³Iµ - macul.ciencias.uchile.cl

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ijijkEiKk lnm3o3pQq_r/q^`CL6J4BTDN 271327aGCI4jN CI4`0 A 6s83CI4$t

f1 u f2A g TD:=<LN CI4O276fA 6f0 .P<>6JC \J^`CL6J4BTDN 271327aGCI4K<LNvTD:7TDCL6w<>.xZaG276s832RA 6f832713:5271y0 A 6983C

P z 27627._adTD43aGC|0 .P<>6JCs~t u N 276JCL8327aGCI4[0MCL1 r u r′<;.P<L4[NvTD4B8E<>6J:7TP<L4N 2G254B832d0 A 6983C u f1 u f2

t"13254B0M25:78BT@SI<>aG276s832I\CL1N 2~J6 TD:7TDUL6]tA 6w<27.@T@0J432e254|27./.@A Hs<>1|H25C>ZaG78B1BTD:5CcN 2Q83C9N CI4R.DCI4R0 A 6983CI4;N 27./0 .P<>6JC0w<>1E<f.DCI4h:7Aw<>.D254r + r′ = 2a

t"2769AJ22a

254RA 6w<:5CL6J4B8E<>6s832 z ae1E<L:7TDUL6276s8B132 f1 u f2~\

698B13C9NvA b=:=<>aGCI4eA 643TD4B8327ae<N 2f:5CCL1ZN 276w<LNJ<L4j:=<>1B83254BTP<>6JC tJ:5CL6c27."CL1BT@H276c27627._:5276s8B13CN 2j.P<*JHIA 1E<;H25CLaG78B1BTD:=1BHCN 2.P<f1325:78E<9AJ2A 6J2.DCI4eN CI4h5=3>

f1 u f2\/-4e:7.P<>13CsAJ227.[5~9=GLLXN 2.P<27.@T@0J4E2Q254

a\ 2=<

2c.P<fNvTD4B8E<>6J:7TP<276s8B132.DCI4hN CI4* ¡C>Z:5CI4=tJ276s83CL6J:5254

¢£¥¤R¦¨§L£(©(£[¤R£Eª«&¬ = b =√

a2 − c2 . x®L¯B°s°±²°

³´Lµ

³´! "$#&%(')+*,*.-0/21143657*.-89589):/28;3<*.)!%(89-89)=*2;.P<JHIA 1E<?>>I\@>;432hN 25NvAJ:52;9AJ2

2a = r + r′ =√

(c + x)2 + y2 +√

(c − x)2 + y2

=√

(c2 + x2 + y2) + 2cx +√

(c2 + x2 + y2) − 2cx .-/.D27SL<>6JN Cf<>.]:7Aw<LNv1E<LN Ce432hCLg 8BTD276J22(c2 + x2 + y2) + 2

√

(c2 + x2 + y2) + 2cx ·√

(c2 + x2 + y2) − 2cx = 4a2 ,CG432=<vt2a2 − (c2 + x2 + y2) =

√

(c2 + x2 + y2) + 2cx ·√

(c2 + x2 + y2) − 2cx

=√

(c2 + x2 + y2)2 − 4c2x2 .-/.D27SL<>6JN Ce6sAJ27SI<>aG276s832Q<>.:7Aw<LNv1E<LN Ce9AJ25NJ<

4a4 − 4a2(c2 + x2 + y2) + (c2 + x2 + y2)2 = (c2 + x2 + y2)2 − 4c2x2 ,CG432=<vt−c2x2 = a4 − a2c2 − a2x2 − a2y2 .

AX4E<>6JN CG.P<Q1327.P<L:7TDUL6c2 = a2 − b2 4E2hN 25NvAJ:52RJ6w<>.@aG276s832

x2

a2+

y2

b2= 1 . z >>I\@>$

"$#&%(')+*,*.-0/21143657*.-89589)B'C1D#(8;3*E)F 8B1E<13270 13254327698E<L:7TDUL6HG 8BT@. u :5CLaIG 6 N 2 .P<27.@T@0J432254276 8371BadT@6JCI4N 2:5C9CL13N 276w<LNJ<L40 CL.P<>13254

(r, φ)tw83CLae<>6JN CG:5CLaGCGCL1BT@H276 <QA 6JCGN 2R.DCI4[ ¡C9:5CI4 z S271jJHIA 1E<?>>I\ ³ ~\ 2h8BTD276J2;9AJ2

x = r cos φ − c ,

y = r sinφ

uy2 = r2 (1 − cos2 φ) . AJ4B8BT@8BA u 276JN C254B8E<L4 2KJv0 13254BTDCL6J254`276e.P<25:7Aw<L:7TDUL6N 2.P<27.@T@0J432n276 :5CCL13N 276w<&ZNJ<L4:=<>1B83254BTP<>6w<L4NJ<LNJ<R0MCL1.P<R25:7Aw<L:7TDUL6

z >>I\@>$~tw432hCLg 8BTD276J2

>

x®L¯B°s°±@L

³Iµ

b2 (r cos φ − c)2 + a2r2 (1 − cos2 φ) = a2b2 ,

b2r2 cos2 φ − 2b2rc cosφ + b2c2 + a2r2 − a2r2 cos2 φ = a2b2 ,CG432=<vta2r2 = (a2 − c2) b2 + 2b2cr cos φ + (a2 − b2) r2 cos2 φ

= b4 + 2b2√

a2 − b2 r cosφ + (a2 − b2) r2 cos2 φ

=[

b2 +√

a2 − b2 r cos φ]2

.

-J8B1E?@btw432hN 25NvAJ:52;9AJ2ar = b2 +

√

a2 − b2 r cos φ . z >>I\ ³ 698B13C9NvA b=:=<>aGCI4X.DCI4O0w<>1EF>aG278B13CI4r0 ≡ b2

az >>I\

uε ≡ c

a=

√

1 − b2

a2z 2KJ :5276s8B1BTD:7TDNJ<LNw . z >>I\ 9^`CL6V254B8E<L4jN 2~J6 TD:7TDCL6J254X.P<25:7Aw<L:7TDUL6 z >>I\ ³ [4E2R0 AJ25N 2;2543:71BT@g T@1KN 2h.P< ¡CL1Bae<

1

r=

1

r0(1 − ε cos φ ) . z >>I\

Z <L4jN CI4[ ¡CL1Bae<L4jaeFL4OAJ43Aw<>.D254O0w<>1E<13270 13254327698E<>1KA 6w<d27.@T@0J432|43CL6

T²c^`CCL13N 276w<LNJ<L4K:=<>1B83254BTP<>6w<L4x, y z :5CL627.CL1BT@H276<>.]:5276s8B13Cs

x2

a2+

y2

b2= 1 , z >>I\

N CL6JN 2h.DCI4K0w<>1EF>aG278B13CI4a u b

13270 132543276s8E<>6<.DCI4K4E27adT92254KaeaG276s832I\T@T²c^`CCL13N 276w<LNJ<L4j0MCL.P<>13254

r, φ z :5CL6c27.CL1BT@H276V276cA 6JCdN 2h.DCI4[ ¡C:5CI4¥1

r=

1

r0(1 − ε cos φ ) . z >>I\ ´

<L4K25:7Aw<L:7TDCL6J254 z >>I\ u z >>I\ 9`1327.P<L:7TDCL6w<>6.DCI4O0w<>1EF>aG278B13CI4 a, b:5CL6

r0, ε\

³Iµ >2=<>aGCI4X<>.@HIA 6w<L4j0 13CL0 TD25NJ<LN 254R<LNvTD:7TDCL6w<>.D254KN 2;.P<L4j27.@T@0J43254=\^`CL.DC99AJ27aGCI4h27./CL1BT@H276276A 6JCN 2Q.DCI4X C:5CI4 u 432=< r1

.P<fNvTD4B8E<>6J:7TP< ad?@6 T@ae< z ~~x®9B$u r2.P<fNvTD4B8E<>6J:7TP<faeF+JT@ae< z 3®sB$;276s8B132d27./CL1BT@H276 u .P<f27.@T@0J432I\ z W*C :5CL6v A 6JNvT@1;27.0 271BTxZH25C u <>0 CLH25C:5CL6 .DCI4h4327adT92B254*aG276JCL1 u ae.P<>g 1E<L4$tJ27.]4327adTD2B2hae1BT@8BaG78BTD:5C

a =r1 + r2

2

u 27.4E27adTD22RaG276JCL1K27.0 13CLaG25NvTDCdH25CLaG78B1BTD:5Cb =

√r1r2N 27.<>0 CLH25C u 0M271BT@H25C \

<d2KJ :5276s8B1BTD:7TDNJ<LN"t 2768371BadT@6JCI4KN 2r1 u r2

t 254ε =

r2 − r1

r1 + r2.

^`CL6JC:7TD276JN C|27.MF>132=<;N 2jA 6f:7?@13:7A .DC u .P<L4y25:7Aw<L:7TDCL6J254`276f:5C9CL13N 276w<LNJ<L4[:=<>1B83254BTP<&Z6w<L4*N 2QA 6 :7?@13:7A .DC u N 2QA 6w< 27.@T@0J4E2It_N 27aAJ254B8B132dsAJ2Q27.F>132=< AN 2|A 6w<f27.@T@0J432S9TD276J2NJ<LNJ<0 CL1

A = πab ,N CL6JN 2a u b

43CL6c4BAJ4j4E27adTD2254=\CI438B1E<>1327aGCI4*J6w<>.@aG276s832G9AJ24BTA 6 1E1BaG2769832276.P<d27.@T@0J432Itw276983CL6J:5254*8E<>.1E1EFd6J25:5254¥<>1BTP<>aG276s8320 CL1K27.]CL8B13Cd C:5C \<>1E<cN 27aGCI4B8B1E<>1|254B8E<f0 13CL0 TD25NJ<LNn:5CL6J43TDN 271327aGCI4|N CI4R0 A 6s83CI4

P u P ′43CLg 132.P<27.@T@0J432ItT@6vJ6 T@83254BT@ae<>.@aG276s832R:52713:=<>6JCI4=\ 2=<>6

R1 u R′

1

.DCI4/1E>I\~\^`CLaGCR1 + R2 = R′

1 + R′

2 = 2at"432N 25NvAJ:52d9AJ2

PA = P ′B\]CI4R8B1BTPF>6 HIA .DCI4

PAP ′ uP ′BP

43CL6c259A T@SL<>[email protected]>6 HIA .DCI4β u δ

43CL6T@HIAw<>.D254$\ 2713Cδ = α

t9.@AJ27HCβ = α

\-4B8E<G .@8BT@ae<|1327.P<L:7TDUL6T@ad0 .@TD:=<Q9AJ2K.DCI4y1E1j0MCL1O.P<27.@T@0J432I\

@®L¯c°s°± ^`CL6J4BTDN 27132;.P< A 6J:7TDUL6V4BT@HIA TD2769832 z 276c:5CCL13N 276w<LNJ<L4K0MCL.P<>13254¥

1

r= 1 − ε cos φ.

1E<&wsAJ2r(φ)

0w<>1E<ε = 0

t \vtD>I\ u >I\v\

ijijkhk jq m3rjr/qr rpQm3rO<L4EF>6JN CI4322760 1325:7TD4E<L4aG25NvTD:7TDCL6J254eN 2f.P<0MCI4BTD:7TDUL6 N 2.DCI40 .P<>6J278E<L4132=<>.@T@b$<LNJ<L4G0MCL1

u :JC`1E<J2ItCw<>6 6J254h270 .D271 z > ´ > E> j0 AJN Cf254B8E<>g .D25:5271*.P<L4X43T@HIA TD276s83254*.D2 u 254K0w<>1E<N 2543:71BT@g T@1K27.]aGC$S9T@adTD276983C N 2R.DCI4O0 .P<>6J278E<L4*<>.@1325N 25N CL1XN 27. CL. >I\ ]CI4K0 .P<>6J278E<L4K432;a|AJ27S276<>.@1325N 25N CL1*N 27. CL.276UL1Bg T@8E<L4K27.@?@0 8BTD:=<L4=t_254B8E<>6JN CG27. CL.276A 6JCdN 2h.DCI4O C:5CI4=\³ \ ^[<LNJ<0 .P<>6J278E<gw<>1B132;F>132=<L4jT@HIAw<>.D254K276c8BTD27ad0MCI4OT@HIAw<>.D254=\v\ -/. :7Aw<LNv1E<LN CN 27._0M271B?DC9N CN 2KA 6 0 .P<>6J278E<<>.@1325N 25N CL1[N 27. CL. 254y0 13CL0MCL13:7TDCL6w<>."<>. :7A g CN 27.]4327adTD2B2hae<&BCL1jN 2;4BA8B1E1BTDCI4

³Iµ <CL1R.DCf9AJ24E<>gM27aGCI4=tw.P< 0 1BT@aG271E<f.D2 u N 2 h270 .D271h6JCf0 AJ25N 243271h:5CLad0 .D278E<>aG276s832G:5C>Z1B1325:78E<v\ TN CI4j:7AJ271B0MCI4X43CL6.@T@g 13254*2KJ :5270 83Cd0 CL1K.P<Q AJ271Bb$<GN 2;T@6s83271E<L:5:7TDUL6 9AJ2;[email protected]$t_276983CL6J:5254h54B83CI4RN 27g 276aGC=S2713432N 2ae<>6J271E<sAJ227.:527698B13CN 2|ae<L4¥<L4*4E2ae<>6983276 Hs<27613270MCI43C z C 276aGC=ST@adTD276s83CA 6 Tx ¡CL1BaG2$~\ F 4E2=<vt_27. CL.6J25:5254¥<>1BTP<>aG276s8328E<>ag TD768BTD276J2Q9AJ2254B8E<>1*aGC=STD76JN CI432I\ T@627a|gw<>1BHC t]4BTD276JN Cf.P<eae<L4E<fN 27. CL.aA u4BA 0M271BTDCL1`<h.P<;N 2j.DCI4y0 .P<>6J278E<L4=t27.w:5276s8B13CN 2Kae<L4E<L4 CL.s0 .P<>6J278E<Q2543276J:7TP<>.@aG2769832*:5CLT@6vZ:7TDN 2y:5CL6.P<K0 CI43TD:7TDUL6N 27. CL. u 27625432 :=<L43CX13254BA .@8E<K43271A 6w<ja|A u g AJ276w<*<>0 13CJvT@ae<L:7TDUL64BA 0MCL6J271j9AJ2h27. CL.]254B8EFd27613270MCI43Cd276A 6JCGN 2R.DCI4[ ¡C9:5CI4ON 2R.P<G27.@T@0J432I\ z <ae<L4E<N 27. CL.]254jN 2

1, 99 · 1030 HdadTD276s8B1E<L4*sAJ2R.P<N 2;.P< [TD271B1E<G254jN 25, 98 · 1024 HJ\

g_<>.X:5CLaGCn432VN 27aGCI4B8B13U 276 27.*227ad0 .DC >VN 2.P<4E25:5:7TDUL6 \ ³ t27. J25: JCN 2VsAJ2A 60 .P<>6J278E< gw<>1B132cF>132=<L4dT@HIAw<>.D254G276 8BTD27ad0MCI4dT@HIAw<>.D254G254d259A T@SL<>.D2769832V< N 25:[email protected]<>6 HIA .P<>1 ~ 6JCQSI<>1B?P<Q276e A 6J:7TDUL6N 27. 8BTD27ad0MC \v-4B83C<Q4BA S27bRT@ad0 .@TD:=<Q9AJ2X27.83CL139AJ22B2713:7TDN Cf0 CL1R27. CL.4ECLg 132Q27.0 .P<>6J278E<254X6sA .DC tM.DCfsAJ2<f4BAS27bQT@ad0 .@TD:=<9AJ2.P<f AJ271Bb$<27698B132G.DCI4|N CI4:7AJ271B0 CI4N 27g 2 43271|<.DCV.P<>1BHCcN 2G.P<.@?@6J2=<VsAJ2.DCI4|A 6J2I\-/6CL8B1E<L4*0w<>.P<>g 1E<L4=t .P< 4327HIA 6JNJ<G.D2 u N 2 h270 .D271RT@ad0 .@TD:=<fsAJ2.P<G AJ271Bb$< 276s8B132d27. CL. u 27.0 .P<>6J278E<dN 27gM2h43271j1E<LNvTP<>.(\ <Q AJ271Bb$<QHI1E<5ST@8E<L:7TDCL6w<>.(tw0MCL1O.DCd8E<>6s83C tw254jN 2h.P< ¡CL1Bae<~F (~r ) = −f(r) r .

:$ <d8E<>g .P<e<LN>A 6s8E<daAJ254B8B1E<f<>.@HIA 6JCI4KN 2;.DCI4K0w<>1EF>aG278B13CI4K0w<>1E<G.DCI4j0 .P<>6J278E<L4X0 271B83276J2~Z:7TD276983254X<>.]4BTD4B8327ae<G43CL.P<>1=\ ª£ £E¤R¦ £(©(£j¤ «$¬ £¥¬ «I« E£Eª ¬¦E¦ ¡ª ¦ ª E¦&ª ¢

¢! "$# d£E¬% ¬ ¦ « &' (*)+ 7, 60 · 106 &' ,*&.- 7 &*&/ 3, 28 · 10230 £Eª ¢ &'1+2,2( 1, 94 · 107 &' &.&*3 3 ,2(/ 4, 83 · 10244 ¦ £¥¬ ¬ 5 ' &*&*& 3, 16 · 107 &' & 5 3 6 5, 98 · 1024

¬ £ 5 '7-*,2( 5, 94 · 107 &' &.8*( 1 - 5 / 6, 37 · 10239*:; ¦<£E¬ -'7,2&., 3, 74 · 108 &' &*=.) 15 )/ 1, 90 · 1027 ¬ª« 8'7-*->= 9, 30 · 108 &' &-*- 2 ,28/ 5, 67 · 1026? ¬ ª« 5 8'7, 5 ) 2, 66 · 109 &' &*=.3 0 =.3/ 8, 80 · 1025@ £ ; ªs« (.&' 5 &.8 5, 20 · 109 &' &.&*) 1 =.3/ 1, 03 · 1026 &ª (*8' 3*& 7, 82 · 109 &' ,2=.3 17 &+/ 5, 4 · 1024

]CI4 <L4B8B13UL6JCLaGCI4=t0w<>1E<aG25NvT@1 NvTD4B8E<>6J:7TP<L4=t/ 1325:7AJ27698327aG276s832AJ4E<>6 .P<¯BAJ >DCB$EAGFIQ¡3A]\¨<v\@\9AX6w<fA]\¨<v\"254hT@HIAw<>.y<>.4E27adTD22aeA]\¨<v\<H1, 495 · 1011 ac\<R2KJ :5276s8B1BTD:7TDNJ<LN

ε = (r1−r2)/(r1 +r2)N 2[.P<*ae6J278E<L4254/gw<L438E<>6s8320M25sAJ2Iw<vt 43TD276JN Cd4BAJ4OUL1Bg T@8E<L4=t 0MCL1O.DC8E<>6s83C tJ:=<L43T]:7T@13:7A .P<>13254=\ zKJ .MNvT@g AL3<>1jA 6w<27.@T@0J432:5CL6A 6w<2KJ :5276s8B1BTD:7TDNJ<LN

ε = 0, 05254;NvTx ?D:7T@.(t43UL.DCadT@1EF>6JN CL.P<vtNJ<>13432d:7AJ27698E<cN 2G9AJ2NvTxw27132;N 2RA 6:7?@13:7A .DC \

³Iµ T<L4BA adT@aGCI4j9AJ2ItJ2760 1BT@aG271E<G<>0 13C JT@ae<L:7TDUL6]t_.P<L4jUL1Bg T@8E<L4jN 2;.DCI4O0 .P<>6J278E<L4K43CL6c:7T@13:7AvZ.P<>13254=t9276983CL6J:5254=tJ<h0w<>1B8BT@1ON 2X.P<;832713:5271E<.D2 u N 2h270 .D271=t90 CN 27aGCI4[276J:5CL6s8B1E<>1OA 6w<2KJ0 13254BTDUL60w<>1E<.P<ae<>HI6 T@8BAJNcN 2h.P< AJ271Bb$<HI1E<=S9T@8E<L:7TDCL6w<>.(\ -/6V2~ ¡25:783C

<832713:5271E<.D2 u N 2 h270 .D271KNvTD:52;9AJ2T 2 = Kr3 ,

N CL6JN 2K254A 6w<j:5CL6J4B8E<>69832 z .P<jadTD4Bae<O0w<>1E<K83C9N CI4.DCI40 .P<>6J278E<L4~ u r

25427.91E<LNvTDCXN 2 .P<jUL1Bg T@8E< :7T@13:7A .P<>1 E\-._1E<LNvTDC

rN 2K.P<|UL1Bg T@8E<vt.P<hS27.DC:7TDNJ<LN

v u 27. 0 271B?DCN C T254B8EF>6f1327.P<L:7TDCL6w<LN CI4[0 CL1

T =2πr

v.

CL1fCL8B1E<0w<>1B832c.P<ae<>HI6 T@8BAJN N 2c.P< AJ271Bb$< HI1E<=S9T@8E<L:7TDCL6w<>.RN 27gM2c:5CLT@6J:7TDNvT@1f:5CL6 .P< AJ271Bb$<:5276s8B1B?@0M278E<vtwCG432=<vtf(r) =

mv2

r,N CL6JN 2

m254K.P<eae<L4E< N 27.0 .P<>6J278E<v\;A*4E<>6JN Cf254B8E<L4*8B13254*25:7Aw<L:7TDCL6J254$t N 25430 23<>6JN C

v u Tt 432276J:7AJ276s8B1E<

f(r) =4π2m

K

1

r2.

<:5CL6J4B8E<>6s832Kt N 2G<L:7AJ2713N Cc< .P<e832713:5271E<.D2 u N 2 h270 .D271=tM6JCN 270 276JN 2dN 26 T@6 HIA 6w<f0 13C>Z0 TD25NJ<LNN 2h.DCI4X0 .P<>6J278E<L4=tw0 2713C 4BT]0MC9Nv1B?P<dN 270M276JN 271XN 2Q<>.@HIA 6w<d0 13CL0 TD25NJ<LNN 27. CL.(\_-4jaeFL4CeaG276JCI4R27STDN 276s832ItM0MCL1K1E<>b=CL6J254hN 2Q4BT@aG278B1B?P<vt_9AJ2Q4BT.P<G AJ271Bb$<GHI1E<=S9T@8E<L:7TDCL6w<>.N 270M276JN 2N 2.P<Qae<L4E<

mN 27."0 .P<>6J278E<vtw276s83CL6J:5254KN 27gM2R83276J271O.P<QadTD4Bae<N 270M276JN 276J:7TP<dN 2R.P<Qae<L4E<N 27. CL.

M\J-438E< G .@8BT@ae<GCLgJ43271BSI<L:7TDUL6V4BA HITD27132;254E:71BT@g T@1

4π2

K= GM ,

N CL6JN 2G254hA 6w<f69AJ27SL<:5CL6J438E<>6s832e9AJ2G<JCL1E<254hT@6JN 270M276JNvTD276s832 N 2d.P<L4hae<L4E<L4|N 27. CL. uN 27.]0 .P<>6J278E<v\ =R2h254B8E<Qae<>6J271E<d432hN 25NvAJ:52|sAJ2R.P< AJ271Bb$<HI1E<5ST@8E<L:7TDCL6w<>.27698B132;N CI4Oae<L4E<L4

mu MtJ43270w<>1E<LNJ<L4O0MCL1jA 6w<NvTD4B8E<>6J:7TP<

rtv254

~F (~r ) = −GmM

r2r .

³Iµ < G .@8BT@ae<25:7Aw<L:7TDUL6 0 1325:7TD4E<>aG276s832254|.P<" ~ s;®L= C(s7KFAn¯ Aw>7E5 " 9| C(A4BTD276JN C

G.P<<[email protected]<>ae<LNJ<n3$AJDC($A C|®LB$E

= C(s7A_ " \ AX4¥<>6JN CdA 6w<gw<>.P<>6 b$<GN 2;83CL1Z4BTDUL6 z S271hJHIA 1E<>>I\ 9~t^[<=S276JNvTD4 27627.<IJC > ´! Iµ adTDNvTDUG276c27..P<>g CL1E<>83CL1BTDC \-/.GNvTD4B0MCI4BT@8BT@SC 2KJ0M271BT@aG276s8E<>.d AJ227.d4BTxZHIA TD276s832 N CI4Oae<L4E<L4m432h:5CL.DC:=<>6V276c.DCI42KJ8B1327aGCI4ON 2*A 6w<gw<>1B1E<1B?@HITDNJ6 N 2543N 2X27._:5276s8B13CN 2jA 6<>.P<>a|g 132 z [email protected] 2 83CL134BTDUL6_9AJ2 432`4BALB278E<[J1BaG27aG2769832ON 27.:7TD27.DC \"-/./4BTD4B8327ae<e0 AJ25N 2HIT@1E<>1*.@T@g 1327aG276vZ832y276d27.0 .P<>6JC JCL1BT@b=CL698E<>.(tL0 2713C*8BTD276J2yA 6w<CL1BTD276s8E<L:7TDUL6 0w<>1E<f.P<f:7Aw<>.254B8EFf276259A T@.@TxZg 1BTDC \

@®L¯°°s± T.P<L4Xae<L4E<L4

m4324E<L:=<>6VN 27.25sA T@.@T@g 1BTDC t_0 CL1*227ad0 .DC t_NJF>6JN CL.D254*A 6w<d0M25sAJ2Iw<GS27.DC9:7TDNJ<LNN 2O13CL8E<L:7TDUL6]t9276983CL6J:5254y27._4BTD4B8327ae<h:5CLaG276 b$<>1EF<hCI43:[email protected]<>1 83276 TD76JN CI4E2jA 6 A#L¯ " s C(IE~KF$A_\

J 0w<>1B8BT@1XN 27.0 271B?DCN CeN 2254B832;aGC$S9T@adTD276983CfCI4E:[email protected]<>83CL1BTDC 432;0 AJ25N 2N 25NvAJ:7T@1*.P<G:5CL6J4B8E<>69832N 213254B8BT@8BAJ:7TDUL6N 2K83CL134BTDUL6 z C|4E2=<vt927._83CL13sAJ2*sAJ2X22713:52*27. [email protected]|N 2X83CL134BTDUL6f:7Aw<>6JN CQ254B832*432jHIT@1E<276cA 6w<ae<>HI6 T@8BAJNα~\

<dCL1BTD27698E<L:7TDUL6N 27.]0M76JNvA .DCG432hN 25N 25:78E<e:5CL6A 6c1E1BT@[email protected]<v\-/.]2KJv0 271BT@aG276983CdadTD4BaGC432*132=<>.@T@b$<276N CI4j278E<>0w<L4=\v/1BT@aG2713CG432R:5CL.DC:=<>6N CI4Oae<L4E<L4

M276V.P<L4X0 CI4BTD:7TDCL6J254A u 432h1327HITD4B8B1E<e.P<GCL1BTD27698E<L:7TDUL6N 2|.P<GSL<>1BT@[email protected]<v\ "AJ27HC 4E2;13CL8E<>6.P<L4jae<L4E<

M< .P<L4X0MCI4BTD:7TDCL6J254

B u 432S9AJ27.@S2d< 1327HITD4B8B1E<>1R.P<fCL1BTD276s8E<L:7TDUL6 N 2Q.P<eSL<>1BT@[email protected]<v\ J 0w<>1B8BT@1hN 2.P<|SI<>1BTP<L:7TDUL6N 2K.P<QCL1BTD27698E<L:7TDUL6cN 2X.P<|SI<>1BT@[email protected]<276254B8E<L4[N CI4àG25NvTD:7TDCL6J254j432X0 AJ25N 2RN 25NvAJ:7T@1`.P< AJ271Bb$<d27698B132m u M

\-/.2KJv0M271BT@aG276s83Cf254jgw<L4B8E<>69832|NvTx ?D:7T@. u <d9AJ2h.P<Q AJ271Bb$<GN 2|<>8B1E<L:5:7TDUL6 N 2;.P<L4XN CI4jae<L4E<L4254Ga|A u 0M259AJ2Iw<v\ CL1f2B27ad0 .DC tA 6w< ae<L4¥<

m = 20 H 254f<>8B1E<>?DNJ<n0MCL1eCL8B1E< N 2ae<L4E<

M = 150 H:5CL6A 6w< AJ271Bb$<QN 2*4EUL.DC

2, 3.5 · 10−5 · g W z 254B83C254=tv27."0M2543CN 2*A 6w<Qae<L4E<QN 2³ adT@.@.@T@HI1E<>aGCI4¥O:7Aw<>6JN CG.P<43270w<>1E<L:7TDUL6N 2h4BAJ4K:5276s8B13CI4*254jN 2 :7ac\-/.]SI<>.DCL1X<L:78BAw<>.@aG276s832Q<L:5270 8E<LN CG0w<>1E<d27."SI<>.DCL1KN 2

G254

G = (6, 673 ± 0, 003) · 10−11

a 3

HG4 2 .

^`CL6JC9:7TDN CI4;27.0M271B?DC9N C u 27./1E<LNvTDCN 2.P<fUL1Bg T@8E<f83271B13254B8B132G<>.@1325N 25N CL1;N 27. CL. uAJ4E<>6JN CG27.]SI<>.DCL1jN 2G1325:7TD76VNJ<LN C t_N [email protected]<ae<L4E<N 27. CL.(\

³Iµ ijijk|k /m3o r/q

-/6254B8E< 4325:5:7TDUL6 <>6w<>.@T@b$<>1327aGCI4f<>.@HIA 6JCI4e<L4B0M25:783CI4dN 27.OaGC=ST@adTD276s83CnN 25C " C~Q9AJ2CL1Bg T@8E<>6 HI1E<=S9T@8E<L:7TDCL6w<>.@aG2769832h276s83CL1B6JC<hA 6 CLgvB2783C;ae<L43T@SC \ J .@HC|254A 6f4E<>837.@T@832K43Tw4BAGae<L4E<m254aAJ:JCaG276JCL1`9AJ2j.P<Rae<L4¥<

MN 27._CLgvB2783C<>.@1325N 25N CL1[N 27._:7Aw<>. CL1Bg T@8E<v\9-6e0w<>1B8BTD:7A .P<>1=t.DCI4h13254BA .@8E<LN CI4;N 2.P< 0 1325432769832G4325:5:7TDUL60MC9Nv1EF>6 <>0 .@TD:=<>13432G<>./aGC$S9T@adTD276983CVN 2Q.DCI4h0 .P<>6J278E<L4<>.@1325N 25N CL1N 27. CL.(t]N 2G.P<L4h.@A 6w<L4<>.@1325N 25N CL1QN 2d.DCI4h0 .P<>6J278E<L4 u N 2.DCI44E<>837.@T@83254|<>1B8BTxw:=<>.D254<>.@1325N 25N CL1XN 2R.P< [TD271B1E<v\ <L4O8B1E< u 25:783CL1BTP<L4*N 2h83CN CI4j254B83CI4jCLgvB2783CI4j:5CL1B13254B0MCL6JN 276<d27.@T@0J43254z u :7?@13:7A .DCI4¥~t_Cd432=<vtJ<d:5UL6 TD:=<L4j:5CL62KJv:527698B1BTD:7TDNJ<LN 254 0 ≤ ε < 1

\-/6 .P< 4E25:5:7TDUL6>>I\@> 432V1327.P<L:7TDCL6w<>13CL6 .DCI4f0w<>1EF>aG278B13CI4

r0 u εN 2A 6w<27.@T@0J432 :5CL6 27.4327adTD22Raea|g TD76:5CL6c27."0 271BT@H25C r1 = r ¤R¦ ª u 27.<>0 CLH25C

r2 = r ¤ N [email protected]<v\ J :5CL6s8BT@6sAw<L:7TDUL6254B8BAJNvTP<>1327aGCI4h.P< N 270 276JN 276J:7TP<N 2254B83CI4*0w<>1EF>aG278B13CI4N 2G.P<L4 E$A DC(A C¥s G =¡A C(dN 2G.P<cUL1Bg T@8E<254B0M25:7?xw:=<>aG276s832N 27.aGCLaG276983C<>6 HIA .P<>1CL1Bg T@8E<>.` u N 2R.P<276J271BHI?P<d83CL8E<>. E

\<f:5CL6JNvTD:7TDUL6

m MT@ad0 .@TD:=<fsAJ2.P<Gae<L4¥<eae.@aG276s832G4E2|ae<>83276JNv1EFf27613270 CI43Ce276A 6JCGN 2R.DCI4O ¡C9:5CI4ON 2;.P<27.@T@0J432It .@A Hs<>1K276c9AJ2hA g TD:=<>1327aGCI4K27.CL1BT@H276]\

<Q AJ271Bb$<d43CLg 132R.P<0w<>1B8B?D:7A .P<m z 27.]4¥<>837.@T@832$`S9TD276J2|NJ<LNJ<0 CL1

~F (r) = −GmM

r2r .-4B8E<K AJ271Bb$<vtI9AJ2`254/:5CL6J43271BSL<>8BT@SI<vtLNJ<*CL1BTxZH276<QA 6w<d276J271BHI?P<d0MCL83276J:7TP<>.

U(r) = −GmM

r.

Ts276|:7TD271B83COT@6J4B8E<>69832.P<[0MCI4BTD:7TDUL6 u S27.DC:7TxZNJ<LNGN 2[.P<*0w<>1B8B?D:7A .P<m254

~r u ~vtI276983CL6J:5254.P<d276J271BHI?P<83CL8E<>.N 27.43TD4B8327ae<43271EF

E = K + U =1

2mv2 − G

mM

r.

@®L¯°°±° 2=<>6

P1 u P2.DCI4h0 A 6983CI4|:5CL1B13254B0MCL6JNvTD276s83254Q<>./0 271BT@H25C u <>0 CLH25C t"1325430 25:78BT@SI<>aG276s832 z S271JHIA 1E<,>>I\~\^`CLaGC.P<276J271BHI?P<V83CL8E<>.O432f:5CL6J43271BSI<V5438E<VN 27gM2 4E271T@HIAw<>.j<

E8E<>6983C 276 27.<>0 CLH25CG:5CLaGCe276V27."0 271BT@H25C twCG432=<vt

E =1

2mv2

1 − GmM

r1=

1

2mv2

2 − GmM

r2. z >>I\ µ

^`CLaGCV.P< AJ271Bb$<VsAJ2 <L:78 Gw<V43CLg 132m

254|:527698B1E<>.`432e8BTD276J2e9AJ2d8E<>a|g TD76 27.yaGCLaG276s83C<>6 HIA .P<>1 z 13254B0 25:783CG<>."CL1BT@H276_`432R:5CL6J43271BSI<v\v-/SI<>.@Aw<>6JN CG27.MaGCLaG276s83CG<>6 HIA .P<>1K27627.]<>0 CLH25C

³Iµ´

u 0M271BT@H25C tJ432h276J:7AJ27698B1E<` = mr1v1 = mr2v2 . z >>I\

=*2|254B8E<25:7Aw<L:7TDUL6432hCLg 8BTD276J2v1 =

`

mr1uv2 =

`

mr2.

AJ4B8BT@8BA u 276JN Ce254B83Ce276 z >>I\ µ [432h276J:7AJ276s8B1E<E =

1

2m

`2

r21

− GmM

r1z >>I\@>

uE =

1

2m

`2

r22

− GmM

r2. z >>I\@>>$

=*2|254B8E<L4jN CI4j25:7Aw<L:7TDCL6J254X432hN 25NvAJ:522mEr2

1 = `2 − 2Gm2Mr1

u2mEr2

2 = `2 − 2Gm2Mr2 .K254B8EF>6JN CL.P<L4X432hCLg 8BTD276J2

E(r2 − r1)(r2 + r1) = −GmM(r2 − r1)

E = −GmM

r1 + r2= −G

mM

2a,

CG432=<vt 27.]4327adTD2B2hae>I\@> ³

-/6J:5CL6s8B1327aGCI4;<JCL1E<eA 6w<d1327.P<L:7TDUL6 276s8B132b z 27.4327adTD2B2;aG276JCL1*N 2;.P<GUL1Bg T@8E6 HIA .P<>1 u 276J271BHI?P<cN 27.y4E<>837.@T@832I\-438E<432GCLg 8BTD276J2e13254B8E<>6JN C.P<L4|25:7Aw<L:7TDCL6J254 z >>I\@>>$hN 2z >>I\@>

`2

2mr21

− `2

2mr22

− GmM

r1+ G

mM

r2= 0

`2

(

1

r21

− 1

r22

)

= 2Gm2M

(

1

r1− 1

r2

)

`2 r22 − r2

1

r21r

22

= 2Gm2Mr1 − r2

r1r2

³IµIµ

`2 r2 + r1

r1r2= 2Gm2M

`2 2a

b2= 2Gm2M ,CG432=<vt

b2 =a`2

Gm2M= − `2

2Em. z >>I\@>

<T@ad0 CL1B8E<>6J:7TP< N 2 .P<L425:7Aw<L:7TDCL6J254 z >>I\@> ³ u z >>I\@> |1E<LNvTD:=<2769AJ2 43CL6 54B8E<L4Q.P<L49AJ21327.P<L:7TDCL6w<>6V.P<L4j:5CL6J4B8E<>6983254XN 2RaGC$S9T@adTD276983C N 2;.P<UL1Bg T@8E<d:5CL6V4BA ¡CL1Bae<H25CLaG78B1BTD:=<v\F 8B1E<c1327.P<L:7TDUL6 T@ad0MCL1B8E<>6s832432GCLg 8BTD276J2 AJ4E<>6JN C.P<c4327HIA 6JNJ< ]2 u N 2 h270 .D271 :5CLaGCV27.aGCLaG276s83C <>6 HIA .P<>1432c:5CL6J43271BSI< 4E28BTD276J29AJ2c.P<n0w<>1B8B?D:7A .P<

mgw<>1B132F>132=<L4fT@HIAw<>.D254 2768BTD27ad0 CI4[T@HIAw<>.D254=\ <:=<>698BTDNJ<LNcN 2hF>132=<QsAJ2R27."4E<>837.@T@832 gw<>1B132R276A 6T@6s83271BSI<>.DCdN 2*8BTD27ad0 C

dt.DC0 CN 27aGCI4j27SI<>.@Aw<>1K:7Aw<>6JN CG54B832;4E2h276J:7AJ276s8B1E<G27627.<>0 CLH25C

dA =1

2r2v2dt =

`

2mdt .

=*2Q254B8E<e1327.P<L:7TDUL6432|N 25NvAJ:529AJ227./F>132=<G83CL8E<>.(tA = πab

t_.P<e0w<>1B8B?D:7A .P<m.DC gw<>1B132276A 68BTD27ad0 C

TNJ<LN CG0 CL1

πab =`

2mT ,254jN 25:7T@1=t

T 2 =4m2π2a2b2

`2.

AX4E<>6JN C z >>I\@> [432hCLg 8BTD276J2It J6w<>.@aG2769832ItT 2 =

4π2

GMa3 , z >>I\@> 9

13254BA .@8E<LN CK9AJ26JCO254]CL8B13Cj9AJ2/.P<`832713:5271E<O.D2 u N 2 h270 .D271=\$2713CjCLgJ43271BS2 sAJ2 <JCL1E<j:5CL6JC9:527aGCI4.P<d:5CL6J4B8E<>6s832N 2R0 13CL0 CL13:7TDCL6w<>.@TDNJ<LN276s8B132T 2 u a3 \

=R25432=<>aGCI41325:=<>.D:=<>1sAJ2`.P<L425:7Aw<L:7TDCL6J254 z >>I\@> u z >>I\@> 94ECL6Q43UL.DC*SIF>.@TDNJ<L4/276d27..@?@adT@832m M

\9^yAw<>6JN C.P<L4 N CI4yae<L4E<L4y43CL6eN 27.wadTD43aGC|CL13N 276e.P<L4`25:7Aw<L:7TDCL6J254`N 27g 276f:5CL1B1327HIT@13432$~\.@AJ4B8B1327aGCI4X27."AJ43CdN 2h.P<L4j25:7Aw<L:7TDCL6J254R<>6s83271BTDCL13254K132543CL.@STD276JN CdA 6c0 13CLg .D27ae<

TM.P< [TD271B1E<:5CL6J4E271BSL<>1E<Q4BA276J271BHI?P<Q83CL8E<>.(tv0 2713CQ0M2713NvTD271E<.P<|adT@8E<LNN 2R4BA:=<>6s8BTDNJ<LNN 2aGCLaG276s83CG<>6 HIA .P<>1[13254B0M25:783CQ<>. CL. z 0MCL1`2B27ad0 .DC tv:=<>a|g TPF>6JN CL.D2Rg 1BAJ43:=<>aG2769832*.P<|NvT@1325:5:7TDUL6N 2;4BAS27.DC:7TDNJ<LNw~t

³Iµ <XAJR8E<>6s83C 432;<L:52713:=<>1B?P<G<>. CL.g_>^yAwF>.943271B?P<j.P<KNvTD438E<>6J:7TP<KN 2`<>.D2B<>adTD276983CXaeF+JvT@ae< z <>0 CLH25CsN 2.P< [TD271B1E<K2764BA|69AJ27SL<UL1Bg T@8E<:$>^yAwF>.]43271B?P<d27.".P<>1BHCGN 27.<IJCG276c25432h:=<L43CNw>^yAwF>.83276JNv1B?P<9AJ2Q43271h27.F>6 HIA .DC276 sAJ2432SI<>1B?P<eg 1BAJ4E:=<>aG276s832.P<fNvT@1325:5:7TDUL6nN 2|.P<[TD271B1E<

A 0 CL6 Hs<d9AJ2hT@6 TD:7TP<>.@aG2769832;.P<dUL1Bg T@8E<d254K:7T@13:7A .P<>1=\

2=<R27.1E<LNvTDCN 2d.P<UL1Bg T@8E<c:7T@13:7A .P<>1Q276n83CL1B6JCV<>. CL. z 254B8E<>aGCI4|43A 0 CL6 TD276JN CsAJ2d27.:5276s8B13CdN 2*ae<L4E<L4ON 27."4BTD4B8327ae<:5CLT@6J:7TDN 2h:5CL6.P<|0MCI4BTD:7TDUL6N 27. CL.²~\ <UL1Bg T@8E<Q:7T@13:7A .P<>1j254yA 6:=<L43Cc0w<>1B8BTD:7A .P<>1QN 2e27.@T@0J432d0w<>1E<c.P<:7Aw<>.

a = b = R\ 2=<>6

a′ u b′.DCI4|4327adTD2B254hae. z N 254B0 AJ54KN 2w<>gM271B.D2;:=<>a|g TP<LN Ceg 1BAJ43:=<>aG276s832Q4BAcNvT@1325:5:7TDUL6_~\^`CLaGCd.P<d276J271BHI?P<dN 2;.P< [TD271B1E<d6JCd:=<>a|g TP<vt_432R8BTD276J2;9AJ2

a′ = a = R\=R2h.P<25:7Aw<L:7TDUL6

z >>I\@> O432hN 25NvAJ:52sAJ2;27.4327adTD22RaG276JCL1*NvTD4BadT@6sA T@1EF <.P<adT@8E<LN"tJCe432=<vt b′ = b/2 = R/2\<>1E<e27."0 271BT@H25C u <>0MCLH25CG432h276J:7AJ276s8B1E<

r1 = a′ −√

a′2 − b′2 = R −√

R2 − R2/4 = R2 −

√3

2

ur2 = a′ +

√

a′2 − b′2 = R2 +

√3

2.^`CLaGCc27.0M271B?DC9N Cc4EUL.DCN 270 276JN 2GN 2

az 832713:5271E<e.D2 u N 2 h270 .D271¥ u a6JCf:=<>aZg TP<vtO432 276J:7AJ276s8B1E< 9AJ227.|<IJCN 2.P<

[TD271B1E< 276 43A6sAJ27SI< UL1Bg T@8E< 4327HIA T@1B?P<4BTD276JN CGN 2 Nv?P<L4=\<ae<>HI6 T@8BAJN N 27.jaGCLaG276s83C<>6 HIA .P<>1S9TD276J2hNJ<LNJ<0MCL1[.P<Qae<>HI6 T@8BAJNcN 27."aGC>ZaG276s83C .@T@6J2=<>.[0 CL1Q27.[g 1E<>b=C \/^`CLaGC27.aGCLaG276s83C.@T@6J2=<>.J6JCh:=<>a|g TP<vt90w<>1E<hNvTD4ZadT@6sA [email protected]<>6 HIA .P<>1*<G.P<adTxZ8E<LNcN 27g 27aGCI4XNvTD4BadT@6sA T@1X27.]g 1E<>b=Ce<Q.P<adT@8E<LN"\+=R2/.P<`JHIA 1E>I\O432 N 254B0 13276JN 2T@6 aG25NvTP<>8E<>aG276s832fsAJ2d27. F>6 HIA .DCcN 27gM243271

α = 60\

x®L¯B°s°±

³ ijijkk rojmcr5r _o 2=<

~r27."S25:783CL1K0 CI4BTD:7TDUL6N 2

m\ < AJ271Bb$<GsAJ2;<L:78 Gw<d4ECLg 132

m254

~F (r) = −GmM

r2r , z >>I\@>

u .P<d276J271BHI?P<d0 CL83276J:7TP<>.<L4EC9:7TP<LNJ<G<25438E<Q AJ271Bb$<254U(r) = −G

mM

r. z >>I\@>

J :5CL698BT@6sAw<L:7TDUL6aGCI4B8B1E<>1327aGCI4K9AJ2h:7Aw<>6JN Cd.P<276J271BHI?P<G0 CL83276J:7TP<>.254O:527698B1E<>.(tw254jN 25:7T@1=t43UL.DChN 270 276JN 2ON 2[.P<Xae<>HI6 T@8BAJNGN 27. S25:783CL1~r u 6JChN 2O4BAdNvT@1325:5:7TDUL6]t276983CL6J:5254 27.v0 13CLg .D27ae<hN 2N 2783271BadT@6w<>1j.P<8B1E837.@T@832R432*0 AJ25N 2R1325NvAJ:7T@1K<A 60 13CLg .D27ae<QA 6 TDNvT@aG276J4BTDCL6w<>.(\-4d27STDN 276s8329AJ2 4BT`.P<276J271BHI?P<0 CL83276J:7TP<>.jN 2 A 6w<0w<>1B8B?D:7A .P< 254Q:527698B1E<>.(t276s83CL6J:5254d27.:=<>ad0 CdN 2X AJ271Bb$<L4[H276J271E<LN CL1KN 27."0 CL83276J:7TP<>.(t 254`1E<LNvTP<>.(\JCL1jCL8B1E<Q0w<>1B832ItvA 6w<| AJ271Bb$<Q1E<LNvTP<>.6JC|2B2713:52j83CL139AJ2O13254B0M25:783C<>.wCL1BT@H276]\ Tw27.w83CL139AJ2O13254B0M25:783C<>._CL1BT@H276f254 6sA .DC ts27._aGCLaG276s83C<>6 HIA .P<>1 ~ N 2.P<0w<>1B8B?D:7A .P<n6JC0 AJ25N 2<>.@83271E<>13432It .@AJ27HC 27.jaGCLaG276983C <>6 HIA .P<>1 z 0w<>1E<A 6w<0w<>1B8B?D:7A .P<aGC=STD76JN CI432;276A 60MCL83276J:7TP<>.:5276s8B1E<>.²O254[A 6w<d:5CL6J4B8E<>6s832|N 2RaGC=ST@adTD276s83C z T@HIAw<>.sAJ2It0MCL1G227ad0 .DC t.P< 276J271BHI?P<83CL8E<>.²~\ XAJ227.OaGCLaG276s83C<>6 HIA .P<>1G4E2=<A 6w< :5CL6J4B8E<>6s832cN 2aGC=ST@adTD276s83CR4BT@HI6 Txw:=<O9AJ2/6 T.P<[ae<>HI6 T@8BAJN6 T.P<ONvT@1325:5:7TDUL6QN 27.saGCLaG276983C*<>6 HIA .P<>1:=<>a|g TP<>6Q<aG25NvTDNJ<;9AJ2[8B1E<>6J43:7A 1B132j27.J8BTD27ad0MC \ <>ag TD76 4E<>gM27aGCI49AJ2[.P<RS27.DC:7TDNJ<LNGN 2O.P<*0w<>1B8B?D:7A .P<;2544BTD27ad0 1320M271B0 276JNvTD:7A .P<>1<>.saGCLaG276983C*<>6 HIA .P<>1=\L^`CLaGCj.P<ONvT@1325:5:7TDUL6N 27.aGCLaG276s83CR<>6 HIA .P<>16JC:=<>a|g TP<432j:5CL6J:7.@A u 2*sAJ2j27.waGC=ST@adTD276s83CN 2O.P<h0w<>1B8B?D:7A .P<;6J25:5254E<>1BTP<>aG276s832RN 27gM2O8B1E<>6J43:7A 1B1BT@1276cA 60 .P<>6JC \

=*27g TDN C <.P<QT@ad0 CL1B8E<>6J:7TP<eN 2h254B832h13254BA .@8E<LN CG.DCSCL.@S27aGCI4*<Q1327ae<>13:=<>1 A ¯ A CDA 7 " EDA CB " " /~¯ 7$R¥(~ 3X~`5 " LeDA C[B L¡ " ± A CDA_7¡ "EA C " A_ =~BE ¯BA C(L5¯ 5~3 " LDC¡7¯ " 3¥ B>C( " L~x®9DA " 7¯J ¥¯ > #" 37¯Jn " GLeDA C($A9®L¯ " L 9 " LC¡7¯ " 3¥ B>C( C( "L~@®sDA AJ¯BA_3 &L¥Q¯ A_V3$A C($A C sdG =Q(DA C(I± A_c3A 5B5¯ DA 7s " QA C~~LX~ 7¯ h " G =Q(DA C(sj¯ A LDC¡7¯ " GA ¯BA CDA_5 " I "~¡~ 3 CA_7¯¥3dDA¯ A #" $A_s±

-/./0w<>0 27./9AJ2XAJ27Hs<f27.aGCLaG276s83CV<>6 HIA .P<>1;:5CLaGC:5CL6J4B8E<>69832GN 2QaGC=ST@adTD276s83Cc254h43T@[email protected]<>1h<>.0w<>0 27.N 2|:5CL6J4B8E<>69832N 2|aGC=ST@adTD276s83C9AJ2OAJ27Hs<d.P<e276J271BHI?P<G83CL8E<>. M<>a|gMCI4X4ECL6Vae<>HI6 T@8BAJN 254sAJ2ItN 2X<>.@HIA 6w<hae<>6J271E<vts25438EF>6 N 2783271BadT@6w<LNJ<L4y0 CL1.P<L4y:5CL6JNvTD:7TDCL6J254 T@6 TD:7TP<>.D254`N 27.J0 13CLg .D27ae<v\ A 0 CL6 Hs<>aGCI4=t 0 CL1G:5CL6J43T@HIA TD276s832It 9AJ2:5CL6JC:527aGCI4G.P< 276J271BHI?P<83CL8E<>.Et.P<ae<>HI6 T@8BAJN N 27.aGCLaG276s83C<>6 HIA .P<>1

`N 27.43TD4B8327ae6JCe276V27.]sAJ2;8B1E<>6J43:7A 1B132;27.]aGC$S9T@adTD276983C \

³ >=*254E:5CLad0 CL6 Hs<>aGCI4.P<jS27.DC9:7TDNJ<LN

~v(~r )N 2y.P<O0w<>1Z8B?D:7A .P<vt:7Aw<>6JN [email protected]<432*276J:7AJ276s8B1E<27627._.@A Hs<>1

~rt276fA 6w<:5CLad0MCL6J276s832R1E<LNvTP<>. u A 6w<Q:5CLad0 CL6J27698320 271B0M276JNvTD:7A .P<>1*<

r z S271OJHIA 1E<?>>I\ ´ ~v(~r ) = vr r + vφ φ . z >>I\@> ´

J :=F r u φ43CL6|S25:783CL13254A 6 T@8E<>1BTDCI4 >27.s0 1BT@aG2713C*276.P<NvT@1325:5:7TDUL6 1E<LNvTP<>.(t]27.4327HIA 6JN Cc0 271B0M276JNvTD:7A .P<>1<

rtv0M2713CG276c27.]0 .P<>6JCGN 2R.P<d8B1E6 HIA .P<>1KN 2h.P<Q0w<>1B8B?D:7A .P<e254

` = r mvφ ,CG432=<vtJA 6w<dS27b:5CL6JC:7TDN C`.P<GS27.DC9:7TDNJ<LN

vφ9AJ25NJ<dN 2783271BadT@6w<LNJ<e0 CL1j.P<eNvTD4B8E<>6J:7TP<

rN 2h.P<0w<>1B8B?D:7A .P<e<>.]:5276s8B13C

vφ(r) =`

mr. z >>I\@> µ

=*2Q.DC<>6s83271BTDCL1:5CL6J:7.@A T@aGCI4;sAJ2Q27.0 13CLg .D27ae<9AJ25NJ< 254E276J:7TP<>.@[email protected]<>aGCI4254B8E<>g .D25:5271K:5ULaGCSI<>1B?P<|.P<NvTD438E<>6J:7TP<r276f A 6J:7TDUL6cN 27."8BTD27ad0MC \AX6w<|S27b;:5CL6JC:7TDN C

r(t)t6JC43UL.DCR432y:5CL6JC:52`.P<XS27.DC9:7TDNJ<LNQ1E<LNvTP<>.

vr = rtL4BT@6JCX9AJ2`8E<>a|g TD76]tLAJ4E<>6JN C*.P<X25:7Aw<L:7TDUL6 z >>I\@> µ ~t.P< S27.DC9:7TDNJ<LN

vφ\ J 43A S27btO:5CL6JC:7TDNJ< .P<S27.DC9:7TDNJ<LN 276 A 6J:7TDUL6 N 27.h8BTD27ad0 C0MC9N 27aGCI4$tT@6s8327HI1E<>6JN C t CLg 83276J271j.P<0MCI4BTD:7TDUL6

~r (t)N 2R.P<0w<>1B8B?D:7A .P<d276c83CN CT@6J4B8E<>69832I\

=*2.P< NvTD43:7AJ4BTDUL6<>6s83271BTDCL1;432N 254B0 13276JN 2sAJ2QN 27g 27aGCI4R:527698B1E<>1h.P< <>83276J:7TDUL6276132543CL.@S271*27.aGC=ST@adTD276s83C 1E<LNvTP<>.N 2h.P<Q0w<>1B8B?D:7A .P<GN 2Rae<L4¥<m\

<>g 27aGCI4O9AJ2X.P<|276J271BHI?P<83CL8E<>."254`.P<|4BA ae<QN 2X.P<|276J271BHI?P<:7T@6J78BTD:=. E = K + U . z >>I\@> CL1*CL8B1E<0w<>1B832It .P<276J271BHI?P<G:7T@6J78BTD:=<dS9TD276J2;NJ<LNJ<d0 CL1

K =1

2mv2 =

1

2m(

v2r + v2

φ

)

, z >>I\ ³ sAJ2It AJ4¥<>6JN Cr = vr u .P<d25:7Aw<L:7TDUL6 z >>I\@> µ ~tw9AJ25NJ<dN 2R.P<Q ¡CL1Bae<

K =1

2mv2 =

1

2m

(

r2 +`2

m2r2

)

. z >>I\ ³ >$ AJ4B8BT@8BA u 276JN Ce254B832;13254BA .@8E<LN CG276 z >>I\@> O4E2hCLg 8BTD276J2

E =1

2mr2 +

(

`2

2mr2+ U(r)

)

. z >>I\ ³I³

³ I³-4B8E<c254;A 6w<c25:7Aw<L:7TDUL69AJ2G43UL.DCVN 270 276JN 2GN 27.y1E<LNvTDC

rt]N 2d.P<S27.DC:7TDNJ<LNn1E<LNvTP<>.

r u :5CL6J4Z8E<>6s83254eN 2caGC=ST@adTD276s83C \`-4B8E< 25:7Aw<L:7TDUL6]t0MCL1d.DC8E<>6983C t4320 AJ25N 21327T@6983271B0 13278E<>1f:5CLaGC.P<:5CL1B13254B0 CL6JNvTD2769832<>. 0 13CLg .D27ae<N 2GA 6w<c0w<>1B8B?D:7A .P<VN 2eae<L4E<m u 276J271BHI?P<V83CL8E<>. E

t9AJ2e432a|AJ27S2|276VA 6w<NvT@aG276J4BTDUL6 z NJ<LNJ<0MCL1O.P<d:5CCL13N 276w<LNJ< r`276A 6c0 CL83276J:7TP<>.NJ<LN Cd0 CL1

U £ (r) = U(r) +`2

2mr2. z >>I\ ³

=*2Q254B8E<Gae<>6J271E< J27aGCI4*1325NvAJ:7TDN C27.0 13CLg .D27ae<fCL1BT@HIT@6w<>.<eA 6V0 13CLg .D27ae< A 6 TDNvT@aG276J4BTDCL6w<>. 27.N 2RA 6w<0w<>1B8B?D:7A .P<GN 2Rae<L4¥<m<& 25:78E<LNJ<0MCL1OA 6w<Q AJ271Bb$<

F £ (r) = −dU £ (r)

dr= −G

mM

r2+

`2

mr3. z >>I\ ³ 9

U £ (r)[email protected]<>ae<K27. CDA 7 " 5B2C&t$adTD27698B1E<L49AJ2 `2/(2mr2)

25427.v<[email protected]<>ae<LN C CDA 7 "EDA C ~¯I®I\^yAw<>6JN C27.aGCLaG276s83C<>6 HIA .P<>1;254h:52713C t"27.0MCL83276J:7TP<>.2~ ¡25:78BT@SC:5CLT@6J:7TDN 2d:5CL6 27.0 CL83276J:7TP<>.CL1BT@HIT@6w<>.(\<d25:7Aw<L:7TDUL6 z >>I\ ³I³ `0 CN 27aGCI4j2543:71BT@g [email protected]<e:5CLaGC

E =1

2mr2 + U £ (r) .

=*271BT@SIF>6JN CL.P<d1325430 25:783CG<>.]8BTD27ad0MCG432hCLg 8BTD276J20 = mr r +

dU £dr

r , z >>I\ ³ N 2yN CL6JN 2It>NvT@S9TDNvTD276JN CX0MCL1r u AJ4E<>6JN C z >>I\ ³ 9~t>276J:5CL698B1E<>aGCI4A 6w<j25:7Aw<L:7TDUL6dN 2aGC$S9T@adTD276983C0w<>1E<

r(t)

mr = −GmM

r2+

`2

mr3. z >>I\ ³ -4O254B8E<.P<Q25:7Aw<L:7TDUL6VsAJ2*[email protected]<0 13U JT@ae<4325:5:7TDUL60w<>1E<Q276J:5CL698B1E<>1j.P<L4[8B1E837.@T@83254=\

Z=R2 <L:7AJ2713N CX<[.DCKNvTD43:7A 8BTDN CjaeFL4<>1B1BT@gw<vt$0w<>1E<j254B8E<>g .D25:5271:5ULaGCj432:5CLad0MCL1B8E<O27.S25:783CL1

~rtN 27g 27aGCI4`132543CL.@S271[27._0 13CLg .D27ae<QN 2KA 6w<;0w<>1B8B?D:7A .P<QN 2Xae<L4E<

maGC=STD76JN CI432X27627. 0 CL83276J:7TP<>.NJ<LN Cd0 CL1K.P<25:7Aw<L:7TDUL6 z >>I\ ³

U £ (r) = −GMm

r+

`2

2mr2. z >>I\ ³´

<e276J271BHI?P< 83CL8E<>.E = K + U u 27.aGCLaG276983Cc<>6 HIA .P<>1 `

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³ J ^[<L43C

` = 0\

^`CL6J4BTDN 271327aGCI4jg 1327S27aG276s832;27.":=<L43CN 2*aGCLaG276s83CG<>6 HIA .P<>1` = 0

\ -4B832*SL<>.DCL1[0w<>1E<Q27.aGCLaG276983C<>6 HIA .P<>1 T@ad0 .@TD:=<|sAJ2[.P<h0w<>1B8B?D:7A .P<h6sA 6J:=<;8BTD276J2OA 6w<RS27.DC:7TDNJ<LNG8E<>6 H276J:7TP<>.(tCd432=<vtv.P<d0w<>1B8B?D:7A .P<d4BTD27ad0 132;432RaAJ27S2<Q.DCd.P<>1BHCeN 2R.P<1325:78E<d9AJ2RA 6J2M

:5CL6m\

<|JHIA 1E< >>I\ µ a|AJ25438B1E<d27.M0 CL83276J:7TP<>.U £ (r)

sAJ2Itv276254B832R:=<L43C t :5CLT@6J:7TDN 2h:5CL6U(r)

\ T".P<276J271BHI?P<dN 2R.P<Q0w<>1B8B?D:7A .P<e254E < 0

tJ276983CL6J:5254j.P<QaeF+JvT@ae<dNvTD4B8E<>6J:7TP<G<Q.P<9AJ2h4320 AJ25N 2|<>.D23<>1j254r0\

@®L¯°°s±OCDA 7 " =B>C & LB ` = 0±

T/.P<0w<>1B8B?D:7A .P<V276nA 6nT@6J4B8E<>69832e254B8EF276r0t"276s83CL6J:5254Q4BAn276J271BHI?P<83CL8E<>.

E:5CLT@6J:7TDN 2:5CL6 .P<276J271BHI?P<0 CL83276J:7TP<>.

U(r0) u .P< 276J271BHI?P< :7T@6J78BTD:=<vt/0MCL1d.DC8E<>6s83C t254G6sA .P< .P<e0w<>1B8B?D:7A .P<432276J:7AJ276s8B1E<276 13270MCI43C \ T@6 27agw<>1BHC t"43CLg 132Q.P< 0w<>1B8B?D:7A .P<<L:78 Gw<eA 6w< AJ271Bb$.276 r06JC254R6sA .P<v\4=R2J25:JC t.P<f0M276JNvTD276s832f254h0MCI4BT@8BT@SL<vt].@AJ27HCc43CLg 132

m<L:78 Gw<A 6w< AJ271Bb$<6J27Hs<>8BT@SI<9AJ2d.P<w<L:52<L:527.D271E<>1 w<L:7TP<|27. CL1BT@H276]\ J aG25NvTDNJ<Q9AJ2j8B1E<>6J43:7A 1B132*27.w8BTD27ad0MC;.P<NvTD4B8E<>6J:7TP<276s8B132

mu MNvTD4BadT@6sA T@1EF 0 13CLHI13254BT@SI<>aG276s832I\<e276J271BHI?P< :7T@6J78BTD:=<eN 2

m z u 0MCL1*:5CL6J43T@HIA TD276s83227._aGU9NvA .DCN 2K.P<;S27.DC9:7TDNJ<LN 1E<LNvTP<>.²~t9:5CLaGC8E<>a|g TD76f.P<; AJ271Bb$<<>8B1E<L:78BT@SL<|T@1EF>6f<>A aG276vZ8E<>6JN C \T@6w<>.@aG276s832ItJN 25430 AJ54ON 2X8B1E<>6J4E:7A 1B1BTDN CA 6f8BTD27ad0MCQJ6 T@83C t.P<ae<L4E<[email protected]<>1EF<>."CL1BT@H276]t83276 TD276JN CA 6w<276J271BHI?P<Q:7T@6J78BTD:=<QT@6vJ6 T@8E<v\ z WXCN 27gM27aGCI4[0 1325C9:7A 0w<>1B6JCI4jN 2~Zae<L4BTP<LN C0MCL1254B832eT@6vJ6 T@83C9AJ2 <>0w<>1325:7TDU \ F gsSTP<>aG276s8326 T@6 HG 6 0MCL83276J:7TP<>. ?D4BTD:5CV254N 2R.P<Q ¡CL1Bae< −GmM/r

w<L438E<r = 0

\ C9N CI4K.DCI4jCLgvB2783CI4O ?D4BTD:5CI4[8BTD276J276VA 68E<>ae<IJC t

³

u <faeFL4R8E<>13NJ<>1;:7Aw<>6JN Cc.P< NvTD438E<>6J:7TP<276s8B132.DCI4hCLgvB2783CI4;254RaG276JCL1;9AJ2.P<43A ae< N 24BAJ4 1E<LNvTDCI4 Et .P<T@6s83271E<L:5:7TDUL6:=<>a|g TP<eN 2;:=<>1EFL:783271=\ ^[<L43C

` 6= 0\

<JHIA 1E< >>I\ a|AJ254B8B1E<e27."0 CL83276J:7TP<>.U(r)

t 27."0MCL83276J:7TP<>.:5276s8B1B?x A HC`2/(2mr2) u 27.0MCL83276J:7TP<>.y2~ ¡25:78BT@SC

U £ (r)0w<>1E<VN CI4;SI<>.DCL13254N 27.yaGCLaG276s83C <>6 HIA .P<>1 z `1 > `2

~\]-4254B832G .@8BT@aGC tv27._0MCL83276J:7TP<>."2~ 25:78BT@SC tv7."9AJ2X254G 8BT@. 0w<>1E<Q<>6w<>.@T@b$<>1j27.M:5CLad0MCL1B8E<>adTD276s83CN 2R.P<SI<>1BTP<>g .D2r = |~r | 276 A 6J:7TDUL6VN 27.]8BTD27ad0MC \

@®L¯°°s± jCDA_7¡ " ¥DA C ~¯L®sff=B>C & LB>& " L3¥ s " dIGA C($A®L¯ " L IPC A C(L A_ Aw¯ " > ±^`CL6J4BTDN 271327aGCI4;6sAJ27SI<>aG276s832dA 6w<f0w<>1B8B?D:7A .P<:5CL6n276J271BHI?P<

E < 0 z S271*JHIA 1E< >>I\@> ~\<;0w<>1B8B?D:7A .P<vt9276 254B832K:=<L43C ts254B8EFh13254B8B1BT@6 HITDNJ<<haGC$S2713432X27698B132r1 u r2

\ z <>1E<27.w:=<L43CN 27.]aGC$S9T@adTD276983C N 2R.P<d8BTD271B1E<e<>.@1325N 25N CL1XN 27.43CL.]27.]S25:783CL1~r1325:5CL1B132;A 6w<8B1EHI6 T@8BAJN 254 r1 u r2

:5CL1B13254B0MCL6JN 276<.P<NvTD4B8E<>6J:7TP<Qad?@6 T@ae1B8B?D:7A .P< 254K = E − U(r)

t 43TD276JN CKr = E − U £ (r)

.P<276J271BHI?P<:7T@6J78BTD:=<1E<LNvTP<>. u Kt = U £ (r) − U(r).P<V276J271BHI?P<:7T@6J78BTD:=<8E<>6 H276J:7TP<>.(\^yAw<>6JN C.P<c0w<>1B8B?D:7A .P<432f276J:7AJ276s8B1E< 27627.y0M271BT@H25Cr1

C276 27.<>[email protected]<;6JCh8BTD276J2K276J271BHI?P<|:7T@6J78BTD:=<;1E<LNvTP<>.(ts0 2713C|4B?J8BTD276J2jA 6w<|276J271BHI?P<;:7T@6J78BTD:=<

³ 8E<>6 H276J:7TP<>.(\WXCL832d9AJ2.P<276J271BHI?P<:7T@6J78BTD:=<c8E<>6 H276J:7TP<>.y254hae0MCLH25C \

@®L¯9B°°±° OCDA_7¡ " =B>C & I ` 6= 0±

T .P<h276J271BHI?P<|2540MCI4BT@8BT@SI< z E > 0~t276983CL6J:5254 .P<R0w<>1B8B?D:7A .P<;6JC;254B8EFR.@T@Hs<LNJ<.P<R0w<>1B8B?D:7AvZ.P<e432Q<L:52713:=< w<L4B8E<eA 6w<fNvTD4B8E<>6J:7TP<Gad?@6 T@ae.D2B<eT@6JN 2~J6 TDNJ<>aG276s832It"0w<>1E<6JCRSCL.@S271=\ z - 4B8E<L4/8B1EZg .D27ae<_.DCI4X:5CLaG278E<L4*9AJ2;0 13C=STD276J276N 2h AJ271E<GN 27.4BTD4B8327ae<e43CL.P<>1X43CL6A 62B27ad0 .DCeN 28E<>.D254e8B1E6JN Cn.P<276J271BHI?P<n83CL8E<>.X254 2KJ <L:78E<>aG2769832:52713C t/.P<n8B1EZ1BTP<e8E<>ad0 C:5C254;<L:5CL8E<LNJ1EF>gMCL.P<v\

^yAw<>6JN C.P<V276J271BHI?P<V83CL8E<>.[254|T@HIAw<>.O<>.yad?@6 T@aGCN 27.`0 CL83276J:7TP<>.[2~ 25:78BT@SC t276983CL6J:5254.P<0w<>1B8B?D:7A .P<*6JCR8BTD276J2O276J271BHI?P<h:7T@6J78BTD:=<*1E<LNvTP<>.(tL0M2713Ch4B?vA 6w<R276J271BHI?P<R:7T@6J78BTD:=<R8E<>6 H276J:7TP<>. 8E<>."8B1E1KN 27.]0 13CLg .D27ae<GHI1E<5ST@8E<>83CL1BTDC \

ijijkhk jr Mojq r mjq q/mEo_r/q-/6 254B8E<4E25:5:7TDUL6 <>6w<>.@T@b$<>1327aGCI4;.P<L4h0MCI4BT@g .D254h8B1E837.@T@832GN 2ae<L4¥< m:7Aw<>6JN Cd254[<>8B1E<>?DN CHI1E<=S9T@8E<L:7TDCL6w<>.@aG2769832;0MCL1[A 6:7AJ271B0MCQae<L4BT@SCN 2*ae<L4E<

MN 2R<L:7AJ2713N Cd<.P<.D2 u N 2RHI1E<=S9T@8E<L:7TDUL6A 6 T@S27134E<>.(\

³ =R27.O<>6w<>.@TD43TD4 J25: JC276.P<4325:5:7TDUL6<>6983271BTDCL1 z S271Q25:7Aw<L:7TDUL6 >>I\ ³ 432fN 254B0 13276JN 29AJ2N 27g 27aGCI4X254B8BAJNvTP<>1j.P<25:7Aw<L:7TDUL6N 2;aGC=ST@adTD276s83C

mr = −GmM

r2+

`2

mr3. z >>I\ ³Iµ

-/./aGCLaG276s83CV<>6 HIA .P<>1`254RA 6w< :5CL6J438E<>6s832GN 2QaGC=ST@adTD276s83C u STD276J2dN 2783271BadT@6w<LNJ<0 CL1R.P<L4:5CL6JNvTD:7TDCL6J254 T@6 TD:7TP<>.D254yN 27.J0 13CLg .D27ae<vts254N 25:7T@1=t0w<>1E<RA 6d0 13CLg .D27ae<;276d0w<>1B8BTD:7A .P<>1y6JC;:=<>a|g TP<4BASL<>.DCL1*<QaG25NvTDNJ<G9AJ2R8B1E<>6J43:7A 1B132;27.]8BTD27ad0MC \ <>g 27aGCI4/9AJ2[<>.9aG276JCI4<>.@HIA 6w<L4N 2`.P<L44ECL.@AJ:7TDCL6J254/N 2y254B8E<*25:7Aw<L:7TDUL6dN 27gM276Q4327127.@T@0J43254=\-/6V:5CCL13N 276w<LNJ<L4j0MCL.P<>13254j.P<25:7Aw<L:7TDUL6N 2RA 6w<G27.@T@0J432;254

1

r=

1

r0(1 − ε cos φ ) .

-4B83C 4BA HITD27132 sAJ2Ity276 .@A Hs<>1N 2 <>6w<>.@T@b$<>1fA 6w<25:7Aw<L:7TDUL6 0w<>1E<r276 A 6J:7TDUL6 N 2

tt 43271B?P<aG2CL1Rg AJ43:=<>1*A 6w<f25:7Aw<L:7TDUL60w<>1E<

w = w(φ) ≡ 1/r276 A 6J:7TDUL6N 2

φt_0 AJ254X8E<>.25:7Aw<L:7TDUL60 13CLgw<>g .D27aG276s832Q432=<d4BT@ad0 .D2I\

K2=<>.@TD:527aGCI4X2543CI4j:=<>a|g TDCI4*N 2RSI<>1BTP<>g .D2I\ 276J27aGCI4r =

d

dtr =

d

dt

1

w= − 1

w2

dw(φ)

dt= − 1

w2

dw

dφ

dφ

dt.

2713Cdφ

dt= φ =

`

mr2=

`w2

m,.@AJ27HC

r = − `

m

dw

dφ.

=*271BT@SI<>6JN Cd69AJ27SL<>aG2769832|13254B0M25:783Ce<>."8BTD27ad0MCG432hCLg 8BTD276J2r = − `

m

(

d

dt

dw

dφ

)

= − `

m

(

d

dφ

dw

dφ

)

dφ

dt= − `

m

d2w

dφ2

dφ

dt= − `2

m2w2 d2w

dφ2.

AJ4B8BT@8BA u 276JN Ce254B8E<G2KJ0 13254BTDUL6c276 z >>I\ ³Iµ u AJ4¥<>6JN CGsAJ2 w = 1/rtw432hCLg 8BTD276J2

−`2

mw2 d2w

dφ2= −GmMw2 +

`2

mw3 ,

CG432=<vtd2w

dφ2+ w =

1

r0. z >>I\ ³

-/6c.P<d25:7Aw<L:7TDUL6<>6983271BTDCL1X432RT@698B13C9NvALBCr0N 2~J6 TDN Ce0 CL1

r0 ≡ `2

GMm2.

³ ´<>.v:5CLaGCX43CI430 25: wF>gw<>aGCI4=t$.P<K25:7Aw<L:7TDUL6dNvTx 2713276J:7TP<>.0w<>1E<

w276 A 6J:7TDUL6QN 2

φt$2~ ¡25:78BT@SI<>aG276s832254Oa|A u 4BT@ad0 .D2I\ T27.".P<LN CGN 271325:JCG AJ25432R69A .DC t .P<d43CL.@AJ:7TDUL643271B?P<

w(φ) = A cos(φ) ,N CL6JN 2A254A 6w<O:5CL6J4B8E<>69832I\ z -/[email protected]<` A 6J:7TDUL6G5DA_ 276|.@A Hs<>1y3>=DA_43UL.DCj1325N 2~J6J2 27..@A Hs<>1N 2543N 2e27.`:7Aw<>.`254B8E<>aGCI4adTDNvTD276JN C.DCI4QF>6 HIA .DCI4=\ |-4:7.P<>13C9AJ2 <>.[<>HI1327Hs<>1B.D2e.P<V:5CL6J4B8E<>6s832

1/r0<c.P< G .@8BT@ae<2KJ0 13254BTDUL6]t432 276J:7AJ276s8B1E<A 6w<43CL.@AJ:7TDUL6 N 2 z >>I\ ³ ~\CL1$G .@8BT@aGC t/4BTy276.@A Hs<>1KN 2

AT@698B13C9NvAJ:7T@aGCI4

εNJ<LN CG0 CL1

ε = −Ar0 ,276s83CL6J:5254X.P<2KJ0 13254BTDUL6V0w<>1E<r(φ) = 1/w(φ)

9AJ25NJ<dN 2R.P<Q ¡CL1Bae<1

r(φ)=

1

r0(1 − ε cos(φ) ) . z >>I\

X27aGCI4276J:5CL6s8B1E<LN C*A 6w<X2KJ0 13254BTDUL60w<>1E<j.P<KNvTD4B8E<>6J:7TP<r276| A 6J:7TDUL6N 27.F>6 HIA .DCX0MCL.P<>1=\L^`CLaGC254OAJ4BAw<>.]0w<>1E<25:7Aw<L:7TDCL6J254*NvTx 2713276J:7TP<>.D254XN 2h4327HIA 6JN CdHI1E<LN C t .P<G43CL.@AJ:7TDUL6H276J271E<>.8BTD276J2;N CI4:5CL6J4B8E<>6s83254 z 276 254B832:=<L43C

r0 u εG9AJ2N 27g 276 N 2783271BadT@13432n< 0w<>1B8BT@1N 2.P<L4:5CL6JNvTD:7TDCL6J254T@6 TD:7TP<>.D254=\

<25:7Aw<L:7TDUL6 z >>I\ ~tN 2<L:7AJ2713N C<>.ySL<>.DCL1N [email protected]]t:5CL1B13254B0MCL6JN 2f<V.P<L4QNvTD4B8BT@6s8E<L44325:5:7TDCL6J254K:5UL6 TD:=<L4 ε = 0

:5CL1B13254B0MCL6JN 2|<QA 6V:7?@13:7A .DC0 < ε < 1

:5CL1B13254B0MCL6JN 2|<QA 6w<d27.@T@0J432ε = 1

:5CL1B13254B0MCL6JN 2|<QA 6w<0w<>1EF>gMCL.P<1 < ε

:5CL1B1325430 CL6JN 2|<dA 6w< T@0 71BgMCL.P<

ijijk|k lnm p p _rojm Ro #ojm<Q.D2 u N 2RHI1E<=S9T@8E<L:7TDUL6A 6 T@S27134¥<>.N 2RWX27YO83CL66JCI4ONJ<.P< AJ271Bb$<N 2;<>8B1E<L:5:7TDUL6cN 27g TDN CG<.P<HI1E<5S25NJ<LNsAJ2;<L:78 Gw<d27698B132|N CI4Oae<L4¥<L4=t 276V2~ ¡25:783C t

~F (~r ) = −GmM

r2r

254O.P<Q AJ271Bb$<dsAJ2;<L:78 Gw<G43CLg 132h.P<Qae<L4E<mN 27g TDN Ce<Q.P<dae<L4E<

MN CL6JN 2

r254j.P<43270w<>1E<L:7TDUL6276s8B132;.P<L4jae<L4E<L4 u r

254OA 6S25:783CL1KA 6 T@8E<>1BTDCe9AJ2;<>0 A 6s8E<GN 2M

<m\-4G:5CL69S276 TD276s832T@6s8B13CNvAJ:[email protected]:5CL6J:5270 83CnN 2:=<>ad0MC HI1E<5ST@8E<L:7TDCL6w<>.K0w<>1E<N 2543:71BT@g [email protected]~ 25:783Cd9AJ2*A 6w<Qae<L4E<

MT@6s8B13CNvAJ:52;2764BA276s83CL1B6JC \ =R2~J6 [email protected] ®LB =KC(7¡$A_ " s

" G>7 M0MCL1

~g(~r ) ≡ −GM

r2r .

³ Iµ T27627..@A Hs<>1

~rN 2|2543832;:=<>ad0 C HI1E<5ST@8E<L:7TDCL6w<>.H276J271E<LN Ce0MCL1K.P<ae<L4¥<

M:5CL.DC9:=<>aGCI4XA 6w<ae<L4E<

mt 276s83CL6J:5254*43CLg 132

m<L:78BAw<>1EFA 6w<Q AJ271Bb$<

~F = m~g(~r ) .AK6:=<>ad0 C*254A 6Q2769832[sAJ2y254B8EFKN 2~J6 TDN CR276Q83C9N CI4.DCI40 A 6s83CI4/N 27.v254B0w<L:7TDCRsAJ2 6JCI4T@6s832713254E<v\^`CLaGC

~g254OA 6cS25:783CL1=tw27.:=<>ad0 CHI1E<=S9T@8E<L:7TDCL6w<>.432hNvTD:52h9AJ2;254OA 6 3I >B>C(L~¡ " \WXCh4EUL.DC;2KJTD4383276G:=<>ad0 CI4/S25:783CL1BTP<>.D254$t8E<>a|g TD76 2KJTD4B83276 :=<>ad0 CI4254E:=<>.P<>13254=tI83276J43CL1BTP<>.D254=t2783:I\@\AK6G2B27ad0 .DChN 2[A 6G:=<>ad0MC;2543:=<>.P<>1254/.P<*8327ad0M271E<>8BA 1E<v\ A 0MCL6 Hs<>aGCI4 9AJ2O276d83C9N CI4/.DCI40 A 6s83CI4hN 2A 6w<G0 TD27b$<e0 AJ25N [email protected]<G8327ad0 271E<>8BA 1E<

Tt 276s83CL6J:5254

T (~r )N 2~J6J2A 6:=<>ad0MC2543:=<>.P<>1 z 27.:=<>ad0MCN 2Q8327ad0 271E<>8BA 1E<*N 2.P< 0 TD27b$<v\ XAJ2QA 6:=<>ad0MCf4E2=<GS25:783CL1BTP<>.4BT@HI6 Txw:=<sAJ2;276c:=<LNJ<d0 A 6s83CG6JCd4EUL.DCd8BTD276J2hA 6w<Qae<>HI6 T@8BAJNV43T@6JCdsAJ2h8E<>ag TD76VA 6w<NvT@1325:5:7TDUL6]\-/.:=<>ad0MCGHI1E<=S9T@8E<L:7TDCL6w<>./4E<>8BTD4B ¡<L:52;27.0 1BT@6J:7T@0 TDC N 24BA 0 271B0MCI4BTD:7TDUL6J43TNvTD4B8BT@6s8E<L4*ae<L4E<L4

m1tm2

tJ\5\5\5tmN

A g TD:=<LN CI4`276G.DCI4 .@A Hs<>13254~r1t~r2tw\5\5\5t

~rNH276J271E<>6:=<>ad0MCI4HI1E<5ST@8E<L:7TDCL6w<>.D254

~g1t~g2t"\5\5\5t

~gNt 276983CL6J:5254X27.]:=<>ad0MCdHI1E<5ST@8E<L:7TDCL6w<>.27627."0 A 6s83C

~r43271EF

~g(~r ) =∑

j

~gj(~r − ~rj) = −G∑

j

mj

|~r − ~rj|3(~r − ~rj) .

^yAw<>6JN Cc4328BTD276J2A 6 :=<>ad0MCN 2Q AJ271Bb$<L4;:5CL6J43271BSI<>8BT@SC t"13254BA .@8E<fa|A u :5CL6sS276 TD2769832eT@6vZ8B13C9NvAJ:[email protected]:5CL6J:5270 83C N 2e276J271BHI?P<V0MCL83276J:7TP<>.(\-.y:=<>ad0MCVHI1E<=S9T@8E<L:7TDCL6w<>.O254Q:5CL6J43271BSL<>8BT@SC u276A 6 :=<>0 ?@8BA .DCc<>6s83271BTDCL1hN 27aGCI4B8B1E<>aGCI4|sAJ2.P<e276J271BHI?P< 0MCL83276J:7TP<>./N 2QN CI4*ae<L4E<L4M u m

t43270w<>1E<LNJ<L4j0MCL1OA 6w<dNvTD4B8E<>6J:7TP<rtvSTD276J2;NJ<LN Cd0MCL1

U(r) = −GmM

r.

<:5CL6J4B8E<>6s832G<LNvT@8BT@SI<<>1Bg T@8B1E<>1BTP<432w<f27.D27HITDN CN 2Qae<>6J271E<sAJ2Q.P<f276J271BHI?P<f0 CL83276J:7TP<>.4E2=<6sA .P<f:7Aw<>6JN Cf.P<L4*N CI4*ae<L4E<L4*254B8EF>643270w<>1E<LNJ<L4*0MCL1XA 6w< NvTD4B8E<>6J:7TP< T@6vJ6 T@8E< z r −→ ∞ ~\ <1327.P<L:7TDUL6<>6s83271BTDCL1*4BA HITD27132;N 2~J6 T@1KCL8B13Ce:=<>ad0 C t A 6c:=<>ad0MCG2543:=<>.P<>1=tJ0 CL1φ(r) = −G

M

r.

φ(~r )432|[email protected]<>ae<e27. CA_7¡ " ®I $ C(C(I¥¡QN 2|.P<Gae<L4E< M

\ T432:5CL.DC9:=<GA 6w<Gae<L4E<m27627.0 CL83276J:7TP<>.]HI1E<=S9T@8E<>83CL1BTDC

φN 2RA 6w<ae<L4E<

Mt 276s83CL6J:5254j.P<G276J271BHI?P<0 CL83276J:7TP<>.N 27.4BTD4B8327ae<254

U(r) = mφ(r) ,N CL6JN 2r254j.P<d43270w<>1E<L:7TDUL6V27698B132;.P<L4Oae<L4E<L4

m u M\-/.0 CL83276J:7TP<>.HI1E<5ST@8E<L:7TDCL6w<>.8E<>a|g TD76n4E<>8BTD4B ¡<L:5227.0 1BT@6J:7T@0 TDCN 2|43A 0 271B0MCI4BTD:7TDUL6 4BTNvTD4BZ8BT@6s8E<L4ae<L4E<L4

m1tm2

ts\5\5\5tmN

A g TD:=<LN CI4276Q.DCI4.@A Hs<>13254~r1t~r2t\5\5\5t

~rNH276J271E<>60 CL83276J:7TP<>.D254HI1E<5ST@8E<L:7TDCL6w<>.D254

φ1tφ2t"\5\5\5t

φNtJ276983CL6J:5254K27.]0MCL83276J:7TP<>.]HI1E<=S9T@8E<L:7TDCL6w<>.83CL8E<>.276V27."0 A 6s83C

~r43271EFφ(~r ) =

∑

j

φj(~r − ~rj) = −G∑

j

mj

|~r − ~rj|.

³ [email protected]:5CL6J:5270 83CI4X<>6983271BTDCL13254X:5CL6N CI4O0 13CLg .D27ae<L4=\

^`CL6J4BTDN 27132nN CI4ae<L4E<L4$t`N 2 SI<>.DCL1m u 2m

tsAJ2 432 276J:7AJ276s8B1E<>6 43270w<>1E<LN CI4 0 CL1 A 6w<NvTD4B8E<>6J:7TP<2a z S271OJHIA 1E<G<LN>A 6s8E<~\

<K^[<>.D:7A .D2Q27.SI<>.DCL1*N 2|.P<G276J271BHI?P<e0 CL83276J:7TP<>.HI1E<5ST@8E<L:7TDCL6w<>.yN 2dA 6w<fae<L4E<m′

A g TD:=<LNJ<27627.]0 A 6s83CdaG25NvTDC 276s8B132h.P<L4KN CI4=\ @®L¯°°s±²°°g_ &^yAwF>.]254j.P<Q AJ271Bb$<dHI1E<5ST@8E<L:7TDCL6w<>.2B2713:7TDNJ<d43CLg 132m′

:$ &^yAwF>.]254j.P< > " =7 s¥53 ve =R270M276JN 271EFd54B8E<dN 2R.P<GNvT@1325:5:7TDUL6

<d276J271BHI?P<d0MCL83276J:7TP<>.N 2RA 6w<ae<L4¥<m′

A g TD:=<LNJ<G276V27."0 A 6s83C :5276s8B1E<>.254U = m′φ = −G

mm′

a− G

2mm′

a= −G

3mm′

a.

<Q AJ271Bb$<dsAJ2;<L:78 Gw<G43CLg 132m′

254~F = m′~g = m′

[

−Gm

a2x + G

2m

a2x

]

= −Gmm′

a2x .

<S27.DC9:7TDNJ<LNN 2*2543:=<>0 2*254`.P<;ad?@6 T@ae<S27.DC:7TDNJ<LNsAJ2*N 27gM2XNJ<>13432R<|.P<;0w<>1B8B?D:7A .P<Q0w<>1E<Q9AJ2432e<>[email protected]<>6JN C w<L4B8E<T@6vJ6 T@83C z 432d2543:=<>0M2GN 27. 4BTD438327ae<~\]<>1E<VsAJ2.P<ae<L4E< m′2543:=<>0M24BA276J271BHI?P<N 27g 2f432716JCc6J27Hs<>8BT@SI<v\<Vad?@6 T@ae<276J271BHI?P<V9AJ2eN 27gM2G83276J271254|0MCL1|.DCV8E<>6s83C

E = 0twC 432=<vtJ.P<d276J271BHI?P< :7T@6J78BTD:=< N 27g 243271KN 2;T@HIAw<>.ae<>HI6 T@8BAJN9AJ2;.P<e276J271BHI?P<G0MCL83276J:7TP<>.0 2713CGN 2h43T@HI6JCG:5CL6s8B1E<>1BTDC

K =1

2m′v2

e = G3mm′

a.

T@6w<>.@aG2769832It0w<>1E<;.P<;S27.DC9:7TDNJ<LN N 2K2543:=<>0M2Its9AJ2j6JCN 270 276JN 2*N 2j.P<|NvT@1325:5:7TDUL6]t9432K276J:7AJ276s8B1E<

ve =

√

6Gm

a.

^`CL6J4BTDN 27132|N CI4K0w<>1B8B?D:7A .P<L4=t_N 2;ae<L4E<L4m u M

twsAJ2hT@6 TD:7TP<>.@aG2769832254B8EF>6V276V13270MCI43C u 43270w<&Z1E<LNJ<L4h0 CL1hA 6w<NvTD438E<>6J:7TP<fa|A u HI1E<>6JN 2 z T@6vJ6 T@8E<~\4=R27a|AJ254B8B132esAJ2276n:7Aw<>.D9A TD271hT@6J4B8E<>6s8324BAS27.DC9:7TDNJ<LNV1327.P<>8BT@SL<GN 2;<L:52713:=<>adTD276s83C<>8B1BT@g A T@g .D2<Q.P<G<>8B1E<L:5:7TDUL6cHI1E<=S9T@8E<L:7TDCL6w<>.254

v =

√

2G(m + M)

D,

N CL6JN 2D254O.P<GNvTD4B8E<>6J:7TP<d9AJ2R.P<L4j43270w<>1E<v\

[email protected]<QN 27.M:527698B13CdN 2*ae<L4E<L4=\ T"276:7TD271B83CQT@6J4B8E<>69832R.P<L4[S27.DC9:7TDNJ<LN 254ON 2.P<L4ON CI4[ae<L4E<L4 z CLgJ43271BSI<LNJ<L4jN 2543N 2R27.]:5276s8B13CdN 2Rae<L4E<L4~ 4ECL6 ~vm u ~vMtv254B8E<L4O:7A ad0 .D276c:5CL6

m~vm = −M~vM .

<>1E<G.P<276J271BHI?P<G:7T@6J78BTD:=<dN 2|<>agw<L4j0w<>1B8B?D:7A .P<L4X432hCLg 8BTD276J2K =

1

2mv2

m +1

2Mv2

M =1

2mv2

m

(

1 +m

M

)

.

6 TD:7TP<>.@aG2769832h8E<>6s83C.P<Q276J271BHI?P<Q0MCL83276J:7TP<>. z .P<Q43270w<>1E<L:7TDUL6c254`HI1E<>6JN 2$`:5CLaGC.P<Q:7T@6J78BTD:=<43CL66sA .P<L4=\^`CLaGCc.P<276J271BHI?P<83CL8E<>.432:5CL6J43271BSI<vt"254B8E<N 27g 271EF4E271h6sA .P<27683CN CT@6J4B8E<>6s832I\^yAw<>6JN CG.P<L4O0w<>1B8B?D:7A .P<L4X254B8EF>6V43270w<>1E<LNJ<L4K0 CL1OA 6w<dNvTD438E<>6J:7TP<D.P<276J271BHI?P<G0 CL83276J:7TP<>.254

U = −GmM

D,

u ty0MCL1f.DC 8E<>6983C t[.P< 276J271BHI?P<:7T@6J78BTD:=< N 27gM243271N 2.P<nadTD43ae< ae<>HI6 T@8BAJN 0M2713CN 24BT@HI6JC:5CL6s8B1E<>1BTDC K =

1

2mv2

m

(

1 +m

M

)

= GmM

D.

=*25430 23<>6JN Cvm

432hCLg 8BTD276J2v2m =

2GM2

D(m + M).

<S27.DC:7TDNJ<LN1327.P<>8BT@SL<G254OT@HIAw<>.<v = vm + vM

tv.@AJ27HC

v =

√

2GM2

D(m + M)+

m

M

√

2GM2

D(m + M)=

√

2G(m + M)

D.

>

ijijkRk lnm jq r/m Mo Gm m3r r Qm

K254BA .@8E<*:5CL69S276 TD276s832Its276d254B832`0 A 6983C tLaGCI4B8B1E<>1/g 1327S27aG276s832O.P<XHI1E<>6G<>6w<>.DCLHI?P<h2KJvTD4B83276s832276s8B132R.P<; AJ271Bb$<QN 2XHI1E<5S25NJ<LN276s8B132hN CI4àe<L4E<L4 u .P<; AJ271Bb$<27.D5:78B1BTD:=<27698B132RN CI4`:=<>1BHs<L4 q1 uq2\

<R AJ271Bb$<RHI1E<5ST@8E<>83CL1BTP<43CLg 132O.P<Rae<L4E< ³ tN 27g TDN CV<f.P< ae<L4E< >S9TD276J2NJ<LNJ<0MCL1R.P<" ~ ¯ Aw>7E5 " s*®LB =KC(7 F$A s | C(A

~F = −Gm1m2

r2r

N CL6JN 2r254yA 6eS25:783CL1 A 6 T@8E<>1BTDCQ9AJ2X<>0 A 6s8E<N 2h.P<Q0w<>1B8B?D:7A .P< >|<Q.P< ³ \-/.O:=<>ad0 C u 0MCL83276J:7TP<>.[HI1E<5ST@8E<L:7TDCL6w<>.K<L43C>Z:7TP<LN Ce<A 6w<Qae<L4E<

M254

~g(~r ) = −GM

r2r

uφ(~r ) = −G

M

r.

<1327.P<L:7TDUL627698B132~g u φ

254

φ(~r ) = φ(~r0) −∫ r

r0

~g(~r ) · ~dr .

J .]:5CL.DC:=<>1OA 6w<ae<L4E< m276V27.]:=<>ad0MCHI1E<&ZS9T@8E<L:7TDCL6w<>.hN 2A 6w< ae<L4¥<Mty276983CL6J:5254.P< AJ271Bb$<*43CLg 132

m u 4BA276J271BHI?P<*0 CL83276J:7TP<>.vSTD2~Z6J276VNJ<LNJ<L4j0MCL1~F (~r ) = m~g(~r )

uU(~r) = mφ(~r) .

< AJ271Bb$< 27.D25:78B13CI4B8EF>8BTD:=< 43CLg 132.P<:=<>1BHs< ³ t/N 27g TDN C <.P< :=<>1BHs< >STD276J2NJ<LNJ<0 CL1O.P< " ~ 9/I¯ " L

~F =1

4πε0

q1q2

r2r

N CL6JN 2r254 A 6 S25:783CL1yA 6 T@8E<>1BTDCQ9AJ2K<>0 A 6s8E<N 2R.P<0w<>1B8B?D:7A .P< >;<.P< ³ \-/.:=<>ad0 C u 0MCL83276J:7TP<>. 27.D25:78B13CI4B8EF>8BTD:5C<L43C>Z:7TP<LN Ce<QA 6w<d:=<>1BHs<

Q254

~E(~r ) =1

4πε0

Q

r2r

uφ(~r ) =

1

4πε0

Q

r.

<1327.P<L:7TDUL6V27698B132 ~E u φ254

φ(~r ) = φ(~r0) −∫ r

r0

~E(~r ) · ~dr .

J . :5CL.DC:=<>1/A 6w<R:=<>1BHs< q276G27. :=<>ad0 CRHI1E<=S9TxZ8E<L:7TDCL6w<>.vN 2À 6w<*:=<>1BHs<

Qt>276s83CL6J:5254/.P<K AJ271Zb$<43CLg 132

q u 4BAn276J271BHI?P<0MCL83276J:7TP<>./S9TD276J276NJ<LNJ<L4O0 CL1~F (~r ) = q ~E(~r )

uU(~r) = qφ(~r) .

F gJ43271BSL<L:7TDCL6J254< <L4R25:7Aw<L:7TDCL6J254R27.D25:78B13CI4B8EF>8BTD:=<L4;4E2 w<>6 NJ<LN Cf276 27.43TD4B8327ae<eN 2A 6 TDNJ<LN 254 \ <A 6 TDNJ<LNcN 2R.P<G:=<>1BHs<d276V254B832h4BTD438327ae<N 2hA 6 TDNJ<LN 254K254j27./I¯ " L ^! \

I³g_-/6 .P<h27.D25:78B13CI4B8EF>8BTD:=<Q27. (<L:783CL1

1/(4πε0)AJ27Hs<;27.w13CL. sAJ2

GAJ27Hs<;276 .P<hHI1E<5ST@8E<L:7TDUL6]\

:$ AX6w<NvTx 2713276J:7TP<e276s8B132|.P<HI1E<5ST@8E<L:7TDUL6 u .P<d27.D25:78B13CI4B8EF>8BTD:=< 254K9AJ2R.P<Q AJ271Bb$<GN 2hHI1E<=S9TxZ8E<L:7TDUL6f276s8B132*N CI4 ae<L4E<L4 T@HIAw<>.D254[254y<>8B1E<L:78BT@SI<;adTD27698B1E<L4`9AJ2O.P<|N 2KN CI4`:=<>1BHs<L4 T@HIAw<>.D254254`13270 A .D4BT@SI<v\ -4B83C432X132w2B<276A 6:=<>a|g TDCdN 2h43T@HI6JCQ276c<>.@HIA 6w<L4ON 2*.P<L4O25:7Aw<L:7TDCL6J254=\Nw F 8B1E<VNvTx 2713276J:7TP<T@ad0MCL1B8E<>6s8322GT@69832713254E<>6s832254QsAJ2 276.P<V27.D25:78B13CI4B8EF>8BTD:=<.P<L4:=<>1BHs<L40 AJ25N 276G432710MCI4BT@8BT@SI<L4 u 6J27Hs<>8BT@SI<L4=\I-4B832 J25: JC;NJ<*CL1BT@H276 <X6sA aG2713CI4E<L4y4BT@8BAw<L:7TDCL6J254 u ¡276JULaG276JCI4T@6s832713254E<>6983254sAJ2y6JC*0 AJ25N 276dNJ<>13432y27627.v:=<L43C*HI1E<5ST@8E<>83CL1BTDC z 0 CL1227ad0 .DC t27.]:5CL6J:5270 83C N 2 L " ¯BA C¯ " ~\

ijijk|k p Ro #ojm Qr jq r/q o 9=R27a|AJ254B8B132sAJ2

d

dx

(√a + bx

)

=b

2√

a + bx,

N CL6JN 2a u b

43CL6c:5CL6J4B8E<>6983254=\_^`CLaGCe:5CL13CL.P<>1BTDCeN 2;254B832h2B2713:7TD:7TDCG432R8BTD276J2;9AJ2∫ x2

x1

1√a + bx

dx =2

b

√a + bx

∣

∣

∣

∣

x2

x1

=2

b[√

a + bx2 −√

a + bx1] .

^`CL6J4BTDN 27132;A 6w<d:=FL4E:=<>1E<G254 ¡71BTD:=<dN 2h1E<LNvTDCRN 2eN 276J4BTDNJ<LNN 2Gae<L4E<cA 6 Tx CL1BaG2I\ T .P<ae<&Z4E<83CL8E<>.KN 2.P<:=FL43:=<>1E< 254

Mt276983CL6J:5254e4BAN 276J4BTDNJ<LNc4BA 0M271w:7TP<>.N 2hae<L4E<4E271EF

σ =M

4πR2.

=*254E2=<>aGCI4e276J:5CL6s8B1E<>127.j0MCL83276J:7TP<>.jHI1E<5ST@8E<&Z:7TDCL6w<>.φ276 27.*0 A 6s83C

PN 27g TDN C <n.P<:=FL4Z:=<>1E<254B 71BTD:=<v\v<>1E<[email protected] 1BT@aG2713Cd27SI<>.@Aw<>aGCI4.P<|:5CL698B1BT@g AJ:7TDUL6cN 2jA 6<>6 T@[email protected] 2R<>6J:JC

R dθsAJ2y ¡CL1Bae<KA 6GF>6 HIA .DCtheta

:5CL6.P<K.@?@6J2=<*9AJ2A 6J2P:5CL627.:527698B13C N 2.P<d254 ¡271E<v\ <eae<L4E<N 27.<>6 T@[email protected] = σ (2πR sin θ) R dθ . @®L¯°°s±²° L

CN CI4].DCI4"0 A 6983CI4N 27.9<>6 T@[email protected]>6<`.P<àdTD4Bae<ONvTD4B8E<>6J:7TP<D =

√

(R sin θ)2 + (r − R cos θ)2N 27.]0 A [email protected]<:5CL6s8B1BT@g AJ:7TDUL6 N 27.<>6 T@[email protected]<>."0MCL83276J:7TP<>.HI1E<=S9T@8E<>83CL1BTDC 254dφ = −G

dm

D= −2πGσR2 sin θ dθ

√

(R sin θ)2 + (r − R cos θ)2

= −2πGσR2 sin θ dθ√R2 + r2 − 2rR cos θ

.

A ae<>6JN CG.P<d:5CL698B1BT@g AJ:7TDUL6N 2h83CN CI4O.DCI4K<>6 T@[email protected] 83276J27aGCI4K27.]0MCL83276J:7TP<>.]83CL8E<>. φ =

∫ π

0dφ = −2πGσR2

∫ π

0

sin θ dθ√R2 + r2 − 2rR cos θ

.

<T@6s8327HI1E<>.O.P<V0 CN 27aGCI427SL<>.@Aw<>1d:5CL6A 6 :=<>ag TDCN 2eSI<>1BTP<>g .D2I\ =R276JCL8327aGCI4cos θ

0MCL1s\-/6s83CL6J:5254

s = cos θds

dθ= − sin θ ,CG432=<

sin θ dθ = ds .F gJ43271BSL<>6JN CsAJ2V4BT

θ = 0276983CL6J:5254

s = 1 u 4BTθ = π

t 276s83CL6J:5254s = −1

t/0 CN 27aGCI413252543:71BT@g [email protected]<QT@698327HI1E<>. I =

∫ π

0

sin θ dθ√R2 + r2 − 2rR cos θ

= −∫

−1

1

ds√R2 + r2 − 2rRs

.

AX4E<>6JN C<JCL1E<27. 132543A .@8E<LN CVN 27. 2B2713:7TD:7TDCc0 13CL0 AJ254B83C<>.T@6 TD:7TP<>1Q.P<0 132543276s832 4325:5:7TDUL6 z :5CL6a = R2 + r2 t b = −2rR

tx1 = 1 u x2 = −1

~t 432hCLg 8BTD276J2I = −

∫

−1

1

ds√R2 + r2 − 2rRs

= − 2

−2rR

[

√

R2 + r2 + (−2rR)(−1) −√

R2 + r2 + (−2Rr) · 1]

=1

rR[(R + r) − |R − r|]

=XTD438BT@6 HIA TD276JN Ce:5CL6V:7A TDNJ<LN Cd.DCI4KN CI4K:=<L43CI4r > R u R < r

t 432hCLg 8BTD276J2|sAJ2I =

2R

4BTr < R

2r

4BTr > R

.

=*2 254B8E<ae<>6J271E<vt0w<>1E<c27.0MCL83276J:7TP<>.yN 2eA 6w<:=FL43:=<>1E<c254B 71BTD:=<VN 2d1E<LNvTDCR u ae<L4E< M

432276J:7AJ276s8B1E<e.P<2KJ0 13254BTDUL6φ(r) =

−GMR

0w<>1E<R > r

−GMr

0w<>1E<r > R

.

J .@HIA 6w<L4XCLgJ43271BSL<L:7TDCL6J254KT@ad0 CL1B8E<>6983254K1325430 25:783CG<254B832;13254BA .@8E<LN C

<<>1E<r > R

t CG432=<vt :7Aw<>6JN C 27."0 A 6s83CP432;276J:7AJ276s8B1E<d AJ271E<GN 2R.P<d:=FL43:=<>1E<d254 ¡71BTD:=<vt27. 0MCL83276J:7TP<>.wHI1E<=S9T@8E<L:7TDCL6w<>.MsAJ2K54B8E<R22713:52X254/TDN 76s8BTD:=<<R.P<h9AJ2j432 9A g TD25432KCLg 83276 TDN C4BT"83CNJ<Q.P<Qae<L4E<dN 2R.P<G:=FL43:=<>1E<d432RA g TD:=<d276c27.CL1BT@H276]\

g_ T[27.[0 CL83276J:7TP<>.j<& AJ271E<25427.àdTD4BaGC<>.`N 2fA 6w<ae<L4E<V0 A 6s8BAw<>.(t/276983CL6J:5254d8E<>a|g TD7627.:=<>ad0MCHI1E<5ST@8E<L:7TDCL6w<>. .DC4E271EFvt]C432=<vtM.P< AJ271Bb$<sAJ2.P<c:=FL43:=<>1E<254 ¡71BTD:=<22713:5271EF43CLg 132RA 6w<dae<L4E<m z 4BT]54B8E<d432h276J:7AJ27698B1E<G276V27.]2KJv83271BTDCL1¥[43271EF

~F = −GmM

r2r .

:$-/6n27.T@6983271BTDCL1|N 2.P<:=FL43:=<>1E<254 ¡71BTD:=<f27./0 CL83276J:7TP<>./HI1E<=S9T@8E<L:7TDCL6w<>.`254h:5CL6J4B8E<>6s832 z 6JCN 270M276JN 2dN 2Q.P<f0 CI4BTD:7TDUL6_~\M-4383C4BT@HI6 Txw:=<sAJ227.:=<>ad0MCfHI1E<5ST@8E<L:7TDCL6w<>.`< ?/254R6sA .DC tCf432=<vt 4BT:5CL.DC:=<>aGCI4*A 6w< ae<L4E<m27627.T@6983271BTDCL1=t"43CLg [email protected]< .P< :=FL43:=<>1E<f254 ¡71BTD:=<G6JC2B2713:5271EF6 T@6 HIA 6w<Q AJ271Bb$<HI1E<5ST@8E<L:7TDCL6w<>.(\

<>1E<GA 6w<d:=FL43:=<>1E<d254 ¡71BTD:=<N 2h1E<LNvTDCR u ae<L4E< M

432R8BTD276J2

~g(~r ) =

−GM2 r

; ¬ r > R

0 ; ¬ r < R

φ(~r ) =

−GM ; ¬ r > R −GM

R

; ¬ r ≤ R < JHIA 1E< >>I\@> aAJ254B8B1E<A 6HI1EF&w:5C N 2 .P<T@6s83276J4BTDNJ<LN N 27.|:=<>ad0 CHI1E<=S9T@8E<L:7TDCL6w<>. u 27.0 CL83276J:7TP<>.0w<>1E<QA 6w<d:=FL43:=<>1E<d254 ¡71BTD:=<v\

x®I¯9Bc°°±°

J :5CL698BT@6sAw<L:7TDUL6 0 132543276s8E<>1327aGCI4|A 6w<N 27aGCI4B8B1E<L:7TDUL6<>.@83271B6w<>8BT@SI<N 2dsAJ2QA 6w<:=FL43:=<>1E<254 71BTD:=<d6JCd22713:52h6 T@6 HG 6:=<>ad0 CdHI1E<=S9T@8E<L:7TDCL6w<>.<>.]T@6s83271BTDCL1*N [email protected]<v\J .@HIA 6JCI4 0 1327.@T@adT@6w<>13254 J .fT@698327134325:78E<>1A 6F>6 HIA .DC z T@6vJ6 T@83254BT@ae<>.² dθ

:5CL6eA 6f:7?@13:7A .DC|N 21E<LNvTDCrtv27.".P<>1BHCGN 27.<>13:5Cd254

ds = r dθ z S271JHIA 1E< >>I\@> 9~\ T 8E<>.M<>13:5CQ432 T@6J:7.@T@6w< R276eA 6F>6 HIA .DCβtM27..P<>1BHCN 27.<>13:5C z .@?@6J2=<f4327HIaG276vZ8E<LNJ<O<JCL1E<d254

ds = rdθ/ cosβ\ @®L¯°°s±²°

-4B83CI4j:5CL6J:5270 83CI4X432R0 AJ25N 276cH276J271E<>.@T@b$<>1j0w<&Z1E<$A9®L¯ " LOF " s>276d8B13254/NvT@aG276J4BTDCL6J254=\s<&Z1E<[email protected] 6w<c254 ¡271E<N 2G1E<LNvTDCr u:5CL6J4BTDN 271327aGCI4A 6 F>132=<

A z sAJ2 0 AJ25N 2 832~Z6J271O CL1Bae<dT@1B1327HIA .P<>1¥O43CLg 132R.P<d43A 0 271w:7TD2I\ TA 6 T@aGCI4O83C9N CI4`.DCI4[0 A 6983CI4jN 27.M0 271B?@aG278B13CdN 27.F>132=< :5CL6 27.;:5276s8B13C CLg 83276J27aGCI4VA 6$A®L¯ " F " sI\-/6 CL1Bae< <>6wF>.DCLHs< <.DC sAJ2 C:7A 1B132 0w<>1E<F>6 HIA .DCI4V276 A 6 0 .P<>6JC tjN 2~J6 T@aGCI4c27.|F>6 HIAvZ.DC 4EUL.@TDN C0MCL1Ω ≡ A/r2 \[-/.;F>6 HIA .DC 43UL.@TxZN C|:5CLad0 .D2783C z 276e27._254B0w<L:7TDC|8B1BTDNvT@aG276J4BTDCL6w<>.²254=t 0MCL1f.DC 8E<>6983C t

4π\ J .P< T@6sS27134E<vtO43TK832~Z6J27aGCI4jA 6F>6 HIA .DC 43UL.@TDN CdT@6vJ6 T@83254BT@ae<>.

dΩ2T@6s832713:5270 8E<>aGCI4O254B832j:5CL6eA 6w<;254 ¡271E<hN 2O1E<LNvTDC

rt_27.F>132=< N 2~J6 TDN Ce0MCL1K.P<GT@6s8327134325:5:7TDUL643271EF

dA = r2 dΩ\ T 27. F>132=<;.P<hT@6J:7.@T@6w<>aGCI4[276GA 6F>6 HIA .DC

βt4BAF>132=<|4E271EF

dA = r2 dΩ/ cos β\

@®L¯°°s±²°CL.@S9TD276JN Cf<Q.P<d:=FL43:=<>1E<d254 ¡71BTD:=<N 2h1E<LNvTDC

Ru ae<L4E< MtL27SI<>.@AJ27aGCI4/27.v:=<>ad0MC*HI1E<5ST@8E<L:7TDC>Z6w<>.276A 60 A 6s83C

P9AJ2h432h276J:7AJ27698B1E<G276c4BAT@6s83271BTDCL1 z S271jJHIA 1E<?>>I\@> ~\^`CL6J4BTDN 271327aGCI4dA 6F>6 HIA .DC43UL.@TDN C

dΩN 2543N 227."0 A 6s83C

P u w<L:7TP<Q.DCI4KN CI4[.P<LN CI4=\ -/.F>6 HIAvZ.DC43UL.@TDN CQT@698327134325:78E<<;.P<Q:=FL43:=<>1E<254 ¡71BTD:=<|276.DCI4/.@A Hs<>13254A u B z S271/JHIA 1E< >>I\@> ~\<*T@6vZ:7.@T@6w<L:7TDUL6JN 2[.P<L4F>132=<L4276

A u B254276Q27.vadTD4ZaGC F>6 HIA .DC

β u <G9AJ2 J F 254jA 68B1BTPF>6 HIA .DCTD43UI43:527.D254=\-/.`F>132=<sAJ2e27.`F>6 HIA .DC43UL.@TDN CcT@6vZ8327134325:78E<|276A254

r21 dΩ/ cos β

adTD276s8B1E<L4[9AJ2276B254

r22 dΩ/ cos β

\ 2=<σ.P<nN 276J4BTDNJ<LN4BA 0 271w:7TP<>.N 2hae<L4E<N 2h.P<:=FL43:=<>1E<G254 71BTD:=<v\ @®L¯°°s±²°

-/.:=<>ad0MCdHI1E<5ST@8E<L:7TDCL6w<>.9AJ2h27.F>132=<AH276J271E<d276

P254

~gA =

[

σr21 dΩ

cos β

]

1

r21

n =σ dΩ

cos βn .

J :=F n254yA 6fS25:783CL1[A 6 T@8E<>1BTDCsAJ2R<>0 A 6s8E<N 2

P<

A\9-/6f ¡CL1Bae<<>6wF>.DCLHs<4E2KCLg 8BTD276J2RsAJ2*27.:=<>ad0 CdHI1E<=S9T@8E<L:7TDCL6w<>.sAJ2|27.F>132=<

BH276J271E<G276

P254

~gB =

[

σr22 dΩ

cos β

]

1

r22

(−n) = −σ dΩ

cos βn.

2hCLgJ43271BSL<Q9AJ2R27.M:=<>ad0MCQHI1E<5ST@8E<L:7TDCL6w<>.N 2*.P<L4jF>132=<L4A u B

432R:=<>6J:527.P<2KJ <L:78E<>aG2769832I\ ]CadTD4BaGCcC:7A 1B132G:5CL6 :7Aw<>.D9A TD271|CL8B13CVF>6 HIA .DCc43UL.@TDN C \D=*2G.P<NvTD43:7AJ4BTDUL6<>6983271BTDCL1432d:5CL6J:7.@A u 2sAJ2<>.yT@6983271BTDCL1=t27.`:=<>ad0MCVHI1E<=S9T@8E<L:7TDCL6w<>.OH276J271E<LN C0MCL1|A 6w<:=FL43:=<>1E<254|6J25:5254¥<>1BTP<>aG276s8326sA .DC \

ijijk|k p Ro #ojm Qr r/q Mo qjm3o-/SI<>.@AJ27aGCI4j27. :=<>ad0MCQHI1E<5ST@8E<L:7TDCL6w<>.276c<>.@HG 60 A 6s83CG<>."2KJ83271BTDCL1ON 2*A 6w<Q254 ¡271E<|43UL.@TDNJ<N 2Oae<L4E<

M u 1E<LNvTDC R\I<>1E<|[email protected];254y:5CL6sS276 TD2769832K0M276J4E<>1 9AJ2O.P<h254 ¡271E<h254B8EF;:5CLad0 AJ254B8E<;N 2a|AJ:w<L4[:=FL43:=<>1E<L4`254 ¡71BTD:=<L4 z :5CLaGC|A 6w<;:[email protected]<~\y<|4E<>g 27aGCI4ysAJ2X27.w:=<>ad0MC;HI1E<5ST@8E<L:7TDCL6w<>.N 2G:=<LNJ<:=FL43:=<>1E<254|27.adTD4BaGCc9AJ2d27. 9AJ2d432dCLg 8BTD276J2 <>.:5CL6J:527698B1E<>183CNJ<f.P<ae<L4E<N 2.P<:=FL43:=<>1E<c276 27.:527698B13C \<4BA ae<N 2d83CNJ<L4h.P<L4:=FL43:=<>1E<L4 z 9AJ2G254h.P<254B 271E<43UL.@TDNJ<~t0 CL1h.DC8E<>6s83C t H276J271E<>1EF A 6:=<>ad0MCfHI1E<5ST@8E<L:7TDCL6w<>.T@HIAw<>.<>.N 2;A 6w< ae<L4E< z 25sA T@SI<>.D276s832d<G.P<e4BA ae<N 2K.P<L4 ae<L4E<L4yN 2K83C9NJ<L4y.P<L4`:=FL43:=<>1E<L4¥/:5CL6J:527698B1E<LNJ<276 27. :5276s8B13C \ F 432=<vt94BTw27._0 A 6s83C

P254B8EF<>.2KJ83271BTDCL1XN 2R.P<d254 ¡271E<vtJ<QA 6w<dNvTD4B8E<>6J:7TP<

rN 27.:5276s8B13C t_27.]:=<>ad0 CGHI1E<5ST@8E<L:7TDCL6w<>.43271EF

~g(r) = −GM

r2r .

AK6G1E<>b=CL6w<>adTD276983CQ<>6wF>.DCLHC|0 271BadT@832X276J:5CL6s8B1E<>1`27.J0 CL83276J:7TP<>._HI1E<5ST@8E<L:7TDCL6w<>. z 0w<>1E<hA 6G0 A 6s83C276V27.]2KJ83271BTDCL1¥φ(r) = −GM/r

\

´ T[27.`0 A 6s83C

P432f276J:7AJ276s8B1E<<>.[T@6s83271BTDCL1dN 2.P<;254 ¡271E<;432j0 AJ25N 2K0 13C:525N 271yN 2KA 6w<;ae<>6J271E<0w<>1325:7TDNJ<v\.=*T@S9TDNJ<>aGCI4j6sAJ27SI<>aG276s832.P<254 271E<276 69A aG2713CI4E<L4O:=FL43:=<>1E<L4[254 ¡71BTD:=<L4=\ C9NJ<L4`.P<L4:=FL43:=<>1E<L4X254B 71BTD:=<L4X:5CL6VA 6V1E<LNvTDCeae.*:=<>ad0MC HI1E<5ST@8E<L:7TDCL6w<>.276P z 0 AJ254 P

254B8EF|<>.JT@6s83271BTDCL1[N [email protected]<L4¥~\<L4:=<>0w<L4X:5CL61E<LNvTDCeaG276JCL1*sAJ2r.P<L4X0 CN 27aGCI4:5CL6J:5276s8B1E<>1j276 27.M:5276s8B13C \v-/.M:=<>ad0MC;HI1E<5ST@8E<&Z:7TDCL6w<>.]0 CL1K.DC8E<>6s83Ce4E271EF

~g(r) = −Gm

r2r ,

@®L¯°°s±²° N CL6JN 2m254O.P<ae<L4E<N 2h.P<L4K:=FL43:=<>1E<L4jT@6983271BTDCL13254=tw254B83Cd254$t

m = Mr3

R3.

=*2ae<>6J271E<f<>6wF>.DCLHs< 4E2;0 13C:525N 2|0w<>1E< 276J:5CL6s8B1E<>1h27.0 CL83276J:7TP<>.HI1E<5ST@8E<>83CL1BTDCc<>.T@6983271BTDCL1RN 2A 6w<d254 ¡271E<43UL.@TDNJ<v\ "<>1E<A 6w<G254 271E<4EUL.@TDNJ< JCLaGCLH76J2=<vtwN 2h1E<LNvTDC

R u ae<L4E< Mt 432R8BTD276J2

~g(~r ) =

−GM2 r

; ¬ r > R −GMr

R3 r ; ¬ r < R

φ(~r ) =

−GM ; ¬ r > R

−GM2R

(

3 −2

R2

) ; ¬ r ≤ R < JHIA 1E< >>I\@> µ aAJ254B8B1E<A 6HI1EF&w:5C N 2 .P<T@6s83276J4BTDNJ<LN N 27.|:=<>ad0 CHI1E<=S9T@8E<L:7TDCL6w<>. u 27.0 CL83276J:7TP<>.0w<>1E<QA 6w<d:=FL43:=<>1E<d254 ¡71BTD:=<v\

x®I¯9Bc°°±°

Iµ *.-7)!%&5895 *.57% 8 5* #(8 % * 338

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g u 27.w1E<LNvTDC;83271B13254B8B132R\

CLg 132hA 6w<ae<L4E<m:5CL.DC:=<LNJ<d276.P<d4BA 0M271w:7TD2;N 2R.P< [TD271B1E<G<L:78 Gw<A 6w< AJ271Bb$<T@HIAw<>./<

mg\_CL1hCL8B1E<G0w<>1B832ItM27.:=<>ad0 C HI1E<5ST@8E<L:7TDCL6w<>.N 2|.P<[TD271B1E<e0w<>1E<GA 60 A 6s83C43CLg 1324BA4BA 0M271w:7TD2f432G0 AJ25N 2f27SL<>.@Aw<>14BA 0MCL6 TD276JN C9AJ2G83CNJ<V.P<ae<L4E<VN 2e.P< [TD271B1E<25438EFc27627.:5276s8B13C tw254KN 25:7T@1=t

|~g(R)| =GM

R2,N CL6JN 2

M254R.P< ae<L4E<f83CL8E<>.N 2.P< 8BTD271B1E<v\^`CL6JC:7TD276JN C27.:=<>ad0 CHs<>1BST@8E<L:7TDCL6w<>.0MC9N 27aGCI427SL<>.@Aw<>1j.P< AJ271Bb$<9AJ2|<L:78 Gw<d43CLg 132RA 6w<dae<L4E<

m 254B8E<254

m|~g(R)| =GMm

R2.

HIAw<>.P<>6JN Ce.P<L4jN CI4j2KJv0 13254BTDCL6J254j0w<>1E<.P< AJ271Bb$<d432hCLg 8BTD276J2;0w<>1E<g.P<2KJv0 13254BTDUL6

g =GM

R2.

-4B8E<25:7Aw<L:7TDUL6n1327.P<L:7TDCL6w<.P<c<L:527.D271E<L:7TDUL6 N 2d.P<fHI1E<=S25NJ<LNgH t µ >da>4 2 :5CL6 .P<:5CL6J4B8E<>6s832HI1E<5ST@8E<L:7TDCL6w<>.

G u .P<ae<L4EaG276s832I\

<dN 276J4BTDNJ<LNcaG25NvTP<dN 2h.P< [TD271B1E<dSTD276J2hNJ<LNJ<0MCL1ρ =

M

V=

3M

4πR3=

3g

4πGR

=3 · 981, 0

4 π 6, 67 · 10−8 6, 4 · 108' 5, 5

#E¤ 3

.

z <>1E<132~ ¡2713276J:7TP< s.P<N 276J4BTDNJ<LNfN 27. TD271B13C tsAJ2X254y27._0 1BT@6J:7T@0w<>.M27.D27aG276983CN 27.w66GJ:7.D25CQ83271B13254Z8B132ItJ254jN 2 ´ t µ QH>:7a 3 \

ijijkEi;k m3r jq>I\ G 0 T@83271O8BTD276J2RN C:52*.@A 6w<L4O:5CL6JC9:7TDNJ<L4=t :7Aw<>8B13CdN 2*.P<L4[:7Aw<>.D254` AJ2713CL6cN 2543:7A g TD271B8E<L4O0MCL1

<>.@[email protected] \w-4B83CI4j43CL6.DCI4j1E<LNvTDCI4 u 0 271B?DCN CI4jN 2h.P<L4O0 1BT@aG271E<L4X:7Aw<>8B13C

z Ra z Nv?P<L4~> µ \ t Iµ ³I³ \ >It ´ ´ >I\ vt2>

>I\ ´L³ \ ´ t@> < F g 25N 25:5276254B83CI4j:7Aw<>8B13CG4E<>837.@T@83254*<Q.P<832713:5271E<.D2 u N 2 h270 .D271 g_O^`CL6V254383CI4jNJ<>83CI4 u 27.]SL<>.DCL1jN 2 G

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A u B43CL6 27.@?@0 8BTD:=<L4=tM4BTD276JN C

R u 4Rt 13254Z0M25:78BT@SL<>aG2769832Itw4BAJ4O2B254Oae^yAwF>.254j27.]:7AJC:7TD276s832276s8B132;4BAJ4K276J271BHI?P<L4jaG25:=F>6 TD:=<L4X83CL8E<>.D254 g_ >^yAwF>.254j27.]:7AJC:7TD276s832276s8B132;4BAJ4j0 271B?DCN CI4 :$ XAJh0 AJ25N 2;N 25:7T@13432|N 27.]:7AJC9:7TD2769832|276s8B1324BAJ4[aGCLaG276s83CI4R<>6 HIA .P<>13254

v\ AX64E<>837.@T@832G<>1B8BTxw:7TP<>.1325:5CL1B132A 6w<f8B1E1 ³ Ra 0MCL1R276J:7T@ae<cN 2.P<4BA 0M271w:7TD2R83271B13254B8B132I\< >^yAwF>.43271EF4BAS27.DC:7TDNJ<LN g_ >^`CL6V9AJ* 1325:7AJ276J:7TP<HIT@1E<>1EFe<>.@1325N 25N CL1XN 2h.P< [TD271B1E<

J\ =RCI44E<>837.@T@83254/<>1B8BTxw:7TP<>.D254N 2yT@HIAw<>.ae<&Z4E<CL1Bg T@8E<>6 <>.@1325N 25N CL1N 2dA 6n0 .P<>6J278E<v\S1

4E2a|AJ27S2 276 A 6w< UL1Bg T@8E<:7T@13:7AvZ.P<>1VN 2 1E<LNvTDC108 a u 0M271B?DC9N C N 2

Nv?P<L4=tS2

432Va|AJ27S2 276 A 6w< UL1Bg T@8E<27.@?@0 8BTD:=<N 2G1E<LNvTDCI4rmin = 108 a u

rmax = 2 · 108 a \< J 0w<>1B8BT@1 N 2O.DCI4 NJ<>83CI40w<>1E<R.P<hUL1Bg TxZ8E<:7T@13:7A .P<>1=t:=<>.D:7A .D2d.P<ae<L4E<N 27.0 .P<&Z6J278E<v\g_`-/6J:7AJ27698B13227."0M271B?DCN CGN 2S2\

@®L¯°°±° :$ >^yAwF>.4E<>837.@T@832;254jaeFL4OS27.DCLb<>."0w<L4E<>1j0MCL1

B >^yAwF>."8BTD276J2hae.D:7A .D227.]:7AJC9:7TD2769832|276s8B132;.P<L4OS27.DC9:7TDNJ<LN 254*9AJ2

S28BTD276J2;276

B u 276 A\2$ XAJyae<>6 TDCLg 1E<*N 27g 271B?P<XCL13N 276w<>1/27.90 AJ254383CXN 2yae<>6JN C*0w<>1E<K0MCL6J271

S2276Q.P<KUL1Bg T@8E<N 2

S1

v\ AX6c4E<>837.@T@832|254X®sB$¥C(7¡$A_L~¡43T]0w<>1E<A 6VCLgJ43271BSI<LN CL1jIBCd43CLg 132|.P< [TD271B1E<d254B832h6JC432*aAJ27S2I\ <UL1Bg T@8E<N 2R8E<>."4E<>837.@T@832R6J25:5254E<>1BTP<>aG276s832N 27g 271EF:5CLT@6J:7TDNvT@1X:5CL627."0 .P<>6JCN 27.]-:7Aw<LN CL1X83271B13254B8B132I\ =R2783271BadT@6J2|27."1E<LNvTDCeN 2h.P<UL1Bg T@8E<GN 27.]4E<>837.@T@832I\

2>

X254B0 AJ254B8E< R ' 42.000

ac\v\ J 9AJNvTD438E<>6J:7TP<N 2.P< [TD271B1E<N 27gM2 :5CL.DC9:=<>13432 A 6 :7AJ271B0MC276 .P< .@?@6J2=< NvT@1BT@HITDNJ<

w<L:7TP<d27. CL.N 2h8E<>.ae<>6J271E<esAJ2R.P<e<>8B1E<L:5:7TDUL6HI1E<=S9T@8E<L:7TDCL6w<>.43CL.P<>1K:5CL698B1E<>1B13254B832<.P<<>8B1E<L:5:7TDUL6N 2Q.P< [TD271B1E<G-/. CL.25438EF <15 · 107 RaN 2NvTD4B8E<>6J:7TP.@TD:52;27.0 13CLg .D27ae<|T@6J:7.@A u 276JN C27._2~ ¡25:783C;T@6s8B13CNvAJ:7TDN CQ0 CL1y.P<h13CL8E<L:7TDUL6N 2j.P< [TD271B1E<<>.@1325N 25N CL1N 27. CL. z C432=<vt8E<>6s83C.P<[TD271B1E<:5CLaGCd27.":7AJ271B0 CQHIT@1E<>6V<>.@1325N 25N CL1jN 27. CL."A 6w<S27bh<>.<IJCs~\<>1E<d276J:5CL6s8B1E<>1*.P<43CL.@AJ:7TDUL6 z <>0 13C JT@ae<LNJ<jAJ432h27. J25: JCeN 2h9AJ2 MT /MS 1

\X254B0 AJ254B8E<

r ' R

(

1 − 3

√

MT

3MS

)

254`.P<43CL.@AJ:7TDUL6sAJ2*432X8BTD276J2*43T 27.M:7AJ271B0 C432*276J:7AJ276s8B1E<27698B132R.P< [TD271B1E.D254*43T432h0M271BadT@8329AJ2r432=<dae837.@T@83254A u B

HIT@1E<>6 <>.@1325N 25N CL1eN 2f.P< [TD271B1E< 276.P<adTD4Bae<UL1Bg T@8E< :7T@13:7A .P<>1z N 21E<LNvTDC R

~tM0M2713CA 6JCV<>./.P<LN CcCL0 AJ254B83CVN 2.P<[TD271B1E<13254B0M25:783CV<>.CL8B13C \ 2dN 25432=<T@69832713:5270 8E<>1X27.]4E<>837.@T@832B:5CL6cA 60 13C u 25:[email protected]<>6 b$<LN CeN 254EN 2 A

\J- 4B8BAJNvTD2;27."0 13CLg .D27ae<2hT@6JNvTD9AJ2;S27.DC:7TDNJ<LN u NvT@1325:5:7TDUL62769AJ2;N 27g 2;.P<>6 b$<>13432|27.0 13C u 25:[email protected]<>1E<d.DCLHI1E<>1X43ACLgvB278BT@SC \ =R;<>.]aG276JCI4 d4ECL.@AJ:7TDCL6J254KNvTD4B8BT@698E<L4=\µ \ -/. 0M271BT@H25C t90 A 6983CaeFL4`0 13UJvT@aGCQN 2X.P<QUL1Bg T@8E<QN 2KA 64E<>837.@T@832Itv432 w<>[email protected]<Q< ³ Ra N 2.P<4BA 0M271w:7TD2h83271B13254B8B132It u 27.<>0MCLH25C t 0 A 6s83CeaeFL4K<>.D23<LN C tw< ³ Rac\< >^yAwF>.254j27.]4327adTD2B2hae837.@T@832 g_ >^yAwF>.254O.P<2KJ :5276s8B1BTD:7TDNJ<LNN 2h4BAcUL1Bg T@8E<:$ T27.]4¥<>837.@T@832h8BTD276J2hA 6w<ae<L4E<N 2 > RHJt =:7AwF>.254j4BAc276J271BHI?P<83CL8E<>.Nw >^yAwF>.254j4BAS27.DC:7TDNJ<LNc276V27.<>0MCLH25C2$ >^yAwF>.254j4BAS27.DC:7TDNJ<LNc276V27."0M271BT@H25C ~ >^yAwF>.254j4BAaGCLaG276983C <>6 HIA .P<>1 \ AX6w<0 259AJ2Iw<ae<L4E<

m:=<L2w<L:7TP<27. CL."0w<>1B8BTD276JN CdN 27."13270MCI43CN 2543N 2RA 6w<NvTD4B8E<>6J:7TP<T@HIAw<>.<>.1E<LNvTDCN 2Q.P< UL1Bg T@8E< 83271B13254B8B132I\[email protected] 2Q:=<>?DNJ< AJ4E<>6JN C43UL.DC.P<L4O.D2 u 254KN 2 h270 .D271=\

> \ 2GNvTD:52 sAJ2e.P<UL1Bg T@8E<cN 2eA 64E<>837.@T@832 254v " ¡>¥ A_7 F$AJ¡3 4BT0w<L4E<VNvTP<>1BTP<>aG276s832f0MCL1.DCI4adTD4BaGCI4Q.@A Hs<>13254d<c.P<VadTD4Bae< JCL1E<v\ A 0MCL6 TD276JN C sAJ2 .P<<>.@8BA 1E<Vad?@6 T@ae<N 2eA 6

2>>

4E<>837.@T@832 254|N 2 ³ a 0 CL143CLg 132G.P<43A 0 271w:7TD2e83271B13254B8B132It276J:7AJ27698B132e.DCI4|1E<LNvTDCI4N 283CNJ<L4O.P<L4jCL1Bg T@8E<L4 J27.@TDCI4BT@6J:713UL6 TD:=<L4X:7T@13:7A .P<>13254$\>>I\ ^`CL6J4BTDN 27132;.P<Q0w<>1EF>g CL.P<

y = ax2 \<`-/6J:7AJ27698B13243A C:5C \g_ ]1E<L4B.P<LN 2d27.4BTD4B8327ae<fN 2Q:5C9CL13N 276w<LNJ<L4|N 2|ae<>6J271E<9AJ227.6sAJ27SCCL1BT@H276:5CLT@6J:7TDNJ<:5CL6c27." ¡C:5Cd:=<>.D:7A .P<LN CG276c.P<Q0w<>1B832|<~\:$y 6s8B13CNvA b=:=<d:5C9CL13N 276w<LNJ<L4K0 CL.P<>13254(r, φ)

tJadTDNvTD276JN CφN 2543N 2h27."ad?@6 T@aGCdN 2R.P<Q0w<&Z1EF>gMCL.P1EF>g CL.P<esAJ25NJ<N 2h.P< ¡CL1Bae<

1

r=

1

s(1 − ε cos φ):5CL6

ε = 1 u s<>.@HG 6cSI<>.DCL1j:5CL6J4B8E<>6s832I\

> ³ \ AX6 4¥<>837.@T@832HIT@1E< 43CLg 13227.[- :7Aw<LN CL1=t276 27.OadTD4BaGC 43276s8BTDN C9AJ2f.P< [TD271B1E<vt/276 A 6w<8B1E837.@T@832e276 UL1Bg T@8E<H25C254B8E<L:7TDCL6w<>1BTP<v\-/. :5CJ27832eN 27.`4E<>837.@T@832G254:=<>0w<>b N 2e<L:527.D271E<>1B.DC:5CL6 A 6w<<L:527.D271E<L:7TDUL6a = 50

a>4 2 \ $-/6f9AJjT@6J438E<>6s83254 u NvA 1E<>6s832R:7AwF>6s83CQ8BTD27ad0MC4E2N 27gM2760 13276JN 271K.DCI4OaGCL83CL13254K0w<>1E<.DCLHI1E<>1K27."0 13CL0MUI4BT@83C> v\ ^[<>.D:7A .D2G27. 0 271B?DCN CN 2d13CL8E<L:7TDUL6N 2.P< "A 6w<276s83CL1B6JC<.P<[TD271B1E<c4¥<>g TD276JN CcsAJ2d27.1E<LNvTDCGN 2h43AcUL1Bg T@8E<d254 tS25:5254X27."1E<LNvTDCGN 2h.P< [TD271B1E<v\> J\ AX6:5CLaG278E< N 2 ae<L4¥<

α m432NvT@1BT@H2 z :=<L2|1E<LNvTP<>.@aG276s832w<&Z:7TP< 27. 43CL.(\ F gJ43271BSL<L:7TDCL6J254 <L4Z8B13CL6JULadTD:=<L40 271BadT@83276 254B8E<>g .D2~Z:52719AJ2/.P<O276J271BHI?P<jaG25:=F>6 TD:=<O83C>Z8E<>. N 27.y:5CLaG278E<254;69A .P<vt254;N 2~Z:7T@1=t

E = 0\ -.X:5CLaG278E<[email protected]<:5CL698B1E< 276sAJ4=t":7A u <fae<&Z4E< 254

m\ A 0MCL6 Hs<>aGCI4;<LN 27aeFL49AJ2*.P<Q8B1E1=tN 2j1E<LNvTDC

R0\ J :5CL6J432~Z:7AJ276J:7TP< N 27.: JC99AJ2It 27.:5CLaG278E6nA 6n43CL.DC<[email protected]<>ae<>1327aGCI4 276 69AJ4=\ @®L¯9B°°± L

v ^[<>.D:7A .D2h.P<Q1E<>0 TDN 27bv0 u 27.]0M271B?DC9N CdN 2 276sAJ4X<>6983254KN 2h.P<:5CL.@TD43TDUL6]\

$ ^[<>.D:7A .D2Q.P<f276J271BHI?P<faG25:=F>6 TD:=<N 2 276sAJ4h2764BA UL1Bg T@8E<<>6s83254hN 2:JC:=<>1h:5CL6 27.:5CLaG278E<v\

2> ³$ ^[<>.D:7A .D2Q.P< S27.DC9:7TDNJ<LN1E<LNvTP<>. u 27.aGCLaG276s83Cc<>6 HIA .P<>1;N 2 276 69AJ4 T@6 aG25NvTP<>8E<&ZaG2769832;N 254B0 AJ54KN 2R.P<G:5CL.@TD4BTDUL6]\J =R2783271BadT@6J2*.P<|276J271BHI?P<aG25:=F>6 TD:=<N 2 276 6sAJ4 u 2KJv0 1354327.P<Q276f8371BadT@6JCI4ON 2 m

tαu v0

\> =R27a|AJ25438B132jsAJ2`.P<RUL1Bg T@8E<hN 2 276 69AJ4 254/27.@?@0 8BTD:=1E<h.DCI4 S276sAJ4BTP<>6JCI4 O4E2 w<h<L:5CL1B8E<LN C|C;<>.P<>1BHs<LN C|<R:=<>AJ4E<N 27.]: JC99AJ2;:5CL6V27.]:5CLaG278E<v\

X254B0 AJ254B8E<Q0w<>1B832R ~ <1E<>b=UL6N 27.]0 271B?DCN CdN 2 276 69AJ4 u 276sAJ4K254T ′

T=

[

1 + α

1 + 4α

]3/2

.

> v\ AX60 13C u 25:78BT@.`N 2dae<L4E< m432.P<>6 b$<8E<>6 H276J:7TP<>.@aG2769832<f.P<4BA 0M271w:7TD2GN 2d.P<[TD271B1E<v\ A 0MCL6 TD276JN CRsAJ2y6JC wT@132 z :5CLaGC*276Q.P< "A 6w<~tL:=<>.D:7A .P<>1.P<j1E<>0 TDN 27b

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t9:5CL61, α <

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> ´ \ =RCI40w<>1B8B?D:7A .P<L4N 2`T@HIAw<>. ae<L4E<R432À 6J276daG25NvTP<>6vZ832A 6w<:7AJ2713NJ<TDN 2=<>. N 2d.DCL6 HIT@8BAJNh\-/.0w<>1;254<>8B1E<>?DN C|HI1E<5ST@8E<L:7TDCL6w<>.@aG276s832*0 CL1 A 6e0 .P<>6J278E<;N 2ae<L4E<

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2

3.

CL1KCL8B1E<0w<>1B832It 0w<>1E<d.P<UL1Bg T@8E<G:7T@13:7A .P<>1U = −2K1

t Ce432=<vtE1 = U + K1 = U − U

2=

U

2= −G

Mm

2R2.

2R8BTD276J2E2

E1=

U + K2

U/2=

2

3.

=R2;<L9A ?]432hN 25NvAJ:52|9AJ2K2 = −2U/3

\ "AJ27HCK2

K1=

−2U/3

−U/2=

4

3=

v22

v21

.

2>

z 2$<>1E<0w<L4E<>1;N 2.P<fULg T@8E<27.@?@0 8BTD:=<c< .P<:7T@13:7A .P<>1=t]27.4¥<>837.@T@832 S2N 27g 2It":7Aw<>6JN Cc4E2Q276vZ:7AJ27698B1E<e0w<L4¥<>6JN C 0MCL1R27.0 A 6983C

Btw0 13276JN 271*.DCI4*aGCL83CL13254 u ¡ 13276w<>1 w<L4B8E<egw<&3<>1X.P<S27.DC:7TDNJ<LNcN 2

v2<

v1\

<G8B1E< u 25:783CL1BTP<N 2|.P<eae<L4E< m9AJ2:=<L2 w<L:7TP<e27. CL.254*27..@?@adT@832N 2|A 6w< 27.@T@0J432Q2769AJ2|27.4327adTD22*aG276JCL1

b8BTD276JN 2h<Q:52713C \v-/625432*:=<L43CQ.P<|8B1E6JN C27. CL.]276fA 6JCdN 2*.DCI4O2KJv8B1327aGCI4=\ -/."4327adTD2B2*ae.]27.@T@0J432R254 a = R/2

t N CL6JN 2R = aT254`27.M1E<LNvTDC z T@HIAw<>.]<>."4327adTD22*ae1jN 2*.P< [TD271B1E<v\=R2*<L:7AJ2713N C<.P<Q832713:5271E<.D2 u N 2 h270 .D271

Tm

TT=

(

a

aT

)3/2

= 2−3/2 = 0, 3536 .

-/.[8BTD27ad0 Ct09AJ2N 27aGCL1E<.P<Vae<L4¥<

[email protected]<>1d<>.O43CL.[254.P<adT@8E<LNN 27.`0M271B?DCN CN 24BAUL1Bg T@8E<vtwCd432=<vt

t0 =Tm

2=

TT

2· 0, 3536 ' 64, 5 ¢ .

z < 2=< M.P<;ae<L4E<N 27. CL.(t276983CL6J:5254=tT@HIAw<>.P<>6JN CQ.P<h AJ271Bb$<;HI1E<=S9T@8E<L:7TDCL6w<>.]:5CL6f.P<; AJ271Bb$<:527698B1B?@0 278E<

−GMm

R20

r = −mv20

R0r

432hCLg 8BTD276J2v20 =

GM

R0.

z g_ <276J271BHI?P<GaG2=F>6 TD:=<dN 2 2769AJ4 z <>6s83254XN 2h.P<:5CL.@TD4BTDUL6_[254Ei = −GMm

R0

2

+1

2mv2

0 = −GMm

2R20

.

z :$c^`CLaGCG27.]:5CLaG278E< z :7Aw<>6JN Ce254B8EFQ.D2BCI4¥`432RaAJ27S2h1E<LNvTP<>.@aG276s832 w<L:7TP<d27.]43CL.(tv6JCd8BTD276J2aGCLaG276983Cc<>6 HIA .P<>1 z 13254B0M25:783C<>./CL1BT@H276n27627. CL.²~\"AJ27HC27.aGCLaG276s83CV<>6 HIA .P<>1;N 2276 69AJ4K254j27."adTD4BaGCesAJ2h27.N 2 2769AJ4L = R0mv0 .

³-4383CV6JCI4Q0M271BadT@832f276J:5CL698B1E<>1d.P<V:5CLad0MCL6J276s832

θN 2e.P<VS27.DC:7TDNJ<LN N 2 276 69AJ4*AJ4B83CN 254B0 AJ54KN 2h.P<:5CL.@TD4BTDUL6]\J-/.aGCLaG276s83C <>6 HIA .P<>1 AJ4383CdN 254B0 AJ254jN 2h.P<:5CL.@TD4BTDUL6254

L = R0(m + αm)vθ .^`CLaGCG27."aGCLaG276983C <>6 HIA .P<>1*432h:5CL6J43271BSI<432hN 25NvAJ:52|9AJ2vθ =

v0

1 + α.

<:5CL6J4E271BSL<L:7TDUL6VN 27."aGCLaG276983CG.@T@6J2=<>.276.P<NvT@1325:5:7TDUL61E<LNvTP<>. w13432h:7AJ276s8E<9AJ2h.P<dT@6983271E<L:5:7TDUL6276s8B132 276sAJ4 u 27.:5CLaG278E<e4ECL6 AJ271Bb$<L4XT@6s83271B6w<L4 u tw0 CL1j.DCe8E<>6s83C t0w<>1E<:=<>.D:7A .P<>1[.P<hS27.DC:7TDNJ<LNfN 27.M:5CLaG278E<|0 CN 27aGCI4`T@HI6JCL1E<>1`27.M2~ 25:783CT@6s8B13CNvAJ:7TDN C0MCL1.P<QT@6983271E<L:5:7TDUL627698B132;27.:5CLaG278E6983254fN 2V.P<:5CL.@TD4BTDUL6 T@HI6JCL1E<>6JN C27.*2~ 25:783CT@698B13C9NvAJ:7TDN Ce0 CL1 2769AJ4¥~\ 2;N 25NvAJ:52;9AJ2

v2C =

2GM

R0.

J 0 .@TD:=<>aGCI4|<JCL1E<.P<:5CL6J43271BSI<L:7TDUL6 N 27./aGCLaG276983Cc.@T@6J2=<>. <f.DC.P<>1BHCcN 2.P<NvT@1325:5:7TDUL61E<LNvTP<>.αmvC = (m + αm)vr ,N CL6JN 2

vr254O.P<QS27.DC:7TDNJ<LNVN 2 276 6sAJ4 AJ4B83CdN 254B0 AJ54jN 2h.P<:5CL.@TD43TDUL6]\ 2;N 25NvAJ:52;9AJ2

vr =α

1 + αvC .

z Nw <276J271BHI?P<GaG25:=F>6 TD:=<GN 2 276 69AJ4 z .P<d27SI<>.@Aw<>aGCI4/AJ4B83CdN 254B0 AJ54jN 27.:JCsAJ2$[254Ef = U + K = −GMm(1 + α)

R0+

1

2m(1 + α)(v2

θ + v2r )

= −(1 + α)GMm

R0+

1

2(1 + α)

[

2

(

α

1 + α

)2

+1

(1 + α)2

]

GMm

R0

= −GMm

2R0

(

1 + 4α

1 + α

)

= Ei

(

1 + 4α

1 + α

)

.

z 2$f.P<hUL1Bg T@8E<|N 2 276 6sAJ4`CLgsSTP<>aG276s832 u <R6JC;254 A 6e:7?@13:7A .DC \^`CLaGC|.P<h276J271BHI?P<;254 6J27Hs<>8BT@SL<N 27gM2Itv0 CL1O.DCG8E<>6s83C tw4E271j27.@?@0 8BTD:=<v\ 2h8BTD276J2;9AJ2Ei

Ef=

af

ai=

af

R0.

³ >J 9A ? ai u af

43CL6f.DCI4`4E27adTD2254àeaG276s832I\ 2hN 25NvAJ:52;sAJ2af = R0

Ei

Ef= R0

1 + α

1 + 4α.

z ~ A*4E<>6JN CX.P<j832713:5271E<*.D2 u N 2 h270 .D271/0MC9N 27aGCI4:=<>.D:7A .P<>1.P<X1E<>b=UL6N 27.0M271B?DC9N CRN 2 276 6sAJ4u 276sAJ4 T ′

T=

(

af

r0

)3/2

=

(

1 + α

1 + 4α

)3/2

.

z < < AJ271Bb$<6J278E<d9AJ2;<L:78 Gw<d43CLg 132;.P<Q0w<>1B8B?D:7A .P<daeFL4j:52713:=<>6w<[email protected]<>aG7aGCI4B.P< >ItJ254~F1 = −GMm

R2r + T r .

< AJ271Bb$<6J278E<d9AJ2;<L:78 Gw<d43CLg 132;.P<CL8B1E<0w<>1B8B?D:7A .P< z ³ [254~F2 = − GMm

(R + h)2r − T r .

^`CLaGCh27. [email protected]<L4A 6J2O254T@6J2KJv83276J4BT@g .D2Its<>agw<L4/0w<>1B8B?D:7A .P<L4 <L:527.D271E<>6e:5CL6d.P<XadTD4Bae<<L:527.D271E<L:7TDUL6~a = −ar

\ 2h8BTD276J2~F1 = −mar = −GMm

R2r + T r .

u~F2 = −mar = − GMm

(R + h)2r − T r .

=R254B0M2B<>6JN C.P<Q83276J43TDUL6VN 27. [email protected]:7AJ27698B1E<G9AJ2

T =GMm

2

(

1

R2− 1

(R + h)2

)

.

^`CL6h R

432hCLg 8BTD276J2h.P<d2KJv0 13254BTDUL6T =

GMmh

R3.

³I³z g_<>1E<c9AJ2G.P<:7AJ2713NJ<V6JCc9AJ25N 2G83276J4E<c.P<f AJ271Bb$<HI1E<5ST@8E<L:7TDCL6w<>.O276s8B132e.P<L4;0w<>1B8B?D:7A .P<L4N 27gM2h:5CLT@6J:7TDNvT@1X:5CL6

Tt CG4E2=<vt

T =Gmm

h2.

=R254B0M2B<>6JN CmN 2R.P<L4XN CI4CG .@8BT@ae<L4K25:7Aw<L:7TDCL6J254X432h276J:7AJ276s8B1E<

m = M

(

h

R

)3

.

-/."8B1E<>gw<&C9AJ2RN 27g 27aGCI4[132=<>.@T@b$<>1j254`T@6JN 270M276JNvTD276s832;N 27.":=<>adT@6JCd9AJ2*27.@T 3<>aGCI4[0w<>1E<[email protected]<>1.P<;ae<L4E<mN 27.M:5276s8B13CN 2X.P<;gw<L4E<N 2X.P<|4327adTD254B 271E< w<L4B8E<;27. T@6vJ6 T@83C \v-/.@T 3<>aGCI4yA 6:=<>adT@6JC1325:783C0w<>1E<>.D27.DC<f.P<gw<L4E2I\ 2=<54B832d27.22

x u N 276JCL8327aGCI4Q0MCL1 y<>.`2B2e9AJ2:5CLT@6J:7TDN 2Q:5CL627.2B2|N 24BT@aG278B1B?P< N 2;.P<4327adTD254 ¡271E< z S271OJHIA 1E<~\ 2=< ~F (x)

.P<R AJ271Bb$<;HI1E<=S9T@8E<L:7TDCL6w<>."9AJ2<L:78 Gw<4ECLg 132V.P< ae<L4E<m

:7Aw<>6JN C254Z8E<432G276J:7AJ27698B1E<c43CLg 132G27.y22x276n.P<0 CI4BTD:7TDUL6

x\[-438E<n AJ271Bb$<83276JNv1EFN CI4:5CLad0 CL6J276983254

~F (x) = −Fx(x)x + Fy(x)y . @®L¯9B°°± L

<j:5CLad0MCL6J276s832yN 2.P<[ AJ271Bb$<O6JCj6JCI4T@69832713254E<j0 AJ254<>.927SI<>.@Aw<>127.98B1E<>gw<&CX:5CL6|.P<j2KJv0 13254BTDUL6

W = −∫

∞

0

~F (x) · (dx x)

432CLgJ43271BSI< sAJ28E<>.j:5CLad0MCL6J276s8322540 271B0M276JNvTD:7A .P<>1f<>.jN 254B0 .P<>b$<>adTD276983C u t0MCL1d.DC 8E<>6s83C t6JCeJHIA 1E<<>.27SL<>.@Aw<>1h27.0 13CNvAJ:783C 0 A 6983C \"<>1E<f27SI<>.@Aw<>1Fx(x)

:5CL.DC99AJ27aGCI4RA 6w<f4327HIA 6JNJ<4327adTD254 ¡271E<TDN 76s8BTD:=<n276 .P<0w<>1B832cT@6v 271BTDCL1 z S271JHIA 1E< .@?@6J2=<0 A 6s832=<LNJ<~\ CL1e4BT@aG278B1B?P<.P< AJ271Bb$<d9AJ2h2B2713:52;254B8E<4327HIA 6JNJ<d254B 271E<254~F ′(x) = −Fx(x)x − Fy(x)y ,

Cd432=<vt.P<:5CLad0MCL6J276s832x254[.P<adTD4Bae<vtv0 2713C.P<Q:5CLad0MCL6J276s832

y:=<>a|g TP<dN 2h43T@HI6JC \-.]2~ 25:783CN 2h.P<L4jN CI4j4327adTD254B 271E<L4j254

~F + ~F ′ = −2Fx(x)x .

³ 2713C t_0 CL1*CL8B1E<d0w<>1B832Itw.P<L4KN CI4*4327adTD254 ¡271E<L4O ¡CL1Bae<>6VA 6w<G254B 271E<e:5CLad0 .D278E<GN 2;ae<L4¥<

2Mt u0w<>1E<d25432h:=<L43CG4¥<>g 27aGCI4j9AJ2R.P<Q AJ271Bb$<254 z S271K4325:5:7TDUL6,>>I\

m~g(~x ) =

−G(2M)m

2 x ; ¬ x > R

−G(2M)mR3 x

; ¬ x < R HIAw<>.P<>6JN Ce.P<L4jN CI4j2KJv0 13254BTDCL6J254K432h276J:7AJ27698B1E<G9AJ2

Fx(x) =

GMm

2

; ¬ x > R GMm

R3

; ¬ x < R -4B83CR2542KJ <L:78E<>aG2769832O27.adTD43aGCX132543A .@8E<LN C*9AJ2`432 83276JNv1B?P<*0w<>1E<KA 6w<*254 ¡271E<K43UL.@TDNJ<*:5CLad0 .D278E<N 2Xae<L4¥<M u 1E<LNvTDC R

\CN 27aGCI4O276s83CL6J:5254OAJ4¥<>1y.DCI4`13254BA .@8E<LN CI4O9AJ2X4E2K276J:5CL6s8B1E<>13CL6276 .P<4325:5:7TDUL6 >>I\ 0w<>1E<e.P<G254 ¡271E<G43UL.@TDNJ<v\ -/.8B1E<>gw<&C .DCG0MCN 27aGCI4X27SI<>.@Aw<>1XAJ4¥<>6JN C 27.0 CL83276J:7TP<>.HI1E<5ST@8E<>83CL1BTDCφ\-/.[0 CL83276J:7TP<>.[HI1E<=S9T@8E<>83CL1BTDC z :5CL6 27.`:52713C N 27.`0 CL83276J:7TP<>.O27627.yT@6vJ6 T@83CsQ<>.:5276s8B13CeN 2;A 6w<254 271E<4EUL.@TDNJ<d254

φ(0) = −3GM

2R.

"AJ27HCe27.@8B1E<>gw<&BCd0w<>1E<[email protected]<>1O.P<ae<L4E<mN 27.CL1BT@H276<>.T@6vJ6 T@83CG254

W = m3GM

2R.

CLg 132.P<G0w<>1B8B?D:7A .P<m6JCf4E22B2713:52Q6 T@6 HIA 6w<G AJ271Bb$<f:7Aw<>6JN C432Q276J:7AJ276s8B1E<<>.T@6s83271BTDCL1;N 27.:=<L43sAJ27832I\ T

v0254.P<S27.DC:7TDNJ<LN:[email protected]<n<.P< 4BA 0M271w:7TD2It/276s83CL6J:5254 27.O8BTD27ad0 CN 28B1E<5S254B?P<254

t0 = 2R/v0\9<>1E<:=<>.D:7A .P<>1O.P<S27.DC9:7TDNJ<LN

v0w<L:527aGCI4[AJ4ECQN 27. J25:JCdN 2R9AJ2K.P<276J271BHI?P<GN 27gM2h:5CL6J43271BSL<>134E2I\ <d276J271BHI?P<dN 2

m276c27.0 A 6s83CGN 2h0w<>1B8BTDNJ<G254

Ei = −GMm

3R,adTD276s8B1E<L4*sAJ2h:7Aw<>6JN [email protected]<G<Q.P<4BA 0M271w:7TD2;254

Ef = −GMm

R+

1

2mv2

0 .HIAw<>.P<>6JN Ce.P<L4jN CI4j2KJv0 13254BTDCL6J254K432hN 25NvAJ:52|sAJ2v20 =

4GM

3R.

J .]T@6983271BTDCL1*N 27.:=<L439AJ27832h6JC w6983C t_4BAVS27.DC9:7TDNJ<LN432Rae<>6s8BTD276J2Q:5CL6s8E<>69832I\w<>1E<G27."8BTD27ad0MCGN 2h8B1E<5S2543?P<d432hCLg 8BTD276J2t0 =

√

3R3

GM.

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