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Table E.2
List of mathematical variables.
| Var. | Description | Ref. |
|
|
||
| Au | the astronomical unit | Sect. 3.2 |
| A | attitude matrix | Eq. (8) |
| a | galactocentric acceleration vector | Sect. 6.1.4 |
| a | attitude parameters | Sect. 3.3 |
| an | quaternion coefficient for Bn(t) | Eq. (10) |
| Bn(t) | B-spline function | Sect. B.1 |
| B | iteration matrix | Sect. 4.5 |
| b | right-hand side of normal equations | Eq. (30) |
| bG(t) | barycentric coordinate of Gaia at time t | Eq. (4) |
| C | celestial reference system | Sect. 3.1 |
| C | constraint matrix for the calibration parameters | Eq. (22) |
| c | calibration parameters | Sect. 3.4 |
| D | non-linear Charge Distortion Model | Eq. (D.1) |
| d | update vector | Sect. 4.5 |
| E | expectation operator | Eq. (23) |
| e(t) | CCD exposure function | Appendix D.3 |
| e | error vector | Sect. 4.5 |
| f | field index ( ± 1 for preceding/following) | Eq. (13) |
| f | detector coordinates | Eq. (1) |
| fP, fF | preceding, following viewing direction | Fig. 2 |
| G | Gaia broadband magnitude | Sect. 3.4 |
| g | gate index | Eq. (13) |
| g | global parameters | Sect. 3.4 |
| h | auxiliary data, e.g., ephemerides | Eq. (7) |
| I | the identity matrix | Sect. 4.5 |
| i | subscript denoting some source | Sect. 3.2 |
| j | “short” calibration time interval index | Eq. (15) |
| K | no. of TDI periods for CCD integration | Sect. 3.5 |
| K | preconditioner matrix | Sect. 4.5 |
| k | TDI index in CCD pixel stream | Sect. 3.5 |
| k | “long” calibration time interval index | Eq. (15) |
| k | iteration index | Eq. (40) |
| L | Line Spread Function | Eq. (23) |
|
shifted Legendre polynomial of degree r | Eq. (15) |
| l | subscript denoting some observation | Sect. 3.6 |
| ℓ | left index in a knot sequence | Sect. B.2 |
| M | spline order – M = 4 for cubic spline | App. B |
| N | number of degrees of freedom for a spline | App. B |
| N | normal equations matrix | Eq. (30) |
| n | attitude parameter index | Eq. (10) |
| n | CCD index | Eq. (14) |
| n | dimension of the normal matrix N | Eq. (35) |
| n | nuisance parameters | Eq. (1) |
| p i | normal-triad component | Eq. (5) |
| Q | objective function to be minimized | Eq. (24) |
| q | attitude quaternion | Eq. (10) |
| q i | normal-triad component | Eq. (5) |
| r | degree of the small-scale calibration polynomial | Eq. (15) |
| r i | normal-triad component | Eq. (5) |
| R l | residual of observation l | Sect. 3.6 |
| S | Scanning Reference System | Sect. 3.1 |
| s | local scale factor | Eq. (D.3) |
| s | astrometric (“source”) parameters | Sect. 3.2 |
| s i | the astrometric parameters for source i | Sect. 3.2 |
| T | light integration (exposure) time on CCD | Sect. 3.4, D.3 |
| t k | time for sample k | Sect. 3.5 |
| t l | time for observation l | Sect. 3.5 |
| tep | reference epoch for astrometric parameters | Sect. 3.2 |
| tfr | reference epoch for frame rotator | Sect. 6.1 |
| tP | reference epoch for non-rotating ICRS sources | Sect. 6.1.4 |
| tPM | reference epoch for moving ICRS sources | Sect. 6.1.4 |
| U | relegation factor for primary-star selection | Sect. 6.2.2 |
| u i | proper direction to source i | Sect. 3.2 |
|
geometric direction to source i | Sect. 3.2 |
| V | nullspace of the normal matrix | Sect. 4.5 |
| W i | weight matrix for source i | Eq. (57) |
| w l | downweighting factor for observation l | Sect. 3.6, 5.1.2 |
| W l | statistical weight of observation l | Sect. 3.6 |
| x | differential parameter vector | Sect. 4.3 |
| [ X Y Z ] | celestial reference system (ICRS or CoMRS) | Sect. 3.1 |
| [ x y z ] | Scanning Reference System (SRS) | Fig. 2 |
| α | flux (image parameter) | Eq. (23) |
| α i | right ascension of source i at tep | Sect. 3.2 |
| β | background level (image parameter) | Eq. (23) |
| Γc | (conventional) basic angle | Eq. (13) |
| γ | PPN curvature parameter | Eq. (88) |
| δ i | declination of source i at tep | Sect. 3.2 |
| ϵ a | excess attitude noise | Eq. (28) |
| ϵ i | excess source noise (for source i) | Eq. (28) |
| ϵ l | excess noise in observation l | Eq. (28) |
| ε | orientation correction (frame rotator) | Sect. 6.1 |
| ζ | across-scan (AC) field angle | Eq. (12) |
| η | along-scan (AL) field angle | Eq. (13) |
| κ | along-scan (AL) pixel coordinate | Eq. (23) |
| (image location) | ||
| λ | regularization parameter for the attitude | Eq. (81) |
| update | ||
| λ k | intensity for CCD sample values | Eq. (23) |
| μ | across-scan (AC) pixel coordinate | Sect. 3.4 |
| (image location) | ||
| μ α ∗ i | proper motion in α ( × cosδi) for source i | Sect. 3.2 |
| μ δi | proper motion in δ for source i | Sect. 3.2 |
| μ ri | radial proper motion for source i | Sect. 3.2 |
| μ 0 | proper motion due to galactocentric | Sect. 6.1.4 |
| acceleration | ||
| ν | number of degrees of freedom | Sect. 5.1.2 |
| ϖ i | parallax for source i | Sect. 3.2 |
| ϱ | normalized RSE error of astrometric | Sect. 7.2.2 |
| parameters | ||
| σ l | formal standard uncertainty for observation l | Sect. 3.6 |
| τ n | attitude spline knot n | Sect. B.1 |
| ϕ | along-scan instrument angle | Eq. (12) |
| ω | spin correction (frame rotator) | Sect. 6.1 |
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