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DETERMINATION OF UNCERTAINTIES ON THE MICRO-GUIDE USE

Ronald Charles Tanguay and Tom Teague have recently published in Sky&Telescope two excellent papers on the handling of the Micro Guide eyepiece of Celestron and its use to determine angular separations and the positioning angle in double stars. However very little has been written about the uncertainties of the measures by means of this eyepiece and for this reason I have developed the detailed calculation of the uncertainties in the Tom's "Standard" method.

When using the Micro Guide (MG) eyepiece the first thing to do is to calculate the scale constant for the focal lenght used. To do this the drift time must be timed in order that a reference star crosses the whole eyepiece linear scale. Since this process is a RANDOM phenomenon due to the different observer's handling of the stopwach, it is evident that the more readings you take the better it will be the estimate of the "real" value of the drift time. The formula used to calculate the scale constant (SC) in arcseconds is the following:

(Eq.1)

Where,
, is the mean of all drift time lectures
, is the conversion factor between the stpowatch time and the sideral time and
is the used reference star declination

Once all the values in the previous equation are introduced we obtain the SC for a characteristic focal. However the use of the previous expression has an uncertainty associated that we should know to be able to determine the total uncertainty of the measures with the MG.

According to incertainties/errors theory propagation the combined uncertainty/error of the SC would be given by the following equation.

(Eq.2)

Where,
is the uncertainty of the time drift and as it is a RANDOM PROCESS its value will be given by the standard deviation of the different measures. Therefore when increasing the number of measures it should diminish the uncertainty..
is the uncertainty of this correction factor and we will assign it the value of 0.0001
is the uncertainty of the reference star chosen to calibrate the MG. With the purpose of standardize the calibrations, the Tycho 2 catalog should be used ,whose uncertainties are of 25 miliarcseconds (mas) and finally,
is the combined uncertainty of the scale constant

Let us see a real example of how to make the calculation. In July 4 2001 I made several measures to calculate the scale constant of my MK-65+Barlow Meade 124, using gamma UMI as reference star and according to Tycho 2 catalogue its declination (J2000) is of 71° 50' 02.318". The 20 measures in seconds were the following:

54.82,54.47,54.24,54.89,55.04,55.05,54.87,54.99,54.69,55.01,54.85,54.77, 54.91,54.74,54.89,55.09,55.01, 54.96,54.99,54.82
Therefore =0.21 s. This uncertainty corresponds to Type A errors and all the other ones to Type B. In this calculation the stopwatch intrinsic error is not taken into account since it is smaller in several orders of magnitude than the answer of any observer.

Substituting the values in (Eq.2) we obtain a SC=4.29, with = 1.61E-02. If we put the uncertainty in a relative way, then SC=4.29±0.38% (with coverage factor k=1).

Once SC is known to calculate the angular separation (SEP) of an couple of stars we should multiply this value for the number of scale divisions (SDiv) of the MG, which means,

SEP = SDiv · SC (Eq.3)

Therefore the total angular separation uncertainty will be:

(Eq.4)

Where,
it is the reading uncertainty of the MG linear scale.

A concept that should be kept very clear is that the reading of the number of MG scale divisions (SDiv) is not a random process. Even if we take many different measures of a pair, that doesn't mean that the precision improves. The maximum precision obtained is determined by the MG precision scale and in this sense the seeing doesn't act as a random process but as an disturbance effect of the measure that should be eliminated. There are two forms of solving this problem, the first one is to use devices of adaptative optics that already exist in the amateur market, and the second one is to wait a nights when the seeing is very good.
Therefore and following the recommendations of all the metrology laboratories, the biggest precision that we can get is of ±0.5 divisions. According to the recommendations of the uncertainties analysis, the standard uncertainty in the measure of the linear scale of the MG will be , that means, 0.3 divisions.

Although some observers say that they can perceive differences of 0.1 divisions, this goes against the recommendations of all international organisms of metrology and in my opinion resolving 0.5 divisions is very complicated unless the seeing is really low.
For example, with the previous scale constant, 0.5 divisions correspond to an angular size of 2", which is quite a habitual seeing in my observation sites. This situation would be worse for small scale constant and therefore more sensitive to the seeing.

As an example and with the previous telescope configuration, measuring the angular separation of beta Cygnus obtain the value of SDiv=8 divisions. Therefore the total uncertainty in the measure of the angular separation is:

SEP=34".3±1".2 (k=1), or in relative uncertainty SEP=34".3±3.5% (k=1),
SEP=34".3±2".4 (k=2), or in relative uncertainty SEP=34".3±7.0% (k=2)

This means, the uncertainty () in the calculation of the angular separation (SEP) is of 3.5% with a coverage factor k=1, which represents a really acceptable value.

To measure the position angle of the double star the protactor scale of the eyepiece is used with minimum divisions of 5º and following international recommendations the resolution in the reading is of 2.5 degrees. I ignore why the manufacturer has adopted such minimum scale so high, but in my opinion and with low seeing the scale resolution can be of 1º taking the readings carefully, and of 2º with no difficulty. Therefore the standard uncertainty (Type B) in the angle measure of the protactor scale will be, , that means, 0.6 degrees.

To measure the position angle accurately it is FUNDAMENTAL to place the axis of the linear scale parallel to the pair axis. However when the scale constant is big and/or the pair separation is small it is very dificult to achieve. How do we know that we are doing it right?. The solution is to repeat the measure several times REPOSITIONING the axis of the linear scale in each measure. The lack of parallelism in each measure between the axis of the linear scale of the MG and that of the pair, will originate variations in the position angle in a RANDOM way, contrary to the measure of the angular separation, and therefore its uncertainty (Type A) DIMINISHES as we increase the number of measures.

Let's suppose that we are measuring the position angle of a pair with a big separation and/or we have a small constant scale in which we obtain the following five values:
230º, 230º, 230º, 230 º, 230º, with mean=230º and with null standard deviation
Does it mean that our uncertainty is null?. Obviously not. As the pair is very open you can achive the alignment of the linear scale with such a precision that the error induced in the position angle is inferior to the angular resolution of the MG (1º-2º). In this case the total uncertainty will be:
, and the final result will be 230º±0º.6 (with k=1) or 230º±1º.2 (with k=2)

Let's imagine now a pair with the same position angle but whose separation is very small and/or big constant scale. In this case the alignment of the linear scale with the pair axis will be very difficult of achieving, and for different reasons there will be variations among the measures that will originate changes in the value of the position angle. Let's suppose the following five values:
228º, 231º, 230º, 229º, 232º with mean=230º and standard deviation=1º.6
Now the total uncertainty of the measure will be:
, and the final result will be 230º±1º.7 (with k=1) or 230º±3º.4 (with k=2)