Orbital effects on time delay interferometry for TianQin
Ming-Yue Zhou, Xin-Chun Hu, Bobing Ye, Shoucun Hu, Dong-Dong Zhu, Xuefeng Zhang, Wei Su, Yan Wang
OOrbital effects on time delay interferometry for TianQin
Ming-Yue Zhou, Xin-Chun Hu, Bobing Ye, Shoucun Hu,
3, 4
Dong-Dong Zhu, Xuefeng Zhang, Wei Su, and Yan Wang ∗ MOE Key Laboratory of Fundamental Physical Quantities Measurements,Hubei Key Laboratory of Gravitation and Quantum Physics,PGMF, Department of Astronomy, and School of Physics,Huazhong University of Science and Technology, Wuhan 430074, China TianQin Research Center for Gravitational Physics & School of Physics and Astronomy,Sun Yat-sen University (Zhuhai Campus), Zhuhai 519082, China CAS Key Laboratory of Planetary Sciences, Purple Mountain Observatory,Chinese Academy of Sciences, Nanjing 210033, China CAS Center for Excellence in Comparative Planetology, Hefei 230026, China (Dated: February 23, 2021)The proposed space-borne laser interferometric gravitational wave (GW) observatory TianQinadopts a geocentric orbit for its nearly equilateral triangular constellation formed by three identicaldrag-free satellites. The geocentric distance of each satellite is ≈ . × km, which makes thearmlengths of the interferometer be ≈ . × km. It is aimed to detect the GWs in 0 . − − ∼ − s for the first-generation TDIand ∼ − s for the second-generation TDI, respectively. While the second-generation TDI isguaranteed to be valid for TianQin, the first-generation TDI is possible to be competent for GWsignal detection with improved stabilization of the laser frequency noise in the concerned GWfrequencies. ∗ [email protected] a r X i v : . [ a s t r o - ph . I M ] F e b I. INTRODUCTION
The detection of gravitational waves (GWs) from the coalescence of a stellar-mass black hole binary (GW150914)by advanced LIGO detectors [1] has opened up the era of observational GW astronomy. During the first two observingruns (O1 and O2) [2] and the first half of O3 [3] of advanced LIGO and advanced Virgo, more than 50 compact binarycoalescences, including the first observation of binary neutron star inspiral (GW170817) [4], have been detected.Both advanced LIGO and advanced Virgo will reach their design sensitivities in the coming years which can boostthe detection of GW events to higher rate. In addition, the underground cryogenic GW telescope KAGRA [5] hasrecently joined in the advanced ground-based detector network.The observational window of GW astronomy will be broadened to millihertz range (0.1 mHz–1 Hz) by the proposedspace-borne laser interferometers, such as Laser Interferometer Space Antenna (LISA) [6], TianQin [7], DECIGO[8], ASTROD-GW [9], gLISA [10], Taiji (ALIA descoped) [11] and BBO [12]. Among these, LISA has been com-prehensively studied and developed for more than three decades [13]. In 2017, it has been selected as ESA’s L-3mission Cosmic Vision programme with the theme of “the Gravitational Universe”. LISA is scheduled for launchin early 2030s and will be operated concurrently with ESA’s next generation Advanced Telescope for High ENergyAstrophysics (Athena). The latter will tremendously enhance the follow-up X-ray observation of the electromagneticcounterparts of LISA’s GW source candidates [14]. The successful flight of LISA Pathfinder has demonstrated thefeasibility of the key technologies, such as gravity reference system and space laser interferometry, to be implementedin LISA [15, 16]. The recent GRACE Follow-on mission has demonstrated the technologies to be used in the laserranging interferometer (LRI) of LISA [17].Similar to LISA, TianQin is comprised of three identical drag-free satellites that form a nearly equilateral triangularconstellation [7]. Each pair of satellites is linked by two one-way infrared laser beams which can be used, together withthe intra-satellite laser links, to synthesize up to three Michelson interferometers. Distinct from LISA, TianQin adoptsa geocentric orbit with an altitude of 10 km from the geocenter, hence the armlength of each side of the triangleis approximately 1 . × km. The detector plane formed by the three satellites faces to the galactic white dwarfbinary RX J0806+1527 [18] (see Fig. 1). The guiding center of the constellation coincides with the geocenter, andthe period of each satellite orbiting around the Earth is 3.65 days. Analytic approximation of the orbit coordinatesfor each satellite and the strain output of a Michelson interferometer for arbitrary incoming GWs are studied in[19]. A series of study on TianQin’s orbit and constellation, including constellation stability optimization [20], orbitalorientation and radius selection [21], eclipse avoidance [22], and the Earth-Moon’s gravity disturbance evaluation [23],have been conducted. The recent progresses in the investigations of both science case and technological realizationfor TianQin can be found in [24–34]. A brief summary can be found in [35].The GW sources in the mHz frequency regime are rich, which include coalescing supermassive black hole binaries(SMBHBs), ultracompact binaries in the Galaxy, extreme mass ratio inspirals (EMRIs), stochastic GW background,etc [36, 37]. For TianQin, preliminary studies on the detection rate of the SMBHBs [25], the associated parameterestimation accuracy based on the inspiral signals [24], and testing the no-hair theorem [26] and constraining themodified gravity [27] with the post-merger ringdown signals have been carried out. The prospects for detectinggalactic double white dwarfs [30], EMRIs [29] and stellar-mass black hole binaries [28] with TianQin have beeninvestigated.Unlike the ground-based interferometers, the armlengths of a space-borne interferometer are unequal and varying intime. Therefore, the common mode laser frequency (or phase) noise, which is 7 − α, β, γ, ζ ), the eight-pulse combinations, such as unequal-arm Michelson ( X, Y, Z ), Relay (
U, V, W ), Monitor (
E, F, G )and Beacon (
P, Q, R ) (see examples in Fig. 4) [39, 47], and the optimal combinations (
A, E, T ) [48].All data combinations are aimed to make the lengths of the two symmetric interference paths in the synthesizedinterferometric measurements nearly equal. However, in the reality, this equality cannot be exactly satisfied dueto the orbital dynamics of each satellite. The second-generation TDI has been proposed to further account for therotation of the constellation and the linear variation of armlengths [44, 45, 49], which improves the length equalityby judiciously splicing the first-generation interference paths [50]. This results in more data combinations than thefirst-generation TDI.The result from
Synthetic LISA , based on the analytic approximation of LISA spacecraft’s orbits [51], shows thatthe time differences of the symmetric interference paths are 10 − s and 10 − s for the first- and second-generationTDI, hence the latter must be adopted in LISA data analysis to comfortably cancel out the laser phase noises [52].This is further confirmed by the detailed numerical simulations, based on the orbits optimized with CGC 2.7 ephemeris[53], of various first- and second-generation TDI data combinations for (e)LISA [54, 55]. Similar investigations havealso been conducted for ASTROD-GW [56, 57] and Taiji [58].In this work, we simulate the time differences of the symmetric interference paths of various TDI data combinationsfor TianQin. The optical paths are evaluated based on the numerical orbits of TianQin’s satellites that have beenoptimized to meet the orbital stability requirements imposed by the long range space laser interferometry. Four typesof the first-generation TDI data combinations show time differences of ∼ − s which makes them competent forGW signal detection for the frequencies (cid:46) − Hz and (cid:38) − Hz given the stabilization of the laser frequency noiseof 10 Hz / √ Hz in concerned frequency range. With an ample margin, the second-generation TDI data combinationswith time differences of ∼ − s are warranted to reduce the laser frequency noise well below the secondary noises.The rest of the paper is organized as follows. In section II, we discuss the orbit optimization for TianQin’s satellites.The resulting numerical orbits interpolated by Chebyshev polynomials are subsequently used in the simulations of thetime differences of the symmetric interference paths for various first- and second-generation TDI data combinationsin section III A and III B, respectively. The paper is concluded in section IV. II. CONSTELLATION OPTIMIZATION FOR TIANQIN
The orbit of each satellite is primarily determined by the monopole gravitational field of the Earth. Besides, theperturbation from the multipole terms and relativistic post-Newtonian correction of the Earth gravitational field, themonople gravitational field of the Moon, the Sun, the major planets in the solar system, Pluto and large asteroidswill also contribute. Dispersion from the Earth atmosphere and solar radiation are ignored due to the implementationof the drag-free control for the satellite platform. The minor eccentricity of the nominal Keplerian orbit along withtime-dependent perturbation forces will induce variation of the armlengths, and flexing and breathing of the triangularconstellation. On the other hand, the high precision space laser interferometry imposes requirements on the stabilityof the triangular constellation [7, 20]: (a) the armlength variation less than 0 . × ( √ × ) km; (b) the breathingangle (subtended by two arms) variation less than 0 . ◦ during the first two years and less than 0 . ◦ during the fiveyears mission lifetime; (c) the relative range rate (Doppler velocity) less than 5 m / s during the first two years andless than 10 m / s in five years. A. Calculation of the satellite orbits
In this work, the geocentric ecliptic coordinate system ( x, y, z ) shown in Fig. 1 has been adopted in the calculationand optimization of the orbits for TianQin’s satellites. This choice is different from [19] in which, for the sakeof calculating the antenna response of TianQin to GWs, the heliocentric ecliptic coordinate system is used. Thegravitational field of the Earth is described in the Earth-fixed reference coordinate system WGS84 [59].From the six initial Keplerian elements σ = ( a k , e k , i k , Ω , ω, M k ) of the k -th ( k ∈ { , , } ) satellite, we can obtainthe initial Cartesian coordinates in the geocentric ecliptic coordinate system as follows [60]: x k = l a k (cos E k − e k ) + l a k √ − e k sin E k ,y k = m a k (cos E k − e k ) + m a k √ − e k sin E k ,z k = n a k (cos E k − e k ) + n a k √ − e k sin E k . (1)Here a k ≈ . × km is the semi-major axis, and e k ≈ E k can beobtained by the Newton’s iteration method [61]: E n +1 ,k = E n,k + M k − E n,k + e k sin E n,k − e k cos E n,k , ( n = 0 , , , · · · ) , (2)where n is the index of the iteration and E ,k = M k . We set the mean anomaly M k = (cid:36) k ( t − t ) + 120 ◦ × ( k −
1) + 60 ◦ in order to arrange the three satellites into a nearly equilateral triangle constellation. (cid:36) k = (cid:112) GM ⊕ /a k representsthe angular velocity of the satellite, where M ⊕ is the Earth mass. The surrogate variables ( l , , m , , n , ) in Eq. 1are defined as follows: l = cos Ω cos ω − sin Ω sin ω cos i k ,m = sin Ω cos ω + cos Ω sin ω cos i k ,n = sin ω sin i k ; l = − cos Ω sin ω − sin Ω cos ω cos i k ,m = − sin Ω sin ω + cos Ω cos ω cos i k ,n = cos ω sin i k . (3)Here we set the argument of periapsis (the angle between ascending node and perigee) ω = 0. The three satellitesform a detector plane (subtended by x (cid:48) and y (cid:48) axes in Fig. 1), the normal of which ( z (cid:48) axis) points toward the referencesource RX J0806+1527. It thus indicates that the inclination of the detector plane i k = 94 . ◦ and the longitude ofthe ascending node (the angle between x axis and x (cid:48) axis) Ω = 210 . ◦ .Finally, through coordinate transformation, we obtain the expression for the initial positions of the satellites ingeocentric equatorial coordinates. Here we choose obliquity of the ecliptic ε = 23 ◦ (cid:48)(cid:48) . x y z S S S x z y i Ecliptic plane
J0806+1527
FIG. 1. ( x, y, z ) represents the geocentric ecliptic coordinate system, x axis points toward the mean equinox of J2000. ( x (cid:48) , y (cid:48) , z (cid:48) )represents the orbit coordinate system of the detector, x (cid:48) axis points toward the ascending node of the satellites and z (cid:48) axispoints toward the reference source RX J0806+1527. Ω is the angle between x and x (cid:48) , i is the inclination of the detector plane. The subsequent evolution of the satellites’ orbits, after setting up the initial condition, is determined by the dynamicequation [59] (for clarity we ignore the satellite index k hereafter), ¨r = ¨r NB + ¨r NS + ¨r PN , (4)where r represents the position vector of the satellite relative to the geocenter. The Newtonian body term is ¨r NB = − GM ⊕ r /r − (cid:88) i =1 Gm i ( ∆ i / ∆ i + r i /r i ) , ( i = 1 , , , · · · , . (5)Here G is the gravitational constant. m i , in the ascending order of i , denotes the masses of the seven major planets,Pluto, the Sun and the Moon. r i and ∆ i = r − r i are the position vectors of the i -th perturbing body relative to thegeocenter and the satellite, respectively. r i = | r i | and ∆ i = | ∆ i | . The positions of the planets, the Sun, the Moonare extracted from the ephemeris DE421 [62]. ¨r NS includes the sectorial and tesseral harmonic terms of the Earth’snon-spherical gravitational perturbation, respectively. The first-order post-Newtonian correction term [63–65] is ¨r PN = GM ⊕ c r (cid:2)(cid:0) GM ⊕ /r − v (cid:1) ( r /r ) + 4 ( r · ˙ r ) (˙ r /r ) (cid:3) , (6) TABLE I. The values of the elements of the optimized orbit ¯ σ ∗ for the three satellites at the initial epoch t = MJD 64104.5.Semi-major axis Inclination Eccentricity a (km) i ( ◦ ) e S1 99995.0717 94.7473 2 . × − S2 100010.8914 95.7094 0S3 99992.5623 94.6469 2 . × − where v = | ˙ r | , and c is the speed of light.Once set the orbit elements at an initial epoch, for example t = MJD 64104.5 (May 22 2034 12:00:00 TDB),we can numerically integrate Eq. 4 by Runge-Kutta 7(8) method to obtain r and ˙r at subsequent time stamps t i ( i = 1 , , · · · , N ), where N = T / ∆ t and T = 5 yr. The integration time stepsize ∆ t = 1 hr, which is chosensuch that the total computational error (combining the accumulated round-off error from arithmetic operations ofdouble-precision float number and the truncation error of Runge-Kutta algorithm) approaches its minimum. B. The constellation optimization
In order to fulfill the aforementioned requirements on orbit stability, we try to find a set of initial orbit elements inthe vicinity of the fiducial ones such that during the mission lifetime the armlength variation measured by the sumof squared length differences of the three arms reaches a global minimum in the parameter space. It has been provedeffective in practice to only search in the nine dimensional parameter space spanned by ¯ σ = ( a k , i k , e k ), k ∈ { , , } [66]. Thus, the constellation optimization problem can be formulated as follows,¯ σ ∗ = arg min ¯ σ ∈ D O (¯ σ ) = arg min ¯ σ ∈ D (cid:88) k =1 N (cid:88) i =1 ( L k,i − L k, ) , (7)where L k,i represents the length of the arm opposite to the k -th satellite at time t i , and L k, is the armlength at theinitial epoch. D is the search space of the optimization with the ranges: a k ∈ [0 . , . × km, e k ∈ [0 , × − ], i k ∈ [93 . ◦ , . ◦ ].The objective function O (¯ σ ) may have a highly multi-modal landscape owning a forest of local minima. Thereforeany deterministic local optimizer would locate a suboptimal solution. On the other hand, a direct grid search will becomputationally prohibitive due to the high dimension (dim( D ) = 9). Instead, the optimization methods includingsome randomness can be used to effectively pinpoint the global minimizer in O (¯ σ ). In this work, we use one of thestochastic optimization methods based on emulating biological groups, namely the particle swarm optimization (PSO)[67]. PSO has been applied in many fields [68], including gravitational wave data analysis for detecting and estimatingcompact binary coalescence signals in a network of ground-based laser interferometers [69, 70] and continuous wavesin pulsar timing arrays [71, 72].The resulting ¯ σ ∗ for each satellite is listed in Table I. The variations of the armlengths, the variations of breathingangles, Doppler velocities and pointing deviation (from the direction of RX J0806+1527) of the satellite constellationduring 5-yr mission lifetime are shown in Fig. 2. We can see that the optimized orbits satisfy the aforementionedrequirements on the constellation stability with some margins. Note that, although the constellation optimizationmethod is different from the three-step scheme adopted in [20], the results, in terms of the initial elements andperformance of constellation optimization is virtually consistent (see Fig. 1 in [20]). C. Interpolation of the orbit
The orbit coordinates of the satellites can be represented as finite sequences with sampling interval of ∆ t . As it willbecome more clear in Sec. III, the orbit positions and velocities at arbitrary time points within ( t i , t i +1 ) are needed ingenerating the TDI data combinations. In this case, we use Chebyshev polynomials to approximate the orbit of eachsatellite at t ∈ ( t i , t i +1 ). This method is stable and has been widely used in high precision ephemeris interpolation,such as the DE series ephemerides developed by Jet Propulsion Laboratory (JPL) [73]. Here, Chebyshev polynomialsup to 15 orders are adopted and the sampling interval of the fitted data is 0.5 day with interpolation precision of ∼ − km. The weight of position and velocity is 1 : 0 .
4, which turns out to be the optimal choice to calculate thepositions of the planets in the solar system [73]. The function approximation algorithm finds the best-fit coefficientsof Chebyshev polynomials by minimizing the variance of the residuals between the data and the model [74]. (a) (b) (c) (d)
FIG. 2. ( a ) The variations of armlengths; ( b ) The variations of breathing angles; ( c ) Doppler velocities; ( d ) Pointing deviationfrom J0806+1527 during 5-yr mission lifetime. III. SIMULATION OF TIME DIFFERENCES FOR TDI DATA COMBINATIONS
In this work, we assume that the lasers on the two optical benches housed in a satellite are locked in phase, so weonly consider the inter-satellite fractional frequency measurements ‘ y ’ which account for the cancellation of the laserfrequency noise. Following [47], the naming convention of the interference arms is shown in Fig. 3. A. Time differences for the first-generation TDI
The first-generation TDI has 15 interference data combinations, namely ( α, β, γ ), (
X, Y, Z ), (
U, V, W ), (
E, F, G ),and (
P, Q, R ). Five representative combinations are shown in Fig. 4. The other subtypes are different in the startingsatellite of light paths and their combinations can be obtained by cyclic permutation of the indices for satellites:1 → → →
1. As an example, the Doppler data for the Sagnac-type α combination is as follows: α = y , + y , + y − y − y , (cid:48) − y , (cid:48) (cid:48) , (8) S1 L S2 S3 L L L L L FIG. 3. The labels of the satellites and the interference arms. α X PEU
FIG. 4. Schematic diagram of the first-generation TDI combinations. where ‘,’ marks the time delay of the laser beam traveling along an arm. y , = y ( t − L − L ), which representsthe time-delayed fractional frequency fluctuation time series measured at reception by S3 with transmission fromS1 along arm 2. Here and hereafter, we set c = 1. As in Fig. 4, the clockwise (1-2-3-1) and the counter clockwise(1-3-2-1) light paths interfere at S1 at time t . While, their initial times of emission at S1 are t − L (cid:48) − L (cid:48) − L (cid:48) and t − L − L − L , respectively. Inserting the fractional frequency fluctuation C i ( t ) of the laser on S i into Eq. 8, weobtain: δC α ( t ) =[ C ( t − L − L − L ) − C ( t − L − L )] + [ C ( t − L − L ) − C ( t − L )]+ [ C ( t − L ) − C ( t ))] − [ C ( t − L (cid:48) ) − C ( t )] − [ C ( t − L (cid:48) − L (cid:48) ) − C ( t − L (cid:48) )] − [ C ( t − L (cid:48) − L (cid:48) − L (cid:48) ) − C ( t − L (cid:48) − L (cid:48) )]= C ( t − L − L − L ) − C ( t − L (cid:48) − L (cid:48) − L (cid:48) ) . (9)The first-generation TDI assumes a fixed detector constellation in space and the armlengths are unequal but constant,thus Eq. 9 can be canceled out exactly. However, in reality, the armlengths are time varying due to the actual rotatingand flexing of the constellation. To the first order of L l ( t ), Eq. 9 can be expanded as: δC α ( t ) (cid:39) ˙ C ( t )[ − L + ˙ L ( L + L ) − L + ˙ L L − L + L (cid:48) − ˙ L (cid:48) ( L (cid:48) + L (cid:48) ) + L (cid:48) − ˙ L (cid:48) ( L (cid:48) ) + L (cid:48) ] (10)where both L l and ˙ L l are evaluated at time t . When only the zero order of L l ( t ) is concerned, as in the first-generationTDI, Eq. 10 vanishes. The data combinations that have this nature are called L-closed [50]. Therefore, the level oflaser frequency noise cancellation is determined by how well we can synthesize equal-length virtual paths, such asthe two paths for the α combination, in the construction of virtual interferometers. In other words, from Eq. 9, wecan see that the magnitude of the residual laser noise can be measured by the time difference of the two interferencepaths, which, for the α combination, is∆ t α = L , + L , + L − L (cid:48) − L (cid:48) , (cid:48) − L (cid:48) , (cid:48) (cid:48) . (11)Here, L , = L ( t − L ( t ) − L ( t )) is the length of L at time t − L ( t ) − L ( t ). Through the armlengths obtainedin Sec. II B, we can calculate the time differences of interference paths, which can be used to analyze the level of thelaser frequency noise cancellation. Fig. 5 presents the result for the α combination in the five years mission lifetime.We can see the net time difference of the two opposite paths induced by the Sagnac effect ∆ t Sag = 4 (cid:126) Ω · (cid:126)A [49, 75].With the area | (cid:126)A | = √ L / L (cid:39) √ × km) and the angular velocity of the constellation | (cid:126) Ω | = 1 / .
65 cycle / dayfor TianQin, ∆ t Sag (cid:39) . × − s. Therefore, the result of this data combination is about three orders of magnitudeworse than the other first-generation data combinations below. FIG. 5. Simulation of the time difference for the first-generation TDI α combination. Following the same procedure, the time differences for the other types can be written as:∆ t X = L (cid:48) , (cid:48) + L , (cid:48) + L , (cid:48) + L (cid:48) − L − L (cid:48) , − L (cid:48) , (cid:48) − L , (cid:48) (cid:48) , ∆ t U = L (cid:48) , (cid:48) (cid:48) + L (cid:48) , (cid:48) + L (cid:48) , + L − L (cid:48) − L (cid:48) , (cid:48) − L (cid:48) , (cid:48) (cid:48) − L , (cid:48) (cid:48) (cid:48) , ∆ t P = L (cid:48) , (cid:48) + L (cid:48) , + L , − L (cid:48) , + L , (cid:48) − L (cid:48) , (cid:48) − L , (cid:48) (cid:48) − L , (cid:48) (cid:48) , ∆ t E = L (cid:48) , + L , + L − L (cid:48) − L (cid:48) , (cid:48) − L , (cid:48) (cid:48) + L (cid:48) , (cid:48) − L , (cid:48) . (12)The corresponding results are shown in Fig. 6. The time differences for the first-generation TDI X , U , P and E datacombinations are ∼ − s.The TDI data combinations can be considered effective when, taking the α combination as an example, | δ ˜ C α ( f ) | ≤ S α ( f ). Here, | δ ˜ C α ( f ) | = 4 π f | ˜ C | δt α , (13)which can be obtained by using the derivative theorem of Fourier transform for Eq. 9 [52]. δt α ≡ sup {| ∆ t α ( t ) |} . S α ( f ) = [4 sin (3 πf L ) + 24 sin ( πf L )] S accel y + 6 S opt y is the secondary noise power spectral density (PSD) for the α combination [39], where S accel y = 2 . × − ( f / − Hz − and S opt y = 4 . × − ( f / Hz − are the PSDsof the acceleration noise and the optical path noise [39] assuming the one-sided amplitude spectra of the accelerationnoise and the optical path noise of a single link are 1 × − m s − / √ Hz and 1 pm / √ Hz, respectively, for TianQin[7, 19]. | ˜ C | = 1 . × − Hz − is the fractional laser frequency fluctuation (corresponding to the raw laser frequencynoise of 10 Hz / √ Hz) [7, 39], which is assumed to be white in the concerned frequency range. Then, we find that therequired time difference δt α ≤ . × − s. (a) (b) (c) (d) FIG. 6. Simulation of the time differences for the first-generation TDI X , U , P and E combinations. Similarly, for the X , U , P , and E combinations with the corresponding secondary noise PSDs as follows, S X ( f ) = [4 sin (4 πf L ) + 32 sin (2 πf L )] S accel y + 16 sin (2 πf L ) S opt y , (14) S U ( f ) = [8 sin (3 πf L ) + 12 sin (2 πf L ) + 24 sin ( πf L )] S accel y + [4 sin (3 πf L ) + 8 sin (2 πf L ) + 4 sin ( πf L )] S opt y , (15) S P ( f ) = [4 sin (2 πf L ) + 32 sin ( πf L )] S accel y + [8 sin (2 πf L ) + 8 sin ( πf L )] S opt y , (16) S E ( f ) = S P ( f ) . (17)The required time differences are shown in the right panel of Fig. 7.The PSDs of the secondary noises and the residual laser frequency noises for the X , U , P , and E combinations areshown in the left panel of Fig. 7. For the latter, the actual time differences δt from Fig. 6 are used. We can see thatthe first-generation TDI data combinations cannot suppress the laser frequency noises below the secondary noises for10 − (cid:46) f (cid:46) − Hz; while they are effective at the frequencies f (cid:46) − Hz and f (cid:38) − Hz.0 -4 -3 -2 -1 0 1-50-48-46-44-42-40-38-36 (a) -4 -3 -2 -1 0 1-11-10-9-8-7-6 (b)
FIG. 7. Left panel shows the secondary noise (solid line) and the residual laser frequency noise (dotted line) for the first-generation X , U , P and E combinations. The lines for P and E are overlapped, only the ones for P are shown. Right panelshows the required time differences for the corresponding combinations. B. Time differences for the second-generation TDI
The second-generation TDI can suppress the laser frequency noise further by canceling the L l and L l ˙ L m terms inEq. 10. The data combinations that have this nature are called ˙ L -closed , for which the interference paths can befound through splicing the first-generation interference paths [44]. Taking the α combination as an example, a newpath 1 (1-3-2-1-2-3-1) can be synthesized by splicing the path 1 (1-3-2-1) and path 2 (1-2-3-1) of the first-generation α ; a new path 2 (1-2-3-1-3-2-1) is in a reversed order of the satellites. Specifically, this process can be expressed as: −−−−−→ ←−−−−− (cid:48) (cid:48) (cid:48) + −−−−−→ (cid:48) (cid:48) (cid:48) ←−−−−− = ⇒ −−−−−−→ (cid:48) (cid:48) (cid:48) ←−−−−−− (cid:48) (cid:48) (cid:48) ( α − y (cid:48) (cid:48) (cid:48) + y (cid:48) (cid:48) (cid:48) + y (cid:48) (cid:48) (cid:48) + y (cid:48) (cid:48) + y (cid:48) + y − y − y − y − y − y (cid:48) − y (cid:48) (cid:48) . (18)Here, y = y ( t − L ( t − L ( t )) − L ( t )). Note that we use ‘;’ here to represent the time delay in the secondgeneration. Two laser beams passing along ‘ −→ ’ and ‘ ←− ’ interfere at time t . The numbers below each arrowrepresent the interference arms, the subscripts on the left hand side represent the indices of satellites. On the righthand side, −−−−−−→ (cid:48) (cid:48) (cid:48) represents a laser beam passing along the arms from left to right; while, ←−−−−−− (cid:48) (cid:48) (cid:48) from right toleft. α : −−−−−→ ←−−−−− (cid:48) (cid:48) (cid:48) represents the paths of two laser beams for the first-generation TDI, then by reversing weobtain α : −−−−−→ (cid:48) (cid:48) (cid:48) ←−−−−− . The path α is spliced with the path α and result in the path α −
1. Here, ‘[ ]’indicates where we insert α into α [50].From Eq. 18, we can write the time difference of the combination α − t α − = L (cid:48) (cid:48) (cid:48) + L (cid:48) (cid:48) (cid:48) + L (cid:48) (cid:48) (cid:48) + L (cid:48) ;1 (cid:48) (cid:48) + L (cid:48) ;2 (cid:48) + L (cid:48) − L − L − L − L (cid:48) ;213 − L (cid:48) ;2 (cid:48) − L (cid:48) ;1 (cid:48) (cid:48) . (19)Similarly, the path α : −−−−−→ (cid:48) ↑ (cid:48) (cid:48) ←−−−−− can be split at the position of ‘ ↑ ’. This results in a new path −−−→ (cid:48) (cid:48) ←−−−−− −→ (cid:48) , which makes S2 as the starting and ending satellite. Stitching it with α , we can obtain anew combination: −−−−−→ ←−−−−− (cid:48) (cid:48) (cid:48) + −−−→ (cid:48) (cid:48) ←−−−−− −→ (cid:48) = ⇒ −−−−→ (cid:48) (cid:48) ←− −−→ (cid:48) ]3 ←−−− (cid:48) (cid:48) (cid:48) ( α − y (cid:48) (cid:48) + y (cid:48) (cid:48) + y (cid:48) + y − y − y − y + y (cid:48) + y (cid:48) − y (cid:48) − y (cid:48) (cid:48) − y (cid:48) (cid:48) (cid:48) . (20)1Here, following the advancement rule introduced in [50], we adopt the advancement indices ¯ l . For example, y (cid:48) = y ( t + L (cid:48) ( t − L ( t )) − L ( t )) . (21)In addition, the path α can be split to make S3 as the starting and ending satellite, which gives: −−−−−→ ←−−−−− (cid:48) (cid:48) (cid:48) + −→ (cid:48) ←−−−−− −−−→ (cid:48) (cid:48) = ⇒ −−→ (cid:48) ←− −−−−→ (cid:48) (cid:48) ]13 ←−−− (cid:48) (cid:48) (cid:48) ( α − . (22)Fig. 8 gives the smallest (right panel) and the largest (left panel) time differences for the second-generation α -typecombinations. (a) (b) FIG. 8. The time differences for the second-generation α -type combinations. For the other second-generation TDI combinations, the results are: X : −−−−−−→ (cid:48) (cid:48) ←−−−−−− (cid:48) (cid:48) + −−−−−−→ (cid:48) (cid:48) ←−−−−−− (cid:48) (cid:48) = ⇒ −−−−−−−−→ (cid:48) (cid:48) [22 (cid:48) (cid:48) ←−−−−−−−− (cid:48) (cid:48) ]33 (cid:48) (cid:48) X − , −−−−−−→ (cid:48) (cid:48) [3 (cid:48) ←−−− (cid:48) (cid:48) −−→ (cid:48) ] ←−−− (cid:48) (cid:48) X − , −−−−−−→ (cid:48) (cid:48) (cid:48) ←−−− (cid:48) (cid:48) −−→ (cid:48) ←−−− (cid:48) (cid:48) ( X − . (23)The path X : −−−−−−→ (cid:48) (cid:48) ←−−−−−− (cid:48) (cid:48) is spliced with its reversed version X : −−−−−−→ (cid:48) (cid:48) ←−−−−−− (cid:48) (cid:48) , resulting in thepath X −
1. Also, the path X : −−−−−−−→ (cid:48) ↑ (cid:48) ←−−−−−− (cid:48) (cid:48) can be split at the position of ↑ , and generate the newpath X : −→ (cid:48) ←−−− (cid:48) (cid:48) −→ (cid:48)
3, which is spliced with the path X to form the path X −
3. The path X can generatethe new path X : −→ (cid:48) ←−−− (cid:48) (cid:48) −→ (cid:48) , which is spliced with the path X to form the path X −
2. Among these threecombinations, X − X − X − X combinations forTianQin. The smallest (right) and largest (left) time differences are shown in Fig. 9. As we will see below, the timedifferences of X combinations are the largest comparing with in the α , U , P and E combinations. This is becausewhen inserting the frequency noise C i ( t ) into the combinations of ‘ y ’ for the X (the expression similar to Eq. 18) andexpanding in L l further than the canceled L l and L l ˙ L m terms, the next leading terms read L ¨ L m and the sums ofthese terms for the X combinations are larger than the other combinations.2 (a) (b) FIG. 9. The time differences for the second-generation X -type combinations. U : −−−−−−→ (cid:48) (cid:48) (cid:48) ←−−−−−− (cid:48) (cid:48) (cid:48) + −−−−−−→ (cid:48) (cid:48) (cid:48) ←−−−−−− (cid:48) (cid:48) (cid:48) = ⇒ −−−−→ (cid:48) (cid:48) (cid:48) ←−−−−−− (cid:48) (cid:48) (cid:48) [3 (cid:48) (cid:48) −−−−→ (cid:48) (cid:48) (cid:48) ←−− (cid:48) ]1 ( U − , −−−−→ (cid:48) (cid:48) (cid:48) ←−− (cid:48) (cid:48) −−−−→ [11 (cid:48) (cid:48) (cid:48) ←−−−−−−− (cid:48) (cid:48) (cid:48) ]1 (cid:48) U − , −−−−−−−−−→ (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) ←−−−− (cid:48) (cid:48) (cid:48) −→ ←−−−− (cid:48) (cid:48) (cid:48) U − . (24)The path U : −−−−−−→ (cid:48) (cid:48) (cid:48) ←−−−−−− (cid:48) (cid:48) (cid:48) is spliced with its reversed version U : −−−−−−→ (cid:48) (cid:48) (cid:48) ←−−−−−− (cid:48) (cid:48) (cid:48) , resulting in thepath U −
2. Also, the path U : −−−−−−−→ ↑ (cid:48) (cid:48) (cid:48) ←−−−−−−− (cid:48) ↑ (cid:48) (cid:48) can be split at the position of ↑ , and generate the newpath U : −−−→ (cid:48) (cid:48) (cid:48) ←−−−− (cid:48) (cid:48) (cid:48) −→ U : ←−− (cid:48) (cid:48) −−−−→ (cid:48) (cid:48) (cid:48) ←− (cid:48) . U is spliced with the path U to form the path U − U − U and U . All the U -type combinations are four-beam interferometers. Here, four directionalarms 2 (cid:48) , (cid:48) , (cid:48) , U -type combinations. P : −−−−−→ (cid:48) (cid:48) ←− (cid:48) −→ ←−−−−− (cid:48) + −−−−−→ (cid:48) ←− −→ (cid:48) ←−−−−− (cid:48) (cid:48) = ⇒ −−−−→ (cid:48) (cid:48) (cid:48) ←− −→ (cid:48) ←−− (cid:48) (cid:48) −→ ←− (cid:48) −→ ←−− (cid:48)
12 ( P − , −−−→ (cid:48) [1 ←− (cid:48) −→ ←−− (cid:48) −−→ (cid:48) (cid:48) ] ←− −→ (cid:48) ←−− (cid:48) (cid:48) ( P − , −−→ (cid:48) ←− −−−→ [3 (cid:48) (cid:48) ←− (cid:48) −→ ←−− (cid:48) −→ (cid:48) ←−− (cid:48) (cid:48) ( P − . (25)The path P : −−−−−→ (cid:48) (cid:48) ←− (cid:48) −→ ←−−−−− (cid:48) is spliced with its reversed version P : −−−−−→ (cid:48) ←− −→ (cid:48) ←−−−−− (cid:48) (cid:48) , resultingin the path P −
3. Also, the path P : −−−−−→ ↑ (cid:48) ←− −→ (cid:48) ←−−−−− (cid:48) (cid:48) can be split at the position of ↑ , and generatethe new path P : −→ (cid:48) ←− −→ (cid:48) ←−− (cid:48) (cid:48) −→
21, which is spliced with the path P to form P −
1. From P we can derive P : −→ ←− (cid:48) −→ ←−− (cid:48) −−→ (cid:48) (cid:48) , which is spliced with the path P to form P −
2. Fig. 11 gives the smallest (right panel)and the largest (left panel) time differences for the second-generation P -type combinations.3 (a) (b) FIG. 10. The time differences for the second-generation U -type combinations. (a) (b) FIG. 11. The time differences for the second-generation P -type combinations. E : −−−−−→ (cid:48) ←−−−−− (cid:48) (cid:48) −→ (cid:48) ←− + −→ ←− (cid:48) −−−−−→ (cid:48) (cid:48) ←−−−−− (cid:48) = ⇒ −−−→ (cid:48) ←−− (cid:48) (cid:48) −→ (cid:48) ←− −−→ (cid:48) (cid:48) ←−− (cid:48) −→ ←− (cid:48) ( E − , −−−−→ (cid:48) [11 (cid:48) (cid:48) ←−− (cid:48) −→ ←− (cid:48) ] −→ ←−− (cid:48) (cid:48) −→ (cid:48) ←− E − , −−→ (cid:48) (cid:48) ←− [3 −−→ (cid:48) ←−− (cid:48) (cid:48) −→ (cid:48) ] −−→ (cid:48) −→ ←− (cid:48) ( E − . (26)By reversing the interference arm of E : −−−−−→ (cid:48) ←−−−−− (cid:48) (cid:48) −→ (cid:48) ←− , we can obtain E : −→ ←− (cid:48) −−−−−→ (cid:48) (cid:48) ←−−−−− (cid:48) . E : −→ ←−− (cid:48) ↑ −−−−−→ (cid:48) (cid:48) ←−−−−− (cid:48) can be split at the position of ↑ , and generate the new path E : −−→ (cid:48) (cid:48) ←−− (cid:48) −→ ←− (cid:48) , whichis spliced with the path E to form the path of E − E −
2. Also, the path E can generate the new pathof E : ←− −−→ (cid:48) ←−− (cid:48) (cid:48) ←− (cid:48) , which is spliced with the path of E to form the path E −
3. The combinations of E -typeand P -type both form eight-beam interferometers. Fig. 12 gives the smallest (right panel) and the largest (left panel)4time differences for the second-generation E -type combinations. (a) (b) FIG. 12. The time differences for the second-generation E -type combinations. As we can see from the results above, the eight-beam interferometers are better than the other second-generationTDI combinations. The time differences of all the second-generation TDI data combinations are below 5 × − s.Similar to Fig. 7, the results for the second-generation combinations, taking α − X − S α − ( f ) = 4 sin (3 πf L ) S α ( f ) and S X − ( f ) = 4 sin (4 πf L ) S X ( f ) [76]are adopted. We can see that the second-generation data combinations are guaranteed to suppress the laser frequencynoise well below the secondary noises for TianQin in the concerned frequencies. -4 -3 -2 -1 0 1-60-55-50-45-40-35 (a) -4 -3 -2 -1 0 1-14-12-10-8-6 (b) FIG. 13. Left panel shows the secondary noise (solid line) and the residual laser frequency noise (dotted line) for thesecond-generation α − X − IV. CONCLUSION AND DISCUSSIONS
Using the numerically optimized orbit that shows realistic features of the satellite constellation, we investigated thetime differences of the symmetric interference paths, as a measure of residual laser frequency noise, of various first- andsecond-generation TDI data combinations for TianQin. We found that while the second-generation TDI with a typicaltime differences of 10 − s is guaranteed to be valid for laser frequency noise suppression, the first-generation TDIis possible to be competent for GW signals at frequencies (cid:46) − Hz and (cid:38) − Hz (given the raw laser frequencynoise of 10 Hz / √ Hz), which cover the coalescence of the SMBHB with a redshifted total mass > M (cid:12) [24] for theformer or the inspiral of the stellar-mass black hole binary (e.g., GW150914) (cid:46) ≈ / √ Hz in 10 − − − Hz. This further stabilization of laser frequency can be possiblyachieved through suppressing the fluctuation of the Pound-Drever-Hall error signal offset, thermal stabilization at thezero-crossing temperature of the optical bench coefficient of thermal expansion, upgrading the automatic functionsof the digital controller, etc. [77–79]. The first-generation TDI combinations, when implemented, will simplify thedata analysis procedure and reduce the computational cost, since they employ half of the single-link data than thecorresponding second-generation ones.The current work serves as our first step towards building an end-to-end TDI simulation for TianQin. Here, weonly discussed the inter-satellite measurements ‘ y ’. In reality, when each laser on the two optical benches housed in asatellite are unlocked or the acceleration noises of the optical benches are concerned, the intra-satellite measurements‘ z ’ need to be included in the simulation, although which are not subjected to the orbital effects of the satellitesdiscussed here. The simulated TDI data combinations, realistic noises from various subsystems, interplanetary andrelativistic effects on the optical path, etc. will be taken as the input of subsequent GW data analysis for variousastrophysical sources. These will be investigated in our future study. ACKNOWLEDGMENTS
Y.W. is supported by the National Natural Science Foundation of China (NSFC) under Grants No. 11973024 andNo. 11690021, and Guangdong Major Project of Basic and Applied Basic Research (Grant No. 2019B030302001).The contribution of S.C.H. to this paper is supported by NSFC under Grants No. 11873098. X.Z. is supported byNSFC Grant No. 11805287. W.S. acknowledges the support from the NSFC under Grant No. 11803008 and NationalKey Research and Development Program of China (Grant No. 2020YFC2201201). [1] B. P. Abbott and et al. (LIGO Scientific Collaboration and Virgo Collaboration). Observation of Gravitational Waves froma Binary Black Hole Merger.
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