Phase-difference measurement in iVLBI and application in the measurement of lunar rotation
- Ming Chen^{1}Email author,
- Nobuyuki Kawano^{1},
- Kun Shang^{1},
- Jing Sun^{1},
- Qinghui Liu^{1},
- Fuyuhiko Kikuchi^{2} and
- Jinsong Ping^{1}
https://doi.org/10.5047/eps.2011.02.005
© The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB. 2011
Received: 12 May 2010
Accepted: 7 February 2011
Published: 21 June 2011
Abstract
Radio waves emitted from two or more landing units on the lunar surface are received by an antenna at the Earth station, and the range differences between these landing units are measured with an error of several millimeters. The phase differences between the oscillators of these landing units are monitored via an orbiter that orbits around the Moon. We have developed a simple roundtrip method to obtain these phase differences and also propose a method of calibrating the system delay on the ground. In order to observe the rotation of the Moon and monitor the phase differences effectively, we designed the position of the landing units and the orbit of the orbiter. Further, we concurrently analyzed the characteristics of the common view period. The error of the system was analyzed and to have a high accuracy. The results show that inverse VLBI technology can be used to measure the rotation of the Moon and new scientific results can be obtained.
Key words
iVLBI lunar rotation roundtrip method NBV orbit design common view period1. Introduction
Both the conventional Very Long Baseline Interferome-try (VLBI) technique and laser ranging have certain limitations in their ability for ground-based geodetic observation of the Moon, with an accuracy reaching up to a few centimeters. Regarding laser ranging technology, the observation period is limited as it cannot be used to observe the Moon during a full or new Moon. In comparison, the VLBI technique is associated with an error caused by the propagation media. The instrument cannot be corrected with high accuracy and, therefore, the accuracy of the delay is limited to almost 0.1 ns. Kawano et al. (1999) proposed that the tides and rotational variations of the Moon or planets could be observed with greater accuracy using a new method, called the “Inverse VLBI (iVLBI)”. This method enables geodetic measurement of the Moon with an error of a few millimeters, which is the same as that for observations from the Earth. Following further development of the concept and design of the iVLBI, as well as demonstration of its applications in science, the two-way method was recommended for synchronizing the reference frequency that was generated by the landing units (Kawano et al., 2009). A different direct ranging method was proposed to monitor the standard frequencies (Kawano et al., 2010). The iVLBI technology was also selected as a candidate research project in SELENE 2. We are aware that the time-transfer and ranging experiment between the ground and a satellite has been performed in the ETS-VIII experiment (Takahashi et al., 2008). Precise time-transfer is a reference technique applied in order to realize the iVLBI system. The phase difference between the signals of the two landing units is an essential parameter and needs to be measured as it directly influences the accuracy of the result. In this article, we propose a solution to the phase-difference measurement.
This article comprises eight sections. Section 1 is the introduction; Section 2 discusses the principle of iVLBI; Section 3 describes the phase-difference measurement of the two landers and its accuracy; examples of frequency selection and link budget are shown in Section 4; Section 5 presents a design of the position of the landers on the lunar surface and the orbit of the orbiter; Section 6 includes an analysis of the accuracy of the delay derived by the iVLBI observation; in Section 7, we demonstrate the possible application of the iVLBI technology in scientific research. The article closes with the conclusion and discussion in Section 8.
2. Principle of iVLBI
The observation system of the iVLBI consists of three sub-systems. These include (1) two or more landing units on the Moon, (2) an orbiter orbiting around the Moon, and (3) a ground station. Phase differences among radio waves that are transmitted on the Moon are monitored through the orbiter. An antenna/receiver on the ground station receives radio waves from all the landing units on the Moon simultaneously, and phase differences can be measured among the radio waves received at the ground station. The range differences among the landing units and the antenna of the ground station can be obtained after correcting for the phase differences among radio waves transmitted from the landing units on the Moon. Tides and rotational variations of the Moon can be detected with a higher accuracy using the iVLBI system than is currently possible with VLBI observation or lunar laser ranging on the Earth. The accuracy of the range difference between two landing units and the antenna on the Earth depends on the measurement error of the phase difference between the two landing units.
3. Phase-Difference Measurements of the Two Landers
3.1 Phase-difference measurements between a roundtrip signal without carrier recovery and return signal in a satellite
The phase of is proportional to the lower frequency instead of the RF (radio frequency).
For lander 2, Eq. (9) can also be derived; however, the delays for lander 2, which are and , are different from and and, therefore, and cannot be estimated separately from Eq. (9). In order to determine , calibration on the ground is required.
There is a method, in principle, to determine the separately, and this is called a phase calibrator. A pulse is inserted at the front end and produces many frequency components after a band-pass filter. The phases are measured by the tone signal that is produced by the pulse at near-signal frequencies, and the delay produced in the instruments can also be calculated. This instrument is, of course, kept at a constant temperature. However, in reality, it is rather difficult to determine separately with high accuracy, although the instrumental total delay variation can be monitored to a certain extent. This could help predict the separately.
3.2 Calibration of the instrumental delay on the ground by using a common frequency standard
For example, if R is 50∼100 m, f^{ i } = 2.2 GHz, f^{0} = 40 MHz, and a phase measurement error is of a few degrees, the can be estimated with an error of a few degrees from Eq. (10b). The measurements for i = 1 , 2, 3 are carried out at the same time. The phases at three frequencies are corrected by the observed , and the three phases at these frequencies give the correct range.
The first term provides the range difference; the fourth term is obtained from Eq. (8) after determining the range; the second and third terms represent the phases caused by the delay difference in the landers. These terms should be well calibrated. The fourth term is rather small compared with the third one. In particular, the change of the second term with temperature is the main source of error. The characteristics with temperature should be obtained prior to this calculation.
Equation (10c) indicates that an original signal is generated at a satellite, and the signal arrives at the Earth through a lander. In order to correct the resultant delay from the satellite to the lander, precise ranging is conducted by the NBV (narrow-band VLBI). From this perspective, a frequency standard in a lander only plays the role of a simple frequency converter. The common signal transmitted from the satellite goes through a lander and arrives at the Earth. The delay (or phase) from the satellite to the lander is estimated separately and is corrected for the total delay of the path from the satellite to the lander.
A measurement test for can be conducted after the calibration of the instrumental delay on the ground by using independent frequency standards. Although the measured phase varies in this test measurement because fluctuates with time, the correct distance R can be estimated after has been corrected by using observed phases at three frequencies.
3.3 The effect of a phase variation of the frequency standard in the satellite
Since can be determined with an accuracy of about 10^{°}, the accuracy of iVLBI depends on the stability of Particular sources of error are phase changes of RF frequency devices and cables with changes in temperature. Several steps are undertaken in order to minimize these errors. These devices should be installed in a temperature-controlled box. Their characteristics at a specific temperature should be measured carefully on the ground prior to launch. After launch, the temperature should be monitored again. Further calibration can be done for large variations in temperature.
3.4 Phase difference between the signals received in the orbiter from the two landers
The effect of the delay and phase instability of the frequency standard in the satellite is cancelled in Eq. (12b). The first term was previously obtained by calibration on the ground. Equation (12b) suggests that the delay changes and phase instability of the frequency standard in the satellite do not directly affect the phase difference of the signals from the two landers.
3.5 A simple roundtrip system with carrier recovery
This circuit produces a pure signal that has the same frequency and phase as the signal received by the antenna. There are two groups of output signals. As shown in Fig. 5, one group of signals is mixed with the standard frequency on the lander. This is called a mixed signal. The other group of signals is the transferred pure signal. The turnaround ratio is given by k/l, and this is called a transferred signal. From the transferred signal, the Doppler shift can be obtained on the orbiter with high accuracy. Thereafter, the standard frequency difference of the orbiter and the lander can be obtained from the mixed and transferred signals. In the following, we describe how the phase value caused by the frequency difference can be calculated.
4. Delay Determination and an Example of the Link Budget
4.1 Phase difference between signals received by the antenna on the ground
The second term can be calculated by Eq. (12b). The last term is the result of the phase noise of the frequency standard for the satellite. There exists a time difference between the path from the orbiter-lander 1-antenna and the orbiter-lander 2-antenna. The value of this term is miniscule, as can be inferred from Eq. (11); therefore, it can be ignored.
4.2 Delay determination by narrow-band VLBI
Ambiguity and resolution of delay.
Frequency | 100 KHz | 4 MHz | 80 MHz | 2,280 MHz |
---|---|---|---|---|
(PRN: 100 KHz) | (f_{2} − f_{1}) | (f_{3} − f_{1}) | (f_{3}) | |
Ambiguity of delay | 0.01 s | 125 ns | 6.25 ns | 0.22 ns |
Measurement of resolution of phase | — | 4° (both at two frequencies) | 4° (both at two frequencies) | 4° |
Resolution of delay | 100 ns | ±3.93 ns | ±0.20 ns | 5ps |
In Table 1, we assume the chip rate of the PRN code to be 100 KHz and the code length to be 1024 bits. The measurement resolution of PRN is 1 /100. Therefore, the final resolution of the time delay can be accurate up to 5 ps, which indicates 1.5 mm in terms of distance.
4.3 An example of the link budget for a simple roundtrip system with carrier recovery
Link budget.
Link stations | Sat. → Lander | Lander → Sat. | Lander → Earth | |
---|---|---|---|---|
Frequency | 2.2 GHz | 2.2 GHz | 2.2 GHz | |
Transmission | ||||
Transmission power | dBW | 6 | 0 | −3 |
Feeder loss | dB | −2 | −2 | −2 |
Power distribution loss | dB | 0 | −4.1 | −4.1 |
Antenna gain | dBi | −2 | −2 | −2 |
EIRP | dBW | 2 | −8.1 | −11.1 |
Pointing loss | dB | 0 | 0 | 0 |
Propagation | ||||
Basic transmission loss | dB | −168.8 | −168.8 | −210.7 |
Absorption by atmos. | dB | 0 | 0 | 0 |
Absorption by the rain | dB | 0 | 0 | 0 |
Polarization loss | dB | 0 | 0 | 0 |
Receiving | ||||
Pointing loss | dB | 0 | 0 | 0 |
Antenna gain | dBi | −2 | −2 | 62.1 |
Feeder loss | dB | −2 | −2 | −2 |
Received level | dBW | −170.8 | −181.0 | −161.7 |
Tsys | dBK | 26.8 | 26.8 | 24.6 |
Noise power density | dBW | −201.8 | −201.8 | −204.0 |
Received C/N_{0} | dBHz | 31.0 | 20.8 | 42.3 |
BW | dBHz | −33 | −10 | −10 |
C/N | dB | -2.0 | 10.8 | 32.3 |
Real [C/N ] | 0.6257 | |||
Power distribution loss | dB | −4.1 | ||
RMS phase | deg | 4.7 | 0.03 |
If we consider that a solar cell produces about several watts/100 cm^{2} and a lander has dimensions of 30 × 30 cm^{2}, the power consumption would be limited to several watts. The link budget is satisfied within this limitation.
5. Design the Positions of the Landers and the Orbit of the Orbiter
5.1 Elevation limit
It is necessary to take into account two conditions when we consider the elevation limit to the observations of a satellite. These constitute an obstacle on the lunar surface as well as to the antenna pattern of a landing unit. If we assume that the unit is not placed very far from the lander, the lander becomes an obstacle after a rover carries the unit. The elevation angle is determined to be about 15^{°} when the unit is 10 m away from the lander at a height of 2.6 m, and 100 m away from rocks that are at a height of 26 m.
With regard to the antenna pattern, a patch antenna or a cross-dipole antenna is considered here. A patch antenna with a wide beam width can be designed, for example, ∓75° at about −6 dB to the maximum (+6 dB). The beam width of a cross-dipole antenna has a rather narrow beam width of ∓60° at −7.4 dB and ∓70° at −11 dB to the maximum (+2 dB). With these data in mind, the elevation limit of 15^{°} would be reasonable if a few dB of the margin in the link budget are known. With regard to the beam width of an antenna onboard an orbiter, the elongation between the two landers is considered to be less than 53^{°} when the orbital height of an orbiter and the distance between the two landers are both 2,000 km. Therefore, it becomes easy to design an antenna that satisfies the condition.
5.2 Position of a lander
We consider two landing units on the lunar surface. The following constraints are taken into consideration when designing the position of the lander.
- (1)
In order to communicate with the Earth, the landers need to be placed on the nearside of the Moon.
- (2)
To measure the rotation of the Moon, the landers should be placed on the lunar surface along a longitude, to the extent possible.
- (3)
To measure the libration of the Moon, the landers need to be located on the lunar surface along a latitude—as much as possible. Therefore, the landers are symmetrically placed on the surface about the point (r_{L}, 0, 0) in the body-fixed coordinate, and r_{L} is the radius of the Moon.
- (4)
The distance between the two landers needs to be greater than the radius, and it should certainly be less than the diameter of the Moon. If the distance is too large, the improvement in the accuracy of the libration obtained becomes quite limited, but the common view period between two landing units and the orbiter will decrease critically. Therefore, an approximate distance of about 2,000 km is selected as the distance between the two landers.
Considering all of the items shown above, the position of the landers on the lunar surface can be designed as:
Lander 1: longitude =−30°, latitude = +25°.
Lander 2: longitude = +30°, latitude =−25°.
The distance between the two landers is, then, about 2,154 km.
5.3 The orbital parameters of the orbiter
Based on the positions of the two landers shown above, we design the orbit of the orbiter. Our concern was for the common view period between the orbiter and the two landers. When designing the orbit, several rules need be followed:
Rule 1: There exists a common view period during every pass;
Rule 2: The mean value of the common view period should be as large as possible;
Rule 3: The difference of the common view period among the passes should be as small as possible.
By following these rules, we can choose the best orbit.
In terms of the observation of rotation of the Moon, in order to obtain as uniform a common view period as possible, we select the circular orbit, that is, e = 0. Then, only the semi-major-axis ‘a’ (or the height of the orbit) and the inclination ‘i’ (relative to the lunar equator) need to be specified.
We determine the height of the orbiter now, where:
- (1)
It is assumed that the half-cone angle of the antenna onboard the orbiter is smaller than 70^{°}. In order to encompass the Moon, the minimal height of the orbiter needs to be greater than 112 km.
- (2)
It is assumed that the minimal elevation of each antenna onboard the lander is 15°. In order to obtain the common view period, the minimal height of the orbiter should be h_{min} =1,070 km. It is important that the common view period lasts for several minutes or longer. The higher orbit will enable this longer common view period to be obtained. Taking into account these conditions, we select the possible height of the orbiter as 1,500, 1,800, 2,000, 2,200, and 2,500 km to analyze the findings.
The inclination of the orbit also needs to be determined. Given the rotation of the Moon, only the scope of i ∈ [0, 90]° needs to be considered. Therefore, we consider inclinations of 0°,15°,30°,45°,60°,75°, and 90°.
The orbit and common view period.
Inclin.(deg)/ height(km) | 1500 | 1800 | 2000 | 2200 | 2500 |
---|---|---|---|---|---|
0 | 160 | 140 | 129 | 119 | 107 |
10.14 | 18.07 | 23.55 | 29.20 | 37.96 | |
0.35 | 0.26 | 0.50 | 0.40 | 0.19 | |
1622.4 | 2529.8 | 3038.0 | 3475.8 | 4061.7 | |
15 | 160 | 140 | 129 | 119 | 107 |
6.86 | 15.34 | 21.03 | 26.76 | 35.74 | |
3.53 | 4.00 | 4.52 | 5.12 | 5.99 | |
30 | 54 | 64 | 69 | 71 | 75 |
8.89 | 15.48 | 19.72 | 24.41 | 30.91 | |
3.36 | 9.85 | 12.26 | 14.46 | 18.18 | |
45 | 34 | 41 | 44 | 43 | 47 |
9.88 | 16.85 | 21.45 | 26.79 | 32.49 | |
9.88 | 14.43 | 17.10 | 20.13 | 24.85 | |
60 | 29 | 34 | 35 | 37 | 36 |
8.93 | 16.18 | 20.89 | 25.32 | 33.53 | |
7.25 | 11.43 | 13.80 | 16.69 | 20.37 | |
75 | 29 | 33 | 35 | 35 | 35 |
8.10 | 14.67 | 18.91 | 23.57 | 30.46 | |
4.20 | 6.91 | 8.79 | 10.80 | 13.59 | |
90 | 32 | 37 | 37 | 36 | 37 |
7.34 | 12.84 | 17.41 | 22.11 | 28.14 | |
3.31 | 5.96 | 7.17 | 8.17 | 11.48 |
6. Time Delay Error Analysis
6.1 The phase variation of the system
From Eqs. (13), (14), and (12b), it is apparent that the error of the time delay between the landers and the antenna is influenced by the error of , and ^{ 2 }π(f^{1-}f^{0}). [(R_{S-L2} - R_{S-L1})/c]. The can be obtained on the ground by correlating it with the signals that were received. This error may increase up to 4°. The is obtained on the orbiter and then transmitted to the Earth by telemetry. It is the difference of , which is the result of the correlation of the roundtrip, and local return signals on the orbiter. Therefore, this error is considered to be . 2π(f_{ i } − f_{0}) · [(R_{S-L2} − R_{S-L1})/c] is caused by the range difference between the two landers and the orbiter. The term is calculated by Eq. (12b). We can consider the error to be 4^{°}.
The two terms of and arise when the signals propagate through the Earth’s atmosphere. For a distance of 2,000 km on the lunar surface, the elongation of the signal path on the ground is less than 0.3°. Assuming a same-beam observation and a single-layer screen model of troposphere phase variations in the Kolmogorov turbulence, the travelling time is about 5 s. The phase variation after the signals from the two landers have been correlated is about 2.4^{°} (RMS). For the ionosphere, the phase variation includes two components: short- and long-term variations. For the short-term variation, an increase in the integration time can decrease the error. For the long-term variation, the GIM (global ionospheric model) can be used to correct the delay. Because the separation angle between the ground antenna and the two landers is small, the error can be decreased to several millimeters (Liu et al., 2009). If more than one frequency signal can be obtained, for example X-band signal, more accurate results can be derived.
The term of is caused by the nonlinear aspect of the phase-frequency characteristic of the antenna. The phase variation is about 0.2^{°} in the effective bandwidth of each channel (Liu et al., 2010).
Considering all of these terms, it can be deduced that the total phase variation reaches approximately 13°; that is, the error of the range difference may reach 5 mm. We can also conclude that the error of the time delay is about 20 picoseconds.
6.2 Required frequency stability of a frequency standard and phase variation with temperature change
This system needs a stable frequency standard. Since the phase variations caused by the frequency standard of the satellite are nearly cancelled, they should not exceed a few degrees of phase-measurement accuracy during the roundtrip time. Taking the roundtrip time to be less than 10 ms, the phase error can be roughly estimated by Eq. (11). Taking f^{ i } and τ as 2.28 GHz and 0.01 s, respectively, the phase variations are small, and is expected to be 1 × 10^{−9} at 10 ms. On the other hand, the frequency standard of the lander never yields phase variations that are more than a few degrees during the integration time in order to measure the phase difference with an error of less than a few degrees. If we assume the frequency f^{ i } of the standard and integration time τ in Eq. (11) to be 40 MHz and 10 s, respectively, then would be 4 × 10^{−11}.
The most severe stipulation for frequency standards is the requirement to continuously track the phase difference between the frequency standards of the satellite and the landers. The phase difference is obtained by . The phase difference cannot be measured for the interval from one pass to the consecutive pass. The phase change for the interval is to be estimated without 2π ambiguity. Since the interval is about 4 h, the is required to be less than 1 × 10^{−12}. The rubidium frequency standard satisfies this condition.
The output frequency and frequency stability of the frequency standard vary with supplied voltage and the temperature around the frequency standard. Therefore, delay or phase characteristics of all devices of the system also vary slightly with these parameters. It is inevitable to both stabilize and monitor these parameters.
7. Lunar Science by the Inverse VLBI
The inverse VLBI observations provide information about the Earth and the Moon, especially with regard to their rotation. Applying the high-accuracy observation from the iVLBI, the results that are available at present in lunar science can be improved up on. The internal structure and material properties of the Moon are to be deduced from external evidence, and the deepest regions are those that are least understood. The iVLBI provides information with regard to the Moon’s tidal response, tidal dissipation, and interactions at the core/mantle interface.
Solid-body tides are raised on the Moon by the gravitational attraction of the Earth. These lunar tidal variations can be represented as sums of periodic components. All of the tidal components > 1 cm occur at about every month or half a month (Williams and Dickey, 2003). These tides can be detected based on the tidal displacements of the landing units.
For the Moon, any description of the three-dimensional rotation requires three Euler angles that can change with time. Two angles describe the orientation of the pole and the orthogonal equator plane, while the third angle generates the rotation about the pole. The angular variations are called physical librations, and the periods of the physical libration terms vary from half a month to almost 273 years (Williams and Dickey, 2003). However, most of these periods last for less than 1 year. Therefore, in order to observe these terms, the lifetime of the iVLBI system should be maintained for more than 1 year.
8. Conclusion and Discussion
The conventional VLBI technique or lunar laser ranging has limited application in the ground-based geodetic observation of the Moon, with an error of a few centimeters. The “Inverse VLBI” has been proposed as a breakthrough method to circumvent such limitations. This article describes the system. The most critical aspect of the technique is how to obtain the phase difference measurements of the two landers. We propose a simple roundtrip signal method to obtain this phase difference. Then, the calibration of the instrumental delay on the ground is provided. The time delay is determined by the NBV method, and an example of the link budget is shown. We also report on the design of the position of the landers and the orbit of the orbiter based on the necessity of the observation, and the common view period is analyzed. After analyzing this error, we conclude that the range difference can be obtained with an error of several millimeters. Hence, the iVLBI technique can achieve geodetic observation with only an error of a few millimeters, while the error of geodetic observation in the conventional VLBI will be magnified with the distance between object and stations. In order to realize the iVLBI system, new equipment for implementing the iVLBI should be developed. In addition, testing on the ground should be carried out prior to launch. Phase noise changes with temperature should also be carefully researched for this method.
We hope that a series of new results can be obtained by using the iVLBI technology. Simulation analyses on solving the parameters of the lunar rotation are expected to be executed in the near future. It is also apparent that the suitable range can be obtained after calibrating the time delay of the landers.
This method can be applied to observe the rotation of the Mars as well as other planets. It is influenced very minimally by the distance between the planet and the Earth. If the planet is farther away from the Earth, elongation of the two signal passes received on the ground will be very minimal, and the influence of the Earth’s atmosphere will be decreased to a very low level. More high-accuracy observations can be facilitated under such conditions.
Declarations
Acknowledgments
This work was supported by the programs of Chinese Academy of Sciences for Visiting Professorship for Senior International Scientists, Hundred Talent Project of Chinese Academy of Sciences, National Natural Science Foundation of China (Grant Nos. 10973031 and 11043004), and the RISE/SELENE Project of the National Astronomical Observatory of Japan.
Authors’ Affiliations
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