National Radio Astronomy Observatory, P. O. Box O, Socorro, NM 87801 USA; Email:

National Optical Astronomy Observatories, P. O. Box 26732, Tucson, AZ 85719 USA; Email:

Australia Telescope National Facility, P. O. Box 76, Epping, NSW 1710 Australia; Email:

UCO/Lick Observatory, University of California, Santa Cruz, CA 95064 USA; Email:

In Paper I, Greisen & Calabretta (2003) describe a generalized method
for specifying the coordinates of FITS data samples. Following that
general method, Calabretta & Greisen (2003) in Paper II describe
detailed conventions for defining celestial coordinates as they are
projected onto a two-dimensional plane. The present paper extends the
discussion to the spectral coordinates of wavelength, frequency, and
velocity. World coordinate functions are defined for spectral axes
sampled evenly in wavelength, frequency, or velocity, evenly in the
logarithm of wavelength or frequency, as projected by ideal dispersing
elements, and as specified by a lookup table. Papers I and II have
been accepted into the FITS standard by the North American, Japanese
and European FITS Committees; we expect the present work to be
accepted as well. The full text of the proposed standards can be
found at http://www.aoc.nrao.edu/~egreisen.

Greisen & Calabretta (2003, ``Paper I'') describes the computation
of the world or physical coordinates as a multi-step process. The
vector of pixel offsets from the reference point is multiplied by a
linear transformation matrix and then scaled to physical units.
Mathematically, this is given by

The basic ``spectral'' coordinates are frequency, wavelength, and
Doppler relativistic velocity. There are several other coordinates
which are proportional to one of these, including wavenumber, energy,
and ``radio'' and ``optical'' conventional velocities. Let us
consider the case in which an axis is linearly sampled in spectral
variable , but is to be expressed in terms of variable . We may
restrict to the basic types since all others are linearly
proportional to one of them. Let us also introduce an intermediate
variable which is the basic variable associated with . The
relationship between and is then
with inverse
. The statement that an axis is linearly sampled in
simply means that

(2) |

(3) |

- Compute once and using Equation (3) and then compute at using Equation (2).
- Compute from using the set of non-linear relationships between the basic spectral coordinates..
- Compute from using the set of linear relationships between the basic and secondary spectral coordinates.

Dispersion coordinates for UV, optical, and IR spectra at nm are commonly given as wavelengths in air rather than in vacuum. The relationship between these is given by and causes a relative difference of around 0.03%. The conversion between wavelengths in air and wavelengths in vacuum adds another step in the chain described above.

Paper III presents a full set of codes to be used in keyword `
CTYPE` for spectral coordinate types and for the non-linear
algorithms involved, including air wavelengths. Keywords `
RESTFRQ` and ` RESTWAV` are reserved to give the line rest
frequency (in Hz) or wavelength (in m) needed for the conversion
between frequency/wavelength and velocity.

One common form of spectral data is produced by imaging the light from
a disperser, such as a prism, grating, or grism, as illustrated in
Figure 1. Paper III presents the full mathematics by
which the wavelength and the spacing at the detector
may be related. The basic grism equation is given by

(4) |

There are numerous instances in which a physical coordinate is well defined at each pixel along an image axis, but the relationship of the coordinate values between pixels cannot be described by a simple functional form. Observations of the same object made at an arbitrary set of frequencies or times are the simplest examples. In addition, the calibration of some spectrographs is represented best by a list of wavelengths for each pixel on the spectral axis.

Fully separable, one-dimensional axes of this type may be represented
by an algorithm, ` -TAB`, defined in Paper III. A FITS binary
table containing only one row is used. The coordinates are given by a
vector of values in a single cell, optionally accompanied by a second
indexing vector in a second cell within the row. The parameters
required by ` -TAB` are the table extension name, the table version
number, the table level number, the column name for the coordinate
vector, and the column name for the optional indexing vector. The
character-valued generic keyword ` PS_` is introduced to
provide the three character-valued parameters of this algorithm. The
coordinate value is found by first evaluating Equation (1) and
adding the reference value. The result is used as a value to be
looked up in the vector of values found in the indexing vector cell. The
corresponding position in the vector of values in the coordinate
vector cell then provides the actual coordinate. If the indexing vector
is omitted, the value found with Equation (1) is used as a
direct index for the coordinate vector.

The ` -TAB` algorithm described above is then generalized to cases
in which the coordinates on axes are dependent on each other, but
the indexing vectors are independent. In this case, the values of the
coordinates are contained in one column of the (one-row) table as an
array of dimensions
, where is the
number of indexing values on axis . The indexing vector for axis
, if present, will occupy a separate column and will contain
values in a one-dimensional array. An additional parameter is
required for each of the coordinates to give the axis number
within the coordinate array.

Paper I has defined a general framework to describe world coordinates in the FITS format; Paper II has extended that framework to describe ideal celestial coordinate representations. Paper III, summarized here, extends the discussion to ideal spectral coordinates and introduces a general table lookup algorithm. All three papers are well on their way to becoming part of the IAU FITS Standard.

Calabretta, M. R. et al. 2003, Representations of distortions in FITS world coordinate systems, in preparation, (``Paper IV'')

Calabretta, M. R. & Greisen, E. W. 2003, A&A, accepted (``Paper II'')

Greisen, E. W. & Calabretta, M. R. 2003, A&A, accepted (``Paper I'')

Greisen, E. W., Valdes, F. G., Calabretta, M. R., & Allen, S. A. 2003, A&A, in preparation (``Paper III'')

- ... Greisen
^{1} - The National Radio Astronomy Observatory is a facility of the (U.S.) National Science Foundation operated under cooperative agreement by Associated Universities, Inc.
- ... Valdes
^{2} - The National Optical Astronomy Observatory is a facility of the (U.S.) National Science Foundation operated under cooperative agreement by Associated Universities for Research in Astronomy, Inc.
- ... Calabretta
^{3} - The Australia Telescope is funded by the Commonwealth of Australia for operation as a National Facility managed by CSIRO.
- ... Allen
^{4} - UCO/Lick Observatory is operated by the University of California.

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