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Relativistic Effects in the Rotation of Dwarf Planets and Asteroids

Vladimir V. Pashkevich
Vladimir V. Pashkevich
 oraz 
Andrey N. Vershkov
Andrey N. Vershkov
   | 21 paź 2022
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Data publikacji: 21 paź 2022
Zakres stron: 158 - 184
Otrzymano: 26 lis 2021
Przyjęty: 28 wrz 2022
DOI: https://doi.org/10.2478/arsa-2022-0008
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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
Triangle used to define the direction of the angular velocity vector of the geodetic rotation

Figure 1a.

Reference system used to define orientation of the body under study (Archinal et al., 2018)
Reference system used to define orientation of the body under study (Archinal et al., 2018)

Figure 2.

Geodetic rotation of the Sun, the Moon, the planets, and dwarf planets (Ceres and Pluto) of the Solar System in the longitude of the descending node (left side) and in the absolute value of the geodetic rotation vector of the parameters of their orientation (right side)
Geodetic rotation of the Sun, the Moon, the planets, and dwarf planets (Ceres and Pluto) of the Solar System in the longitude of the descending node (left side) and in the absolute value of the geodetic rotation vector of the parameters of their orientation (right side)

Figure 3.

Geodetic rotation of the Pluto–Charon System (without their mutual influence on each other), Pluto (with taking into account the perturbations from Charon) and Charon (with taking into account the perturbations from Pluto) in the longitude of the descending node (left side) and in the absolute value of the geodetic rotation vector of the parameters of their orientation (right side)
Geodetic rotation of the Pluto–Charon System (without their mutual influence on each other), Pluto (with taking into account the perturbations from Charon) and Charon (with taking into account the perturbations from Pluto) in the longitude of the descending node (left side) and in the absolute value of the geodetic rotation vector of the parameters of their orientation (right side)

Figure 3a.

The values of the velocities of the change in geodetic rotations for Pluto and Charon (without their mutual influence on each other) (top row) and Pluto+ (with taking into account the perturbations from Charon) and Charon+ (with taking into account the perturbations from Pluto) (bottom row) in ecliptic Euler angles (the red line in the graphs shows a secular trend)
The values of the velocities of the change in geodetic rotations for Pluto and Charon (without their mutual influence on each other) (top row) and Pluto+ (with taking into account the perturbations from Charon) and Charon+ (with taking into account the perturbations from Pluto) (bottom row) in ecliptic Euler angles (the red line in the graphs shows a secular trend)

Figure 4.

Geodetic rotation of the Earth, the Moon, Mars, Ceres, Jupiter, and asteroids of 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)
Geodetic rotation of the Earth, the Moon, Mars, Ceres, Jupiter, and asteroids of 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)

Figure 5.

The orbits of the studied asteroids and the dwarf planet Ceres relative to the Sun, and the planets of the Earth, Mars, and Jupiter (L’vov et al., 2012)
The orbits of the studied asteroids and the dwarf planet Ceres relative to the Sun, and the planets of the Earth, Mars, and Jupiter (L’vov et al., 2012)

Secular terms of the geodetic rotation for the Solar System bodies under study, calculated for ecliptic Euler angles (part 1/2)

Itokawa (25143)e = 0.280i = 1.62 Eros (433)e = 0.223i = 10.83 Gaspra (951)e = 0.174i = 4.11 Vesta (4)e = 0.088i = 7.14 Steins (2867)e = 0.146i = 9.93
a (km) 198 094 516 218 138 719 330 494 569 353 354 672 353 580 460
ΔψIα (μas) ΔψI (μas) ΔψI (μas) ΔψI (μas) ΔψI (μas)
t −30888468.4680 −7539806.8764 −2644643.7323 −2563687.1111 95713.4424
t2 17712332.5604 16089.7169 5462.4226 −23519.8462 365452.1188
ΔΘI (μas) ΔΘI (μas) ΔΘI (μas) ΔΘI (μas) ΔΘI (μas)
t −299340.7816 1158037.6874 - 9902.5537 −191566.5107 −345887.4810
t2 11292.2763 48406.3346 −21824.0040 15481.8592 21837.5697
ΔφI (μas) ΔφI (μas) ΔφI (μas) ΔφI (μas) ΔφI (μas)
t −20532653.8884 −934897.6652 −207819.5424 382841.2435 2351901.1773
t2 17697440.4989 −53118.6194 −5333.5745 29110.7501 371810.7571

Secular terms of the geodetic rotation for the Solar System bodies under study, calculated for ecliptic Euler angles (part 2/2)

Lutetia (21)e = 0.163i = 3.06 Ceres (1)e = 0.078i =10.59 Pallas (2)e = 0.230i = 34.85 Ida (243)e = 0.043i = 1.13 Europa (52)e = 0.111i = 7.48
a (km) 364 359 304 413 801 038 415 041 593 428 085 277 463 012 430
ΔψI (μas) ΔψI (μas) ΔψI (μas) ΔψI (μas) ΔψI (μas)
t2 −2117178.9998 −3360178.0926 −1398863.8492 −1330838.6427 −1057008.1802
t2 2431.1845 10775.8519 45720.2961 −16354.9403 −4910.3016
ΔΘI (μas) ΔΘI (μas) ΔΘI (μas) ΔΘI (μas) ΔΘI (μas)
t −78659.8041 −8577.2633 −832986.4619 12929.7503 −86060.4985
t2 7415.8840 36307.2412 −34963.2327 5290.8816 −21571.6021
ΔφI (μas) ΔφI (μas) ΔφI (μas) ΔφI (μas) ΔφI (μas)
t 82943.9848 1891371.5898 −366135.5559 61556.5113 −147420.0367
t2 3717.5078 −10758.5439 75110.2601 −17648.9762 15432.2955
Davida (511)e = 0.188i = 15.94 Pluto(134340)e = 0.249i = 17.12 Charon (P I)e = 0.00005i = 0 Pluto(with taking into account the perturbations from Charon Charon(with taking into account the perturbations from Pluto
a (km) 473 341 349 5 900 898 409 19 591)
ΔψI (μas) ΔψI (μas) ΔψI (μas) ΔψI (μas) ΔψI (μas)
t −1093309.1477 −2196.6026 −2196.6029 1328.9252 24690.9143
t2 −16061.0721 1233.0913 1233.0910 1235.6518 1214.0777
ΔΘI (μas) ΔΘI (μas) ΔΘI (μas) ΔΘI (μas) ΔΘI (μas)
t −293147.0102 −229.9056 −229.9063 7460.6486 58441.2641
t2 8887.6360 128.4493 128.4467 128.5626 129.3395
ΔφI (μas) ΔφI (μas) ΔφI (μas) ΔφI (μas) ΔφI (μas)
t 68969.1272 −606.8179 −606.8175 87.6110 4694.8347
t2 58587.1184 340.9066 340.9119 343.2894 367.0618

The rotational elements of dwarf planets of the Solar System (α0, δ0, W) and their secular terms of the geodetic rotation

Dwarf planetsThe rotational elements Ceres (1)7 Pluto (134340)8 Charon (P I )9 Pluto without Charon10 Charon without Pluto11
α0 (°) 291.418 132.993 132.993 132.993 132.993
Δα0I (°)
T 4.54×10-6 −9.40×10−8 −1.09×10−6 5.60 ×10−8 5.60×10−8
T2 2.5l×10−7 −3.15×10−9 −3.09 ×10−9 −3.14×10−9 −3.14×10−9
δ0 (°) 66.764 −6.163 −6.l63 −6.163 −6.163
Δδ0I (°)
T l.36×10−5 −1.88×10−7 −1.37×10−6 −1.04×10−8 −1.04×10−8
T2 −l.92×10−8 5.90×10−10 5.71×10−10 5.88×10−10 5.88×10−10
W (°) 170.65 302.695 122.695 302.695 122.695
d 952.1532 56.3625225 56.3625225 56.3625225 56.362523
ΔW1 (°)
T −4.40×10−5 −2.20×10−8 −2.52×10−7 1.28×10−8 1.28×10−8
T2 −2.31×10−7 −7.l5×10−10 −6.20×10−10 −7.18×10−10 −7.l8×10−10

The parameters of the investigation of the geodetic rotation for the bodies under study

The body The time span (years) Spacing Date of the ephemeris, rotation, and orbital periods
Itokawa (25143) 900 (from AD1599 12 December 00:00 to AD2500 29 December 23:40) 2 h 20 m Aug 17 06:26:37 2021 12.13 hrs, 1.52 yrs
Eros (433) 900 (from AD1599 12 December 00:00 to AD2500 30 December 00:00) 2 h 00 m Aug 26 10:57:04 2021 5.270 hrs, 1.76 yrs
Gaspra (951) 900 (from AD1599 12 December 00:00 to AD2500 30 December 00:00) 2 h 00 m Sep 2 10:02:44 2021 7.042 hrs, 3.285 yrs
Vesta (4) 900 (from AD1599 12 December 00:00 to AD2500 30 December 00:00) 2 h 00 m Aug 25 05:22:33 2021 5.342 hrs, 3.63 yrs
Steins (2867) 900 (from AD1599 12 December 00:00 to AD2500 30 December 00:00) 2 h 00 m Aug 23 00:38:28 2021 6.049 hrs, 3.64 yrs
Lutetia (21) 900 (from AD1599 12 December 00:00 to AD2500 29 December 23:30) 2 h 30 m Aug 31 12:56:30 2021 8.168 hrs, 3.80 yrs
Ceres (1) 900 (from AD1599 12 December 00:00 to AD2500 29 December 23:30) 2 h 30 m Aug 19 05:24:26 2021 9.074 hrs, 4.60 yrs
Pallas (2) 900 (from AD1599 12 December 00:00 to AD2500 29 December 23:30) 2 h 30 m Aug 24 11:27:29 2021 7.813 hrs, 4.61 yrs
Ida (243) 900 (from AD1599 12 December 00:00 to AD2500 30 December 00:00) 2 h 00 m Aug 29 08:59:22 2021 4.633 hrs, 4.84 yrs
Europa (52) 900 (from AD1599 12 December 00:00 to AD2500 30 December 00:00) 2 h 00 m Aug 28 13:25:25 2021 5.6304 hrs, 5.451 yrs
Davida (511) 900 (from AD1599 12 December 00:00 to AD2500 30 December 00:00) 2 h 00 m Aug 30 11:09:54 2021 5.131 hrs, 5.628 yrs
Pluto (134340) 400 (from AD1700 07 January 00:00 to AD2099 31 December 00:00) 1 d Oct 13 08:08:45 2021 −6.387 days, 247.921 yrs
Charon (Pluto: I) 400 (from AD1700 07 January 00:00 to AD2099 31 December 00:00) 1 d Oct 13 07:52:41 2021 6.387 days, 6.387 days

The rotational elements of the asteroids of the Solar System (α0, δ0, W) and their secular terms of the geodetic rotation

The asteroidsThe rotational elements Itokawa (25143) Eros (433) Gaspra (951) Vesta (4) Steins (2867)
α0 (°) 90.53 11.35 9.47 309.031 91
Δα0I (°)
T 2.27×10−5 2.11×10−4 6.99×10−5 4.20×10−5 1.33×10−6
T2 −6.08×10−7 −9.81×10−8 −4.29×10−8 6.89×10−8 −1.83×10−7
δ0 (°) −66.30 17.22 26.7 42.235 −62
Δδ0 I (°)
T 3.32×10−6 5.22×10−5 2.86×10−5 2.25×10−5 9.59×10−6
T2 1.57×10−7 1.06×10−7 4.91×10−8 −2.16×10−8 −5.34×10−8
W (°) 0 326.07 83.67 285.39 321.76
d 712.143 1639.38865 1226.91149 1617.33294 1428.09917
ΔWI (°)
T 3.08×10−4 −1.30×10−4 −6.32×10−5 −7.78×10−5 6.39×10−5
T2 −5.97×10−7 −1.10×10−7 9.82×10−9 −2.07×10−8 −1.41×10−7
The asteroidsThe rotational elements Lutetia (21) Pallas (2) Ida (243) Europa (52) Davida (511)
α0 (°) 52 33 168.76 257 297
Δα0 I (°)
T 5.73×10−5 2.72×10−5 1.59×10−5 2.48×10−5 2.53×10−5
T2 −1.45×10−9 −1.49×10−7 −2.63×10−7 1.81×10−8 4.45×10−8
δ0 (°) 12 −3 −87.12 12 5
Δδ0 I (°) T 1.65×10−5 3.47×10−5 −1.45×10−5 −2.51×10−7 1.35×10−5
T2 −2.16×10−8 4.88×10−8 −1.89×10−8 5.83×10−8 −1.61×10−8
W (°) 94 38 274.05 55 268.1
d 1057.7515 1105.8036 1864.62801 1534.64722 1684.41935
ΔWI (°)
T −2.80×10−6 1.53×10−6 5.16×10−5 −2.60×10−5 −1.34×10−5
T2 9.85×10−9 1.67×10−7 −2.70×10−7 3.13×10−8 1.40×10−7

The periodic terms of the geodetic rotation for the Solar System bodies under study, calculated for the rotational elements (α0, δ0, W) (part 2/3)

Body Angle Period Argument Coefficient of cos (Argument) (μas) Coefficient of sin (Argument) (μas)
Itokawa(25143)e = 0.280i = 1.62 Δθ0 II 1.5139 yrs λlto 41.7577 −75.8886t −70.3291 +153.0315t
Δδ0 II 1.5139 yrs λlto 4.7516 −4.1915t −8.0811 +5.5740t
ΔWII 1.5139 yrs λlto 508.0721 −548.2598t −876.1647 +1515.4770t
Ceres (1)e = 0.078i = 10.59 Δα0 II 4.6049 yrs λCer 3.7269 +3.7663t 27.5024 +28.9016t
Δδ0 II 4.6049 yrs λCer 11.0371 – 1.1300t 82.7509 – 6.8295t
ΔWII 4.6049 yrs λCer −35.7425 −1.0690t −267.3755 −13.4006t
Pluto(134340)e = 0.249i = 17.12 Δ0 II 247.9673 yrs λ9 45.7326 +76.8920t −47.1527 −1.6917t
123.9837 yrs 2λ9 −9.9483 −4.5637t 3.8725 +35.3736t
82.6558 yrs 3λ9 1.0709 −7.1765t 1.5396 −7.2298t
61.9918 yrs 4λ9 −0.0787 +1.2146t −0.7342 +0.8507t
Δδ0 II 247.9673 yrs λ9 −8.5420 −14.3663t 8.8011 +0.2718t
123.9837 yrs 2λ9 1.8556 +0.8591t −0.7277 −6.6002t
82.6558 yrs 3λ9 −0.1998 +1.3409t −0.2840 +1.3547t
61.9918 yrs 4λ9 −0.0153 −0.2299t 0.1363 −0.1606t
ΔWII 247.9673 yrs λ9 10.4692 +17.6031t −10.7952 −0.3926t
123.9837 yrs 2λ9 −2.2822 −1.0427t 0.8893 +8.1005t
82.6558 yrs 3λ9 0.2481 −1.6437t 0.3529 −1.6581t
61.9918 yrs 4λ9 −0.0188 +0.2775t −0.1695 +0.1943t
Charon (P I)e = 5×10−5i = 0 Δ0 II 247.9673 yrs λ9 45.7326 +76.8918t −47.1527 −1.6915t
123.9837 yrs 2λ9 −9.9484 −4.5638t 3.8725 +35.3736t
82.6558 yrs 3λ9 1.0709 −7.1766t 1.5396 −7.2299t
61.9918 yrs 4λ9 −0.0787 +1.2146t −0.7342 +0.8506t
Δδ0 II 247.9673 yrs λ9 −8.5420 −14.3667t 8.8012 +0.2719t
123.9837 yrs 2λ9 1.8556+0.8590t −0.7277-6.6002t
82.6558 yrs 3λ9 −0.1998 +1.3408t −0.2840 +1.3546t
61.9918 yrs 4λ9 −0.0153-0.2299t 0.1363 −0.1607t
ΔWII 247.9673 yrs λ9 10.4691 +17.6022t −10.7951 −0.3932t
123.9837 yrs 2λ9 −2.2821 −1.0423t 0.8892 +8.1005t
82.6558 yrs 3λ9 0.2481 −1.6440t 0.3529 −1.6581t
61.9918 yrs 4λ9 −0.0188 +0.2776t −0.1695 +0.1942t

The periodic terms of the geodetic rotation for the Solar System bodies under study, calculated for ecliptic Euler angles (part 1/3)

Body Angle Period Argument Coefficient of cos (Argument) (μas) Coefficient of sin (Argument) (μas)
Pallas (2)e = 0.230i = 34.85 ΔψII 4.6133 yrs λPal 88.9576 −25.2003t −697.0760 +199.7455t
ΔθII 4.6133 yrs λPal 52.7890 −7.0685t −414.3066 +58.2399t
ΔψII 4.6133 yrs λPal 23.4799 −14.6468t −183.4599 +113.8270t
Vesta (4)e = 0.088i = 7.14 ΔψII 3.6299 yrs λVes 134.5261 +7.4573t −372.1918 −20.2328t
ΔθII 3.6299 yrs λVes 10.0519 −1.2577t −27.8049 +3.5120t
ΔφII 3.6299 yrs λVes −20.0867 −3.8164t 55.6047 +10.4824t
Lutetia (21)e = 0.163i = 3.06 ΔψII 3.8012 yrs λLut 434.7883 +9.1964t −439.8789 −8.0052t
ΔθII 3.8012 yrs λLut 16.1543 −2.6762t −16.3367 +2.7483t
ΔφII 3.8012 yrs λLut −17.0374 −1.9309t 17.2406 +1.9029t
Europa (52)e = 0.111i = 7.48 ΔψII 5.4539 yrs λEur −210.3443 +31.2243t −204.6667 +30.6463t
ΔθII 5.4539 yrs λEur −17.0471 −5.7352t −16.5686 −5.6809t
ΔφII 5.4539 yrs λEur −29.4063 +10.6478t −28.6229 +10.4089t
Ida (243) e = 0.043i = 1.13 ΔψII 4.8428 yrs λIda 123.9993 −23.6262t 54.0423 −10.4464t
ΔθII 4.8428 yrs λIda −1.1923 −0.7028t −0.5184 −0.31056t
ΔφII 4.8428 yrs λIda −5.7846 +4.4539t −2.5247 +1.9428t
Eros(433)e = 0.223i = 10.83 ΔψII 1.7609 yrs λEro −1173.7477 +7.8618t −738.0132 +5.7199t
ΔθII 1.7609 yrs λEro 180.2847 −15.5085t 113.3614 −9.8689t
ΔφII 1.7609 yrs λEro −145.5448 −16.1855t −91.5093 −10.0787t
Davida (511)e = 0.188i = 15.94 ΔψII 5.6626 yrs λDav 80.8186 −9.5533t 514.4072 −73.0080t
ΔθII 5.6626 yrs λDav 21.6921 −4.5331t 138.1255 −31.6875t
ΔφII 5.6626 yrs λDav −4.9923 −7.8146t −31.8137 −49.1951t
Gaspra (951)e = 0.174i = 4.11 ΔψII 3.2853 yrs λGas −707.3531 +6.8524t 79.2730 −0.7751t
ΔθII 3.2853 yrs λGas 2.6585 −11.6876t −0.2882 +1.3092t
ΔφII 3.2853 yrs λGas −55.5945 −2.5442t 6.2352 +0.2873t
Steins (2867)e = 0.146i = 4.11 ΔψII 3.6421 yrs λSte −32.2178 −100.5467t −13.2111 −41.5365t
ΔθII 3.6421 yrs λSte −6.0932 +236.4883t 53.5204 −36.6313t
ΔφII 3.6421 yrs λSte −5.9565 − 1658.3287t −361.3345 +176.9216t

The periodic terms of the geodetic rotation for the Solar System bodies under study, calculated for ecliptic Euler angles (part 2/3)

Body Angle Period Argument Coefficient of cos (Argument) (μas) Coefficient of sin (Argument) (μas)
Itokawa(25143)e = 0.280i = 1.62 ΔψII 1.5139 yrs λlto −1720.2459 +4498.8779t 2888.8876 −7571.1834t
ΔθII 1.5139 yrs λlto −14.2316 +15.8766t 24.0302 −42.9370t
ΔφII 1.5139 yrs λlto −1250.3802 +4020.0291t 2077.0710 −6195.7015t
Ceres (1)e = 0.078i = 10.59 ΔψII 4.6049 yrs λCer −75.8046 +6.1151t −568.1727 +34.5235t
ΔθII 4.6049 yrs λCer −0.1761 +1.6365t −1.4920 +12.2860t
ΔφII 4.6049 yrs λCer 42.6647 −3.6571t 319.9076 −20.9923t
Pluto(134340)e = 0.249i = 17.12 ΔαII 247.9673 yrs λ9 −49.8707 −83.8508t 51.4174 +1.8306t
123.9837 yrs 2λ9 10.8477 +4.9787t −4.2243 −38.5721t
82.6558 yrs 3λ9 −1.1678 +7.8260t −1.6777 +7.8854t
61.9918 yrs 4λ9 0.0860 −1.3255t 0.8004 −0.9282t
ΔθII 247.9673 yrs λ9 −5.2163 −8.7663t 5.3841 +0.2352t
123.9837 yrs 2λ9 1.1372 +0.5141t −0.4375 −4.0414t
82.6558 yrs 3λ9 −0.1224 +0.8181t −0.1790 +0.8205t
61.9918 yrs 4λ9 0.0084 −0.1356t 0.0846 −0.0954t
ΔφII 247.9673 yrs λ9 −13.7787 −23.1663t 14.2049 +0. 4989t
123.9837 yrs 2λ9 2.9922 +1.3778t −1.1645 −10.6541t
82.6558 yrs 3λ9 −0.3196 +2.1615t −0.4629 +2.1758t
61.9918 yrs 4λ9 0.0230 −0.3669t 0.2197 −0.2570t
Charon(P I)e = 5×10−5i = 0 ΔψII 247.9673 yrs λ9 −49.8707 −83.8508t 51.4173 +1.8305t
123.9837 yrs 2λ9 10.8477 +4.9788t −4.2243 −38.5721t
82.6558 yrs 3λ9 −1.1678 +7.8260t −1.6777 +7.8854t
61.9918 yrs 4λ9 0.0860 −1.3255t 0.8004 −0.9282t
ΔθII 247.9673 yrs λ9 −5.2162 −8.7658t 5.3841 +0.2350t
123.9837 yrs 2λ9 1.1372 +0.5143t −0.4375 −4.0414t
82.6558 yrs 3λ9 −0.1224 +0.8182t −0.1790 +0.8206t
61.9918 yrs 4λ9 0.0084 −0.1356t 0.0846 −0.0953t
ΔφII 247.9673 yrs λ9 −13.7788 −23.1672t 14.2049 +0.4983t
123.9837 yrs 2λ9 2.9923 +1.3783t −1.1645 −10.6541t
82.6558 yrs 3λ9 −0.3197 +2.1612t −0.4629 +2.1758t
61.9918 yrs 4λ9 0.0230 −0.3667t 0.2197 −0.2571t

The periodic terms of the geodetic rotation for the Solar System bodies under study, calculated for ecliptic Euler angles (part 3/3)

Body Angle Period Argument Coefficient of cos (Argument) (μas) Coefficient of sin (Argument) (μas)
Pluto(with taking into account the perturbations from Charon) ΔψII 6.3868d λPl + λ9 −0.0544 +0.0719t 0.0712 −0.0255t
6.3877d DPl −0.0188 +0.0038t −0.0347 +0.0054t
247.9673 yrs λ9 −49.8834 −83.9692t 51.4126 +1.8707t
6.3872d λPl −0.0008 −0.0145t −0.0018 −0.0144t
123.9837 yrs 2λ9 10.8431 +4.9539t −4.2304 −38.6559t
82.6558 yrs 3λ9 −1.1614 +7.8894t −1.6786 +7.8874t
61.9918 yrs 4λ9 0.0866 −1.3409t 0.8022 −0.9077t
ΔθII 6.3868d λPl + λ9 −0.0437 −0.0024t 0.0511 +0.0275t
6.3877d DPl 0.0303 −0.0344t 0.0389 +0.0028t
247.9673 yrs λ9 −5.2145 −8.7615t 5.3824 +0.2222t
6.3872d λPl 0.0028 +0.0133t 0.0020 +0.0220t
123.9837 yrs 2λ9 1.1405 +0.5279t −0.4392 −4.0521t
82.6558 yrs 3λ9 −0.1210 +0.8317t −0.1787 +0.8270t
61.9918 yrs 4λ9 0.0091 −0.1340t 0.0857 −0.0882t
ΔφII 6.3868d λPl + λ9 −0.0570 +0.0584t −0.0075 +0.0511t
6.3877d DPl 0.0357 +0.0004t −0.0432 +0.0184t
247.9673 yrs λ9 −13.7681 −23.1282t 14.2099 +0.5103t
6.3872d λPl 0.0018 +0.0146t −0.0026 −0.0243t
123.9837 yrs 2λ9 2.9929 +1.3861t −1.1620 −10.6384t
82.6558 yrs 3λ9 −0.3195 +2.1537t −0.4628 +2.1762t
61.9918 yrs 4λ9 0.0228 −0.3633t 0.2196 −0.2575t
Charon(with taking into account the perturba–tions from Pluto) ΔψII 6.3868d λ91 + λ9 −0.5192 +0.6870t 0.6803 −0.2432t
6.3877d D91 −0.1800 +0.0360t −0.3312 +0.0516t
6.3872d λ91 −0.0077 −0.1384t −0.0172 −0.1373t
247.9673 yrs λ9 −49.7551 −82.7024t 51.4769 +1.5174t
123.9837 yrs 2λ9 10.8429 +5.2419t −4.1450 −37.7086t
82.6558 yrs 3λ9 −1.2110 +7.2077t −1.6608 +7.8435t
61.9918 yrs 4λ9 0.0778 −1.1786t 0.7764 −1.1215t
ΔθII 6.3868d λ91 + λ9 −0.4173 −0.0233t 0.4877 +0.2628t
6.3877d D91 0.2891 −0.3283t 0.3717 +0.0270t
6.3872d λ91 0.0272 +0.1270t 0.0193 +0.2097t
247.9673 yrs λ9 −5.2392 −8.9526t 5.4239 +0.3446t
123.9837 yrs 2λ9 1.1235 +0.3709t −0.4330 −4.0066t
82.6558 yrs 3λ9 −0.1418 +0.7045t −0.1869 +0.7648t
61.9918 yrs 4λ9 0.0030 −0.1487t 0.0761 −0.1640t
ΔφII 6.3868d λ91 + λ9 −0.5445 +0.5577t −0.0714 +0.4881t
6.3877d D91 0.3412 +0.0041t −0.4126 +0.1755t
6.3872d λ91 0.0174 +0.1398t −0.0248 −0.2323t
247.9673 yrs λ9 −13.8080 −23.5456t 14.2437 +0.5339t
123.9837 yrs 2λ9 2.9738 +1.2625t −1.1718 −10.8312t
82.6558 yrs 3λ9 −0.3110 +2.2409t −0.4733 +2.1292t
61.9918 yrs 4λ9 0.0221 −0.3851t 0.2192 −0.2414t

The periodic terms of the geodetic rotation for the Solar System bodies under study, calculated for the rotational elements (α0, δ0, W) (part 3/3)

Body Angle Periods Argument Coefficient of cos (Argument) (μas) Coefficient of sin (Argument) (μas)
Pluto(with taking into account the perturbations from Charon) Δα0 II 6.3868 d λPl + λ9 0.0611 −0.0630t −0.0782 +0.0144t
6.3877 d DPl 0.0077 +0.0068t 0.0192 −0.0056t
247.9673 yrs λ9 45.7434 +76.9955t −47.1480 −1.7233t
123.9837 yrs 2λ9 −9.9455 −4.5471t 3.8788 +35.4534t
6.3872 d λP1 −0.0001 +0.0089t 0.0010 0.0062t
82.6558 yrs 3λ9 1.0648 −7.2372t 1.5401 −7.2340t
61.9918 yrs 4λ9 −0.0794 +1.2282t −0.7362 +0.8298t
Δδ0 II 6.3868 d λP1 + λ9 0.0270 +0.0218t −0.0295 −0.0332t
6.3877 d DP1 −0.0340 +0.0339t −0.0466 −0.0012t
247.9673 yrs λ9 −8.5472 −14.4029t 8.8015 +0.2951t
123.9837 yrs 2λ9 1.8521 +0.8428t −0.7291 −6.6243t
6.3872d λP1 −0.0029 −0.0166t −0.0024 −0.0025t
82.6558 yrs 3λ9 −0.1992 +1.3479t −0.2841 +1.3504t
61.9918 yrs 4λ9 −0.0147 +0.2367t −0.1359 +0.1591t
ΔWII 6.3868 d λP1 + λ 9 −0.0294 +0.0237t −0.0435 +0.0625t
6.3877 d DPl 0.0439 −0.0003t −0.0277 +0.0157t
247.9673 yrs λ 9 10.4859 +17.6982t −10.7878 -0.4001t
123.9837 yrs 2λ9 −2.2797 −1.0270t 0.8948 +8.1528t
6.3872 d λPl 0.0021 +0.0212t −0.0018 -0.0181t
82.6558 yrs 3λ9 0.2452 -1.6813t 0.3534 -1.6603t
61.9918 yrs 4λ9 −0.0192 +0.2890t −0.1704 +0.1846t
Charon(with taking into account the perturbations from Pluto) Δα0 II 6.3868 d λ91 + λ9 0.5835 −0.6002t −0.7471 +0.1377t
6.3877 d D91 0.0739 +0.0653t 0.1835 −0.0536t
6.3872 d λ91 −0.0012 +0.0851t 0.0094 0.0596t
247.9673 yrs λ9 45.6369 +75.9256t −47.2173 −1.4465t
123.9837 yrs 2λ9 −9.9400 −4.7545t 3.8009 +34.5981t
82.6558 yrs 3λ9 1.1150 −6.5950t 1.5269 −7.1762t
61.9918 yrs 4λ9 −0.0698 +1.0883t −0.7105 +1.0422t
Δδ0 II 6.3868d λ91 + λ9 0.2579 +0.2086t −0.2816 −0.3171t
6.3877d D91 −0.3251 +0.3235t −0.4451 −0.0118t
6.3872d λ91 −0.0281 −0.1590t −0.0231 −0.2377t
247.9673 yrs λ9 −8.4887 −13.8767t 8.7793 +0.0823t
123.9837 yrs 2λ9 1.8674 +1.0673t −0.7105 −6.3992t
61.9918 yrs 4λ9 0.0183 −0.1776t 0.1378 −0.1475t
82.6558 yrs 3λ9 −0.1930 +1.2817t −0.2719 +1.3966t
ΔWII 6.3868d λ91 + λ9 0.2805 +0.2267t −0.4154 −0.5972t
6.3877 d D91 −0.4190 −0.0028t −0.2645 −0.1498t
6.3872 d λ91 −0.0203 −0.2027t −0.0171 −0.1727t
247.9673 yrs λ9 10.3847 +16.6751t −10.7867 −0.2098t
123.9837 yrs 2λ9 −2.2978 −1.2805t 0.8435 +7.5052t
82.6558 yrs 3λ9 0.2783 −1.2621t 0.3346 −1.6827t
61.9918 yrs 4λ9 −0.0156 +0.1888t −0.1581 +0.3053t

The periodic terms of the geodetic rotation for the Solar System bodies under study, calculated for the rotational elements (α0, δ0, W) (part 1/3)

Body Angle Period Argument Coefficient of cos (Argument) (μas) Coefficient of sin (Argument) (μas)
Pallas (2)e = 0.230i = 34.85 Δα0 II 4.6133 yrs λPal −62.3217α +20.3863t 488.1305 −160.8216t
Δδ0 II 4.6133 yrs λPal −79.2069 +15.0390t 621.2777 −121.2805t
ΔWII 4.6133 yrs λPal −3.2991 −6.9178t 26.3696 +52.6044t
Vesta (4)e = 0.088i = 7.14 Δα0 II 3.6299 yrs λVes −79.3188 −5.5469t 219.4540 +15.1126t
Δδ0 II 3.6299 yrs λVes −42.5756 −0.7573t 117.7886 +1.9666t
ΔWII 3.6299 yrs λVes 146.9691 +6.2171t −406.5890 −16.7824t
Lutetia (21)e = 0.163i = 3.06 Δα0 II 3.8012 yrs λLut −423.8069 −9.7248t 428.7706 +8.5707t
Δδ0 II 3.8012 yrs λLut −122.1336 +0.3414t 123.5570 −0.7030t
ΔWII 3.8012 yrs λLut 20.7078 −0.9744t −20.9469 +1.0484t
Europa (52)e = 0.111i = 7.48 Δα0 II 5.4539 yrs λEur 177.5560 −25.4369t 172.7614 −24.9603t
Δδ0 II 5.4539 yrs λEur −1.8759 +8.4951t −1.8435 +8.3894t
ΔWII 5.4539 yrs λEur −186.1907 +33.7301t −181.1749 +33.0627t
Ida (243)e = 0.043i = 1.13 Δα0 II 4.8428 yrs λIda −53.6870 +28.6795t −23.4218 +12.6767t
Δδ0 II 4.8428 yrs λIda 48.4735 −9.1614t 21.1260 −4.0508t
ΔWII 4.8428 yrs λIda −173.5102 +54.8385t −75.6477 +24.2165t
Eros (433)e = 0.223i = 10.83 Δα0 II 1.7609 yrs λEro 1180.4497 −13.8623t 742.2291 −9.4965t
Δδ0 II 1.7609 yrs λEro 292.3515 +11.1627t 183.8167 +6.8238t
ΔWII 1.7609 yrs λEro −726.0456 −10.5342t −456.5093 −6.1414t
Davida (511)e = 0.188i = 15.94 Δα0 II 5.6626 yrs λDav −67.2810 +7.5571t −428.2301 +58.3470t
Δδ0 II 5.6626 yrs λDav −35.8472 +6.1664t −228.2203 +44.2293t
ΔWII 5.6626 yrs λDav 35.8691 −12.6102t 228.2665 −85.8956t
Gaspra (951)e = 0.174i = 4.11 Δα0 II 3.2853 yrs λGas 673.5156 −12.0004t −75.4762 +1.3514t
Δδ0 II 3.2853 yrs λGas 275.1211 +7.9210t −30.8416 −0.8844t
ΔWII 3.2853 yrs λGas −608.4609 +5.2720t 68.1928 −0.5941t
Steins (2867)e = 0.146i = 9.93 Δα0 II 3.6421 yrs λSte 6.5886 −26.7418t −7.6696 +13.8217t
Δδ0 II 3.6421 yrs λSte 5.8465 −236.2914t −53.4095 +36.2043t
ΔWII 3.6421 yrs λSte 31.9757 −1581.7149t −354.9374 +230.5293t
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