# FITS Astrometry Calibration

A detailed description of a form used by Munipack to store astrometry calibration in FITS header.

- Keywords and values strictly conforms FITS standard recommendations
- The calibration is fully compatible with other astronomical software.
- The detailed list of astrometry reference stars and primary fitting parameters is provided.
- The image data itself are untouched by the calibration.
- The affine transformation is used for description of all translation, rotation, scale, mirroring, etc.

## Astrometry Keywords

All parameters describing the precious astrometry calibration are presented in a table that follows. For parameters description, see Astrometry Overview.

The background description of representation of spherical coordinates in FITS is defined in the article Greisen and Calabretta: A & A, 395, 1061-1075 (2002). See also links in FITS Support Office.

Note, that the set of parameters used by Munipack is different to set of ones in standard FITS header. For direct manipulation with standard WCS keywords, FITS utilities can be used.

Parameters of astrometry calibration
FITS key | Parameter | Description | Units
--- | --- | --- | ---
CTYPE1 |  | projection type along horizontal axis[a] |
CTYPE2 |  | projection type along vertical axis[a] |
CRPIX1 | xc | horizontal coordinate of reference pixel[b] | pix
CRPIX2 | yc | vertical coordinate of reference pixel[b] | pix
CRVAL1 | αc | longitude-like spherical coordinate related to CRPIX1 | deg
CRVAL2 | δc | latitude-like spherical coordinate related to CRPIX2 | deg
CD1_1 | - c ⋅ cos φ | element of matrix of rotation[c] | deg/pix
CD1_2 | - c ⋅ sin φ | element of matrix of rotation[c] | deg/pix
CD2_1 | - c ⋅ sin φ | element of matrix of rotation[c] | deg/pix
CD2_2 | c ⋅ cos φ | element of matrix of rotation[c] | deg/pix
CRDER1 |  | typical uncertainty in horizontal direction[d] | deg
CRDER2 |  | typical uncertainty in vertical direction[d] | deg
CRUNIT1 |  | units of spherical coordinates[e] |
CRUNIT2 |  | units of spherical coordinates[e] |

[a] The projection type is the same for both axis. Currently only Gnomonic projection (coded as RA---TAN, DEC--TAN) is implemented.

[b] Reference pixel is located at centre of an image (the origin around which the image is rotated). The centre is by default determined from image sizes: CRPIX1 = NAXIS1 / 2 (widht / 2), CRPIX2 = NAXIS2 /2 (height / 2).

[c] φ is an angle of rotation about the reference point (centre). One is an analogy to the position angle (PA), but the PA has origin on twelve and also the opposite direction. c is the image scale in degrees per pixel.

[d] RMS (root mean square) is used for both directions.

[e] Degrees are always used.

## Details Of Astrometry

Munipack adds related comments to the header which includes some additional information describing astrometry calibration in detail.

Astrometry comments are included within the "bracket" lines:

    === Astrometry Solution by Munipack ===
    ..
    ..
    === End of Astrometry Solution by Munipack ===

The form is designed for simple machine recognition and removing.

The items are parts of the comments:

Type
  Absolute (against to an astrometry catalogue) or relative (against to an another image). The second case also induces use of pixels in all coordinates.
Reference catalogue
  Identification of photometry catalogue. Given by EXTNAME of catalogue or <DESCRIPTION> field in <RESOURCE> part of VOtable.
Projection
  Human readable type of projection
Number of objects used
  How many object has been used to estimate the calibration.
RMS
  Root mean square of residuals between catalogue and estimated coordinates.
Residual Table
  First, second columns are equatorial coordinates of reference stars, third, fourth are coordinates on image and last two columns are differences for catalogue and computed coordinates.

## Example

An example of calibration included in FITS header follows (with removed irrelevant lines).

    NAXIS   =                    2 / number of axes
    NAXIS1  =                  768 / length of data axis
    NAXIS2  =                  512 / length of data axis
    CTYPE1  = 'RA---TAN'           / the coordinate type for the first axis
    CTYPE2  = 'DEC--TAN'           / the coordinate type for the second axis
    CRVAL1  = 3.3069918316990805E+02 / [deg] first axis value at reference pixel
    CRVAL2  = 4.2273184170171220E+01 / [deg] second axis value at reference pixel
    CRDER1  =             3.96E-05 / [deg] random error in first axis
    CRDER2  =             3.97E-05 / [deg] random error in second axis
    CUNIT1  = 'deg     '           / units of first axis
    CUNIT2  = 'deg     '           / units of second axis
    CRPIX1  = 3.8400000000000000E+02 / x-coordinate of reference pixel
    CRPIX2  = 2.5600000000000000E+02 / y-coordinate of reference pixel
    CD1_1   = -1.8251021731683297E-04 / partial of first axis coordinate w.r.t. x
    CD1_2   = 2.1990049474366615E-06 / partial of first axis coordinate w.r.t. y
    CD2_1   = 2.1990049474366615E-06 / partial of second axis coordinate w.r.t. x
    CD2_2   = 1.8251021731683297E-04 / partial of second axis coordinate w.r.t. y
    COMMENT === Astrometry Solution by Munipack ===
    COMMENT Type: absolute
    COMMENT Reference catalogue: UCAC5 Catalogue (Zacharias+, 2017)
    COMMENT Projection: GNOMONIC
    COMMENT Objects used = 44
    COMMENT RMS =      142.3699873     [mas]
    COMMENT Scale =      657.0844718     +-   2.0E-02 [mas/pix]
    COMMENT cos(pa) =    0.9999274227 +-   2.1E-06
    COMMENT sin(pa) =    0.0120477932 +-   1.7E-04
    COMMENT Position Angle (pa)  =    0.6903044013 +-   9.9E-03 [deg]
    COMMENT Reflexion =     F
    COMMENT Alpha center projection (CRVAL1) =  330.6991831699 +-   2.5E-06 [deg]
    COMMENT Delta center projection (CRVAL2) =   42.2731841702 +-   3.9E-06 [deg]
    COMMENT Horizontal center (CRPIX1) =   384.000 [pix]
    COMMENT Vertical   center (CRPIX2) =   256.000 [pix]
    COMMENT Catalogue RA,DEC [deg]        Data X,Y [pix]     Residuals [mas]
    COMMENT  330.68963830  42.26674250  422.700  220.245  -282.0E+00   878.9E-03
    COMMENT  330.68923240  42.27652500  424.821  273.784  -174.0E+00    22.5E+00
    COMMENT  330.67244740  42.31684500  494.557  493.933   412.9E+00   -20.5E+00
    ...
    COMMENT  330.61571920  42.30084060  724.180  403.675  -108.1E+00   -47.7E+00
    COMMENT  330.61550500  42.24693730  721.816  108.630  -131.7E+00  -224.3E+00
    COMMENT  330.78309090  42.26153810   42.777  196.604   169.4E+00   -91.0E+00
    COMMENT  330.68781030  42.24563480  428.254  104.291    32.5E+00   151.8E+00
    COMMENT  330.78027980  42.29270640   56.484  366.877   110.6E+00   124.0E+00
    COMMENT  330.63015890  42.26125810  663.472  187.155  -264.1E+00   165.2E+00
    COMMENT  330.68914300  42.23852450  422.511   65.362   -50.4E+00   178.9E+00
    COMMENT Munipack 0.5.4, (C) 1997-2013 F.Hroch (hroch@physics.muni.cz),
    COMMENT http://munipack.physics.muni.cz, Masaryk University, Brno, CZ.
    COMMENT === End of Astrometry Solution by Munipack ===

In this example, the calibration can be easy interpreted as:

- The image has 768 × 512 pixels according to NAXIS1, NAXIS2 keywords.
- CRPIX1 = 768/2, CRPIX2 = 512/2 (at centre of the image size)
- The Gnomonic projection is used (Both CTYPEs contains TAN).
- Centre of projection (CRVAL1, CRVAL2): α = 330.6991832° ± 2.5E-06°, δ = 42.2731842° ± 3.9E-06°
- Scale: 1/√(CD1_1² + CD1_2²) ≅ 5478.7 pix/°, c=√(CD1_1² + CD1_2²) = 1.825e-04 deg/pix (note sin²φ + cos²φ = 1), 3600*c ≅ 0.657 arcsec/pixel.
- Rotation around central pixel: atan(CD1_2/CD1_1) ≅ 0.7°, 1 radian = 180°/π ≅ 57.3°
- CD1_1 < 0 means that Right Ascension increase to left (breaking Cartesian habit)
- There is no reflection.
- Positions of stars has uncertainty about CRDER1,2 ≅ 0.14 arcsec on 1-σ level (68% of stars will fall to the circular neighbourhood with this radius).

## See Also

Astrometry Manual, Astrometry Overview