Analyses of Laser Propagation Noises for TianQin Gravitational Wave Observatory Based on the Global Magnetosphere MHD Simulations
Wei Su, Yan Wang, Chen Zhou, Lingfeng Lu, Ze-Bing Zhou, T. Li, Tong Shi, Xin-Chun Hu, Ming-Yue Zhou, Ming Wang, Hsien-Chi Yeh, Han Wang, P.F. Chen
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Analyses of Laser Propagation Noises for TianQin Gravitational Wave Observatory Based on theGlobal Magnetosphere MHD Simulations
Wei Su ( 苏 威 ), Yan Wang ( 王 炎 ), Chen Zhou ( 周 晨 ), Lingfeng Lu, Ze-Bing Zhou, T. Li,
4, 5
Tong Shi, Xin-Chun Hu, Ming-Yue Zhou, Ming Wang, Hsien-Chi Yeh, Han Wang,
1, 8 and P.F. Chen
4, 5 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 Department of Space Physics, School of Electronic Information, Wuhan University, Wuhan 430072, China Southwestern Institute of Physics, Chengdu 610041, China School of Astronomy and Space Science, Nanjing University, Nanjing 210023, China Key Laboratory of Modern Astronomy & Astrophysics, Nanjing University, China Department of Climate and Space Sciences and Engineering, University of Michigan, Ann Arbor, MI 48109, USA Institute of Space Weather, School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing210044, China TianQin Research Center for Gravitational Physics & School of Physics and Astronomy, Sun Yat-sen University, Zhuhai 519082, China
ABSTRACTTianQin is a proposed space-borne gravitational wave (GW) observatory composed of three identicalsatellites orbiting around the geocenter with a radius of 10 km. It aims at detecting GWs in thefrequency range of 0.1 mHz – 1 Hz. The detection of GW relies on the high precision measurementof optical path length at 10 − m level. The dispersion of space plasma can lead to the optical pathdifference (OPD, ∆ l ) along the propagation of laser beams between any pair of satellites. Here, westudy the OPD noises for TianQin. The Space Weather Modeling Framework is used to simulate theinteraction between the Earth magnetosphere and solar wind. From the simulations, we extract themagnetic field and plasma parameters on the orbits of TianQin at four relative positions of the satelliteconstellation in the Earth magnetosphere. We calculate the OPD noise for single link, Michelsoncombination, and Time-Delay Interferometry (TDI) combinations ( α and X ). For single link andMichelson interferometer, the maxima of | ∆ l | are on the order of 1 pm. For the TDI combinations,these can be suppressed to about 0.004 and 0.008 pm for α and X . The OPD noise of the Michelsoncombination is colored in the concerned frequency range; while the ones for the TDI combinations areapproximately white. Furthermore, we calculate the ratio of the equivalent strain of the OPD noiseto that of TQ, and find that the OPD noises for the TDI combinations can be neglected in the mostsensitive frequency range of f (cid:46) . Keywords: plasmas, gravitational waves, Sun: solar wind INTRODUCTIONThe first direct detection of gravitational waves (GWs) by the advanced Laser Interferometer Gravitational-WaveObservatory (LIGO) opens up the era of the GW astronomy (Abbott et al. 2016). So far, more than fifty GW eventsgenerated by the coalescences of stellar-mass black hole binaries and double neutron stars have been detected bythe advanced LIGO and advance Virgo (Abbott et al. 2019, 2020). The underground and cryogenic detector KAGRA(Somiya 2012) has recently started joint observation with the advanced LIGO and advance Virgo. Due to the terrestrialnoises, the ground-based detectors are most sensitive to the GW signals in the acoustic band ( (cid:38)
10 Hz).Several space-borne missions, e.g., LISA (Amaro-Seoane et al. 2017), TianQin (TQ; Luo et al. 2016), Taiji (ALIAdescoped; Gong et al. 2015), ASTROD-GW (Ni 1998), gLISA (Tinto et al. 2015), BBO (Cutler & Harms 2006) and
Corresponding author: Wang Y., Zhou C., Lu [email protected], [email protected], [email protected] a r X i v : . [ a s t r o - ph . S R ] F e b Su et al.
DECIGO (Kawamura et al. 2011), have been proposed to explore the abundant GWs sources in the mHz band, whichcan be used to deepen our understandings in fundamental physics, astrophysics and cosmology.Both LISA and TQ are in nearly equilateral triangular constellations which are formed by three drag-free satellitesinterconnected by infrared laser beams. The heterodyne transponder-type laser interferometers are used to measurethe relative displacements of the test masses (TMs) with the accuracy of 10 − m / Hz / in mHz. This constellationforms up to three Michelson-type interferometers. Different from LISA, TQ’s satellites will be deployed in a geocentricorbit with an altitude of 10 km from the geocenter and the distances between each pair of satellites ≈ . × km(Luo et al. 2016). The detector’s plane formed by three satellites is optimized to detect the continuous GW signalsfrom the candidate ultracompact white-dwarf binary RX J0806.3+1527 (Israel et al. 2002). Currently, both sciencecases (Feng et al. 2019; Wang et al. 2019; Shi et al. 2019; Bao et al. 2019; Liu et al. 2020; Fan et al. 2020; Huanget al. 2020) and technological realizations (Luo et al. 2020; Ye et al. 2019; Tan et al. 2020; Yang et al. 2020; Su et al.2020; Lu et al. 2020) have been under intensive investigations for TQ. A brief summary of TQ’s recent progress canbe found in Mei et al. (2020).The space plasma contributes as the main source of environmental noises for space-borne GW detectors. For example,when the laser beams propagate in the space plasma the dispersion effect can lead to time delay and optical pathdifference (OPD) between different beams and produce additional noise for the relative displacement measurement.Since the plasma frequency ω p in the space environment is much larger than the electron gyrofrequency, OPD is mainlycaused by the total electron content (TEC) along each laser beam. Moreover, the space magnetic field can induce thetime variation of the polarization of electromagnetic (EM) waves, and the interaction between the space magnetic fieldand the test masses can generate additional non-conservative forces on the test masses (Hanson et al. 2003; Schumaker2003; Su et al. 2020).Space environment parameters, e.g., magnetic field, and density, vary significantly in time and space (Su et al. 2016;Wang et al. 2018), which can be categorized, in descending characteristic sizes, into three spatial scales: global scale,magnetohydrodynamic (MHD) scale, and plasma scale. In the global scale, the solar wind interacts with the Earth’smagnetic dipole field to form the structures such as bow shocks, magnetoheath, magnetopause, magnetotail, etc. (Luet al. 2015; Wang et al. 2016). The density and magnetic field in different structures are essentially different. As thesatellites orbiting around the Earth, the laser beam between two satellites passes through different structures, thus theOPD caused by the space plasma will be a function of time. Furthermore, because the solar wind is changing in realtime, the shapes and properties (e.g., magnetic field, and density) of these structures are also evolving. In the MHDscale, instabilities, such as Kelvin-Helmholtz (K-H) instability, can cause variations of density and magnetic field at themagnetosphere boundary layer (Hasegawa et al. 2004). In the plasma scale, the plasma waves, such as electromagneticion cyclotron (EMIC) waves (Allen et al. 2015), ultra-low-frequency (ULF) waves (Soucek et al. 2015; Takahashi et al.2018), kinetic Alfv´en waves (KAWs) (Zhao et al. 2014), etc., are widely found in the solar wind, magnetosheath, andmagnetosphere. Turbulence exists at scales ranging from MHD to plasma (He et al. 2012; Sahraoui et al. 2013; Huanget al. 2018). Besides, the eruption events from the Sun (e.g., coronal mass ejections, and coronal shocks, Su et al.2015, 2016) and Earth magetosphere (e.g., magnetic reconnections, Huang et al. 2012; Takahashi et al. 2018; Zhouet al. 2019) can lead to variations of density and magnetic field in multiple scales. In this work, we evaluate the effectof the OPD noise rooted from the space plasma at the global and MHD scales on the detection of GWs for TQ.This paper is organized as follows. The theory of EM wave propagation in space plasma is briefly summarized inSection 2. In Section 3, we introduce the MHD model, i.e., the Space Weather Modeling Framework (SWMF; T´othet al. 2005), which is adopted in this work. The calculation and results of the OPD noise are presented in Section 4.Section 5 discusses the impact of our work on the detection of GWs for TQ. Our paper is concluded in Section 6. ELECTROMAGNETIC WAVE PROPAGATION IN SPACE PLASMAFor a train of EM wave with a frequency f (angular frequency ω ) propagating in the cold magnetized plasma of theEarth magnetosphere and solar wind, the refractive index µ can be described by the Appleton-Hartree (A-H) equation(Hutchinson 2002): µ = 1 − X − iZ − Y T − X − iZ ) ± (cid:113) Y L + Y T − X − iZ ) , (1)where i represents the imaginary unit. X , Y , Z are defined as: X = ω p ω , Y = ω B ω , Z = νω . (2) aser propagation noise for TQ ω p , ω B and ν are the plasma frequency, gyrofrequency, and electron collision frequency, respectively: ω p = e Nmε , ω B = eBm , (3)where m is the electron mass, e is the elementary charge, N is the electron number density, ε is the vacuum electricpermittivity, and B is the background magnetic field strength. In Equation (1), Y T = Y cos θ and Y L = Y sin θ , where θ is the angle between the propagation direction of the EM wave and the direction of the background magnetic field.Since the electron collision frequency ν of the plasma in the magnetosphere and solar wind are on the order of10 − s − , which are much lower than the frequency of the diode-pumped Nd:YAG laser ( ≈ . × s − ) used forTQ, the space plasma can be considered as collisionless. Thus, Z can be ignored. Besides, take the typical electronnumber density to be 5 cm − and the typical magnetic strength to be 5 nT at the geocentric distance of 10 km, ω p and ω B in the magnetosphere and solar wind are on the order of 10 rad s − and 10 rad s − , respectively, both aremuch lower than ω . Therefore, Equation (1) can be simplified as, µ = 1 − X . (4)The group refractive index µ g can be deduced as: µ g = ∂ ( µω ) ∂ω = ∂ω √ − X∂ω = 2 ω (cid:113) ω − ω p = 1 √ − X = 1 µ ≈ X KN f , (5)where K = e / (4 π mε ) = 80 . s − .The time ( τ ) that takes EM waves propagating a distance of L in space plasma is: τ = (cid:90) L d sv g = (cid:90) L d sc/µ g , (6)where c is the speed of light in vacuum, L ≈ . × m for TQ. The time delay (∆ τ ) relative to the vacuum case is:∆ τ = 1 c (cid:90) L (1 + KN f )d s − Lc = K cf (cid:90) L N d s . (7)Here, (cid:82) L N d s is called TEC. According to Equation (7), the OPD can be calculated as:∆ l = c ∆ τ = K f (cid:90) L N d s . (8)Equation (8) shows that the OPD noise ∆ l of a single arm between two satellites is determined by the integratedelectron number density along the laser link. MHD SIMULATIONAccording to Section 2, in order to study the OPD noise for TQ, we need to obtain the distributions of the electronnumber density in the vicinity of the laser links in Fig. 2 (see also Fig. 1 in Su et al. (2020)), which requires theglobal MHD simulations of the Earth magnetosphere. In this work, we adopt the Space Weather Modeling Framework(SWMF) to simulate the interaction between the solar wind and the Earth magnetosphere (T´oth et al. 2005). SWMFhas been thoroughly validated in the study of the Earth magnetosphere (Zhang et al. 2007; Welling & Ridley 2010;Dimmock & Nykyri 2013), and it has been used widely (Lu et al. 2015; Wang et al. 2016; Takahashi et al. 2018).The simulation can be requested on the Community Coordinated Modeling Center (CCMC) which is done by theSWMF/Block-Adaptive-Tree-Solarwind-Roe-Upwind-Scheme (BATSRUS).The real time solar wind parameters observed by the Advanced Composition Explorer (ACE; Stone et al. 1998) aretaken as the simulation inputs, which include the ion number density n i , z component of magnetic field B z , and solarwind dynamic pressure P dyn as illustrated by Fig. 1. The time range of the inputs is from 2008-05-01 00:00 UT to2008-05-04 24:00 UT with a temporal resolution of 1 min. The input data are the same as Su et al. (2020). The rangesof the Geocentric Solar Magnetospheric (GSM) coordinates in the simulation domain are − R E < x < R E ( R E Su et al. the radius of the Earth) and | y | = | z | < R E , which contain the bow shock, magnetopause and magnetotail of theEarth. In the region where | x | , | y | , | z | < R E , the vicinity of the dayside magnetopause and the near-tail has thefinest resolution of 0.25 R E , the resolution of the rest region is 0.5 R E . The output parameters of the simulationcontain the magnetic field ( B x , B y , B z ), the plasma parameters (e.g., bulk flow velocity v x , v y , v z , number density ofion n i , pressure P ), and electric current ( J x , J y , J z ). The output parameters in the GSM coordinates are convertedto the Geocentric Solar Ecliptic (GSE) coordinates in the following calculation. Generally, the plasma in the solarwind and magnetosphere is quasi-neutral at the MHD scale, and the number densities of election n e and ion n i areapproximately equal. This has been confirmed by several observations (Zhang et al. 2007; Welling & Ridley 2010).Therefore, we simply use n i outputted from the simulation as n e in the calculation of the OPD noise.In the GSE coordinates, we define the open angle between the Sun-Earth vector and the projection of the normal ofthe detector plane on the ecliptic plane as φ s , which shows an annual variation from 0 ◦ to 360 ◦ (Su et al. 2020) andis equal to 120.5 ◦ at the spring equinox (Hu et al. 2018). In order to describe the relative position of the geometricstructure of the Earth magnetosphere and the TQ’s constellation conveniently, φ s is transformed to its correspondingacute angle ϕ s hereafter. ϕ s ranges from 0 ◦ to 75.5 ◦ in TQ’s observation time intervals, while it ranges from 14.5 ◦ to90 ◦ in non-observation time intervals (see Fig. 4 and the associated text in Su et al. (2020) for details). ϕ s can beapproximately regarded as a constant during one orbit period of the TQ satellite around the Earth (3.65 days) (Suet al. 2020). In the following sections, we focus on the OPD noises at four typical positions with ϕ s = 0 ◦ , 30 ◦ , 60 ◦ ,and 90 ◦ . RESULTS4.1.
Laser links in magnetosphere and OPD noise
Taking the simulation at 2008-05-03 20:00 UT as an example, the electron number density distributions on thedetector’s planes (at ϕ s = 0 ◦ , ◦ , ◦ , ◦ ) are shown on the left columns of Fig. 2, in which ξ is the intersectionline between the orbit plane and the ecliptic plane, ζ is along the intersection of the detector’s plane and a planeperpendicular to ξ . ζ is approximately vertical to the ecliptic plane since the angle between the ecliptic plane and thenormal of the detector’s plane is only 4.7 ◦ (Hu et al. 2018). The Earth magnetosheath is the downstream of the bowshock, therefore its electron number density is higher than the ones in the solar wind and the magnetosphere. As shownin Fig. 2, the boundary of the magnetosheath on the sunside and earthside are the bow shock and magnetopause,respectively. The geometric structures of the magentosphere on the four detector’s planes are different. For ϕ s = 90 ◦ ,the nose of the bow shock is located at ξ ≈ R E measured from the geocenter. For ϕ s = 0 ◦ , the magnetopause andbow shock are approximately circular and they are located at ≈ R E and (cid:38) R E , respectively.With the time-varying positions of three TQ’s satellites (S1, S2, and S3), the laser links can be obtained. In Fig.2, the laser links S1–S2, S2–S3, and S3–S1 are represented as blue, orange, and green lines, respectively. Note thatthe initial position of S1 is located at ξ = 15 . E and ζ = 0 R E . For ϕ s = 60 ◦ and 90 ◦ , S1–S2 and S3–S1 willpass through the solar wind, bow shock, magnetosheath, magnetopause and magnetosphere; While S2–S3 that passesthrough the magnetotail is almost enclosed in the magnetosphere. We obtain the number density distributions alongthese three laser links, shown in the corresponding colors in the right column of Fig. 2, by interpolating the values ofnumber densities on the grid of the simulation domain. The number density characteristics of the regions, such as thesolar wind (moderate), magnetosheath (high), and magnetosphere (low), are also revealed here.The laser links sweep across the annulus formed by the inscribed circle and the circumcircle (red circle) of thedetector’s triangle as the satellites orbiting around the geocenter. We calculate the OPD noise ∆ l based on Equation(8) and the satellites’ orbits. The integration of the number density along each laser beam shows spatial and temporalvariations due to the changes of the positions and directions of the laser beams in the magnetosphere and the evolutionof the geometrical shapes and the number densities of the structures. During one revolution of the satellites aroundthe Earth, the time series of ∆ l of S1–S2 for ϕ s = 0 ◦ , 30 ◦ , 60 ◦ , 90 ◦ are shown in Fig. 3. For ϕ s = 0 ◦ , the correlationcoefficient between the time series of P dyn and ∆ l is 0.83, which is about two to three times larger than the ones for ϕ s = 30 ◦ (0.48), 60 ◦ (0.33), and 90 ◦ (0.28). For ϕ s = 30 ◦ , 60 ◦ , 90 ◦ , the amplitude of the OPD noise reaches 1.2 pm atthe position when the laser beam passes through the magnetosheath on the dayside (around 300 ◦ in Fig. 3), while ∆ l is only about 0.05 pm at the position where the laser beam passes through the magnetotail on the nightside (around120 ◦ in Fig. 3). These results indicate that the variation of ∆ l for ϕ s = 0 ◦ is mainly due to the evolution of P dyn intime, whereas the variations of ∆ l for ϕ s = 30 ◦ , 60 ◦ , 90 ◦ are mainly due to the fact that the number density in themagnetosheath is much higher than that in the magnetotail. aser propagation noise for TQ OPD noise for single link and Michelson interferometer
Fig. 4 shows the amplitude spectral densities (ASDs) of the time series of ∆ l for the single links. Here, we have usedSavitzky-Golay filter (Savitzky & Golay 1964) to smooth the ASDs before fitting them by a single power law function.Note that the finest spatial resolution of the simulation is 0.25 R E and the speed of TQ’s satellites is about 2 km s − ,it takes each satellite about 800 s to move between two grid points. So that the ASDs of the OPD noise at range of f > /
800 Hz can be underestimated. In fact, this underestimation has been shown in Fig. 4, where there is a kneepoint at f ≈ /
800 Hz and the spectra become steeper when f (cid:38) /
800 Hz. Here, only the ASDs of the OPD noiseat range of f (cid:46) /
800 Hz are used in the fitting of spectral index. The best-fit spectral indices for ϕ s = 0 ◦ , ◦ , ◦ and 90 ◦ , shown as the red dashed lines, are -0.718, -0.577, -0.567 and -0.588, respectively. The corresponding spectralamplitudes at 1 mHz read 0.760, 0.651, 0.587 and 0.573 pm/ √ Hz, respectively.Michelson-type interferometer sketched in Fig. 5a has been used as the fiducial data combination to study the sciencepotential and data analysis for space-borne detectors (Feng et al. 2019; Wang et al. 2019; Liu et al. 2020; Huang et al.2020). Its response and sensitivity to arbitrary incoming GWs for TQ have been studied in Hu et al. (2018). Wedenote the OPD noise that is produced during the propagation of the EW wave sent from spacecraft i and receivedby satellite j as ∆ l ij (Prince et al. 2002). For a Michelson-type interferometer centered on S1, the OPD noise of twointerferometer arms (S1–S2 and S1–S3) are ∆ l , ∆ l , ∆ l and ∆ l . From Equation (8), ∆ l = K/ f (cid:82) S S N d s ,and similarly for ∆ l , ∆ l and ∆ l . Since (cid:82) ji N d s = (cid:82) ij N d s , and the light propagation time between a pair ofsatellites ( ≈ . l = ∆ l and ∆ l =∆ l . Therefore, the OPD noise for a Michelson-type combinations can be written as follows,∆ l = 2(∆ l − ∆ l ) . (9)From Equation (9), we can calculate the time series of ∆ l for a Michelson combination during one revolution of thesatellites around the Earth. From Fig. 6, we can see that the maxima of | ∆ l | for the Michelson combination is about3 pm. The typical amplitudes are magnified due to four combinations of the OPD noises of the single links.Shown in Fig. 7 are the ASDs of ∆ l for the Michelson interferometer for ϕ s = 0 ◦ , 30 ◦ , 60 ◦ , and 90 ◦ . Similarly, wefit the spectral profiles with power-law functions for the Michelson interferometer, which are shown as the red dashedlines in Fig. 7. The best-fit spectral indices for ϕ s = 0 ◦ , ◦ , ◦ , and 90 ◦ are -0.887, -0.544, -0.626 and -0.683. Thecorresponding spectral amplitudes at 1 mHz read 0.752, 1.167, 1.400 and 1.512 pm/ √ Hz, respectively.4.3.
OPD noise for TDI combinations
In order to eliminate the otherwise overwhelming laser phase noise, the time delay interferometry (TDI) has beendevised for the data processing of space-borne interferometric GW detectors (Armstrong et al. 1999; Estabrook et al.2000; Tinto & Dhurandhar 2014). There are various data combinations for the TDI (Tinto & Dhurandhar 2014). Inthis work, we focus on the α and X data combinations, shown in Fig. 5b and 5c, as the typical examples of thesix-pulse and eight-pulse combinations of the first-generation TDI, respectively.Consider the OPD noise accumulated along the laser propagation in space plasma, the phase fluctuation of the laserthat is sent from satellite i and received by satellite j can be expressed as follows,Φ ij ( t ) = φ ij ( t ) + h ij ( t ) + n ij ( t ) + s ij ( t ) , (10)where φ ij ( t ) is the laser phase noise to be canceled by TDI, h ij ( t ) is the GW signal, and n ij ( t ) is the total nonlaserphase noise (Hellings 2001). s ij ( t ) is the phase noise associated with the OPD noise ∆ l ij ( t ), s ij ( t ) = 2 π ∆ l ij ( t ) /λ , (11)where λ = 1064 nm for the Nd:YAG laser used by TQ.Set L ij as the distance between satellites i and j and c = 1 hereafter, the total phase noise due to space plasma forthe α combination s α can be written as, s α = s ( t − L − L ) + s ( t − L ) + s ( t ) − s ( t − L − L ) − s ( t − L ) − s ( t ) . (12)Similarly, for the X combination, s X can be written as, s X = s ( t − L − L − L ) + s ( t − L − L ) + s ( t − L ) + s ( t ) − s ( t − L − L − L ) − s ( t − L − L ) − s ( t − L ) − s ( t ) . (13) Su et al.
For the nearly equilateral triangular constellation of TQ, L ij = L ≈ . L is much smaller than thetemporal resolution of the MHD simulation, ∆ t = 60 s. As mentioned in Section 4.2, (cid:82) ij N d s = (cid:82) ji N d s , so that s ij ( t ) = s ji ( t ). Thus, s α is reduced to s α = s ( t − L ) + s ( t ) − s ( t ) − s ( t − L ) . (14)And s X is reduced to s X = s ( t − L ) + s ( t − L ) + s ( t − L ) + s ( t ) − s ( t − L ) − s ( t ) − s ( t − L ) − s ( t − L ) . (15)As in the single link case, we can obtain s ij ( t ) for every 60 s. s ij at delayed times can be obtained by linearinterpolation, i.e., s ij ( t − δt ) = s ij ( t ) + ( s ij ( t − ∆ t ) − s ij ( t ))( δt/ ∆ t ). In this way, Equations (14) and (15) can bemodified as follows, s α = ( s ( t − ∆ t ) − s ( t )) 2 L ∆ t − ( s ( t − ∆ t ) − s ( t )) 2 L ∆ t , (16) s X = 2 s α . (17)From Equation (17), we can see that the OPD noise reduction for the α combination is a factor of two betterthan that for the X combination. This is because the laser beam will pass through one of the arms twice for the X combination, but only once for the α combination (see Fig. 5). Combining Equations (11), (16), and (17), we cancalculate the OPD noises for the α and X combinations as shown in Fig. 8. The maxima of | ∆ l | for the α and X combinations are about 0.004 and 0.008 pm, respectively, which are about two orders of magnitude smaller than thatfor the Michelson combination. This indicates that TDI can significantly suppress the common-mode OPD noise.Fig. 9 shows the ASDs of ∆ l for the α and X combinations. Similar to the single link and Michelson combination,the spectra of the ASDs become steeper when f (cid:38) /
800 Hz. The best-fit spectral indices for the α ( X ) combinationare 0.108, 0.399, 0.341, and 0.309 for ϕ s = 0 ◦ , ◦ , ◦ , and 90 ◦ ; The corresponding spectral amplitudes at 1 mHzread 0.003 (0.005), 0.004 (0.008), 0.005 (0.010), and 0.005 (0.010) pm/ √ Hz at 1 mHz, respectively. DISCUSSIONS5.1.
The impact of OPD noise on the sensitivity
The equivalent strain noise ASD √ S n for the Michelson combination is as follows (Hu et al. 2018), S n = S xn + S an (1 + 10 − Hz f ) , (18)where, (cid:112) S xn is the equivalent strain noise ASD of the displacement measurement, (cid:112) S an is the equivalent strain noisedue to residual acceleration. In order to compare the OPD noises with TQ’s equivalent strain noise ASD √ S n , wecalculate the equivalent strain of the OPD noises as ∆ l/L . Using the best-fit spectra of the Michelson, α , and X combinations, we calculate the ratio between (∆ l/L ) and √ S n , and the results are shown in Fig. 10.For the Michelson combination, the maximum of (∆ l/L ) / √ S n is about 0.29 at ≈
10 mHz. (∆ l/L ) / √ S n increaseswith increasing frequency f when f (cid:46)
10 mHz. This is due to the fact that √ S n is dominated, in this frequency range,by the acceleration noise with (cid:112) S an ∝ f − , the spectral index of which is less than the ones of ∆ l for the Michelsoncombination. (∆ l/L ) / √ S n decreases with f when f (cid:38)
10 mHz. This is because √ S n is dominated by the positionnoise which is approximately white in this frequency range.For the TDI combinations of the α and X types, (∆ l/L ) / √ S n at f = 10 mHz are about 0.01 and 0.02, respectively,which are much smaller than that for the Michelson combination. However, as the spectral index of ∆ l is slightlylarger than 0 for the TDI combinations, (∆ l/L ) / √ S n increase smoothly in the frequency range of 10 − < f < l/L ) / √ S n from different ϕ s are about 0.035 and 0.07 at the transfer frequency f ∗ = 0 .
28 Hz (Huet al. 2018) for the α and X combinations, respectively. The results suggest that the TDI combinations can efficientlysuppress the common-mode OPD noise produced during the laser propagation in space plasma for the most sensitivefrequency range of TQ. However, due to the limited high frequency reach of our simulation, the results in Fig. 10,which are obtained by extrapolating the red dashed lines in Fig. 9, may overestimate or underestimate the OPD noiseswhen f (cid:38) /
800 Hz. Further investigation with higher spatial and temporal resolution is needed. aser propagation noise for TQ
Space weather
The solar wind dynamic pressure P dyn is the most important parameter that determines the Earth magnetosphere’sgeometric structures and distributions of electron number density (Lu et al. 2015; Wang et al. 2016). The more stronglythe magnetosphere is compressed by the solar wind, and the larger the number density will be (Lu et al. 2015), whichleads to the larger amplitude of the OPD noise. From the OMNI data of the solar wind (King & Papitashvili 2005),we obtain P dyn with the value of 2 . ± . P dyn = 2 . ± . P dyn during the total solar cycle. Consider thatboth the period of P dyn and solar magnetic cycle are about 22 years and TQ is proposed to launch at early 2030s, P dyn that is obtained 22 years before the real operation is a good approximation. We get P dyn data from 2008 to 2012(about 22 years before the lanuch) on OMNI website, and find that the time when P dyn is larger than 3 nPa and 5nPa accounts for only about 5% and 1% of the total period. Besides, the Earth magnetoshpere encounters the solareruptions, e.g. interplanetary shocks and coronal mass ejections, P dyn can reach or even exceed 5 nPa in these cases.The laser propagation noises in the cases of solar eruptions are expected in future works. Furthermore, the SWMFmodel is an MHD model, only the physical processes at global and the MHD scale can be revealed. On the other hand,the plasma-scale physical processes, such as plasma waves nor turbulences, cannot be revealed by the SWMF model.For the impact of plasma scale physical process on the OPD noise, the hybrid or particle-in-cell (PIC) simulations areneeded.Besides obtaining the TEC from MHD simulations, it is also possible to derive the TEC from real-time observations.In order to obtain the TEC along the laser propagation path, we can transmit signals with two frequencies to reduce∆ l , the dual-frequency scheme has been used in the Compass system (Yang et al. 2011), the Gravity Recovery andClimate Experiment (GRACE) and GRACE Follow-on (GRACE-FO) (Tapley et al. 2004; Landerer et al. 2020). Fordual-frequency scheme, there are two general methods, one is differential group delay, the other is differential carrierphase. Differential group time delay measures the time delay difference of two EM waves with different frequencies,and differential carrier phase measures the phase difference of two EM waves with different frequencies. The accuracyof differential group delay method is lower than that of differential carrier phase, but it can measure the absolute valueof TEC. For the next-generation space-borne GW detectors, e.g., DECIGO (Kawamura et al. 2011), the proposedstrain sensitivity is about five orders of magnitude lower than that of TQ, and the difference of the electron densitybetween the solar wind at 1 AU and that around the TQ orbit with a geocentric altitude of 10 km is generally nomore than one order of magnitude. Thus, the laser propagation noise will become a dominating environmental noisefor DECIGO, and the laser ranging scheme with the dual-frequency laser will become a necessity. CONCLUSIONSDispersion can cause the OPD noise when the laser beams propagate in space plasma. In this work, the Appleton-Hartree equation, the orbits of TQ satellites and the global magnetosphere simulation based on the SWMF are usedto analyze the OPD noise ∆ l for TQ at four typical relative positions of the detector’s planes with ϕ s = 0 ◦ , ◦ , ◦ ,and 90 ◦ .The maxima of | ∆ l | for the single link and the Michelson combination are about 1 and 3 pm, respectively. The maximaof | ∆ l | can be reduced to about 0.004 and 0.008 pm for the TDI combination α and X , respectively. Furthermore, wecalculate the ratio between the equivalent strain of the OPD noise and the one proposed for TQ, i.e., (∆ l/L ) / √ S n , forthe Michelson combination and the TDI combinations ( α and X ). We find that in the frequency range of f (cid:46) . f (cid:38) . Su et al.
ACKNOWLEDGMENTSThe simulations used in this work are provided by the Community Coordinated Modeling Center (CCMC) atGoddard Space Flight Center through their public Runs on the Request system. This work is carried out using theSWMF and BATSRUS tools developed at the University of Michigan’s Center for Space Environment Modeling. Thiswork is supported by the National Natural Science Foundation of China (NSFC) under grants 11803008, 42004156and 11961131002, National Key Research and Development Program of China (Grant 2020YFC2201201), and JiangsuNSF under grant BK20171108. Y.W. is supported by NSFC under grants 11973024 and 11690021, and GuangdongMajor Project of Basic and Applied Basic Research (Grant No. 2019B030302001). S.T. was supported by NASA grant80NSSC17K0453. Z.Z.B. is supported by NSFC under grants 11727814 and 11975105.REFERENCES
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The input parameters observed by the ACE which include the ion number density n i (top panel), the space magneticfield B z in GSM coordinate (middle panel). The solar wind dynamic pressure P dyn (bottom panel) is derived from the observationof the ACE. aser propagation noise for TQ ( R E ) S1S2S3 s = 0 N ( c m ) ( R E ) S1S2S3 s = 30 N ( c m ) ( R E ) S1S2S3 s = 60 N ( c m ) -20 -10 0 10 20 (R E ) ( R E ) S1S2S3 s = 90 N ( c m ) N ( c m ) S1-S2S2-S3S3-S1 N ( c m ) S1-S2S2-S3S3-S1 N ( c m ) S1-S2S2-S3S3-S1
L (normalized) N ( c m ) S1-S2S2-S3S3-S1
Figure 2.
The left panels are the electron number density distributions on the orbit planes with ϕ s = 0 ◦ , ◦ , ◦ , ◦ . ξ is theintersection line between the orbit plane and the ecliptic, ζ is perpendicular to the ecliptic, the red circle is the orbit of TQ’ssatellites. The laser link S1 – S2, S2 – S3, and S3 – S1 are represented by blue, orange, and green lines, respectively. The rightpanels are the distributions of the electron number densities along these three laser beams. Different colors correspond to thelaser beams with the same colors in the left panels. Su et al.
Degree (°) l ( p m ) s = 0 s = 30 s = 60 s = 90 Figure 3.
The OPD noises for a single link during a 3.65-day orbit around the Earth on the orbit planes with ϕ s = 0 ◦ , 30 ◦ ,60 ◦ , and 90 ◦ . aser propagation noise for TQ l ( p m / H z ) index = -0.718 s = 0 index = -0.577 s = 30 f (mHz) l ( p m / H z ) index = -0.567 s = 60 f (mHz) index = -0.588 s = 90 Figure 4.
The ASDs of the OPD noises for a single link on the orbit planes with ϕ s = 0 ◦ , 30 ◦ , 60 ◦ , and 90 ◦ . The orangedashed line is the proposed displacement measurement accuracy of TQ (1 pm/ √ Hz) (Luo et al. 2016). Red dashed lines are thebest-fit spectra of the ASDs. α XM(a) (c)(b)
Figure 5.
Schematic diagrams of Michelson, α , and X combinations. Su et al.
Degree (°) l ( p m ) s = 0 s = 30 s = 60 s = 90 Figure 6.
The OPD noises for a Michelson combination during a 3.65-day orbit around the Earth. The blue, orange, green,and red curves are the OPD noises on the detector’s planes with ϕ s = 0 ◦ , ◦ , ◦ , and 90 ◦ . aser propagation noise for TQ l ( p m / H z ) index = -0.887 s = 0 index = -0.544 s = 30 f (mHz) l ( p m / H z ) index = -0.626 s = 60 f (mHz) index = -0.683 s = 90 Figure 7.
The ASDs of the OPD noises for the Michelson combination on the orbit planes with ϕ s = 0 ◦ , 30 ◦ , 60 ◦ , and 90 ◦ .The orange dashed line is the proposed displacement measurement accuracy of TQ (1 pm/ √ Hz) (Luo et al. 2016). Red dashedlines are the best-fit spectra of the ASDs. Su et al. l ( p m ) s = 0 l ( p m ) s = 30 l ( p m ) s = 60 Degree (°) l ( p m ) s = 90 l ( p m ) s = 0X l ( p m ) s = 30 l ( p m ) s = 60 Degree (°) l ( p m ) s = 90 Figure 8.
The OPD noises distributions for the α (left panels) and X (right panels) combinations on the orbit planes with ϕ s = 0 ◦ , 30 ◦ , 60 ◦ , and 90 ◦ . aser propagation noise for TQ l ( p m / H z ) s = 0 index = 0.108 l ( p m / H z ) s = 30 index = 0.399 l ( p m / H z ) s = 60 index = 0.341 f (mHz) l ( p m / H z ) s = 90 index = 0.309 l ( p m / H z ) s = 0 index = 0.108 X l ( p m / H z ) s = 30 index = 0.399 l ( p m / H z ) s = 60 index = 0.341 f (mHz) l ( p m / H z ) s = 90 index = 0.309 Figure 9.
The ASDs of the OPD noises for the α (left panels) and X (right panels) combinations on the orbit planes with ϕ s = 0 ◦ , 30 ◦ , 60 ◦ , and 90 ◦ . The orange dashed line is the proposed displacement measurement accuracy of TQ (1 pm/ √ Hz)(Luo et al. 2016). Red dashed lines are the best-fit spectra of the ASDs. Su et al. f (Hz) ( l / L ) / S x Michelson s = 0 s = 30 s = 60 s = 90 f (Hz) s = 0 s = 30 s = 60 s = 90 f (Hz) X s = 0 s = 30 s = 60 s = 90 Figure 10.
The ratios between the equivalent strain of the OPD noises and that of TQ, i.e., (∆ l/L ) / √ S n , for the Michelsoncombination (left panel), the TDI α combination (middle panel) and the TDI X combination (right panel) on the orbit planeswith ϕ s = 0 ◦ , 30 ◦ , 60 ◦ , and 90 ◦◦