III. 2MASS Details

3. Data Processing

e. Position Reconstruction

Position Reconstruction Technique

Table 3: Formulation of Least-Squares Problem

The total sum "S" of weighted differences squared can be broken down into three components:

(a) frame-to-frame differences

	(Eq. III.3.e.1)

n_f = number of frames in scan
n_kj = number of points common to frames k and j
We_kji = inverse variance weighting for i^th common point between frames k and j
USx_ki = U-scan x coordinate of frame k source associated with point i of the kj set of common points
USx_ji = U-scan x coordinate of frame j source associated with point i of the kj set of common points
USy_ki = U-scan y coordinate of frame k source associated with point i of the kj set of common points
USy_ji = U-scan y coordinate of frame j source associated with point i of the kj set of common points;

(b) frame-to-reference differences

	(Eq. III.3.e.2)

n_f = number of frames in scan
r_k = number of reference stars matched to frame k
Wr_ki = inverse variance weighting for i^th reference match to a frame k source
USx_ki = U-scan x coordinate of frame k source associated with i^th reference match within frame
USy_ki = U-scan y coordinate of frame k source associated with i^th reference match within frame
USxR_ki = U-scan x coordinate of i^th reference star matched within frame k
USyR_ki = U-scan y coordinate of i^th reference star matched within frame k;

and,

(c) relative a priori frame-to-frame differences

	(Eq. III.3.e.3)

n_f = number of frames in scan (only J2 frames used)
n_p = number of artificial pegs in each frame-to-frame overlap
W_ap = arbitrarily small weighting factor
USxP_ki = U-scan x coordinate of peg from frame k associated with i^th common point of the (k-1, k) set
USyP_ki = U-scan y coordinate of peg from frame k associated with i^th common point of the (k-1, k) set
USxP_(k-1)i = U-scan x coordinate of peg from frame k-1 associated with i^th common point of the (k-1, k) set
USyP_(k-1)i = U-scan y coordinate of peg from frame k-1 associated with i^th common point of the (k-1, k) set.

The third set of differences is weighted very lightly and is intended to bridge gaps. Three artificial stars or "pegs" are added between each J2 frame and the next. The "pegs" are placed in a triangular pattern within the a priori frame-to-frame overlaps, with one set between J2 frames 1 and 2, a second set between frames 2 and 3 and so on. With all frames in their a priori positions, these differences are all zero.

In order to formulate the required set of these simultaneous equations,

	(Eq. III.3.e.4)

	(Eq. III.3.e.5)

S_FF = frame-to-frame sums
S_FR = frame-to-reference sums
S_AP = a priori J2-frame-to-J2-frame sums
P_i = parameter i
n_P = total number of parameters,

partial derivatives of S with respect to each of the 563 adjustment parameters are set to zero.