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The operational requirements for the pointing control system are expressed in terms of point- ing accuracy and pointing stability. After manufacturing the telescope system, the decision was made to combine the pointing stability requirement with the image quality requirements to an overall image size requirement.

Pointing accuracy is defined as the fidelity with which the pointing control system after acquiring a suitable target (or set of targets), initially positions the science object relative to the TA boresight at the onset of observation.

As illustrated in Figure 2.3, pointing accuracy (under both on-axis tracking and offset track- ing) will be quantified by establishing the average value of the set of measured angular dis- placements between the TA boresight location and 10 “instantaneous” science object location samples taken immediately following a 5-second settling time upon initiation of track.

Figure 2.3. Definition of pointing accuracy and pointing stability, after [NASA SPO 2000]. Initial object location

Average of initial 10 science object locations

Pointing accuracy

Angular displacement of initial object location to the TA boresight location TA boresight

location

Pointing stability

Radial RMS value of image deviation from centroid of over a period from 5 ms to an hour

Pointing stability is defined as the measure of relative motion of a science object of interest in the TA focal plane, over the observation period of time during which TA attitude is con- trolled. Specifically, under both on-axis and offset tracking operations, it is defined as the image deviation from the centroid of the image imprint in the focal plane, measured as the RMS value of the deviation over any period from 5 milliseconds up to an hour of continuous pointing at the same object. This is also illustrated in Figure 2.3.

The Image quality requirement is expressed in terms of the diameter of a circle enclosing 80% of the energy from a stellar image at a wavelength of 0.633 µm at the infrared focus. Both, the pointing stability and image quality contribute to the size of an image. The various error sources and limitations that influence the pointing accuracy and image size are listed in Table 2.1. These errors and limitations for the SOFIA telescope are broken down in detail in the image quality budget [Erdmann 2002] and the pointing stability budget [Kaercher 2000b]. The pointing stability budget is further broken down into contributions caused by the aircraft environment and contributions within the servo subsystem. The pointing stability budget is shown in Appendix A.

Table 2.1. Error sources and limitations affecting the image size under tracking

Pointing accuracy Image size

Pointing stability Image quality

Misalignment of focal plane and imagers

- due to mounting inaccuracies, including non-orthogonality - due to gravity

- due to slow temperature and pressure variations in flight

Scale factor error of gyros Centroid errors

Vibrations of telescope structure (under 70 Hz)

- induced by aircraft vibrations

- induced by aero-acoustic vibrations

Random drift of gyroscopes and its estimation process

Centroid errors and contributions from tracking subsystem

Diffraction

Imperfections of manufactured optical elements

Optical aberrations - due to gravity

- due to slow temperature and pressure variations in flight

Shear layer seeing, atmospheric seeing (for ground observations) Vibrations of telescope structure (above 70 Hz)

The size of an image is characterized by its point spread function (PSF). The PSF is the distri- bution of light intensity in the image of a point source and is wavelength dependent. For a perfect telescope with a circular aperture, the PSF is described with an Airy function [Hecht 2002]. The PSF can be approximated by a two-dimensional Gaussian intensity distribution with a mean value at zero and the standard deviation σ for both independent variables.

Assuming the Gaussian distribution, the relation between the pointing stability requirement which is expressed as the radial RMS value of the centroid motion rRMS and the image quality

requirement which is expressed as the diameter D(80%) that encloses 80% of the energy in the image can be described with [Erickson 2000]:

Diameter of 80% encircled energy: D(80%)=2.54r_{RMS (2.1)}

The angular resolution of the SOFIA telescope (Figure 1.3) is described with the diameter

D(50%) that encloses 50% of the energy in the image. For a Gaussian distribution it is

roughly equal to the Full Width at Half Maximum (FWHM) [Erickson 2000] and is given by: Diameter of 50% encircled energy: D(50%)≈FWHM =1.67r_{RMS (2.2)}

The standard deviation of the Gaussian distribution is related to the radial RMS by

2

σ

=

RMS

r . Figure 2.4 shows the Gaussian PSF for σ = 1 for both independent variables

and the three described definitions of image size.

Figure 2.4. Definitions of image size requirements.

The operational requirement values for pointing accuracy, pointing stability and image quality are listed in Table 2.2. The requirement values for the image quality and pointing stability expressed are combined to the image size requirement as root sum square.

Table 2.2. Key operational requirements for the pointing control system [NASA SPO 2000].

Pointing accuracy On-axis tracking with

WFI 6 arcsec for visible source brighter than mv=13

FFI 3 arcsec for visible source brighter than mv=13

FPI 0.5 arcsec for visible source brighter than mv=16

Offset tracking with all

imagers On axis tracking value plus 0.06 arcsec per 1 arcmin offset 80% encircled energy 50% encircled energy 2 Radial RMS σ=1 y x D(80%) D(50%) 2rRMS σ=1

Pointing stability On-axis tracking for a period of one hour with

WFI 3 arcsec for visible source brighter than mv=13

FFI 0.8 arcsec for visible source brighter than mv=13

FPI 0.2 arcsec for visible sources brighter than mv=16

Offset tracking with all imagers

On axis tracking value plus 0.06 arcsec per 1 arcmin offset

Image quality 1.5 arcsec diameter of circle enclosing 80% energy at a wavelength of 0.663 µm

Image size Diameter of circle enclosing 80% energy at a wavelength of 0.663 µm Initial operations 5.3 arcsec

(Combined requirement of pointing stability and

image size) First science plus 3 years 1.5 arcsec