Mon. Not. R. Astron. Soc. 345, 981–984 (2003)

The absolute magnitude distribution of trans-Neptunian objects

David W. Hughes�Department of Physics and Astronomy, The University, Sheffield S3 7RH

Accepted 2003 July 14. Received 2003 July 10; in original form 2003 January 6

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

Key words: comets: general – Kuiper Belt.

1 I N T RO D U C T I O N

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 diam-eters of 2300 and 1250 km respectively, down to a limit imposed bypresent-day telescopic power. The numbers of bodies detected so far,602 when writing, confirm that the trans-Neptunian proto-cometarybelt proposed by Kuiper (1951) and Edgeworth (1943) has beenfound. This paper investigates whether the known trans-Neptunianobjects (TNOs) have the same absolute magnitude distribution ascomets.

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

m = m0 − 2.5 log

[pRr 2φ(α)

2.25 × 1016 R2�2

], (1)

where m0 is the apparent magnitude of the Sun (−27.1), pR is thegeometric albedo of the TNO, r(km) is the effective radius of theTNO, φ(α) is the phase function [which, at opposition, α = 0◦, issuch that φ(0) = 1], R(au) is the heliocentric distance and �(au) isthe geocentric distance. Equation (1) can be simplified to

m = H + 5 log � + 5 log R − 2.5 log φ(α), (2)

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

and the absolute magnitude H were related by

�E-mail: d.hughes@sheffield.ac.uk

log(

0.25d2 pR

) = 11.642 − 0.4H. (3)

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

log d(m) = 6.60 − 0.2H.

If the TNOs have the average albedo of cometary nuclei, i.e. around0.04, the relationship becomes

log d(m) = 6.821 − 0.2H. (4)

The brighter members of known cometary populations have his-torically been assumed to have an absolute magnitude distributiongiven by

C = baH �H, (5)

where C is the number of comets that have absolute magnitudesin the range H to H + �H . The quantities b and a are usuallytaken to be constants. Notice that, in the cometary context, we arediscussing the absolute magnitude of the integrated dust and gascoma of an active comet, and not the absolute magnitude of thecometary nucleus.

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

N = baH (loge a)−1. (6)

Normally log10 N is plotted as a function of H. Here

log10 N = log10

[b(loge a)−1

] + H log10 a. (7)

The quantity log10 a in equation (7) is referred to as the absolutemagnitude distribution index. Hughes (2001) analysed the absolutemagnitude distribution of long-period comets. These comets havevisited the inner solar system very few times, have undergone verylittle decay, and are expected to have their primordial magnitudedistribution index. This index was found to be log10 a = 0.359 ±0.009. A similar magnitude distribution index was found to applyto those short-period comets with perihelion distances, q, greaterthan 2.0 au (see Hughes 2002). These, like the long-period comets,

C© 2003 RAS

982 D. W. Hughes

have suffered little from decay. [In the past, many populations havebeen also been quantified by the mass distribution index, s. Herethe number of bodies with mass greater than M is taken to be pro-portional to M (1−s). Note that s = 1 + (5/3)log10 a. Comets havethus been found to have a mass distribution index of 1.60 ± 0.02,a value indicative of a population produced by planetesimal accre-tion as opposed to collisional fragmentation (see Daniels & Hughes1981).]

We must stress that the indices given above have been obtained bythe analysis of the number distribution of the absolute magnitudes ofactive comets, and therefore rely on equations such as equation (4)above being strictly valid, i.e. there being a firm relationship betweencomet nucleus size and absolute magnitude of the integrated dust andgas coma of that comet when it is close to the Sun and active. We arefortunately getting to the stage where the Hubble Space Telescopeand some of the largest ground-based telescopes are being used toobserve 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 surfacearea do not have to be made. The number of accurately knowndiameters 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-periodcomets and those short-period comets with q > 2.0 au are all re-lated, and these objects are all essentially primordial, then they areall expected to have the same magnitude and mass distribution in-dices. This is especially true if the TNOs are the reservoir that feedsthe low-inclination Jupiter-family cometary population, as first sug-gested by Fernandez (1980, 1985).

2 T R A N S - N E P T U N I A N O B J E C T S

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

The perihelion distance, aphelion distance and absolute magni-tude distributions of the 602 TNOs listed when writing are shownin Figs 1, 2 and 3. (Note that Pluto and Quaoar were not includedin the list at that time.) We are not going to go into details about theorbits [see Levison & Weissman (1999) for more information andillustrations]. The known TNOs seem to be in three main dynami-cal groupings, the first consisting of bodies that are trapped in a 2:3

Figure 1. The perihelion distance distribution of the 602 TNOs listed onthe web site http://cfa-www.harvard.edu/cfa/ps/lists/TNOs.html.

Figure 2. The aphelion distance distribution of the 602 TNOs listed on theweb site http://cfa-www.harvard.edu/cfa/ps/lists/TNOs.html.

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

mean motion resonance with Neptune, these having orbits similar tothat of Pluto. The second group are in near-circular orbits and havea range of perihelion distances, and the third group are similar tothe second group but have larger eccentricities. (Edgeworth–KuiperBelt formation models predict that many TNOs in the 3:4, 3:5 and1:2 mean motion resonances of Neptune wait to be discovered.)The perihelion distribution of known TNOs shown in Fig. 1 peaks ataround 41 au, and the aphelion distance distribution (Fig. 2) peaks ataround 46 au. For the second and third dynamical groups mentionedabove, these two values should increase in the future as fainter andmore distant TNOs are found. Remember that Neptune and Plutoare at mean solar distances of 30.1 and 39.5 au respectively.

In the present paper we are concerned with the absolute magni-tude distribution, and this is shown for known TNOs in Fig. 3. Ifequation (5) is applicable, the gradient of the histogram in Fig. 3should have a value of log10 a. Fitting equation (5) to the 4.4 < H <

6.8 region of the data indicates that log10 a = 0.66 ± 0.06 (i.e. s =2.1 ± 0.1). This absolute magnitude range contains the largest TNOs

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The absolute magnitude distribution of TNOs 983

(870 > d > 290 km), the assumption being that many of the fainterand smaller H > 6.8, d < 290 km TNOs are yet to be discovered. It isclear that the TNO value of log10 a is significantly different from thecometary value. An s = 2.1 mass distribution index has often beentaken to signify a population produced by collisional fragmentation(see for example Hughes 1994).

Interestingly, Gladman et al. (2001) analysed the apparent (as op-posed to absolute) magnitude distribution of the bright TNOs knownat the time. (As the orbital periodicity of TNOs is above 250 yr, theapparent magnitudes of specific TNOs have hardly changed sincediscovery.) Assuming a power-law heliocentric distance distribu-tion, Gladman et al. (2001) converted their apparent magnitude dis-tribution into a differential TNO size index, quoting a resulting indexof q = 4.4 ± 0.3. Here the number of TNOs with diameters betweenD and D + dD was taken to be proportional to D−q dD. In terms ofthe above-mentioned mass distribution index, s,

q = (3s − 2)

(see, for example, Hughes 1982). Gladman et al. (2001) thereforeconclude that s = 2.13 ± 0.10 a result that is to all intents and pur-poses indistinguishable from the one quoted in this paper, obtainedusing the absolute magnitudes of TNOs.

Another approach to the problem of ascertaining the distributionindex of TNOs is to use the cumulative absolute magnitude data andequation (7) as opposed to the non-cumulative data and equation (6).In the cumulative approach the logarithm of the number of TNOshaving magnitudes less than a specific H-value (i.e. the numberbrighter than that value) is plotted as a function of H, and the gradientof the low-H, linear region of the plot is calculated using a fittingprogram.

The non-cumulative data shown in Fig. 3 were such that only162 out of the total of 602 TNOs were in the H < 6.8, d > 290km range that was deemed to be free of observational selection ef-fects. Using a cumulative approach, these bright TNOs could beanalysed in more detail. In the preliminary analysis presented inthis paper, it was decided that there were sufficient bright TNOsto be divided into six groups. In an attempt to distinguish betweeninner Edgeworth–Kuiper Belt TNOs and more distant TNOs, thedata set was divided into six different ‘perihelion distance ranges’,each containing similar numbers of TNOs. Needless to say, theTNOs could have been divided up according to orbital inclinations,eccentricities, major axes, etc., but it was thought that periheliongroups might have more relevance when it came to comparing TNOswith comets. Levison & Stern (2001), for example, dividedTNOs into resonance and non-resonance groups. The latter containsTNOs with semi-major axes greater than 42.5 au and eccentricitiesless than 0.2 (thus hopefully avoiding TNOs in 4:5, 3:4, 2:3, 3:5and 1:2 Neptune mean motion resonances, and the members of thescattered disc). Of the 80 non-resonanceTNOs in Levison & Stern’s2000 October sample, they found that the large TNOs (H < 6.5,diameter bigger than about 340 km) tended to have higher orbitalinclinations than the smaller TNOs. They concluded that the low-inclination TNOs were more likely to be primordial. Unfortunatelytheir data set was not large enough for them to investigate absolutemagnitude distribution indices.

The results of our analysis are listed in Table 1. No standarddeviations are quoted for the log10 a values, as the cumulative datapoints used in the graph plotting routine are non-independent. As isto be expected, the log10 a values for the six perihelion range subsetscluster around the single result obtained for the non-cumulativedata. The mean of the six values given in Table 1 is log10 a =0.61 ± 0.05, in comparison with the log10 a = 0.66 ± 0.06 from

Table 1. The absolute magnitude distribution (i.e. log10 a values) forTNOs having perihelion distances in different ranges. These rangeshave been chosen to contain similar numbers of TNOs. i.e. about100 in each. Equation (7) was fitted to the cumulative data. The plotsindicated that bright TNOs with absolute magnitudes less the kneevalue of Hk were such that the log N versus H curve was linear,the gradient being equal to the quoted log10 a value. The number of‘bright’ TNOs fitted in each case is given by N k.

Perihelion Knee value log10 a N k

range (au) Hk

25.88–33.66 7.1 ± 0.2 0.49 2833.72–38.42 7.1 ± 0.2 0.53 4438.50–40.55 6.8 ± 0.2 0.61 3140.56–41.64 6.8 ± 0.2 0.58 3241.64–43.07 7.1 ± 0.2 0.71 4943.08–47.17 6.7 ± 0.3 0.76 30

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 criteriathat depend on limiting magnitudes are reasonably similar for eachperihelion 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 byincompleteness as H gets larger and larger than the knee value.

The second conclusion is that the distribution of log10 a values isnot random. These log10 a values seemingly increase steadily withperihelion distance, over the 26 < q < 47 au range.

3 C O N C L U S I O N S

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 regionsof the outer solar system, the number of known TNOs will increase,even though the depth of the survey does not change. An example ofthis comes from the work of Trujillo & Brown (2002). They state:‘It is very likely that there are more big Kuiper Belt Objects likeQuaoar. We looked at only 5 per cent of the entire sky before findingQuaoar. So there could be 20 Quaoars out there and we wouldn’thave seen them yet. It is also likely that a few Plutos are out therewaiting to be discovered.’

Surveys such as these should increase the N k-values for periheliongroup analyses similar to that shown in Table 1, but should notchange the log10 a (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 ofMillis et al. (2002) will increase, and fainter and small TNOs willbe discovered. This should lead to an increase in both the N k- andthe H k-values for the perihelion group analyses. The accuracy ofthe resulting log10 a values should increase, and, more importantly,one will be able to ascertain whether small TNOs have the samelog10 a values as large ones.

If loci of constant q are plotted on a TNO semi-major axis ver-sus eccentricity graph, one can see that the vast majority of the25.88 < q < 38.42 au objects are in the 2:3 mean motion resonancewith Neptune. These have been found to have the lowest log10 avalues. TNOs in the higher perihelion distance groups listed in Ta-ble 1 tend to be in the 40 < a < 50 au region, with the eccentricityprogressively decreasing as the perihelion distance increases.

The variation of log10 a with perihelion distance is certainly notexpected to be monotonic. Kenyon & Windhorst (2001) invoke

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984 D. W. Hughes

Olbers’ paradox. Extrapolating from the large TNO results, theynote that unless log10 a (a quantity they refer to as α) were lessthan 0.48 in the 2 µm < d < 2 km region of the TNO size distribu-tion, there would be so many small TNOs in the Edgeworth–KuiperBelt that there would be a band of reflected light around the eclipticproduced by scattered sunlight.

Let me end by emphasizing three points.

(i) An average log10 a = 0.66 ± 0.06 (i.e. s = 2.1 ± 0.1) for theabsolute magnitude distribution of TNOs is not unexpected. Thisis similar to the 0.5–0.75 values for apparent magnitude log10 afound by, for example, Gladman et al. (1998), Jewitt, Luu & Trujillo(1998), Luu & Jewitt (1998) and Chiang & Brown (1999).

(ii) The absolute magnitude distribution index, log10 a, varieswith perihelion distance, being smaller for the TNOs in the 25.88< q < 38.42 au region (these mainly being objects in the 2:3 meanmotion resonance with Neptune) and larger for the TNOs beyond theν8 resonance. As more and more TNOs are discovered, we mightthen be able to tell whether the log10 a variability illustrated inTable 1 and Fig. 4 represents a gradual trend, or whether we havespecific log10 a values for specific perihelion distance regions (anddynamical groups). If the latter is the case, the ‘smooth’ variabilityexhibited in Fig. 4 is then illusory and is just due to the uncertaintiesengendered by the limited data set that is available.

(iii) The magnitude distribution index of bright short-periodcomets (log10 a = 0.359 ± 0.009, mass distribution index s = 1.60 ±0.02), and the magnitude distribution index of the known brightTNOs (log10 a = 0.66 ± 0.06, s = 2.1 ± 0.1) are completely dif-ferent. This could be regarded as a major stumbling block to thecommonly held belief that the Edgeworth–Kuiper Belt is the sourceregion of the short-period comets, especially as the general paradigmis that the primordial accretion process leads to an s ≈ 1.65 mass dis-tribution index, which subsequently changes because of collisionalfragmentation into an s ≈ 2.00 distribution (see Daniels & Hughes

Figure 4. The variation of the absolute magnitude distribution index, log10

a, of the large d > 290 km TNOs, plotted as a function of perihelion distance(see Table 1).

1981). If this is true, the ‘parent’ TNOs are expected to have a lowers-value than the ‘off-spring’ comets, and not vice versa as observed.There is, however, one obvious snag with using the difference in dis-tribution indices as a reason to break the family relationship betweenTNOs and short-period comets. At the present time the median di-ameters of the known members of these two populations (220 and2.9 km respectively) differ by a factor of about 75. The fact that weknow the mass distribution index of the very large d > 290 km TNOsdoes not mean that we have any idea as to the mass distribution indexof the much small kilometre-sized TNOs.

AC K N OW L E D G M E N T S

I thank Alan Fitzsimmons for both his encouragement and his mosthelpful comments.

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This paper has been typeset from a TEX/LATEX file prepared by the author.

C© 2003 RAS, MNRAS 345, 981–984