PRECISION POINTING OF IBEX-Lo OBSERVATIONS

The Star Sensor calibrations were performed in a low light calibration facility of the magnetospheric and ionospheric research laboratory of the University of New Hampshire, which contains an Integrating Sphere light source (by Sphere Optics) that is calibrated to NIST standards within a clean dark room. The optics is set up on an optical bench that is isolated from the floor. In order to use the Integrating Sphere, which produces homogenous light over a 10 cm diameter opening, a cover plate with a 4 mm diameter pinhole centered on the opening was added to simulate individual stars.

The boresight of the Star Sensor was designed and fabricated as perpendicular to the baseplate of the IBEX-Lo sensor, which also defines the boresight of the IBEX-Lo collimator. Through careful choices of machining techniques the mounting of the Star Sensor within IBEX-Lo was controlled such that the collimator boresight is perpendicular to the baseplate to ±0066.

In order to calibrate the pointing of the Star Sensor relative to an absolute boresight reference that is perpendicular to its mounting plate, a precision jig for the calibration facility was designed, whose perpendicularity could be adjusted and tested on the optical bench with a laser level. A schematic view of the test setup is shown in Figure 3. The Star Sensor is mounted on a baseplate in one of the three height positions (1, 2, 3) that are separated by Δh = 14 cm and with position 2 such that the Star Sensor boresight points exactly at the artificial star aperture when correct alignment of the mount is achieved (shown in the top part of Figure 3). The separation between the artificial star aperture and the pinhole aperture of the Star Sensor is l = 248 cm, thus leading to elevation pointing of +323 ± 002 (1), 0° (2), and −323 ± 002 (3). The jig is attached to a precision rotation table whose rotation axis represents the z-axis and is centered on the pinhole aperture of the Star Sensor so that the Star Sensor can be rotated relative to the artificial star to simulate the spacecraft spin. For exact alignment, the rotation table is set to 0°.

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To achieve alignment a laser level that emits laser light along the same axis to both sides is mounted between the Integrating Sphere and the Star Sensor mount (shown in the center and bottom parts of Figure 3). The laser beam adjusts itself to exactly level. The laser level is adjusted in y and z directions so that it is exactly centered on the artificial star aperture. A precision polished metal plate with four markings at equal distance from the center of the metal plate is placed into position 2 of the Star Sensor mount. Now the mount is adjusted in the y and z directions so that the laser beam is centered between the four center markings. Finally, the mount is adjusted in angle in the x−y and in the x−z plane so that the laser beam is exactly reflected back into the beam exit of the laser level. These alignments are achieved to ±0.5 mm so that an overall alignment accuracy and thus Star Sensor reference in perpendicularity to the mounting plate of ⩽0018 was achieved.

For consistency we used a mirror that was placed at the position of the polished plate which reflects the laser beam back on itself. Hence we double the travel distance of the laser beam, and thus, we doubled the accuracy.

To calibrate the Star Sensor for its angular response the orientation of the Star Sensor is changed in increments of 05 with the rotation table when the Star Sensor is mounted in either of the three height positions to allow for the simulation of three different elevations. Figure 4 shows the result from such a scan for the FM and the FS in comparison with an illumination equivalent to a magnitude 1.5 star. The measurements were obtained with engineering model electronics, which allows an amplifier output range from 0 to 10 V. On the angle scale, 0° refers to exact alignment of the boresight with the line of sight to the artificial star. The results in Figure 4 demonstrate that the alignment of the Star Sensor was highly reproducible, even between different Star Sensor models. The reproducibility was also achieved before and after vibration testing of the Star Sensor.

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For exact boresight pointing, the center position in spin angle was verified to be accurate within ±0065. The separation of the legs of the split-V aperture in the center position (2) in the mount, i.e., 0° elevation, was found as 8365 ± 0039. Including the fact that the two legs are inclined at 144 relative to the center line, the elevation pointing was found to be accurate to ±0077.

Including the mechanical tolerance stack-up of the collimator boresight of 0066 and the uncertainty in the knowledge of the alignment of the Star Sensor in the calibration mount of 0018, the accuracies of the alignment of the Star Sensor and the collimator boresight are known to within 0094 in spin angle and 0102 in elevation.

Both Star Sensor models have been calibrated for a wide range of star magnitudes that include the magnitudes of the visible outer planets (Mars, Jupiter, and Saturn). This calibration provides the necessary information to set the bias voltage of the PMT tube adequately in flight and to help identify bright stars in the Star Sensor signals in flight. It should be noted that the star magnitudes are given as magnitudes in the visual spectral band based on the color temperature of the lamp in the Integrating Sphere. To relate the results to actual star magnitudes the spectral curves of individual stars and the spectral curve of the PMT need to be factored in. Because the Star Sensor will be solely used for pointing purposes, no absolute calibration taking into account these spectral sensitivities has been attempted. The result shows the wide dynamic range that the Star Sensor can operate in. In additional tests with a different light source, which was less well calibrated than the Integrating Sphere, it was verified that the Star Sensor is even capable of reliable observations of the half-moon (magnitude ∼ − 10 mag) with PMT voltages between 200 and 250 V, which are now used in flight for moon viewing time periods.

The "first light" observations from the Star Sensor demonstrated that the instrument is working excellently and very close to expectations. This is illustrated in Figure 6 where an observation from a selected orbit is compared to the signal predicted by the Star Sensor Simulation Program (Appendix A). Several effects can obscure Star Sensor data: the signal from the Star Sensor can be saturated by the light of the Moon or Earth, when they are close to the FoV; and the synchronization of the spin pulse can be lost during a portion of the orbit if either Earth or the Moon blind the spacecraft Star Tracker. The Star Sensor signals are used to correct pointing information when the spin pulse is lost.

Examples of such effects in the Star Sensor data are shown in Figure 5. The upper two panels and the lower left panel show the Star Sensor signal as a function of spin phase (vertical) and time (horizontal) in a color-coded representation, and the lower right panel presents an example of the data from orbit 33 in a more detailed three-dimensional view.

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In a scenario that is straightforward to analyze, a point source should show up as a pair of horizontal "rails," which correspond to the double peak of the star over time. Diffuse background should show up as broad horizontal bands. Any departures from strictly horizontal structures mean that either a time-related phenomenon (like an object moving through the FoV) is observed or a problem with spin synchronization exists. The synchronization problems are illustrated in the upper left panel, where the horizontal structures are transformed into oblique wavy ones—and the waviness is similar for all elements of the picture. Only at the very end of this orbit, at the far right in the panel, the synchronization was restored. In such a case, the observations of point sources from the Star Sensor can potentially be used to despin the observations provided that at least one clearly visible star is in the FoV.

If the Star Sensor is to be used also to independently determine the spacecraft spin axis orientation, at least two well separable stars are required. The upper right panel shows an example of a good orbit for this purpose: two stars are visible against a relatively weak background in the upper portion of the panel. There is even a possibility to resolve a third star, but this one is seen against a bright background strongly varying with spin phase, which makes using it for the spin axis pointing determination problematic. However, the remaining two stars are sufficient for the task. Only at the end of the orbit, strong reflected stray light from the Earth enters into detector and swamps the two good stars.

The lower left panel illustrates a case of heavy contamination by the glow of the Earth (the big red structure in the upper portion of the panel) and Moon (the oval structure). There are also two bright stars shining through the background and the Moon glow. At about 2/3 into the orbit interval the Star Sensor was temporarily switched into the Moon Mode (with appropriate reduction of the PMT voltage) because of the expected passage of the Moon through the FoV. This time period is visible as the vertical black strip. And indeed, the Moon signal was registered, as manifested by the two lines visible against the dark background cutting through the oval. The Moon shows a component of motion through the FoV from lower to higher elevation in the S/C reference system. As a consequence, the angular distance between the two peaks is increasing with time. During this particular orbit, Earth was also within the FoV simultaneously with the Moon, as evidenced by the strong signal visible during the Moon Mode time period. The Earth signal is saturated even in the Moon Mode, nevertheless Earth is resolved and visible as a compact object (since it gives two distinct peaks). Because of the orbital motion of IBEX, Earth is crossing the FoV downward (i.e., its elevation decreasing, in contrast to the Moon), as can be established from the fact that the distance between the two peaks is decreasing with time. It is evident that Earth and the Moon move also in azimuth, as indicated by the fact that the imaginary center lines between their double peaks are not horizontal, as are the signals from the two stars. Since the synchronization is maintained in this orbit, the Star Sensor provides us with time-resolved positional information on the objects moving through the FoV. Even under such challenging conditions, the signal from the two good stars at the beginning and the end of the orbit could be used for the axis pointing determination.

This qualitative discussion is offered to justify that given the complex and varying composition of the data we decided not to develop an algorithm for an automatic selection of suitable data intervals for the analysis. Instead, we rely on human judgment supported by simulation.

Determination of orientation of the optical axis of the IBEX-Lo instrument from the Star Sensor observations requires two steps: extraction of positions of point-like objects (stars, planets) from the telemetry data and identification of these objects with stars or planets listed in a stellar catalog or planetary ephemerides for the time of the observation. Typical reference objects most frequently observed by the Star Sensor will evidently be bright stars, but also the naked-eye outer planets and even the Moon can be considered as additional reference sources.

The point-like stars are always observed against a comparably strong diffuse sky background. The most important components of the background are the zodiacal light, diffuse Galactic light, and a huge number of faint, unresolved stars (Leinert et al. 1998). Hence the signal registered by the Star Sensor is composed of signals from the objects of interest for the Star Sensor on top of an extended background in the spectral sensitivity range of the detector.

The method used to determine the positions of stars from the IBEX telemetry data is inverse to the method used to simulate the Star Sensor response function, described in Appendix A. The Star Sensor data collected during one entire orbit typically include a few hundred histograms (scan data blocks). To identify stars, we look for double pulses that are considerably stronger than the neighboring sky background level (Figure 6) and do not have other strong stars nearby. Strong neighbor stars in the FoV modify the shape of the pulses of the target star. Because the distortions of each of the two peaks usually differ, the distance between the two peaks is changed, which impacts both the azimuth and elevation obtained from such a star. The selection of the candidate peak pairs is finalized through human involvement because of the varying and complex character of the signal, as discussed in the preceding section.

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Processing of observations from a given orbit begins with selection of a subset of data blocks free from synchronization issues and saturated reflected light. Then, the histogram portions with the peak pairs are determined. The latter choice is maintained for all suitable data blocks from the given orbit because the stability of the signal is very good for orbits in full spin synchronization. Only peaks stronger than a preselected level of 0.15 V are analyzed.

With the appropriate pairs of peaks selected, the local sky background level (approximated by straight line, Figure 7) is subtracted from the sensor signal, as illustrated in Figures 6 and 7. Based on the residual signal, the center position of each peak is determined through the calculation of its center of mass. Because the FWHM of the slit width is 14, each peak typically extends over seven angular bins. After identifying the maximum bin of a peak, the center of mass is obtained from the seven bins centered on this bin, using bin centers as angular coordinates.

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We obtain the spin angles α₁, α₂ for both peaks of a pair. Before further processing of the selected pair we check whether it meets the criteria of the split-V aperture geometry, i.e.,

$$\begin{equation} 5\mbox{$.\!\!^\circ $}73 \le \alpha _2 - \alpha _1 \le 9\mbox{$.\!\!^\circ $}87. \end{equation} \tag{ 1 }$$

If valid, the azimuth (spin angle) of the identified star α_star in the spacecraft system is derived as

$$\begin{equation} \alpha _{\mathrm{star}} = \left(\alpha _1 + \alpha _2\right)/2. \end{equation} \tag{ 2 }$$

We do not require here that the peaks are equal in height because they still might be affected by unaccounted background and photocathode inhomogeneities.

With the azimuths of the two peaks and the azimuth of the star, the elevation of the star δ_star can be calculated from the formula

$$\begin{equation} \delta _{\mathrm{star}} = \arcsin \left[\frac{\tan \left(\left(\alpha _2 - \alpha _1 - \sigma _V \right)/2\right)}{\tan \left(\beta _V \right)} \right], \end{equation} \tag{ 3 }$$

where β_V = 144 is the tilt angle of each slit in the split-V aperture relative to the symmetry line and σ_V = 84 is the angular separation of the slits at 0° elevation.

A detailed comparison of the spin angles (azimuths) of stars determined from the telemetry with the spin angles expected from simulation showed that the spin angles of the stars in the S/C frame are shifted relative to the positions expected from simulations by an angle that increases with the spin angle. We identified two reasons for this effect: (1) a constant shift by ∼03 is introduced by the Star Sensor amplifier and (2) a linear "scaling" effect in the onboard signal formation process exists. The scaling effect comes up because of a finite resolution of the coding of the duration of the clock ticks used for bin width determination, indicated in Section 2. Here, we develop a correction factor, f_s, that is the ratio of the actual bin width divided by the 05 bin width. The actual bin width can be solved for using a housekeeping register k that is used to solve for the time interval (in seconds) τ_s = (k + 192)/14, 400 between the interruptions that separate each of the 720-spin bins. The actual bin width (in degrees) is then Δϕ_s = 360°τ_s/T_CEU, where T_CEU is the spin period of the S/C stored in the memory of the onboard computer and measured on a spin-by-spin basis. The correction factor f_s = Δϕ_s/05 or, combining factors

$$\begin{equation} f_s = \left(k + 192\right)\frac{720}{14{,}400}\frac{1}{T_{\mathrm{CEU}}}. \end{equation} \tag{ 4 }$$

The corrections for the instrumental shift (03) and scaling of the spin phases are implemented directly in the program processing the Star Sensor telemetry data.

The method for the star position determination has been extensively tested on the signal obtained from simulations. A series of 50 cycles of the Star Sensor signal simulation/scanned star position findings was performed for a few IBEX orbits (spin axis positions). Since the simulations include statistical fluctuations of the intensity of the signal, the results were statistically distributed, but the mean position of the identified stars was found to be only slightly different from the actual star positions used for the simulation. For example, for IBEX orbit 22 the error in right ascension was only ∼002 and in declination ∼005. In this way we made sure that the implemented method could determine the star positions with sufficient accuracy after the input data had been appropriately corrected.

An illustration of quality of the correction defined in Equation (4) is shown in Figure 8, where histograms of the differences between the orbit-averaged observed and simulated spin angles (azimuths) of the stars in the S/C frame are shown. The red histogram of pre-correction differences is not symmetrical and features an extended wing toward the negative differences. The differences between the simulated and post-correction azimuths (blue in the figure) are much more symmetric and the histogram is narrower. This histogram is shown in a finer scale in the right-hand panel of Figure 11.

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Once the positions of the stars in the spacecraft reference system are determined, they have to be identified with reference stars from the star catalog. The identification is based on a comparison of the telemetry data with simulation results (Figure 6). The simulation of the signal expected from the telemetry is calculated for a given UTC time and spin axis position obtained from the Star Tracker signal processed by the IBEX Science Operations Center (ISOC; Schwadron et al. 2009). Based on the simulation results, a short list of candidate stars is returned whose positions are then compared with the observed star positions. Details of the simulation program are given in Appendix A. Even though a software module to identify the stars from the observations was developed and successfully tested, the manual identification turned out to be more practical in situations when the signal is affected by one of the unexpected complexities discussed in the preceding sections.

If at least two stars for one orbit can be identified with star catalog entries, it is possible to calculate the spin axis orientation. Determining the axis pointing requires finding a matrix $\mathsf {R}^{bi}$ that transforms the orientation of the spacecraft reference system, in which we have obtained the star positions, into the target inertial celestial reference system. In our case, the target reference system is the equatorial system and the method adopted is the TRIAD algorithm (Shuster & Oh 1981). There is relatively little robustness in the process of determining the S/C attitude using observations of just two stars because there is no redundancy of information. Nonetheless, this method offers a very valuable way to compare the Star Sensor's attitude determination with that of the ACS system. Furthermore, the resulting spin axis pointings can be statistically analyzed to validate the actual pointing of the Star Sensor with that of IBEX-Lo in spacecraft coordinates. Finally, it is demonstrated that the Star Sensor could provide the spacecraft attitude determination if such a need would arise.

In order to determine the spin axis orientation with the use of the TRIAD algorithm, the positions of the two stars must be determined quite accurately, which is satisfied by the selection criteria for the stars as described above. In practice, such conditions are not always fulfilled and thus there are orbits for which no determination of the spin axis position is possible because of the lack of suitable stars.

The relation between the measured star position vectors in the S/C system $\bm v_{1b}$, $\bm v_{2b}$ and their position vectors in the equatorial system $\bm v_{1i}$, $\bm v_{2i}$ is

$$\begin{equation} \bm v_{1b} = \mathsf {R}^{bi}\bm v_{1i}\quad\hbox{ and }\quad \bm v_{2b} = \mathsf {R}^{bi} \bm v_{2i}, \end{equation} \tag{ 5 }$$

which must be solved for the transformation matrix $\mathsf {R}^{bi}$. The matrix algorithm is based on the construction of two triads $\bm t_b$ and $\bm t_i$ of orthonormal unit vectors using the observed and reference star position vectors $\bm v_{1i}$, $\bm v_{2i}$, $\bm v_{1b}$, $\bm v_{2b}$ (Shuster & Oh 1981):

$$\begin{equation} \bm t_b = \left[\bm t_{1b}, \bm t_{2b}, \bm t_{3b}\right]\quad \hbox{ and }\quad \bm t_i = \left[\bm t_{1i}, \bm t_{2i}, \bm t_{3i} \right], \end{equation} \tag{ 6 }$$

where

$$\begin{eqnarray} \bm t_{1i} &=& \bm v_{1i}\nonumber \\ \bm t_{2i} &=& \frac{\bm v_{1i} \times \bm v_{2i}}{\left|\bm v_{1i} \times \bm v_{2i}\right|} \\ \bm t_{3i} &=& \bm t_{1i} \times \bm t_{2i} \nonumber \end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray} \bm t_{1b} &=& \bm v_{1b}\nonumber \\ \bm t_{2b} &=& \frac{\bm v_{1b} \times \bm v_{2b}}{\left|\bm v_{1b} \times \bm v_{2b}\right|} \\ \bm t_{3b} &=& \bm t_{1b} \times \bm t_{2b} \nonumber \end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray} {[}\bm t_{1i}, \bm t_{2i}, \bm t_{3i}] &=& \mathsf {R}^{it} \mathsf {I} = \mathsf {R}^{it} \nonumber \\ {[}\bm t_{1b}, \bm t_{2b}, \bm t_{3b}] &=& \mathsf {R}^{bt} \mathsf {I} = \mathsf {R}^{bt}. \end{eqnarray} \tag{ 9 }$$

Finally,

$$\begin{eqnarray} \mathsf {R}^{bi} &=& \mathsf {R}^{bt}(\mathsf {R}^{it})^{\mathrm{T}} \nonumber \\ &=& \left[\bm t_{1b}, \bm t_{2b}, \bm t_{3b}\right] \left[\bm t_{1i}, \bm t_{2i}, \bm t_{3i} \right]^{\mathrm{T}} \end{eqnarray} \tag{ 10 }$$

represents the transformation from the equatorial system into the S/C system. The inverse transformation is the transposed matrix:

$$\begin{equation} \mathsf {R}^{ib} = (\mathsf {R}^{bi} )^{\mathrm{T}}. \end{equation} \tag{ 11 }$$

The IBEX spin axis is the z-axis of the S/C reference system. Therefore, the transformation matrix allows the calculation of the spin axis $\bm v_{zb} = [0, 0, 1 ]$ in the equatorial system as

$$\begin{equation} \bm v_{zi} = (\mathsf {R}^{bi})^{\mathrm{T}}\bm v_{zb}. \end{equation} \tag{ 12 }$$

The method was implemented numerically and successfully applied to determine the spin axis pointing of IBEX during the first two years of science operations.

Pointing of the IBEX spin axis is determined by the ISOC based on data from the IBEX ACS. Experience from more than two years of operations showed that when the observation conditions are favorable, the ACS performs with high precision and according to specifications, enabling a very accurate determination of spin axis pointing. The spin axis is very stable during an entire orbit, with a very small precession amplitude of ∼002, i.e., comparable to the typical precision of the ACS system. The high stability of the IBEX spin axis enables us to adopt one solution for the entire duration of each orbit and to perform a reliable statistical comparison with solutions obtained by the Star Sensor, as discussed in the following sections.

As a first verification step for the IBEX-Lo boresight in the spacecraft system, we analyzed the positions of all identified stars in the spacecraft reference system for systematic deviations from the expected locations. For most of the suitable orbits, positions of two stars could be determined and the IBEX spin axis calculated. For a few orbits, the positions for either just one or as many as three stars could be determined. In total, 69 stars were processed. The positions of stars in the S/C reference system were retrieved separately from each suitable telemetry block. Examples of the determinations of these positions are shown in Figure 9.

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The positions of identified stars are somewhat distributed in both coordinates, with standard deviations varying from orbit to orbit. An example of statistical fluctuations of the differences between the simulated and actually measured azimuths and elevations for the star SAO 246574 observed in Orbit 73 is shown in Figure 10. The positions were stable and consistent. The median value of standard deviations in the phase angle was 002 and in elevation 006. Such a behavior of a greater uncertainty in elevation than in spin angle was expected, as discussed in the previous section.

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The angular separation between stars found for a given orbit is very stable. The distribution of angular separations for an example for orbit 73 has a standard deviation of less than ∼004.

We calculated the mean positions of all identified stars, followed by the mean deviations of these positions from the simulated values based on the spin axis positions obtained from ISOC (Schwadron et al. 2009) using the spacecraft ACS. Histograms of the differences between the simulated and observed positions of the stars in the S/C reference system are shown in Figure 11. The uncertainty from the mean is given by $\sigma /\sqrt{N}$, where N = 69 is the number of observed stars. Therefore, the mean deviation in the phase angle is equal to −00155 ± 0017, while the mean deviation in elevation is −00179 ± 0037. About 68% of the results in elevation and about 85% results in spin angle are within ±02 from 0. We conclude that there is no statistically significant deviation of the Star Sensor boresight from the direction perpendicular to the ACS-determined spin axis to within 0037, nor in spin angle relative to the spin pulse to within 0017.

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We conclude that there is no statistically significant deviation of the FoV from the Star Sensor and thus the IBEX-Lo pointing other than expected from the observed deflection of the spin angle from spin axis and from the ground-measured uncertainty in the IBEX-Lo mounting. In addition, the Star Sensor signal can be used to correct the spin-phase information with high precision when bright objects, such as the Earth and Moon, blind the Star Tracker causing inaccuracies in the ACS data.

For all orbits with at least two clearly identified stars, we have calculated the pointing of the IBEX spin axis. We followed two paths: (1) we calculated the spin orientation separately for all spin blocks available from a given orbit and (2) we calculated the spin axis from the mean star positions averaged over a given orbit. Subsequently, we calculated the deviation of our pointing from the ACS pointing.

The representation of the results of the spin axis determination from Star Sensor observations is presented in the S/C reference system in Figure 18 in Appendix B. The S/C reference frame is oriented with the ACS-derived spin axis at the pole, and IBEX-Lo directed in the Y direction. Typically, determinations for a given orbit form elongated streaks in the S/C reference system because the Star Sensor's accuracy for a star's elevation is lower than that of its azimuth. The quality of the IBEX pointing determinations varies between orbits because of the varying measurement conditions between orbits: relative position of the stars (differences in the azimuth and in the elevation), signal-to-background ratio of the stars, intensity and inhomogeneity of the background, presence of the stray light, and the finite accuracy of the projection of the stars azimuth on the Star Sensor 720-bin histogram.

The positions of the spin axis as obtained from individual data blocks for all processed orbits are shown in the left panel of Figure 12 and the average positions for these orbits are shown in the right panel. The orbit-averaged spin axis pointings determined by the Star Sensor are almost uniformly distributed around the pole. The small void region between 60° and 330° azimuth is due to the season during the year when no sufficiently bright star pairs can be observed. Importantly, almost all the center lines through the uncertainty ellipses pass very close to the pole in the plot.

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The spread between the ACS-derived spin axis and that derived by the Star Sensor is between 024 and 028, which roughly agrees with the pre-launch measurements of the Star Sensor boresight relative to that of IBEX-Lo. From this, we conclude that there is no systematic statistically significant deviation of the IBEX-Lo boresight from the direction perpendicular to the ACS-derived spin axis. The spread obtained from the analysis of the Star Sensor data is due to the observing conditions and follows a yearly pattern caused by the positions of individual stars. This is illustrated in Figure 13 where differences in right ascension and declination of the spin axis between the ACS and Star Sensor determinations are shown as yearly time series based on orbit number. The accuracy of the spin axis pointing determined by the Star Sensor shows that it could be used for spacecraft attitude determination for almost all orbits in case of a Star Tracker failure, i.e., the Star Sensor indeed provides good redundancy to the spacecraft system.

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