The Shapes and Spins of Kuiper Belt Objects Lacerda, Pedro

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

2

Detectability o f

K B O lig h tcu rv es

ABSTRAC T

We p resen t a statistic al stu dy of th e detec tability of lig h tc u rves of K u ip er B elt objec ts (K B O s). S om e K u ip er B elt objec ts disp lay lig h tc u rves th at ap p ear ” fl at” , i.e., th ere are n o sig n ifi c an t brig h tn ess variation s with in th e p h otom etric u n c ertain ties. U n der th e assu m p tion th at K B O lig h tc u rves are m ain ly du e to sh ap e, th e lack of brig h tn ess variation s m ay be du e to (1 ) th e objec ts h ave very n early sp h eric al sh ap es, or (2 ) th eir rotation ax es c oin c ide with th e lin e of sig h t. We in vestig ate th e relative im p ortan c e of th ese two eff ec ts an d relate it to th e observed frac tion of “ fl at” lig h tc u rves. T h is stu dy su g g ests th at th e frac tion of K B O s with detec table brig h tn ess variation s m ay p rovide c lu es abou t th e sh ap e distribu tion of th ese objec ts. A lth ou g h th e c u rren t database of rotation al p rop erties of K B O s is still in su ffi c ien t to draw an y statistic ally m ean in g fu l c on c lu sion s, we ex p ec t th at, with a larg er dataset, th is m eth od will p rovide a u sefu l test for c an didate K B O sh ap e distribu tion s.

22 Detec ta b ility o f K B O lig h tc u rv es

2.1

In tr o d u c tio n

T

he K u iper B elt holds a larg e popu lation of small objects which are thou g ht to be remnants of the protosolar nebu la (J ewitt & L u u 1 9 9 3 ). The B elt is also the most lik ely orig in of other ou ter solar system objects su ch as P lu to-C haron, Triton, and the short-period comets; its stu dy shou ld therefore provide clu es to the u nderstanding of the processes that shaped ou r solar system. M ore than 65 0 K u iper B elt objects (K B O s) are k nown to date and a total of abou t 1 05

objects larg er than 5 0 k m are thou g ht to orbit the S u n beyond N eptu ne (J ewitt & L u u 2000).

O ne of the most fu ndamental ways to stu dy physical properties of K B O s is throu g h their lig htcu rves. L ig htcu rves show periodic brig htness variations du e to rotation, since, as the K B O rotates in space, its cross-section as projected in the plane of the sk y will vary du e to its non-spherical shape, resu lting in periodic brig htness variations (see Fig . 2.1 ). A well-sampled lig htcu rve will thu s yield the rotation period of the K B O , and the lig htcu rve amplitu de has information on the K B O ’s shape. This techniq u e is commonly u sed in planetary astronomy, and has been developed ex tensively for the pu rpose of determining the shapes, internal density stru ctu res, rotational states, and su rface properties of atmosphereless bodies. These properties in tu rn provide clu es to their formation and collisional environment.

Althou g h lig htcu rves stu dies have been carried ou t rou tinely for asteroids and planetary satellites, the nu mber of K B O lig htcu rves is still meag er, with few of su ffi cient q u ality for analysis (see Table 2.1 ). This is du e to the fact that most K B O s are faint objects, with apparent red mag nitu de of mR∼ 23 (Tru jillo

et a l. 2001 ), rendering it very diffi cu lt to detect small amplitu de chang es in their brig htness. O ne of the few hig h q u ality lig htcu rves is that of (20000) Varu na, which shows an amplitu de of ∆m = 0.4 2 ± 0.02 mag and a period of P_{ro t= 6.3 4 4 2±0.0002 hrs (J ewitt & S heppard 2002). O nly recently have su rveys} started to yield sig nifi cant nu mbers of K B O s brig ht enou g h for detailed stu dies (J ewitt et a l. 1 9 9 8 ).

Another diffi cu lty associated with the measu rement of the amplitu de of a lig htcu rve is the one of determining the period of the variation. If no periodicity is apparent in the data, any small variations in the brig htness of an object mu st be du e to noise. Fu rthermore, a precise measu rement of the amplitu de of the lig htcu rve req u ires a complete coverag e of the rotational phase. Therefore, any conclu sion based on amplitu des of lig htcu rves mu st assu me that their periods have been determined and confi rmed by well sampled phase plots of the data.

The S hap es an d S p in s of KBOs 23 ∆m mag time a c b c ∆m = 2.5log(πab/πcb) = 2.5log(a/b)

Figure 2 .1 –The lig htcurv e of an ellip soidal K B O ob serv ed at asp ect ang le θ = π/2. C ross-sections and lig htcurv e are rep resented for one full rotation of the K B O . The am p litude, ∆m, of the lig htcurv e is determ ined for this p articular case. S ee tex t for the g eneral ex p ression.

Ta b le 2 .1 –K B O s w ith m easured lig htcurv es. N am e C lassa _Hb ∆mc Pd So u rc ee [m a g ] [m a g ] [h rs] 1 9 9 3 SC C 6 .9 <0 .0 4 RT9 9 1 9 9 4 TB P 7 .1 0 .3 6 .5 RT9 9 1 9 9 6 TL66 S 5 .4 <0 .0 6 RT9 9 1 9 9 6 TP66 P 6 .8 <0 .1 2 RT9 9 1 9 9 4 VK 8 C 7 .0 0 .4 2 9 .0 RT9 9 1 9 9 6 TO 66 C 4 .5 0 .1 6 .2 5 H a 0 0 (2 0 0 0 0 ) Va ru n a C 3 .7 0 .4 2 6 .3 4 J S0 2 1 9 9 5 QY9 P 7 .5 0 .6 7 .0 RT9 9 1 9 9 6 RQ2 0 C 7 .0 — RT9 9 1 9 9 6 TS66 C 6 .4 <0 .1 6 RT9 9 1 9 9 6 TQ66 C 7 .0 <0 .2 2 RT9 9 1 9 9 7 CS2 9 C 5 .2 <0 .2 RT9 9 1 9 9 9 TD 1 0 S 8 .8 0 .6 8 5 .8 Co 0 0 a

dy n a m ic a l c la ss (C= c la ssic a l K B O , P= plu tin o , S= sc a ttered K B O )

b a b so lu te m a g n itu de c lig h tc u rv e a m plitu de d spin perio d e

RT9 9 = Ro m a n ish in & Teg ler (1 9 9 9 ), H a 0 0 = H a in a u t (2 0 0 0 ), Co 0 0 = Co n so lm a g n o et a l. (2 0 0 0 ), J S0 2 = J ew itt & Sh eppa rd (2 0 0 2 )

24 Detectab ility o f K B O lig h tcu rv es

Symbo l D esc rip tio n

a ≥ b ≥ c a x es o f ellip so id a l K B O ˜

a ≥˜b ≥˜c _{n o rma liz ed a x es o f K B O (˜b = 1 )} θ a sp ec t a n g le

∆mmin min imu m d etec ta ble lig htc u rv e a mp litu d e

θmin a sp ec t a n g le a t w hich ∆m = ∆mmin

K _{1 0}0.8∆ m_min

Ta b le 2 .2 –_{Used symbols} and notation.

2.2

D e fi n itio n s a n d A ssu m p tio n s

The observed brightness variations in KBO lightcurves can be due to: · eclipsing binary KBOs

· surface albedo variations · irregular shape

In general the brightness variations will arise from some combination of these three factors, but the preponderance of each eff ect among KBOs is still not known. In the following calculations we exclude the fi rst two factors and assume that shape is the sole origin of KBO brightness variations. We further assume that KBO shapes can be approximated by triaxial ellipsoids, and thus expect a typical KBO lightcurve to show a set of 2 maxima and 2 minima for each full rotation (see Fig. 2.1 ). Table 2.2 summariz es the used symbols and notations. The listed quantities are defi ned in the text.

The detailed assumptions of our model are as follows:

1 . T h e K B O sh ap e is a triax ial ellip so id . This is the shape assumed by a rotating body in hydrostatic equilibrium (Chandrasekhar 1 9 6 9 ). There are reasons to believe that KBOs might have a “rubble pile” structure (Farinella et al. 1 9 8 1 ), justifying the approximation even further.

2. T h e albed o is co n stan t o v er su rface. A lthough albedo variegation can in principle explain any given lightcurve (R ussell 1 9 0 6 ), the large scale bright-ness variations are generally attributed to the object’s irregular shape (Burns & Tedesco 1 9 7 9 ).

3. All ax is o rien tatio n s are eq u ally p ro bab le. G iven that we have no knowledge of preferred spin vector orientation, this is the most reasonable a p rio ri assumption.

4. T h e K B O is in a state o f sim p le ro tatio n aro u n d th e sh o rtest ax is (th e ax is o f m ax im u m m o m en t o f in ertia). This is likely since the damping timescale of a complex rotation (e.g., precession), ∼ 1 03

yr, (Burns & S afronov 1 9 7 3), (H arris 1 9 9 4) is smaller than the estimated time between collisions (1 07

– 1 01 1

The S hapes and S pins of KBOs 25 5. The KBO is observed at zero phase angle (α = 0). It has been shown from asteroid data that lightcurve amplitudes seem to increase linearly with phase angle,

A(θ, 0) = A(θ, α)/(1 + mα) ,

where θ is the aspect angle, α is the phase angle and m is a coeffi cient which depends on surface composition. The aspect angle is defined as the angle between the line of sight and the spin axis of the KBO (see Fig. 2.2a), and the phase angle is the Sun-object-E arth angle. The mean values of m found for different asteroid classes are m(S) = 0.030, m(C) = 0.015, m(M) = 0.013, where S, C, and M are asteroid classes (see (Michalowsky 1993). Since KBO are distant objects the phase angle will always be small. E ven allowing m to be one order of magnitude higher than that of asteroids the increase in the lightcurve amplitude will not exceed 1% .

6. The brightness of the KBO is proportional to its cross-section area (geo-metric scattering law ). This is a good approximation for KBOs because (1) most KBOs are too small to hold an atmosphere, and (2) the fact that they are observed at very small phase angles reduces the influence of scattering on the lightcurve amplitude (Magnusson 1989).

The KBOs will be represented by triaxial ellipsoids of axes a ≥ b ≥ c rotating around the short axis c (see Fig. 2.2b). In order to avoid any scaling factors we normalize all axes by b, thus obtaining a new set of parameters ˜a, ˜b and ˜c given by

˜

a = a/b , ˜b = 1 , ˜c = c/b . (2.1) As defined, ˜a and ˜c can assume values 1 ≤ ˜a < ∞ and 0 < ˜c ≤ 1. N ote that the parameters ˜a and ˜c are dimensionless.

The orientation of the spin axis of the KBO relative to the line of sight will be defined in spherical coordinates (θ, φ), with the line of sight (oriented from the object to the observer) being the z-axis, or polar axis, and the angle θ being the polar angle (see Fig. 2.2a). The solution is independent of the azimuthal angle φ, which would be measured in the plane perpendicular to the line of sight, between an arbitrary direction and the projection of the spin axis on the same plane. The observation geometry is parameterized by the aspect angle, which in this coordinate system corresponds to θ.

As the object rotates, its cross-section area S will vary periodically between Smax and Smin (see Fig. 2.1). These areas are simply a function of a, b, c and

26 Detectability of KBO lightcurves

φ

θ

θ

min

l

ine

of

sight

c

a

b

s

a)

b)

s

Figure 2.2 – a) A spherical coordinate system is used to represent the observing geometry. The line of sight (oriented from the object to the observer) is the polar ax is and the az imuthal ax is is arbitrary in the plane orthogonal to the polar ax is. θ and φ are the spherical angular coordinates of the spin ax is ~s. In this coordinate system the aspect angle is given by θ. The “ non-detectability” cone, with semi-vertical angle θmin, is represented in grey. If the

spin ax is lies within this cone the brightness variations due to changing cross-section will be smaller than photometric errors, rendering it impossible to detect brightness variations. b) The picture represents an ellipsoidal KBO with ax es a ≥ b ≥ c.

from the quantities ˜a, ˜c and θ and is given by ∆m = 2.5 log s ˜ a2_{c os}2_{θ + ˜a}2_c˜2_sin2_θ ˜ a2_{c os}2_{θ + ˜c}2_sin2_θ . (2.2)

2.3

“ F la t” L ig h tc u rv e s

It is c lear from E q n . (2.2) that u n d er c ertain c on d ition s, ∆m will be z ero, i.e., the K B O will ex hibit a fl at lig htc u rve. These sp ec ial c on d ition s in volve the shap e of the objec t an d the observation g eom etry , an d are d esc ribed q u an titatively below. Tak in g in to ac c ou n t p hotom etric error bars will brin g this “ fl atn ess” threshold to a fi n ite valu e, ∆mm in, a m in im u m d etec table am p litu d e below which brig htn ess

variation c an n ot be asc ertain ed .

The two fac tors that in fl u en c e the am p litu d e of a K B O lig htc u rve are: 1. S p h e r ic ity For a g iven ellip soid al K B O of ax es ratios ˜a an d ˜c the lig htc u rve am p litu d e will be larg est when θ = π/2 an d sm allest when θ = 0 or π. At θ = π/2, E q n . (2.2) bec om es

∆m = 2.5 log ˜a . (2.3 ) E ven at θ = π/2, havin g a m in im u m d etec table am p litu d e, ∆mm in, p u ts c on

The S ha p es a n d S p in s o f K B O s 27

Figure 2 .3 –Illustration of a rotating e llipsoid at diff e re nt aspe c t ang le s. A q uarte r of a full rotation is re pre se nte d. R otational ph ase of e llipsoid inc re asing from top to bottom and θ de c re asing from le ft to rig h t. T is th e pe riod of rotation. Ax e s ratios are ˜a= 1.2 and ˜c= 0.9 .

This constraint is thus ˜

a < 1 00.4 ∆ m_min

⇒“flat” lightcurve . (2.4 ) 2. O b serv a tio n g eo m etryIf the rotation axis is nearly aligned with the line of sight, i.e., if the aspect angle is suffi ciently small, the object’s projected cross-section will hardly change with rotation, yielding no detectable brightness variations (see Fig. 2.3). The finite accuracy of the photometry defines a min-imum aspect angle, θmin, within which the lightcurve will appear flat within

the uncertainties. This angle rotated around the line of sight generates the “non-detectability cone” (see Fig. 2.2a), with the solid angle

28 Detec tab ility of KBO lig htc u rv es

Any aspect angle θ which satisfies θ < θmin falls within the “non-detectability

cone” and results in a non-detectable lightcurve amplitude. Therefore, the prob-ability that the lightcurve will be flat due to observing geometry is

p˜a,˜c(non-detection) = 2 × Ω(θ min)

4π = 1 − cos θmin (2.6 ) p˜a,˜c(detection) = cos θmin. (2.7)

The factor of 2 accounts for the fact that the axis might be pointing towards or away from the observer and still give rise to the same observations, and the 4π in the denominator represents all possible axis orientations.

From Eqn. (2.2) we can write cos θmin as a function of ˜a and ˜c,

cos θmin= Ψ (˜a, ˜c) = s ˜ c2_(˜a2 − K) ˜ c2_(˜a2 − K) + ˜a2_{(K − 1)}, (2.8)

where K = 100.8 ∆mmin_{. The function Ψ (˜a, ˜c), represented in Fig. 2.4, is the} probability of detecting brightness variation from a given ellipsoid of axes ratios (˜a, ˜c). It is a geometry weighting function. For ˜a in [1,√K] we have Ψ (˜a, ˜c) = 0 by definition, since in this case the KBO satisfies Eqn. (2.4) and its lightcurve amplitude will not be detected irrespective of the aspect angle. It is clear from Fig. 2.4 that it is more likely to detect brightness variation from an elongated body.

2.4

D etectab ility o f Lightcurves

In order to generate a “non-flat” lightcurve, the KBO has to satisfy both the shape and observation geometry conditions. M athematically this means that the probability of detecting brightness variation from a KBO is a function of the probabilities of the KBO satisfying both the sphericity and observing geometry conditions.

We will assume that it is possible to represent the shape distribution of KBOs by two independent probability density functions, f (˜a) and g(˜c), defined as

The Shapes and Spins of KBOs 29

Figure 2.4 – The function Ψ (˜a,c) (E qn. 2.8 ).˜ This plot assumes photometric errors ∆mmin= 0.15 mag. The detection probability is z ero w hen ˜a <100.4 ∆ mmin≈ 1.15 .

∆mmin), where both the shape and observation geometry constraints are taken

into account, p(∆m > ∆mmin) = Z 1 0 Z ∞ 1 Ψ(˜a, ˜c)f (˜a)g(˜c) d˜a d˜c. (2.11) The right hand side of this equation represents the probability of observing a given KBO with axes ratios between (˜a, ˜c) and (˜a + d˜a, ˜c + d˜c), at a large enough aspect angle, integrated for all possible axes ratios. This is also the probability of detecting brightness variation for an observed KBO.

The lower limit of integration for ˜a in Eqn. (2.11) can be replaced by √K, with K defined as in Eqn. (2.8), since Ψ(˜a, ˜c) is zero for ˜a in [1,√K]. In fact, this is how the sphericity constraint is taken into account.

P rovided that we know the value of p(∆m > ∆mmin) Eqn. (2.11) can test

candidate distributions f (˜a) and g(˜c) for the shape distribution of KBOs. The best estimate for p(∆m > ∆mmin) is given by the ratio of “non-flat” lightcurves

(ND) to the total number of measured lightcurves (N ), i.e.,

p(∆m > ∆mmin) ≈

ND

N . (2.12)

Because N is not the total number of KBOs there will be an error associated with this estimate. S ince we do not know the distributions f (˜a) and g(˜c) we will assume that the outcome of an observation can be described by a binomial distri-bution of probability p(∆m > ∆mmin). This is a good approximation given that

30 Detectability of KBO lightcurves

the hypergeometric distribution should be used since we will not unintention-ally observe the same object more than once (sampling without replacement). H owever, since the total number of KBOs (which is not known with certainty) is much larger than any sample of lightcurves, any eff ects of repeated sampling will be negligible, thereby justifying the binomial approximation. This simplification allows us to calculate the upper (p+) and lower (p₋) limits for p(∆m > ∆mmin)

at any given confidence level, C. These values, known as the C lopper– Pearson confidence limits, can be found solving the following equations by trial and error (Barlow 1989), N X r=ND+1 P¡r; p+(∆m > ∆mmin), N¢ = C + 1 2 (2.13) ND₋₁ X r=0 P¡r; p₋(∆m > ∆mmin), N¢ = C + 1 2 , (2.14) (see Table 2.2 for notation) where C is the desired confidence level and P (r; p, N ) is the binomial probability of detecting r lightcurves out of N observations, each lightcurve having a detection probability p. U sing the values in Table 2.1 and ∆mmin= 0.15 mag we have ND= 5 and N = 13 which yields

p(∆m > ∆mmin) = 0.38+0.18_−0.15

at a C = 0.68 (1σ) confidence level. At C = 0.997 (3σ) we have p(∆m > ∆mmin) = 0.38+0.41_{−0.3 1}.

The value of p(∆m > ∆mmin) could be smaller since some of the flat lightcurves

might not have been published.

N ote that for moderately elongated ellipsoids (small ˜a) the function Ψ(˜a, ˜c) is almost insensitive to the parameter ˜c (see Fig. 2.4), in which case the axisymmet-ric approximation with respect to ˜a can be made yielding ˜c ≈ 1. Equation (2.11) then has only one unknown parameter, f (˜a).

p(∆m > ∆mmin) ≈ Z ˜ama x √ K Ψ(˜a, 1)f (˜a) d˜a ≈ 0.38+0.41 −0.3 1. (2.15)

If we assume the function f (˜a) to be gaussian, we can use Eqn. (4.3) to de-termine its mean µ and standard deviation σ, after proper normalization to satisfy Eqn. (2.9). The result is represented in Fig. 2.5, where we show all possible pairs of (µ,σ) that would satisfy a given p(∆m > ∆mmin). For

exam-ple, the line labeled ”0.38”identifies all possible pairs of (µ,σ) that give rise to p(∆m > ∆mmin) = 0.38, the line labeled ”0.56”all possible pairs of (µ,σ) that

The Shapes and Spins of KBOs 31

Figure 2.5 – Contour plot of the theoretical probabilities of detecting brightness variation in K B O s (assuming ∆mmin= 0.15 mag), drawn from gaussian shape distributions parameterized

by µ and σ (respectively the mean and spread of the distributions). The solid lines represent the observed ratio of “ non-fl at” lightcurves (at 0.3 8) and 0.6 8 confi dence limits (at 0.23 and 0.56 respectively).

Clearly, with the present number of lightcurves the uncertainties are too large to draw any relevant conclusions on the shape distribution of KBOs. With a larger dataset, this formulation will allow us to compare the distribution of KBO shapes with that of the main belt asteroids. The latter has been shown to resemble, to some extent, that of fragments of high-velocity impacts (Catullo et al.1984). It deviates at large asteroid sizes that have presumably relaxed to equilibrium figures. A comparison of f (˜a) with asteroidal shapes should tell us, at the very least, whether KBO shapes are collisionally derived, as opposed to being accretional products.

The usefulness of this method is that, with more data, it would allow us to derive such quantitative parameters as the mean and standard deviation of the KBO shape distribution, if we assume a priori some intrinsic form for this distribution. The method’s strength is that it relies solely on the detectability of lightcurve amplitudes, which is more robust than other lightcurve parameters.

This paper focuses on the influence of the observation geometry and KBO shapes in the results of lightcurve measurements. In which direction would our conclusions change with the inclusion of albedo variegation and/ or binary KBOs?

num-32 Detectability of KBO lightcurves

ber of elongated objects by attributing all brightness fluctuations to asphericity. Binary KBOs would influence the results in different ways depending on the orientation of the binary system’s orbital plane, on the size ratio of the components, and on the individual shapes and spin axis orientations of the primary and secondary. For example, an elongated KBO observed equator-on would have its lightcurve flattened by a nearly spherical moon orbiting in the plane of the sky, whereas two spherical KBOs orbiting each other would generate a lightcurve if the binary would be observed edge-on.

These effects are not straightforward to quantify analytically and might re-quire a different approach. We intend to incorporate them in a future study. Also, with a larger sample of lightcurves it would be useful to apply this model to subgroups of KBOs based on dynamics, size, etc.

2.5

C on clusion s

We derived an expression for the probability of detecting brightness variations from an ellipsoidal KBO, as a function of its shape and minimum detectable amplitude. This expression takes into account the probability that a “flat” lightcurve is caused by observing geometry.

Our model can yield such quantitative parameters as the mean and standard deviation of the KBO shape distribution, if we assume a priori an intrinsic form for this distribution. It concerns solely the statistical probability of detecting brightness variation from objects drawn from these distributions, given a min-imum detectable lightcurve amplitude. The method relies on the assumption that albedo variegation and eclipsing binaries play a secondary role in the de-tection of KBO lightcurves. The effect of disregarding albedo variegation in our model is that we might overestimate the fraction of elongated objects. Binaries in turn could influence the result in both directions depending on the geometry of the problem, and on the physical properties of the constituents. We intend to incorporate these effects in a future, more detailed study.

A ck n ow led gm en ts

The Shapes and Spins of KBOs 33

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