The absolute magnitude distribution of trans-Neptunian objects

Abstract

It is shown that the known trans-Neptunian objects (TNOs) have an absolute magnitude distribution index that increases as a function of orbital perihelion distance. In no perihelion range is the TNO index the same as that found for known short-period comets. However, the fact that the median diameters of the known members of these two populations (220 and 2.9 km respectively) differ by a factor of about 75 means that very small TNOs and short-period comets might still be related.

comets: general, Kuiper Belt

1 Introduction

The trans-Neptunian region of the solar system (see, for example, Williams 1997; Levison & Weissman 1999) contains many bodies, these ranging in size from Pluto and Quaoar (2002 LM₆₀), with diameters of 2300 and 1250 km respectively, down to a limit imposed by present-day telescopic power. The numbers of bodies detected so far, 602 when writing, confirm that the trans-Neptunian proto-cometary belt proposed by Kuiper (1951) and Edgeworth (1943) has been found. This paper investigates whether the known trans-Neptunian objects (TNOs) have the same absolute magnitude distribution as comets.

Following Sheppard & Jewitt (2002), the apparent magnitude, m, of a TNO is given by  

(1)

where m₀ is the apparent magnitude of the Sun (−27.1), p_R is the geometric albedo of the TNO, r(km) is the effective radius of the TNO, φ(α) is the phase function [which, at opposition, α= 0°, is such that φ(0) = 1], R(au) is the heliocentric distance and Δ(au) is the geocentric distance. Equation (1) can be simplified to  

(2)

where the absolute magnitude H is clearly equal to the apparent magnitude the TNO would have if it were observed in an idealistic situation in which it was 1 au from the Earth, 1 au from the Sun and at a zero phase angle. Knowing H and the albedo of the TNO, one can easily calculate the mean cross-sectional area and the mean diameter. [Brian Marsden (private communication) noted that the absolute magnitude of a typical TNO was known to an accuracy of about ±0.5.]Zellner & Bowell (1977) followed Russell (1916) and proposed that the diameter d(m), the surface geometric albedo p_R and the absolute magnitude H were related by  

(3)

For the average albedo of asteroids (p_R= 0.125), this becomes  

If the TNOs have the average albedo of cometary nuclei, i.e. around 0.04, the relationship becomes  

(4)

The brighter members of known cometary populations have historically been assumed to have an absolute magnitude distribution given by  

(5)

where C is the number of comets that have absolute magnitudes in the range H to H+ΔH. The quantities b and a are usually taken to be constants. Notice that, in the cometary context, we are discussing the absolute magnitude of the integrated dust and gas coma of an active comet, and not the absolute magnitude of the cometary nucleus.

The cumulative number, N, of comets brighter than magnitude H can be obtained by integrating C from absolute magnitudes H to H=−∞. So  

(6)

Normally log₁₀N is plotted as a function of H. Here  

(7)

The quantity log₁₀a in equation (7) is referred to as the absolute magnitude distribution index. Hughes (2001) analysed the absolute magnitude distribution of long-period comets. These comets have visited the inner solar system very few times, have undergone very little decay, and are expected to have their primordial magnitude distribution index. This index was found to be log₁₀a= 0.359 ± 0.009. A similar magnitude distribution index was found to apply to those short-period comets with perihelion distances, q, greater than 2.0 au (see Hughes 2002). These, like the long-period comets, have suffered little from decay. [In the past, many populations have been also been quantified by the mass distribution index, s. Here the number of bodies with mass greater than M is taken to be proportional to M^(1−s). Note that s= 1 + (5/3)log₁₀a. Comets have thus been found to have a mass distribution index of 1.60 ± 0.02, a value indicative of a population produced by planetesimal accretion as opposed to collisional fragmentation (see Daniels & Hughes 1981).]

We must stress that the indices given above have been obtained by the analysis of the number distribution of the absolute magnitudes of active comets, and therefore rely on equations such as equation (4) above being strictly valid, i.e. there being a firm relationship between comet nucleus size and absolute magnitude of the integrated dust and gas coma of that comet when it is close to the Sun and active. We are fortunately getting to the stage where the Hubble Space Telescope and some of the largest ground-based telescopes are being used to observe the bare nuclei of comets at large heliocentric distances. Here the cometary nucleus is not obscured by a surrounding coma, and proportionalities between coma brightness and nucleus surface area do not have to be made. The number of accurately known diameters is slowly growing: see, for example, Lamy et al. (2000), Lincandro et al. (2000), Lowry & Fitzsimmons (2001) and Lowry, Fitzsimmons & Collander-Brown (2003).

If the TNOs in the Edgeworth–Kuiper Belt, Centaurs, long-period comets and those short-period comets with q > 2.0 au are all related, and these objects are all essentially primordial, then they are all expected to have the same magnitude and mass distribution indices. This is especially true if the TNOs are the reservoir that feeds the low-inclination Jupiter-family cometary population, as first suggested by Fernández (1980, 1985).

2 Trans Neptunian Objects

A continually updated list of known TNOs together with their physical and orbital parameters is maintained on the web site http://cfa-www.harvard.edu/cfa/ps/lists/TNOs.html.

The perihelion distance, aphelion distance and absolute magnitude distributions of the 602 TNOs listed when writing are shown in Figs 1, 2 and 3. (Note that Pluto and Quaoar were not included in the list at that time.) We are not going to go into details about the orbits [see Levison & Weissman (1999) for more information and illustrations]. The known TNOs seem to be in three main dynamical groupings, the first consisting of bodies that are trapped in a 2:3 mean motion resonance with Neptune, these having orbits similar to that of Pluto. The second group are in near-circular orbits and have a range of perihelion distances, and the third group are similar to the second group but have larger eccentricities. (Edgeworth–Kuiper Belt formation models predict that many TNOs in the 3:4, 3:5 and 1:2 mean motion resonances of Neptune wait to be discovered.) The perihelion distribution of known TNOs shown in Fig. 1 peaks at around 41 au, and the aphelion distance distribution (Fig. 2) peaks at around 46 au. For the second and third dynamical groups mentioned above, these two values should increase in the future as fainter and more distant TNOs are found. Remember that Neptune and Pluto are at mean solar distances of 30.1 and 39.5 au respectively.

Figure 1.

The perihelion distance distribution of the 602 TNOs listed on the web site http://cfa-www.harvard.edu/cfa/ps/lists/TNOs.html.

Figure 2.

The aphelion distance distribution of the 602 TNOs listed on the web site http://cfa-www.harvard.edu/cfa/ps/lists/TNOs.html.

Figure 3.

The absolute magnitude distribution of the 602 TNOs listed on the web site http://cfa-www.harvard.edu/cfa/ps/lists/TNOs.html. The application of equation (3), assuming a geometric albedo of 0.04, indicates that TNOs with absolute magnitudes of 4, 5, 6, 7, 8, 9 and 10 have diameters of about 1050, 660, 420, 260, 170, 105 and 66 km respectively. The ordinate represents the logarithm of the number of TNOs with absolute magnitudes in the range H to H+ 0.2.

In the present paper we are concerned with the absolute magnitude distribution, and this is shown for known TNOs in Fig. 3. If equation (5) is applicable, the gradient of the histogram in Fig. 3 should have a value of log₁₀a. Fitting equation (5) to the 4.4 < H < 6.8 region of the data indicates that log₁₀a= 0.66 ± 0.06 (i.e. s= 2.1 ± 0.1). This absolute magnitude range contains the largest TNOs (870 > d > 290 km), the assumption being that many of the fainter and smaller H > 6.8, d < 290 km TNOs are yet to be discovered. It is clear that the TNO value of log₁₀a is significantly different from the cometary value. An s= 2.1 mass distribution index has often been taken to signify a population produced by collisional fragmentation (see for example Hughes 1994).

Interestingly, Gladman et al. (2001) analysed the apparent (as opposed to absolute) magnitude distribution of the bright TNOs known at the time. (As the orbital periodicity of TNOs is above 250 yr, the apparent magnitudes of specific TNOs have hardly changed since discovery.) Assuming a power-law heliocentric distance distribution, Gladman et al. (2001) converted their apparent magnitude distribution into a differential TNO size index, quoting a resulting index of q= 4.4 ± 0.3. Here the number of TNOs with diameters between D and D+ dD was taken to be proportional to D^−qdD. In terms of the above-mentioned mass distribution index, s,  

(see, for example, Hughes 1982). Gladman et al. (2001) therefore conclude that s= 2.13 ± 0.10 a result that is to all intents and purposes indistinguishable from the one quoted in this paper, obtained using the absolute magnitudes of TNOs.

Another approach to the problem of ascertaining the distribution index of TNOs is to use the cumulative absolute magnitude data and equation (7) as opposed to the non-cumulative data and equation (6). In the cumulative approach the logarithm of the number of TNOs having magnitudes less than a specific H-value (i.e. the number brighter than that value) is plotted as a function of H, and the gradient of the low-H, linear region of the plot is calculated using a fitting program.

The non-cumulative data shown in Fig. 3 were such that only 162 out of the total of 602 TNOs were in the H < 6.8, d > 290 km range that was deemed to be free of observational selection effects. Using a cumulative approach, these bright TNOs could be analysed in more detail. In the preliminary analysis presented in this paper, it was decided that there were sufficient bright TNOs to be divided into six groups. In an attempt to distinguish between inner Edgeworth–Kuiper Belt TNOs and more distant TNOs, the data set was divided into six different ‘perihelion distance ranges’, each containing similar numbers of TNOs. Needless to say, the TNOs could have been divided up according to orbital inclinations, eccentricities, major axes, etc., but it was thought that perihelion groups might have more relevance when it came to comparing TNOs with comets. Levison & Stern (2001), for example, divided TNOs into resonance and non-resonance groups. The latter contains TNOs with semi-major axes greater than 42.5 au and eccentricities less than 0.2 (thus hopefully avoiding TNOs in 4:5, 3:4, 2:3, 3:5 and 1:2 Neptune mean motion resonances, and the members of the scattered disc). Of the 80 non-resonanceTNOs in Levison & Stern's 2000 October sample, they found that the large TNOs (H < 6.5, diameter bigger than about 340 km) tended to have higher orbital inclinations than the smaller TNOs. They concluded that the low-inclination TNOs were more likely to be primordial. Unfortunately their data set was not large enough for them to investigate absolute magnitude distribution indices.

The results of our analysis are listed in Table 1. No standard deviations are quoted for the log₁₀a values, as the cumulative data points used in the graph plotting routine are non-independent. As is to be expected, the log₁₀a values for the six perihelion range subsets cluster around the single result obtained for the non-cumulative data. The mean of the six values given in Table 1 is log₁₀a= 0.61 ± 0.05, in comparison with the log₁₀a= 0.66 ± 0.06 from the non-cumulative data. Two things are clear from Table 1. First, the knee values, H_k, of the cumulative data curves are very similar. This is taken to indicate that those observational selection criteria that depend on limiting magnitudes are reasonably similar for each perihelion group. We concluded that the knee value separates low-H, large-diameter, linear, ‘complete TNO data’ from high-H, small-diameter data that are progressively affected more and more by incompleteness as H gets larger and larger than the knee value.

Table 1.

The absolute magnitude distribution (i.e. log₁₀a values) for TNOs having perihelion distances in different ranges. These ranges have been chosen to contain similar numbers of TNOs. i.e. about 100 in each. Equation (7) was fitted to the cumulative data. The plots indicated that bright TNOs with absolute magnitudes less the knee value of H_k were such that the log N versus H curve was linear, the gradient being equal to the quoted log₁₀a value. The number of ‘bright’ TNOs fitted in each case is given by N_k.

The second conclusion is that the distribution of log₁₀a values is not random. These log₁₀a values seemingly increase steadily with perihelion distance, over the 26 < q < 47 au range.

3 Conclusions

Two things should happen in the future. As today's mid-sized (2 m) telescopes are used to survey more and more of the ecliptic regions of the outer solar system, the number of known TNOs will increase, even though the depth of the survey does not change. An example of this comes from the work of Trujillo & Brown (2002). They state: ‘It is very likely that there are more big Kuiper Belt Objects like Quaoar. We looked at only 5 per cent of the entire sky before finding Quaoar. So there could be 20 Quaoars out there and we wouldn’t have seen them yet. It is also likely that a few Plutos are out there waiting to be discovered.'

Surveys such as these should increase the N_k-values for perihelion group analyses similar to that shown in Table 1, but should not change the log₁₀a (gradient) values or the H_k (knee) values.

As larger and larger telescopes are used to survey the Edgeworth–Kuiper Belt, the limiting magnitudes of surveys such as that of Millis et al. (2002) will increase, and fainter and small TNOs will be discovered. This should lead to an increase in both the N_k- and the H_k-values for the perihelion group analyses. The accuracy of the resulting log₁₀a values should increase, and, more importantly, one will be able to ascertain whether small TNOs have the same log₁₀a values as large ones.

If loci of constant q are plotted on a TNO semi-major axis versus eccentricity graph, one can see that the vast majority of the 25.88 < q < 38.42 au objects are in the 2:3 mean motion resonance with Neptune. These have been found to have the lowest log₁₀a values. TNOs in the higher perihelion distance groups listed in Table 1 tend to be in the 40 < a < 50 au region, with the eccentricity progressively decreasing as the perihelion distance increases.

The variation of log₁₀a with perihelion distance is certainly not expected to be monotonic. Kenyon & Windhorst (2001) invoke Olbers' paradox. Extrapolating from the large TNO results, they note that unless log₁₀a (a quantity they refer to as α) were less than 0.48 in the 2 μm < d < 2 km region of the TNO size distribution, there would be so many small TNOs in the Edgeworth–Kuiper Belt that there would be a band of reflected light around the ecliptic produced by scattered sunlight.

Let me end by emphasizing three points.

• (i)

  An average log₁₀a= 0.66 ± 0.06 (i.e. s= 2.1 ± 0.1) for the absolute magnitude distribution of TNOs is not unexpected. This is similar to the 0.5–0.75 values for apparent magnitude log₁₀a found by, for example, Gladman et al. (1998), Jewitt, Luu & Trujillo (1998), Luu & Jewitt (1998) and Chiang & Brown (1999).

• (ii)

  The absolute magnitude distribution index, log₁₀a, varies with perihelion distance, being smaller for the TNOs in the 25.88 < q < 38.42 au region (these mainly being objects in the 2:3 mean motion resonance with Neptune) and larger for the TNOs beyond the ν₈ resonance. As more and more TNOs are discovered, we might then be able to tell whether the log₁₀a variability illustrated in Table 1 and Fig. 4 represents a gradual trend, or whether we have specific log₁₀a values for specific perihelion distance regions (and dynamical groups). If the latter is the case, the ‘smooth’ variability exhibited in Fig. 4 is then illusory and is just due to the uncertainties engendered by the limited data set that is available.

• (iii)

  The magnitude distribution index of bright short-period comets (log₁₀a= 0.359 ± 0.009, mass distribution index s= 1.60 ± 0.02), and the magnitude distribution index of the known bright TNOs (log₁₀a= 0.66 ± 0.06, s= 2.1 ± 0.1) are completely different. This could be regarded as a major stumbling block to the commonly held belief that the Edgeworth–Kuiper Belt is the source region of the short-period comets, especially as the general paradigm is that the primordial accretion process leads to an s≈ 1.65 mass distribution index, which subsequently changes because of collisional fragmentation into an s≈ 2.00 distribution (see Daniels & Hughes 1981). If this is true, the ‘parent’ TNOs are expected to have a lower s-value than the ‘off-spring’ comets, and not vice versa as observed. There is, however, one obvious snag with using the difference in distribution indices as a reason to break the family relationship between TNOs and short-period comets. At the present time the median diameters of the known members of these two populations (220 and 2.9 km respectively) differ by a factor of about 75. The fact that we know the mass distribution index of the very large d > 290 km TNOs does not mean that we have any idea as to the mass distribution index of the much small kilometre-sized TNOs.

Figure 4.

The variation of the absolute magnitude distribution index, log₁₀a, of the large d > 290 km TNOs, plotted as a function of perihelion distance (see Table 1).

Acknowledgments

I thank Alan Fitzsimmons for both his encouragement and his most helpful comments.