ACTA ASTRONOMICA
Vol. 37 (1987) pp. 79–88
by
Toruñ Radio Astronomy Observatory,
Nicolaus Copernicus University,
ul. Chopina 12/18, PL87100 Toruñ, Poland
Received June 10, 1986
ABSTRACT The diurnal rotation of the, sky, when viewed in the horizon system of coordinates, exhibits a singularity at the zenith where the angular azimuthal speed and acceleration are infinite. This prevents all telescopes of the altitudeazimuth mounting, no matter how good is their performance, to be able to track celestial objects in a small region of the sky about the zenith. A simple algorithm to determine the shape and size of this blind spot is presented. An example is given for the projected 32m radio telescope to be located near Toruñ. 
It is known that both radio and optical telescopes with the altitudeazimuth mounting fail to follow the diurnal rotational motion of the sky in a close neighbourhood of the zenith. The extent and shape of this blind circumzenithal spot depends in a complicated way on the maximum velocity and acceleration which can be attained by the azimuth drive. Since, in practice, the spot is small in size this inability of tracking is a rather minor disadvantage of the mounting. Nevertheless, it is clearly important for the telescope designer, as well as for the user, to be aware of its existence and nature. Possibly the best insight into nature of this drawback is gained through noting that the closer a celestial object passes the zenith, the less time it needs to swing between two arbitrarily chosen vertical circles, which define the angular range a telescope must rotate.
Some formulae for estimation of the dimensions of the blind spot have been given by Watson (1978), but his equations are all expressed in the azimuthzenith distance coordinate system, what makes them inconvenient for certain practical applications. In particular, this is the case with all problems related to analysis of the time behaviour of the telescope while traversing the spot during tracking an object with a critical declination. In this paper we present a simple algorithm to compute all important quantities related to the blind spot and expressed as functions of the hour angle (or sidereal time) and declination. Most of these equations were originally developed, prior to familiarizing ourselves with the Watson's work, for an investigation of the blind spot of the projected 32m radio telescope of the Toruñ Radio Astronomy Observatory (Borkowski 1986). Therefore, all illustratory examples concern this telescope and geographical location of the Observatory. It will be noted, however, that the examples are pretty typical and easy to rescale.
The next two sections contain the above mentioned algorithm. In section 4. we present further study which is intended to validate the assumption underlying the algorithm that other theoretically possible limitations are unimportant for a typical radio telescope and in particular can be safely neglected with the Toruñ design. Another set of formulae is given for the case when the azimuth drive acceleration may be a liiiting factor.
Most of textbooks on spherical and practical astronomy give formulae necessary to transform coordinates of the equatorial system, i.e. the declination δ and hour angle t, to the horizon system, i.e. the zenith distance z and azimuth A, given the observer's geographical latitude φ. They can be conveniently condensed into the following equations:
z(t) = arccos(sinφsinδ + cosφcosδcos t),  (1) 
A(t) = arctg[sin t/(sinφcos t – cosφtgδ)],  (2) 
p(t) = arctg[–sin t/(sinδcos t – cosδtgφ)],  (3) 
which are in the form of functions of the hour angle. Both the azimuth and the parallactic angle, p, are defined to lie in the range from –π to π (westward positive) centered at the observer's meridian (south of the zenith), the quadrant being determined by relative signs of the numerator and denominator of the arctangent function arguments (Eqs. 2 and 3).
We denote the rate of change, or velocity, by placing a dot above the corresponding coordinate [to avoid awful setting of the dotted symbols in html language this web version has the time derivatives or velocities distinguished by a prime mark (′)]. By differentiating the three above functions with respect to t we have for velocities
z′ = cosφsin A,  (4) 
 (5) 
p′ = cosφcos A/sin z,  (6) 
which, by virtue of Eqs. (1) and (2), can be understood as functions of time. Here, natural units are assumed, i.e. rad/rad, which can be converted to deg./min. through a multiplicatory factor of 0.25 (more precisely, this factor should be increased still by about 0.3% to account for the difference between solar and sidereal time).
The Eq. (4) discloses that the absolute changes of the zenith distance are never faster than these of the hour angle, while velocities in the azimuth and parallactic angle may reach arbitrarily high values as z approaches 0. Since the parallactic angle carries somewhat different meaning and lesser importance and, in fact, its behavior near to the zenith resembles very much that of the azimuth, it becomes clear that the most serious impact on the blind spot arises from azimuth drive limitations, and in particular from its finite velocity.
The rate of change of the azimuth (Eq. 5) can be shown (see also Fig; 1) to possess an absolute maximum at the transit, where its value is
A′ = cosδ/sin(φ – δ) .  (7) 
The circumpolar objects, as evidenced by this expression, have both positive and negative azimuthal velocities. The change of sign takes place at hour angles equal to ±arccos(tgφ/tgδ), the angles at which A′ = 0. An altitudeazimuth telescope tracking the diurnal motion of an object with a fixed declination will keep pace with it until a point, east of the meridian, where A′ of Eq. (5) reaches the maximum allowable speed of the telescope, V. To keep our notation simple it is assumed that this limiting value of the azimuth drive is signed according to the actual direction of object's motion during the meridian passage: V = V sign(φ – δ).

Considerations on the actual sky velocity, and specifically at the meridian (Eq. 7), immediately lead to the conclusion that limitation to the tracking refers only to objects whose declination lies in the range
 (8) 
This inequality can serve to determine the northsouth extent of the blind spot. Objects with declination satisfying this condition begin to move faster than the telescope is able to, at the moment for which the hour angle is
 (9) 
obtained by solving Eq. (1), subject to the condition A′ = V, for t = t_{o}. Though Eq. (9) is exact, admittedly it is not simple enough for quick and easy reference. If, however, a small loss in accuracy is admissible this equation can be much simplified to
 (10) 
which exhibits extrema of about –0.5/V at the azimuths of –π/4 and –3π/4, corresponding to objects' declination of
δ = φ – cosφ/(2V).  (11) 
While Eq. (9) gives an exact eastern boundary of the blind spot, the above approximations may be used for rapid estimates, whose accuracy will probably satisfy even a quite demending steering program designer. Assuming that while lagging (over the spot) the telescope rotates with its highest speed in the azimuth and keeps due pace with the tracked object in the altitude, the increasing angular separation between the object and the telescope direction can be expressed by
 (12) 
A_{o} = A(t_{o}) + π[1 – sign(φ – δ)].  (13) 
At this point it is of no real importance that the definition (13) includes a 2π shift of azimuth. It is meant for a later use where it will effectively help to avoid problems resulting from a discontinuity inherent in the conventional definition of this coordinate when the transit of objects with δ > φ is involved. It poses no difficulty to observe that the separation (12) is greatest at t = –t_{o}, so that
θ_{max} = 2arcsin[cosδsin t_{o} sin(Vt_{o} – A_{o})/sin A_{o}].  (14) 
From this moment the telescope regains the lost distance and eventually catches up with the object at some hour angle greater than –t_{o} (west of the meridian). For the time of reacquisition, at the western boundary of the blind spot, the following transcendental equation obviously holds
t_{+} = t_{o} + [A(t_{+}) – A_{o}]/V.  (15) 
It is of moment to note that V given above is slew velocity and may sometimes be greater than the maximum tracking velocity. The solution that follows is clearly affected by distinguishing between V and, say, V_{slew} in Eq. (15).
In solving for the hour angle of the western edge of the blind spot, t_{+}, we adopted the Newton's method. The following guess for start proves very efficient:
 (16) 
 (17) 
to yield results that are overestimated by less than 1 second of time (as deduced from numerical computations). Since a real telescope misses objects during traversing the spot for somewhat longer time than that implied by t_{+} – t_{o} of Eq. (15) (allowence must be made for the finite deceleration with which the telescope slows down its highest speed to match the actual sky rotation at the reapproach), we find ourselves on the safe side accepting the solution (17).

Formula (17) completes our determination of the zenithal blind spot in the hour angledeclination coordinate system. The presented algorithm gives the optimal solution, i.e. the smallest spot. In fact, the spot has a peculiar shape, as depicted in Fig. 2. It will be noted, however, that the shape varies significantly with the choice of other schemes of driving the telescope through it. Worthy mentioning is, for instance, the procedure of symmetrizing the blind spot as described by Watson (1978), which results in entirely different shape. We observe that our algorithm would produce the western boundary of such a spot if we had chosen deliberately t_{o} = 0 in Eqs. (13) and (15). We have found that, as could be expected, the symmetric spot is slightly wider in the eastwest direction than the optimal one. At the farfromzenith end the widening is the largest (but less than 20% for examined La Palma and Toruñ telescope cases) and vanishes as declination nears that of the zenith. The extra time needed for pass over the symmetric spot is necessarily unimportant since it will usually be well below 1 minute of tracking time. It should be born in mind, however, that the procedure requires the telescope control programs to incorporate in one or another way the a priori limits, which are not directly related to the actual motions of the telescope and tracked object.
To get a practical example of the use of our algorithm consider a telescope with maximum speed in the azimuthal axis overriding the diurnal sky rotation by a factor of 120 (V = 120, or 30°/min.), and located at the geographical latitude φ = 53.1°. Such a telescope cannot track astronomical objects with declination between 52.8° and 53.4° (Eq. 8), beginning with hour angles (t_{o}, Eq. 9) of –0h0.96m corresponding to the declination of 52.95° (Eqs. (10) and (11) give –0h0.95m and 52.96°, respectively). The eastwest extent of the spot (t_{+} – t_{o} calculated from Eqs. 15 to 17) has a maximum at δ = φ of about 6 min. (this value can also be rapidly approximated by t_{1} from Eq. (16), where A_{o} = –π/2 and t_{o} = 0). The complete limits for tracking of this example are presented in Fig. 2, which has been computed using Eqs. (9), (13), (16) and (17) with (2) and (5) as user defined functions in FORTRAN programming language. In the scale of this figure an approximate solution based on Eq. (10) is indistinguishable from this actually displayed.
In radio astronomy the radio telescope directivities allow to see sources which are sometimes considerably away from the direction of maximum gain. This narrows the region of ineffective tracking, assuming some loss of antenna gain can be tolerated. In this case, the limits for effective tracking can be found by using the actual angular mispointing as expressed by Eq. (12). Given the tolerable θ(t) < θ_{max} (Eq. 14), Eq. (12) can be solved for t = t_{oNew} and t = t_{+New} to yield the new limits (provided there are any, i.e. the solutions satisfy the inequality t_{o} < t_{oNew} < t_{+New} < t_{+}).
There are number of other theoretically possible limiting factors to the tracking near the zenith. These concern the finite azimuth drive acceleration, and the velocity and acceleration of the altitude and field rotation drives. This section is devoted to their brief discussion to conclude that in practice the blind spot is determined by the azimuth drive alone.
The sky acceleration in the azimuth is
A′′ = –cos^{2}φsin(2A) (1 + ctg z tgφ/cos A + 2 ctg^{2}z)/2.  (18) 
This equation is quadratic in ctg z and, after solving with the use of the normal quadratic formula, can be used to construct an equal acceleration contour (Fig. 3). In a close vicinity of the zenith the contour can be conveniently approximated by

 (19) 
 (20) 
where A′′ is set to a, the maximum azimuth drive acceleration. The contour due to acceleration (20) intersects that for the velocity (10) at (for illustration see Fig. 3)
 (21) 
 (22) 
Eqs. (21) and (22) are interpreted to determine approximately the region around the zenith where the blind spot due to acceleration limit is larger in the eastwest direction than the velocity spot. Analysed examples entitle us to add again that, although the Eqs. (19) to (22) are all approximations, they are nevertheless very accurate near the zenith due to the smallness of the zenith distances and hour angles involved.

On passing to a conerete example note that in practice radio astronomy telescopes have accelerations in both axes of the order of, but frequently exeeeding, 1°/s^{2} which corresponds to about 3.3·10^{6} of natural units (rad/rad^{2}). Assuming this value for the limit of attainable acceleration and our earlier V = 120 gives –1^{s} for t* (Eq. 22) and a belt 0.26"cosφ in width in declination centered at the zenith (Eq .21). This is evidently negligibly small region. Restricting the tracking acceleration to 0.02°/s^{2} enlarges the region of dominant acceleration limit to –0.7^{m} and 9.2'cosφ, respectively. The eastern edge of the combined spot is in general the outer envelope of the two contours and its shape can easily be inferred from Fig. 3, where two contours of each type are plotted superimposed in the same coordinate system. It is clear now that in most practical cases the acceleration contour lies almost entirely inside the velocity contour. Be it desirable, however, the simplest way to incorporate the acceleration limit into our algorithm is to set t_{o} of Eq. (9) at t* for declinations enveloped by the two of Eqs. (21). Noteworthy, this problem disappears altogether if earlier mentioned procedure of symmetrizing the blind spot is adopted in a steering telescope control program.
As already noted, Eq. (4) shows that velocity in the zenith distance (or altitude) is less than or equal to 1 rad/rad, or l°/4 min., which is well below any reasonable technical limit of the altitude drive. Likewise, the acceleration in this coordinate, z′′ = A′ cos A cosφ, never exceeds the value of the azimuth velocity. This can be interpreted to mean that in practice the spot generated by the altitude acceleration limit lies wholly inside the azimuth velocity spot. Finally, since sin p = sin A cosφ/cosδ and δ is close to φ, the spots due to limits of the field rotation drive closely resemble respective spots due to the azimuth drive. The point about these parallactic angle spots is especially relevant to optical telescopes.
The blind spot around the zenith is determined primarily by the limit of the azimuth drive velocity and in some cases its acceleration. For radio telescopes the acceleration is practically never of real moment. The easy to program algorithm for estimation of the size and shape of the spot due to azimuth velocity is presented in section 3. Aside of basic functions, Eqs. (2) and (5), it consists of the following formulae: (9) for the eastern contour of the spot, and (13), (15) and (17) for its extent. For a telescope designer, simplifications including and following Eq. (10) and those given in section 4 might prove practically more useful than the exact solutions. Examples are given in section 3 and 4. The reader is advised to consult also the work of Watson (1978), which deals with the same problem but through a different approach.
Acknowledgements. The author thanks Prof. G.H.A. Cole (University of
Hull) for drawing his attention to the paper by F. Watson. This work
was in part financially supported through the Government research
problem RPB Nr RR.I 11/2.
REFERENCES
Borkowski, K. M., 1986, Post. Astron., 34, 201 (in Polish).
Watson, F. G., 1978, Mon. Not. R. astr. Soc., 183, 277.