The Shapes and Spins of Kuiper Belt Objects
Lacerda, Pedro
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Lacerda, P. (2005, February 17). The Shapes and Spins of Kuiper Belt Objects. Retrieved
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CHAPTER
3
The sha p e d istrib u tio n
ABSTRAC T
If we assu me that the period ic brig htn ess variation s in a K u iper B elt lig htc u rve are d etermin ed on ly by their aspheric al shapes an d the observ-in g g eometry (n o spobserv-in rate bias is c on sid ered ), the frac tion of d etec table K u iper B elt lig htc u rves an d the lig htc u rve amplitu d e d istribu tion c an be u sed to c on strain the shapes of K u iper B elt objec ts. T he resu lts in d ic ate that most K u iper B elt objec ts (∼ 8 5 % ) have shapes that are c lose to spher-ic al (a/b ≤ 1.5 ), bu t there is a small bu t sig n ifi c an t frac tion (∼ 12 % ) pos-sessin g hig hly aspheric al shapes (a/b ≥ 1.7 ). T he d istribu tion c an n ot be well fi tted by a g au ssian an d is mu ch better approx imated by a power law.
36 The sha p e d istrib u tio n
3.1
In tr o d u c tio n
S
incetheir d isc overy in 1 992, the K u iper B elt objec ts (K B O s) have attrac ted a g reat d eal of in terest in plan etary astron omy bec au se of the in formation they mig ht c on tain . Thou g ht to be a relic from the orig in al protoplan etary d isk , they are ex pec ted to still bear sig n atu res of their orig in an d evolu tion . In partic u lar, they are believed to be mu ch less evolved than other k n own solar system objec ts, an d thu s mig ht show plan etary formation at an early stag e.A lthou g h it has been a d ec ad e sin c e their d isc overy, n ot mu ch is k n own abou t the K B O s physic al properties, main ly bec au se most are too fain t (red mag n i-tu d e mR ≥20) for d etailed stu d ies. M ost of the ex istin g d ata are broad ban d photometry, with a few low-resolu tion optic al an d n ear-IR spec tra. B road ban d photometry in d ic ates that the K B O s possess d iverse c olors, ran g in g from n eu -tral to very red (L u u & J ewitt 1 996; Teg ler & R oman ishin 2000; J ewitt & L u u 2001 ). The low-resolu tion K B O spec tra are u su ally featu reless, althou g h a few show weak 2 µm water ic e absorption (B rown , C ru ik shan k & Pen d leton 1 999; J ewitt & L u u 2001 ). S ome broad ban d photometric d ata have been obtain ed for the pu rpose of stu d yin g K B O rotation al properties, an d althou g h reliable lig htc u rves are still sparse, the sample is su ffi c ien t for d etailed an alysis. In this paper we c ollec t d ata from reliable lig htc u rves an d ex amin e their implic ation s for the shape d istribu tion of K B O s.
3.2
O b se r v a tio n s
The S hapes and S pins of K B O s 37
Ta b le 3 .1 –KBO rotational p rop erties
KBO P [hr] ∆m Reference
KBOs lig h tc u r ves c o n sid er ed to h a ve∆m < 0.15 m a g
1993 S C – 0.04 RT99, D M cG 97 1994 TB – - S J 02 1996 G Q 21 – < 0.10 S J 02 1996 TL 66 – 0.06 RT99, L J 98 1996 TP 66 – 0.12 RT99, C B99 1997 C S 29 – < 0.08 S J 02 1998 H K151 – < 0.15 S J 02 1998 VG 4 4 – < 0.10 S J 02 (C haos) 1998 W H 24 – < 0.10 S J 02 1999 D E 9 – < 0.10 S J 02 (4 7171) 1999 TC 3 6 – < 0.10 S J 03 (H uya) 2000 E B173 – < 0.06 S J 02 2000 Y W 13 4 – < 0.10 S J 03 2001 C Z 3 1 – < 0.20 S J 02 2001 F P 185 – < 0.10 S J 03 2001 F Z 173 – < 0.06 S J 02 2001 KD 77 – < 0.10 S J 03 (28978) Ixion 2001 KX 76 – < 0.10 S J 03 ,O01 2001 Q F 298 – < 0.10 S J 03 (4 23 01) 2001 U R163 – < 0.10 S J 03 (4 23 55) 2002 C R4 6 – < 0.10 S J 03 (5563 6) 2002 TX 3 00 16.24 ± 0.08 0.08 ± 0.02 S J 03 (5563 7) 2002 U X 25 – < 0.10 S J 03 (5563 8) 2002 VE 95 – < 0.10 S J 03 KBO lig h tc u r ves c o n sid er ed to h a ve∆m ≥ 0.15 m a g
1995 Q Y 9 0.60 S J 02, RT99 (24 83 5) 1995 S M 55 8.08 ± 0.03 0.19 ± 0.05 S J 03 1996 TO66 – 0.26 ± 0.03 S J 03 , H 00 1998 S M 165 0.4 5 S J 02, R01 1998 BU 4 8 9.8 ± 0.1 0.68 ± 0.04 S J 02 12.6 ± 0.1 1999 KR16 11.858 ± 0.002 0.18 ± 0.04 S J 02 11.680 ± 0.002 2000 G N 171 8.3 29 ± 0.005 0.61 ± 0.03 S J 02 (Varuna) 2000 W R106 6.3 4 0.4 2 ± 0.03 S J 02 2003 A Z 84 13 .4 2 ± 0.05 0.14 ± 0.03 S J 03
C B99 = C ollander-Brown et a l. 1999, D M cG 97 = D avies, M cBride & G reen 1997, H 00 = H ainaut et a l. 2000, L J 98 = L uu & J ewitt 1998, O01 = Ortiz et a l. 2001, R01
38 The shape distribution lightcurves is then:
f (∆m ≥ 0.15) = 9/33 = 27% . (3.1) All 33 KBOs and their lightcurve parameters are listed in Table 1; the 9 KBOs considered in this work as having periodic lightcurves are clustered at the bottom of the Table. Assuming that the lightcurves are modulated by aspherical shapes and are therefore double-peaked, the periods range from 6 to 12 hrs.
3.3
D iscussion
Lacerda & Luu (2003) show that the fraction of detectable KBO lightcurves can be used to infer these objects shape distribution, if certain (reasonable) assumptions are made. The assumptions are
1. A spheric ity . The lightcurve modulations are assumed to arise from an aspherical shape, taken to be a triaxial ellipsoid with axes a ≥ b ≥ c. The minimum detectable lightcurve amplitude ∆mmin must be as large as the photometric error, or it will not be detected.
2. Observ ation g eom etry . The angle between the KBO spin axis and the line of sight – the aspect angle θ – should also be large enough to give rise to a detectable lightcurve amplitude, i.e., larger than ∆mmin.
Here we adopt SJ02’s photometric error of 0.15 mag, i.e., ∆mmin = 0.15. If a lightcurve does not show periodic modulations, it is assumed that this is because neither the asphericity nor the observation geometry criterion is satisfied. N o spin rate bias is considered here.
W ith these assumptions, the probability p of detecting a lightcurve can be written as (Lacerda & Luu 2003)
p(∆m > ∆mmin) = Z 1 0 Z ∞ 1 Ψ(˜a, ˜c) f (˜a) g(˜c) d˜a d˜c. (3 .2 ) wh ere, for th e sak e of bein g c on c ise, we d efi n e ˜a = a/ b, ˜c = c/ b, an d Ψ(˜a, ˜c) is th e probability of d etec tin g brig h tn ess variation s from a g iven ellipsoid of ax is ratio (˜a, ˜c). Ψ(˜a, ˜c) is g iven by Ψ(˜a, ˜c) = s ˜ c2(˜a2 − K) ˜ c2(˜a2 − K) + ˜a2(K − 1 ), (3 .3 ) wh ere K = 1 00.8 ∆ mmin
. T h e rig h t h an d sid e of E q n . (3 .2 ) represen ts th e proba-bility of observin g a g iven K B O with ax is ratios between (˜a, ˜c) an d (˜a+d˜a, ˜c+d˜c) at a larg e en ou g h aspec t an g le, in teg rated over all possible ax is ratios. For m od -erately elon g ated ellipsoid s (sm all ˜a), th e fu n c tion Ψ(˜a, ˜c) is alm ost in d epen d en t of ˜c. If we fu rth er assu m e ˜c ≈ 1 , th en g(˜c) is ≈ 1 , an d E q n . (3 .2 ) bec om es
p(∆m > ∆mm in) ≈ Z ∞
1
The S ha p es a n d S p in s o f K B O s 39 0 0.1 0.2 0.3 0.4 0.5 0.6 s ig m a 1 1.1 1.2 1.3 1.4 m e a n 0.09 0.13 0.19 0.27 0.37 0.46 0.55
Figure 3 .1 –The p ro bability p as a fu nc tio n o f the m ean µ and stand ard d eviatio n σ o f a
g au ssian ˜a d istribu tio n. The thic k blac k line rep resents all µ −σ p airs that g ive rise to p = 0.27 .
The shad ed areas im m ed iately ad jac ent to the line rep resent the µ and σ valu es within the 1σ lim its, the nex t shad ed areas o u tward rep resent the 2σ lim its, and the o u term o st shad ed area the 3σ lim its. The nu m ber to the left o f each bo u nd ary line ind ic ates the p valu e c o rresp o nd ing to that line. The c irc le with the c ro ss m arks the best-fi t µ − σ: µ = 1.00, σ = 0.24 .
From Eqn. (3.1), p(∆m > ∆mmin) = 0.27 . S ince the data did not sample the entire KBO population, there are necessarily errors associated with p. The 1-, 2- and 3σ error bars on p can be calculated based on the C lopper-P earson confidence limits (L acerda & L uu 2003):
1σ p = 0.27+ 0.10 −0.08
2σ p = 0.27+ 0.19−0.14 (3.5 ) 3σ p = 0.27+ 0.28−0.18
W ith p known, the problem then becomes inverting Eqn. (3.4) to determine the shape distribution f (˜a). This can be done if f (˜a) is assumed to be a simple analytical function.
3.3.1 G a u ssia n d istr ib u tio n
40 The shape distrib u tion 1 2 3 4 5 ab 0.2 0.4 0.6 0.8 1 c u m u la ti v e fr a c ti o n
Figure 3.2 –Cumulative fractions of K B O s as a function of a/b = ˜a. The solid line
cor-responds to the best-fit gaussian (µ = 1.00, σ = 0.24), the dashed line a gaussian with µ = 1.22, σ = 0.24. The data are from Table 1, with vertical error bars calculated from Poisson statistics. The horiz ontal error bars are calculated from 1σ deviation from the average
aspect angle of 6 0o
, assuming that the aspect angle is uniformly distributed in sin θ. N ote: our x-axis is a/b, which is the inverse of S J 02’s x-axis, b/a.
by p’s 1-, 2-, and 3σ error bars. The entire shaded regions thus represent all possible combinations of µ − σ that are consistent with p = 0.27+0.28−0.18.
We can constrain µ − σ further by making use of the observed axis ratios ˜a. U sing the data from Table 1, for each detected lightcurve, the observed ∆m is converted to the axis ratio ˜a by using the relation (Lacerda & Luu 2003)
∆m = 2.5 log s ˜ a2cos2θ + ˜a2c˜2sin2 θ ˜ a2cos2θ + ˜c2sin2θ , (3.6 ) and by assuming an average aspect angle θ = 6 0o. (θ = 6 0o is the average angle if θ is distributed uniformly in sin θ). The observed cumulative fractions of KBOs, as a function of ˜a, are plotted in Fig. 3.2. These can then be compared with the cumulative fractions predicted by each allowed µ − σ pair to yield the best-fit gaussian. A grid search is performed through all possible µ − σ pairs allowed by Eqn. (3.5); using χ2as the comparison criterion, the best-fit gaussian is given by
The Shapes and Spins of KBOs 41 1 2 3 4 5 ab 0.2 0.4 0.6 0.8 1 c u m u la ti v e fr a c ti o n
Figure 3.3 –Cumulative fractions of KBOs as a function of a/b = ˜a. Same as F ig. 3.2, but
this time with the dotted line representing a gaussian with µ = 1.00, σ = 0.11, the dashed line a gaussian with µ = 1.00, σ = 0.42.
µ since µ represents the mean ˜a, defined to be ≥ 1. The best-fit µ − σ is also marked in Fig. 3.1.
The goodness of the fit is shown graphically in Fig. 3.2 where we plot the observed cumulative fractions of KBOs, as a function of ˜a = a/b, with the cumulative fractions predicted by the best-fit gaussian (Eqn. 3.7). The fit is good for ˜a ≤ 1.5 but poor at larger ˜a’s. The Figure also shows that if we increase µ to µ = 1.22 (1 standard deviation away from the best-fit µ), the theoretical curve comes closer to fitting the larger ˜a’s, but misses practically all the data points. [N ote: our x-axis is ˜a = a/b, which is the inverse of SJ 02’s x-axis, b/a].
In Fig. 3.3 we try fitting the data with gaussians of diff erent widths (σ = 0.11 and σ = 0.42, both being 1 standard deviation away from the best-fit σ), while keeping µ fixed at µ = 1.00. The fit off ered by σ = 0.11 is much poorer than those seen in Fig. 3.2; σ = 0.42 comes closer to fitting all the data, but still misses all the data points.
42 The shape distribution
KBOs have axis ratios a/b ≥ 1.72 (larger than 3σ from the mean), while the data indicate that ∼ 12% have axis ratios in this range. Increasing µ to µ = 1.22 (dashed line in Fig. 3.2) reduces some of this skewness but does not significantly improve the fit at larger a/b’s.
Keeping µ fixed and decreasing σ to σ = 0.11 (dotted line in Fig. 3.3) worsens the fit, as expected. Keeping µ fixed and increasing to σ = 0.42 (dashed line in Fig. 3.3) arguably produces the best fit yet (as judged by eye) since it tries to fit all the data points and does so equally well for all of them (or equally badly, depending on one’s point of view). In short, none of the gaussians presented in Figs. 3.2 and 3.3 offers a good fit to the data. The conclusion to draw from the Figures is that the KBO shape distribution is not well approximated by a gaussian.
This is because the KBO shape distribution has two characteristics that can-not be met simultaneously by a standard gaussian: (1) a large fraction (∼ 85%) has shapes that are close to spherical (a/b ≤ 1.5), yet (2) there is a significant tail to the distribution (∼ 12%) that has highly aspherical shapes (a/b ≥ 1.7). In other words, most KBOs are nearly spherical, but a signicant fraction is not. We note that, using a smaller data set, SJ02 came to the conclusion that a broad gaussian was needed to fit their available data. This is roughly consistent with our result here. With the benefit of a larger data sample, and the additional constraint from the detection probability p, we are able to improve SJ02’s con-clusion: the best description of the shape distribution is actually more like a moderately narrow peak with a long tail.
3.3.2 Powe r law distribution
Considering how poorly a gaussian fits the shape distribution, we try approxi-mating f (˜a) with a power law, f (˜a) ∝ ˜a−q. The solution is shown graphically in Fig. 3.4, where the thick horiz ontal line represents p = 0.27, and the solid black curve represents the detection probability p as a function of q:
p(˜a)d˜a = ˜a−qd˜a. (3.8) In Eqn. (3.8), p(˜a)d˜a is the fraction of a KBO with axis ratios in the range ˜a to ˜
a + d˜a. The probability is normaliz ed so that the integral of p(˜a) from ˜a = 1 to ˜
a = 5 is equal to 1. The shaded areas represent all possible values of p within its 1-, 2-, and 3σ error bars, so the allowed values of q are those that lie within these shaded areas. We note that the horiz ontal line and the curved line intersect at q = 6.7.
The Shapes and Spins of KBOs 43 -5 0 5 10 15 20 q 0 0.2 0.4 0.6 0.8 d e te c ti o n p ro b a b ili ty
Figure 3.4 –The thick horizontal line represents all values of q that that give rise to p = 0.27.
The shaded areas immediately adjacent to the line represent the p’s within the 1σ limits, the next shaded areas outward the 2σ limits, and the outermost shaded area the 3σ limits. The solid curve represents p as a function of the exponent q (E q n. 3.8 ). The intersections of the shaded areas and the curve satisfy both E q ns. (3.2) and (3.8 ). The vertical line marks q = 6.4.
1 2 3 4 5 ab 0 .2 0 .4 0 .6 0 .8 1 c u m u la ti v e fr a c ti o n
Figure 3 .5 –Cumulativ e frac tion s of K B O s as a fun c tion of ˜a. S ame as F ig . 3 .2 , b ut th is
44 The sha p e d istrib u tio n 1 1.5 2 2.5 3 ab 0.5 1 1.5 2 2.5 3 3.5 4 fH a b L
Figure 3.6 –Distribution of axis ratio ˜a. The dashed line is the q = 6.4 power law (from E q n.
3.9 ), the solid line a gaussian with µ = 1.00, σ = 0.24 (from E q n. 3.7). f (˜a) is normaliz ed to
1 between ˜a= 1 and ˜a= 5.
perform ed throu g h all possible valu es of q; u sin g χ2as the com parison criterion , we obtain the best-fi t q:
q = 6.4 ± 1.4 (1σ error bars). (3 .9 ) It is reassu rin g that χ2 statistics yield q = 6.4 as the best fi t, as this is very close to the q = 6.7 fou n d in depen den tly by the L acerda & L u u m ethod. T he fi ts off ered by q = 5.0, 6.4 an d 7.8 are shown in F ig . 3 .5. It can be seen that the power laws g en erally fi t the data better than the g au ssian s. T he shape distribu tion f (˜a) as a g au ssian an d a power law is shown in F ig . 3 .6.
The S hapes and S pins of K B O s 45 KBO shapes and spin rates (high spin rate → large ∆m). The data sample is as yet too small to confirm such a trend (e.g., see Fig. 13 of SJ02).
3.4
S u m m a ry
We have applied the method described in Lacerda & Luu (2003) to the available KBO lightcurve data to constrain the KBO shape distribution. The method as-sumes that the detectability of KBO lightcurves depends only on the KBO shape and the observing geometry; it does not take into account any spin freq uency effect (e.g., the bias against very slow spinners). The results can be summarized as follows:
1. With 9 out of 33 reliable KBO lightcurves showing periodic brightness variations, the fraction of detectable KBO lightcurves is f (∆m ≥ 0.15) = 0.27. This implies that the probability of detecting a KBO lightcurve is p = 0.27+0.28
−0.1 8 (3σ error bars).
2. The KBO shape distribution has a steep peak at small axis ratios and drops off q uickly to form a long tail: most of the distribution (∼ 85% ) has shapes that are close to spherical (a/b ≤ 1.5), yet (2) there is also a significant fraction (∼ 12% ) that has highly aspherical shapes (a/b ≥ 1.7). 3. Fitting the KBO a/b distribution with a gaussian yields the best-fit mean µ = 1.00(+ 0.22) and standard deviation σ = 0.24+0.1 8−0.1 3 (1σ error bars). H owever, this gaussian is strongly skewed toward small axis ratios (a/b ≤ 1.5), and offers a bad fit for larger axis ratios. Increasing the standard deviation reduces the skewness, but then all data points are fitted eq ually poorly.
4. The KBO a/b distribution is better fitted with power law distributions of the form f (a/b) ∼ (a/b)−q, with the best-fit exponent q = 6.4 ± 1.4 (1σ error bars).
A c k n o w le d g m e n ts
46 The shape distribution
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