Anisotropy of Long-period Comets EPSC2020-265 Explained by Their Formation Process Arika Higuchi1,2 1 University of Occupational and Environmental Health, Japan (2020.7.1-) 2 RISE Project, National Astronomical Observatory of Japan (-2020.6.30) Published 2020.8.26 in AJ, available at https://doi.org/10.3847/1538-3881/aba94d
Abstract Long-period comets coming from the Oort cloud are Sun thought to be planetesimals formed in the planetary region Galactic plane on the ecliptic plane. We have investigated the orbital evolution of these bodies due to the Galactic tide. We extended our previous work and derived analytical Earth The Oort cloud solutions to the Galactic longitude and latitude of the direction of aphelion. Using the analytical solutions, we predict that Oort cloud comets returning to the planetary region are concentrated on the ecliptic plane and a second plane, which we call the “empty ecliptic.” Introduction The Oort cloud comets are planetesimals formed in the protoplanetary disk and initially on the ecliptic plane with the perihelion distances q in the planetary region. (1) Outward transportation by planetary scattering
• qi in the planetary region and ei~1. • on the ecliptic (inclination ii ~ 60 deg.)
(2) Evolution into the spherical structure by perturbations from the Galaxy
Dominant perturbation: The vertical component of the Galactic tide
z: the position of the comet in the Galactic coordinates
• q outside the planetary region. • i changed.
(3) Observed as long-period comets when q come back to the planetary region. Aim of this paper
n The Galactic longitude and latitude of the direction of aphelion, L and B (or those of perihelion), are used to evaluate the distribution of observed long-period comets (e.g., Biermann et al. (1983), Luest (1984), Delsemme (1986, 1987), Matese & Whitmire (1996), Matese et al. (1999)). n Based on the standard formation scenario, the Oort cloud comets are planetesimals formed in the protoplanetary disk and initially on the ecliptic plane with the perihelion distances near the giant planets (e.g., Dones et al. 2004). n As long as the Oort cloud is not completely destroyed by close stellar encounters, the memory of the initial distribution can be found as anisotropies in the present distribution. n We investigate the evolution of the aphelia of comets initially on the ecliptic plane under the axisymmetric approximation of the Galactic tide with the same procedure as in Higuchi et al. (2007). n Using the analytical solutions, we predict the distribution of long-period comets on the L−B plane.
• Biermann, L., Huebner, W. F., and Lust, R., “Aphelion Clustering of ``New'' Comets: Star Tracks through Oort's Cloud”, Proceedings of the National Academy of Science, vol. 80, no. 16. pp. 5151–5155, 1983. doi: 10.1073/pnas.80.16.5151. • Luest, R., “The distribution of the aphelion directions of long-period comets”, Astronomy and Astrophysics, vol. 141, no. 1. pp. 94–100, 1984. • Delsemme, A. H., “Cometary evidence for a solar companion?”, in The Galaxy and the Solar System, 1986, pp. 173–203. • Delsemme, A. H., “Galactic Tides Affect the Oort Cloud - an Observational Confirmation”, Astronomy and Astrophysics, vol. 187. p. 913, 1987. • Matese, J. J., Whitman, P. G., and Whitmire, D. P., “Cometary Evidence of a Massive Body in the Outer Oort Clouds”, Icarus, vol. 141, no. 2. pp. 354–366, 1999. doi: 10.1006/icar.1999.6177. • Dones, L., Weissman, P. R., Levison, H. F., and Duncan, M. J., “Oort cloud formation and dynamics”, in Comets II, 2004, p. 153. • Higuchi, A., Kokubo, E., Kinoshita, H., and Mukai, T., “Orbital Evolution of Planetesimals due to the Galactic Tide: Formation of the Comet Cloud”, The Astronomical Journal, vol. 134, no. 4. pp. 1693– 1706, 2007. doi: 10.1086/521815. Anisotropy of Long-period Comets 3
Using the orbital elements, the unit vector of the direction of aphelion in the Galactic coordinates rQ is written as
Qx cos ! cos ⌦ +sin! sin ⌦ cos i AnisotropyAnisotropy of Long-period ofAnisotropy Long-period Comets of Comets Long-period Comets 3 3 3 rQ = Q = cos ! sin ⌦ sin ! cos ⌦ cos i (1) 0 y 1 0 1 UsingUsing the orbital the orbital elements,Using elements, the the unit orbital the vector unit elements, vector of the of direction the the unit directionQz vector of aphelion of of aphelion thesin in! direction thesin ini Galactic the of Galactic aphelion coordinates coordinates in ther GalacticQ isr writtenQ is coordinates written as as rQ is written as Anisotropy ofAnisotropy Long-periodB ofC Long-period CometsB Comets C 3 3 Then L and sin B are written as @ A @ A Qx Qx cos ! coscos⌦!Q+sincosx ⌦!+sinsin ⌦!coscossin!i⌦coscos⌦i+sin! sin ⌦ cos i Qy Using the orbitalUsing elements, the orbital the unit elements, vectorrQ = ther ofQQ the= unity direction=Q vector=cos of ofr the! aphelionsin=cos direction⌦!Lsin=sin in atan⌦=! the ofcossin aphelion Galactic⌦!coscos=i⌦ in⌦cos coordinates+ thei✓, GalacticrQ coordinatesis written asrQ is(1) written(1) as (2) (1) 0 10 y01 0Q 0Q y 1 0 Qcos ! sin1 ⌦ 1sin ! cos ⌦ cos i1 ✓ x ◆ Qz Qz sin ! sinsini!Qsin i sin ! sin i Qx B CB cosBQC! cosB⌦ +sincossin!z!sinBcos=⌦Q⌦cos+sin=i !sinsinC! ⌦sincosCi, i Derivations are here -> https://doi.org/10.3847/1538(3) -3881/aba94d @ AnalyticalA @ x B solutionsC Bz A C Then LThenandL sinandB are sin B writtenare written as as @ A @ @ A @ A A whereThenrQL =andQ siny B =arerQ written= cosQ!y assin=⌦ sincos! cos! sin⌦⌦cos isin ! cos ⌦ cos i (1) (1) 0 1 0 0 1 Q0y Q 1 1 L = atan = y⌦ + ✓, Qy (2) Qz • sinTheQ!Lsin Galactic=i atan sinlongitude! sinL=sin=i⌦! atan+andcos✓i, latitude= of⌦ the+ ✓, direction of the aphelion,(2) L and B: (2) z Qx Qatanx for cos ! < 0 B C B B C✓✓ =B ◆✓ ◆ cos ! C Qx C (4) @ A @ @ A @ sin ! cosA✓i ◆ A Then L and sinThenB areL writtenand sin asB are written as sin B =sinQBz = Qz⇡sin=+! atansinsini,! sin i, for cos ! > 0. (3) (3) ( sin B cos= Q! z = sin ! sin i, (3) Qy Qy where where where L = atan L = atan⌦ +2.2.✓, Conserved= ⌦ + ✓ quantities, (2) (2) i: inclination Qx Qx ✓ sin ◆! cossini ! cos✓i ◆ ω: argument of perihelion atan atancos ! for cosfor! sin< cos!0!cos cossin0!. !>cos0i. moving under the approximated( ( Galactic cos ! potential cos ! ⇡ + is atan given as for cos ! > 0. where where ( cos ! *all in the Galactic coordinates 2.2. Conserved2.2. Conserved quantities quantitiesGM atan sin ! cos i atanforsin ! coscosH!i2.2.<=0 Conservedfor cos ! R,< quantities0 (5) AssumeAssume that the that Galactic the Galactic tide✓ is= tide much is much smaller• ✓cos smaller=The than! time the than-averaged solar thecos ! gravity. solarh idisturbing gravity. The2a time-averaged Thefunction time-averaged that Hamiltonian arises Hamiltonian from the of(4) a vertical body of a body component(4) of the Galactic tide Assume that⇡ the+ atan Galacticsin(! tidecos⇡ +i is atan muchforsin cos smaller!!cos>i 0. thanfor cos the! solar> 0. gravity. The time-averaged Hamiltonian of a body movingmoving under under thewhere approximated theG approximatedis the gravitational Galactic( Galactic potential constant, potentialcos is! given M is is givenas theHeislercos solar as!, J. and mass, Tremaine,a is S., the“The influence semimajor of the Galactic axis tidal of field the on body,the Oort comet and cloudR is”, Icarus the, vol. disturbing 65, no. 1. pp. 13–26, 1986. doi: 10.1016/0019-1035(86)90060-6. moving under the approximated Galactic potential is given as function 2.2. Conserved quantities 2.2. Conserved quantitiesGM GM2 a: semimajor axis H = H = ⌫0 R, 2 2R, GM2 2 2 (5) (5) Assume that the Galactic tide is muchh smalleri h Ri 2= thana the2a sin solar Hi gravity.1= e +5 The e sin time-averagedR, ! , Hamiltonian of a body (6)e: eccentricity(5) Assume that the Galactic tide is much smaller than the solar gravity. 4 Theh time-averagedi 2a Hamiltonian of a body movingwhere underwhereG theismoving the approximatedG gravitationalis under the gravitational the Galactic approximated constant, constant, potentialM GalacticisM the is solaris given the potential mass, solar as mass,a isis given thea is semimajor as the semimajor axis of axis the of body, the body, and R andis theR is disturbing the disturbing wherewhere⌫0 =Gpis4 the⇡G⇢ gravitationalis the vertical constant, frequencyM is and the⇢ solaris the mass, totala is density the semimajor in the solar axis neighborhood of the body, and (e.g.,R isHeisler the disturbing & functionfunction 2 GM2 GM Tremainefunction1986). From Equation⌫0 2 (⌫6)2 andH Lagrange’s=2 2 planetary2 R, equation for da/dt,weknowa is constant.(5) Then we R =HR ==a sin0 ai2 sinh1 2R,iie 1+5 ee22⌫a2+5sin e!2 sin, 2 ! , (5) (6) (6) introduce a new simplifiedh Hamiltonian:i4 42a R = 0 a2 sin2 i 1 e2 +5e2 sin2 ! , (6) where G is the gravitational constant,•MTwois the Patterns solar mass, of evolution 2a4is the semimajor2of L and 2 B2 axis and of the the ratio body, of and theirR isperiods: the disturbing* In both case, the period of the wherewhereG is thewhere⌫0 gravitational= p⌫04⇡=G⇢p4is⇡ constant,G the⇢ is vertical theM verticalis frequency the solar frequency and mass, ⇢ andaisis the the⇢ is total semimajor the densitytotal density axis in of the the in solar body,the solar neighborhood and neighborhoodR is the disturbing (e.g., (e.g.,HeislerHeisler & & c =sin i 1 e +5e sin ! . (7) function where ⌫0 = p4⇡G⇢ is the vertical frequency and ⇢ is the total density in the solar neighborhood (e.g.,oscillationHeisler of & q is the same as PB. functionTremaineTremaine1986).1986 From). EquationFrom Equation (6) and (6) Lagrange’s and Lagrange’sPattern planetary 21: L planetarycirculates, equation equation B oscillates for da/dt for around,weknowda/dt,weknow 0 (anda isω constant.libratesa is constant.) Then Then we we The simplified z-component2 of the angular⌫0 2 momentum,2 2 which2 is a2 conserved quantity under thePL/ axisymmetricPB 〜 2 approx- introduceintroduce a new a simplified newTremaine simplified Hamiltonian:1986 Hamiltonian:).⌫ From0 2 EquationR2 = (26a)sin and2 i Lagrange’s12 e +5 planetarye sin ! , equation for da/dt,weknowa is constant.(6) Then we imationintroduce of the apotential,R new= simplifieda is writtensin i Hamiltonian:1 as4e +5e sin ! , (6) * These relations are established 4 P2attern 2 2: 2 L changes22 2little,2 2B oscillates (and ω circulates) PL/PB 〜 ∞ where ⌫ = p4⇡G⇢ is the verticalc =sin frequencyc =sini 1 andie 1+5⇢ isee the +5sin totale!sin. density! 2. in the solar neighborhood (e.g.,(7)Heisler(7) & where ⌫ = p4⇡G⇢ is0 the vertical frequency and ⇢ is the total density j in= the21 solare cos2 neighborhoodi. 2 2 (e.g., Heisler & only (8)for ei〜1. 0 c =sin i 1 e +5e sin ! . (7) The simplifiedTheTremaine simplifiedz-component1986z-component). From of the Equation of angular the angular ( momentum,6) and momentum, Lagrange’s which which is planetary a conserved is a conserved equation quantity quantity for underda/dt under,weknow the axisymmetric the axisymmetrica is constant. approx- approx- Then we Tremaine 1986). From EquationSubstituting (6) and Equation Lagrange’s (8) into planetary Equation equation (7), one for canda/dtp draw,weknow equi-Hamiltoniana is constant. curves Then on the we! e plane for given c imation ofintroduce the potential, a newThe is simplified simplifiedwritten as Hamiltonian:z-component of the angular momentum, which is a conserved quantity under the axisymmetric approx- introduce a newimation simplified of theand potential,Hamiltonian:j using Equation is written (7 as). From the Hamiltonian curves, we can learn the overall behavior of ! and e without solving imation of the potential, isj written= j1= as2 e21cosei.22cos i. 2 2 (8) (8) the equation of motion. For2 somec2 cases,=sin2 equi-Hamiltoniani 12 e +5e sin curves! . circulate with ! and for other cases they(7) librate c =sin i 1 e +5 e sin ! . j = 1 e2 cos i. (7) (8) SubstitutingSubstituting Equationaround Equation (!8)= into 90 (8) Equationor into 270 Equation. ( This7), one libration ( 7), can onep draw canisp essentially equi-Hamiltoniandraw equi-Hamiltonian the von Zeipel-Lidov-Kozai curves curves on the on! the mechanisme!planee plane for given(von for Zeipelgivenc c1910; Kozai The simplified z-component of the angular momentum, which is a conserved quantity under the axisymmetric approx- The simplifiedand j usingandz-componentj Equationusing Equation ( of7).Substituting the From ( angular7). the From Hamiltonian momentum, Equation the Hamiltonian (8 curves,which) into curves, Equationis we a conserved can we learn ( can7), onethe learn quantity overall can thep draw under overall behavior equi-Hamiltonian the behavior axisymmetric of ! and of !eandwithout curves approx-e without on solving the solving! e plane for given c imation of1962 the; potential,Lidov 1962 is; Ito written & Ohtsuka as 2019). The condition for circulation is to have a solution to e for ! = 0, i.e., imationthe of equation thethe potential, equation of motion. is of written motion.and Forj using as some For Equation cases, some cases, equi-Hamiltonian (7). equi-Hamiltonian From the Hamiltonian curves curves circulate curves, circulate with we! canwithand learn! forand the other for overall other cases behavior cases they theylibrate of ! librateand e without solving j = c1+ j2e2
4 4 [au]
4 q goodPanels for (3)ai =2 and (4)10 inau. Figure For 6aishow=5 the10 behaviorau, the of disagreementL and sin B isfor largera =2 than10 30au, for respectively, some 100 bodies and beyond panels 4.5(7) Gyrand ⇥ ⇥ i ⇠ but still much better than that in ⌦ or i . Therefore, we conclude that the⇥ relation4 among q 10000, L, and sin B in Table (8) in Figure 6 are the same as panels (3)ResultE and (4),: respectively, Relation but of for Baiand=5 L10inau. the Both planetaryL 10 and sin B are region almost 1000 ⇥ 10000 1, for any m, is safely satisfied for q in the observable region. 1 independent of q for q<50 au. The agreement of the results of numerical calculations and the [au] 0 analytical 1 solutions 2 is 3 4 4.5 q 100 time [109 yr] 4 4 1000 good for a =2 10 au. For a =5 10 au, the disagreement is larger than 30 for some 10000 bodies beyond 4.5 Gyr i i [au]
q 1 In the planetary region, L and B satisfy equation (1) or (2). 10 ⇥ ⇥ 4.4. The Empty Ecliptic ⇠ 100 0.8 1000 but still much better than that in ⌦ or iE. Therefore, we conclude that the relation among [au] q, L, and sin B in Table
[au] 0.6
q 1 Anisotropy of Long-period Comets q 10 0 11 1 2 3 4 4.5 0.4 100 Based on the standard scenario of the formationAnisotropy of the of Long-period Oort cloud, Comets the initial orbital elements of the11 Oort cloud9 1, for any m, is safely satisfied for q in theNumber observable of returns to region. time [10 yr] k = even 0.2
k = odd B 1 0 10 the planetary region k (including initial condition) 1 0 1 2 3 4 4.5 comets are restricted as follows; q 30 au to be near a giant planet, i i = 60 and ⌦ sin ⌦ = 186 to be on the we confirmed that these oppositei evolutions seen in ⌦ and L are due to the radiali component of the Galactici tide that 9 we confirmed that these. opposite evolutions seen in ⌦ and L are due to the radial component of the Galactic-0.2 tide that time [10 yr] ' ' 0.8 1 breaks the conservation of j. In panels (6) and (8), the disagreement that arises from the shifts of the periods is-0.4 0.6 quite ecliptic plane. Alsobreaks!i theis uniformly conservation of distributedj. Inpattern panels 4.4.1 (6) for and 0The (8),! Empty thei
2 2 2 2 sin 0.4 0 1 2 3 4 4.5 From the above comparisons, we conclude that the analytical solutions are basically useful for describing 0.8 the orbital 2 -0.24 9 0.2 time [10 yr]
tan B =sin(L ⌦ ) tan i . B (100) evolution, except for i and ⌦ of comets in the observablei regioni (i.e., q 10 au). For2 bodies with a =5 10 0.6 au, 4 evolution, except for i and ⌦ of comets in the observable region. (i.e., q 10 au). For bodies with B a-0.4=5 10 au, comets are restricted as follows; qi 30 aurelation to be satisfied near a giant planet, ii . i = 60 and ⌦i ⇥⌦ 0 = 186 to be on the . eq. (1) eq. (2) sin 0.4 360-0.6 ⇥ the small di↵erences between the periods given by the analytical solutions and the' ones obtained from the' numericalsin -0.2 the small di↵erences between the periods given by the analytical solutions and the ones obtained from 0.2 the numerical -0.8 ecliptic plane. Also ! is uniformly distributed for 0 ! < 360 , L is given by Equation (2),B -0.4 and sin B , in order to The relationintegrations between pileiL upand and are sin quiteB in large Table at t 1 definesa few Gyr. two This planesi could be in understood thei Galactic simply coordinates. as the results of As the 300L shifts 0 i and Bii are assumed
sin -1 integrations pile up and are quite large at t a few Gyr. This could be understood simply as the results-0.6 of the shifts ⇠ -0.2 0 1 2 3 4 4.5 ⇠ 9 of oscillation/circulation of orbital evolution. Therefore, the time evolution normalized by the periods obtained 240-0.8 by time [10 yr] beto on be the on the ecliptic eclipticof plane, oscillation/circulation plane is given the points by of that orbital satisfy evolution. 0 Therefore,L<360 the and time sin evolutionB given normalized by Equation by the periods (100-0.4 obtained) for L bydrawThe hatched a area curve indicate ”in the planetary region” -1 -0.6 numerical integrations is well reproduced by the analytical solutions. 180 360 0 1 2 3 4 4.5 numerical integrations is well reproduced by the analytical solutions. ………… (1) [deg] 9 on the L sin B plane, by definition. Anothertan setBi =sin( of pointsLi for⌦ an) tan oddi m. that satisfy 0 L deg 180 In this section,In this we section, apply thewe apply analytical the analytical solutions derivedsolutions in derived Section in2 Sectionto fictional2 to observable fictional observable long-period long-period [deg] comets comets [ 0 1 2 3 4 4.5 L 240 to be on the ecliptic plane the points that satisfy 0 L<360 and sin B given by Equation (100) for L draw a curve9 tan B = sin(L ⌦ ) tan i . ………… (2) L 300 time (101)[10 yr] entering theentering planetary the planetaryregion from region the Oort from cloud. the Oort We cloud. assume We that assume the comets that the initially comets have initially very have elongated very elongated orbits120 orbits 4 180 (4) a =2x10 au [deg] i L 1240 on the L givensin byB planetaryplane, by scattering definition. on the Another ecliptic plane. set For of thesepoints comets, for an setting odde m 1that is a good satisfy approximation 0 L< and360 60 it and sin B draw given by planetary scattering on the ecliptic plane. For these comets,i setting ei 1 is a good approximationq =10 au, =310 and deg it The empty ecliptic 0.8 120 i i ' q =30 au, =210 deg ' i i 180 We call thismakes plane themakes solutions as the “the simple. solutions empty We simple. call ecliptic,” this We approximation call this since approximation it as is the not quasi-rectilinear as initially the quasi-rectilinear populated approximation. approximation. and Since this the plane Since planetary [deg] 0.6 0 the and planetary the ecliptic are a curve that defines a second plane, which is formed by a rotation of the ecliptic around the GalacticL pole by 180 : 60 0 1 2 3 4 4.5 0.4 9 scatteringscattering does not give does high not inclinations give high inclinations (see Appendix), (see Appendix), we assume weii assume= i =i 60i = iand=⌦ 60i =and⌦ ⌦=i 186= ⌦ to= be 186 on 120 to the be on the time [10 yr] 0.2 time [Gyr] symmetrical about the plane perpendicular to the Galactic plane through the intersection ofthe 0 ecliptic4 plane and B (4) ai=2x10 au ecliptic plane.ecliptic Using plane. this Using approximation, this approximation, we compare we the compare periods the of periods , ⌦, and of L, ⌦and, and investigateL and investigate the relation the among relation0 60 1 0 among 1 2 3 4 4.5 sin 9 q =10 au, =310 deg time [10 yr] tan B = sin(L ⌦ ) tan i . -0.2 0.8 i i (101) the Galactic plane (just like the focus and the empty focus of an ellipse). If L and B of long-periodq =30 au, =210 deg comets in the them especially for comets in the observable region. 0 i 4 i them especially for comets in the observable region. -0.4 0.6 (4) ai=2x10 au 1 0 1 2 3 4 4.5 9 -0.6 0.4 q =10 au, =310 deg time [10 yr] observable region are concentrated on these two planes, it would constitute observational evidence 0.8 i thati the comets qi=30 au, i=210 deg -0.8 0.2 4 0.6 (4) ai=2x10 au We call this plane as “the empty ecliptic,” since it is4.1. notPreparation initially populated and this4 planeB 1 and the ecliptic are 4.1. Preparation -1 0 were on the ecliptic plane at t = 0. Comets with relatively small values such as a 10 au, whichsin 0.4 satisfyq =10 au, =310 deg P =4.5 Gyr 0.8 0 i i -0.2 qi=30 au, i=210 deg 0.22 2 2 time [109 yr] ⇠ 2 2 0.6 2 symmetricalIn about this section,In the this we plane section, first derive perpendicular we first the derive explicit the expression to explicit the expression Galactic for ↵0, ↵1 for, and plane↵0,↵↵2 1(eq.(, through and70↵))2 (eq.( and the their70)) intersection and relation, theirk relation,, w of, andB k-0.4 the, ww1 ecliptic, and w2 plane and (i.e., m = 1 for present), are predicted to be on the empty ecliptic for their first return to the1 planetary 0 2 region. sin 0.4 2 2 -0.6 (eqs. (32), (37), and (51)) under the quasi-rectilinear approximation, which gives 0 1 https://ssd.jpl.nasa.gov/ query.cgi Discussion: Distribution of observed long-period comets 22 from JPL/NASA, comets with a > 1000 au || e ≧Higuchi1 and q > 1 au • On the L - sin B plane: To evaluate the concentrations• onThe the distribution two planes, weof definethe longitude a new angle " as Empty Ecliptic Ecliptic 1 ε defined by tan B tan " = . (104) sin(L ⌦ ) For the ecliptic and empty ecliptic planes,ε = " 60= degi : Ecliptic= 60 and " = i = 60 , respectively. The solid and dashed 0.5 ε = -60 deg : Empty Ecliptic =-80 =-60 curves in Figure 7 show the ecliptic and empty ecliptic planes, respectively. Curves for " = 0, 30 , and 80 are also =-30 ± ± ’Oumuamua shown in thin dashed curves in Figure 7. Figure 8 shows the distribution of ". There are two sharp peaks not exactly at the ecliptic or empty ecliptic plane but near them. For an isotropic distribution, it would be flat in ". B B Galactic plane, =0 0 sin Borisov sin =30 =80 =60 -0.5 E’ E 80 -1 0 60 120 180 240 300 360 L [deg] 60 (+ ’Oumuamua■ +Borisov■) L [deg] Two sharp peaks near the ecliptic or empty ecliptic plane. *An isotropic distribution would 40 be flat in ε. 20 0 -90 -60 -30 0 30 60 90 ε [deg] Figure 8. Distribution of " defined by equation (104) for all bodies in Figure 7 except U1. Filled bars show bodies with e>1. Panel (1) in Figure 9 shows the distribution of sin B. The depletions around b = 0 and b = 90 found by Luest ± (1984) and Delsemme (1987) is seen (note that b = B), however, the shape of the distribution depends on the regions of L0 = L ⌦ . Panels (2)-(4) in Figure 9 show the distribution of sin B for regions blue ( 30