Relativistic Effects in the Rotation of Dwarf Planets and Asteroids

The effect of the geodetic rotation is the most significant relativistic effect in the rotation of the celestial bodies. This effect includes two relativistic effects: geodetic precession (De Sitter, 1916) and geodetic nutation (Fukushima, 1991), which are secular and periodical changes in the direction of the axis of rotation of a celestial body as a result of the parallel transfer of the angular momentum vector of the body along its orbit in curved space–time, respectively.

This article is a continuation of our previous investigations (Eroshkin and Pashkevich, 2007, 2009), (Pashkevich, 2016), (Pashkevich and Vershkov, 2019, 2020) of the geodetic rotation for the Solar System bodies. As a result, 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. 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. Thus, a more detailed study of the relativistic effects in the rotation of the bodies in the Solar System becomes relevant and interesting.

The main purpose of this investigation is to study the relativistic effect of the geodetic rotation for dwarf planets and asteroids (Appendix A: Table 1) in the Solar System with known values of their rotation parameters. Applying the method of studying the geodetic rotation of any bodies in the Solar System with long-term ephemeris (Pashkevich, 2016), for the first time, the secular and periodic terms of the geodetic rotation of these celestial bodies in ecliptic Euler angles relative to their proper coordinate systems and in their rotational elements relative to the fixed equator of the Earth and the vernal equinox (at the epoch J2000.0) are calculated.

This research is devoted to the study of the most significant relativistic effect in the rotational motion of dwarf planets (Ceres, Pluto with its satellite Charon) and asteroids (Pallas, Vesta, Lutetia, Europa, Ida, Eros, Davida, Gaspra, Steins, and Itokawa) in the Solar System with known values of their rotation parameters (Archinal et al., 2018). It is a geodetic rotation effect, which includes two relativistic effects: geodetic precession and geodetic nutation. The first of these is a systematic or secular effect, and the second is a periodic effect. The problem of geodetic (relativistic) rotation of the investigated bodies was studied with respect to their proper coordinate system (Archinal et al., 2018).¹

It is well known that the angular velocity vector of the geodetic rotation of the body under study, which is the most essential relativistic component of the body rotational motion around the proper center of mass, is defined by the following expression (Eroshkin, 2005): 1 σ¯=1c2∑jGmj| R¯−R¯j |3(R¯−R¯j)×(32R¯˙−2R¯˙j),

Here c is the velocity of light; G is the gravitational constant; m_j is the mass of a perturbing body j; R¯ and R¯˙ are the vectors of the barycentric position and velocity of the investigated body, respectively; R¯j and R¯˙j are the vectors of the barycentric position and velocity of the perturbing bodies j, respectively. The symbol × means a vector cross product; the subscript j correspond to the perturbing bodies (the Moon, the planets, dwarf planet Pluto, Charon,² and the Sun, excluding the body under study from this set). The relativistic angular velocity vector for any Solar System bodies is calculated as follows: 2 ω¯R=ω¯+σ¯.

Here ω¯ is Newtonian angular velocity vector for any Solar System bodies.

Calculations of the geodetic rotation of each body under study were carried out using data on the positions, velocities, and orbital elements of the bodies of the Solar System from ephemeris at all time intervals of their existence (Appendix A: Table 1). JPL DE431/ LE431 fundamental ephemeris (Folkner et al., 2014) were used for the Sun, the Moon, and the planets of the Solar System. For other investigated bodies of the Solar System with known rotation parameters (Archinal et al., 2018), the ephemeris data samples were formed from the Horizons On-Line Ephemeris System (Giorgini et al., 2001). The rotation parameters for dwarf planets (Ceres, Pluto with its satellite Charon) and asteroids (Appendix A: Table 1) in the Solar System were taken from the article by Archinal et al. (2018). Table 1 (see Appendix A) provides information for each body under study about their rotation and orbital periods and about the step and time interval of the studies.

In this study, the method for studying the geodetic rotation of any bodies in the Solar System (Pashkevich, 2016) using long-time ephemeris was applied. The calculation of the geodetic rotations of dwarf planets and asteroids in the Solar System with known values of their rotation parameters were carried out with respect to the proper coordinate systems of these celestial bodies under study (Archinal et al., 2018), whose origin coincides with their centres of mass:

for the parameters of their orientation (α₀, δ₀, W) relative to the stationary equator of the Earth of the epoch J2000.0 (ICRF) (Ma et al., 1998) and the vernal equinox of the epoch J2000.0;

Figure 1.

The expressions of the geodetic rotation velocities of bodies in the Solar System are defined in ψ, θ, φ angles (left, 1) (Pashkevich, 2016) and in α₀, δ₀, W angles (right, 1) (Pashkevich and Vershkov, 2020) as follows: 3 Δψ˙=−σ1sinφ+σ2cosφsinθΔθ˙=−σ1cosφ+σ2sinφΔφ˙=σ3−Δψ˙cosθ },Δα˙0=σ1sinW+σ2cosWcosδ0Δδ˙0=−σ1cosW+σ2sinWΔW˙=σ3−Δα˙0sinδ0 }.

Here ψ is the longitude of the descending node of epoch J2000 of the body equator; θ is the inclination of the body equator to the fixed ecliptic J2000; φ is the proper rotation angle of the body between the descending node of epoch J2000 and point B, where the prime meridian crosses the equator of the body (see Figure 1) (or for the case when the equator of the body figure coincides with the equator of the body rotation: between the descending node of epoch J2000 and the principal axis of the minimum moment of inertia); α₀ is the right ascension of the north pole of rotation of the body; δ₀ is the declination of the north pole of rotation of the body; angle W = QB specifies the location of the prime meridian of the body,³ which is measured along the equator of the body in easterly direction with respect to the north pole of body from the node Q (located at right ascension 90° + α₀) of the equator of the body on the standard Earth equator of epoch J2000 to the point B, where the prime meridian crosses the equator of the body (see Figure 1) (recommended values of the constants in the expressions for α₀, δ₀ and W are given by Archinal et al. (2018)); Δψ˙=ψ˙r−ψ˙ , Δθ˙=θ˙r−θ˙ , Δφ˙=φ˙r−φ˙ , Δα˙0=α˙0r−α˙0 , Δδ˙0=δ˙0r−δ˙0 , ΔW˙=W˙r−W˙ are the differences of the relativistic and Newtonian angles of rotation of the investigated body, respectively; the dot denotes differentiation with respect to time; σ₁, σ₂, σ₃ are projections of the angular velocity vector of the geodetic rotation of the body under study σ¯ on the coordinate axes in its own body-centric reference frame (Figure 1a), given by Archinal et al. (2018).

Figure 1a.

The geodetic rotation velocities for each of the investigated Solar System bodies are determined over various time spans with different time spacing (Appendix A: Table 1) by the least-squares method and spectral analysis (Pashkevich, 2016). The expressions for the secular and periodic terms of the velocities of the geodetic rotation of the body can be represented in the following form: 4 Δx˙=∑n=1NΔx˙ntn−1+∑t∑k=0M(Δx˙Clkcos(vl0+vl1t)+Δx˙Slksin(vl0+vl1t))tk, where Δx˙n are the coefficients of the secular terms; Δx˙Slk , Δx˙Clk are the coefficients of the periodic terms; x˙=ψ˙,θ˙,φ˙,α˙0,δ˙0,W˙ ; ν_l0, ν_l1 are phases and frequencies of the body under study, which are combinations of the corresponding Delaunay arguments (Smart, 1953) and the mean longitudes of the perturbing bodies; the summation index l is the number of added periodic terms and its value changes for each body under study; t is the time from standard epoch JD 2451545.0, i.e. January 1, 2000 12 hours TDB (Archinal et al., 2018), in Appendix A: Table 1 the time spans and steps for the studies of the geodetic rotation of the bodies are presented; N and M are approximation parameters.

As a result of calculations by the least squares method, the values of the approximation parameters for providing the best approximation of the geodetic rotation were obtained: N = 2 and M = 1.

After analytical integration (4) Δx=∫Δx˙dt , the expressions for the secular Δx₁ and periodic Δx₁₁ terms of the body geodetic rotation are obtained: 5 Δx=ΔxI+ΔxII=∑n=1NΔxntn+∑l∑k=0M(ΔxClkcos(vl0+Δvl1t)+ΔxSlksin(vl0+vl1t))tk, where x = ψ, θ, φ, α₀, δ₀, W; Δxn=Δx˙nn are the coefficients of the secular terms, and the coefficients for sine Δx_Slk and cosine Δx_Clk periodic terms are calculated using the “Cascade” method (Pashkevich, 2016).

The absolute value of the angular velocity vector of the geodetic rotation of the body under study is presented by the following expression: 6 | σ¯|=σX2+σY2+σZ2=σ12+σ22+σ32.

Here σ_X, σ_Y, σ_Z are the components of the geocentric vector of the angular velocity of the geodetic rotation of a body (Pashkevich, 2016); as defined above, (σ₁, σ₂, σ₂ are reduced (Pashkevich, 2016) components of the body-centric vector of the angular velocity of the geodetic rotation of a body. Equation (6) is true, because the magnitude of the vector does not depend on the coordinate system in which its projections are considered.

The absolute value of the geodetic rotation vector for the Solar System bodies in the parameters of their orientation is presented by the following expression: 7 ΔΩ=Δσ02+Δδ02+ΔW2.

In this study, for the first time, the values of the most significant secular (Appendix A: Tables 2 and 3) and periodic (Appendix A: Tables 4 and 5) terms of geodetic rotation were calculated for dwarf planets of the Solar System with known rotation parameters (Ceres, Pluto, and Charon).

In Appendix A: Tables 2 and 4 shows the angles of rotation (α₀, δ₀, W) of dwarf planets in the Solar System (Archinal et al., 2018) and the most significant secular (Δα_0I, Δδ_0I, ΔW₁) and periodic (Δα_0II, Δδ_0II, ΔW_II) terms of their geodetic rotation (Figures 2 and 3, right side) calculated in this study; Tables 3 and 5 shows the most significant secular (Δψ_I, Δθ_I, Δφ_I) and periodic (Δψ_II, Δθ_II, Δφ_II) terms of geodetic rotation of the dwarf planets in the Solar System for ecliptic Euler angles (Figures 2 and 3, left side) relative to their proper coordinate systems calculated in this study.

Figure 2.

Figure 3.

As is known from previous articles by the authors Pashkevich (2016) and Pashkevich and Vershkov (2019), the geodetic rotation of the Sun and the major planets of the Solar System in ecliptic Euler angles (Figure 2, left side) have been ranging from −870.02 μas per thousand years to −426″.45 per thousand years. The value of the geodetic precession of Ceres (−3″.36 per thousand years) obtained in this study is between the values of the geodetic precessions of Mars (−7″.11 per thousand years) and Jupiter (−0″.21 per thousand years) (Pashkevich and Vershkov, 2019), which is in good agreement with the location of its orbit in the asteroid belt between the orbits of these planets.

Ceres geodetic nutation value (periodic term with the period 4.6049 years) is calculated in ecliptic Euler angles (Appendix A: Table 5) and in their angles of rotation (Appendix A: Table 4).

The obtained values of the geodetic rotations of Pluto and Charon without mutual influence (that is, with the influence only from the Sun, Moon, and planets) are quite close (Figure 3, Pluto–Charon) (Figure 3a, Pluto, Charon) (Appendix A: Tables 2, 3, 4, and 5). In this case, the difference between Pluto and Charon values of the geodetic precessions⁴ Δψ₁ is 0.0003 μas per thousand years (Appendix A: Table 3). This is due to the same distance between Pluto and Charon from the Sun, which in this case has the greatest effect on the dwarf planets under study. In this case, the negative value of the geodetic precession corresponds to their prograde rotation along the heliocentric orbit.

Figure 3a.

The values of geodetic precessions for Pluto and Charon, obtained taking into account the mutual influence of Pluto and Charon on each other, differ significantly from each other (Figure 3, Pluto, Charon) (Appendix A: Tables 2 and 3). This is explained by the fact that Pluto is more massive than Charon (Charon’s mass is ~ 0.1 Pluto’s mass). This is clearly seen from the values of the periodic harmonics of the geodetic nutation of these bodies (Appendix A: Tables 4 and 5) with the periods 6.3868 days, 6.3877 days, and 6.3872 days, which determine the periodic mutual influences of these celestial bodies on each other. So, the harmonic values (with the periods 6.3868 days, 6.3877 days, and 6.3872 days) contained in Pluto’s geodetic nutation are 10 times less than the corresponding harmonic values contained in Charon’s geodetic nutation (Figure 3a, Pluto+, Charon+). Thus, Pluto’s influence on Charon is significantly greater than Charon’s influence on Pluto.

The positive value of the geodetic precession corresponds to their retrograde orbital rotation around the common barycenter of the Pluto–Charon system. In absolute value, the geodetic precession of Charon (24690.9 μas per thousand years) (Appendix A: Table 3) exceeds similar values of the geodetic precessions for Pluto (1328.9 μas per thousand years) (Appendix A: Table 3), Neptune (−3903.9 μas per thousand years), and Uranus (–11924.6 μas per thousand years) (Pashkevich and Vershkov, 2019).

The velocities of the geodetic rotation for Pluto and Charon without their mutual influence on each other (Figure 3a, Pluto, Charon) are 30 and 300 times less than corresponding velocities for Pluto and Charon with taking into account their mutual influence on each other (Figure 3a, Pluto+, Charon+) in ecliptic Euler angles. Consequently, in this case, the influence of Pluto and Charon on each other’s geodetic rotation turned out to be greater than the influence of the Sun on their geodetic rotation.

Figure 4.

In this study, for the first time, the values of the most significant secular (Appendix A: Tables 2a and 3) and periodic (Appendix A: Tables 4 and 5) terms of geodetic rotation were calculated for asteroids (Pallas, Vesta, Lutetia, Europa, Ida, Eros, Davida, Gaspra, Steins, and Itokawa) of the Solar System with known rotation parameters (Figures 4 and 5).⁵

Figure 5.

In Appendix A: Tables 2a and 4 show the angles of rotation (α₀, δ₀, W) of asteroids in the Solar System (Archinal et al., 2018) and the most significant secular (Δα_0I, Δδ_0I, ΔW_I) and periodic (Δα_0II, Δδ_0II, ΔW_II) terms of their geodetic rotation (Figure 4, right side) calculated in this study; Tables 3 and 5 show the most significant secular (Δψ_I, ΔΘ_I Δφ_I) and periodic (Δψ_II, Δθ_II, Δφ_II) terms of geodetic rotation of the asteroids in the Solar System for ecliptic Euler angles (Figure 4, left side) relative to their proper coordinate systems calculated in this study.

The value of the geodetic precession of Itokawa asteroid obtained in this study is −30″.89 per thousand years (Appendix A: Table 3); thus, it is 1.6 times higher than similar values (Pashkevich et al., 2019) for the Earth (−19″.19 per thousand years) and the Moon (−19″.49 per thousand years). This is due to the large elongation of the heliocentric orbit of Itokawa (Figure 5). Indeed, (despite the fact that the value of the semi-major axis of the heliocentric orbit of Itokawa is greater than the corresponding values for the Earth and the Moon, and the aphelion of its orbit is located farther from the Sun than the aphelion of the orbit of Mars) due to the large eccentricity (e = 0.280) (Appendix A: Table 3), the perihelion of its orbit is closer to the Sun than the perihelion of the heliocentric orbits of the Earth and the Moon. Therefore, the Sun has a greater influence on this asteroid than on the Earth and the Moon. The value of the geodetic nutation for Itokawa (Appendix A: Table 5) is the largest among the studied asteroids. This is due to the large lengthening of its orbit (Figure 5) and the influence on it from the planets Mars and the Earth.

The value of the geodetic precession of Eros (−7″.54 per thousand years) calculated in this research is between the values of the geodetic precessions of the Earth (−19″.19 per thousand years), the Moon (−19″.49 per thousand years), and Mars (−7″.11 per thousand years) (Pashkevich et al., 2019), which is in good agreement with the location of its orbit in the space between the orbits of these planets (Figure 5).

The orbit of the asteroid Gaspra is between the orbits of Mars and Vesta. The value of its geodetic precession calculated in this study is –2″.64 per thousand years.

The asteroid Vesta has a geodetic precession in absolute value slightly less than that of Gaspra and amounts to −2″.56 per thousand years.

The value of the geodetic precession of the next Steins asteroid is positive and amounts to 0”.10 per thousand years.

Thus, for the asteroids Vesta and Steins, which have similar semimajor axes of their orbits, the values of their geodetic precession Δψ₁ (Appendix A: Table 3) differ significantly from each other and have the opposite sign. If you look at the orientations of their rotation axes, you can see significant differences: the ecliptic coordinates of the North Pole of Vesta (λ is the longitude, β is the latitude) λ = 330°.83, β = 57°.73 (Russell et al., 2013), and the North Pole of Steins has λ = 250°.0, β = −89°.0 (Lamy et. al., 2008). Since the ecliptic latitude of the North Pole of Steins is negative, and its rotation around its axis is positive (Appendix A: Table 2a) (Archinal et al., 2018), this means that, unlike Vesta, it has a reverse rotation relative to the North Pole of the ecliptic. This circumstance explains the positive value of the Steins geodetic precession.

Despite the fact that Vesta is the first in mass among asteroids, its relativistic influence on the asteroid Steins, which has a relatively close orbit with it (Appendix A: Table 3), is not too great. Thus, the change in the value of Steins’ geodetic rotation from the relativistic influence of Vesta is −1.6 × 10^-4 μas per thousand years in the longitude of the node, −1.6 × 10^-5 μas per thousand years in the inclination, and 8.0 × 10^-5 μas per thousand years in the proper rotation angle.

The next farthest from the Sun is the investigated asteroid Lutetia. The value of its geodetic precession calculated in this study is −2″.12 per thousand years.

Further, there are two massive bodies of Ceres and Pallas, their mutual relativistic influence on each other is also small. So, the change in the geodetic rotation of Ceres from the relativistic influence of Pallas is −7.5 × 10^-4 μas per thousand years in the longitude of the node, −3.8 × 10^-5 μas per thousand years in the inclination, and 6.9 × 10^-4 μas per thousand years in the proper rotation angle; and the change in the geodetic rotation of Pallas due to Ceres relativistic influence is equal −5.4 × 10^-4 μas per thousand years in the longitude of the node, −6.0 × 10^-5 μas per thousand years in the inclination, and 6.0 × 10^-4 μas per thousand years in the proper rotation angle.

According to the results of this study, the values of the geodetic precession for the asteroids Pallas, Ida, Europa, and Davida are −1″.40 per thousand years, −1″.33 per thousand years, − 1”.06 per thousand years, and −1”.09 for a thousand years, respectively.

In the present research, the most significant relativistic effect of geodetic rotation in the rotation motions for dwarf planets (Ceres, Pluto, and Charon) and asteroids (Pallas, Vesta, Lutetia, Europa, Ida, Eros, Davida, Gaspra, Steins, and Itokawa) in the Solar System with known quantities of their rotation parameters was investigated. For the first time, the secular (geodetic precession) and periodic (geodetic nutation) terms of the geodetic rotation of these celestial bodies were calculated in ecliptic Euler angles relative to their proper coordinate systems and in their rotational elements relative to the stationary equator of the Earth and the vernal equinox (for the epoch J2000.0). The mean longitudes for Ceres and for asteroids were computed in this investigation using an algorithm elaborated by Pashkevich (see Appendix B) based on the spectral analysis method (Jenkins, Watts, 1969).

We have shown that the calculated values of geodetic precession are influenced not only by the orbital parameters of the body under study, the distance of these bodies from the disturbing bodies and the values of the masses of the disturbing bodies, but also the spatial nature of their proper rotation, determined by the orientation of the North Pole of the body under study relative to the reference plane (in this study, this plane is the ecliptic of the J2000.0 epoch). As a result, it becomes clear whether the proper rotation is prograde or retrograde relative to the orbital motion, and, accordingly, whether the value of the geodetic precession is negative or positive, and explains some variations in the geodetic precession values of the studied celestial bodies close in position to their orbits.

Pluto’s geodetic nutation value with taking into account the perturbations from Charon is 10 times less than Charon’s geodetic nutation value with taking into account the perturbations from Pluto. Thus, Pluto’s influence on Charon is significantly greater than Charon’s influence on Pluto.

The influence of Pluto and Charon on the geodetic rotation of each other turned out to be greater than the influence of the Sun.

The obtained analytical values for the geodetic rotation of the studied celestial bodies can be used for the numerical study of their rotation in the relativistic approximation, and also used to estimate the influence of relativistic effects on the orbital–rotational dynamics of exoplanetary systems bodies.⁶