Les surprises de la montée capillaire

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Les surprises de la montée capillaire

Joachim Delannoy

To cite this version:

Joachim Delannoy. Les surprises de la montée capillaire. Matière Molle [cond-mat.soft]. Sorbonne Université, 2019. Français. �NNT : 2019SORUS484�. �tel-02867154v2�

THÈSE DE DOCTORAT

DE SORBONNE UNIVERSITÉ

Spécialité : Physique des liquides

École doctorale no564 : Physique en Île-de-France

réalisée au laboratoire de

Physique et Mécanique des Milieux Hétérogènes

sous la direction de David QUÉRÉ

présentée par

Joachim DELANNOY

pour obtenir le grade de :

DOCTEUR DE SORBONNE UNIVERSITÉ Sujet de la thèse :

Les surprises de la montée capillaire

soutenue le 20 septembre 2019

devant le jury composé de :

M. Dominic VELLA Rapporteur

M. Benjamin DOLLET Rapporteur

Mme Emmanuelle RIO Examinatrice

M. Thierry LENGAGNE Examinateur

M. Pierre-Yves LAGRÉE Examinateur

M. Christophe CLANET Codirecteur de thèse

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S@> H2 ;`M/ K2`+B /Ƕû;v2` iQmi2b H2b bQB`û2b /m H#Q p2+ i2b /Bb+Qm`b HB#2`iB`2b /2 THmb 2M THmb HQM; m 7m` 2i ¨ K2bm`2 /2b p2``2b /2 TbiBbX `KM/ 2i _QKûQ UK2`+B mbbB TQm` iQM #m`2mV- #`pQ TQm` iQmb pQb D2mt /2 KQib- bQmp2Mi MmHb KBb T`7QBb i`ĕb #QMb Ĝ pQi`2 +?m/ /m KQmHBM /ǶM/û pHBi #B2M mM2 i?ĕb2 bm` H +TBHH`Biû 5 Jii?Bb- +Ƕ2bi +QQH /2 pQB` mM HH2KM/ p2+ mM T2m /2 bivH2- K2`+B TQm` H ;2biBQM /2 K2b BKT`2bbBQMb m #m`2tX J`iBM- K2`+B TQm` i2b `?mKb ``M;ûb U2i iQM +`K2H +`KûVX >2m`2mb2K2Mi [m2 iQmb +2b DQv2mt /Q+iQ`Mib bQMi dz2M+/`ûbǴ T` /2b dzT2`KM2MiǴ UH2b T`BpBHû;Bûb- +2mt [mB HǶQM 2m H2 TQbi2XXXV [mB MǶ2M bQMi Tb KQBMb ;`û#H2b- K2`+B CmbiBM2 /2 7B`2 `û;M2` HǶQ`/`2 2i H bû+m`Biû U2i TQm` iǶāi`2 #iim2 p2+ KQB bm` +2ii2 BKT`BKMi2 j.VX J2`+B J`+ TQm` iQM B/2 bm` *Qt@oQBMQpX MiQMBM +Ƕ2bi bvKT /2 +QMiBMm2` ¨ p2MB` T`2M/`2 [m2H[m2b #Bĕ`2b p2+ MQmbX J2`+B TQm` iQmb i2b +QMb2BHb T`iB[m2b bm` H2b `Qm;2b /2 H `2+?2`+?2 T`BbB2MM2X ú`B+ K2`+B TQm` i2b `2K`[m2b 7`M+?2- [mB 7QMi #B2M `û~û+?B` UD2 T2Mb2 K2 `TT2H2` 2M+Q`2 mM #QM KQK2Mi /m dziB` /2 iQMiQM ú`B+Ǵ ¨ h2H@pBp Ĝ ?2BM S@>- :bT`/ 2i :#vVX .ûbQHû Gm`2ii2- D2 MǶB Tb `ûmbbB ¨ i`Qmp2` mM2 T`ĕK TQm` p2MB` ¨ i2b `ûmMBQMb /2 +QMi` /Mb2X PHBpB- MF2- CmHB2M- avHpBM +ǶûiBi iQmDQm`b +?Qm2ii2 /2 pQmb +`QBb2` mt bûKBMB`2b +7ûX 1M}M K2`+B m /2mt THmb M+B2Mb- >Mb 2i 1iB2MM2- /2 MQmb KQMi`2` [m2 KāK2 p2+ mM ;2 pûMû`#H2 QM T2mi +QMiBMm2` ¨ T`iB+BT2` bMb T`Q#HĕK2 ¨ H pB2 /ǶmM H#Q 5

"B2M bȿ`- H pB2 ¨ S`Bb MǶm`Bi Tb ûiû mbbB bvKT bMb iQmb H2b ;2Mb m /2?Q`b /m H#QX CǶB H +?M+2 /ǶpQB` mM #QM ;`QmT2 /ǶKBb iQmDQm`b KQiBpûb TQm` Tb KH /2 i`m+b #B2M /û#BH2b- /2TmBb H2b T2iBi2b bQ`iB2b /2 +Qm`b2 ¨ TB2/ U¨ oBM+2MM2b- H2b /BKM+?2 KiBM-/Mb H #Qm2V mt THmb HQM;m2b `M/QMMû2b 2M mMB7Q`K2 Zm2b?mX J2`+B sp TQm` pQB` Q`@ ;MBbû +2 bmT2` iQm` /2 JBMQ`[m2 U+Qm+Qm "B/Qm 2i aFV TmBb /2 KǶpQB` ?û#2`;û T2M/Mi mM M UKQBMb mM2 T2iBi2 /BxBM2 /ǶB`"M"VX JBKB2- K2`+B TQm` H2b r22F2M/b 2M *?KT;M2 2i H2b THMb +mHim`2Hb T`BbB2Mb Ĝ DǶ2bTĕ`2 [m2 im pB2M/`b ¨ :`2MQ#H2 7B`2 [m2H[m2b i`BHb /2 i2KTb 2M i2KTbX HB i #2HH2 +?2p2Hm`2 #HQM/2 2i i2b ;Qȿib /2 T`BM+2bb2 UTiiû \V pQMi K2 KM[m2` 5 J2`+B 2M+Q`2 TQm` H `2H2+im`2 2tT`2bb /2 K T`2KBĕ`2 T`iB2 /2 i?ĕb2 5 J2`+B `i?m` 2i *H2K /2 KǶpQB` bQmi2Mm /Mb KQM KQ/2 /2 pB2 /2 i?ûb`/ dzT`û+B`2Ǵ Ubm`iQmi *H2KVX GmHm- DǶ2bTĕ`2 [mǶQM TQm`` b2 `27B`2 [m2H[m2b T2iBi2b bQ`iB2b Hb+Q@oQb;B2MM2bX Jii?B2m QM b2 `2i`Qmp2 2M SQHQ;M2 #B2MiƬi cVX LB+Q- +ǶûiBi p`BK2Mi bvKT /2 T`i;2` i +QHQ+Ƕ T2M/Mi /2mt Mb- i2b ?BbiQB`2b /2 "2`+v 2i mM2 #QMM2 ;`Qbb2 /Qb2 /2 `2Tb bvKT@ i?B[m2b U+Qm+Qm H2tVX qMQm 2i aQT?B2- KāK2 bB QM MǶ2bi DKBb /BbTQb bBKmHiMûK2Mi-+ǶûiBi +?Qm2ii2 /2 Tbb2` [m2H[m2b bQB`û2b +?2x pQmb 5 CmDm 2i *BbbQm K2`+B TQm` H2b T`ĕKb D2mt /2 bQ+Bûiû- H2b /ŗM2`b- HǶ?û#2`;2K2Mi /Ƕm`;2M+2 2i H2b UMQK#`2mb2bV +Q``2+iBQMb Q`i?Q@ ;`T?B[m2bX J2`+B ¨ H i2K 2b+H/2- J`B2- 1KK- Jt- m`ûHB2M- GQmBb- JtBK2 TQm` pQB` û;vû K2b HmM/B bQB`b 2i +QMi`B#mû ¨ H T`ûb2`piBQM /2 K2b +T+Biûb T?vbB[m2bX H@ #M 2i aû#- 2M+Q`2 K2`+B TQm` iQmi H2 i2KTb [m2 pQmb +QMb+`2x ¨ 7B`2 iQm`M2` H2 ;`QmT2 dz+QKTĕiǴ 2i TQm` H2b bmT2` +KTb ¨ aBMi *`ûTBM Ĝ H T`Q+?BM2 7QBb pQmb m`2x T2mi āi`2 mM T2m THmb /2 +?M+2 ¨ H +QBM+?2 U?2BM GQmBbV 5 G2b bûM+2b ¨ JBHHi ûiB2Mi mbbB mM #QM KQv2M /2 b2 7Q`+2` ¨ bQ`iB` /m #m`2m ¨ Rd?jy- K2`+B C2D2- :QFv- *?`HQii2-m/`2v- 2i Zm2MiBM TQm` KǶpQB` bbm`û bMb 7BHH2 KāK2 bB D2 ;`BKT2 i`QT /2 7+2X J2`+B mbbB mt Q`;MBbi2m`b /2 HǶa 6HpB2M- .Dû- *?`BbiQT?2 U"X 2i JXV- _QKBM 2i Jm/ /2

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oQBH Ï 7Bi #2m+QmT /2 K2`+Bb- D2 MǶB Tb i`Qmpû bb2x /2 bvMQMvK2b TQm` `2M/`2 iQmb +2b KQib THmb DQHBbX p2+ +2ii2 i?ĕb2- D2 iQm`M2 2M}M H T;2 /ǶmM2 #QMM2 /BxBM2 /ǶMMû2b /Ƕûim/2b 5 *Ƕ2bi /B{+BH2 /2 /B`2 [m2 D2 bmBb /2p2Mm THmb bû`B2mt KBb ;`+2 ¨ pQmb-Ï  p`BK2Mi ûiû mM ;`M/ THBbB` /2 p2MB` m #QmHQi +2b /2`MBĕ`2b MMû2b HQ`b , J1_*A 55

RX lM2 +Q``2+iBQM BKTQbû2 i`ĕb i`ĕb ;Hĕ`2 p2+ G_h1sX

h#H2 /2b KiBĕ`2b

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A JûMBb[m2 2i HB;M2 /2 +QMi+i kN

R aBM;mH`Biû BMBiBH2 /2 H KQMiû2 +TBHHB`2 , H2 /Qm#H2 D2m /2 H pBb+QbBiû jR

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h"G1 .1a JhAĔ_1a

d JQmp2K2Mi Q`B2Miû T` /2b TBHB2`b bvKûi`B[m2b R9R

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observation and comparison. In my manuscript of the Book of the Balance of Wisdom there remain only a few leaves of it, the contents of which cause the loss of the rest to be greatly lamented. Notwithstanding the multiplied errors of the copyist, omissions, and inaccuracies of every kind, which prevent me from giving the text itself, it is possible to perceive what is being treated of, and I shall cite here and there fragments of ble phrases, which have guided me in the following exposition. The first of the remaining leaves, after a few unintelligible words, which evidently belong to a phrase commenced upon the ceding leaf, contains a figure representing an instrument devised by 'Abu-r-Raihan for the determination of specific gravities. I

Form of the Conical Instrument of 'Abu-r-Raihan.

E e r a LOL.e Neck of the Instrument. b .bJI Perforation.

/e I | c UT].. I)_C (s3 ' ._ JS1 Tube in the Form

of a Water-pipe.

d LEji3 , Handle of the Instrument. e YJ%Sf e Mouth of the Instrument.

f "\ X _ Place of the Bowl [of the .J-?fa/w7 ,- /^--- . Balance].

give an exact copy of it,1 with all the explanations which company it. A mere inspection of it will suffice to show that we have here to do with an instrument made for determining the volumes of different heavy bodies immersed in the water which fills a part of the cavity, by means of the weight of the water displaced by these bodies, which is ascertained by conducting it through a lateral tube into the bowl of a balance. The tion given by our author of the use of this instrument, which he calls oLk.JIt 'J ' 1 'J9 t, '"the conical instrument of Raihan," confirms this idea; but he adds that the instrument is very difficult to manage, since very often the water remains suspended in the lateral tube, dropping from it little by little into

the scale of the balance: X 'i '2 or- ". . ' ' L . ' cjl .

Capillary action was accordingly known to the Arabs; and

our author asserts that 'Abu-r-Raihan had ascertained that, if

the lateral tube had a circular flexure given it, was made shorter than a semicircle, and was pierced with holes, the

water would flow readily through it, no more remaining in the

tube than just enough to moisten its inner surface: ~i< 3

jit (j lj I sLt 4y; . A. 5 .& L

54 _{N. Khanikoff,}

b a

c

6B;m`2 k ,  *Qmp2`im`2 /ǶmM2 û/BiBQM /2 ReeN /2 HǶQmp`;2 Lim`HBb >BbiQ`B /2 SHBM2 HǶM+B2MX # a+?ûK /ǶmM2 ?Q`HQ;2 ?v/`mHB[m2- 2ti`Bi /2 HǶQmp`;2 SM2mKiBF /Ƕ>û`QM /ǶH2tM/`B2- 2ti`Bi /2 (k)X + G2 /2MbBKĕi`2 /Ƕ#m@`@_B?M- 2ti`Bi /2 (j)X

G2 /QKBM2 /Ƕûim/2 [mB b2 +QMb+`2 mt 2z2ib /2 H i2MbBQM bmT2`}+B2HH2 2bi TT2Hû +TBHH`BiûX *2 /QKBM2 /2 `2+?2`+?2 `2;`QmT2- i`ĕb ;ûMû`H2K2Mi- HǶûim/2 /2b HB[mB/2b mt û+?2HH2b Qɍ H2b BMi2`7+2b DQm2Mi mM `ƬH2 T`ûTQM/û`MiX G +TBHH`Biû T2`K2i /Ƕ2tTHB[m2` MQM b2mH2K2Mi H 7Q`K2 /2b #mHH2b 2i /2b ;Qmii2b KBb mbbB H 7Q`KiBQM /ǶmM2 KQmbb2- H 7Q`+2 /?ûbBp2 /ǶmM }HK /Ƕ2m BMbû`û 2Mi`2 /2mt bbB2ii2b Qm H +T+Biû /2 +2`iBMb BMb2+i2b ¨ K`+?2` bm` HǶ2mX

b

a c

6B;m`2 j ,  :Qmii2b /Ƕ2m /ûTQbû2b T` H `Qbû2 KiBMH2 bm` mM #`BM /Ƕ?2`#2 Tv`ûMû2MX *`û/Bi T?QiQ , *?`BbiQT?2 J2`H2iX # G2 ;2``Bb UT`7QBb TT2Hû BKT`QT`2K2Mi `B;Mû2 /Ƕ2mV miBHBb2 H i2MbBQM bmT2`}+B2HH2 TQm` b2 /ûTH+2` bm` HǶ2mX *`û/Bi T?QiQ , AM;Q `M/iX + G 7Q`K2 /2b }HKb /2 bpQM- mM mi`2 2z2i /2 H i2MbBQM bmT2`}+B2HH2 U2ti`Bi /2 iQvb7#X+QKVX

G2 i2`K2 +TBHH`Biû T`QpB2Mi /2 HǶ2tTû`B2M+2 [mB +QMbBbi2 ¨ i`2KT2` mM im#2 bb2x }M /Mb mM p2``2 /Ƕ2mX G2 HB[mB/2 T`Q;`2bb2 HQ`b /Mb H2 im#2- KQMiMi bTQMiMûK2Mi ¨ /2b ?mi2m`b [mB T2mp2Mi ii2BM/`2 THmbB2m`b +2MiBKĕi`2bX *Ƕ2bi 2M +QKK2MiMi +2ii2 2t@ Tû`B2M+2 [m2 H2 Ki?ûKiB+B2M CQ?M qHHBb BMi`Q/mBi H MQiBQM /2 +TBHH`Biû 2M Red9 (9)-[mM/ BH /û+`Bi TQm` H T`2KBĕ`2 7QBb mM im#2 +TBHHB`2 , dz im#2 Q7 2ti`2K2 i?BMM2bb r?B+? ;HbbrQ`F2`b +HH  +TBHH`v- bBM+2 Bi Bb bQ /`rM Qmi i?i Bi b+`+2Hv 2t+22/b i?2 bBx2 Q7  ?B`ǴX *2 ivT2 /2 im#2 /QMM2 HQ`b bQM MQK ¨ H KQMiû2 +TBHHB`2- mM2 2tTû`B2M+2 7QM/@ K2MiH2 [m2 MQmb `2i`Qmp2`QMb iQmi m HQM; /2 +2 KMmb+`BiX *Ƕ2bi mbbB mM T?ûMQKĕM2 [m2 MQmb +ƬiQvQMb 7`û[m2KK2Mi TmBb[m2 H KQMiû2 +TBHHB`2 T`iB+BT2 ¨ 7B`2 `2KQMi2` H bĕp2 Dmb[mǶm 72mBHH;2 /2b `#`2b- T2`K2i /2 7B`2 /2b dz+M`/bǴ 2M i`2KTMi mM KQ`+2m /2 bm+`2 /Mb bQM +7û Qɍ +mb2 /2b BM}Hi`iBQMb /Ƕ2m m TB2/ /2b Km`bX

GûQM`/ /2 oBM+B b2K#H2 āi`2 H2 T`2KB2` ¨ /û+`B`2 H KQMiû2 /ǶmM HB[mB/2 ¨ HǶBMiû`B2m` /ǶmM2 #M/2 /2 iBbbm m /û#mi /m soAĕK2_{bBĕ+H2 (8) UK2MiBQMMû /Mb H i`/m+iBQM /m K@</sub> Mmb+`Bi :- û+`Bi mt 2MpB`QMb /2 R8RyV 2i :mBHHmK2 GB#`B /Mb (e) HmB ii`B#m2 H2b T`2KBĕ`2b Q#b2`piBQMb /2 KQMiû2b +TBHHB`2b /Mb /2b im#2b U#B2M [m2 H2b KMmb+`Bib K2MiBQMMûb B2Mi ûiû T2`/mb /2TmBbVX o2`b H }M /m soAAĕK2<sub>bBĕ+H2- QM `2i`Qmp2 /2 MQK#`2mb2b ûim/2b} +QM+2`MMi H +TBHH`BiûX *2b Tm#HB+iBQMb- MHvbû2b /Mb mM `iB+H2 /2 JBHHBM;iQM (d)-`2~ĕi2Mi HǶBMiû`āi /2 H +QKKmMmiû b+B2MiB}[m2 TQm` +QKT`2M/`2 H2b 2z2ib i`Qm#HMib /2 H i2MbBQM bmT2`}+B2HH2X 1M Rej3- H2 T?vbBQHQ;Bbi2 BiHB2M :BQpMMB H7QMbQ "Q`2HHB `ûHBb2 mM2 ûim/2 UTm#HBû2 2M RedyV /QMi H2b `ûbmHiib KQMi`2Mi [m2 H ?mi2m` /Ƕ2m /Mb mM im#2 +TBHHB`2 2bi BMp2`b2K2Mi T`QTQ`iBQMM2HH2 m `vQM /m im#2 (3)X h`ĕb Dmbi2K2Mi- BH `2K`[m2 /ûD¨ [m2 HǶ2m KQMi2 ¨ H KāK2 ?mi2m` [m2HH2 [m2 bQBi HǶBM+HBMBbQM /m im#2 2i [m2 H2 HB[mB/2 KQMi2 THmb pBi2 /Mb mM im#2 T`ûH#H2K2Mi KQmBHHû (d) Ĝ mM2 `2K`[m2 ûiQMM2K2Mi Hm+B/2- bm` mM TQBMi [mB 72` HǶQ#D2i /m T`2KB2` +?TBi`2 /2 +2 KMmb+`BiX m KāK2 KQK2Mi- /2b ûim/2b bBKBHB`2b bQMi `ûHBbû2b 2M 6`M+2 T` H2 T?vbB+B2M C+[m2b _Q@ ?mHi (N)- T` H2 T?BHQbQT?2 2i Kû/2+BM Mû2`HM/Bb >2M/`BF .2 _Qv m Svb@"b (Ry) BMbB [m2 T` H2 +?BKBbi2 #`BiMMB[m2 _Q#2`i "QvH2 U+QKK2 /û+`Bi T` >QQF2 /Mb (RR)VX

1M Rdy3- 6`M+Bb >mFb#22 HǶM+B2M UTQm` H2 /BbiBM;m2` /2 bQM M2p2m 6`M+Bb >mFb#22 H2 C2mM2V Tm#HB2 mM `iB+H2 BMiBimHû dza2p2`H 1tT2`BK2Mib iQm+?BM; i?2 b22KBM; aTQMiM2Qmb

ALh_P.l*hAPL

b+2Mi Q7 qi2`Ǵ /Mb H2[m2H BH T`ûb2Mi2 [mi`2 2tTû`B2M+2b bm` H KQMiû2 +TBHHB`2 (Rk)X AH ûim/B2 /2b HB[mB/2b p`Bûb U+QKK2 /2 HǶ2m- /2 H HB[m2m` /2 pBM Qm HǶ?mBH2 /2 iû`û#2Mi?BM2V `2KQMi2` 2Mi`2 /2mt TH[m2b /2 p2``2 i`ĕb Hû;ĕ`2K2Mi 2bT+û2b T` /2b 72mBHH2b /2 TTB2` Ĝ [mB T`2M/`QMi THmb i`/ H2 MQK /2 +2HHmH2b /2 >2H2@a?rX AH Q#b2`p2 [mǶ2M m;K2MiMi H2 MQK#`2 /2 72mBHH2b /2 TTB2` [mB 2bT+2Mi H2b TH[m2b /2 p2``2- H2 HB[mB/2 KQMi2 /2 KQBMb 2M KQBMb ?miX AH MQi2 mbbB [m2 HǶûTBbb2m` /2b im#2b MǶ Tb /ǶBM~m2M+2 bm` H KQMiû2 /m HB[mB/2 Ĝ mM 2z2i HQ`b bmbT2+iû /Ƕāi`2 ¨ HǶQ`B;BM2 /2 H KQMiû2 +TBHHB`2X 1M RdRN- H2 Kû/2+BM CK2b Cm`BM `2K`[m2 HmB mbbB [m2 H ?mi2m` ii2BMi2 T` H2 HB[mB/2 HQ`b /ǶmM2 KQMiû2 +TBHHB`2 p`B2 +QKK2 HǶBMp2`b2 /m `vQM /m im#2 2i 2M /û/mBi [m2 H2 pQHmK2 }MH /2 H +QHQMM2 /2 HB[mB/2 2bi T`QTQ`iBQMM2H m /BKĕi`2 /m im#2 (Rj)X AH bm;;ĕ`2 BMbB- TQm` H T`2KBĕ`2 7QBb- mM2 2tTHB+iBQM /2 H KQMiû2 +TBHHB`2 ¨ T`iB` /2 H 7Q`+2 HBû2 ¨ HǶ2tBbi2M+2 /ǶmM2 HB;M2 /2 +QMi+iX

JH;`û H2b 2zQ`ib TQm` 2tTHB[m2` H KQMiû2 +TBHHB`2- BH 7m/` MûMKQBMb ii2M/`2 T`2b[mǶmM bBĕ+H2 TQm` [m2 bQB2Mi i?ûQ`Bbûb +QMp2M#H2K2Mi H2b 2z2ib /2 H i2MbBQM bmT2`}@ +B2HH2X G2 T?ûMQKĕM2 û+?TT2 MQiKK2Mi ¨ L2riQM 2i MǶ2bi 2tTHB[mû T` GTH+2 [mǶ2M R3ye (R9)X

6Q`K2 /ǶmM2 ;Qmii2 HB[mB/2

G +TBHH`Biû 2bi /m2 mt BMi2`+iBQMb KQHû+mHB`2b m b2BM /m HB[mB/2X .Mb mM HB[mB/2-HǶûM2`;B2 /ǶBMi2`+iBQM U- [mB `2T`ûb2Mi2 HǶ#Bbb2K2Mi /ǶûM2`;B2 /ǶmM2 KQHû+mH2 HQ`b[mǶ2HH2 2bi 2MiQm`û2 T` b2b pQBbBM2b- 2bi THmb 7Q`i2 [m2 HǶ;BiiBQM i?2`KB[m2 ⇠ kT Ĝ +QMi`B`2@ K2Mi mt ;xX lM2 KQHû+mH2 /2 HB[mB/2 b2` /QM+ 2MiQm`û2 /ǶmM KtBKmK /2 b2b pQBbBM2bX AMp2`b2K2Mi- BH 2bi /û7pQ`#H2 TQm` mM2 KQHû+mH2 /Ƕāi`2 HQ+HBbû2 ¨ HǶBMi2`7+2 Qɍ HǶûM2`;B2 /2 H KQHû+mH2 MǶ2bi #Bbbû2 [m2 /Ƕ2MpB`QM U/2 ¨ +mb2 /2b BMi2`+iBQMb dzKM[mMi2bǴX lM2 m;K2MiiBQM mMBiB`2 /2 HǶB`2 /2 HǶBMi2`7+2 +Qȿi2 HQ`b mM2 ûM2`;B2 γ ⇠ U/2a2_{- Qɍ}

a2bi H iBHH2 +`+iû`BbiB[m2 /ǶmM2 KQHû+mH2 Ĝ mM2 iBHH2 /2 HǶQ`/`2 /2 HǶM;bi`ƺKX

b a

6B;m`2 9 , AMi2`+iBQMb KQHû+mHB`2bX  úii ;x2mt , HǶ;BiiBQM i?2`KB[m2 2bi THmb 7Q`i2 [m2 H2b BMi2`+iBQMb 2Mi`2 H2b KQHû+mH2bX # úii HB[mB/2 Qm bQHB/2 , H2b BMi2`+iBQMb KQHû+mHB`2b- BM/B[mû2b T` /2b ~ĕ+?2b p2`i2b- bbm`2Mi H +Q?ûbBQM /2b KQHû+mH2bX § HǶBMi2`7+2- H2b KQHû+mH2b bQMi KQBMb bi#BHBbû2b ¨ +mb2 /2b BMi2`+iBQMb KM[mMi2b U2M `Qm;2VX

LQmb p2MQMb B+B /2 /û}MB` H i2MbBQM /2 bm`7+2 γ /m HB[mB/2 [mB `2T`ûb2Mi2 H p`BiBQM /ǶûM2`;B2 E /m bvbiĕK2 HQ`b[mǶQM m;K2Mi2 HǶB`2 A /2 bQM BMi2`7+2 ,

γ= ∂E ∂_A " T,Ω,n URV GǶûM2`;B2 /2 bm`7+2 /ǶmM HB[mB/2 2bi /QM+ 2tT`BKû2 2M CfK2 <sub>Qm- THmb</sub> 7`û[m2KK2Mi-2M KLfKX 1HH2 p`B2 /2 γ ⇡ 20 KLfK TQm` /2b ?mBH2b- bi#BHBbû2b T` /2b BMi2`+iBQMb /2 oM /2` qHb- ¨ γ ⇠ 500 KLfK TQm` /2b Kûimt HB[mB/2b +QKK2 H2 K2`+m`2 [mB QMi /2b BMi2`+iBQMb BMi2`iQKB[m2b i`ĕb 7Q`i2bX G i2MbBQM bmT2`}+B2HH2 /2 HǶ2m γ ⇡ 70 KLfK 2bi `2HiBp2K2Mi ûH2pû2 ¨ +mb2 /2b HBBbQMb ?v/`Q;ĕM2 [mB `2M7Q`+2Mi H2b BMi2`+iBQMb 2Mi`2 KQHû+mH2bX LQmb `2TQ`iQMb /Mb H i#H2 R H2b i2MbBQMb bmT2`}+B2HH2b /2b HB[mB/2b K2MiBQMMûb /Mb H2b +?TBi`2b bmBpMibX

GB[mB/2 >mBH2 1i?MQH 1m bpQMM2mb2 1m 1m bm+`û2 J2`+m`2

γ UKLfKV kR kk _⇡k8 dk _⇡d8 93d

h#H2 R , h2MbBQM /2 bm`7+2 /2b HB[mB/2b miBHBbûb ¨ mM2 i2KTû`im`2 /2 kyê*X aQm`+2 , GM;2Ƕb >M/#QQF Q7 *?2KBbi`v (R8)X

G bBimiBQM H THmb bi#H2 2bi /QM+ +2HH2 Qɍ H bm`7+2 /ǶmM HB[mB/2 2bi KBMBKH2X lM HB[mB/2 BbQHû bǶ``M;2 Mim`2HH2K2Mi b2HQM mM2 7Q`K2 bT?û`B[m2 [mB +Q``2bTQM/ m KBMBKmK /2 bm`7+2X *Ƕ2bi H2 +b T` 2t2KTH2 /2b T2iBi2b ;Qmii2b /2 THmB2 [mB QMi mM2 7Q`K2 bT?û`B[m2X *2ii2 7Q`K2 2bi bi#H2 Dmb[mǶ¨ +2 [mǶmM2 mi`2 +QMi`BMi2 pB2MM2 /û7Q`K2` H ;Qmii2X am` h2``2- HǶ++ûHû`iBQM /2 H ;`pBiû g HBKBi2 H iBHH2 /2 +2b bT?ĕ`2b KQHH2bX LQmb `2i`QmpQMb H iBHH2 m@/2H¨ /2 H[m2HH2 mM2 ;Qmii2 /2 iBHH2 +`+iû`BbiB[m2 r MǶm` THmb mM2 7Q`K2 bT?û`B[m2 2M +QKT`Mi bQM ûM2`;B2 /2 bm`7+2 ⇠ γr2 <sub>p2+ bQM ûM2`;B2</sub>

TQi2MiB2HH2 /2 T2bMi2m` ⇠ ρgr4<sub>- Qɍ ρ 2bi H Kbb2 pQHmKB[m2 /m HB[mB/2X GǶû[mBHB#`2 2Mi`2</sub>

i2MbBQM bmT2`}+B2HH2 2i T2bMi2m` }t2 H iBHH2 KtBKmK /2 H ;Qmii2 [m2 MQmb MQiQMb κ−1 ,

κ−1₌✓ γ ρg ◆1/2 UkV G HQM;m2m` κ−1 2bi TT2Hû2 HQM;m2m` +TBHHB`2 /m HB[mB/2X *Ƕ2bi H HQM;m2m` +`+@ iû`BbiB[m2 2M /2bbQmb /2 H[m2HH2 mM HB[mB/2 M2 bǶû+QmH2` Tb bQmb bQM T`QT`2 TQB/b KBb THmiƬi ¨ +mb2 /2b 2z2ib /2 i2MbBQM bmT2`}+B2HH2X SQm` HǶ2m- H HQM;m2m` +TBHHB`2 pmi κ−1 <sub>⇡ j KK- mM2 HQM;m2m` bb2x 7KBHBĕ`2- TmBb[mǶ2HH2 }t2 T` 2t2KTH2 H iBHH2 /ǶmM2</sub>

;Qmii2 [mB +QmH2 H2 HQM; /ǶmM2 pBi`2 Qm HǶûTBbb2m` /ǶmM2 ~[m2 /Ƕ2mX G HQM;m2m` +TBH@ HB`2 +?M;2 2M 7QM+iBQM /2 H i2MbBQM /2 bm`7+2 2i /2 H Kbb2 pQHmKB[m2 /m HB[mB/2 KBb mbbB /2 HǶ++ûHû`iBQM /2 H T2bMi2m`X .Mb /2b 2MpB`QMM2K2Mib 2M KB+`Q;`pBiû- +QKK2 /Mb H biiBQM bTiBH2 BMi2`MiBQMH2- BH /2pB2Mi TQbbB#H2 /ǶQ#b2`p2` /2b ;Qmii2b bT?û`B[m2b #2m+QmT THmb ;`M/2b- TQmpMi ii2BM/`2 THmbB2m`b +2MiBKĕi`2bX

ALh_P.l*hAPL

b

a

c

h"G1 .1a JhAĔ_1a

6B;m`2 8 , AMi2`+iBQMb KQHû+mHB`2bX  úii ;x2mt , HǶ;BiiBQM i?2`KB[m2 2bi THmb 7Q`i2 [m2 H2b BMi2`+iBQMb 2Mi`2 H2b KQHû+mH2bX # úii HB[mB/2 Qm bQHB/2 , H2b BMi2`+iBQMb KQHû+mHB`2b- BM/B[mû2b T` /2b ~ĕ+?2b p2`i2b- bbm`2Mi H +Q?ûbBQM /2b KQHû+mH2bX § HǶBMi2`7+2- H2b KQHû+mH2b bQMi KQBMb bi#BHBbû2b ¨ +mb2 /2b BMi2`+iBQMb KM[mMi2b U2M `Qm;2VX

LQmb p2MQMb B+B /2 /û}MB` H i2MbBQM /2 bm`7+2 /m HB[mB/2 [mB `2T`ûb2Mi2 H p`BiBQM /ǶûM2`;B2 /m bvbiĕK2 HQ`b[mǶQM m;K2Mi2 HǶB`2 /2 bQM BMi2`7+2 ,

URV GǶûM2`;B2 /2 bm`7+2 /ǶmM HB[mB/2 2bi /QM+ 2tT`BKû2 2M CfK Qɍ- THmb 7`û[m2KK2Mi- 2M KLfKX 1HH2 p`B2 /2 KLfK U/2 HǶQ`/`2 /2 V TQm` /2b ?mBH2b [mB bQMi bi#BHBbû2b T` /2b BMi2`+iBQMb /2 oM /2` qHb ¨ KLfK TQm` /2b Kûimt HB[mB/2b +QKK2 H2 K2`+m`2 [mB QMi /2b BMi2`+iBQMb BMi2`iQKB[m2b i`ĕb 7Q`i2b- /Ƕ2MpB`QM mM 2oX G i2MbBQM bmT2`}+B2HH2 /2 HǶ2m KLfK 2bi `2HiBp2K2Mi ûH2pû2 ¨ +mb2 /2b HBBbQMb ?v/`Q;ĕM2 [mB `2M7Q`+2Mi H2b BMi2`+iBQMb 2Mi`2 KQHû+mH2bX LQmb `2TQ`iQMb /Mb H i#H2 R H2b i2MbBQMb bmT2`}+B2HH2b /2b HB[mB/2b K2MiBQMMûb /Mb H2b +?TBi`2b bmBpMibX

GB[mB/2 >mBH2 1i?MQH 1m bpQMM2mb2 1m 1m bm+`û2 J2`+m`2

UKLfKV kR kk k8 dk d8 93d

h#H2 R , h2MbBQM /2 bm`7+2 /2b HB[mB/2b miBHBbûb ¨ mM2 i2KTû`im`2 /2 kyê*X aQm`+2 , GM;2Ƕb >M/#QQF Q7 *?2KBbi`v (R8)X

G bBimiBQM H THmb bi#H2 2bi /QM+ +2HH2 Qɍ H bm`7+2 /2 HǶBMi2`7+2 /2 HB[mB/2 2bi KBMBKmKX GQ`b[m2 H2 HB[mB/2 2bi BbQHû- BH T`2M/ mM2 7Q`K2 bT?û`B[m2 [mB +Q``2bTQM/ mM pQHmK2 /2 bm`7+2 KBMBKH2X lM2 T2iBi ;Qmii2 /Ƕ2m /QM+ H 7Q`K2 /ǶmM2 bT?ĕ`2 Dmb[mǶ¨ +2 [mǶmM2 mi`2 +QMi`BMi2 pB2MM2 /û7Q`K2` H ;Qmii2X am` h2``2- HǶ++ûHû`iBQM /2 H ;`pBiû HBKBi2 H iBHH2 /2b ;Qmii2bX PM `2i`Qmp2 H iBHH2 m /2H¨ /2 H[m2HH2 mM2 ;Qmii2 /2 iBHH2 +`+iû`BbiB[m2 MǶm` THmb mM2 7Q`K2 bT?û`B[m2 2M +QKT`Mi HǶûM2`;B2 /2 bm`7+2 2i HǶûM2`;B2 TQi2MiB2HH2 /2 T2bMi2m` /2 H ;Qmii2 - Qɍ 2bi H Kbb2 pQHmKB[m2 /m HB[mB/2X GǶû[mBHB#`2 2Mi`2 H i2MbBQM bmT2`}+B2HH2 2i H T2bMi2m` }t2 H iBHH2 KtBKmK /2 H ;Qmii2 [m2 MQmb MQiQMb ,

κ−1 UkV

N

6B;m`2 8 ,  lM2 ;Qmii2 /Ƕ2m TQbû2 bm` mM2 bm`7+2 ?v/`QT?Q#2 ;`/2 mM2 7Q`K2 bT?û`B[m2 iMi [m2 b iBHH2 +`+iû`BbiB[m2 2bi BM7û`B2m`2 ¨ H HQM;m2m` +TBHHB`2X # GǶbi`QMmi2 E2pBM 6Q`/ Q#b2`p2 mM2 ;Qmii2 /Ƕ2m +2MiBKûi`B[m2 HQ`b /2 HǶ1tTû/BiBQM j9 /2 H biiBQM bTiBH2 BMi2`MiBQMH2X *`û/Bi T?QiQ , LaX + lM mi`2 bTiBQMmi2- H2 +TBiBM2 >//Q+F- bǶûiQMM2 ¨ Dmbi2 iBi`2 /2 pQB` H2 r?BbFv bǶû+?TT2` /m p2``2 KBb M2 /2p`Bi Tb āi`2 bm`T`Bb /2 pQB` H2 HB[mB/2 7Q`K2` mM2 ;Qmii2 /2 H iBHH2 /2 b KBMX aQm`+2 , PM  K`+?û bm` H GmM2- >2`;ûX

PM T2mi mbbB 2tTHB[m2` +2b 7Q`K2b T` mM2 MHQ;B2 p2+ /2b TBM;QmBMbX hQmi +QKK2 H2b KQHû+mH2b /2 HB[mB/2- H2 KM+?Qi 2KT2`2m` T`û7ĕ`2 āi`2 2MiQm`û /2 b2b pQBbBMb U+?m/bV [m2 /2 HǶB` 2tiû`B2m` U7`QB/VX GQ`b[mǶBHb ii2M/2Mi ¨ THmbB2m`b- 2i 2M Tû`BQ/2 /2 ;`M/ 7`QB/- H2b KM+?Qib 2KT2`2m`b b2 `2;`QmT2Mi /2 KMBĕ`2 ¨ KBMBKBb2` H2m` BMi2`7+2 p2+ HǶB`X AH 7Q`K2Mi BMbB mM +2`+H2 UH bm`7+2 /2 Tû`BKĕi`2 KBMBKH 2M /2mt /BK2MbBQMbV }M /2 /BKBMm2` HǶû+?M;2 i?2`KB[m2 m KtBKmKX _2K`[mQMb mbbB [m2 H2b KM+?Qib ¨ HǶ2tiû`B2m` /m +2`+H2 bQMi /û7pQ`Bbûb UTmBb[m2 H2m` /Qb 2bi 2tTQbû m +Ƭiû dz7`QB/ǴVX AH pQMi /QM+ 2bbv2` /2 TQmbb2` H2m`b +K`/2b TQm` `2Mi`2` /Mb H2 +2`+H2 TQm` TQmpQB`-2mt mbbB- b2 K2ii`2 m +?m/- 2t2`ÏMi mM2 +2`iBM2 T`2bbBQM ¨ HǶBMiû`B2m` /2 HǶKb /2 TBM;QmBMb (Re- Rd)X

§ HǶBMiû`B2m` /2b ;Qmii2b U+QKK2 /2b #mHH2bV- H i2MbBQM /2 bm`7+2 +`û2- 2HH2 mbbB- mM2 bm`T`2bbBQMX GǶû+`i /2 T`2bbBQM ∆p BM/mBi T` H +Qm`#m`2 /2 HǶBMi2`7+2  ûiû Q#i2Mm2 T` GTH+2 2M +H+mHMi H2 i`pBH /2b 7Q`+2b /2 T`2bbBQM 2i /2b 7Q`+2b +TBHHB`2b HQ`b /ǶmM /ûTH+2K2Mi BM}MBiûbBKH /2 HǶBMi2`7+2 Ĝ mM2 /ûKQMbi`iBQM [m2 HǶQM T2mi `2i`Qmp2` T` 2t2KTH2 /Mb H2 HBp`2 S?vbB+H +?2KBbi`v Q7 bm`7+2b (R3)X G2 i?ûQ`ĕK2 [mB `2HB2 H bm`T`2bbBQM ∆p ¨ HǶBMiû`B2m` /ǶmM2 bm`7+2 /2 `vQMb /2 +Qm`#m`2 r1 2i r2 2bi ûMQM+û 2M R3ye , ∆p =Ƙγ✓ 1 r1 + 1 r2 ◆ UjV *2ii2 û[miBQM 2bi 7QM/K2MiH2 TQm` +QKT`2M/`2 H2b û+QmH2K2Mib +TBHHB`2bX GǶû[miBQM

j- TT2Hû2 HQB /2 GTH+2 UQm û[miBQM /2 uQmM;@GTH+2 2M M;HBbV- 2tTHB+Bi2 HǶû+`i /2 T`2bbBQM 2Mi`2 H2b /2mt +Ƭiûb /ǶmM2 BMi2`7+2 +Qm`#û2X GǶBMi2`7+2 2Mi`2 /2mt HB[mB/2b m `2TQb  /QM+ mM2 +Qm`#m`2 KQv2MM2 MmHH2 Ĝ 2i 7Q`K2 mM2 bm`7+2 KBMBKH2 2Mi`2 H2b /2mt ~mB/2bX AMp2`b2K2Mi- /Mb H2b KBHB2mt +QM}Mûb- HǶBMi2`7+2 2bi 7Q`i2K2Mi +Qm`#û2 T` H ;ûQKûi`B2 /m bm#bi`i +2 [mB +QM/mBi ¨ /2b û+QmH2K2Mib [mB T2mp2Mi TT`ŗi`2 +QKK2 bTQMiMûb /Mb H2 HB[mB/2X

6B;m`2 e , G dziQ`im2Ǵ- mM2 bi`iû;B2 /2b KM+?Qib 2KT2`2m`b 2M h2``2 /ûHB2 TQm` Hmii2` +QMi`2 H2 7`QB/X G 7Q`K2 +B`+mHB`2 /m ;`QmT2 T2`K2i /2 KBMBKBb2` HǶû+?M;2 i?2`KB[m2 p2+ HǶB` }M /2 ;`/2` mM2 i2KTû`im`2 +QM7Q`i#H2 m +2Mi`2 /m ;`QmT2X 1ti`Bi /m K;xBM2 :ûQX *`û/Bi T?QiQ , LB+QHb *QBHH`/X

AMi2`+iBQMb 2Mi`2 i`QBb T?b2b

GQ`b[m2 H2 HB[mB/2 2bi 2MiQm`û /ǶmM ;x- BH T`2M/ mM2 7Q`K2 bT?û`B[m2 }M /2 KBMBKBb2` b bm`7+2 HB#`2X G bBimiBQM /2pB2Mi mM T2m THmb +QKTH2t2 2M T`ûb2M+2 /ǶmM2 i`QBbBĕK2 T?b2- HQ`b[m2 H2 HB[mB/2 2Mi`2 m +QMi+i /ǶmM bQHB/2 T` 2t2KTH2X _2T`2MQMb mM2 2t@ Tû`B2M+2 bBKTH2 [mB +QMbBbi2 ¨ /ûTQb2` mM2 ;Qmii2 /2 HB[mB/2 bm` mM2 bm`7+2X SHmbB2m`b bBimiBQMb T2mp2Mi HQ`b āi`2 Q#b2`pû2bX URV GQ`b[mǶQM 7Bi iQK#2` mM2 ;Qmii2 /Ƕ?mBH2 bm` mM THM /2 i`pBH- H2 HB[mB/2 bǶûiH2 +QKTHĕi2K2Mi bm` H bm`7+2X UkV 1M /ûTQbMi mM2 ;Qmii2 /Ƕ2m bm` mM2 +Qmp2`im`2 THbiB}û2- H ;Qmii2 bǶûiH2 Hû;ĕ`2K2Mi TmBb b2 bi#BHBb2 2M T`2MMi mM2 7Q`K2 /2 +HQii2 bT?û`B[m2X UjV .Mb +2`iBMb +b mM T2m bTû+Bmt- T` 2t2KTH2 HQ`b[mǶQM p2`b2 [m2H[m2b ;Qmii2b /2 K2`+m`2 m bQH- Qɍ 2M /ûTQbMi /2 HǶ2m bm` mM2 TH[m2 i`ĕb +?m/2- H ;Qmii2 M2 bǶûiH2 Tb 2i ;`/2 mM2 7Q`K2 bT?û`B[m2X G2 HB[mB/2 T2mi HQ`b dz`QmH2`Ǵ bm` H bm`7+2 bQHB/2X

ALh_P.l*hAPL

b

a c

6B;m`2 d ,  :Qmii2 /Ƕ?mBH2 bm` mM2 TH[m2 2M bBHB+BmK T`ĕb [m2H[m2b DQm`bX G2b B`BbiBQMb KQMi`2Mi [m2 H2 HB[mB/2 bǶ2bi ûiHû Dmb[mǶ¨ ii2BM/`2 mM2 ûTBbb2m` /2 [m2H[m2b +2MiBM2b /2 MMQKĕi`2bX # :Qmii2 /Ƕ2m /ûTQbû2 bm` /m TH2tB;HbbX + :Qmii2b /2 K2`+m`2 bm` /m #QBbX

GǶûim/2 /2 HǶûiH2K2Mi UQm MQMV /ǶmM HB[mB/2 /ûTQbû bm` mM bm#bi`i bQHB/2- bQmp2Mi TT2Hû2 ûim/2 /m KQmBHH;2- 2bi mM2 #`M+?2 BKTQ`iMi2 /2 H +TBHH`BiûX G2 KQmBHH;2 DQm2 mM `ƬH2 BKTQ`iMi /Mb /2 MQK#`2mt /QKBM2b BM/mbi`B2Hb U+QbKûiB[m2b- BM/mbi`B2 Tûi`QHBĕ`2- û`QMmiB[m2- Kû/2+BM2V 2i /Mb #2m+QmT /2 bvbiĕK2b #BQHQ;B[m2bX .Mb +2`@ iBMb +b- BH 2bi miBH2 /2 7pQ`Bb2` H2 KQmBHH;2- T` 2t2KTH2 TQm` T2`K2ii`2 ¨ mM2 +`ĕK2 /2 TûMûi`2` iQmi2 H bm`7+2 bm` H[m2HH2 2HH2 2bi /ûTQbû2 Qm ¨ mM2 ?mBH2 /2 Hm#`B}2` +Q`@ `2+i2K2Mi H2 7QM/ /ǶmM2 TQāH2X .Mb /Ƕmi`2b +b- BH b2` T`û7û`#H2 /ǶûpBi2` [m2 H2 HB[mB/2 bǶûiH2 bm` H2 bQHB/2- }M /2 ;`/2` mM KMi2m b2+ HQ`b[mǶBH TH2mi Qm TQm` 2KTā+?2` H +QMiKBMiBQM /ǶmM T`Q/mBi HBK2MiB`2 T` H bm`7+2 /2 bQM 2K#HH;2X G2b T`QT`Bûiûb /2 KQmBHH;2 /Bzĕ`2Mi 2M 7QM+iBQM /m HB[mB/2 2i /m bmTTQ`i miBHBbû Ĝ mM 2z2i i`ĕb #B2M BHHmbi`û T` H2 i#H2m /2 aQT?B2 EBi+?BM; U};m`2 3V [mB 2tTHQBi2 H2b `2M/mb pBbm2Hb /2 +BM[ ivT2b /2 T2BMim`2 bm` mM bmTTQ`i 2M TQHv+`#QMi2X

6B;m`2 3 , aQT?B2 EBi+?BM;X aMb iBi`2 UJB``Q`BM;V- kyRN- :H2`B2 Ab#2HH2 :QmMQ/X 3k t eR +KX G2b /Bzû`2Mi2b T2BMim`2b miBHBbû2b U;Qm+?2- +`vHB[m2- ?mBH2- 2M+`2 /2 *?BM2- ;2bbQV bm` H2 bmTTQ`i 2M TQHv+`#QMi2 /QMM2Mi /2b `2M/mb `2K`[m#H2K2Mi /Bzû`2MibX *`û/Bi T?QiQ , _2#2++ 6Mm2H2X

GǶûim/2 /m KQmBHH;2 2bi mM bmD2i bb2x `û+2Mi 2i 7Bi 2M+Q`2 mDQm`/Ƕ?mB HǶQ#D2i /2 MQK#`2mt i`pmt- +QKK2 2M iûKQB;M2 T` 2t2KTH2 H2 ;`M/ BMiû`āi TQ`iû ¨ H `2pm2 /2 SB2``2 :BHH2b /2 :2MM2b bm` H2 KQmBHH;2 (RN)- +Biû2 THmb /2 dyyy 7QBb /2TmBb RN38X G2 #`BiMMB[m2 h?QKb uQmM; 2bi H2 T`2KB2` ¨ i?ûQ`Bb2` H2b BMi2`+iBQMb m MBp2m /2 H HB;M2 /2 +QMi+i 2Mi`2 mM HB[mB/2 2i mM bQHB/2 /Mb mM `iB+H2 Tm#HBû 2M R3y8 (ky)X GQ`b[mǶmM2 ;Qmii2 HB[mB/2 bǶûiH2 bm` mM2 bm`7+2- H2 HB[mB/2 7Q`K2 mM2 BMi2`7+2 p2+ H2 bQHB/2 2i H T?b2 ;x2mb2 KBb bmTT`BK2 mbbB mM2 BMi2`7+2 2Mi`2 H2 bQHB/2 2i H2 ;xX LQmb pQMb pm [m2 H2b BMi2`+iBQMb KQHû+mHB`2b +QM/mBb2Mi ¨ HǶTT`BiBQM /ǶmM2 i2MbBQM /2 bm`7+2 γX AH 2M 2bi /2 KāK2 TQm` mM bQHB/2 [mB TQbbĕ/2 HmB mbbB mM2 ûM2`;B2 /2 bm`7+2 [m2 MQmb MQiQMb γsX

*QKK2 TQm` H2b HB[mB/2b- γs/ûT2M/ /2b HBBbQMb +?BKB[m2b 2Mi`2 H2b KQHû+mH2b /m bQHB/2 2i

p`B2 /Ƕ2MpB`QM 20 KLfK TQm` /2b bm`7+2b /2 dz#bb2 ûM2`;B2Ǵ U+QKK2 H2b THbiB[m2bV ¨ THmb /2 Ryyy KLfK TQm` /2b bm`7+2b /2 dz?mi2 ûM2`;B2Ǵ U+QKK2 H2b KûimtVX G2 /ûTH+2K2Mi /2 H HB;M2 /2 +QMi+i +`û2 mM2 BMi2`7+2 2Mi`2 H2 HB[mB/2 2i H2 bQHB/2 +`+iû`Bbû2 2HH2 mbbB T` mM2 ûM2`;B2 /2 bm`7+2 γls[mB /ûT2M/ /2b {MBiûb 2Mi`2 H2 HB[mB/2 2i H2 bQHB/2X SQm` +?[m2

+QK#BMBbQM bQHB/2@HB[mB/2- H2b BMi2`+iBQMb 2Mi`2 H2b /2mt T?b2b 2Mi`ŗM2Mi HǶTT`BiBQM /ǶmM M;H2 /2 +QMi+i 2Mi`2 H bm`7+2 HB[mB/2 2i H bm`7+2 bQHB/2X LQmb TQmpQMb `2i`Qmp2` +2i M;H2 2M Q#b2`pMi HǶû[mBHB#`2 HQ+H m MBp2m /2 H HB;M2 /2 +QMi+i U};m`2 NVX

α

R _α cos α dx

dx

b a

6B;m`2 N , a+?ûK /ǶmM2 ;Qmii2 /ûTQbû2 bm` mM bQHB/2X G2 HB[mB/2 2bi +QHQ`û 2M #H2m- H2 bQHB/2 2bi `2T`ûb2Miû T` H HB;M2 MQB`2 2i H T?b2 ;x2mb2 2bi 2M #HM+X  § HǶû+?2HH2 /ǶmM2 ;Qmii2- H2 bvbiĕK2 2bi T`Kûi`û T` H2 `vQM R /2 H ;Qmii2- bQM pQHmK2 Ω 2i HǶM;H2 /2 +QMi+i α 2Mi`2 H2 HB[mB/2 2i H2 bQHB/2X # m MBp2m /2 H HB;M2 i`BTH2- b2mH HǶM;H2 /2 +QMi+i BMi2`pB2MiX

GQ`b[mǶQM TQb2 mM2 ;Qmii2 bm` mM2 bm`7+2 bQHB/2 THi2- 2HH2 T`2M/ mM 7Q`K2 /2 +HQii2 bT?û`B[m2 /QMi H2 pQHmK2 Ω 2bi }tû T` bQM `vQM R 2i bQM M;H2 /2 +QMi+i p2+ H2 bQHB/2 α Ub+?ûK NVX LQmb TQmpQMb +H+mH2` H pH2m` /Ƕû[mBHB#`2 /2 HǶM;H2 α /2 /2mt KMBĕ`2b /Bzû`2Mi2bX GQ+H2K2Mi- HQ`b[m2 H HB;M2 /2 +QMi+i b2 /ûTH+2 /ǶmM2 [mMiBiû /x-2HH2 +`û2 mM2 BMi2`7+2 bQHB/2@HB[mB/2 /2 iBHH2 /x 2i mM2 BMi2`7+2 HB[mB/2@;x /2 HQM;m2m` /x +Qb α- H2 /ûTH+2K2Mi bmTT`BK2 mbbB mM2 BMi2`7+2 bQHB/2@;x /ǶmM2 iBHH2 /xX GQ`b /ǶmM /ûTH+2K2Mi BM}MBiûbBKH /2 H HB;M2 i`BTH2- HǶûM2`;B2 /m bvbiĕK2 T` mMBiû /2 HQM;m2m` e p`B2 /QM+ /ǶmM2 [mMiBiû /e = (γ +Qb α + γls− γs)/x Ub+?ûK N#VX AH 2bi mbbB TQbbB#H2

/2 `BbQMM2` ¨ HǶû+?2HH2 /2 H ;Qmii2X 1M Mû;HB;2Mi H ;`pBiû- HǶûM2`;B2 /2 H ;Qmii2 pmi E_{= γA + (γ}ls− γs)S Qɍ A 2i S `2T`ûb2Mi2Mi HǶB`2 /2 H +HQii2 bT?û`B[m2 2i /2 H bm`7+2

/2 +QMi+i bQHB/2@HB[mB/2X

ALh_P.l*hAPL

GQ`b[m2 |γs− γls| < γ- BH 2tBbi2 mM M;H2 /2 +QMi+i θ TQm` H2[m2H H HB;M2 i`BTH2 b2

bi#BHBb2X PM T`H2 HQ`b /2 KQmBHH;2 T`iB2H- Q#b2`pû T` 2t2KTH2 bm` H T?QiQ d#X lM2 ;Qmii2 /Ƕ2m b2` bQmp2Mi 2M bBimiBQM /2 KQmBHH;2 T`iB2H ¨ +mb2 /2 H i2MbBQM /2 bm`7+2 ûH2pû2 /2 +2 HB[mB/2X G KBMBKBbiBQM /2 e- /QMM2 mM2 +QM/BiBQM /Ƕû[mBHB#`2 TQm` /e//x = 0 HQ`b[m2 α = θ p2+ ,

+Qb θ = Ƙγs− γls

γ U9V

*2ii2 TT`Q+?2 2bi û[mBpH2Mi2 ¨ H KBMBKBbiBQM /2 HǶûM2`;B2 /2 bm`7+2 E TQm` mM2 ;Qmii2 /2 pQHmK2 Ω UpQB` HǶMM2t2 VX

.Mb H2 +b Qɍ H i2MbBQM /2 bm`7+2 /m HB[mB/2 2bi 7B#H2 Qm HQ`b[m2 +2HH2 /m bQHB/2 2bi ;`M/2- BH 2bi TQbbB#H2 [m2 γs− γls > γX .Mb +2 +b- /e//x < 0 [m2H [m2 bQBi HǶM;H2

/2 +QMi+iX G2 HB[mB/2 bǶûiH2 HQ`b Dmb[mǶ¨ `2+Qmp`B` 2MiBĕ`2K2Mi H2 bQHB/2- +QKK2 TQm` H ;Qmii2 /2 H };m`2 d#X *2ii2 bBimiBQM /2 KQmBHH;2 iQiH 2bi mM T?ûMQKĕM2 7`û[m2Mi TQm` H2b ?mBH2b [mB QMi mM2 7+?2mb2 i2M/M+2 ¨ `2+Qmp`B` iQmi2b H2b bm`7+2b bm` H2b[m2HH2b 2HH2b bQMi /ûTQbû2bX

G2 +b γs− γls < γ +Q``2bTQM/ ¨ mM2 bBimiBQM Qɍ H2b KQHû+mH2b /m bQHB/2 bQMi THmb

bi#BHBbû2b T` H2b KQHû+mH2b /m HB[mB/2 [m2 T` /2b KQHû+mH2b /m KāK2 bQHB/2- +2 [mB MǶ2bi 2M T`iB[m2 DKBb Q#b2`pûX .2b bBimiBQMb /2 KQmBHH;2 MmH T2mp2Mi iQmi27QBb āi`2 TT`Q+?û2b Qɍ KāK2 ii2BMi2b 2M DQmMi bm` H `m;QbBiû /2b bm`7+2b Qm `ûmbbBbbMi ¨ BMbû`2` mM +QmbbBM /ǶB` 2Mi`2 H2 HB[mB/2 2i H bm`7+2 Ĝ T` 2t2KTH2 2M +?mzMi H bm`7+2X LQmb Q#i2MQMb HQ`b mM2 bm`7+2 [m2 HǶQM T2mi [mHB}2` /Ƕû`QT?BH2 TmBb[mǶ2HH2 T`û7ĕ`2 H2 +QMi+i /m ;x m +QMi+i /m HB[mB/2X

1M T`iB[m2- BH 2bi T`2b[m2 BKTQbbB#H2 /2 K2bm`2` /B`2+i2K2Mi H i2MbBQM /2 bm`7+2 /ǶmM bQHB/2- QM p /QM+ THmiƬi K2bm`2` H i2MbBQM /2 bm`7+2 /m HB[mB/2 b2mH- TmBb HǶM;H2 /2 +QMi+i 2Mi`2 H2 HB[mB/2 2i H2 bQHB/2 Ĝ /2b Kûi?Q/2b /2 K2bm`2b bQMi T`ûb2Miû2b 2M /ûiBH /Mb H2 HBp`2 :Qmii2b- #mHH2b- T2`H2b 2i QM/2b (kR)X *2b /2mt ;`M/2m`b T2`K2ii2Mi /2 /û+`B`2 H2b BMi2`+iBQMb 2Mi`2 H2 HB[mB/2 2i H2 bQHB/2 2i /Ƕûim/B2` H2b +QM/BiBQMb /ǶûiH2K2MiX .Mb H2 +b Qɍ +Qb θ > 0- +Ƕ2bi@¨@/B`2 bB γs> γls HǶM;H2 2bi B;m 2i QM T`H2 ;ûMû`H2K2Mi /ǶmM HB[mB/2

dzKQmBHHMiǴ Qm +QKTHĕi2K2Mi KQmBHHMi bB θ = 0X AMp2`b2K2Mi- bB γs < γls Uθ > π/2V- QM

T`H2 /2 +QMi+i T2m KQmBHHMi- pQB` MQM KQmBHHMi HQ`b[mǶQM Q#b2`p2 mM M;H2 /2 +QMi+i i`ĕb ;`M/ U;ûMû`H2K2Mi θ > 150êVX GQ`b[m2 H2 HB[mB/2 +QMbB/û`û 2bi HǶ2m- H bm`7+2 2bi /Bi2 ?v/`QT?Q#2 bB θ > 90ê UQɍ bmT2` ?v/`QT?Q#2 2M +b /2 MQM KQmBHH;2VX SQm` /2b ;`M/b M;H2b /2 +QMi+i- H2b ;Qmii2b /2pB2MM2Mi i`ĕb KQ#BH2b bm` H bm`7+2X *Ƕ2bi H2 +b /m K2`+m`2 [mB T`ûb2Mi2 mM M;H2 /2 +QMi+i ûH2pû p2+ T`2b[m2 iQmb H2b bmTTQ`ibX G2b ;Qmii2b /2 K2`+m`2 pQMi /QM+ `QmH2` i`ĕb 7+BH2K2Mi bm` H bm`7+2- bMb /Qmi2 H `BbQM T`BM+BTH2 /2 bQM M+B2MM2 /ûMQKBMiBQM , H2 pB7@`;2MiX

JQMiû2 +TBHHB`2 , û[mBHB#`2

_2T`2MQMb HǶ2tTû`B2M+2 +HbbB[m2 /ǶmM2 KQMiû2 +TBHHB`2 2M i`2KTMi i`QBb im#2b /2 `vQMb /Bzû`2Mib /Mb mM2 ?mBH2 bBHB+QM2 +QHQ`û2 2M `Qm;2X LQmb T?QiQ;`T?BQMb bm` H };m`2 Ry HǶûii }MH /m bvbiĕK2X

5

mm

6B;m`2 Ry , >mi2m` /Ƕû[mBHB#`2 /ǶmM2 ?mBH2 bBHB+QM2 U+QHQ`û2 2M `Qm;2V /Mb /2b im#2b /2 `vQMb /2 THmb 2M THmb ;`QbX

GǶ?mBH2 `2KQMi2 ¨ HǶBMiû`B2m` /2b im#2b Dmb[mǶ¨ ii2BM/`2 mM2 ?mi2m` /2 HǶQ`/`2 /m +2MiBKĕi`2- [mB /ûT2M/ /m `vQM /m im#2 miBHBbûX GQ`b[mǶQM KĕM2 mM im#2 +TBHHB`2 /2 `vQM r m +QMi+i /ǶmM HB[mB/2- +2HmB@+B T2mi KQmBHH2` mM2 bm`7+2 A ¨ HǶBMiû`B2m` /m im#2-HǶûM2`;B2 /m bvbiĕK2 p`B2 HQ`b /ǶmM2 [mMiBiû Ec = (γls− γs)A = −2πγr +Qb θz- Qɍ z 2bi

H ?mi2m` ii2BMi2 T` H2 KûMBb[m2X aB H2 HB[mB/2 2bi KQmBHHMi TQm` H2 Kiû`Bm +QMbiBimiB7 /m im#2 U+Qb θ > 0V- BH 2bi iB`û ¨ HǶBMiû`B2m` /m im#2 2i KQMi2 Dmb[mǶ¨ +2 [m2 H T2bMi2m` pB2MM2 biQTT2` H T`Q;`2bbBQM /m KûMBb[m2X GǶûM2`;B2 TQi2MiB2HH2 /2 T2bMi2m` /2 H +QHQMM2 /2 HB[mB/2 pmi Ep = πr2ρgz2/2 UbB H2 im#2 2bi p2`iB+HVX LQmb TQmpQMb HQ`b +H+mH2` H

?mi2m` h 2M û+`BpMi H +QM/BiBQM /Ƕû[mBHB#`2 /(Ec+ Ep)//z = 0 TQm` z = h ,

h= 2γ+Qb θ

ρgr U8V

G ?mi2m` h 2bi TT2Hû2 ?mi2m` /2 Cm`BM 2M ?QKK;2 mt i`pmt /2 +2 bpMi M;HBb bm` H KQMiû2 +TBHHB`2X G HQB 8 2bi mM2 û[miBQM 7QM/K2MiH2 TQm` H2 `2bi2 /2 MQi`2 ûim/2X LQmb pQvQMb /Ƕ#Q`/ [m2 H ?mi2m` h 2bi BMp2`b2K2Mi T`QTQ`iBQMM2HH2 m `vQM /m im#2 r- +2 [mB 2tTHB[m2 H2b /Bzû`2M+2b Q#b2`pû2b bm` H };m`2 RyX G ?mi2m` /Ƕû[mBHB#`2 /ûT2M/ mbbB /2 T`Kĕi`2b T`QT`2b m HB[mB/2 +QKK2 b i2MbBQM bmT2`}+B2HH2 2i b Kbb2 pQHmKB[m2X lM2 ?mBH2- /2 i2MbBQM bmT2`}+B2HH2 γ ⇡ 20 KLfK- KQMi2` /QM+ KQBMb ?mi [m2 /2 HǶ2m- /2 i2MbBQM bmT2`}+B2HH2 γ ⇡ 70 KLfK- ¨ M;H2 /2 +QMi+i /QMMûX G Kbb2 pQHmKB[m2 BM~m2M+2 mbbB H ?mi2m` /m KûMBb[m2 ¨ H }M /2 HǶ2tTû`B2M+2X .Mb mM KāK2 im#2- HǶûi?MQH U/2 Kbb2 pQHmKB[m2 ρ = 789 F;fK3V KQMi2` /QM+ mM T2m THmb

?mi [mǶmM2 ?mBH2 bBHB+QM2 Uρ ⇡ 970 F;fK3_{V HQ`b [m2 +2b /2mt HB[mB/2b QMi mM2 i2MbBQM}

bmT2`}+B2HH2 bBKBHB`2 Uγ ⇡ 20 KLfKVX G ?mi2m` /m HB[mB/2 /ûT2M/ mbbB /2 bQM {MBiû p2+ H2 bQHB/2 [m2 i`/mBi H2 i2`K2 +Qb θX lM KāK2 HB[mB/2 TQm`` /QM+ KQMi2` THmb Qm KQBMb ?mi /Mb /2b im#2b /2 `vQMb B/2MiB[m2b 2M 7QM+iBQM /m Kiû`Bm [mB +QMbiBim2 H2 im#2X GQ`b[m2 +Qb θ 2bi Mû;iB7- QM bǶii2M/ KāK2 ¨ pQB` H2 HB[mB/2 /2b+2M/`2 ¨ HǶBMiû`B2m` /m im#2- mM2 T`QT`Bûiû /ûD¨ `2K`[mû2 T` Cm`BM p2+ /m dzpB7@`;2MiǴ U[mB+FbBHp2` 2M M;HBbV (kk)X LQmb ûim/B2`QMb +2b +b T2m +QKKmMb /2 dz/2b+2Mi2Ǵ +TBHHB`2 /Mb H b2+QM/2 T`iB2

ALh_P.l*hAPL /2 MQi`2 ûim/2X

SHmb ;ûMû`H2K2Mi- H HQB /2 Cm`BM /û+`Bi H ?mi2m` /Ƕû[mBHB#`2 ii2BMi2 T` mM HB[mB/2 /Mb mM KBHB2m TQ`2mt [m2H+QM[m2X G2 `vQM r `2T`ûb2Mi2 HQ`b mM `vQM KQv2M /2b THmb T2iBib BMi2`biB+2b /Mb H2 KBHB2m TQ`2mtX .Mb H2b KBHB2mt ;`MmHB`2b +QKK2 /Mb mM KQ`+2m /2 bm+`2- +Ƕ2bi H iBHH2 /ǶmM ;`BM [mB /ûi2`KBM2 H2 `vQM +`+iû`BbiB[m2 ¨ T`2M/`2 2M +QKTi2X *Ƕ2bi BMbB H KQMiû2 +TBHHB`2 [mB }t2 H iBHH2 KtBKH2 /2b +?i2mt /2 b#H2 ?mKB/2 TmBb[m2 +Ƕ2bi HǶ2m [mB bbm`2 H +Q?ûbBQM 2Mi`2 H2b ;`BMb /2 b#H2X G iBHH2 KB+`QKûi`B[m2 /2b ;`BMb /2 b#H2 }t2 H ?mi2m` KtBKH2 ii2BMi2 T` HǶ2m [mB pmi [m2H[m2b /BxBM2b /2 +2MiBKĕi`2bX .2 KāK2- HQ`b[mǶQM TBM+2 /2mt TH[m2b 2M p2``2 HǶmM2 +QMi`2 HǶmi`2- QM +`ûû mM +QBM /Mb H2[m2H H2 HB[mB/2 `2KQMi2 +QKK2 T?QiQ;`T?Bû bm` H };m`2 RR#X G2 +QM}M2K2Mi 2bi /ȿ ¨ HǶû+`i2K2Mi 2Mi`2 H2b /2mt TH[m2b /2 p2``2- 2i HǶBMi2`7+2 2bi HQ`b mM2 ?vT2`#QH2 [mB bǶQ#b2`p2 /B`2+i2K2Mi 2M pm2 /2 +ƬiûX

b a

≈ 2 cm 5 mm

6B;m`2 RR ,  lM +?i2m /2 b#H2 û;vTiB2MX 1ti`Bi /m }HK biû`Bt 2i P#ûHBt , JBbbBQM *HûQTi`2X

# lM2 bQHmiBQM /Ƕûi?MQH U+QHQ`û2 2M #H2mV 2Mi`2 /2mt TH[m2b /2 p2``2 TH[mû2b HǶmM2 +QMi`2 HǶmi`2

¨ ;m+?2 2i Hû;ĕ`2K2Mi û+`iû2b ¨ /`QBi2X

JQMiû2 +TBHHB`2 , /vMKB[m2

lM /2mtBĕK2 pQH2i /Ƕûim/2 /2 H KQMiû2 +TBHHB`2 +QM+2`M2 H /vMKB[m2 /m 7`QMi /2 HB[mB/2X _2T`2MQMb HǶ2tTû`B2M+2 /2 KQMiû2 +TBHHB`2 T`ûb2Miû2 bm` H };m`2 Ry- 2M bǶBMiû`2b@ bMi +2ii2 7QBb ¨ H /vMKB[m2 /ǶBKT`û;MiBQM /m HB[mB/2X SQm` +2H- MQmb T?QiQ;`T?BQMb mM2 ?mBH2 bBHB+QM2 +QHQ`û2- 8y 7QBb THmb pBb[m2mb2 [m2 HǶ2m m +Qm`b /ǶmM2 KQMiû2 +TBHHB`2X G2b T?QiQ;`T?B2b bQMi T`Bb2b 2M /Qm#HMi ¨ +?[m2 7QBb HǶBMi2`pHH2 /2 i2KTb bm` H };m`2 Rk TQm` TQmpQB` Q#b2`p2` THmb T`û+BbûK2Mi H KQMiû2 /m HB[mB/2X

0 s 0.3 s 0.6 s 1.3 s 2.5 s 5 s 10 s 20 s

3

mm

6B;m`2 Rk , S?QiQ;`T?B2b /2b T`2KBĕ`2b b2+QM/2b /2 H KQMiû2 +TBHHB`2 /ǶmM2 ?mBH2 bBHB+QM2 /Mb mM im#2 2M p2``2X

G2 HB[mB/2 KQMi2- iB`û T` H 7Q`+2 +TBHHB`2- TmBb bǶ``āi2 ¨ H ?mi2m` /2 Cm`BM T`ĕb [m2H[m2b KBMmi2bX G2 KûMBb[m2 T`Q;`2bb2 `TB/2K2Mi m /û#mi /2 HǶ2tTû`B2M+2- TmBb `@ H2MiBi T2iBi ¨ T2iBiX AH 7mi T` 2t2KTH2 2MpB`QM yXj b2+QM/2 m HB[mB/2 TQm` T`+Qm`B` H2b /2mt T`2KB2`b KBHHBKĕi`2b TmBb mM2 b2+QM/2 2MiBĕ`2 TQm` T`+Qm`B` H2b /2mt bmBpMibX *2 T?ûMQKĕM2  ûiû `2Tû`û bb2x iƬi T` H2b 2tTû`BK2Mii2m`b [mB QMi `2K`[mû [m2 H2 7`QMi /2 HB[mB/2 T`Q;`2bb2 +QKK2 H `+BM2 /m i2KTb HQ`b[m2 H ;`pBiû 2bi Mû;HB;2#H2 Ĝ +2 [mB 2bi H2 +b HQ`b[m2 H2 KûMBb[m2 2bi HQBM /2 H ?mi2m` /2 Cm`BMX *2b Q#b2`piBQMb bQMi +2T2M/Mi +QKTH2t2b ¨ i?ûQ`Bb2` TmBb[m2 +Ƕ2bi H 7`B+iBQM pBb[m2mb2 /m HB[mB/2 [mB pB2Mi `H2MiB` H2 KûMBb[m2X AH 7m/` /QM+ ii2M/`2 [m2 bQBi TQTmH`Bbû2 H +ûHĕ#`2 HQB /2 SQBb2mBHH2 2M R399 (kj) TQm` +QKT`2M/`2 +2ii2 /vMKB[m2 /Ƕû+QmH2K2MiX

m /û#mi /m ssĕK2 _{bBĕ+H2 /2b +?2`+?2m`b +QKK2M+2Mi ¨ i?ûQ`Bb2` +Q``2+i2K2Mi H} /vMKB[m2 /2 H KQMiû2 +TBHHB`2X 1M RNye- "2HH 2i *K2`QM `ûHBb2Mi /2b 2tTû`B2M+2b /ǶBKT`û;MiBQM /Mb /2b im#2b +TBHHB`2b KBMi2Mmb ?Q`BxQMiH2K2Mi p2+ /2 HǶ2m- /2 HǶûi?MQH 2i /m #2MxĕM2 (k9)X AHb B/2MiB}2Mi TQm` H T`2KBĕ`2 7QBb H 7Q`K2 /2 HǶû[miBQM /2 T`QT;iBQM 2M `+BM2 /m i2KTb , z ⇠ t1/2 <sub>2i `2K`[m2Mi [m2 +2ii2 HQB `2bi2 TTHB+#H2</sub>

/Mb /2b KBHB2mt TQ`2mt i2Hb [m2 /m TTB2` #mp`/ Qm /2b bQHb `;BH2mtX AHb T`QTQb2Mi HQ`b mM2 Kûi?Q/2 TQm` K2bm`2` H2 +Q2{+B2Mi /2 T`QT;iBQM 2i Q#iB2MM2Mi /2b `ûbmHiib /Bzû`2Mib TQm` H2b i`QBb HB[mB/2b miBHBbûb Ĝ mM `ûbmHii [mǶBHb M2 T`pB2MM2Mi Tb ¨ 2tTHB[m2`X Zm2H[m2b MMû2b THmb i`/- Gm+b U2M RNR3V (k8) 2i qb?#m`M U2M RNkRV (ke) Tm#HB2Mi /2mt `iB+H2b i`ĕb bBKBHB`2b UH2 T`2KB2` 2M HH2KM/ 2i H2 b2+QM/ 2M M;HBbV [mB /û+`Bp2Mi H /vMKB[m2 /ǶBKT`û;MiBQM /Mb mM im#2X *2b /2mt ûim/2b bǶTTmB2Mi bm` H2b `ûbmHiib /2 "2HH 2i *K2`QM- BMbB [m2 bm` H HQB /2 SQBb2mBHH2- 2i KQ/ûHBb2Mi /2 KMBĕ`2 bb2x +QKTHĕi2 H /vMKB[m2 /2 T`Q;`2bbBQM /m KûMBb[m2 /Mb mM im#2 +TBHHB`2X lM2 7QBb HǶû+QmH2K2Mi ûi#HBi /Mb H2 im#2- H 7`B+iBQM 2t2`+û2 T` H2 im#2 bm` H +QHQMM2 /2 HB[mB/2 pmi Fp =

8πηz ˙z UQɍ ˙z `2T`ûb2Mi2 H pBi2bb2 /m KûMBb[m2V 2i pB2Mi `H2MiB` HǶpM+û2 /m HB[mB/2 Ĝ QM TQm`` `2i`Qmp2` H /û`BpiBQM /2 H 7`B+iBQM /2 SQBb2mBHH2 /Mb T`2b[m2 iQmb H2b Qmp`;2b /Ƕ?v/`Q/vMKB[m2 +QKK2 T` 2t2KTH2 >v/`Q/vMKB[m2 T?vbB[m2 (kd)X GǶû[mBHB#`2 2Mi`2 H 7`B+iBQM pBb[m2mb2 2i H 7Q`+2 +TBHHB`2 Fc = 2πrγ+Qb θ /QMM2 H T`Q;`2bbBQM /m

ALh_P.l*hAPL

+QMMm2 bQmb H2 MQK /2 HQB /2 qb?#m`M ,

z2 = γr+Qb θ

2η t UeV

GǶû[miBQM e- Q#i2Mm2 T` Gm+b 2i qb?#m`M- b2` `2i2Mm2 bQmb H2 MQK /2 dzHQB /2 qb?@ #m`MǴ Ĝ mM2 /ûMQKBMiBQM bb2x +`m2HH2 TQm` Gm+b- /QMi HǶ`iB+H2  TQm`iMi ûiû Tm#HBû i`QBb Mb pMi +2HmB /2 qb?#m`M- 2bi mDQm`/Ƕ?mB +Biû 2MpB`QM /Bt 7QBb KQBMb bQmp2MiXXX G HQB /2 qb?#m`M /û+`Bi H T`Q;`2bbBQM /m KûMBb[m2 HB[mB/2 /Mb mM im#2 /2 T2iBi /BKĕi`2-H2 KûMBb[m2 `/BKĕi`2-H2MiBi ¨ +mb2 /2 H 7`B+iBQM pBb[m2mb2 /2 THmb 2M THmb BKTQ`iMi2 /Mb /BKĕi`2-H2 im#2 2i H pBi2bb2 /m 7`QMi /2 HB[mB/2 /BKBMm2 +QKK2 ˙z ⇠ 1/zX LQmb `2i`QmpQMb #B2M H HQB /Ƕû+?2HH2 Q#b2`pû2 T` "2HH 2i *K2`QM z ⇠ t1/2- bbQ`iB2 /ǶmM +Q2{+B2Mi [mB /ûT2M/

¨ H 7QBb /2b T`QT`Bûiûb /m HB[mB/2 UH pBb+QbBiû η- H i2MbBQM /2 bm`7+2 γV- /2b T`Kĕi`2b ;ûQKûi`B[m2b /m im#2 UH2 `vQM rV 2i /m `2pāi2K2Mi UM;H2 /2 +QMi+i θV Ĝ +QKK2 H2 `2K`[m2 Gm+b /Mb HǶBMi`Q/m+iBQM /2 bQM `iB+H2- +2b MQK#`2mt T`Kĕi`2b `2M/2Mi H KQ/ûHBbiBQM `2HiBp2K2Mi /B{+BH2- +2 [mB 2tTHB[m2 bMb /Qmi2 HǶBMi2`pHH2 /2 i2KTb `2HiB@ p2K2Mi HQM; U/Bt MbV 2Mi`2 H2 KQ/ĕH2 i?ûQ`B[m2 2i H2b Q#b2`piBQMb 2tTû`BK2MiH2b /2 "2HH 2i *K2`QMX 10 cm 1 min 3 h 20 h b a 1 s 4 s 12 s 100 s 1 cm

6B;m`2 Rj , G HQB /2 qb?#m`M /û+`Bi i`ĕb #B2M mM ;`M/ MQK#`2 /2 T?ûMQKĕM2b /ǶBKT`û;MiBQMX

 b+2MbBQM +TBHHB`2 /Mb mM #HQ+ /2 i2``2X aQm`+2 , K¨+Q- JiBĕ`2b ¨ +QMbi`mB`2X # lM2 i+?2

/Ƕ2M+`2 bǶûiH2 /Mb mM2 72mBHH2 /2 TTB2` p2+ mM2 /vMKB[m2 [mB pû`B}2 mbbB H HQB /2 qb?#m`MX G HQB /2 T`QT;iBQM z ⇠ t1/2 <sub>`ûbBbi2 ¨ /2b ;ûQKûi`B2b THmb +QKTH2t2b 2M+Q`2 2i MQmb</sub>

`2i`QmpQMb +2b /vMKB[m2b /ǶBKT`û;MiBQM TQm` /2b Kiû`Bmt mbbB /Bp2`b [m2 /2b iBbbmb-/m b#H2- iBbbmb-/m TTB2` Qm /2b }#`2b i2tiBH2b (k3- kN- jy- jR)X JāK2 H /BbiM+2 BK#B#û2 T` /m i?û +?m/ /Mb /2b +QQFB2b pû`B}2 H HQB /2 qb?#m`M- +QKK2 KQMi`û T` H2 +?2`+?2m` G2M 6Bb?2` 2M +H+mHMi H2 i2KTb dzQTiBKHǴ TQm` i`2KT2` mM #Bb+mBi /Mb mM2 ibb2 /2 i?û (jk)X G2 KQ/ĕH2 /2 qb?#m`M /û+`Bi mbbB HǶBKT`û;MiBQM /ǶmM HB[mB/2 /Mb mM KBHB2m

TQ`2mt [m2H+QM[m2- H2 HB[mB/2 p /Bzmb2` /Mb iQmi2b H2b /B`2+iBQMb- 2i H2 +Q2{+B2Mi /2 T`QTQ`iBQMMHBiû [mB `2HB2 H /BbiM+2 T`+Qm`m2 T` H2 7`QMi /2 HB[mB/2 m i2KTb /ǶûiH2K2Mi T2mi āi`2 bbBKBHû ¨ mM +Q2{+B2Mi /2 dz/BzmbBQMǴ [mB /ûT2M/ /2 H ;ûQKûi`B2 /2b TQ`2b 2i /2b BMi2`+iBQMb 2Mi`2 H2 bQHB/2 2i H2 HB[mB/2X *2 +Q2{+B2Mi /2 /BzmbBQM- 7+BH2K2Mi K2bm`#H2-2bi miBHBbû TQm` /ûi2`KBM2` H2 `vQM +`+iû`BbiB[m2 /2b TQ`2b ¨ HǶBMiû`B2m` /ǶmM KBHB2m +QKTH2t2 (jj- j9)X

AMp2`b2K2Mi [m2H[m2b bBimiBQMb bTû+B}[m2b [mB û+?TT2Mi ¨ H /vMKB[m2 /û+`Bi2 T` qb?#m`M QMi ûiû `2TQ`iû2b `û+2KK2MiX .Mb /2b im#2b +TBHHB`2b /2 b2+iBQM p`B#H2- H T`Q;`2bbBQM /m HB[mB/2 b2 7Bi p2+ mM2 /vMKB[m2 [mB /ûT2M/ /2 H ;ûQKûi`B2 /m im#2 (j8-je)- /2b ;ûQKûi`B2b QTiBKBbû2b T2`K2ii2Mi HQ`b /Ƕ++ûHû`2` H2 /ûTH+2K2Mi (jd)X .Mb /2b +Mmt T`ûb2MiMi /2b +QBMb- HQ`b[mǶBH MǶv  THmb /2 /BbiM+2 +`+iû`BbiB[m2- H /vMKB[m2 `2TQ`iû2 2bi z(t) ⇠ t1/3 <sub>(j3)X G2b KBHB2mt TQ`2mt p2+ mM2 H`;2 /Bbi`B#miBQM /2 TQ`2b bQMi</sub>

mbbB BKT`û;Mûb p2+ mM2 /vMKB[m2 /Bzû`2Mi2- i2HH2 [m2 z ⇠ t1/4 _(jN)X

1M Qmi`2- H HQB /2 qb?#m`M T`û/Bi mM2 pBi2bb2 [mB /Bp2`;2 ¨ i2KTb +Qm`i- mM2 bBM@ ;mH`Biû [m2 MQmb ûim/B2`QMb /Mb H T`2KBĕ`2 T`iB2 /2 +2 KMmb+`BiX GQ`b /ǶmM2 KQMiû2 +TBHHB`2- H /vMKB[m2 /2 qb?#m`M 2bi HBKBiû2 T` H ;`pBiû- [mB biQTT2 H T`Q;`2bbBQM /m KûMBb[m2X 1M +?M;2Mi HǶBM+HBMBbQM /m im#2- T` 2t2KTH2 2M KBMi2MMi H2 im#2 ?Q@ `BxQMiH2K2Mi- H /vMKB[m2 2M `+BM2 /m i2KTb T2mi āi`2 Q#b2`pû2 bm` THmbB2m`b /BxBM2b /2 +2MiBKĕi`2b TmBb[m2 H ;`pBiû M2 pB2Mi THmb biQTT2` H2 HB[mB/2X AH 2tBbi2 HQ`b /Ƕmi`2b Kû+MBbK2b U+QKK2 HǶûpTQ`iBQMV [mB T2mp2Mi #HQ[m2` HǶBKT`û;MiBQM- mM T?ûMQKĕM2 [m2 MQmb `2i`Qmp2`QMb /Mb H i`QBbBĕK2 T`iB2 /2 +2 KMmb+`BiX

P#D2+iB7 /2 H i?ĕb2

G KQMiû2 +TBHHB`2 2bi mM T?ûMQKĕM2 i`ĕb `B+?2- [mB  7Bi HǶQ#D2i /2 MQK#`2mb2b ûim/2b /2TmBb THmb /2 i`QBb bBĕ+H2bX G2b i`pmt bm` H +TBHH`Biû bQMi MQK#`2mt- TQm`@ iMi [m2H[m2b [m2biBQMb T2`bBbi2Mi bm` H2b Kû+MBbK2b /ǶBKT`û;MiBQM /Mb /2b KBHB2mt +QM}MûbX LQmb `2pB2M/`QMb /Mb +2 KMmb+`Bi bm` +2b T`Q#HĕK2b ¨ T`iB` /ǶmM2 TT`Q+?2 2tTû`BK2MiH2 +2Mi`û2 bm` H KQMiû2 +TBHHB`2 Ĝ mM2 2tTû`B2M+2 [mB MQmb `ûb2`p2 2M+Q`2 [m2H[m2b bm`T`Bb2b- KH;`û THmb /2 jyy Mb /Ƕûim/2X

.Mb mM2 T`2KBĕ`2 T`iB2- MQmb MQmb BMiû`2bb2`QMb mt BMi2`+iBQMb 2Mi`2 H2 HB[mB/2 2i H2 bQHB/2 m MBp2m /2 H HB;M2 /2 +QMi+iX GǶûim/2 /2 H /BbbBTiBQM pBb[m2mb2 m MBp2m /m KûMBb[m2 MQmb T2`K2ii` /2 `2p2MB` bm` H bBM;mH`Biû /m KQ/ĕH2 /2 qb?#m`M ¨ i2KTb +Qm`i /Mb H2 +?TBi`2 RX LQmb T`QTQbQMb 2MbmBi2 mM KQ/ĕH2 TQm` /û+`B`2 H /vMKB[m2 /ǶûiH2K2Mi /ǶmM TQMi +TBHHB`2 2Mi`2 mM `ûb2`pQB` /2 HB[mB/2 2i mM bQHB/2 /Mb H2 +?TBi`2 kX .Mb mM2 b2+QM/2 T`iB2- MQmb MQmb BMiû`2bb2`QMb ¨ /2b bm`7+2b mt[m2HH2b MQmb TTHB@ [mQMb mM `2pāi2K2Mi bTû+BH- 2Mi`ŗMMi THmb /Ƕ{MBiûb 2Mi`2 H2 bQHB/2 2i HǶB` [mǶ2Mi`2 H2 bQHB/2 2i HǶ2mX *2b bm`7+2b û`QT?BH2b T`ûb2Mi2Mi /2b T`iB+mH`Biûb [m2 MQmb /ûiBHH2`QMb m +Qm`b /2 /2mt +?TBi`2bX LQmb ûim/B2`QMb H /vMKB[m2 /Ƕû+QmH2K2Mi /Mb /2b im#2b +TBHHB`2b û`QT?BH2b m +?TBi`2 jX LQmb i2bi2`QMb 2MbmBi2 H2b T`QT`Bûiûb /2b +2b bm`7+2b HQ`b[mǶ2HH2b bQMi BKK2`;û2b /Mb mM HB[mB/2 m +?TBi`2 9X G /2`MBĕ`2 T`iB2 +QM+2`M2 /2b

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G2b T`iB2b /ûp2HQTTû2b bQMi BM/ûT2M/Mi2b 2i +?[m2 +?TBi`2 `2TQb2 bm` mM2 2tTû@ `B2M+2 /Bzû`2Mi2X 1M pQmb bQm?BiMi #QMM2 H2+im`2 5

S`2KBĕ`2 T`iB2

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aBM;mH`Biû BMBiBH2 /2 H KQMiû2

+TBHHB`2 , H2 /Qm#H2 D2m /2 H

pBb+QbBiû

*2 T`2KB2` +?TBi`2 2MpBb;2 HǶ2tTû`B2M+2 /2 KQMiû2 +TBHHB`2 /Mb b THmb bBKTH2 2tT`2bbBQM , mM im#2 2M p2``2- T`7Bi2K2Mi HBbb2 2i `B;B/2- [mǶQM KĕM2 p2`iB+H2K2Mi m +QMi+i /ǶmM HB[mB/2 pBb[m2mt 2i KQmBHHMiX LQmb MQmb ii2M/QMb ¨ +2 [m2 H T`Q;`2bbBQM /m KûMBb[m2 bQBi +HbbB[m2K2Mi /û+`Bi2 T` H2 KQ/ĕH2 /2 qb?#m`M- [mB T`ûpQBi mM2 T`Q@ ;`2bbBQM T`QTQ`iBQMM2HH2 ¨ H `+BM2 /m i2KTb- mM2 /vMKB[m2 bBM;mHBĕ`2 ¨ i2KTb +Qm`iX LQmb `2p2MQMb bm` H2b T`2KB2`b BMbiMib /2 +2ii2 2tTû`B2M+2 TQm` H2p2` +2ii2 bBM;mH`Biû-2M ûim/BMi bBM;mH`Biû-2M T`iB+mHB2` HǶ2z2i /ǶmM }HK /2 Hm#`B}+iBQM ¨ HǶBMiû`B2m` /m im#2X LQmb M@ HvbQMb 2MbmBi2 H /BbbBTiBQM /m HB[mB/2 m MBp2m /2 H HB;M2 /2 +QMi+i TQm` KQ/ûHBb2` H /vMKB[m2 BMBiBH2 /m KûMBb[m2X 1M}M MQmb 2tTHQBiQMb H2 KQ/ĕH2 /ûp2HQTTû [mB MQmb T2`@ K2i /2 T`û/B`2 H /vMKB[m2 ;HQ#H2 2i /2 `2i`Qmp2` HǶûpQHmiBQM /2 H 7Q`K2 /m KûMBb[m2 m +Qm`b /ǶmM2 KQMiû2 +TBHHB`2X aQKKB`2 RXR G bBM;mH`Biû /m KQ/ĕH2 /2 qb?#m`M X X X X X X X X X X X X X X X jj RXk AM~m2M+2 /ǶmM }HK /2 Hm#`B}+iBQM X X X X X X X X X X X X X X X X X X j9 RXj .BbbBTiBQM m MBp2m /2 H HB;M2 /2 +QMi+i X X X X X X X X X X X X jd RX9 oBi2bb2 BMBiBH2 TQm` mM HB[mB/2 pBb[m2mt X X X X X X X X X X X X X X 9y RX8 6Q`K2b /m KûMBb[m2 2i /vMKB[m2 K+`Qb+QTB[m2 X X X X X X X X 9k RXe *QM+HmbBQM X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X 9d

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