2 Effects of non-plane-parallelism of floating mirror upon vertical direction

Generally, the upper plane of a floating mirror will not be parallel to the corresponding plane tangent to the fluid, and its normal will be tilted at an angle $\Delta z$ with respect to the zenith Z. This situation is depicted in Fig. 1, showing a section of mirror M on fluid F, perpendicular to the mirror's principal (horizontal) lines.

  

Figure 1: Position of a floating mirror inclined by an angle $\Delta z$ with reference to zenith, and its position after rotation through $180^\circ$

An object S (a star) at a zenith distance z with relation to the normal to the fluid OZ (direction to true zenith), will be observed at an angle $z + \Delta z$ with relation to the normal to the mirror (M) in position I (see Fig. 1). If the mirror is rotated through $180^\circ$ (M'), in position II, the angle observed to S will be $z - \Delta z$. Its average value will then be z, and hence it will be free from any error from the tilt of the mirror to the horizontal plane.

When the mirror rotates on the surface of a fluid about O, its normal circumscribes the surface of a cone having the vertex angle $2\Delta z$, whose axis is directed to the true zenith Z. If with prismatic astrolabes (Danjon's and the like), a floating mirror were used instead of the surface of mercury, this would revolve simultaneously with the instrument's (upper) rotatable part, and the observed zenith distances would then be $z_0 + \Delta z = 90^\circ - h_0 + \Delta z$= const., where h₀ is the instrument's altitude referred to a horizontal surface. The basic principle of the method of equal altitudes would thus be maintained. With circumzenithal-type of astrolabes the mercury's surface is stationary and centred at the instrument's rotational axis. If with these instruments the floating mirror (only) were coupled to the upper rotatable part of the instrument, and revolved with it, the above condition would also be fulfilled. If, with a PZT, the mirror were rotated through $180^\circ$ (once) between exposures, the observation would likewise be referred to the true zenith, Z. However, generally, with regard to the family of planes perpendicular to the mirror in O, the deviation ($\Delta z'$) of the rays reflected from the mirror is not constant. This is clear from Fig. 2.

  

Figure 2: Depending on angle of rotation $\Omega$, mirror's inclination $\Delta z$ is characterized by a spherical triangle

Here the (semi-) straight lines k₀, k and $k_{\rm H}$ form a trihedron with the vertex at O; k₀ and k lie in the plane of the mirror, k₀ being a principal (horizontal) line, and $k_{\rm H}$ being produced by the intersection of a horizontal plane with the plane perpendicular to the mirror passing through k. The corresponding spherical triangle to the trihedron is consequently right-angled, and the deviation $\Delta z'$ in the direction $\Omega$ (the angle of rotation of the mirror) measured from k₀ is

$$\tan \Delta z' = \sin\Omega\, .\, \tan\Delta z .$$ (1)

A maximum deviation $\Delta z'$ occurs in sections perpendicular to the mirror's principal lines (in the direction of its sloping lines, such as in Fig. 1); and $\Delta z' = 0$ in sections passing through these lines. From Eq. (1) it is also clear that the deviation in the observed zenith distance is $-\Delta z'$ for the angle of rotation $\Omega + 180^\circ$, as we have stated earlier.

Let us derive the error dz' in $\Delta z'$ due to variations d$\Omega$ in the angle of rotation $\Omega$ (occuring from free-play in mirror's coupling with the rotational part of the instrument, or eventually, from an incorrect setting of $\Omega$ through $180^\circ$, e.g.). Differentiating (1), we obtain: ${\rm d } z' = \cos^2\Delta z' .\cos\Omega .\tan\Delta z.{\rm d}\Omega$. Assuming that $\Delta z$ and $\Delta z'$ are small quantities of the first order (this condition can be fulfilled, as will be seen later), the formula, correct to second order terms (the neglected terms being of third order and less), is:

$${\rm d}z' = \Delta z\, .\, \cos\Omega\, .\, {\rm d}\Omega .$$ (2)

The error dz' is at its maximum, if $\Omega = 0$, i.e. $\Delta z.\Delta \Omega$ (when the plane perpendicular to the mirror passes through its principal line); and it is zero if $\Omega = 90^\circ$ (when the respective perpendicular plane is in the direction of the mirror's sloping line). From this it follows (for the error to be minimized or eliminated), that the mirror's sloping line should be oriented: (a) in the direction of the vertical plane of observation (optical axis) with astrolabe-type of instruments, (b) in an east-west direction with a PZT.

If we require that the error in zenith distance d$z' \leq 0.01''$, it is necessary to orient the mirror with an accuracy given in Table 1. The values of $\Delta\Omega$ were computed here according to Eq. (2), and $\overline{\Delta\Omega}$ is the corresponding linear displacement tangent to the circle of radius R = 100 mm.

   Table 1: Necessary accuracy in orientation of the mirror: $\Delta\Omega$ (angular); and $\overline{\Delta\Omega}$ (linear at a radius of 100 mm) for some values of its inclination $\Delta z$ and certain rotation angles $\Omega$ for the error in zenith distance dz' to be $\leq 0.01''$

From Table 1 it is obvious that the tolerances in the mirror's orientation are moderate, if $\Omega = 90^\circ$ (the mirror's sloping line being in the direction of observation): $\overline{\Delta\Omega}$ is then about 2 mm for mirror inclination $\Delta z = 60''$. However, if $\Omega= 0^\circ$, the tolerances are quite severe. The latter circumstance provides the ability to make practical adjustment of the direction of the mirror. If we first determine (by means of an autocollimation method, for instance) the mirror's position in the direction of its principal line ($\Omega = 0$), where the changes dz are the maximum with varying $\Omega$, we then set it into the direction of its sloping line by rotating it by $90^\circ$. It also therefore follows that the mirror's rotational axis has to be adequately centred. Otherwise, its fluctuations exceeding the above tolerances would also affect the accuracy of mirror's orientation.