Analysis of the J2 relativistic effect on the performance of on-board atomic clocks

Abstract

The GNSS satellite atomic clock is subjected to appreciable relativistic effects during its in-orbit operation. These effects include drift due to the gravitational redshift and time dilation due to the relative velocity, the periodic effects of the orbital eccentricity, the oblateness of the Earth’s gravity field, and the tidal potential of the Sun, Moon and other planets. The accuracy of navigation, positioning and timing of satellite navigation systems is thus affected. The small terms, including those relating to the Earth’s oblateness and the tidal potential of other planets, are usually neglected to estimate satellite clock. Inevitably, residual relativistic errors are introduced into the final solutions. Therefore, we comprehensively analyze the characteristics of relativistic effects on satellite clocks of the Beidou navigation satellite system, Galileo and Global Positioning System, including the combined relativistic effect, periodic relativistic effect, J2 relativistic effect and the tidal potential of other planets. The characteristics and performance of the satellite clock before and after the correction for J2 relativistic effect are evaluated. It is found that the amplitude of the satellite clock half-orbital periodic term is reduced by the correction, and the fitting residuals reduce by 6%. The Hadamard variance sequence tends to be stable. Meanwhile, J2 effect correction is applied to satellite clock prediction and improving the accuracy of the quadratic polynomial model by 9% and stability by 11%. Additionally, the results of using the spectral analysis model with and without adding the J2 period are improved.

Introduction

Global navigation satellite systems (GNSSs) aim to provide all-weather, all-time, high-precision navigation, positioning and timing services for global users. Because on-board atomic clocks provide an accurate and stable frequency source for navigation signal generation and system ranging. The atomic clock is a core component of the payload of a navigation satellite and its performance directly determines the user service accuracy. Therefore, high accuracy and stability atomic clock is helpful to achieve high-accuracy positioning, navigation and timing.

Theoretically, positioning can be solved as long as the times of the ground user and the space segment are consistent. However, because of relativistic effects, the signal transmission chain, and clock noise, the frequency of the atomic clock observed from the ground is shifted, resulting in inconsistency between the terrestrial time and on-board time. One important factor contributing to this is the combined effect of the gravitational redshift and time dilation due to the relative velocity, refer to Ashby (1994) for a detailed discussion. The two aspects compensate for each other at the critical altitude of 3167.4 km. Due to the height (H ≈ 21,528.0 km) of the medium earth orbit (MEO) satellites in the Beidou navigation satellite system (BDS), the satellite clock is ahead of the BDS receiver clock on the ground. That is to say, time passes more quickly on the navigation satellite than on the Earth owing to the Earth’s gravitational field, in contrast to what happens when applying the special relativistic speed correction (time dilation). The combined effect causes an apparent speeding up of BDS MEO clocks of approximately 39,617.8 ns/day, with additional periodic errors of tens of nanoseconds, if they are not corrected. The satellite clock on the ground before the launch must be delayed with respect to another clock located at the same position, so that both clocks are synchronized once the satellite is in orbit. Ashby and Allan (1979) proposed theoretical of setting up such a coordinate clock network using electromagnetic signals and revealed the practical importance of relativity at a global coordinate time scale. Ashby (1994) discussed the influence of three relativistic effects, namely time dilation, the Sagnac effect and the gravitational frequency shift, on the Global positioning system (GPS), a time synchronization system, communication and geodesy. Kouba (2002) discussed and derived a time transformation method for the GPS in detail. The constant drift (caused by the gravitational redshift and time dilation due to the relative velocity) and periodic effect (depending on the orbit eccentricity) are called conventional relativistic effects because they are well known and widely used in the field of satellite navigation and positioning. Ashby (2003) introduced and discussed relativistic principles and effects that must be considered for the GPS, i.e., the constancy of the speed of light, the equivalence principle, the Sagnac effect, time dilation, gravitational frequency shifts, and the relativity of synchronization. He provided a strict derivation and demonstration of the time transformation formula, which is worthy of further study and discussion.

At present, GNSS analysis centers around the world make conventional relativistic corrections when generating satellite clock solutions. Users need to use the same relativistic correction method in recovering the actual on-board time. However, the conventional correction method does not completely eliminate the relativistic effects of satellite clocks and uncorrected errors in the residuals are retained in the clock solutions, destroy the linear characteristics of the satellite clock solutions and make it difficult for users to accurately analyze the characteristics and performance of clocks. The interpolation and prediction of clocks are thus affected.

The high-performance atomic clocks currently used for satellites are sensitive to the slight influence of the surrounding environment, and it is thus important to test relativistic effects. Similar to the research of Ashby (2003), Kouba (2004) demonstrated the relativistic effect on GPS satellite clocks in detail and deduced an approximate analytic formula of the J2 relativistic effect. The existence of the J2 relativistic effect on the periodic characteristics of the satellite clock was then verified using International GNSS Service (IGS) final satellite clock solutions. Steigenberger et al. (2015) and Wang et al. (2015) applied a J2 effect correction to the periodic characteristic analysis of the Galileo satellite clocks and the stability analysis of the BDS-2 satellite clocks, respectively. Han and Cai (2019) derived a correction formula for the effects of the Earth’s oblateness (J2 term) and the tidal potential of the Sun and Moon on the satellite clocks and proposed that these corrections be applied when evaluating the performance of the BDS hydrogen clock. Kouba (2019) calculated the fitted residuals of Galileo passive hydrogen maser (PHM) clocks using linear and quadratic polynomial models and found that the characteristics of the fitting residuals varying with time for both models were similar to the J2 relativistic effect and that the fitting residuals of both models decreased after correcting for the J2 relativistic effect. It is thus concluded that the linearity of satellite clocks can be improved through correction for the J2 relativistic effect. Formichella (2018) analyzed the influence of J2 relativity on the frequency and periodic characteristics of GPS satellite clocks. Formichella et al. (2019, 2021) also conducted time–frequency analysis of the long-term data of Galileo satellite clocks, evaluated the periodic characteristics of clock frequency and verified the J2 relativistic effect.

Scholars have verified the validity of the J2 relativistic correction model using GPS and Galileo satellite clocks, demonstrating that the correction effectively reduces the periodic error in the satellite clock solutions and improves the linear characteristics of satellite clocks. As for satellite clock prediction, because most models rely on prior data and the application of the J2 relativistic correction provides “purer” source data, it is considered that the accuracy of clock prediction can be improved through such correction which has not been directly verified by scholars and is the focus of the present study.

Additionally, time transmission is affected by the rotation of the Earth. In an inertial system, the times of two ground clocks are not synchronized which move with approximately the same linear velocity of the Earth’s rotation in the inertial system.

The above effect is known as the Sagnac effect, which is an effect of special relativity attributable to the Earth’s rotation. Ashby and Allan (1979) proposed that rate changes arising from the velocity and gravitational potential of a transported clock used for synchronization of a network must be accounted for when synchronizing remote clocks. Allan et al. (1985) verified the Sagnac effect using the GPS common-view method and proposed an approach for calculating the time difference between clocks. If the time synchronization link goes around the equator once, the time difference between the first and last clocks is approximately 207 ns. Ashby (2003, 2004) gave a more detailed explanation and introduced the transformation from an inertial frame to a rotating frame of reference in Minkowski spacetime. In the case that the communication link is between the satellite and ground receiver, as shown in Fig. 1, the real path of the satellite downlink signal is τ₂ instead of τ₁, and the real path of the uplink signal is τ₃ instead of τ₄, owing to the Earth’s rotation. Therefore, the two-way satellite time and frequency transfer will be invalid if not corrected. However, the Sagnac effect is automatically included if the receiver motion is accounted for owing to the Earth’s rotation during signal propagation.

Fig. 1

Signal transmission link between the satellite and ground receivers

Currently, satellite positioning systems are based on Newtonian spacetime and require relativistic correction to ensure correctness. A relativistic positioning system (RPS) is a four-dimensional system that allows us to know the proper coordinates of every event in a given domain without delay. Galileo’s SYstème de POsitionnement Relativiste project, introduced by Pascual-Sánchez (2007), is a typical relativistic positioning system. Coll et al. (2006) explained the basic features of RPSs in a two-dimensional spacetime and obtained an analytic expression of the emission coordinates associated with such a system. The transformation between emission coordinates and inertial coordinates in Minkowski spacetime has been obtained for arbitrary configurations of the emitters, as explained by Coll et al. (2010). On this basis, Colmenero et al. (2021) computed the perturbed orbits of satellites considering a metric that takes into account the gravitational effects of the Earth (including the Earth’s oblateness), Moon, and Sun.

On the basis of the existing theories and methods, we advance the previous ideas and systematically analyze the influences and characteristics of various relativistic effects on the satellite clocks of the BDS, Galileo and the GPS. We then discuss in depth the J2 relativistic effect on the periodic characteristics, fitting residuals and stability of satellite clocks. Finally, the feasibility of correcting for the J2 relativistic effect in satellite clock prediction is analyzed. The conclusion of the paper summarizes the experimental results and discusses potential future development and application.

Models and algorithms

After a short introduction to the relativistic effects acting on GNSSs, this section briefly reviews high-precision time-conversion theory and the derivation of various relativistic correction formulas for satellite clocks. The strict formula for converting the on-board time T_sv to the terrestrial time t, based on an International Astronomical Union (IAU) resolution and rigorously derived by IERS (2010) and Nelson (2011), is

$$ \frac{{{\text{d}}T_{{{\text{sv}}}} }}{{{\text{d}}t}} = 1 - \frac{1}{{c^{2} }}\left[ {V\left( {x,y,z} \right) - W_{0} + \Delta V\left( {x,y,z} \right) + \frac{{{\varvec{v}}^{2} }}{2}} \right] $$

(1)

where V is the Earth’s gravitational potential at the clock position (x, y, z), composed of a central term and a disturbing potential R. ΔV is the sum of the tidal potentials of other bodies (mainly the Sun and Moon). W₀ is the geoidal potential at mean sea level (i.e., on the geoid). v is the clock velocity in an inertial geocentric coordinate system and c is the speed of light. The detailed formula for V is

$$ V(x,y,z) = \frac{{{\text{GM}}}}{r} - R(x,y,z) $$

(2)

where $r = \sqrt {x^{2} + y^{2} + z^{2} }$ is the radius of the satellite orbit, GM = 3.986004418 × 10¹⁴ m³/s² is geocentric gravitational constant. By integrating (1) and using Kepler’s law, a high-precision time-conversion formula supporting subpicosecond (10⁻¹⁸ clock frequency) accuracy and including Kepler orbit parameters is obtained [as described by Kouba (2004)] as

$$ t = T_{{{\text{sv}}}} - \Delta t^{{{\text{rel}}}} = T_{{{\text{sv}}}} - \frac{{{\text{GM}}}}{{c^{2} }}\left\{ {\frac{1}{{R_{0} }} \cdot t + \int {\left( {\frac{1}{2a} - \frac{2}{r} + \frac{R}{{{\text{GM}}}} - \frac{\Delta V}{{{\text{GM}}}}} \right){\text{d}}t} } \right\} $$

(3)

At the picosecond (10⁻¹⁶ clock frequency) precision level, the effect of the gravitational potential ΔV due to planets except the Earth can be ignored. Moreover, the main contribution to the disturbing potential R caused by the Earth’s oblateness is from the first J2 term, with the other terms being almost two orders of magnitude smaller and thus negligible. This simplified magnitude is acceptable for satellite navigation systems.

Thus, as introduced by Kouba (2004), without considering the perturbation of orbital parameters caused by the Earth’s oblateness, using only the mean values of the Kepler orbit parameters (i.e., the orbit semi-major axis a₀, orbit eccentricity e₀ and osculating eccentric (angular) anomaly E₀), and integrating (3), a numerical formula is written as

$$ \Delta t_{0}^{{{\text{rel}}}} = \Delta t_{{{\text{con}},0}} + \Delta t_{0}^{{{\text{per}}}} = \frac{{{\text{GM}}}}{{c^{2} }}\left( {\frac{1}{{R_{0} }} - \frac{3}{{2a_{0} }}} \right) \cdot t - \frac{{2\sqrt {a_{0} {\text{GM}}} }}{{c^{2} }}e_{0} \sin E_{0} $$

(4)

where $R_{0} = 6363672.6\;{\text{m}}$ is the geopotential scale factor and t is the time interval measured by a ground clock, which can be calculated from geocentric gravitational constant GM and geoidal potential W₀. The first term $\Delta t_{{{\text{con}},0}}$ is the constant clock phase drift in the measurement time interval t, which mainly relates to the semi-major axis of the satellite orbit. The corresponding hardware frequency $\Delta f_{{{\text{con}},0}}$, compensating for the satellite clock drift, is obtained as

$$ \Delta f_{{{\text{con}},0}} = \frac{{{\text{GM}}}}{{c^{2} }}\left( {\frac{1}{{R_{0} }} - \frac{3}{{2a_{0} }}} \right) $$

(5)

This frequency was referred to as the constant clock rate offset by Kouba (2004). We obtain the BDS MEO satellite hardware frequency offset $\Delta f_{{{\text{con}},0}} = 458.54 \times 10^{ - 12}$, which corresponds to a clock drift of 39,617.8 ns/day.

Additionally, the drift term $\Delta t_{{{\text{con}},0}}$ in (4) is calculated from the nominal value of the semi-major axis a₀ for each GNSS. However, there is a bias between the actual semi-major axis and the nominal value in the construction of a satellite constellation. If the correction accuracy of (4) is required to be 0.01 ns/day, the parameter accuracy of the actual orbital semi-major axis must be controlled within 15 m. Therefore, the average length of the semi-major axis in the broadcast ephemeris is often used in calculation to ensure that the drift term error is less than 0.1 ns/day. The actual phase shift term $\Delta t_{{{\text{con}}}}$ is calculated as

$$ \Delta t_{{{\text{con}}}} = \Delta t_{{{\text{con}},0}} + \delta \Delta t_{{{\text{con}}}} = \frac{{{\text{GM}}}}{{c^{2} }}\left( {\frac{1}{{R_{0} }} - \frac{3}{{2a_{0} }}} \right) \cdot t + \frac{3}{2}{\text{GM}}\frac{{\Delta a_{0} }}{{\left( {a_{0} c} \right)^{2} }} \cdot t $$

(6)

where $\Delta a_{0} = a_{0} - a_{n}$ is the difference between the actual mean value a₀ and the nominal value a_n of the satellite semi-major axis.

The second term (periodic term) $\Delta t_{0}^{{{\text{per}}}}$ in (4) depending on the orbital eccentricity e has periodic characteristics and is referred to as the periodic relativistic effect (PRE). When precise orbit parameters (vectors r and v of the satellite position and velocity) are used, the PRE is calculated as

$$ \Delta t^{{{\text{per}}}} = \frac{{ - 2{\varvec{r}} \cdot {\varvec{v}}}}{{c^{2} }} $$

(7)

which is equivalent to the second term $\Delta t_{0}^{{{\text{per}}}}$ in (4).

The above is the conventional relativistic correction model. However, the Earth oblateness J2 term slightly perturbs the mean orbital parameters in a short period ($\Delta a\left( {J_{2} } \right)$, $\Delta e\left( {J_{2} } \right)$ and $\Delta E\left( {J_{2} } \right)$) and leads to a certain error in the relativistic correction using (4). Kouba (2004) wrote the correction formula for the relativistic effect due to the J2 term as

$$ \delta \Delta t^{{{\text{rel}}}} = \frac{1}{2}\frac{{a_{E}^{2} J_{2} }}{{a_{0}^{2} c^{2} }}\left[ {3\sqrt {{\text{GM}}a_{0} } \sin^{2} i\sin \left( {2u} \right) + \frac{{7{\text{GM}}}}{{a_{0} }}\left( {1 - \frac{3}{2}\sin^{2} i} \right) \cdot t} \right] $$

(8)

where $a_{{\text{E}}} = 6378137\;m$ is the Earth’s equatorial radius, i is the orbit inclination, and $u = \omega + f$ is the sum of the orbit perigee angle ω and true anomaly f, also known as the latitude angle. The coefficient $J_{2} = 1.0826359 \times 10^{ - 3}$ is the dynamical form factor of the Earth.

The second term (drift term) in (8) is rather small, being approximately 0.1 ns/day for MEO, and it is readily absorbed into much larger satellite clock hardware drifts. The first term (periodic term) in the equation determines the amplitude of the periodic variation of the J2 relativistic effect, and it is reflected in satellite clock solutions calculated on the ground. The period of this term is equal to half of the orbital period, whereas the amplitude depends on the flatness of the Earth and the inclination of the satellite orbit. Therefore, the general treatment method is to ignore the influence of the drift term, such that (8) is often rewritten as

$$ \delta \Delta t_{0}^{{{\text{rel}}}} = K_{{{\text{GNSS}}}} \sin \left( {2u} \right) = \frac{{3\sqrt {{\text{GM}}} J_{2} a_{E}^{2} \sin^{2} i}}{{2a_{0}^{3/2} c^{2} }} \cdot \sin \left( {2u} \right) $$

(9)

where K_GNSS is a coefficient that determines the magnitude of the amplitude and is determined by the navigation system.

Equation (8) is well known and the theoretical basis of many studies, but there is a sign difference between the formulas in these studies, mainly owing to the different orders of time conversion. If t is converted to T_sv, the sign is positive; otherwise, it is negative.

Equation (9) is the phase offset due to the J2 relativistic effect, and the frequency offset is the time derivative of the phase offset. Therefore, the J2 relativistic effect on the apparent frequency offset of the satellite clock $\delta \Delta f_{0}^{{{\text{rel}}}}$ is obtained by differentiating (9), i.e.,

$$ \delta \Delta f_{0}^{{{\text{rel}}}} = \frac{{{\text{d}}\left( {\delta \Delta t_{0}^{{{\text{rel}}}} } \right)}}{{{\text{d}}t}} = K_{{{\text{GNSS}}}} \cdot 2\cos (2u) \cdot \frac{{{\text{d}}u}}{{{\text{d}}t}} $$

(10)

where the latitude angle $u = \frac{2\pi t}{T}$ is a time-dependent orbital parameter based on the first-order approximation (of a circular orbit). The orbital period T is given by

$$ T = 2\pi \sqrt {\frac{{a_{0}^{3} }}{{{\text{GM}}}}} $$

(11)

Equation (10) is rewritten by combining the orbital period T and angle u as

$$ \delta \Delta f_{0}^{{{\text{rel}}}} = B\cos \left( {2u} \right) = \frac{{3{\text{GM}}J_{2} a_{E}^{2} \sin^{2} i}}{{a_{0}^{3} c^{2} }}\cos \left( {2u} \right) $$

(12)

giving the theoretical value of the amplitude of the J2 signal affecting the apparent satellite clock frequency, which is approximately 1.82 × 10⁻¹⁴ Hz for BDS MEO satellite clocks and 1.55 × 10⁻¹⁴ Hz for Galileo satellite clocks. The value is even higher than the stability at 86400 s of current high-performance hydrogen clocks (on the order of magnitude of 10⁻¹⁵), and the correction for the J2 relativistic effect on these clocks should thus be considered.

So far, we have obtained an approximate analytical formula for the relativistic correction of the satellite clock:

$$ \Delta t^{{{\text{rel}}}} = \Delta t_{0}^{{{\text{rel}}}} + \delta \Delta t_{{{\text{con}}}} + \delta \Delta t^{{{\text{rel}}}} = \Delta t_{{{\text{con}},0}} + \Delta t_{0}^{{{\text{per}}}} + \delta \Delta t_{{{\text{con}}}} + \delta \Delta t^{{{\text{rel}}}} $$

(13)

in which the effect of the gravitational potential ΔV due to celestial bodies other than the Earth is still ignored. The relativistic effect of the Sun and Moon must be considered if a time conversion is to be carried out with higher accuracy. According to the derivation given by Han and Cai (2019), the integral formulas of the effects of the Sun and Moon on the satellite clock frequency ($\Delta f_{{\text{S}}}$ and $\Delta f_{{\text{M}}}$) are given directly as

$$ \Delta f_{{\text{S}}} = A_{{\text{S}}} \int {\left( {\cos 2\theta_{{\text{S}}} + \frac{1}{3}} \right){\text{d}}t} = - \frac{{3\mu_{{\text{S}}} a^{2} }}{{4c^{2} r_{{{\text{SE}}}}^{3} }}\int {\left( {\cos 2\theta_{{\text{S}}} + \frac{1}{3}} \right){\text{d}}t} $$

(14)

$$ \Delta f_{{\text{M}}} = A_{{\text{M}}} \int {\left( {\cos 2\theta_{{\text{M}}} + \frac{1}{3}} \right){\text{d}}t} = - \frac{{3\mu_{{\text{M}}} a^{2} }}{{4c^{2} r_{{{\text{ME}}}}^{3} }}\int {\left( {\cos 2\theta_{{\text{M}}} + \frac{1}{3}} \right){\text{d}}t} $$

(15)

where $\mu_{{\text{S}}}$ and $\mu_{{\text{M}}}$ are, respectively, the gravitational constants of the Sun and Moon. $\theta_{{\text{S}}}$ and $\theta_{{\text{M}}}$ are, respectively, the angles of intersection between the satellite and the Sun and the satellite and the Moon. However, these two (subpicosecond) effects are 1–2 orders of magnitude smaller than J2 relativistic effects and far less than the current nominal accuracy of an on-board atomic clock.

Data analysis and experimental results

In this section, we evaluate the numerical and periodic properties of various relativistic effects on the on-board atomic clocks of GNSSs (especially the BDS and Galileo). Additionally, the analysis focuses on the more pronounced and not yet corrected J2 relativistic effects on the satellite clock characteristics. The work includes an analysis of periodic characteristics using the fast Fourier transform, least-squares polynomial fitting, and assessment of the satellite clock stability using the Hadamard variance. Two prediction models, namely quadratic polynomial model (QPM) and spectral analysis model (SAM), are used to predict the satellite clocks before and after correcting for the J2 signal and to analyze the J2 relativity effect on the satellite clock prediction. As an example, we use data for June 1, 2020 to analyze the characteristics of the relativity effect for one day.

Analysis of the characteristics of different relativistic effects

As previously described, on-board atomic clocks are subject to appreciable relativistic effects when operating in orbit. These effects include the hardware drift due to the gravitational redshift and time dilation due to the relative velocity (called the combined relativistic effect (CRE)), effects having periodic characteristics (collectively called the PRE), the effect of the Earth’s oblateness (called the J2 relativistic effect), and the currently negligible effect of the gravitational tidal potential of the Sun and Moon.

CRE

The CRE can induce tens of microseconds of clock drift in one day. For GNSS MEO satellites, the drift is nearly 40 microseconds and mainly depends on the semi-major axis of the satellite orbit. The GNSS satellite clock frequency is shifted in proportion to compensate for the CRE.

The daily phase offset $\Delta t_{{{\text{con}},0}}$ and frequency offset $\Delta f_{{{\text{con}},0}}$ of the on-board clock can be calculated using (4) and (5) and the known radius of the satellite orbit. According to the nominal value of the semi-major axis a of the satellite orbit and the geocentric gravitational constant GM, the theoretical CRE can be calculated for the GPS, Galileo and BDS. The results are given in Table 1.

Table 1 Theoretical CRE on the satellite clock, where a is the nominal value of the semi-major axis

The results in Table 1 are calculated using only the nominal GNSS orbit radius. However, owing to several aspects of the construction of a satellite constellation, the actual semi-major axis of satellite orbit is usually different from the nominal value; the error of the CRE $\delta \Delta t_{{{\text{con}}}}$ can be calculated using the second term in (6). Taking selected MEO satellites of BDS-3 as an example, the upper subplot in Fig. 2 shows the deviation $\Delta A$, which is the difference between the mean value of real semi-major axis and the nominal value, and the lower subplot shows the deviation in the satellite clock hardware drift term $\delta \Delta t_{{{\text{con}}}}$ due to $\Delta A$. It is seen that the maximum deviation of the semi-major axis exceeds 50 m, in the case of satellite C19, and the corresponding deviation of the hardware frequency drift is approximately − 0.05 ns. In fact, for all BDS satellites, the maximum deviation of the semi-major axis is approximately 7700 m, and the corresponding deviation of hardware frequency drift is 2.488 ns. It is thus necessary to use actual orbit parameters in the ephemeris to calculate the CRE in avoiding this error.

Fig. 2

Deviation of the semi-major axis (top) and hardware frequency drift (bottom) of selected BDS-3 satellites in 1 day

PRE

GNSS satellite clocks are also affected by the obvious PRE, which depends on the orbital eccentricity e and can be calculated using (5) or the second term in (4). According to (4), the amplitude of the periodic relativity $\Delta t^{{{\text{per}}}}$ is proportional to e. Taking the example of BDS-2 satellites, the eccentricity exceeds 0.004 for selected satellites in inclined geosynchronous orbit (IGSO); thus, the amplitude of $\Delta t^{{{\text{per}}}}$ can be 10 ns regularly and even 27 ns. For most BDS-3 MEO satellites, the eccentricity is usually less than 0.0007, which makes the amplitude of $\Delta t^{{{\text{per}}}}$ less than 1.7 ns. The magnitude of the PRE is shown in Fig. 3 for selected BDS satellites, including geosynchronous orbit (GEO), IGSO, and MEO satellites.

Fig. 3

Periodical relativistic effects for selected BDS satellites

The PRE is also observed for Galileo satellites. In particular, two Galileo satellites (E14 and E18) in the test state are in extended elliptical orbits with a high orbit eccentricity of approximately 0.166 and the amplitude of the PRE is approximately 400 ns. Meanwhile, the orbit eccentricity of the other Galileo satellites is less than 0.0005, and the amplitude of $\Delta t^{{{\text{per}}}}$ is less than 1.0 ns. The PRE of the two “special” Galileo satellites and a selection of properly operating satellites on DOY 153 of 2020 are shown in Fig. 4.

Fig. 4

Periodical relativistic effect of selected Galileo satellites

As a conventional relativistic effect correction, the PRE is corrected by GNSS analysis centers when estimating satellite clock solutions to recover the original satellite clocks as precisely as possible and maintain their linear characteristics. Therefore, the performance of the satellite clock can be analyzed and evaluated clearly and the accuracy of interpolation and extrapolation can be guaranteed. However, users need to make the same correction to ensure consistency with the on-board time.

J2 relativistic effect

It is important to note that the above conventional relativistic corrections do not fully describe the relativistic effect on GNSS satellite clocks. One error, which is caused by the Earth’s oblateness and cannot be ignored in high-precision time transfer (on the order of picoseconds), is the J2 relativistic effect. The approximate analytical formula was given by (8), and the amplitude of $\delta \Delta t^{{{\text{rel}}}}$ is obviously positively correlated with the inclination and height of the orbit.

By substituting the constant J₂ and the nominal value i of the orbital inclination for each navigation satellite system into Eq. (9), the coefficient K_GNSS is obtained as given in Table 2. The semi-major axis a is given in Table 1. Of course, the real orbital parameters in the satellite broadcast ephemeris need to be used in obtaining the actual value of the J2 relativistic effect.

Table 2 GNSS coefficient $K_{{{\text{GNSS}}}}$, where i is the nominal value of orbital inclination

Figure 5 shows the J2 relativistic effect on BDS GEO, IGSO and MEO satellite clocks. Owing to the high orbital height and almost zero orbital inclination of GEO satellites, the amplitude of the J2 relativistic effect is less than 0.001 ns. However, the IGSO satellites are obviously affected by the J2 relativistic effect, with the amplitude being approximately 0.04 ns. This is because the IGSO satellites have large orbital inclination even though they have the same altitude as the GEO satellites. The J2 relativistic effect is most evident on the MEO satellites, with the amplitude being approximately 0.067 ns. It is also seen that the period of the J2 relativity effect is nearly half of the satellite orbit period, with the MEO satellites having an orbital period of 12.9 h and the IGSO satellites having an orbital period of 23.9 h.

Fig. 5

J2 relativistic effects on BDS GEO, IGSO and MEO satellites

The periodic characteristics of the J2 relativistic effect on Galileo satellites are almost the same as those on the BDS MEO satellites except for the two backup satellites in a test state (E14 and E18). The variation curve of the J2 relativistic effect is different from the sinusoidal curve of the other Galileo satellites owing to their special orbit characteristics, as shown in Fig. 6. The affected times in the two half cycles for the J2 relativistic effect are not equal (one being 5.35 h and the other being 7.58 h) owing to the uneven satellite velocities in the two half cycles, resulting in different rates of the latitude angle, i.e., u in (8).

Fig. 6

J2 relativistic effects on two Galileo backup satellites

The J2 relativistic effect is not considered in conventional relativistic corrections and is reflected as an error in the final satellite clock solutions, which affects the performance of the satellite clocks. We thus evaluate aspects of the satellite clock performance before and after J2 relativistic effect correction.

Tidal potential of the Sun and Moon

The improved formula (13) mentioned above still cannot accurately correct the effect of the tidal potential of the Sun and Moon because it is too small to be ignored. The effect on the satellite clock frequency is described by (14) and (15). These formulas show that the effect is positively correlated with the altitude of the satellite orbit. For the BDS MEO satellites, we take the nominal value a = 27,906.1 km, the theoretical values of the coefficients are $A_{{\text{S}}} \approx 2.6 \times 10^{ - 16}$ and $A_{{\text{M}}} \approx 5.3 \times 10^{ - 16}$. For BDS GEO and IGSO satellites, we take a = 42,162.2 km to obtain the theoretical values of $A_{{\text{S}}} \approx 5.9 \times 10^{ - 16}$ and $A_{{\text{M}}} \approx 1.2 \times 10^{ - 15}$. This effect for GEO and IGSO satellites is thus approximately 2.5 times that for MEO satellites.

We calculated the effect of the tidal potential of the Sun and Moon on the frequency of BDS and Galileo satellite clocks in 2020, as shown in Fig. 7. The mean amplitudes of the effect on GPS, Galileo and BDS satellite clocks are given in Table 3. Meanwhile, the amplitude of the J2 relativistic effect on the frequencies of satellite clocks were calculated according to (12) as given in the same table. In the table, $df$ is the relativistic effect on the frequency.

Fig. 7

Relativistic effect due to the tidal potential of the Sun (left) and Moon (right). The left and right axes, respectively, indicate the influence and the distance between the Earth and the celestial body

Table 3 Relativistic effect due to the Sun and Moon on the GNSS satellite clock frequency

We conclude from Fig. 7 and Table 3 that the effects of the tidal potential of the Sun and Moon on GEO and IGSO satellites, having a high orbital height, are approximately 5 × 10⁻¹⁶ and 1.5 × 10⁻¹⁵, respectively. MEO satellites are less affected, with the amplitudes being approximately 2.5 × 10⁻¹⁶ and 6.5 × 10⁻¹⁶, respectively. Furthermore, both of the effects exhibit periodic variations that relate to the Earth–Sun and Earth–Moon distances. The periods of the effects of the Sun and Moon are approximately 1 year and 28 days, respectively. The influence of the Sun reaches its maximum in June and July and the influence of the Moon has multiple peaks in 1 year. Additionally, we see that the two special Galileo satellites (E14 and E18) have not only fixed periodic variations relating to distance but also large-scale fluctuations because the eccentricity of these satellites results in a large variation in the orbital altitude within an orbital period.

The effects of the Sun and Moon are 1 to 2 orders of magnitude smaller than the J2 relativistic effect, making them difficult to detect. Owing to their small amplitudes, we did not investigate these effects on satellite clock characteristics in depth. If a more stable satellite clock (i.e., an optical clock) is used in the future, this relativistic correction may need to be carefully considered.

J2 relativistic effect on GNSS satellite clocks

Table 3 shows that the J2 relativistic effect on the frequency of an on-board atomic clock is as high as 10⁻¹⁴. The frequency stability of current high-performance on-board satellite clocks has reached the same order of magnitude such that the J2 relativistic effect is clearly reflected. Therefore, J2 relativistic effects should be considered when evaluating the performance of satellite clocks. The following sections focus on three aspects of the J2 relativistic effect on the characteristics and performance of GNSS satellite clocks, namely the satellite clock periodic characteristics, satellite clock fitting residuals and satellite clock stability. Additionally, the J2 relativistic effect on satellite clock prediction is analyzed. Precise satellite clock products from DOY 110 to 165 of 2020, which were provided by the Wuhan University analysis center, are adopted.

Periodic characteristics

Owing to the effects of the space environment, the use of a non-optimal satellite model and orbital estimation errors, some periodic signals inevitably appear in the satellite clock solution calculated by the IGS analysis center. The period and amplitude of the signals are obtained using the fast Fourier transform to analyze the spectrum of satellite clock fitting residuals (Wang et al. 2021). Periodic characteristics of the satellite clocks are shown in Fig. 8, which presents spectrogram for Galileo, BDS, and GPS satellites before and after the correction for the J2 relativistic effect (where the results before and after the correction are, respectively, denoted CLK and J2 in the legend).

Fig. 8

Spectrogram of BDS, Galileo and GPS satellites before and after correction for the J2 relativistic effect (presented in blue and red, respectively)

A comparison of the results of the two spectral analyses in Fig. 8 reveals that the periodic characteristics of satellite clocks change appreciably with correction for the J2 relativistic effect. The amplitude of the satellite half-orbital periodic term (i.e., 6.4 h for BDS satellites, 7.0 h for Galileo satellites and 5.9 h for GPS satellites) is weakened, but there remain residual parts, which may relate to the orbital disturbance caused by lunar gravity (Kouba 2019, 2021). At the same time, it is seen that the J2 relativistic effect indeed exists in the satellite clock solution and can be corrected for using the correction formula.

Fitting residuals

To investigate the J2 relativistic effect on satellite clock fitting residuals, we calculated the residuals of BDS, GPS and Galileo clocks before and after correcting for the J2 relativistic effect. The statistical results are presented in Fig. 9. The blue and red points, respectively, represent the fitting residuals before and after correction and the yellow bar is the percentage improvement.

Fig. 9

Variations of satellite clock fitting residuals after the correction for the J2 relativistic effect

The following results were obtained by correcting for the J2 relativistic effect. (1) The fitting residuals of the Galileo satellites equipped with passive hydrogen maser clocks are significantly reduced by 5% to 16%. (2) For BDS satellites, the BDS-2 satellite clocks are hardly improved by the correction for the J2 relativistic effect. This result may relate to the performance of the on-board atomic clock. The fitting residuals of BDS-3 satellites reduce more obviously by 1% to 9%. (3) For GPS satellites, the fitting residuals reduce for IIIA satellites, but the improvement is negligible for IIR and IIR-M satellites, and the improvement is irregular for IIF satellites, which is consistent with the spectral analysis results in Fig. 8. We speculate that these results also relate to the performance of the satellite clocks.

Hadamard variance and stability

The previous sections showed that the performance of satellite clocks is indeed improved by correcting for the J2 relativistic effect. We believe that the correction can also improve the stability of satellite clocks in the time domain. Hadamard variance is commonly used to determine the stability and precision of oscillators and continuous measurements, and it is usually applied in stability analysis. Galileo and BDS satellites equipped with high-performance hydrogen clocks were selected for the present experiment. We also selected two GPS satellites with appreciable changes in the amplitude of the periodic term and calculated the fitting residual after correction for the J2 relativistic effect. The Hadamard variance diagrams of the selected satellites are shown in Fig. 10.

Fig. 10

Hadamard variance sequence diagrams of selected satellites before and after correcting for the J2 relativistic effect (where blue and red, respectively, represent the Hadamard variance sequence of the satellite clock before and after correcting for the J2 relativistic effect)

Carson and Haljasmaa (2015) described how a sinusoidal frequency variation with period T results in a bump in the Hadamard deviation, visible at an averaging time of approximately T/2. Therefore, the correlation region corresponding to Hadamard deviation will bump up when there is an obvious signal in the satellite clock. This bump can be used to determine signal characteristics and assess the stability and performance of atomic clocks. For the selected GNSS satellites, the two most obvious periodic terms in the satellite clock observed from the ground are T and T/2 (where T is the orbital period), which are analyzed in Fig. 8. This leads to two bumps at T/2 and T/4 in the Hadamard variance of the GNSS satellite clocks.

When the J2 relativistic effect is corrected, the bump in the Hadamard variance at the averaging time of T/4 is notably weakened. Especially for the satellite whose T/2 periodic term is almost eliminated after the correction (i.e., E25), the bump at the averaging time of approximately T/4 almost disappears. Therefore, the Hadamard variance of such a clock approaches the theoretical value of that with almost only white noise. It is evident that the J2 relativistic effect cannot be ignored in assessing the stability of satellite clocks and comparing it with the manufacturer’s calibrations.

For the more concerned stability indicators, the stability at 10,000 s (between T/4 and T/2) of the BDS, Galileo and selected GPS satellite clocks undergoes an average improvement of 9%, 15% and 20%, respectively. However, the change in stability at 86,400 s (beyond T/2) is not obvious.

Table 4 gives the fitting residuals and stability at 10,000 s for selected satellites before and after correcting for the J2 relativistic effect. In this table, ‘PRE’ is the amplitude of the PRE, ‘CLK’ and ‘J2’ are, respectively, the results with and without the J2 relativistic effect correction, and ‘percentage’ is the magnitude of the improvement after correction. It is seen that after correcting for the J2 relativistic effect, the fitting residuals and stability at 10,000 s improve to varying degrees for each satellite clock.

Table 4 Changes in the satellite clock fitting residual error and stability at 10,000 s before and after correcting for the J2 relativistic effect

J2 relativistic effect on clock prediction

According to the previous analysis, the satellite clock estimated on the ground contains the J2 relativistic effect, which can be regarded as a periodic signal tightly coupled with the satellite orbit. It is thus speculated that the correction affects the prediction of satellite clocks using the QPM and SAM.

The SAM is a widely used satellite clock prediction model based on the QPM with one or more periodic terms and is expressed as

$$ x_{i} = a_{0} + a_{1} \left( {t_{i} - t_{0} } \right) + a_{2} \left( {t_{i} - t_{0} } \right)^{2} + \sum\limits_{n = 1}^{m} {A_{n} \sin \left( {2\pi f_{n} \left( {t_{i} - t_{0} } \right)} \right) + \Delta i} $$

(16)

where x_i denotes the clock data at the observation epoch t_i, t₀ is the reference epoch, and a₀, a₁ and a₂ are three parameters of the satellite clock model, respectively, referred to as the clock offset (phase), clock velocity (frequency) and clock drift (frequency drift). A_n and f_n denote the amplitude and frequency of the clock period, respectively. m is the number of periods. Δi is the residual of the prediction model. Generally, parameters f_n and m in (16) are determined first through spectral analysis, and other parameters, such as a₀, a₁, a₂ and A_n, are then estimated through least-squares fitting. Although we usually use only the main periodic terms in spectral analysis (e.g., those of 12.8 and 6.4 h for the BDS and those of 14.0 and 7.0 h for Galileo), the period should be obtained as accurately as possible (Huang et al. 2018 and Zhao et al. 2021).

The above is also the basis of our conjecture: the accuracy of the prediction results of the QPM and SAM can be improved if the J2 relativistic effect correction effectively weakens or eliminates the amplitude of the periodic term whose period is half of the orbital period in the satellite clock data. Even removing the half-orbital periodic term of the SAM hardly reduces the prediction accuracy of the model.

To verify the conjecture, the Galileo precise satellite clock solution on day 155 of 2020 was adopted, and four experimental schemes were designed for comparative analysis, including six processing strategies. The details are given in Table 5. The six strategies were applied to predict the 24-h clock for all Galileo satellites. The prediction residuals of selected satellites are shown in Fig. 11. The legend within in the figure indicates the type of method via the format of “clock data used—prediction model (Periodic term not included in the SAM).” For example, “J2-SA (Without J2)” indicates that the clock data after the correction for the J2 relativistic effect are adopted, and the SAM without the J2 period (i.e., 7.0 h for Galileo) is used for prediction. The variations of the root mean square (RMS) and standard deviation (STD) of the clock prediction results before and after correcting for the J2 relativistic effect are shown in Fig. 12.

Table 5 Detailed description of the experimental schemes

Fig. 11

Residuals of the 24-h satellite clock prediction for the four schemes

Fig. 12

Comparison of the clock prediction results for the four schemes

Figure 11 is explained in detail as follows. Different colors represent different prediction models. Red represents the QPM, magenta the SAM including the true periodic term, and blue the SAM without the period of the J2 relativistic effect. Different marker shapes distinguish the data sources. Triangles and circles indicate satellite clock data before and after correcting for the J2 relativistic effect, respectively. Figure 11 shows that the three models that have been corrected for the J2 relativistic effect (circles) outperform those without correction (triangles), have smaller prediction residuals and are more stable. Especially when the time exceeds 6 h, the residuals after correction are closer to zero whereas the uncorrected prediction residuals deviates more evidently with time.

Comparing and analyzing the prediction results of the four schemes in Fig. 12 reveals that the correction for the J2 relativistic effect using the QPM is most effective in improving the prediction accuracy of the satellite clocks. Predictions using corrected J2 relativistic effects satellite clock data show a 9% and 11% improvement in RMS and STD than the uncorrected results. If the SAM is used for prediction, the accuracy of some satellites improves after the J2 relativistic effect correction. The average RMS increases by 6%, 4% and 7% and the average STD increases by 10%, 4% and 11% for two strategies for using corrected data and uncorrected data in Schemes 2, 3 and 4, respectively. However, the prediction accuracy of some satellite clocks even deteriorates, indicating the need for further research and optimization of the prediction methods.

Conclusions and discussion

We introduced and analyzed various relativistic effects on the performance of satellite clocks. The frequency shift due to the CRE is approximately 4.5 × 10⁻¹⁰ Hz. Meanwhile, the phase shift is approximately 39.62 μs/day for BDS MEO satellites and approximately 40.80 μs/day for Galileo satellites, but there is a small bias due to differences in the semi-major axis of the actual orbit. Satellites in orbit are also affected by the PRE, which has a nanosecond magnitude and mainly depends on the orbital eccentricity. Additionally, the J2 relativistic effect due to the Earth’s oblateness and the effect of the tidal potential of the Sun and Moon were analyzed. The former has a sub-nanosecond effect on the MEO satellite clock in GNSS whereas the latter is 1–2 orders of magnitude weaker than the former. Neither is corrected when the satellite clock is estimated, which is reflected in the final product of the GNSS analysis center.

We then verified the J2 relativistic effect on the characteristics and performance of satellite clocks. After the correction for this effect, the amplitude of the satellite half-orbital periodic term is weakened and the fitting residuals of BDS-3 and Galileo satellite clocks decrease by 5% and 8% on average. The Hadamard variance of satellite clocks decreases appreciably for a particular averaging time that is equal to a quarter of the satellite orbit period.

The application of the J2 relativistic effect correction in satellite clock prediction was verified. The results obtained using a QPM improved by 9% in terms of the RMS and by 11% in terms of the STD on average after the correction, and the effect using a SAM was slightly weaker. Additionally, the improvement of the J2 relativistic effect correction for different fitting times, prediction times and the long-term accuracy of prediction variations need to be further explored.

Meanwhile, given that the J2 relativistic effect on GNSS MEO satellites is approximately 0.067 ns, the corresponding positional error is as large as 2 cm. It should be discussed further how this effect can be detected with current hydrogen clocks and even future optical clocks.