The relativistic effect of the geodetic precession, first considered by Willem de Sitter in 1916 (De Sitter, 1916), is a secular change in the direction of the axis of rotation of a celestial body as a result of a parallel transfer of the angular momentum vector of the body along its orbit in curved space–time.
In our previous investigation (Pashkevich and Vershkov, 2020), the main effects of the relativistic rotation for the inner satellites of Jupiter (J14, J5, J15, and J16) were studied. As results, foregoing study showed that the values of the geodetic precession can be significant not only for objects orbiting around super-massive central relativistic bodies, but also for bodies with a short distance to the central body, for example, close satellites of giant planets.
Contemporary development of cosmonautics, especially in the implementation of projects such as “Gravity Probe B” (Everitt et al. 2011) for giant planets, will make it possible, after a few years, to obtain from observations the magnitude of the geodetic precession of their inner satellites. For example, it is necessary to launch a similar artificial satellite into the orbit of one of these satellites. The range of the obtained (Pashkevich and Vershkov, 2020) theoretical values of the geodetic precession of the inner satellites of Jupiter varies from −13″.37255 per year to −52″.95725 per year. Thus, theoretically, within 1 year after the implementation of such a project, it will allow testing the general theory of relativity. Then, the artificial satellite can be transferred to the orbit of the next inner satellite of Jupiter and the experiment is repeated.
Thus, a more detailed study of the relativistic effects in the rotation of the planets and their satellites in the Solar System becomes relevant and interesting.
Modeling the orbital–rotational dynamics of bodies of exoplanetary systems is best done using a planetary system with well-known parameters of the motion of its bodies. New research methods have made it possible to obtain high-precision long-term ephemerides of the orbital–rotational motion of many bodies in the Solar System, thus the Solar System is a good model for studying the rotational dynamics of exoplanetary systems. Based on studies of the Solar System bodies with well-known parameters of motion, it is possible to reveal patterns in the distribution and influence of relativistic effects on the orbital–rotational dynamics of exoplanetary systems’ bodies.
In this article, the geodetic precession values for Mars satellites in Euler angles are taken from our previous study (Pashkevich and Vershkov, 2019), and the geodetic precession values for the inner satellites of Jupiter are taken from our previous study (Pashkevich and Vershkov, 2020).
The main aims of this research are:
to improve the geodetic precession values for the Sun, and the Solar System planets in the Euler angles relative to their proper coordinate systems and in the absolute value of the geodetic rotation angular velocity vector; to improve the geodetic precession values for the Moon in the perturbing terms of the physical libration relative to her proper coordinate systems and in the absolute value of the geodetic rotation angular velocity vector; to calculate, for the first time, the values of the geodetic precession for the Sun, all the Solar System planets, the Moon, and Mars satellites in their rotational elements; to obtain new additional and corrected values of the relativistic influence of Martian satellites (M1 and M2) on Mars; to calculate, for the first time, the values of the geodetic precession for the other planetary satellites with known quantities of the rotational elements (Galilean moons of Jupiter: J1–J4, satellites of Saturn: S1–S6, S8–S18, satellites of Uranus: U1–U15, and satellites of Neptune: N1, N3–N8) in the Euler angles relative to their proper coordinate systems and in their rotational elements.
The structure of this article is as follows:
In this investigation, we studied the most significant relativistic effect in rotational motion for the Sun, all planets and their satellites in the Solar System with known rotation parameters (Archinal et al. 2011, 2018). This effect is geodetic precession, which is the systematic or secular effect of the studied body’s geodetic rotation.
The method for studying the geodetic rotation of any Solar System bodies using long-time ephemeris will be applied (Pashkevich, 2016):
The problem of the geodetic (relativistic) rotation of the Solar System bodies is studied with respect to the proper coordinate systems of the bodies. The values of the velocities of the geodetic rotation of the Solar System bodies are determined by using the ephemeris. The most essential terms of the geodetic precession are found by means of the least-squares method.
The expressions for the velocities of the geodetic rotation of bodies in the Solar System for the parameters of their orientation (
Triangle used to define the direction of the angular velocity vector of the geodetic rotation for any body of the Solar System
Here
As can be seen from equation (2), the geodetic rotation of a body under study depends only on the masses of the disturbing bodies and on the distance to them and does not depend on the mass of the body itself. Therefore, the magnitude of the vector of the geodetic rotation of the body under study
In order to eliminate the singularity cos−1
In this investigation, the expressions (3) are used to study the geodetic rotation of the Earth, for which cos
The expressions of the geodetic rotation velocities are defined in the perturbing terms of the physical librations (
Here,
After analytical integration (6), the expressions for the secular terms of the body’s geodetic rotation are obtained:
The absolute value of the angular velocity vector of the geodetic rotation of the body under study is presented by the following expression:
Here,
The absolute value of the geodetic rotation velocity vector for the Solar System bodies in the parameters of their orientation is presented by the following expression:
In our previous investigation (Eroshkin, and Pashkevich, 2007), the absolute geodetic precession magnitudes of the angular velocity vector
In the research of Klioner et al. (2009), the magnitude of the geodetic precession was obtained only for some Solar System bodies (Mercury, Venus, the Earth, the Moon and Mars). Comparison of the magnitude of the geodetic precession of our studies with those of Klioner et al. (2009) showed that the results obtained for the same bodies in our research have the same order of magnitude (Appendix: Table 2).
In our previous study (Pashkevich and Vershkov, 2019), the geodetic precession values in Euler angles for the Sun, the Moon, and the Solar System planets were calculated by using values of the rotation elements (Seidelmann et al. 2005). In this investigation, the geodetic precession values in Euler angles for these bodies have been improved (Appendix: Table 2a) by using updated values of the rotation elements (Archinal et al. 2011, 2018).
Geodetic precession of the Sun and the planets of the Solar System in Euler angles (Figure 2, left side) ranges from −870.28 μas per thousand years (for the Sun) to –425″.61 per thousand years (for Mercury) (Appendix: Table 2a).
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) (
The values of geodetic precession for the Earth and the Moon are very close (Appendix: Table 2a). Therefore, in Figure 2, the point for the Moon overlaps with the point for the Earth (point of the red color). This is due to the large distance of the Moon from the Earth; as a result, the Sun has a greater influence on the Earth (comparable to its influence on the Moon) than the Moon. Thus, if we exclude the influence of the Moon on the Earth, then the value of its geodetic precession will be the same as for the Earth with the influence of the Moon – 19″.19 per thousand years (Appendix: Table 2a). It is also possible to evaluate the quantity of the influence of the Moon on the geodetic rotation of the Earth; if we exclude the influence of the Sun on the Earth, then the value of its geodetic precession will be −0″.005 per thousand years (Appendix: Table 2a).
In this study (subsection 4.1), the quantity of the inverse influence of the Earth on the geodetic rotation of the Moon, which is −0″.30 per thousand years, was also calculated (Appendix: Table 3).
As a result of this study, the secular terms of the geodetic rotation of the planets and the Sun (Figure 2, right side) in the elements of their rotation were calculated for the first time (Appendix: Table 2b).
Appendix: Table 2, 2a, 2b and Figure 2 show that all planets of the Solar System are characterized by a decrease in their absolute value of the geodetic rotation with an increase in their distance from the central body, which confirms the main property of equation (2) for the longitude of the descending node and for the absolute value of the vector of the geodetic rotation of the parameters of their orientation.
The study (Pashkevich and Vershkov, 2020) of the rotational dynamics of the inner satellites of Jupiter (Metis (J16), Adrastea (J15), Amalthea (J5), and Thebe (J14)) showed that the quantity of the relativistic geodetic rotation can be significant not only for relativistic objects, but, under certain conditions, also for ordinary satellites of planets, such as close satellites of giant planets. For all satellites of the planets of the Solar System (except for the satellites listed above, whose geodetic precession values were obtained in our previous studies), the secular terms of their geodetic rotation have been determined here for the first time.
In this research, the values of the secular terms of the geodetic rotation for the satellites of planets were first determined in the Euler angles relative to their proper coordinate systems (Appendix: Table 3 and Figures 3–7, left side) and in their rotational elements (Appendix: Table 3a and Figures 3–7, right side).
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) (
Geodetic precession velocity of the satellites of Jupiter in the longitude of the descending node (left side) and in the absolute value of the velocity vector of the geodetic rotation of the parameters of their orientation (right side) (a is the length of the satellite orbit’s semi-major axis)
Geodetic precession velocity of the satellites of Saturn in the longitude of the descending node (left side) and in the absolute value of the velocity vector of the geodetic rotation of the parameters of their orientation (right side) (a is the length of the satellite orbit’s semi-major axis)
Geodetic precession velocity of the satellites of Uranus in the longitude of the descending node (left side) and in the absolute value of the velocity vector of the geodetic rotation of the parameters of their orientation (right side) (a is the length of the satellite orbit’s semi-major axis)
Geodetic precession velocity of the satellites of Neptune in the longitude of the descending node (left side) and in the absolute value of the velocity vector of the geodetic rotation of the parameters of their orientation (right side) (a is the length of the satellite orbit’s semi-major axis)
Geodetic precession of the Moon is −19″.49 per thousand years (Appendix: Tables 2a, 3).
The value of the geodetic precession of the Moon (E1) is close to the value of the geodetic precession of the Earth (Figure 2, left side) (−19″.19 per thousand years; Appendix: Table 2a). This is due to the large distance of the Moon from the Earth; as a result, the Sun has a greater influence on the Moon (comparable to its influence on the Earth) than the Earth. Thus, if we exclude the influence of the Earth on the Moon, then the value of its geodetic precession will be the same as for the Earth, that is, −19″.19 per thousand years (Appendix: Table 3). It is also possible to evaluate the quantity of the influence of the Earth on the geodetic rotation of the Moon; if we exclude the influence of the Sun on the Moon, then the value of its geodetic precession will be −0″.30 per thousand years (Appendix: Table 3).
In this study (Section 3), the quantity of the inverse influence of the Moon on the geodetic rotation of the Earth was also calculated, which was −0″.005 per thousand years (Appendix: Table 2a).
Geodetic precession of the satellites of Mars (Pashkevich and Vershkov, 2019) ranges from −27″.68 per thousand years (for Deimos) to −209″.31 per thousand years (for Phobos) (Figure 3, left side and Appendix: Table 3).
In the Mars satellite system, the values Δ
Here, for completeness, we present the values obtained in our previous studies and in this investigation. We also investigated the mutual relativistic influence of Martian satellites on each other (Pashkevich and Vershkov, 2019) and on Mars (in this study, we obtained additional values for the relativistic influence on Mars separately related to Phobos and separately related to Deimos, as well as new corrected values for the relativistic influence on Mars from both Martian satellites):
the change in the geodetic rotation of Deimos due to the relativistic influence of Phobos is equal to −0.22 μas per thousand years in the longitude of the node, −9.3× 10−6 μas per thousand years in the inclination, and 0.12 μas per thousand years in the proper rotation angle (Pashkevich and Vershkov, 2019); the change in the geodetic rotation of Phobos due to the relativistic influence of Deimos is equal to −5.3×10−2 μas per thousand years in the longitude of the node, 6.2×10−6 μas per thousand years in the inclination, and 2.9×10−2 μas per thousand years in the proper rotation angle (Pashkevich and Vershkov, 2019); the change in the geodetic rotation of Mars due to the relativistic influence of Phobos is equal to −4.48 μas per thousand years in the longitude of the node, −9.0×10−4 μas per thousand years in the inclination, and 2.50 μas per thousand years in the proper rotation angle; the change in the geodetic rotation of Mars due to the relativistic influence of Deimos is equal to −6.2×10−2 μas per thousand years in the longitude of the node, 3.5×10−5 μas per thousand years in the inclination, and 3.4×10−2 μas per thousand years in the proper rotation angle; and the change in the geodetic rotation of Mars due to relativistic influence of Phobos and Deimos is equal to −4.54 μas per thousand years in the longitude of the node, −8.6× 10−4 μas per thousand years in the inclination, and 2.54 μas per thousand years in the proper rotation angle.
Geodetic precession of Jupiter’s moons runs from −64″.00 per thousand years (for Callisto) to −52,957″.25 per thousand years (for Metis) (Figure 4, left side and Appendix: Table 3).
It is found that there are objects in the Solar System with significant geodetic rotation comparable to their main rotation (Archinal et al. 2018). The values of geodetic precession of the inner satellites of Jupiter (Metis (J16), Adrastea (J15), Amalthea (J5), and Thebe (J14)) (Pashkevich and Vershkov, 2020) turned out to be comparable to the values of their precession (Appendix: Table 3a, red color) (Archinal et al. 2018). These values are, on average, 105 times higher than the value of geodetic precession of Jupiter itself (Figure 2, left side and Appendix: Table 2a) and 100 times higher than the value of geodetic precession of the Mercury, which is the closest planet to the Sun in the Solar System (Figure 2, left side and Appendix: Table 2a).
The next of the studied satellites in terms of distance from Jupiter is the group of Jupiter’s Galilean moons (Io (J1), Europa (J2), Ganymede (J3), and Calisto (J4)). Studies have shown that Io (J1) and Europa (J2) have geodetic precession values that are 6 and 2 times higher, respectively, than the geodetic precession value for Mercury (Figure 2, left side and Appendix: Table 2a), and the geodetic precession value for Ganymede (J3) is 1.7 times higher than that for Venus (Figure 2, left side and Appendix: Table 2a). The value of the geodetic precession of Calisto (J4) is 3 times greater than that of the Earth (Figure 2, left side and Appendix: Table 2a).
Geodetic precession of Saturn’s moons runs from −0″.02 per thousand years (for Phoebe) to −232″.74 per thousand years (for Pan) (Figure 5, left side and Appendix: Table 3).
In the satellite system of Saturn, the geodetic precession values of the group of inner satellites closest to it (Pan (S18), Atlas (S15), Prometheus (S16), Pandora (S17), Epimetheus (S11), and Janus (S10)) exceed the geodetic precession value of Venus (Appendix: Table 2a). For the satellites of this group and Mimas (S1), the geodetic precession values turned out to be comparable to their precession values (Appendix: Table 3a in red color).
The geodetic precession values of the next largest satellite group (Mimas (S1)–Telesto (13)) exceed the geodetic precession value of the Earth (Appendix: Table 2a).
The moons of Saturn Telesto (S13) and Calypso (S14) have close orbits and synchronously rotate relative to each other. Consequently, the theoretically predicted values of their geodetic precession should be close. As follows from the main property of equation (2), this value should be slightly less than the geodetic precession of Telesto (−28″.61 per thousand years; Appendix: Table 3) and much more than that of Dione (−17″.25 per thousand years; Appendix: Table 3), between the orbits of which the orbit of Calypso is located. However, the obtained value of the geodetic precession of Calypso (S14), that is, 0″.28 per thousand years (Figure 5, left side), is two orders of magnitude less in absolute value than that of Telesto (S13) and has the opposite sign compared to similar values of other satellites of Saturn (Appendix: Table 3). The discovered feature of the geodetic rotation for this satellite is probably related to the inaccuracy of its rotation parameters (Archinal et al. 2018)4 and their incompatibility with the used ephemeris (Giorgini et al. 1996).
Indeed, as the experiment showed, if we replace the polar rotation parameters of Calypso (
The values of the geodetic precessions of the moons of Saturn Dione (S4) and Helena (S12) are comparable in magnitude with the geodetic precession of the Earth (Figure 2, left side and Appendix: Table 2a) and the satellite of Rhea (S5) with the geodetic precession of Mars (Figure 2, left side and Appendix: Table 2a). For Titan (S6) and Iapetus (S8), the values of these quantities exceed those of Jupiter, and for Phoebe (S9), they are less than that of Saturn, but greater than that of Uranus (Figure 2, left side and Appendix: Table 2a).
Geodetic precession of Uranus satellites ranges from 1″.57 per thousand years (for Oberon) to 737″.38 per thousand years (for Cordeli) (Figure 6, left side and Appendix: Table 3).
A distinctive feature in the system of Uranus satellites is the positive value of geodetic precession of all satellites under study (Ariel (U1), Umbriel (U2), Titania (U3), Oberon (U4), Miranda (U5), Cordelia (U6), Ophelia (U7), Bianca (U8), Cressida (U9), Desdemona (U10), Juliet (U11), Portia (U12), Rosalind (U13), Belinda (U14) and Puck (U15)) (Appendix: Table 3). This feature is due to their reverse rotation.
Geodetic precession of Neptune’s moons runs from 43″.45 per thousand years (for Triton) to −6670″.30 per thousand years (for Naiad) (Figure 7, left side and Appendix: Table 3).
In the system of the investigated satellites of Neptune (Triton (N1), Naiad (N3), Thalassa (N4), Despina (N5), Galatea (N6), Larisa (N7), Proteus (N8)), Triton is the most interesting. This satellite, like the satellites of Uranus, has a positive value of the geodetic precession and reverse rotation (Appendix: Table 3). The value of the geodetic precession of other satellites of Neptune turned out to be on average an order of magnitude higher than the value of the geodetic precession of Mercury, which is the closest planet to the Sun in the Solar System (Figure 2, left side and Appendix: Table 2a). This is due to the greater influence of Neptune on them as the central body than the influence on Mercury from the more massive central body of the Sun.
The obtained analytical expressions for the parameters of the geodetic rotation of all satellites of the planets of the Solar System can be used to numerically study their rotation in the relativistic approximation.
Appendix: Tables 2–3a and Figures 3–7, right side show that the absolute value of the geodetic rotation of the central body is always less than that of its satellites.
The theoretical investigations of the relativistic effects in the rotational motions for the Sun, all planets of the Solar System, and their satellites with known quantities of their rotational elements (E1, M1, M2, J1–J5, J14–J16, S1–S6, S8–S18, U1–U15, N1, N3–N8) were carried out.
As a result, the most significant secular terms of the geodetic rotation have been improved
for the Sun and the Solar System planets, in the Euler angles relative to their proper coordinate systems and in the absolute value of the geodetic rotation angular velocity vector and for the Moon (E1) in the perturbing terms of the physical libration relative to her proper coordinate systems and in the absolute value of the geodetic rotation angular velocity vector.
The values of the geodetic precession were first calculated
for the Sun, all the Solar System planets, the Moon (E1), and satellites of Mars (M1, M2) in their rotational elements and for Galilean moons of Jupiter (J1–J4), satellites of Saturn (S1–S6, S8–S18), satellites of Uranus (U1–U15), and satellites of Neptune (N1, N3–N8), in the Euler angles relative to their proper coordinate systems and in their rotational elements.
The values of geodetic rotation were determined
for the Earth and for the Moon (E1) without taking into account the perturbations from the Sun; for the Earth without taking into account the perturbations from the Moon (E1); and for the Moon (E1) without taking into account the perturbations from the Earth.
Additional values were determined for the relativistic influence on Mars separately related to Phobos (M1) and separately related to Deimos (M2), as well as new corrected values for the relativistic influence on Mars from both Martian satellites.
The largest values of the geodetic rotation of bodies in the Solar System were found in Jovian satellites system (Appendix: Tables 2, 3, and 3a). Further, in decreasing order, these values were found in the satellite systems of Saturn, Neptune, Uranus, and Mars, for Mercury, for Venus, for the Moon, for the Earth, for Mars, for Jupiter, for Saturn, for Uranus, for Neptune, and for the Sun (Appendix: Table 2). First of all, these are the inner satellites of Jupiter: Metis (J16), Adrastea (J15), Amalthea (J5), and Thebe (J14) and satellites of Saturn: Pan (S18), Atlas (S15), Prometheus (S16), Pandora (S17), Epimetheus (S11), Janus (S10), and Mimas (S1), whose values of geodetic precession are comparable to the values of their precession (Appendix: Table 3a, red color).
Such an arrangement of the geodetic precession values of the angular velocity vector differs somewhat from the location of the geodetic precession values in the longitudes of the descending nodes of the bodies under study. Thus, the largest values of geodetic precession in longitude of the descending node (Appendix: Tables 2a and 3) were found in the satellite system of Jupiter, then, in descending order of these values, follow the satellites of the Neptune system, the satellites of the Uranus system, the planet of Mercury, the Saturn satellite system, the Mars satellite system, the planet of Venus, the Moon, and the planets of the Earth, Mars, Jupiter, Saturn, Uranus, Neptune and the Sun.
For all studied objects of the Solar System, a characteristic pattern has been revealed:
a decrease in their absolute value of the geodetic precession with an increase in their distance from the central body (it is the main property of the formula for the angular velocity vector of the geodetic rotation of a body under study, which confirms for the longitude of the descending node and for the absolute value of the vector of the geodetic rotation of the parameters of their orientation) and an absolute value of the geodetic rotation of the central body is always less than that of its satellites.
The obtained analytical values for the geodetic precession for the Sun, all the Solar System planets and their satellites can be used to numerically study their rotation in the relativistic approximation and as an estimate of the influence of relativistic effects on the orbital–rotational dynamics of bodies of exoplanetary systems.
The results of this study can also be used to test the general theory of relativity in the implementation of space projects like “Gravity Probe B” (Everitt et al. 2011).
In the future, it is planned to expand our studies of the relativistic effect of geodetic rotation for other bodies of the Solar System (dwarf planets and asteroids). Also, our studies will be expanded for all investigated bodies of the Solar System to obtain the values of the most significant periodic terms of their geodetic nutation.