A new post-Newtonian long-term precession model for the Earth
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MNRAS 507, 3690–3697 (2021) https://doi.org/10.1093/mnras/stab2396
A new post-Newtonian long-term precession model for the Earth
K. Tang ,1‹ M. Soffel,2‹ J. H. Tao3 and Z. H. Tang3,4
1 Key Laboratory of Planetary Sciences, Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai 200030, China
2 Lohrmann Observatory, Dresden Technical University, Dresden 01062, Germany
3 Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai 200030, China
4 School of Astronomy and Space Science, University of Chinese Academy of Sciences, Beijing 100049, China
Accepted 2021 August 9. Received 2021 August 7; in original form 2021 April 9
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ABSTRACT
Long-term precession represents the secular motion of the Earth’s axis for a long time interval. In 2015, we calculated the
Earth’s long-term precession in a relativistic framework. However, our previous works involving the ecliptic in the Geocentric
Celestial Reference System are deficient, because the natural definition of this ecliptic is still ambiguous. To obtain a long-term
precession model in accordance with general relativity, improvements are made including the following: all calculations are
no longer carried out in the reference related to any ecliptic; a new hybrid integrator is designed and used for this precession
model; ecliptic-independent precession parameters are calculated and provided. A detailed comparison analysis is performed
to estimate the importance of various relativistic influences on the precession parameters. Valid within the interval of ±1 Myr
around J2000.0, a consistent post-Newtonian long-term precession model for the Earth has been achieved and is presented here.
Key words: relativistic processes – astrometry – ephemerides – reference systems.
term precession model covers periods longer than 100 centuries and
1 I N T RO D U C T I O N
could reflect the realistic secular motion of the ecliptic and the equator
The rotational dynamics of the Earth is illustrated by precession, for a longer time-span. Beyond that, there are some other numerical
nutation, polar motion, and its spin. Knowledge of these is necessary solutions for the long-term motion of the Earth. Quinn, Tremaine &
for many fundamental astronomy and geodesy applications. With Duncan (1991) calculated the evolution of the Solar system and the
the development of modern technology, the accuracy of astrometric Earth’s spin direction over 3 Myr. Laskar, Joutel & Boudin (1993)
observations could attain a very high level. Currently, Very Long and Laskar et al. (2004) used a more realistic model to obtain the
Baseline Interferometry (VLBI) determinations of Earth’s rotation long-term precession of the Earth, and Laskar et al. (2011) provided
variations and the coordinates of terrestrial sites and celestial objects a solution of Earth’s orbital motion over 250 Myr. However, all these
are made with estimated accuracies of about ± 0.2 milliarcsecond previous works were traditionally modelled in a Newtonian way and
(mas) or better.1 A better theoretical model of these issues needs only considered the dominant relativistic corrections.
to be formulated in Einstein’s general relativity, at least in its first A post-Newtonian precession model with a long time-span for the
post-Newtonian approximation. Earth has been provided by Tang et al. (2015a, b). The evolution
Precession describes a smooth long-term variation of the celestial of the Solar system was obtained in the J2000 ecliptic and equinox
Earth’s orientation. The P03 solution (Capitaine, Wallace & Chapront reference system, while the rotation of the Earth was calculated
2003) has been adopted as the International Astronomical Union in the corresponding geocentric reference system. Especially, part
(IAU) 2006 precession. It provides polynomial expressions of various of Earth’s rotation was treated in a rigorous relativistic framework,
precession quantities up to the fifth degree in barycentric dynamical based on the post-Newtonian theory of Earth’s rotation by Klioner,
time (TDB), or terrestrial time (TT) in practice. Also, it is known to Gerlach & Soffel (2010). Approximate expressions for the precession
be very accurate near the epoch J2000.0. However, this current IAU of the ecliptic and the precession of the equator were provided, valid
precession model did not result from a rigorous relativistic treatment in the interval ±1 Myr around J2000.0.
(Capitaine 2010). However, the long-term precession model of Tang et al. (2015a,
Moreover, the IAU 2006 precession theory has been valid for b) is still not totally consistent with relativity. Its calculations were
several centuries, but it diverges quickly from the numerical solution executed in the reference to an ecliptic. The ecliptic in the Barycentric
for more distant epochs (Vondrák, Capitaine & Wallace 2011). An Celestial Reference System (BCRS) was explicitly defined in the IAU
extension of the IAU 2006 precession model to scales of several resolutions, but the concept of an ecliptic in the Geocentric Celestial
thousand centuries was provided by Vondrák et al. (2011). This long- Reference System (GCRS), which is reconciled with relativity, is
still not clear (Capitaine & Soffel 2015). Precession of the equator
quantities provided by Tang et al. (2015a, b), such as lunisolar
E-mail: tangkai@shao.ac.cn (KT); michael.soffel@tu-dresden.de (MS) precession, correlate with the concept of ecliptic and equinox in
1 https://www.iers.org/IERS/EN/Science/Techniques/vlbi.html the GCRS. Furthermore, the spatial orientation of the BCRS, as well
C The Author(s) 2021.
Published by Oxford University Press on behalf of Royal Astronomical Society. This is an Open Access article distributed under the terms of the Creative
Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium,
provided the original work is properly cited.A new PN long-term precession model for the Earth 3691
system can be split into three parts:
H = HKep + Hpos + Hmom , (1)
Here, HKep describes the Keplerian motion of the bodies around the
Sun, Hpos is the part only related to the heliocentric positions and
Hmom depends only on the barycentric momenta. In our dynamical
model, the Moon is treated as a separate object. Its orbit is around
the Earth, but meanwhile it is affected by perturbations from the Sun
and other planets. It is inappropriate to calculate the motion of the
Moon in HKep as a Keplerian orbit relative to the Sun, like other
major planets. Thus, we take the treatment for the close encounter
issue from Chambers (1999): the term with regard to the mutual
interaction between the Earth and Moon is moved from Hpos to HKep .
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This makes the new HKep no longer integrable, as it contains the
three-body problem involving the Earth, Moon and Sun. This issue
would be handled in the mini simulation of our hybrid integrator by
Figure 1. Structure of the hybrid integrator. using a conventional integrator.
For the post-Newtonian effects from the monopole mass moments
as the derived orientation of the GCRS, could be realized by the of bodies, the part from the Sun (Schwarzschild terms) is treated as in
International Celestial Reference Frame (ICRF). The ecliptic has Saha & Tremaine (1994). The related post-Newtonian Hamiltonian
lost its importance and is no longer needed as a reference in modern could be split into three parts that have the same form of equation (1),
astronomy (Capitaine & Soffel 2015). and it is calculated by our symplectic integrator. However, the
As alluded to above, we decided to provide a new post-Newtonian relativistic effects from monopole mass moments of other planets
long-term precession model for the Earth as a continuation of our are omitted here. These relativistic effects are smaller than that of
previous works (Tang et al. 2015a, b). The reference does not refer the Sun. The corresponding post-Newtonian Hamiltonian contains
to any ecliptic. The translational motion of the Solar system bodies lots of cross-terms that mix coordinate and momentum dependences
is obtained in the BCRS whose spatial axes are oriented by the (Damour, Soffel & Xu 1991). Hence, it is not easily decomposed
International Celestial Reference System (ICRS), and the rotational into the form of equation (1) that can be calculated by the symplectic
motion of the Earth is calculated in the corresponding GCRS. The integrator. We used another integrator to assess the magnitude of
solutions for ecliptic-independent parameters are provided. The new these relativistic effects from the planets. and this is discussed in
long-term precession hereto is in agreement with relativity. Section 4.
In the following sections, we present a model of the post- The numerical integration for the Solar system was performed with
Newtonian long-term precession for the Earth. Section 2 outlines a SABA4 scheme symplectic integrator (Laskar & Robutel 2001).
the numerical model we use. In Section 3, analytical approximations The integration started at t0 = 245 1545.0 (J2000.0, TDB) and went
of the precession parameters are provided, and comparisons with to ±1 Myr. The step size used for calculating the motion of the Solar
other long-term precession solutions and our previous works are system bodies was 1 d. The initial positions and velocities of the
made. Section 4 concerns the analysis of the relativistic influences planets and the values of planetary masses were taken from the JPL
on the precession. Finally, an overview of the long-term precession DE430 ephemeris (Folkner et al. 2014).2 Other constants used here
models and possible future improvements is given in Section 5. are listed in Table 1. To reduce the accumulation of round-off error,
the algorithm compensated summation was adopted in the program.
2 NUMERICAL SOLUTION
The precession of the Earth is calculated by a hybrid integrator. 2.2 The evolution of the Earth–Moon System
This integrator is composed of two parts: a global simulation obtains The evolution of the Earth–Moon System is calculated by the mini
the translational motion of the Solar system bodies by a symplectic simulation of our hybrid integrator. The calculations in this part
integrator; a mini simulation is special for the evolution of the Earth– contain the translational motions of the Earth and the Moon, the
Moon system using a Bulirsch–Stoer integrator. The structure of this rotational motion of the Earth and different time-scales.
hybrid integrator is shown in Fig. 1 and described below. As we mentioned in Section 2.1, the translational motions of
the Earth and the Moon are obtained here, caused by the mutual
2.1 The translational motion of the Solar system bodies and simultaneous interactions among the Earth, Moon and Sun.
The perturbations on the Earth and the Moon from other planets
The translational motion of the Solar system bodies is modelled in are still considered in the global simulation. On account of this,
the BCRS, whose origin is at the Solar system barycentre, and axes all calculations in the mini simulation are accomplished during the
are oriented according to the ICRS, using TDB as a time-scale. The integration phase of HKep by the global simulation. To obtain a more
dynamical model comprises the Sun, all eight planets of the Solar realistic evolution of the Earth–Moon system, we add a force on the
system, Pluto and the Moon. The influences from the quadrupole Moon associated with the tidal dissipation, modelled by Mignard
moment of the Sun and the Sun’s mass loss are all taken into account. (1979), Touma & Wisdon (1994).
For relativistic terms in the laws of translational motion, here we shall The rotational motion of the Earth is also acquired by the
consider a simple model where only the post-Newtonian terms from mini simulation, relying on the post-Newtonian theory of Earth’s
the monopole mass moments of all bodies are supposed not to vanish.
The democratic heliocentric method (Duncan, Levison & Lee
1998) is used as our symplectic algorithm. The Hamiltonian of the 2 https://ssd.jpl.nasa.gov/?planet eph export
MNRAS 507, 3690–3697 (2021)3692 K. Tang et al.
Table 1. Astronomical constants.
Symbol Value Quantity Reference
c 2.99792458 × 108 m s−1 Speed of light Luzum et al. (2011)
LG 6.969290134 × 10−10 1-d(TT)/d(TCG) Luzum et al. (2011)
LB 1.550519768 × 10−8 1-d(TDB)/d(TCB) Luzum et al. (2011)
TDB0 −6.55 × 10−5 s TDB-TCB at Luzum et al. (2011)
T0 = 244 3144.5003725
au 1.49597870700 × 1011 m Astronomical unit Luzum et al. (2011)
aS 696 000 000 m Equatorial radius of the Sun Laskar et al. (2004)
aE 6.3781366 × 106 m Equatorial radius of the Earth Luzum et al. (2011)
J2S 2.0 × 10−7 Dynamical form factor of the Sun Petit & Luzum (2010)
k2 0.305 k2 of the Earth Laskar et al. (2004)
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t 639 s Time lag of the Earth Laskar et al. (2004)
7 × 10−14 yr−1 Rate of the Sun’s mass loss Quinn et al. (1991)
rotation by Klioner et al. (2010). Earth’s rotation is modelled in TCG, which is computed by the mini simulation. To integrate the
the corresponding GCRS, which is constrained by the IAU 2000 equations of rotational motion, a correct relativistic scaling of various
transformation between BCRS and GCRS. The geocentric coordinate parameters is dealt with as follows (see Klioner et al. 2010):
time (TCG) is used as the coordinate time. The GCRS spatial
(i) for translational motion of the Solar system bodies, all related
coordinates can be transformed to those of a terrestrial reference
variables are TDB-compatible;
system (TRS) by applying the rotation R3 (ϕ)R1 (θ )R3 (ψ). The Euler
(ii) the relativistic torque (1/l!)abc MbL GcL is computed by
angles ϕ, θ and ψ define the orientation of the Earth’s pole in
TDB-compatible parameters;
the GCRS. The expansion of the gravitational field with potential
(iii) for Earth’s rotation, the differential equation is integrated
coefficients Clm and Slm is obtained in the TRS.
using TCG-compatible variables.
In calculating the Earth’s rotational motion, the Earth is treated as a
rigid body and a correction is taken on account of its tidal dissipation. To improve the accuracy of the Earth–Moon system’s motion, the
According to Klioner et al. (2003), the post-Newtonian equation of mini simulation was performed with a Bulirsch–Stoer integrator and
Earth’s rotation is the time-step was set as one-tenth of that for the global simulation.
d ab b 1 The coefficients of the Earth’s gravity field in the TRS were computed
C ω = abc MbL GcL + La (C, ω, iner ) + LaTD , (2)
dT l! from the GEM2008 (Pavlis et al. 2012) normalized coefficients:
l
where T is TCG time, C = C ab is the post-Newtonian tensor of C20 = −1082.626173852223 × 10−6 ,
inertia and ω = ωa is the angular velocity of the post-Newtonian C22 = 1.574615325722917 × 10−6 ,
Tisserand axis (Klioner 1996), which can be obtained from ϕ, θ and S22 = −0.9038727891965667 × 10−6 .
ψ and their angular velocities. The terms on the right-hand side are
torques, described as follows. The moments of inertia were taken from Tang et al. (2015a):
(i) The relativistic torque (1/l!)abc MbL GcL consists of the A = 1.799538227025858 × 10−15 MS au2 ,
lunisolar and planetary torques acting on the oblate Earth. ML are the B = 1.799577876994722 × 10−15 MS au2 ,
multipole moments of the Earth’s gravitational field in the GCRS, C = 1.805468786696834 × 10−15 MS au2 .
and can be obtained from Clm and Slm . GL are the multipole moments
of the external tidal gravito-electric field defined in the GCRS. All SMART97 (Bretagnon et al. 1998) only provided the initial Euler
formulae to compute ML and GL can be found in Klioner et al. (2003, angles referring to the J2000 ecliptic and equinox. After rotating
2010). Here we only consider the torques given by the Sun and the them to the GCRS, we can obtain the initial conditions for t0 =
Moon, and we calculate the term with l = 2. 245 1545.0 (TCG):
(ii) The additional torque La depends on C, ω, together with the
ψ(t0 ) = −0.768533505667843 rad,
angular velocity iner indicating the relativistic precessions. The
formula of La depending on geodetic precession can be found in θ(t0 ) = 3.89681853496466 × 10−5 rad,
Klioner et al. (2010). ϕ(t0 ) = 5.66349465843964 rad,
(iii) The torque LaTD is the tidal torque acting on the Earth, given ψ̇(t0 ) = 0.00685783866492344 rad d−1 ,
by Touma & Wisdon (1994). This is an extra term added into the
θ̇(t0 ) = −1.24771253460864 × 10−7 rad d−1 ,
equations of rotational motion by Klioner et al. (2003).
ϕ̇(t0 ) = 6.2935296485402 rad d−1 .
The calculation of Earth’s rotational motion involves various
quantities defined in several reference systems, including BCRS, Other constants are listed in Table 1.
GCRS and TRS. It is inevitable that we will deal with different time-
scales and relativistic scaling of the parameters. TDB is set as the
3 LONG-TERM PRECESSION MODEL
basic time of our hybrid integrator and used in the global simulation.
Other different time-scales, including Barycentric Coordinate Times
3.1 Precession parameters
(TCB), TCG and TT, will appear in our calculation of Earth’s rotation
and can be acquired according to Irwin & Fukushima (1999) and After calculating the translational motion of the Solar system bodies
Klioner (2008). Especially, there is an integral between TCB and and the rotational motion of the Earth, we can obtain the Euler angles
MNRAS 507, 3690–3697 (2021)A new PN long-term precession model for the Earth 3693
107
Table 2. The periodic terms in XA andYA .
4
i XA YA Pi (yr)
2 Ci (arcsec) Si (arcsec) Ci (arcsec) Si (arcsec)
(arcsec)
1 2266 81 536 75 101 −1244 25 691
0
2 −5837 1586 1442 5414 69 080
3 −3167 −621 −609 2917 72 630
-2 4 722 −2594 −2310 −675 233 673
5 −896 −664 −574 824 190 438
-1 -0.5 0 0.5 1
6 911 663 619 −882 49 213
104 7 −497 4218 −296 −467 1 027 090
8 312 711 622 −286 67 472
15 9 291 -596 −522 −280 65 582
(arcsec)
10 419 −115 −81 −392 25 327
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10
11 428 −391 481 −213 481 960
12 396 62 46 −353 26 060
5
13 406 −3790 51 364 1 309 223
14 398 −2203 378 171 725 900
0
15 −429 1322 −482 17 569 083
-1 -0.5 0 0.5 1
16 −237 235 165 206 172 955
Time (Myr, TCG)
17 −276 −100 −89 252 15 785
18 −44 291 252 60 63 767
Figure 2. Numerical results for Euler angles ψ and θ from −1 to 1 Myr. 19 36 280 23 −76 377 687
20 −218 −37 −61 179 76 486
21 122 −122 −79 −126 26 691
directly: ϕ represents an intrinsic rotation around the Earth’s pole
with a speed about 2π rad/d. The numerical results of ψ and θ over
105
the whole time-span are depicted in Fig. 2. Because the curves of ψ 2
and θ are not smooth, their solutions are unsuitable to be expressed
1
XA (arcsec)
by the function of periodic terms. We chose two other parameters
XA and YA as the precession parameters for our long-term precession
0
model. XA and YA are the precession part of the coordinates of the
post-Newtonian Tisserand axis unit vector in the GCRS. Here the -1
coordinates X and Y with respect to the GCRS can be evaluated
from the unit vector coordinates (0, 0, 1) with respect to the TRS by -2
-1 -0.5 0 0.5 1
simply reversing the transformation between the GCRS and the TRS
105
mentioned in Section 2.2: 2
⎛ ⎞ ⎛ ⎞
X 0 1
YA (arcsec)
⎝ Y ⎠ = R3 (−ψ)R1 (−θ ) ⎝0⎠ . (3)
− 1 0
To find the analytical approximations for the precession parame- -1
ters, we performed some algorithms on our numerical results. First,
polynomial curve fitting was applied to remove the linear drift of the -2
-1 -0.5 0 0.5 1
data. Second, numerical analysis of fundamental frequency (Laskar, Time (Myr, TCG)
Froeschlé & Celletti 1992) was used to extract the secular terms. The
expressions for the precession parameters XA and YA are presented
Figure 3. Solution of precession parameters XA and YA from −1 to 1 Myr.
as
XA = 4962. 50 + 0. 00188T 3.2 Comparison with other long-term precessions
21
We compared our numerical solutions with other long-term preces-
+ [Ci cos(2πT /Pi ) + Si sin(2πT /Pi )], sions. The orbital solution for the Earth over the past 3 Myr was
i=1
obtained by Quinn et al. (1991). Laskar et al. (2004) provided the
YA = −73826. 89 − 0. 000402T values of the longitude of perihelion from moving equinox ω• and
21
the obliquity A from −50 to 21 Myr around J2000.0.3 We used
+ [Ci cos(2πT /Pi ) + Si sin(2πT /Pi )], (4) their results to calculate the precession parameters XA and YA . The
i=1
relations of all these parameters were referred to Vondrák et al. (2011)
where T is the elapsed time in Julian TCG year since J2000.0 (TCG), and Laskar et al. (1993). Fig. 4 gives the comparisons between these
and the cosine/sine amplitudes Ci and Si of the periods Pi are listed two results for XA and YA over 1 Myr in negative time from J2000.0.
in Table 2. Fig. 3 shows the solution of precession parameters XA Top panels of Fig. 4 display our solution as the solid line and the
and YA in the interval ±1 Myr around J2000.0. Their curves appear
to be smooth and periodic. The major period 25 691 yr is related to
the main precession frequency given by Laskar et al. (2004). 3 http://vo.imcce.fr/insola/earth/online/earth/earth.html
MNRAS 507, 3690–3697 (2021)3694 K. Tang et al.
105 on a fixed ecliptic ωA . All these numerical calculations for the Earth’s
2
our solution
long-term precession were achieved in a relativistic framework.
other solutions However, our previous works involving the ecliptic in the GCRS
1 are deficient, because the natural definition of this ecliptic is still
ambiguous (Capitaine & Soffel 2015). The precession of the equator
X A (arcsec)
parameters concerning this ecliptic, such as ψ A and ωA , are also
0
ill-defined in the relativistic framework. To make our work more
coincident with relativity, we re-obtained the long-term precession
-1 model independent of any ecliptic. The alterations cover the follow-
ing.
-2 (i) Reference system: the BCRS, which is oriented according
-1 -0.8 -0.6 -0.4 -0.2 0
to the ICRS axes, was chosen as the global reference system,
4000
instead of the previous J2000 ecliptic and equinox system. The
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X A (arcsec)
2000
corresponding GCRS was derived from this BCRS based on the
0
IAU 2000 Resolution B1.3.
-2000
(ii) Dynamical model: in our previous works, the relativistic effect
-4000
-1 -0.8 -0.6 -0.4 -0.2 0 only from the mass of the Sun was considered in translational motion.
Time (Myr, TCG) To enrich the assessment process, more relativistic influences were
analysed, especially the post-Newtonian effects from the masses of
105 planets. This is discussed in Section 4.
2
our solution (iii) Numerical integrator: a hybrid integrator was used to improve
other solutions
the accuracy of the integration. The calculations for the evolution of
1
the Solar system and Earth’s rotation could no longer be performed
Y A (arcsec)
respectively as in our previous works.
0 (iv) Precession parameters: XA and YA were chosen as the long-
term precession parameters. They were derived from the Euler angles
-1 ψ and θ describing the rotations between the GCRS and the TRS
using equation (3). Therefore, XA and YA have clear definitions in
the relativistic framework.
-2
-1 -0.8 -0.6 -0.4 -0.2 0
The solutions of XA and YA were also computed using our previous
4000 numerical results (Tang et al. 2015a). We let the old and new
Y A (arcsec)
2000 integrators cross-check each other, to avoid any numerical instability.
0
The relative differences in XA and YA are below 0.005 over the length
-2000
of the integration time.
-4000
-1 -0.8 -0.6 -0.4 -0.2 0
Time (Myr, TCG)
4 R E L AT I V I S T I C I N F L U E N C E S
Figure 4. Top panel: comparisons between our numerical solution (solid Our long-term precession model for the Earth was achieved in a
line) and the results calculated from Quinn et al. (1991) and Laskar et al. relativistic framework. To estimate the importance of various contri-
(2004) (dotted line) for X and Y over the past 1 Myr from J2000.0. Bottom butions from relativity, we appraised the post-Newtonian effects on
panel: the difference between these two results (dotted line). the precession parameters using our program, which could calculate
the precession for both the Newtonian and the post-Newtonian cases.
results from Quinn et al. (1991) and Laskar et al. (2004) as the dotted We first repeated the calculations of the Newtonian translational
line. As is apparent in the figure, the agreement between the two motion of the Solar system bodies and rotational motion of the
results is good. This indicates the correctness of our computational Earth. Then the post-Newtonian numerical solutions were computed
process. The detailed disparities of the two calculations are given in using another code. All the relativistic effects could be switched
the bottom panels and the relative differences are about 0.03. These on/off independently of each other. The relativistic influences on
discrepancies are mostly attributed to different numerical models precession parameters XA and YA are close to each other, with similar
of these long-term precessions, for instance, different dynamical magnitudes and periods. So in this section, we only offer the figures
models of the Solar system, numerical calculation methods, initial of the relativistic influences on XA .
conditions, etc. When calculating the translational motion of the Solar system
bodies, we only took into account the relativistic effect due to the
mass of the Sun in our hybrid integrator. This relativistic influence
3.3 Comparison with our previous works
on precession parameter XA is illustrated in Fig. 5 (top panel). On
In our previous work (Tang et al. 2015a), we calculated and provided XA , it has a similar shape as that on the precession of the ecliptic
the expressions for the precession of the ecliptic parameters PA and parameters PA and QA , given in Tang et al. (2015a). It imparts a
QA , general precession pA and obliquity A . Because pA and A perturbation on the orbit of the Earth and indirectly affects Earth’s
mix the motion of the equator in the GCRS and the motion of the rotation through the changes of torques on the Earth. This influence
ecliptic of date, we used other appropriate parameters to present is slight within ±0.2 Myr, and increases rapidly with a longer time-
the precession of the equator in Tang et al. (2015b): the lunisolar span. Its peak amplitude is around 1500 arcsec over 1 Myr. Because of
precession in longitude ψ A , and the inclination of the moving equator the limitation of the symplectic integrator, we used a Bulirsch–Stoer
MNRAS 507, 3690–3697 (2021)A new PN long-term precession model for the Earth 3695
103 104
2 1
XA (arcsec)
XA (arcsec)
1
0
0
-1
-1 -1 -0.5 0 0.5 1
1
XA (arcsec)
-2
-1 -0.5 0 0.5 1
0
4
XA (arcsec)
2 -1
-1 -0.5 0 0.5 1
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1
0
XA (arcsec)
-2 0
-4
-1 -0.5 0 0.5 1 -1
Time (Myr, TCG) -1 -0.5 0 0.5 1
Time (Myr, TCG)
Figure 5. The post-Newtonian effects from the mass of the Sun (top panel)
and from the masses of other planets (bottom panel) on XA from −1 to 1 Myr. Figure 6. The relativistic effects of the geodetic precession (top panel), the
post-Newtonian torque (middle panel) and the relativistic scaling and time-
scales (bottom panel) on XA from −1 to 1 Myr.
integrator to assess the magnitude of the relativistic effects from the
masses of other planets. The Einstein–Infeld–Hoffmann equations of geodetic precession’s influence. For this reason, the error caused
motion have been employed to recalculate the translational motion of by the traditional way of treating geodetic precession is commonly
all bodies. Fig. 5 (bottom panel) shows the total of these relativistic tolerated.
effects on XA . Although the effects increase, their values lie within 4 We also finished a series of numerical calculations to estimate the
arcsec over 1 Myr. These are small compared with the effect from the magnitude of other relativistic effects that people usually neglect.
Sun and could be omitted in our case. In the post-Newtonian model Fig. 6 represents the relativistic influences of the post-Newtonian
for translational motion, other relativistic effects from multipole torque (middle panel), the relativistic scaling and time-scales (bottom
moments and spin dipoles of the Solar system bodies are tiny, and panel). Their curves are indistinguishable from that of geodetic
will be discussed in a future work. precession, but with extremely small amplitudes, less than 1 arcsec
Our calculation about the rotational motion of the Earth followed in this time-span. These relativistic effects are tiny compared with
the Earth rotation theory of Klioner et al. (2010), which is consistent geodetic precession. Analogous estimations for these relativistic
with relativity. This approach differs from the traditional way: solve effects can be found in Klioner et al. (2010) and Tang et al. (2015a).
the purely Newtonian equations of rotational motion, and then add the It should be noted that the contributions to the influences on pre-
precomputed relativistic corrections to it. Geodetic precession was cession parameters above are complicated. The fitting of parameters
naturally taken into account as an additional torque in equation (2). to observations is usually used to distinguish various models and
The full post-Newtonian torques were calculated using symmetric to understand where such differences come from, but this is not fit
and trace-free Cartesian (STF) tensors ML and GL . The treatment for our case. Here, a comparison of the purely Newtonian result
of several reference systems was referred to the relevant IAU with integration where relativistic terms are switched on is just an
resolutions. For each TDB time, the corresponding TCB, TT and expedient to estimate the magnitude of these relativistic effects.
TCG times were obtained. The relativistic scaling of all relevant All comparisons above allow us to conclude that the relativistic
quantities was treated properly, as in Klioner et al. (2010). effects on the Earth’s precession accumulate with time and show
Geodetic precession is widely known as the dominant relativistic obvious periodical features related to the main period of precession
effect in Earth’s rotation. In Tang et al. (2015b), it is easy to parameters. The post-Newtonian effects related to the mass of the
recognize its influence on the traditional lunisolar precession ψ A . Sun, including the geodetic precession, have great impacts on Earth’s
Geodetic precession leads the slope of the curve ψ A with the well- precession, while other relativistic effects are relatively tiny.
known amount of 2 arcsec per century, but the appearance of this
influence is veiled on precession parameters XA and YA . Fig. 6 (top
5 OV E RV I E W O F T H E L O N G - T E R M
panel) demonstrates the influence of geodetic precession on XA . This
PRECESSION MODELS
influence accumulates with time and reaches about 7750 arcsec at
±1 Myr around J2000.0. Its main periodic term is 25 468 yr, close to As we said at the beginning, all other previous long-term precession
that of XA . Here, using the additional torque is a rigorous and natural solutions were not achieved in a relativistic framework. For the
way to take geodetic precession into account (Klioner et al. 2010). precession model of Vondrák et al. (2011), the solution was obtained
A code has been written in order to attempt to distinguish between through the Mercury 6 package (Chambers 1999) for translational
this treatment and the approach of Laskar et al. (1993), which added motions of bodies and the La93 solution (Laskar et al. 1993) for
a constant correction into the related equations of Earth’s rotation. Earth’s rotation. The Mercury 6 package did not consider relativity.
Two different approaches could induce the deviation on XA and YA , Laskar et al. (1993) added a geodetic precession constant into
approximately 20 arcsec in ±1 Myr. It is near 0.3 per cent of the the related equations. According to our analysis in Section 4, the
MNRAS 507, 3690–3697 (2021)3696 K. Tang et al.
relativistic effects are still slight in the valid time interval of the planets, and two different treatments of geodetic precession. For the
model of Vondrák et al. (2011): ±200 millennia from J2000. This time-span ±1 Myr around J2000.0, the post-Newtonian effects from
implies that the deficiencies of relativistic treatment in Vondrák et al. the mass of the Sun, including the geodetic precession, have great
(2011) are endurable. This is also the reason why we chose a longer effects on the Earth’s precession, while other relativistic effects are
integration time for more obvious relativistic influences. tiny.
Our precession solution covering ±1 Myr is inevitably imperfect Compared with all previous long-term precession models based
because of the uncertain parameters in the dynamical model, such on a Newtonian approach, this post-Newtonian long-term precession
as the Earth’s dynamical ellipticity and the dissipative effects at model has been achieved to be consistent with general relativity. To
a remote epoch. The predictive knowledge of those values cannot fulfil a complete model of Earth’s precession with high accuracy,
be extended to a time-span over 400 millennia without increasing further tasks need to be dealt with in the future: more effects in the
the uncertainty considerably (Vondrák et al. 2011). Furthermore, to dynamical model should be considered, and the parameters should
achieve a precession model close to reality, Vondrák et al. (2011) have be fitted to the high-precision observational data.
resorted to the following tactics. The values of precession parameters
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were replaced by those from the IAU 2006 precession inside the
interval ±2000 yr around J2000.0, and several additional corrections AC K N OW L E D G E M E N T S
were applied to the solution based on the IAU 2006 model. Hence, for We wish particularly to thank Dr Jan Vondrák for his advice on
the interval ±200 millennia from J2000.0, the long-term precession the long-term precession model, Dr Enrico Gerlach for his help
model of Vondrák et al. (2011) is recommended. with the numerical calculations, and Professor Chongming Xu and
Although we have constructed a long-term precession model for Professor Xuejun Wu for their valuable comments. This work was
the Earth in the post-Newtonian approximation of general relativity, supported by the B-type Strategic Priority Program of the Chinese
there is still room for improvement. To make this model complete Academy of Sciences (Grant No. XDB41000000), the National
and close to reality, some points should be taken into account in Natural Science Foundation of China (Grant No. 11503067), and
future work. the Pre-research Project on Civil Aerospace Technologies funded by
the China National Space Administration (Grant No. D020303).
(i) Consideration of the dynamical model should be sufficient,
especially regarding perturbations from dissipative effects, such as
body tides, core-mantle friction, atmospheric tides, mantle con- CONFLICT OF INTEREST
vection, climate friction, etc. Those effects of non-rigidity are
significantly larger than the relativistic effects and would cause the No potential conflict of interest was reported by the authors.
inaccuracy of the model.
(ii) The initial conditions and parameters in the dynamical model DATA AVA I L A B I L I T Y
need to be fitted to real observations or adjusted in agreement with
the IAU model. The data underlying this article are available in the article and in its
(iii) The numerical integration time may be limited to reduce the online supplementary material.
uncertainty associated with the physical parameters.
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