The timing of visual observations :

1/ Personal equation and accuracy

If you want to make an visual observation of any scientific interest, it is necessary to subtract your reaction time (personal equation = P.E.) from your timings and to give the estimated accuracy of your measures. Very often, observers underestimate their P.E. and the uncertainty of their results. Here are some cues to estimate them, at least crudely.

Everyone has his own P.E. : at each timing, you must subtract it to find the "true" time. But the P.E. varies a little at every timing : this fluctuation generates an uncertainty in the "true" time and decreases the accuracy of the timing.

Here is a simple method to measure the P.E. with a digital stopwatch. With a piece of paper, hide the digits of seconds : SS.ss, and start the stopwatch. As soon as you see the minute digit changing, stop your stopwatch and look at the hidden digits : you have a first measurement of your P.E. Repeat the measure a number of times to find the most probable value and its usual variation.

When doing your experiments, be careful to isolate yourself from noise, and especially rhythmic noises, since one can unconsciously take up and follow the beat.

As an example, I made about 30 tests of this method (on 2001 May 04 from UT 09 35 to 10 15), P.E. measured in 1/100th s :

40 37 46 43 43 43 37 56 40 37

46 40 37 40 40 37 144 40 68 40

34 59 40 43 40 53 34 43 34 31 31

So, I discarded this value and did one more test, which one can't do when monitoring a real event !

Well, from these values, it is obvious that this stopwatch is running by 1/30th of second ; we find :

31(x2) 34(x3) 37(x5) 40(x8) 43(x5) 46(x2) ...All these values are spaced by about 0.03s . Test your own stopwatch, one never knows ! You have to take it in account for any timing made with such a peculiar stopwatch.

On the other hand, if you observe that the most frequent value is 0.4s, it could be better to estimate the P.E. to 0.4s, with an accuracy of +0.3/-0.1s . Then when you get a timing t , the "true" time is t - 0.4s, -0.3/ +0.1s (= between t - 0.7s and t - 0.3s).

If you prefer to make some statistic computation, the mean of the 30 "correct" measurements is 0.42s (rounded from 0.417s), the estimated standard deviation is 0.08s (rounded from 0.081s).

[more advanced : however it is not valid to use a gaussian law (or rather here a Student's law) for a random variable which is not at all gaussian. The reaction time law is bounded and strongly skewed, because nerve impulse travels at a finite speed of a few m/s between brain and finger. Furthermore, the distribution of reaction time could be bi-modal, with values clustered around two main ones, a fast one and a slower other one. And it depends if you are tense or inattentive, or tired ... Hence a symetric error bar is almost certainly wrong]

[more advanced : draw a graph of the values. If there's a hint of multiple clustering : i/ make more tests, ii/ you could try to fit a log-gaussian law to each cluster]

Remember that there are causes of unaccuracy other than the intrinsic fluctuation of P.E. . For example, if the stopwatch rounds the values to multiples of .03s it introduces a further unaccuracy by at least .015s

[more advanced : according the way it rounds, to the nearest value, or to the smaller, or greater one. It still is NOT a gaussian variable, but a uniform one with constant probability].

You can estimate crudely the total uncertainty as the sum of the spans :

Yes, but it requires more hardware. The best one could use an artificial star seen through an eyepiece, lighting on and off at random times, while it is properly recorded (for example with a chart recorder) with your pressing on a button.

According to your time-keeping source(*) and your method of recording, you can have more uncertainties to take into account.

All these uncertainties increase the error bar !

You might also be surprised by an unexpected phenomenon (blinks, occultation of an unknown double star, etc..), and have a supplementary time lag : do not underestimate the resulting uncertainty!

[more advanced : For the variables that are gaussian, it is right to compute the quadratic sum of their standard deviations, and then the 95% confidence error bar. For the others variables, it is conservative to add the half-span to the previous 95% (or 0.975 single-sided probability) error bar].

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(*) 2 short stories, both outside Europe : 15 years ago, I discovered that a local phone time-signal was exactly 1 minute fast. A decade ago in another country, two observers synchronized their stopwatch from local radio braodcasting station. The observations were positive, but timed 15 seconds late ...

A positive asteroidal occultation observation requires TWO absolute timings :

- the disappearance (or immersion)

- the reappearance ( or emersion ).

It is necessary to link what you see to a reliable time-keeping source. You generally need a medium (a temporary time-keeper, very often a stopwatch or a tape recorder) to make this link, so there are several sources of possible errors :

- your reaction time (personal equation),

- the unaccuracy of the medium,

- the comparison of the medium with the time source,

- the uncertainty of the time-keeping source itself.

Every point needs attention and testing.

Listen to the bips of a time source, take up the rhythm (3 or 4 beats are enough) and press the button synchronously with the following beat (just like a musician in a band).

Start the stopwatch at a precise time and let it run. Hours later, stop the running stopwatch in the same way, and compare its measured time with the precise time elapsed. This will give you a correction factor if you need it.

Example : suppose you start the stopwatch at UT 19 36 04.00 and stop it one day later at UT 19 36 04.00 , and then the stopwatch says it measured 24h00m01.55s. Then your correction factor is 86400/86401.55 = 0.999982 . It means almost nothing if you run such a stopwatch 10 minutes, but it must be taken into account if it runs 1 hour or more.

Also, there could be a time delay when you press the "start button" ( as much as 0.16s in one experiment).

[More advanced : You still can use the stopwatch if it varies at constant rate but you have to take it into account]

Test it in the same way, by measuring (many times) precise short rounds (of 1 minute for instance). You should find that a measured minute is between 59.95s and 1m00.05s .

[More advanced : Your mean error is the standard deviation (sd) of the differences between precise and measured durations. Note that the mean of these differences should be zero : it is approximately OK if the absolute value of this mean is less than 2 sd divided by the square root of the number of measurements (the so-called standard error, for it is here a gaussian variate).]

Experiments show that digital LCD very often have a time delay (increasing with cold). To test it, hide the digits of tenths and hundredths of seconds with a piece of paper. Start the stopwatch in rhythm with a reliable time-keeping source. Then (time source turned off not to be disturbed) look at the display, take up rhythm of the seconds digit, and stop the watch. The interval measured should be an integer number of seconds : generally it is not. Take note of the difference (only fractions of a second, generally positive, maybe negative sometimes). Repeat this test many times (for example 50 times). Then compute the mean and standard deviation of these numbers. The mean is an estimate of the time delay of your display. The standard error of this estimation is equal to the standard deviation divided by the square root of the number of experiments (~7 if you made 50 tests). If not negligible, take it into account for the estimation of your P.E.

EXAMPLE : with the stopwatch used in PART ONE, I found a mean time delay of 0.05s , the standard deviation is 0.035s, the standard error is (for 50 tests) 0.035/7 = 0.005s . This stopwatch has a time delay of 0.05s ± 0.01 at 2 sd level. Thus my P.E. is 0.40 - 0.05 = 0.35s

There are TWO kinds of stopwatches :

NOTE : (one timing is enough only for regular -no grazing - lunar occultation)

There are TWO methods to proceed :

- start the stopwatch before and stop it when you see the phenomenon.

The risk with the first method is to stop accidentally the running clock BEFORE the observation. Then reverse to second method ! Make a subtraction instead of an addition ...

The error comes from the variability of your personal equation (maybe greater if the phenomenon takes you by surprise) and the uncertainty of your comparison with precise time-source which is (in 95% of occurences) 2 times your mean error (see above).

If the stopwatch has memories, use them to repeat several times the comparison with precise time one or a few minutes later. Then compute the mean of timings (you can't do that without memories : here is a drawback of simple stopwatches).

SQRT[ 0.25² + 0.07² ] =0.2596 and SQRT[ 0.15² + 0.07² ] = 0.1655 hence an error bar +0.26 / -0.17s ]

You need TWO simple stopwatches (and it can be confusing if you're timing a very short occultation : here is another drawback) or use a spared memory to take the reappearance in intermediate timing. Reduce each timing separately, as indicated above.

Don't forget that you may estimate differently your P.E. at disappearnce (you might be surprised) and at reappearance (you might be tense and attentive, since reappearance comes soon after disappearance). Some observers prefer to time the disappearance with one stopwatch, and the duration of occultation with the other. The drawback is that both timings could suffer from the "surprise lag" effect.

Only one stopwatch is enough to carry all the work to be done. Start the stopwatch BEFORE the beginning of observation according to your time-keeping source (radio signal or reliable phone time service) and do all the timings with the "intermediate time" button. Spare some memories to compare again your stopwatch with the time-keeping source AFTER the observation is done.

NOTE : If you have only one memory, time the disappearance with it and the reappearance by stopping the stopwatch. Or start the stopwatch at immersion, take intermediate time at emersion, and compare with time source when stopping.

[More advanced : If you took some intermediate times in comparison with the time-keeping source before and after the observation, you can compute a regression line and have a better accuracy for the event timings.The best accuracy is reached when the observation time is at midtime between the start and the end of the stopwatch run. The timing of events are isolated estimates (not averaged ones) from the regression line : the uncertainty is more important than for the estimation of a mean ]

In this standard method, one marks vocally the event while simultaneously recording time signals. Radio signals like DCF or WWV give a bip every second.

If you have no adequate radio receiver, you can use a stopwatch (in the "countdown" mode) which sounds a bip every 5 or 10 seconds and start it according to a reliable time signal (like the hourly signals of some broadcasting stations, or wired phone time service. DO NOT trust mobil phone or FM band stations before testing them : sometimes the signals could come from satellite relay with unknown time delay).

After the recording is made, one must read it up to time the events. Listen to the tape and count the recorded bips to know which one is just before and which one is just after every mark of events. Thus you know (in the favourable case) the second of the event (don't forget to subtract your P.E.). Now you must refine :

Use a stopwatch to time some of the recorded bips AND the mark of the event. Start the stopwatch as usual by taking up the rhythm of the recorded bips (not possible if one bip every 10 seconds), and stop it at the mark of event. THEN, don't forget that the stopwatch stops NOT at the time of the mark BUT AFTER YOUR TIME DELAY ! To retrieve the correct time of event, you have to subtract TWO times the P.E. from the stopwatch indication : a first time to find the time of the mark, and a second time to find the time of event.

It is much better to use a stopwatch with memories to read your tape, so you can time a bip BEFORE the mark, the mark itself, and a bip AFTER the mark. The tape speed is varying a little at every reading (see above !), due to changing motor temperature, rewinding of the tape, etc ... measuring the time between bips of known interval allows to correct the timing of the event mark from this factor.

(Note : this example comes from a real experiment made today -20020520- with an old cassette record : the 1998.03.21 HIP 28954 occultation by 39 Laetitia when I was NOT surprised at disappearance, thanks to a Martin Federspield update.)

Of course you can average many readings of your tape.

[More advanced : The uncertainty of the P.E. in listening is divided by the square root of the number of readings, BUT NOT the uncertainty of P.E. at the time of event, which is recorded and occured only one time ]

If you have no stopwatch (try to have one) you might test the "ear method" : if you recorded a bip every second, you know the time of marks at better than 1s . You then can try to estimate if the mark is near the preceding bip, or near the following bip, or in the middle (it it is not coincident with a bip). So you can have a refined time with an additional uncertainty of 0.25s (here, there is no listening P.E. , and no averaging is possible).You also could try to use simultaneously stopwatch and tape timing.

All this can take a lot of time ! It is the price to pay for scientific accuracy ...

Sources of error are :

1/ Persistency at disappearance (the retina -?- property which allows us to see movies),

2/ Variability of P.E.,

3/ Uncertainty in the signals of the time-keeping source,

4/ Uncertainty in the start (or stop) in rhythm of the stopwatch,

5/ Uncertainty in the rounding of digits of the stopwatch,

6/ Variation of the running stopwatch or of the tape speed.

1/ is unknown, 2/ is the main source of error, 3/ should be negligible, 4/ & 5/ are small, 6/ is to take into account. You can compound all them quadratically and take an error half-bar of at least 2 standard deviations ...

[More advanced : Take notice that such uncertainties are not gaussian ]