On the Nature of the Anti-Tail of Comet Kohoutek (1973F)
zcAavs 27, 135-146 (1976)
On the Nature of the Anti-Tail of Comet Kohoutek (1973f). I1. Comparison of the Working Model with Ground-Based Photographic Observations
ZDENEK SEKANINA Center for Astrophysics, Harvard College Observatory and Smithsonian Astrophysical Observatory, Cambridge, Massachusetts 02138
AND
FREEMAN D. MILLER Department of Astronomy, University of Michigan, Ann Arbor, Michigan 48104
Received May 13, 1975; revised June 3, 1975
Photographic observations of the anti- tail of Comet Kohoutek ( 1973f ), obtaitled at the Cerro Tololo Inter-American Observatory, are photometrically reduced and the results compared with a recently formulated working model of the anti- tail. The applied technique of photometric reduction is described, and the radial and transverse profiles of the anti-tail, corrected for the effects of atmospheric extinction and the variable intensity of the ambient sky, are derived. Brightness variations in time are also studied, The most important result reached so far is a quantitative confirmation of the previously suggested hypothesis, arguing that dust particles in the anti-tail suffered a significant loss in radius due to evaporation near the perihelion passage. We find that only particles initially larger than 100-150/~m in diameter (at an assumed density of I gcm -3) survived. Numerically, however, this result is still preliminary, because the dynamical effect associated with particle evaporation remains to be explained. It is also tentatively suggested that. the emission rate of the dust from the comet was probably comparable with the rates derived earlier for Comets Arend-Roland (1957 III) and Bennett (1970 II).
I. INTRODUCTION then tested quantitatively. The photo- graphic material studied here was secured A working model was recently formu- under the supervision of the second author. lated for the anti-tail of Comet Kohoutek The photometric reduction and physical (Sekanina, 1974; referred to hereafter as interpretation are due to the first author. Paper I). The model was based on the Finson-Probstein (1968) theory of dust II. TttE MICHIGAN PROGRAM OF comets and fitted reasonably well the semi- PHOTOGRAPHIC OBSERVATIONS OF COMETS quantitative descriptions of the anti-tail by a number of observers, including the The plates described in Section III form Skylab III astronauts. In the present paper, part of a program of comet observations the radial and transverse brightness pro- initiated by the second author in 1950 with files and the time variations in the surface the Curtis-Schmidt telescope (61/91cm, brightness of the anti-tail are established f/3.5, 96':6mm -~) of the University of from plates of the comet taken at the Cerro Michigan. ]in addition to unfiltered "blue" Tololo Inter-American Observatory, near plates, exposures have been taken through La Serena, Chile, and the working model is red filters to suppress the CO t emissions of Copyright (c) 1976 by Academic Press, Inc. 135 All rights of reproduction in any form reserved. Printed in Great Britain 136 SEKANINA AND MILLER
Type I tails relative to the scattered Still- telescope between January 16 and Feb- light of the Type II tails, on which the ruary 15, 1974. Five plates, taken on former are often superposed. The extent to January 23, 24, and 26 and February 12 which the suppression is successful has and 15, are 098-02 panchromatic enmlsions varied from one comet to another (Miller, combined with an RG1 filter to provide 1958, 1962). The development of a model maximum sensitivity near 661~(1A and of dust tails by Finson and Probstein (1968) to cut off the visible spectrum below has stimulated interest in the study of the A ~ 5900A. The photometric system thus Michigan material, and analyses for Comets obtained is similar to the R system of Seki-Lines 1962 III (Jambor, 19731 and Johnson and Mitchell (1962). The other Bennett 1970 I] (Sekanina and Miller, seven plates, taken on January 16, 17, 20, 1973) have already appeared. 21, 23, 25, and 27, are standard 103a-O From the first, the comet-program plates emulsions without filter. The red-sensitive have been photometrically calibrated with plates are of primary interest, both a 14-tube spot sensitometer designed by because they span a longer interval of E. B. Weston; when the telescope was time and because they show the anti-tail transferred from Michigan to Cerro Tololo, better. The guiding of the ,lanuary 24 the sensitometer accompanied it. Since the plate was somewhat imperfect, so only the precision of the sensitometer calibration is other four plates, two in January an(I two critical, the photometric scale was meas- in February, have been photometricall.v ured photoelectrically by W. Liller in 1951 studied. The four observations are listed and 1955 and by S. M. Simkin in 1965. The in Table 1. average deviation from the mean scale of Of much concern has been the very low the three independent measurements for altitude of the comet. At the end of the each spot is ±0.04mag; a more detailed exposure, the comet was only slightly over discussion of the calibration tests can be 1() ° above the horizon on plates 15520, tbund in a paper by Miller (1967). A calibra- 15687, and 157(i13 and about 7 ° above the tion exposure is made for each comet horizon on 15554! Thanks to the excellent photograph through the same filter used in sky at Cerro Tololo, the plates are still the comet exposure and on a plate taken photometrically valuable, but the very from the same box and developed with the low altitude of the comet is a source of two comet [)late. The length of the calibration potential complications for the photometry exposure is the same as that, for the comet. of the plates, namely the effects of dif- ferential refi'action and strongly variable sky brightness. In addition, atmospheric III. THE (~ERRO TOLOLO PHOTOGRAPHS extinction, which is severe at these alti- A total of 12 calibrated plates were taken tudes even in the red section of the at Cerro Tololo with the Curtis-Schmidt spectrum, also needs to be accounted for.
TABLE I
(~ERRO TOLOLO ]~E|)-SENSITIVE PHOTOGfCAPHS OF (~OMET ]%OHOUTEK (098-02 EMULSION "tVITHAN RGI FILTEU)
C,omet's posit, ion 1950.0 Phas(' Date UT /1 r angle Altitude at Exl)oSur(~ Platte 1974 R.A. I)eel. (AU) (AU) ~ midexposure" (mm)
15520 Jan. 23.0495 23n40~. 67 +@ 46~5 O.853 O.837 71 ~2 1316 20 15554 Jan. 26.0614 0n07 .m3(~ +3c 12:2 0.894 (k909 66.2 I 1~8 50 15687 Feb. 12.0572 lh54m21 +12~01:9 1.257 1.278 45 ~ 16~3 60 15703 Feb. 15.0566 2h07T40 +12 57~7 1.334 1.338 43'14 16~2 60
" Not corrected tbr t~ depressed horizon. THE ANTI-TAIL OF COMET KOHOUTEK (1973f). II 137
The differential-refraction effect has been quantitatively analyzed following the procedure described in Appendix A. PLATE 15 Numerically, refraction has been found to affect the angular distance from the nucleus on the four plates by less than 1%, and the position angle, by no more than 0.1 ° at an angular distance of 1 ° from the nucleus and no more than 0.2 ° at 2 +. The effect of variable sky brightness proved considerably more severe. It so happened that the brightest section of the anti-tail was directed almost exactly toward the horizon on all the plates, so that the maximum brightness gradient of the sky projected unforeshortened along the anti-tuil. In addition, all exposures began during astronomical twilight (with the Sun's depression between 15 + and 17°), and almost the whole 15520 exposure was taken during twilight (with the Sun's depression at midexposure 17+). To demon- strate the extent of interference, we I000 mention that the relative gradient of the sky brightness on plates 15520 nad 15554 /• reached a maximum rate of 7% per degree S(;~,LE of altitude, while at the same time the intrinsic surface brightness of the anti-tail 0 I0' 20 ~ 50 / Fro. 1. A two-dimensional scan of plate 15687. amounted to only some 4-6% of the sky Scanning aperture is 9'.'7 by 9'.'7. The anti-tail brightness at 15' from the nucleus and points to the bottom of the scan, and the regular about 1-2% at 25". dust tail to the top. The numbers indicate the relative surface brightness, uncorrected for the contribution from the sky. IV. THE TECHNIQUE OF PHOTOMETRIC REDUCTION known positions have been used to fix the The technique for reducing the plates position angles. The position of the comet was essentially dictated by the trouble- on the plates was measured relative to the some observing conditions. Because of the ambient stars and reduced routinely with complicated character of the sky-bright- the use of the method of dependences. The ness variations near the horizon, it was result has also been checked by computing virtually impossible to use the standard the expected topocentric position of the two-dimensional scans to map the sky comet from Marsden's (1974) definitive "noise" over the area occupied by the orbit. The measured and computed posi- anti-tail (see Fig. 1). Preferable, because tions agree within 2" on plate 15520, more revealing, are one-dimensional radial 7" on 15554, 8" on 15687, and 13" on 15703. tracings, passing through the comet's These residuals can be accounted for nucleus and effectively covering the whole entirely by the uncertainty in the location anti-tail. Each radial scan is defined by on the plates of the nucleus in the strongly the position angle, and the "noise" varia- overexposed coma, which is almost 2' in tions in the respective section of the anti- diameter. The diffuse image of the comet's tail can be reasonably interpolated from head is also responsible for an uncertainty the known field brightness outside the in the distance scale of the scans, amount- anti-tail (see Fig. 2). Nearby stars of ing to several percent at 10' from the 138 SEKANINA AND MILLER
12C ! PLATE 15687
p-L
120 ,J ++ w w z I 5 tD g
U I1©
~ IC©
~J II© m m W c~
i i I I , ~__ I . 6(' 40' 20' 0 20' 40' ANGULAR DISTANCE FROM NUCLEUS
Fro. '2. Three radial scans of plat(' 15687. Scannin~ aperture is 578 by 5"8. Note the variat)le sky brightness (dashe CALIBRATION STAR D 4} 1 PLATE 15554 w ~- 158 5 ~) !20 D ! 0 0 I I I I i 50" 20' ;3" 0 I0" 20" ~0" 40" ANGULAR O~STANCE FRO~ CENTER OF STAR 1RAIL FIG. 3. An example of a scan across a calibration star's trail. Scanning slit is 1';0 by 19'.'3; the long side of the slit is parallel to the trail. The dashed line is the adopted sky brightness. 140 SEKANINA AND MILLER TABLE 11 ( ~ALIBRATION STARS Apparent Star's position 1950.0 ll magnitude Calibration <)utside Earth's star Plate I{ .A. 1)eel. atmosphere A 15520 23h42mlT.~90 +0~40'46':8 13.:14 B 15520 23h44maTe I 6 +0°48'10".'7 13.04 (~ 15554 0n()5~53~42 +3 ~ 18' l 7'.'2 12.95 D 15554 0n06m50~05 +3°22'55'.'5 13.49 E 15687 I "53m30741 +12°13'10':9 13.29 F 15687 I n55~07-'.83 + 12':'03'40':8 1:1.12 G 15703 2"06m01~57 + l 2°50'32'.'7 13.93 H 15703 2"1)6m01~88 + 12°50'48".'6 14.54 J 15703 2h08m07786 +13°23'45".'5 13.69 + T i I --~ .... VI. THE P~ESULTS PLATE 15520 The photometric procedure described in the previous sections has been applied to the four plates listed in Table I. The absolute units used for the surface bright- ness are S~0(red ), the equivalent number of stars per square degree of the apparent R magnitude 10. Figures 4-7 show the distribution of light in the anti-tail on the four nights as a function of position angle ~o • a: (transverse profiles). A comparison with 55 5' L w the expected profiles predicted from the working model shows a certain degree of t > similarity. However, the anti-tail appears ---primarily near the nucleus to bc broader than anticipated, 1)erhaps because of the effect of nonzero ejection velocities IO • • 5o' of the dust particles, which was not taken .. WORKING into account in the formulation of thc '.. MODEL model. Also, Fig. 4 suggests a rather '". . strong dependence of the position angle 230 270 280 290 5OO 310 of the maximum brightness on the distance POSITION ANGLE (deg) from the nucleus. On the othcr han(t, the Fl(;. 4. The distribution, in position angle, of observed positions of the brightness maxi- surface brightness in the anti-tail of Comet rna on the two February [)l~tes agree with Kohoutek on plat(" 15520 at several angular the model within a few degrees, an(1 the (listanees t'rmn the )tucleus. The surface bright- observed profiles at large distances from hess, expressed in '~',o units (the (~(tuivalent the nucleus (at)out 30') on l)late 156S7 arc )tumber of magnitude 10 stars pet" sqllare degree), fitted by the calculate(1 profilcs fairly well refers to a color system similar to that of ,Iohnson's R magnitudes and is corrected for indee(I. atmospheric absorption (air mass = 0). For the The variations in the maximun~ surface sake of comparison, the profile predict(~(l by the brightness of the anti-tail with angular working mode| is shown, in relative units, at the distance from the nucleus arc displayed in t)ott i I i i i T 1 I T 7 PLATE 15554 L PLATE 15687 I • ~. 5.6' ~ 1oo / \ ,,8 ~ g g- uJ N Q: <" ->GP // ...... \250 I ": "'".. "'-. 50.6' ,, 506' ...... i;-:::o WORKING .: :: ""'.. MODEL \\\ MODEL : 370' L l : I i J ", ~ l • .;o .'o .g ~'o 300' ..... 310' 260 270 280 290 300 5JO POSITION ANGLE (de{J) POSITION ANGLE (deg) FIG. 5. The distribution, in position angle, of FIG. 6. The distribution, in position angle, of surface brightness in the anti-tail of Comet surface brightness in the anti-tail of Comet Kohoutek on plate 15554 at several angular Kohoutek on plate 15687 at several angular distances from the nucleus. For details, see the distances from the nucleus. For details, see the caption to Fig. 4. caption to Fig. 4. inverse 2.8 power law at distances from reached a maximum sometime around the nucleus exceeding 20', but the surface February 1, and was almost constant or brightness drops more slowly at smaller somewhat subsiding in mid-February. distances. In the next section, we submit a Comparison of the observations with the tentative interpretation of this effect in working model shows some signs of terms of a variable population index s of resemblance to the predicted curves. The the differential particle mass distribution fit is better for the asteroidal-type phase m-Sdm in the mass range under considera- law in late January and for a constant tion, a case not considered in Paper I. phase effect in mid-February. Finally, Fig. 9 is a plot of the maximum Because of the differences between the surface brightness of the anti-tail as a observations and the working model in function of time and distance from the general and the variability of the radial nucleus. The total error is estimated at gradient in particular, the relation between ~:20% to ~30% at 15' from the nucleus the surface-brightness distribution and the and at probably more than :~50% at 30'. mass-loss rate of the dust from the comet Unfortunately, the 17-day gap between is more complicated than suggested in the second and third observations--the Paper I and will have to be reassessed by period of interference from the Moon-- means of an improved model. We note, makes the actual curve of time variations however, that a cursory comparison of the rather indeterminate, and the fits shown Cerro Tololo plates with the Lowell in the figure are just one of many possi- Observatory prints (for the latter, see bilities. However, we can be reasonably Paper I) suggests that the surface-bright- certain that the brightness was increasing ness ratio of the anti-tail to the sky is fairly rapidly in late January, must have likely to be lower than was adopted in 142 SEKANINA AI~D MILLER I I .... 7 7 T- I - PLATE 15705 "~ ~ ~ \ WORKING MODEL | o ~ 5 6' ~~.... "~.. 8 I~ ){\\ i o , \ 2 ,, w uJ N ~ g ~- ~ ~ ", I ~ 20 I' 4 ,~ I 25 O' [ U io I0 14 m g \.o, [ c) I : •'••WORKING MODEL 260 27'O 280 290 500 510 5 6 7 8 9 I0 0 50 50 60 POSITION ANGLE (deg) ANGULAR DISTANCE FROM NUCLEUS (ercmin) b'[('. 7. The distribution, in position angle, ()t" Ibm. 8. The maxhnum surface brightness in surface brightness in the anti-tail of Comet the anti-tail of Comet Kohoutek on the four Kohoutek on plate 15703 at several angular plates as a function of the angular distance from distances from the nucleus. For details, see the the nucleus. The inverse 2.8 power law predicted caption to Fig. 4. by the working model is also plotted. Paper I by a factor of at least 20, and it scans emerges from a model tbrmulated by thus appears that the emission rate of the Finson and Probstein (1968) for small (lust should be reduced by a comparable particle-ejection velocities. The model factor. This conclusion would be in line relates the modified surface density of the with a suggestion put tbrward in Paper I emitted particles (which is proportional to that the apparent order-of-magnitude dis- the surface brightness) to their emission crepancy between the dust-output rates flux and size distribution and to the area from Comet Kohoutek on the one hand they occupy in the plane of the sky. As long and from Comets Arend-Roland and as we consider only larger particles, for Bennett on the other was due to the which the acceleration ratio 1 - tx of solar inaccessibility, at the time of completion radiation pressure to solar" gravit.y does not of Paper I, of a reliable source for' absolute exceed ~0.01, the relative surface bright- calibration and is therefore not real. A ness in a particular direction away from proper account of evaporation, however, the nucleus is related to 1 -/~ in a simple will somewhat increase the final estimate way. First, the radial direction is practic- for the loss of mass of the (lust from the ally identical with a synchrone. Second, comet. the projected area occupied by the par- ticles is proportional to 1 - F. Third, for a constant particle density, the particle VII. PRELIMINARY PHYSICAL diameter ~ is inversely proportional to INTERPRETATION OF THE OBSERVEI) l --/x because the scattering efficiency of IR,ADIAL PROFILES OF THE ANTI-TAIL the large particles tbr radiation pressure is The physical significance of" tile pro- practically independent of the particle gressively increasing gradient of the radial size. And fourth, the distance R of the THE ANTI-TAIL OF COMET KOttOUTEK (1973f). II 143 ill [/~lllllllllll II I[lll Illlll'I Consider an initial (preevaporation) / )~/ ~'\ .~ distribution function of particle diameters of the form _o t /" ,( \\ ~.__~_8.1 g(3) d8 - 3-" d3, (;~) / / "~. ~* / "~ ~ 113' where u is a constant. Consider further i / // ///--~ -- -- that, as a result of intense solar heating over a limited period of time, an outer layer/18/2 in thickness is lost by evapora- tion from each particle. This amount is assumed to be independent of the particle -- / / w size for material of given structure and // > composition. A particle of initial diameter / ~"+~ ~ e-. 30.6' c.) /( e: 8 now has a smaller diameter, 7+ = 8 -/13, and the particle-size distribution (3) / / ~ ~* ~._ 57.0' changes to g(7+)dT+ ~ (7+ +/18)-Udy. (4) ..'""'" WORKING MODEL The logarithmic slope of this distribution, ..'"'~ (a) = I0 -O'OI2a t(7+) = -0 log e g(7+)/3 log e 7+, (5) I f t I I r I i i i i i i f i i [ i i i [ I i i i i i i i i i 26 31 5 I0 15 20 1974. JANUARY FEBRUARY depends on the diameter 7+ and is related DATE to the initial exponent u by Fw. 9. The maximum surface brightness of t(7+) -- u/[1 + (/18/y)]. (6) the anti-tail of Comet Kohoutek at several angular distances from the nucleus as a function Since I/=: C(1 -/~)-~ (where C is virtually of time. The dashed curves show only one of a constant for particles of constant density, many possible fits to the observations. For com- and 1 -/~ now refers to y rather than to 8), parison, relative brightness variations in time we can write (6) as predicted by the working model are plotted for two idealized cases: the asteroidal-type phase ~/t = (1/,~) + (/1~/Cu) (1 - t~). (v) law ~ = l0 -°.°12: (a is the phase angle), and a constant phase effect (/) ----const. On the other hand, we now have I ~ y~ g(7+), so that the radial brightness gradient is large particles from the nucleus is propor- 01ogeI(R)_ 31ogeI(7+) 31og¢7+ W tional to 1-/~. The relative surface 0 log~ R 0 log~ 9' 0 log~ R brightness at distance R is then = 5 - t(fl. (s) I(R) ~f(1 - t~) (1 - t~) -~, (1) Deriving t(7+) tbr a number of angular distances on each of the scanned profiles, where f(1-/4 is related to the particle- assigning the proper value of 1-t~ (de- size distribution function g(8) as follows: pending not only on the distance R, but f(1 --U)~84g(8) ~ (1 -- ~[),)--4g(8). (2) also on the time of observation and on the position angle of the scan), and plotting We established in Paper I that there was the resulting pairs 1 - t~ versus 1/t, we can an excess of heavier particles in the anti- test the hypothesis of particle evaporation tail, which we interpreted as possible by comparing such a plot with the linear evidence for a significant loss of radius of relation predicted by (7). The plot, based the dust particles due to evaporation on the four plates and including only the near the perihelion passage. With a sections of the anti-tail where its relative sufficient body of observational informa- brightness exceeded 2% of the sky inten- tion, we can now test this hypothesis sity (to avoid unrealistic gradients from more quantitatively. very faint areas), is exhibited in Fig. 10. 144 SEKANINA AND MILLER 04 o • o o J /X 0 3 • .~o~ o.,,~ 0 I/t O& t 03 a" g~% U e oe •°& o o i~ :o ° o ~ " o 15520 t • 15554 A 15687 02 ,D 15705 5 ~ 6 o o o02 o o04 o o06 oo08 o OlO I-/x FK(:. 10. A plot of particle acceleration I --/~ versus the h)garithmie gradient t of the size distribution of the vaporizing particles in the anti-tail. A least-squares solution indicates a con> The table also includes the population plete absence of nonlinearity (the mean index s of the preevaporation differential error of the qtmdratic term came out 4.5 particle-mass distribution m-'~dm, which times as large as the term itself) and relates to the particle-size exponent u provides numerical values tbr two im- of (a), portant characteristics of the population , (u ~ 2)/a, (.(~) of large (lust particles in Comet Kohoutek : the exponent u of the initial particle-size and the acceleration ratio (1 -/*)m,~ of the distribution (3) and the evaporation loss in largest particles that evaporated com- the particle diameter AS. The two quanti- pletely (i.e., whose 8 = 28). ties derived from both the separate plates Taken at face value, the data of Table [lI and the whole set are listed in Table Ill. appear to indicate rather consistently that TABLE Ill (~|{ARA(!TERISTICN OF THI~; DUST PARTI('IA,;S (AT AN ASSUMED I)ENSITY OF I gClll 3) Initial size Initial l~]vapoi'~tion loss in Initial repulsive force for distribution populati(m particle diameter, A8 largest completely ew~porat,ed Plat(, exponent, ~ index, s (/*m) particles, ( 1 -- F) .... 1552O 4.72±0.16 2.24±0.05 103± 1(} 0.0113 ± 0.001 | 15554 5.17 _ 0.27 2.39 ± 0.09 142 ± 13 0.0082 ± 0.0007 15687 4.79 + (L09 2.26 ± O.03 131 ± 8 0.0089 ± 0.0005 157O3 5.06 ± 0.07 2.35 ± 0.02 147 ± ~i 0.0080 ± 0.0003 All 4.81 ± 0.06 2.27 ± 0.02 118 ± 4 0.0099 ± 0.0003 THE ANTI-TAIL OF COMET KOItOUTEK (1973f). II 145 the population index is s ~ 2.3 (and the Denoting the refraction-affected dis- size-distribution exponent u _~ 5) and that tance of point (32,82) as 4" and realizing only those particles initially larger than that 100-150tLm in diameter (at an assumed cos 1[(82 + 48,) + (a; + ,%)] density of 1 g cm-3) survived the exposure to p solar heat. Comparison with Fig. 7 of cos ½(8; +8,.') z. cos81, t 1 t Paper I suggests that the effective vapor- (~2 - ~;)2 cos 2 ~(a, + 82) ization heat of the anti-tail particles +(8 2-8;) 2=A '2, (A2) amounts to about 46 kcal mole -~ . However, an uncertainty is involved in these results, 148= = 4811 -~ 182 = 8;I, owing to the fact that particle evaporation A 2 -- 4 t2 "~ 24"(A -- A'), also affects the particle's future dynamics because it implies a change in the magni- we find that the refraction correction in tude of the radiation pressure. Therefore, distance amounts to the present interpretation is only pre- liminary, and the derived loss rate of particle radius due to evaporation, as well ×cosZS(+(82--8~)(ASz--481)]. (A3) as the actual emission rate of the dust, is Similarly, the refraction-free position angle still subject to further analysis. The p of the point (%, 82) relative to the nucleus improved model of the anti-tail will have is to account for the dynamical effect of particle evaporation (variable 1 - t~). tanp = (32 + A32) - (8; + 48,) × eos~[(8~1 + A81)+ (32 + 482)]. (A4) With the approximations (A2), the APPENDIX A relation between the refraction-free and Effects of Differential Refraction on the refraction-affected position angles p Distance and Direction in the Anti-Tail and p" is expressed as at Low Altitudes tanp -: tanp" Because of the sharp increase in the ×(l+A~2--~x2_c~40:, 482 -~1~8' I'.' refi'action correction of the spherical co- ordinates at low altitudes, it is desirable (A5) to establish the effects of refraction on the distance and directional scales in the and, withp andp" in degrees, the refraction anti-tail. correction in direction becomes Let 31, 81 be the refraction-free co- p -- p' := 28.°65 sin 2p" ordinates of the comet's nucleus; %, 8> - 4=, - those of a point in the anti-tail; and, X similarly, a~, 32, 32, 82, the corresponding coordinates affected by refraction. The (A6) corrections for refraction are, therefore, APPENDIX B 43 i = a~ -- a~, /18 i = 8 i -- 8~ (i = 1, 2). Since we consider only small arcs, then for the Calibration of the Ambient Sky Brightness refraction-free angular distance A of the from the Trail of a Star of Known point (32,82) from the nucleus, we can Magnitude write We assume that a star's image on a A'- = [(~ + A=:) -- (~; + A~,)V calibrated plate, guided on a moving object (comet), appears in the form of a × cos2 ½[(8; + z18,) + (32 + A82)] trail of length L and characteristic width + [(a; + 48~) - (8/+ 481)V. W (arcsec). We further assume that a (A1) photometric scan across a section of the 146 SEKANINA AND MILLER trail is available (such as the one in Fig. 3) or, in terms of the equivalent number of and that the diaphragm used is a slit of stars of magnitude 10 per square degree, length l and width w (aresee), where Sl0 :- 1.296 × l0 ll-°'4us l -; L and w ~ W. Converting the photo- graphic densities across the measure