Focal-Plane Astrometry

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+ +

+
+ Next: Numerical Methods +
+Up: EXPLANATION AND EXAMPLES +
+ Previous: Radial Velocity and Light-Time Corrections +

+Focal-Plane Astrometry +

+The relationship between the position of a star image in +the focal plane of a telescope and the star's celestial +coordinates is usually described in terms of the tangent plane +or gnomonic projection. This is the projection produced +by a pin-hole camera and is a good approximation to the projection +geometry of a traditional large f-ratio astrographic refractor. +SLALIB includes a group of routines which transform +star positions between their observed places on the celestial +sphere and their $[\,x,y\,]$ coordinates in the tangent plane. The +spherical coordinate system does not have to be $[\,\alpha,\delta\,]$ but +usually is. The so-called standard coordinates of a star +are the tangent plane $[\,x,y\,]$ , in radians, with respect to an origin +at the tangent point, with the y-axis pointing north and +the x-axis pointing east (in the direction of increasing $\alpha$ ). +The factor relating the standard coordinates to +the actual $[\,x,y\,]$ coordinates in, say, millimetres is simply +the focal length of the telescope. +

+Given the $[\,\alpha,\delta\,]$ of the plate centre (the tangent point) +and the $[\,\alpha,\delta\,]$ of a star within the field, the standard +coordinates can be determined by calling +sla_S2TP +(single precision) or +sla_DS2TP +(double precision). The reverse transformation, where the + $[\,x,y\,]$ is known and we wish to find the $[\,\alpha,\delta\,]$ , is carried out by calling +sla_TP2S +or +sla_DTP2S. +Occasionally we know the both the $[\,x,y\,]$ and the $[\,\alpha,\delta\,]$ of a +star and need to deduce the $[\,\alpha,\delta\,]$ of the tangent point; +this can be done by calling +sla_TPS2C +or +sla_DTPS2C. +(All of these transformations apply not just to $[\,\alpha,\delta\,]$ but to +other spherical coordinate systems, of course.) +Equivalent (and faster) +routines are provided which work directly in $[\,x,y,z\,]$ instead of +spherical coordinates: +sla_V2TP and +sla_DV2TP, +sla_TP2V and +sla_DTP2V, +sla_TPV2C and +sla_DTPV2C. +

+Even at the best of times, the tangent plane projection is merely an +approximation. Some telescopes and cameras exhibit considerable pincushion +or barrel distortion and some have a curved focal surface. +For example, neither Schmidt cameras nor (especially) +large reflecting telescopes with wide-field corrector lenses +are adequately modelled by tangent-plane geometry. In such +cases, however, it is still possible to do most of the work +using the (mathematically convenient) tangent-plane +projection by inserting an extra step which applies or +removes the distortion peculiar to the system concerned. +A simple r₁=r₀(1+Kr₀²) law works well in the +majority of cases; r₀ is the radial distance in the +tangent plane, r₁ is the radial distance after adding +the distortion, and K is a constant which depends on the +telescope ( $\theta$ is unaffected). The routine +sla_PCD +applies the distortion to an $[\,x,y\,]$ and +sla_UNPCD +removes it. For $[\,x,y\,]$ in radians, K values range from -1/3 for the +tiny amount of barrel distortion in Schmidt geometry to several +hundred for the serious pincushion distortion +produced by wide-field correctors in big reflecting telescopes +(the AAT prime focus triplet corrector is about K=+178.6). +

+SLALIB includes a group of routines which can be put together +to build a simple plate-reduction program. The heart of the group is +sla_FITXY, +which fits a linear model to relate two sets of $[\,x,y\,]$ coordinates, +in the case of a plate reduction the measured positions of the +images of a set of +reference stars and the standard +coordinates derived from their catalogue positions. The +model is of the form: + +

+x_p = a + bx_m + cy_m +

+ +

+y_p = d + ex_m + fy_m +

+where the p subscript indicates ``predicted'' coordinates +(the model's approximation to the ideal ``expected'' coordinates) and the +m subscript indicates ``measured coordinates''. The +six coefficients a-f can optionally be +constrained to represent a ``solid body rotation'' free of +any squash or shear distortions. Without this constraint +the model can, to some extent, accommodate effects like refraction, +allowing mean places to be used directly and +avoiding the extra complications of a +full mean-apparent-observed transformation for each star. +Having obtained the linear model, +sla_PXY +can be used to process the set of measured and expected +coordinates, giving the predicted coordinates and determining +the RMS residuals in x and y. +The routine +sla_XY2XY +transforms one $[\,x,y\,]$ into another using the linear model. A model +can be inverted by calling +sla_INVF, +and decomposed into zero points, scales, x/y nonperpendicularity +and orientation by calling +sla_DCMPF. +

+ +

+
+ Next: Numerical Methods +
+Up: EXPLANATION AND EXAMPLES +
+ Previous: Radial Velocity and Light-Time Corrections +

+SLALIB --- Positional Astronomy Library
Starlink User Note 67
P. T. Wallace
12 October 1999
E-mail:ptw@star.rl.ac.uk +

+ + -- cgit