Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.mso.anu.edu.au/pfrancis/comets/comets.pdf Äàòà èçìåíåíèÿ: Tue Sep 6 07:41:32 2005 Äàòà èíäåêñèðîâàíèÿ: Fri Feb 28 00:14:10 2014 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: stellar evolution

Draft version September 5, 2005
A Preprint typ eset using L TEX style emulateap j v. 11/26/04

THE DEMOGRAPHICS OF LONG-PERIOD COMETS
Paul J. Francis
Research School of Astronomy and Astrophysics, the Australian National University, Mt STromlo Observatory, Cotter Road, Weston ACT 2611, Australia Draft version September 5, 2005

ABSTRACT The absolute magnitude and perihelion distributions of long-perio d comets are derived, using data from the Lincoln Near-Earth Asteroid Research (LINEAR) survey. The results are surprising in three ways. Firstly, the flux of comets through the inner solar system is much lower than some previous estimates. Secondly, the expected rise in comet numbers to larger perihelia is not seen. Thirdly, the number of comets per unit absolute magnitude do es not significantly rise to fainter magnitudes. These results imply that the Oort cloud contains many fewer comets than some previous estimates, that small long-period comets collide with the Earth to o infrequently to be a plausible source of Tunguska-style impacts, and that some physical pro cess must have prevented small icy planetesmals from reaching the Oort cloud, or have rendered them unobservable. A tight limit is placed on the space density of interstellar comets, but the predicted space density is lower still. The number of long-period comets that will be discovered by telescopes such as SkyMapper, Pan-Starrs and LSST is predicted, and the optimum observing strategy discussed. Subject headings: comets: general -- Oort Cloud -- solar system: formation
1. INTRODUCTION

The Oort cloud (Oort 1950) remains the most mysterious part of our solar system, primarily because it cannot be directly observed. Our only observational clues to the size, shape, mass and composition of the Oort cloud come from observations of long-perio d comets. The demographics of observed long-perio d comets have been the starting point of almost all attempts to mo del the Oort cloud (eg. Oort 1950; Weissman 1996; Wiegert & Tremaine 1999; Dones et al. 2004). Until the last ten or so years, the vast ma jority of comets were discovered by systematic eyeball searches, using small telescopes (Hughes 2001). These surveys have been highly effective at identifying large samples of comets, and in deriving their orbital parameters. They do, however, have three ma jor drawbacks: · Unknown selection function: it is very unclear how often different parts of the sky are surveyed, and to what depth. Surveys are clearly more sensitive to comets with bright absolute magnitudes and perihelia close to the Earth, but the strength of this effect is very hard to estimate (Everhart 1967a,b). · Limited range of comets observed: eyeball surveys find few comets fainter than an absolute magnitude of 10 and with perihelia beyond 3AU. · Po orly defined photometry: these surveys quote the "total brightness" of a comet. Total cometary magnitudes are notoriously unreliable. They are typically measured by defocussing a standard star to the same apparent size as the comet, but this apparent size is heavily dependent on observing conditions and observational set-up. Despite these drawbacks, many attempts have been made to derive the basic parameters of the long perio d
Electronic address: pfrancis@mso.anu.edu.au

comet population from eyeball-selected historic samples. The most heroic and influential attempt was that of Everhart (1967b). Everhart carried out an exhaustive analysis of the historical circumstances in which comets were discovered, over a 127 year perio d. He developed a model for the sensitivity of the human eye, and used it to calculate the period over which a given historical comet could have been seen. This was then used to estimate the completeness of the comet sample: if a given type of comet was typically seen early in its visibility window, surveys should be complete for this type of comet. If the mean time to find a given type of comet is, however, comparable to the length of the estimated visibility window, the completeness is probably low. Using this method, Everhart estimated that for every comet seen, another 31 were missed. A more mo dest and recent attempt was that of Hughes (2001). He restricted himself to the brightest and nearest comets, for which he claimed (on the basis of discovery trends) historical surveys were highly complete. The statistics of these comets were simply extrapolated to larger perihelia and fainter absolute magnitudes, with no correction for observational incompleteness. As one would expect, the flux of long-perio d comets through the inner solar system estimated by Hughes (2001) is much lower that that estimated by Everhart (1967b). Despite these attempts, several basic questions about the demographics of long-perio d comets remain unresolved. One question concerns small comets (Brandt et al. 1996): those with nuclear radii less than 1km (absolute magnitudes H 10). Extrapolating the Everhart data implies that there should be a large population of such comets. Hughes (2001) was unable to tell whether his mo del predicted a large population of such comets or not. A second question concerns the number of comets per unit perihelion. Everhart (1967b) found that this number rises from the sun out to 1AU, but was unable to determine whether it keeps rising at larger perihelia.


2 Hughes (2001) found no significant rise, but had large enough error bars to bracket both of Everhart's possibilities. The observational situation has changed radically in the last few years. The advent of large format sensitive CCDs has allowed automated surveys to supplant eyeball searches as the main mechanism for finding new long-perio d comets1 . Most long perio d comets are now being found as by-pro ducts of various automated searches for near-Earth ob jects, such as the Lincoln NearEarth Asteroid Research (LINEAR) pro ject (Stokes et al. 2000), the Catalina Sky Survey2 , LONEOS3 and NEAT4 (Pravdo et al. 1999). Many are also found by spacebased coronagraphs as they approach very close to the Sun (Biesecker et al. 2002), though these are mostly fragments of recently disintegrated larger comets (Sekanina & Cho das 2004). In this paper, I attempt to deduce the statistical properties of the long-perio d comet population from one of these CCD surveys: the LINEAR survey. This has a far better defined selection criterion than any historical eyeball survey, and extends to much larger perihelia and fainter absolute magnitudes. It thus allows both an independent check and an extension of previous estimates of the long-period comet population. Near Earth asteroid (NEO) surveys are not optimized for comet detection. While they find many long-perio d comets, they do not publish their raw data, nor all the details one would like of their exact detection algorithms and sky coverage. In particular, they do not publish ongoing photometry of the comets they discover. Nonetheless, enough information is available to make a first pass at estimating the true population of long-period comets from their data. There have been previous attempts to use these surveys to detemine the true populations of NEOs (eg. Jedicke et al. 2003) and dormant comets (Levison et al. 2002), but this paper is the first attempt of which I am aware to do this for active comets. In the next few years, the situation should further improve, with the advent of a new generation of wide-field survey telescopes, such as SkyMapper5 , Pan-Starrs (Hodapp et al. 2004) and Gaia (Perryman et al. 2001). These surveys will predominantly find comets much fainter and more distant than historical surveys. The analysis in this paper allows a first estimate of just how many longperio d comets these surveys can find, and how best to identify them. I start off by defining a sample of comets drawn from the LINEAR sample, and examining its properties, which are very different from those of eyeball samples (§ 2). A mo del of the long-perio d comet population is then generated (§ 3) based on and extrapolating the historical eyeball-selected surveys. A Monte-Carlo simulation of this comet population as it would be observed by LINEAR is then developed (§ 4). The results are compared to the observed sample in § 5: I find that the Hughes mo del is quite a good fit to the data, but that the Everhart model is not. I derive my own best-fit mo del of
1 2 3 4 5

long-perio d comet demographics. The consequences of this new mo del are many: I examine them in § 6 before drawing conclusions in § 7.
2. THE COMET SAMPLE

Of the several near-Earth asteroid surveys now underway, the Lincoln Near-Earth Asteroid Research (LINEAR) pro ject (Stokes et al. 2000) was most suitable for constraining the long-perio d comet population. This is because: · They discover more comets than any other single survey. · They publish sky charts on their web page6 showing the area of the sky observed during each lunation, with the point-source magnitude limit reached at each lo cation. · Their sky coverage and magnitude limit is relatively simple and uniform across this perio d. The comet sample was defined as follows: 1. The comet has an orbital perio d longer than 200 years. 2. The comet reached perihelion between 2000 Jan 1 and 2002 Dec 31. 3. The comet was either tween these dates, or by LINEAR between ready been discovered ered prior to 2000 Jan discovered by LINEAR becould have been discovered these dates had it not alby someone else, or discov1.

The 2000-2002 date range was chosen because comet details (from the Catalog of Cometary Orbits, Marsden & Williams 2003) and sky-maps (including limiting magnitudes) are available. Marsden & Williams (2003) listed 25 comets as having been discovered or co-discovered by LINEAR which met our criteria. I needed, however, to add two additional sub-samples: · Comets discovered prior to 2000, but which reach perihelion in the perio d 2000-2002, and which could have been first discovered by LINEAR within this perio d, had they not already been found. · Comets discovered in 2000-2002 inclusive by other surveys, but which would subsequently have been seen by LINEAR during this perio d. Potential members of the two additional sub-samples were selected from Marsden & Williams (2003). Each candidate was checked for its detectability by LINEAR, using the ephemerides and predicted magnitudes generated by the Minor Planet Center7 . The predicted positions and brightnesses were compared to the maps of LINEAR sky coverage. These maps show only the integrated coverage per lunation, not the night-by-night or
6 7

http://comethunter.de/ http://www.lpl.arizona.edu/css/ http://asteroid.lowell.edu/asteroid/loneos/loneos.html http://neat.jpl.nasa.gov/ http://www.mso.anu.edu.au/skymapper/

http://www.ll.mit.edu/LINEAR/ http://cfa-www.harvard.edu/iau/mpc.html


3 hour-by-hour coverage, but most of these comets move slowly enough that this shouldn't much matter. This pro cess added another 27 comets to our sample. 8 had been detected by LINEAR during 1999, but reached perihelion in 2000 or 2001. Most of the remainder were first identified by other near-Earth asteroid surveys, particularly the Catalina Sky Survey, LONEOS and NEAT. For every comet in our final sample, the original discovery details (as distributed by the Central Bureau of Astronomical Telegrams) were checked. From these, the discovery date, discovery magnitude Hdis and discovery circumstances were noted. The discovery magnitudes are total magnitudes (m1). It is not clear how reliable and homogeneous these magnitudes are, but no better source of CCD photometry is available. They are based on CCD observations by professional astronomers of typically barely resolved ob jects, and so should be go o d to 0.5mag. Absolute magnitudes H were computed from these discovery magnitudes Hdis . The standard equation was used: (1) Hdis = H +5 log10 +2.5n log10 r (eg. Whipple 1978), where Hdis is the observed total magnitude at discovery and n a power-law parameterization of the dependence on helio centric distance. As is conventional for solar system work, the absolute magnitude is defined as the observed magnitude if the ob ject were at a distance of 1 AU from both the Earth and the Sun. Following Whipple (1978), the dynamically new and old comets were treated differently (the new ones are much brighter at large helio centric radii, at least on their way in). A comet is classed as dynamically new if its original semi-ma jor axis a is > 10,000 AU, old if a < 10,000 AU, and undetermined if the orbit class in Marsden & Williams (2003) is II or worse. For new comets, n = 2.44 was used if they are seen pre-perihelion and n = 3.35 if seen afterward. For old comets, the values are 5.0 and 3.5 respectively. The canonical value of n = 4 is used for comets of undetermined orbit type. This is uncertain both because real comets show a dispersion in n, and because the n values in Whipple (1978) are based on observations at smaller helio centric distances. It is, however, self-consistent with the analysis used in our Monte-Carlo simulations. Our sample is listed in Table 1.
2.1. Properties of the Sample
TABLE 1 The LINEAR Long-Period Comet Sample Name C/1999 C/1999 C/1999 C/1999 C/1999 C/1999 C/1999 C/1999 C/1999 C/1999 C/1999 C/1999 C/2000 C/2000 C/2000 C/2000 C/2000 C/2000 C/2000 C/2000 C/2000 C/2000 C/2000 C/2000 C/2001 C/2001 C/2001 C/2001 C/2001 C/2001 C/2001 C/2001 C/2001 C/2001 C/2001 C/2001 C/2001 C/2001 C/2001 C/2002 C/2002 C/2002 C/2002 C/2002 C/2002 C/2002 C/2002 C/2002 C/2002 C/2002 C/2002 F1 J2 K5 K8 L3 N4 S4 T1 T2 T3 U4 Y1 A1 CT54 H1 J1 O1 OF8 SV74 U5 W1 WM1 Y1 Y2 A1 A2 B1 B2 C1 G1 HT50 K3 K5 N2 RX14 S1 U6 W1 X1 B2 B3 C2 E2 H2 K2 L9 O4 P1 Q2 Q5 U2 q (AU) 5.7869 7.1098 3.2558 4.2005 1.9889 5.5047 0.7651 1.1717 3.0374 5.3657 4.9153 3.0912 9.7431 3.1561 3.6366 2.4371 5.9218 2.1731 3.5416 3.4852 0.3212 0.5553 7.9747 2.7687 2.4062 0.7790 2.9280 5.3065 5.1046 8.2356 2.7921 3.0601 5.1843 2.6686 2.0576 3.7500 4.4064 2.3995 1.6976 3.8430 6.0525 3.2538 1.4664 1.6348 5.2378 7.0316 0.7762 6.5307 1.3062 1.2430 1.2086 H 7.82 6.39 9.75 6.33 10.21 9.99 7.84 4.36 6.05 5.12 7.60 9.80 8.13 10.71 10.29 12.62 7.03 14.07 9.40 9.88 10.44 6.81 9.54 9.65 10.71 14.22 11.14 5.60 10.30 7.45 3.15 9.80 8.10 5.77 6.06 11.36 7.42 14.04 12.54 10.12 7.92 8.88 10.34 13.29 7.62 5.60 13.59 8.55 16.09 16.59 14.63 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1/aa 000038 000019 000024 000681 013741 000068 000720 000173 000596 000231 000037 000044 000044 000051 ·· · 0.001406 0.000037 0.000048 0.000090 0.000358 ·· · -0.000459 0.000063 0.001934 0.005738 0.000447 0.000071 0.000187 0.000020 0.000024 0.000878 0.000072 0.000029 0.000455 0.000776 0.018168 0.000998 ·· · 0.002285 ·· · ·· · 0.000393 0.000173 0.004024 ·· · 0.000035 -0.000772 0.002023 ·· · 0.000058 0.001075 Orbit class 1A 1A 1A 1A 1B 1A II II 1A 1B 1A 1A 1A 1A 1A 1A 1B 1A 1A II 1A 1B 2A II 1B 1B 1A 1A 1A 1B 1A 1A 1A 1A 2A 1B 1B 2A 2A 2A 2A 1B 1B
b

The LINEAR sample has very different properties from historical samples (as typified by the Everhart sample). Figure 1 shows that the LINEAR sample extends around 4 magnitudes deeper, and to much larger perihelia. The overlap is small: only 5 of the LINEAR comets lie within the absolute magnitude and perihelion region sampled by historical samples. Analysis of the discovery telegrams indicates that almost all of the comets in the sample were originally identified as moving point sources. They were posted on the Near Earth Ob ject (NEO) confirmation page8 at the Minor Planet Center. Follow-up observations then determined that the sources were spatially extended and hence comets. 77% were discovered before reaching per8

a Where available, this is the recipro cal of the original semima jor axis, ie. before planetary perturbations. Taken from Marsden & Williams (2003) b Quality flag for the orbit determination. Original semima jor axes only available for classes 1A thru 2B.

http://cfa-www.harvard.edu/iau/NEO/ToConfirm.html

ihelion (Fig 2), and 73% were were first detected when more than 3AU from the Sun. The necessity for follow-up potentially introduces two sources of incompleteness into this sample. Firstly, some fraction of ob jects posted on the NEO confirmation page are never followed up in enough detail to determine whether they are comets or not. Timothy Spahr kindly provided records of all ob jects posted to the NEO confirmation page in 2000-2002. Only 11% of these were not followed-up well enough to determine an orbit: this


4

Fig. 1.-- The location of our comet sample in the absolute magnitude vs perihelion plane, as compared to the Everhart (1967) sample.

Fig. 2.-- The time interval between the discovery of a comet and its perihelion passage (solid line). Negative values indicate that the comet was discovered before passing perihelion. For comparison, the dotted line shows our prediction of this distribution, from the best-fit model comet population.


5 places an upper limit on the fractional incompleteness of our sample due to failed follow-up. This is probably a conservative upper limit: most of these lost ob jects were most likely either not real to begin with or fastmoving ob jects only visible for a short window of time. Secondly, some comets might have been inactive at these large helio centric distances, and hence classified as minor planets. The minor planet centre database was checked for non-cometary ob jects on long-perio d, highly eccentric orbits, but only one was found which reached perihelion within the period 2000-2002: 2002 RN109. Thus this to o is not a ma jor source of incompleteness. It also shows that most comets down to the LINEAR magnitude limit are still active out to 10AU from the Sun. Fig 3 shows an intriguing correlation between perihelion distance and semi-ma jor axis in the LINEAR sample. This correlation was first noted by Marsden & Sekanina (1973) at smaller perihelia. They suggested that it was a selection effect. Dynamically new comets are brighter at large helio centric radii (eg. Whipple 1978), presumably due to extra outgassing at large heliocentric distances from their relatively pristine surfaces, due perhaps to CO2 or a water ice phase transition. The Whipple data did not extend to distances beyond 4AU from the Sun. If this trend continues to larger heliocentric distances, however, it would make dynamically new comets far brighter than older comets with the same absolute magnitude. This could thus, in principle, bias the sample heavily towards new comets, and hence larger semi-ma jor axes. The distribution of perihelion positions is shown in Fig 4. The comets are weakly concentrated at intermediate galactic latitudes (Fig 5), consistent with the galactic tide playing a ma jor part in making them observable (Matese & Whitmire 1996). The galactic latitude distribution is not, however, significantly different from the predictions of a best-fit mo del (§ 5.3) assuming a random distribution, as measured by the KolmogorovSmirnov (KS) test or the Kuiper statistic. There are also no significant great-circle alignments (Horner & Evans 2002).
3. MODEL OF THE LONG PERIOD COMET POPULATION

perihelion distribution as an unbroken straight line: dn = 1 + Aq , dq (2)

where A = 1 gives a reasonable fit to the Everhart (1967b) distribution. The Wiegert & Tremaine (1999) slope is shallower, but only shown out to 3AU. The distribution of semi-ma jor axes a makes no significant difference to the conclusions, as the comets are all very close to being on parabolic orbits. I chose to randomly class 40% of comets as dynamically new and given them all a =20,000, while the remainder were given a value of a randomly and uniformly distributed between 1000 and 20,000 AU. The time of perihelion passage, orbital inclination and perihelion direction were randomly chosen to give a uniform distribution on the celestial sphere. This ignores possible great-circle alignments (Horner & Evans 2002) and galactic tidal effects (Matese & Whitmire 1996; Matese & Lissauer 2004).
3.2. Absolute Magnitudes Everhart (1967b) found that the absolute magnitude distribution of comets was best fit by a broken powerlaw, with the break at H 6. Hughes (2001) also found a break at about the same absolute magnitude, but was unable to decide whether it was a real break or simply the effect of increasingly incomplete samples at fainter magnitudes. To bracket the possibilities, I use a broken power-law of the form.

dn dH

b(H -Hb ) , H < Hb f (H -Hb ) , H > Hb ,

(3)

The LINEAR comet sample was compared against a Monte-Carlo simulation of the long perio d comet distribution. In this section I discuss the parameters used in this simulation.
3.1. Orbital Parameters The Everhart (1967b) and Hughes (2001) studies are based on comets with a limited range of perihelion distances q and hence give only weak constraints on this distribution. Everhart found a factor of two rise in the number of comets per unit perihelion between 0 and 1 AU, but beyond that the data are consistent either with a continuing rise or a flat distribution. Hughes found no significant trend in number of comets against perihelion, but his data are quite consistent with such a trend. On theoretical grounds, however, a gentle rise in the number of comets as a function of perihelion is expected, as comets diffuse into the solar system past the barrier of giant planet perturbations (eg. Tsuji 1992; Wiegert & Tremaine 1999). As a first guess, I chose to mo del the

where Hb is the break magnitude, b is the bright end slope and f is the faint end slope. Everhart gives Hb = 6, b = 3.65 and f = 1.82. Hughes gives Hb = 6.5 and b = 2.2. If I assume that his observed break is real and not an artefact of sample incompleteness, his plots imply a faint end slope of f = 1.07, which I adopt to bracket the possibilities. This version of the Hughes formulation thus predicts dramatically fewer faint comets, as would be expected as these are the ones for which Everhart applies the largest incompleteness correction fraction.
3.3. Comet Flux

Everhart (1967b) and Hughes (2001) give different estimates of the long-perio d comet flux through the inner solar system. Everhart estimates a flux of 8000 comets with H < 10.9 and q < 4 over 127 years. Hughes estimates a flux of 0.53 comets per year brighter than H = 6.5 per unit perihelion. I ran the simulations using both.
4. MONTE-CARLO SIMULATION

For a given mo del comet population, the aim is to simulate the observable properties of a sample that matches the selection effects of the LINEAR sample. The simulation starts off by generating a set of comets that reach perihelion within a three year perio d. The comets are randomly generated using the mo del distributions in the previous section. The mo del extends down to H = 19 and out to q = 15. I generated two mo del populations: one using the Everhart absolute magnitude distribution and flux, the other


6

Fig. 3.-- The perihelia and semi-ma jor axes of all comets in the LINEAR sample with class 1A or 1B orbit determinations. The distribution remains largely unchanged if comets with less well determined orbits are included.

Fig. 4.-- The ecliptic coordinates of the perihelia of comets in the LINEAR sample. Solid triangles are comets which class 1a or 1b orbits which are dynamically new (as defined in the text).


7

Fig. 5.-- The galactic latitude distribution of the perihelion directions of the LINEAR sample comets (solid line). The dotted line is the predicted distribution from our best-fit model (§ 5.3).


8 using the Hughes absolute magnitude distribution and flux. In the Everhart model 260,000 comets are generated, while only 4,000 are needed in the Hughes mo del. The position of each comet is then calculated at 24 hour intervals throughout the three year period, and its helio centric distance r, distance from the Earth , apparent celestial co ordinates and apparent angular velo city written to file. Pure elliptical orbits are used: no attempt is made to allow for planetary perturbations. At each lo cation, the apparent total magnitude is then calculated. Comets are notoriously variable in how rapidly their apparent magnitude varies as a function of helio centric distance. I parameterize this, as is conventional, using Equation 1. Two values of n are randomly assigned to each comet: one for before perihelion and another for after. For the 40% of our simulated comets that I set as dynamically new, the pre-perihelion value of n is chosen from a Gaussian distribution of mean 2.44 and standard deviation 0.3. Post-perihelion, the mean is 3.35 with a scatter of 0.27. For the remaining comets, the pre-perihelion numbers are 5.0 with a scatter of 0.8, and after perihelion 3.5 with a scatter of 0.5. All these values are taken from Whipple (1978). At large distances from the Sun, cometary activity will presumably stop, and a bare nucleus will have n = 2. The near-ubiquitous detection of fuzz around the LINEAR comets implies, however, that this only happens further from the Sun than our mo dels reach. This approach can only be a rough approximation to the real radial brightness dependence. The value of n for an individual comet is typically time dependent, and all the tabulated values are for comets within 3 AU of the Sun, whereas our simulation tracks them out beyond 10AU. In addition, comets show o ccasional flares above and beyond this power-law behavior, which I have not attempted to mo del. Such flares might intro duce an amplification bias, with comets being pushed over the detection threshold. As we will see, however, the slope of the absolute magnitude distribution is so gentle that this is unlikely to be a ma jor effect.
4.1. Converting total magnitudes to point-source

equivalent magnitudes The apparent total magnitude of each simulated comet can now be calculated at any given point in its orbit. Unfortunately, in any CCD-based survey, it is the peak surface brightness of the coma that determines whether something has been seen, not the total magnitude. The LINEAR skymaps, furthermore, list only the magnitude limit for a point source at any given lo cation on the sky (typically around 19). As discussed in the intro duction, total cometary magnitudes are notoriously unreliable. Quantitative studies prefer more reproducible and physically meaningful parameters such as Af (eg. A'Hearn et al. 1995). Unfortunately, not enough long perio d comets have been studied in this way to derive the Af distribution. We are therefore forced to attempt some conversion between total magnitudes and point-source equivalent magnitudes. For bright and near-by comets, this correction can be as large as 5 magnitudes (eg. Fern´ dez et al. 1999). an The comets in the LINEAR sample were, however, typically first seen when very faint (Fig 6), and were generally mistaken for point sources in the initial observation. We

might therefore expect the correction factor to be much smaller, at least when close to the detection threshold. The histogram of detection magnitudes (Fig 6) climbs steeply down to Hdis 19, and then falls off fast (the one comet discovered when fainter than 20th mag was found by Spacewatch, which has a fainter magnitude limit). This fall-off o ccurs at almost exactly the same magnitudes as LINEAR's point source limit, which ranged from around 18 to 20. I thus conclude that near the LINEAR detection threshold, total magnitudes and point source equivalent magnitudes are similar. When generating mo ck samples, it is only the magnitude near the detection threshold that determines whether or not a given mo del comet is included in the mo ck catalog. The exact value of this correction value was set iteratively. I initially guess that the point-source equivalent (PSE) magnitude and total magnitude (TM) are the same, and run the simulations of the comet sample. I use the mo del that best fits the data (§ 5.3) to calculate the predicted discovery magnitude distribution, and compare this to the observed distribution. I then tweak the PSE-TM correction to bring the histograms into agreement. The best match is obtained when PSE-TM= 0.5 ± 0.5 (Fig 6). I use a value of 0.5 throughout this paper, except where otherwise noted. The predicted magnitude is corrected for the effects of trailing. LINEAR exposure times vary from 3 to 12 sec: the latter was used in the correction as it minimised the predicted number of very faint comets. 2 seeing (FWHM) was assumed. Trailing makes very little difference, except for the very faintest comets. LINEAR uses unfiltered CCD magnitudes while the historical surveys use unfiltered visual magnitudes. These will be somewhat different, due to the different wavelength sensitivity of the human eye and of the LINEAR CCDs, but the discrepancy should only be a few tenths of a magnitude at most, and hence is not a dominant source of error. Anoher possible worry: the absolute magnitudes I quote for the comets in the LINEAR sample (Table 1) are derived from total magnitudes measured when the comets were barely resolved and far from the Sun, using a mo del for the helio centric brightness variation. The absolute magnitudes fit by Everhart and Hughes are based on observations of highly extended comets observed close to the Sun and Earth. These are thus very different quantities, and might well be systematically different, if there is some error in our heliocentric brightness correction, if total magnitudes for barely resolved comets are systematicaly different from total magnitudes for greatly extended comets, or if there is some systematic bias in the discovery magnitudes reported to the central bureau of astronomical telegrams. To test this, I picked out the five comets in the LINEAR sample which were discovered when far (more than 3.5 AU) from the Sun, but which subsequently passed close enough to the Earth and Sun for traditional small telescope visual magnitude estimates (within 2AU). Dan Green kindly provided me with compilations of visual magnitude estimates of these comets while they were close to the Earth and Sun, taken from the archives of the International Comet Quarterly9 . These visual/small telescope magnitude estimates should be broadly com9

http://cfa-www.harvard.edu/icq/icq.html


9

Fig. 6.-- The predicted (dotted) and observed (solid) distributions of discovery magnitudes.

parable to the data on which the Everhart and Hughes papers were based. I then compared the predicted magnitudes when close to the sun (based on the discovery magnitude and the mo del in this paper) with the tabulated observations. There was a considerable scatter in the measured visual magnitudes for each comet: I simply averaged all visual small telescope magnitudes made when the comet was as close as possible to 1AU from both Sun and Earth. My predicted magnitudes were consistent with the observed values, albeit with a large scatter. The mean difference (predicted magnitude minus observed magnitude) was 0.4 ± 0.7, where the error indicates the 1 dispersion of the mean. This is not, alas, a strong constraint, but does indicate that the two magnitude scales are not grossly discrepant.
4.2. LINEAR's Sky Coverage

The final step is to determine whether LINEAR imaged a part of the sky in which the comet was detectable and within its magnitude limit. The published LINEAR skymaps show that during each dark perio d in 2000-2002, they attempted to survey the region whose midnight hour angle is in the range -7 < HA < 7, and in the declination range -30 < < +80 . In winter months with go o d weather, they surveyed more than 90% of this whole region down to a point-source m