Revisiting Earth tide parameters used in the development of planetary and lunar ephemeris

Abstract

Earth tide effects play important roles in evolution of the Earth–Moon system. At present, a Five-Time-Delay tide model developed by the JPL ephemeris group is widely adopted in the development of high-precision planetary and lunar ephemerides. However, a significant discrepancy between tide models of Five-Time-Delay type and a conventional model suggested by the International Earth Rotation and Reference Systems Service (IERS2010 model) used in fields of geodesy and geophysics exists. The paper aims to investigate possible causes of the above-mentioned discrepancy and derives an Earth tide model closer to the IERS2010 model. We found that the discrepancy between the IERS2010 model and tide models of Five-Time-Delay type was mainly caused by an error in computation of diurnal tidal parameters in previous tide models of Five-Time-Delay type. After correcting tidal parameters caused by the error, we obtain a new parametrization of Five-Time-Delay model with a weighted least square method. The difference of an updated model and the IERS2010 model can be reduced to the level of less than one-tenth of the IERS2010 model. The discrepancy between models of Five-Time-Delay type and IERS2010 model is dominated by resonance phenomena at diurnal band, and by cross-over terms and retrograde terms induced by dynamical effects of ocean tides at low-frequency and semi-diurnal bands.

Appendix A Derivation of Eq. 2

The definitions of $H_f$ and $\theta _f$ in the IERS conventions (Petit and Luzum 2010) are consistent with that of Cartwright and Tayler (1971). Therefore, we can derive Eq. 2 of this work straightforwardly from Eq. 12 of Cartwright and Tayler (1971). For the component of degree l and order m at the field point ($R_\textrm{e}, \theta , \uplambda$), the tidal-raising potential reads

$$\begin{aligned} V_{n}^{m}(R_\textrm{e}, \theta , \uplambda ,t) = g_\textrm{e} c^{m*}_{n}(t) W^{m}_{n}(\theta , \uplambda ) \,, \end{aligned}$$

(A1)

where $g_\textrm{e} = GM_\oplus / R_\textrm{e}^2$ and

$$\begin{aligned} W^{m}_{n}(\theta , \uplambda )= & {} (-1)^m \sqrt{4\pi (2-\delta _{0m})}Y^{m}_{n}(\theta , \uplambda ) \nonumber \\= & {} (-1)^m \sqrt{4\pi (2-\delta _{0m})}{\bar{P}}_{nm}(\cos \theta ) e^{i m \uplambda } \,. \end{aligned}$$

(A2)

$c^{m*}_{n}(t)$ is the conjugate of the complex number

$$\begin{aligned} c^{m}_{n}(t) = \sum \limits _{f} H_f \left\{ \begin{array}{l} \cos \theta _f - i \sin \theta _f \,\,\, \textrm{for}\, n - m \, \textrm{even}\\ \sin \theta _f + i \cos \theta _f \,\,\, \textrm{for}\, n - m \, \textrm{odd} \\ \end{array} \right. \end{aligned}$$

(A3)

The additional potential $\Delta V_{nm}(R_\textrm{e}, \theta , \uplambda ,t)$ due to the deformation of the Earth can be defined as $k_{nm} V^{m}_{n}(R_\textrm{e}, \theta , \uplambda ,t)$. Outside the Earth($r>R_\textrm{e}$), by Dirichlet’s theorem (Lambeck 1980),

$$\begin{aligned} \Delta V_{n}^{m}(R_\textrm{e}, \theta , \uplambda ,t) =(R_\textrm{e}/r)^{n+1} k_{nm} V_{n}^{m}(R_\textrm{e}, \theta , \uplambda ,t) \end{aligned}$$

(A4)

By inserting Eqs. A1–A3 into Eq. A4 and reshaping Eq. A4 into the following form:

$$\begin{aligned} \Delta V_{n}^{m}(R_\textrm{e}, \theta , \uplambda ,t) = \frac{GM_\oplus }{r} (\frac{R_\textrm{e}}{r})^{n} (\Delta {\bar{C}}_{nm} - i \Delta {\bar{S}}_{nm})Y^{m}_{n}(\theta , \uplambda ) \,, \end{aligned}$$

(A5)

one can obtain Eq. 2 with $k_{nm} = \delta k_f$ for $n=2$.