地球接近天体(1566)Icarusとその同一起源候補天体2007 MK6の近赤外測光観測 櫻井 友里1,  浦川 聖太郎2,  高橋 隼3,  谷川 智康4,  中村 小百合5,  はしもと じょーじ1  

1岡山大学大学院自然科学研究科,  2日本スペースガード協会,  3兵庫県立大学,  4三田祥雲館高校,  5岡山大学理学部  

概要

2007 MK6 は Icarus から分裂した？

測光 スペクトル型の推定

まとめ  

観測 西はりま天文台で Icarus と 2007 MK6 の近赤外 3色同時測光観測をおこなった．Icarus と MK6の 近赤外測光観測は世界初である．近赤外の色を比較 することで，Icarus と 2007 MK6 の分裂仮説 (Ohtsuka et al. 2007)について考察した．

Ohtsuka et al. (2007) ・軌道進化計算に基づいて， 　MK6 の母天体は Icarus 　である可能性を指摘 (1566)Icarus ・自転周期 : 2.2726時間 　(Warner 2015)

　- 小惑星が自転により分裂 　　する限界周期 : 2.2時間 ・直径 : 1km 　- YORP効果が働く可能性 ・スペクトル型 : S / Q 　(Gehrels et al. 1970, Pravec et al. 2010)

自転周期と分裂  小惑星の直径が200m以上で， 自転周期が2.2時間より短いものは存在しない

ラブルパイル構造  　　　↓  自転が速くなると  遠心力で分裂する  

Urakawa  et  al.  (2014)

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Figure 3. Plot of the asteroid rotation period, P, vs. their effective diameter, D. The legend of “LCDB” shows the data in the Asteroid Light Curve Database (LCDB)with quality code U ! 2. The filled circle indicates the mean diameter and rotational period of 2011 XA3. The open marks show the diameter and rotational periodfor asteroids that have the possibility of being more than 200 m in diameter.

respectively. An effective diameter of asteroids D (in kilometer)is described as

D = 1329 × 10−HV /5p−1/2V , (3)

where pV is the geometric albedo. Assuming the albedo of0.209 ± 0.008 for S-complex (Pravec et al. 2012) and 0.297 ±0.131 for V-type (Usui et al. 2013), we found the diameter of225 ± 97 m (S-complex) and 166 ± 63 m (V-type), respectively.

Next, we estimate the axial ratio of 2011 XA3. There is arelation between the light curve amplitude and the phase angleas follows:

A(0) = A(α)1 + mα

. (4)

Here, A(α) is the light curve amplitude at the α◦ phase angleand m is the slope coefficient. In the case of an S-type asteroid,the m value is 0.03 (Zappala et al. 1990). Moreover, the m valuedepends on the surface roughness (Ohba et al. 2003). The mvalue is ∼0.02 when we adopt the surface roughness 18◦ ofVesta as the representative for V-type (Li et al. 2013). Assumingthat the axial ratio of 2011 XA3 is projected on the plane of thesky, the light curve amplitude is described through the lowerlimit to the true axial ratio of the body as

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!a

b

", (5)

where a and b are, respectively, normalized long and shortaxis lengths. The amplitude of 2011 XA3 is 0.68 mag andthe light curve data are obtained at the phase angle of ∼40◦.Therefore, the ratio of the long axis length to the short axislength is larger than 1.3 (S-complex) and 1.4 (V-type). Theobtained physical properties of 2011 XA3 are summarized inTable 3. The construction of fast-rotating asteroids is thoughtto be a monolith. We confirm the validity for 2011 XA3. If2011 XA3 is preserved only by self-gravity, the critical bulkdensity, ρ (in g cm−3), for a rubble pile asteroid can bewritten as

ρ =#

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(1 + A(0)), (6)

Table 3Physical Properties of 2011 XA3

Rotational Period 0.0304 ± 0.0003 days

g′ − r ′ 0.581 ± 0.067a

r ′ − i′ 0.171 ± 0.036a

i′ − z′ −0.120 ± 0.045a

g′ − r ′ 0.639 ± 0.070b

r ′ − i′ 0.164 ± 0.088b

i′ − z′ −0.079 ± 0.054b

Taxonomy S-complex | V-type

Absolute magnitude 20.56 ± 0.43 | 20.84 ± 0.25

Diameter 225 ± 97 m | 166 ± 63 m

Axial ratio ! 1.3 | ! 1.4

Notes.a Color index from phase 0.6 to 0.8.b Averaged color index except for between phase 0.6 and 0.8.

where P is the rotational period in hours (Pravec & Harris 2000).Substituting the rotational period and the light curve amplitudeof 2011 XA3 in Equation (6), the bulk density, ρ, becomes26.3 g cm−3. In addition to it, a more precise calculation, usinga theory for cohesionless elastic–plastic solid bodies (Holsapple2004), gives a lower limit on the bulk density of 20.5 g cm−3

(P. Pravec 2014, private communication). These are incrediblevalues as material of asteroids. Therefore, 2011 XA3 is not arubble pile but evidently a monolithic asteroid.

We show the rotational period and the effective diameter inFigure 3 for registered asteroids in the LCDB and 2011 XA3.The effective diameter of 2011 XA3 on Figure 3 corresponds tothe mean value between the diameter of the assumed S-complexasteroids and the diameter of the assumed V-type asteroids.The range of errors designates the upper limit diameter of theassumed S-complex asteroids and the lower limit diameter ofthe assumed V-type asteroids. Moreover, the rotational periodsand effective diameters of 2001 OE84, 2001 FE90, 2001 VF2,

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Supplementary Figure 1: A parent asteroid consisting of small component pieces that canbe pulled apart without tensile resistance is spun up to the critical fission frequency by theYarkovsky-O’Keefe-Radzievskii-Paddack (YORP) effect, forms a proto-binary system whichsubsequently disrupts under its own internal dynamics and becomes an asteroid pair. This is aconceptual sketch based on important model assumptions/limitations.

www.nature.com/ nature 1

YORP$$o�RADIATIVE SPIN-UP AND SPIN-DOWN OF SMALL ASTEROIDS 3

FIG. 1. Spin-up of an asymmetrical asteroid. The asteroid is modeled as asphere with two wedges attached to its equator. The asteroid is a blackbody, sothat it absorbs all the sunlight falling upon it. The solar energy is reemitted asthermal radiation, which causes a net torque on the asteroid.

FIG. 2. The radiative forces on the wedges. Sunlight (wavy arrows) comesin horizontally from the left. Each surface absorbs the same amount of sunlightand reradiates in the infrared according to Lambert’s law. The net momentumof the photons departing the surface is normal to the surface (thick arrows). Byaction–reaction, the wedge is kicked in the opposite direction. There is a nettorque about the rotation axis because each force has the same magnitude butdifferent directions.

solar torques will cancel themselves as the Sun sees the silhou-ette in all aspects.

PSEUDO-ASTEROIDS

The thermal torques from the YORP effect are computed be-low on the five hypothetical asteroids Pseudo-gaspra, Pseudo-ida, Pseudo-eros, Pseudo-phobos, and Pseudo-deimos. They areso-called because their shapes are based on spherical harmonicexpansions of the real shapes of asteroids 951 Gaspra, 243 Ida,and 433 Eros, as well as Phobos and Deimos (the two moonsof Mars). Their sizes are allowed to vary, but the shapes anddensities stay the same. Each is a blackbody. They are takento be in circular orbits about the Sun. Pseudo-gaspra orbits theSun at 2.21 AU, just as the real Gaspra does; likewise Pseudo-ida orbits at 2.86 AU while Pseudo-eros orbits at 1.46 AU, likethe actual asteroids do. Pseudo-phobos and Pseudo-deimos aretaken to circle the Sun at 3 AU, in order to place them in themain asteroid belt. These objects have been chosen because nu-merical shapes have been determined for them by P. Thomas(Thomas et al. 1994, 1996, Rubincam et al. 1995) and others(see Acknowledgments). Hence their shapes will be more re-alistic for (if not necessarily representative of) small asteroidsthan artificially concocted shapes.Each pseudo-asteroid’s shape is given by

r =lmax!

lmiAlmiYlmi (θ, λ), (1)

where the spherical harmonics are

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Ylm2(θ, λ) = Plm(cos θ ) sinmλ,

with Plm(cos θ ) being the associated Legendre polynomial ofdegree l and order m, while θ is colatitude and λ is east longi-tude. Here the Almi are the shape coefficients derived from innerproducts of spherical harmonics with the numerical shapes.The thermal torques are computed numerically by finding

the thermal force on 5◦ × 5◦ squares of the spherical harmonicshape. The Sun shines on an asteroid with strength

FS = FE"aErS

#2, (2)

where rS is the asteroid’s distance from the Sun, and FE =1378 W m−2 at the Earth’s distance aE = 1 AU. The amount ofsolar power deposited on a surface element with area dA is thenSdA = (rS · N)FSdA, where rS is the unit vector pointing fromthe asteroid to the Sun, N is the unit vector normal to the surfaceelement, and S is the insolation. If each surface element is in aradiative steady state with respect to the insolation, the black-body temperature T of each square will be given by σT 4 = S,

RADIATIVE SPIN-UP AND SPIN-DOWN OF SMALL ASTEROIDS 3

FIG. 1. Spin-up of an asymmetrical asteroid. The asteroid is modeled as asphere with two wedges attached to its equator. The asteroid is a blackbody, sothat it absorbs all the sunlight falling upon it. The solar energy is reemitted asthermal radiation, which causes a net torque on the asteroid.

FIG. 2. The radiative forces on the wedges. Sunlight (wavy arrows) comesin horizontally from the left. Each surface absorbs the same amount of sunlightand reradiates in the infrared according to Lambert’s law. The net momentumof the photons departing the surface is normal to the surface (thick arrows). Byaction–reaction, the wedge is kicked in the opposite direction. There is a nettorque about the rotation axis because each force has the same magnitude butdifferent directions.

solar torques will cancel themselves as the Sun sees the silhou-ette in all aspects.

PSEUDO-ASTEROIDS

The thermal torques from the YORP effect are computed be-low on the five hypothetical asteroids Pseudo-gaspra, Pseudo-ida, Pseudo-eros, Pseudo-phobos, and Pseudo-deimos. They areso-called because their shapes are based on spherical harmonicexpansions of the real shapes of asteroids 951 Gaspra, 243 Ida,and 433 Eros, as well as Phobos and Deimos (the two moonsof Mars). Their sizes are allowed to vary, but the shapes anddensities stay the same. Each is a blackbody. They are takento be in circular orbits about the Sun. Pseudo-gaspra orbits theSun at 2.21 AU, just as the real Gaspra does; likewise Pseudo-ida orbits at 2.86 AU while Pseudo-eros orbits at 1.46 AU, likethe actual asteroids do. Pseudo-phobos and Pseudo-deimos aretaken to circle the Sun at 3 AU, in order to place them in themain asteroid belt. These objects have been chosen because nu-merical shapes have been determined for them by P. Thomas(Thomas et al. 1994, 1996, Rubincam et al. 1995) and others(see Acknowledgments). Hence their shapes will be more re-alistic for (if not necessarily representative of) small asteroidsthan artificially concocted shapes.Each pseudo-asteroid’s shape is given by

r =lmax!

lmiAlmiYlmi (θ, λ), (1)

where the spherical harmonics are

Ylm1(θ, λ) = Plm(cos θ ) cosmλ

Ylm2(θ, λ) = Plm(cos θ ) sinmλ,

with Plm(cos θ ) being the associated Legendre polynomial ofdegree l and order m, while θ is colatitude and λ is east longi-tude. Here the Almi are the shape coefficients derived from innerproducts of spherical harmonics with the numerical shapes.The thermal torques are computed numerically by finding

the thermal force on 5◦ × 5◦ squares of the spherical harmonicshape. The Sun shines on an asteroid with strength

FS = FE"aErS

#2, (2)

where rS is the asteroid’s distance from the Sun, and FE =1378 W m−2 at the Earth’s distance aE = 1 AU. The amount ofsolar power deposited on a surface element with area dA is thenSdA = (rS · N)FSdA, where rS is the unit vector pointing fromthe asteroid to the Sun, N is the unit vector normal to the surfaceelement, and S is the insolation. If each surface element is in aradiative steady state with respect to the insolation, the black-body temperature T of each square will be given by σT 4 = S,

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16年9月11日日曜日

・観測日時 　- Icarus : 2015年6月21日 12:12-15:53 (UT) 　- MK6 : 2016年6月17日 12:22-16:15 (UT) ・観測機器 　- 西はりま天文台 　　なゆた望遠鏡(口径2m) 　- 3色同時近赤外撮像装置 NIC 　　- Jバンド : 1.253μm 　　- Hバンド : 1.632μm 　　- Ksバンド : 2.146μm 　　- 視野 : 2.7’ × 2.7’ ・観測方法 　-対象天体と標準星を交互に撮像 　　-標準星の明るさを使い,対象天体の明るさを決定

取得データ

・1次処理 　- ダーク引き, フラット割り, 重ね合わせ, etc. ・測光 　- IRAFを用いたアパーチャー測光 　- アパーチャー半径 　　- Icarus : 25, MK6 : 15 ・標準星 　- 2MASS(近赤外星表カタログ)

1次処理後のIcarus Jバンド FWHM : 8.5 ピクセル

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西はりま天文台での近赤外測光観測の結果によると， Icarus と 2007 MK6 の近赤外の色(J-H, H-Ks)は S型に近い． Icarus と MK6 の色は誤差の範囲で一致．

参考文献

　赤 : Icarus，青 : 2007 MK6，黒 : Sykes et al. 2000 ・西はりまでの近赤外測光観測 　- Icarus : S型，MK6 : S型(?) ・OAOでの可視光測光観測 (浦川 他 PPS07-P07 参照) 　- Icarus : S型，MK6 : V型

測光結果 ・Icarus　　　　　　　　　・MK6

・14225942-0113366　 ・14301571+3942435

Gehrels, T., et al. (1970) Minor Planets and Related Objects. IV. Asteroid (1566) Icarus. Astron. J. 75, 186-195. Ohtsuka, K., et al. (2007) Apollo Asteroids 1566 Icarus and 2007 MK6:Icarus Family Members? Astrophysical J. 668, 71-74. Pravec, P., et al. (2010) Formation of Asteroid pairs by rotational fission. Nature 466, 1085-1088. Sykes, M. V., et al. (2000) The 2MASS Asteroid and Comet Survey. Icarus 146, 161-175. Urakawa, S., et al. (2014) Fast Rotation of a Subkilometer-sized Near-Earth Object 2011 XA3. Astron. J. 147, 121-128. Warner, B. D. (2015) Near-Earth Asteroid Lightcurve Analysis at CS3-Palmer Divide Station: 2015 March-June. The Minor Planet Bulletin, Vol.42, No 4, 256-266

青  :  J  ,  緑  :  H  ,  赤  :  Ks