Irv Bromberg
In reply to this post by Lance Latham
Dear Calendar People:

It is understood that the larger proportion of variability of the  
duration of the lunation is due to the extra approximately 2 days 5  
hours that Moon takes over and above a sidereal revolution to get back  
into conjunction with Sun.  The speed of Moon for that extra segment  
varies, mainly depending on the lunar distance from Earth, and to a  
lesser degree mainly on the Earth's rate of motion, which in turns  
depends on its distance from Sun.

I attempted an analysis of the variability of the lunar sidereal  
revolution, expecting to find only a minor amount of variability, such  
as plus or minus 15 to 30 minutes.  My method was to implement an  
iterative function that searches for a specified lunar ecliptic  
longitude after a given moment, then I used that to list each moment  
that Moon reached zero degrees ecliptic longitude for 1000 lunar  
revolutions, then expressed the duration of the sidereal revolution in  
terms of hours and minutes in excess of 27 days and plotted the  
distribution of values as a cumulative percentile chart.

To my surprise, the lunar sidereal revolution appears to vary over a  
range of 6 hours, which is 4/9 of the total 13+1/2 hour variability of  
the duration of the lunation (from New Moon to New Moon) !

I've posted this analysis as a 60 KB PDF which is only accessible via  
the following URL:

<http://individual.utoronto.ca/kalendis/hebrew/ 
Lunation_Sidereal_Percentiles.pdf>

Have I done something wrong?
Is this degree of variability to be expected?  If so, then what  
explains it?

-- Irv Bromberg, Toronto, Canada

<http://www.sym454.org/>
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Re: New Moon arithmetic

Palmen, KEV (Karl)
In reply to this post by Irv Bromberg
Dear Irv and Calendar People

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:[hidden email]]On Behalf Of Irv Bromberg
Sent: 16 March 2006 22:48
To: [hidden email]
Subject: Re: New Moon arithmetic


Note that Delta T is not a factor in the accuracy here, even though
included, because the duration of each lunation is from one lunar
conjunction to the next, over which time Delta T would hardly change.

KARL SAYS: Nevertheless the change of the Delta T over a lunation is sufficiently large to reverse the long term change in the mean lunation.

Karl

08(01(17 till noon
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Re: New Moon arithmetic

Irv Bromberg
Dear Calendar People:

Further to my posting of the chart showing the variation of the lunar
sidereal month, it occurred to me that perhaps some of the monthly
variations are due to the lunar latitude, because I am measuring the
duration of the lunar revolution according to the ecliptic longitude.  
When Moon's latitude is moving away or towards the ecliptic, a portion
of the lunar motion is not projected as a change in ecliptic longitude.
  The most direct projection of the lunar motion onto the ecliptic
occurs, as I understand it, when Moon is near either orbital node or
near either extreme of latitude.

Therefore, I checked but found NO CORRELATION between the lunar
latitude and the length of the lunar sidereal month (taking the lunar
latitude when Moon is at zero degrees ecliptic longitude on each
revolution).  Perhaps this is not a good way to detect such a
relationship, however.

-- Irv Bromberg, Toronto, Canada

<http://www.sym454.org/>
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Re: New Moon arithmetic

Irv Bromberg
In reply to this post by Palmen, KEV (Karl)
On Mar 17, 2006, at 03:51, Palmen, KEV (Karl) wrote:
> BROMBERG wrote:
> New chart posted today:  "The Duration of the Lunation", as the number
> of hours and minutes in excess of 29 days versus the cumulative
> percentage, for past, present, and future:
> <http://individual.utoronto.ca/kalendis/hebrew/rect.htm#pct>

and later...

> Note that Delta T is not a factor in the accuracy here, even though
> included, because the duration of each lunation is from one lunar
> conjunction to the next, over which time Delta T would hardly change.

> KARL SAYS: Nevertheless the change of the Delta T over a lunation is
> sufficiently large to reverse the long term change in the mean
> lunation.

BROMBERG replies:

Perhaps it is not clear how I generated the chart.

For the present era, I took this month's lunar conjunction moment (UT)
as the center point.
Starting from 1000 lunations in the past and continuing to 1000
lunations into the future, I generated a list of 2000 actual lunation
durations.
They were all 29 days plus a fraction of a day, so I subtracted 29 from
all, and expressed the fraction as hours and minutes.
I sorted the list, ascending, to produce 2000 lunation durations sorted
from shortest to longest.
I plotted row number / 2000 expressed as a percentage on the Y axis
(labeled "Cumulative Percentile") against the hours and minutes of the
corresponding lunation length in excess of 29 days.
I connected the points with straight line interpolation, and hid the
display of the points themselves, so as not to clutter the chart.

The chart shows a variation range of 13 1/2 hours between the shortest
and longest lunations in the present era.
It also shows that the variation was slightly wider 5000 years ago due
to the greater Earth orbital eccentricity.
It also shows that the variation will be slightly narrower 5000 years
from now due to the lesser Earth orbital eccentricity.

According to my estimate, as per the arithmetic posted on the Rectified
Hebrew Calendar web page, the mean synodic month length, in terms of
mean solar days, will be almost 4.5 seconds shorter 5000 years from now
than it was 5000 years ago.

There is absolutely no way, with or without Delta T correction, that
anybody could visually notice that the "future" curve on my chart is
shifted by 4.5 seconds towards shorter lunations than is the "past"
curve, in the presence of the 13-hour inherent variations of the
duration of the lunations.  It is simply too small a difference to
shift even one pixel on the chart.

The progressively shorter mean synodic month is very important when the
length of each month is being simply summed to estimate a mean lunar
conjunction moment, or a molad, or similar cumulative value.  But as an
individual lunar month length, the secular change in the lunation
interval is truly negligible, amounting to less than 30 microseconds
out of a value that has a median close to 29 days 12 hours 30 minutes.

-- Irv Bromberg, Toronto, Canada

<http://www.sym454.org/>
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Re: New Moon arithmetic

Tom Peters-2
In reply to this post by Irv Bromberg
> Dear Calendar People:

> I attempted an analysis of the variability of the lunar sidereal
> revolution, expecting to find only a minor amount of variability, such
> as plus or minus 15 to 30 minutes.  My method was to implement an
> iterative function that searches for a specified lunar ecliptic
> longitude after a given moment, then I used that to list each moment
> that Moon reached zero degrees ecliptic longitude for 1000 lunar
> revolutions, then expressed the duration of the sidereal revolution in
> terms of hours and minutes in excess of 27 days and plotted the
> distribution of values as a cumulative percentile chart.
>
> To my surprise, the lunar sidereal revolution appears to vary over a
> range of 6 hours, which is 4/9 of the total 13+1/2 hour variability of
> the duration of the lunation (from New Moon to New Moon) !

> Is this degree of variability to be expected?  If so, then what
> explains it?

The speed of the Moon varies with its anomalistic period (27.55455 days).
The sidereal month is shorter (27.32166).  Also from one month to the next
the solar contribution changes.

--
    Tom Peters
12