Microgravity experiment on the collisional
behavior of saturnian ring particles
Daniel Heißelmann, Jürgen Blum, Helen J. Fraser, Kristin Wolling
Icarus, Volume 206, Issue 2, April 2010, Pages 424-430

名古屋大大学2010年4月22日木曜日

地球環境科学専攻

M1

藤田幸浩

1. Introduction
・土星の環
メインリング
無数の粒子で構成されている
大部分が氷を主成分とする粒子（1〜10 cm）
NASA/JPL/Space Science Institute環がもつ特徴的な構造は衛星や小型衛星による摂動が関係

軌道が乱されることで離心率やリングの厚さが増加

この乱れを中和するために低速度（〜0.5cm/s）の非弾性衝突が頻繁に起こり、k.e が散逸する
再び軌道が円形化され、厚さは制限される

土星の環の進化には氷粒子の衝突が大きく関係
2010年4月22日木曜日

1. Introduction
これまでの研究
Bridges et al.(1984)

氷平板への氷球衝突実験

衝突速度、ターゲット表面の霜の量が増加するにつれて反発係数が減少
Supulver et al.(1995)
研磨した氷球の斜め衝突実験

Higa et al.(1996,1998)
氷ターゲットへのサイズの異なる氷球の衝突実験

その他にも、理論・数値シミュレーションが数多く行われてきた
本研究では土星環の粒子間で起こる低速度衝突の研究に対して2つの新しい実験手法を検証する
・Parabolic-ﬂight experiment・Drop-tower experiment
2010年4月22日木曜日

074501-3

2. Experimental setup

Salter et al.

＜Parabolic-ﬂight experiment＞
氷球の二体衝突を低速度で行う
微小重力

発射装置
発射装置

重力加速度 - 0.01G0 程度
継続時間 - 22 s
※G0：地上での重力加速度
Salter et al.

・氷球試料

Rev. Sci. Instrum. 80, 074501 !2009"

FIG. 1. !Color online" A computer aided design schematic showin
through the center of the experiment chamber. The particle storage
蒸留水を型に入れて冷却!1" sits on top of the thermal reservoir !2" built of copper. The
storage unit rises up and down with a cork-screw system !3" to rota
サイズ ｰ 15 [mm]tional particles into the fixed line of fire. A copper shield !4" prot
六方晶氷
storage device from thermal irradiation. The two “firing” pistonsparticle acceleration system stretch from opposite sides beginning
the side flanges and continuing toward the center of the chamber !
・観察whole system is chilled through contact with a copper tubing !6" thtains fps
a flow of liquid nitrogen. The entire instrument is situated w
高速度ビデオカメラで記録：107
vacuum chamber !7". Collisions are monitored from above with theacquisition system via a transparent viewport on the top flange.
ビームスプリッター付き

olor online" A computer aided design schematic showing a cut
e center of the experiment chamber. The particle storage device
2010年4月22日木曜日
topof the thermal reservoir !2" built of copper. The particle

the initial flight, but was later replaced by two synchro

074501-3

2. Experimental setup

Salter et al.

＜Parabolic-ﬂight experiment＞

Rev. Sci. Instrum. 80, 074501 !2009"

Salter et al.

氷球の二体衝突を低速度で行う
微小重力

発射装置
発射装置

重力加速度 - 0.01G0 程度
継続時間 - 22 s
※G0：地上での重力加速度
Salter et al.

・氷球試料

Rev. Sci. Instrum. 80, 074501 !2009"

FIG. 1. !Color online" A computer aided design schematic showin
Color online" A computer
aided design schematic showing a cut through the center of the experiment chamber. The particle storage
蒸留水を型に入れて冷却
he center of the experiment chamber. The particle storage device !1" sits on top of the thermal reservoir !2" built of copper. Then top of the thermal reservoir !2" built of copper. The particle
storage unit rises up and down with a cork-screw system !3" to rota
サイズ ｰ 15 [mm]it rises up and down with a cork-screw system !3" to rotate addi氷球試料
tionalparticles
into
the
fixed
line
of
fire. A copper shield !4" prot
icles into the fixed line of fire. A copper shield !4" protects the
六方晶氷
evice from thermal irradiation.
The two “firing” pistons of the storage device from thermal irradiation. The two “firing” pistonscceleration system stretch from opposite sides beginning outside particle acceleration system stretch from opposite sides beginning
anges and continuing
・観察toward the center of the chamber !5". The the side flanges and continuing toward the center of the chamber !
tem is chilled through contact with a copper tubing !6" that con- whole system is chilled through contact with a copper tubing !6" thw of liquid nitrogen. The entire instrument is situated within atains fps
a flow of liquid nitrogen. The entire instrument is situated w
高速度ビデオカメラで記録：107
hamber !7". Collisions aremonitored from above with the image
vacuum chamber !7". Collisions are monitored from above with then system via a transparent viewport on the top flange.
acquisition system via a transparent viewport on the top flange.
ビームスプリッター付きalflight,was lateraided
replaced
two synchronizedoloronline"but
A computer
design by
schematic
showing a cut
dc motors
in a master-and-slave
configuel centerof the operating
experiment chamber.
The particle storage
device2010年4月22日木曜日
topofreservoir!2" builtof copper.
The particle
Sec.
II theC". thermal
For thepurposeof theexperiments,thethe initial flight, but was later replaced by two synchro(Supulver et al., 1995). Higa et al. (1996, 1998) conducted impact
experiments of different-sized ice spheres ðr ¼ 0:14—3:6 cmÞ intoa larger ice target. They found that the coefficient of restitution decreases with increasing particle size and is independent of
temperature.
A major limitation of all these experimental studies (Bridges
et al., 1984; Hatzes et al., 1988; Higa et al., 1996, 1998) is that they
only treat impacts into a large body of infinite mass and infinitesimal surface curvature. In the case of the pendulum experiments
an additional problem is that the mass and moment of inertia of
the ice sphere are not realistic as they reflect those of the sample-pendulum-system instead of that of the individual projectile.
However, the results obtained by Bridges et al. (1984) are commonly used in numerical simulations of Saturn’s rings.
In this work we present two novel experimental methods forinvestigating collisions of saturnian ring particles. In Section 2
the two experimental setups for collisions of individual ice particles and ensembles of particles are described. The results of our
microgravity experiment campaigns are presented in Section 3,
and their application
to Saturn’s dense rings is given in Section 4.
-5

Salter et al. (2009).
Fourty-one image sequences of colliding ice spheres (15 mm
diameter) were recorded during DLR’s 12th Parabolic Flight Campaign in April 2008. The samples were produced by freezing purified water inside silicone molds in a regular kitchen freezer, toproduce 15 mm-sized, hexagonal-ice spheres. After removing themolds from the freezer, the spheres were extracted and stored inside a bath of liquid N2 from where they were individually loaded
into the sample repository, which had been pre-cooled, and cooled
further to temperatures of %80 K. This procedure is required tominimize the amount of frost on the samples’ surfaces. Although
the maximum exposure time of the samples to the humid air
was less than 1 min, some experiments show tiny amounts of surface material (which might be frost) chipped off during thecollisions.
Since liquid N2 is prohibited on board the parabolic-flight aircraft, the cooling procedure had to be stopped before take-off. Consequently, the thermal block warmed slightly prior to take-off, andsubsequently during flight, resulting in sample temperatures ranging from 130 to 180 K.

2. Experimental setup

2.2. Drop-tower experiment

2. Experimental setup＜Drop-tower experiment＞

ガラスビーズの低速度多体衝突
微小重力

重力加速度 - 10 G0程度

継続時間 -For9ours microgravity studies we used two different experimental approaches. The parabolic-flight setup was designed to study
collisions of pairs of individual ice projectiles, whereas the Bremen
drop tower facility experiment is a prototype, built to study a large
number of low-velocity collisions within an ensemble of particles.
Microgravity is required to be able to investigate low-velocity
pair-collisions of free particles with an arbitrary impact parameter(i.e. from central to grazing collisions), because it is the only way toaccess all the translational and rotational degrees of freedom. Parabolic flights offer typically 22 s of weightlessness per parabolawith a residual acceleration of a few times 0.01g 0 , where g 0 is
Earth’s gravitational acceleration. Drop-tower experiments achieve
residual accelerations of better than 10$5 g 0 for up to 9 s.

・試料：ガラスビーズ（10mm）※ 1set - 50個

BremenDrop tower

A quasi-two-dimensional rectangular glass box of 150 & 150 &15 mm3 volume (see Fig. 1) was constructed to investigate a large
number of collisions within an ensemble of %100 (cm-sized) particles. The test-chamber’s walls are made of 2 cm thick glass toensure highly elastic collisions between the sample particles andthe walls. Two sets of up to 50 projectiles, stored opposite eachother, were accelerated into the test chamber with an initial

試料を両端にセット

2.1. Parabolic-flight experiment

長方形のチャンバーに注入（~10 cm/s）
Measurements were conducted on the German Space Agency’s(DLR) 12th Parabolic Flight Campaign, using a setup consisting of
a cryogenic cooled experiment body, placed inside a high-vacuum
3
chamber. The experimental apparatus is composed of a massive
copper block which acts as a thermal reservoir, with a rotating
sample repository covered by a copper shield, which plays both aprotective and cryo-cooling role. The entire system is initiallycooled to roughly 77 K with liquid nitrogen ðN2 Þ spiraling throughan attached pipe system. Up to 34 pairs of cm-sized ice particles
can be stored in the sample repository, which allows us to collide
approximately one pair per parabola. To accelerate the ice particles
into the collision volume, two diametrically opposed pistons areused, driven by two synchronized DC motors, which accelerate
the particles to maximum velocities between 3 and 17 cm s$1 .
These are positioned outside the cryo-vacuum chamber, and enterthe vacuum by differentially pumped feedthroughs. Since the projectiles are driven towards each other from exactly opposite directions, the resulting binary collisions are predominantly centralized,with normalized impact parameters, b=R, close to zero, where b is

150 × 150 × 15 mm

・観察

高速度ビデオカメラで記録：115 fps※他に2つのカメラで概観を撮影
2010年4月22日木曜日

Fig. 1. Single image of an ensemble of cm-sized glass spheres colliding in
microgravity. The black dots on the particles are marks for later use in thedetermination of particle rotation (see additional online material).

3. Results ＜Parabolic-ﬂight experiment＞

arus 206 (2010) 424–430

ビームスプリッターで捉えた２組の画像から三次元のサンプル座標を生成

衝突前、衝突後の速度：反発係数：ε
衝突パラメーター：
R = r1 + r2

r2
Fig. 2. Image sequence of two 15-mm-sized ice spheres colliding at a relative
velocity of 14 cm s"1 . The temporal separation between two consecutive frames is
4
s. The images were captured using a beam-splitter optics. Therefore, each image
107
frame contains two views of the set of particles, which are separated by an angle of
48.8!.

上下画像間の角度差：48.8

2010年4月22日木曜日

r1b

om this, the relative velocities befored v 3.
respectively,
Results and the normalized
e calculated. Additionally, the velocity
ngential to the colliding surfaces (vbe extracted.
t the impact parameters are restricted
ue to the limitations of our experimenPðe "b=R ¼ 0 to 0.5 with a mean valuer¼=
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiP
geometrical considerations show thatðe " !eÞ
are the most frequent ones. Thus theimpact parameters smaller thanion of P (i.p. <0.5) ¼ ðb=RÞ ¼ 0:25 of" rings. From the obt
mpactswhere r b=R ¼ 0:10 giv
P in Saturn’sP !
ðe " !eÞ '
"
e
n
and
coefficients of restitution no evi- 同様に線形的な相関はplained by a linear defwhere r ¼ 0:10 gives the fraction of the data which can be explainedby a linear dependence.
Based on a similar
analysis of the
ween
these
parameters
could
be found あまり見られない
coefficient of restitudcoefficient of restitution e and the relative impact velocity v
2r¼ 0:11; see Fig. 4) a correlation of those can also be ruled out.
is, (rInin
which
wecomputedthelinear(r
v ¼ 0:11; see Fig. 4)
a detailed analysis of the coefficient of restitution in normalðe Þ and tangential direction ðe Þ no evidence for a correlation of
In a detailed analysis
these values with the impact parameter or the impact velocity
could be found. Elaborate investigation showed that also a depenðe? Þ and tangential dFig. 2. Image sequence of two 15-mm-sized ice spheres colliding at a relative
velocity of 14 cm s"1 . The temporal separation between two consecutive frames is
4
s. The images were captured using a beam-splitter optics. Therefore, each image
107
frame contains two views of the set of particles, which are separated by an angle of
48.8!.
0hree
dimensions, and the trajectories were determineds to the0 data. From this, the relative velocities beforen＜Parabolic-ﬂight
and the normalized experiment＞
he- collision, v and v respectively,.
ameter,b=R, were calculated. Additionally, the velocity
Fig. 3. Obtained coefficients of restitution for ice-particle colliFig. 3. Obtained coefficien
s normal and tangential to the colliding surfaces (v ?
impact parameters in the range b=R ¼ 0—0:5. There is no evid
・衝突パラメーターと反発係数の関係
between the twoimpact
variables. parameters in the rpectively) could be extracted.
?
. 3 it is clear that the impact parameters are restricted（相関係数）
between! the"two variables
tral collisions (due to the limitations of our experimenPn
bi
b
!
ð
e
"
e
Þ
"
i
ranging from b=R ¼ 0 to 0.5 with a mean value
i¼0
R
Ri
ffiffiffiffiffi ;rb=R ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi!ffiffiffiffiffiffiffiffiffiffiffiffiffi"
hHowever, simple geometrical considerations show thatPn
Pn b i b 2
2!
yancing collisions are the most frequent ones. Thus thei¼0 ðei " eÞ 'i¼0 Ri " Rhncollisions
with impact parameters smaller than2
eover only a fraction of P (i.p. <0.5) ¼ ðb=RÞ ¼ 0:25 of
i¼0 i
2
2
ally
occurring
impacts
inSaturn’s
rings.
Fromtheobwherer
fraction of the data w
0.10
b=Rgives theb=R ¼ 0:10
plained by a linear dependence. Based on a simila
dact parameters and coefficients of restitution no evin2
ecorrelation between these parameters D.could
427
Heißelmann/ Icarus 206 (2010)
424–430
beet al.
foundimcoefficientof restitution e and the relativei
線形的な相関はあまり見られない
di¼02
orrelation
analysis, in which we computed the linear(r v ¼ 0:11; see Fig. 4) a correlation of those can ayFig. 3. Obtained coefficients of restitution for ice-particle collisions at near-central
impact parameters in the range b=R ¼ 0—0:5. There is no evidence of a correlation
coefficient:
In a detailed analysis of the coefficient of restit
between the two variables.
ðe? Þ and tangential direction ðek Þ no evidence for2d
衝突速度と反発係数の関係!
" facility almost doubled the duration
thesevalues with the impact parameter or the-the Bremen dropP
tower’s
catapultnbb!
i¼0 ðei " eÞ R " R
could be found. Elaborate investigation
showed thye comparedtoaregulardrop
the catapult.
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiusing
ffiffi ;rb=R ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiwithout
ð1Þ

n

i

i

n
i¼0

2

i

n
bi
i¼0 Ri

b
R

2

2
b=R

2

v

?

k

Fig. 4. Obtained coefficients of restitution for ice-particle collisions in the velocity
2010年4月22日木曜日#1

ocity and the impact parameter could be ruled out.
ce, we can treat the coefficient of restitution data in a statisy. One way to do this, is to plot the cumulative number of colwith a coefficient of restitution 6e (Fig. 5a). It turns out that427
rus 206 (2010) 424–430
n be fitted by a uniform distribution (see dashed curve ine¼. The fit provides a mean coefficient of restitution of !
d an overall range of e ¼ 0:06—0:84 (dotted lines in Fig. 5a).
obtained normal coefficient of restitution e? was treated in away. Fig. 5b shows that values of e? fall in the interval
2 (dashed lines). The analysis of our dataset yields a mean
Fig. 5. (a) Normalized cumulative number of ice-particle collisions with coefficientsof restitution 6e for all captured collisions in the parabolic-flight experiments? ¼ 0:41 with an error of the mean re? ¼ 0:04 and individual(crosses). The distribution of e can be fitted by a uniform distribution which is given
ement errors of De? ¼ 0:06. The standard deviation of theby the dashed curve. The mean coefficient of restitution is at e ¼ 0:45 (denoted byual measurements from the mean value was calculated tothe dash-dotted line) with an overall range of e ¼ 0:06—0:84 (dotted lines). The:24 indicating a significant scatter of the values for e? arounddiamonds represent the data obtained from binary collisions of glass beads in theh is not caused by the measurement uncertainties. We therelaboratory mini drop tower. The coefficients of restitution span a range ofe $ 0:35—0:95 (solid curve). (b) Normalized cumulative number of collisions withommend to include a flat statistical distribution of coeffinormal coefficients of restitution 6e\ for all 41 captured experiments. The normalof restitution (see insert in Fig. 5b) rather than a single
component has a mean value of e? ¼ 0:41 % 0:04 with a standard deviation of
the models investigating Saturn’s rings. However, it is likely
individual measurements of 0.24. The measurement error of an individual
wide range of coefficients of restitution observed in our parmeasurement is 0.06 (indicated by the error bars). The insert shows a schematic
flight experiments reflects the anisotropic nature of the ice
distribution of coefficients of restitution that could be included into numericalsimulations.
s’ surface properties, especially the surface roughness or preslightly melted areas altering the collisional properties.
the analysis of the tangential component ek only experiTable 1#1(36 out of 41
with tangential velocities v k P 5 mm s
Summary of the experimental results of the coefficient of restitution of low-velocity
ice-particle collisions. The velocity range was v ? ¼ 4 " " " 22 cm s#1 andments) were taken into account to avoid ek ¼ v 0k =v kv k ¼ 0:5 " " " 8 cm s#1 , respectively.
ching unrealistically large values when the denominator
oaches zero. The mean value of the tangential coefficiente?
ektution was measured to be ek ¼ 1:08 with an error of
re
re
De?
re
re
Dek
e?
ek
19 and individual standard deviation of rek ¼ 1:11, which
0.41
0.04
0.24
0.06
1.08
0.19
1.11
0.38
air agreement with the value found by Supulver et al.
The large individual measurement errors of Dek ¼ 0:38
e? ; ek : mean normal/ tangential coefficient of restitution.
re ; re : error of mean normal/tangential coefficient of restitution.
the small absolute values of v ? ) do not allow any secure
re ; re : standard deviation of individual measurements of normal/tangential
etation, however. Table 1 summarizes our experimental recoefficient of restitution from mean value.
r the normal and tangential coefficient of restitution of binDe? ; Dek : measurement uncertainty of individual measurement of normal/tanisions between cm-sized ice spheres.
gential coefficient of restitution.

3. Results ＜Parabolic-ﬂight experiment＞

・反発係数の積算個数分布

反発係数εの積算個数分布

傾きが一定：εは均一な分布をとる
ε ＝ 0.45

ε ＝ 0.06 〜 0.84

反発係数εの積算個数分布ε ＝ 0.0 〜 0.82?

?
?

Fig. 5. (a) Normalized cumulative number of ice-particle collisions with coefficientsof restitution 6e for all captured collisions in the parabolic-flight experiments
2010年4月22日木曜日

k

k

?

k

k

was binned in logarithmic intervals. It is clearly visible tha
first 2 s is due to the equilibration phase of the ensemble after theequilibration the dissipation is in agreement with Haff’s csample injection, whereas the clear change in slope after '7 s is agases (Haff, 1983) between 2 and '7 s experiment duration
real effect that can either be explained by an increased coefficient
the relation given in Eq. (2) with a coefficient of restituti＜Drop-tower
experiment＞
of restitution towards low relative velocities or the onset of clusterresidual between Haff’s law and the data shows a good ag
ing (Goldhirsch and Zanetti, 1993; Brito and Ernst, 1998; Miller andexperiment time. The strong deviation within the first 2 s isphase after the particle injection, whereas the divergence
Luding, 2004). However, the investigation of the latter exceeds the・ビーズ速度の積算個数分布
explained by a change of the coefficient of restitution
scope of this work.
velocities. (c) The scatter of the individual data points aroun
We performed a detailed analysis of individual binary collisions
a broad velocity distribution.
among the constituents of the ensemble in the drop-tower exper-

3. Results

実験時間のラスト1.5 s付近

Fig. 6. Normalized cumulative number of glass particles with velocity 6v in thetime interval of 7–8.5 s of microgravity duration for the drop-tower experiments.
The dashed line denotes the median particle velocity and the dotted lines indicate
the particle velocities covering 50% of all particles.
2010年4月22日木曜日

連続画像間における全粒子の速度（絶対値）を計算

iment as well as of binary collisions of identicalducted in a recently built laboratory mini droheight. In the latter experiments we observed 1
The measured
inthe velocity range 1—4 cm
s!1 .mm/s
平均の粒子速度3.5titution span a range from e ( 0:35 to 0.95, areuted (see solid curve in Fig. 5a), and show n
50%の粒子 → 2 〜 5 mm/s
impact velocity.
In the drop-tower experiments with the parhave no information about the three-dimensio
particles, so that we are forced to restrict ours
sured two-dimensional projection of the velo
and after the collisions. Splitting the velocity
土星環内の衝突研究でのone-dimensional
components, we can describe
the absolute velocity values by a one-dimension
予想と近い
mann
distribution function (see right curve in Fi
in Fig. 9) with a fixed coefficient of restitution o（Esposito,
2002）
range of coefficientsof restitution,i.e. e ¼ 0:2 &very similar velocity distribution function (lefThe shaded areas in Fig. 8 are the result of a Mtion of a one-dimensional Maxwell–Boltzmann
tion and denote the 2r-range.

was binned in logarithmic intervals. It is clearly visible tha
first 2 s is due to the equilibration phase of the ensemble after theequilibration the dissipation is in agreement with Haff’s csample injection, whereas the clear change in slope after '7 s is agases (Haff, 1983) between 2 and '7 s experiment duration
real effect that can either be explained by an increased coefficient
the relation given in Eq. (2) with a coefficient of restituti＜Drop-tower
experiment＞
of restitution towards low relative velocities or the onset of clusterresidual between Haff’s law and the data shows a good ag
ing (Goldhirsch and Zanetti, 1993; Brito and Ernst, 1998; Miller andexperiment time. The strong deviation within the first 2 s isphase after the particle injection, whereas the divergence
Luding, 2004). However, the investigation of the latter exceeds the・ビーズ速度の積算個数分布
explained by a change of the coefficient of restitution
scope of this work.
velocities. (c) The scatter of the individual data points aroun
We performed a detailed analysis of individual binary collisions
a broad velocity distribution.
among the constituents of the ensemble in the drop-tower exper-

3. Results

実験時間のラスト1.5 s付近

Fig. 6. Normalized cumulative number of glass particles with velocity 6v in thetime interval of 7–8.5 s of microgravity duration for the drop-tower experiments.
The dashed line denotes the median particle velocity and the dotted lines indicate
the particle velocities covering 50% of all particles.
2010年4月22日木曜日

連続画像間における全粒子の速度（絶対値）を計算

iment as well as of binary collisions of identicalducted in a recently built laboratory mini droheight. In the latter experiments we observed 1
The measured
inthe velocity range 1—4 cm
s!1 .mm/s
平均の粒子速度3.5titution span a range from e ( 0:35 to 0.95, areuted (see solid curve in Fig. 5a), and show n
50%の粒子 → 2 〜 5 mm/s
impact velocity.
In the drop-tower experiments with the parhave no information about the three-dimensio
particles, so that we are forced to restrict ours
sured two-dimensional projection of the velo
and after the collisions. Splitting the velocity
土星環内の衝突研究でのone-dimensional
components, we can describe
the absolute velocity values by a one-dimension
予想と近い
mann
distribution function (see right curve in Fi
in Fig. 9) with a fixed coefficient of restitution o（Esposito,
2002）
range of coefficientsof restitution,i.e. e ¼ 0:2 &very similar velocity distribution function (lefThe shaded areas in Fig. 8 are the result of a Mtion of a one-dimensional Maxwell–Boltzmann
tion and denote the 2r-range.

3. Results ＜Drop-tower experiment＞

us 206 (2010) 424–430

・全体の平均粒子速度の減衰

Haﬀ (1983)
熱運動速度が0の粒子流における粒子
速度（二乗平均平方根）の時間変化：ビーズ注入時の速度
：数密度
：断面積
Fig. 7. (a) Velocity decrease of the glass particles in the drop-tower experiment due
2010年4月22日木曜日

coefficient of restitution this clearly plays no role for the glass
beads. Thus, we expect not a single value for the coefficient of restitution for the collisions in Saturn’s rings, but a more or less wide
distribution around a mean value. Thus, also very elastic and very
429
D. Heißelmann et al. / Icarus 206 (2010) 424–430
inelastic collisions concurrently take place in Saturn’s rings. The
experiments performed by Bridges et al. (1984) and Hatzes et al.
gue that surface frost might be responsible for the scatter in the(1988) – although investigating central collisions at lower velocicoefficient of restitution this clearly plays no role for the glass
ties than we did – showed only a weak velocity dependence of ebeads. Thus, we expect not a single value for the coefficient of resabove 1 cm s!1 with values of e # 0:2 and e # 0:5, respectively,
titution for the collisions in Saturn’s rings, but a more or less wide
and thereby enclose the results obtained from our parabolic-flight
distribution around a mean value. Thus, also very elastic and very
experiments. The experiments by Dilley and Crawford (1996) show
inelastic collisions concurrently take place in Saturn’s rings. The
a decrease of the coefficient of restitution with decreasing projecexperiments performed by Bridges et al. (1984) and Hatzes et al.
tile mass and with increasing impact velocities.
These results do※cmサイズのガラス球(1988) – although investigating central
collisions at lower velocinot match those obtained from our parabolic-flight experiments.
ties than we did – showed only a weak velocity dependence of e
Recent works byGrasselli et al. (2009) and Sorace et al. (2009) preabove 1 cm s!1 with values of e # 0:2 and e # 0:5, respectively,
senting the investigation of granular cooling and low-velocity binand thereby enclose the results obtained from our parabolic-flightary collisions of mm-sized spherical particles, respectively, report aexperiments. The experiments by Dilley and Crawford (1996) show
decrease of the coefficients of restitution with decreasing impact
Fig. 8. The coefficients of restitution obtained by analysis of the binary collisions ofa decrease of the coefficient of restitution with decreasing projecvelocities.
This finding is supported by the collision model of Brilcm-sized glass spheres (crosses) can be fitted by a Maxwell–Boltzmann distribution
tile mass and with increasing impact velocities. These results doliantov et al. (2007). This means that it is unlikely that the deviawith a constant coefficient of restitution of e ¼ 0:77 (right shaded area). Using a
not match those obtained from our parabolic-flight experiments.
range of coefficients of restitution ðe ¼ 0:2 & & & 1:0Þ results in a very similar velocitytion of our data points from Haff’s law (see Fig. 7a) for t > 7 s isRecent works by Grasselli et al. (2009) and Sorace et al. (2009) predistribution function (left shaded area). The shaded areas denote the 2r-standarddue to an increase in the coefficient of restitution. Clustering seems
senting the investigation of granular cooling and low-velocity bindeviation derived by a Monte Carlo simulation.
to be a plausible explanation, but needs further exploration.
arycollisionsof mm-sized
spherical
particles,
respectively,
report a
RecentN-bodysimulations
by Porco
et al. (2008)
were carrieddecreaseof athecoefficientsof A
restitutionout
to create
modelof Saturn’sand C ring.withThisdecreasing
model was impact
usedFig. 8. The coefficients of restitution obtained by analysis of the binary collisions ofvelocities.
Thisfindingissupportedbythecollision
modelofBrilcm-sized glass spheres (crosses) can be fitted by a Maxwell–Boltzmann distribution
as input for numerical light-scattering studies trying to match theliantov et al.properties(2007). Thismeans by
thatCassini.
it is unlikelythat thefounddeviawith a constant coefficient of restitution of e ¼ 0:77 (right shaded area). Using aphotometricobserved
The authors
range of coefficients of restitution ðe ¼ 0:2 & & & 1:0Þ results in a very similar velocity
tiontoofreproduce
our data points
from Haff’s law
Fig.
7a) for t codes,> 7 s isthat
their observations
with(seetheirnumerical
distribution function (left shaded area). The shaded areas denote the 2r-standarddue required
to an increasein the coefficient ofrestitution.
seems
they
a velocity-dependent
coefficientofClustering
restitutionofdeviation derived by a Monte Carlo simulation.
tobeaplausible
explanation,
but
needs
further
exploration.
$3.5 times lower than that found by Bridges et al. (1984) and five
Recentsimulations
et al.et(2008)
wereThese
carriedtimes
lowerN-bodythan thoseobtainedby
byPorcoSupulveral. (1995).
out to could
create nota modelof Saturn’s AbyandC ring.
This model
was used
results
be reproduced
ourexperiments.
However,
as presence
input for ofnumerical
light-scattering
trying
tothe
match
the
the
surface frost
or regolith studies
layers willalter
colliphotometricproperties
observed
by Cassini.
Theauthorstrend
foundsionalproperties
towards moreinelastic
behavior.
A similar
to reproduce
theirobservations
with theirnumerical
codes,
isthatknown
from impact
experimentsof cm-sized
spheres
into lunar
theyrequireda velocity-dependent
coefficient
restitution
andmartianregolith
analog material (Colwell
andofTaylor,
1999;of$3.5 times
lowerandthan low-velocity
that found by collision
Bridges et experimentsal. (1984) and offive
Colwell,
2003)
times lower thanobtained
et al.
(1995).
These
high-porosity
dust thoseaggregates(seebye.g.SupulverBlum and
Münch,
1993;
results could
not2007;
be reproduced
experiments. However,
Heißelmann
et al.,
Langkowskibyet oural., 2008).
theof surface frost
or regolithlayers will alterthecolliIn presence
a future microgravity
campaign
the investigationof icecollisional
properties
towards
more inelastic
A similar
sions
will
be extended
to impacts
onto a behavior.
solid ice target
andtrend
at
is
known
from
impactexperimentsofcm-sized
spheres
into
lunar
arbitrary impact angles using a slightly modified parabolic-flightand martian regolith analog material (Colwell and Taylor, 1999;
setup.
Colwell,
and low-velocity
collision
experiments
Using the2003)
setup developed
as a feasibility
study fordrop-tower of
2010年4月22日木曜日

3. Results ＜Drop-tower experiment＞

Maxwell-Boltzmann 速度分布を用いて計算した反発係数の積算個数分布
右：ε＝0.77（一定）左：ε＝0.2 〜 1.0（幅）非常に近い形

２体衝突の一次元速度の絶対値はMaxwell-Boltzmann 速度分布とフィットする

Fig. 9. The obtained absolute values of the one-dimensional velocities of the binary

4. Conclusion
本研究では、微視的な氷物質個々の衝突を研究するために新たな2つの手法で実験を行った＜Parabolic-ﬂight experiment＞

・ 相対速度 6cm/s 以上の２体衝突に適している（near-central impact）= 0 〜 0.5

・ 反発係数と衝突パラメーター、相対速度との相関はあまり見られなかった・ 反発係数： ε ＝ 0.45 、ε ＝ 0.06 〜 0.84
＜Drop-tower experiment＞

・ 1cm/s より小さい相対速度での多体衝突を達成できた・ 速度の時間進化はε=0.64としてHaﬀ(1983)の式でフィッティングできた・ 得られたビーズの速度分布は予想される土星環の速度分布に近い
2010年4月22日木曜日