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Keywords:

  • same-beam VLBI;
  • SELENE;
  • orbit determination;
  • lunar gravity field model

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Same-Beam VLBI Observations in SELENE
  5. 3. Principle for Obtaining Differential Phase Delay
  6. 4. Observation Results
  7. 5. Error Analysis and Correction of Atmospheric and Ionospheric Delays
  8. 6. Contribution of VLBI Data to Orbit Determination and Lunar Gravity Model
  9. 7. Conclusion
  10. References
  11. Supporting Information

[1] The Japanese lunar mission, Selenological and Engineering Explorer (Kaguya), which was successfully launched on 14 September 2007, consists of a main satellite and two small satellites, Rstar and Vstar. Same-beam very long baseline interferometry (VLBI) observations of Rstar and Vstar were performed for 15.4 months from November 2007 to February 2009 using eight VLBI stations. In 2008, S band same-beam VLBI observations totaling 476 h on 179 days were undertaken. The differential phase delays were successfully estimated for most (about 85%) of the same-beam VLBI observation periods. The high success rate was mainly due to the continuous data series measuring the differential correlation phase between Rstar and Vstar. The intrinsic measurement error in the differential phase delay was less than 1 mm RMS for small separation angles and increased to approximately 2.5 mm RMS for the largest separation angles (up to 0.56 deg). The long-term atmospheric and ionospheric delays along the line of sight were reduced to a low level (several tens of milimeters) using the same-beam VLBI observations, and further improved through application of GPS techniques. Combining the eight-station (four Japanese telescopes of VLBI Exploration of Radio Astrometry and four international telescopes) S band same-beam VLBI data with Doppler and range data, the accuracy of the orbit determination was improved from a level of several tens of meters when only using Doppler and range data to a level of 10 m. As a preliminary test of the technique, the coefficient sigma degree variance of the lunar gravity field was compared with and without 4 months of VLBI data included. A significant reduction below around 10 deg (especially for the second degree) was observed when the VLBI data were included. These observations confirm that the VLBI data contribute to improvements in the accuracy of the orbit determination and through this to the lunar gravity field model.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Same-Beam VLBI Observations in SELENE
  5. 3. Principle for Obtaining Differential Phase Delay
  6. 4. Observation Results
  7. 5. Error Analysis and Correction of Atmospheric and Ionospheric Delays
  8. 6. Contribution of VLBI Data to Orbit Determination and Lunar Gravity Model
  9. 7. Conclusion
  10. References
  11. Supporting Information

[2] The lunar gravity field is a fundamental physical quantity for studying the Moon's internal structure and evolution and has been studied by many missions such as Luna 10, Lunar Orbiters from I to V, Apollo 15 and 16, Clementine and Lunar Prospector. Lunar gravity field determination is hampered by two specific geometrical issues: one is the lack of direct observations of the farside gravity signals, and the other is reduced accuracy near the limb, because the conventional two-way Doppler and range measurements are less sensitive in tracking the limb region.

[3] The Japanese lunar mission, SELENE (Kaguya), was successfully launched from the Tanegashima Space Center on 14 September 2007. SELENE (Selenological and Engineering Explorer) consists of a three-axis stabilized main satellite and two small free-flying spin satellites, called Rstar and Vstar [Iwata et al., 2001]. Rstar and Vstar were separated from the main satellite and injected into elliptical lunar orbits of about 100–2400 km and 100–800 km in altitude, respectively. The main satellite is controlled to maintain a nearly circular orbit of about 100 km in altitude (Figure 1). Rstar relays the Doppler signal between the main satellite and the ground station (Usuda Deep Space Center: UDSC) to enable direct measurement of the lunar farside gravity field, which has contributed to improvement of the farside gravity field model [Namiki et al., 2009]. In addition, differential VLBI gives plane-of-sky position differences between the Rstar and Vstar satellites, in contrast to two-way Doppler measurements that give line-of-sight velocity information. The combination of VLBI and Doppler observations are important to precisely determine the orbit of Rstar due to the better 3 dimensional positioning sensitivity achieved, because Rstar serves as a reference for the four-way Doppler observations. In addition, the VLBI observations have the potential to improve the lunar gravity field especially for the limb region. Additionally, the high-altitude orbits and free flying (i.e., no induced accelerations from momentum wheels) of Rstar and Vstar makes it possible to improve the accuracy of the low-degree coefficients of the spherical harmonics [Hanada et al., 2008; Matsumoto et al., 2008]. A highly accurate gravity field model is important for understanding not only the internal structure near the surface by analyzing the high-degree coefficients [Namiki et al., 2009], but also the deep interior through the low-degree terms. For example, one can constrain the density of the lunar core through the moment of inertia by combining the low-degree gravitational harmonics with the amplitudes of the lunar physical librations [Hanada et al., 2008; Matsumoto et al., 2008].

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Figure 1. Conceptual diagram showing the differential VLBI observation method used in the SELENE project.

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[4] Rstar and Vstar each transmit three carriers (2212 MHz (S1), 2218 MHz (S2), 2287 MHz (S3)) in S band and one carrier (8456 MHz (X)) in X band; the differential phase delay (DPD, Δτi, i = S1, S2, S3, X) between Rstar and Vstar is obtained from the relationship between the correlation phase (the phase of the cross correlation function of signals received at two VLBI stations) and frequency of the four carriers. In this case, the difference in correlation phase at each frequency has to be estimated without the cycle ambiguity, and this imposes strict conditions such as the error of the correlation phase must be lower than 4.3 deg RMS [Kono et al., 2003]. To resolve the cycle ambiguity problem, we primarily use same-beam differential VLBI, in which Rstar and Vstar are observed simultaneously within the main beam of the receiving antennas [Liu et al., 2006, 2007a; Kikuchi et al., 2008]. In this situation, the influence of the atmosphere, ionosphere and receivers are almost canceled from the difference in the correlation phases, and the DPD can be determined to a very high accuracy of 1 mm [Kono et al., 2003]. Previously, group delays with accuracies limited to several tens of mm were used for spacecraft tracking [Sagdeyev et al., 1992; Thornton and Border, 2000]. Differential phase delay data with an accuracy of about 3 mm over a timescale of several days has previously been used for spacecraft tracking, where the cycle ambiguity was estimated by using the 10-station VLBA interferometer and imaging algorithms for astronomical sources [Martín-Mur et al., 2006].

[5] In the SELENE VLBI observations, the cycle ambiguity was directly derived from the differential correlation phases at different frequencies for each baseline. In our previous paper [Kikuchi et al., 2009], we reported that the X band DPD can be determined with a very small error of 0.3 mm when the separation angle between Rstar and Vstar is less than 0.1 deg and the baselines are relatively short (2000 km), as is the case when using the four stations of the Japanese VERA (VLBI Exploration of Radio Astrometry) array (Figure 2). This error is more than 1 order of magnitude smaller than former VLBI results. However, the opportunities for X band same-beam VLBI observations in which the separation angle is less than 0.1 deg amount to only 6% during tracking periods (Figure 3), which is insufficient for detecting the global lunar gravity field signal accurately. From March 2008, we thus changed the VLBI observation method, so that the telescope tracks the midpoint of Rstar and Vstar for separation angles up to 0.56 deg. This was because the opportunities for S band same-beam VLBI observations include 69% of the time during tracking periods (Figure 3), and so more same-beam VLBI data have been obtained. This paper reports the results of these observations for the first time.

image

Figure 2. The location of the VLBI stations and baseline lengths used for SELENE VLBI observations.

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Figure 3. The separation angle between the Rstar and Vstar satellites in January 2008. The S band same-beam VLBI observations are possible 69% of the total time, while for X band same-beam VLBI is only possible 6% of the time. It should be noted that here we have only considered the separation angle in estimating the percentage of time same-beam VLBI observations can be undertaken. Rstar and Vstar are sometimes over the lunar farside, and when this occurs same-beam VLBI observations cannot be performed.

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[6] In the following sections we give a general introduction to the SELENE same-beam VLBI observations performed using all eight VLBI stations for the whole period of 15.4 months from November 2007 to 12 February 2009 (when Rstar crashed into the lunar farside). We have analyzed the measurement error in the DPD not only for small separation angles less than 0.1 deg, but also for larger separation angles up to 0.56 deg. We show why the cycle ambiguity can be resolved using same-beam VLBI observations while it cannot usually be resolved for switched VLBI observations under relatively bad conditions (such as large separation angles, or the atmospheric and the ionospheric fluctuations are strong). We introduce the results of orbit determination for longer arc length, and report how the accuracy of the orbit determination was improved by combining the VERA VLBI data and further improved by combining the international-baseline VLBI data. Finally, we report how the lunar gravity field model, especially the accuracy of the low-degree coefficients was improved by combining 4 months of S band same-beam VLBI data with Doppler and range data.

2. Same-Beam VLBI Observations in SELENE

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Same-Beam VLBI Observations in SELENE
  5. 3. Principle for Obtaining Differential Phase Delay
  6. 4. Observation Results
  7. 5. Error Analysis and Correction of Atmospheric and Ionospheric Delays
  8. 6. Contribution of VLBI Data to Orbit Determination and Lunar Gravity Model
  9. 7. Conclusion
  10. References
  11. Supporting Information

[7] Figure 2 shows the VLBI observation network of SELENE, consisting of the four Japanese 20 m telescopes of VERA, located in Mizusawa (MZ), Ogasawara (OG), Ishigaki (IS), and Iriki (IR) [Honma et al., 2003], and four international telescopes, located in Shanghai (SH 25 m, China), Urumqi (UR 25 m, China), Hobart (HO 26 m, Australia) and Wettzell (WZ 20 m, Germany) [Schlüter and Behrend, 2007]. The baseline lengths vary from 796 km (IS-SH) to 12247 km (HO-WZ). VERA was the primary network for the VLBI observations and participated for about 1600 h from November 2007 to February 2009 for observing Rstar, Vstar, the main satellite and quasars. Since February 2009, it has been used to observe Vstar and the main satellite. The four international stations participated for about 200 h in January, May, June, and July 2008 and February 2009 for observing Rstar and Vstar.

[8] From November 2007 to February 2008, we performed differential VLBI observations between Rstar and Vstar as follows: same-beam VLBI observations for the S band and X band signals when the separation angle θ < 0.1 deg and switched VLBI observations for θ > 0.1 deg. In the same-beam VLBI, the telescopes track the midpoint of Rstar and Vstar to simultaneously observe the two satellites. In switched VLBI, the telescopes track Rstar and Vstar alternatively with a switching interval of 2 min. Note that in the case of θ < 0.3 deg, the S band signals from Rstar and Vstar can also be simultaneously observed even if the telescopes track Rstar and Vstar alternatively, which is called an S band same-beam VLBI observation. From the correlation analyses, we found that the cycle ambiguity could not be resolved for many of the periods of switched VLBI observations. The atmospheric and ionospheric fluctuations are expected to be stronger in summer and autumn, increasing the difficulty in resolving the cycle ambiguity in the switched VLBI observations. We thus changed our VLBI observation method to increase the probability of being able to undertake S band same-beam VLBI observations for θ up to 0.56 deg. The requirement for a maximum separation of 0.56 deg was adopted after consideration of not only the phase characteristics and power characteristics of the telescopes [Liu et al., 2007b], but also the correlation results obtained prior to February 2008 for all eight stations. For the observations prior to February 2008, we were able to successfully resolved the cycle ambiguity for all eight stations when θ is less than approximately 0.3 deg in S band through same-beam VLBI observations by alternatively tracking Rstar and Vstar. From March 2008, we thus changed the observation method by tracking the midpoint of Rstar and Vstar even for θ up to 0.56 deg (±0.28 deg) to perform S band same-beam VLBI observations. Of course, X band same-beam VLBI observations are also performed simultaneously when θ < 0.1 deg after February 2008.

[9] As shown in Figure 4, we have performed 476 h of S band same-beam VLBI observations on 179 days in 2008. The average time for same-beam VLBI observations per day was about 2.66 h, but sometimes it exceeded 5 h, when the orbital plane of Rstar and Vstar were in a face-on configuration. Long-term VLBI observations in a face-on configuration have the potential to improve the lunar gravity field model for the limb region. We also performed S band same-beam VLBI observations for separation angles θ < 0.7 deg to confirm the performance of the telescopes in January and February 2009. The analysis of these observations is ongoing.

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Figure 4. The observation time for S band same-beam VLBI observations for SELENE in 2008.

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3. Principle for Obtaining Differential Phase Delay

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Same-Beam VLBI Observations in SELENE
  5. 3. Principle for Obtaining Differential Phase Delay
  6. 4. Observation Results
  7. 5. Error Analysis and Correction of Atmospheric and Ionospheric Delays
  8. 6. Contribution of VLBI Data to Orbit Determination and Lunar Gravity Model
  9. 7. Conclusion
  10. References
  11. Supporting Information

[10] The predicted geometric delay τgeo between the reference and the remote stations was obtained through an iterative process using a finite distance model [Kono et al., 2003; Kikuchi et al., 2008]. The cross correlation between the four respective carriers from Rstar and Vstar at two stations produces 4 × 2 correlation phases. Taking the difference between the correlation phases of Rstar and Vstar, results in four differential correlation phases Δϕi(i = S1, S2, S3, X). The differential correlation phases can be expressed as follows [Kono et al., 2003; Liu et al., 2006, 2007a; Kikuchi et al., 2008],

  • equation image

where fi is the transmission frequency of the carriers and Δτg−k (k = S,X) is the double difference in the residual of the predicted geometric delay, the first difference being between two stations and the second one between Rstar and Vstar. The difference between Δτg−S and Δτg−X mainly originates from the difference in position of the transmitting S band and X band antennas, which is less than 1 mm [Liu et al., 2006]. Because we mainly used S band DPD in the SELENE observations, the difference between Δτg−S and Δτg−X can usually be neglected. Δτatm, Δτion−i, and Δτinst−k are the double differences in the long-term delays of the atmosphere, the ionosphere and receivers, respectively. Ni is an integer ambiguity. The effect of the ionosphere is to advance the wavefront, while the atmosphere and receivers act to delay the wavefront, as the sign of Δτion−i is negative. In addition, Δτion−i is proportional to the inverse square of the frequency, which has an influence on the estimation of Ni. In the SELENE VLBI observations, the long-term double differenced total electron content must be less than 0.42 TECU in order to estimate Ni in S band [Kono et al., 2003; Liu et al., 2006; Kikuchi et al., 2008]. 0.42 TECU corresponds to Δτion−i = 0.035 m in S band. On the other hand, Δτatm and Δτinst−S are only reflected as a bias in Δτres−i; they have no influence on estimating Ni. σi is the short-term doubly differenced correlation phase noise, which is caused by the atmosphere, the ionosphere, and thermal noise in the receivers. Δτres−i is called the residual differential phase delay, which can be estimated from linear equations of (1) after Ni has been determined.

[11] There are three steps for obtaining the S band residual DPD, Δτres−i [Kono et al., 2003; Liu et al., 2006; Kikuchi et al., 2008]: (1) resolve the cycle ambiguity of NS2NS1 in the frequency range between S2 and S1, (2) resolve the cycle ambiguity of NS3NS1 in the frequency range between S3 and S1, and (3) resolve the cycle ambiguity of NS1 for S1. NS2NS1, NS3NS1 and NS1 can be estimated from equations (2)–(4):

  • equation image
  • equation image
  • equation image

where, σS21 = σΔϕ(S2−S1), σS31 = σΔϕ(S3−S1), i.e., only the double differences in correlation phase, the first difference being between Rstar and Vstar and the second one between S1 and S2 or S1 and S3, have influence on σS21 and σS31. Table 1 shows the conditions required to estimate NS2NS1, NS3NS1 and NS1. In same-beam VLBI, σS21 and σS31 are small enough to allow estimation of NS1, NS2 and NS3 for the following reasons. The first reason is that the fluctuations in the atmosphere and ionosphere are similar for the correlation phases of Rstar and Vstar due to the small separation angle, so they largely cancel in the difference between the correlation phases between Rstar and Vstar. The second reason is that the fluctuations in the atmosphere, ionosphere and receiver are similar for the differential correlation phases for S1, S2 and S3 due to the relatively small frequency difference for S1, S2 and S3. The similar fluctuations for S1, S2 and S3 have little influence on σS21 and σS31 as mentioned above. Thus, although the condition σS31 < 4.3 deg RMS is very strict, we were able to successfully estimate NS1, NS2 and NS3 in about 85% of S band same-beam VLBI observation periods, even in the summer (June–August) 2008. These two conditions were usually satisfied in the same-beam VLBI observations because the data series are continuous. In contrast we were not able to correctly estimate NS1 for many of the switched VLBI observations because the data series are not continuous, and the short-term phase fluctuations (durations less than several hundreds seconds), caused by the atmosphere, the ionosphere and the receiver cannot be removed. Δτatm, Δτion−i, and Δτinst−k are typically reduced to a very low level (less than several centimeters) in the same-beam VLBI observations. The residual Δτres−S mainly originates from the predicted geometric delay error Δτg−k, but Δτg−k can be reduced to a low level (several nanoseconds) by using Doppler and range data so that the condition Δτres−S < 83 ns (24.9 m) is satisfied [Kono et al., 2003].

Table 1. Conditions Required to Estimate NS2NS1, NS3NS1, and NS1a
Cycle AmbiguityErrorΔτres−SΔτion−i (S Band)
NS2NS1σS21 <127 deg24.9 m
NS3NS1σS31 <10.2 deg
NS1σS31 <4.3 deg0.035 m

[12] After estimating NS1, NS2 and NS3 the DPD (Δτi) can be obtained from,

  • equation image

where Δτgeo is the difference in τgeo between Rstar and Vstar.

4. Observation Results

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Same-Beam VLBI Observations in SELENE
  5. 3. Principle for Obtaining Differential Phase Delay
  6. 4. Observation Results
  7. 5. Error Analysis and Correction of Atmospheric and Ionospheric Delays
  8. 6. Contribution of VLBI Data to Orbit Determination and Lunar Gravity Model
  9. 7. Conclusion
  10. References
  11. Supporting Information

[13] The receiver of the VLBI systems for SELENE consisted of a front end and a back end. The S band and X band carriers were converted into video signals at frequencies of tens of kHz by using frequency down-converters and by setting suitable local frequencies considering Doppler shifts. The bandwidth was reduced from 2 MHz to 100 kHz using low-pass filters (LPF) with linear-phase characteristics [Liu et al., 2006]. The video-band signals were recorded on a personal computer after a 6 bit A-D conversion with a sampling frequency of 200 kHz. The correlation of the signals received at two VLBI stations was performed in software [Kono et al., 2003; Kikuchi et al., 2008].

[14] Figure 5 shows the spectra of the received signals at eight stations during S band same-beam VLBI observations. As shown in Figures 5a5h, the power signal-to-noise ratio is about 18 dB for the four VERA stations and larger than 22 dB for the international stations when the separation angle is 0.07 deg. The reason for this difference is that the diameter of VERA telescopes is smaller than that of the other telescopes. Another reason is that the LNA (Low Noise Amplifier) for the VERA S band and X band receivers are not cryogenically cooled. Figure 5e also shows the spectrum of the Vstar signal, the bandwidth of which is about 5 Hz. We thus calculated the correlation phase using only the signal within a narrow bandwidth of B = 10 Hz to decrease the influence of the system noise. The influence for B = 10 Hz can be as much as 10 (10 dB) times smaller than that for B = 100 Hz, which was the planned bandwidth when Rstar and Vstar were developed. Figures 5i and 5j show the spectra of the received signals at the Urumqi and Ishigaki stations when the separation angle is 0.53 deg. The signal-to-noise ratio is about 13 dB, which is smaller than that for the smaller separation angle of 0.07 deg due to the antenna patterns of the ground stations. However, this signal-to-noise ratio is still sufficient to obtain the DPD when a narrow bandwidth of 10 Hz is used in the correlation processing. In addition, the frequencies of the receiving signals of Rstar and Vstar shown in Figures 5i and 5j were different from those shown in Figures 5a5h. This was because the data were obtained in different period, and the Doppler shifts were also different.

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Figure 5. The spectra of the received signals at eight stations during S band same-beam VLBI observations. (a–h) The separation angle is 0.07 deg. (i and j) The separation angle is 0.53 deg.

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[15] Figures 6a6c show the correlation phase fluctuations of Rstar and Vstar at S1, S2, S3 and the difference between Rstar and Vstar (for a small separation angle). Figure 6d shows all the data in Figures 6a6c. The data were obtained on 24 May 2008 for the Hobart to Iriki baseline, which has a length of 7828 km. The S band same-beam observation was performed with the separation angle θ ranging between 0.09 and 0.14 deg, as shown in Figure 6e. The correlation phases were obtained with 1 s time resolution and 60 s averages were taken to produce these plots. To analyze the relative short-term fluctuations, it was necessary to first remove long-term trends (which are mainly attributed to errors in the a priori orbit prediction). These were removed by fitting ninth-order polynomials to the correlation phases for each 0.5 h segment separately. The correlation phase fluctuations with amplitudes of approximately 20 deg are mainly caused by the atmosphere and the ionosphere. They are very similar for Rstar and Vstar because the propagation paths from the two satellites to the ground stations are nearly the same. In addition, the profiles of the phase fluctuations at S1, S2 and S3 are also very similar because the difference between the frequencies is not large. These fluctuations were reduced to a low level by taking the difference between the correlation phases of Rstar and Vstar. The RMS of the differences between correlation phase fluctuations of Rstar and Vstar at S1, S2 and S3 are 1.8, 1.9, 1.8 deg, respectively, which satisfy the condition σS31 < 4.3 deg RMS for obtaining an S band DPD. Figure 6e shows the S band DPD, which was in the range from −5.5 to 0.5 km during these observations. Figure 6f shows the fluctuation in the DPD after removing the long-term trend (by fitting ninth-order polynomial). The residual is very small, with an RMS of 0.88 mm. This is typical of the results we obtain in the case of small separation angles. In some observation periods when the atmospheric and ionospheric fluctuations were small, the measurement error in the S band DPD was approximately 0.3 mm.

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Figure 6. (a–c) The correlation phase fluctuations of Rstar and Vstar at S1, S2, and S3 and the difference between Rstar and Vstar, (d) including all data shown in Figures 6a–6c, (e) the separation angle between Rstar and Vstar and the DPD, and (f) the residual DPD. The separation angle was small during these observations.

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[16] Figures 7a7c show the correlation phase fluctuations of Rstar and Vstar at S1, S2, and S3 for an observation made with a large separation angle. The separation angle θ was in the range 0.22 to 0.56 deg, as shown in Figure 7f. The data were obtained on 31 May 2008 on the Urumqi to Wettzell baseline, which has a length of 5356 km. The long-term trends were removed by fitting ninth-order polynomials. The correlation phase fluctuations with an amplitude of about 15 deg are mainly caused by the atmosphere and the ionosphere. They are slightly different for larger separation angles, due to the different propagation paths [Liu et al., 2005]. As shown in Figure 7d, the fluctuations in the differential correlation phases between Rstar and Vstar vary with amplitude and are typically around 10 deg, which is larger than the 4.3 deg RMS limit required to obtain an S band DPD. However, the fluctuations in the differential correlation phases at S1, S2 and S3 are very similar. Figure 7e shows the differences between S2 and S1, S3 and S1, and S3 and S2, which were calculated from the data shown in Figure 7d. As discussed in section 3, the differences shown in Figure 7e correspond to σS21, σS31, and σS32, respectively. Here we measure σS21 = 1.15, σS31 = 0.93, and σS32 = 1.07 deg RMS, which satisfies the condition σS31 < 4.3 deg RMS. The cycle ambiguity NS1, NS2 and NS3 can thus be determined and the S band DPD can be obtained. These results clearly show why same-beam VLBI observations were so important for determining NS1, NS2 and NS3. It can produce continuous correlation phase, which cannot be achieved with switched VLBI observations. The continuous data series is necessary to correct for the short-term phase fluctuations (less than several hundreds seconds), as outlined in section 3. Figure 7f shows the fluctuation in the DPD after removing the long-term trend by fitting the ninth-order polynomial. In this case it has an RMS of 1.55 mm, which is larger than that shown in Figure 6f. This is because the fluctuations in the differential correlation phases at S1, S2, and S3 shown in Figure 7d influence the measurement error in the DPD as shown in equation (5). However, 1.55 mm is still an acceptably small measurement error for the S band DPD.

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Figure 7. (a–c) The correlation phase fluctuations of Rstar and Vstar at S1, S2, and S3 and (d) the difference between Rstar and Vstar, (e) the difference among S1, S2, and S3 calculated from the data shown in Figure 7d, and (f) the separation angle between Rstar and Vstar and the residual DPD. The separation angle was large during these observations.

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[17] Figure 8a shows the S band DPD for three baselines: Ishigaki-Mizusawa (IS-MZ), Mizusawa-Iriki (MZ-IR), and Iriki-Ishigaki (IR-IS). The data were obtained on 24 May 2008. The separation angle θ was between 0.08 and 0.56 deg, as shown in Figure 8d. The fluctuations in the DPD for the three baselines after removing long-term trends (by fitting ninth-order polynomials), are shown in Figure 8b. The DPD data are estimated for each 0.5 h time interval separately, however, the long-term polynomial fitting is performed over periods of longer than 0.5 h. The time period over which each polynomial fitting is undertaken contains at least two of the 0.5 h DPD solution intervals. This means that if the cycle ambiguity is incorrectly determined for each solution interval it is easily detected as it produces an obvious jump in the DPD data between two intervals. Figure 8b shows an example where the cycle ambiguity has been correctly determined for each interval and so there is no jump in the DPD. From Figures 8b and 8d we can also clearly see that the fluctuations in the DPD become large when the separation angles become large. The measurement errors in the DPD for the IS-MZ, MZ-IR, and IR-IS baselines were 2.18, 2.36, and 2.91 mm RMS, respectively. They are typical of the measurement errors obtained from SELENE VLBI observations in relatively bad conditions. The closure DPD for the three baselines, i.e., the sum of DPD of (IS-MZ) + (MZ-IR) + (IR-IS) is calculated. Figure 8c shows that it varies over a small range (−0.011–0.006 m), but is not zero. The reason for this seems to be that in the computation of the closure DPD, the reference station per baseline is sometimes different, causing a slight difference in the epoch to which the datum is referred to. This causes a closure DPD that is slightly different from zero, with a signal that correlates to the actual data signal. Note that the closure DPD in Figure 8c correlates to the data as shown in Figure 8a, either positively (baseline IS-MZ) or negatively (the other two baselines). This might induce some extra residual signal in the data, and this is currently under investigation, but it should be pointed out that the variations in the closure DPD are well below the value of a cycle ambiguity. In summary, we can confirm from the correlation phase fluctuations and the closure DPD that the S band DPD has been successfully estimated, without any cycle ambiguity.

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Figure 8. (a) The differential phase delay (DPD) for the three baselines IS-MZ, MZ-IR, and IR-IS, (b) the residual DPD fluctuations, (c) the closure phase delay for the three baselines, and (d) the separation angle between Rstar and Vstar.

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5. Error Analysis and Correction of Atmospheric and Ionospheric Delays

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Same-Beam VLBI Observations in SELENE
  5. 3. Principle for Obtaining Differential Phase Delay
  6. 4. Observation Results
  7. 5. Error Analysis and Correction of Atmospheric and Ionospheric Delays
  8. 6. Contribution of VLBI Data to Orbit Determination and Lunar Gravity Model
  9. 7. Conclusion
  10. References
  11. Supporting Information

[18] As discussed above, the influence of short-term fluctuations in the atmosphere, the ionosphere and the instruments on the DPD are reflected in σi. σi is less than 1 mm RMS in case of small separation angles between the two satellites and about 2.5 mm RMS for larger separation angles. In addition, the condition σS31 < 4.3 deg RMS in Table 1 was also satisfied. However, the estimated DPD may also be affected by long-term variations in the atmosphere, the ionosphere and the instruments, Δτatm, Δτion−i, Δτinst−k. In this section, we analyze and correct for these long-term changes, and show how their influence can be reduced to a low level by using a combination of same-beam VLBI and GPS techniques.

[19] The long-term instrumental delay Δτinst−k in the S band same-beam VLBI observations can be due to two factors: one is the phase-frequency characteristics of the receivers; the other is phase variations in the main beam of the telescopes. Δτinst−k caused by the first of these factors is about 0.03 mm for S band same-beam observations [Liu et al., 2006], while that caused by the second is about 1.2 mm [Liu et al., 2007b]. The long-term instrumental delay Δτinst−k was very much smaller than the condition of 24.9 m in Table 1.

[20] We have used GPS techniques to correct for the long-term atmospheric delay. At each VLBI station for SELENE observations, at least one GPS receiver is located near the telescope with distances from 40 m to 193 m. The long-term zenith total delay ZTD caused by the atmosphere can be estimated with an accuracy of 10 mm by using GPS techniques [Behrend et al., 2000]. The long-term zenith hydrostatic delay ZHD caused by the hydrostatic atmosphere can be estimated with an accuracy of 1 mm by measuring the surface pressure [Niell, 1996]. The long-term zenith water vapor delay ZWD caused by water vapor can be calculated by using ZWD = ZTDZHD.

[21] Figure 9 shows examples of corrections for the long-term atmospheric and ionospheric delay using this approach. The results for two baselines: a short baseline (Ishigaki-Iriki) with a length of 1037 km, and a longer baseline (Hobart-Ishigaki) with a length of 7331 km are shown. Figure 9a shows that the ZTD was about 2.4 m at Hobart, 2.43 m at Iriki, and 2.58 m at Ishigaki. The elevation angles of Rstar at Hobart, Ishigaki, and Iriki are shown in Figure 9b. Figure 9b also shows the separation angles between Rstar and Vstar, which varied over a range between 0.08 and 0.55 deg. The long-term atmospheric delay along the line of sight τatm−l (where l is either R for Rstar or V for Vstar) were calculated by using Δτatm−l = ZHD * mh + ZWD * mw, where mh and mw are the hydrostatic and wet Herring-Niell Mapping Functions, respectively [Niell, 1996]. As shown in Figure 9c, τatm−R (for Rstar) varied over a range from 2.5 to 2.9 m at Hobart, from 4.1 to 9.2 m at Ishigaki, and from 4.4 to 8.9 m at Iriki during this observation. The variations are primarily caused by the changing elevation angle as shown in Figure 9b. Figures 9e and 9f show the corrected long-term atmospheric delay for the DPD, Δτatm, which is calculated by using the formula Δτatm = (τatm−R−ISτatm−V−IS) − (τatm−R−IRτatm−V−IR). The corrected long-term atmospheric delay Δτatm varies over the range −0.01 to 0.011 m for the Ishigaki-Iriki baseline, and from −0.05 to 0.18 m for the Hobart-Ishigaki baseline. These values are typical of those obtained for correcting the long-term atmospheric delay in the DPD. These results show that long-term atmospheric delays along the line of sight as large as several meters are reduced to level of several centimeters through the same-beam VLBI observations.

image

Figure 9. (a) The zenith total delay (ZTD), (b) the elevation angles for Rstar at three stations and the separation angle between Rstar and Vstar, (c) the atmospheric delay in the line of sight for Rstar, (d) the ionospheric delay in the line of sight for Rstar, and (e and f) the corrected atmospheric and ionospheric delay for the DPD and the DPD on two baselines.

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[22] When both S band and X band same-beam observations are performed, the ionospheric delay along the line of sight, Δτion−i, can be directly calculated from the S band and X band DPDs. For small separation angles, Δτion−i for S band has a bias of about 1 mm and an RMS variation of 0.6 mm. However, because the X band DPD cannot be obtained in S band only same-beam observations, and the error in Δτion−i calculated from S3 and S1 DPDs is large, we have used GPS techniques to correct the long-term ionospheric delay.

[23] The VLBI array used for these observations includes four international stations and also the Ogasawara and Ishigaki stations which are located on isolated islands and because of this we used the CODE (Center for Orbit Determination in Europe) Global Ionosphere Model (GIM) to determine the long-term ionospheric delay correction [Hernadez-Pajares et al., 2008]. On the global scale, the ionospheric TEC distribution and its variation is monitored by the worldwide GPS network and is modeled using an expansion of spherical harmonic functions. This model is released to the public in the format of spherical harmonic function coefficients. Since 2000, the GIM has been provided with 15 deg and orders at 2 hourly intervals. We calculated the slant ionospheric delay with a time interval of 1 min by smoothly interpolating the global TEC model [Ping et al., 2002]. Figure 9d shows the calculated long-term slant ionospheric delays τion−S1 at S1 for the paths of Rstar to the three VLBI stations. This varied from 0.03 to 0.11 m at Hobart, from 0.84 to 1.38 m at Ishigaki, and from 1.05 to 1.42 m at Iriki during this observation period. The corrected long-term ionospheric delay for the DPD, Δτion−S1, which is calculated using the same formula as that used for correcting Δτatm, is shown in Figures 9e and 9f. It varies over a range from −0.0032 to 0.0015 m for the Ishigaki-Iriki baseline, and from −0.0068 to 0.008 m for the Hobart-Ishigaki baseline. These results suggest that most long-term ionospheric delay can be canceled in same-beam VLBI observations, and the S band Δτion−i was smaller than the condition of 0.035 m in Table 1, which is required to resolve the cycle ambiguity. Note that the sunspot number was relatively low from November 2007 to February 2009, and ionospheric fluctuations were relatively small. When the sunspot number becomes large, it may become relatively difficult to resolve the cycle ambiguity.

6. Contribution of VLBI Data to Orbit Determination and Lunar Gravity Model

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Same-Beam VLBI Observations in SELENE
  5. 3. Principle for Obtaining Differential Phase Delay
  6. 4. Observation Results
  7. 5. Error Analysis and Correction of Atmospheric and Ionospheric Delays
  8. 6. Contribution of VLBI Data to Orbit Determination and Lunar Gravity Model
  9. 7. Conclusion
  10. References
  11. Supporting Information

[24] The orbit determination of Rstar and Vstar is performed by using the software GEODYN II/SOLVE [Pavlis et al., 2007; Ullman, 1994]. The new lunar gravity model SGM100g (SELENE Gravity Model), has been determined using SELENE four-way and two-way Doppler and range data obtained before October 2008, combined with historical data such as that from Lunar Orbiters I to V, Apollo 15 and 16, Clementine, LP, and SMART-1. To confirm the positive contribution of VLBI DPD data, we compared orbit determination obtained using only Doppler and range data with that using Doppler and range and S band VLBI DPD data [Goossens et al., 2009]. The atmospheric and ionospheric delays were corrected for the DPD (Δτi) data by using Δτi − Δτatm + Δτion−i as discussed above. Data weights for Doppler, range, and DPD were 0.2 mm/s, 0.7 m and 10 mm, respectively.

[25] The data coverage is shown in Figure 10. The total time spans are 328 min for the two-way Doppler and range data for Rstar, 163 min for Vstar, and 765 min for the DPD data. Orbit errors were evaluated through overlap analysis, where the orbit differences between two overlapping arcs are computed. The two arcs are also shown in Figure 10. Arc 1 spans the time range from 0758 UT, 21 May to 2000 UT, 23 May, while arc 2 covers the interval from 1550 UT, 23 May to 1330 UT, 27 May. The overlapping period is 4.17 h.

image

Figure 10. The data coverage and overlapping arcs used to test the orbit determination.

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[26] The RMS of the orbit differences during the overlap is shown in Table 2. It shows that the orbit consistency is greatly improved by including the DPD data. Without the DPD data, the root-sum-square orbit consistency for Rstar is 70.11 m, and that for Vstar is 50.86 m. Including VERA four-station DPD data, the root-sum-square consistency is improved to 18.99 m for Rstar and 10.75 m for Vstar, and including all eight-station DPD data, the root-sum-square consistency is further improved to 11.11 m for Rstar and 7.62 m for Vstar. The individual directions (radial, along, and cross) may not show clear improvement when the DPD data are included for a variety of reasons. The DPD data are most sensitive in the along and cross track directions, so depending on how the data are weighted, consistency in the radial direction may be reduced (this is the case for Vstar in Table 2). The reduction of cross track consistency for Rstar in Table 2 is thought to be due to the combined affects of data distribution and data weight. However, it should be kept in mind that the improvement in the along track is of a larger order than the reduction of consistency in the cross track, so that in the overall sense, the orbit consistency as expressed in the root-sum-square value, is better when DPD data are included. In addition, while initially Rstar and Vstar showed a different level of orbit consistency when only Doppler and range data were used, their consistency converges to a similar level when DPD data are used. Summarizing, these results show that the international baseline DPD data further improve the accuracy of the orbit determination as expected, clearly confirming the positive contribution of VLBI data to orbit determination.

Table 2. Overlap Results for Rstar and Vstar for Different Data Types
SatelliteData TypeRadial (m)Along (m)Cross (m)Total (m)
RstarDoppler/range29.4363.575.7670.11
 Doppler/range + VERA four-station DPD4.1217.595.8418.99
 Doppler/range + eight-station DPD2.736.888.2811.11
VstarDoppler/range0.8350.772.9850.86
 Doppler/range + VERA four-station DPD3.3110.200.7510.75
 Doppler/range + eight-station DPD3.706.660.177.62

[27] We have also confirmed that the inclusion of VLBI data improves a lunar gravity model. To demonstrate this we compared the following two lunar gravity models. One is obtained using four-way and two-way Doppler and range data of SELENE with the same analysis setup for the force modeling and the data weights as were used by Namiki et al. [2009], but with the data of SELENE nominal mission phase until October 2008 and slightly longer R/Vstar mean arc length of 3.2 days. The other additionally includes S band DPD data for 4 months of January and March–May 2008 with data weight of 1 cm. In order to evaluate lunar gravity model, we use the coefficient sigma degree variance σn, which is defined as

  • equation image

where σ(equation imagenm) and σ(equation imagenm) are the errors of the normalized Selenopotential coefficients of degree n and order m of spherical harmonics [Kaula, 1966; Matsumoto et al., 2008]. Figure 11 shows the coefficient sigma degree variance with and without SELENE VLBI data included in the model. As shown, σn below about degree 10 is reduced for the model including VLBI data, especially for the second degree. The contribution of VLBI data to improving low-degree coefficients was also expected in the prelaunch simulation study [Matsumoto et al., 2008]. Although the present result is still preliminary and the improvement is not so dramatic, the coefficient sigma degree variance will be further reduced when all the VLBI data over the 15.4 months of observations are used and longer arc lengths for R/Vstar are achieved.

image

Figure 11. The coefficient sigma degree variances with and without SELENE VLBI data included in the lunar gravity model.

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[28] The improvement of the low-degree lunar gravity field model places strong constraints upon the deep internal structure of the Moon. Accompanying the improvements in the lunar gravity model which will be achieved by using more SELENE VLBI data, the accuracy of orbit determination for Rstar, Vstar and Main satellite will also be further improved.

7. Conclusion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Same-Beam VLBI Observations in SELENE
  5. 3. Principle for Obtaining Differential Phase Delay
  6. 4. Observation Results
  7. 5. Error Analysis and Correction of Atmospheric and Ionospheric Delays
  8. 6. Contribution of VLBI Data to Orbit Determination and Lunar Gravity Model
  9. 7. Conclusion
  10. References
  11. Supporting Information

[29] SELENE same-beam VLBI observations have been performed for 15.4 months from November 2007 to February 2009 using eight VLBI stations. The VLBI observation method was changed from March 2008 onward so that the telescopes tracked the midpoint of Rstar and Vstar when the separation angle between the two satellites was less than 0.56 deg. S band same-beam VLBI observations with a total duration of 476 h spread over 179 days were undertaken in 2008. The differential phase delays have been successfully estimated in about 85% of the same-beam VLBI observation periods, which is mainly due to the continuous data series of differential correlation phase. The measurement error achieved is less than 1 mm RMS for small separation angles and about 2.5 mm RMS for larger separation angles. The long-term atmospheric delay was reduced to a level of several centimeters through the same-beam VLBI observations, and further corrected using GPS techniques. The long-term ionospheric delay was reduced to a level of several millimeters, which satisfy the condition for resolving the cycle ambiguity. The accuracy of orbit determination was improved from a level of tens of meters when using only Doppler and range data to a level of 10 m when the same-beam VLBI data are also included. The coefficient sigma degree variance below around 10 deg was reduced for the lunar gravity field model when 4 month VLBI data are included especially for the second degree. This confirms the positive contribution of VLBI data to improving the accuracy of orbit determination and the lunar gravity field model. A new lunar gravity field model will be published after including all 15.4 month VLBI data.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Same-Beam VLBI Observations in SELENE
  5. 3. Principle for Obtaining Differential Phase Delay
  6. 4. Observation Results
  7. 5. Error Analysis and Correction of Atmospheric and Ionospheric Delays
  8. 6. Contribution of VLBI Data to Orbit Determination and Lunar Gravity Model
  9. 7. Conclusion
  10. References
  11. Supporting Information
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Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Same-Beam VLBI Observations in SELENE
  5. 3. Principle for Obtaining Differential Phase Delay
  6. 4. Observation Results
  7. 5. Error Analysis and Correction of Atmospheric and Ionospheric Delays
  8. 6. Contribution of VLBI Data to Orbit Determination and Lunar Gravity Model
  9. 7. Conclusion
  10. References
  11. Supporting Information
FilenameFormatSizeDescription
rds5708-sup-0001-t01.txtplain text document0KTab-delimited Table 1.
rds5708-sup-0002-t02.txtplain text document0KTab-delimited Table 2.

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