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Journal of Geodesy

, Volume 93, Issue 11, pp 2421–2428 | Cite as

Assessment of the impact of one-way laser ranging on orbit determination of the Lunar Reconnaissance Orbiter

  • Anno LöcherEmail author
  • Jürgen Kusche
Original Article
  • 384 Downloads

Abstract

From June 2009 to September 2014, ten stations of the International Laser Ranging Service acquired more than 5000 h of one-way ranging data to NASA’s Lunar Reconnaissance Orbiter (LRO). The LRO campaign was intended to support the determination of precise science orbits for which S-band Doppler and range measurements are the primary source. The laser ranges were finally not used for the NASA orbits, since the quality of the radiometric data exceeded the expectations. In this contribution, we explore the potential of the LRO laser ranges using an independent software which is able to process all types of LRO tracking data. It is shown that the laser ranges agree with the radiometric data at the level of 5 cm, provided a proper modeling of the LRO clock error is done, including relativistic effects. This accuracy does not fully attain that of the Doppler data, hence the impact of the laser ranges on orbit determination is small, improving the orbit on average by no more than 7 cm. Additional tests without the Doppler data show, however, that the laser ranges would have been of great use if radiometric tracking to LRO were less successful.

Keywords

Lunar Reconnaissance Orbiter Laser ranging Precise orbit determination 

1 Introduction

The Lunar Reconnaissance Orbiter (LRO, Chin et al. 2007) was launched in 2009 as a pathfinder mission for the resumption of manned lunar spaceflights considered at that time as a priority in the US space programme. Its main objective is the detailed survey of the lunar topography required for the identification of future landing sites. This survey is accomplished by photographic images taken by the Lunar Reconnaissance Orbiter Camera (LROC, Robinson et al. 2010) and by altimetric ranges collected by the Lunar Orbiter Laser Altimeter (LOLA, Smith et al. 2010). The LRO instrument suite is completed by further devices measuring the lunar surface temperature or looking for signs of water ice.

Some of the LRO instruments have a spatial resolution in the range of meters; the precision of the LOLA ranges even is specified with 10 cm. To retain as much as possible of these accuracies in the derived maps and terrain models, highly accurate and consistent orbits are required. The primary source for the orbits are S-band Doppler and range measurements acquired by an LRO-dedicated station at White Sands, New Mexico, and four stations in Hawaii, Australia and Europe associated with the commercial Universal Space Network (USN). The initial strategy of NASA also included to constrain the orbits by altimetric crossovers as practiced successfully in the Mars Global Surveyor mission. Even the first release of the NASA science orbits thus attained a precision of 20 m in total position and 1 m radially, both assessed by an analysis of overlaps of adjacent arcs (Mazarico et al. 2012). After a reprocessing in 2013, the precision is reported to be around 10 and 0.5 m, respectively (Mazarico et al. 2013, 2017), the improvement achieved by using a lunar gravity field from the Gravity Field and Interior Laboratory (GRAIL, Zuber et al. 2013) mission.

The performance of the LRO radio network was initially not believed to enable such accurate orbits. It was decided, therefore, to extend the LOLA capabilities to acquire additional tracking data from the Earth. To this end, a laser optics was mounted onto the LRO radio antenna and connected to the LOLA detector by a fiber optic cable. The spacecraft was thus equipped for one-way ranging from laser stations which are capable to reach the Moon. The first laser shot successfully received from Earth was from the station GO1L located at NASA’s Goddard Space Flight Center (GSFC) on June 30, 2009, less than two weeks after LRO’s launch (McGarry et al. 2013). In the following months, nine more stations joined the laser tracking campaign to LRO which was pursued over more than five years until September 2014.

Since the radiometric data turned out to be better than anticipated, the laser ranges to LRO were never used for the official NASA orbits. The LRO laser campaign thus produced de facto an experimental data set for investigating the properties of a novel data type. As such, the LRO laser ranges were intensively studied by various research groups. A topic repeatedly addressed is their use for characterizing the LRO clock behavior and the related problem of time transfer from station clock to spacecraft clock (Bauer et al. 2017; Mao et al. 2017). Most research, of course, was focused on the potential of the laser ranges for orbit determination. In Bauer et al. (2016) this was studied exclusively with orbits from laser ranges alone, while Buccino et al. (2016) and Mao et al. (2017) also computed orbits with mixed data sets from radio and laser observations. In all studies, the quality of the orbits is assessed by comparisons with the NASA science orbits. For the laser-only orbits, the differences reported vary strongly, from 7 m averaged from four arcs with a length of 1.5 days (Buccino et al. 2016) to 50 m based on 58 arcs each spanning two weeks (Mao et al. 2017). There is also no consensus on how the laser ranges contribute to the mixed solution. Buccino et al. (2016) observe an improvement in the orbit by 2 m, Mao et al. (2017) a degradation by 6 m, though the radial direction is improved by 0.5 m. All these results are affected by considerable uncertainties, since the orbits compared for these assessments differ themselves by 20 m and more from the NASA orbits.

In this work, the laser ranges are studied over the full period and without referring to orbits of external origin. The investigation builds on a set of radio-only orbits computed independently from NASA which is described in detail in Löcher and Kusche (2018). These orbits are close to the NASA science orbits, the averaged difference being 5.56 m in total position and 0.42 m radially. The overall precision, assessed by overlap errors, was found to be 2.81 m and 0.11 m, respectively. The orbits are based on both types of radiometric data, the data weighting being accomplished by a variance component estimation (VCE) on a pass-by-pass basis. To allow for such a procedure, the Doppler range-rates are integrated to biased ranges so that all observations have the same unit of meters. It is made easy by this policy to introduce the laser data as a third type of ranges. The data weights can then again be adjusted by a VCE now applied to all passes of the augmented data set.

The main challenge in using the laser ranges consists in the proper modeling of this observation type. The issues to solve are mostly related to the LRO clock which is affected both by the errors of its oscillator and the relativistic time shift observable for a moving clock in space. In this paper, both these disturbances are treated in a pragmatic way. Since we are interested primarily in precise orbits, no attempt is made to separate the error of the LRO clock from those of the station clocks. The modeling rather aims at the resulting error by estimating one clock polynomial per pass. For the relativistic time shift, a model for near-Earth space is used whose applicability for LRO will be demonstrated.

The paper proceeds in Sect. 2 with the detailed development of the observation model. In Sect. 3, the laser ranges and the chosen model are studied in residual analyses along our radiometric orbits. Section 4, finally, presents the results of orbit determination from the mixed data set. This section also includes some additional computations to assess the accuracies obtainable with less accurate radiometric data.

2 Modeling of the laser data

The laser ranging to LRO is technically a one-way uplink ranging. A laser station transmits laser pulses to the spacecraft and records the transmit times referred to the station clock which is synchronized with UTC. The pulses received by the laser optics and forwarded to the LOLA detector trigger again clock readings, now in the time system of the LRO clock denominated as Mission Elapsed Time (MET). The receive times are later transmitted to Earth via the science telemetry link and matched with the transmit times using preliminary solutions for the LRO orbit and the lag between the time systems involved (Mao et al. 2017). To reduce the data amount and improve the data quality, the matched time tags are finally condensed to normal points with a nominal sampling rate of 5 s.

The normal points stored for public access in the LRO Radio Science Archive are the input data for the present work. As apparent from the above, each of them consists of a station transmit time \(\tau _\mathrm{STT}\) in UTC and a spacecraft receive time \(\tau _\mathrm{LRO}\) in the spacecraft timescale MET. Forming the difference of the time tags results in a raw one-way runtime which can be converted with the light velocity c into a raw one-way range
$$\begin{aligned} \tilde{\rho } = c(\tau _\mathrm{LRO} - \tau _\mathrm{STT}) . \end{aligned}$$
(1)
For use as tracking observation, \(\tilde{\rho }\) must be related to the geometrical range \(\rho \), the euclidean distance between the laser’s reference point at transmit time and the center of mass of LRO when receiving the pulse. Both range types differ by a variety of disturbing quantities to be summarized as follows:
$$\begin{aligned} \tilde{\rho } = \rho + \Delta _\mathrm{Tropo} + \Delta _\mathrm{CoM} + (\Delta _\mathrm{Clock}^\mathrm{LRO}-\Delta _\mathrm{Clock}^\mathrm{STT}) + \Delta _\mathrm{Rel} + \delta .\nonumber \\ \end{aligned}$$
(2)
Here, \(\Delta _\mathrm{Tropo}\) denotes the tropospherical delay of the laser pulse, \(\Delta _\mathrm{CoM}\) a center-of-mass correction, \(\Delta _\mathrm{Clock}^\mathrm{LRO}\) and \(\Delta _\mathrm{Clock}^\mathrm{STT}\) the errors of the LRO clock and the station clock, respectively, and \(\Delta _{Rel}\) the sum of relativistic effects. The \(\delta \) term represents all remaining systematical effects and random errors.

The first two corrections are familiar from SLR analysis to Earth satellites and can be modeled with high accuracy. For the tropospherical correction, we use the standard model Mendes and Pavlis (2004). The center-of-mass correction is computed using the orientation data for the spacecraft and the antenna to which the laser telescope is mounted. The antenna boom is modeled with two hinges and its dimensions derived from the drawings in Tooley (2009). For both corrections, a coarse position of LRO is needed which is taken from the navigation orbit from the GSFC Flight Dynamics Facility.

The clock errors \(\Delta _\mathrm{Clock}^\mathrm{LRO}\) and \(\Delta _\mathrm{Clock}^\mathrm{STT}\) are items which appear exclusively in one-way ranging. Since the knowledge of these individual errors is of minor interest in this study, they are merged to one error
$$\begin{aligned} \Delta _\mathrm{Clock} = \Delta _\mathrm{Clock}^\mathrm{LRO}-\Delta _\mathrm{Clock}^\mathrm{STT} . \end{aligned}$$
(3)
The merged error is modeled by a polynomial of degree 2 estimated separately for each pass. This choice is mainly motivated by the limited stability of the LRO clock which is far below that of the atomic clocks in use at the stations. The degree 1 coefficients will thus show, above all, the drift of the LRO clock, the degree 2 coefficients its drift rate. The constant term of the polynomial can be interpreted as the lag between UTC and MET at the beginning of the pass.
The last correction in Eq. 2, the relativistic term \(\Delta _{Rel}\), also needs some discussion. Indeed, there are two effects at work:
$$\begin{aligned} \Delta _\mathrm{Rel}= \Delta _\mathrm{Rel}^\mathrm{SHP} + \Delta _\mathrm{Rel}^\mathrm{CLK}. \end{aligned}$$
(4)
The first contribution \(\Delta _\mathrm{Rel}^\mathrm{SHP}\) is the delay of an electromagnetic signal caused by the gravitation of celestial bodies. This effect is described by the well-known Shapiro term (Petit and Luzum 2010)
$$\begin{aligned} \Delta _\mathrm{Rel}^\mathrm{SHP} = \sum _{J} \frac{2GM_{J}}{c^{2}} ln \left( \frac{r_{J1} + r_{J2} + \rho }{r_{J1} + r_{J2} - \rho }\right) , \end{aligned}$$
(5)
where \(GM_{J}\) denotes the gravitational constant of the celestial body J, \(r_{J1}\) the distance from the body to the emitting station, \(r_{J2}\) the distance to the receiver, and \(\rho \) the length of the light path.

The second effect \(\Delta _\mathrm{Rel}^\mathrm{CLK}\) is again specific for one-way ranging. It arises from the fact that the clocks involved in one-way ranging move on different paths in space. According to relativity, such clocks have different proper times. Consequently, a range computed from two readings of the clocks is wrong by an offset continuously changing according to the motion of the clocks and the surrounding masses.

This relativistic clock error has been treated differently in previous studies. While it is neglected in Mao et al. (2017), it is accounted for in Buccino et al. (2016) and Bauer et al. (2017), but computational details are given only in the latter reference. The strategy applied there is to refer both clocks to Barycentric Dynamical Time (TDB). For the station clocks, this is done via Terrestrial Time (TT) using the conventional relation between TT and TDB with periodical terms given by Moyer (1981). The proper time of the LRO clock is transformed via Barycentric Coordinate Time (TCB) which differs from TDB only by a constant rate. The transformation from proper time to TCB is accomplished by numerically solving a differential equation given by Moyer (1971).

Though this approach is the recommended one for synchronizing clocks in interplanetary space, we found it more appropriate for LRO to solve the problem within the Earth–Moon system. Instead of the barycentric time TDB, we thus use Geocentric Coordinate Time (TCG) as reference time scale. The station clocks are transformed to that scale again via TT from which TCG deviates by a constant rate \(L_{G}\) (Petit and Luzum 2010):
$$\begin{aligned} \frac{\mathrm{d}t_\mathrm{TT}}{\mathrm{d}t_\mathrm{TCG}} = 1 - L_{G}. \end{aligned}$$
(6)
For the LRO clock, the transformation can be based on a differential equation from, e.g., Petit and Wolf (2005) which directly connects proper time \(\tau \) and TCG:
$$\begin{aligned} \begin{aligned} \frac{\mathrm{d}\tau }{\mathrm{d}t_\mathrm{TCG}}&= 1 - \frac{1}{c^{2}} \Bigl (\Bigr . \frac{\mathbf {v}^{2}}{2} + U_{E}(\mathbf {x}) \\&\quad \,\,+\, V(\mathbf {X}) - V(\mathbf {X}_{E}) - x^{i} \partial _{i} V(\mathbf {X}_{E}) \Bigl .\Bigr ). \end{aligned} \end{aligned}$$
(7)
Here, \(\mathbf {x}\) and \(\mathbf {v}\) denote the geocentric position and velocity of the clock, and \(\mathbf {X}\) and \(\mathbf {X}_{E}\) the barycentric position of the clock and the center of the Earth, respectively. \(U_{E}\) is the potential of the Earth, V the sum of the potentials of the other bodies including the Moon.
Table 1

Magnitude of laser range residuals along radiometric orbits for different setups

Case

Orbit

Relativity applied

Drift rate adjusted

RMS of residuals (m)

1

IGG

No

No

0.48

2

IGG

No

Yes

0.06

3

IGG

Yes

No

0.06

4

IGG

Yes

Yes

0.05

5

NASA science

Yes

Yes

0.10

6

NASA navigation

Yes

Yes

0.15

RMS values averaged from 10536 passes, pass RMS outlier-cleaned with threshold 100 m

Integrating \((\mathrm{d}\tau /\mathrm{d}t_\mathrm{TCG} - 1)\) from a certain time \(t_{0}\) yields the deviation of the LRO clock from TCG up to a constant. Since UTC runs in parallel with TT (except when a leap second is inserted), the deviation of the station clock can be obtained in a similar way by integrating \((\mathrm{d}t_\mathrm{TT}/\mathrm{d}t_\mathrm{TCG} - 1) = -L_{G}\). The effect on the range at time \(t_{i}\) is then given by the difference of the integrals scaled by the light velocity:
$$\begin{aligned} \Delta _\mathrm{Rel}^\mathrm{CLK} = c \int _{t_{0}}^{t_{i}} \left( \frac{\mathrm{d}\tau }{\mathrm{d}t_\mathrm{TCG}} -1 + L_{G} \right) \mathrm{d}t . \end{aligned}$$
(8)
The core of this model, Eq. (7), is recommended for altitudes “up to geosynchronous orbit or slightly above” (Petit and Wolf 2005). This recommendation is clearly violated by our use. The results presented in the following indicate, however, that this approach is still valid at much larger distances, at least at the level of accuracy needed in the present case. In line with the handling of the clock problem, we also compute the Shapiro delay (Eq. 5) for geocentric spacetime, considering only the Earth as disturbing body, not the Sun as recommended for Lunar Laser Ranging analyzed in a barycentric frame (Petit and Luzum 2010).

It should be noted, however, that our LRO processing is quite different from that of SLR to Earth satellites. The orbit computation is performed in a Moon-centered reference frame, since LRO moves in a Moon-centered gravity field. To achieve this change of frame, the station vectors associated to the ranges are reduced by the geocentric position of the Moon. For this, the lunar ephemeris from DE421 (Williams et al. 2008) is used, adapted to the geocentric frame by applying the transformation between TT- and TDB-compatible coordinates given in Petit and Luzum (2010). In order to be consistent with the time scale of the ephemeris, the time tags of the ranges are finally transformed to TDB.

3 Residual analyses along radiometric orbits

To investigate the properties of the laser ranges and the suitability of the observation model developed, we first consider the residuals along our orbits from radiometric data. The analysis is carried out with different setups: with and without applying the relativistic clock correction and with different degrees of the clock polynomial. In the first setup, the observation model is simplified by omitting both the relativistic correction and the clock drift rate as estimable parameter (Case 1). Another two setups each keep one of these features, Case 2 the clock drift rate (omitting relativity), and Case 3 the relativistic correction (with no drift rate adjusted). Case 4 implements the complete model from the previous section.

The results of these analyses given in Table 1 suggest that the laser ranges are in line with the radiometric orbits at a level of 5–6 cm. This level, however, is not attainable without neutralizing the relativistic clock error. With the most simplified model used in Case 1, the residuals are on average as large as 48 cm. This value is reduced by 42 cm by applying the relativistic clock correction (Case 3), and by exactly the same amount when adjusting the drift rate for the clock (Case 2); the complete model applied in Case 4 provides only a slight additional improvement, reducing the residuals just by 1 cm more. All these results together demonstrate quite impressively that the relativistic model based on Eq. (7) is appropriate for LRO, but that, on the other hand, the relativistic effect can be absorbed by a clock polynomial of degree 2 with comparable efficiency.
Fig. 1

Coefficients of clock polynomials adjusted in residual analyses in units of clock parameters. From top to bottom: clock bias, clock drift, clock drift rate. Blue: no relativistic clock correction applied (Case 2), red: relativity applied (Case 4)

Table 2

Observation volumes of the laser stations and station-averaged residuals. Data statistics based on the start/end time tags of normal point files as listed in index.tab in the LRO Radio Science Archive. Residuals from Case 4 in Table 1

Station

Location

Time observed

RMS of residuals (m)

  

h

%

 

MDOL

McDonald, Texas, USA

638.3

13.3

0.05

YARL

Yarragadee, Australia

836.8

16.2

0.04

GODL

Greenbelt, Maryland, USA

418.5

8.1

0.04

MONL

Monument Peak, California, USA

1375.2

26.7

0.06

GO1L

Greenbelt, Maryland, USA

1530.3

29.7

0.06\(^\mathrm{a}\)

HARL

Hartebeesthoek, South Africa

50.0

1.0

0.04

ZIML

Zimmerwald, Switzerland

81.5

1.6

0.02

HERL

Herstmonceux, UK

70.9

1.4

0.11

GRSM

Grasse, France

97.1

1.9

0.03

WETL

Wettzell, Germany

9.2

0.2

0.02

\(^\mathrm{a}\)Until October 2010: 0.08 m, after October 2010: 0.05 m

Figure 1 allows to watch how this effect is absorbed by the clock parameters. The diagrams display the coefficients of the degree 2 polynomial obtained with the uncorrected ranges (Case 2, blue) and with the ranges reduced by the relativistic model (Case 4, red). In the first case, the estimates for the drift rate are spread almost uniformly in a range of \(\pm 9 \times 10^{-15} \; \text {s}/\text {s}^{2}\) (Fig. 1, bottom). A closer look reveals that the estimates ascend in periods of 14 days from the lower to the upper boundary and then suddenly drop down by a change of sign. This conspicuous pattern is absent when the relativistic model is applied. It can thus be identified as the fingerprint of the relativistic clock error—one of the rare signals from relativity which can be isolated unambiguously from satellite data.

The lower degrees of the clock polynomial are less affected by relativity. The clock bias, almost identical in both cases, is governed by the drift of MET with respect to UTC caused by the drift of the LRO clock (Fig. 1, top); the discontinuities occurring are possibly due to offsets applied to MET during the pairing of transmit and receive times (Mao et al. 2017). The slope of the bias between these discontinuities allows to assess the clock drift to be around \(-7 \times 10^{-8} \; \text {s}/\text {s}\). This is confirmed and specified by the dedicated estimates for the clock drift given by the linear term of the polynomial (Fig. 1, center). The shift between the time series depicted is due to the change of the reference time scale in Case 4 and equals the rate \(L_{G}\, (6.97 \times 10^{-10}\), Petit and Luzum 2010) included in the relativistic model.

While our modeling does not allow any statement about the behavior of the station clocks, the overall performance of the stations can be assessed by further analyzing the residuals. Table 2 summarizes their averages for the stations, together with the volumes of observation time. The largest residuals (in a rather homogeneous field) are found for the station Herstmonceux, probably due to a peculiarity in the emitted pulse sequence which is known to cause problems for the LOLA detector (Mao et al. 2017). The most favorable averages result for Zimmerwald, Grasse and Wettzell, in the last case, however, on a poor statistical basis. Another hardware-related issue can be observed at the station GO1L at GSFC, the most productive one in the network. For about one year, until October 2010, the residuals found for this station are on average 8 cm, then decrease to 5 cm. This coincides with the change of the station’s time standard from a cesium to a H-maser clock (Mao et al. 2017), which has obviously led to much more precise records of the transmit times.

Two final analyses will show that the residuals reveal much less about the orbit. For this, the residual computation is carried out along the NASA science orbits (Case 5 in Table 1) and the NASA navigation orbits (Case 6). In the former case, the total average is 10 cm, an increase of 5 cm with respect to our orbits. This increase does not reflect at all the actual difference between the orbits which is 5.56 m (see Sect. 1). The mismatch is even larger with the navigation orbits along which the residuals are on average 15 cm, while the difference to our orbits is 74.70 m. It is obvious that, in both cases, most of the difference is absorbed by the clock polynomials. The residuals are thus nearly blind for large-scale orbit errors which makes them unusable for orbit validation.

4 Orbit determination with combined data set

The size of the residuals found in the previous section may suggest that the laser ranges should be of great use in orbit computation. Indeed, when merged with the radiometric data, they compete with other precise observations which are typically collected in parallel at locations close to the laser stations. Moreover, the radiometric data clearly prevail over them only through their volume. Though the laser network succeeded on many days in observing all possible passes, the observation time during the five years did not exceed 5130 h, while the radio stations acquired 16,567 h. The ratio of data volumes is thus 1 : 3.2, or even 1 : 6.4 when considering that the radio stations produce two types of observations at each epoch.

Against this background, moderate expectations are appropriate for the orbits from the combined data set. As mentioned above, such orbits can easily be computed based on the setup for our radiometric orbits, using the models and the parameter set described in Löcher and Kusche (2018). For the combined solution, the laser data, processed and parametrized as developed in Sect. 2, are added to the observation equations with individual weights according to the accuracies provided with the normal points. The weights of all observation types are refined during the orbit determination process by adjusting variance components for the passes. This procedure also is inherited from the setup for our radio orbits where it has proven to be indispensable to cope with the wide range of quality found in LRO tracking data.
Table 3

Mean RMS of overlap errors of orbit arcs from radio and laser observations, in meters. Nominal arc length is 2.5 days. In italics: difference to radio-only orbits

Mission phase\(^\mathrm{a}\)

Total

Along

Cross

Radial

CO

4.21

3.15

2.31

0.21

 

+0.04

+0.03

+0.01

0.00

NO

2.02

1.45

1.10

0.06

 

0.00

0.00

+0.01

0.00

SM1

2.04

1.50

1.13

0.08

 

0.34

0.34

0.01

0.00

SM2

3.19

2.69

1.19

0.13

 

0.00

+0.01

0.00

0.00

ES1

2.91

2.37

1.23

0.14

 

0.01

0.01

0.01

0.00

ES2

3.49

2.76

1.54

0.13

 

0.01

0.01

0.00

0.00

All phases

2.74

2.15

1.27

0.11

 

0.07

0.08

0.01

0.00

\(^\mathrm{a}\)Mission phases as defined in Löcher and Kusche (2018). CO: June 23, 2009–September 6, 2009, NO: September 6, 2009–September 16, 2010, SM1: September 16, 2010–December 11, 2011, SM2: December 11, 2011–September 15, 2012, ES1: September 15, 2012–September 18, 2013, ES2: September 18, 2013–September 18, 2014

Though the weights from the normal points are in most cases strengthened by the VCE procedure, the overall impact of the laser ranges is, as expected, small. As shown in Table 3, a substantial improvement of the orbit precision can be observed only in the mission phase SM1, the mean overlap error in this phase decreasing by 34 cm. A closer analysis shows that this improvement is due to one single arc, in which laser ranges fill a gap in the radiometric data and override large colored noise in nearby Doppler passes. The positions within the arc thus are shifted on average by 26 m, and the differences to the adjacent arcs decrease by 25 and 22 m, respectively. These two figures nearly account for the whole improvement by the laser ranges, which finally amounts to 7 cm when averaged over the whole mission.
Fig. 2

Pass accuracies from variance component estimation, plotted against the pass length. Top: Doppler ranges, center: two-way ranges, bottom: laser ranges

As stated above, the limited impact of the laser ranges is due to both the quality and the volume of the radiometric data. Figure 2 illustrates the first point showing the pass accuracies for the different range types assessed by the VCE. The lowest errors are attributed to the Doppler ranges, with an average of 3 cm, favored in part by the large number of bias parameters we adjust for this range type. The two-way ranges are, in contrast, the least accurate ones, with an averaged error of 16 cm, but with distinct differences between White Sands (10 cm) and the stations of the USN network (33 cm). The accuracy of the laser ranges, finally, is found to be 8 cm, roughly the mean of the other values. The laser ranges thus mostly outweigh the two-way ranges, but are outweighed themselves by the Doppler ranges. Accordingly, their influence is limited to situations where Doppler data are lacking or distorted by gross errors.
Table 4

Mean RMS of overlap errors from different range types and range type combinations

Range types

Arcs solved

Overlaps

Overlap error (m)

   

Total

Radial

Doppler

882

740

3.10

0.12

Doppler, two-way

883

740

2.81

0.11

Doppler, two-way\(^\mathrm{a}\)

704

427

12.76

2.98

Doppler, laser

880

737

3.01

0.12

Doppler, two-way, laser

883

740

2.74

0.11

Two-way

883

738

22.09

0.58

Two-way\(^\mathrm{b}\)

 

432

19.42

0.52

Two-way, laser

883

740

12.30

0.38

Two-way, laser\(^\mathrm{b}\)

 

432

6.09

0.23

Laser

637

391

82.81

3.25

Laser\(^\mathrm{c}\)

686

428

34.53

1.86

\(^\mathrm{a}\)Coverage reduced to periods with laser data

\(^\mathrm{b}\)Statistics considering only arcs with laser data

\(^\mathrm{c}\)No clock drift rate adjusted

This finding may appear disappointing, but it should be kept in mind that the laser ranging to LRO was intended to support radiometric data with an increased error level. According to the numbers given in Mazarico et al. (2012), the S-band tracking error was overestimated by a factor 3 during the mission planning stage. To learn a lesson about the LRO laser campaign, it would be instructive to know what the laser ranges would have contributed in that case. An approximate answer can be found when considering the errors of the radiometric range types as assessed above. A comparison of the figures shows that the errors of the two-way ranges exceed those of the Doppler ranges by a factor 3 at White Sands, and by a factor 5 on average for all stations. The two-way ranges in the quality provided should thus emulate rather well the Doppler ranges with the anticipated reduced accuracy. The resulting orbit precision can be found in Table 4 summarizing all solutions from the different range types and the combinations of them. With the noise of the two-way ranges, the orbit precision degrades from 3.10 to 22.09 m in total position and from 0.12 to 0.58 m in radial direction. When combining two-way and laser ranges, however, the degradation is only half the size, the precision achieved 12.30 and 0.38 m, respectively. Considering only the arcs for which laser data actually are provided, the precision is even 6.09 and 0.23 m. The orbit quality in these arcs is thus already close to that from the Doppler data with their actual noise, degraded only by a factor 2.

For completeness, Table 4 also includes the solution from laser ranges only. From a total of 883 arcs, only 637 arcs can be solved in this case, with an error of 82.81 m in total position and 3.25 m radially. These figures are reduced to 34.53 and 1.86 m by a more restrictive parametrization omitting the drift rate for the clock. Even these results do not match the supposed quality of the laser ranges, obviously due to the lesser density of the observations. This is easily verified with the radiometric ranges by trimming them exactly to the periods of the laser passes. Only due to the sparser coverage, the overlap errors then increase to similar orders, from 2.81 and 0.11 m obtained with the full data set to 12.76 and 2.98 m with the reduced data.

It is difficult to predict from that, which precision would be achieved, if the laser ranges had, inversely, the coverage of the radiometric data. Considering the accuracies assessed by the VCE, it is obvious, however, that it should lie between the precisions obtained with the two radiometric types. Assuming that, in this interval, the orbit error increases linearly with the range error, a prediction of 12 m for the position vector and 30 cm for the radial direction would be a reasonable choice.

5 Conclusions

With the one-way ranges to LRO, the ILRS network produced a set of tracking data which is unique both for the type of observation and for the distance of the spacecraft. It is shown in this paper that this data set is of very good quality, with a mean error of 5 cm, assessed in a residual analysis along independently derived radiometric orbits. To attain this level of agreement, a proper modeling of the range is needed which is complicated, in particular, by the relativistic delay of the LRO clock. It was demonstrated that this effect can be calculated conveniently using a model for near-Earth space, but that it can be absorbed with similar accuracy by a clock polynomial of degree 2.

The high quality of the laser data turns out to be less effective when compared with the radiometric observations. The laser ranges clearly outperform the radiometric ranges, but do not fully attain the accuracy of the Doppler data. Their impact on the orbits is thus small, limited to situations where they fill gaps in the Doppler data or compensate large Doppler noise. Averaged over the mission, the improvements achieved this way were found to be not larger than 7 cm.

This result does fully agree with the expectations since the LRO laser ranging was intended to support radiometric data with a larger error than actually observed. It could be shown that, in this case, the laser ranges would have contributed significantly to the orbit precision, reducing the effect of the additional noise by half. A further improvement would have been prevented by the lesser density of the laser data.

Notes

Acknowledgements

This research was funded by the German Research Foundation (DFG) within the research unit FOR 1503 “Space-Time Reference Systems for Monitoring Global Change and for Precise Navigation in Space.” We thank three anonymous reviewers for their helpful comments.

References

  1. Bauer S, Hussmann H, Oberst J, Dirkx D, Mao D, Neumann G, Mazarico E, Torrence M, McGarry J, Smith D, Zuber M (2016) Demonstration of orbit determination for the Lunar Reconnaissance Orbiter using one-way laser ranging data. Planet Space Sci 129:32–46.  https://doi.org/10.1016/j.pss.2016.06.005 CrossRefGoogle Scholar
  2. Bauer S, Hussmann H, Oberst J, Dirkx D, Mao D, Neumann G, Mazarico E, Torrence M, McGarry J, Smith D, Zuber M (2017) Analysis of one-way laser ranging data to LRO, time transfer and clock characterization. Icarus 283:38–54.  https://doi.org/10.1016/j.icarus.2016.09.026 CrossRefGoogle Scholar
  3. Buccino D, Seubert J, Asmar S, Park R (2016) Optical ranging measurement with a lunar orbiter: limitations and potential. J Spacecr Rockets 53:457–463.  https://doi.org/10.2514/1.A33415 CrossRefGoogle Scholar
  4. Chin G, Brylow S, Foote M, Garvin J, Kasper J, Keller J, Litvak M, Mitrofanov M, Paige D, Raney K, Robinson M, Sanin A, Smith D, Spence H, Spudis P, Stern S, Zuber M (2007) Lunar Reconnaissance Orbiter overview: the instrument suite and mission. Space Sci Rev 129:391–419.  https://doi.org/10.1007/s11214-007-9153-y CrossRefGoogle Scholar
  5. Löcher A, Kusche J (2018) Precise orbits of the Lunar Reconnaissance Orbiter from radiometric tracking data. J Geod.  https://doi.org/10.1007/s00190-018-1124-4 CrossRefGoogle Scholar
  6. Mao D, McGarry J, Mazarico E, Neumann G, Sun X, Torrence M, Zagwodzki T, Rowlands D, Hoffman E, Horvath J, Golder J, Barker M, Smith D, Zuber M (2017) The laser ranging experiment of the Lunar Reconnaissance Orbiter: five years of operations and data analysis. Icarus 283:55–69.  https://doi.org/10.1016/j.icarus.2016.07.003 CrossRefGoogle Scholar
  7. Mazarico E, Rowlands D, Neumann G, Smith D, Torrence M, Lemoine F, Zuber M (2012) Orbit determination of the Lunar Reconnaissance Orbiter. J Geod 86(3):193–207.  https://doi.org/10.1007/s00190-011-0509-4 CrossRefGoogle Scholar
  8. Mazarico E, Lemoine F, Goossens S, Sabaka T, Nicholas J, Rowlands D, Neumann G, Torrence M, Smith D, Zuber M (2013) Improved precision orbit determination of lunar orbiters from GRAIL-derived lunar gravity models. In: Proceedings of the 23rd AAS/AIAA space flight mechanics meeting, San Diego CA, pp 1125–1141Google Scholar
  9. Mazarico E, Neumann G, Barker M, Goossens S, Smith D, Zuber M (2017) Orbit determination of the Lunar Reconnaissance Orbiter: status after seven years. Planet Space Sci.  https://doi.org/10.1016/j.pss.2017.10.004 CrossRefGoogle Scholar
  10. McGarry J, Sun X, Mao D, Horvath J, Donovan H, Clarke C, Hoffman E, Cheek J, Zagwodzki T, Torrence M, Barker M, Mazarico E, Neumann G, Zuber M (2013) LRO-LR: four years of history making laser ranging. In: 18th international workshop on laser ranging, Fujiyoshida, JapanGoogle Scholar
  11. Mendes V, Pavlis E (2004) High-accuracy zenith delay prediction at optical wavelengths. Geophys Res Lett 31:L14602.  https://doi.org/10.1029/2004GL020308 CrossRefGoogle Scholar
  12. Moyer T (1971) Mathematical formulation of the Double-Precision Orbit Determination Program (DPDOP). NASA JPL CALTEC Technical Report 32-1527Google Scholar
  13. Moyer T (1981) Transformation from proper time on earth to coordinate time in solar system barycentric space-time frame of reference. Part 1. Celestial Mech 23:33–56CrossRefGoogle Scholar
  14. Petit G, Luzum B (2010) IERS conventions. IERS Technical Note 36. Frankfurt am Main: Verlag des Bundesamts für Kartographie und GeodäsieGoogle Scholar
  15. Petit G, Wolf P (2005) Relativistic theory for time comparisons: a review. Metrologia 42:138–144.  https://doi.org/10.1088/0026-1394/42/3/S14 CrossRefGoogle Scholar
  16. Robinson M, Brylow S, Tschimmel M, Humm D, Lawrence S, Thomas P, Denevi B, Bowman-Cisneros E, Zerr J, Ravine M, Caplinger M, Ghaemi F, Schaffner J, Malin M, Mahanti P, Bartels A, Anderson J, Tran T, Eliason E, McEwen A, Turtle E, Jolliff B, Hiesinger H (2010) Lunar Reconnaissance Orbiter Camera (LROC) instrument overview. Space Sci Rev 150:81–124.  https://doi.org/10.1007/s11214-010-9634-2 CrossRefGoogle Scholar
  17. Smith D, Zuber M, Jackson G, Cavanaugh J, Neumann G, Riris H, Sun X, Zellar R, Coltharp C, Connelly J, Katz B, Kleyner I, Liiva P, Matuszeski A, Mazarico E, McGarry J, Novo-Gradac A, Ott M, Peters C, Ramos-Izquierdo L, Ramsey L, Rowlands D, Schmidt S, Scott V, Shaw G, Smith J, Swinski J, Torrence M, Unger G, Yu A, Zagwodzki T (2010) The Lunar Orbiter Laser Altimeter investigation on the Lunar Reconnaissance Orbiter mission. Space Sci Rev 150:209–241.  https://doi.org/10.1007/s11214-009-9512-y CrossRefGoogle Scholar
  18. Tooley C (2009) Lunar Reconnaissance Orbiter mission update. Wernher von Braun Memorial Symposium 2009, Huntsville, Alabama http://astronautical.org/sites/default/files/vonbraun/2009/Von_Braun_Symposium_2009-10-20_4a_Tooley.pdf
  19. Williams J, Boggs D, Folkner W (2008) DE421 Lunar orbit, physical librations, and surface coordinates. JPL Interoffice Memorandum 335-JW,DB,WF-20080314-001Google Scholar
  20. Zuber M, Smith D, Lehman D, Hoffman T, Asmar S, Watkins M (2013) Gravity Recovery and Interior Laboratory (GRAIL): mapping the lunar interior from crust to core. Space Sci Rev 178:3–24.  https://doi.org/10.1007/s11214-012-9952-7 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut für Geodäsie und GeoinformationUniversität BonnBonnGermany

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