A Novel Orbit Determination and Time Synchronization Architecture for a Radio Navigation Satellite Constellation in the Cislunar Environment

2.1 Signal Structure

SS modulation represents the backbone of any GNSS, as it enables simultaneous signal transmission, better time-delay resolution, and good immunity to interference, especially for long SS sequences (Meurer & Antreich, 2017). The ground station transmits 24-Mcps SS signals with a modulation format derived from work by the Consultative Committee for Space Data Systems (CCSDS, 2011, 2013). The modulation is offset quadrature phase-shift keying (OQPSK) with a CDM scheme, where the four SS streams associated with the in-phase (I)-channel are superimposed and then fed to the majority vote logic (CDM-M) used to generate a constant-envelope binary phase-shift keying (BPSK) signal, suitable for operations in nonlinear channels (Donà & Iess, 2017; Donà & Rovelli, 2016; Campa, 2022). For a general overview on SS and code-division multiple access (CDMA), see, for example, the work by Holmes (1982).

The four codes of the I-channel are short Gold codes (2¹⁰ – 1 chips in length) that are also used for onboard carrier signal acquisition. The quadrature (Q)-channel code (same for all four satellites) is a truncated maximum length (TML) sequence generated by a linear feedback shift register with N = 18. The TML sequence has a length of 256 times the length of the Gold code [256 (2¹⁰ – 1) chips] and is synchronous with the Gold code. This feature is used by the onboard transponder to aid in acquisition of the long TML sequence. After the onboard signal acquisition, the ground station can send telecommand data on the relevant I-channel component.

The return link signal generated by the onboard transponder is different depending on the transponder configuration (Table 1), i.e., whether it is in coherent or non-coherent mode. In non-coherent mode, short Gold codes are transmitted to simplify the implementation of the SS acquisition algorithm at the ground station and to acquire the relevant quadrature phase-shift keying (QPSK) telemetry. In coherent mode, two TML sequences (of the same family as the received sequence) are transmitted in the I- and Q-channels. The two downlink TML sequences are synchronized in terms of both chip rates and code epoch with the uplink TML sequence. In this way, the ground station modem can acquire the long TML sequences aiding the SS code search with a preliminary knowledge of the spacecraft distance and the associated turn-around delay. The TML sequence turn-around in coherent mode allows the ranging, Doppler, and code epoch measurements to occur, parallelizing ODTS functions. Finally, QPSK modulation enables a balanced I and Q modulation format (telemetry data on both I and Q components).

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FIGURE 2

Forward link modulation format based on the CDM-M scheme

Note: TC: telecommand; UQPSK: unbalanced quadrature phase-shift keying

This signal structure is not considered adequate for asynchronous time transfer (i.e., with the transponder in non-coherent mode). For this scenario, the proposed solution (not addressed by CCSDS (2011, 2013)) is based on the downlink transmission of the same type of modulation format and codes specified for the uplink. In detail, the proposed solution includes a short Gold code (length of 2¹⁰ – 1 chips) on the I-branch and a TML sequence on the Q-branch that has a length of 256 times the length of the short Gold code [256 (2¹⁰ – 1) chips] and is synchronous with the Gold code. This feature is used by the ground station to aid in acquisition of the long TML sequence. For the uplink, the modulation is unbalanced OQPSK (I/Q power ratio of 10 dB), with telemetry modulation on the I-branch and Q-branch and carrying the TML sequence, used for time transfer operations only.

In some mission phases, for instance, during contingencies, the TT&C link can be configured with pulse code modulation (PCM)/phase-shift keying (PSK)/phase modulation (PM) CCSDS standards (CCSDS, 2021), which significantly widens the ground support beyond the stations dedicated to the LRNS. The satellite in contingency is contacted by a dedicated ground station equipped with a high-directivity (and power) antenna to close the link via an onboard low-gain antenna (LGA). Although the interference between the standard-modulated signal (for the contingent satellite) and the SS signals (for the nominal satellites) must be kept under control, the power unbalance favors the former, which, coupled with the limited interference suppression capability of short Gold codes, suggests the use of mitigation strategies. One option is to employ a different frequency channel for the contingency mode. This approach would require a frequency-flexible transponder, a feature that could also facilitate simultaneous launches by leveraging external ground stations supporting only CCSDS standard modulation during the launch and early orbit phase (LEOP). Alternatively, a proper filtering mechanism in the SS acquisition/demodulation processing could be implemented to suppress the standard signal interference, as proposed by Simone et al. (2010), who analyzed an adaptive cancellation algorithm for jamming rejection in SS receivers.

2.2 Onboard Radio System

Figure 3 shows the proposed architecture for the onboard radio system, which consists of the following:

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FIGURE 3

Onboard radio system architecture

The configuration during nominal operations entails the main branch (Transponder_1, Amplifier_1, and Diplexer_1) connected to the MGA, with the redundant chain (Transponder_2, Amplifier_2, and Diplexer_2) connected either to LGA_1 or LGA_2. Waveguides are indicated as black lines, whereas the red lines indicate connections that can be implemented as coaxial cables to conserve mass. RX: receiver; TX: transmitter

• • One 30-cm steerable, two-degree-of-freedom, medium-gain antenna (MGA) for nominal operations

• • Two LGAs for quasi-omnidirectional coverage when the MGA is not available (as in LEOP or contingency phases)

• • Two redundant K-band dual-mode transponders (hot redundancy for the receiver and cold redundancy for the transmitter)

• • Two traveling-wave-tube amplifiers (TWTAs; 10 W) in cold redundancy

• • The radio frequency distribution assembly (RFDA), including two diplexers and five switches for signal routing to/from antennas

• • Waveguides (in black) to minimize radio frequency (RF) path losses, while coaxial cables (in red) can be used on the transmitter side before signal amplification

This solution is robust, as the mission is not compromised in the case of failure of one transponder or TWTA.

The presence of the coupler in the transmitter chain prior to the 10-W amplification allows easy cross-strapping between the transponders and amplifiers, improving the system reliability. Furthermore, the RFDA allows for the following:

• • During nominal operations, the RFDA ensures that the nominal transponder is connected to the MGA, whereas the redundant transponder (receiver-ON and transmitter-OFF) is connected to one of the two LGAs and is ready for telecommand demodulation.

• • During contingency scenarios, the RFDA keeps each transponder connected to one of the two LGAs in order to ensure near-omnidirectional coverage for ground station contact.

• • The RFDA minimizes the RF losses and the impacts on the link budget through the use of switches instead of couplers.

The K-band TT&C transponder is a flexible unit capable of working in CCSDS PCM/PSK/PM standard modulation and in SS mode, interfacing with the following:

• • The onboard computer (OBC) for sending demodulated telecommands and receiving the telemetry data stream to be modulated and downlinked to the ground station

• • The signal generation unit (SGU), for pseudonoise (PN) code epoch time-stamping and frequency synchronization with the external ultra-stable frequency reference (EUFR), as required by the time transfer implementation

Transponders also feature their own internal oscillator, typically an oven-controlled crystal oscillator (OCXO), for TT&C operations when the EUFR is not required/available. Nonetheless, synchronization of the transponder frequency with the EUFR is highly recommended in SS mode to reduce the signal acquisition frequency range that would otherwise suffer from the OCXO frequency instability; in addition, this synchronization is considered indispensable for the asynchronous time transfer function.

2.3 Technology Survey and Technology Readiness Level

The proposed radio system architecture makes use of several enabling features:

• • The K-band instead of the more commonly used S- or X-band

• • The use of wideband signals, with the current baseline considering an SS chip rate in the range of 24 Mcps

• • SS modulation based on the CDM-M scheme

• • Ground station modulators and receivers enabling MSPA

• • Dual-mode transponders working with CCSDS standards and SS modulation

• • Interference suppression onboard to handle the CCSDS standard signal in the contingency scenario and, if needed, at the ground station to avoid link interruption due to the limited rejection of SS codes

• • Transponder frequency flexibility, recommended for LEOP in case of multiple launches

TT&C technology is well proven and has been demonstrated in-flight, even at the higher Ka-band frequency, as shown by the BepiColombo and JUICE missions (Cappuccio, Notaro et al., 2020; Iess et al., 2021). Their transponders, built by TAS-I, enable the novel 24-Mcps PN modulation scheme, providing two-way range measurements with centimeter-level accuracy at an integration time of only 4 s at 0.3 AU. In tracking passes with favorable weather conditions, the range-rate measurements attained an average accuracy of 0.01 mm/s for a 60-s integration time. Dual-mode transponders (compatible with CCSDS standards and SS modulation) implementing flight frequency flexibility have already been developed for the TT&C link in terrestrial GNSS constellations. The CDM-M scheme has been proposed, studied, and implemented at the breadboard level for onboard application under ESA contract, for both spacecraft (Donà & Iess, 2017) and ground station equipment (ESA-GRST-STU-SOW-0010, 2016).

Algorithms for interference cancellation in SS receivers, based on either adaptive filtering (Simone et al., 2010) or fast Fourier transform solutions, have been proposed and successfully tested at the breadboard level and have now been implemented in TT&C secure transponders. The MSPA, once supported by CDM-M technology, is primarily employed as a new approach for the control and operation of all satellites in a constellation, implementing simultaneous TT&C operation (with telecommand, telemetry, and ranging) for all links with visibility from the ground station.

In the framework of the European System Providing Refueling, Infrastructure and Telecommunications (ESPRIT) project for the upcoming lunar gateway, TAS-I is developing a K-band transceiver (KBT) for communications and tracking, a nodal element of the HALO lunar communication system. The KBT is based on the IDST, a product platform under design and development by TAS-I under ESA contract. The KBT implements frequency flexibility in the range of 60 MHz in the K-band and handles wideband signals with a data rate on the order of 50 Mbps. The digital section will be fully adequate to support the 24-Mcps chip rate of the ATLAS configuration for Moonlight.

Finally, two-degree-of-freedom steerable MGAs are a consolidated space technology, used on the BepiColombo (X-band) and JUICE (dual X- and Ka-band) missions. An off-the-shelf MGA in close alignment with ATLAS ambitions is produced by the Norwegian company Kongsberg (dish diameter: 30 cm, total mass: 12 kg, power consumption: 25 W).

2.4 Link Budget and Data Rate

The primary design feature of the proposed architecture is the MSPA concept, which concurrently enables the implementation of SBI (Section 3.3). The relatively small Earth–Moon distance and the high directivity at the K-band limit the antenna size to approximately 26 cm. However, this size is sufficient to ensure adequate data rates in the radio links. Via a pointing strategy based on the minimum enclosing circle, the pointing loss from ground can be reduced to less than 3 dB for 99% of the time (1.3 dB on average).

Table 2 presents the link budget, which was computed under the assumption of conservative conditions: the link elevation is fixed at 15°, the transmission power is limited to 200 W on ground and 10 W onboard, the slant range is set to the maximum Earth–Moon distance, and the pointing losses are set to worst-case values. In addition, a room-temperature low-noise amplifier is considered on ground. Atmospheric attenuation, quoted from the DSMS Telecommunications Link Design Handbook (2015), refers to a deep space antenna site with a cumulative distribution CD = 0.9 (implying that the attenuation is ≤3 dB for 90% of the time). Choosing a dry site is important not only for attenuation (not negligible for the K-band), but also to reduce tropospheric effects degrading the quality of radiometric observables and to guarantee an electromagnetically clean site. The latter is particularly critical for small dishes, which are prone to radiation disturbances from the surrounding environment. The transmission power from ground can easily be increased by a factor of 2 or 3 (and possibly more) if enhanced uplink performance is desired. However, the onboard RF power is expected to be a limited resource, as we assumed a small-sat platform dedicated to navigation services, in line with the Moonlight concept. Owing to power and pointing requirements, a steerable MGA with a dish diameter of 30 cm has been selected for the spacecraft.

The carrier signal-to-noise ratio (SNR) for the uplink and downlink is 38 dBHz and 35 dBHz, respectively, for each satellite link of the mini-constellation, including a link margin of 3 dB (suggested by the European Cooperation for Space Standardization). The uplink SNR is primarily limited by the power-sharing loss implied by CDM-M, corresponding to 8.5 dB for four (or five) spacecraft, as reported by Donà and Iess (2017). Those losses would become prohibitively large if the constellation was expanded in the future. In that case, the strategy would be to form groups of four or five satellites and to use additional ground antennas in order to simultaneously track each group. This expansion would be low-cost, given the small size of the ground terminal and the use of an existing infrastructure. The obtained signal level is considered sufficient for acquisition, provided that the onboard receiver is compatible with an acquisition window of ±130 kHz. Because of the MSPA, uplink pre-steering in the acquisition phase can only be based on a spacecraft-averaged Doppler shift owing to CDM-M. Pre-steering the carrier frequency eases the acquisition phase because each spacecraft receives a signal with a different Doppler shift (up to 150 kHz in the K-band), and each transponder has a different rest frequency, whereas the uplink carrier is the same for the entire constellation. The signal acquisition proceeds more rapidly if the uplink carrier has a frequency that minimizes, according to some cost function, the difference between the received frequency and the rest frequency for each satellite transponder. For the downlink, the ground station can adjust the frequency independently for each satellite (given the CDMA channel), making pre-steering much more effective.

Telecommand, telemetry, and ranging SNRs are computed under the assumption of the signal modulation scheme suggested by Donà and Iess (2017), namely, unbalanced QPSK with a power unbalance of 1:10 for uplink and OQPSK for downlink. A ranging precision of less than 35 cm in two-way coherent mode can be obtained using a code rate of 24 Mcps, a jitter contribution from the SNR of the uplink I-channel, and either the I- or Q-channel SNR for downlink. The uplink format contains only the long PN code used for ambiguity resolution and coherently generated from the shorter code used for acquisition on the I-channel. Conversely, for downlink, only the long code is used, both for acquisition and ambiguity resolution. Telecommand and telemetry data rates are reported for two possible coding schemes. The target bit error rate is 10⁻⁵ for telecommand and 10⁻⁶ for telemetry. Note that the MSPA concept maximizes the data volume rather than the data rate: it allows simultaneous and continuous communication with all spacecraft. For a case with roughly 22 h of tracking per day, the overall downlink data volume from each satellite is greater than 110 Mbit/day.

In contingency scenarios, the small ground antenna performance is insufficient, and a dedicated larger dish with a diameter of at least 13 m is required. In such cases, the MSPA concept is not necessary, as the link is established with a single satellite at a time, utilizing the onboard LGA and standard modulation, with a residual carrier, as recommended by the CCSDS (2021). The SS signals from a nominally operating spacecraft will be much lower in power (by approximately 34 dB) than the residual carrier from a 13-m ground antenna. Thus, their interference is expected to be negligible. Conversely, the residual carrier signal may interfere with the SS signals, particularly for the uplink, where short Gold codes are employed. This problem can be overcome by using the strategies outlined above.

3.1 Doppler

Doppler measurements rely on the comparison between the frequency of an uplink signal generated by a highly stable frequency standard (typically a hydrogen maser) and the received signal, coherently returned by the onboard transponder. Because the relative frequency shift is equal to the time derivative of the round-trip light-time (dRTLT/dt = Δf/f) (Thornton & Border, 2003), defining range as RTLT/c, range rate and Doppler, being proportional, are often used interchangeably. Range rate, not to be confused with the difference of two range measurements divided by the time interval, is naively interpreted as the radial velocity of the spacecraft. Doppler shifts, derived from phase measurements, are a standard output of ground back-end receivers. Different from Doppler measurements, which depend on the carrier frequency, range rate is independent of the carrier frequency. State-of-the-art deep space tracking systems based on Ka-band links provide range rate accuracies in the range 0.005−0.02 mm/s at 60 s integration times, or, in terms of the Allan deviation (ADEV) (Barnes et al., 1971) of the frequency residuals after orbital fit, 1.7−6.8 × 10⁻¹⁴. At longer time scales (1000 s), the ADEV of frequency residuals often reaches 5 × 10⁻¹⁵, corresponding to 0.0015 mm/s. Water vapor radiometers (WVR) and normal weather conditions are necessary to attain these values (Asmar et al., 2005, Iess, et al., 2018; Cappuccio, Hickey et al., 2020). Doppler is the observable of choice for positioning of orbiters within the planetocentric frame, due to its extreme sensitivity to gravity gradients. For this reason, it deserves special attention in the LRNS.

The error budget of Doppler measurements for the ATLAS radio architecture has been derived from recent data collected by BepiColombo, reported by Iess et al. (2021), and from a previous analysis by Iess et al. (2014). We adjusted thermal noise to the SNR computed in Section 2.4 and neglected the contribution of interplanetary plasma. Each error source in the Doppler error budget has its own spectral properties and a different relationship with the integration time. Because the numerical simulations were carried out using Doppler data sampled at 60 s, the error budget in Table 2 has been computed for a 60-s integration time, scaling the values reported in the literature at a different integration time as needed. Note that some error sources, such as errors in the station locations and Earth orientation parameter (EOP) and mismodeling of Earth solid tides, produce larger ADEVs at longer integration times. However, their contribution, which is roughly proportional to the integration time, does not affect the end-to-end noise budget, even at 1000 s. Thus, the assumption of white noise at 60 s made in the simulations of the OD process is valid. Finally, the very small value of the ground antenna mechanical noise (8 × 10⁻¹⁸) is a consequence of the small size of the dish. By comparison, Asmar et al. (2005) reported a value of 4 × 10⁻¹⁵ at 1000 s for NASA’s Deep Space Network antenna DSS 25, a 34-m dish. The value in Table 2 was derived by scaling linearly from the value reported by Asmar et al. (2015) and accounting for the different integration times (60 s vs. 1000 s).

3.2 Range

Range measurements are affected by random noise (thermal jitter) and systematic effects (range bias). The ranging jitter due to thermal noise depends on the available SNR (see Table 2), in both the uplink and downlink. For an SS ranging system, the jitter is obtained from the following expression of the chip tracking loop (Holmes, 1982):

$\frac{\sigma }{T_C}\cong \sqrt{\frac{N_o{B}_L}{2S}(1+\frac{2N_0{B}_P}{S})}=\sqrt{{\left[\frac{N_o}{S}\right]}_{eq}\frac{B_L}{2}}\hspace{0.17em}\left[\text{chip}\right]$ 1

where:

${\left[\frac{S}{N_0}\right]}_{eq}=\frac{S/N_0}{1+\frac{2N_0{B}_P}{S}}$ 2

is the equivalent signal power over noise power spectral density, T_c is the chip length $(T_c=\frac{1}{\text{chip}\hspace{0.17em}\text{rate}})$, B_P is the pre-detection bandwidth, and B_L is the loop bandwidth. B_P is conservatively taken as 8 times the symbol rate. For the onboard transponder (telecommand based on BPSK modulation on the I-branch at a bit rate of 1.8 kbps with a code rate of r = 1/2), we assumed B_P = 28.8 kHz; for the on-ground receiver (telemetry based on OQPSK modulation at a bit rate of 1.4 kbps with a code rate of r = 1/2), B_P = 11.2 kHz. In downlink, the pre-detection bandwidth is derived from the telemetry symbol rate in one branch, which is assumed to be half of the total symbol rate derived from the link budget.

When the onboard loop bandwidth is much narrower than the on-ground loop bandwidth, the uplink and downlink jitters are summed as the square root of their squared values. Under this unlikely condition, the overall end-to-end one-way ranging jitter can be computed as follows (CCSDS, 2014, Section 2.7.3):

${\left({\sigma }_{rng}\right)}_{E2E}\approx \frac{c}{2}{T}_c\sqrt{{\left(\frac{\sigma }{T_c}\right)}_U^2+{\left(\frac{\sigma }{T_c}\right)}_D^2}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\left[m\right]$ 3

where c is the speed of light. In a more realistic case, when the onboard loop bandwidth is much larger than the on-ground loop bandwidth, the overall end-to-end one-way ranging jitter is independent of the onboard loop bandwidth and can be written as follows (CCSDS, 2014, Section 2.7.3):

${\left({\sigma }_{rng}\right)}_{E2E}=\frac{c}{2}{T}_c\sqrt{\frac{B_{L_{\_}D}}{2}[{\left[\frac{N_o}{S}\right]}_{e{q}_{\_}U}+{\left[\frac{N_o}{S}\right]}_{e{q}_{\_}D}]}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\left[m\right]$ 4

In our case, we have assumed a loop bandwidth of B_{L_U} = 5 Hz for the onboard transponder and a value of B_{L_D} = 0.1 Hz for the ground receiver (considering range observables every 10 s).

In contrast to range-rate measurements, range observables are generally affected by a significant bias, generated by both ground and onboard electronics and antennas. Both biases can be calibrated with high accuracy by means of dedicated hardware. The transponder group delay can be monitored with accuracies of <0.1 ns by means of a calibration circuit embedded in the unit. This is the case of the BepiColombo KaT, a Ka-band transponder that is part of the MORE scientific investigation (Cappuccio, Hickey et al., 2020). The two-way group delay of the ground electronics can also be calibrated with similar accuracies, as demonstrated by Cappuccio, Hickey et al. (2020) for ESA’s deep space station DSA-3 (Malargüe, Argentina). Although the transponder remains a critical element, the delays due to all intervening elements in the end-to-end chain must be calibrated (waveguides, cables, filters, amplifiers, etc.), including possible variations with temperature.

The end-to-end range error budget is reported in Table 3. Although the use of an advanced WVR would improve the tropospheric calibration of the zenith wet delay to 0.5 cm (Linfield et al., 1996), we have assumed a more conservative value of 2 cm, in line with the accuracies provided by the radiometer installed at DSA-3 (Lasagni et al., 2023) and an elevation angle of 30°. We have also used relatively large values for the station location (3 cm), as very long baseline interferometry (VLBI) observations cannot be used with small dishes. This value is compatible with positioning via laser metrology from a fiducial point, assuming that the ATLAS antenna is located close to a large deep space tracking antenna included in the VLBI network or a GNSS station.

3.3 Single-Beam Interferometry

The application of radio interferometry for OD is usually realized by delta-differential one-way ranging (DDOR), where two widely separated ground antennas sequentially track a spacecraft and a quasar. In a different interferometric scheme (Bender, 1994), the phases of a two-way signal received from two spacecraft, whose directions fall within the beamwidth of a single ground antenna, are simultaneously recorded and differenced. This configuration, known as SBI, was originally considered for a network of landers to study the interior structure of the Moon (Bender, 1994; Gregnanin et al., 2012) and Mars (Gregnanin et al. 2014). The SBI observable retains information on the differential range between the two spacecraft, relative to the ground antenna (Figure 1). Common noise sources, such as instabilities in the frequency standard and antenna deformations, are completely suppressed. If the angular separation between the two spacecraft is small (approximately 2°), typical noise rejection is also effective for propagation delays due to media (troposphere, ionosphere, and interplanetary plasma). Although SBI has never been implemented or demonstrated (because of the need to uplink two coherent carriers or CDMA signals from the same deep space antenna), the generation of SBI observables is expected to be straightforward in an MSPA tracking configuration such as the one considered for ATLAS.

Gregnanin et al. (2012) analyzed the performance of a Ka-band system capable of measuring the differential phase between two landers at a fraction of a wavelength (≤0.1 mm). The considered configuration entails a network of three lunar landers separated by a baseline of approximately 1000 km. As the main errors in SBI are, to first order, proportional to the angular separation between the spacecraft, the achievable accuracy for the ATLAS constellation will be lower (approximately 1.3 mm), primarily limited by the incomplete cancellation of the tropospheric and ionospheric path delays. The SBI error budget is evaluated with the same assumptions used for the link budget (see Section 2.4). The reported values for media errors provide an order-of-magnitude evaluation based on the same approach used for the DDOR error budget in CCSDS 500.1-G2 (2019), where average values for GNSS tropospheric and ionospheric calibrations are assumed. Should a WVR be available at the ground site, the residual tropospheric path delay can be significantly reduced. Note that media errors also depend on the tracking elevation (an average pass elevation of 30° is assumed in Table 4).

4.1 Clock Synchronization Methods

Any measure of time desynchronization between distant clocks requires a comparison of their readings at the same coordinate time. This process involves relativistic transformations between proper time τ (the actual time read by each clock) and coordinate time t. However, we report only a simplified formulation here, where de facto proper and coordinate times are assumed to be equivalent. Indeed, relativistic transformations come into play only as deterministic corrections that need to be applied in an operational scenario. With this simplification, we need only to consider the finite speed of light. For a review of time and frequency transfer methods, see the work by Levine (2008).

We have identified two methods for performing ground-to-space time transfer. The first method is similar to the well-established TWSTFT asynchronous technique (Howe et al., 1989), which is routinely used on Earth for comparing distant clocks and generating the UTC time scale. The second method is based on a novel approach in which the comparison is aided by ranging measurements and OD. This synchronous technique does not require an interruption of the two-way coherent link used for navigation and TT&C, therefore allowing, in principle, a greatly increased frequency of desynchronization measurements. In addition, this second implementation is fully compliant with the SS signal, as defined by the CCSDS (2011, 2013), and, for this reason, could be adopted as the baseline solution. Ultimately, both methods rely on the combination of time tags applied onboard and on ground. The time-tagging is triggered by a specific portion of the intervening signals, called the “code epoch,” which is most easily identified as the first chip of the PN ranging sequence. The use of code epoch tagged at time of ground transmit, time of onboard receipt and transmit, and ground receipt has been employed for decades in NASA's TDRSS, although the synchronization accuracy is only at about 1 ms (Sank, 1993).

4.1.1 Asynchronous Mode (TWSTFT)

Asynchronous ground-to-space time transfer relies on non-coherent links (hence, the incompatibility with two-way radiometric measurements), similar to terrestrial TWSTFT, where geostationary satellites are used as relays for signals coming from/to remote terminals that are not in mutual visibility (ITU-R TF.1153-4, 2015). For the Moon, the two remote terminals are in mutual visibility, and the configuration is even simpler. Figure 4 shows the world lines and the scheme of the asynchronous links, along with all of the relevant epochs involved in the measurement. A ground clock at t₁ generates a signal transmitted from the antenna at t₂ after a certain delay in the transmitter channel, primarily due to cables and electronics. The signal reaches the satellite antenna at t₃ and, after another delay in the satellite receiver channel, reaches the satellite time comparator at t₄, where the onboard observable Δτ^B = t₅ − t₄ is produced. In this scheme, t₅ is the epoch at which the onboard SGU, clocked by the EUFR, generates a signal for downlink transmission, as in the uplink process. This signal is transmitted from the onboard antenna at t₆ after a delay in the spacecraft transmitter channel and reaches the ground antenna at t₇; after another delay in the ground receiver channel, the signal reaches the station time comparator at t₈. Finally, the observable quantity Δτ^A = t₁ − t₈ is generated at the Earth tracking station.

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FIGURE 4

World lines and scheme for the asynchronous time transfer method

The two observable quantities, measured onboard and on ground, are Δτ^B = t₅ − t₄ and Δτ^A = t₁ − t₈, respectively. The other terms represent onboard and ground delays $({\Delta }_T^A,{\Delta }_R^A,{\Delta }_T^B,{\Delta }_R^B)$ in the transmitter and receiver path and propagation delays (T₂₃, T₆₇).

Following the formulation given by Duchayne et al. (2009) and Delva et al. (2012) and combining the observables from the two asynchronous links, the desynchronization equation between the two clocks may be written as follows:

$t^B\left(t\right)-t^A\left(t\right)=-\frac{1}{2}\left[(\Delta {\tau }^B-\Delta {\tau }^A)+({\Delta }_T^A-{\Delta }_R^A)+({\Delta }_R^B-{\Delta }_T^B)+(T_{23}-T_{67})\right]$ 5

which shows that the clock desynchronization between ground and space clocks is the average of the two measured observables corrected for differential delays due to electronic and propagation delays (including geometric path delays and additional effects due to the Earth’s troposphere and ionosphere). If t₁ ≈ t₅, the path reciprocity ensures that most errors cancel out in the differentiation procedure, including errors in OD. According to the error budget, accounting for all time error contributors, the end-to-end measurement uncertainty could be as low as ∼ 0.3 ns, remarkably in line with the uncertainty reported in the Bureau International des Poids et Mesures (BIPM) Circular T (BIPM, 2023).

A drawback of the asynchronous method is the need to switch the operational mode of the onboard transponder from coherent to non-coherent, which interrupts the acquisition of radiometric data at the ground station. Therefore, the number of clock comparisons performed during a tracking pass is limited. Although a practical realization of the asynchronous method requires some changes in the hardware of state-of-the-art deep space transponders, the current digital architectures make such modifications easy to implement.

Figure 5 shows a block diagram of the onboard architecture and interfaces between the transponder, SGU, EUFR, and onboard computer (OBC) that allow implementation of the asynchronous TWSTFT method. In this mode, the transponder should be configured to operate in non-coherent mode synchronized in frequency with the EUFR.

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FIGURE 5

Block diagram for the onboard architecture supporting time transfer asynchronous mode based on non-coherent links

The EUFR provides the time reference to the transponder for receiver and telecommand code epoch time-tagging. The OBC uses its internal clock for telemetry frame time-tagging, as the code epoch time-tagging has already occurred. CE: code epoch; CLK: clock; RX: receiver; TM: telemetry; TX: transmitter; XO: internal oscillator

The two separate registers inside the SGU should operate independently to record readings from the SGU counter according to separate trigger signals associated with the code epoch in the received signal and the code epoch in the deep space transponder transmitter code generator, clocked by the EUFR. These readings are collected by the OBC, where they are formatted in the telemetry data stream and sent to the ground station via the TT&C link. On ground, they are used in post-processing to generate the onboard observable Δτ^B to be applied in the desynchronization equation (Equation (5)). The SGU counter clock frequency should be on the order of ∼300 MHz to provide a time-stamp resolution at the level of 3.3 ns. This resolution may then be further improved by an averaging process.

As an alternative to the standard synchronization marker (SM) approach used, for example, in the ESA GAIA mission, the proposed architecture also includes a fast strobe line (receiver code epoch) to trigger the time-stamp operation clocked by a standard crystal oscillator within the OBC. In this case, the triggering signal can be either the receiver code epoch or the SM resulting from the output of a telemetry frame generator; the latched value inside the register is the OBC spacecraft elapsed time (SCET). It is worth noting that the navigation payload time to be synchronized with UTC/TAI is referred to here as the onboard time, which is distinct from the OBC time, indicated here as SCET. The latter is less accurate because of uncertainties in the telemetry chain and/or crystal oscillator instability but could be used in mission phases when the other method cannot be used, or even exploited to initiate the time transfer algorithm for onboard time synchronization.

4.1.2 Synchronous Mode

In any two-way coherent radiometric measurement (range or Doppler), the mathematical model of the light-time considers the time of three participating events (neglecting, for now, the internal spacecraft delays):

• • t₁: epoch of signal transmission from ground

• • t₂: epoch of signal reception onboard the satellite

• • t₃: epoch of signal reception on ground

Two-way range observables represent a measure of the round-trip light-time (RTLT = t₃ − t₁). In vacuum, RTLT × c (the speed of light) provides a measure of the distance traveled by the signal in the uplink and downlink paths. The ground clock directly measures t₃, whereas t₂ and t₁ are retrieved from a backward solution of the light-time equation (Moyer, 2003), a mathematical pillar of any OD code. Instead, the onboard clock has direct access to t₂, measured in terms of its proper time and converted to coordinate time. Therefore, in a two-way coherent link configuration, the ground-to-space clock desynchronization can be accessed by comparing different measurements of the epoch t₂:

$\text{Desynch}\left(t\right)={\hat{t}}_2\left(t\right)-t_2\left(t\right)=(t_3\left(t\right)-\frac{{\rho }_{23}\left(t\right)}{c})-t_2\left(t\right)$ 6

where ${\hat{t}}_2\left(t\right)$ is the onboard time obtained from the OD solution through ρ₂₃/c, the one-way return light-time. t₂(t) are simply the readings of the onboard clock that are then transferred to ground via the telemetry data stream. This difference represents the desynchronization between ground and space clocks. The onboard time-stamping operations are triggered by a code epoch signal, activated by a specific chip of the SS signal. The proposed TML sequence for the uplink SS code has a length of 256 · (2¹⁰ – 1) = 261,888 chips. Assuming a chip rate of ∼ 24 Mcps, the chip length is τ ≈ 41.6 ns, and the code repetition period is T_TML ∼ 10.9 ms.

Two aspects must be carefully considered when this method is used: the capability to solve for the code ambiguity and the precise association of the code epoch arrival to the time-stamping operation carried out by the clock assembly. Data encoding and transmission through the telemetry channel may indeed add a certain delay Δ₂, which, depending on the adopted coding method, may cause Δ₂ to exceed T_TML. However, this delay is largely deterministic and could be calibrated in post-processing, while also taking into account possible thermal effects. Therefore, the important requirement is that the residual uncertainty δΔ₂ ≪ T_TML, a condition that can be easily met. When the signal is received on ground, a similar time-tagging operation is triggered by the received code epoch signal, and the epoch t₃ provides ${\hat{t}}_2$ through the OD process. In parallel, telemetry data for t₂ are downloaded by the satellite. This step may add an additional delay Δ₃, for which the same considerations described for Δ₂ apply. Figure 6 illustrates the transfer method based on the two-way coherent link.

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FIGURE 6

Scheme for time synchronization with the two-way coherent approach

The delays Δ₂ and Δ₃ introduced by data encoding/decoding in the telemetry transmission may exceed the code repetition period. The associated uncertainties δΔ₂ and δΔ₃ must be ≪ T_TML. ${\hat{t}}_2\left(t\right)$ is the onboard time of the receiver code epoch obtained from the OD solution. CE: code epoch; TLM: telemetry; RX: receiver; TX: transmitter; G/S: ground-to-space

The performances of this time synchronization method depend on a) the precision of time-stamping operations (σ_t2 and σ_t3), b) the OD accuracy in the line-of-sight direction (σ_ρ) and c) the accuracy of the onboard and on-ground delay calibration. The Lunar Orbiter Laser Altimeter of the Lunar Reconnaissance Orbiter has achieved a time-stamping precision of approximately 0.5 ns (Bauer et al., 2017), while numerical simulations (see Section 5) show an OD accuracy better than 10 cm along the line of sight, corresponding to 0.3 ns (see Section 5.4). Therefore, we may conservatively assume an overall contribution of the ground-to-space time transfer procedure at the 1-ns level to the time synchronization across the entire lunar constellation. Note that the (much larger) errors in the other two directions (transverse to the line of sight) do not contribute to the time synchronization procedure.

In Equation (6), the contributions from the media and the internal delays at the station and outside the transponder (which need to be accurately calibrated) do not appear explicitly. The media and orbital errors are included in the term ρ₂₃, whereas the internal calibrations also affect t₂. Although appropriate media and internal delay calibrations can keep the error within a few nanoseconds, the media contributions may be strongly reduced even in synchronous mode if the uplink path is also exploited. Indeed, if Equation (6) is also written for the uplink as follows:

$\text{Desynch}\left(t\right)={\hat{t}}_2\left(t\right)-t_2\left(t\right)=(t_1\left(t\right)-\frac{{\rho }_{12}\left(t\right)}{c})-t_2\left(t\right)$ 7

then one can combine and average two separate ground-to-space time comparisons, similar to what has been done for the asynchronous configuration. Considering the world lines in Figure 4, one may recognize that for the synchronous case t₄ = t₅, the combination of the two independent uplink/downlink paths leads to the following:

$\text{Desync}=\frac{1}{2}\left\{(t_1+t_8)-2·t_4+({\Delta }_{\uparrow }-{\Delta }_{\downarrow })+({\Delta }_{\text{Media}\uparrow }-{\Delta }_{\text{Media}\downarrow })+(\Delta T_{GS}-\Delta R_{GS})+(\Delta R_{SC}-\Delta T_{SC})\right\}$ 8

where the up and down arrows refer to the uplink and downlink geometric and media delays, respectively. The last four terms indicate the ground station delays in the transmitting and receiving chain and the onboard delays in the receiving and transmitting chains, respectively. In this way, the coherent method also exploits the benefits from differenced terms for propagation and media delays.

Figure 7 shows a block diagram of the onboard architecture and the interfaces between the onboard transponder, SGU, EUFR, and OBC. All of these units and interfaces enable the time synchronization method based on two-way coherent ranging observables, either exploiting only the downlink path (Equation (6)) or both the uplink and downlink paths (Equation (8)). We recall that in the synchronous mode, the transponder is configured to operate in two-way coherent mode. The code epoch of the TML sequence (receiver code epoch, also used for ranging turn-around) triggers a time-stamp operation by the SGU through a fast time strobe line, clocked by the EUFR. As soon as this trigger signal is received by the SGU, the reading from the SGU counter (corresponding to t₂(t) of Equation (6)) is recorded in a register. This time-stamp information is routed to the OBC and incorporated into a data packet transmitted into the downlink telemetry data channel.

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FIGURE 7

Block diagram for the onboard architecture supporting time transfer in synchronous mode, based on two-way coherent ranging observables

CLK: clock; RX: receiver; TLM: telemetry; TX: transmitter; XO: internal oscillator

Similar to the previous asynchronous time transfer method, the synchronous time transfer solution also includes, as an alternative to the standard SM approach, a fast strobe line (receiver code epoch) to trigger the time-stamp operation clocked by a standard crystal oscillator within the OBC. The latched value inside the register is the OBC SCET with the same functionality described for the asynchronous time transfer method.

4.2 Onboard Clocks

The selection of onboard clocks is a trade-off between performance (accuracy and stability); size, weight, and power (SWaP) and cost; and technology readiness level. Although the MSPA approach allows for nearly continuous contact between each satellite and the tracking stations, the system should be able to still work when contact with Earth is interrupted for a few hours. This requirement rules out the use of microchip-scale atomic clocks, whose drift would result in SISE values of >100 m in a few hours. Deep space atomic clocks (Burt et al., 2021) and Galileo passive hydrogen maser clocks (Steigenberger & Montenbruck, 2017) are the best choices in terms of absolute performance (especially for their long-term stability, <10⁻¹⁴ for a 1-day time scale) but at the cost of a mass no lower than approximately 10 kg. The long-term stability of Galileo rubidium clocks is approximately 10 times worse, but the error contribution after 1 day is still acceptable (∼ 1 m), and the mass (3.3 kg) is still compatible with use onboard a small/minisat. Two other attractive solutions are miniRAFSs and ultra-stable crystal oscillators (USOs), given their low mass and power consumption. Although USOs generally suffer from large drift, the AccuBeat USO, developed for ESA’s JUICE mission to Jupiter, combines an excellent short-term ADEV with good performance at longer time scales (∼10⁻¹² at 10,000 s). Table 5 summarizes a SWaP and performance comparison among RAFS (Orolia), USO (AccuBeat), and miniRAFS (Orolia) (with references to manufacturer fact sheets). A rough estimation of the clock error contribution to the positioning error at different time scales may be estimated as Δp ≈ cτσ_y.

6.1 Chebyshev Polynomials

Chebyshev polynomials constitute a sequence of orthogonal polynomials (with respect to the weight function $1/\sqrt{1-x^2}$ on the interval [−1, 1]), defined as solutions of the Chebyshev differential equation (Rivlin, 2020). Chebyshev polynomials are an advantageous solution to the classic numerical interpolation problem. Their purpose in numerical analysis is to obtain well-behaved and efficiently computed approximations to smooth but complex mathematical functions. Therefore, these polynomials can be applied to represent ephemeris parameters or rotation parameters between reference frames. The recursive calculation of the Chebyshev polynomial coefficients is expressed as follows:

$\begin{array}{c}{T}_0\left(x\right)=1\\ T_1\left(x\right)=x\\ T_{n+1}\left(x\right)=2x{T}_n\left(x\right)-T_{n-1}\left(x\right)\end{array}$ 10

where the x values in the time domain range from −1 to 1. A final function with a larger degree of polynomial expansion and more coefficients can represent curves with higher complexity. The functions are broken up into small intervals, within which the coefficients and the resulting functions are valid. We conduct a least-squares fit of the Chebyshev polynomials into the simulated position of the lunar orbiter decomposed into X, Y, and Z coordinates. Each component of the lunar orbiter position, i.e., X, Y, and Z, is calculated as a sum of Chebyshev terms:

$\left\{XYZ\right\}=\sum _{i=1}^n{a}_i{T}_i\left(x\right)$ 11

where a_i denotes the coefficients from the navigation message.

Figure 9 illustrates the 95th percentile and maximum of three-dimensional (3D) residuals within a full data set covering 10 days, considering 7–12 Chebyshev coefficients (vertical axis) and window lengths of 0.5–3.0 h (horizontal axis) for the orbit of a lunar spacecraft. Chebyshev polynomials of degree six (with seven coefficients) are sufficient to obtain an orbit quality better than 1 m in a 0.5-h window. Increasing the number of Chebyshev coefficients to 10 allows for an accurate orbit representation in a window of 1 h, with the 95th percentile at the level of 0.116 m. A Chebyshev polynomial of degree 11 (comprising 12 coefficients) is sufficient to ensure an orbit reconstruction quality of up to 1 m, valid within 1.5-h intervals. In an analysis of the orbit fit time series, the primary issues arise as the satellite passes over the periselene, leading to a notable deterioration in the quality of orbit reconstruction. Employing 12 Chebyshev coefficients is sufficient to obtain a centimeter-level orbit quality in 75% of the considered 2.5-h windows, except for the windows comprising the periselene passes, where the maximum position errors reach hundreds of meters for single epochs. Assuming an orbit representation using Chebyshev polynomials with 10 coefficients and the validity of the orbit model for 1 h, the position errors do not exceed 60 cm for single epochs during the periselene passes. Therefore, this situation requires a representation with a shorter window for periselene passes than for other parts of the orbital arc. However, at the lunar south pole, signals from a satellite close to the periselene cannot be received; therefore, the periselene representation is of minor importance to fulfill the accuracy requirements. The disadvantage of the Chebyshev polynomial representation is the number of bits (187) that are needed for a single coordinate component. In total, for three position components together with time-stamp information, 595 bits are needed for an 11-coefficient representation.

6.2 Keplerian Representation

To provide a centimeter-level accuracy of the orbit recovery, a model of six Keplerian and nine empirical accelerations is recommended, including constant accelerations in the along-track, cross-track, and radial directions, as well as sine and cosine once-per-revolution corrections in this orthogonal frame. For a 1-h window, 95% of residuals are below 2.9, 5.0, and 2.9 mm in the radial, along-track, and cross-track directions, respectively. As the window increases, the quality of the orbit fit decreases. The median values of the orbit fit residual are at the level of 1, 2, 6, and 203 mm for 1-, 1.5-, 2-, and 4-h intervals, respectively. If the model is reduced to ten parameters, i.e., six Keplerian and four selected empirical parameters that provide the smallest fit error, the quality of the orbit fit is substantially reduced, i.e., the median values of the orbit fit residual are at the level of 22.8, 52.4, 91.1, and 395.0 m for 1-, 1.5-, 2-, and 4-h intervals, respectively. The Keplerian representation of six Keplerian and nine empirical parameters requires 370 bits, and another 31 bits are needed to mark the time stamp of the message, resulting in a total of 401 bits.

Determining a reasonable number of bits for the representation of the navigation message for eccentric orbits is challenging, as it requires a balance between an accurate representation of the lunar orbiter position and the storage space for communication purposes. The time required to update the navigation depends on both the constellation characteristics and the chosen orbit representation option. For Earth GNSSs, the refresh cycles for navigation messages vary from several minutes to daily. In the case of GPS, the refresh cycle is every 2 h; for Galileo, it is shortened to every 20 min. The update frequency for the navigation message depends on the rate at which the quality of the chosen orbit representation deteriorates over time compared with the expected positioning accuracy provided to users. For lunar systems, it is crucial to consider the visibility of satellites from the ground segment, which is critical for data transmission.

6.3 Trade-off for the Representation of Broadcast Orbits

Keplerian parameters with corrections are a natural choice for representing the broadcast ephemerides for a navigation system designed for the Moon, as they require a smaller number of bits when compared with Chebyshev representation. The eccentric orbit design, however, introduces challenges in modeling the orbit representation and in representing the ephemerides. The Keplerian elements must be converted into Cartesian coordinates for orbit extrapolation and for introducing empirical accelerations in the radial, along-track, and cross-track directions; thus, the receiver must be capable of conducting numerical integration, which introduces additional computation complexity.

For orbit representation based on Chebyshev polynomials, the accuracy of the function fit increases with the degree of the polynomial. Hence, the user does not need to receive a full message to obtain a satellite position. However, as the degree of the polynomial received by the user increases, the orbit representation obtained by the user becomes more accurate. Moreover, the transformation parameters between the lunar and celestial reference frames can also be based on Chebyshev polynomials, simplifying the algorithms that must be applied and implemented. Receiving only the first five coefficients of the Chebyshev polynomial provides an orbit representation accuracy of roughly 30 cm in 50% of windows, which is sufficient for most real-time navigation purposes on the Moon.

7.1 Inertial and Kinematically Non-Rotating systems

Moon-centered inertial reference systems are useful for OD of spacecraft in the cislunar environment. Current definitions of inertial systems are based on the International Astronomical Union (IAU) B3 recommendations (Soffel et al., 2003; Petit and Luzum, 2010) and relativistic descriptions of the barycentric celestial reference system (BCRS) and geocentric celestial reference system (GCRS) reported by Damour et al. (1991). The BCRS metric is taken to be kinematically fixed with respect to distant quasars, as defined by Soffel et al. (2003) and following harmonic gauge conditions. A kinematically non-rotating reference system centered on the Earth’s center of mass has been defined in the same type of framework, leading to the definition of the GCRS. Time scales derived from the BCRS and the GCRS metric have been reported by Soffel et al. (2003).

For the Moon, Turyshev et al. (2013) proposed to adapt the GCRS definition to the Moon in the context of the GRAIL mission (recall, however, that the International Astronomical Union is in charge of defining the LCRS). A description of such a Moon-centered inertial system is given in Section 6.6, with a definition of the Moon-centered metric and time scale in Section 6.7. In Section 6.8, an assessment of the accuracy of two possible realizations of a lunar celestial reference system (LCRS) is given by comparing results obtained with DE421 (Folkner et al., 2014) and INPOP19a (Fienga et al., 2019).

7.2 Principal Axis System

The PA system is defined by the PA orientation of the inertia matrix of the Moon with constant tide contributions. Indeed, this is a useful reference system for writing and integrating the rotational equation. Its materialization is realized from a temporal series of the Moon orientation, coming from ephemerides, and described through the rotation Euler angles (ψ: precession angle, θ: nutation angle, φ: proper rotation). The rotational angles are extracted from the lunar ephemerides (Jet Propulsion Lab [JPL] development ephemerides [DE] or Intégrateur Numérique Planétaire de l’Observatoire de Paris [INPOP] ephemerides produced by the Institut de Mécanique Céleste et de Calcul des Ephémérides and Observatoire de la Cote d’Azur). These two ephemerides are similar in terms of weighted root mean square residuals. Let I be a vector in the lunar celestial reference frame (LCRS) and let PA be a vector in the PA reference frame, with Rx and Rz as elementary rotational matrices. The direction and transformation are written as follows (Folkner et al., 2014):

$\begin{array}{cc}PA=R_z\left(\phi \right)R_x\left(\theta \right)R_z\left(\psi \right)I& I=R_z(-\psi )R_x(-\theta )R_z(-\phi )PA\mathrm{.}\end{array}$ 12

7.3 Mean Earth Direction and Rotation System (EPM)

Because the Moon is not a purely triaxial ellipsoid, the average orientation of the PA does not coincide with the ME direction. Figure 10 shows the position of the Earth in the lunar PA reference system over 1 and 6 years. The loops indicate the geometrical libration of the Moon. The average of the loops is not exactly zero, and there is a constant offset. The ME/polar axis reference system is defined by the z-axis as the mean rotational pole. The prime meridian (0° longitude) is defined by the ME direction. The intersection of the lunar equator and prime meridian occurs at what can be called the Moon’s “mean sub-Earth point.” This system is an idealization; a practical attempt to determine these mean directions with high accuracy would depend on the approach and time interval used. This reference system is defined from the numerical ephemerides of the PA system through the following rotation matrix relationship (Folkner et al., 2014):

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FIGURE 10

Direction from the Moon to Earth over 1 year (left) and 6 years (right)

u1, u2, and u3 are cosines of the direction vector of the Earth as seen in the PA reference frame of the Moon (no units).

$ME=R_x(-p_{2c})R_y\left(p_{1c}\right)R_z(-{\tau }_c+I_c^2\frac{{\sigma }_c}{2})PA$ 13

where I_c is the Moon’s moment of inertia, p_1c, p_2c, are constant coordinates of the ecliptic pole in the PA reference frame, and τ_c and σ_c are constant libration angles. The offset between these coordinate systems for a point on the lunar surface is approximately 860 m and depends on the ephemeris and the associated gravitational field. The inverse transformation is as follows:

$I=R_z(-\psi )R_x(-\theta )R_z(-\phi )R_z({\tau }_c-I_c^2\frac{{\sigma }_c}{2})R_y(-p_{1c})R_x(+p_{2c})\text{ME}$ 14

7.4 Lunar Laser Ranging Retroreflector as Lunar Point Control

There are five points on the Moon whose positions are very accurately measured. These points correspond to lunar laser ranging retroreflectors (LLRRs), distributed over the equatorial and mid-latitude regions of the Moon. These LLRRs can be used as control points of a lunar geodetic network, given their high stability over time. The stability of the LLRR positions has been recently discussed by Williams and Boggs (2021), who provided the coordinate rates of the LLRRs. The weighted average rate of the four LLRs is 8.3 ± 3.1 mm/year, corresponding to 20 cm over 25 years. The origin of this shift is not known, but the authors argued that it could be due to mismodeling of rotational motion. Finally, the formal uncertainties in the LLRR positions are approximately 3 cm in the INPOP ephemerides, and the global uncertainty in position is 40 cm for the JPL DE (Williams & Boggs, 2008). These uncertainties of few tens of centimeters represent the internal precision in ephemerides.

A comparison of various ephemerides provides their external accuracy, corresponding to approximately 2 m for EPM (Pavlov, 2020) compared with the DE (Park et al., 2021; Folkner et al., 2014) and INPOP (Viswanathan et al., 2018; Fienga et al., 2019; Fienga et al., 2021) values and 1 m for INPOP and DE440. Besides comparing LLRR coordinates obtained in different PA frames, one can also compare these coordinates with those obtained via other methods, such as altimetry or imaging.

7.5 Proposition of Lunar Reference Systems: A Moon-Centered System (LCRS)

Based on the concept of the GCRS given in the IAU 2000 Resolution B1.3 (Soffel et al., 2003; Petit & Luzum, 2010), we propose to define the LCRS, centered at the Moon’s center of mass, associated with this metric tensor based on the work of Damour et al. (1991), but adapted to the gravitational environment of the Moon. A lunar time scale is also defined. This system will be used for navigation and for integrating the equations of motion at the vicinity of the Moon.

By analogy with the definition of the GCRS metric tensor and the IAU 2000 Resolution B1.3 (Soffel et al., 2003; Petit and Luzum, 2010), we consider the selenocentric metric tensor S_ab with selenocentric coordinates (T, X), where T is the lunar coordinate time (TCL) and X is the coordinate vector of an orbiter in the LCRS. Similarly, in the BCRS, t and x denote the barycentric time scale (barycentric coordinate time [TCB] or barycentric dynamical time [TDB]) and the coordinate vector of an orbiter, respectively. The selenocentric tensor is in the same form as the barycentric or geocentric tensor but with Moon-centered potentials L(T, X) and L^a(T, X) as follows:

$\begin{array}{ccc}{S}_{00}=-1+\frac{2L}{c^2}-\frac{2L^2}{c^4}& S_{0a}=-\frac{4}{c^3}{L}^a& S_{\text{ab}}={\delta }_{\text{ab}}(1+\frac{2L}{c^2})\end{array}$ 15

The potentials L and L^a should be split into two parts: potentials L_S and $L_a^S$ arising from the gravitational action of the Moon and external terms L_ext and L_a due to tidal and inertial effects. The external parts of the metric potentials are assumed to vanish at the selenocenter and admit an expansion into positive powers of X_L. Explicitly, we have the following:

$L=L_S+L_{\text{ext}}+O\left(c^{-4}\right)$ 16

$L^a=L_S^a+L_{\text{ext}}^a=-\frac{G}{2X^3}[X\times A_M]+O\left(c^{-2}\right)$ 17

where A_M is the Moon’s angular momentum. L_S corresponds to the lunar Newtonian gravitational potential, and L_ext is the tidal contribution produced by all solar system bodies (excluding the Moon) estimated at the center of mass of the Moon (origin of the LCRS) at T = TCL. The expression of the potentials L_ext, L_S, and L_a can be found, for example, in the work by Turyshev et al. (2013).

Considering T = TCL, t = TCB, r_M = x − x_M, i.e., the differences between the vectors of the BCRS barycentric position of the orbiter x and the barycentric position of the Moon x_M, with v_M and a_M being the vectors of its barycentric velocity and acceleration, the transformation between the BCRS and LCRS is given explicitly. The expression of the LCRS coordinate vector X from the BCRS x coordinate vector (appearing in the vector r_M = x − x_M) is given by the following:

$X=r_M+\frac{1}{c^2}[\frac{1}{2}{v}_M·(v_M·r_M)+l_{\text{ext}}\left(x_M\right)·r_M+r_M·(a_M·r_M)-\frac{1}{2}{a}_M·r_M^2]+O\left(c^{-4}\right)$ 18

Here, l_ext (X_M) = ∑_A≠M GM_a/r_AM + O(c⁻²), and r_AM = x_A − x_M, the Moon-centered vector of body A in the BCRS.

The time transformation between BCRS time t and LCRS time T is as follows:

$T_L=t-\frac{1}{c^2}[A\left(t\right)+v_M^i{r}_M^i]+\frac{1}{c^4}\times [B\left(t\right)+B^i\left(t\right)r_M^i+B^{\text{ij}}\left(t\right)r_M^j{r}_M^i+C(t,x)]+O\left(c^{-5}\right)$ 19

where $v_M^i,\hspace{0.17em}{r}_M^i$, and $a_M^i$ are the coordinates of the vectors v_M, r_M, and a_M, respectively, with the following:

$\begin{array}{c}\frac{dA\left(t\right)}{\text{dt}}=\frac{1}{2}{v}_M^2+l_{\text{ext}}\left(x_M\right)\\ \frac{dB\left(t\right)}{\text{dt}}=-\frac{1}{8}{v}_M^4-\frac{3}{2}{v}_M^2{l}_{\text{ext}}\left(x_M\right)+4v_M^i{l}_{\text{ext}}^i\left(x_M\right)+\frac{1}{2}{l}_{\text{ext}}^2\left(x_M\right)\\ B^i\left(t\right)=-\frac{1}{2}{v}_M^2{v}_M^i-3v_M^i{l}_{\text{ext}}\left(x_M\right)+4l_{\text{ext}}^i\left(x_M\right)\\ C(t,x)=-\frac{1}{10}{r}_M^2\left({\dot{a}}_M^i{r}_M^i\right)\end{array}$ 20

$Q^a={\delta }_{\text{ai}}[\frac{\partial }{\partial x^i}{l}_{\text{ext}}\left(x_M\right)-a_M^i]$

7.6 Proposition of a Lunar Time Scale for the Moon-Centered System (LCRS)

The UTC time scale is used for time-tagging observations from the ground. For the definition of selenodetic frames or when one uses Earth-based observations for studying the dynamics of the Moon, the UTC time scale is transformed into TDB (Fienga et al., 2009). The GPS time can be used as an intermediate time scale between lunar orbiters and ground-based stations, a natural choice for receivers of Earth GNSS signals in lunar orbit. A recent paper by Ashby and Patla (2024) established a framework for the estimation of clock rates on the Moon.

For time-tagging on the Moon surface or for a constellation orbiting the Moon, we define the TCL, the relativistic time scale at the center of mass of the Moon, similar to the geocentric coordinate time (TCG), the relativistic time scale at the geocenter (Soffel et al., 2003; Petit & Luzum, 2010). Based on the previous equations, we can define TCL as related to TCB in the same way that TCG is related to TCB (Damour et al., 1991):

$\frac{dTCL}{dTCB}=1+\frac{1}{c^2}{\alpha }_L+\frac{1}{c^4}{\beta }_L+O\left(c^{-5}\right)$ 21

with:

${\alpha }_L=-\frac{1}{2}{v}_M^2-\sum _{A\ne L}\frac{{GM}_A}{r_{MA}}$ 22

and:

${\beta }_L=-\frac{1}{8}{v}_M^4+\frac{1}{2}{\left[\sum _{A\ne M}\frac{{\mu }_A}{r_{AM}}\right]}^2+\sum _{A\ne M}\frac{{\mu }_A}{r_{AM}}\{4v_M·v_A-\frac{3}{2}{v}_M^2-2v_A^2+\frac{1}{2}{a}_A·r_{AM}+\frac{1}{2}{\left(\frac{v_A·r_{AM}}{r_{AM}}\right)}^2\sum _{B\ne A}\frac{{\mu }_B}{r_{BA}}\}$ 23

Subscripts A and B denote massive bodies, L corresponds to the Moon, M_A is the mass of body A, r_LA = x_L − x_A (vectorial differences), r_LA is the norm of the vector r_LA, x_A is the barycentric vector of position of the massive body A, and v_A and a_A are the barycentric vectors of velocity and acceleration of the massive body A in TCB. αL is approximately 1.48 × 10⁻⁸ in the work by Turyshev et al. (2013), and as explained by Nelson and Ely (2006) and Turyshev et al. (2013), the β_L term (Equation (21)) is negligible for the present applications.

In the same way that TCL can be defined relative to TCB (Equation (21)), it is also possible to describe TCL relative to TCG in the GCRS. In this case, we have the following:

$\frac{dTCL}{dTCG}=1+\frac{1}{c^2}{\alpha }_{L/C}+\frac{1}{c^4}{\beta }_{L/C}+O\left(c^{-5}\right)$ 24

where α_L/G and β_L/G (which is negligible and whose expression is not reported) have the same definitions as in Equations (22) and (23) but with geocentric positions and velocities:

${\alpha }_{L/G}=-\frac{1}{2}{v}_{\frac{L}{G}}^2-\sum _{A\ne L}\frac{{GM}_A}{r_{LA}}+\sum _{A\ne G}\frac{{GM}_A}{r_{GA}}$ 25

In contrast, terrestrial time (TT) is defined relative to TCG as follows (Petit & Luzum, 2010):

$\frac{dTT}{dTCG}=1-L_G$ 26

with L_G = 6.969290134 10⁻¹⁰. By considering that TCL − TT = (TCL − TCG) + (TCG − TT), with Equation (21) and the current definition of TT (Equation (26)), we can derive the following:

$\frac{dTCL-TT}{dTT}=\frac{1}{1-L_G}\times [L_G+\frac{1}{c^2}{\alpha }_{L/C}+\frac{1}{c^4}{\beta }_{L/c}]+O\left(c^{-5}\right)$ 27

Explicit expressions for the transformations between local lunar-centered reference frame coordinates of an object in the vicinity of the Moon, X(TCL), and the coordinates of the same object in the BCRS relative to the Moon, r_M (TDB) (shown as r_M as in Equation (18)) have been reported by Soffel et al. (2003) and Turyshev et al. (2013).

7.7 Assessment of Realization Accuracy

The LCRS can be realized using different lunar and planetary ephemerides. Here, we consider two sets of ephemerides, namely, DE421 (Williams et al., 2022), which is the present reference for the definition of the lunar ME frame (see Section 6.3), and INPOP19a (Fienga et al., 2019). These two ephemerides differ, as the DE421 model does not account for core–mantle interactions or the shape of the fluid core. More than 11 years of additional observations were also included in the INPOP19a construction. The differences in terms of localization of the Moon relative to the Earth using DE421 and INPOP19a over 5 years are plotted in Figure 11. The maximum difference of approximately 1.5 m is obtained in the Z direction. The ratio between the gravitational mass of the Earth–Moon barycenter obtained with DE421 and with INPOP19a differs from unity by (1.2 ± 10⁻¹¹) × 10⁻⁸.

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FIGURE 11

Differences (in meters) in Moon barycenter positions relative to the Earth estimated with DE421 and INPOP19a in the International Celestial Reference Frame (ICRF)