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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 magnitude limit of better than 19. In bad months, this dropped to a magnitude limit of around 18.5 over 60% of this region, and o ccasionally worse. In the mean month, 72% of this region was surveyed to a visual magnitude limit of 18.5 or better. The exact pattern surveyed was complex and variable: the only constant was that the densest regions of the galactic plane were avoided. Each field was imaged

five times in succession, with 3 ­ 12 sec per exposure, once in every dark perio d. This sky coverage was approximated as follows. Each comet that enters the -7 < HA < 7, -30 < < +80 region at any point, with a point-source equivalent (PSE) magnitude brighter than 19 is considered to have been potentially observable, unless it was within ten degrees of the galactic plane. If a comet is predicted to be detectable for an entire lunation, it is given a 80% chance of having been detected during that lunation. If it was predicted to be visible for less than the whole lunation, it is given a probability of having been detected equal to 80% of the fraction of the lunation for which is was potentially observable. Do es this approximation match the real, more complex selection function? This was tested by manually checking 100 simulated comet ephemerides, containing monthly positions and magnitudes, against the real LINEAR skymaps. The approximation was found to give a number and absolute magnitude distribution of detected comets indistinguishable from the manually checked sample. This approach should slightly overestimate the probability of a comet being observed, as comets could be blended with star or galaxy images. Experience suggests that at this relatively bright magnitude limit, this is only a few percent effect at worst, at least away from the denser regions of the galactic plane, which the survey did not cover. Another possible source of error is sky subtraction. It is unclear exactly how the LINEAR survey do their sky subtraction, but if some of the extended coma emission is included in the sky value, this will artificially suppress the point source equivalent magnitude. The worst case sky subtraction algorithm would be to measure the sky


10 brightness from an annulus close to the comet nucleus. If I assume that the sky is measured only 5 from the nucleus (unlikely), we can used the observed 1/r surface brightness profiles of cometary comae (Jewitt & Meech 1987) to show that this sky subtraction algorithm would reduce the measured comet brightness by 0.2 mag. More plausible sky subtraction schemes would reduce it by less, or not at all. Thus this to o is not a dominant source of error.
5. COMPARISON WITH THE MODELS

In this section, I compare the data to the Monte-Carlo simulations of what LINEAR would have seen over a three year perio d. In the scatter plots, the data are compared to a single run of the simulation. In all histograms and quoted statistics, however, the data are compared against the average or sum (as appropriate) of five Monte-Carlo runs based on the same comet population mo del. This summation should suppress the error due to small number statistics in the simulated samples to well below that of the observed sample.
5.1. Everhart

The observed sample properties are first compared against the Monte-Carlo prediction using the Everhart (1967b) flux normalization and absolute magnitude distribution. Figs 7, 8 and 9 compare the distribution of mo del and observed comets in perihelion q and absolute magnitude H . The upper boundary to the lo cus of points is set by the magnitude limit, and seems a reasonable fit to the data. But the mo del clearly predicts far to o many comets: 2228 as compared to the 52 observed. The discrepancy is primarily at fainter absolute magnitudes: brighter than H 7 the mo del and data are consistent. The worst discrepancy is for comets fainter than H 11: ie. fainter than the data on which Everhart based his mo del. It is thus a test of the power-law extrapolation. The mo del predicts that LINEAR should have seen 1848 comets fainter than H = 11, whereas only 12 were seen. Irrespective of the flux normalization, the shape of the absolute magnitude distribution (Fig 9) is wrong: a KS-test comparison with the observed distribution shows that they are inconsistent with > 99.99% confidence. Is this discrepancy real, or is there some reason why LINEAR would miss faint comets close to the Earth? The mo del comets with H > 11 are predicted to be observable for a median perio d of 52 days, at a median distance from the Earth of = 1.0 AU. Their typical apparent angular velo city is predicted to be 1.1 degrees per day. Their observational properties are thus typical of near Earth ob jects, which LINEAR finds in profusion. It is thus hard to see what selection effect could prevent their detection. Grant Stokes (personal communication) confirms that there is nothing in their data analysis which should preclude the discovery of comets like these. Image trailing and short observability windows do reduce the number of these comets seen, but these are already taken into account by the Monte-Carlo simulation. Could the discrepancy be an artifact of the various approximations made in the mo del? The discrepancy is insensitive to assumptions about the dependence of

brightness or comet number on helio centric distance, as these sources are observed at close to 1AU. One possibility is that I have incorrectly estimated the effective magnitude limit of LINEAR for sources with these apparent total magnitudes. To test this, I repeated the analysis with a magnitude limit set two magnitudes brighter than my best estimate. This reduced the discrepancy but did not remove it: the prediction dropped to 577 observed comets fainter than H = 11, still more than two orders of magnitude above the data. None of the plausible incompletenesses in the data, nor other assumptions in the mo del can come close to removing this discrepancy. I therefore conclude that the Everhart mo del cannot be extrapolated to absolute magnitudes fainter than H = 11. Even at brighter absolute magnitudes, however, there remains a substantial discrepancy. The mo del predicts that 188 comets with H < 11 and q < 4 should have entered the solar system within the three year period, and that 86% of them (162) would have been detected by LINEAR. Only 21 such comets were observed. It is hard to see that LINEAR could have missed many comets this bright passing this close to the Sun. The mo del predicts that these comets should remain visible for a median 208 days (7 lunations), so almost regardless of position on the sky, they should have had several opportunities to be observed. They spend much of this time many magnitudes above the survey detection limit. Indeed, the 21 comets observed with these properties were discovered a median 11 months before perihelion, at a median helio centric distance of 4.3AU, confirming that these are easy targets. The discrepancy o ccurs mostly at the fainter magnitudes within this range: brighter than H 7 there is no significant difference between the Everhart predictions and the LINEAR observations. I conclude, therefore, that the Everhart mo del fails in two ways. Firstly, the quoted normalization of 8,000 comets per 127 years with H < 10.9and q < 4is too high by a factor of 7. Secondly, the faint end slope of the Everhart absolute magnitude relation is much too steep, and immensely over-predicts the number of faint comets. This second conclusion was first reached by Sekanina & Yeomans (1986): the current paper independently confirms their result. Both discrepancies suggest that Everhart overestimated the incompleteness of his sample of long-period comets. The discrepancy go es away where the incompleteness correction is small, but is largest at the faint magnitudes where the correction is large. A discrepancy here is perhaps not surprising, as the correction factors calculated by Everhart were so large: he corrected the 256 observed comets to a flux of 8000: a factor of 31. My analysis reduces this correction factor to only 4. One possible reason for the difference: Everhart calculated the detection threshold for typical historical comet searchers, and assumed that the same threshold applied when searching for comets initially, and when making follow-up observations of known comets. His mo del was validated by noting that the last observations of comets o ccurred close to the time when his mo del suggested that they dropped below detectability. But let us hypothesize that comets just above the detection threshold might be missed as the telescope speeds past during a scan for new comets, even though they could be detected when lo oking hard for an already known comet at a known po-


11

Fig. 7.-- The perihelia and absolute magnitudes of model comets (crosses) and the LINEAR sample (triangles). The model comet population was generated using the Everhart (1967b) flux and absolute magnitude distribution.

Fig. 8.-- The perihelia distributions predicted by three different model comet populations, compared to the observed distribution for the LINEAR sample.


12

Fig. 9.-- The absolute magnitude distributions predicted by three different model comet populations, compared to the observed distribution for the LINEAR sample.


13 sition. This would reduce the length of time over which a given comet could have been detected. Detections would thus have o ccurred earlier in the detectability window, and the correction factor for incompleteness would thus decrease.
5.2. Hughes Model I now compare the data against a Monte-Carlo simulation using my version of the Hughes (2001) absolute magnitude distribution and normalization, combined with the linearly rising perihelion distribution (Figs 8, 9, 10). Once again, the overall envelope of points agrees well with the mo del, giving some confidence that the selection effects have been mo deled correctly. The predictions from the Hughes mo del are in much better agreement with the data. There is no vast excess of faint predicted comets, implying that the break seen in Hughes' data was real, and hence that the flatter faintend slope of the absolute magnitude distribution is more accurate. Neither the perihelion nor absolute magnitude distributions, however, are formally consistent with the observations at the 99% confidence level, as measured by the KS-test. The overall flux normalization is also to o high: the model predicts that LINEAR should have seen 171 comets, rather than the 52 observed. There is no significant discrepancy within the region in which Hughes quoted his flux normalization (0.54 comets per year with H < 6.5 per unit perihelion): the discrepancy is at fainter absolute magnitudes and larger perihelia. Once again the discrepancy is fairly robust against the exact detection limit: dropping the detection threshold by a magnitude reduces the predicted comet numbers to 148 - still too high. How can this discrepancy be addressed? Possible incompletenesses in the LINEAR sample were discussed in § 2.1 and they can at best increase the observed numbers by 20%. The Hughes flux normalization is unlikely to be too low, as it was based on observed counts of very bright comets and made no correction for incompleteness. I therefore tried to improve the match by tinkering with the absolute magnitude and perihelion distributions. 5.3. The Best-Fit Model

marginally acceptable (KS-test gave 8 and 5% probabilities of them coming from the same population as the data) but the joint probability was still uncomfortably low (though the magnitude limit cut means that the two distributions are not independent, so this should not be taken to o seriously). The mo del predicted too many comets with q < 1 and q > 5, and to o few in the middle. Moving the lo cation of the break in the absolute magnitude relation to fainter magnitudes was also a failure: given that the comet flux normalisation is at brighter magnitudes, this simply increased the number of fainter comets still further above the observations. I therefore adopted a different perihelion distribution: one that rises from q = 0 out to q = 2, and is a power-law beyond that. This preserves the Everhart observation of a drop in comet numbers below q = 1, while allowing us to tinker with the distribution further out. The parameterization used was: dn 1+ q, q < 2AU (4) 2.41 â (q/2) , q > 2AU dq where controls the behavior at large perihelia. I then ran a grid of mo dels, varying and the faint end slope of the absolute magnitude distribution (f in Equation 3). Each simulated population was tested against the data in three ways: a KS-test on the perihelion distribution, a KS-test on the absolute magnitude distribution, and a Chi squared test on the overall predicted number of comets. The latter was done for both the observed number of comets and a number 10% higher, to allow for possible incompletenesses. The lowest of these three significance values was used to compute the go o dness-of-fit conutours in Fig 11. Note that these contours include only random errors: the systematic errors are almost certainly larger, especially on . The lowest probability rather than the joint probability was used becuase the three tests are not strictly independent. Quite a tight constraint could be placed on f : f = 1.03 ± 0.09 (95% confidence, not including systematic errors). The constraint on was weaker: = -0.27±0.3. No useful constraint could be placed on the bright-end slope b: in the mo deling I use the Hughes (2001) value of 2.2, but it makes little difference. Note that these slopes are often described in the literature using the parameter: = log10 (slope), so our best-fit faint end slope has = 0.004. The predicted distribution of comets for the best fit mo del is shown in Figs 8, 9 and 12. Mo dels generated using this mo del predict 55 ± 3 (1 ) observed comets, in excellent agreement with the data. This mo del was used to predict the discovery magnitude distribution (Fig 6). This comparison was used to set the equivalent point source vs. total magnitude offset, as described in § 4.1. My preferred mo del thus approximately preserves the faint-end slope and normalization derived by Hughes. A flat perihelion distribution is marginally ruled out, and best fits are obtained for one that rises from the Sun out to q 2 and is either flat or gently falling beyond that.
6. DISCUSSION 6.1. The Comet Flux

I first tried reducing the faint end slope of the absolute magnitude relation. If we assume that the LINEAR sample is 20% incomplete, we need to reduce the faint end slope from 1.07 down to 0.8 to bring the number of predicted comets down to the observed number. Unfortunately, this changes the observed absolute magnitude distribution to o much: a KS test shows that a mo del with this slope predicts an observed H distribution inconsistent with the data with greater than 99.99% confidence. I then tried changing the perihelion distribution. Decreasing the slope A in Eqn 2 to zero brought the predicted number of comets down to 95, but the perihelion distribution is now to o skewed towards small values of q (99.96% confidence). I next tried combining both approaches. Decreasing the faint end slope to 1.0, combined with a flat perihelion distribution, brought the predicted numbers into line with the observed numbers. Both the perihelion and absolute magnitude distributions were individually

The best-fit mo del can be used to estimate the flux of long-perio d comets through the inner solar system.


14

Fig. 10.-- The perihelia and absolute magnitudes of model comets (crosses) and the LINEAR sample (triangles). The model comet population was generated using the Hughes flux and absolute magnitude distribution.

Fig. 11.-- Goodness-of-fit contours as a function of model parameters and f . The solid line is the 95% confidence contour, and the dotted line is the 99% contour.


15

Fig. 12.-- The perihelia and absolute magnitudes of model comets (crosses) and the LINEAR sample (triangles). The model comet population was generated using our best fit model of the comet population.


16 By definition, this mo del uses the Hughes (2001) flux of 0.53 comets per year per unit perihelion, brighter than H = 6.5. Everhart (1967b) estimated a flux of 8000 long-perio d comets per 127 years with H < 10.9 and q < 4. My mo del suggests that the flux of comets with these parameters is much lower: 11 long-perio d comets per year (1600 per 127 years). The mo del suggests that LINEAR is picking up over 60% of comets with these parameters. I estimate a true flux of 37 comets/year with H < 15 and q < 8, of which LINEAR is detecting 40%.
6.2. The Number of Oort Cloud Comets

Many published estimates of the number of comets in the Oort cloud use the Everhart (1967b) comet flux as their starting point, and hence should be revised down. Estimates include Bailey & Stagg (1988), Heisler (1990), Weissman (1996) and Dones et al. (2004). The LINEAR sample includes 22 comets with H < 10.9 and q < 4 over a three year period. 5 of these were dynamically new, 11 dynamically old, and 6 had orbit determinations to o po or to tell. If we assume that the comets without go o d orbit determinations break up between new and old in the same ratio as the other comets, we find a flux of 7 dynamically new comets over the three years. My mo del suggests that the LINEAR sample is 80% complete for finding comets in this range, implying a total flux of dynamically new comets of 3 per year. This corresponds to 0.8 per unit perihelion per year if a uniform perihelion distribution is assumed. This is a factor of 2 lower than was assumed by Heisler (1990) and Dones et al. (2004). Bailey & Stagg (1988) and Weissman (1996) assume long-perio d comet fluxes (not just the dynamically new ones) of 10 per year per AU down to H < 10.9, compared to my value of 2.5. After correction, all these estimates come out roughly the same: 1 ­ 3 â1011 in the outer Oort cloud, down to H < 10.9. If this is extended to fainter absolute magnitudes, I estimate an outer Oort cloud population of 5 â 1011 comets down to H = 17.
6.3. The Mass of Oort Cloud Comets

mass resides in comets with 0 < H < 9. For the Hughes bright-end slope, however, the total mass diverges as you count brighter comets. The bright comets are rare, but their mass go es up faster than their number goes down at bright magnitudes. Thus for the Weissman mass relation, comets with H -5 are 10,000 times more massive than those with H 5. The Hughes bright-end slope, however, suggests that they are only 2,600 times less common, so the total mass in the brighter comets is actually four times greater. For the Everhart brightend slope, however, the brighter comets would be 300,000 times less common. The Everhart slope is thus physically more appealing, as it avoids the need for a bright cut-off. Given the success of both the Hughes normalization and faintend slope in fitting our sample, however, his bright-end slope should perhaps be taken seriously, leading to the prospect of an Oort cloud dominated (in mass terms) by very large comets. In this section, I will cut off the magnitude range of comets at H = -5, but this is arbitrary and will have a large effect on the total Oort cloud mass if the Hughes distribution is assumed. Bernstein et al. (2004) showed that Kuiper Belt ob jects have a break in their mass distribution at sizes of 100 km, which corresponds to H -3 (§ 6.7). This may or may not apply to Oort cloud comets. If I take the Everhart slope, the mean mass of comets down to H = 11 is 1.7 â 1017 g for the Weissman (1996) mass relation, and 5.6 â 1016 g for the Bailey & Stagg (1988) relation. If I take the Hughes slope, however, the average masses rise to 1.1 â 1018 g and 1.2 â 1018 g respectively. These values translate into total outer Oort cloud masses of 2 ­ 40 Earth masses.
6.4. Terrestrial Impact Probabilities

My mo del has a much shallower slope of the absolute magnitude distribution than that of Everhart (1967b), so the average mass of a comet increases. This cancels out the decreased number of Oort cloud comets I predict to give a similar total Oort cloud mass to previous estimates (eg. Weissman 1996; Dones et al. 2004), both of which used a mean mass computed by integrating the Everhart curve. I used two suggested mass-brightness relations to estimate masses from the observed absolute magnitudes: one from Bailey & Stagg (1988) and one from Weissman (1996). Note that this relation is extremely uncertain very few long-perio d comets have had even their nuclear magnitudes measured. The average mass is crucially dependent on the slope of the bright end of the absolute magnitude distribution, which the data in this paper do not constrain. I bracket the possibilities by using both the Everhart and Hughes values (3.54 and 2.2 respectively). I use my own estimate of the faint-end slope. For the Everhart bright-end slope, the total mass converges as you go to brighter magnitudes: the bulk of the

My mo del can be used to calculate the probability of a long-perio d comet impacting the Earth. Steel (1993) calculated that the probability of a given long-perio d comet with q 1AU impacting the Earth is 3 â 10-9 per perihelion passage. My mo del suggests that the flux of comets brighter than H = 19 with q < 1AU is 8 per year. The mean time between comet collisions with the Earth is thus 40 million years: very comparable to the figure calculated by Sekanina & Yeomans (1986), and to the mean time between global extinction events. Most of these comets will, however, be quite small. Using the conversion between absolute magnitude and radius described in § 6.7, a comet with H 19 has a radius of only 20m: to o small to cause a global extinction event. Comets with radii of 1km or greater (H 9) are rarer - the mean time between collisions with comets this large is 150 million years. The shallow faint end slope means that collisions with even small long-perio d comets are rare. If, for example, the Tunguska impact was caused by a comet (eg. Bronshten 2000), it would have a mass of 1011 g (Vasilyev 1998) and hence an absolute magnitude of 18. The probability of such a long-perio d comet impacting the Earth in the last 100 years is thus < 10-5 . The Tunguska impactor must therefore be either asteroidal (eg. Sekanina 1998; Farinella et al. 2001) or associated with a short-period comet (eg. Asher & Steel 1998).
6.5. Detection of Interstel lar Comets


17 No comet has ever been detected with a strongly hyperbolic original orbit (Kresak 1992): ie. a comet that was not gravitationally bound to the solar system. Several authors have discussed this (eg. Sekanina 1976; McGlynn & Chapman 1989; Stern 1990; Sen & Rama 1993), with many claiming that this is surprising. Oort cloud formation models predict that for every comet that reaches the classical outer Oort cloud, a factor = 3--100 more are expelled into interstellar space (eg. Duncan, Quinn & Tremaine 1987). If most stars have planets, and if planetary formation is usually accompanied by comet ejection, then there should be a substantial population of free-floating interstellar comets. By some estimates, we should have expected to have seen one or more such comets by now, passing through the inner solar system. The results in this paper impact upon this question in two ways. Firstly, the non-detection of interstellar comets in the LINEAR sample places an upper limit on their space density. Secondly, most previous estimates of the expected number of interstellar comets relied upon the Everhart (1967b) comet flux: our lower comet flux thus leads to smaller predictions of the interstellar comet density. What limit can we place upon the space density of interstellar comets (those with strongly hyperbolic orbits) from the non-detection of any by LINEAR? I will assume that interstellar comets have the same absolute magnitude distribution that I derive for long perio d comets, and that their apparent brightness varies with helio centric distance in the same way as dynamically new inbound comets in our model. Given these assumptions, and an assumed magnitude limit of 19 (point source equivalent), one can derive the distance r out to which a comet with any given absolute magnitude could have been detected. To convert this into the volume surveyed during the three years of the survey, one must allow for the motion of the comets with respect to the solar system, which can carry new comets into range. Typical motions of nearby stars with respect to the Sun are v 40km s-1 (Garc´ -S´ chez et al. 2001). The volume surveyed in a ia an survey of duration T is thus 43 r + r2 Tv . (5) 3 The pro duct of this equation and the absolute magnitude distribution (Equation 3) was integrated to calculate the number of comets potentially within LINEAR's magnitude limit, for a given assumed space density of interstellar comets (defined as the number of interstellar comets per cubic astronomical unit brighter than H = 11). The integral suggests that the bulk of interstellar comets detected will be those with absolute magnitudes near the break at H = 6.5. The results are quite sensitive to the adopted bright-end slope of the absolute magnitude distribution, being 40% lower for the Everhart slope as compared to the Hughes slope. Not all of these comets will be seen: my mo del suggests that LINEAR finds 70% of the Oort cloud comets within its magnitude limit. The fraction may be lower for interstellar comets because they move faster and hence are not observable for long, but the velo city difference is only < 50%. Furthermore, a larger fraction of interstellar comets will be brght ones seen at large helio centric distances, where the visibility perio d is larger. I adopt V= a conservative 50% detection probability, which should be ample to include comets not being followed up or not having go o d orbit determinations. The mean number of comets seen over the three years is then evaluated as a function of the assumed average density. If more than 5 comets are predicted to have been observed, the Poisson probability of us having not seen any interstellar comets is less than 5%: this is our adopted limit. I thus derive an upper limit on the lo cal space density of interstellar comets of 6 â 10-4 per cubic AU (95% confidence) if the Hughes bright-end slope is assumed. For the Everhart bright-end slope, this limit increases to 9 â 10-4 per cubic AU. These limits are very comparable to the best existing limit: that of Sekanina (1976). Sekanina's limit is, however, based upon Everhart's papers and should thus be regarded with some suspicion. I can extend this calculation by noting that LINEAR has not discovered any interstellar comets in other years. From 1999 through to the end of 2004 their monthly sky coverage (though not available in detail) is at least comparable to that during my sample perio d. These extra three years of data reduce our upper limits to 3 -- 4.5 â10-4 per cubit AU. I now evaluate the expected space density of interstellar comets, given our reduced Oort cloud population estimate. Following Stern (1990), the number density of interstellar comets nism is given by: n
ism

= n

stars

N

comets

,

(6)

where nstars is the lo cal number density of stars, Ncomets is the mean number of outer Oort cloud comets per star, and is the ratio of comets expelled from a solar system to the number ending up in the outer Oort cloud. I adopt nstars 0.1 per cubic parsec, from the 8pc sample of Reid & Gizis (1997): this is consistent with the value used by McGlynn & Chapman (1989) but considerably lower than the value used by Stern (1990). For Ncomets , I assume that the number of comets generated per star is proportional to its mass and metallicity. This may or may not be true, but is consistent with the observed tendency for low mass stars to lack hot Jupiter planets (eg. Grether & Lineweaver 2005, and refs therein). The lo cal stellar population is dominated by low mass dwarfs: the average stellar mass of the nearby stars in the Reid & Gizis (1997) catalogue is only 0.3M . The average metallicity of near-by F and G stars is [Fe/H ] -0.14 (Nordstrom et al. ¨ 2004): no information is available for lo cal dwarf stars, so I assume the same value. Assuming that the outer Oort cloud population derived in § 6.2 is typical of local stars with solar mass and metallicity, I therefore derive Ncomets = 0.3 â 0.72 â 2 â 1011 = 4.3 â 1010 . The most uncertain parameter is : literature values range from 3 ­ 100 (eg. Duncan, Quinn & Tremaine 1987; McGlynn & Chapman 1989; Stern 1990; Weissman 1996). Given these parameters, 1.4 â 10-6 < nism < 4.7 â 10-5 . Thus even in the most optimistic case, the predicted flux is an order of magnitude below current limits. Could future surveys reach these predicted densities? I ran the simulation for five year surveys reaching to deeper magnitude limits. A survey reaching 24th magnitude (perhaps PanStarrs) would place 95% upper limits of 5 â 10-5 , and might hence detect one or two interstellar comets if is very large. Jewitt (2003) reached


18 similar conclusions. A survey reaching 26th magnitude (LSST?) would push the limit down to 1.5 â 10-5 , which would be enough to detect or rule out a large value of . This prediction do es, however, depend crucially on the assumed brightness behavior of comets a long way from the Sun.
6.6. Perihelion Distribution Simple models which assume that Oort cloud comets have an isotropic velo city dispersion imply a constant number of comets per unit perihelion. n-bo dy integrations (eg. Tsuji 1992; Wiegert & Tremaine 1999), in contrast, imply a rising number of comets at larger perihelia, due to the diffusion pro cess of comets past the perturbations of the giant planets. Both Tsuji (1992) and Wiegert & Tremaine (1999), for example, predict an increase in the number of comets per unit perihelion of â2 between q = 2 and q = 8 (though this is an extrapolation of the Wiegert & Tremaine mo del which is only shown out to q = 3). Is this consistent with the LINEAR sample? The best-fit model has the number of comets per unit perihelion (beyond q = 2) going as a power law of index = -0.27 ± 0.3 (95% confidence), ie. a gentle fal l. To get a rise in numbers consistent with the Tsuji (1992) and Wiegert & Tremaine (1999) predictions, we require 0.5, which is inconsistent with our model with 99% confidence. The measured perihelion distribution is, however, somewhat degenerate with the assumed dependence of comet brightness on helio centric distance: n in equation 1. As I have repeatedly noted, the choice of n is an approximation based on extrapolations of observations obtained at much smaller helio centric distances. We can estimate the change in n that would be needed to bring our observations into line with the Tsuji (1992) predictions. We need to drop the predicted brightnesses of comets with q 8 by enough to reduce the observed numbers by 50%, to meet our 95% upper limit. Given our best-fit absolute magnitude relation, this requires that n be increased by n > 0.8. This is quite a small rise - well within the observed scatter of n values seen at lower perihelia. If, for example, I had used the canonical value of n = 4 for all comets, rather than our more complex scheme, this would make dynamically new comets much fainter when far from the Sun, as required (at the expense of the correlation seen in Fig 3). The data in this paper, while suggestive, are not therefore significantly at o dds with the theoretical predictions. A better understanding of helio centric brightness variations when distant from the Sun will be needed to see if this anomaly is real. 6.7. Implications of the Shal low Faint-End Slope The faint-end slope of the absolute magnitude relation derived in this paper (1.01 ± 0.11) is very flat: the number of comets per unit magnitude barely increases as you go fainter. This presumably indicates that small comets are not that much more abundant than large ones. If the absolute magnitudes are converted to nuclear masses, using either the Bailey & Stagg (1988) or Weissman (1996) relations, I find that the differential number of comets

per unit mass m go es as: dn m-1.04±0.1 . (7) dm Brandt et al. (1996) made the case for the existence of small comets: those with nuclei only meters to tens of meters in radius. If we assume the Weissman (1996) relationship between mass and absolute magnitude, and a density of 0.6g cm-3 , 100m radius corresponds to H 15 and 10m to H 20. LINEAR is therefore detecting at least a few comets with nuclei smaller than 100m. Similar small comets are also detected by the LASCO instrument on the SOHO spacecraft (Biesecker et al. 2002), though these are mostly fragments of recently disintegrated larger comets (Sekanina & Chodas 2004). Hughes (2001) was unclear on whether the shallow faint-end slope was real or a selection effect: I confirm that it is real. Sekanina & Yeomans (1986) pointed out that small long-perio d comets must be rare from statistics of comets passing close to the Earth. Meech, Hainaut & Marsden (2004) found a similar paucity of small Jupiter-family comets. In this section, I point out that the shallowness of the faint-end slope is actually quite interesting theoretically. Collisions are rare in the Oort cloud (Stern 1988), so the nuclear size distribution should remain largely unchanged from when the proto-comets were planetesimals expelled from the protoplanetary disk (Duncan, Quinn & Tremaine 1987; Goldreich, Lithwick & Sari 2004). Weidenschilling (1997) mo deled the size distribution of planetesimals in the protoplanetary disk. These models imply that 100m diameter ob jects should outnumber 2km sized ob jects by a factor of 1010 (per unit log radius) in the oldest disks mo deled. Our faint-end slope, however, if combined with the Weissman (1996) mass/absolute magnitude relation, implies a ratio of only 104 . The size distribution of planetesimals may be greatly mo dified by collisions while they lie within the dense environment of the proto-planetary disk. Most models of this collisional evolution also, however, predict much flatter size distributions than we see: they typically predict an increase in comet numbers, even at faint magnitudes, of a factor of 3.2 per magnitude (Dohnanyi 1969; Wetherill & Stewart 1993; Charnoz & Morbidelli 2003). This is much greater than my measurement (1.03 ± 0.09). As noted in § 5.1, our shallow faint-end slope is quite robust to sample incompletenesses and mo del assumptions. The conversion of absolute magnitudes to radii is, however, highly uncertain even at bright magnitudes, let alone the faint absolute magnitudes of relevance here. Theoretical predictions are also quite uncertain (Lissauer 1993; Youdin & Shu 2002), and may be consistent with this slope. Another possibility, however, is that the probability of a planetesimal escaping the protoplanetary disk and reaching the Oort cloud is size-dependent. Stern & Weissman (2001), for example, suggested that collisions between planetesimals would act to circularize their orbits, and would prevent them from escaping into the Oort cloud until the density of planetesimals was greatly depleted (see also Charnoz & Morbidelli 2003). The squarecube law would suggest that this effect is most serious for smaller planetesimals, which may be ground down to


19 dust before escaping. This could thus explain the deficit of small comets. Alternatively, gas drag could play the same role: the largely gaseous nature of the giant planets indicates that the protoplanetary disk was still full of gas when the giant planets formed. No published mo delling currently includes this effect. Another possibility: the square-cube law implies that small comets may lo ose their volatiles faster than large ones. Fading may thus be more severe for these comets, making them harder to detect. A final possibility is that comets have been exposed to high temperatures at some point in their history. The resultant loss of volatiles could thus destroy small comets without much affecting the numbers of large ones. Meech, Hainaut & Marsden (2004) tentatively suggest this as a reason for the lack of Jupiter-family comets with small nuclei. Long perio d comets might have been exposed to high temperatures while still in the proto-planetary disk. Once in the Oort cloud, temperatures are much lower. Nearby supernovae and O-stars may have temporarily heated them enough to remove some volatiles (Stern 2003), but this should not appreciably affect the numbers of small comets. I note in passing that my mo dels are strongly inconsistent with the claims by Frank & Sigwarth (2001) for an immense population of small comets bombarding the Earth. These claims have, however, already been ruled out in many other ways (eg. Bashar & Dessler 1997; Hannegan et al. 1999; Knowles et al. 1999; Harris 2000).
6.8. Implications for Future Surveys

I conclude that telescopes such as SkyMapper, PanStarrs and LSST, each capable of surveying large areas to deeper than 22nd magnitude, should be capable of detecting more than 50 long-period comets per year. A substantial fraction of these are forecast to have perihelia beyond 10AU, though this conclusion relies upon the rather shaky assumptions of how brightness varies with helio centric distance this far out. Thus five years should suffice to build up a quantitatively selected sample of long-perio d comets equal in size to any historical sample.
7. CONCLUSIONS

Six main conclusions can be drawn from this analysis: 1. The outer Oort cloud contains 5 â 1011 comets down to H = 17 ( 2 â 1011 comets down to H = 11). This is 2 -- 10 times fewer comets than previous published estimates. 2. The average mass of these comets is, however, higher than previous estimates. Down to H = 11, the average mass is between 5.6 â 1016 g and 1.2 â 1018 g, leading to a total mass in the outer Oort cloud of 2 -- 40 Earth masses, comparable to or larger than previous estimates. The mass of the Oort cloud may be dominated by a few large comets. 3. Small comets do exist, but are rarer than predicted by many mo dels. This may be because they have difficulty escaping from the protoplanetary disk. The probability of the Earth being hit by a longperio d comet similar in energy to the Tunguska impactor is only one in forty million per year. 4. I place an upper limit on the space density of interstellar comets of 4.5 â 10-4 per cubic AU (95% confidence). This is still an order of magnitude above our revised prediction for the space density of interstellar comets. 5. The number of long-perio d comets per unit perihelion seems to decline, or at best rise slowly beyond 2AU. This do es not agree with theoretical predictions. The discrepancy may be resolved if comets are fainter at large helio centric distances than I assume. 6. Future survey telescopes should be able to assemble samples of several hundred long-perio d comets in a few years of operation. The ma jor weakness in this analysis is in the photometry: in particular in our limited understanding of how the brightness of comets varies with helio centric distance when far from the Sun, and in how to convert ill-defined total magnitudes into more repro ducible and physically meaningful parameters such as Af . The data exist to address these problems, but are not publicly available. I would like to thank Grant Kennedy, Eriita Jones and Chris Weekes for their work on aspects of this paper, Stephen Pravdo and Grant Stokes for responding to email questions about their NEO surveys, Timothy Spahr

The mo del in this paper can be used to guide future automated comet-searches. The most basic conclusion follows from the shallow faint-end slope of the number/absolute magnitude relation. This means that to find more comets, a survey should always try to maximize the area covered rather than going deep in a small area. If LINEAR, for example, exposed for six times as long per field, it would be sensitive to fainter comets, and would hence see 1.5 times more comets per unit area. But it would cover a six times smaller area. The more frequently a survey covers a given area of sky, the higher the probability of a given comet being detected. In Fig 13, however, I show that this is not an enormous effect: decreasing the survey frequency for a hypothetical survey from weekly to biannual only drops the number of detected comets by 20%. The comets lost are primarily those with fainter absolute magnitudes, because their visibility period is small. The small number lost is a direct consequence of the shallow faint-end slope: there are few small comets to lose. In Fig 14, I show the predicted samples that would be found by telescopes using a similar survey technique to LINEAR, but going deeper. The deeper limit essentially increases the absolute magnitude limit reached at all perihelia, and by increasing the time over which comets are visible, also improves the completeness for brighter comets. Over 3 years, a survey to 20th mag would detect 103 comets, to 22nd mag would detect 150 and to 24th mag, 186 comets. This is much fewer than Jewitt (2003) estimated. The discrepancy is probably due to my shallow inferred absolute magnitude and perihelion distributions, which mean that the dramatic increase in sensitivity of these surveys as compared to LINEAR only yields a relatively small increase in sample size.


20

Fig. 13.-- The predicted long-period comet sample found by a hypothetical survey regularly covering the whole accessible sky down to a point-source magnitude limit of 22. Open triangles are the comets that would be found if this region were surveyed once per week. Solid triangles are those that would also have been seen if the region were only surveyed twice a year. The sample was obtained for a hypothetical three year survey and only includes comets reaching perihelion within these three years. No weather losses were included.

Fig. 14.-- The predicted long-period month. Open triangles are the comets those that would also have been seen if l comets reaching perihelion within these

comet sample found by a hypothetical survey regularly covering the whole accessible sky once per that would be found if survey reached a point-source magnitude limit of 24. Solid triangles are imit were only 20. The sample was obtained for a hypothetical three year survey and only includes three years. No weather losses were included.


21 for providing details of ob jects posted to the NEO confirmation page, and Dan Green for provide magnitude measurements from the archives of the International Comet Quarterly. Paul Weissmann was referee on an earlier version of this paper: his detailed comments were invaluable in educating the author (an extragalactic astronomer by training) in comet lore.

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