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

Vladimir V. Pashkevich

Vladimir V. Pashkevich

Central (Pulkovo) Astronomical Observatory of RAS, St. Petersburg
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Pashkevich, Vladimir V.
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Andrey N. Vershkov

Andrey N. Vershkov

Central (Pulkovo) Astronomical Observatory of RAS, St. Petersburg
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Vershkov, Andrey N.
  
22. Apr. 2022
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Online veröffentlicht: 22. Apr. 2022
Seitenbereich: 77 - 109
Eingereicht: 12. Mai 2021
Akzeptiert: 18. März 2022
DOI: https://doi.org/10.2478/arsa-2022-0005
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© 2022 Vladimir V. Pashkevich et al., published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International License.

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)
Name Δψ (″) Δθ (″) Δφ (″) a (km)
t t2 t t2 t t2
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)
t t2 t t2 t t2
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

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

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)
t t2 t t2 t t2
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
Uranus16
Name Δψ (″) Δθ (″) Δφ (″) a (km)
t t2 t t2 t t2
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

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

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)
t t2 t t2 t t2
The Moon (E1)10 −19.4942 −0.0001 −0.0004 −0.0014 0.5117 −0.0144 384,400
without the Earth11 −19.1932 −3×10–5 −0.0005 −0.0014 0.5171 −0.0144 149,597,870
without the Sun12 −0.3014 −4×10–5 3×10–5 −0.0001 −0.0054 −1×10–5 384,400
Mars
Name Δψ (″) Δθ (″) Δφ (″) a (km)
t t2 t t2 t t2
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)
t t2 t t2 t t2
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

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

Name Eroshkin and Pashkevich (2007) Klioner et al. (2009) In this paper a (au)
“ per century “ per century “ per century
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

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)18294,721 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

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

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

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

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

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

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

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)19294,721 α0 (°) 36.41 – 0.036T Δψ (″) 0.2767
δ0 (°) 85.04 – 0.004T Δθ (″) −84.1557
W (°) 153.51 + 190.6742373d Δφ (″) 541.1466
Calypsowith α0 fromTethys α0 (°) 40.66 – 0.036T Δψ (″) −3.2464
δ0 (°) 85.04 – 0.004T Δθ (″) −81.6924
W (°) 153.51 + 190.6742373d Δφ (″) −529.5805
Calypsowith α0, δ0 fromTethys α0 (°) 40.66 – 0.036T Δψ (″) −32.1455
δ0 (°) 83.52 – 0.004T Δθ (″) −90.3406
W (°) 153.51 + 190.6742373d Δφ (″) −502.3581
Calypsowith α0 fromTelesto α0 (°) 50.51 – 0.036T Δψ (″) −9.7437
δ0 (°) 85.04 – 0.004T Δθ (″) −75.4873
W (°) 153.51 + 190.6742373d Δφ (″) −524.7257
Calypsowith δ0 fromTelesto α0 (°) 36.41 – 0.036T Δψ (″) −18.2542
δ0 (°) 84.06 – 0.004T Δθ (″) −90.4678
W (°) 153.51 + 190.6742373d Δφ (″) −514.7784
Calypso20 with α0, δ0 fromTelesto α0 (°) 50.51 – 0.036T Δψ (″) −28.5499
δ0 (°) 84.06 – 0.004T Δθ (″) −80.2670
W (°) 153.51 + 190.6742373d Δφ (″) −507.3759
Calypso20with α0, δ0, Wfrom 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

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)
t t2 t t2 t t2
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)
t t2 t t2 t t2
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
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