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Relativistic Effects in the Rotation of Dwarf Planets and Asteroids Cover

Relativistic Effects in the Rotation of Dwarf Planets and Asteroids

Artificial Satellites
Volume 57 (2022): Issue 3 (September 2022)
By: Vladimir V. Pashkevich and  Andrey N. Vershkov  
Open Access
|Oct 2022

Figures & Tables

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 = 1.62Eros (433)e = 0.223i = 10.83Gaspra (951)e = 0.174i = 4.11Vesta (4)e = 0.088i = 7.14Steins (2867)e = 0.146i = 9.93
a (km)198 094 516218 138 719330 494 569353 354 672353 580 460
ΔψIα (μas)ΔψI (μas)ΔψI (μas)ΔψI (μas)ΔψI (μas)
t−30888468.4680−7539806.8764−2644643.7323−2563687.111195713.4424
t217712332.560416089.71695462.4226−23519.8462365452.1188
ΔΘI (μas)ΔΘI (μas)ΔΘI (μas)ΔΘI (μas)ΔΘI (μas)
t−299340.78161158037.6874 -9902.5537−191566.5107−345887.4810
t211292.276348406.3346−21824.004015481.859221837.5697
ΔφI (μas)ΔφI (μas)ΔφI (μas)ΔφI (μas)ΔφI (μas)
t−20532653.8884−934897.6652−207819.5424382841.24352351901.1773
t217697440.4989−53118.6194−5333.574529110.7501371810.7571

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 = 3.06Ceres (1)e = 0.078i =10.59Pallas (2)e = 0.230i = 34.85Ida (243)e = 0.043i = 1.13Europa (52)e = 0.111i = 7.48
a (km)364 359 304413 801 038415 041 593428 085 277463 012 430
ΔψI (μas)ΔψI (μas)ΔψI (μas)ΔψI (μas)ΔψI (μas)
t2−2117178.9998−3360178.0926−1398863.8492−1330838.6427−1057008.1802
t22431.184510775.851945720.2961−16354.9403−4910.3016
ΔΘI (μas)ΔΘI (μas)ΔΘI (μas)ΔΘI (μas)ΔΘI (μas)
t−78659.8041−8577.2633−832986.461912929.7503−86060.4985
t27415.884036307.2412−34963.23275290.8816−21571.6021
ΔφI (μas)ΔφI (μas)ΔφI (μas)ΔφI (μas)ΔφI (μas)
t82943.98481891371.5898−366135.555961556.5113−147420.0367
t23717.5078−10758.543975110.2601−17648.976215432.2955
Davida (511)e = 0.188i = 15.94Pluto(134340)e = 0.249i = 17.12Charon (P I)e = 0.00005i = 0Pluto(with taking into account the perturbations from CharonCharon(with taking into account the perturbations from Pluto
a (km)473 341 3495 900 898 40919 591)
ΔψI (μas)ΔψI (μas)ΔψI (μas)ΔψI (μas)ΔψI (μas)
t−1093309.1477−2196.6026−2196.60291328.925224690.9143
t2−16061.07211233.09131233.09101235.65181214.0777
ΔΘI (μas)ΔΘI (μas)ΔΘI (μas)ΔΘI (μas)ΔΘI (μas)
t−293147.0102−229.9056−229.90637460.648658441.2641
t28887.6360128.4493128.4467128.5626129.3395
ΔφI (μas)ΔφI (μas)ΔφI (μas)ΔφI (μas)ΔφI (μas)
t68969.1272−606.8179−606.817587.61104694.8347
t258587.1184340.9066340.9119343.2894367.0618

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

Dwarf planetsThe rotational elementsCeres (1)7Pluto (134340)8Charon (P I )9Pluto without Charon10Charon without Pluto11
α0 (°) 291.418132.993132.993132.993132.993
Δα0I (°)
T4.54×10-6−9.40×10−8−1.09×10−65.60 ×10−85.60×10−8
T22.5l×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 (°)
Tl.36×10−5−1.88×10−7−1.37×10−6−1.04×10−8−1.04×10−8
T2−l.92×10−85.90×10−105.71×10−105.88×10−105.88×10−10
W (°) 170.65302.695122.695302.695122.695
d952.153256.362522556.362522556.362522556.362523
ΔW1 (°)
T−4.40×10−5−2.20×10−8−2.52×10−71.28×10−81.28×10−8
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 bodyThe time span (years)SpacingDate 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 mAug 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 mAug 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 mSep 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 mAug 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 mAug 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 mAug 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 mAug 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 mAug 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 mAug 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 mAug 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 mAug 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 dOct 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 dOct 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 elementsItokawa (25143)Eros (433)Gaspra (951)Vesta (4)Steins (2867)
α0 (°) 90.5311.359.47309.03191
Δα0I (°)
T2.27×10−52.11×10−46.99×10−54.20×10−51.33×10−6
T2−6.08×10−7−9.81×10−8−4.29×10−86.89×10−8−1.83×10−7
δ0 (°) −66.3017.2226.742.235−62
Δδ0 I (°)
T3.32×10−65.22×10−52.86×10−52.25×10−59.59×10−6
T21.57×10−71.06×10−74.91×10−8−2.16×10−8−5.34×10−8
W (°) 0326.0783.67285.39321.76
d712.1431639.388651226.911491617.332941428.09917
ΔWI (°)
T3.08×10−4−1.30×10−4−6.32×10−5−7.78×10−56.39×10−5
T2−5.97×10−7−1.10×10−79.82×10−9−2.07×10−8−1.41×10−7
The asteroidsThe rotational elementsLutetia (21)Pallas (2)Ida (243)Europa (52)Davida (511)
α0 (°) 5233168.76257297
Δα0 I (°)
T5.73×10−52.72×10−51.59×10−52.48×10−52.53×10−5
T2−1.45×10−9−1.49×10−7−2.63×10−71.81×10−84.45×10−8
δ0 (°) 12−3−87.12125
Δδ0 I (°)T1.65×10−53.47×10−5−1.45×10−5−2.51×10−71.35×10−5
T2−2.16×10−84.88×10−8−1.89×10−85.83×10−8−1.61×10−8
W (°) 9438274.0555268.1
d1057.75151105.80361864.628011534.647221684.41935
ΔWI (°)
T−2.80×10−61.53×10−65.16×10−5−2.60×10−5−1.34×10−5
T29.85×10−91.67×10−7−2.70×10−73.13×10−81.40×10−7

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)

BodyAnglePeriodArgumentCoefficient of cos (Argument) (μas)Coefficient of sin (Argument) (μas)
Itokawa(25143)e = 0.280i = 1.62Δθ0 II1.5139 yrsλlto41.7577 −75.8886t−70.3291 +153.0315t
Δδ0 II1.5139 yrsλlto4.7516 −4.1915t−8.0811 +5.5740t
ΔWII1.5139 yrsλlto508.0721 −548.2598t−876.1647 +1515.4770t
Ceres (1)e = 0.078i = 10.59Δα0 II4.6049 yrsλCer3.7269 +3.7663t27.5024 +28.9016t
Δδ0 II4.6049 yrsλCer11.0371 – 1.1300t82.7509 – 6.8295t
ΔWII4.6049 yrsλCer−35.7425 −1.0690t−267.3755 −13.4006t
Pluto(134340)e = 0.249i = 17.12Δ0 II247.9673 yrsλ945.7326 +76.8920t−47.1527 −1.6917t
123.9837 yrs2λ9−9.9483 −4.5637t3.8725 +35.3736t
82.6558 yrs3λ91.0709 −7.1765t1.5396 −7.2298t
61.9918 yrs4λ9−0.0787 +1.2146t−0.7342 +0.8507t
Δδ0 II247.9673 yrsλ9−8.5420 −14.3663t8.8011 +0.2718t
123.9837 yrs2λ91.8556 +0.8591t−0.7277 −6.6002t
82.6558 yrs3λ9−0.1998 +1.3409t−0.2840 +1.3547t
61.9918 yrs4λ9−0.0153 −0.2299t0.1363 −0.1606t
ΔWII247.9673 yrsλ910.4692 +17.6031t−10.7952 −0.3926t
123.9837 yrs2λ9−2.2822 −1.0427t0.8893 +8.1005t
82.6558 yrs3λ90.2481 −1.6437t0.3529 −1.6581t
61.9918 yrs4λ9−0.0188 +0.2775t−0.1695 +0.1943t
Charon (P I)e = 5×10−5i = 0Δ0 II247.9673 yrsλ945.7326 +76.8918t−47.1527 −1.6915t
123.9837 yrs2λ9−9.9484 −4.5638t3.8725 +35.3736t
82.6558 yrs3λ91.0709 −7.1766t1.5396 −7.2299t
61.9918 yrs4λ9−0.0787 +1.2146t−0.7342 +0.8506t
Δδ0 II247.9673 yrsλ9−8.5420 −14.3667t8.8012 +0.2719t
123.9837 yrs2λ91.8556+0.8590t−0.7277-6.6002t
82.6558 yrs3λ9−0.1998 +1.3408t−0.2840 +1.3546t
61.9918 yrs4λ9−0.0153-0.2299t0.1363 −0.1607t
ΔWII247.9673 yrsλ910.4691 +17.6022t−10.7951 −0.3932t
123.9837 yrs2λ9−2.2821 −1.0423t0.8892 +8.1005t
82.6558 yrs3λ90.2481 −1.6440t0.3529 −1.6581t
61.9918 yrs4λ9−0.0188 +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)

BodyAnglePeriodArgumentCoefficient of cos (Argument) (μas)Coefficient of sin (Argument) (μas)
Pallas (2)e = 0.230i = 34.85ΔψII4.6133 yrsλPal88.9576 −25.2003t−697.0760 +199.7455t
ΔθII4.6133 yrsλPal52.7890 −7.0685t−414.3066 +58.2399t
ΔψII4.6133 yrsλPal23.4799 −14.6468t−183.4599 +113.8270t
Vesta (4)e = 0.088i = 7.14ΔψII3.6299 yrsλVes134.5261 +7.4573t−372.1918 −20.2328t
ΔθII3.6299 yrsλVes10.0519 −1.2577t−27.8049 +3.5120t
ΔφII3.6299 yrsλVes−20.0867 −3.8164t55.6047 +10.4824t
Lutetia (21)e = 0.163i = 3.06ΔψII3.8012 yrsλLut434.7883 +9.1964t−439.8789 −8.0052t
ΔθII3.8012 yrsλLut16.1543 −2.6762t−16.3367 +2.7483t
ΔφII3.8012 yrsλLut−17.0374 −1.9309t17.2406 +1.9029t
Europa (52)e = 0.111i = 7.48ΔψII5.4539 yrsλEur−210.3443 +31.2243t−204.6667 +30.6463t
ΔθII5.4539 yrsλEur−17.0471 −5.7352t−16.5686 −5.6809t
ΔφII5.4539 yrsλEur−29.4063 +10.6478t−28.6229 +10.4089t
Ida (243) e = 0.043i = 1.13ΔψII4.8428 yrsλIda123.9993 −23.6262t54.0423 −10.4464t
ΔθII4.8428 yrsλIda−1.1923 −0.7028t−0.5184 −0.31056t
ΔφII4.8428 yrsλIda−5.7846 +4.4539t−2.5247 +1.9428t
Eros(433)e = 0.223i = 10.83ΔψII1.7609 yrsλEro−1173.7477 +7.8618t−738.0132 +5.7199t
ΔθII1.7609 yrsλEro180.2847 −15.5085t113.3614 −9.8689t
ΔφII1.7609 yrsλEro−145.5448 −16.1855t−91.5093 −10.0787t
Davida (511)e = 0.188i = 15.94ΔψII5.6626 yrsλDav80.8186 −9.5533t514.4072 −73.0080t
ΔθII5.6626 yrsλDav21.6921 −4.5331t138.1255 −31.6875t
ΔφII5.6626 yrsλDav−4.9923 −7.8146t−31.8137 −49.1951t
Gaspra (951)e = 0.174i = 4.11ΔψII3.2853 yrsλGas−707.3531 +6.8524t79.2730 −0.7751t
ΔθII3.2853 yrsλGas2.6585 −11.6876t−0.2882 +1.3092t
ΔφII3.2853 yrsλGas−55.5945 −2.5442t6.2352 +0.2873t
Steins (2867)e = 0.146i = 4.11ΔψII3.6421 yrsλSte−32.2178 −100.5467t−13.2111 −41.5365t
ΔθII3.6421 yrsλSte−6.0932 +236.4883t53.5204 −36.6313t
ΔφII3.6421 yrsλSte−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)

BodyAnglePeriodArgumentCoefficient of cos (Argument) (μas)Coefficient of sin (Argument) (μas)
Itokawa(25143)e = 0.280i = 1.62ΔψII1.5139 yrsλlto−1720.2459 +4498.8779t2888.8876 −7571.1834t
ΔθII1.5139 yrsλlto−14.2316 +15.8766t24.0302 −42.9370t
ΔφII1.5139 yrsλlto−1250.3802 +4020.0291t2077.0710 −6195.7015t
Ceres (1)e = 0.078i = 10.59ΔψII4.6049 yrsλCer−75.8046 +6.1151t−568.1727 +34.5235t
ΔθII4.6049 yrsλCer−0.1761 +1.6365t−1.4920 +12.2860t
ΔφII4.6049 yrsλCer42.6647 −3.6571t319.9076 −20.9923t
Pluto(134340)e = 0.249i = 17.12ΔαII247.9673 yrsλ9−49.8707 −83.8508t51.4174 +1.8306t
123.9837 yrs2λ910.8477 +4.9787t−4.2243 −38.5721t
82.6558 yrs3λ9−1.1678 +7.8260t−1.6777 +7.8854t
61.9918 yrs4λ90.0860 −1.3255t0.8004 −0.9282t
ΔθII247.9673 yrsλ9−5.2163 −8.7663t5.3841 +0.2352t
123.9837 yrs2λ91.1372 +0.5141t−0.4375 −4.0414t
82.6558 yrs3λ9−0.1224 +0.8181t−0.1790 +0.8205t
61.9918 yrs4λ90.0084 −0.1356t0.0846 −0.0954t
ΔφII247.9673 yrsλ9−13.7787 −23.1663t14.2049 +0. 4989t
123.9837 yrs2λ92.9922 +1.3778t−1.1645 −10.6541t
82.6558 yrs3λ9−0.3196 +2.1615t−0.4629 +2.1758t
61.9918 yrs4λ90.0230 −0.3669t0.2197 −0.2570t
Charon(P I)e = 5×10−5i = 0ΔψII247.9673 yrsλ9−49.8707 −83.8508t51.4173 +1.8305t
123.9837 yrs2λ910.8477 +4.9788t−4.2243 −38.5721t
82.6558 yrs3λ9−1.1678 +7.8260t−1.6777 +7.8854t
61.9918 yrs4λ90.0860 −1.3255t0.8004 −0.9282t
ΔθII247.9673 yrsλ9−5.2162 −8.7658t5.3841 +0.2350t
123.9837 yrs2λ91.1372 +0.5143t−0.4375 −4.0414t
82.6558 yrs3λ9−0.1224 +0.8182t−0.1790 +0.8206t
61.9918 yrs4λ90.0084 −0.1356t0.0846 −0.0953t
ΔφII247.9673 yrsλ9−13.7788 −23.1672t14.2049 +0.4983t
123.9837 yrs2λ92.9923 +1.3783t−1.1645 −10.6541t
82.6558 yrs3λ9−0.3197 +2.1612t−0.4629 +2.1758t
61.9918 yrs4λ90.0230 −0.3667t0.2197 −0.2571t

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

BodyAnglePeriodArgumentCoefficient of cos (Argument) (μas)Coefficient of sin (Argument) (μas)
Pluto(with taking into account the perturbations from Charon)ΔψII6.3868dλPl + λ9−0.0544 +0.0719t0.0712 −0.0255t
6.3877dDPl−0.0188 +0.0038t−0.0347 +0.0054t
247.9673 yrsλ9−49.8834 −83.9692t51.4126 +1.8707t
6.3872dλPl−0.0008 −0.0145t−0.0018 −0.0144t
123.9837 yrs2λ910.8431 +4.9539t−4.2304 −38.6559t
82.6558 yrs3λ9−1.1614 +7.8894t−1.6786 +7.8874t
61.9918 yrs4λ90.0866 −1.3409t0.8022 −0.9077t
ΔθII6.3868dλPl + λ9−0.0437 −0.0024t0.0511 +0.0275t
6.3877dDPl0.0303 −0.0344t0.0389 +0.0028t
247.9673 yrsλ9−5.2145 −8.7615t5.3824 +0.2222t
6.3872dλPl0.0028 +0.0133t0.0020 +0.0220t
123.9837 yrs2λ91.1405 +0.5279t−0.4392 −4.0521t
82.6558 yrs3λ9−0.1210 +0.8317t−0.1787 +0.8270t
61.9918 yrs4λ90.0091 −0.1340t0.0857 −0.0882t
ΔφII6.3868dλPl + λ9−0.0570 +0.0584t−0.0075 +0.0511t
6.3877dDPl0.0357 +0.0004t−0.0432 +0.0184t
247.9673 yrsλ9−13.7681 −23.1282t14.2099 +0.5103t
6.3872dλPl0.0018 +0.0146t−0.0026 −0.0243t
123.9837 yrs2λ92.9929 +1.3861t−1.1620 −10.6384t
82.6558 yrs3λ9−0.3195 +2.1537t−0.4628 +2.1762t
61.9918 yrs4λ90.0228 −0.3633t0.2196 −0.2575t
Charon(with taking into account the perturba–tions from Pluto)ΔψII6.3868dλ91 + λ9−0.5192 +0.6870t0.6803 −0.2432t
6.3877dD91−0.1800 +0.0360t−0.3312 +0.0516t
6.3872dλ91−0.0077 −0.1384t−0.0172 −0.1373t
247.9673 yrsλ9−49.7551 −82.7024t51.4769 +1.5174t
123.9837 yrs2λ910.8429 +5.2419t−4.1450 −37.7086t
82.6558 yrs3λ9−1.2110 +7.2077t−1.6608 +7.8435t
61.9918 yrs4λ90.0778 −1.1786t0.7764 −1.1215t
ΔθII6.3868dλ91 + λ9−0.4173 −0.0233t0.4877 +0.2628t
6.3877dD910.2891 −0.3283t0.3717 +0.0270t
6.3872dλ910.0272 +0.1270t0.0193 +0.2097t
247.9673 yrsλ9−5.2392 −8.9526t5.4239 +0.3446t
123.9837 yrs2λ91.1235 +0.3709t−0.4330 −4.0066t
82.6558 yrs3λ9−0.1418 +0.7045t−0.1869 +0.7648t
61.9918 yrs4λ90.0030 −0.1487t0.0761 −0.1640t
ΔφII6.3868dλ91 + λ9−0.5445 +0.5577t−0.0714 +0.4881t
6.3877dD910.3412 +0.0041t−0.4126 +0.1755t
6.3872dλ910.0174 +0.1398t−0.0248 −0.2323t
247.9673 yrsλ9−13.8080 −23.5456t14.2437 +0.5339t
123.9837 yrs2λ92.9738 +1.2625t−1.1718 −10.8312t
82.6558 yrs3λ9−0.3110 +2.2409t−0.4733 +2.1292t
61.9918 yrs4λ90.0221 −0.3851t0.2192 −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)

BodyAnglePeriodsArgumentCoefficient of cos (Argument) (μas)Coefficient of sin (Argument) (μas)
Pluto(with taking into account the perturbations from Charon)Δα0 II6.3868 dλPl + λ90.0611 −0.0630t−0.0782 +0.0144t
6.3877 dDPl0.0077 +0.0068t0.0192 −0.0056t
247.9673 yrsλ945.7434 +76.9955t−47.1480 −1.7233t
123.9837 yrs2λ9−9.9455 −4.5471t3.8788 +35.4534t
6.3872 dλP1−0.0001 +0.0089t0.0010 0.0062t
82.6558 yrs3λ91.0648 −7.2372t1.5401 −7.2340t
61.9918 yrs4λ9−0.0794 +1.2282t−0.7362 +0.8298t
Δδ0 II6.3868 dλP1 + λ90.0270 +0.0218t−0.0295 −0.0332t
6.3877 dDP1−0.0340 +0.0339t−0.0466 −0.0012t
247.9673 yrsλ9−8.5472 −14.4029t8.8015 +0.2951t
123.9837 yrs2λ91.8521 +0.8428t−0.7291 −6.6243t
6.3872dλP1−0.0029 −0.0166t−0.0024 −0.0025t
82.6558 yrs3λ9−0.1992 +1.3479t−0.2841 +1.3504t
61.9918 yrs4λ9−0.0147 +0.2367t−0.1359 +0.1591t
ΔWII6.3868 dλP1 + λ 9−0.0294 +0.0237t−0.0435 +0.0625t
6.3877 dDPl0.0439 −0.0003t−0.0277 +0.0157t
247.9673 yrsλ 910.4859 +17.6982t−10.7878 -0.4001t
123.9837 yrs2λ9−2.2797 −1.0270t0.8948 +8.1528t
6.3872 dλPl0.0021 +0.0212t−0.0018 -0.0181t
82.6558 yrs3λ90.2452 -1.6813t0.3534 -1.6603t
61.9918 yrs4λ9−0.0192 +0.2890t−0.1704 +0.1846t
Charon(with taking into account the perturbations from Pluto)Δα0 II6.3868 dλ91 + λ90.5835 −0.6002t−0.7471 +0.1377t
6.3877 dD910.0739 +0.0653t0.1835 −0.0536t
6.3872 dλ91−0.0012 +0.0851t0.0094 0.0596t
247.9673 yrsλ945.6369 +75.9256t−47.2173 −1.4465t
123.9837 yrs2λ9−9.9400 −4.7545t3.8009 +34.5981t
82.6558 yrs3λ91.1150 −6.5950t1.5269 −7.1762t
61.9918 yrs4λ9−0.0698 +1.0883t−0.7105 +1.0422t
Δδ0 II6.3868dλ91 + λ90.2579 +0.2086t−0.2816 −0.3171t
6.3877dD91−0.3251 +0.3235t−0.4451 −0.0118t
6.3872dλ91−0.0281 −0.1590t−0.0231 −0.2377t
247.9673 yrsλ9−8.4887 −13.8767t8.7793 +0.0823t
123.9837 yrs2λ91.8674 +1.0673t−0.7105 −6.3992t
61.9918 yrs4λ90.0183 −0.1776t0.1378 −0.1475t
82.6558 yrs3λ9−0.1930 +1.2817t−0.2719 +1.3966t
ΔWII6.3868dλ91 + λ90.2805 +0.2267t−0.4154 −0.5972t
6.3877 dD91−0.4190 −0.0028t−0.2645 −0.1498t
6.3872 dλ91−0.0203 −0.2027t−0.0171 −0.1727t
247.9673 yrsλ910.3847 +16.6751t−10.7867 −0.2098t
123.9837 yrs2λ9−2.2978 −1.2805t0.8435 +7.5052t
82.6558 yrs3λ90.2783 −1.2621t0.3346 −1.6827t
61.9918 yrs4λ9−0.0156 +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)

BodyAnglePeriodArgumentCoefficient of cos (Argument) (μas)Coefficient of sin (Argument) (μas)
Pallas (2)e = 0.230i = 34.85Δα0 II4.6133 yrsλPal−62.3217α +20.3863t488.1305 −160.8216t
Δδ0 II4.6133 yrsλPal−79.2069 +15.0390t621.2777 −121.2805t
ΔWII4.6133 yrsλPal−3.2991 −6.9178t26.3696 +52.6044t
Vesta (4)e = 0.088i = 7.14Δα0 II3.6299 yrsλVes−79.3188 −5.5469t219.4540 +15.1126t
Δδ0 II3.6299 yrsλVes−42.5756 −0.7573t117.7886 +1.9666t
ΔWII3.6299 yrsλVes146.9691 +6.2171t−406.5890 −16.7824t
Lutetia (21)e = 0.163i = 3.06Δα0 II3.8012 yrsλLut−423.8069 −9.7248t428.7706 +8.5707t
Δδ0 II3.8012 yrsλLut−122.1336 +0.3414t123.5570 −0.7030t
ΔWII3.8012 yrsλLut20.7078 −0.9744t−20.9469 +1.0484t
Europa (52)e = 0.111i = 7.48Δα0 II5.4539 yrsλEur177.5560 −25.4369t172.7614 −24.9603t
Δδ0 II5.4539 yrsλEur−1.8759 +8.4951t−1.8435 +8.3894t
ΔWII5.4539 yrsλEur−186.1907 +33.7301t−181.1749 +33.0627t
Ida (243)e = 0.043i = 1.13Δα0 II4.8428 yrsλIda−53.6870 +28.6795t−23.4218 +12.6767t
Δδ0 II4.8428 yrsλIda48.4735 −9.1614t21.1260 −4.0508t
ΔWII4.8428 yrsλIda−173.5102 +54.8385t−75.6477 +24.2165t
Eros (433)e = 0.223i = 10.83Δα0 II1.7609 yrsλEro1180.4497 −13.8623t742.2291 −9.4965t
Δδ0 II1.7609 yrsλEro292.3515 +11.1627t183.8167 +6.8238t
ΔWII1.7609 yrsλEro−726.0456 −10.5342t−456.5093 −6.1414t
Davida (511)e = 0.188i = 15.94Δα0 II5.6626 yrsλDav−67.2810 +7.5571t−428.2301 +58.3470t
Δδ0 II5.6626 yrsλDav−35.8472 +6.1664t−228.2203 +44.2293t
ΔWII5.6626 yrsλDav35.8691 −12.6102t228.2665 −85.8956t
Gaspra (951)e = 0.174i = 4.11Δα0 II3.2853 yrsλGas673.5156 −12.0004t−75.4762 +1.3514t
Δδ0 II3.2853 yrsλGas275.1211 +7.9210t−30.8416 −0.8844t
ΔWII3.2853 yrsλGas−608.4609 +5.2720t68.1928 −0.5941t
Steins (2867)e = 0.146i = 9.93Δα0 II3.6421 yrsλSte6.5886 −26.7418t−7.6696 +13.8217t
Δδ0 II3.6421 yrsλSte5.8465 −236.2914t−53.4095 +36.2043t
ΔWII3.6421 yrsλSte31.9757 −1581.7149t−354.9374 +230.5293t
DOI: https://doi.org/10.2478/arsa-2022-0008 | Journal eISSN: 2083-6104 | Journal ISSN: 1509-3859
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Language: English
Page range: 158 - 184
Submitted on: Nov 26, 2021
Accepted on: Sep 28, 2022
Published on: Oct 21, 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 rotation,
Solar System bodies,
the rotation of the dwarf planets and asteroids,
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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