4 Misalignment measurements at the VLT

4.1 General procedure

The VLT has both Nasmyth and Cassegrain focii. It is optimised to be a Ritchey-Chretien telescope at the Nasmyth focus. The Cassegrain focus has a different aperture ratio from that of the Nasmyth focus. To switch from the Nasmyth to the Cassegrain focus one has to change the distance between the mirrors (done by refocussing with M2) and bend the primary mirror to remove spherical aberration. In its Cassegrain configuration the telescope is no longer aplanatic. Furthermore, the stop is at the secondary mirror. For the measurement of the misalignment using the features of the field astigmatism one therefore has to use the constants B'₀, B'₁ and B'₂ given at the end of Sect. 3.3.

The measurement procedure is then as follows. The telescope is first corrected for coma at the center of the adapter. With B'₀, B'₁ and B'₂ known, the values for $\alpha_{x}$ and $\alpha_{y}$ could, in principle, be obtained from one measurement of $Z_{4}^{{\rm sys}}$ and $Z_{5}^{{\rm sys}}$somewhere in the field of the telescope. But, in large telescopes field independent astigmatism can easily be generated elastically. In addition, the measurements are, for example due to local air effects, not free of noise. Therefore, it will be necessary and more accurate to do measurements at several locations in the field and obtain the values for $\alpha_{x}$ and $\alpha_{y}$ with a least squares fit. We use typically eight measurements at evenly distributed points at the edge of the field. Such eight measurements would at least take fifteen minutes. During this time the VLT optics changes, because of elastic deformations due to changes of the zenith distance, significantly. In particular, the aberrations decentering coma and astigmatism are strongly affected. Therefore, we had to do a closed loop active optics correction at the center before each measurement at the edge and, in addition, had to subtract the variation generated by the change of altitude between the correction at the center and the measurement at the edge.

4.2 Measurement of the accuracy of the method

By changing the misalignment in a well defined way it is possible to estimate the accuracy of this method. The VLT has got the useful feature that the primary mirror can be moved by motors in five degrees of freedom. The only full body movement which cannot be remotely controlled, is a movement in the direction which is perpendicular to the optical and to the altitude axis. This allows that, in particular, the primary mirror can be tilted arbitrarily around its vertex. We can therefore modify the misalignment angle $\Delta \alpha$ between the primary and secondary mirror very accurately. By measuring afterwards $\Delta \alpha$ we can therefore check the accuracy of the method described above and, in addition, the validity of the theoretical parameters B'₀, B'₁ and B'₂.

The expected change $\Delta \alpha$ of the angle between the axes of the primary and secondary mirrors due to a rotation of the primary mirror around its vertex can be deduced from Fig. 4. In an initially perfectly aligned telescope first the primary mirror has been rotated around its vertex by $\Delta \phi_{1}$. Afterwards decentering coma has been corrected by rotating the secondary mirror by $\Delta \beta$around its center of curvature. The axes of the primary and secondary mirrors then intersect at the coma-free point $P_{{\rm cfp}}$. For small angles we get

$$\Delta \beta = -\frac{d_{1} - z}{2f'_{2} + z}\, \Delta \phi_{1}.$$ (83)

The angle between the axes of the primary and secondary mirrors is then

$$\Delta \alpha = \Delta \beta - \Delta \phi_{1}.$$ (84)

The derivation of the change of the angle $\Delta \alpha$ between the axes of the primary and secondary mirrors does, at least for small initial misalignments, not depend on the actual initial state of the telescope. Equation (84) is therefore always correct.

With the VLT parameters one gets

$$\Delta \alpha = -6.64 \Delta \phi_{1}.$$ (85)

4.3 Measurement data

The first mapping was done for the initial setup of the telescope. The second and third mappings were done after tilting the primary mirror around it vertex around two orthogonal axes A and B each time by 20''.

For a rotation of the primary mirror around its vertex by 20'' one expects a change $\Delta \alpha = 132.8''$.

In addition, a rotation around the vertex will shift the whole pattern due to the tilt of the M1 axis, but the shift is only of the order of 0.5'' on the sky and therefore negligible.

The three mappings gave the following results for the x- and y-components of $\alpha$ (all figures in arcseconds).

$\Delta \phi$	$\alpha_{x}$	$\alpha_{y}$	$\Delta \alpha_{x}$	$\Delta \alpha_{y}$	$\Delta \alpha$
0	39.24	149.18
20 A	38.52	285.23	-0.72	136.05	136.05
20 B	179.17	142.67	139.93	-6.51	140.08

The total change $\Delta \alpha$ of the misalignment is defined by $\Delta \alpha = \sqrt{(\Delta \alpha_{x})^{2} + (\Delta \alpha_{y})^{2}}$. The average of the measured changes of the misalignment angles $\Delta \alpha$ between the first and the second configuration on the one hand and the first and the third configuration on the other hand is $\Delta \alpha = 138.07''$. The difference to the expected value $\Delta \alpha = 6.64 \cdot 20'' = 132.8''$ is only 5.3''. If, after a rotation of the secondary mirror around the coma free point a similar accuracy is achieved, the two nodes of the binodal field will only be 7'' apart. At the edge of the field of $0.25^{\circ}$ this would lead to an error in the coefficient of third order astigmatism of $56~ {\rm nm}$, which is negligible.