Keywords

1 Introduction

In this article, the application of accurate novel optical clocks for the determination of temporal variations of the Earth’s gravity field is described. In the past decade, the performance of optical atomic clocks have made spectacular progress in the laboratories of metrological institutes. Over many decades Cs atomic clocks provided microwave frequency standards with superior long-term stability and accuracy, which today are approaching relative uncertainties of 10\(^{-16}\) (Guéna et al. 2017). Nowadays, optical clocks have become 100 times more precise than the best cesium clocks (Godun 2021). An optical atomic clock generates a frequency reference in the form of light stabilized to an atomic transition frequency in the optical frequency range (Sören et al. 2022). The analysis of the clock measurements has to be done in the framework of general relativity (Philipp 2018). Figure 1 depicts the progress of Cs microwave clocks and optical clocks over the last three decades. It should be mentioned the atomic and optical clocks have steadily improved since the emergence of laser-cooled fountain clocks in the early 1990s, but two distinct types of optical atomic clocks, i.e. optical lattice based and based on trapped ion based clocks, currently compete at a fractional frequency uncertainty of approximately 10\(^{-18}\) which corresponds to gravitational potential difference of 0.1 m\(^{2}\)/s\(^{2}\) (for more details, we refer to Alonso et al. (2022)). The optical lattice clocks at Physikalisch-Technische Bundesanstalt (PTB) in Germany are currently upgraded from a fractional frequency uncertainties of few 10\(^{-17}\) (Schwarz et al. 2020) to the low 10\(^{-18}\) regime. As optical lattice clocks approach a fractional frequency uncertainty 10\(^{-20}\) in the near to far future, this will dramatically improve the accuracy for unifying height and the determination of temporal variations of the Earth’s gravity field.

Fig. 1
figure 1

Progress in the relative accuracy of microwave Cs atomic clocks as well as atomic optical clocks (Alonso et al. 2022)

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The concept to obtain the physical height value with the gravitational redshift measurement with clocks is called chronometric levelling which was originally proposed by Bjerhammar (1985). Furthermore, that concept is extended as chronometric geodesy (Delva et al. 2019). In addition, the high-performance optical clocks connected with dedicated frequency or time links are novel promising network in geodesy (Müller et al. 2018; Mehlstäubler et al. 2018). Also, the frequency transfer via optical fibers in optical clock network has been reached the level of 10\(^{-19}\) (Lisdat et al. 2016; Xu et al. 2018; Dix-Matthews et al. 2021) which fulfills the scientific requirement for the comparison of terrestrial optical clocks with 10\(^{-18}\) uncertainty. Based on optical clock networks, the physical height differences are estimated from optical clock measurements by observing the gravitational redshift effect through the ultra-precise comparison of their frequencies which is called relativistic geodesy (Müller et al. 2018; Mehlstäubler et al. 2018). Moreover, transportable clocks (Grotti et al. 2018; Takamoto et al. 2020) which are developed with high accuracy and stability can be exploited for clock network densification (Wu and Müller 2020). Those dense optical clock networks are suitable candidate for the realization of the International Height Reference System (IHRS) and detection of time-variable Earth’s gravity field signals (Wu and Müller 2020, 2021). The development and progress trend of novel optical atomic clocks make it feasible in the near future to recover the temporal long-wavelength of the Earth’s gravity field from space, to establish a frequency-based physical height reference system as well as the unification of geodetic height systems. Several investigations on affecting the optical clock measurements by mass variations in the Earth system or height differences have been run at Institute of Geodesy, Hannover (Voigt et al. 2016; Denker et al. 2018).

The objective of this paper is to present a description of estimation of lower degree and order spherical harmonic coefficients of the Earth’s gravity field using novel optical clock measurements onboard of low Earth orbiters such as GRACE-like and LAGEOS-like missions. Section 2 introduces the concept and methodology of determining temporal long-wavelength variations of Earth’s gravity field by observing the gravitational redshift with optical atomic clocks. Section 3 gives details on the simulation scenarios with different configurations of satellite orbits at different altitudes for estimation of mass variations with optical atomic clocks. Section 4 presents the numerical results. We first show the results of the analysis of nearly 2 years of optical clock measurements onboard LAGEOS-1 and LAGEOS-2 for the determination of temporal variations of lower degree and order spherical harmonic coefficients. Then, the mass variations from optical clocks onboard a GRACE-like satellite mission is addressed. Finally, the combined solution is discussed.

2 Temporal Variations of the Earth’s Gravity Field from Clock Measurements

2.1 Methodology

The optical clocks measure the gravitational redshift (GRS) within a gravitational potential field. According to general relativity theory (GRT), the optical clocks readings reflect the effect of a potential field on frequency. In GRT, it is essential to distinguish between proper time which is locally measurable and coordinate time which is based on convention. In fact, an ideal clock observes local time as proper time (Soffel and Langhans 2013; Müller et al. 2008).

The relation between proper (relative) time \(\tau \) of an atomic clock within a potential field W such as Earth’s gravity field and coordinate time t at point s can be written as (Mai 2013; Mai and Müller 2013):

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \frac{d\tau_s}{dt}=\sqrt{1-\frac{2W_s}{c^2}-\frac{v_s}{c^2}} =1-\frac{W_s}{c^2}-\frac{v_s}{2c^2}+\varepsilon \left({c^{-4}} \right). \end{array} \end{aligned} $$
(1)

\(W_s\) represents the gravitational potential at point s which depends only on the positions within the potential field in an Earth-fixed system, \(v_s\) is the clock velocity, c is the speed of light as fixed value and \(\varepsilon \left ( {c^{-4}} \right )\) stands for omitting higher order terms. The relativistic time dilation according to Eq. (1) is closely related to the relativistic red shift. Equation (1) can also be applied for a second clock position, replacing s by p. By assuming that the velocities of two stations (or rovers) were precisely determined via Global Navigation Satellite Systems (GNSS), the relativistic time dilation between two optical clocks is then obtained as:

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \begin{array}{ll} \dfrac{d\tau _s }{d\tau _p }&=\left( {1-\dfrac{W_s }{c^2}} \right)/\left( {1-\dfrac{W_p }{c^2}} \right)=\left( {1-\dfrac{W_s }{c^2}} \right)\left( {1+\dfrac{W_p }{c^2}} \right)\\ &\quad +\varepsilon \left( {c^{-4}} \right). \end{array} \end{array} \end{aligned} $$
(2)

For a clock located on the Earth surface, W also includes the effect due to Earth rotation, then called gravity potential, whereas in satellites just reflects the gravitational potential. Since the proper frequency is inversely proportional to the proper time, the following Eq. (3) can be used to derive the relativistic red shift observation equation for two optical clocks as:

$$\displaystyle \begin{aligned} {} 1-\frac{f_p}{f_s}=1-\frac{d\tau_s}{d\tau_p} = \frac{W_s - W_p}{c^2}+\varepsilon \left({c^{-4}} \right) \end{aligned} $$
(3)

or

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \frac{f_s-f_p}{f_s}=\frac{\Delta f}{f_s}=\frac{W_s-W_p}{c^2}+\varepsilon \left({c^{-4}} \right)=\frac{\Delta W}{c^2}+\varepsilon \left({c^{-4}} \right) \end{array} \end{aligned} $$
(4)

where \(f_s\) and \(f_p\) are the proper frequencies of an electromagnetic signal as observed at two points s and p. By multiplying the relative frequency difference \(\frac {\Delta f}{f_s}\) by \(c^2\) and define it as \(\frac {\Delta f^\ast }{f_s}\), Eq. (4) is simplified as:

$$\displaystyle \begin{aligned} {} \frac{\Delta f^\ast}{f_s }=\Delta W+\varepsilon \left( {c^{-4}} \right). \end{aligned} $$
(5)

Equation (5) is the backbone formula which relates the frequency differences and gravitational potential differences where a fractional frequency difference of one part in 10\(^{18 }\) corresponds to about 0.1 m\(^{2}\)/s\(^{2}\) in terms of gravitational potential differences.

2.2 Setup of Optical Clock Observation Equations for the Estimation of Temporal Gravity Field Variations

The gravitational red shift effect which is observed by an optical clock onboard a low earth orbiter is directly related to the gravitational potential difference. Based on this new measurement technique, for the first time in geodesy, it is possible to directly observe the gravitational potential differences. Based on Eq. (5), the optical clock observations as gravitational potential differences between two points s and p can be written as:

$$\displaystyle \begin{aligned} \frac{\Delta f^\ast}{f_s }=W_p \left( {r,\lambda ,\phi ;t} \right)-W_s \left( {r,\lambda ,\phi ;t} \right)+\varepsilon \left( {c^{-4}} \right) \end{aligned}$$

where r, \(\lambda \), \(\phi \) represents the spherical coordinates i.e. radial distance, longitude and latitude of point along the satellite orbit at time t.

On the other hand, the disturbing potential at point s can be formulated as:

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} W\left( {r,\lambda ,\phi ;t} \right)=\frac{GM}{R}\sum_{n=0}^{n_{\max } } {\left( {\frac{R}{r}} \right)} ^{n+1}\sum_{m=0}^n {\left( {\bar {c}_{\textit{nm}} \left( t \right)\cos \left( {m\lambda } \right)+\bar {s}_{\textit{nm}} \left( t \right)\sin \left( {m\lambda } \right)} \right)P_{\textit{nm}} \left( {\sin \phi } \right)} \end{array} \end{aligned} $$
(6)

where G, M and R are the gravitational constant, mass of the Earth and the reference radius of Earth. \(P_{nm} \left ( {\sin \phi } \right )\) is the associated Legendre polynomial of degree and order n and m at latitude \(\phi \) and \(\bar {c}_{nm} \left ( t \right )\) and \(\bar {s}_{nm} \left ( t \right )\) are the normalized geopotential coefficients at time t.

The objective of this paper is to see the performance of optical clocks onboard low Earth orbiters such as LAGEOS- and GRACE-like missions for the estimation of lower degree and order spherical harmonic coefficients of the Earth’s gravity field. Figure 2 depicts a schematic diagram of clocks onboard of LAGEOS- and GRACE-like satellite missions with different altitudes and configuration for the estimation of lower degree/order harmonic coefficients. Therefore, here we simplify Eq. (7) up to degree and order 2:

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} W\left( {r,\lambda ,\phi ;t} \right)=\frac{GMR^2}{r^3}\left( {\begin{array}{l} \bar {c}_{20} \left( t \right)P_{20}\left( {\sin \phi } \right)+ \\ \bar{c}_{21} \left( t \right)\cos \lambda P_{21} \left( {\sin \phi } \right)+ \bar {s}_{21} \left( t \right)\sin \lambda P_{21} \left( {\sin \phi } \right)+ \\ \bar {c}_{22} \left( t \right)\cos \left( {2\lambda } \right)P_{22} \left( {\sin \phi } \right)+\bar {s}_{22} \left( t \right)\sin \left( {2\lambda } \right)P_{22} \left( {\sin \phi } \right) \\ \end{array}} \right). \end{array} \end{aligned} $$
(7)
Fig. 2
figure 2

Schematic diagram of clocks onboard of LAGEOS- and GRACE-like satellite missions with different altitudes and configuration for the estimation of lower degree/order harmonic coefficients

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The gravitational disturbing potential attenuates with respect to altitude as shown in Fig. 3. The gravitational disturbing potential has a value of 282.57 m\(^{2}\)/s\(^{2}\) at an altitude of zero, at an altitude of a GRACE-like mission (450 km), it is 210.04 m\(^{2}\)/s\(^{2}\), and at an altitude of geo-stationary satellites (35,786 km), it is 0.52 m\(^{2}\)/s\(^{2}\). With further improvement of optical clock uncertainties into the 10\(^{-18 }\) to 10\(^{-19 }\) regimes, the higher satellite altitudes, e.g., the geostationary orbit are good choices for the establishment of a reference optical atomic clocks in space.

Fig. 3
figure 3

The gravitational disturbing potential attenuation at different altitudes. (a): disturbing potential at altitude of zero with a mean value of 282.57 m\(^{2}\)/s\(^{2}\), (b): disturbing potential at an altitude of a GRACE-like mission of 450 km with a mean value of 210.04 m\(^{2}\)/s\(^{2}\), (c): disturbing potential at the altitude of 35,786 km for geo-stationary satellites with a mean value of 0.52 m\(^{2}\)/s\(^{2}\)

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The observation equations for optical clock observations i.e. the gravitational potential differences along the satellite orbits with the sampling rate of \(\Delta t\) can be written as:

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \begin{array}{l} \left( {{\begin{array}{*{20}c} {\Delta W\left( {t_0 } \right)} \hfill \\ \hspace{-1.5pt}\vdots \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \\ {\Delta W\left( {t_n } \right)} \hfill \\ \end{array} }} \right) =\left( {{\begin{array}{*{20}c} {P_{20} \left( {t_0 } \right)} \hfill & {\cos \lambda P_{21} \left( {t_0 } \right)} \hfill & {\sin \lambda P_{21} \left( {t_0 } \right)} \hfill & {\cos 2\lambda P_{22} \left( {t_0 } \right)} \hfill & {\sin 2\lambda P_{22} \left( {t_0 } \right)} \hfill \\ \hspace{-8pt}\vdots \hfill\hfill & \hspace{-15pt}\vdots \hfill \hfill & \hspace{-15pt}\vdots \hfill \hfill & \hspace{-15pt}\vdots \hfill \hfill & \hspace{-15pt}\vdots \hfill \hfill \\ {P_{20} \left( {t_n } \right)} \hfill & {\cos \lambda P_{21} \left( {t_n } \right)} \hfill & {\sin \lambda P_{21} \left( {t_n } \right)} \hfill & {\cos 2\lambda P_{22} \left( {t_n } \right)} \hfill & {\cos 2\lambda P_{22} \left( {t_n } \right)} \hfill \\ \end{array} }} \right) {\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}\left( {{\begin{array}{*{20}c} {\Delta \bar {c}_{20} \left( {t_0 } \right)} \hfill \\ {\Delta \bar {c}_{21} \left( {t_0 } \right)} \hfill \\ {\Delta \bar {s}_{21} \left( {t_0 } \right)} \hfill \\ {\Delta \bar {c}_{22} \left( {t_0 } \right)} \hfill \\ {\begin{array}{l} \Delta \bar {s}_{22} \left( {t_0 } \right) \\ \Delta \bar {c}_{20} \left( {t_1 } \right) \\ \end{array}} \hfill \\ \hspace{-9pt} \vdots \hfill \hfill \\ \end{array} }} \right) \\ \end{array} \end{array} \end{aligned} $$
(8)

or

$$\displaystyle \begin{aligned} {} \Delta {\mathbf{W}}={\mathbf{A}}\Delta {\mathbf{x}}+\varepsilon \end{aligned} $$
(9)

where \(\Delta {\mathbf {W}}\) is the vector of the observables as potential difference of dimension \(\left ( n \right )\), \({\mathbf {A}}\) is the design matrix of dimension \(\left ( {n\times m} \right )\) and \(\Delta {\mathbf {x}}\) represents the monthly spherical harmonic coefficients of dimension \(\left ( m \right )\).

The overall set of spherical harmonic coefficients as unknowns in Eq. (9) is estimated by least-squares adjustment.

2.3 Optical Atomic Clock Noise

For this study, the stochastic model for optical clock observations along the satellite orbit is considered as white noise with zero mean and known variance \(\sigma _c^2\) as:

$$\displaystyle \begin{aligned} {} {\mathbf{E}}\left( {\varepsilon \left( t \right)} \right)={{\mathbf{0}}};\quad {\mathbf{C}}_c =\sigma _c^2 {\mathbf{I}}. \end{aligned} $$
(10)

The operator \({\mathbf {E}}\left ( {\varepsilon \left ( t \right )} \right )\) is the expectation of optical clock noise and \({\mathbf {C}}_c\) is the diagonal matrix with known variance \(\sigma _c^2\) and unit matrix \({\mathbf {I}}\).

It should be mentioned that different averaging periods of 15 min or 30 min along the satellite orbit are assumed to achieve clock accuracies of 10\(^{-18}\) or 10\(^{-19}\).

3 Simulation Scenarios for Estimating Mass Variations with Optical Atomic Clocks

Figure 4 depicts the simulation chain of optical clock observations along the satellite orbits as potential differences and the recovery of the Earth gravity field based on those observations. Based on constant degree-2 Stokes coefficients and a-priori secular and annual variations from SLR monthly gravity solutions, the coefficients are synthesized. The optical clock measurement are computed as gravitational potential differences along the satellite orbits. In the second step, white noise for the optical clock measurements is added. Monthly gravity field solutions are determined from the gravitational potential differences. In the final step, a-posteriori secular and annual variations of the degree- and order-2 coefficients are estimated by least-squares adjustment. The zonal coefficient c\(_{20}\) represents the dynamic flattening of the Earth. The temporal variations of that coefficient reflect the hydrostatic balance between gravitational and centrifugal force variations as global scale mass redistribution. The temporal variations of the tesseral harmonic coefficients c\(_{21}\), s\(_{21}\) represent the Earth’s principal figure axis variations related to polar motion or rotational deformation. The sectorial c\(_{22}\), s\(_{22}\) coefficients describe the flattening of the equator.

Fig. 4
figure 4

Flowchart of the simulation of optical clock observations as potential differences and recovery of the Earth gravity field

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3.1 Data

To simulate the optical clock measurements along the satellite orbits, the geodetic satellite missions LAGEOS-1, LAGEOS-2 and GRACE-FO are utilized. Table 1 summarizes the orbital parameters such as altitude, inclination and revolution of satellites. Figure 5 depicts the periodic altitude variations of satellite orbit LAGEOS-1 for three days. For this study, two years of real satellite orbits from Sep. 2018 to Aug. 2020 are used. Due to GRACE and GRACE-FO orbit designs, specific configuration and polar gaps, the lower degrees of the Earth’s gravity field can not be estimated with good accuracy. Therefore, the low degree monthly gravity field coefficients are taken from satellite laser ranging (SLR) observations such as LAGEOS-1 and LAGEOS-2 (Cheng et al. 2013) to be used as a-priori values for this study.

Fig. 5
figure 5

Altitude variations of satellite orbit LAGEOS-1 for three days

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Table 1 Orbital information of geodetic satellite missions as used for the simulation of optical clock measurements
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The gravitational potential differences observed by optical clocks along the satellite orbits LAGEOS-1, LAGEOS-2 and GRACE-FO are computed based on Eqs. (9) and (10). Table 2 shows the different white noise cases for the simulation scenarios of the optical clocks measurements.

Table 2 White noise cases with different average time, different frequency uncertainties and corresponding potential differences used in this simulation study
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4 Numerical Results

Temporal variations i.e. seasonal variations and secular trend of spherical harmonic coefficients up to degree and order 2 from 24 months noise-free SLR observations is shown in Fig. 6. Figure 7 depicts the temporal variations of spherical harmonic coefficients up to degree and order 2, i.e. \(\Delta \bar {c}_{20} ,\;\Delta \bar {c}_{21} ,\;\Delta \bar {s}_{21} ,\;\Delta \bar {c}_{22} ,\;\Delta \bar {s}_{22} \), for 24 months which are estimated based on two years of optical clock observations along LAGEOS-1 and LAGEOS-2 orbits considering different clock uncertainties and different averaging times of 10, 2 and 60 min.

Fig. 6
figure 6

Temporal variations of spherical harmonic coefficients up to degree and order 2 from 24 months noise-free SLR observations

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Fig. 7
figure 7

Estimated spherical harmonic coefficients of degree/order 2 from optical clock measurements along LAGEOS-1 and LAGEOS-2 orbits, from top to bottom corresponding to cases 1–4

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An averaging time of 60 min is needed to achieve frequency uncertainties of 1\(\times \)10\(^{-19}\). With these measurements, the spherical harmonic coefficients \(\Delta \bar {c}_{20} ,\;\Delta \bar {s}_{22} \) can accurately be estimated and are comparable to SLR-derived monthly gravity field solutions. However, for the averaging times of 2, 10 and 60 min with fractional frequency uncertainties of 4.52\(\times \)10\(^{-18}\), 4.08\(\times \)10\(^{-18}\) and 1\(\times \)10\(^{-18}\) which correspond to gravitational potential differences of 0.452, 0.408 and 0.100 m\(^{2}\)/s\(^{2}\), the temporal variations of\(\;\Delta \bar {c}_{21} ,\;\Delta \bar {s}_{21} ,\;\Delta \bar {c}_{22}\) can not as precisely be estimated as with SLR.

The temporal variations of spherical harmonic coefficients up to degree and order 2 from 24 months optical clock observations onboard GRACE-FO is shown in Fig. 8. Again, the monthly solutions are estimated with least-squares adjustment for different frequency uncertainties and different averaging times of 10, 2 and 60 min. The averaging times of 2 and 10 min are selected to demonstrate the performance of optical clocks onboard low earth orbiters for precise determination of temporal long-wavelength variations of the Earth’s gravity field. With an averaging time of 60 min to achieve the frequency uncertainties of 1\(\times \)10\(^{-19}\) along the satellite orbit GRACE-FO, the spherical harmonic coefficients \(\Delta \bar {c}_{20} ,\;\Delta \bar {s}_{21} ,\;\Delta \bar {s}_{22} \) can be accurately estimated and are comparable to SLR-derived monthly gravity field solutions. But the temporal variations of \(\Delta \bar {c}_{21} , \Delta \bar {c}_{22}\) are not obtained accurate enough. The same holds for the other GRACE cases, where a poorer clock performance has been assumed.

Fig. 8
figure 8

Estimated spherical harmonic coefficients of degree/order 2 from optical clock measurements along GRACE-FO satellite orbits, from top to bottom corresponding to cases 1–4

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Figure 9 shows the temporal variations of spherical harmonic coefficients up to degree and order 2 from 24 months of optical clock observations along the orbits of LAGEOS-1, LAGEOS-2 and GRACE-FO. The monthly solutions are estimated for the same cases as before. For the averaging time of 60 min enabling frequency uncertainties of 1\(\times \)10\(^{-19}\), the spherical harmonic coefficients \(\Delta \bar {c}_{20} ,\;\Delta \bar {s}_{21} ,\;\Delta \bar {s}_{22}\) are accurately obtained and comparable to SLR-derived monthly gravity field solutions. But the temporal variations of \(\Delta \bar {c}_{21} , \Delta \bar {c}_{22}\) are less accurately obtained than the SLR monthly gravity field solutions. For taveraging times of 2, 10 and 60 min with fractional frequency uncertainties of 4.52\(\times \)10\(^{-18}\), 4.08\(\times \)10\(^{-18}\) and 1\(\times \)10\(^{-18}\), the temporal variations of \(\Delta \bar {c}_{22}\) is improved relative to the GRACE-FO case, but \(\Delta \bar {c}_{21}\) is still worse.

Fig. 9
figure 9

Estimated spherical harmonic coefficients of degree/order 2 from optical clock measurements along LAGEOS-1, LAGEOS-2 and GRACE-FO orbits, from top to bottom corresponding to cases 1–4

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5 Conclusions

Changes of the low-degree spherical harmonic coefficients, such as the zonal term \(\bar {c}_{20}\), reflect significant mass variations in the Earth system. Nowadays, SLR observations, e.g., from LAGEOS-1 and LAGEOS-2 are routinely used for the estimation of temporal variations of lower degree/order spherical harmonic coefficients. Moreover, as the low-degree zonal coefficients of the Earth’s gravity field are poorly recovered with GRACE and GRACE-FO satellite missions, their temporal variations are taken from SLR observations to supplement the GRACE and GRACE-FO estimates. In future, also optical lattice clocks onboard of low earth orbiters have the potential to determine temporal variations of those low-degree gravity field coefficients with good accuracy. Different configurations of satellite orbits such as GRACE-FO, LAGEOS-1 and LAGEOS-2 between 400 and 6000 km with certain assumptions on the optical clock errors have been studied to quantify this application. Optical clocks with instabilities of 1.0\(\times \)10\(^{-19}\) in 60 min can reach the SLR accuracy in the future.

Assuming some progress in the development of optical atomic clocks in the future, the precise determination of temporal long-wavelength variations of the Earth’s gravity field from space is possible.