Effect of Earth-Moon's gravity on TianQin's range acceleration noise
Xuefeng Zhang, Chengjian Luo, Lei Jiao, Bobing Ye, Huimin Yuan, Lin Cai, Defeng Gu, Jianwei Mei, Jun Luo
aa r X i v : . [ g r- q c ] F e b Effect of Earth-Moon’s gravity on TianQin’s range acceleration noise
Xuefeng Zhang, ∗ Chengjian Luo, † Lei Jiao, Bobing Ye, HuiminYuan, Lin Cai, ‡ Defeng Gu, Jianwei Mei, and Jun Luo
1, 2 TianQin Research Center for Gravitational Physics & School of Physics andAstronomy, Sun Yat-sen University (Zhuhai Campus), Zhuhai 519082, P. R. China Center for Gravitational Experiments, School of Physics, MOE Key Laboratory of FundamentalPhysical Quantities Measurement & Hubei Key Laboratory of Gravitation and QuantumPhysics, PGMF, Huazhong University of Science and Technology, Wuhan 430074, P. R. China (Dated: February 17, 2021)TianQin is a proposed space gravitational-wave detection mission using circular high Earth or-bits. The geocentric concept has raised questions about the disturbing effect of the nearby gravityfield of the Earth-Moon system on the highly-sensitive intersatellite ranging measurements. Herewe examine the issue through high precision numerical orbit simulation with detailed gravity-fieldmodels. By evaluating range accelerations between distant free-falling test masses, the study showsthat the majority of the Earth-Moon’s gravity disturbances are not in TianQin’s detection frequencyband above 10 − Hz, and hence present no showstoppers to the mission.
I. INTRODUCTION
The current TianQin design assumes high Earth or-bits with an orbital radius of 10 km [1]. The nearlyequilateral-triangle constellation stands almost verticallyto the ecliptic. High precision laser ranging interferom-etry tracks distance changes between well-protected testmasses (TM) in separate drag-free controlled satellites,within a preliminary frequency range of 10 − − ∗ [email protected] † [email protected] ‡ [email protected] optimization [10, 11] and control to meet the stability re-quirements of the science payloads. On small scales, theperturbations impinge on TMs’ geodesic motion undernearly pure gravity. Since space-based detectors accu-rately measure arm-length variations between TMs, theyrespond not only to GWs (radiation zone), but equallywell to Newtonian gravity fields (near zone), in targetedfrequency bands. Appearing as environmental noise, thelatter should be avoided or mitigated.Ideally, gravitational perturbations in space shouldonly manifest as long-term and slow changes in inter-spacecraft displacement measurements. If there exists aproper separation in the frequencies of gravity-field fluc-tuations and GWs, then the GW signals, superimposedon top of a smooth and slow-varying background, can beextracted ([12], Sec 2.1.1). Therefore the GW detectionrelies heavily on the “quietness” of the ambient gravity-field environment in the measurement band.The problem of environmental gravity disturbanceswere recognized early on in designing ground-based de-tectors [13], and hence is not unique for space missions,where the problem is thought to be much less severe. Inground-based detectors, Newtonian or gravity-gradientnoise caused by terrestrial gravity fluctuations poses alimitation to sensitivity improvement below ∼
10 Hz[14]. Multiple strategies have been developed to effec-tively mitigate such noise, and the techniques have a ma-jor influence on designing next-generation ground-baseddetectors.If not handled properly in space GW detection, dis-turbing gravity fields may induce excessive “orbitalnoise” that encroaches on the sensitivity curve, causing asituation somewhat similar to galactic foreground noisein lower frequencies [15]. The potential risk has drawnattentions for TianQin, and may raise concern for othergeocentric concepts as well. With regard to LISA [16],the majority of the effect is expected to be out of the sen-sitive frequency band because of its heliocentric yearlyorbits and being placed far away from the Earth-Moonsystem ( ∼ ◦ trailing angle, ∼ × km).In general, gravity disturbances in space constitute animportant potential noise source for inter-spacecraft mea-surements. In this work, we aim to determine the am-plitudes and frequencies of the disturbances for Tian-Qin’s orbit, and quantitatively evaluate the impact onTianQin’s acceleration noise requirement. The forwardmodeling takes into account a variety of main gravita-tional perturbations including the gravity fields of theEarth (static and tidal), the Moon, and the Sun, as wellas other solar-system bodies. It requires realistic andaccurate orbit propagation that is also used in perfor-mance assessment and data analysis of gravity mappingmissions such as GRACE [17], GRACE Follow-On [18],GOCE [19], and GRAIL [20]. However, for TianQin, aproblem with insufficiency of double precision arithmetichas emerged owing to the high measurement accuracyrequirement over the long baseline. To tackle the issue,an earlier attempt was made in [21], where analyticalexpansions of perturbed orbits were derived. Unfortu-nately, the approach cannot handle complicated gravityfield models, and only the Earth’s static gravity field wasconsidered without realistic Earth’s rotation (precession,nutation, etc), the Earth’s tides, and third bodies. It mo-tivated us to take a fully numerical approach to be shownin this paper. For other works regarding environmentalmagnetic and plasmic effects on TianQin, one can referto, e.g., [22–24].This is our third paper of the concept study series onTianQin’s orbit and constellation. It is based on the pre-vious work of orbit optimization and constellation stabil-ity [10, 11], and shifts the attention to small-scale orbitalmotion through much refined simulation. The paper isorganized as follows. In Section II, three types of inter-satellite observables are analysed, and the range acceler-ation is chosen for evaluating the impact. In Section III,we describe the high precision orbit propagator, detailedforce models, and orbital parameters used in the assess-ment. Section IV presents the amplitude spectral density(ASD) results of the calculated range accelerations. Atthe end, the conclusions are made in Section V. II. OBSERVABLES AND CRITERIA
For the evaluation purpose, the numerical simulationshould provide an observable accuracy better than theinstrumental measurement noise level. The selectable in-tersatellite observables include the (instantaneous) range,range-rate, and range acceleration. Mathematically, theyare interchangeable by differentiation and integration.But their numerical calculations require different compu-tational resources. Here we estimate the magnitudes oftheir numerical ranges (numbers of significant digits re-quired) for TianQin. First, the range between two satel-lites is given by ρ = | r − r | , (1) where r , denotes the position vector of each satelliterelative to the Earth’s center. Taking the baseline 1 . × m and the displacement measurement noise 1 × − m/Hz / [1], the numerical representation of the rangeobservable requires at least 20 digits, exceeding the 16digits of the double-precision format (64 bits). Second,the range rate reads˙ ρ = ˆ e · (˙ r − ˙ r ) , (2)with the unit vector ˆ e = ( r − r ) /ρ . The relativevelocities between the TianQin satellites is expected to bewithin ± × − m/s/Hz / ( ∼ πf × − m/Hz / atthe crossover frequency f ∼ − Hz of the displacementand residual acceleration noises [1]), the dynamical rangeof ˙ ρ takes up about 15 digits. Third, differentiating therange rate yields the equation for the range acceleration:¨ ρ = ˆ e · (¨ r − ¨ r ) + 1 ρ (cid:0) | ˙ r − ˙ r | − ˙ ρ (cid:1) , (3)where, on the right-hand side, the first term representsprojected differential acceleration, and the second termcentrifugal acceleration. The gravitational accelerationof one TianQin satellite is in the order of 10 − m/s .This is 13 order of magnitude greater than the residualacceleration noise level of one TM, i.e., 1 × − m/s [1].For either ˙ ρ or ¨ ρ , if one takes into account that numericalerrors compounded over time may occupy 2-3 digits, andredundant numerical accuracy another 1-2 digits, thenthe requirement would exceed 16 digits. Therefore, thecommonly used double-precision arithmetic is insufficientin representing the intersatellite observables, and the as-sociated roundoff error becomes a bottleneck for precisionimprovement (cf. Fig. 1).Among the three observables, the range accelerationappears more favorable for taking up less digits in numer-ical computation. In the frequency domain, accelerationand displacement can be easily converted. For evaluatinggravity disturbances in space, we henceforward adopt therange acceleration as the main observable (cf. [25]), anddirectly compare its ASD with the intersatellite residualacceleration noise requirement √ × − m/s /Hz / at 10 − − − Hz as the criteria, which is simply √ − Hz in the future [1, 16].
III. SIMULATION AND FORCE MODELS
The evaluation requires careful calculation and mod-eling of satellite orbits and gravity fields. The accuracyof numerical integration must surpass the noise require-ment √ × − m/s /Hz / of the range accelerationobservable by at least one order of magnitude. The forcemodeling should be sufficiently detailed and up-to-date toreflect as many significant gravity disturbances as possi-ble, particularly those that may enter the detection band. A. Quadruple Precision Orbit Propagation
There exist a few strategies to tackle the inadequacy ofdouble precision. A straightforward way is by extendingto 34 significant digits with quadruple precision arith-metic (128 bits). The potential downsides are low exe-cution speed and heavy programming workload. Follow-ing this “brute force” approach, the TQPOP (TianQinQuadruple Precision Orbit Propagator) program basedon MATLAB has been developed so as to evaluate therange acceleration at < − m/s /Hz / levels. Thequadruple precision data type is applied to all the nec-essary aspects of the program, including parameter in-puts, ephemeris data outputs, reference frame transfor-mations, time conversion, numerical integration, forcemodels, etc. For the nearly circular high orbits, the in-tegrator uses the 8th-order embedded Prince-Dormand(DP87) method [27] with a constant step size of 50 sec-onds (Nyquist frequency 10 − Hz). The algorithm pro-vides a relative truncation error of < − (more than20 significant digits) in both satellite positions and veloc-ities. Thereby the range acceleration error is estimatedto be < − m/s and well below 10 − m/s . Theroundoff error due to finite digits is approximately 10 − m/s /Hz / , and no longer poses a limiting factor (seeFig. 1), which otherwise would overwhelm gravity fieldsignals in the case of double precision. To mitigate lowefficiency of quadruple precision calculations, great effortwas made on optimizing code execution to have reducedthe run time significantly. Other quadruple precision or-bit simulations can be found in, e.g., [28, 29]. B. Detailed Force Models
As the satellites are drag-free controlled, we only con-sider pure free-fall orbits of the TMs in order to focuson the gravitational perturbative effects. Excluding non-gravitational forces, the force models implemented aresummarized in Table I. The types of the gravity fieldmodels are comparable with those used in the Earth’sgravity field determination in satellite missions such asGRACE [17, 30] and GOCE [19].The solar system ephemeris uses DE430 [31] includingall eight planets and the Moon. The effect from the mainbelt asteroids is estimated not to enter LISA’s detectionband [32] , nor TianQin’s due to the shorter arm-length.Hence they are not included in the simulation.For the Earth’s orientation, the International Astro-nomical Union (IAU) 2006 precession and IAU 2000Anutation models [33] are used with the help of the Stan-dards Of Fundamental Astronomy (SOFA) software col-lection [34]. The Earth’s polar motion adopts the IERSEarth Orientation Parameters (EOP) 14 C04 data series[35].Earth’s non-spherical static gravity field is providedby the EGM2008 model [36], following the recommen-dation of IERS (2010) [33]. The normalized spherical -4 -3 -2 Frequency [Hz] -35 -30 -25 -20 -15 -10 R ange a cc e l e r a t i on AS D [ m s - H z - / ] Quadruple precisionDouble precisionAcc noise req 1.73e-15
FIG. 1. The ASD of the range acceleration ¨ ρ between twosatellites in circular orbits of the radius 10 km and separatedby 120 ◦ in phase. The orbits are integrated with a constantstep size of 50 s and under the central force of the Earth’spoint mass. The roundoff error of the quadruple precisionarithmetic is at the level of 10 − m/s /Hz / (at 10 − Hz).The curve tilts up toward low frequencies due to accumulationof roundoff errors over time. For comparison, the roundofferror of double precision is also shown, but at a much higherlevel of 10 − m/s /Hz / (at 10 − Hz), hence not sufficientfor the accuracy requirement. harmonic coefficients ( C nm , S nm ) are kept up to the12th degree and order. High-degree terms decay rapidlywith increasing radius as 1 /r n +1 . Our numerical testsand perturbation analysis shows that the effect from the9th-degree gravity field has already dropped below 10 − m/s /Hz / . The contribution from the 12th degree sinksdeeper to the level of 10 − m/s /Hz / . Hence we deemit safe to truncate at the degree and order 12.Temporal variations of the Earth’s gravity field areadded as corrections to the spherical harmonic coeffi-cients. To model the Earth’s tidal effects, we have fol-lowed IERS (2010) [33] and taken into account solidEarth tides (an-elastic), ocean tides, solid Earth poletide, and ocean pole tide, as specified in Table I. Thewidely-used ocean tide model FES2004 [37] includes long-period (Ω , Ω , S a , S sa , M m , M f , M tm , M sqm ), diurnal( Q , O , P , K ), semi-diurnal (2 N , N , M , S , K ),and quarter-diurnal ( M ) waves. The coefficients up tothe degree and order 10 are used. Additionally, atmo-spheric tides are incorporated, though their effect is smallcompared to the solid Earth and ocean tides. The associ-ated model [38] consists of the diurnal and semi-diurnalwaves S and S in the highest frequency constituents.The correction is made up to the degree 8 and order 5.The non-tidal temporal gravity changes have been esti-mated to be orders of magnitude smaller than the staticgravity [39], and will be discussed elsewhere.Moon’s liberation varies about ± ◦ , and is provided byDE430 ([31], IIE). For the Moon’s static gravity field, weuse GL0660B [20] up to the degree and order 7, and the TABLE I. The list of force models implemented in the simu-lation.Models SpecificationsSolar system ephemeris JPL DE430 [31]Earth’s precession & nutation IAU 2006/2000A [33]Earth’s polar motion EOP 14 C04 [35]Earth’s static gravity field EGM2008 ( n = 12) [36]Solid Earth tides IERS (2010) [33]Ocean tides FES2004 ( n = 10) [37]Solid Earth pole tide IERS (2010) [33]Ocean pole tide Desai (2003) [33]Atmospheric tides Biancale & Bode (2003) [38]Moon’s libration JPL DE430 [31]Moon’s static gravity field GL0660B ( n = 7) [20]Sun’s orientation IAU [40], Table 1Sun’s J IAU [40], Table 1relativistic effect post-Newtonian [41] effect of the 7th degree and order alone is below 10 − m/s /Hz / . The model was obtained from the GRAIL(Gravity Recovery and Interior Laboratory) mission [20]with improved low-degree harmonics. The lunar tide isnot included, since the effect is quite small and (semi-)monthly periodic, hence out of the detection band, own-ing to the Moon’s tidal locking with the Earth. The val-ues of the Sun’s oblateness J and orientation are takenfrom [40] (see also [31], Table 9). Moreover, relativisticeffect is added as post-Newtonian correction terms to theequations of motion [41]. The effect is slow varying andexpected to be outside the detection band. C. Orbital Parameters
The initial orbital parameters are given in Table II.The integration lasts for one observation window of threemonths [1], that is, from 06 Jun. to 04 Sep. 2004 for 90days, when the orbital plane is facing the Sun within ± ◦ . The year 2004 is chosen without preference but totake advantage of the available EOP observation data,which is more accurate than prediction in 2030s. Ourtests have shown that the dominant spectral behaviordoes not depend on a specific year chosen.To make the simulation more realistic, we use theoptimized initial orbital elements in Table II that canmeet TianQin’s constellation stability requirement (e.g.,the breathing angles within 60 ± . ◦ ) for three months[10, 11]. The optimization removes linear drift in thearm-lengths and breathing angles, and prevents thenearly equilateral-triangle constellation from having se-vere distortion. The initial eccentricities are set to ze-ros to keep the orbits almost circular. Note that evenif one starts with less optimized initial orbital elements(e.g., the nominal values, a = 10 km, etc), the domi-nant spectral behavior of three months (cf. Fig. 2) willbe unaffected. TABLE II. The initial orbital elements of the TianQin con-stellation in the J2000-based Earth-centered equatorial coor-dinate system at the epoch 06 Jun. 2004, 00:00:00 UTC forthe evaluation purposes. Here a denotes the semimajor axis, e the eccentricity, i the inclination, Ω the longitude of as-cending node, ω the argument of periapsis, and ν ini the trulyanomaly. a e i Ω ω ν ini SC1 100000 . . ◦ . ◦ ◦ ◦ SC2 100009 . . ◦ . ◦ ◦ ◦ SC3 99995 . . ◦ . ◦ ◦ ◦ IV. SPECTRAL RESULTS
It should be emphasized that the purpose of this workis to determine the frequency-domain effects (especially > − Hz) of various gravity disturbances on the rangeacceleration observable, and it concerns less about theabsolute accuracy of an integrated orbit, which may driftaway from true values over long time scales outside thefrequency band of interest.
A. Total Effect
The overall result of the range acceleration ASD is pre-sented in Fig. 2 for the arm SC1-SC2 using the models ofTable I assembled together in the simulation. The plotsfor the other two arms severely overlap with the first one,hence not presented for clarity.In the frequency domain, the gravity field signals aredominating below 10 − Hz, and roll off rapidly in am-plitudes toward high frequencies, and intersect with thelower end of the range acceleration noise requirement at1 × − Hz The steep fall-off roughly follows a powerlaw of ∼ . × − m/s /Hz / × (0 . /f ) near f = 1 × − Hz. The plot demonstrates that the effectof the gravity field models in Table I does not enter thedetection band > − Hz. Note that the slanted partof the ASD curve ( < − m/s /Hz / and > − Hz,marked by “Numerical error” in Fig. 2) is an artifact ofnumerical interpolation of the EOP data.The frequency-domain behavior somewhat resemblesthe one in the intersatellite laser ranging measurementresult of the GRACE Follow-On mission [18], whichalso shows a steep fall-off, but at a higher frequency( ∼ × − Hz) because of its low orbit altitude of ap-proximately 500 km.
B. Effect Breakdown
Now we examine various contributions to the totalrange acceleration ASD of the arm SC1-SC2. The re-sult is presented in Fig. 3.
FIG. 2. The range acceleration ASD of two TianQin satellitesSC1 and SC2, calculated in quadruple precision and with themodels of Table I and step size 50s using 90 days of data.The orbital period 3.6 days corresponds to 3 . × − Hz.The plots for SC1-SC3 and SC2-SC3 are nearly identical tothe one shown above.
The Earth’s non-spherical static gravity field with de-grees n ≥ × − Hz in the totalASD, indicated by the overlapping of the blue and redcurves. The effect decreases rapidly toward high frequen-cies and impinges on the noise requirement at 10 − Hz.At the high altitude of TianQin, high-degree harmonicsare effectively attenuated.The contribution from the Moon’s non-spherical staticgravity field ( n ≥
2) is minute and only sticks out in thelow-frequency region. One may have expected so sincethe Moon is a slowly rotating body. The same argumentalso justifies the omission of the lunar tides in the simula-tion. Nevertheless, the Moon’s point mass and its orbitalmotion play an important role, largely accounting for thetotal ASD below 4 × − Hz.The effects of relativity and Earth’s tidal gravity field(solid Earth, oceanic, pole, and atmospheric, cf. Ta-ble I) are considerably smaller than the total effect, andboth peak at low frequencies away from the detectionband. These low frequency disturbances show no signif-icant coupling into high frequencies, and do not inducepronounced range acceleration response above 10 − Hz.
C. Model Errors
The models inevitably contain errors. To estimatetheir effect, a straightforward way is to determinewhether discrepancies between different models can sig-nificantly alter the spectral result in Fig. 2.For a cross-check, we test another set of gravity fieldmodels shown in Table III where several replacementsare made to Table I. The substitute models are deemedless accurate than the corresponding more recent onesin Table I, and thus can mimic model errors (see, e.g., -6 -5 -4 -3 -2 Frequency [Hz] -20 -15 -10 -5 T Q r ange a cc e l e r a t i on AS D [ m s - H z - / ] Earth static n=3-12Earth tidesMoon static n=2-7RelativisticMoon pt. massTotal Tab1 90d 50sAcc noise req 1.73e-15
FIG. 3. Components of the range acceleration ASD of twoTianQin satellites SC1 and SC2. The total ASD in Fig. 2 isduplicated in dotted red curve for comparison. [42] ). In Fig. 4, both ASD results show good agree-ment with each other, and the difference ∆¨ ρ is well belowthe noise requirement. In addition, the spectral behaviorabove 10 − m/s /Hz / is also confirmed by running theflight-qualified, open source program GMAT [43] in dou-ble precision. Hence the overall frequency-domain behav-ior appears to be robust, which instills more confidencein the results. TABLE III. The list of replacement force models to Table Iused for spectrum comparison.Models SpecificationsSolar system ephemeris JPL DE405 [44]Earth’s precession & nutation IAU 1976/1980 [41]Earth’s static gravity field EGM96 ( n = 12) [45]Moon’s libration JPL DE405 [44]Moon’s static gravity field LP165P ( n = 7) [46] V. CONCLUSION
The TianQin mission, to be deployed in a high Earthorbit, shares technological similarities with low-Earthgravimetry missions using satellite-to-satellite tracking.They diverge on a key point that the Earth’s gravity fieldsignals targeted in gravimetry missions become environ-mental noise in TianQin’s GW detection. Hence Tian-Qin must keep a safe distance from the Earth by flyinghigh enough, so as to push the Earth’s gravity field in-terference out of the detection band. This work has beendevoted to evaluating and examining this type of effect,and two main conclusions can be drawn here.1. With the orbital radius of 10 km for TianQin, thecurrent models show that the effect of the Earth-Moon’sgravity field dominates at low frequencies, and that the -6 -5 -4 -3 -2 Frequency [Hz] -20 -15 -10 -5 T Q r ange a cc e l e r a t i on AS D [ m s - H z - / ] Tab1+3 90d 50sTab1 90d 50sDifferenceAcc noise req 1.73e-15
FIG. 4. The range acceleration ASD of two TianQin satellitesSC1 and SC2 using the models of Table III for replacementto Table I (Tab1+3, blue). The ASD in Fig. 2 is duplicatedin dashed red curve (Tab1) for comparison. The ASD of theirdifference ∆¨ ρ is marked by cyan curve. amplitude rolls off rapidly toward high frequencies andintersects with the range acceleration noise requirement( √ × − m/s /Hz / ) at 10 − Hz, right on the lowerend of the preliminary detection band. To provide morecontext, the gravity field signals from GRACE-FO laserranging interferometry along a ∼
200 km baseline fallsoff at about 4 × − Hz [18] with an orbit altitude of ∼
500 km.2. The high-precision numerical simulations help torule out the majority of the perturbing gravity sources for TianQin, including the Sun’s point mass and its J , thesolar system planets’ point masses (under their orbitalmotion), the Earth’s static gravity (with its rotation),the Earth’s tidal gravity changes (solid Earth, oceanic,pole, and atmospheric), Moon’s static gravity, relativisticeffect, etc (cf. Table I). These effects are slowly varying,not entering the detection band, and present no show-stoppers for TianQin. The Newtonian gravity-field en-vironment at a distance of 10 km from the Earth isexpected to be fairly “quiet” for TianQin.The results can provide useful inputs and guidelinesto several aspects of the mission concept studies, suchas orbit selection, noise reduction, and data processing.For future works, further refined gravity models will beexplored to identify other possible noise sources. Onanother note, the strong low-frequency gravity field sig-nals ( < − Hz) illustrated in Fig 2, which carry long-wavelength gravity information, may find potential ap-plications in geodesy and geophysics [47]. This may helpto enrich TianQin’s secondary science output.
ACKNOWLEDGMENTS
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