CHAPTER XXI.

THE ADJUSTMENTS OF THE EQUATORIAL.

As the equatorial is _par excellence_ the amateur’s instrument, and as in setting up an equatorial it is important that the several adjustments should be correctly made, they are here dwelt upon as briefly as possible. They are six in number.

1. The inclination of the polar axis must be the same as that of the pole of the heavens.

2. The declination circle must read 0° when the telescope is at right angles to the polar axis.

3. The polar axis must be placed in the meridian.

4. The optic axis of the telescope, or line of collimation must be at right angles to the declination axis, so that it describes a great circle on moving about that axis.

5. The declination axis must be at right angles to the polar axis, in order that the telescope shall describe true meridians about that axis.

6. The hour circle must read 0h. 0m. 0sec. when the telescope is in the meridian.

When these are correctly made the line of collimation will, on being turned about the declination axis, describe great circles through the pole, or meridians, and when moved about the polar axis, true parallels of declination; and the circles will give the true readings of the apparent declination, and hour angles from the meridian.

To make these adjustments, the telescope is set up by means of a compass and protractor, or otherwise in an approximately correct position, the declination circle put so as to read nearly 90° when the telescope points to the pole, and the hour circle reading 0h. 0m. 0sec. when the telescope is pointing south.

First, then, to find the error in _altitude_ of the polar axis.

Take any star from the Nautical Almanac of known declination on or near the meridian, and put an eyepiece with cross wires in it in the telescope, and bring the star to the centre of the field as shown by the wires. Then read the declination circle, note the reading down and correct it for atmospheric refraction, according to the altitude[18] of the star by the table given in the Nautical Almanac, turn the telescope on the polar axis round half a circle so that the telescope comes on the other side of the pier. The telescope is then moved on its declination axis until the same star is brought to the centre of the field, and the circle read as before and corrected. The mean of the two readings is then found, and this is the declination of the star as measured from the equator of the instrument, and its difference from the true declination given by the almanac is the error of the instrumental equator and of course, also of the pole at right angles to it.

It is obvious that if the declination circle were already adjusted to zero, when the telescope was pointing to the equator of the instrument, one observation of declination would determine the error in question; and it is to eliminate the _index error_ of the circle, as it is called, that the two observations are taken in such a manner that the index error increases one reading just as much as it decreases the other, so that the mean is the true instrumental declination.

_Index Error._—From what has just been stated it follows that half the difference of the two readings is the index error, which can be at once corrected by the screws moving the vernier, giving correction No. 2.

To correct the error in altitude of the pole, the circle is then set to the declination of the star given by the almanac, corrected for refraction, and the telescope brought above or below the star as the error may be, and the polar axis carrying the telescope is moved by the setting screws, until the star is in the centre of the field.

3rd Adjustment.—A single observation of any known star, about 6 hours to the east or west will give the error of the polar axis east and west, the difference between the observed and true declination being this error, and it can be corrected in the same manner as the last. These observations should be repeated, and stars in different parts of the heavens observed, in order to eliminate errors of division of the circle until the necessary accuracy is obtained.

For example:

Observed dec. of Capella 43° 50´ 30˝ Telescope west. 47° 0´ 0˝ Telescope east. ——— ——— ——— 2) 90° 50´ 30˝ ——— ——— ——— 45° 25´ 15˝ 47° 0´ 0˝ Error due to refraction 0° 0´ 7˝ 43° 50´ 30˝ ——— ——— ——— ——— ——— ——— Instrumental declination 45° 25´ 8˝ 2) 3° 9´ 30˝ True declination 45° 52´ 0˝ ——— ——— ——— ——— ——— ——— Index error 1° 34´ 45˝ 26´ 52˝

This indicates that the pole of the instrument is pointing below the true pole, and index error 1° 34´ 45˝.

Observed declination of Pollux 6h. west 28° 19´ 18˝ Refraction 0° 0´ 46˝ ——— ——— ——— 28° 18´ 3˝ True declination 28° 20´ 10˝ ——— ——— ——— 0° 1´ 38˝

This shows the pole to be 1´ 38˝ east of true pole.

4th Adjustment.—For the estimation and correction of the third error, that of collimation, an equatorial star is brought to the centre of the field of the telescope, the time by a clock noted, and the hour circle read. The polar axis is then turned through half a circle, and the star observed with the telescope on the opposite side (say the west) of the pier, the time noted, and the hour circle read. Subtract the first reading from the second (plus twenty-four hours if necessary) and subtract the time elapsed between them, and the result should be exactly twelve hours, and half the difference between it and twelve hours is the error in question. If it is more than twelve hours the angle between the object end of the telescope and the declination axis is acute, and if less then it is obtuse. This error can then be corrected by the proper screws. A little consideration will show, that if the angle between the object end of the telescope and the declination axis be acute, and the telescope is on the east side of the pier, and pointing to a star, say on the meridian, the hour circle will not read so much as it would do if the line perpendicular to the declination axis were pointing to the meridian. When the telescope is on the wrest side of the pier, the circle will read higher for the same reason, and therefore the difference between the angle through which the hour circle is moved and 180° is equal to double the angle between the line perpendicular to the declination axis and the collimation axis of the telescope; allowance being made for the star’s motion.

For example γ Virginis, Dec. 0° 46´·5.

Time by clock. Hour circle reading. 11h. 23m. 52s. 11h. 55m. 30s. Telescope east. 11h. 31m. 55s. 24h. 8m. 24s. Telescope west. ———— ———— ———— ———— ———— —————— 8m. 3s. 12h. 12m. 54s. 8m. 3s. ———— —————— 2) 4m. 51s. ———— —————— Collimation error at 2m. 25·5s. dec. 46´·5 angle between object glass and declination axis acute.

If this error is not corrected, it must be added when the telescope is on the east side of the pier, and subtracted when on the west.[19]

5th Adjustment.—Place a striding level on the pivots of the declination axis and bring the bubble to zero by turning the polar axis; read off the hour circle and note it; then reverse the declination axis east and west and replace the level; bring the bubble to zero and again read the circle. The readings should show the axis to be turned through half a circle, and the difference shows the error.

If the second reading minus the first be more than half a circle or 12 hours, it shows that the pivot at the east at the first observation is too high, and therefore in bringing the declination axis level, the first reading of the hour circle is diminished from its proper amount and increased on the axis being reversed.

To adjust the error, find half the difference of circle readings and apply it, with the proper sign, to each of the two circle readings, which will then differ by exactly twelve hours; bring the circles to read one of the corrected readings and alter the declination axis until the bubble of the level comes to zero. If the pivots of the declination axis are not exposed, so that the level can be applied, the following method must be adopted:—Fasten a small level on any part of the declination axis or its belongings, say on the top of the counterpoise weight; bring the axis apparently horizontal and the bubble to zero; turn the telescope on the declination axis, so that by the turning of the counterpoise the level comes below it; if then the bubble is at zero, the axis of the level is parallel to the declination axis, and both are horizontal, and if not it is clear that neither of these conditions holds; therefore bring the bubble to zero by the two motions of the level with reference to the counterpoise and the motion of the declination axis on the polar axis, so that the error is equally corrected between them; repeat the proceeding until the level is parallel with the axis, when it will show when the axis is horizontal as well as the striding level.

For example:—

Hour circle reading when } 11h. 57m. 57s. Telescope east. declination axis is horizontal. } 23h. 59m. 47s. Telescope west. ———— ———— ———— 12h. 1m. 50s. Error 0h. 1m. 50s.

Or this error can be found and corrected without a level by taking two observations of a star of large declination in the same manner as in estimating the collimation error, for example:—

Η URSÆ MAJORIS.

Time by clock. Hour circle reading. 12h. 8m. 57s. 0h. 28m. 44s. Telescope east. 12h. 18m. 53s. 12h. 46m. 42s. Telescope west. ———— ———— ———— ———— ———— ———— 9m. 56s. 12h. 17m. 58s. 9m. 56s. ———— ———— 2) 8m. 2s. ———— ———— Error of hour circle due to error of 4m. 1s. inclination of axes[20]

6th Adjustment.—Bring the declination axis to a horizontal position with a level and set the hour circle to zero, or obtain the sidereal time from the nearest observatory, or again find it from the solar time by the tables, and correct it for the longitude of the place (subtracting the longitude reduced to time when the place is west and adding when east of the time-giving observatory) and set a clock or watch to it. Take the time of transit of a known star near the meridian and then the sidereal time by the clock at transit minus the right ascension of the star will give the hour angle past the meridian, and its difference from the circle reading is the index error, which is easily corrected by the vernier. If the star is east of the meridian the time must be subtracted from the right ascension to give the circle reading.

In the above examples we have assumed, for the sake of better illustration, that the hour circle is divided into twenty-four hours, but more usually they are divided into two halves of twelve hours each. A movement through half a circle, therefore, brings the hour circle to the same reading again instead of producing a difference of twelve hours, as in the above example.

When the equatorial is once properly in adjustment, not only can the co-ordinates of a celestial body be observed with accuracy when the time is known, but a planet or other body can easily be found in the day-time. The object is found by the two circles—the declination circle and the hour or right-ascension circle. The declination of the required object being given, the telescope is set by the circle to the proper angle with the equator. The R.A. of the object is then subtracted from the sidereal time, or that time plus twenty-four hours, which will give the distance of the object from the meridian, and to this distance the hour-circle is set. The object should then be in the field of the telescope, or at least in that of the finder. We subtract the star’s R.A. from the sidereal time because the clock shows the time since the first point of Aries passed the meridian, and the star passes the meridian later by just its R.A., so that if the time is 2_h._, or the first point of Aries has passed 2_h._ ago, a star of 1_h._ R.A., or transiting 1_h._ after that point, will have passed the meridian 2_h._ - 1_h._ = 1_h._ ago; so if we set the telescope 1_h._ west of the meridian we shall find the star. The moment the object is found the telescope is clamped in declination, and the clock thrown into gear, so that the star may be followed and observed for any length of time.

Footnote 18:

The altitude of the star in this case is its declination plus the co-latitude of the place, but this only applies when the star is on the meridian. When the altitude of a star in another situation is required, it is found sufficiently accurately by means of a globe. A sextant, if at hand, will of course give it at once.

Footnote 19:

Since the velocity of the star varies as the cosine of the declination, the error of collimation at the equator = 2m. 25·5s. cos. 0° 45´·5 = 2m. 25·08s.; and for non-equatorial stars, 2m. 25·08s. sec. dec.

Footnote 20:

This error varies as the tangent of the declination, and therefore to find the constant for the instrument, in case the parts do not admit of easy adjustment, we divide 4m. 1s. by 1·18 the tan. of Dec. of η Ursæ Majoris, giving 3 min. 28 sec.