2.5 Analysis kit used for studying the residuals
2.5.5 Satellite ground tracks
The projection of the orbit of a satellite onto the surface of the earth is called ground track (Curtis, 2005; Seeber, 2003). When the direction of the ground track of a satellite is from the south to the north pole, the path covered is calledascending arc, whereas the descending arc is the ground track covered by the satellite from the north to the south pole. In Fig. 2.7, an ascending arc of the ground track covered by thegracesatellites is plotted for 7 days. The ground track of a satellite
Figure 2.7: Ground-track coverage of the earth by the grace satellites from december 1st to 7th 2008. The ground-track is plotted for the ascending orbit (upward direction of flight) of the spacecraft.
is given by its geographical coordinates (λ, φ, h). The geographical coordinates are computed from the cartesian coordinates of the satellites (rx,ry,rz) given in the
earth centered earth fixed (ecef) frame of reference as φ= tan
rz
qr2x + r2y
,
λ= tan
ry rx
,
h=
qr2x + r2y cos(φ) .
2.16
For an ascending arc, the change in φ, i.e. (dφ), for two consecutive epochs must be >0◦. For adescending arc, dφfor any two consecutive epochs must be<0◦. Analyzing the observations on the ground-track representation is helpful when their geographical variations are of interest. Here, it is useful to analyze weekly, fortnighly or monthly observations. For the analysis of long-term geographical variations, it is advised to chunk the observations into small time intervals, otherwise too much overlap of information can be misleading in the interpretation of the analysis results.
Analysis of K-band range-rate
residu-als
Inter-satellite pointing errors in range-rate resid-uals
range-rate residuals
In this chapter, an analysis of the range-rate residuals is presented with focus on the satellite pointing errors. The two spacecraft need pointing information in order to measure the precise range changes as well as for the gravity field determination.
Due to the use of pointing information in the gravity field recovery process, related errors also propagate to the gravity field solutions through range-rate information.
In this chapter, these pointing errors are studied by analyzing the attitude data (used to derive pointing information). Further, the range-rate residuals are analyzed to identify the pointing errors present in them.
3.1 What is inter-satellite pointing?
Inter-satellite pointing is a part of theattitude andorbitcontrolsystem (aocs).
Theaocsis responsible for the attitude determination and its control in the satellite.
The sensors used for determination of the attitude in grace are the star camera assembly (sca), the inertialmeasurementunit (imu) and thecoarse andearth sun sensors (cess), where main attitude determination sensors in grace are the star cameras.
The cess are used for the initial acquisition of the attitude but they are not as accurate as the star cameras. The accuracy of the cess ranges between ≈ 5◦ - 10◦ for theearth-vector and≈3◦- 6◦for thesun-vector depending on the orbit geometry (Herman et al.,2004). IMUs provide the angular rates of the spaceraft along three axes and angular accelerations are provided as a side-product by the accelerometer along with the linear accelerations. Attitude of a spacecraft is controlled by the thrusters and the magnetic torquer rods used along the three pointing axes roll, pitch and yaw (see Fig. 3.1 (a.)).
The pointing between the two satellites is a fundamental requirement of a satellite-to-satellite tracking mission. Perfect pointing is required to measure the precise range between the two satellites. In grace, the range between the two satellites is defined asthe distance between the phase centers of the K-band ranging assembly
of the two satellites. For precise range measurements, the phase center should be aligned with the line-of-sight (los), but in reality there is an offset between the phase center and thelos (see Fig. 3.2). This could be due to external and internal perturbations on the satellite. Therefore, the measured range and range-rate obser-vations are corrected to get their values along the losvector (Herman et al.,2004).
These corrections are called antenna phase center offset corrections (aoc) which are discussed in detail in Section 3.3.
The maintenance of the pointing between the two satellites in their orbit is one of the most difficult and important tasks. The variations in the pointing angles should be kept minimum as they directly affect the measured range observations.
There are dead bands defined for the maximum allowed pointing variations between the two satellites. These dead bands are defined for the roll, pitch and yaw axis, respectively, see Table 3.1 (b.).
(a.)
Pointing angles dead band limit (mrad)
Roll 2
Pitch 4
Yaw 4 (since 2002)
4.4 (since October 2007) 4.8 (since June 2008) 5.2 (since January 2012) (b.)
Figure 3.1: (a.) Depiction of roll, pitch and yaw axes in the gracespacecraft. (b.) The dead band limits for the three pointing angles in grace(Herman et al., 2004;
Herman and Steinhoff, 2012)
Since the pointing information of the satellites is derived from the attitude data, the precision of the attitude data is crucial for precise pointing. The precision of the attitude is also important for the gravity field determination because it directly affects the precision of the range observations in two ways. First, the range ob-servations are corrected for these pointing imperfections, so called aoc, using the attitude data (see Fig. 3.2). These corrections are applied to the range observations in post-processing to get these observations along thelosof the two spacecraft. Sec-ond, the range observations are also corrected for the effects of the non-gravitational accelerations acting on the satellite. Accelerometers are used to measure the
non-(a)PCVs
LOS GRAC
E-A
COM
GRAC E-B KBR phase center
AOC(θ)
(b)Zoomed-inPCVs
LOS to GRACE-B
θi θi+1 θi+2
t
it
i+1t
i+2G R AC E-A
COM
KBR phase center
Figure 3.2: (a.) The antenna phase center offset (aoc) is defined by the angle (Θ) between the two KBR phase centers and the center of mass (com) of the two satellites. (b.) Variations in angle (Θ) at every epoch because the angle varies with the change of satellite orientation.
gravitational accelerations acting on the satellites. The accelerometer observations, which are provided in the instrument frame (here it is the Accelerometer frame), are rotated into the eci frame of reference using the attitude data. Then, the orbit is corrected for the effect of non-gravitational forces acting on the satellites. Therefore, an understanding of the precision of the attitude is important in order to understand the effects of pointing on the range observations.