The separation of the echelle orders in the spatial direction by the
cross-disperser complicates the calculation of the wavelength
parameterization of high-dispersion IUE spectra. NEWSIPS departs from
IUESIPS in seeking a 1:1 correspondence between dispersion parameters
and physical properties of the spectro-optical system. The goal of such
a representation is to identify each term with optical properties of
the spectrograph and to prevent these physical effects from introducing
cross-terms that could complicate the estimate of errors in the
wavelength solution (for a full discussion of this concept, see
Smith 1990a, 1990b). In the raw geometry this is not possible
because the rotation of the order produces a correlation of the line and
sample positions for a wavelength. For low-dispersion images this
problem is largely solved by de-rotating the image. In the
high-dispersion geometry the dispersion axis is also dependent on the
echelle order: the precise angle is equal to the tangent of the
dispersing powers of the echelle and cross-dispersing grating. Because
this factor varies as 1/*m*_{ech}, the dispersion axis slowly rotates
and produces a splaying of the orders (see Chapter
7.3.1). In order to
place all echelle orders along a common pseudo-dispersion axis, the
order-splaying is removed as part of a single resampling step in the
*GEOM* module. This removes the second and last of the important
cross-coupling terms between the two axes in the original raw space.
Thus the *GEOM* resampling forces the echelle orders to fall along a
common *sample* axis (*s*) and to be separated by a difference in
*line* positions (*l*) on the high-dispersion SI. This operation
produces a small tilt of a monochromatic image on an order which is
usually ignored because its effect on the spectral instrumental profile
is small.

The representation of the dispersion parameters in the rectilinear (*s*,
*l*) high-dispersion SI geometry can be expressed as a Taylor expansion
of the grating equation in terms of the quantity .The equations for the dispersion solution for sample and line positions
are:

(2) |

(3) |

In practice higher-order terms in the Taylor expansion of the dispersion
solution are small for the IUE gratings and need not be considered.
However, the quadratic term is still significant for the IUE grating
geometry (Smith 1990a, 1990b). Furthermore, although the cubic and
quartic terms are not significant in the Taylor expansion of the grating
equation, such terms *do* arise from the electro-optical distortions
within the IUE cameras. These high-order terms (quadratic, cubic, and
quartic) have been determined empirically as a function of echelle order
and incorporated within the *GEOM* module as additional terms, as in
the linearization process performed in low dispersion (Chapter
8.1). The
result of these *GEOM* corrections is an SI with very nearly linear
relation between wavelength and sample position in each order.