Relativistic time transfer for a Mars lander: from Areocentric Coordinate Time to Barycentric Coordinate Time

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1 Research in Astron. Astrophys. 217 Vol. X No. XX, (L A TEX: 88.tex; printed on May 27, 217; 1:56) Research in Astronomy and Astrophysics Relativistic time transfer for a Mars lander: from Areocentric Coordinate Time to Barycentric Coordinate Time Wen-Zheng Yang, De-Wang Xu, Qing-Shan Yu, Jie Liu and Yi Xie School of Astronomy and Space Science, Nanjing University, Nanjing 2193, China; yixie@nju.edu.cn Key Laboratory of Modern Astronomy and Astrophysics, Nanjing University, Ministry of Education, Nanjing 2193, China Received ; accepted Abstract As the second step of relativistic time transfer for a Mars lander, we investigate the transformation between the Areocentric Coordinate Time (TCA) and Barycentric Coordinate Time (TCB) in the framework of the IAU Resolutions. TCA is a local time scale for Mars, which is analogous to the Geocentric Coordinate Time (TCG) for the Earth. This transformation has two parts: contributions associated with gravitational bodies and those depending on the position of the lander. After setting the instability of the onboard clock as 1 13 and uncertainty of the time is about 3.2 microsecond after one Earth year, we find that the contributions of Sun, Mars, Jupiter and Saturn in the leading term associated with the bodies can reach the level exceeding the threshold and must be taken into account. Other terms can be safely ignored in this transformation for a Mars lander. Key words: reference systems time space vehicles 1 INTRODUCTION Time transfer plays an important role in every deep space mission. Einstein s general relativity (GR) (Iorio 215; Debono & Smoot 216) has been an inevitable part of the time transfer in both theoretical senses (Misner et al. 1973; Landau & Lifshitz 1975) and practical ways (Petit & Wolf 25; Nelson 27, 211). In the framework of GR, there two kinds of times: proper times and coordinate times, which are connected by a -dimensional spacetime interval, so that clock synchronization and time/frequency transfer for deep space tracking and navigation have been considerably changed (Moyer 23; Thornton & Border 25). In the future, time transfer might be used to test GR and alternative theories of gravity (e.g., Wolf et al. 29; Christophe et al. 29; Schiller et al. 29; Deng & Xie 213; Christophe et al. 212; Angélil et al. 21; Delva et al. 215). In this work, as a continuing work of the time transfer for a Mars lander (Xu et al. 216), we will focus on the relativistic transformation from Areocentric Coordinate Time (TCA) to Barycentric Coordinate Time (TCB), where TCA is the time coordinate of the Areocentric Celestial Reference System (ACRS). ACRS is introduced to describe the events happened in the vicinity of Mars and its definition is consistent with the International Astronomical Union (IAU) 2 Resolutions (Soffel et al. 23) and 26 Resolution B2 1 (see Xu et al. 216, for more details). This transformation is the second Supported by the National Natural Science Foundation of China. 1

2 2 W.-Z. Yang et al. step for the whole procedure of time transfer. Although this work is devoted to the time transfer for a Mars lander, the major part of the transformation from TCA to TCB only depends on the position and velocity of the gravitational bodies in the Solar system and it can also be applied to a Mars orbiter (Deng 212; Pan & Xie 213, 21, 215). In the practical realization of the time transfer link, we will also consider GR effects on the propagation of the electromagnetic waves from Mars to the Earth and vice-versa in the post-newtonian field of the Solar System and the effects of the post-newtonian field of Mars itself on the Mars probes (Iorio 21, 26, 29; Oberst et al. 212; Turyshev et al. 21), which will be one of our next moves. Following engineering settings of the previous work (Xu et al. 216), we also consider a clock onboard the Mars lander with instability at the level of 1 13 so that uncertainty of the time is about 3.2 microsecond (µs) after linear drifting for one (Earth) year. Therefore, any effects below these two thresholds will be dropped. The paper is organized as follows. In Section 2, we describe the relativistic transformation between TCA and TCB based on IAU 2 Resolutions (Soffel et al. 23). This transformation has two parts: contributions associated with the gravitational bodied in the Solar system and terms directly depending on the position of the lander. Sections 3 and are respectively devoted to examining these two contributions. We summarize the results in Section 5. 2 TRANSFORMATION FROM TCA TO TCB According to the IAU Resolutions (Soffel et al. 23), the post-newtonian transformation between TCA and TCB can be written in a linear form as TCB TCA = P 1 +L 1 +P 2 +L 2. (1) Here, P 1 and P 2 are the contributions associated with the gravitational bodies in the Solar System and they are independent of the position and velocity of the Mars lander. P 1,2 are in the forms of integrals as (Soffel et al. 23) t [ v 2 ] P 1 = c 2 dt t 2 +Ū (x ), (2) t [ 1 P 2 = c dt t 8 v v2 Ū (x ) vi Ūi (x ) 2Ū2 1 ] (x ), (3) where c is the speed of light; t is the coordinate time in the scale of TCB; x and v are respectively the position and velocity of Mars in the Barycentric Celestial Reference System (BCRS). In equations (2) and (3), Ū and Ūi are respectively the Newtonian gravitational potential and a vector potential, both of which are generated by the gravitational bodies except Mars; and their expressions read as Ū (x ) = A Ū i (x ) = A GM A r A, () GM A r A v i A, (5) wherem A is the mass of body A;vA i is the velocity of body A in the BCRS;r is the distance between A Mars and body A. Since the distance between bodies is much larger than the characteristic lengths of them in the Solar System, the non-spherical parts of their gravitational fields can be neglected in this case. It is worth mentioning that all of the variables in the integrands ofp 1 andp 2 are functions of TCB, although such a dependence is not shown explicitly.

3 Time transfer from TCA to TCB 3 Unlike P 1 and P 2, the rest parts of TCB-TCA, L 1 and L 2, directly depend on the position of the Mars lander as L 1 = c 2 v r L, (6) L 2 = c [ v Ū (x ) ]v r L, (7) where r L x L x and x L is the position of the lander in the BCRS. 3 CONTRIBUTIONS OFP 1 AND P 2 In order to calculate the contributions of P 1 and P 2, we have to know the positions and velocities of bodies in the Solar System. In this investigation, we consider the Sun, eight planets and three biggest asteroids (Ceres, Pallas and Vesta). We read their positions and velocities by DE31 ephemeris (Folkner et al. 21) from the starting time 223-Jan-1 to the end point 22-Jan-1. However, the time coordinate of DE31 is the Barycentric Dynamical Time (TDB), instead of TCB, but they have a linear relation as (Petit & Luzum 21) dtdb = (1 L B )dtcb, (8) where L B = is a defining constant. Hence, we can rewrite equations (2) and (3) as (Soffel et al. 23) 1 TDB [ v 2 ] P 1 = c 2 dtdb (1 L B ) TDB 2 +Ū (x ), (9) 1 TDB [ 1 P 2 = c dtdb (1 L B ) TDB 8 v v2 Ū (x ) vi Ūi (x ) 1 ] 2Ū2 (x ), (1) where all variables in the above integrands are functions of TDB and can be read directly from DE31. Figure 1 shows components of P 1 contributed by various bodies and their overall values. We find that P 1 can increase to about.3 s after one year and the contributions of Sun, Mars, Jupiter and Saturn can reach the level more than 3µs. Figure 2 shows components ofp 2 contributed by various bodies and their overall values. Since its overall contributions is about 1 nanosecond (ns) much less than our threshold, we can safely ignore P 2 in this investigation. Like the work of Xu et al. (216), we also estimate the contribution of the hypothetical Planet Nine or Telisto (Batygin & Brown 216; Brown & Batygin 216; Mustill et al. 216; Iorio 212, 21, 217) onp 1 as about 1 ns, which can be neglected. If the lander has to process the calculation of P 1 for autonomous time transfer, it would be necessary to construct an analytic form to approximatep 1 with sufficient accuracy because the capacity of the onboard computer is expected to be very limited. In principle, we can apply the strategy for constructing time transformations TB-TT by using an analytical planetary ephemeris (Fairhead & Bretagnon 199) and TCB-TCG by Chebyshev polynomials fitting (Fukushima 1995; Irwin & Fukushima 1999; Fukushima 21) or by nonlinear harmonic decomposition (Harada & Fukushima 23). Although the detailed investigation on this will be left as one of our next moves, we will study the properties ofp 1 in order to provide some clues for future works. Following the work of Pan & Xie (213), we take an nth polynomial to fit P 1 as n P 1 a j t j, (11) j= and then perform Fourier analysis on its post-fit residuals. In the future, this preliminary procedure will be replaced with more sophisticated one, such as the approach of the works (Kudryavtsev 27, 216). Table 1 shows the coefficients a j of the fitted nth polynomial and figure 3 represents the post-fit residuals. It is found that, for the cases that n = and n = 5, the residuals can decrease to about 3 µs

4 W.-Z. Yang et al. and they can even decrease to severalµs duringtfrom 1 d to 3 d. Figure shows frequency spectra of these post-fit residuals. The spectra with n = 1 and n = 3 have distinct peaks, but these peaks are absent in the others. As demonstrated by the work of Pan & Xie (215), their spike-like and tips-like shapes do not mean any discontinuity at those points but they are caused by the low resolution of the figure. P 1 (s) Sun 1 5. Venus Moon Ceres Vesta P 1 (s) Saturn Neptune Mercury 1-8 Earth 1-7 Mars 1-2 Pallas 1-11 Jupiter 1-5 Uranus ALL Fig. 1 Components of P 1 contributed by various bodies and their overall values (marked by ALL ). CONTRIBUTIONS OFL 1 AND L 2 The terms L 1 and L 2 are different from P 1 and P 2 by depending on positions of the lander and their values will not accumulate with time. As a straightforward estimation, we can find that L 1 c 2 v r L =.9 µs, (12) L 2 c 7 2 v2 v r L = s. (13) Therefore, both of them are below the threshold we set and can be safely ignored in this case. In the above estimation, we take r L as the radius of the Mars for a lander. When an orbiter of Mars is considered,r L can be larger and makesl 1 close to or bigger than the threshold so that the term ofl 1 has to be included in the relativistic transformation between TCA and TCB. 5 CONCLUSIONS In this paper, we continue the work of the time transfer for a Mars lander (Xu et al. 216) and investigate its second step that is the relativistic transformation between TCA and TCB. This transformation has

5 Time transfer from TCA to TCB 5 P 2 (s) Sun Venus Moon 1-17 Ceres P 2 (s) Vesta Saturn Neptune Mercury 1-16 Earth 1-15 Mars 1-1 Pallas 1-19 Jupiter 1-13 Uranus ALL 1-9 Fig. 2 Components of P 2 contributed by various bodies and their overall values (marked by ALL ). Table 1 Coefficients of the fitted nth polynomial. n a j Value 1 a a a a a 2 3 a a a a 3 a a a a a 5 a a a a a a 5

6 6 W.-Z. Yang et al. Residual of P 1 (s) n= n=1 n=2 n=3 n=5 Fig. 3 Post-fit residuals of P 1 for nth polynomial where n = 1,,5. log 1 A (s) n= log 1 f (rad s -1 ) Fig. Frequency spectra of the post-fit residuals ofp 1 fornth polynomial wheren = 1,,5, where A and f denote amplitude and frequency. n=1 n=2 n=3 n=5 two parts:p 1,2 terms associated with the gravitational bodies [see equations (2) and (3)] andl 1,2 terms depending on the position of the lander [see equations (6) and (7)]. After setting the instability of the onboard clock as 1 13, we find that the contributions of Sun, Mars, Jupiter and Saturn in P 1 can reach the level exceeding the threshold and must be taken into account. Other terms,p 2 and L 1,2 can be safely ignored in this transformation for a Mars lander. Besides a part of the procedure of time transfer, such a transformation between TCA and TCB might also be used in the model of Doppler tracking of a lander and an orbiter of Mars (Deng & Xie 21; Zhang et al. 21; Yang et al. 21; Xie & Huang 215), since the time coordinate of their equations

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