precision of telescope pointing for outdoor targets

JOURNAL OF TIllE OPTICAL. SOCIi-TX OF AMERICA o

Precision of Telescope Pointing for Outdoor Targets

FRANciS E. WASHER AND HELEN BRUBAKER WILLIAMSNational Bureau of Standards, Washington, D. C.

(Received April 3, 1946)

The probable error of a single pointing (PE 8) is measured for a single telescope with a varietyof targets. This investigation shows that, although some change in PE, with distance doesoccur, the distribution of PE, as a function of distance can usually be neglected and a valueof 0.62 second assigned as a practical average. The values of PE, for an indoor target usuallyshow a small variation from one experienced observer to another, and from right to left eyeof the same observer. There is also a measurable systematic difference in pointing between theright and left eyes of the same observer. In outdoor pointing a long period error or drift isusually superposed upon the short period errors.

I. INTRODUCTION

W HEN pointings are made with a telescope,V the probable error of a single setting is of

interest. In this paper, the error, associated withthe optical characteristics of the problem, is in-vestigated. The errors present mast be eitheraccidental or systematic. The study of the acci-lental errors is emphasized herein. These acci-

dental errors arise partly from the conditionsprevailing along the optical path and partlyfrom limitations on the part of the observer. Theformer are greater in magnitude. In outdoorpointing, the image of the target formed by theobjective and seen through the ocular is con-stantly oscillating with respect to the cross-hairsmounted in the focal plane of the objective.Setting the cross-hairs into apparent coincidence

TABLE I. Probable error for a single observation (PE,) forindoor pointing with target in the focal plane of a

collimator.

PE.Aperture Filter x; seconds

27.5 K-3 50 0.24A 50 .23B 50 .22C 50 .42

21.4 K-3 50 0.21A 50 .19B 50 .21

40 K-3 100 0.2427.5 K-3 50 .2421.4 K-3 50 .2115 K-3 50 .2210 K-3' 50 .276.8 K-3 50 .28

Total number of independent observations 700Average PE.,= 0.244 second.

with the image of the target formed by theobjective and seen through the ocular is, for themost part, a matter of deciding upon the meanposition of the image of the target and settingthe cross-hairs thereon. Even when the imageof the target appears stationary, an error ofpointing still persists and is measurable.

In the present work, the pointing errors for agiven telescope have been measured for a numberof outdoor targets and an average value deter-mined for the probable error of a single pointing.The figure so determined is of interest in that itis the limit of precision that can be attained inoutdoor pointing under average daylight condi-tions and over a terrain that consists partly ofpark area and partly of city residential and busi-ness section. When, therefore, measurementsthat involve pointing accuracy are being made,under similar conditions, it is profitless toattempt, by further refinement of the equip-ment, to better the precision if this limit hasalready been reached. For outdoor pointing, theaverage value of the probable error of a singlesetting found here is 0.62 second.

II. APPARATUS AND METHOD OFMEASUREMENT

The apparatus consists simply of a telescopeand a means of varying the pointing withoutdisturbing the telescope. This is accomplished byplacing in front of the telescope objective a weakprism capable of rotation about its vertical axis.Since the deviation of a parallel beam of lightcaused by a prism is a function of the angle ofincidence on the prism surface, the image of the

400

VOCIX I I" 36, '\C'M11lo I l .1uTIIY, i

PI RCISION OF TELESCOPE POINTING

target can be made to move with respect to thetelescope cross-hairs by rotating the prism.

A Berger level telescope with an effectiveaperture of 39 millimeters and a magnificationof 37 diameters is used in the experiment. It isequipped with cross-hairs intersecting at 90 inthe center of the field, the cross-hairs beingoriented to appear as an X. The telescope issupported in a special holder fixed to a saddlewhich fits on the ways of an optical bench. Theholder itself is capable of micrometric adjust-ment which is used in making the initial settingof the image of the target on the intersection ofthe cross-hairs. However after the telescope isproperly aimed, these controls are left undis-turbed during the course of a run.

The weak prism has a clear aperture of 70 mil-limeters. The angle between its surface is 4minutes. The prism is so oriented in a holderwhich is capable of rotation about a vertical axisthat the prism axis is parallel to the axis of rota-tion. The prism system is carried by a saddleplaced on the optical bench in front of thetelescope. The prism is set initially so that itssurface makes an angle of approximately 300with the line of sight when the image of the targetis on the cross-hairs of the telescope. A planemirror is fixed to the prism holder in such mannerthat the observer can with the aid of an auxiliarytelescope and cross-hair note any angular dis-placement of the prism by means of a scalemounted behind the observer.

To make an observation the prism is oscillatedabout its vertical axis causing the image of thetarget to sweep from right to left of the cross-hairs. The amplitude of oscillation is successivelydecreased until the image of the target appearsto coincide with the cross-hairs in the telescopeeyepiece. The position of the prism is then notedwith the aid of the auxiliary telescope and scale.A series of ten such observations is taken and the.probable error of a single pointing determinedfrom these data. Five such 10-groups are usuallytaken in succession to constitute a single run.

The instrument is calibrated by measuring theangular displacement of the target image betweentwo extreme points on the scale. This measure-ment is facilitated by using the micrometeradjustments on the telescope holder by meansof which the ttal angle can be measured to the

TABLE II. Description of targets.

RangeName meters Nature of target

Collimator 3 Vertical wire in focal plane of col-limator.

Radio 100 Vertical metal column.Tilden 323 Six-foot vertical vent pipe two inches

in width painted white.Sedgwick 445 Six-foot vertical vent pipe two inches

in width painted white.Broadmoor 703 Ball-tipped finial.Ordway 905 Six-foot vertical wooden strip 14

inches in width, painted white,fastened to brick wall.

Klingle 1153 Six-foot vertical wooden strip sixinches in width painted red andwhite, 1 inch strip down center.

Swank 1399 Vertical narrow finial.Majestic 2722 Vertical pipe painted white.Baptist 3000 Finial on spire.Unity 3080 Finial on spire.Howard 4505 Finial on spire.Clock 4779 Narrow window on clock tower.Trinity 5533 Gold cross on dome.Capitol 7554 Vertical member attached to dome.Bradbury 13,623 Supporting column of water tank.

nearest half-second. Since the total angle isis usually 40 or more seconds, the calibrationerror probably does not exceed one percent fromthis source. A second source of error arises fromnon-linearity of the relation between prism rota-tion and angle of deviation. However, for thesmall displacements which occur, it is believedthat the error resulting from this cause does notexceed one percent. Thus the total error ofcalibration is not likely to be in excess of twopercent. Since this is the maximum errorthroughout the entire experiment, its effect uponany conclusions drawn can be considered neg-ligible. Usually the scale is placed at such dis-tance from the mirror that the angular equivalentof one scale division is less than one-fifth themagnitude of the least probable error of a singlepointing.

The probable error of a single pointing, PE,is computed from the approximate formula

0.8453k E rPE, (in seconds) =

[n(n-1)] (1)

where n is the number of observations, Fr is thesum of the residuals, and k is the calibrationconstant of the prism for conversion of mil-limeters observed on the scale to seconds devia-tion of the light beam caused by the prism. The

401

F. E. WASHER AND H. B. WILLIAMS

observations are considered 10 at a time, and forn=10 Eq. (1) can be written,

PE = 0.0891kEr. (2)

III. RESULTS OF MEASUREMENT

1. Indoor Target

While the main aim of the present work is thedetermination of PE, for outdoor targets, it wasconsidered desirable to determine first the PE,for the given telescope with an indoor target.Thus when, as is later found, the PE. for anoutdoor target is greater than the value obtainedwith an indoor target, one can be sure that thelimit on PE, is being placed by the conditions ofoutdoor pointing and not by limitations of thetelescope itself. Accordingly the experiment wasfirst performed indoors with a vertical wiremounted in the focal plane of a collimator as

TABLE III. Values of PE, for 6 sets of 50 observationseach, considered 10 at a time, made by observer W.

10-group PE, PE,Time number seconds Pp 50-set PP

1 2 3 4 5 66-14-44 1 0.58 -40.091:00 P.m. - 2 .34 .05

3 .44 .07 0.45 -4-0.034 .44 .075 .47 .07

6-16-44 6 .46 .073:00 P.M. 7 .57 .09

8 .65 .10 .51 .049 .44 .07

10 .45 .07

6-17-44 11 .61 .108:50 A.M. 12 .52 .08

13 .53 .08 .56 .0414 .64 .1015 .48 .08

7-1-44 16 .59 .091:00 P.M. 17 .72 .11

18 .75 .12 .67 .0519 .81 .1320 .48 .08

8-17-44 21 .82 .132:20 P.Mi. 22 .77 .12

23 .52 .08 .63 .0424 .75 .1225 .29 .05

8-18-44 26 .45 .072:50 P.M. 27 .58 .09

28 .59 .09 .69 .0529 1.18 .1930 .65 .10

Av. 0.58 .02

target. The collimator used has a focal length ofone meter and a clear aperture of 75 millimeters.The diameter of the vertical wire subtended anangle of 16 seconds in the true field.

These indoor observations were made forseveral apertures and for several colors. Thevarious apertures were obtained with a series ofcircular stops of varying diameters. The color ofthe light was changed by interposing filtersbetween the illuminating lamp and the targetin the collimator. The following Wratten filterswere used,-No. 9 (yellow) K-3; No. 25 (red) A;No. 58 (green) B; and No. 49 (blue ) C. Theresults of measurement are shown in Table I.It is clear from the table that, with the exceptionof observations made with the C filter and thosemade at-apertures 10 and 6.8 mm with the K-3filter, the accuracy of pointing with this telescopeis not significantly changed for the range ofapertures or filters studied. Even for the excep-tions it is possible that the decrease in accuracyresults more from the greatly decreased bright-ness of the visual field than from a definite cor-relation with color or aperture. We shall thereforeassume that, for the conditions under whichthese measurements are made, the accuracy ofpointing for this telescope is not affected byaperture or by color of light and that the value,PE,= 0.24 second, is the optimum value thatcan be expected for indoor pointing with thisinstrument.

2. Outdoor Targets

The outdoor test objects consist of a numberof distant targets that are readily discerniblefrom the laboratory. A brief description' of thetest objects is given in Table II. For complete-ness, the indoor target, collimator, is included.

Several series of observations were made oneach target. In most cases measurements weremade independently by two observers. The fullaperture, 39 mm, of the telescope was used forall observations. Occasionally a red filter wasused over the eyepiece, but no attempt is madeto differentiate the results of measurement forthis condition from the great mass of data takenwith no filter. This is justified on the basis of the

1 This description of the test objects is taken chieflyfrom an unpublished report, "Surveyed Ranges" preparedby L. W. Scott, April 17, 1944.

402

PRECISION OF TELESCOPE POINTING

.0 1

1 2 3 4 5 6_ 50-SET NUMBER

FIG. 1. Variation in PE. for a given target. There are values of PE, obtained ontarget Swank distant 1399 meters. The five determinations made in a single 50-setare shown by X's. The average PE. for 50-set is shown by a horizontal bar. Thegrand average of 0.58 second is indicated by the horizontal line.

results of measurement shown in Table I andfurther justified in that no significant differencewas found for observations with and without ared filter on an outdoor target.

(a) Single Observer with Single Target

For a given target, the values of PE, spreadover a wide range. Not only is this true for ob-servations taken on different days, but it is alsotrue for observations taken on the same day. Toshow the manner in which the values of PE,fluctuate, Table III lists the results of observa-tion on a single outdoor target by a single ob-server. Fifty observations are made in a singlerun of approximately 30-minute duration, andthese observations are considered as five con-secutive 10-groups for each of which the valueof PE8 is computed in order to minimize theeffect of possible long period fluctuations. Ac-cordingly, each value of PEB listed in column 3is the probable error of a single observation asdetermined from a series of 10 settings. Thesimple average of PE, for each five 10-groups islisted under column 5. Consideration of thevalues listed shows that the average value ofPEB for five 10-groups has nearly as much vari-ation from one 50-set to another as is shownfrom one 10-group to another within the limitsof a single 50-set.

It is perhaps advisable to consider at thispoint whether or not each of the listed values of

PEB is equally valid. The rule followed in thecourse of these experiments is to compute PE,from five 10-groups and compare their residualswith the value of P, obtained from the formula,2

Pc h2P/3[2(n- 1)]I. (3)

In the above expression, P is approximately theprobable error of the probable error, P and nis the number of observations. Since the numberof observations in any one group is always 10in this experiment, this expression is sufficientlywell approximated by

Pp = -JPB 8/2n (4)

which may be used for n = 1lOm, where m is thenumber of 10 groups whose P, is wanted.

Values of P, for each 10-group are given incolumn 4 of Table III. The departures of thesingle values of PE, from the average PEB forfive 10-groups do not appear excessive whencompared to values of Pp so it may be assumedthat all values occurring in a single 50-set areequally valid and that it is reasonable to averagethem.

The manner in which the values of PEB fluc-tuate is shown graphically in Fig. 1. Here eachvalue of PEB is plotted and the average for each50-set is shown by a horizontal bar. The valueof P for the average PE, of each of the five

2 A. G. Worthing and J. Geffner, Treatment of Experi-mental Data (John Wiley & Sons, Inc., New York, 1943),p. 197.

a)C

0.

z0

z

403

. ^^


FABLE IV. Probable error of a single pointing for each oftwo observers With distant targets.

Range Observer W Observer B AverageTarget meters PE, n PE. it PE, ,

Collimator 2 0.24 700 0.24 700Radio 100 0.45 50 .45 50Tilden 323 .46 100 .42 150 .44 250Sedgwick 445 .36 200 .50 200 .43 400

Broadmoor 703 .57 300 .57 300Ordway 905 .61 300 .44 100 .57 400Klingle 1153 .60 200 .57 100 .59 300Swank 1399 .58 300 .58 200 .58 500

Majestic 2722 .51 150 .52 100 .51 250Baptist 3000 .73 200 .73 200Unity 3080 .75 600 .67 200 .73 800Howard 4505 .90 50 .72 50 .81 100

Clock 4779 .66 500 .71 150 .67 650Trinity 5553 .65 200 .65 200Capitol 7554 .92 100 .69 100 .80 200Bradbury 13,624 .86 50 .97 50 .92 100

Average 0.64 2750 0.60 950 0.62 4700

10-groups is indicated by the length of the ver-tical line intersecting the bar. The solid hori-zontal line at value, PE 5=0.58, is the averagePE, derived from 300 observations considered asthirty 10-groups.. Although the departures of the50-set averages from the grand average of PE,are in some cases greater than one would estimatefrom the values of P, it seems likely that thebest value of PE, is the simple average. Ac-cordingly the accepted value of PE. with thistarget for this observer is taken as 0.58i0.02second.

In further justification of this procedure oftaking a single average of such a widely varyingset of values of PE,, it must be remembered thatthe purpose of this investigation is the deter-mination of the average error of pointing thatmay be expected under average seeing condi-tions. It has accordingly not been consideredprofitable to attempt, in this study, any corre-lation between pointing accuracy and state ofthe weather or degree of visibility. It is of coursevery probable that the accuracy of pointing isaffected by such conditions and that the widevariation in the values of PE. is a result ofchanging weather and visibility. However, sincethese quantities (lo not admit of control andsince the main objective of this study is theleternuination of the average error that is to be

expected under average conditions, no extensiverecords of such variables have been made.

(b) Two Observers with Many Targets

In view of the wide variation of PE, shownin Section (a), it seemed advisable to guardagainst too much reliance on the judgment of asingle observer. Therefore when the range of theinvestigation was extended to several outdoortargets at varying distances, the observationswere made by two observers identified as ob-server W and observer B.3 The results of meas-urement are assembled in Table IV. The resultsreported under observers W and B represent theaverage PE, values where both observers havemade observations on a given target. The weightgiven to each PE, in determining the final PE.is it.

There are several items worthy of note in thisexhibit. First, the values of PE, for a giventarget do not differ greatly from one observerto the other when n is large. Second, if it beassumed that any effect on PE, resulting fromvariation in distance can be neglected, the grandaverage of 2750 observations by W differs byonly 0.04 second from the grand average of 1950observations by B. The differences that existbetween the PE. values for the two observerson a given target when n is small are no greater

TABLE V. Probable error of a single telescope pointing as afunction of distance.

PE,0 Cl C2

Distance sec- sec- see- P5 0 -C1 0 -C2Target meters it onds onds onds seconds seconds seconds

Colliiimator 2 600 0.25 0.26 0.41 40.01 -0.01 -0.16Radio 100 50 .45 .40 .45 .03 .05 .00Tilden 323 250 .44 .47 .49 .01 -. 03 -. 05Sedgwick 445 400 .43 .49 .50 .01 -. 06 -. 07

Broadmoor 703 300 .57 .52 .53 .02 .05 .04Ordway 905 400 .57 .54 .54 .02 .03 .03Klingle 1153 300 .59 .57 .56 .02 .02 .03Swank 1399 500 .58 .59 .57 .01 -. 01 .01

Majestic 2722 250 .51 .66 .64 .02 -. 15 -. 13Baptist 3000 200 .73 .67 .65 .03 .06 .08Unity 3080 800 .73 .67 .65 .01 .06 .08Howard 4505 100 .81 .72 .70 .04 .09 .11

Clock 4779 650 .67 .73 .71 .01 -. 06 -. 04Trinity 5533 200 .65 .75 .74 .02 -. 10 -. 09Capitol 7554 200 .80 .79 .79 .03 .01 .01Bradbury 13,624 100 .92 .89 .92 .05 .03 .00

o is the value derived from observations, Cl =0.064dl/4 +0. 19 second,and C-3 =0.0044d1//2+0.41 second.

Observer W is F. 1x. \Vasher an(l observer B is Mrs.H. B. Williams.

4()4

PRP ECISION OF TELESCOPE POINTING

5000 10000 15000RANGE IN METERS

FIG. 2. PE, versus distance. The points indicate observed values. The solid curve is the relationcomputed from the formula PE.=0.064d"+0.19.

than those that sometimes occur between succes-sive groups for a single observer. The averagePE, for 4700 observations by two observers is0.62 second. If therefore, one wishes to neglectany distance effect as being too elusive or toosmall to determine with accuracy, one may saythat the probable error of pointing for an outdoortarget for a single observation is

PE,=0.62O0.01 second.

3. Effect of Distance

In the preceding section it has been assumedthat PE, is independent of the distance sepa-rating observer and target and a value wasassigned to it on that basis. This procedure isjustified practically because of the seeminglyrandom scattering of the PE, values with respectto distance. Also, so many variables enter intopointing accuracy that the effect of distance isscreened by the greater influence of factors thatare not subject to control such as state of theweather and conditions of visibility.

Notwithstanding these handicaps it is none-theless intriguing to examine the probable formof an equation connecting PE, and distance andto note how closely the predicted values ap-proximate to the observed values. To this end,least squares solutions were made for each oftwo assumptions, first that the probable errormay be expressed in either some such form as

PE, = bdi+a (5)or second as

PE 8= bdI+a, (6)

where a and b are arbitrary constants, and d isthe distance separating target and observer. Ineach solution, the same fifteen averaged valuesof PE, covering the range from 100 to 13,624meters were used. Evaluation of the constants inEq. (5) yielded the relation

PE, = 0.064di+ 0.19. (7)

Values of PE, calculated on this basis are listedin Table V under the column headed C1. Evalu-ation of the constants in Eq. (6) yielded therelation

PE, = 0.0044d1+0.41. (8)

Values of PE, calculated on this basis are listedin Table V under the column headed C2.

To facilitate comparison of observed andpredicted values, the differences are tabulatedunder 0-C 1 , and 0-C 2. It is noteworthy thatthese differences for the majority of the targetsare of the same order of magnitude as P com-puted according to Eq. (4). There is little tofavor either solution for targets distant 100meters or more. However, the agreement fortarget collimator is much better with Eq. (7)than with Eq. (8), although the PE, value forthis target was not used in obtaining the least-squares solution for either. On the basis, there-fore, Eq. (7) appears to be the better of the twosolutions.

In Fig. 2, the values of PE. predicted by Eq.(7) are plotted as a function of distance. Theobserved values of PE, are shown as circles. Itis evident that on the whole there is fairly goodagreement. In contemplating the points of dis-

1.20

.80.

z0C-)

In"Ia. .40

0

405

F. EI. WASIER AND H. B. WILLIAMS

TAULE VI. Consistency of observations for each of two observers with distant targets.

Observer V Observer B

Range (PE)Av (PEm)A C C -(PEm-)Av (PE)AV (PE )Av C C-(PEm)A,Target meters second second second second second second second second

Radio 100 0.45 0.14 0.86 0.72Tilden 323 0.49 0.15 0.14 -0.01 .42 .13 .62 .49Sedgwick 445 .36 .11 .16 .05 .50 .16 .63 .47Broadnoor 703 .57 .18 .70 .52Ordway 905 .62 .20 .29 .09 .44 .14 .50 .36

Klingle 1153 .62 .20 .30 .10 .57 .18 .42 .24Swank 1399 .55 .17 .31 .14 .58 .18 .42 .24Majestic 2722 .51 .16 .16 .00 .52 .16 .21 .05Baptist 3000 .73 .23 .59 .36Unity 3080 .75 .24 .34 .10 .67 .21 .63 .42

Howard 4505 .90 .28 .30 .02 .72 .23 .45 .22Clock 4779 .67 .21 .42 .21 .71 .22 .53 .31Trinity 5533 .65 .21 .40 .19Capitol 7554 .92 .29 .31 .02 .69 .22 .54 .32Bradbury 13,624 .86 .27 .37 .10

Average .66 .12 .29 .08 .58 .18 .55 .36

agreement, it may be mentioned there may nothave been enough variation in the viewingconditions to bring about a truly average valuefor each target. Thus, Trinity was either clearlyvisible, in which case good values of PE, wereobtained, or it could not be seen at all. Hence thevalue of 0.65 probably is somewhat on the lowside. The data on target Majestic were takenunder unusually good seeing conditions. A pre-ponderant amount of the data on Tilden andSedgwick was taken on cool cloudy days in theforenoon, and, since under these conditions lowvalues of PE, were usually obtained, the reportedvalues are doubtless a little lower than the aver-ages for all conditions of visibility.

4. Consistency

It was early noted that, when a set of 50 ob-servations comprising five 10-groups is made, themean values of deviations in the line of sight forthe various 10-groups in a 50-set are not con-sistent. In other words, the means of the 10-groups appeared to drift or oscillate as if a longslow period of change was superposed on theshort period changes that occur in a 10-group.For example, such a change would be introducedby variation in a horizontal air temperaturegradient perpendicular to the line of sight. Totest for the existence of such long period error,each of the five 10-group means in a 50-set is

treated as a single observation and the probableerror, C, of sch a single mean is computed fromthe formula

0.8453C= 5 r,

[n(n- 1) ] (9)

which for n = 5 becomes

C=0.1890 Z r. (10)

If the distribution of the observations containedin the 50-set is normal, thee the value of Cshould be approximately equal to the averageprobable error of the mean, (PEm)Av, for the five10-groups which is obtained from the formula

(PEm,)AV =PE)AY10

(it)

The values of (PE,,,AV and C for some of thetargets are given in Table VI where, however,these data are averaged for a number of 50-sets.Values are also listed of the difference C - (PEm)AVto facilitate comparison. If no long period error ispresent, the values of C - (PEm)A, ought to berelatively small and the average over all thetargets ought to be approximately zero. However,the table shows that C is greater than (PEm)AYin all cases for observer B and in 10 out of 12cases for observer W. Therefore it may be statedthat a long period error does exist. For observerW, the average value of C-(PEm)Av is 0.08

406


second and for observer B, the value is 0.36second.

The quantity C-PEm is very much greaterfor observer B than for observer W, and since thevalues of PE, are closely comparable for both,this indicates the presence of yet another errorin addition to drift. In seeking an explanation forthis large difference between observers W and B,it was recalled that observer B used the left andright eyes for alternate 10-groups in a 50-setwhile observer W used the right eye only. At thetime this practice was initiated it was believedthat no difference should arise therefrom. How-ever, on analyzing the averages of the means forright and left eyes, a systematic difference in thedirection of pointing for right and left eyesappears to be definitely present for observer B.This difference, which is designated OD - OS,while showing some difference one 50-set toanother, is generally positive. To show that thisdifference is persistent, values of OD -OS forobserver B for twenty-three 50-sets are shownin Table VII. The values are given in chrono-logical order, and the total elapsed time betweenrun 1 and run 23 is three weeks. The values ofOD-OS do not appear to be constant for ob-server B, but in 18 out of 23 cases, it is a positivequantity. The average value for the 23 runs is0.82 second. It may be mentioned that systematicdifferences between the settings made by theright and left eye are evident in tables of datashown in a paper by Labitzke4 which deals withbisection errors. On the basis of Labitzke's work,it appears that the average of settings made byeither eye is likely to show a systematic errorbut that the magnitude is not the same for eacheye; thus an OD - OS value exists for each in-dividual although it is improbable that the valuewill be the same for all individuals.

5. Observer Differences with anIndoor Target

Having noted the pointing differences forright and left eye with outdoor targets in thecase of observer B, it seemed worth while toinvestigate this matter further using the indoortarget to eliminate the uncertainty arising from

I P. Labitzke, "Untersuchung fiber psychologisch- undphysiologische Bisektionsfehler," Zeits. f. Instrumenten-kunde 44, 61 and 165 (1924).

the long air column present with the outdoortarget. A series of observations was made byeach of six different observers5 at various aper-tures of the viewing telescope ranging from thefull aperture of 39 mm to an aperture of 6.8 mm.Whereas the outdoor observations by observerB usually had three 10-groups of left-eye pointingto two 10-groups of right-eye pointing, theindoor work involved an equal number of10-groups for each eye. Accordingly, each ob-server made at least sixty observations in groupsof ten alternating the use of right and left eye.

(a) Probable Error of a Single Pointing

The values of PE, for the six observers arelisted in Table VIII. PEOD and PEos are theprobable errors of single pointings with right andleft eyes, respectively, and PE. is the average forthe two eyes. It may be noted in this table andin Fig. 3 that there is usually a difference betweenthe two eyes in the magnitude of the probable

TABLE VII. Values obtained by observer B arranged inchronological order. Each run contains 50 observations.

Target

Broadmoor

*

Baptist

*Unity

BaptistMajesticTildenSedgwick

CapitolHowardMajestic

Tilden

Broadmoor

(PE*)AVRun sec.

1 0.562 .583 .574 .705 .58

(OD -OS)AVsec.

1.761.851.811.94

-.31

6 .53 1.217 .79 .978 .73 1.229 .81 -. 55

10 .65 .35

11 .88 .1312 .59 .2613 .41 1.5114 .53 1.62.15 .70 .04

16 .42 2.0117 .33 .7918 .54 .4619 .72 -. 3820 .46 -. 09

21 .42 .2022 .44 1.5423 .44 .76

C sec.

0.80.84.83.88.42

.55

.75

.55

.76

.95

.52

.38

.71.70.54

.91

.36

.44

.45

.04

.45

.70

.45Average 0.83

* Runs 5 and 9 are ot definitely marked OD and OS (hence thesevalues of OD -OS are not as reliable as the others).

5 The four additional observers cooperating in this workare L. W. Tilton, T; F. A. Case, C; E. H. Samuels, E;and F. C. Samuels, F.

__

407


C

_ - - _-

L. \I~~~~~~~~~~~~~~~~~~~~~~~~' - - E -r3il''\ I

I I

/ - _ F -

20 60 40APERTURE IN MILLIMETERS

FIG. 3. Probable error of a single pointing (PE,) versusaperture of the viewing telescope. Values of PE. for right-eye pointing are marked 0 and those for left-eye pointingare marked X.

error of a single pointing. Whether this dif-ference is sufficiently great to be significant inmost cases in problematical, since the averagedeparture of PEOD or PEos from PE, for allobservations listed is 8.5 percent, and the uncer-tainty in PEOD or PEos may be as great as 9.4percent using Eq. (4) and n=30.

However, it seems reasonable that each eyeshould be considered as a separate individual,and on this assumption perhaps some significancemay be attached to this difference. In any event,it is interesting to note that the lowest PE, is notalways obtained with the eye most generally usedfor observing by each individual. For example,while observers C and W, who normally observewith left and right eye, respectively, do havelower probable errors for those eyes, observerB who generally observes with the left eye andobservers T, E, and F, who normally use theright eye, all have a little greater error at mostapertures when observing with the eye normallyused.

It should be pointed out that observers E andF, whose probable errors range much higher thanthose of other observers and whose differences inprobable error for right and left eye are morepronounced, were at this time very inexperienced.Observers W, T, B, and C, the most experienced,have generally the lowest and least erraticprobable errors. These patterns, then, of in-dividual differences in observation are goodevidence that experience is conducive to pre-cision in pointing.

In the upper graph, Fig. 4, the probable errorsof a single observation of pointing made withright and left eyes are averaged for each in-dividual. Here may be seen more strikingly thedifference in the magnitude of the probableerror between the more experienced and theinexperienced observers. For most observers the

TABLE VIII. Individual differences in probable error of asingle observation of pointing with right and left eye.

Numberof obser-

Observer Aperture vations (PEoD)AV (PEos)Av (PES)AV

B 39.0 180 0.33 0.39 0.3627.5 60 .22 .31 .2721.4 60 .48 .63 .5615.0 120 .37 .37 .3710.0 120 .33 .28 .316.8 120 .35 .37 .36

W 39.0 120 0.31 0.29 0.3027.5 60 .29 .29 .2921.4 60 .26 .29 .2715.0 60 .33 .38 .356.8 60 .41 .47 .44

T 39.0 60 0.36 0.30 0.3327.5 60 .31 .35 .3321.4 60 .42 .30 .3615.0 60 .39 .41 .4010.0 60 .53 .47 .506.8 60 .73 .64 .68

C 39.0 60 0.45 0.35 0.4027.5 60 .41 .35 .3815.0 60 .53 .47 .5010.0 60 .50 .44 .476.8 60 .46 .54 .50

1X' 39.0 120 0.79 0.82 0.8127.5 60 .94 .77 .8621.4 60 1.66 1.54 1.6015.0 60 1.53 .88 1.1210.0 60 .86 1.19 1.026.8 60 .90 .94 .92

F 39.0 60 0.61 0.55 0.5827.5 60 .54 .42 .48

-21.4 60 .93 .55 .7415.0 60 1.02 .87 .9410.0 60 1.20 1.18 1.196.8 60 1.55 .94 1.24

2

2

20 10

.6 _ _ B

.2 -- I

.8 -

.4 -

0 2 I

.6 _ _ W

.2 - -

.8 - - - - -

.4-.0~~~

.6 - - T

.2 - - I

.8 - - - - -

.4 - -

0 - t10 20 30 40

.

408

-9

IN3

II

I

10


2.0-

1.6

1.2 - _1.0

SOA

. .4

0.

10 20 30 40APERTURE IN MILLIMETERS

FIG. 4. The probable error of a single pointing (PE,)versus aperture of viewing telescope (right- and left-eyeobservations averaged). The upper graph shows theobserver differences in probable error; the lower graphillustrates the average probable error of the experiencedobservers, W, T, C, and B.

probable error tends to decrease slightly whenthe aperture is decreased from 39 millimeters to27.5 millimeters. Then the probable error beginsto increase gradually as the aperture is decreasedprogressively from 27.5 to 6.8 millimeters. Thisdecrease in probable error followed by an increasewhen the aperture is stopped down progressivelywas also found by Tilton 6 in his investigation ofpointing accuracy in connection with prism sizein minimum deviation refractometry. The lowergraph of Fig. 4, in which the probable errors ateach aperture are averaged for the four moreexperienced observers, shows this gradual in-crease in probable error as a function of aperture.While the decrease in aperture may have someeffect on increasing the probable error, it - ispossible that this effect may be caused by thedecrease in illumination or in contrast.

(b) Pointing Differences

Successive 10-groups of observations weremade with right and left eyes alternately and it

6 L. W. Tilton, "Prism size and orientation in minimumdeviation refractometry," Nat. Bur. Stand. J. Research 6,59 (1931) RP 262.

TABLE IX. Pointing differences for right and left eye in atypical run.

10-group Eye Departure of 10-group meannumber used from 60-set mean seconds

1 Left +0.042 Right -. 183 Left +.354 Right -. 245 Left +.166 Right -. 10

was found that the mean of the 10-group settingsfor the right eye differed from the mean of the10-group settings for the left eye; that is, theobservers tended to point more to one side of thetarget with one eye than with the other. This isillustrated in Table IX, where the summary of atypical observation sheet is shown. For the runshown, the average departure from the mean is

TABLE X. Individual eye differences in pointing and theirprobable errors.

Ob- Aperture MD MS R OD -OSserver mm sec. sec. sec. sec.

B 39.0 i0.03 -4:0.04 -10.05 0.6127.5 .04 .06 .07 1.0921.4 .09 .12 .14 1.2615.0 .05 .05 .07 -. 0110.0 .04 .04 .06 -. 086.8 .04 .05 .07 -. 34

W 39.0 0.04 0.04 0.05 0.1527.5 .05 .05 .08 .2221.4 .05 .05 .07 .4115.0 .06 .07 .09 .286.8 .08 .09 .12 .06

T 39.0 0.06 0.05 0.09 -0.5727.5 .06 .06 .09 - 26214 08 06 10 -. 2915 0 07 .08 .10 -. 1410.0 .10 .09 .13 -1.236.8 .13 .12 .18 -. 88

C 39.0 0.08 0.06 0.10 -0.1627.5 .08 .06 .10 .0315.0 .10 .09 .13 .3010.0 .09 .08 .12 -. 356.8 .08 .10 .13 .17

E 39.0 0.10 0.11 0.15 -1.1327.5 .17 .14 .22 .0521.4 .31 .28 .42 -1.1515.0 .28 .17 .32 -1.2010.0 .16 .22 .27 -1.406.8 .17 .17 .24 -. 25

F 39.0 0.11 0.10 0.15 -0.4327.5 .10 .08 .13 -. 2321.4 .17 .10 .20 -. 7715.0 .18 .16 .24 -. 1610.0 .22 .21 .31 -. 356.8 .28 .17 .33 .25

J1

409

F. E. WASHER AND H . B. WILLIAMS

C

I E

- -- F

-2 1I z I "1 I I - -10 20 30 40 10 20 30 40APERTURE IN MILLIMETERS

FIG. 5. Right- and left-eye differences (OD-OS) in point-ing as a function of aperture for individual observers.

+0.18 second with the left eye and -0.17 secondwith the right eye; these values indicate anaverage eye difference of 0.35 second. Theseeye differences are tabulated under the columnOD-OS in Table X for each observer. AID and

s are the probable errors of the mean for rightand left eye, respectively. R is the probable errorof the difference in the means for right and lefteyes. It may be seen in Table X that in 28 out of34 instances, the values of OD -OS are greaterthan R; moreover the value of OD - OS rangesfrom 1 to 17 times as great as R and in 12instances is 4 or more times greater than R,which indicates definitely that there is a realdifference between the settings made with theright and left eye. There is no attempt here toexplain the cause of this difference but merely togive evidence thereof. Figure 5 illustrates valuesof OD -OS for each observer. It is interesting tonote that observer T has all negative values forthe differences and observers E and F havepredominately negative, while observer W hasall positive values and observers B and C have amajority of positive values.

IV. HISTORICAL SUMMARY

The errors inherent in telescope pointing havebeen the subject for numerous experiments in the

past. Such investigations have been primarilyconcerned with proving the existence of a relationbetween the probable error of a single pointingand the magnification of the telescope used. Theresults of a Dumber of these investigations areshown in Table XI. The mean error m of a singlepointing, which is the coefficient of error pre-ferred by most Continental writers, in terms ofthe magnification M in accordance with theformula deduced by each author is shown.Although some authors report several constants

TABLE XI. Results of earlier investigations arranged inchronological order.

PE,mn PE. (M =37)

Date Author seconds seconds seconds

1834 S. Stampfera 15/M 10/M 0.271856 F. Hartnerb 10/M 6.7/M .181879 C. M. v. Bauernfeindc 15/M 10/M .271885 C. A. Voglerd 50/M 34/M .911886 R. Wagner- 8/M 5.4/M .151901 J. Adamczikf 20/M 13.4/M .361910 H. Hlohennerg 33/M 22/M .601910 F. Hartnerh 30/M 20/M .551911 E. Hammer 50/M 34/M .911912 Klauser-Lohni 30/M 20/M .551912 M. Nabiuerk 8/M 5.4/M .151915 A. Noetzlil 3.5/1VM 3.4/-VM .391923 F. Klempau'n 6/1M 4.0OVM .661923 F. Kleinpaurn 29/M 19.6/M .531924 Wandhoff' 40/M 27/M ,731924 K. Ludemann' 7/1/M 4.7 VM .771925 SteinschligerP 40/M 27/M .73

24 0.4 0.27 0.271926 A. Pelzerq -+ - 16 +- .43 +-

M C2L C2L C2L1940 P. Werkmeister 6.5/M 4.4VM .72

Average PE. for all authors except Pelzer 0.52PE, found here 0.62

S. Stampfer, "Uber die Genauigkeit des Visierens bei Winkel-messungen," Jahrbilcher des k. k. polylechischen Islifuls in Wien(1934), Vol. 18, p. 231.

b F. Hartner, Handbuch der iederen Geoddsie (Wien, 1856), p. 102.C. M. v. Bauernfeind, Eleimzents des Verenessungsknde (Stuttgart,

1879), Vol. 1, p. 110.d C. A. Vogler, Lehrbuch der Pakischen Geoietrie (Braunschweig,

1885), Vol. I, p. 81.R. Wagner, "Uber die mit dem Reichbachschen Distanzmesser

erreichbare Genauigkeit und einige Errterungen ber die Fehlerur-sachen desselben" Zeits. f. Vermessungswesen 49 (1886).

f J. Adamczik, Compendiunm der Geoddsie (Leipzig und Wien, 1901).p. 58.

g H. Holienner, Geoddsie (Leipzig und Berlin, 1910), p. 27.h F. Hartner, J. Wastler, and E. Dolelal, Hand- nnd Lehrbuch der*iederen Geoddsie (Wien, 1910), Vol. I, p. 255.

i E. Hammer, Lehrbuch der elementaren praktischen Geometrie (Leipzigund Berlin, 1911), Vol. I, p. 270.

i Klauser-Lohn, Lehrbuch der Vereinessngskunde (Wien, 1912), p. 20.k M. Nbauer, Grndziige der Geoddsie (Leipzig und Berlin, 1915), p.90.

I A. Noetzli, "Untersuchung die Genauigkeit des Zielens mit Fern-rohren," Zeits. f. Instrumentenkunde 35, 65 and 89 (1915).m Fr. Klempau, "ber die Beziehungen zwischen Winkel-, Nonien-,und Zielgenauigkeit," Allgemeine Vermessungs-Nachrichten 317 (1923).

- Wandhoff and F. Kgler, Taschenbuch feir Berg-und Hiillenlete(Berlin, 1924), p. 771.

K. Lidemann, "Grundlagen fr den Voranschlag der Genauigkeits-leistung von einigen Theodoliten bei der Kleindreieckmessung und beifeinen Zugmessungen," Zeits. f. Instrumentenkunde 44, 558 (1924).

P SteinechlAger, "Untersuchuing eines 8-cm-Schraubenmikroscop-'rheodolites," Allgemeine Vermessungs-Nachrichten 445 (1925).

q See reference 8 in text.r P. Werkmeister, Geoddtische Istremnente (Leipzig, 1940), Vol. I, p.22.

410

0

0

0

2~~~~

I - - - - -

-I _- - - - -

0---

2 1 TX I


to cover different conditions of visibility, thesegiven here are the averages for the conditionsdescribed by the various authors. Each meanerror has been transformed into probable errorof a single pointing by multiplying by 0.6745 inorder to make possible a comparison with thevalue obtained by the present authors. Finallythe values of PE. are calculated for magnification37. It is worthy of note that the average of thosevalues of PE, in excess of 0.20 is 0.60 secondwhich practically equals the 0.62 found here.The values under 0.20 can legitimately beneglected for it is very probable that theseauthors obtained their values for indoor targetsor else for very near outdoor targets under idealconditions. However, even if the three values

7 J. W. Mellor, Higher Mathematics for Students ofPhysics and Chemistry (Longmans, Green and Company,London, 1909), p. 524.

less than 0.20 are included the average for all(the Pelzer value is perforce omitted from thisaverage) is 0.52 second which is still remarkablyclose to the value of 0.62 second. Pelzer has aterm dependent upon brightness in his formulawhich, judging from the results reported in hispaper, would yield values considerably in excessof the value 0.62 second for the lighting condi-tions prevailing during the present work.

The great variety of formulas as well as thevariation in the constants for the same type offormula tends to strengthen further the beliefof the present authors that for magnifications inexcess of 20 diameters there is no correlationbetween magnification and outdoor pointingaccuracy, for objects at a great distance.

8 A. Pelzer, "ber den Einfluss der Lichtstarke vonTheodolit- und Nivellier-fernrohren auf den mittlerenZielfehler," Zeits. f. Instrumentenkunde 46, 354 (1926).

.411

precision of telescope pointing for outdoor targets

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