Test of gravitational red-shift based on tri-frequency combination of microwave frequency links

Eur. Phys. J. C (2021) 81:634 https://doi.org/10.1140/epjc/s10052-021-09415-y Regular Article - Theoretical Physics Test of gravitational red-shift based on tri-frequency combination of microwave frequency links Xiao Sun1,2 , Wen-Bin Shen1,2,a , Ziyu Shen3 , Chenghui Cai1, Wei Xu1, Pengfei Zhang1 1 Department of Geophysics, School of Geodesy and Geomatics/Key Laboratory of Geospace Environment and Geodesy of Ministry of Education, Wuhan University, Wuhan, China 2 State Key Laboratory of Information Engineering in Surveying, Mapping and Remote Sensing, Wuhan University, Wuhan, China 3 Hubei University of Science and Technology, Xianning, China Received: 8 February 2021 / Accepted: 4 July 2021 © The Author(s) 2021 Abstract Over the decades, testing gravitational red-shift (GRS) based on microwave links has made great process, including the GPA experiment, the planned Atomic Clock Ensemble in Space mission, and the China Space Station (CSS). Until now, the formulations of microwave links are almost all based on the time comparison. However, there are advantages of using frequency comparison instead of time comparison to test GRS. Here we develop a tri-frequency combination method based on the measurements of the frequency shifts of three independent microwave links between a space station and a ground station. Aiming at the frequency links’ accuracy of 3 × 10−16 , we should consider various effects, including the Doppler effect, second-order Doppler effect, atmospheric frequency shift, tidal effects, refraction caused by the atmosphere, and Shapiro effect, with accuracy levels of tens of centimeters. The CSS will complete construction in 2022, and the formulation proposed in this study will enable us to test GRS at an accuracy level of at least 2 × 10−6 , which is one order higher than the present accuracy level of 7 × 10−5 . 1 Introduction General relativity theory (GRT) [1] has three classic predictions: Mercury precession, light deflection and gravitational red-shift (GRS). The first and second ones have been confirmed respectively by Einstein himself [1] and a group led by Eddington [2], but the GRS was not tested until 1960. The first direct experimental verifications of GRS are the series of Pound–Rebka–Snider experiments during 1960– 1965 [3,4], who observed the shift using a Mössbauer emitter and absorber at the Jefferson Physical Laboratory tower a e-mail: (corresponding author) 0123456789().: V,-vol at Harvard University. Later, there is an around-the-world experiment. Four cesium beam clocks were used to fly around the world on commercial jet flights during several days in October 1971, and they flew in opposite directions while recording the time differences [5]. Additionally, other types of experiments measure the shift of spectral lines in the Sun’s gravitational field since 1960 [6]. A solar redshift experiment carried out with the Galileo space probe tested the GRS to 1% accuracy [7]. The most famous test was obtained by the Gravity Probe A (GPA) mission in June 1976, which launched a hydrogen maser onboard a sounding rocket to a height of 10,000 km [8]. During its flight, frequency comparisons were conducted between the maser on the rocket and a corresponding maser on the ground. The uncertainty on the measurement of GRT is 7 × 10−5 [9]. Until now, the most precise indirect tests were performed by eccentric Galileo satellites. The tests were based on the satellites GSAT-0201 and GSAT-0202 of the European Global Navigation Satellite System (GNSS) Galileo, which were accidentally delivered on elliptic instead of circular orbits. Two research teams simultaneously published papers with precision of (0.19 ± 2.48) × 10−5 [10] and (4.5 ± 3.1) × 10−5 [11]. To further improve the precision of testing GRS, scientists can obtain benefits from some space missions, such as the Atomic Clock Ensemble in Space (ACES), the Space– Time Explorer and QUantum Equivalence Space Test (STEQUEST) [12], China Space Station (CSS) mission. The ACES experiment [13–16], which will be installed onboard the International Space Station (ISS), is an ESACNES mission mainly planned to test the GRS. Equipped with atomic clocks of fractional frequency instability and inaccuracy of (1−3)×10−16 , it aims to test the GRS at a level of 2 × 10−6 [14,15], which is one and a half orders of magnitude higher than the GPA experiment. The main onboard instruments are an active hydrogen maser (SHM) and a 123 634 Page 2 of 16 cold cesium atomic clock (PHARAO). The PHARAO clock √ reaches a fractional frequency stability of 1.1 × 10−13 τ , where τ is the integration time in seconds, and an accuracy of a few parts in 1016 [15]. Meanwhile, SHM demonstrates a fractional frequency instability of 1.5 × 10−15 after 10,000 s of integration time. ACES enables frequency/time comparisons between ISS and ground stations by using two kinds of independent time & frequency transfer links (Microwave Links (MWL) and European Laser Timing (ELT) optical link) to test GRT and develop various applications, for instance in geodesy and time & frequency metrology [14,15]. These science objectives are closely related to the MWL performance [17], and its performance plays a key role in this study. MWL uses the one uplink and two downlinks to transfer time and frequency and will perform with time deviation better than 0.3 ps at 300 s, 7 ps at 1 day, and 23 ps at 10 days of integration time [18]. Similar with ACES mission, CSS will also be equipped with an active hydrogen maser, a cold microwave atom clock and an optical atomic clock, being able to transmit microwave links and optical links. The relevant frequency parameters related to the hydrogen maser and the cold microwave atom clock are at the same level with those of ACES, but the frequency stability of the optical atomic clock will be 8 × 10−18 /day [19,20], which will greatly improve the accuracy of testing GRS. Since CSS is under construction and the detailed report of CSS payloads have not been officially released, our paper will focus on general performances of ACES payloads and MWL, which could be applied to CSS time and frequency experiments. Concerning the ACES mission, some studies have addressed the test of GRS based on time comparison [17,21,22], but there are few publications related to the frequency comparison. Frequency links and comparisons will be available on CSS mission. Compared with time comparison, frequency comparison has the following advantages: (1) it will not be influenced by phase ambiguity because the latter is only relevant in ranging but not in frequency comparison measurements; (2) it can determine the gravitational potential within a short time interval, while for time comparison, we must accumulate data to solve the time changing rate to deduce the GRS value. However, the accuracy of measuring the short-term frequency is largely constrained, which implies that we must also accumulate observations to obtain results with higher accu (...truncated)