Table 6.

Present‐Day Secular Rates of Change of Earth Rotation Rate ${\omega }_E$, Obliquity $\epsilon$, and Earth‐Moon Orbital Parameters $a$ (Semi‐Major Axis), $e$ (Eccentricity), and $i$ (Inclination), due to Tides on Earth, Tides Within the Moon, and Lunar Core‐Mantle Boundary Effects (CMB) ^{a , b}

Constituent	$\frac{da}{dt}\left(\frac{mm}{year}\right)$	$\frac{de}{dt}\left(\frac{10^{-12}}{year}\right)$	$\frac{di}{dt}\left(\frac{\mu as}{year}\right)$	$\frac{d\epsilon }{dt}\left(\frac{\mu as}{year}\right)$	$\frac{d{\omega }_E}{dt}\left(\frac{″}{centur{y}^2}\right)$
Long‐period tides
${\mathrm{M}}_f$	−0.161	0.0057	0.0019	−0.528	0
${\mathrm{M}}_m$	−0.0178	−0.416	0	0	0
Diurnal tides
${\mathrm{K}}_1$	0	0	0	−9.51	−93.7
${\mathrm{O}}_1$	4.98	−0.176	−0.0596	8.83	−73.8
${\mathrm{P}}_1$	0	0	0	1.36	−11.4
${\mathrm{Q}}_1$	0.283	2.20	−0.0023	0.335	−2.80
$\mathrm{\Omega }$ (${\mathrm{K}}_1$ nodal)	0	0	−0.172	0.0154	−1.72
$L+F$ (${\mathrm{O}}_1$ nodal)	0.181	−0.0064	0.266	0.0241	−2.68
Semi‐diurnal tides
${\mathrm{K}}_2$	0	0	0	−1.10	−10.9
${\mathrm{M}}_2$	31.3	−1.11	−0.375	8.37	−930
${\mathrm{S}}_2$	0	0	0	1.32	−147
${\mathrm{N}}_2$	2.23	17.4	−0.0178	0.397	−44.2
$\mathrm{\Omega }$ (${\mathrm{K}}_2$ nodal)	0	0	−0.0489	−0.0452	−0.978
$L+F$ (${\mathrm{M}}_2$ nodal)	0.0429	−0.0015	0.0633	−0.0590	−1.27
Tides within the Moon c
–	−0.387 (−0.471)	−5.49 (−6.9)	−0.459 (−0.51)
Lunar core‐mantle boundary (CMB) c
–	−0.0143 (−0.015)	0 (0)	−0.0429 (−0.04)

^a Shown are results for 14 constituents of tides on Earth, as included in our orbital dynamics model.

^b Effects of lunar tides and lunar core‐mantle boundary (CMB) computed respectively from Equations (40), (41), (42) and (46), (47), (48), and parameter values used in this study.

^c DE440 values are given in parentheses.