The chronometric method allows a significant increase in accuracy over the ring method. Of comparable antiquity, it requires the addition to the telescope of an external position circle or dial, as well as a single wire or thread mounted at the focus of the optical system. A motor-driven mount is, if not an absolute necessity, at any rate highly desirable. Since position angles are measured directly with the circle, the chronometric method is a hybrid technique rather than a pure transit method. The sole purpose of the timed transits is to obtain differences in right ascension, from which it follows that no calibration exercise is necessary.

An ordinary crosswire eyepiece will serve admirably as the basis of the micrometer. If no such eyepiece is available, a single thread or wire can be mounted in the focal plane of a positive eyepiece, preferably one having a relatively short focal length. The thread must be as fine as possible, ideally no more than 15 microns in diameter. Various materials have been suggested, including nylon or spider's thread. In order to render such materials visible against the dark sky background, some means of illuminating either the field or the thread is essential. A small torch bulb or light-emitting diode may be installed near the objective or inside the eyepiece or Barlow lens. A potentiometer can also be provided so as to enable the observer to vary the level of illumination. Alternatively, at the cost of some degree of precision, the need for a source of illumination may be dispensed with altogether by making the wire relatively thick. I have used a length of 5-amp fuse wire for this purpose. The wire must be stretched diametrically across the field stop and glued in position. The most difficult part of fitting the wire is to keep it under tension so as to ensure that it is perfectly straight. Even then, it is likely to prove rather a crude substitute for an illuminated thread or field.

The position circle or dial can be made from an ordinary 360° protractor, which is fitted to the focusing mount. It must be carefully centred on the eyepiece, to which a pointer or vernier index is attached. The dial must be capable of adjustment by rotation about the optical axis. It is graduated anticlockwise unless the optical system reverses the field, in which case the dial should be graduated in the opposite sense.

Although there is no need to calibrate the micrometer, it is necessary to establish the circle reading that corresponds to north (0°) before measurement begins. One way of achieving this is to find a star near the equator and allow it to drift across the field of view, rotating the eyepiece until the star accurately trails the single thread. Then, leaving the eyepiece undisturbed, adjust the position circle until the pointer indicates a reading of 270° (west). Provided the circle is correctly graduated, it will follow that the zero reading indicates celestial north. By this method, position angles of double stars can be read directly from the PA dial without the need for any correction. However, it is practically impossible to exclude all sources of error in such a home-made device. Quite apart from any defects in the protractor itself, it is unlikely to be perfectly centred on the optical axis. In order to overcome such sources of error, Courtot1 has recommended the following alternative approach. Adjust the web so that a star drifts along it when the motor is stopped, and note the reading on the dial. Then rotate the eyepiece through 180°, so as to minimise the effects of any centring error, and repeat the process, this time subtracting 180° from the reading. Proceed in this way until you have gathered six readings, and take the mean. The difference between the result and 90° gives the north angle.

Let us illustrate the procedure by reference to Courtot's own example. Suppose that by repeatedly drifting a star along the web we obtain the following circle readings:

East |
92° 2 |
West |
273° 3 |

92° 5 |
273° 0 | ||

92° 3 |
273° 1 | ||

Mean |
92° 33 |
273° 13 |

Subtract 180° from the mean west result: 273.13° -180° = 93°13.

Hence the overall mean is (92°33 + 93°13)/2 = 92°73.

Since this corresponds to the true position angle 90°, the north angle is 92°73 - 90° = 2°73.

This angle is a correction which will be applied to all subsequent circle readings.

To obtain the position angle of a double star, carefully rotate the eyepiece until the wire is precisely parallel to the pair's axis and note the reading of the PA dial. Then reverse the pointer through 180° and take another measurement. The entire process should be repeated until a total of at least six readings have been obtained. Of these, half will have to be adjusted by 180°. Take the overall mean, remembering to correct for any north angle.

The observer obtains the separation of the pair by timing transits across the wire. At least 20 such timings should be made. There are several variations in the procedure. The simplest way is to set the wire exactly north-south, so that the interval in the times of passage across the wire of the two components corresponds to the difference in RA. The separation is then given by:

sin 6

in which t is the mean interval in seconds, S is the declination of the pair and 6 its position angle.

For example, on the night of 2001 August 26, I measured the well-known pair 61 Cyg, with the following results: 6= 149°9, t = 1.3384 seconds.

Since the declination of 61 Cyg is 38°75, the separation, p, is given by (12.12):

sin 149.9

In the case of pairs having a PA close to 0° or 180°, both components will transit the wire more or less simultaneously. There are two ways of overcoming this difficulty. One is to set the web at exactly 45° to the direction of drift (see Figure 12.3), remembering to take into account the north angle. Then, assuming the web is orientated in PA 135°/315° as shown in Figure 12.3, the separation is given by:

If the web is orientated in PA 45°/225°, the separation is: 15.0411 x t x cos 8

cos 9-sin 9

Courtot1 has suggested an alternative procedure in which the web is placed approximately perpendicular to the pair's axis. The angle, i, between the wire and the direction of drift is read from the circle (making allowance for any north angle). It is positive, increasing from east through south and so on (Figure 12.4). With the telescope clamped a short distance west of the pair, use a stopwatch to measure the time taken for both components to cross the thread. Repeat the process at least ten times, noting the results to two decimal places. Then reverse the wire 180° and take another ten timings.

Figure 12.4. Using Courtot's method with the wire approximately perpendicular to the orientation of the pair.

Figure 12.4. Using Courtot's method with the wire approximately perpendicular to the orientation of the pair.

These timings, together with the declination of the pair and the position angle already determined from the PA dial, enable the observer to deduce the separation, p, of the two components:

Using Courtot's own example, suppose that the mean transit interval, t, is 2.386 seconds and the declination of the star is 25°25. The position angle, 0, has already been measured as 223°04. Let us further suppose that for the purpose of timing the transits, the web was set with a circle reading of 135°, which corresponds to 45° starting from east. After subtracting the north angle, 2°73, we find that the web was actually set at i = 45 - 2.73 = 42°27 from east. Then, applying equation (12.15):

= (15.0411 x 2.386 x cos 25.25 x tan 42.27 x tan 223.04)

P = (sin 223.04x(1 + tan 42.27x tan 223.04)) = -21"8 .

The negative value of p merely indicates that the companion is west of (preceding) the primary, and the minus sign is therefore ignored.

The main advantage of the chronometric method is that it has greater accuracy than the ring method and can handle pairs down to a separation of as little as 15''. By the careful use of Courtot's variation, this limit may be reduced still further - perhaps even below 10''. Because each transit lasts only a few seconds, it is a relatively quick technique. The reduction procedure, while still somewhat elaborate, is far simpler than the ring method, although the advent of modern electronics has greatly reduced this difficulty in respect of both techniques.

The principal disadvantage of the chronometric method is that for reliable results it demands the use of an equatorial mount. Indeed, it is highly sensitive to misalignment of the mount. If significant errors in position angle are to be avoided, the mount must be accurately set on the celestial pole, with an error of 1' or less. It follows that the chronometric method is better suited to permanently mounted telescopes than to the portable instruments favoured by many amateurs. Another drawback is that the use of a fine filament necessitates the provision of some form of field or web illumination, which in turn necessarily reduces the working magnitude threshold of the telescope.

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