The continual changing of the parallactic angle is known as field rotation and it is the main difficulty in measuring double stars with alt-azimuth mounted telescopes. The difficulty lies not so much in the fact that the orientation of the field is continually changing, but in the rate at which it is changing. The rate of field rotation, therefore, needs to be evaluated to determine the feasibility of being able to measuring the position angle accurately
The rate at which the parallactic angle is changing, i.e. the instantaneous rate of field rotation, can be found by differentiating the above equation for the parallactic angle. Hence, dq 15 cos2q(tan $ cos 8 cos H - Esin 8) dH (tan $ cos 8- sin 8 cos H)2
15(tan $ cos H - sin 8) sin2 H + (tan $ cos 8-sin 8 cos H)2
The constant, 15, converts the rate to degrees per hour. The second form of the equation enables the rate of field rotation to be found without having to find the parallactic angle.
Evaluating the derivative we find that the rate of field rotation peaks when the star crosses the meridian, i.e. when H = 0. Furthermore, the higher the star's culmination, i.e. the smaller the difference between 8 and j, the greater will be its rate of field rotation when it crosses the meridian. The maximum rate of field rotation, therefore, occurs when a star passes through the zenith. This implies that the worst time to observe a double is when it is best placed for observing! Consequently, there is a spherical cap around the zenith in which the rates of field rotation are too great to enable accurate measurements to be made. Field rotation rates close to the zenith can reach hundreds of degrees per hour. However, such high rates can only be sustained for very short periods (as they clearly cannot rotate more than 360° in 24 hours) after which they reduce to low rates again.
Conversely, the rate of field rotation is zero when the star crosses the prime vertical, i.e. when the star is due east and again when it is due west. Obviously, only stars with declinations that lie between the observer's latitude and the celestial equator will cross the prime vertical. Hence, the best times to observe double stars, as far as field rotation rates are concerned, are when the stars are in the eastern and western regions of the sky.
The average rate of field rotation is, not too surprisingly, 15° per hour. This is half the rate of 30° per hour at which the hour hand of a clock turns. A rotation rate of 360° would be a very high rate, yet it is the rate at which the minute hand of a clock turns.
The problem, then, lies not in whether the rotation rate is too great to make a position angle measurement, but in whether the observation can be timed with sufficient accuracy, i.e. in recording the time when the companion was at that particular zenithal positional angle. For a star with a field rotation rate of 15° per hour, the time of the zenithal position angle measurement would have to be made to an accuracy of 12 seconds; that is to say that the time will have to be noted within 12 seconds of having set the position angle on the micrometer if an accuracy of 0.1 arcminutes is to be achieved. This is because the position angle would have rotated 0.1 arcminutes in 24 seconds and after 12 seconds the position angle will be nearer the next tenth of a degree. In practice one would set the positional angle and then note the time before taking the positional angle reading.
The rate of field rotation that can be tolerated will depend upon how accurately the observation can be timed. If it is done manually and we assume that the time can be read off the clock within 10 seconds of making the position angle setting then we have an upper limit on the rate of field rotation of 18? per hour. Field rotation rates less than this are typically found in the eastern and western sections of the sky. If the time is recorded electronically then much higher rates can be tolerated and the "no go" area around the zenith could be reduced considerably.
The highest rate of field rotation, in degrees per hour, that can be tolerated is just 180? divided by the number of seconds it takes to note the time of the observation, or conversely, divide 180? by the field rotation rate to determine the time limit.
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