### Experimental geodetic satellite - Ajisai

The Experimental Geodetic Satellite Ajisai (Figure 1) was launched on 12 August 1986 by the National Space Development Agency of Japan (NASDA), currently reorganized as the Japan Aerospace Exploration Agency (JAXA). Objective of the mission is the accurate position determination of fiducial points on the Japanese Islands. The satellite is equipped with 1,436 corner cube reflectors (CCR) for satellite laser ranging (SLR), arranged in the form of 15 rings around the symmetry axis. Ajisai is also equipped with 318 mirrors, used in the past to determine the direction to the satellite (Sasaki and Hashimoto 1987). The mirrors have also been used for photometric measurements of Ajisai spin (Otsubo et al. 1998). The satellite is placed in a quasi-circular orbit of altitude 1,490 km and inclination of 50°.

The positions of the 120 CCR panels are presented on Figure 1. Each panel contains 12 CCRs except for the one mounted on the central ring - with 8 CCRs only. The system elements of the satellite are presented on Figure 2. The corner cube reflectors are made of fused silica for 532-nm green light, and the back faces of the reflectors are un-coated.

Ajisai was launched with an initial spin period of 1.5 s (Hashimoto et al. 2012) and is currently the fastest spinning SLR satellite in orbit. The specific construction of the spherical body (laminated sheets of aluminum foil) minimizes the interaction with the Earth’s magnetic field and prevents loss of the rotational energy. The orientation of the spacecraft was set to be parallel to the Earth’s spin axis and is stabilized by the passive nutation dumper (Figure 2).

### Satellite laser ranging

Satellite laser ranging (SLR) measures the distance to satellites equipped with retroreflectors. The laser pulses transmitted from the SLR station are reflected by the spacecraft back to the receiver telescope of the ground system. The time-of-flight measurements of the laser pulses are recalculated into an optical range to the satellite. The range measurements are used for precise orbit determination (POD), study of tectonic plate motion, as well as determination of Earth’s gravity field, Earth rotation parameters, and geocenter position (Moore and Wang 2003). The SLR data can be used for deriving the product GM of the universal gravitational constant G with the mass M of the Earth (Dunn et al. 1999).

During a pass of a satellite over an SLR station, the laser range measurements are collected. After the pass, the predicted range trend is fitted to the measured range values by adjusting the time bias and range bias estimations. As the next step, the range residuals are formed by calculating the difference between the observed and predicted (computed) range values, O-C. In the course of this process, fitting functions (orbital function or low degree polynomials) are used in order to remove the systematic trends from the distribution of the range residual data. The final step of the post-processing is the formation of the so-called normal points (NP).

### Normal point formation with sigma filter

The development of the SLR technology and the use of the 10 to 15 Hz repetition rate lasers significantly increased the amount of the range measurements per pass compared to the early SLR systems. In order to reduce the computer time required for the orbital analysis of the large data sets, while retaining the information content of the individual range measurements, a data compression technique has been introduced to create normal points. At the 5th Laser Ranging Instrumentation Workshop held at Herstmonceux in 1984, a procedure was recommended for the formation of normal points from the full rate ranging data (Recommendation 84A: ‘On SLR normal-point generation and data exchange’). The calculated range residuals of the pass are clipped with an iterative sigma rejection filter. The retained data set is divided into short time slots (30 s for Ajisai); the arithmetic mean of the range residuals of each bin is calculated and used as the reference for the normal point.

The on-site produced normal points are the principal SLR data product and express the one-way optical distance between the reference point of the SLR ground station and the mean reflection point of the satellite (Figure 3A).

In order to deliver an effective distance between the ground SLR system and the satellite’s center of mass (which represents its orbital motion), the normal points are corrected for the systematic errors such as the atmospheric refraction delay (Mendes and Pavlis 2004), the calibration system delay, and for the center-of-mass offset. The center-of-mass correction (CoM correction - Figure 3A) is the one-way distance to be added to a normal point at the precise orbit determination stage and can be determined from the prelaunch measurements or by the analysis of a theoretical satellite’s response function (Arnold 1979). The prelaunch tests give the center-of-mass correction for Ajisai of 1,010 mm (Sasaki and Hashimoto 1987). At the 9th International Laser Ranging Workshop in Canberra in 1994, papers were presented by J. J. Degnan and by R. Neubert which theoretically investigated the reflection signature of the LAGEOS-type satellites and calculated the correction from the mean reflection point to the center of mass of the spherical satellites.

Otsubo and Appleby (2003) analyzed the temporal spread of the optical pulse signals due to a reflection from the multiple onboard reflectors and modeled the response function of LAGEOS, Ajisai, and the Etalons. The obtained center-of-mass corrections depend on the ranging system configuration and observation policy at the terrestrial stations and vary by about 1 cm for LAGEOS and 5 cm for Ajisai and Etalons. Otsubo et al. (2000) demonstrated that the center-of-mass correction for Ajisai depends on the orientation of the satellite and can vary between 985 and 1,030 mm for a Gaussian profile of the incoming laser pulse with FWHM (full width at half maximum) of 200 ps.

The optical range measured by SLR is longer from the physical value due to the delay of a laser pulse passing through the glassy corner cube reflectors. The optical distance to a CCR refers to the optical reflection point and can be corrected by the range correction ΔR in order to represent the physical distance to the center of the CCR’s external surface. The range correction ΔR is defined as \( \varDelta R=L\sqrt{n^2-{ \sin}^2\theta } \), where *L* is the height of CCR (17.15 mm for Ajisai), *n* is the refractive index of the material (1.46 for fused silica and 532-nm wavelength) and *θ* is the incident angle between the laser vector (vector from the satellite’s center directed towards the ground SLR system) and the optical axis of a retroreflector (Fitzmaurice et al. 1977, Figure 3B).

In this paper, we propose a ‘reflector filter’ which allows us to calculate a physical distance to the single CCR panels of Ajisai. The physical range values are corrected by the center-of-mass vector (vector length along the laser range direction = *R*⋅cos(*θ*)) in order to determine the distance between the ground SLR system and the satellite’s center of mass (Figure 3A).

### SLR measurements to Ajisai at Graz observatory

Since 8 October 2003, the Graz SLR station (Austria) operates with a 2-kHz repetition-rate Nd:Van laser of 10-ps FWHM pulse duration (Kirchner and Koidl 2004). The detection system is based on the Compensated Single Photoelectron Avalanche Detector (C-SPAD) (Kirchner and Koidl 1995) and responds to the first incoming photon. The energy-dependent range bias of C-SPAD is reduced to about 10 ps (time walk compensation), thus the measured range does not depend on the energy of the incoming signal.

Figure 4 presents the range residuals of Ajisai obtained by the Graz SLR system at different configuration stages. The three data samples mark the improvements in the amount and the accuracy of the range measurements. The example Figure 4A was measured on 23 August 1986 (11 days after Ajisai was launched) with 2.5-Hz repetition-rate laser - the small amount of the data points gives RMS of 36.8 mm. The case B shows the measurements obtained on 17 January 1996 with the 10-Hz repetition-rate laser. The increase in the amount of measurements per pass improves the RMS (11.8 mm) and allows for the analysis of the statistical parameters of the data distribution. The 2-kHz repetition-rate laser (case C: pass measured on 17 January 2006) greatly increased the number of the range measurements per second. The high amount of the accurate data and the very short laser pulse length not only improve the statistics of the obtained range residuals (RMS = 8.81 mm) but also allow to distinguish between the laser echoes given by the single CCR panels of the satellite (Figure 4C: visible as the down-peaks of the data points).

Figure 5A presents a 5 s part of the Ajisai pass measured by Graz on 17 January 2006. The selected data span shows the range measurements to the CCR panels of the ring R_{−3} (Figure 1), and the observed mm-scale modulation of the range residuals (amplitude of approximately 25 mm) is caused by the rotation of the satellite (spin period of 1.995 s).

The presented range residuals are determined during an iterative 2.5 sigma elimination process and give RMS of 8.81 mm. The statistical parameters of the range residual distribution can be obtained with the smoothing algorithm derived by Sinclair (1993). The distribution function is calculated with a strong smoothing coefficient of 15 mm which assures that all of the leading range residuals will be used during the process of normal point formation. The determined histogram function (Figure 5B) gives peak = −0.78 mm, full width at half maximum (FWHM) = 41.7 mm, and leading edge at half maximum (LEHM) = −21.2 mm; the determined leading edge corresponds to the front face of the spherical satellite.

Due to the fast spin of Ajisai, the distance between the nearest CCR panel and the satellite’s front face is quickly changing - this effect is represented as the V-shape distribution of the laser echoes on Figure 5C. The situation when the incident angle *θ* between the laser vector and the central axis of the panel (Figure 3B) is the smallest defines the minimum deviation of the observed CCR panel (MD). The epoch times of the minimum deviation events throughout the pass can be predicted by the complete attitude model of Ajisai. The spin axis orientation of the satellite (in J2000.0 celestial reference frame) during a single pass can be predicted with RMS of 0.128° (Kucharski et al. 2010b, 2013a). The spin period and the rotational phase angle of Ajisai can be obtained during the post-processing of a complete pass with RMS of 84 μs (42 ppm) for the spin period and with RMS of 0.48° for the phase angle (Kucharski et al. 2010a).

The range coordinate of the minimum deviation (MD) can be calculated as an arithmetic mean of the range residuals located around the predicted MD epoch (data bin width of 20 ms) and clipped by an iterative 2.2 sigma filter. In the example case (Figure 5C), the range coordinate of the MD event is calculated as the mean of 23 selected points (return rate of 57.5%) and is equal to −11.7 mm, RMS = 2.29 mm.