New moon: Difference between revisions
m →Approximate formula: Fix ups, my bad |
Tom Peters (talk | contribs) →Explanation of the formulae: TP: because nobody laughed, replaced whimsical reference to a book cover by the simple explanation |
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==Approximate formula== |
==Approximate formula== |
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The time interval between New Moons - a ''lunation'' - is variable. The mean time between New Moons, the [[synodic month]], is about 29.53... days. An approximate formula to compute the mean moments of New Moon ([[conjunction]] between Sun and Moon) for successive months is: |
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An approximate formula for calculating the occurence of the'' 'Nth' New Moon (''''N'''')'' (the moment of astronomical conjunction) is<!-- {{fact}} see "explanation" section below --> given in units known as [[Terrestrial Time]] (TT), which by current conventions is calculated with a base universal time from the instant of the new years day in the current millenium, which is to say from the year two thousandths (2000 [[AD]]) very first second. The formula gives only what is known as the ''average time'', for orbital perturbations (slight variations) will cause a similar slight error from an astronomically exact moment. The offset or adjustment termed''' 'd' '''(for difference) is: |
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: |
:<math>d = 5.597661 + 29.5305888610 \times N + (102.026 \times 10^{-12})\times N^2</math>, |
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where |
where '''N''' is an integer, starting with 0 for the first New Moon in the year 2000, and that is incremented by 1 for each successive synodic month; and the result '''d''' is the number of days (and fractions) since 2000-01-01 00:00:00 reckoned in the time scale known as [[Terrestrial Time]] ('''TT''') used in [[ephemerides]]. |
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(Note that the number 29.53.. is the [[synodic month]].) |
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<!-- To: TomPeters... |
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Since you didn't close the equation above, introducing this one into the discussion is premature, without at least applying the same symbol'' 'd' ''or some context linking it to the above. I can but presume you meant to convert the date and time from TT to Universal Time, take d less the new equation. |
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| ⚫ | |||
:<math>-0.000739 - (235 \times 10^{-12})\times N^2</math> days |
:<math>-0.000739 - (235 \times 10^{-12})\times N^2</math> days |
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| ⚫ | Periodic perturbations change the time of true conjunction from these mean values. For all New Moons between 1601 and 2401, the maximum difference is 0.592 days = 14h13m in either direction. The duration of a lunation (''i.e.'' the time from New Moon to the next New Moon) varies in this period between 29.272 and 29.833 days, i.e. −0.259d = 6h12m shorter, or +0.302d = 7h15m longer than average <ref>Roger W. Sinnott: "How Long Is a Lunar Month?", Sky&Telescope Nov.1993 pp.76..77</ref>. This range is smaller than the difference between mean and true conjunction, because during one lunation the periodic terms cannot all change to their maximum opposite value. |
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So say what you mean, and remember what you write should be something that a kid in Junior high should be able to follow the gist of. Hence, I'm removing this equation temporarily. Also, below you are switching terms to mean, and used average above. We both know that mean is but one kind o f average, so I urge consistancy. I introduced (forshadowed) this point, but I'm budding out for you to take the example of my dumbing down and apply in the following. Best regards User:fabartus |
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---> |
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| ⚫ | Periodic perturbations change the time of true conjunction from these mean values. For all New Moons between 1601 and 2401, the maximum difference is 0.592 days = 14h13m in either direction. The duration of a lunation (from New Moon to the next New Moon) varies in this period between 29.272 and 29.833 days, i.e. |
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See the article on the [[full moon cycle]] for a fairly simple method to compute the moment of New Moon more accurately. |
See the article on the [[full moon cycle]] for a fairly simple method to compute the moment of New Moon more accurately. |
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==Explanation of the formulae== |
==Explanation of the formulae== |
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The moment of mean conjunction can easily be computed from an expression for the |
The moment of mean conjunction can easily be computed from an expression for the mean ecliptic longitude of the Moon minus the mean ecliptic longitude of the Sun (Delauney parameter '''D'''). [[Jean Meeus]] gave formulae to compute this in his popular ''Astronomical Formulae for Calculators'' based on the ephemerides of Brown and Newcomb (ca. 1900); and in his ''Astronomical Algorithms'' <ref>formula 47.1 in Jean Meeus (1991): "Astronomical Algorithms" (1st ed.) ISBN 0-943396-35-2</ref> based on the ELP2000-85 <ref>M.Chapront-Touzé, J.Chapront (1988): "ELP2000-85: a semianalytical lunar ephemeris adequate for historical times". Astron.Astrophys. '''190''', 342..352</ref>. These are now outdated: Chapront ''et al.'' (2002) <ref>J.Chapront, M.Chapront-Touzé, G.Francou (2002): "[http://aanda.u-strasbg.fr:2002/articles/aa/abs/2002/20/aa2201/aa2201.html A new determination of lunar orbital parameters, precession constant, and tidal acceleration from LLR measurements]". Astronomy & Astrophysics '''387''', 700–709</ref> published improved parameters. Also Meeus's formula uses a fractional variable to allow computation of the four main phases, and uses a second variable for the secular terms. For the convenience of the reader, the formula given above is based on Chapront's latest parameters and expressed with a single integer variable, and the following additional terms have been added: |
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constant term: |
constant term: |
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* Like Meeus, applied the constant terms of the [[aberration of light]]<ref> |
* Like Meeus, applied the constant terms of the [[aberration of light]]<ref>which, for solar system bodies, is the correction for light time; the constant term in the expressions is just the mean distance divided by the speed of light: this yields an angle in radians that the apparent position of the body lags behind its computed geometric position.</ref> to obtain the apparent difference in ecliptic longitudes: |
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:Sun: +20.496" <ref>Derived Constant #14 from from the IAU (1976) System of Astronomical Constants (proceedings of IAU Sixteenth General Assembly (1976): Trans. IAU XVIB p.58 (1977)); or any astronomical almanac; or ''e.g.'' [http://www.kayelaby.npl.co.uk/general_physics/2_7/2_7_2.html]</ref> |
:Sun: +20.496" <ref>Derived Constant #14 from from the IAU (1976) System of Astronomical Constants (proceedings of IAU Sixteenth General Assembly (1976): Trans. IAU XVIB p.58 (1977)); or any astronomical almanac; or ''e.g.'' [http://www.kayelaby.npl.co.uk/general_physics/2_7/2_7_2.html]</ref> |
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: +102.026E−12*N<sup>2</sup> days. |
: +102.026E−12*N<sup>2</sup> days. |
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* For UT: analysis of historical observations show that ΔT has a long-term increase of +31 s/cy<sup>2</sup>. |
* For UT: analysis of historical observations show that ΔT has a long-term increase of +31 s/cy<sup>2</sup> <ref>F.R. Stephenson, Historical Eclipses and Earth's Rotation. Cambridge University Press 1997. ISBN 0-521-46194-4 . p.507, eq.14.3</ref>. Converted to days and lunations <ref>31 s / (86400 s/d) / [ (36525 d/cy) / (29.530... d/lunation) ]<sup>2</sup> </ref>, the correction from ET to UT becomes : |
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: −235E−12*N<sup>2</sup> days. |
: −235E−12*N<sup>2</sup> days. |
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The theoretical tidal contribution to ΔT is about +42 s/cy<sup>2</sup>; |
The theoretical tidal contribution to ΔT is about +42 s/cy<sup>2</sup> <ref>Stephenson 1997 ''op.cit.'' p.38 eq.2.8</ref>; the smaller observed value is thought to be mostly due to changes in the shape of the Earth <ref>Stephenson 1997 ''op.cit.'' par.14.8</ref>. Because the discrepancy is not fully explained, uncertainty of our prediction of UT (rotation angle of the Earth) may be as large as the difference between these values: 11 s/cy<sup>2</sup>. The error in the position of the Moon itself is only maybe 0.5 "/cy<sup>2</sup> <ref>from differerences of various earlier determinations of the tidal acceleration, see e.g. Stephenson 1997 ''op.cit.'' par.2.2.3</ref>, or (because the apparent mean angular velocity of the Moon is about 0.5"/s), 1 s/cy<sup>2</sup> in the time of conjunction with the Sun. |
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==Religious use== |
==Religious use== |
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Revision as of 14:41, 21 December 2006
- This article is about the lunar phase; for other uses, see New Moon (disambiguation).
The New Moon is the lunar phase that occurs when the Moon, in its monthly orbital motion around Earth, lies between Earth and the Sun, and is therefore in conjunction with the Sun as seen from Earth. At this time, the illuminated half of the Moon faces directly toward the Sun, and the dark or unilluminated portion of the Moon faces directly toward Earth, so that the Moon is invisible as seen from Earth.
New Moon is often considered to occur at the time of the appearance of the first visible crescent of the Moon, after conjunction with the Sun. This takes place over the western horizon in a brief period between sunset and moonset, and therefore the precise time and even the date of the appearance of the New Moon by this definition will be influenced by the geographical location of the observer. The astronomical New Moon, sometimes known as the dark moon to avoid confusion, occurs by definition at the moment of conjunction in ecliptic longitude with the Sun, when the Moon is invisible from the Earth. This moment is unique and does not depend on location, and under certain circumstances it may be coincident with a solar eclipse.
The New Moon is the beginning of the month in lunar calendars such as the Muslim calendar, and in lunisolar calendars such as the Hebrew calendar, Hindu calendars, Buddhist calendar, and Chinese calendar.
Approximate formula
The time interval between New Moons - a lunation - is variable. The mean time between New Moons, the synodic month, is about 29.53... days. An approximate formula to compute the mean moments of New Moon (conjunction between Sun and Moon) for successive months is:
- ,
where N is an integer, starting with 0 for the first New Moon in the year 2000, and that is incremented by 1 for each successive synodic month; and the result d is the number of days (and fractions) since 2000-01-01 00:00:00 reckoned in the time scale known as Terrestrial Time (TT) used in ephemerides.
To obtain this moment expressed in Universal Time (UT, world clock time), add the result of following approximate correction to the result d obtained above:
- days
Periodic perturbations change the time of true conjunction from these mean values. For all New Moons between 1601 and 2401, the maximum difference is 0.592 days = 14h13m in either direction. The duration of a lunation (i.e. the time from New Moon to the next New Moon) varies in this period between 29.272 and 29.833 days, i.e. −0.259d = 6h12m shorter, or +0.302d = 7h15m longer than average [1]. This range is smaller than the difference between mean and true conjunction, because during one lunation the periodic terms cannot all change to their maximum opposite value.
See the article on the full moon cycle for a fairly simple method to compute the moment of New Moon more accurately.
The long-term error of the formula is approximately: 1*cy*cy seconds in TT, and 11*cy*cy seconds in UT (cy is centuries since 2000; see section Explanation of the formulae for details.)
Explanation of the formulae
The moment of mean conjunction can easily be computed from an expression for the mean ecliptic longitude of the Moon minus the mean ecliptic longitude of the Sun (Delauney parameter D). Jean Meeus gave formulae to compute this in his popular Astronomical Formulae for Calculators based on the ephemerides of Brown and Newcomb (ca. 1900); and in his Astronomical Algorithms [2] based on the ELP2000-85 [3]. These are now outdated: Chapront et al. (2002) [4] published improved parameters. Also Meeus's formula uses a fractional variable to allow computation of the four main phases, and uses a second variable for the secular terms. For the convenience of the reader, the formula given above is based on Chapront's latest parameters and expressed with a single integer variable, and the following additional terms have been added:
constant term:
- Like Meeus, applied the constant terms of the aberration of light[5] to obtain the apparent difference in ecliptic longitudes:
- Sun: +20.496" [6]
- Moon: −0.704" [7]
- Correction in conjunction: −0.000451 days. [8]
- For UT: at 1 Jan. 2000, ΔT (= TT - UT ) was +63.83 s [9]; hence the correction for the clock time UT = TT - ΔT of the conjunction is:
- −0.000739 days.
quadratic term:
- In ELP2000–85 (see Chapront et alii 1988), D has a quadratic term of −5.8681"×T2; expressed in lunations N, this yields a correction of +87.403E–12*N2 [10] days to the time of conjunction. The term includes a tidal contribution of 0.5×(−23.8946 "/cy2). The most current estimate from Lunar Laser Ranging for the acceleration is (see Chapront et alii 2002): (−25.858 ±0.003) "/cy2. Therefore the new quadratic term of D is = -6.8498"×T2 [11]. Indeed the polynomial provided by Chapront et alii (2002) provides the same value (their Table 4). This translates to a correction of +14.622E−12*N2 days to the time of conjunction; the quadratic term now is:
- +102.026E−12*N2 days.
- For UT: analysis of historical observations show that ΔT has a long-term increase of +31 s/cy2 [12]. Converted to days and lunations [13], the correction from ET to UT becomes :
- −235E−12*N2 days.
The theoretical tidal contribution to ΔT is about +42 s/cy2 [14]; the smaller observed value is thought to be mostly due to changes in the shape of the Earth [15]. Because the discrepancy is not fully explained, uncertainty of our prediction of UT (rotation angle of the Earth) may be as large as the difference between these values: 11 s/cy2. The error in the position of the Moon itself is only maybe 0.5 "/cy2 [16], or (because the apparent mean angular velocity of the Moon is about 0.5"/s), 1 s/cy2 in the time of conjunction with the Sun.
Religious use
The Islamic calendar has retained an observational definition of the New Moon, marking the new month when the first Crescent Moon is actually seen, and making it impossible to be certain in advance of when a specific month will begin (in particular, the exact date on which Ramadan will begin is not known in advance). In Saudi Arabia, if the weather is cloudy when the New Moon is expected, observers are sent up in airplanes. In Iran a special committee receives observations of every new moon to determine the beginning of each month. This committee uses one hundred groups of observers.
The New Moon is the beginning of the month in the Chinese calendar. Some Buddhist Chinese keep a vegetarian diet on the New Moon and Full Moon each month.
Trivia
- In the anime and manga InuYasha, this is the time where the hanyō Inuyasha loses his yōkai powers temporarily and turns full human for the night.
Literature
- ^ Roger W. Sinnott: "How Long Is a Lunar Month?", Sky&Telescope Nov.1993 pp.76..77
- ^ formula 47.1 in Jean Meeus (1991): "Astronomical Algorithms" (1st ed.) ISBN 0-943396-35-2
- ^ M.Chapront-Touzé, J.Chapront (1988): "ELP2000-85: a semianalytical lunar ephemeris adequate for historical times". Astron.Astrophys. 190, 342..352
- ^ J.Chapront, M.Chapront-Touzé, G.Francou (2002): "A new determination of lunar orbital parameters, precession constant, and tidal acceleration from LLR measurements". Astronomy & Astrophysics 387, 700–709
- ^ which, for solar system bodies, is the correction for light time; the constant term in the expressions is just the mean distance divided by the speed of light: this yields an angle in radians that the apparent position of the body lags behind its computed geometric position.
- ^ Derived Constant #14 from from the IAU (1976) System of Astronomical Constants (proceedings of IAU Sixteenth General Assembly (1976): Trans. IAU XVIB p.58 (1977)); or any astronomical almanac; or e.g. [1]
- ^ formula in: G.M.Clemence, J.G.Porter, D.H.Sadler (1952): "Aberration in the lunar ephemeris", Astronomical Journal 57(5) (#1198) pp.46..47 [2]; but computed with the conventional value of 384400 km for the mean distance which gives a different rounding in the last digit.
- ^ Apparent mean solar longitude is -20.496" from mean geometric longitude; apparent mean lunar longitude -0.704" from mean geometric longitude; correction to D = Moon - Sun is -0.704" + 20.496" = +19.792" that the apparent Moon is ahead of the apparent Sun; divided by 360*3600"/circle is 1.527..E-5 part of a circle; multiplied by 29.53... days for the Moon to travel a full circle with respect to the Sun is 0.000451 days that the apparent Moon reaches the apparent Sun ahead of time.
- ^ see e.g. [3] ; the IERS is the official source for these numbers; they provide TAI-UTC here and UT1-UTC here; ΔT = 32.184s + (TAI-UTC) - (UT1-UTC)
- ^ delay is − (−5.8681") / (60×60×360 "/circle) / (36525/29.530... lunations per Julian century)2 × (29.530... days/lunation) days
- ^ −5.8681" + 0.5×(−25.858 − −23.8946)
- ^ F.R. Stephenson, Historical Eclipses and Earth's Rotation. Cambridge University Press 1997. ISBN 0-521-46194-4 . p.507, eq.14.3
- ^ 31 s / (86400 s/d) / [ (36525 d/cy) / (29.530... d/lunation) ]2
- ^ Stephenson 1997 op.cit. p.38 eq.2.8
- ^ Stephenson 1997 op.cit. par.14.8
- ^ from differerences of various earlier determinations of the tidal acceleration, see e.g. Stephenson 1997 op.cit. par.2.2.3
See also