Two birds with one stone: Simultaneous realization of Lunar Coordinate Time and lunar geoid time with a single orbital clock

Open Access

Table 2

Comparison of nominal mean orbital elements ( ${\bar{σ}}_{p}$ $Mathematical equation: ${{\bar \sigma }_{\rm{p}}}$$ ) and mean orbital elements ( ${\bar{σ}}_{p}^{*}$ $Mathematical equation: $\bar \sigma _{\rm{p}}^*$$ ) from our numerical simulations with σ = {a, e, i}.

Number	ā_p [km]	ī_p	${\bar{a}}_{p}^{}$ $Mathematical equation: $\bar a_{\rm{p}}^ $$ [km]	${\bar{e}}_{p}^{}$ $Mathematical equation: $\bar e_{\rm{p}}^ $$	${\bar{i}}_{p}^{}$ $Mathematical equation: $\bar i_{\rm{p}}^ $$	∆^∗ [ns]	∆f^∗	$Δ_{c}^{}$ $Mathematical equation: $\Delta _{\rm{c}}^$$ [ns]	$Δ f_{c}^{}$ $Mathematical equation: $\Delta f_{\rm{c}}^$$
1	2606.2658	0	2606.1094	0.0039	1.907^◦	50	1.6 × 10⁻¹⁵	−6	−2 × 10⁻¹⁶
2	2606.1186	25^◦	2605.9063	0.0056	23.395^◦	70	2.3 × 10⁻¹⁵	−13	−4 × 10⁻¹⁶
3	2605.7163	54.736^◦	2605.3468	0.0046	53.475^◦	140	4.4 × 10⁻¹⁵	−9	−3 × 10⁻¹⁶
4	2605.4477	85^◦	2604.9803	0.0040	84.715^◦	190	6.0 × 10⁻¹⁵	9	3 × 10⁻¹⁶

Notes. In our simulations, we adopted each initial state as $σ_{p, 0} = {\bar{σ}}_{p}$ $Mathematical equation: ${\sigma _{{\rm{p}},0}} = {{\bar \sigma }_{\rm{p}}}$$ . The resulting and corrected frequency offsets and de-synchronizations are also listed. The de-synchronizations ∆^∗ and $Δ_{c}^{*}$ $Mathematical equation: $\Delta _{\rm{c}}^*$$ are shown as their values at the end of a year.

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