Modeling Main Reflector Deformations of VLBI Radio Telescopes

The receiving properties of VLBI radio telescopes mainly depend on the surface quality of the parabolic main reflector and the stability of the focal length. Temperature, snow loads, wind, insolation, but also dead load deform the main reflector surface. The deformation of the main reflector leads to deviations from the geometry of an ideal, rotationally symmetric paraboloid. Such deformations have been investigated in detail since the 1980s using methods of close-range photogrammetry, holography and laser scanning. The assumed model of a rotationally symmetric paraboloid is justified for most of the so-called legacy radio telescopes. However, radio telescopes that fulfill the increased requirements of the agenda VLBI2010 and GGOS are usually equipped with an improved main reflector design known as Ring-Focus-Paraboloid. Such telescopes are called VGOS radio telescopes and form the backbone of the next generation of the geodetic VLBI system. In contrast to a rotationally symmetric paraboloid, a ring-focus-paraboloid describes a complex geometrical shape and does not belong to the well-known class of surfaces of the second order (quadric).

In order to validate the surface quality of the main reflector of VGOS-specified radio telescopes, the Laboratory for Industrial Metrology has derived an innovative mathematical model to describe the geometry of a ring-focus-paraboloid. This model has been generalized and represents the complex geometry of a double-elliptical ring-focus-paraboloid. An Error-in-Variables (EIV) model formulated by a Sequential Quadratic Programming (SQP) algorithm is used to estimate the shape parameters rigorously. Introducing constraints to the model yields the simplest shape type: a rotationally symmetric paraboloid. For that reason, the model is unified and is suitable to describe the main reflector geometry of many legacy radio telescopes but also to parameterize the improved reflector geometry of most of the VGOS-specified radio telescopes. Hence, the strength of the derived model is the universal applicability in reverse engineering and the framework of main reflector deformation measurements. Moreover, the model of the double-elliptical ring-focus paraboloid is complex and exceeds the designt geometry of radio telescopes. These additional model parameters can be used to interprete obtained surface deviations geometrically.

At the Onsala Space Observatory (Sweden), two radio telescopes, that are identical in construction, have been installed in 2017. Both telescopes are called the Onsala Twin Telescopes (OTT). As for many VGOS-specified radio telescopes, the geometry of the main reflectors is designed as a ring-focus paraboloid. In order to validate the quality and reliability of the main reflectors, the surface was measured by the manufacturer MT Mechatronics (Mainz, Germany). Due to the high accuracy requirements, the measurements were performed by close-range photogrammetry. The uncertainty of an observed spatial position is about 5 µm + 5 µm/m. This remarkably high accuracy is required to detect and to identify deviations of the surface with respect to the designed geometry. The result of the measurements form the basis for readjusting the main reflector surface of the OTT. The RMS of the final adjustments was well below 100 µm and meets the VGOS specifications.

Due to main reflector rotations around the elevation axis of the VLBI radio telescope, the position of center of mass of the main reflector is shifted and elevation-dependent deformations of the main reflector occur. These deformations change the focal length and - among other - affect the receiving properties of a VLBI radio telescope. Investigations of legacy VLBI radio telescopes show discrepancies on the centimeter level which, if unconsidered, bias the global position and the scale of the reference frame systematically. In order to study the deformation behavior of compact VGOS-specified VLBI radio telescopes, the surface of the main reflector as well as the sub reflector position was photogrammetrically observed in several elevation positions at Onsala. For the first time, the surface measurements are carried out by an unmanned aerial vehicle (UAV). In total, 21 measurement campaigns were performed and the variation of the focal length of one of the Onsala Twin Telescopes was derived. The OTT-design is more compact and stable, and in comparison to legacy radio telescopes the obtained variation of the focal length is about 10 times smaller. For the telescope under investigation, an elevation-dependent correction function was derived to compensate for variation of the signal path. The maximum signal path variation is about 500 µm and yields a time delay of less than 2 ps.