Geodetic Precession of the Sun, Solar System Planets, and their Satellites

Vladimir V. Pashkevich; Andrey N. Vershkov

doi:10.2478/arsa-2022-0005

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Geodetic Precession of the Sun, Solar System Planets, and their Satellites

Artificial Satellites

Volume 57 (2022): Issue 1 (March 2022)

By: Vladimir V. Pashkevich and Andrey N. Vershkov

Open Access

|Apr 2022

Figures & Tables

Triangle used to define the direction of the angular velocity vector of the geodetic rotation for any body of the Solar System

Geodetic precession velocity for the Sun, the Moon, and the planets of the Solar System in the longitude of the descending node (left side) and in the absolute value of the velocity vector of the geodetic rotation of the parameters of their orientation (right side) (a is the length of the planetary orbit’s semi-major axis)

Geodetic precession velocity of the satellites of Mars in the longitude of the descending node (left side) and in the absolute value of the velocity vector of the geodetic rotation of the parameters of their orientation (right side) (a is the length of the satellite orbit’s semi-major axis)

Geodetic precession velocity of the satellites of Jupiter in the longitude of the descending node (left side) and in the absolute value of the velocity vector of the geodetic rotation of the parameters of their orientation (right side) (a is the length of the satellite orbit’s semi-major axis)

Geodetic precession velocity of the satellites of Saturn in the longitude of the descending node (left side) and in the absolute value of the velocity vector of the geodetic rotation of the parameters of their orientation (right side) (a is the length of the satellite orbit’s semi-major axis)

Geodetic precession velocity of the satellites of Uranus in the longitude of the descending node (left side) and in the absolute value of the velocity vector of the geodetic rotation of the parameters of their orientation (right side) (a is the length of the satellite orbit’s semi-major axis)

Geodetic precession velocity of the satellites of Neptune in the longitude of the descending node (left side) and in the absolute value of the velocity vector of the geodetic rotation of the parameters of their orientation (right side) (a is the length of the satellite orbit’s semi-major axis)

Table 1.

The time spans and steps for the studies of the geodetic precession of the bodies

Satellites	Time span (years)	Spacing
The Earth
The Moon (E1)	2000 (from AD1000 01 Jan. to AD3000 01 Jan.)	1 day 00 h 00 min
Mars
Phobos (M1) Deimos (M2)	900 (from AD1600 01 Jan. to AD2499 14 Oct.)	09 h 30 min
Jupiter
Metis (J16) Adrastea (J15)	400 (from AD1799 19 Dec. to AD2200 13 Jan.)	42 min
Amalthea (J5)	1000 (from AD1600 07 Feb. to AD2599 06 Dec.)	01 h 00 min
Thebe (J14)	400 (from AD1799 19 Dec. to AD2200 13 Jan.)	01 h 30 min
Io (J1) Europa (J2) Ganimede (J3) Callisto (J4)	1000 (from AD1600 07 Feb. to AD2599 07 Dec.)	04 h 15 min
Saturn
Pan (S18) Atlas (S15)	100 (from AD1949 27 Dec. to AD2050 09 Jan.)	01 h 20 min
Prometheus (S16)	100 (from AD1949 27 Dec. to AD2050 09 Jan.)	01 h 00 min
Pandora (S17)	100 (from AD1949 27 Dec. to AD2050 09 Jan.)	01 h 20 min
Epimetheus (S11) Janus (S10)	100 (from AD1949 27 Dec. to AD2050 09 Jan.)	01 h 40 min
Mimas (S1)	300 (from AD1849 29 Dec. to AD2150 07 Jan.)	02 h 00 min
Enceladus (S2)	300 (from AD1849 29 Dec. to AD2150 07 Jan.)	03 h 00 min
Tethys (S3) Telesto (S13) Calypso (S14)	300 (from AD1849 29 Dec. to AD2150 07 Jan.)	04 h 30 min
Dione (S4) Helene (S12)	300 (from AD1849 29 Dec. to AD2150 07 Jan.)	06 h 30 min
Rhea (S5)	300 (from AD1849 29 Dec. to AD2150 07 Jan.)	10 h 50 min
Titan (S6)	300 (from AD1849 29 Dec. to AD2150 07 Jan.)	1 day 14 h 20 min
Iapetus (S8)	300 (from AD1849 29 Dec. to AD2150 07 Jan.)	7 days 22 h 00 min
Phoebe (S9)	300 (from AD1849 29 Dec. to AD2150 07 Jan.)	5 days 10 h 00 min
Uranus
Cordelia (U6)	80 (from AD1980 02 Jan. to AD2059 31 Dec.)	48 min
Ophelia (U7)	80 (from AD1980 02 Jan. to AD2059 31 Dec.)	50 min
Bianca (U8)	80 (from AD1980 02 Jan. to AD2059 31 Dec.)	01 h 00 min
Cressida (U9) Desdemona (U10)	80 (from AD1980 02 Jan. to AD2059 31 Dec.)	01 h 05 min
Juliet (U11) Portia (U12)	80 (from AD1980 02 Jan. to AD2059 31 Dec.)	01 h 10 min
Rosalind (U13)	80 (from AD1980 02 Jan. to AD2059 31 Dec.)	01 h 20 min
Belinda (U14)	80 (from AD1980 02 Jan. to AD2059 31 Dec.)	01 h 30 min
Puck (U15)	80 (from AD1980 02 Jan. to AD2059 31 Dec.)	01 h 50 min
Miranda (U5)	1000 (from AD1599 08 Dec. to AD2600 12 Jan.)	03 h 20 min
Ariel (U1)	1000 (from AD1599 08 Dec. to AD2600 12 Jan.)	06 h 00 min
Umbriel (U2)	1000 (from AD1599 08 Dec. to AD2600 12 Jan.)	10 h 00 min
Titania (U3)	1000 (from AD1599 08 Dec. to AD2600 12 Jan.)	20 h 00 min
Oberon (U4)	1000 (from AD1599 08 Dec. to AD2600 12 Jan.)	1 day 07 h 40 min
Neptune
Naiad (N3)	100 (from AD1950 02 Jan. to AD2049 30 Dec.)	42 min
Thalassa (N4)	100 (from AD1950 02 Jan. to AD2049 30 Dec.)	45 min
Despina (N5)	100 (from AD1950 02 Jan. to AD2049 30 Dec.)	48 min
Galatea (N6)	100 (from AD1950 02 Jan. to AD2049 30 Dec.)	01 h 00 min
Larissa (N7)	100 (from AD1950 02 Jan. to AD2049 30 Dec.)	01 h 20 min
Proteus (N8)	100 (from AD1950 02 Jan. to AD2049 30 Dec.)	02 h 40 min
Triton (N1)	1000 (from AD1599 04 Dec. to AD2599 31 Dec.)	13 h 20 min
The bodies	Time span (years)	Spacing
The Sun and the planets	2000 (from AD1000 01 Jan. to AD3000 01 Jan.)	1 day 00 h 00 min

Table 2.

Magnitudes of the geodetic precession for the Sun, the Solar System planets and for each planet system, its satellite with the largest geodetic precession, calculated for the angular velocity vector $| \bar{σ} |$ of the geodetic rotation of the body under study

Name	Eroshkin and Pashkevich (2007)	Klioner et al. (2009)	In this paper	a (au)
Name	“ per century	“ per century	“ per century	a (au)
The Sun	0.0001		0.0000692	0
Mercury	21.4905	21.43	21.4902924	0.387098
Venus	4.3124	4.32	4.3123523	0.723330
The Earth	1.9199	1.92	1.9198805	1.000001
The Moon (E1)	1.9495	1.95	1.9494951
Mars	0.6756	0.68	0.6754500	1.523679
Phobos (M1)			30.6419590
Jupiter	0.0312		0.0311851	5.202603
Metis (J16)			2653.6443645
Saturn	0.0069		0.0068507	9.554910
Pan (S18)			390.9201274
Uranus	0.0012		0.0011949	19.218446
Cordelia (U6)			276.4934392
Neptune	0.0004		0.0003876	30.110387
Naiad (N3)			381.1538211

1 1 astronomical unit (au) = 149,597,870.7 km

(taken from Horizons On-Line Ephemeris System; Giorgini et al. 1996); a is the length of the planetary orbit’s semi-major axis.

Table 2a.

Secular terms of the geodetic rotation for the Sun and the Solar System planets, calculated for the Euler angles, and for the Moon, calculated for the perturbing terms of the physical libration

	The Sun	Mercury a (au) = 0.387098	Venus a (au) = 0.723330	The Moon a (km) = 384400
	Δψ (μas)	Δψ (μas)	Δψ (μas)	Δτ (μas)
t	−870.2788	−425,606,984.4341	−155,952,178.4711	−19,494,198.9139
t²	1.3770	−33,155.9302	−687,024.3196	−77.7041
	Δθ (μas)	Δθ (μas)	Δθ (μas)	Δρ (μas)
t	−1.8970	−43,920.9632	−740,253.4678	−413.2193
t²	0.0809	504.4556	60,179.7955	−1436.3972
	Δφ (μas)	Δφ (μas)	Δφ (μas)	Δ(Iσ) (μas)
t	179.6136	213,919,825.1563	112,930,676.1063	511,726.8500
t²	−1.3915	−3798.8818	687,231.8895	−14,383.0938
	The Earth⁶ a (au) = 1.000001	without the Moon⁷	without the Sun⁸	Mars a (au) = 1.523679
	Δψ (μas)	Δψ (μas)	Δψ (μas)	Δψ (μas)
t	−19,199,865.4438	−19,194,966.2971	−5289.2214	−7,125,692.1811
t²	49,150.8059	49,136.4217	11.7867	10,109.0014
	Δθ (μas)	Δθ (μas)	Δθ (μas)	Δθ (μas)
t	−4127.7653	−4127.7520	−9.5398	127,569.2300
t²	−1878.6778	−1878.5492	−0.2078	−1098.6657
	Δφ (μas)	Δφ (μas)	Δφ (μas)	Δφ (μas)
t	1174.6090	1172.9236	−1.3267	414,234.7545
t²	−53,414.8819	−53,399.9048	−12.2192	−11,846.4356
	Jupiter a (au) = 5.202603	Saturn a (au) = 9.554910	Uranus a (au) = 19.218446	Neptune a (au) = 30.110387
	Δψ (μas)	Δψ (μas)	Δψ (μas)	Δψ (μas)
t	−212,778.4891	−67,171.5760	−11,949.3883	−3902.8771
t²	3097.9909	−54.6002	−21.3019	4.3541
	Δθ (μas)	Δθ (μas)	Δθ (μas)	Δθ (μas)
t	−5974.5301	−2892.9323	−161.0625	−118.6838
t²	133.7664	−27.8319	1.4159	0.1154
	Δφ (μas)	Δφ (μas)	Δφ (μas)	Δφ (μas)
t	−99,066.0037	−1440.3592	10.3345	32.9359
t²	−3118.0679	137.5508	−1.0611	0.7553

1 a is the length of the planetary orbit’s semi-major axis;

1 au = 149,597,870.7 km (from Horizons On-Line Ephemeris System; Giorgini et al. 1996);

t is the time in Julian thousand years.

Table 2b.

The rotational elements of the Sun and its planets and their secular terms of the geodetic rotation

Name, a (au)	Archinal et al. (2018)		Present paper	T	T²
The Sun	α0 (°)	286.13	Δα₀ (″)	1×10^–5	−3×10^–10
0	δ0 (°)	63.87	Δδ₀ (″)	1×10^–5	−2×10^–9
	W(°)	84.176+14.1844000d	ΔW (″)	−0.0001	1×10^–11
Mercury	α₀ (°)	281.0103−0.0328T	Δα₀ (″)	8.5439	0.0015
0.387098	δ₀ (°)	61.4155−0.0049T	Δδ₀ (″)	3.2367	−0.0047
	W(°)	329.5988+6.1385108d	ΔW (″)	−28.3505	−0.0018
Venus	α₀ (°)	272.76	Δα₀ (″)	0.2342	0.0016
0.723330	δ₀ (°)	67.16	Δδ0 (″)	0.3331	−0.0001
	W(°)	160.20−1.4813688d	ΔW (″)	−4.5144	−0.0014
Name	Archinal et al. (2011)^⁹		Present paper	T	T²
The Earth	α₀ (°)	0.00–0.641T	cos δ₀Δα₀ (″)	0.0426	−3×10^–5
1.000001	δ₀ (°)	90.00–0.557T	Δδ₀ (″)	0.7622	−0.0002
	W(°)	190.147+360.9856235d	Δα₀+ ΔW₀ (″)	−1.7614	0.0001
Name	Archinal et al. (2018)		Present paper	T	T²
Mars	α₀ (°)	317.269202–0.10927547T	Δα₀ (″)	0.3972	−0.0001
1.523679	δ₀ (°)	54.432516–0.05827105T	Δδ₀ (″)	0.1991	−0.0002
	W(°)	176.049863+350.891982443297d	ΔW (″)	−0.9273	0.0001
Jupiter	α₀ (°)	268.056595–0.006499T	Δα₀ (″)	0.0023	−4×10^–6
5.202603	δ₀ (°)	64.495303+0.002413T	Δδ₀ (″)	0.0003	−1×10^–6
	W(°)	284.95+870.5360000d	ΔW (″)	−0.0332	3×10⁻⁶
Saturn	α₀ (°)	40.589–0.036T	Δα₀ (″)	0.0199	−1×10^–5
9.554910	δ₀ (°)	83.537–0.004T	Δδ₀ (″)	0.0023	1×10–⁶
	W(°)	38.90+810.7939024d	ΔW (″)	−0.0258	1×10^–5
Uranus	α₀ (°)	257.311	Δα₀ (″)	0.0012	2×10^–7
19.218446	δ₀ (°)	−15.175	Δδ₀ (″)	−0.0001	−3×10^–8
	W(°)	203.81–501.1600928d	ΔW (″)	0.0002	2×10^–8
Neptune	α₀ (°)	299.36	Δα₀ (″)	0.0002	−1×10^–8
30.110387	δ₀ (°)	43.46	Δδ₀ (″)	0.0001	2×10^–8
30.110387	W(°)	249.978+541.1397757d	ΔW (″)	−0.0005	1×10^–7

1 a is the length of the satellite orbit’s semi-major axis;

1 au = 149,597,870.7 km (from Horizons On-Line Ephemeris System; Giorgini et al. 1996);

T is the time in Julian centuries years;

d is the time in days from standard epoch is JD 2451545.0, that is, 2000 January 1, 12 h TDB.

Table 3.

Secular terms of the geodetic rotation for the satellites of the Solar System planets, calculated for the Euler angles (part 1/4)

The Earth
Name	Δτ (″)		Δρ (″)		Δ(Iσ) (″)		a (km)
Name	t	t²	t	t²	t	t²	a (km)
The Moon (E1)^¹⁰	−19.4942	−0.0001	−0.0004	−0.0014	0.5117	−0.0144	384,400
without the Earth^¹¹	−19.1932	−3×10^–5	−0.0005	−0.0014	0.5171	−0.0144	149,597,870
without the Sun^¹²	−0.3014	−4×10^–5	3×10^–5	−0.0001	−0.0054	−1×10^–5	384,400
Mars
Name	Δψ (″)		Δθ (″)		Δφ (″)		a (km)
Name	t	t²	t	t²	t	t²	a (km)
Phobos (M1)^¹³	−209.3145	0.0411	0.1096	−0.0800	113.6015	−0.0202	9376
Deimos (M2) ^¹³	−27.6800	0.0145	0.1189	−0.0057	11.8433	−0.0124	23,458
Jupiter
Name	Δψ (″)		Δθ (″)		Δφ (″)		a (km)
Name	t	t²	t	t²	t	t²	a (km)
Metis (J16) ^¹⁴	−52,957.2516	−20.0929	−0.4232	−3.9838	26,460.9380	19.8859	128,000
Adrastea (J15) ^¹⁴	−51,932.8456	−19.7509	−0.4151	−3.9067	25,949.0709	19.5347	129,000
Amalthea (J5) ^¹⁴	−22,118.2274	−0.7460	−0.0923	4.7351	11,055.1784	0.5755	181,400
Thebe (J14) ^¹⁴	−13,372.5500	−2.8287	−2.4703	37.7619	6693.8317	2.8902	221,900
Io (J1)	−2682.6602	−0.2122	−0.1196	−0.1392	1342.6373	0.2016	421,800

Table 3.

Secular terms of the geodetic rotation for the satellites of the Solar System planets, calculated for the Euler angles (part 2/4)

Jupiter (continue)
Name	Δψ (″)		Δθ (″)		Δφ (″)		a (km)
Name	t	t²	t	t²	t	t²	a (km)
Europa (J2)	−840.1721	0.0710	−0.0242	−0.0184	420.3792	−0.0663	671,100
Ganimede (J3)	−261.5694	−0.0066	−0.0112	−0.0131	130.7141	0.0084	1,070,400
Callisto (J4)	−63.9972	0.0399	−0.0102	−0.0022	31.8543	−0.0360	1,882,700
Saturn
Name	Δψ (″)		Δθ (″)		Δφ (″)		a (km)
Name	t	t²	t	t²	t	t²	a (km)
Pan (S18)	−232.7364	−1.9587	−657.4148	−3.6720	−3639.3938	2.4248	133,585
Atlas (S15)	−212.0393	−1.8264	−608.6535	−3.4048	−3378.7169	2.1959	137,774
Prometheus S16)	−205.5740	−1.8422	−590.0954	−3.3490	−3275.6980	2.1078	139,429
Pandora (S17)	−197.1983	−1.6079	−566.0443	−3.1597	−3142.2080	2.2945	141,810
Epimetheus (S11)	−167.4579	−1.7650	−479.4434	−2.3182	−2660.0806	2.2733	151,422
Janus (S10)	−167.8324	−1.8772	−479.5517	−2.6082	−2659.7131	2.0168	151,472
Mimas (S1)	−100.5028	−1.0302	−285.8875	−1.9636	−1600.2855	1.2301	185,539
Enceladus (S2)	−54.3148	−0.4696	−154.5777	−0.9223	−857.7077	0.5695	238,042
Tethys (S3)	−31.5081	−0.8095	−90.1979	−0.9998	−503.0932	0.8855	294,672
Telesto (S13)	−28.6117	−0.4386	−80.2283	−0.9827	−507.3979	0.5456	294,720
Calypso (S14)^¹⁵	0.2767	0.9812	−84.1557	−0.6733	−532.3406	−0.7741	294,721

Table 3.

Secular terms of the geodetic rotation for the satellites of the Solar System planets, calculated for the Euler angles (part 3/4)

Saturn (continue)
Name	Δψ (″)		Δθ (″)		Δφ (″)		a (km)
Name	t	t²	t	t²	t	t²	a (km)
Dione (S4)	−17.2533	−0.1941	−48.8066	−0.3387	−270.7718	0.2360	377,415
Helene (S12)	−19.1048	−0.3180	−49.2847	−0.3515	−269.0649	0.3216	377,444
Rhea (S5)	−7.4990	−0.0869	−21.2180	−0.2158	−117.4761	0.1224	527,068
Titan (S6)	−1.2430	−0.3765	−2.6297	−0.2605	−14.1293	0.3897	1,221,865
Iapetus (S8)	−0.9239	−0.6512	−0.3113	0.0624	−0.1668	0.6569	3,560,854
Phoebe (S9)	−0.0214	−0.0148	0.0005	0.0104	−0.0046	0.0181	12,947,918
Uranus^¹⁶
Name	Δψ (″)		Δθ (″)		Δφ (″)		a (km)
Name	t	t²	t	t²	t	t²	a (km)
Cordelia (U6)	737.3755	−5.7294	−0.0321	−4.5750	−2743.4732	−0.2103	49,800
Ophelia (U7)	607.0945	−3.4941	−0.0225	−3.1162	−2258.9890	−0.3097	53,800
Bianca (U8)	478.2575	−3.0865	−0.0291	−2.1307	−1779.4917	−0.1653	59,200
Cressida (U9)	429.6241	−2.5533	−0.0104	−2.3131	−1598.6213	−0.2149	61,800
Desdemona (U10)	414.5513	−2.0570	−0.0060	−1.5043	−1542.6289	−0.2530	62,700
Juliet (U11)	387.6458	−2.2473	−0.0075	−1.8160	−1442.4253	−0.1466	64,400
Portia (U12)	362.6995	−2.2701	0.0121	−2.2402	−1349.5799	−0.1988	66,100
Rosalind (U13)	314.9000	0.0858	0.0312	−2.2876	−1172.0590	−0.3380	69,900
Belinda (U14)	262.1423	−1.4551	0.0062	−1.2547	−975.4948	−0.1107	75,300

Table 3.

Secular terms of the geodetic rotation for the satellites of the Solar System planets, calculated for the Euler angles (part 4/4)

Uranus (continue)
Name	Δψ (″)		Δθ (″)		Δφ (″)		a (km)
Name	t	t²	t	t²	t	t²	a (km)
Puck (U15)	187.8308	−4.1306	−0.0069	−0.4330	−698.6046	−0.0686	86,000
Miranda (U5)	67.0376	0.3657	−0.4716	−0.0004	−249.4464	0.0003	129,900
Ariel (U1)	25.7442	0.0158	−0.2705	−0.0105	−95.1556	−0.0012	190,900
Umbriel (U2)	11.2105	−0.0032	−0.0603	−0.0418	−41.5373	0.0025	266,000
Titania (U3)	3.2515	0.0270	−0.0165	−0.0204	−12.0532	−0.0030	436,300
Oberon (U4)	1.5706	−0.0019	−0.0084	−0.0129	−5.8282	0.0000	583,500
Neptune
Name	Δψ (″)		Δθ (″)		Δφ (″)		a (km)
Name	t	t²	t	t²	t	t²	a (km)
Naiad (N3)	−6670.3047	−14.7325	−5.1794	−55.2350	3809.5037	64.7022	48,227
Thalassa (N4)	−6092.0806	−8.2229	−4.2733	−17.4608	3467.5926	58.1521	50,074
Despina (N5)	−5405.6116	−6.2192	−3.6995	−15.2733	3076.8866	50.3420	52,526
Galatea (N6)	−3576.1301	−0.6725	−2.4523	−9.8477	2035.4328	30.7942	61,953
Larissa (N7)	−2328.1449	1.1397	−1.4266	−7.7510	1325.4696	20.5667	73,548
Proteus (N8)	−716.2634	−5.9168	−0.7619	−16.4520	409.4543	17.8405	117,646
Triton (N1) ^¹⁷	43.4450	0.0594	0.8107	−0.1928	−25.3711	−0.2680	354,759

1 t is the time in Julian thousand years.

Table 3a.

The rotational elements of the satellites of the Solar System planets and their secular terms of the geodetic rotation (part 1/6)

The Earth
Name, a (km)	Archinal et al. (2011)		Present paper	T	T²
The Moon (E1) 384,400	α₀ (°)	269.9949 + 0.0031T	Δα₀ (°)	2×10^–5	−2×10^–7
	δ₀ (°)	66.5392 + 0.0130T	Δδ₀ (°)	3×10^–8	1×10^–8
	W (°)	38.3213+13.17635815d−1.4×10^–12d ²	ΔW (°)	−0.0006	1×10^–7
Mars
Name, a (km)	Archinal et al. (2011)		Present paper	T	T²
Phobos (M1) 9376	α₀ (°)	317.67071657 − 0.10844326T	Δα₀ (°)	0.0033	−2×10^–7
	δ₀ (°)	52.88627266 − 0.06134706T	Δδ₀ (°)	0.0017	−1×10^–6
	W (°)	34.9964842535 + 1128.8447592d	ΔW (°)	−0.0047	1×10^–6
Deimos (M2) 23,458	α₀ (°)	316.65705808 − 0.10518014T	Δα₀ (°)	0.0004	−1×10^–8
	δ₀ (°)	53.50992033 − 0.05979094T	Δδ₀ (°)	0.0002	−2×10^–7
	W (°)	79.39932954 + 285.16188899d	ΔW (°)	−0.0007	2×10^–6
Jupiter
Name, a (km)	Archinal et al. (2018)		Pashkevich et al. (2020)	T	T²
Metis (J16) 128,000	α₀ (°)	268.05 − 0.009T	Δα₀ (°)	0.1241	−7×10^–5
	δ₀ (°)	64.49 + 0.003T	Δδ₀ (°)	−0.0199	−4×10^–5
	W (°)	346.09 + 1221.2547301d	ΔW (°)	−0.8469	6×10^–5
Adrastea (J15) 129,000	α₀ (°)	268.05 − 0.009T	Δα₀ (°)	0.1217	−6×10^–5
	δ₀ (°)	64.49 + 0.003T	Δδ₀ (°)	−0.0195	−4×10^–5
	W (°)	33.29 + 1206.9986602d	ΔW (°)	−0.8306	6×10^–5
Amalthea (J5) 181,400	α₀ (°)	268.05 − 0.009T	Δα₀ (°)	0.0518	−3×10^–5
	δ0 (°)	64.49 + 0.003T	Δδ₀ (°)	−0.0083	−2×10^–5
	W (°)	231.67 + 722.6314560d	ΔW (°)	−0.3536	3×10^–5
Thebe (J14) 221,900	α0 (°)	268.05 − 0.009T	Δα₀ (°)	0.0312	−2×10^–5
	δ₀ (°)	64.49 + 0.003T	Δδ₀ (°)	−0.0050	−2×10^–5
	W (°)	8.56 + 533.7004100d	ΔW (°)	−0.2133	1×10^–5
Name, a (km)	Archinal et al. (2018)		Present paper	T	T²
Io (J1) 421,800	α₀ (°)	268.05 − 0.009T	Δα₀ (°)	0.0063	−4×10-6
	δ0 (°)	64.50 + 0.003T	Δδ₀ (°)	−0.0010	−2×10^–6
	W (°)	200.39 + 203.4889538d	ΔW (°)	−0.0428	3×10^–6
Europa (J2) 671,100	α0 (°)	268.08 − 0.009T	Δα₀ (°)	0.0019	2×10^–7
	δ0 (°)	64.51 + 0.003T	Δδ₀ (°)	−0.0003	−7×10^–7
	W (°)	36.022 + 101.3747235d	ΔW (°)	−0.0134	−2×10^–7

Table 3a.

The rotational elements of the satellites of the Solar System planets and their secular terms of the geodetic rotation (part 2/6)

Jupiter (continue)
Name, a (km)	Archinal et al. (2018)		Present paper	T	T²
Ganimede (J3) 1,070,400	α₀ (°)	268.20 – 0.009T	Δα₀ (°)	0.0006	2×10^–7
	δ₀ (°)	64.57 + 0.003T	Δδ₀ (°)	−0.0001	−1×10^–7
	W (°)	44.064 + 50.3176081d	ΔW (°)	−0.0042	−2×10^–7
Callisto (J4) 1,882,700	α₀ (°)	268.72 – 0.009T	Δα₀ (°)	0.0001	7×10^–7
	δ₀ (°)	64.83 + 0.003T	Δδ₀ (°)	−1×10^–5	−2×10^–7
	W (°)	259.51 + 21.5710715d	ΔW (°)	−0.0010	−6×10^–7
Saturn
Name, a (km)	Archinal et al. (2018)		Present paper	T	T²
Pan (S18) 133,585	α₀ (°)	40.6 – 0.036T	Δα₀ (°)	−0.0829	−5×10^–5
	δ₀ (°)	83.5 – 0.004T	Δδ₀ (°)	0.0160	7×10^–6
	W (°)	48.8 + 626.0440000d	ΔW (°)	−0.0244	5×10^–5
Atlas (S15) 137,774	α₀ (°)	40.58 – 0.036T	Δα₀ (°)	−0.0775	−4×10^–5
	δ₀ (°)	83.53 – 0.004T	Δδ₀ (°)	0.0147	7×10^–6
	W (°)	137.88 + 598.3060000d	ΔW (°)	−0.0220	4×10^–5
Prometheus (S16) 139,429	α₀ (°)	40.58 – 0.036T	Δα₀ (°)	−0.0752	−4×10^–5
	δ₀ (°)	83.53 – 0.004T	Δδ₀ (°)	0.0143	7×10^–6
	W (°)	296.14 + 587.289000d	ΔW (°)	−0.0213	4×10^–5
Pandora (S17) 141,810	α₀ (°)	40.58 – 0.036T	Δα₀ (°)	−0.0721	−4×10^–5
	δ₀ (°)	83.53 – 0.004T	Δδ₀ (°)	0.0137	6×10^–6
	W (°)	162.92 + 572.7891000d	ΔW (°)	−0.0205	4×10^–5
Epimetheus (S11) 151,422	α₀ (°)	40.58 – 0.036T	Δα₀ (°)	−0.0610	−1×10^–5
	δ₀ (°)	83.52 – 0.004T	Δδ₀ (°)	0.0116	8×10^–6
	W (°)	293.87 + 518.4907239d	ΔW (°)	−0.0173	1×10^–5
Janus (S10) 151,472	α₀ (°)	40.58 – 0.036T	Δα₀ (°)	−0.0609	−2×10^–5
	δ₀ (°)	83.52 – 0.004T	Δδ₀ (°)	0.0116	7×10^–6
	W (°)	58.83 + 518.2359876d	ΔW (°)	−0.0175	2×10^–5
Mimas (S1) 185,539	α₀ (°)	40.66 – 0.036T	Δα₀ (°)	−0.0376	−3×10^–5
	δ₀ (°)	83.52 – 0.004T	Δδ₀ (°)	0.0068	5×10^–6
	W (°)	333.46 + 381.9945550J	ΔW (°)	−0.0096	3×10^–5
Enceladus (S2) 238,042	α₀ (°)	40.66 – 0.036T	Δα₀ (°)	−0.0196	−1×10^–5
	δ₀ (°)	83.52 – 0.004T	Δδ₀ (°)	0.0038	2×10^–6
	W (°)	6.32 + 262.7318996d	ΔW (°)	−0.0057	1×10^–5

Table 3a.

The rotational elements of the satellites of the Solar System planets and their secular terms of the geodetic rotation (part 3/6)

Saturn (continue)
Name, a (km)	Archinal et al. (2018)		Present paper	T	T²
Tethys (S3) 294,672	α0 (°)	40.66 – 0.036T	Δα₀ (°)	−0.0116	5×10^–6
	δ₀ (°)	83.52 – 0.004T	Δδ₀ (°)	0.0022	3×10^–6
	W (°)	8.95 + 190.6979085d	ΔW (°)	−0.0032	−5×10^–6
Telesto (S13) 294,720	α₀ (°)	50.51 – 0.036T	Δα₀ (°)	−0.0084	−1×10^–5
	δ₀ (°)	84.06 – 0.004T	Δδ₀ (°)	0.0021	2×10^–6
	W (°)	56.88 + 190.6979332d	ΔW (°)	−0.0064	1×10^–5
Calypso (S14)^¹⁸ 294,721	α₀ (°)	36.41 – 0.036T	Δα₀ (°)	−0.0193	−2×10^–5
	δ₀ (°)	85.04 – 0.004T	Δδ₀ (°)	0.0016	6×10^–8
	W (°)	153.51 + 190.6742373d	ΔW (°)	0.0045	2×10^–5
Dione (S4) 377,415	α₀ (°)	40.66 – 0.036T	Δα₀ (°)	−0.0062	−4×10^–6
	δ₀ (°)	83.52 – 0.004T	Δδ₀ (°)	0.0012	7×10^–7
	W (°)	357.6 + 131.5349316d	ΔW (°)	−0.0018	4×10^–6
Helene (S12) 377,444	α₀ (°)	40.85 – 0.036T	Δα₀ (°)	−0.0058	−3×10^–6
	δ0 (°)	83.34 – 0.004T	Δδ₀ (°)	0.0012	9×10^–7
	W (°)	245.12 + 131.6174056d	ΔW (°)	−0.0021	3×10^–6
Rhea (S5) 527,068	α0 (°)	40.38 – 0.036T	Δα₀ (°)	−0.0027	−2×10^–6
	δ₀ (°)	83.55 – 0.004T	Δδ₀ (°)	0.0005	4×10^–7
	W (°)	235.16 + 79.6900478d	ΔW (°)	−0.0007	2×10^–6
Titan (S6) 1,221,865	α0 (°)	39.4827	Δα₀ (°)	−0.0003	−9×10^–7
	δ₀ (°)	83.4279	Δδ₀ (°)	0.0001	9×10^–7
	W (°)	186.5855 + 22.5769768d	ΔW (°)	−0.0001	1×10^–6
Iapetus (S8) 3,560,854	α₀ (°)	318.16 – 3.949T	Δα₀ (°)	−3×10^–5	2×10^–6
	δ₀ (°)	75.03 – 1.143T	Δδ₀ (°)	8×10^–6	3×10^–7
	W (°)	355.2 + 4.5379572d	ΔW (°)	2×10^–6	−2×10^–6
Phoebe (S9) 12,947,918	α₀ (°)	356.90	Δα0 (°)	6×10^–7	2×10^–7
	δ₀ (°)	77.80	Δδ₀ (°)	2×10^–7	5×10^–9
	W (°)	178.58 + 931.639d	ΔW (°)	−1×10^–6	−1×10^–7

Table 3a.

The rotational elements of the satellites of the Solar System planets and their secular terms of the geodetic rotation (part 4/6)

Uranus
Name, a (km)	Archinal et al. (2018)		Present paper	T	T²
Cordelia (U6) 49,800	α₀ (°)	257.31	Δα₀ (°)	−0.0209	2×10^–5
	δ₀ (°)	−15.18	Δδ₀ (°)	0.0018	1×10^–5
	W (°)	127.69 – 1074.5205730d	ΔW (°)	−0.0789	5×10^–7
Ophelia (U7) 53,800	α₀ (°)	257.31	Δα₀ (°)	−0.0172	1×10^–5
	δ₀ (°)	−15.18	Δδ₀ (°)	0.0015	8×10^–6
	W (°)	130.35 – 956.4068150d	ΔW (°)	−0.0650	1×10^–6
Bianca (U8) 59,200	α₀ (°)	257.31	Δα₀ (°)	−0.0136	9×10^–6
	δ₀ (°)	−15.18	Δδ₀ (°)	0.0012	5×10^–6
	W (°)	105.46 – 828.3914760d	ΔW (°)	−0.0512	5×10^–7
Cressida (U9) 61,800	α₀ (°)	257.31	Δα₀ (°)	−0.0122	8×10^–6
	δ₀ (°)	−15.18	Δδ₀ (°)	0.0010	6×10^–6
	W (°)	59.16 – 776.5816320d	ΔW (°)	−0.0460	4×10^–7
Desdemona (U10) 62,700	α₀ (°)	257.31	Δα₀ (°)	−0.0118	6×10^–6
	δ₀ (°)	−15.18	Δδ₀ (°)	0.0010	4×10^–6
	W (°)	95.08 – 760.0531690d	ΔW (°)	−0.0444	4×10^–7
Juliet (U11) 64,400	α₀ (°)	257.31	Δα₀ (°)	−0.0110	7×10^–6
	δ₀ (°)	−15.18	Δδ₀ (°)	0.0009	4×10^–6
	W (°)	302.56 – 730.1253660d	ΔW (°)	−0.0415	5×10^–7
Portia (U12) 66,100	α₀ (°)	257.31	Δα₀ (°)	−0.0103	7×10^–6
	δ₀ (°)	−15.18	Δδ₀ (°)	0.0009	5×10^–6
	W (°)	25.03 – 701.4865870d	ΔW (°)	−0.0388	4×10^–7
Rosalind (U13) 69,900	α₀ (°)	257.31	Δα₀ (°)	−0.0089	8×10^–7
	δ₀ (°)	−15.18	Δδ₀ (°)	0.0008	6×10^–6
	W (°)	314.90 – 644.6311260d	ΔW (°)	−0.0337	2×10^–6
Belinda (U14) 75,300	α₀ (°)	257.31	Δα₀ (°)	−0.0074	4×10^–6
	δ₀ (°)	−15.18	Δδ₀ (°)	0.0006	3×10^–6
	W (°)	297.46 – 577.3628170d	ΔW (°)	−0.0281	3×10^–7
Puck (U15) 86,000	α₀ (°)	257.31	Δα₀ (°)	−0.0053	1×10^–5
	δ₀ (°)	−15.18	Δδ₀ (°)	0.0005	−3×10^–7
	W (°)	91.24 – 472.5450690d	ΔW (°)	−0.0201	−1×10–⁶
Miranda (U5) 129,900	α₀ (°)	257.43	Δα₀ (°)	−0.0019	−1×10^–6
	δ₀ (°)	−15.08	Δδ₀ (°)	0.0002	9×10^–8
	W (°)	30.70 – 254.6906892d	ΔW (°)	−0.0071	1×10^–7

Table 3a.

The rotational elements of the satellites of the Solar System planets and their secular terms of the geodetic rotation (part 5/6)

Uranus (continue)
Name, a (km)	Archinal et al. (2018)		Present paper	T	T²
Ariel (U1) 190,900	α₀ (°)	257.43	Δα₀ (°)	−0.0007	−4×10^–8
	δ₀ (°)	−15.10	Δδ₀ (°)	0.0001	3×10^–8
	W (°)	156.22 – 142.8356681d	ΔW (°)	−0.0027	1×10^–8
Umbriel (U2) 266,000	α₀ (°)	257.43	Δα₀ (°)	−0.0003	2×10^–8
	δ₀ (°)	−15.10	Δδ₀ (°)	3×10^–5	1×10^–7
	W (°)	108.05 – 86.8688923d	ΔW (°)	−0.0012	−7×10^–9
Titania (U3) 436,300	α₀ (°)	257.43	Δα₀ (°)	−0.0001	−6×10^–8
	δ₀ (°)	−15.10	Δδ₀ (°)	8×10^–6	6×10^–8
	W (°)	77.74 – 41.3514316d	ΔW (°)	−0.0003	2×10^–8
Oberon (U4) 583,500	α₀ (°)	257.43	Δα₀ (°)	−4×10^–5	8×10^–9
	δ₀ (°)	−15.10	Δδ₀ (°)	4×10^–6	4×10^–8
	W (°)	6.77 – 26.7394932d	ΔW (°)	−0.0002	−2×10^–11

Table 3a.

The rotational elements of the satellites of the Solar System planets and their secular terms of the geodetic rotation (part 6/6)

Neptune
Name, a (km)	Archinal et al. (2018)		Present paper	T	T²
Naiad (N3) 48,227	α₀ (°)	299.36	Δα₀ (°)	0.1098	0.0002
	δ₀ (°)	43.36	Δδ₀ (°)	0.0361	0.0005
	W (°)	254.06 + 1222.8441209d	ΔW (°)	−0.1311	0.0001
Thalassa (N4) 50,074	α₀ (°)	299.36	Δα₀ (°)	0.1006	0.0002
	δ₀ (°)	43.45	Δδ₀ (°)	0.0331	0.0004
	W (°)	102.06 + 1155.7555612d	ΔW (°)	−0.1209	0.0002
Despina (N5) 52,526	α₀ (°)	299.36	Δα₀ (°)	0.0892	0.0001
	δ₀ (°)	43.45	Δδ₀ (°)	0.0294	0.0003
	W (°)	306.51 + 1075.7341562d	ΔW (°)	−0.1073	0.0002
Galatea (N6) 61,953	α₀ (°)	299.36	Δα₀ (°)	0.0590	0.0001
	δ₀ (°)	43.43	Δδ₀ (°)	0.0194	0.0002
	W (°)	258.09 + 839.6597686d	ΔW (°)	−0.0710	0.0001
Larissa (N7) 73,548	α₀ (°)	299.36	Δα₀ (°)	0.0385	0.0001
	δ₀ (°)	43.41	Δδ₀ (°)	0.0126	0.0001
	W (°)	179.41 + 649.0534470d	ΔW (°)	−0.0462	0.0001
Proteus (N8) 117,646	α₀ (°)	299.27	Δα₀ (°)	0.0119	6×10^–6
	δ₀ (°)	42.91	Δδ₀ (°)	0.0039	0.0001
	W (°)	93.38 + 320.7654228d	ΔW (°)	−0.0141	5×10^–5
Triton (N1) 354,759	α₀ (°)	299.36	Δα₀ (°)	−0.0005	−8×10^–6
	δ₀ (°)	41.17	Δδ₀ (°)	−0.0002	1×10^–6
	W (°)	296.53 – 61.2572637d	ΔW (°)	0.0007	5×10^–6

1 T is the time in Julian centuries years;

d is the time in days from standard epoch,

which is JD 2451545.0, that is, 2000 January 1, 12 hours TDB;

a is the length of the satellite orbit’s semi-major axis.

Table 4.

Variation of the rotational elements for Calypso and comparison with near satellites for their secular terms of the geodetic rotation in Euler angles

Name, a (km)	Archinal et al. (2018)		Present paper	t
Tethys (S3) 294,672	α₀ (°)	40.66 – 0.036T	Δψ (″)	−31.5081
	δ₀ (°)	83.52 – 0.004T	Δθ (″)	−90.1979
	W (°)	8.95 + 190.6979085d	Δφ (″)	541.1442
Telesto (S13) 294,720	α₀ (°)	50.51 – 0.036T	Δψ (″)	−28.6117
	δ₀ (°)	84.06 – 0.004T	Δθ (″)	−80.2283
	W (°)	56.88 + 190.6979332d	Δφ (″)	541.1450
Calypso (S14)^¹⁹ 294,721	α₀ (°)	36.41 – 0.036T	Δψ (″)	0.2767
	δ₀ (°)	85.04 – 0.004T	Δθ (″)	−84.1557
	W (°)	153.51 + 190.6742373d	Δφ (″)	541.1466
Calypso with α₀ from Tethys	α₀ (°)	40.66 – 0.036T	Δψ (″)	−3.2464
	δ₀ (°)	85.04 – 0.004T	Δθ (″)	−81.6924
	W (°)	153.51 + 190.6742373d	Δφ (″)	−529.5805
Calypso with α₀, δ₀ from Tethys	α₀ (°)	40.66 – 0.036T	Δψ (″)	−32.1455
	δ₀ (°)	83.52 – 0.004T	Δθ (″)	−90.3406
	W (°)	153.51 + 190.6742373d	Δφ (″)	−502.3581
Calypso with α₀ from Telesto	α₀ (°)	50.51 – 0.036T	Δψ (″)	−9.7437
	δ₀ (°)	85.04 – 0.004T	Δθ (″)	−75.4873
	W (°)	153.51 + 190.6742373d	Δφ (″)	−524.7257
Calypso with δ₀ from Telesto	α₀ (°)	36.41 – 0.036T	Δψ (″)	−18.2542
	δ₀ (°)	84.06 – 0.004T	Δθ (″)	−90.4678
	W (°)	153.51 + 190.6742373d	Δφ (″)	−514.7784
Calypso^²⁰ with α₀, δ₀ from Telesto	α₀ (°)	50.51 – 0.036T	Δψ (″)	−28.5499
	δ₀ (°)	84.06 – 0.004T	Δθ (″)	−80.2670
	W (°)	153.51 + 190.6742373d	Δφ (″)	−507.3759
Calypso^²⁰ with α₀, δ₀, W from Telesto	α₀ (°)	50.51 – 0.036T	Δψ (″)	−28.5499
	δ₀ (°)	84.06 – 0.004T	Δθ (″)	−80.2670
	W (°)	56.88 + 190.6979332d	Δφ (″)	−507.3759

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DOI: https://doi.org/10.2478/arsa-2022-0005 | Journal eISSN: 2083-6104 | Journal ISSN: 1509-3859 (formerly 0208-841X)

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Language: English

Page range: 77 - 109

Submitted on: May 12, 2021

Accepted on: Mar 18, 2022

Published on: Apr 22, 2022

Published by: Polish Academy of Sciences, Space Research Centre

In partnership with: Paradigm Publishing Services

Keywords:

relativistic effects,

geodetic precession,

Solar System bodies,

planetary satellites,

exoplanetary systems bodies

Related subjects:

Geosciences,

Geosciences, other

© 2022 Vladimir V. Pashkevich, Andrey N. Vershkov, published by Polish Academy of Sciences, Space Research Centre
This work is licensed under the Creative Commons Attribution 4.0 License.

Volume 57 (2022): Issue 1 (March 2022)

Geodetic Precession of the Sun, Solar System Planets, and their Satellites

Figures & Tables

Figure 1.

Figure 2.

Figure 3.

Figure 4.

Figure 5.

Figure 6.

Figure 7.

Table 1.

Table 2.

Table 2a.

Table 2b.

Table 3.

Table 3.

Table 3.

Table 3.

Table 3a.

Table 3a.

Table 3a.

Table 3a.

Table 3a.

Table 3a.

Table 4.

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