Wirtanen

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Planetary and Space Science 52 (2004) 573 – 580

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The e!ect of using di!erent scale lengths on the production rates of Comet 46P/Wirtanen U. Finka;∗ , M.R. Combib b Space

a Lunar and Planetary Laboratory, The University of Arizona, Tucson AZ 85721, USA Physics Research Laboratory, Department of Atmospheric, Oceanic, and Space Sciences; University of Michigan, Ann Arbor, MI 48109-2143, USA

Received 15 September 2003; accepted 4 December 2003

Abstract The variations in production rates for Comet 46P/Wirtanen for the species H2 O and the parents of C2 and CN are examined from the point of view of a variety of commonly used scale lengths. The calculations are carried out as a function of heliocentric distance. It is shown that, by using a common set of scale lengths, the results of various investigators can be brought into acceptable accord. The resulting production rates of H2 O and the parents of C2 and CN versus heliocentric distance are recalculated and plotted versus the heliocentric distance rH . The curves show reasonable agreement with a slope of ∼ rH−4 . The water production rate near perihelion of 46P/Wirtanen is close to 2 × 1028 mol s−1 . ? 2003 Elsevier Ltd. All rights reserved.

1. Introduction The selection of Comet 46P/Wirtanen as the target for Rosetta prompted a considerable e!ort by a number of groups to obtain data for this comet near its perihelion passage of 14 March 1997. In order to discuss the outgassing properties of Wirtanen, a small workshop was held on 13 October 1998 in Cologne, Germany. Several investigators, including the =rst author of this paper, gave summary presentations of the production rates of H2 O and the parents of CN, C2 and NH2 . It became apparent that there were discrepancies in the production rates presented by the various investigators that went beyond inherent observational errors. It looked very much that these discrepancies were caused by the use of di!erent scale lengths. The calculations presented here were performed to investigate this possibility and to compare the production rates by various observers using common Haser scale lengths. This problem is not con=ned to Wirtanen but also a!ects the intercomparison of production rates for many other comets. As it was the subject of relatively intensive studies, Wirtanen a!ords the opportunity to investigate these

∗

Corresponding author. E-mail address: [email protected] (U. Fink).

0032-0633/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.pss.2003.12.002

problems, although it was a somewhat weak comet with an unfavourable apparition. 2. The problem Several models are available to convert measured cometary Auxes of a given species into a production rate. The simplest is the Haser model (Haser, 1957). Physically, more realistic models are the vectorial model (Festou, 1981), the average random walk model and the Monte Carlo model (Combi and Delsemme, 1980). None includes collisional e!ects between molecules or molecules and dust. Intercomparison between these models has shown that the largest e!ect on the derived production rates does not come as much from the choice of models as from the choice of the lifetimes of the species (; s−1 ), their outAow velocities (v; km s−1 ) or their equivalent Haser scale lengths (; km). A case in point is the reduction of Wirtanen C2 and CN data of Farnham and Schleicher (1998) by Schulz et al. (1998) using both a Haser and a vectorial model, resulting in di!erences of only 10 –20% between the two methods. To investigate the e!ects of using various scale lengths, we concentrated on the Haser model because it is easier and faster to calculate for a variety of scale length choices, heliocentric distances and apertures. The scale lengths used

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Table 1 Scale lengths used by investigators for production rate determinations for Comet 46P/Wirtanen

H2 O=OH

CN

prt

dtr

prt

dtr

prt

dtr

prt

dtr

prt

dtr

prt

dtr

116 000 —

21 900 28 000

300 000 320 000

16 000 58 000

110 000 58 000

1000 —

60 000 —

— 4900

— 62 000

5800 —

434 000 —

160 000

13 000

210 000

22 000

66 000

2800

27 000

—

—

50 000

150 000

— 159 000 —

24 225 — 18 300

400 000 — 300 000

34 000 — —

65 000 — —

3400 — —

140 000 — —

4505 — —

62 000 — —

— — —

— — —

Schleicher et al. (1987) 41 000 Fink et al. (1991) 80 000 Fink and Hicks (1996) A’Hearn et al. (1995) 24 000 Randall et al. (1992) Schulz et al. (1993, 1994, 1998) — Stern et al. (1998) 86 400 Jockers et al. (1998) —

C2

C3

NH2

NH

prt: parent; dtr: daughter. The =rst three teams scale their parent and daughter lengths with heliocentric distance as rH2 . Schulz et al. (1994, 1998) and Stern et al. (1998) scale only the daughter scale lengths as rH2 . Both use a heliocentric distance dependence of the parent outAow velocity of v=0:85rH−0:5 . The parent scale length is thus equivalently scaled as rH1:5 . Jockers et al. (1998) assumed the daughter scale length as 300 000 and =tted a parent scale length to a CN image of Wirtanen.

are shown in Table 1. The scale lengths of Schleicher et al. (1987) were those used for many years in the cometary photometry programme started by A’Hearn (A’Hearn and Millis, 1980). These scale lengths were changed substantially in the compendium of photometric observations of 85 comets (A’Hearn et al., 1995), a fact that appears to have been missed by a number of investigators. The new scale lengths of A’Hearn et al. (1995) are cited in an abstract by Randall et al. (1992) using observations of Comet Austin (1989c1). These observations were carried out in 1990 during solar maximum. It is possible that this caused the photodissociation rates to be higher (especially for any dissociations a!ected by solar L) and thus led to relatively shorter lifetimes and scale lengths. The scale lengths of Fink et al. (1991) are based on the analysis of a sizable amount of long-slit spectroscopic data of P/Halley. Jockers et al. (1998) determined his scale lengths from a =t to a CN Wirtanen image. Schulz et al. (1998) use the vectorial model. Their lifetimes come from an analysis of CCD images of Comet Wilson (Schulz et al., 1993, 1994). The reported lifetimes of both Schulz et al. (1998) and Stern et al. (1998) were converted to scale lengths, using the parent molecule outAow dependence of v = 0:85rH−0:5 km s−1 ; as used by those authors. Thus, for example, for C2 we get  = 40 000rH2 (s) × 0:85rH−0:5 (km s−1 ) = 34 000rH1:5 (km): Their parent outAow velocity dependence is theoretically predicted for moderate H2 O production rates (Q = 1028 – 1030 mol s−1 ) and for heliocentric distances in the range 0.5 –2:5 AU (Combi, 1989). To clarify our calculations of production rates calculated from observed emission Auxes, which in turn are produced by resonance Auorescence, we give a brief de=nition of terms. In our terminology, the production rate Q(s−1 ) for the parent of a species is related to the Aux (F, in erg s−1 cm2

or photon s−1 m2 ) of the daughter species measured at the telescope within an observing aperture by Q=

F × 4 2 × Haser correction × vd : g0 × d rH2 r2

(1)

H

Here, g0 is the Auorescence eKciency at 1 AU for a given emission band (in erg s−1 mol or photon s−1 ). This must be scaled inversely as the square of the heliocentric distance (in AU). d is the daughter scale length at 1 AU scaled as rH2 (in AU) and vd is the daughter outAow velocity (km s−1 ). All authors in Table 1 used a daughter velocity of 1 km s−1 and all of them scaled it as rH2 . Production rates between di!erent investigators A and B can be written as Q(A) Haser correction(A) vd (A) d (B) = × × : (2) Q(B) Haser correction(B) vd (B) d (A) L for their Fink et al. (1998) use the OI 1 D line at 6300 A water production determination. The production rate of water from this line is given by QH2 O = 16 × F × 4 2 × Haser correction (parent H2 O only)

(Festou and Feldman, 1981; Fink and Hicks, 1996). The factor of 16 depends on the branching ratio of H2 O → O(1 D) and can vary between about 12 and 16 depending on solar UV activity (DiSanti and Fink, 1991). To keep the reductions on a consistent basis, a constant factor of 16 has been used by Fink and Hicks (1996) and Fink et al. (1998). The next goal was to see if there was a signi=cant heliocentric distance (rH ) dependence that could be introduced by using di!erent scale lengths or, more signi=cantly, heliocentric dependent scale lengths. The ratio in expression (2) was thus calculated as a function of rH . The actual heliocentric and geocentric distances for Wirtanen spanning distances of 3:50 AU from the Sun to perihelion at 1:06 AU were used. These are given in the =rst column of Table 2.

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575

Table 2 Examples of Wirtanen C2 Haser corrections for an aperture of 30 × 2:5 using various scale lengths

Date

rH (AU)

(AU)

Fink et al. p = 58 000 d = 58 000

Schleicher et al. p = 16 000 d = 110 000

A’Hearn et al. p = 22 000 d = 66 000

Schulz et al. p = 34 000rH1:5 d = 65 000

1996 Mar 21 Jun 12 Jul 19 Aug 21 Sep 12 Oct 18 Nov 15 Dec 09

3.50 3.00 2.80 2.50 2.40 2.00 1.80 1.60

4.07 2.45 1.95 1.54 1.50 1.59 1.69 1.73

1267.88 1795.41 2103.66 2138.61 1940.57 929.18 582.84 375.85

789.42 1107.84 1293.04 1314.06 1195.16 583.71 371.71 243.68

632.46 888.94 1038.23 1055.18 959.29 466.94 296.60 193.92

536.43 799.89 960.02 1021.48 944.22 494.08 326.51 222.75

1997 Jan 02 Jan 30 Mar 14

1.40 1.20 1.06

1.72 1.64 1.52

243.01 158.48 120.09

160.57 107.00 82.37

127.41 84.64 65.02

153.22 107.03 85.50

The range of heliocentric distances is considerably larger than that covered by the observations, but makes it easier to recognise potential trends with heliocentric distances. Since the Haser corrections also depend on the geocentric distances, a comet-speci=c bias (i.e. for Wirtanen) may be in our graphs, but values for a realistic comet were required. To check whether there was a strong aperture e!ect in our ratios, aperture slit lengths of 6 ; 12 ; 30 and 60 and widths of 2:5 and 5 were also run. A sample calculation is given in Table 2 for C2 . In this table, the Haser corrections are shown for the scale lengths used by the listed authors. For illustrative purposes, the calculations were performed for a slit aperture of 30 × 2:5 , an aperture commonly used in spectroscopy. It was assumed that the sky level had been perfectly subtracted. Fig. 1 plots the ratios of the production rates for C2 , Qothers =QFink et al: 1998 as de=ned in Eq. (2). For example, for a given measured Aux of C2 , the scale lengths used by Schleicher et al. (1987) give a production rate a factor of ∼3:0 lower than if the scale lengths of Fink et al. (1991) had been used. The numbers plotted in the =gures can be thought of as correction factors to be applied to the Fink et al. (1998) Wirtanen data if Fink et al. had used the scale lengths of the investigators in Table 2. There is no judgement as to which are the correct scale lengths. This has long been a subject of debate in the literature. We simply examine the e!ects when di!erent scale lengths are used. The =gure demonstrates that sizable variations in production rates can occur, solely arising from di!erences in the scale lengths. A slight heliocentric distance dependence is seen for the scale lengths of A’Hearn et al. (1995) and Schleicher et al. (1987). The scale lengths of Schulz et al. (1998) show a much stronger heliocentric distance dependence. This is to be expected because the parent scale lengths of Schulz

Fig. 1. Ratios of production rates for C2 , Qothers =QFink et al: (1991) as given by Eq. (2) in the text. The aperture used for these calculations is 30×2:5 and it is assumed that the sky has been subtracted perfectly. The numbers plotted can be thought of as correction factors to be applied to the Fink et al. (1998) Wirtanen production rates if Fink et al. had used the scale lengths of the other investigators in Table 2 instead of their own.

et al. were scaled as rH1:5 while all others used a power-of-2 dependence. The slope of the Schulz et al. correction factors is rH−0:45 . To guard against possible aperture e!ects, the production rate ratios were also calculated for slit apertures of 2:5 × 6:0 , 2:5 × 12:0 and 2:5 × 60 as well as circular apertures corresponding to the slit lengths. A slight e!ect on the ratios not exceeding 5 –10% was found for these di!erent apertures. The dominant e!ect was the choice of the scale lengths used for calculating the Haser corrections.

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U. Fink, M.R. Combi / Planetary and Space Science 52 (2004) 573 – 580

the correction factors to be applied to the data of Farnham and Schleicher (1998) if the scale length of ‘others’ had been used. This was done because there is a reasonable consensus among investigators that the dissociation lifetime for H2 O is close to 80 000 s near solar minimum, while at solar maximum it is probably closer to 60 000 s. In addition to the scale lengths of Fink et al. (1991), the case was also calculated that allowed for a heliocentric parent outAow velocity of v = 0:85rH−0:5 km s−1 (used by Stern et al., 1998). This is equivalent to using a parent scale length for H2 O of: Fig. 2. Similar to Fig. 1 but showing the production rate ratios for CN. The Fink et al. (1991) scale lengths result in production rates a factor of 1.18 higher than those of Schleicher et al. (1987) and about a factor of 1.7 higher than the scale lengths of A’Hearn et al. (1995).

80 000rH2 × 0:85rH−0:5 = 68 000rH1:5 (km): This curve is labelled ‘Fink modi=ed’ in Fig. 3. Again as expected, this case exhibits a slope with respect to a parent scale length scaling of rH2 . Fig. 3 clearly shows that the discrepancy of roughly a factor of 2.5 between the H2 O production rates determined by Fink et al. (1998) and Farnham and Schleicher (1998) is almost exclusively due to di!erent scale lengths used. 3. Applying the corrections The observations for Wirtanen published in the May 1998 issue of Astronomy and Astrophysics can now be placed on a common basis. The water production rates reported are listed in Table 3. Also listed in Table 3 are the corrected H2 O production rates. The =rst corrected H2 O column uses the relatively simpler scale lengths laws: p = 80 000rH2

Fig. 3. Results for OH similar to Figs. 1 and 2. For this =gure, the displayed ratio is Qothers =QA
Hearn et al: (1995) . Also, this =gure is calculated for a 30 -diameter circular aperture. The =gure shows the correction factors to be applied to the data of Farnham and Schleicher (1998) if the scale lengths of ‘others’ had been used. The curve Fink et al. (1991) modi=ed uses a heliocentric velocity scaling law of rH−0:5 .

A similar calculation was carried out for CN, and its results are shown in Fig. 2. The general trend is the same but the di!erences are not as severe because the scale lengths used by various investigators are closer. Again, the use of an rH1:5 scaling law for the parent scale length by Schulz et al. (1998) results in a correction factor slope of approximately rH−0:40 . Fig. 3 shows the results for the important molecule OH, from which the production rate of water is calculated, which dominates the activity of a comet. Rather than using a rectangular slit aperture, this =gure is calculated for a circular aperture of 30 diameter. In this =gure, the displayed ratio is Qothers =QA
Hearn et al: 1995 . In other words, the =gure shows

and

d = 160 000rH2 :

This assumes a dissociation lifetime for H2 O of 80 000 s and an average parent and daughter outAow velocity of 1 km s−1 with no variation with heliocentric distance. The second corrected H2 O column in Table 3 assumes the parent velocity varies as v = 0:85rH−0:5 km s−1 ; while the daughter outAow velocity is 1 km s−1 . For most comets with water production rates in the range of 1028 –1030 mol s−1 , the heliocentric dependence of the parent outAow speed can be taken as v = 0:85rH−0:5 km s−1 (Combi, 1989). This breaks down for very active comets such as Hale–Bopp as well as for relatively weak comets such as Wirtanen where the water production rate is greater than 1028 mol s−1 only near perihelion. The heating e!ect from the photochemistry that increases the parent gas outAow speed near perihelion does not occur to any appreciable extent (Combi, 1989). Thus for Wirtanen a constant outAow

U. Fink, M.R. Combi / Planetary and Space Science 52 (2004) 573 – 580

577

Table 3 Comparison of water production rates (QH2 O ; 1025 mol s−1 ) for Comet Wirtanen

Method

rH (AU)

(AU)

Reported QH2 O or OH

QH 2 O correctedb v = const:

QH 2 O correctedc v = 0:85 rH−0:5

Pre-perihelion 1996 Aug 26 1997 Jan 15 1997 Jan 18–Feb 04 1997 Feb 10 1997 Feb 12 1997 Feb 15 1997 Feb 19 1997 Mar 02– 06 1997 Mar 05

Stern et al. Stern et al. Fink et al. Bertaux Farnham and Schl. Farnham and Schl. Crovisier Fink et al. Farnham and Schl.

HST OH HST OH Spec OI SOHO H Phot OH Phot OH Radio OH Spec OI Phot OH

2.47 1.31 1.19 1.15 1.14 1.13 1.11 1.07 1.07

1.51 1.69 1.64 1.61 1.60 1.59 1.58 1.54 1.54

26:5 ± 5 510 ± 3 1985 ± 500 700 ± 350 740 ± 260a 776 ± 110a ¡ 1500a 2915 ± 600 710 ± 300a

43 648 1985 1000 1726 1799 ¡ 1760 2915 1640

26.5 510 1575 1000 1482 1512 ¡ 1760 2429 1445

Post-perihelion 1997 Mar 29 –Apr 02 1997 Apr 29 –May 03 1997 Jun 01– 02

Fink et al. Fink et al. Fink et al.

Spec OI Spec OI Spec OI

1.08 1.24 1.47

1.52 1.64 1.93

2295 ± 650 856 ± 300 310 ± 250

2295 856 310

1897 603 221

a Reported

OH production rates. These were converted to H2 O production rates by using a branching ratio for OH of 0.85. p = 80 000rH2 , d = 160 000rH2 . c Using  = 68 000r 1:5 ,  = 160 000r 2 . p d H H b Using

speed as listed in Table 3 in the column labelled ‘v =const:’, or possibly a very weak dependence with heliocentric distance, is probably to be preferred. A further point needs to be made for the corrections for Table 3. The correction factors are not the same as plotted in Fig. 3, although they are in general similar. To calculate the corrections for Farnham and Schleicher (1998), their reported circular apertures of 75 ; 56 and 110 were used. Fink et al. (1998) subtract the sky adjacent to their object aperture. Thus some of the comet’s emission signal is also subtracted from the object aperture. This is taken into account in their Haser correction. Thus the Haser correction factors for Fink et al. (1998) are di!erent from those that assume a separate sky measurement far from the comet. The corrected H2 O production rates of the various investigators are plotted versus heliocentric distance in Fig. 4. With these corrections, the original considerably discordant reported results—e.g. Fink et al. (1998) and Farnham and Schleicher (1998) di!ered by about a factor of 3—have been brought into reasonable agreement. The agreement is quite remarkable considering the relative faintness of P/Wirtanen and the di!erent methods used: OH spectroscopic Hubble Space Telescope measurements, OH ground-based photometry, OH radio observations, spacecraft H measurements and OI 1 D ground-based measurements. A rough =t to the data results in a slope of approximately 4.3. Maximum deviations from this =t are about a factor of about 1.5. If the parent scale length p = 68 000rH1:5

Fig. 4. P/Wirtanen water production rate versus heliocentric distance. Values plotted are from column 7 of Table 3 which uses p =80 000rH2 km for the parent scale length. Approximate dashed line =t gives a slope of rH−4:3 .

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U. Fink, M.R. Combi / Planetary and Space Science 52 (2004) 573 – 580

Table 4 Comparison of Comet Wirtanen C2 =CN production rates (1024 mol s−1 ) and ratios by various investigators

Date

rH (AU)

(AU)

Qcorrected a

Qreported QC 2

QCN

C2 =CN

Q C2

QCN

C2 =CN

Schulz et al. (1998) 1996 Sep 10 –15 1996 Oct 15 –18 1996 Nov 14 –15 1996 Dec 08–12

2.34 2.04 1.81 1.60

1.49 1.59 1.69 1.73

— 0.48 0.92 5.87

0.69 1.92 3.49 10.17

— 0.25 0.26 0.58

— 0.45 0.82 5.03

0.61 1.61 2.85 8.05

— 0.28 0.29 0.63

Farnham and Schleicher (1998) 1997 Feb 12–14 1997 Feb 15 –21 1997 Mar 5 –12 1997 Jun 04 –18 1997 Jul 01–18

1.14 1.13 1.07 1.49 1.72

1.60 1.59 1.54 1.96 2.30

16.2 19.1 27.5 7.94 5.62

15.9 17.8 26.9 7.08 3.98

1.02 1.07 1.02 1.12 1.41

16.2 19.1 27.5 7.94 5.62

15.9 17.8 26.9 7.08 3.98

1.02 1.07 1.02 1.12 1.41

Fink et al. (1998) 1997 Jan 28–Feb 04 1997 Mar 02– 06 1997 Mar 29 –Apr 02 1997 Apr 29 –May 03 1997 Jun 01– 02

1.19 1.07 1.08 1.24 1.47

1.64 1.54 1.52 1.64 1.93

40.1 56.0 57.7 23.4 12.7

24.4 31.3 33.1 16.1 12.9

1.64 1.79 1.74 1.45 0.98

16.8 23.4 24.1 9.79 5.3

13.3 17.0 18.0 8.75 7.0

1.26 1.38 1.34 1.12 0.76

Jockers et al. (1998) 1997 Mar 11

1.07

1.52

—

29

—

—

21

—

a Corrected

production rates (Q) use the scale lengths of Farnham and Schleicher for C2 and CN to provide a common basis for intercomparison.

is used, the slope of the production rate becomes steeper (∼rH−6:3 ). This causes considerable disagreement with the heliocentric production rate dependence of C2 and CN presented below, and gives circumstantial supporting evidence to the use of a constant parent outAow velocity for P/Wirtanen. The situation for C2 and CN is considered in Table 4 and Figs. 5 and 6. Again using common scale lengths, the initial discrepancies disappear where, for example, the Fink et al. (1998) C2 production rates are a factor of 3 above Farnham and Schleicher (1998). For both C2 and CN, the scale lengths of A’Hearn et al. (1995) have been used. This is not meant to indicate a preference for these scale lengths— a single common set was required for intercomparison. Both Farnham and Schleicher (1998) and Fink et al. (1998) agree reasonably well on a ratio of QC2 =QCN of the order of 1.0. Schulz et al. (1998) at larger heliocentric distances get a considerably lower ratio of ∼0:30. We note that the production rate ratios of QC2 =QCN as a function of heliocentric distance have been the subject of a number of earlier investigations. Newburn and Spinrad (1989) in their spectrophotometry of 25 comets reported a change in the C2 =CN production rate ratio with heliocentric distance. On the other hand, Cochran et al. (1992) using the McDonald faint comet survey of 17 comets observed no such trend. For Comet P/Halley, Fink (1994) found no change for the QC2 =QCN ratio for a heliocentric distance

range of 0.84 –2:52 AU. Perhaps the best test case is Comet Hale-Bopp for which good signal-to-noise observations could be obtained over a very large range in heliocentric distance. Schleicher et al. (1997) obtained production rates for C2 and CN for heliocentric distances of 7.1–1:2 AU. Within their measurement error, the production rate curves of C2 and CN versus heliocentric distance can be overlaid. The same goes for their P/Halley data, which are shown on the same plot. Rauer et al. (1997) observed Hale–Bopp spectroscopically for heliocentric distances of 4.6, 4.0, 3.3, 3.1 and 2:9 AU. They obtained QC2 =QCN ratios of 3.0, 1.2, 1.6, 1.0 and 1.7, showing essentially no trend with heliocentric distances within their error estimates. We cannot rule out completely that weak comets could possibly show a variation of the QC2 =QCN ratio with heliocentric distance. However, their weak emission intensities make it diKcult to measure the production rates for these comets at larger heliocentric distances. The diKculty of measuring C2 accurately at large distances can be clearly seen in Fig. 1 of Schulz et al. (1998). It may be possible that this contributed to their low QC2 =QCN ratio of ∼0:30 at a distance near 2:0 AU. If the C2 production rates of Schulz et al. (1998) are given somewhat lower weight, the slope of the C2 production rate with heliocentric distances is ∼4:0 in Fig. 5. Similarly for the CN production rate in Fig. 6, we obtain a slope of ∼3:7. Within errors, the C2 and CN production rate curves can be

U. Fink, M.R. Combi / Planetary and Space Science 52 (2004) 573 – 580

Fig. 5. Production rate of the parent of C2 versus heliocentric distance. Giving a lower weight to the two points near 2 AU gives an approximate =t of rH−4:0 . Error bars are similar to those in Fig. 4.

overlaid. The same can be said for the water production rate of Fig. 4, although these data possibly indicate a slightly steeper slope.

579

Fig. 6. Production rate of the parent of CN versus heliocentric distance. Approximate slope is rH−3:7 . Error bars are similar to those in Fig. 4.

values should probably be preferred over the used 80 000 km, although that value is correct for the OI calculation. References

4. Conclusions In summary, the production rates of H2 O and the parents of C2 and CN could be brought into very acceptable agreement using common Haser scale lengths. It must be emphasised again that this is good relative agreement only. Since there is still no consensus on the ‘best’ scale lengths, the absolute production rates can easily be o! by a factor of 3. In particular for water, a common scale length of 80 000 km and an outAow velocity of 1 km s−1 were used. Using the calculation by Combi and Delsemme (1980), the theoretical lifetime of 82 400 s for H2 O at solar minimum and ∼60 000 s at solar maximum can be converted into an equivalent Haser scale length for the OH daughter via their average random walk model. This results in equivalent Haser scale lengths for H2 O and its daughter OH of 52 000 km and 266 000 km (vd = 1:33 km s−1 ) and 37 000 km and 160 000 km (vd = 1:34 km s−1 ) for solar minimum and maximum, respectively. For Haser model calculations, these

A’Hearn, M.F., Millis, R., 1980. Abundance correlations among comets. Astron. J. 85, 1528–1537. A’Hearn, M.F., Millis, R., Schleicher, D., Osip, D., Birch, P., 1995. The ensemble properties of comets: results from narrowband photometry of 85 comets, 1976 –1992. Icarus 118, 223–270. Bertaux, J.-L., 1997. IAU Circ. 6565. Cochran, A., Barker, E., Ramseyer, T., Storrs, A., 1992. The McDonald observatory faint comet survey: gas production in 17 comets. Icarus 98, 151–162. Combi, M., 1989. The outAow speed of the coma of Halley’s comet. Icarus 81, 41–50. Combi, M., Delsemme, A., 1980. Neutral cometary atmosphere. I. An average random walk model for photodissociation in comets. Astrophys. J. 237, 633–640. Crovisier, J., Biver, N., Bockelee-Morvan, D., Colom, P., Gerard, E., Rauer, H., 2003. Radio observations of the OH radical in comets 46P/Wirtanen and 81P/Wild 2. Planet Space Sci., (submitted for publication). DiSanti, M.A., Fink, U., 1991. A composition comparison between comets P/Halley and P/Brorsen–Metcalf. Icarus 91, 105–111. Farnham, T., Schleicher, D., 1998. Narrowband photometric results for comet 46P/Wirtanen. Astron. Astrophys. 335, L50–L55.

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