Invented by the Croatian Jesuit astronomer Roger Boscovich (1711-87), this is an elegant method of measuring differences in right ascension and declination. In its true form, the ring micrometer comprises a flat opaque ring mounted at the focus of the telescope objective. Using a stopwatch, the observer times transits of double stars across the ring. The times at which the components cross the inner and outer peripheries of the ring, together with the declination of the primary component and the known value in arcseconds of the ring diameter, contain all the information necessary to calculate the rectangular coordinates of the pair (i.e. the differences in right ascension and declination separating the two stars), from which it is then possible to derive its polar coordinates (p, 6).

It cannot be denied that the mathematical process of reducing the results is somewhat cumbersome, and must have been almost prohibitively tedious in the days of slide rules and logarithm tables, but the advent of modern electronics has banished such difficulties forever. The observer who makes good use of a computer or programmable calculator need not be deterred by the mathematical complexities which are, in any case, more apparent than real.

Commercially made ring micrometers are no longer obtainable, and the construction of a good one is not for the faint-hearted. My own, manufactured by Carl Zeiss Jena, consists of a metal ring mounted on a centrally perforated glass diaphragm which is fitted at the focus of a positive eyepiece. Happily for those who prefer not to undertake their own precision engineering, it is not actually necessary to have a purpose-made ring micrometer. All that is required is an eyepiece having minimal field curvature and an accurately circular field stop. It is the latter which serves as the micrometer. Some modern eyepieces, though of acceptable optical quality, have plastic field stops that may not be truly circular. Select a good-quality eyepiece with a flat field and a metal field stop. It is possible to flatten the field by incorporating a Barlow lens into the optical train.

The first step is to calibrate the eyepiece by determining the radius of its field in arcseconds. A simple method of doing this is to time how many seconds of mean solar time it takes a star of declination ¿to drift across the field diametrically, multiplying the result by 7.5205 cos 8. Even the mean of a number of such timings, however, is unlikely to be very accurate, since the observer has no way of being sure that the star has passed through the exact centre of the field of view, as opposed to trailing a chord.

A more reliable calibration method is to use a pair of stars having declinations which have been determined to a high degree of precision. The Tycho-2 catalogue will yield plenty of suitable candidates. In order to minimise the effects of timing errors, choose stars of relatively high declination, between 60 and 75° north or south of the celestial equator. The difference in dec-

Figure 12.1. Timing the transits of a wide pair of stars to determine the accurate diameter of a field stop or ring.

lination of the two stars should be slightly less than the diameter of the field stop or ring. Their separation in right ascension is less important, but should obviously not be inconveniently large.

The two stars are allowed to drift across the field, so that one star, N, describes a chord near the north edge of the field and the other, S, near the south edge. The times at which each star enters and leaves the field are recorded using a stopwatch (Figure 12.1). A cheap electronic sports watch with lap counter will be found perfectly adequate.

Two angles, X and Y, are required in order to calculate the precise radius of the field stop in arcseconds. Suppose that star N, of known declination SN, enters and leaves the field at N and N2> respectively, and star S, of declination SS, enters and leaves at Sj and S2. Let AS be the difference in declination between the two stars. Then:

from which the radius of the field, R, may be derived as follows:

Take the mean of not fewer than 30 transits. For the greatest possible accuracy, allow for the effects of differential refraction (see Chapter 22).

The procedure for measuring a double star is as follows. Set and clamp the telescope just west of the pair to be measured, so that the object's diurnal motion will carry both components, A and B, across the field as far as possible from its centre (Figure 12.2); they should both transit the field near the same (north or south) edge unless they are very widely separated in declination, in which case they may pass on opposite sides of the centre of the field. The importance of ensuring that the stars pass close to the north or south edge is that it minimises the impact of timing errors upon AS. However, it should not be carried to extremes, as the precise moment of ingress or egress of a star that merely grazes the field edge will eventually become impossible to pinpoint.

The first transit should be used as a "reconnaissance" to determine and record the sequence of appearances and disappearances. On subsequent transits, the observer uses a stopwatch to obtain the times (Aj, A2, and Bj, B2) at which each star enters and leaves the field; these times are noted in tabular form, as shown in Table 12.1.

Figure 12.2. Using the eyepiece field stop or ring to measure a pair of stars by transits.

Table 12.1. A specimen observation of £ I 57 made on | |||

1997, October 27. The three transits |
are individually | ||

numbered in the top row |
of the table. |
Also recorded in each | |

column is the portion of the field in which the transit took | |||

place (i.e. north or south, 1 |
as the case 2 |
may be) |
3 |

(N field) |
(N field) |
(S field) | |

A1 0.00 |
0.00 |
Bi |
0.00 |

B1 29.36 |
30.95 |
A, |
3.56 |

B2 276.77 |
270.68 |
a2 |
251.80 |

A2 279.81 |
275.06 |
b2 |
281.69 |

In order to calculate the position angle, 9, and separation, p, of the pair, it is first necessary to determine the differences in right ascension, a, and declination, S, between the two components. The time at which each star transits the centre of the field is given by the mean of the times at which it enters and leaves. Hence the difference, Aa, in RA between the two stars, A and B, is given by:

The result is expressed as a time difference. At a later stage, after we have ascertained the individual declinations of both components, we will be able to convert Aa to its great circle equivalent, in seconds of arc.

In order to obtain the difference in declination, AS, between the two stars, we first need to ascertain the distance, D, in declination between the centre of the field and each of the stars, A and B:

where the angles jk and /B are given by the following equations:

The difference in declination between the two objects is then given by:

Table 12.2. How to assign a position angle to its correct quadrant. Note that for the purpose of using this table, the sign (+ or -) of A8 is always taken from a transit carried out in the northern half of the field; otherwise the signs must be reversed. Aa A8 Quadrant 0 =

The value of DB is added to DA when the stars are on opposite sides of the centre of the field and subtracted from it when, as is more usual, they are on the same side. Note that in the latter case, the sign (positive or negative) of ASvaries according to whether the north or south portion of the field is used. When both stars pass to the north of the field centre and AS is positive, B lies south of A; a negative result indicates the contrary. When both stars pass to the south of the field centre, the rule is reversed.

Since only the declination, SA, of the main component, A, is usually known in advance, the declination, SB, of the secondary component, B, must initially be given the same value for a first approximation. Once a preliminary value has been derived for AS, the result is added to or subtracted from SA (as the case may be) to obtain a refined value for SB, from which sin /B and thence AS may be recalculated.

We are now in a position to convert Aainto arcsec-onds. To do this, multiply by 15.0411 cos S, where Sis the mean declination of both stars.

Having thus obtained final values for Aa and AS, we use simple Pythagorean trigonometry to work out the polar coordinates, p and 0 and:

When calculating 0, it is necessary to allow for the quadrant in which the companion (B) star lies by applying the appropriate correction, as shown in Table 12.2.

The first transit of the star X I 57 recorded in Table 12.1 provides a convenient practical example. We can see that the difference in right ascension, Aa, is given by (12.1):

Simple Techniques of Measurement (143

= 13.16 seconds

Let us now calculate the difference in declination between the two components. The first step is to find the angles ya and /B. Consulting our catalogue, we find that the declination (2000) of X I 57 is +66°7333 (this refers to the A component). The radius of the ring used to make the observation was 916'.' Therefore:

A 916

from which it follows that ya itself must be 65.16. By the same method, we find sin yb to be 0.8024, and yb = 53.36 (note that at this stage, in the absence of an accurate figure, we have had to treat the declination of B, SB, being equivalent to that of A, SA).

Applying equations (12.5) and (12.6), the distance in declination of A from the centre of the field is:

It therefore follows that according to this preliminary calculation, the difference in declination between the two stars is:

Since this transit took place north of the field centre, the minus symbol in the answer tells us that B lies north of A. Now, X I 57 is a northern hemisphere pair. Hence, in order to obtain B's declination, we need to add 161'.'86, or 0°0450, to that of A:

(If your calculator does not have a facility for automatically converting degrees, minutes and seconds into decimal degrees, simply find the total number of arc-seconds and divide by 3600.)

We are now in a position to refine our results by recalculating AS, substituting the new value for SB in equation (12.8). This gives a final figure of 163'.'63. We also convert our Aa figure into arcseconds, using the mean declination of both stars:

After repeating this process for each of the other transits, means are taken of Aa and AS. In this particular case, the results are Aa = 78'.'37 and AS = 164'.'85.

Applying equation (12.10), we obtain the position angle: 8 = 25°4 and from equation (12.11), the separation is p = 182'.'5.

Since Aa is positive (B following A) and B lies north of A, we see from Table 12.2 that in this particular case B is in the first quadrant (0-90°), and no further correction to 8 is necessary.

According to the WDS, this pair was actually measured by the Hipparcos satellite with the following results (1991): p = 182'.'4; 8 = 25°. It will be seen that our figures, which are based upon observations made in 1997, are remarkably close. This is certainly a fluke. As a rule, even a large number of transits is unlikely to produce results as seemingly impressive as these. In practice, if you can consistently get within 1° in position angle and 1'' in separation, you will be doing very well indeed. In this particular case, the Zeiss ring micrometer was used on two nights to time six transits across the inner and outer edges of the ring, with the following overall result:

The position angle result is in full agreement with the Hipparcos figure, whereas the separation result differs from Hipparcos by less than 1%. This is fairly typical of the level of performance to be expected from the ring method.

For maximum accuracy, a total of not fewer than 10 transits should be taken, preferably spread over several nights. It is good practice to take half the transits near the north edge of the field and the rest near its south edge, taking care not to apply the wrong sign (plus or minus) when calculating AS. If you have a proper ring micrometer, record the times of appearance and reappearance at its outer and inner edges. In that way, you will be able to refine your results slightly by taking the mean of twice as many timings during each transit. My own experience, as can be seen from the example of X I 57, suggests that in this way it should be possible to obtain results to within about 1'' of the true position. Although this is nowhere near good enough for measuring close doubles, it is perfectly acceptable for pairs wider than about 100''.

The rather involved mathematical process of reduction may seem daunting at first sight, but it need not be either laborious or complex if the observer uses a programmable calculator or computer. Once such a device has been programmed to carry out the tedious computations, results can be obtained almost as quickly as the raw timings can be keyed in.

The particular advantages of the ring method are that it requires no special apparatus beyond a stopwatch, needs no form of clock drive or field illumination, can be used with an alt-azimuth telescope as well as an equatorial and is capable of producing consistently accurate results on very wide pairs (separation greater than 100''). It may be worth bearing in mind that although wide and faint doubles lack the glamour of close and fast-moving binaries, they are probably in even greater need of measurement.

The drawbacks of the method, apart from the restriction of its accurate use to very wide pairs, are the rather time-consuming nature of the observations and the elaborate process of reduction. These, although they are greatly reduced by the use of a computer or programmable calculator, can never be entirely eliminated. A Delphi 5 program to carry out this reduction, written by Michael Greaney, is available on the accompanying CD-ROM.

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