Impact of orbital orientations and radii on TianQin constellation stability
IImpact of orbital orientations and radii on TianQinconstellation stability
Zhuangbin Tan, Bobing Ye, Xuefeng Zhang
TianQin Reseach Center for Gravitational Physics and School of Physics andAstronomy, Sun Yat-sen University (Zhuhai Campus), Zhuhai 519082, P.R. ChinaE-mail: [email protected]
Abstract.
TianQin is a proposed space-based gravitational-wave observatorymission to be deployed in high circular Earth orbits. The equilateral-triangleconstellation, with a nearly fixed orientation, can be distorted primarily under thelunisolar perturbations. To accommodate science payload requirements, one mustoptimize the orbits to stabilize the configuration in terms of arm-length, relativevelocity, and breathing angle variations. In this work, we present an efficientoptimization method and investigate how changing the two main design factors, i.e.,the orbital orientation and radius, impacts the constellation stability through single-variable studies. Thereby, one can arrive at the ranges of the orbital parameters thatare comparatively more stable, which may assist future refined orbit design.
1. Introduction
TianQin is a geocentric space-based low-frequency gravitational-wave observatorymission consisting of three drag-free satellites in a nearly equilateral-triangleconstellation [1, 2]. The inter-satellite measurement and inertial reference scheme isbased on LISA [3]. The current mission design proposes an orbital radius of 10 km, andsets the orbital plane roughly perpendicular to the ecliptic plane (see Fig. 1) and facing averification source RX J0806.3 + a r X i v : . [ g r- q c ] D ec ianQin constellation stability Figure 1.
An illustration of the TianQin constellation comprising three satellitesSC1-3 (figure reproduced from [1]). The direction to the reference source J0806 isshown. those mentioned above (also cf. [5]) in both the orientation and the radius. Hence theconcept deserves careful investigations in its own right.Due to perturbing gravitational forces in space, the geometry of the TianQinconstellation will deviate from the nominal equilateral triangle. Particularly, the long-term perturbation effects can cause persistent drift in the relative positions betweenthe satellites, altering the overall configuration in the long run. If not restrained,the induced changes in the detector’s arm-lengths and subtended angles, as well asthe relative velocities between the satellites, will exceed the capacity of the sciencepayloads (e.g., laser pointing, phase readout, laser frequency noise removal, etc.).Therefore, requirements on the constellation stability must be imposed to meet thenormal working conditions of the observatory [16]. In fact, to alleviate pressure onprecision instrumentation on-board, it would be desirable to have deviation from theequilateral triangle as small as possible. To summarize, acquiring stable orbits are ofgreat necessity and importance to the mission.In the area of orbit stability and optimization, the heliocentric LISA-like design[17, 16, 18] has been extensively studied with both analytic [19, 20, 21, 22] and numericalmethods [23, 24, 25, 26, 27, 28, 29], over a variety of issues of perturbation analysis,arm-length and trailing-angle selection, injection error requirement, etc. Optimizedorbits for the ASTROD-GW mission [30] at the Sun-Earth L3, L4, and L5 Lagrangepoints have been obtained through tuning the average orbital periods and eccentricities[31, 32, 33, 6]. The two-week evolution of the GEOGRAWI/gLISA constellation betweentwo station-keeping maneuvers has been analyzed in [12]. Regarding optimization,at least two types of methods are commonly used. In the cost-function method,one minimizes a set of carefully chosen performance measures. The perturbationcompensation method is to apply orbital parameter offsets to compensate for long-termperturbative effects [34, 31, 35]. In our previous work [36] for TianQin, a combinedapproach was developed, and seven sets of stable orbits were found with detectorpointings spreading over the ecliptic plane, in addition to an earlier example presentedin [1] (also [37]). It has been suspected that the constellation tends to be more stablewith orbital planes roughly vertical to the ecliptic. Here we will put this speculation to ianQin constellation stability
2. Simulation setup
We simulate the satellite orbits by the open-source, flight-qualified program, GeneralMission Analysis Tool (GMAT) [39]. As in [36], the force models include a 10 × a , i , Ω, ν ini , as well as the initial time t , which we will discuss in the nextsection. Table 1.
The nominal orbital elements of the TianQin constellation in the J2000-based Earth-centered ecliptic coordinate system (EarthMJ2000Ec). a e i Ω ω ∆ ν ν ini SC1, 2, 3 10 km 0 94 . ◦ . ◦ ◦ ◦ ◦ , ◦ , ◦
3. Optimization method
The combined approach used in our previous work [36] consists of two mains steps. In thefirst step, one iteratively adjusts the initial orbital elements at a fixed initial time so thatthe mean semi-major axes, inclinations and longitudes of ascending nodes of the threesatellites can be kept the same. In the second step, one refines the orbital elements bynumerically searching for minimums of a cost function which encodes five-year stability ianQin constellation stability Figure 2.
Mean eccentricities of one-year orbits, averaged over SC1, 2, 3 with theinitial elements given in Table 1, and started at different initial times through out 2034.The curve indicates strong correlation with the lunar position such that it attains localminimums when the Earth-to-Moon vector is nearly perpendicular to the orbital plane,e.g., 22 May 2034 (day 142). performance. In practice, the first step runs fast, while the second suffers from longcalculation time, which may take a few days depending on the numerical methods used.Hence we deem the second step inefficient for future engineering applications, as well asfor the large-scale search to be performed in this work.To overcome this difficulty, we propose a new method that expands on the first stepof the previous approach and dispenses with the slow “blind search” of the cost function’soptima. The idea is inspired by two observations we have made.
First , the position ofthe Moon (also the Sun, but less prominent), at an initial time t = t , relative to theorbital plane affects the optimization processes. It was found that for certain initialdates, such as 22 May 2034, by matching the three mean semi-major axes alone, oneis able to stabilize the triangle to the required level without taking further effort. Thistypically happens when the Earth-to-Moon vector aligns almost perpendicularly to theorbital plane at t . To account for this phenomenon, one can look into the eccentricityevolution of the individual orbits since the stability strongly depends on the magnitudeof the mean eccentricities [36]. In Fig. 2 we show how the mean eccentricity of one-yearpropagation, averaged over SC1, 2, 3 of Table 1, changes with different inputs of t . Onecan clearly see a monthly modulation due to the Moon. Therefore, by tuning the initialtime, one can make the orbits more circular and thus obtain a more stable configuration. Second , starting from optimized initial elements, the orbits can be backward propagatedfor a few months while still maintaining the same level of stability (see Fig. 3). It allowsone to extend lengths of usable orbits without the need of re-optimizing. Based on thesetwo observations, we have decided to introduce the initial time t into the pool of theoptimization variables as the major improvement in our new method.The new optimization method is divided into two main steps: ianQin constellation stability Figure 3.
Breathing angle evolutions for a set of optimized orbits (P1 of [36], Tables4, 5 therein) backward propagated for one year from the initial time ( t = 0). Theoptimization is performed after t = 0. The requirement of ± . ◦ is marked by dashlines. The orbits in this case can be extended reversely for months without failing therequirement. Step 1 . For one-year propagation, one applies the following iteration formulas tooffset the initial elements of the three satellites so that the resulting mean elementscan be tuned to the desired nominal values [36]. This effectively compensates for long-term linear drifts in the arm-lengths and breathing angles caused by, predominantly,the lunisolar perturbation. a new = a nom ¯ a old a old , (1) e new = 0 , (2) i new = i nom ¯ i old i old , (3)Ω new = Ω old + (Ω nom − ¯Ω old ) , (4) ω new = 0 . (5)Here for nearly circular orbits, the initial eccentricities and arguments of periapsides areset to zeros. In addition, one adjusts the true anomalies to evenly position the threesatellites along the circle: ν ini1 = 60 ◦ , ν ini2 = 180 ◦ , ν ini3 = 300 ◦ , (6) ν new1 = ν old1 , (7) ν new2 = ν old2 + [120 ◦ − ¯ u old21 ] , ¯ u old21 := ( ν + ω − ν − ω ) old , (8) ν new3 = ν old3 + [240 ◦ − ¯ u old31 ] , ¯ u old31 := ( ν + ω − ν − ω ) old . (9) Step 2 . One repeats the Step 1 for an array of initial times. For our test purposes,we have sampled t ’s over the course of one year (1 Jan. 2034 - 31 Dec. 2034) andset one day apart between two adjacent times. This enables us to take into accountthe combined effect of initial positions of the Moon and the Sun. Then we compareall the orbit evolutions from different t ’s. As discussed earlier, the essence of the Step ianQin constellation stability Figure 4.
Breathing angle variations of one-year orbits obtained from the Step 1,averaged over SC1, 2, 3 with the nominal (mean) elements given in Table 1, and startedat different initial times through out 2034. The curve indicates strong correlation withthe lunar position such that it attains local minimums when the Earth-to-Moon vectoris nearly perpendicular to the orbital plane, e.g., 22 May 2034 (day 142). The optimizedvariation at the level of ∼ . ◦ is consistent with [36] using a different method. t ’s.However, for our single-variable studies, we request that in all the cases the initialtimes be picked within the same month. This is relaxed from requesting the same initialdate, which does not affect the comparisons given the second observation we made inthis section (Fig. 3). In accordance, only the local minimum (such as in Fig. 4) withinthe requested month, not the entire year, will be used in the comparisons.As a comment, the entire algorithm is deterministic, and particularly, we havemanaged to implement the second step in parallel computing. For optimizing one setof orbital parameters, the method takes on average 2 hours to complete on 30 cores,greatly reduced from a few days of the previous approach. This is also achieved withoutcompromising the optimizing capability, which can be seen, for instance, in the case ofFig. 4. Furthermore in the Appendix, we show that the new method generates resultsconsistent with the particle swarm optimization method (Fig. A1).
4. Optimization results
Our study cases involve three orbital parameters, i , Ω, and a . To make the trends clear,we only vary one parameter at a time. Without preference, the starting epoch is set in ianQin constellation stability To show the outcome of changing the orbital orientation, four separate cases areinvestigated with the orbital parameters given in Table 2. More specifically, weconsider two orthogonal orbital planes with both prograde and retrograde orbits. Theycorrespond to four different values of Ω with 90 ◦ apart. For each Ω, we shift theinclination from 0 ◦ to 180 ◦ by ∆ i = 5 ◦ intervals, that is, sampling along a 180 ◦ arcwith the radius 10 km. Table 2.
The (mean) orbital parameters used for studying the impact of orbitalorientations. a i ∆ i Ω10 km 0 − ◦ ◦ ◦ , 120 ◦ , 210 ◦ , 300 ◦ Figure 5.
Impact of orbital orientations on the constellation stability, i.e., one-yearvariations of arm-lengths, relative velocities, and breathing angles, averaged over SC1,2, 3, for Ω = 30 ◦ , ◦ , ◦ , ◦ . The sampling interval is 5 ◦ in the inclination. The results are shown in Fig. 5 for the four cases. One can make a few observations ianQin constellation stability Figure 6.
One-year eccentricity evolutions of the retrograde orbit (red, i = 135 ◦ ,Ω = 30 ◦ ) and the prograde orbit (blue, i = 45 ◦ , Ω = 210 ◦ ) lying in the same plane.The common initial elements are a = 10 km, e = 0, ω = 0 ◦ , and ν = 0 ◦ at the epoch22 May, 2034 12:00:00 UTC (EarthMJ2000Ec). The disparity of the magnitudes isevident. here. First , the four curves follow a similar trend. Changing Ω appears to have arelatively small impact on the constellation stability, which one may expect from anapproximate rotational symmetry of the averaged perturbation about the z -axis in theecliptic coordinate system. The differences in the curves, such as the trough locationsand depths, can be attributed to the 5 ◦ misalignment of the Moon’s orbital plane fromthe ecliptic plane, and the combined initial lunisolar position relative to the orbitalplanes. Second , the inclinations i ≈ ◦ and i ≈ ◦ stand out and exhibit favorablestability performance. It shows that, as we suspected, “standing” constellations areindeed comparatively more stable. Third , inclined orbits moving in the retrograde direction with respect to the Moon’sorbit ( i > ◦ ) generally result in more stable constellations than those in the progradedirection ( i < ◦ ). This also reflects from the fact that the curves in Fig. 5 are lopsidedand asymmetric about the mid-line i ≈ ◦ . This behavior is related to irregular naturalsatellites, which are more commonly found in retrograde orbits. Relevant studies canbe seen, e.g., in [42, 43]. Here, to quickly see why retrograde constellations are morestable, one can compare eccentricity fluctuations of individual orbits rotating in eitherdirection. Indeed, as in Fig. 6, we illustrate that in the same orbital plane retrogradeorbits tend to have lower eccentricities than the prograde ones. Fourth , orbital planes close to the ecliptic ( i ≈ ◦ , ◦ ) perform unfavorably dueto larger distortion per orbit by the Moon’s attraction. ianQin constellation stability Focused on the current TianQin orientation, i.e., the prograde orbits (w.r.t. the Earth)facing J0806 (see Table 1), we exhibit the dependence on the orbital radius from 0 . × km to 1 . × km in Fig. 7. Figure 7.
Impact of orbital radii on the constellation stability, i.e., variationsof arm-lengths, relative velocities, and breathing angles, for the orbital plane facingJ0806 (Ω = 210 . ◦ , i = 94 . ◦ ). The sampling interval is 1000 km. Table 3.
The orbital period ratios between the satellites and the Moon, and thecorresponding orbital radii where the resonance occurs. T sat /T lun T sat (day) 3.4 3.9 4.6 5.5 6.8Radius (10 km) 9.6 10.5 11.6 13.1 15.3 Generally, the stability tends to worsen as the radius increases because of theMoon’s attraction. On top of this trend, several peaks also show up owing to orbitalresonance with the Moon. They take place at the orbital period ratio T sat /T lun = 1 / /
7, 1 /
6, 1 /
5, 1 /
4. The associated nonlinear effect severely undermines the constellationstability and cannot be mitigated to nearby non-resonant levels by the optimization ianQin constellation stability a = 10 km for TianQindoes not fall in the resonant regions. Our result agrees with [37] by a different method.
5. Concluding remarks
In this work, we have studied the influence of choosing different orbital orientationsand radii on the one-year constellation stability. By using a new efficient optimizationmethod, we can identify the ranges of the orbital parameters that show comparativelybetter stability. Three main conclusions can be drawn here.1. The constellation can persist more stably with “standing” (either progradeor retrograde w.r.t the Earth) orbital planes, and inclined retrograde (w.r.t the Moon)orbits with inclinations ∼ ◦ , relative to the ecliptic plane. The dynamics of the latteris related to the irregular moons of the outer planets. Here we recall that OMEGA alsoadopts retrograde orbits [8]. In contrast, orbital planes close to the ecliptic would suffermore severely from the lunar disturbance.2. For a given inclination, altering the longitude of the right ascension has onlya small impact on the stability. It allows a flexibility of re-orienting the detector forpossible enhancement of the science output.3. The stability tends to degrade as the orbital radius increases. For instance,to keep the breathing angle variation within ± . ◦ , the orbital radius is not to exceed1 . × km. Additionally, the regions resonating with the Moon’s orbit should alsobe avoided (Table 3).The findings provide support to the initial TianQin design ( i = 94 . ◦ , a = 10 km) [1] and our speculation made in the introduction. From the perspective ofthe constellation stability, the selectable ranges of orbital orientations and radii arerather broad for TianQin, permitting further adjustment according to engineering andtechnological needs. In addition to the optimized orbits found in [36], other stableoptions, such as the inclined retrograde orbits, have been identified and may be furtherevaluated, from other aspects, as potential backups to TianQin. For future work,systematic refinement of TianQin geocentric orbit design will be carried out, wheremore environmental factors, such as solar eclipses [44], the Earth-Moon’s gravity field[45], and solar illumination, are taken into account. Acknowledgements
The authors thank Dong Qiao, Jianwei Mei, Yi-Ming Hu, Jihe Wang, Defeng Gu, YunheMeng, Jinxiu Zhang, and Jun Luo for helpful discussion and comment. Our gratitudeextends to the developers of GMAT. The work is supported by NSFC 11805287 and11690022. ianQin constellation stability Figure A1.
Comparison of breathing angle variations at different inclinationsobtained from PSO and the new method of Sec. 3.
Appendix A. Results compared with particle swarm optimization
Particle swarm optimization (PSO) features global stochastic search of optimal solutionsby a population of candidates, and has been widely used in computational science[46, 47]. For its implementation, we have adopted a similar cost function from [36]and a swarm of 60 particles. Applied at different inclinations along the arc Ω = 210 ◦ , a = 10 km, both methods yield consistent results and a similar trend in Fig. A1, hencedemonstrating the effectiveness of our new, and more efficient, method. References [1] Luo J et al ., 2016 TianQin: a space-borne gravitational wave detector
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