Nutation (astronomy)

This article deals with precession under fluctuating external influence. For the movement of the figure axis of a freely rotating body see Nutation (Physics) .

Nutation of the earth's axis for the period from 2015 to 2033

The astronomical nutation (from Latin nutare 'nod') is the relatively rapidly fluctuating part of the precession of the earth's axis in space under the influence of the sun and moon . The strongest of these fluctuations is caused by the precession of the lunar orbit , whose nodal line revolves with a period of 18.6 years. The amplitude of this component is

• ± 9.2 ″ perpendicular to the ecliptic and
• ± 6.8 ″ parallel to the ecliptic.

Other components of nutation have amplitudes below 1 "and shorter periods.

Together with the direction of the earth's axis, the orientation of the equatorial coordinate system for the star locations also changes .

The nutation of the earth's axis was discovered in 1728 by James Bradley , when he was doing precise analyzes of star coordinates . The cause could only be clarified 20 years later. The aberration of light, also discovered by Bradley, is similarly large.

Influence on the star coordinates

Since the earth's axis and ecliptic define the astronomical coordinate system, the coordinates of all celestial bodies change with the direction of the earth's axis. For the approximately 2000 fundamental stars - on which most measurements in the sky are based - the star locations are calculated in advance, taking precession and nutation into account, and published in astronomical yearbooks or on the Internet . The most important of these yearbooks is called Apparent Places of Fundamental Stars and is published annually in advance by the Astronomisches Recheninstitut (ARI) in Heidelberg.

However, the influence of the “short-period” nutation components with periods of less than 35 days is not taken into account in the case of stars whose locations are tabulated at intervals of ten days; they have to be calculated using auxiliary tables or small additional programs and added to the published star words.

The influence of the pole movement , however, is attached to the measurements themselves, as is the time correction dUT1 of the earth's rotation .

Calculation model

The different periods of nutation

To calculate the nutation of the earth, models have been published by the IAU . The positions of the moon and sun are taken into account (IAU 1980 Theory of Nutation) , with the latest model also the planetary positions (IAU 2000A Theory of Nutation) . The 1980 theory achieves an accuracy of 0.0001 ″, which is sufficient for most astronomical applications.

The basis is formed by the periodic elements of nutation with their different period durations (see graphic).

Be

• T is the number of Julian centuries with ${\displaystyle T={\frac {JDE-2451545}{36525}}}$
• ${\displaystyle \Delta \psi }$the nutation of the length and
• ${\displaystyle \Delta \epsilon }$that of the crooked .

JDE traditionally means Julian date counted according to ephemeris time , which can be transferred almost identically to the date and time value in TDB . For a conversion from UTC , the value of Delta T is particularly required, which in 2010 is about 61 seconds.

The amplitudes are:

• ± 9.2 ″ for ${\displaystyle \Delta \epsilon }$
• ± 17.2 ″ for (= ± 6.8 ″ / sin ) ${\displaystyle \Delta \psi }$ ${\displaystyle \epsilon }$

with the mean current inclination of the ecliptic . ${\displaystyle \epsilon =23^{\circ }\,26'}$

For the further calculation, five influencing variables are required:

• Average elongation of the moon

${\displaystyle D={\frac {1^{\circ }}{3600}}\left({1072260{,}703692+1602961601{,}2090T-6{,}3706{T^{2}}+0{,}006593{T^{3}}-0{,}00003169{T^{4}}}\right)}$

• Mean anomaly of the sun

${\displaystyle M={\frac {1^{\circ }}{3600}}\left({1287104{,}793048+129596581{,}0481T-0{,}5532{T^{2}}+0{,}000136{T^{3}}-0{,}00001149{T^{4}}}\right)}$

• Mean anomaly of the moon

${\displaystyle M'={\frac {1^{\circ }}{3600}}\left({485868{,}249036+1717915923{,}2178T+31{,}8792{T^{2}}+0{,}051635{T^{3}}-0{,}00024470{T^{4}}}\right)}$

• Middle argument of the perigee

${\displaystyle F={\frac {1^{\circ }}{3600}}\left({335779{,}526232+1739527262{,}8478T-12{,}7512{T^{2}}-0{,}001037{T^{3}}+0{,}00000417{T^{4}}}\right)}$

• Mean length of the ascending node of the lunar orbit

${\displaystyle \Omega ={\frac {1^{\circ }}{3600}}\left({450160{,}398036-6962890{,}5431T+7{,}4722{T^{2}}+0{,}007702{T^{3}}-0{,}00005939{T^{4}}}\right)}$.

For further calculation it is recommended to reduce the angle to the value range 0 °… 360 °.

With the following table the individual expressions are summed up as follows:

${\displaystyle \Delta \psi =10^{-4}\cdot \sum \limits _{i}\left(S_{i}'+S_{i}''\cdot T\right)\cdot \sin \left(D_{i}\cdot D+M_{i}\cdot M+M_{i}'\cdot M'+F_{i}\cdot F+\Omega _{i}\cdot \Omega \right)}$

${\displaystyle \Delta \varepsilon =10^{-4}\cdot \sum \limits _{i}\left(C_{i}'+C_{i}''\cdot T\right)\cdot \cos \left(D_{i}\cdot D+M_{i}\cdot M+M_{i}'\cdot M'+F_{i}\cdot F+\Omega _{i}\cdot \Omega \right)}$

If sines and cosines require arguments in radians , the values ​​of the previous calculations must be converted into radians . The result ( ) is available in arc seconds . ${\displaystyle \Delta \psi ,\Delta \varepsilon }$

Table of the IAU 1980 Theory of Nutation

${\displaystyle i}$	Periodic arguments					Non-periodic arguments
	${\displaystyle M_{i}'}$	${\displaystyle M_{i}}$	${\displaystyle F_{i}}$	${\displaystyle D_{i}}$	${\displaystyle \Omega _{i}}$	Longitude ${\displaystyle (10^{-4}\mathrm {arcsec} )}$		Latitude ${\displaystyle (10^{-4}\mathrm {arcsec} )}$
						${\displaystyle S_{i}'}$	${\displaystyle S_{i}''}$	${\displaystyle C_{i}'}$	${\displaystyle C_{i}''}$
1	0	0	0	0	1	−171996	−174.2	92025	8.9
2	0	0	2	−2	2	−13187	−1.6	5736	−3.1
3	0	0	2	0	2	−2274	−0.2	977	−0.5
4th	0	0	0	0	2	2062	0.2	−895	0.5
5	0	1	0	0	0	1426	−3.4	54	−0.1
6th	1	0	0	0	0	712	0.1	−7	0.0
7th	0	1	2	−2	2	−517	1.2	224	−0.6
8th	0	0	2	0	1	−386	−0.4	200	0.0
9	1	0	2	0	2	−301	0.0	129	−0.1
10	0	−1	2	−2	2	217	−0.5	−95	0.3
11	1	0	0	−2	0	−158	0.0	−1	0.0
12	0	0	2	−2	1	129	0.1	−70	0.0
13	−1	0	2	0	2	123	0.0	−53	0.0
14th	0	0	0	2	0	63	0.0	−2	0.0
15th	1	0	0	0	1	63	0.1	−33	0.0
16	−1	0	2	2	2	−59	0.0	26th	0.0
17th	−1	0	0	0	1	−58	−0.1	32	0.0
18th	1	0	2	0	1	−51	0.0	27	0.0
19th	2	0	0	−2	0	48	0.0	1	0.0
20th	−2	0	2	0	1	46	0.0	−24	0.0
21st	0	0	2	2	2	−38	0.0	16	0.0
22nd	2	0	2	0	2	−31	0.0	13	0.0
23	2	0	0	0	0	29	0.0	−1	0.0
24	1	0	2	−2	2	29	0.0	−12	0.0
25th	0	0	2	0	0	26th	0.0	−1	0.0
26th	0	0	2	−2	0	−22	0.0	0	0.0
27	−1	0	2	0	1	21st	0.0	−10	0.0
28	0	2	0	0	0	17th	−0.1	0	0.0
29	0	2	2	−2	2	−16	0.1	7th	0.0
30th	−1	0	0	2	1	16	0.0	−8	0.0
31	0	1	0	0	1	−15	0.0	9	0.0
32	1	0	0	−2	1	−13	0.0	7th	0.0
33	0	−1	0	0	1	−12	0.0	6th	0.0
34	2	0	−2	0	0	11	0.0	0	0.0
35	−1	0	2	2	1	−10	0.0	5	0.0
36	1	0	2	2	2	−8	0.0	3	0.0
37	1	1	0	−2	0	−7	0.0	0	0.0
38	0	1	2	0	2	7th	0.0	−3	0.0
39	0	−1	2	0	2	−7	0.0	3	0.0
40	0	0	2	2	1	−7	0.0	3	0.0
41	−2	0	0	2	1	−6	0.0	3	0.0
42	1	0	0	2	0	6th	0.0	0	0.0
43	2	0	2	−2	2	6th	0.0	−3	0.0
44	0	0	0	2	1	−6	0.0	3	0.0
45	1	0	2	−2	1	6th	0.0	−3	0.0
46	0	−1	2	−2	1	−5	0.0	3	0.0
47	0	0	0	−2	1	−5	0.0	3	0.0
48	1	−1	0	0	0	5	0.0	0	0.0
49	2	0	2	0	1	−5	0.0	3	0.0
50	2	0	0	−2	1	4th	0.0	−2	0.0
51	0	1	2	−2	1	4th	0.0	−2	0.0
52	1	0	0	−1	0	−4	0.0	0	0.0
53	0	1	0	−2	0	−4	0.0	0	0.0
54	1	0	−2	0	0	4th	0.0	0	0.0
55	0	0	0	1	0	−4	0.0	0	0.0
56	−2	0	2	0	2	−3	0.0	1	0.0
57	1	−1	0	−1	0	−3	0.0	0	0.0
58	1	1	0	0	0	−3	0.0	0	0.0
59	1	0	2	0	0	3	0.0	0	0.0
60	1	−1	2	0	2	−3	0.0	1	0.0
61	−1	−1	2	2	2	−3	0.0	1	0.0
62	3	0	2	0	2	−3	0.0	1	0.0
63	0	−1	2	2	2	−3	0.0	1	0.0
64	0	−2	2	−2	1	−2	0.0	1	0.0
65	−2	0	0	0	1	−2	0.0	1	0.0
66	1	1	2	0	2	2	0.0	−1	0.0
67	−1	0	2	−2	1	−2	0.0	1	0.0
68	2	0	0	0	1	2	0.0	−1	0.0
69	1	0	0	0	2	−2	0.0	1	0.0
70	3	0	0	0	0	2	0.0	0	0.0
71	0	0	2	1	2	2	0.0	−1	0.0
72	−1	0	2	4th	2	−2	0.0	1	0.0
73	2	0	−2	0	1	1	0.0	0	0.0
74	2	1	0	−2	0	1	0.0	0	0.0
75	0	0	−2	2	1	1	0.0	0	0.0
76	0	1	−2	2	0	−1	0.0	0	0.0
77	0	1	0	0	2	1	0.0	0	0.0
78	−1	0	0	1	1	1	0.0	0	0.0
79	0	1	2	−2	0	−1	0.0	0	0.0
80	−1	0	0	0	2	1	0.0	−1	0.0
81	1	0	0	−4	0	−1	0.0	0	0.0
82	−2	0	2	2	2	1	0.0	−1	0.0
83	2	0	0	−4	0	−1	0.0	0	0.0
84	1	1	2	−2	2	1	0.0	−1	0.0
85	1	0	2	2	1	−1	0.0	1	0.0
86	−2	0	2	4th	2	−1	0.0	1	0.0
87	−1	0	4th	0	2	1	0.0	0	0.0
88	1	−1	0	−2	0	1	0.0	0	0.0
89	2	0	2	−2	1	1	0.0	−1	0.0
90	2	0	2	2	2	−1	0.0	0	0.0
91	1	0	0	2	1	−1	0.0	0	0.0
92	0	0	4th	−2	2	1	0.0	0	0.0
93	3	0	2	−2	2	1	0.0	0	0.0
94	1	0	2	−2	0	−1	0.0	0	0.0
95	0	1	2	0	1	1	0.0	0	0.0
96	−1	−1	0	2	1	1	0.0	0	0.0
97	0	0	−2	0	1	−1	0.0	0	0.0
98	0	0	2	−1	2	−1	0.0	0	0.0
99	0	1	0	2	0	−1	0.0	0	0.0
100	1	0	−2	−2	0	−1	0.0	0	0.0
101	0	−1	2	0	1	−1	0.0	0	0.0
102	1	1	0	−2	1	−1	0.0	0	0.0
103	1	0	−2	2	0	−1	0.0	0	0.0
104	2	0	0	2	0	1	0.0	0	0.0
105	0	0	2	4th	2	−1	0.0	0	0.0
106	0	1	0	1	0	1	0.0	0	0.0

Sample values

Date ( TDB) ${\displaystyle 0^{h}}$	${\displaystyle T}$	${\displaystyle D[^{\circ }]}$	${\displaystyle M[^{\circ }]}$	${\displaystyle M'[^{\circ }]}$	${\displaystyle F[^{\circ }]}$	${\displaystyle \Omega [^{\circ }]}$	${\displaystyle \Delta \psi ['']}$	${\displaystyle \Delta \epsilon ['']}$
June 20, 1964	−0.355331964408	120.2126	165.9158	130.9535	116.1496	92.30525	−17.3256	−0.787239
August 17, 1967	−0.323764544832	136.1463	222.3130	74.89018	249.5905	31.24952	−7.41725	7,88539
3/12/2080	0.801930184805	250.9860	66.25406	135.1452	227.5530	14.00364	−3.70677	9.33751
12/13/1924	−0.750513347023	198.9391	339.7613	190.8491	323.7067	136.6408	−12.4542	−7.33544
11/4/2047	0.478398357290	192.9045	299.4156	186.1198	136.3227	279.7574	15.2424	1.67236
June 28, 1974	−0.255126625599	98.35445	173.2129	68.82716	295.5717	258.4943	17.0891	−2.25946
5/15/2032	0.323682409309	62.98143	129.7884	155.8435	257.2649	218,9989	10.0856	−7.39013
January 25, 2083	0.830650239562	79.08177	20.14875	160.3232	65.14120	318.4552	12.3513	6.7399
8/26/2048	0.486502395619	201.3662	231.1533	93.35776	92.21031	264.0831	18.1016	−0.434817
September 7, 1940	−0.593169062286	59.17461	244.0062	35.36172	32.78325	192.3151	4.16406	−8.59891

Approximate calculation

If you only consider the terms up to degree 1 for the five polynomials and only use the first four table rows (these have the highest coefficients), a simplified formula can be derived. For this purpose, the argument of each sine and cosine value, which is a linear combination of the five input values, is explicitly calculated. The conversion to rad is also done, as most implementations require this information.

${\displaystyle A=\left({\begin{array}{ccccc}0&0&0&0&1\\0&0&2&-2&2\\0&0&2&0&2\\0&0&0&0&2\\\end{array}}\right)\cdot \left({\begin{array}{rr}{485868{,}249036}&{1717915923{,}2178}\\{1287104{,}793048}&{129596581{,}0481}\\{335779{,}526232}&{1739527262{,}8478}\\{1072260{,}703692}&{1602961601{,}2090}\\{450160{,}398036}&{-6962890{,}5431}\\\end{array}}\right)\cdot \left({\begin{array}{rr}1\\T\\\end{array}}\right)\cdot {\frac {\pi }{3600\cdot 180}}=\left({\begin{array}{r}2{,}18243920-33{,}7570460T\\-2{,}77624462+1256{,}66393T\\7{,}62068856+16799{,}4182T\\4{,}36487839-67{,}5140919T\end{array}}\right)}$

The nutation then results from

${\displaystyle \Delta \psi =10^{-4}\cdot \left({\begin{array}{r}-171996-174{,}2T\\-13187-1{,}6T\\-2274-0{,}2T\\2062+0{,}2T\end{array}}\right)\cdot \sin A}$

${\displaystyle \Delta \epsilon =10^{-4}\cdot \left({\begin{array}{r}92025+8{,}9T\\5736-3{,}1T\\977-0{,}5T\\-895+0{,}5T\\\end{array}}\right)\cdot \cos A}$

where the sine and cosine from a vector are defined as the vector of the sine and cosine values. The subsequent scalar product then multiplies the values ​​by the coefficients from the table.

The error is for | T | <1 (i.e. from 1900 to 2100) at a maximum of 0.33 "and a maximum of 0.09". ${\displaystyle \Delta \psi }$ ${\displaystyle \Delta \epsilon }$

The IAU 2000 Theory of Nutation

The theory from 2000 consists of a lunisolar and a planetary table. Here, the accuracy can be increased to arc seconds; however, these tables have about 470 terms. The exact position of the planets is also required. ${\displaystyle 0{,}3\cdot 10^{-6}}$

literature

• Wolfgang Vollmann. Changing star locations. In: Hermann Mucke (Hrsg.): Modern astronomical phenomenology. 20th Sternfreunde Seminar, 1992/93. Zeiss Planetarium of the City of Vienna and Austrian Astronomical Association 1992, pp. 55–102 ( weblink, July 27, 2010 )

Web links

• Precession and Nutation - Description with clear graphics

Individual evidence

1. ^ PK Seidelmann: 1980 IAU Theory of Nutation: The final report of the IAU Working Group on Nutation. In: Celestial Mechanics. 27, 1982, pp. 79-106, doi : 10.1007 / BF01228952 .
2. ↑ Standards of Fundamental Astronomy FORTRAN Library: SOFA Library Issue 2009-02-01 for Fortran 77: Complete List
3. ^ IERS Conventions Center