text: 4 Qi Hao et al. (i+1,j-1), (i+1,j), (i+1,j+1), are 8-connected with current pixel (i,j)), it means that there is an edge between them. The longest acyclic path of two nodes is deﬁned as the ﬁlament spine. One can ﬁnd the spine without changing the topological structure of the ﬁlament skeleton. The main algorithm is described as following. The algorithm is pictorially shown by a sample ﬁlament skeleton example in Figure 1. Firstly, for each binary image with a single ﬁlament skeleton, we have to ﬁnd all the nodes (i.e., pixels) of the ﬁlament skeleton and assign index numbers to them. Since a real observed ﬁlament is relatively large and complex, in order to explain our method clearly, here we take an artiﬁcial ﬁlament skeleton as an example, which has the same typical features as real ones. As the model example shown in Figure 1(a), ﬁlament skeleton nodes are the pixels whose values are equal to 1 (the values of the background pixels are equal to 0). Figure 1(b) shows that each ﬁlament skeleton node is assigned with a unique index number. This example has 23 nodes in total. The algorithm goes through each pixel starting at the lower left corner from bottom to top and from left to right, as indicated by the arrows shown in Figure 1(b). × Secondly, we build an N N connection matrix that records the edge information between each nodes, where N is the number of the ﬁlament skeleton nodes. If the ith node and the jth node are 8-connected, the values of the corresponding coordinates (i, j) and (j, i) in connection matrix are 1. It means that there is an edge between them, namely, a connection exists between the two nodes. Otherwise the value is 0, which indicates that the two nodes have no edge between them. In addition, self connecting nodes are not allowed here. Therefore, we ignore the diagonal of the connection matrix and set these values as 0 for convenience. Next, for each node of the ﬁlament skeleton, we check its connectivity with other nodes and update the corresponding values in the connection matrix. Figure 1(c) shows only four corner parts × of the whole connection matrix since the size of the whole matrix is 23 23, which is too large to be shown completely. For instance, we can see that the node No. 2 just connects with nodes Nos. 1 and 3 in Figure 1(b), so the corresponding values of the coordinates (1, 2), (2, 1), (2, 3) and (3, 2) in connection matrix are 1. Meanwhile, the other coordinates in the second row and the second column are 0, which means that these nodes are not connected with node No. 2. Thirdly, we have to ﬁnd the vertex nodes (i.e., pixels in the ends of the ﬁlament skeleton). Vertex nodes are deﬁned as those nodes that only have one connected neighbour, namely, each of them has only one edge associated with it. The vertex nodes are easy to be found in the connection matrix since the corresponding row or column has only one item whose value is equal to 1. The example in Figure 1(b) shows the vertex nodes in yellow pixels, i.e., nodes Nos. 1, 5, 14, 21, and 23. Fourthly, we use the shortest-path method of the graph theory to ﬁnd all the paths from one vertex node to other vertex nodes. The longest acyclic path is the spine. There are 5 vertex nodes so that the program ﬁnds the shortest-path between every 2 of the 5 vertex nodes and obtains 10 paths. Among them, the longest path has 13 edges that are from node No. 1 to node No. 23. This path is determined as the spine of the example ﬁlament as shown in Figure 1(b) with blue color, including the two vertex nodes No. 1 and No. 23 that are treated as the footpoints of the ﬁlament. The shortest-path method determines the single-source shortest path from one node to all other nodes × in the graph represented by the N N connection matrix, where N is the number of nodes. It can get the spine as thin as one pixel. Here, we use the Dijkstra algorithm (Dijkstra 1959) to ﬁnd the shortest paths.

text: Qi

text: Hao

text: et

text: al.

text: are

text: with

text: current

text: pixel

text: (i,j)),

text: it

text: means

text: that

text: there

text: is

text: an

text: edge

text: between

text: them.

text: The

text: longest

text: acyclic

text: path

text: of

text: two

text: nodes

text: is

text: deﬁned

text: as

text: the

text: ﬁlament

text: spine.

text: One

text: can

text: ﬁnd

text: the

text: spine

text: without

text: changing

text: the

text: topological

text: structure

text: of

text: the

text: ﬁlament

text: skeleton.

text: The

text: main

text: algorithm

text: is

text: described

text: as

text: following.

text: The

text: algorithm

text: is

text: pictorially

text: shown

text: by

text: a

text: sample

text: ﬁlament

text: skeleton

text: example

text: in

text: Figure

text: Firstly,

text: for

text: each

text: binary

text: image

text: with

text: a

text: single

text: ﬁlament

text: skeleton,

text: we

text: have

text: to

text: ﬁnd

text: all

text: the

text: nodes

text: (i.e.,

text: pixels)

text: of

text: the

text: ﬁlament

text: skeleton

text: and

text: assign

text: index

text: numbers

text: to

text: them.

text: Since

text: a

text: real

text: observed

text: ﬁlament

text: is

text: relatively

text: large

text: and

text: complex,

text: in

text: order

text: to

text: explain

text: our

text: method

text: clearly,

text: here

text: we

text: take

text: an

text: artiﬁcial

text: ﬁlament

text: skeleton

text: as

text: an

text: example,

text: which

text: has

text: the

text: same

text: typical

text: features

text: as

text: real

text: ones.

text: As

text: the

text: model

text: example

text: shown

text: in

text: Figure

text: ﬁlament

text: skeleton

text: nodes

text: are

text: the

text: pixels

text: whose

text: values

text: are

text: equal

text: to

text: (the

text: values

text: of

text: the

text: background

text: pixels

text: are

text: equal

text: to

text: Figure

text: shows

text: that

text: each

text: ﬁlament

text: skeleton

text: node

text: is

text: assigned

text: with

text: a

text: unique

text: index

text: number.

text: This

text: example

text: has

text: nodes

text: in

text: total.

text: The

text: algorithm

text: goes

text: through

text: each

text: pixel

text: starting

text: at

text: the

text: lower

text: left

text: corner

text: from

text: bottom

text: to

text: top

text: and

text: from

text: left

text: to

text: right,

text: as

text: indicated

text: by

text: the

text: arrows

text: shown

text: in

text: Figure

text: Secondly,

text: we

text: build

text: an

text: N

text: N

text: connection

text: matrix

text: that

text: records

text: the

text: edge

text: information

text: between

text: each

text: nodes,

text: where

text: N

text: is

text: the

text: number

text: of

text: the

text: ﬁlament

text: skeleton

text: nodes.

text: If

text: the

text: ith

text: node

text: and

text: the

text: jth

text: node

text: are

text: the

text: values

text: of

text: the

text: corresponding

text: coordinates

text: (i,

text: j)

text: and

text: (j,

text: i)

text: in

text: connection

text: matrix

text: are

text: It

text: means

text: that

text: there

text: is

text: an

text: edge

text: between

text: them,

text: namely,

text: a

text: connection

text: exists

text: between

text: the

text: two

text: nodes.

text: Otherwise

text: the

text: value

text: is

text: which

text: indicates

text: that

text: the

text: two

text: nodes

text: have

text: no

text: edge

text: between

text: them.

text: In

text: addition,

text: self

text: connecting

text: nodes

text: are

text: not

text: allowed

text: here.

text: Therefore,

text: we

text: ignore

text: the

text: diagonal

text: of

text: the

text: connection

text: matrix

text: and

text: set

text: these

text: values

text: as

text: for

text: convenience.

text: Next,

text: for

text: each

text: node

text: of

text: the

text: ﬁlament

text: skeleton,

text: we

text: check

text: its

text: connectivity

text: with

text: other

text: nodes

text: and

text: update

text: the

text: corresponding

text: values

text: in

text: the

text: connection

text: matrix.

text: Figure

text: shows

text: only

text: four

text: corner

text: parts

text: of

text: the

text: whole

text: connection

text: matrix

text: since

text: the

text: size

text: of

text: the

text: whole

text: matrix

text: is

text: which

text: is

text: too

text: large

text: to

text: be

text: shown

text: completely.

text: For

text: instance,

text: we

text: can

text: see

text: that

text: the

text: node

text: No.

text: just

text: connects

text: with

text: nodes

text: Nos.

text: and

text: in

text: Figure

text: so

text: the

text: corresponding

text: values

text: of

text: the

text: coordinates

text: and

text: in

text: connection

text: matrix

text: are

text: Meanwhile,

text: the

text: other

text: coordinates

text: in

text: the

text: second

text: row

text: and

text: the

text: second

text: column

text: are

text: which

text: means

text: that

text: these

text: nodes

text: are

text: not

text: connected

text: with

text: node

text: No.

text: Thirdly,

text: we

text: have

text: to

text: ﬁnd

text: the

text: vertex

text: nodes

text: (i.e.,

text: pixels

text: in

text: the

text: ends

text: of

text: the

text: ﬁlament

text: skeleton).

text: Vertex

text: nodes

text: are

text: deﬁned

text: as

text: those

text: nodes

text: that

text: only

text: have

text: one

text: connected

text: neighbour,

text: namely,

text: each

text: of

text: them

text: has

text: only

text: one

text: edge

text: associated

text: with

text: it.

text: The

text: vertex

text: nodes

text: are

text: easy

text: to

text: be

text: found

text: in

text: the

text: connection

text: matrix

text: since

text: the

text: corresponding

text: row

text: or

text: column

text: has

text: only

text: one

text: item

text: whose

text: value

text: is

text: equal

text: to

text: The

text: example

text: in

text: Figure

text: shows

text: the

text: vertex

text: nodes

text: in

text: yellow

text: pixels,

text: i.e.,

text: nodes

text: Nos.

text: and

text: Fourthly,

text: we

text: use

text: the

text: shortest-path

text: method

text: of

text: the

text: graph

text: theory

text: to

text: ﬁnd

text: all

text: the

text: paths

text: from

text: one

text: vertex

text: node

text: to

text: other

text: vertex

text: nodes.

text: The

text: longest

text: acyclic

text: path

text: is

text: the

text: spine.

text: There

text: are

text: vertex

text: nodes

text: so

text: that

text: the

text: program

text: ﬁnds

text: the

text: shortest-path

text: between

text: every

text: of

text: the

text: vertex

text: nodes

text: and

text: obtains

text: paths.

text: Among

text: them,

text: the

text: longest

text: path

text: has

text: edges

text: that

text: are

text: from

text: node

text: No.

text: to

text: node

text: No.

text: This

text: path

text: is

text: determined

text: as

text: the

text: spine

text: of

text: the

text: example

text: ﬁlament

text: as

text: shown

text: in

text: Figure

text: with

text: blue

text: color,

text: including

text: the

text: two

text: vertex

text: nodes

text: No.

text: and

text: No.

text: that

text: are

text: treated

text: as

text: the

text: footpoints

text: of

text: the

text: ﬁlament.

text: The

text: shortest-path

text: method

text: determines

text: the

text: single-source

text: shortest

text: path

text: from

text: one

text: node

text: to

text: all

text: other

text: nodes

text: in

text: the

text: graph

text: represented

text: by

text: the

text: N

text: N

text: connection

text: matrix,

text: where

text: N

text: is

text: the

text: number

text: of

text: nodes.

text: It

text: can

text: get

text: the

text: spine

text: as

text: thin

text: as

text: one

text: pixel.

text: Here,

text: we

text: use

text: the

text: Dijkstra

text: algorithm

text: (Dijkstra

text: to

text: ﬁnd

text: the

text: shortest

text: paths.


text: 6 • • • (CKM: a) (CKM: b) (CKM: c) • • • (PMNS: a) (PMNS: b) (PMNS: c) ± FIG. 3: Schematic shapes of J for the CKM and PMNS triangles. All of them have one angle with δ = δ in our CKM PMNS scenario. (cid:104) (cid:105) IV. RELATING THE PHASE OF X TO δ AND δ CKM PMNS Even though the physical argument presented in the previous section is enough for relating δ and δ , here PMNS CKM we present a scheme in detail how they are connected. The common phase in (cid:104)X(cid:105) = V eiδ will appear in the (ij) element of the quark or lepton mass matrix with a form, A (eiδ)n if the singlet Xn is located in the (ij) element. We ij work in the bases where M(d) and M(e) are already diagonalized, and the left-hand unitary matrices diagonalizing M(u) and M(ν) are the CKM and PMNS matrices, respectively. The parameters of mass matrix M(u) combine to produce the phase in the CKM matrix. For a complex mass matrix M(u), there are 18 real parameters. It can be diagonalized by a bi-unitary transformation by UL and UR. Thus, there are 36 parameters to be considered initially from M(u), UL and UR. The left- and right-hand sides of the diagonaliza- † tion relation, UL M(u)UR = eiα(real diagonal mass matrix), have the same overall phase, corresponding to the baryon number conservation. Disregarding the baryon number, by making Det.M(u) = real so that the diagonalized masses † are real, we consider 35 real parameters in the diagonalizing conditions. The condition UL M(u)UR= (real diagonal mass matrix) gives 18 relations. With these conditions imposed, then there remain 17 independent parameters from 35. Out of 17, ﬁve (since the overall phase cannot be used) can be removed by redeﬁning L- and R-handed u-type quark phases. Thus, there remain 12 independent parameters. Out of 12, 11 parameters remain as physical ones, three real u-type quark masses, four angles of UL, and four angles of UR. But, parameters in UR are hidden at low ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

font: default

text: (CKM:

text: a)

text: (CKM:

text: b)

text: (CKM:

text: c)

text: (PMNS:

text: a)

text: (PMNS:

text: b)

text: (PMNS:

text: c)

label: caption

text: FIG.

text: Schematic

text: shapes

text: of

text: J

text: for

text: the

text: CKM

text: and

text: PMNS

text: triangles.

text: All

text: of

text: them

text: have

text: one

text: angle

text: with

text: δ

text: δ

text: in

text: our

text: CKM

text: PMNS

text: scenario.

label: section

text: IV.

text: RELATING

text: THE

text: PHASE

text: OF

text: X

text: TO

text: δ

text: AND

text: δ

text: CKM

text: PMNS

text: Even

text: though

text: the

text: physical

text: argument

text: presented

text: in

text: the

text: previous

text: section

text: is

text: enough

text: for

text: relating

text: δ

text: and

text: δ

text: here

text: PMNS

text: CKM

text: we

text: present

text: a

text: scheme

text: in

text: detail

text: how

text: they

text: are

text: connected.

text: The

text: common

text: phase

text: in

text: V

text: eiδ

text: will

text: appear

text: in

text: the

text: (ij)

text: element

text: of

text: the

text: quark

text: or

text: lepton

text: mass

text: matrix

text: with

text: a

text: form,

text: A

text: (eiδ)n

text: if

text: the

text: singlet

text: Xn

text: is

text: located

text: in

text: the

text: (ij)

text: element.

text: We

text: ij

text: work

text: in

text: the

text: bases

text: where

text: M(d)

text: and

text: M(e)

text: are

text: already

text: diagonalized,

text: and

text: the

text: left-hand

text: unitary

text: matrices

text: diagonalizing

text: M(u)

text: and

text: M(ν)

text: are

text: the

text: CKM

text: and

text: PMNS

text: matrices,

text: respectively.

text: The

text: parameters

text: of

text: mass

text: matrix

text: M(u)

text: combine

text: to

text: produce

text: the

text: phase

text: in

text: the

text: CKM

text: matrix.

text: For

text: a

text: complex

text: mass

text: matrix

text: M(u),

text: there

text: are

text: real

text: parameters.

text: It

text: can

text: be

text: diagonalized

text: by

text: a

text: bi-unitary

text: transformation

text: by

text: UL

text: and

text: UR.

text: Thus,

text: there

text: are

text: parameters

text: to

text: be

text: considered

text: initially

text: from

text: M(u),

text: UL

text: and

text: UR.

text: The

text: left-

text: and

text: right-hand

text: sides

text: of

text: the

text: diagonaliza-

text: tion

text: relation,

text: UL

text: M(u)UR

text: eiα(real

text: diagonal

text: mass

text: matrix),

text: have

text: the

text: same

text: overall

text: phase,

text: corresponding

text: to

text: the

text: baryon

text: number

text: conservation.

text: Disregarding

text: the

text: baryon

text: number,

text: by

text: making

text: Det.M(u)

text: real

text: so

text: that

text: the

text: diagonalized

text: masses

text: are

text: real,

text: we

text: consider

text: real

text: parameters

text: in

text: the

text: diagonalizing

text: conditions.

text: The

text: condition

text: UL

text: M(u)UR=

text: (real

text: diagonal

text: mass

text: matrix)

text: gives

text: relations.

text: With

text: these

text: conditions

text: imposed,

text: then

text: there

text: remain

text: independent

text: parameters

text: from

text: Out

text: of

text: ﬁve

text: (since

text: the

text: overall

text: phase

text: cannot

text: be

text: used)

text: can

text: be

text: removed

text: by

text: redeﬁning

text: L-

text: and

text: R-handed

text: u-type

text: quark

text: phases.

text: Thus,

text: there

text: remain

text: independent

text: parameters.

text: Out

text: of

text: parameters

text: remain

text: as

text: physical

text: ones,

text: three

text: real

text: u-type

text: quark

text: masses,

text: four

text: angles

text: of

text: UL,

text: and

text: four

text: angles

text: of

text: UR.

text: But,

text: parameters

text: in

text: UR

text: are

text: hidden

text: at

text: low

text: ##LTFigure##

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text: ##LTLine##

text: ##LTLine##


text: 10 a dodeca-symmetry, JHEP 1102 (2011) 097 [arXiv:1005.4684 [hep-ph]]. [21] L. M. Krauss and F. Wilczek, Discrete gauge symmetry in continuum theories, Phys. Rev. Lett. 62 (1989) 1221 [doi: 10.1103/PhysRevLett.62.1221]. [22] L. E. Ibanez and G. G. Ross, Discrete gauge symmetries and the origin of baryon and lepton number conservation in supersymmetric versions of the standard model, Nucl. Phys. B 368 (1992) 3 [doi: 10.1016/0550-3213(92)90195-H]. [23] S. M. Barr and D. Seckel, Planck scale corrections to axion models, Phys. Rev. D 46 (1992) 539 [doi: 10.1103/Phys- RevD.46.539]; M. Kamionkowski and J. March-Russell, Planck scale physics and the Peccei-Quinn mechanism, Phys. Lett. B 282 (1992) 137 [hep-th/9202003]; R. Holman, S. D. H. Hsu, T. W. Kephart, E. W. Kolb, R. Watkins, and L. M. Widrow, Solutions to the strong CP problem in a world with gravity , Phys. Lett. B 282 (1992) 132 [hep-ph/9203206]; B. A. Dobrescu, The Strong CP problem versus Planck scale physics , Phys. Rev. D 55 (1997) 5826 [hep-ph/9609221]. [24] J. E. Kim, Natural Higgs-ﬂavor-democracy solution of the µ problem of supersymmetry and the QCD axion, Phys. Rev. Lett. 111 (2013) 031801 [arXiv:1303.1822 [hep-ph]]; Abelian discrete symmetries Z and Z from string orbifolds, Phys. Lett. B N nR 726 (2013) 450 [arXiv:1308.0344 [hep-th]]. [25] T. D. Lee, A theory of spontaneous T violation, Phys. Rev. D 8 (1973) 1226 [doi: 10.1103/PhysRevD.8.1226]. [26] A.E. Nelson, Naturally weak CP violation, Phys. Lett. B 136 (1984) 387 [doi: 10.1016/0370-2693(84)92025-2]; S.M. Barr, Solving the strong CP problem without the Peccei-Quinn symmetry, Phys. Rev. Lett. 53 (1984) 329 [doi: 10.1103/Phys- RevLett.53.329]. [27] J.E. Kim, The CKM matrix with maximal CP violation from Z(12) symmetry, Phys. Lett. B 704 (1911) 360 [arXiv:1109.0995 [hep-ph]]. [28] S. M. Barr, A new symmetry breaking pattern for SO(10) and proton decay, Phys. Lett. B 112 (1982) 219 [doi: 10.1016/0370- 2693(82)90966-2]. [29] J. P. Derendinger, J. E. Kim, and D. V. Nanopoulos, Anti-SU(5), Phys. Lett. B 139 (1984) 170 [doi:10.1016/0370- 2693(84)91238-3 ]. × [30] I. Antoniadis, J. R. Ellis, J .S. Hagelin, and D. V. Nanopoulos, The ﬂipped SU(5) U(1) string model revamped, Phys. Lett. B 231 (1989) 65 [doi: 10.1016/0370-2693(89)90115-9]. [31] J. E. Kim and B. Kyae, Flipped SU(5) from Z − orbifold with Wilson line, Nucl. Phys. B 770 (2007) 47 [arXiv: hep- 12 I th/0608086]. [32] R.D.Peccei and H.R. Quinn, Constraints imposed by CP conservation in the presence of instantons, Phys. Rev. Lett. 16 (1977) 1791 [doi: 10.1103/PhysRevD.16.1791]. For a recent review, see, H. Baer, K-Y. Choi, J. E. Kim, and L. Roszkowski, Dark matter production in the early Universe: beyond the thermal WIMP paradigm, Phys. Rep. 555 (2015) 1 [arXiv:1407.0017 [hep-ph]]. [33] J. E. Kim, The CKM matrix with maximal CP violation from Z symmetry, Phys. Lett. B 704 (2011) 360 [arXiv: 1109.0995 12 [hep-ph]]; H. Georgi, A. E. Nelson, and M. Shin, CP violation and Fritzsch mass matrices, Phys. Lett. B 150 (1985) 306 [doi: 10.1016/0370-2693(85)91016-0]. [34] T.W.B. Kibble, Topology of cosmic strings and domains, J. Phys. A9 (1976) 1387 [doi: 10.1088/0305-4470/9/8/029]. [35] L-L. Chau and W-Y. Keung, Comments on the parametrization of the Kobayashi-Maskawa matrix, Phys. Rev. Lett. 53 (1984) 1802 [doi: 10.1103/PhysRevLett.53.1802]. Multiplying an overall phase, see, L. Maiani, CP violation in purely lefthanded weak interactions, Phys. Lett. B 62 (1976) 183 [doi: 10.1016/0370-2693(76)90500-1] and New currents, Proc. 1977 Int. Symp. on Lepton and Photon Interactions at High Energies (DESY, Hamburgh, 1977), p. 867. [36] H. M. Georgi and S. L. Glashow, Unity of all elementary particle forces, Phys. Rev. Lett. 32 (1974) 438 [doi: 10.1103/Phys- RevLett.32.438]. [37] See, however, Z. Kakushadze and S.H.H. Tye, Three family SO(10) grand uniﬁcation in string theory, Phys. Rev. Lett. 77 (1996) 2612 [arXiv: hep-th/9605221]. [38] L. Autonne, Sur les matrices hypohermitiennes et sur les matrices unitaires, Ann. Univ. Lyon 38 (1915) 1–77; T. Takagi, On an algebraic problem related to an analytic theorem of Carath´eodory and Fej´er and on an allied theorem of Landau, Japan. J. Math. 1 (1925) 83–93. [39] J.E. Kim, to be published.

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text: – 11 – sophisticated asteroid shape to construct synthetic lightcurve that introduces formidable time-consuming computation, the synthetic objects were assumed relaxed triaxial ellipsoids having equal axes in b and c and rotating around the principle-axis a. Therefore, the lightcurves can be written as (cid:118) (cid:117) (cid:34)(cid:18) (cid:19) (cid:35) (cid:117) (cid:116) b 2 − m = 2.5 log 1 + 1 cos2(2πφ) sin2 θ, (4) 10 a and the amplitude (∆m) can be written as (cid:34) (cid:35) (cid:18) (cid:19) − 1/2 b 2 ∆m = 2.5 log cos2 θ + sin2 θ , (5) 10 a where φ is rotational phase and θ is the angle between line of sight and spin vector (Lacerda & Luu 2003; Lacerda & Jewitt 2007). The cadences of the synthetic lightcurves were chosen to be identical with the survey observations and the numbers of synthetic measurements were assigned according to the number of detections of PTF-detected asteroids with diﬀerent apparent magnitudes. We generated 402,000 synthetic lightcurves that uniformly distribute in frequency ≤ ≤ ≤ ≤ of 0 f 12 rev/day, in pole orientation of 10 θ 90 degree and in axis ratio of ≤ ≤ 0.1 b/a 1. Then, these synthetic lightcurves were analyzed by the aforementioned second-order Fourier series method to search for assigned periods. We deﬁned a successful period determination as : (a) the derived period is within 5% of the original period, (b) the folded lightcurve of the derived period has double peaks or double dips and (c) the assigned photometric error is smaller than the derived amplitude. Therefore, the “PTF-P”-like objects could not be picked up in the simulation. ##LTLine## ##LTLine## ##LTLine##

text: sophisticated

text: asteroid

text: shape

text: to

text: construct

text: synthetic

text: lightcurve

text: that

text: introduces

text: formidable

text: time-consuming

text: computation,

text: the

text: synthetic

text: objects

text: were

text: assumed

text: relaxed

text: triaxial

text: ellipsoids

text: having

text: equal

text: axes

text: in

text: b

text: and

text: c

text: and

text: rotating

text: around

text: the

text: principle-axis

text: a.

text: Therefore,

text: the

text: lightcurves

text: can

text: be

text: written

text: as

label: equation

text: b

text: m

text: log

text: θ,

text: a

text: and

text: the

text: amplitude

text: (∆m)

text: can

text: be

text: written

text: as

text: b

text: ∆m

text: log

text: θ

text: θ

text: a

text: where

text: φ

text: is

text: rotational

text: phase

text: and

text: θ

text: is

text: the

text: angle

text: between

text: line

text: of

text: sight

text: and

text: spin

text: vector

text: (Lacerda

text: Luu

text: Lacerda

text: Jewitt

text: The

text: cadences

text: of

text: the

text: synthetic

text: lightcurves

text: were

text: chosen

text: to

text: be

text: identical

text: with

text: the

text: survey

text: observations

text: and

text: the

text: numbers

text: of

text: synthetic

text: measurements

text: were

text: assigned

text: according

text: to

text: the

text: number

text: of

text: detections

text: of

text: PTF-detected

text: asteroids

text: with

text: diﬀerent

text: apparent

text: magnitudes.

text: We

text: generated

text: synthetic

text: lightcurves

text: that

text: uniformly

text: distribute

text: in

text: frequency

text: of

text: f

text: rev/day,

text: in

text: pole

text: orientation

text: of

text: θ

text: degree

text: and

text: in

text: axis

text: ratio

text: of

text: b/a

text: Then,

text: these

text: synthetic

text: lightcurves

text: were

text: analyzed

text: by

text: the

text: aforementioned

text: second-order

text: Fourier

text: series

text: method

text: to

text: search

text: for

text: assigned

text: periods.

text: We

text: deﬁned

text: a

text: successful

text: period

text: determination

text: as

text: (a)

text: the

text: derived

text: period

text: is

text: within

text: of

text: the

text: original

text: period,

text: (b)

text: the

text: folded

text: lightcurve

text: of

text: the

text: derived

text: period

text: has

text: double

text: peaks

text: or

text: double

text: dips

text: and

text: (c)

text: the

text: assigned

text: photometric

text: error

text: is

text: smaller

text: than

text: the

text: derived

text: amplitude.

text: Therefore,

text: the

text: “PTF-P”-like

text: objects

text: could

text: not

text: be

text: picked

text: up

text: in

text: the

text: simulation.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: – 15 – ≤ ≤ due to a greater ability in detecting 2 f 5 rev/day (Harris, Pravec & Warner 2012), we do not see this situation in the spin-rate distributions for various intervals of ∆m (i.e., roughly ﬂat for asteroids with ∆m < 0.2 and 0.2 < ∆m < 0.4 mag). Moreover, most of ≥ ≤ ≤ ≥ asteroids with ∆m 0.4 mag are of 2 f 5 rev/day and only a few at f 6 rev/day that consists with the sense of lower spin-rate limit for large-amplitude asteroid (see Fig. 8). Moreover, our detection simulation shows roughly fair detection rate for f > 2 rev/day. This indicates that the quasi-Maxwellian distribution for D3-15 asteroids is not a result of ≤ ≤ favoring to detect asteroids of 2 f 5 rev/day. When we look at the spin-rate distribution of asteroids with D < 3 km in Fig. 11, we notice an obvious number decrease at f = 6 rev/day both in PTF-U2s and debiased PTF-U2s. This number drop still remains when applying a amplitude limit of ∆m > 0.2 mag. Since the YORP eﬀect timescale is relatively shorter for asteroids of D < 3 km and moreover the spin-rate limit is lower for large-amplitude asteroids, we believe that more large-amplitude asteroids of D < 3 km have been spun up to reach their spin-rate limit and broken up when comparing with D3-15 asteroids that results in a number decrease at f = 6 rev/day. Note that the small numbers at f < 3 rev/day in PTF-U2s is due to the lower detection rate for small-sized asteroids with long period. However, this deﬁciency becomes more comparable after debiasing. Our result does not aﬀected much by applying a brighter limiting magnitude of 19 mag (see bottom rows in Fig. 10 and Fig. 11), such that our conclusion remains the same. 4.3.3. Spin Rate vs. Spectral Type In PTF-U2 and LCDB data sets, there are 478 C-type, 928 S-type, and 136 V-type asteroids according to SDSS colors. When we look at the plot of diameter vs. rotation period (left panel in Fig. 12) and the plot of spin rate vs. lightcurve amplitude (middle panel) for the C-, S-, and V-types (upper, middle and lower panels, respectively), all of

text: due

text: to

text: a

text: greater

text: ability

text: in

text: detecting

text: f

text: rev/day

text: (Harris,

text: Pravec

text: Warner

text: we

text: do

text: not

text: see

text: this

text: situation

text: in

text: the

text: spin-rate

text: distributions

text: for

text: various

text: intervals

text: of

text: ∆m

text: (i.e.,

text: roughly

text: ﬂat

text: for

text: asteroids

text: with

text: ∆m

text: and

text: ∆m

text: mag).

text: Moreover,

text: most

text: of

text: asteroids

text: with

text: ∆m

text: mag

text: are

text: of

text: f

text: rev/day

text: and

text: only

text: a

text: few

text: at

text: f

text: rev/day

text: that

text: consists

text: with

text: the

text: sense

text: of

text: lower

text: spin-rate

text: limit

text: for

text: large-amplitude

text: asteroid

text: (see

text: Fig.

text: Moreover,

text: our

text: detection

text: simulation

text: shows

text: roughly

text: fair

text: detection

text: rate

text: for

text: f

text: rev/day.

text: This

text: indicates

text: that

text: the

text: quasi-Maxwellian

text: distribution

text: for

text: asteroids

text: is

text: not

text: a

text: result

text: of

text: favoring

text: to

text: detect

text: asteroids

text: of

text: f

text: rev/day.

text: When

text: we

text: look

text: at

text: the

text: spin-rate

text: distribution

text: of

text: asteroids

text: with

text: D

text: km

text: in

text: Fig.

text: we

text: notice

text: an

text: obvious

text: number

text: decrease

text: at

text: f

text: rev/day

text: both

text: in

text: and

text: debiased

text: This

text: number

text: drop

text: still

text: remains

text: when

text: applying

text: a

text: amplitude

text: limit

text: of

text: ∆m

text: mag.

text: Since

text: the

text: YORP

text: eﬀect

text: timescale

text: is

text: relatively

text: shorter

text: for

text: asteroids

text: of

text: D

text: km

text: and

text: moreover

text: the

text: spin-rate

text: limit

text: is

text: lower

text: for

text: large-amplitude

text: asteroids,

text: we

text: believe

text: that

text: more

text: large-amplitude

text: asteroids

text: of

text: D

text: km

text: have

text: been

text: spun

text: up

text: to

text: reach

text: their

text: spin-rate

text: limit

text: and

text: broken

text: up

text: when

text: comparing

text: with

text: asteroids

text: that

text: results

text: in

text: a

text: number

text: decrease

text: at

text: f

text: rev/day.

text: Note

text: that

text: the

text: small

text: numbers

text: at

text: f

text: rev/day

text: in

text: is

text: due

text: to

text: the

text: lower

text: detection

text: rate

text: for

text: small-sized

text: asteroids

text: with

text: long

text: period.

text: However,

text: this

text: deﬁciency

text: becomes

text: more

text: comparable

text: after

text: debiasing.

text: Our

text: result

text: does

text: not

text: aﬀected

text: much

text: by

text: applying

text: a

text: brighter

text: limiting

text: magnitude

text: of

text: mag

text: (see

text: bottom

text: rows

text: in

text: Fig.

text: and

text: Fig.

text: such

text: that

text: our

text: conclusion

text: remains

text: the

text: same.

text: Spin

text: Rate

text: vs.

text: Spectral

text: Type

text: In

text: and

text: LCDB

text: data

text: sets,

text: there

text: are

text: C-type,

text: S-type,

text: and

text: V-type

text: asteroids

text: according

text: to

text: SDSS

text: colors.

text: When

text: we

text: look

text: at

text: the

text: plot

text: of

text: diameter

text: vs.

text: rotation

text: period

text: (left

text: panel

text: in

text: Fig.

text: and

text: the

text: plot

text: of

text: spin

text: rate

text: vs.

text: lightcurve

text: amplitude

text: (middle

text: panel)

text: for

text: the

text: C-,

text: S-,

text: and

text: V-types

text: (upper,

text: middle

text: and

text: lower

text: panels,

text: respectively),

text: all

text: of


text: – 16 – them occupy similar regions in rotation-period or spin-rate spaces. Because of the limiting magnitude of the observations and the various geometric albedo values involved, the asteroids have diﬀerent diameter ranges. We note that C- and V-type asteroids seldom have objects with P > 100 hr and P > 20 hr, respectively. We see the S-type and V-type show clear boundaries at ρ = 2 g/cm3, and the C-type seems to have a boundary at ρ = 1.5 g/cm3 with a small group extending to ρ = 2 g/cm3. Therefore, the C-type asteroids have much fewer objects with f > 8 rev/day. Since M-type and E-type asteroids cannot be separated from C-type asteroids by SDSS colors, the small group in the C-type asteroids is properly due to those indistinguishable M-type and E-type asteroids, which have larger bulk density with respect to C-type. Despite the small group of C-type, what we see here is in a good agreement with the general sense of a lower bulk density for C-type relative to S-/V-type. For a comparison without possible incompleteness at the large/small-sized ends, we controlled our samples in the range of 3 < D < 20 km to generate spin-rate distributions for these three asteroid types (right panel of Fig. 12), in which there were 229, 657, and 81 objects, respectively. The overall shapes do not appear to be the same, but all distributions ≤ ≤ peak at 2 f 5 rev/day. Besides, the C-type distribution is more Maxwellian-like. The − Kolmogorov-Smirnov test on the spin-rate distribution gives p values of 0.01, 0.003, and 0.026 for each pairing of CS, CV and SV types, respectively. This indicates that their spin-rate distributions might be diﬀerent to each other. However, this preliminary study needs to be conﬁrmed when the samples are more complete in the parameter space. 5. Summary Two surveys for asteroid rotation periods have been carried out by using the iPTF in January 6-9 and February 20-23 of 2014. We obtained 1,438 rotation periods from this ≥ campaign having quality codes U 2, and most of them are associated with main-belt asteroids. There are 53 survey objects that match objects in the LCDB, and the derived

text: them

text: occupy

text: similar

text: regions

text: in

text: rotation-period

text: or

text: spin-rate

text: spaces.

text: Because

text: of

text: the

text: limiting

text: magnitude

text: of

text: the

text: observations

text: and

text: the

text: various

text: geometric

text: albedo

text: values

text: involved,

text: the

text: asteroids

text: have

text: diﬀerent

text: diameter

text: ranges.

text: We

text: note

text: that

text: C-

text: and

text: V-type

text: asteroids

text: seldom

text: have

text: objects

text: with

text: P

text: hr

text: and

text: P

text: hr,

text: respectively.

text: We

text: see

text: the

text: S-type

text: and

text: V-type

text: show

text: clear

text: boundaries

text: at

text: ρ

text: and

text: the

text: C-type

text: seems

text: to

text: have

text: a

text: boundary

text: at

text: ρ

text: with

text: a

text: small

text: group

text: extending

text: to

text: ρ

text: Therefore,

text: the

text: C-type

text: asteroids

text: have

text: much

text: fewer

text: objects

text: with

text: f

text: rev/day.

text: Since

text: M-type

text: and

text: E-type

text: asteroids

text: cannot

text: be

text: separated

text: from

text: C-type

text: asteroids

text: by

text: SDSS

text: colors,

text: the

text: small

text: group

text: in

text: the

text: C-type

text: asteroids

text: is

text: properly

text: due

text: to

text: those

text: indistinguishable

text: M-type

text: and

text: E-type

text: asteroids,

text: which

text: have

text: larger

text: bulk

text: density

text: with

text: respect

text: to

text: C-type.

text: Despite

text: the

text: small

text: group

text: of

text: C-type,

text: what

text: we

text: see

text: here

text: is

text: in

text: a

text: good

text: agreement

text: with

text: the

text: general

text: sense

text: of

text: a

text: lower

text: bulk

text: density

text: for

text: C-type

text: relative

text: to

text: S-/V-type.

text: For

text: a

text: comparison

text: without

text: possible

text: incompleteness

text: at

text: the

text: large/small-sized

text: ends,

text: we

text: controlled

text: our

text: samples

text: in

text: the

text: range

text: of

text: D

text: km

text: to

text: generate

text: spin-rate

text: distributions

text: for

text: these

text: three

text: asteroid

text: types

text: (right

text: panel

text: of

text: Fig.

text: in

text: which

text: there

text: were

text: and

text: objects,

text: respectively.

text: The

text: overall

text: shapes

text: do

text: not

text: appear

text: to

text: be

text: the

text: same,

text: but

text: all

text: distributions

text: peak

text: at

text: f

text: rev/day.

text: Besides,

text: the

text: C-type

text: distribution

text: is

text: more

text: Maxwellian-like.

text: The

text: Kolmogorov-Smirnov

text: test

text: on

text: the

text: spin-rate

text: distribution

text: gives

text: p

text: values

text: of

text: and

text: for

text: each

text: pairing

text: of

text: CS,

text: CV

text: and

text: SV

text: types,

text: respectively.

text: This

text: indicates

text: that

text: their

text: spin-rate

text: distributions

text: might

text: be

text: diﬀerent

text: to

text: each

text: other.

text: However,

text: this

text: preliminary

text: study

text: needs

text: to

text: be

text: conﬁrmed

text: when

text: the

text: samples

text: are

text: more

text: complete

text: in

text: the

text: parameter

text: space.

text: Summary

text: Two

text: surveys

text: for

text: asteroid

text: rotation

text: periods

text: have

text: been

text: carried

text: out

text: by

text: using

text: the

text: iPTF

text: in

text: January

text: and

text: February

text: of

text: We

text: obtained

text: rotation

text: periods

text: from

text: this

text: campaign

text: having

text: quality

text: codes

text: U

text: and

text: most

text: of

text: them

text: are

text: associated

text: with

text: main-belt

text: asteroids.

text: There

text: are

text: survey

text: objects

text: that

text: match

text: objects

text: in

text: the

text: LCDB,

text: and

text: the

text: derived


text: – 20 – Lacerda, P., & Luu, J. 2003, Icarus, 161, 174 Lacerda, P., & Jewitt, D. C. 2007, AJ, 133, 1393 Laher, R. R., Surace, J., Grillmair, C. J., et al. 2014, PASP, 126, 674 Law, N. M., Kulkarni, S. R., Dekany, R. G., et al. 2009, PASP, 121, 1395 Law, N. M., Dekany, R. G., Rahmer, G., et al. 2010, Proc. SPIE, 7735 Mainzer, A., Grav, T., Bauer, J., et al. 2011, ApJ, 743, 156 Margot, J. L., Nolan, M. C., Benner, L. A. M., et al. 2002, Science, 296, 1445 ˇ Masiero, J., Jedicke, R., Durech, J., et al. 2009, Icarus, 204, 145 Masiero, J. R., Mainzer, A. K., Grav, T., et al. 2011, ApJ, 741, 68 Ofek, E. O., Laher, R., Law, N., et al. 2012a, PASP, 124, 62 Ofek, E. O., Laher, R., Surace, J., et al. 2012b, PASP, 124, 854 Polishook, D., Ofek, E. O., Waszczak, A., et al. 2012, MNRAS, 421, 2094 Pravec, P., & Harris, A. W. 2000, Icarus, 148, 12 ˇ Pravec, P., Kuˇsnir´ak, P., Sarounova´, L., et al. 2002, Asteroids, Comets, and Meteors: ACM 2002, 500, 743 Pravec, P., Scheirich, P., Kuˇsnira´k, P., et al. 2006, Icarus, 181, 63 Pravec, P., Harris, A. W., Vokrouhlicky´, D., et al. 2008, Icarus, 197, 497 Pray, D. P., Galad, A., Husarik, M., & Oey, J. 2008, Minor Planet Bulletin, 35, 34 Rau, A., Kulkarni, S. R., Law, N. M., et al. 2009, PASP, 121, 1334

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text: al.

text: PASP,


text: )gam( R – 35 – 19.5 c4547 U=2 2109..08 c4655 U=2 1199..43 c4997 U=3 2200..42 c5104 U=2 20.0 c5529 U=2 222222222221111000101000009998...............460000085520855 PcPcPc567===501555453...565053519373 UUU===322 222222222222211000001010000199...............4466808005222050.0PcPcP56===65026514....3728352144690.4 0.6 0.UU8==221.022222211222111110000019900199999................44608066222889570.0PcPcP56===75076407....5823952578710.4 0.6 0.UU8==221.0222222222222222222221100010001001001101019.....................4446666800800802222280.0PcPcP56===781033611...13207.11969430.4 0.6 0.UU8==221.0222222221112211000010009990099...............4466000546622880.0PcPcP57===81075391....3426521879160.4 0.6 0.UU8==221.0 19.0 19.5 20.0 P=4.192 0.0 0.2 0.4 0.6 0.8 1.0 Phase Fig. 14.— Same as Fig. 13 for other 15 PTF-U2 asteroids. ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## 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text: – 49 – Table 2—Continued (cid:52) (cid:52) Obj ID Designation a e i Ω ω D r α HR n m PTFR Period (hr) m U ± ± ± 29584 (29584) 1998 FQ60 3.11 0.17 0.84 357.5 85.7 9.6w2.85 1.86 0.48 13.58 0.11 4 94 17.35 0.01 4.66 0.02 0.20 3 ± ± ± 29804 (29804) 1999 CH90 2.40 0.18 2.05 330.4 26.0 6.5w2.77 1.78 2.16 14.84 0.09 2 43 18.54 0.03 3.19 0.03 0.16 2 ± ± ± 29955 (29955) 1999 JE90 3.08 0.04 10.90 116.3 73.2 5.4w3.05 2.07 3.03 13.78 0.15 4 71 18.08 0.02 3.98 0.03 0.24 3 ± ± ± 30168 (30168) 2000 GG66 2.41 0.18 1.61 235.1 54.7 2.4 2.84 1.87 2.74 15.09 0.10 2 47 18.93 0.05 5.68 0.16 0.21 2 ± ± ± 30467 (30467) 2000 OV14 3.06 0.05 8.04 306.9 52.2 8.1w3.18 2.21 3.88 13.30 0.21 4 99 17.86 0.02 5.93 0.04 0.60 3 ± ± ± 30491 (30491) 2000 QJ38 3.19 0.17 2.17 133.0 316.7 8.7w2.70 1.75 6.60 13.86 0.11 4 69 17.71 0.02 3.60 0.04 0.12 2 ± ± ± 30567 (30567) 2001 OR90 2.36 0.13 3.00 218.8 163.1 1.5 2.51 1.53 4.68 16.13 0.12 4 66 19.42 0.07 3.62 0.03 0.33 2 ± ± ± 30772 (30772) 1986 RJ1 2.38 0.25 5.15 343.0 355.9 6.3w2.97 1.98 0.92 14.86 0.17 4 59 18.83 0.05 8.07 0.07 0.52 3 ± ± ± 31234 (31234) 1998 CL1 3.06 0.18 2.21 254.3 225.6 5.5 2.57 1.59 3.50 15.05 0.11 4 73 18.44 0.03 3.56 0.03 0.18 2 ± ± ± 31291 (31291) 1998 FH64 3.16 0.14 2.69 138.5 51.3 7.6w3.01 2.03 2.35 14.23 0.15 4 67 18.40 0.03 3.62 0.04 0.12 2 ± ± ± 31427 (31427) 1999 BS5 2.43 0.14 0.66 260.7 290.6 2.0 2.32 1.35 3.75 15.49 0.23 4 82 18.23 0.03 5.36 0.03 0.61 3 ± ± ± 31641 (31641) 1999 GW34 2.44 0.13 1.21 278.4 215.6 2.7w2.13 1.15 1.74 14.94 0.20 4 80 17.00 0.01 2.82 0.01 0.54 3 ± ± ± 31716 (31716) 1999 JJ57 3.04 0.15 3.45 162.3 89.4 5.9w3.07 2.09 2.15 14.34 0.12 4 77 18.61 0.04 4.59 0.02 0.22 3 ± ± ± 31774 (31774) 1999 JW121 3.04 0.12 2.60 144.9 97.0 5.4w3.00 2.01 0.75 14.55 0.09 4 87 18.57 0.04 3.17 0.02 0.27 2 ± ± ± 31818 (31818) 1999 RM135 3.22 0.06 22.64 340.0 170.0 19.5w3.02 2.04 1.69 12.20 0.15 3 66 16.22 0.01 5.36 0.03 0.51 3 ± ± ± 31862 (31862) 2000 EY70 2.38 0.14 8.65 106.8 245.9 2.3w2.52 1.56 5.70 15.09 0.22 4 71 18.50 0.03 5.71 0.03 0.62 3 ± ± ± 31888 (31888) 2000 FM35 2.35 0.14 7.31 159.7 143.8 2.4 2.64 1.67 3.77 15.14 0.12 4 98 18.75 0.04 5.13 0.03 0.26 3 ± ± ± 31978 (31978) 2000 HA14 2.37 0.18 2.02 15.0 222.4 2.3 2.47 1.56 11.19 15.22 0.12 3 61 18.80 0.04 3.42 0.02 0.27 2 ± ± ± 32089 (32089) 2000 KO28 2.40 0.15 1.29 171.5 70.4 1.9 2.56 1.60 5.73 15.62 0.19 4 73 19.12 0.06 6.62 0.05 0.70 3 ± ± ± 32145b (32145) 2000 LE30 2.42 0.18 9.02 105.9 139.4 4.4w2.64 1.68 5.11 14.67 0.25 2 57 18.30 0.03 9.14 0.09 0.85 3 ± ± ± 32215 (32215) 2000 OG16 3.12 0.20 2.67 101.7 229.5 8.3w3.65 2.73 6.35 14.19 0.28 2 49 19.65 0.08 8.00 0.07 0.90 3 ± ± ± 32254 (32254) 2000 OR51 3.22 0.07 13.32 290.2 54.9 15.5w3.34 2.36 2.42 12.92 0.14 4 81 17.66 0.02 5.42 0.03 0.33 3 ± ± ± 32290 (32290) 2000 QH5 2.45 0.13 12.88 344.1 221.9 6.9w2.24 1.25 2.43 14.45 0.10 3 38 16.92 0.01 3.42 0.01 0.28 2 ± ± ± 32354 (32354) 2000 QN119 3.15 0.07 9.49 313.3 260.5 9.0w3.13 2.22 8.00 13.67 0.15 4 69 18.56 0.04 3.93 0.03 0.18 2 ± ± ± 33121 (33121) 1998 BR15 2.54 0.11 9.25 105.2 202.1 4.2w2.81 1.85 5.42 14.69 0.28 1 26 18.68 0.04 6.23 0.17 0.80 2 ± ± ± 33165 (33165) 1998 EO2 3.19 0.25 2.19 143.0 200.7 10.8w3.94 2.96 1.16 13.59 0.14 4 81 19.05 0.05 17.78 0.34 0.41 3 ± ± ± 33197 (33197) 1998 FA52 3.17 0.11 2.97 105.4 123.7 7.6w3.29 2.32 2.60 14.35 0.13 4 86 18.99 0.05 6.71 0.05 0.32 3 ± ± ± 33329 (33329) 1998 RY77 2.30 0.11 5.27 152.1 51.7 3.5w2.13 1.15 3.49 14.62 0.10 4 83 16.65 0.01 3.45 0.01 0.21 3 ± ± ± 34154 (34154) 2000 QC20 2.48 0.15 3.16 140.0 124.2 2.3 2.79 1.81 2.91 15.24 0.22 4 76 19.04 0.07 6.62 0.05 0.72 3 ± ± ± 34217 (34217) 2000 QA78 2.48 0.18 6.11 344.1 357.9 3.3w2.91 1.93 0.85 14.17 0.10 4 74 18.04 0.03 7.33 0.06 0.25 3 ± ± ± 34226 (34226) 2000 QM88 2.54 0.23 4.38 270.9 74.7 5.0 2.80 1.83 3.38 14.53 0.15 4 81 18.28 0.03 3.25 0.01 0.26 3 ± ± ± 34319 (34319) 2000 QD193 3.10 0.16 3.41 106.4 168.0 6.5w3.54 2.57 3.13 14.64 0.21 4 60 19.75 0.09 8.57 0.15 0.57 2 ##LTLine## ##LTLine## ##LTLine##

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text: – 61 – Table 2—Continued (cid:52) (cid:52) Obj ID Designation a e i Ω ω D r α HR n m PTFR Period (hr) m U ± ± ± 96014 (96014) 2004 PD7 2.45 0.19 3.15 152.6 189.7 1.4 2.90 1.91 0.75 16.31 0.16 4 54 20.14 0.12 3.56 0.03 0.55 2 ± ± ± 96069 (96069) 6060 P-L 2.53 0.15 3.80 342.6 45.5 2.6w2.48 1.54 8.50 15.20 0.07 2 52 18.65 0.03 2.20 0.02 0.20 2 ± ± ± 96796 (96796) 1999 RQ124 2.66 0.17 12.24 318.5 41.6 3.2 2.87 1.97 9.69 15.50 0.33 4 61 19.80 0.08 5.96 0.04 0.92 3 ± ± ± 96848 (96848) 1999 RL221 2.58 0.24 10.28 147.5 184.6 4.4 3.21 2.22 0.80 14.78 0.11 2 48 19.17 0.06 6.36 0.20 0.23 2 ± ± ± 96974 (96974) 1999 TL194 2.73 0.15 7.56 286.2 131.0 5.5w2.45 1.48 4.42 15.25 0.11 4 71 18.41 0.04 3.47 0.03 0.27 2 ± ± ± 97029 (97029) 1999 TT323 2.69 0.16 12.03 295.4 48.1 4.0 2.94 1.99 5.32 14.97 0.14 4 68 19.22 0.07 3.32 0.02 0.32 2 ± ± ± 97050 (97050) 1999 UD45 2.68 0.20 10.71 299.1 52.8 5.7w2.93 1.98 6.22 15.09 0.15 4 75 19.49 0.07 6.11 0.11 0.51 2 ± ± ± 97512 (97512) 2000 CV118 2.86 0.08 1.15 318.2 53.4 3.8 2.89 1.92 3.84 15.84 0.18 4 66 19.82 0.10 9.06 0.08 0.55 2 ± ± ± 98442 (98442) 2000 UT55 2.59 0.17 4.59 151.7 344.0 2.5w2.15 1.17 5.25 15.06 0.10 2 27 17.47 0.01 2.87 0.03 0.17 2 ± ± ± 98505 (98505) 2000 VC16 2.57 0.11 6.14 98.5 70.1 4.5 2.37 1.47 12.06 14.73 0.06 1 26 18.12 0.02 3.05 0.11 0.20 2 ± ± ± 98746 (98746) 2000 YQ49 2.61 0.17 3.14 273.3 108.7 2.8 2.81 1.83 3.48 15.74 0.14 4 70 19.57 0.08 10.11 0.21 0.42 3 ± ± ± 98786 (98786) 2000 YR96 2.66 0.12 6.74 114.0 6.6 3.3 2.36 1.38 4.13 15.44 0.20 4 81 18.42 0.03 17.45 0.31 0.63 3 ± ± ± 99013 (99013) 2001 DX64 2.66 0.30 7.13 314.9 347.1 3.2w3.30 2.31 1.80 14.51 0.14 4 82 19.12 0.05 3.54 0.01 0.38 3 ± ± ± 99014 (99014) 2001 DG65 2.66 0.05 11.23 333.0 70.4 3.2w2.70 1.71 1.88 15.05 0.11 4 80 18.58 0.04 14.55 0.45 0.22 2 ± ± ± 99163 (99163) 2001 FO140 2.72 0.12 6.07 107.4 354.6 3.4 2.40 1.46 9.17 15.34 0.30 4 78 18.68 0.04 13.15 0.18 0.94 3 ± ± ± 99172 (99172) 2001 FG160 2.70 0.03 5.46 328.4 63.9 3.8 2.74 1.75 1.52 15.14 0.11 4 83 18.74 0.04 7.06 0.10 0.25 2 ± ± ± 99175 (99175) 2001 FY163 2.72 0.10 1.85 138.8 249.0 3.2 2.84 1.85 1.75 15.51 0.12 4 84 19.15 0.06 3.36 0.02 0.28 3 ± ± ± 99189 (99189) 2001 FV190 2.71 0.06 8.46 114.8 340.4 4.5w2.56 1.62 8.08 15.61 0.22 4 71 19.21 0.06 11.16 0.13 0.89 3 ± ± ± 99264 (99264) 2001 OP42 2.36 0.17 4.05 293.1 335.5 1.6 2.71 1.74 3.51 15.99 0.13 4 68 19.67 0.09 2.95 0.04 0.34 2 ± ± ± 99302 (99302) 2001 RU92 3.07 0.13 2.45 265.1 146.6 5.5 3.05 2.07 3.30 15.04 0.10 4 70 19.38 0.06 5.65 0.07 0.21 2 ± ± ± 99760b (99760) 2002 JN100 2.62 0.10 15.53 118.4 15.6 5.6w2.35 1.40 7.27 14.71 0.28 4 81 17.78 0.02 7.50 0.06 0.85 3 ± ± ± A0390 (100390) 1995 WK13 2.29 0.14 6.08 101.4 161.3 1.6 2.50 1.59 11.12 16.03 0.32 3 43 19.77 0.08 8.42 0.15 0.82 2 ± ± ± A0685 (100685) 1997 YH2 2.17 0.12 5.34 99.8 29.7 1.2 1.92 0.96 8.46 16.57 0.06 2 50 18.42 0.04 2.50 0.03 0.18 2 ± ± ± A1174 (101174) 1998 SZ4 2.35 0.18 3.28 125.8 245.9 1.8w2.37 1.41 5.43 16.13 0.35 3 61 19.18 0.06 9.06 0.08 1.00 3 ± ± ± A1383 (101383) 1998 UK23 2.35 0.12 13.39 303.4 50.3 2.0w2.48 1.53 7.84 15.48 0.09 1 28 18.90 0.05 2.57 0.11 0.35 2 ± ± ± A1978 (101978) 1999 RZ52 2.21 0.14 4.43 335.0 13.5 1.4 2.39 1.45 8.83 16.25 0.15 4 65 19.53 0.06 5.42 0.06 0.30 2 ± ± ± A2032 (102032) 1999 RC106 2.20 0.19 5.02 281.6 2.2 2.3w2.61 1.64 4.20 15.52 0.23 4 70 19.03 0.06 5.65 0.03 0.74 3 ± ± ± A2333 (102333) 1999 TK112 2.22 0.15 1.59 314.4 2.4 1.1 2.53 1.57 6.60 16.85 0.27 3 52 20.31 0.13 7.16 0.05 1.04 3 ± ± ± A2358 (102358) 1999 TN134 2.58 0.15 4.01 0.4 267.7 2.4w2.68 1.69 1.53 15.37 0.19 4 86 18.86 0.05 18.11 0.34 0.58 3 ± ± ± A3375b (103375) 2000 AT114 2.31 0.05 6.08 121.9 22.9 1.2 2.20 1.22 5.35 16.68 0.26 4 75 19.21 0.06 7.33 0.06 0.79 3 ± ± ± A3424 (103424) 2000 AY155 2.29 0.12 2.38 301.2 152.7 1.6w2.04 1.10 10.66 16.07 0.09 4 78 18.47 0.03 4.51 0.04 0.12 2 ± ± ± A3708 (103708) 2000 CD84 2.80 0.20 2.34 324.5 267.6 3.3 2.60 1.61 0.78 15.39 0.11 4 78 18.45 0.04 5.36 0.06 0.21 2 ##LTLine## ##LTLine## ##LTLine##

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text: – 63 – Table 2—Continued (cid:52) (cid:52) Obj ID Designation a e i Ω ω D r α HR n m PTFR Period (hr) m U ± ± ± B4074 (114074) 2002 VF33 3.05 0.20 2.69 94.6 309.7 5.2 2.72 1.74 3.39 15.20 0.09 3 59 18.87 0.05 3.93 0.05 0.23 3 ± ± ± B4132 (114132) 2002 VH54 2.90 0.01 2.10 30.0 32.5 5.2 2.90 1.92 1.16 15.17 0.11 4 75 19.06 0.06 2.70 0.02 0.33 2 ± ± ± B4398 (114398) 2002 YT13 2.37 0.22 1.56 261.3 131.8 1.4w2.17 1.20 4.63 16.45 0.22 4 91 18.86 0.05 6.11 0.04 0.69 3 ± ± ± B4419 (114419) 2002 YQ31 2.39 0.21 4.16 300.6 100.8 1.6w2.18 1.23 9.25 15.87 0.25 3 62 18.58 0.04 4.97 0.03 0.59 3 ± ± ± B4508 (114508) 2003 AS82 2.39 0.22 4.04 114.3 244.4 4.3w2.51 1.54 4.58 15.83 0.14 4 66 18.96 0.05 14.33 0.61 0.25 2 ± ± ± B4546 (114546) 2003 BY29 2.38 0.15 1.97 311.2 115.3 1.4w2.11 1.14 5.96 16.73 0.29 4 72 19.07 0.06 8.73 0.08 0.80 3 ± ± ± B4574 (114574) 2003 BL68 3.13 0.19 19.13 147.9 98.1 7.2w3.03 2.04 1.24 14.09 0.16 4 80 18.33 0.03 5.36 0.03 0.40 3 ± ± ± B4756 (114756) 2003 HC45 2.53 0.18 1.60 93.7 205.1 2.3 2.98 2.04 6.40 16.20 0.16 4 50 20.53 0.15 3.18 0.03 0.48 2 ± ± ± B4915 (114915) 2003 QU34 2.67 0.17 12.90 150.4 252.7 2.5 2.75 1.76 0.61 15.99 0.23 4 73 19.54 0.08 16.55 0.28 0.74 3 ± ± ± B4999 (114999) 2003 QR74 2.66 0.11 7.96 309.8 103.4 3.2 2.65 1.67 3.46 15.51 0.28 2 37 19.02 0.06 5.42 0.03 0.80 2 ± ± ± B5152 (115152) 2003 SA68 2.73 0.19 2.16 207.6 183.2 3.3 2.85 1.87 4.50 15.43 0.12 4 71 19.45 0.07 5.65 0.03 0.32 3 ± ± ± B5504 (115504) 2003 UH29 2.80 0.08 0.57 76.2 325.1 4.4w2.72 1.75 3.97 16.18 0.16 4 63 19.91 0.10 10.11 0.31 0.59 2 ± ± ± B5887 (115887) 2003 VT1 2.75 0.07 7.75 307.9 198.5 2.7 2.55 1.57 3.62 15.89 0.33 4 75 19.29 0.06 12.31 0.16 1.07 2 ± ± ± B6232 (116232) 2003 YW5 2.75 0.03 6.52 138.9 191.8 3.0 2.83 1.84 1.26 15.66 0.14 4 72 19.39 0.07 2.88 0.03 0.29 2 ± ± ± B6503 (116503) 2004 BN23 2.89 0.05 3.11 123.8 303.5 4.0 2.87 1.89 1.13 15.76 0.20 4 75 19.54 0.08 11.85 0.14 0.67 2 ± ± ± B6588 (116588) 2004 BJ98 2.26 0.15 6.95 157.2 153.8 1.4 2.57 1.58 1.26 16.30 0.23 4 67 19.54 0.07 5.61 0.03 0.70 3 ± ± ± B6652 (116652) 2004 CX24 2.28 0.18 5.40 297.6 329.3 1.6 2.60 1.65 6.46 16.04 0.24 4 64 19.96 0.13 3.66 0.04 0.75 2 ± ± ± B6653 (116653) 2004 CJ29 2.86 0.08 0.99 294.2 53.6 4.3 3.07 2.08 0.58 15.61 0.20 4 75 19.85 0.09 8.81 0.08 0.55 3 ± ± ± B6944 (116944) 2004 GK44 2.27 0.13 1.00 198.9 101.0 1.0 2.50 1.51 0.75 17.09 0.14 4 67 19.97 0.09 2.75 0.02 0.32 2 ± ± ± B7352 (117352) 2004 XQ69 2.66 0.20 2.54 295.2 67.7 2.8 2.75 1.77 3.60 15.77 0.25 4 63 19.50 0.07 6.53 0.04 0.86 3 ± ± ± B7545 (117545) 2005 EN5 2.78 0.14 7.88 144.2 135.7 3.1 2.97 1.98 0.93 15.56 0.19 4 74 19.56 0.08 5.08 0.03 0.60 3 ± ± ± B8219 (118219) 1996 HT20 2.41 0.09 1.19 135.2 357.2 4.4w2.21 1.24 4.11 15.88 0.12 4 75 18.34 0.03 4.82 0.02 0.16 3 ± ± ± B8555 (118555) 2000 FH4 2.37 0.15 1.29 311.4 293.0 1.2 2.58 1.61 3.31 16.71 0.25 4 72 20.07 0.11 6.96 0.05 0.76 3 ± ± ± B8594 (118594) 2000 GZ89 2.37 0.07 7.59 109.7 130.9 1.5 2.45 1.50 7.50 16.09 0.26 4 92 19.46 0.08 5.33 0.03 0.87 3 ± ± ± B8756 (118756) 2000 QK200 3.03 0.14 14.67 156.1 111.9 5.1 3.17 2.18 1.20 15.24 0.16 4 64 19.68 0.09 11.16 0.13 0.38 2 ± ± ± B8873 (118873) 2000 TY14 2.56 0.08 1.11 21.7 69.4 2.5 2.38 1.42 7.57 16.03 0.16 4 75 19.18 0.06 3.12 0.01 0.39 3 ± ± ± B9173 (119173) 2001 QL49 2.41 0.17 2.86 122.0 271.5 1.6 2.28 1.31 5.61 16.01 0.12 4 73 18.82 0.04 11.29 0.27 0.29 3 ± ± ± B9184 (119184) 2001 QK87 2.43 0.17 11.64 287.1 135.5 3.8w2.14 1.17 5.26 15.61 0.10 4 81 17.98 0.02 3.42 0.02 0.21 3 ± ± ± B9403 (119403) 2001 TU62 2.43 0.14 2.34 168.3 289.1 1.5w2.20 1.21 3.32 16.14 0.11 4 78 18.60 0.04 8.65 0.16 0.30 3 ± ± ± C0071 (120071) 2003 DM16 3.08 0.03 9.98 153.2 7.0 5.1 3.00 2.01 1.37 15.21 0.16 4 82 19.32 0.07 9.70 0.10 0.45 3 ± ± ± C0556 (120556) 1995 CG10 2.42 0.21 2.83 167.5 185.6 1.4 2.87 1.88 1.19 16.29 0.31 4 46 20.15 0.10 5.85 0.04 1.03 3 ± ± ± C0559 (120559) 1995 EB2 2.43 0.15 2.02 183.0 247.9 1.3 2.29 1.31 5.96 16.39 0.11 4 64 19.15 0.06 2.77 0.02 0.25 2 ##LTLine## ##LTLine## ##LTLine##

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text: PTFR

text: Period

text: (hr)

text: m

text: U

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text: – 64 – Table 2—Continued (cid:52) (cid:52) Obj ID Designation a e i Ω ω D r α HR n m PTFR Period (hr) m U ± ± ± C0785 (120785) 1998 FT3 2.54 0.17 1.97 15.5 72.6 2.1 2.15 1.24 13.06 16.35 0.19 4 70 19.24 0.06 4.62 0.02 0.50 3 ± ± ± C0815 (120815) 1998 HB42 2.59 0.19 5.56 157.1 164.7 2.3 3.09 2.10 1.00 16.17 0.12 4 63 20.33 0.12 2.25 0.01 0.35 2 ± ± ± C0960 (120960) 1998 VC3 2.37 0.20 0.79 312.7 44.8 1.2 2.49 1.51 4.15 16.66 0.25 4 54 19.87 0.10 6.36 0.04 0.91 3 ± ± ± C1017b (121017) 1999 BG4 2.40 0.10 2.43 351.6 203.1 3.8w2.21 1.22 2.19 16.19 0.19 3 62 18.62 0.04 9.23 0.09 0.78 3 ± ± ± C2154 (122154) 2000 JS59 2.38 0.13 5.18 318.0 4.7 2.1 2.70 1.72 3.90 15.41 0.12 4 71 19.13 0.06 3.53 0.03 0.29 2 ± ± ± C2350 (122350) 2000 QB42 2.56 0.35 9.58 336.8 92.0 3.2 2.16 1.17 1.02 15.51 0.12 4 83 17.50 0.02 7.27 0.05 0.34 3 ± ± ± C2415 (122415) 2000 QF94 2.55 0.17 5.99 296.5 91.4 2.2 2.43 1.46 3.32 16.32 0.14 4 74 19.20 0.06 5.96 0.04 0.45 3 ± ± ± C2561 (122561) 2000 RP2 2.52 0.13 1.29 11.5 39.1 4.0 2.57 1.59 2.33 14.99 0.09 4 81 18.32 0.03 3.50 0.03 0.20 2 ± ± ± C2603 (122603) 2000 RM40 2.52 0.15 8.11 303.9 69.1 4.0 2.74 1.77 4.29 15.02 0.09 4 79 18.79 0.05 4.40 0.04 0.22 2 ± ± ± C2702 (122702) 2000 SV24 2.55 0.17 3.02 326.3 81.2 2.9 2.59 1.60 0.58 15.67 0.14 4 79 18.73 0.04 6.91 0.05 0.42 3 ± ± ± C2975 (122975) 2000 SE230 2.60 0.23 2.10 38.5 7.2 2.1 2.31 1.36 8.36 16.37 0.33 4 76 19.36 0.07 4.73 0.02 1.07 3 ± ± ± C3241 (123241) 2000 UL60 2.57 0.15 2.07 308.9 80.0 2.5 2.53 1.60 8.68 15.99 0.15 4 63 19.58 0.09 3.04 0.02 0.42 2 ± ± ± C4200 (124200) 2001 OM81 2.47 0.36 8.73 150.7 260.0 2.4 2.33 1.34 0.64 15.15 0.17 4 98 17.75 0.02 9.50 0.10 0.43 3 ± ± ± C4297 (124297) 2001 QE57 2.38 0.13 6.64 129.0 201.9 1.7 2.61 1.63 3.46 15.85 0.26 3 48 19.51 0.09 5.78 0.11 0.77 3 ± ± ± C4393 (124393) 2001 QM186 2.34 0.14 1.82 291.3 330.4 2.8w2.58 1.64 7.48 16.69 0.18 4 63 20.33 0.13 3.89 0.05 0.57 2 ± ± ± C4432 (124432) 2001 QJ247 2.38 0.21 2.21 89.9 268.8 1.4 2.82 1.84 1.77 16.22 0.22 4 68 19.97 0.11 15.48 0.48 0.69 3 ± ± ± C4497 (124497) 2001 RF45 2.43 0.14 7.43 156.5 257.2 3.9w2.42 1.43 2.88 15.19 0.11 4 81 18.14 0.03 7.33 0.06 0.21 3 ± ± ± C4525 (124525) 2001 RE81 2.39 0.20 2.37 298.0 63.7 3.2w2.75 1.77 3.00 16.15 0.12 4 69 19.86 0.10 4.07 0.03 0.36 2 ± ± ± C4575 (124575) 2001 SJ12 2.34 0.08 4.84 176.6 89.2 3.9w2.42 1.45 4.73 16.56 0.29 2 31 19.60 0.09 6.81 0.15 0.86 2 ± ± ± C4627 (124627) 2001 SF60 2.39 0.10 3.87 275.2 114.6 1.4 2.48 1.50 4.84 16.29 0.19 4 69 19.56 0.08 9.23 0.09 0.71 3 ± ± ± C4856 (124856) 2001 TS22 2.44 0.21 1.45 317.0 93.1 1.1 2.41 1.43 2.50 16.80 0.17 4 71 19.71 0.09 2.71 0.02 0.49 3 ± ± ± C4866b (124866) 2001 TM30 2.41 0.14 2.28 264.6 283.7 1.5 2.28 1.30 3.19 16.11 0.30 4 71 18.76 0.05 5.11 0.03 0.83 3 ± ± ± C4934 (124934) 2001 TA76 2.38 0.20 2.44 85.8 259.9 1.5 2.86 1.87 1.59 16.09 0.17 2 34 19.92 0.11 4.92 0.03 0.69 2 ± ± ± C5083 (125083) 2001 UM12 2.46 0.19 1.89 330.2 85.2 1.0 2.19 1.25 10.18 16.99 0.11 2 49 19.81 0.09 5.03 0.08 0.35 2 ± ± ± C5310b (125310) 2001 VN32 2.46 0.17 2.15 303.8 119.5 2.1w2.35 1.37 3.80 15.79 0.40 4 78 18.71 0.04 6.15 0.04 1.26 3 ± ± ± C5390 (125390) 2001 VZ84 2.38 0.10 6.18 320.8 325.2 1.6 2.53 1.55 1.54 16.05 0.18 4 71 19.26 0.07 9.14 0.09 0.53 3 ± ± ± C5539 (125539) 2001 WN90 3.15 0.13 4.49 314.4 183.5 7.4w2.75 1.77 3.79 14.48 0.10 4 78 18.30 0.03 8.07 0.21 0.21 2 ± ± ± C5579 (125579) 2001 XT24 2.49 0.10 6.42 317.2 124.0 2.2w2.36 1.38 3.76 15.70 0.09 4 77 18.52 0.04 2.92 0.02 0.17 2 ± ± ± C5793 (125793) 2001 XC153 2.43 0.20 2.05 144.9 208.6 1.5 2.85 1.86 1.48 16.20 0.20 4 53 19.99 0.12 5.11 0.05 0.63 2 ± ± ± C5828 (125828) 2001 XT174 2.51 0.10 1.38 171.7 244.9 2.2 2.34 1.37 3.51 16.27 0.22 4 74 19.00 0.06 5.85 0.07 0.54 3 ± ± ± C6236 (126236) 2002 AC59 2.53 0.16 8.17 296.0 172.3 1.9w2.12 1.15 4.54 15.94 0.25 4 68 18.27 0.03 9.80 0.10 0.72 3 ∗ ± ± ± C6334 (126334) 2002 AW152 2.54 0.05 5.52 118.3 37.2 2.7 2.44 1.49 7.40 15.82 0.17 4 78 19.13 0.06 3.50 0.01 0.47 3 ##LTLine## ##LTLine## ##LTLine##

text: Table

text: Obj

text: ID

text: Designation

text: a

text: e

text: i

text: Ω

text: ω

text: D

text: r

text: α

text: HR

text: n

text: m

text: PTFR

text: Period

text: (hr)

text: m

text: U

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##