Effects of lunisolar perturbations on TianQin constellation: An analytical model

TianQin is a proposed space-based gravitational-wave observatory mission that critically relies on the stability of an equilateral-triangle constellation. Comprising three satellites in high Earth orbits of a ${10}^5\mathrm{km}$ radius, this constellation’s geometric configuration is significantly affected by gravitational perturbations, primarily originating from the Moon and the Sun. In this paper, we present an analytical model to quantify the effects of lunisolar perturbations on the TianQin constellation, derived using Lagrange’s planetary equations. The model provides expressions for three kinematic indicators of the constellation: arm’s lengths, relative line-of-sight velocities, and breathing angles. Analysis of these indicators reveals that lunisolar perturbations can distort the constellation triangle, resulting in three distinct variations: linear drift, bias, and fluctuation. Furthermore, it is shown that these distortions can be optimized to display solely fluctuating behavior, under certain predefined conditions. These results can serve as the theoretical foundation for numerical simulations and offer insights for engineering a stable constellation in the future.

Figure 1

Depiction of the TianQin constellation in the geocentric ecliptic coordinate system. The ecliptic plane is spanned by the $x$ and $y$ axes, with the $x$ axis directed toward the vernal equinox. Also illustrated is the orbital coordinate system $\{X,Y,Z\}$ for SC1, where the $X$ axis points toward the perigee of the satellite’s orbit, and the $Z$ axis (not shown) is perpendicular to the orbital plane. The angles $i$, $\mathrm{\Omega }$, $\omega$, and $\nu$ denote the orbital inclination, longitude of ascending node, argument of perigee, and true anomaly, respectively. Specifically, $i=94.7°$ and $\mathrm{\Omega }=210.4°$ are set to orient the TianQin detector plane toward the reference source, the white-dwarf binary RX $\mathrm{J}0806.3+1527$.

Figure 2

Time evolution of deviations between the simplified (SNM) and high-precision (HPNM) numerical models for satellite positions and three indicators. SNM incorporates the point-mass gravity fields of the Earth, Moon, and Sun, as well as the Earth’s $J_2$. HPNM additionally considers the higher-degree nonspherical gravity fields of the Earth and the point-mass gravity fields of other planets (for further details, see Appendix pp2-s3). In these plots, red corresponds to SC1, $v_{23}$, $L_{23}$, or ${\alpha }_1$; green represents SC2, $v_{31}$, $L_{31}$, or ${\alpha }_2$; and blue indicates SC3, $v_{12}$, $L_{12}$, or ${\alpha }_3$. The initial orbital elements used are the same as those presented in Table 6.

Figure 3

Time evolution of three kinematic indicators for the analytical model compared to the numerical model. The left panel displays subplots illustrating variations in arm’s lengths ($L_{ij}$), relative velocities ($v_{ij}$), and breathing angles (${\alpha }_k$), respectively. Colors in each subplot represent the numerical model (blue for $L_{12}$, $v_{12}$, or ${\alpha }_3$; red for $L_{23}$, $v_{23}$, or ${\alpha }_1$; green for $L_{31}$, $v_{31}$, or ${\alpha }_2$), while black denotes the analytical model (only $v_{12}$ is plotted for clarity in relative velocities). The right panel illustrates the temporal evolution of discrepancies between the analytical and numerical models for these three indicators.