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.

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.

FITS key | Parameter | Description | Units |
---|---|---|---|

CTYPE1 | projection type along horizontal axis^{[a]} | ||

CTYPE2 | projection type along vertical axis^{[a]} | ||

CRPIX1 | x_{c} | horizontal coordinate of reference pixel^{[b]} | pix |

CRPIX2 | y_{c} | vertical coordinate of reference pixel^{[b]} | pix |

CRVAL1 | α_{c} | longitude-like spherical coordinate related to CRPIX1 | deg |

CRVAL2 | δ_{c} | 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.

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.

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 sin²*φ*+ cos²*φ*= 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).