Setting an Equatorial Head
abstracted from Amateur Astronomer's Handbook by J.B. Sidgwick  Faber & Faber ©1955
Introduction.
The successive stages in the setting up of an equatorial are:
(1) approximate orientation of the polar axis in altitude and azimuth;
(2) zero setting of the Dec circle index or vernier;
(3) final adjustment of the polar axis in altitude;
(4) final adjustment of the polar axis in azimuth;
(5) zero setting of the hour circle index or vernier;
(6) checking collimation: perpendicularity of the optical and Dec axes;
(7) checking that the polar and Dec axes are mutually perpendicular.
These adjustments between them ensure that the polar axis is parallel to the Earth's axis of rotation, and that the Dec axis is perpendicular both to the polar axis and to the plane containing the optical axis. The order in which the operations are carried out is important: if the mounting is not fitted with circles, stages (2) and (5) are of course omitted; if, on the other hand, either of the circles is used in determining a particular error, its index error must first be determined; with one exception the methods described here for carrying out stages (3) and (4) require at most the Dec circle, while the hour circle is employed in stages (6) and (7).
Most small mountings make no provision for the adjustment of the optical axis relative to the Dec axis, or the Dec axis relative to the polar axis, these being attended to by the makers. In the case of new instruments, at least, it is safe to assume that they are correct, and stages (6) and (7) may be omitted..
If any particular stage involves a large shift in the position of the head, it is likely that prior adjustments will have been affected, and these must therefore be repeated. A large change in the azimuth of the head in stage (4), for example, will also affect the adjustment made in stage (3); stages (3) and (4) must therefore be repeated as fine adjustments after the first coarse adjustment of (4).
A variety of methods are available to choose from at each stage; these vary among themselves in accuracy and in convenience and speed of operation. It is assumed in the following instructions that a HP and a LP ocular are available (if possible both, and certainly the former, provided with crosswebs), and that from the observing site there is a clear view of the pole and the meridian point of the equator.
Approximate orientation of polar axis
(i) Azimuth:
(a) The equatorial head aligned on a meridian mark on the ground with the aid of a prismatic compass, the current magnetic variation being taken from the Ordnance Survey sheet of the district in which the observing site is situated.
(b) Meridian marked by the shadow of a plummet at true noon (i.e. equation of time applied to LMT).
(c) Meridian defined by two plummets aligned upon Polaris at upper or lower culmination.
(d) With hour circle reading 0h 0m 0s _or, in the absence of circles with the Dec axis made horizontal with a striding level _ bring a star of known RA, to the centre of a LP field when LST = RA by rotating the whole head in azimuth.
(ii) Altitude:
(a) The latitude of the observing site, correct to the smallest fraction of a degree to which the latitude scale on the mounting can be read, is obtained from a 1inch O.S. map. The base of the head having been levelled, this angle is then set on the latitude scale.
(b) If the mounting is not fited with a latitude scale, employ a homemade clinometer.
Note: Before making any adjustments of the head, the base supporting it must be shown to be at least approximately horizontal by levels placed on it in, and at right angles to, the meridian.
Index error of Declination circle
(a) Select any fairly bright star near to the meridian (whose altitude, therefore, will vary by a negligible amount during a few minutes). With the telescope E of the pier, bring the star to the intersection of the webs of the HP ocular; clamp, and read the Dec circle (d'). Reverse the telescope, return the star to the crosswires, and reread the Dec circle (d'').
Assuming that d' differs from d'', adjust the vernier or index to read (d' + d'') / 2 when the star is at the intersection of the webs, or is bisected by the horizontal web if the webs have been oriented NS and EW. Repeat, until the readings E and W of the pier agree, or treat any residual error as a correction to be applied, with a change of sign, to all Dec circle readings. It is adviseable, as a check, to repeat the procedure with a star of widely different Dec from the first. A distant and welldefined terrestrial object could equally well be used.
(b) The base supporting the head having been levelled both NS and EW, mark the head's position in azimuth. Then turn it through 180°, so that the upper end of the polar axis is lying due S of the lower. Bring Polaris, at its upper or lower culmination, to the intersection of the webs; clamp, and read the Dec vernier. Reverse the telescope and repeat. As in (a), if the two readings do not agree, set the vernier to their mean value when Polaris is at the centre of the field, and repeat.
(c) Schaeberle's Method. A method that can be used by day, but is not otherwise convenient. Remove the OG cell and ocular; attach a plummet to the intersection of a wooden crossbar laid symmetrically over the top of the telescope. Adjust the tube so that the plummet falls centrally through the drawtube opening; the tube is then directed at the zenith. Set the Dec vernier to read the latitude of the observing site; reverse the telescope; readjust its position relative to the plummet and reread the vernier. If the two readings are different, set the vernier to their mean. Repeat until the circle reading is the same on either side of the pier, the plummet falling centrally through the ocular flange. Note the index error, if any, and apply it as a correction to all subsequent Dec circle readings. Alternatively, two brackets may be fixed to the tube by collars as in the diagram below, CD being parallel to the optical axis; a plumet suspended from point A on one bracket will pass centrally through the hole B in the other, provided the telescope is vertical CA = DB.
Alternatively, again, the telescope may be pointed to the zenith by placing a spiritlevel across the cell of the objective and centring the bubble in both EW and NS directions. Given an accurately constructed level the results are likely to be as accurate as those obtained from the use of a plummet, which in practice is not easy to manipulate with precision.
Adjustment of the polar axis in altitude
(a) Without circles:
The North Celestial Pole lies within about 3m of RA of the line joining Polaris and eta UMa. At ±6h ST of the culmination of eta UMa, therefore, the altitudes of Polaris and the NCP are very nearly equal.
Set the Dec axis in the vertical plane through the polar axis, using the plummet, and clamp the polar axis. Adjust the elevation of the polar axis until Polaris can be brought to the intersection of the webs by rotation of the Dec axis only..
The inaccuracy in the derived altitude which is involved by assuming that the two stars and the NCP lie on the same hour circle is of the order 50": elevation too small if eta UMa is W of the pole, too great if it is E.
(b) Without circles:
A quick method of correcting both the elevation and the azimuth of the polar axis in a single operation.
Bring Polaris to the centre of the LP field and clamp the Dec axis. Rotate the polar axis through a wide angle, and note whether the image has moved relative to the webs. Reduce any such movement to a minimum by adjustment of the telescope in Dec; carry out the fine adjustment with the HP ocular. The tube and the polar axis are now virtually parallel.
Clamp both axes and adjust the head bodily (elevation and azimuth of the polar axis) so as to bring the NCP to centre of the field. The pole lies at a distance of 58' from Polaris in the direction of eta UMa. The adjustment can be made by eye estimation (the field diameter being known), with the aid of a micrometer, or by eye in conjunction with a map of the polar star field.
North Polar Sequence  Epoch 1950
Features stars within 10° of the North Celestial Pole, based on data
from the "General Catalogue of 33 342 Stars for the Epoch 1950", by B.Boss,
published by the Carnegie Institution of Washington D.C., 1938. Stars down
to magnitude +6.55 are given, of spectral types A, B, F and early G (only
when no other star available). Variables and naked eye doubles are not used.
The sequence was compiled and originally drawn out by Alastair McBeath, March
1988.
Stars brighter than magnitude 6.1
Stars fainter than magnitude 6.1
Key
1 
2.1 
B8 
2 
4.4 
G5 
3 
4.44 
A0 
4 
5.34 
F2 
5 
5.38 
A0 
6 
5.39 
B3 
7 
5.55 
A2 
8 
5.62 
F0 
9 
5.64 
A3 

10 
5.69 
A0 
11 
5.73 
G0 
12 
5.84 
F5 
13 
5.86 
A3 
14 
5.92 
A2 
15 
5.99 
A0 
16 
6.02 
A2 

17 
6.13 
A0 
18 
6.15 
A2 
19 
6.16 
A2 
20 
6.17 
A0 
21 
6.2 
B9 
22 
6.26 
A0 
23 
6.26 
F0 
24 
6.28 
F0 
25 
6.33 
F2 

26 
6.34 
A2 
27 
6.38 
F5 
28 
6.39 
A0 
29 
6.39 
A2 
30 
6.4 
F8 
31 
6.48 
A2 
32 
6.53 
F2 
33 
6.54 
A5 
34 
6.55 
A0 

(c) Without circles:
Orient the webs of the LP ocular NS, EW by trailing an equatorial star near the meridian (star over web, or web over star with Dec axis clamped)..
Select a pair of stars in the NE or NW sky, differing in RA by several hours and in Dec by at most a few arcmins. Set the f member of the pair at the intersection of the webs and clamp the Dec axis. Rotate the polar axis to bring the p to the NS web. Knowing the difference in Dec of the two stars, estimate whether it is too far N or S in the field. Then:
p star too far N    both stars E of meridian    elevation of polar axis too great
p star too far S    both stars E of meridian    elevation of polar axis too small
p star too far N    both stars W of meridian    elevation of polar axis too small
p star too far S    both stars W of meridian    elevation of polar axis too great
Adjust the elevation of the axis by half the amount required to bring the p star to its estimated correct position on the NS web. Correct, if necessary, the orientation of the webs, and repeat the procedure first with the LP and finally with the HP ocular; in the latter case it may be necessary to use another pair of stars, of smaller Dec difference.
The practical inconvenience of this method lies in the necessity of selecting pairs of stars of closely similar Dec, which involves reference to one of the more comprehensive star catalogues.
(d) Without circles:
Observe a nearequatorial star when on the meridian and when at equal distances E and W. At the first (E) observation bring the star to the intersection of the webs, oriented NS and EW, and clamp the Dec axis.
The second and third observations consist in noting whether the star has shifted N or S from the EW web and, if so, estimating by how much.
Then:
If field position of star at meridian is midway between its positions when E and W of the meridian: elevation of polar axis correct.
If position at meridian is N of the mean of its positions when E and W of the meridian: elevation of polar axis is too great.
If position at meridian is S of the mean of its positions when E and W of the meridian: elevation of polar axis is too small.
Having reduced the error, and if necessary reoriented the EW web, repeat the observations at increased intervals before and after culmination.
(e) Without circles:
The most generally satisfactory method, and one capable of revealing very small errors in the elevation of the polar axis, is the following:
Select two stars about 6h E and W of the meridian respectively, and in about 45º N Dec. Orient the HP ocular with its webs NS and EW by trailing an equatorial star on the meridian.
Bring the E star to the intersection of the webs, clamp Dec and follow the star with the driving clock or RA slow motion; note whether it tends to drift N or S from the EW web. If the elevation of the axis is nearly correct this drift will be slow, and some time will pass before it becomes perceptible.
Repeat the observation with the W star. Then:
If E star drifts N    elevation of polar axis too great.
If W star drifts S    elevation of polar axis too great.
If E star drifts S    elevation of polar axis too small.
If W star drifts N    elevation of polar axis too small.
Correction of the elevation of the axis must be carried out by trial and error.
(f) With circles
Select a star of known Dec, d, and, to minimise the effects of refraction and of the still only approximate azimuth adjustment, small zenith distance.
Set d on the circle, the index error having been eliminated or allowed for, and clamp the Dec axis.
Adjust the elevation of the polar axis so that the star, when on or very close to the meridian, can be brought to the intersection of the webs of the HP ocular by sweeping in RA only.
Reverse the telescope and repeat; finally check with another star.
(g) With circles:
With the setup described under section (c) Index error of Declination circle, set the Dec axis horizontal by means of a striding level, and clamp the polar axis.
Rotate the telescope about the Dec axis to make the plummet fall centrally through the hole in the lower bracket.
The Dec circle reading should now be the latitude of the observing site. If it is not, set this reading on the Dec circle and bring the plummet back to the centre of the lower bracket hole by adjusting the elevation of the polar axis.
Alternatively, a spiritlevel may be used in the manner described in section (c) Index error of Declination circle.
Adjustment of the polar axis in azimuth
(a) Without circles:
The same assumption is made as in section (a) Adjustment of the polar axis in altitude. The Dec axis is set horizontal with a striding level, and the polar axis clamped.
At the upper or lower culmination of eta UMa, the azimuth of the head is adjusted so that Polaris can be brought to the intersection of the webs by the Dec slow motion.
If the observation is made at upper culmination, the N end of the axis will lie about 50" too far W; if at the lower, 50" too far E.
(b) Without circles:
See section (b) Adjustment of polar axis in altitude.
(c) Without circles:
Orient the webs NS, EW on an equatorial star at the meridian. Select two nearequatorial stars of known RA & DEC, situated at equal distances (about 30m) E and W of the meridian, and differing in Dec by not more than a few arcmins.
Bring the E star to the intersection of the webs and clamp the Dec axis. Swing the telescope westward in RA to bring the second star into the field. Knowing the Dec diff, estimate whether it lies too far N or S in the field.
Then:
If W star is too far S: polar axis is lying NESW.
If W star is too far N: polar axis is lying NWSE.
Reduce the error by half, by means of azimuth adjustment of the polar axis. Repeat with a second pair of stars, differing in Dec from the first. Finally, repeat with pairs of stars of increasing RA diff until no error is detectable over arcs of 6h to 8h. This method is open to the same practical objection as that mentioned under section (c) Adjustment of polar axis in altitude.
A single star, observed E and again W of the meridian, could of course be used instead of a pair of stars, but this procedure takes longer. In the early stages of correcting the azimuth it is sufficient to note the N or S drift of a single star (preferably near the zenith) when crossing the meridian.
(d) Without circles:
Employs the same principle as section (c) Adjustment of polar axis in altitude; but the first setting of the telescope in Dec is made on a star at the meridian, and the comparisons are made with stars at equal distances E and W of the meridian. Then:
If E star lies too far N    polar axis is lying NESW.
If W star lies too far S    polar axis is lying NESW.
If E star lies too far S    polar axis is lying NWSE.
If W star lies too far N    polar axis is lying NWSE.
(e) Without circles:
Orient the webs NS, EW on an equatorial star at the meridian. Select two stars of known RA and DEC, and Dec diff of a few arcmins, situated respectively in S Dec and high N Dec.
When they are in the neighbourhood of the meridian, clamp the telescope in RA slightly ahead of the p star and time its transit at the NS web.
Rotate the telescope about the Dec axis to pick the f star, and time its transit. Then:
If transit of f star occurs too early: polar axis is lying NESW.
If transit of f star occurs too late: polar axis is lying NWSE.
(f) With circles:
Select a star of known RA and DEC, situated roughly midway between the meridian and the E or W horizon, and between the zenith and the horizon.
Set the Dec circle to d, and clamp. Adjust the azimuth of the polar axis so that the star can be brought to the intersection of the webs by motion in RA alone.
Repeat with a second star, situated in approximately the same position across the meridian, making the final adjustments with the HP ocular.
(g) With circles:
A method of great precision (given finely and accurately divided circles), which requires the prior elimination, or determination, of the hour circle index error (see Index error of the hour circle). It involves the observation of a circumpolar star of known RA and DEC. A few minutes before upper or lower culmination note the time of its transit at the NS web of the HP ocular (T1), and the hour circle reading (t1). Reverse the telescope and note T2, t2.
Then:
where a = required azimuth in arcsecs,
t’1, t’2 = the values of t1, t2 corrected for index error,
= clock correction,
= latitude of the observing site.
The sign of Dec in the denominator of the last term is + for lower culminations,  for upper. If the derived value of a is +, the polar axis is oriented NWSE; if , NESW.
Knowing the precise value of the required adjustment, and not merely its direction, involves a great saving of time, since the micrometer web can be laid off a" from some reference point on the horizon, and the whole head rotated in azimuth until it is returned to the web, thus eliminating an often rather lengthy process of trial and error.
Index error of the hour circle
(a) Set the Dec axis horizontal by means of a striding level (bubble central in both positions of the level), and clamp the polar axis. Set the hour circle index or vernier to zero; if two verniers are fitted, one reading LST and the other RA, their zero readings must coincide.
(b) Calculate the exact time of culmination of a star whose RA is known. Just before culmination, bring it to the intersection of the webs in a HP ocular, clamp both axes, and follow it with slow motion in RA. At the calculated instant of meridian transit stop following the star; set the hour circle vernier to zero, and the RA vernier, if present, to the RA of the star.
(c) Setup as described in section (a) Index error of Declination circle. Note the hour circle reading, t. Reverse the telescope and reread the hour circle, t'. If t and t' differ, either (5), (6) or (7) ref. Introduction, may require adjustment. Assuming that the two latter are in fact correct, the index error is:
which can either be removed by setting the circle to the reading
and adjusting the vernier to read 24h 0m 0s, or can be applied with a change of sign as a constant to all hour circle readings.
Collimation
With circles:
Select an equatorial star near the meridian; clamp the telescope ahead of it, and time its transit at the NS web (T1). Read the hour circle (t1).
Reverse the telescope; again clamp it ahead of the star, time its transit at the NS web (T2) and read the hour circle (t2).
Then if the collimation error is zero, T2  T1 = t2  t1.
If T2  T1 is greater / less than t2  t1,the angle between the OG end of the tube and the Dec axis is more than / less than a right angle. The collimation error:
may be applied as a correction, or may be removed if the requisite adjustment is provided. With small instruments, errors of collimation can be corrected by inserting tinfoil or thin metal shims inside one end of the cradle, with frequent trials on a star. The procedure is bad for the temper, but practicable.
If the difference between (T2  T1) and (t2  t1) is expressed in seconds of time, and b=length of the cradle in inches, the distance that the end of the cradle must be displaced from its existing position is, in inches:
Declination and polar axes perpendicular
(a) With circles:
Set the Dec axis horizontal with a striding level, and read the hour circle (t1). Reverse the telescope, level the axis, then reread the hour circle (t2). Then it should be that t1 = t2 (12h graduation) or t1 = t2±12h (24h graduation).
Most small mountings make no provision for the correction of this error, and a faulty mounting should be returned to the makers or to any optical firm specialising in repair work.
(b) With circles:
This error may be distinguished from collimation error by observations of the type described in the section on Collimation. Both errors will have the effect of introducing inequality between (T2  T1) and (t2  t1). But if the cause of the discrepancy is lack of perpendicularity between the instrumental axes, the error will be a function of Dec d; more precisely, of tand. Therefore repeat the observations with a star not less than 45º from the equator. The magnitude of the error is:
Limits of accuracy required
The more accurate of the methods described are capable of reducing the angle between the polar axis and the parallel to the Earth's axis passing through the mounting, to less then 1'. Providing this angle does not exceed 2' or 3' the setting will be perfectly good enough for ordinary visual observation under all circumstances.
Even for photographic work a divergence between the polar and terrestial axes not exceeding a few arcmins is permissible. The effect of such divergence is to make the photographic plate rotate slowly about the image of the guide star, so that the images of all other stars in the field will not be points but arcs; the lengths of these arcs are a function of their distance from the guide star, and of the guide star from the pole. From the curve in the figure below may be read the maximum permisssible error in the setting of the polar axis at various polar distances, assuming that a length of trail of 0.0025 ins for an image situated 1.5 ins from the guide star in the focal plane, during an exposure of 2h, is tolerable. The figure reveals the approximate values of the permissible divergence of the polar axis, using the same criterion, for a variety of exposure times.
For micrometer work involving the measurement of position angles, as for photography, the tolerance for the orientation of the polar axis decreases rapidly with decreasing polar distance. Such work demands the greatest possible precision in the adjustment of the polar axis, which must be directed at the true pole  i.e. the effect of refraction must be taken into account (see section on Refraction). Precision of this order is only obtainable if the instrument is fitted with finely divided circles, or by photographic methods based on star trails.
Effect of refraction
The effect of atmospheric refraction is to increase the observed zenith distance of a star as it approaches culmination, and to decrease it as it passes from culmination towards setting. Hence a precisely adjusted polar axis would necessitate southward adjustments in Dec while following a star E of the meridian, and northward adjustments when the star is W of the meridian. At the same time the clock rate is falsified, since refraction keeps the star apparently above the horizon for a slightly longer period than it in fact is.
For photographic and ordinary visual observation it is convenient to effect a partial compensation of this by increasing the elevation of the polar axis slightly, so as to direct it at a fictitous pole as affected by refraction. The effects of refraction are virtually nil over an area of sky centred on the meridian in (for our latitudes) Dec about 20º N, whose radius is about 30º. Hence, if as a final adjustment the elevation of the polar axis is increased enough to keep, say, Arcturus, Aldebaran, or Algeiba at the intersection of the webs without use of Dec slow motion  or, alternatively, to use one of these stars in the first place  then the effect of refraction will be largely eliminated over a wide area of the southern sky.
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