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

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

Figure 1.

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

Figure 2.

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 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)

Figure 3.

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 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)

Figure 4.

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 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)

Figure 5.

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 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)

Figure 6.

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 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)

Figure 7.

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)
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)

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

Jupiter (continue)
NameΔψ (″)Δθ (″)Δφ (″)a (km)
tt2tt2tt2
Europa (J2)−840.17210.0710−0.0242−0.0184420.3792−0.0663671,100
Ganimede (J3)−261.5694−0.0066−0.0112−0.0131130.71410.00841,070,400
Callisto (J4)−63.99720.0399−0.0102−0.002231.8543−0.03601,882,700
Saturn
NameΔψ (″)Δθ (″)Δφ (″)a (km)
tt2tt2tt2
Pan (S18)−232.7364−1.9587−657.4148−3.6720−3639.39382.4248133,585
Atlas (S15)−212.0393−1.8264−608.6535−3.4048−3378.71692.1959137,774
Prometheus S16)−205.5740−1.8422−590.0954−3.3490−3275.69802.1078139,429
Pandora (S17)−197.1983−1.6079−566.0443−3.1597−3142.20802.2945141,810
Epimetheus (S11)−167.4579−1.7650−479.4434−2.3182−2660.08062.2733151,422
Janus (S10)−167.8324−1.8772−479.5517−2.6082−2659.71312.0168151,472
Mimas (S1)−100.5028−1.0302−285.8875−1.9636−1600.28551.2301185,539
Enceladus (S2)−54.3148−0.4696−154.5777−0.9223−857.70770.5695238,042
Tethys (S3)−31.5081−0.8095−90.1979−0.9998−503.09320.8855294,672
Telesto (S13)−28.6117−0.4386−80.2283−0.9827−507.39790.5456294,720
Calypso (S14)150.27670.9812−84.1557−0.6733−532.3406−0.7741294,721

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 paperTT2
Ganimede(J3)1,070,400α0 (°)268.20 – 0.009TΔα0 (°)0.00062×10–7
δ0 (°)64.57 + 0.003TΔδ0 (°)−0.0001−1×10–7
W (°)44.064 + 50.3176081dΔW (°)−0.0042−2×10–7
Callisto(J4)1,882,700α0 (°)268.72 – 0.009TΔα0 (°)0.00017×10–7
δ0 (°)64.83 + 0.003TΔδ0 (°)−1×10–5−2×10–7
W (°)259.51 + 21.5710715dΔW (°)−0.0010−6×10–7
Saturn
Name, a (km)Archinal et al. (2018)Present paperTT2
Pan(S18)133,585α0 (°)40.6 – 0.036TΔα0 (°)−0.0829−5×10–5
δ0 (°)83.5 – 0.004TΔδ0 (°)0.01607×10–6
W (°)48.8 + 626.0440000dΔW (°)−0.02445×10–5
Atlas(S15)137,774α0 (°)40.58 – 0.036TΔα0 (°)−0.0775−4×10–5
δ0 (°)83.53 – 0.004TΔδ0 (°)0.01477×10–6
W (°)137.88 + 598.3060000dΔW (°)−0.02204×10–5
Prometheus(S16)139,429α0 (°)40.58 – 0.036TΔα0 (°)−0.0752−4×10–5
δ0 (°)83.53 – 0.004TΔδ0 (°)0.01437×10–6
W (°)296.14 + 587.289000dΔW (°)−0.02134×10–5
Pandora(S17)141,810α0 (°)40.58 – 0.036TΔα0 (°)−0.0721−4×10–5
δ0 (°)83.53 – 0.004TΔδ0 (°)0.01376×10–6
W (°)162.92 + 572.7891000dΔW (°)−0.02054×10–5
Epimetheus(S11)151,422α0 (°)40.58 – 0.036TΔα0 (°)−0.0610−1×10–5
δ0 (°)83.52 – 0.004TΔδ0 (°)0.01168×10–6
W (°)293.87 + 518.4907239dΔW (°)−0.01731×10–5
Janus(S10)151,472α0 (°)40.58 – 0.036TΔα0 (°)−0.0609−2×10–5
δ0 (°)83.52 – 0.004TΔδ0 (°)0.01167×10–6
W (°)58.83 + 518.2359876dΔW (°)−0.01752×10–5
Mimas(S1)185,539α0 (°)40.66 – 0.036TΔα0 (°)−0.0376−3×10–5
δ0 (°)83.52 – 0.004TΔδ0 (°)0.00685×10–6
W (°)333.46 + 381.9945550JΔW (°)−0.00963×10–5
Enceladus(S2)238,042α0 (°)40.66 – 0.036TΔα0 (°)−0.0196−1×10–5
δ0 (°)83.52 – 0.004TΔδ0 (°)0.00382×10–6
W (°)6.32 + 262.7318996dΔW (°)−0.00571×10–5

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

Saturn (continue)
NameΔψ (″)Δθ (″)Δφ (″)a (km)
tt2tt2tt2
Dione (S4)−17.2533−0.1941−48.8066−0.3387−270.77180.2360377,415
Helene (S12)−19.1048−0.3180−49.2847−0.3515−269.06490.3216377,444
Rhea (S5)−7.4990−0.0869−21.2180−0.2158−117.47610.1224527,068
Titan (S6)−1.2430−0.3765−2.6297−0.2605−14.12930.38971,221,865
Iapetus (S8)−0.9239−0.6512−0.31130.0624−0.16680.65693,560,854
Phoebe (S9)−0.0214−0.01480.00050.0104−0.00460.018112,947,918
Uranus16
NameΔψ (″)Δθ (″)Δφ (″)a (km)
tt2tt2tt2
Cordelia (U6)737.3755−5.7294−0.0321−4.5750−2743.4732−0.210349,800
Ophelia (U7)607.0945−3.4941−0.0225−3.1162−2258.9890−0.309753,800
Bianca (U8)478.2575−3.0865−0.0291−2.1307−1779.4917−0.165359,200
Cressida (U9)429.6241−2.5533−0.0104−2.3131−1598.6213−0.214961,800
Desdemona (U10)414.5513−2.0570−0.0060−1.5043−1542.6289−0.253062,700
Juliet (U11)387.6458−2.2473−0.0075−1.8160−1442.4253−0.146664,400
Portia (U12)362.6995−2.27010.0121−2.2402−1349.5799−0.198866,100
Rosalind (U13)314.90000.08580.0312−2.2876−1172.0590−0.338069,900
Belinda (U14)262.1423−1.45510.0062−1.2547−975.4948−0.110775,300

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 paperTT2
The Moon(E1)384,400α0 (°)269.9949 + 0.0031TΔα0 (°)2×10–5−2×10–7
δ0 (°)66.5392 + 0.0130TΔδ0 (°)3×10–81×10–8
W (°)38.3213+13.17635815d−1.4×10–12d 2ΔW (°)−0.00061×10–7
Mars
Name, a (km)Archinal et al. (2011)Present paperTT2
Phobos(M1)9376α0 (°)317.67071657 − 0.10844326TΔα0 (°)0.0033−2×10–7
δ0 (°)52.88627266 − 0.06134706TΔδ0 (°)0.0017−1×10–6
W (°)34.9964842535 + 1128.8447592dΔW (°)−0.00471×10–6
Deimos(M2)23,458α0 (°)316.65705808 − 0.10518014TΔα0 (°)0.0004−1×10–8
δ0 (°)53.50992033 − 0.05979094TΔδ0 (°)0.0002−2×10–7
W (°)79.39932954 + 285.16188899dΔW (°)−0.00072×10–6
Jupiter
Name, a (km)Archinal et al. (2018)Pashkevich et al. (2020)TT2
Metis(J16)128,000α0 (°)268.05 − 0.009TΔα0 (°)0.1241−7×10–5
δ0 (°)64.49 + 0.003TΔδ0 (°)−0.0199−4×10–5
W (°)346.09 + 1221.2547301dΔW (°)−0.84696×10–5
Adrastea(J15)129,000α0 (°)268.05 − 0.009TΔα0 (°)0.1217−6×10–5
δ0 (°)64.49 + 0.003TΔδ0 (°)−0.0195−4×10–5
W (°)33.29 + 1206.9986602dΔW (°)−0.83066×10–5
Amalthea(J5)181,400α0 (°)268.05 − 0.009TΔα0 (°)0.0518−3×10–5
δ0 (°)64.49 + 0.003TΔδ0 (°)−0.0083−2×10–5
W (°)231.67 + 722.6314560dΔW (°)−0.35363×10–5
Thebe(J14)221,900α0 (°)268.05 − 0.009TΔα0 (°)0.0312−2×10–5
δ0 (°)64.49 + 0.003TΔδ0 (°)−0.0050−2×10–5
W (°)8.56 + 533.7004100dΔW (°)−0.21331×10–5
Name, a (km)Archinal et al. (2018)Present paperTT2
Io(J1)421,800α0 (°)268.05 − 0.009TΔα0 (°)0.0063−4×10-6
δ0 (°)64.50 + 0.003TΔδ0 (°)−0.0010−2×10–6
W (°)200.39 + 203.4889538dΔW (°)−0.04283×10–6
Europa(J2)671,100α0 (°)268.08 − 0.009TΔα0 (°)0.00192×10–7
δ0 (°)64.51 + 0.003TΔδ0 (°)−0.0003−7×10–7
W (°)36.022 + 101.3747235dΔW (°)−0.0134−2×10–7

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

The Earth
NameΔτ (″)Δρ (″)Δ(Iσ) (″)a (km)
tt2tt2tt2
The Moon (E1)10−19.4942−0.0001−0.0004−0.00140.5117−0.0144384,400
without the Earth11−19.1932−3×10–5−0.0005−0.00140.5171−0.0144149,597,870
without the Sun12−0.3014−4×10–53×10–5−0.0001−0.0054−1×10–5384,400
Mars
NameΔψ (″)Δθ (″)Δφ (″)a (km)
tt2tt2tt2
Phobos (M1)13−209.31450.04110.1096−0.0800113.6015−0.02029376
Deimos (M2) 13−27.68000.01450.1189−0.005711.8433−0.012423,458
Jupiter
NameΔψ (″)Δθ (″)Δφ (″)a (km)
tt2tt2tt2
Metis (J16) 14−52,957.2516−20.0929−0.4232−3.983826,460.938019.8859128,000
Adrastea (J15) 14−51,932.8456−19.7509−0.4151−3.906725,949.070919.5347129,000
Amalthea (J5) 14−22,118.2274−0.7460−0.09234.735111,055.17840.5755181,400
Thebe (J14) 14−13,372.5500−2.8287−2.470337.76196693.83172.8902221,900
Io (J1)−2682.6602−0.2122−0.1196−0.13921342.63730.2016421,800

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 | σ¯| of the geodetic rotation of the body under study

NameEroshkin and Pashkevich (2007)Klioner et al. (2009)In this papera (au)
“ per century“ per century“ per century
The Sun0.0001 0.00006920
Mercury21.490521.4321.49029240.387098
Venus4.31244.324.31235230.723330
The Earth1.91991.921.91988051.000001
The Moon (E1)1.94951.951.9494951
Mars0.67560.680.67545001.523679
Phobos (M1) 30.6419590
Jupiter0.0312 0.03118515.202603
Metis (J16) 2653.6443645
Saturn0.0069 0.00685079.554910
Pan (S18) 390.9201274
Uranus0.0012 0.001194919.218446
Cordelia (U6) 276.4934392
Neptune0.0004 0.000387630.110387
Naiad (N3) 381.1538211

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 paperTT2
Tethys(S3)294,672α0 (°)40.66 – 0.036TΔα0 (°)−0.01165×10–6
δ0 (°)83.52 – 0.004TΔδ0 (°)0.00223×10–6
W (°)8.95 + 190.6979085dΔW (°)−0.0032−5×10–6
Telesto(S13)294,720α0 (°)50.51 – 0.036TΔα0 (°)−0.0084−1×10–5
δ0 (°)84.06 – 0.004TΔδ0 (°)0.00212×10–6
W (°)56.88 + 190.6979332dΔW (°)−0.00641×10–5
Calypso(S14)18294,721α0 (°)36.41 – 0.036TΔα0 (°)−0.0193−2×10–5
δ0 (°)85.04 – 0.004TΔδ0 (°)0.00166×10–8
W (°)153.51 + 190.6742373dΔW (°)0.00452×10–5
Dione(S4)377,415α0 (°)40.66 – 0.036TΔα0 (°)−0.0062−4×10–6
δ0 (°)83.52 – 0.004TΔδ0 (°)0.00127×10–7
W (°)357.6 + 131.5349316dΔW (°)−0.00184×10–6
Helene(S12)377,444α0 (°)40.85 – 0.036TΔα0 (°)−0.0058−3×10–6
δ0 (°)83.34 – 0.004TΔδ0 (°)0.00129×10–7
W (°)245.12 + 131.6174056dΔW (°)−0.00213×10–6
Rhea(S5)527,068α0 (°)40.38 – 0.036TΔα0 (°)−0.0027−2×10–6
δ0 (°)83.55 – 0.004TΔδ0 (°)0.00054×10–7
W (°)235.16 + 79.6900478dΔW (°)−0.00072×10–6
Titan(S6)1,221,865α0 (°)39.4827Δα0 (°)−0.0003−9×10–7
δ0 (°)83.4279Δδ0 (°)0.00019×10–7
W (°)186.5855 + 22.5769768dΔW (°)−0.00011×10–6
Iapetus(S8)3,560,854α0 (°)318.16 – 3.949TΔα0 (°)−3×10–52×10–6
δ0 (°)75.03 – 1.143TΔδ0 (°)8×10–63×10–7
W (°)355.2 + 4.5379572dΔW (°)2×10–6−2×10–6
Phoebe(S9)12,947,918α0 (°)356.90Δα0 (°)6×10–72×10–7
δ0 (°)77.80Δδ0 (°)2×10–75×10–9
W (°)178.58 + 931.639dΔW (°)−1×10–6−1×10–7

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 paperTT2
Naiad(N3)48,227α0 (°)299.36Δα0 (°)0.10980.0002
δ0 (°)43.36Δδ0 (°)0.03610.0005
W (°)254.06 + 1222.8441209dΔW (°)−0.13110.0001
Thalassa(N4)50,074α0 (°)299.36Δα0 (°)0.10060.0002
δ0 (°)43.45Δδ0 (°)0.03310.0004
W (°)102.06 + 1155.7555612dΔW (°)−0.12090.0002
Despina(N5)52,526α0 (°)299.36Δα0 (°)0.08920.0001
δ0 (°)43.45Δδ0 (°)0.02940.0003
W (°)306.51 + 1075.7341562dΔW (°)−0.10730.0002
Galatea(N6)61,953α0 (°)299.36Δα0 (°)0.05900.0001
δ0 (°)43.43Δδ0 (°)0.01940.0002
W (°)258.09 + 839.6597686dΔW (°)−0.07100.0001
Larissa(N7)73,548α0 (°)299.36Δα0 (°)0.03850.0001
δ0 (°)43.41Δδ0 (°)0.01260.0001
W (°)179.41 + 649.0534470dΔW (°)−0.04620.0001
Proteus(N8)117,646α0 (°)299.27Δα0 (°)0.01196×10–6
δ0 (°)42.91Δδ0 (°)0.00390.0001
W (°)93.38 + 320.7654228dΔW (°)−0.01415×10–5
Triton(N1)354,759α0 (°)299.36Δα0 (°)−0.0005−8×10–6
δ0 (°)41.17Δδ0 (°)−0.00021×10–6
W (°)296.53 – 61.2572637dΔW (°)0.00075×10–6

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 paperTT2
Cordelia(U6)49,800α0 (°)257.31Δα0 (°)−0.02092×10–5
δ0 (°)−15.18Δδ0 (°)0.00181×10–5
W (°)127.69 – 1074.5205730dΔW (°)−0.07895×10–7
Ophelia(U7)53,800α0 (°)257.31Δα0 (°)−0.01721×10–5
δ0 (°)−15.18Δδ0 (°)0.00158×10–6
W (°)130.35 – 956.4068150dΔW (°)−0.06501×10–6
Bianca(U8)59,200α0 (°)257.31Δα0 (°)−0.01369×10–6
δ0 (°)−15.18Δδ0 (°)0.00125×10–6
W (°)105.46 – 828.3914760dΔW (°)−0.05125×10–7
Cressida(U9)61,800α0 (°)257.31Δα0 (°)−0.01228×10–6
δ0 (°)−15.18Δδ0 (°)0.00106×10–6
W (°)59.16 – 776.5816320dΔW (°)−0.04604×10–7
Desdemona(U10)62,700α0 (°)257.31Δα0 (°)−0.01186×10–6
δ0 (°)−15.18Δδ0 (°)0.00104×10–6
W (°)95.08 – 760.0531690dΔW (°)−0.04444×10–7
Juliet(U11)64,400α0 (°)257.31Δα0 (°)−0.01107×10–6
δ0 (°)−15.18Δδ0 (°)0.00094×10–6
W (°)302.56 – 730.1253660dΔW (°)−0.04155×10–7
Portia(U12)66,100α0 (°)257.31Δα0 (°)−0.01037×10–6
δ0 (°)−15.18Δδ0 (°)0.00095×10–6
W (°)25.03 – 701.4865870dΔW (°)−0.03884×10–7
Rosalind(U13)69,900α0 (°)257.31Δα0 (°)−0.00898×10–7
δ0 (°)−15.18Δδ0 (°)0.00086×10–6
W (°)314.90 – 644.6311260dΔW (°)−0.03372×10–6
Belinda(U14)75,300α0 (°)257.31Δα0 (°)−0.00744×10–6
δ0 (°)−15.18Δδ0 (°)0.00063×10–6
W (°)297.46 – 577.3628170dΔW (°)−0.02813×10–7
Puck(U15)86,000α0 (°)257.31Δα0 (°)−0.00531×10–5
δ0 (°)−15.18Δδ0 (°)0.0005−3×10–7
W (°)91.24 – 472.5450690dΔW (°)−0.0201−1×10–6
Miranda(U5)129,900α0 (°)257.43Δα0 (°)−0.0019−1×10–6
δ0 (°)−15.08Δδ0 (°)0.00029×10–8
W (°)30.70 – 254.6906892dΔW (°)−0.00711×10–7

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 paperTT2
Ariel(U1)190,900α0 (°)257.43Δα0 (°)−0.0007−4×10–8
δ0 (°)−15.10Δδ0 (°)0.00013×10–8
W (°)156.22 – 142.8356681dΔW (°)−0.00271×10–8
Umbriel(U2)266,000α0 (°)257.43Δα0 (°)−0.00032×10–8
δ0 (°)−15.10Δδ0 (°)3×10–51×10–7
W (°)108.05 – 86.8688923dΔW (°)−0.0012−7×10–9
Titania(U3)436,300α0 (°)257.43Δα0 (°)−0.0001−6×10–8
δ0 (°)−15.10Δδ0 (°)8×10–66×10–8
W (°)77.74 – 41.3514316dΔW (°)−0.00032×10–8
Oberon(U4)583,500α0 (°)257.43Δα0 (°)−4×10–58×10–9
δ0 (°)−15.10Δδ0 (°)4×10–64×10–8
W (°)6.77 – 26.7394932dΔW (°)−0.0002−2×10–11

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

SatellitesTime 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 bodiesTime span (years)Spacing
The Sun and the planets2000 (from AD1000 01 Jan. to AD3000 01 Jan.)1 day 00 h 00 min

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 SunMercury a(au) = 0.387098Venus a(au) = 0.723330The 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
t21.3770−33,155.9302−687,024.3196−77.7041
Δθ (μas)Δθ (μas)Δθ (μas)Δρ (μas)
t−1.8970−43,920.9632−740,253.4678−413.2193
t20.0809504.455660,179.7955−1436.3972
Δφ (μas)Δφ (μas)Δφ (μas)Δ(Iσ) (μas)
t179.6136213,919,825.1563112,930,676.1063511,726.8500
t2−1.3915−3798.8818687,231.8895−14,383.0938
The Earth6 a(au) = 1.000001without the Moon7without the Sun8Mars 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
t249,150.805949,136.421711.786710,109.0014
Δθ (μas)Δθ (μas)Δθ (μas)Δθ (μas)
t−4127.7653−4127.7520−9.5398127,569.2300
t2−1878.6778−1878.5492−0.2078−1098.6657
Δφ (μas)Δφ (μas)Δφ (μas)Δφ (μas)
t1174.60901172.9236−1.3267414,234.7545
t2−53,414.8819−53,399.9048−12.2192−11,846.4356
Jupiter a(au) = 5.202603Saturn a(au) = 9.554910Uranus a(au) = 19.218446Neptune a(au) = 30.110387
Δψ (μas)Δψ (μas)Δψ (μas)Δψ (μas)
t−212,778.4891−67,171.5760−11,949.3883−3902.8771
t23097.9909−54.6002−21.30194.3541
Δθ (μas)Δθ (μas)Δθ (μas)Δθ (μas)
t−5974.5301−2892.9323−161.0625−118.6838
t2133.7664−27.83191.41590.1154
Δφ (μas)Δφ (μas)Δφ (μas)Δφ (μas)
t−99,066.0037−1440.359210.334532.9359
t2−3118.0679137.5508−1.06110.7553

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 paperTT2
The Sunα0 (°)286.13Δα0 (″)1×10–5−3×10–10
0δ0 (°)63.87Δδ0 (″)1×10–5−2×10–9
W(°)84.176+14.1844000dΔW (″)−0.00011×10–11
Mercuryα0 (°)281.0103−0.0328TΔα0 (″)8.54390.0015
0.387098δ0 (°)61.4155−0.0049TΔδ0 (″)3.2367−0.0047
W(°)329.5988+6.1385108dΔW (″)−28.3505−0.0018
Venusα0 (°)272.76Δα0 (″)0.23420.0016
0.723330δ0 (°)67.16Δδ0 (″)0.3331−0.0001
W(°)160.20−1.4813688dΔW (″)−4.5144−0.0014
NameArchinal et al. (2011)9Present paperTT2
The Earthα0 (°)0.00–0.641Tcos δ0Δα0 (″)0.0426−3×10–5
1.000001δ0 (°)90.00–0.557TΔδ0 (″)0.7622−0.0002
W(°)190.147+360.9856235dΔα0+ ΔW0 (″)−1.76140.0001
NameArchinal et al. (2018)Present paperTT2
Marsα0 (°)317.269202–0.10927547TΔα0 (″)0.3972−0.0001
1.523679δ0 (°)54.432516–0.05827105TΔδ0 (″)0.1991−0.0002
W(°)176.049863+350.891982443297dΔW (″)−0.92730.0001
Jupiterα0 (°)268.056595–0.006499TΔα0 (″)0.0023−4×10–6
5.202603δ0 (°)64.495303+0.002413TΔδ0 (″)0.0003−1×10–6
W(°)284.95+870.5360000dΔW (″)−0.03323×10−6
Saturnα0 (°)40.589–0.036TΔα0 (″)0.0199−1×10–5
9.554910δ0 (°)83.537–0.004TΔδ0 (″)0.00231×10–6
W(°)38.90+810.7939024dΔW (″)−0.02581×10–5
Uranusα0 (°)257.311Δα0 (″)0.00122×10–7
19.218446δ0 (°)−15.175Δδ0 (″)−0.0001−3×10–8
W(°)203.81–501.1600928dΔW (″)0.00022×10–8
Neptuneα0 (°)299.36Δα0 (″)0.0002−1×10–8
30.110387δ0 (°)43.46Δδ0 (″)0.00012×10–8
30.110387W(°)249.978+541.1397757dΔW (″)−0.00051×10–7

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 papert
Tethys(S3)294,672α0 (°)40.66 – 0.036TΔψ (″)−31.5081
δ0 (°)83.52 – 0.004TΔθ (″)−90.1979
W (°)8.95 + 190.6979085dΔφ (″)541.1442
Telesto(S13)294,720α0 (°)50.51 – 0.036TΔψ (″)−28.6117
δ0 (°)84.06 – 0.004TΔθ (″)−80.2283
W (°)56.88 + 190.6979332dΔφ (″)541.1450
Calypso(S14)19294,721α0 (°)36.41 – 0.036TΔψ (″)0.2767
δ0 (°)85.04 – 0.004TΔθ (″)−84.1557
W (°)153.51 + 190.6742373dΔφ (″)541.1466
Calypsowith α0 fromTethysα0 (°)40.66 – 0.036TΔψ (″)−3.2464
δ0 (°)85.04 – 0.004TΔθ (″)−81.6924
W (°)153.51 + 190.6742373dΔφ (″)−529.5805
Calypsowith α0, δ0 fromTethysα0 (°)40.66 – 0.036TΔψ (″)−32.1455
δ0 (°)83.52 – 0.004TΔθ (″)−90.3406
W (°)153.51 + 190.6742373dΔφ (″)−502.3581
Calypsowith α0 fromTelestoα0 (°)50.51 – 0.036TΔψ (″)−9.7437
δ0 (°)85.04 – 0.004TΔθ (″)−75.4873
W (°)153.51 + 190.6742373dΔφ (″)−524.7257
Calypsowith δ0 fromTelestoα0 (°)36.41 – 0.036TΔψ (″)−18.2542
δ0 (°)84.06 – 0.004TΔθ (″)−90.4678
W (°)153.51 + 190.6742373dΔφ (″)−514.7784
Calypso20 with α0, δ0 fromTelestoα0 (°)50.51 – 0.036TΔψ (″)−28.5499
δ0 (°)84.06 – 0.004TΔθ (″)−80.2670
W (°)153.51 + 190.6742373dΔφ (″)−507.3759
Calypso20with α0, δ0, Wfrom Telestoα0 (°)50.51 – 0.036TΔψ (″)−28.5499
δ0 (°)84.06 – 0.004TΔθ (″)−80.2670
W (°)56.88 + 190.6979332dΔφ (″)−507.3759

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

Uranus (continue)
NameΔψ (″)Δθ (″)Δφ (″)a (km)
tt2tt2tt2
Puck (U15)187.8308−4.1306−0.0069−0.4330−698.6046−0.068686,000
Miranda (U5)67.03760.3657−0.4716−0.0004−249.44640.0003129,900
Ariel (U1)25.74420.0158−0.2705−0.0105−95.1556−0.0012190,900
Umbriel (U2)11.2105−0.0032−0.0603−0.0418−41.53730.0025266,000
Titania (U3)3.25150.0270−0.0165−0.0204−12.0532−0.0030436,300
Oberon (U4)1.5706−0.0019−0.0084−0.0129−5.82820.0000583,500
Neptune
NameΔψ (″)Δθ (″)Δφ (″)a (km)
tt2tt2tt2
Naiad (N3)−6670.3047−14.7325−5.1794−55.23503809.503764.702248,227
Thalassa (N4)−6092.0806−8.2229−4.2733−17.46083467.592658.152150,074
Despina (N5)−5405.6116−6.2192−3.6995−15.27333076.886650.342052,526
Galatea (N6)−3576.1301−0.6725−2.4523−9.84772035.432830.794261,953
Larissa (N7)−2328.14491.1397−1.4266−7.75101325.469620.566773,548
Proteus (N8)−716.2634−5.9168−0.7619−16.4520409.454317.8405117,646
Triton (N1) 1743.44500.05940.8107−0.1928−25.3711−0.2680354,759
DOI: https://doi.org/10.2478/arsa-2022-0005 | Journal eISSN: 2083-6104 | Journal ISSN: 1509-3859
Journal RSS Feed
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
Publication frequency: 4 issues per year
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.

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