3-Dimensional dynamics of the galactic bulge Soto Vicencio, M.H.

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(1)3-Dimensional dynamics of the galactic bulge Soto Vicencio, M.H.. Citation Soto Vicencio, M. H. (2010, March 24). 3-Dimensional dynamics of the galactic bulge. Retrieved from https://hdl.handle.net/1887/15120 Version:. Corrected Publisher’s Version. License:. Licence agreement concerning inclusion of doctoral thesis in the Institutional Repository of the University of Leiden. Downloaded from:. https://hdl.handle.net/1887/15120. Note: To cite this publication please use the final published version (if applicable)..

(2) CHAPTER 4. A Schwarzschild model for three minor-axis ﬁelds % +CHC # #I=>?9B HC 69 GI6A=HH98 ,<9 GHI8M C: H<9 ;5@57H=7 6I@;9 =G =BH9F9GH=B; :CF 5 BIA69F C: F95 GCBG CK9J9F =B GD=H9 C: =HG DFCL=A=HM A5BM C: =HG 7<5F57H9F=GH=7G GH=@@ F9A5=B J9=@98 IB?BCKB CF IB89H9FA=B98 8I9 HC H<9 @=A=H5H=CB C: CIF C6G9FJ5H=CBG 5B8 H97<B=EI9G /9 5F9 9A65F?98 CB 5 DFC>97H HC 89H9FA=B9 H<9 8=A9BG=CB5@ ?=B9A5H=7G =B G9J9F5@ @CK :CF9;FCIB8 9LH=B7H=CB K=B8CKG C: H<9 ;5@57H=7 6I@;9 ,<=G 7<5DH9F DF9G9BHG 5 GH9@@5F?=B9A5H=7 AC89@ K<=7< <5G 699B GD97=5@@M 89J9@CD98 HC F9DFC 8I79 H<9 DFCD9F ACH=CB D<CHCA9HF=7 D5F5@@5L C6G9FJ98 8=GHF=6IH=CB C: GH5FG =B H<F99 9@8G 57FCGG H<9 ;5@57H=7 A=BCF 5L=G ;F=8 C: G9@:7CBG=GH9BH +7<K5FNG7<=@8 AC89@G <5G 699B FIB /9 <5J9 :C 7IG98 CB H<9 89H9FA=B5H=CB C: =ADCFH5BH 65F D5F5A9H9FG GI7< 5G 65F =B7@=B5H=CB φbar 5B8 65F D5HH9FB GD998 Ωb /9 <5J9 :CIB8 7CBG=GH9B7M 69HK99B CIF 69GH H F9GI@HG 5B8 H<CG9 :CF F979BH GHI8=9G 'IF AC89@G GI;;9GH H<5H H<9 ACGH DFC656@9 G79B5F=C HC F9DFC8I79 CIF 85H5 7CA9G :FCA 5 65F FCH5H=B; 5H km s−1 kpc−1 5B8 5 65F =B7@=B5H=CB C: 30◦ −40◦ K=H< =HG B95F 9B8 =B H<9 FGH ;5@57H=7 EI58F5BH.

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(14) *=7< %7/=@@=5A I@@6F=;<H 9H 5@ G=A=@5F HF9B8G <5J9 699B 5@GC :CIB8 K=H< <=;<F9GC@IH=CB GD97HF5 =B CH<9F α9@9A9BHG 5G ' 2C775@= 9H 5@ ,<9 =ADCFH5B79 C: H<9G9 α9@9A9BHG =G F9@5H98 HC H<9 :CFA5H=CB H=A9G75@9 C: H<9 ;5@57H=7 6I@;9 α9@9A9BHG 5F9 DFC8I798 8IF=B; H<9 9LD@CG=CB C: +& ,MD9 !! 6M G<CFH @=J98 A5GG=J9 GH5FG H=A9G75@9G < 107 M95FG K<=@9 +&9 ,MD9 !5 9LD@CG=CB DFC8I79 A5=B@M =FCB ,<9 @5HH9F 5F9 75IG98 6M DFC;9B=HCFG H<5H 9JC@J9 CB 5 H=A9G75@9 5H @95GH

(15) CF89F C: A5;B=HI89 @CB;9F > 108 M95FG ,=BG@9M

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(18) +7<K5FNG7<=@8 AC89@ :CF H<F99 A=BCF5L=G 9@8G. . ?=B9A5H=7G C6G9FJ98 =B 6I@;9 9@8G +=;B=75BH 7CFF9@5H=CB 69HK99B HF5BGJ9FG5@ DFCD9F ACH=CB 5B8 F58=5@ J9@C7=H=9G <5G 699B :CIB8 =B /TG GH5FG K=H< 5 GI889B 7<5B;9 =B H<9 A9H5@@=7=HM G75@9 5FCIB8 89L G99 <5DH9F +9J9F5@ 8MB5A=75@ G9@:7CBG=GH9BH AC89@G <5J9 699B 89J9@CD98 K<=7< HFM HC F9DFC 8I79 H<9 GH9@@5F DFCD9FH=9G =B H<9 %=@?M /5M 5B8 CH<9F ;5@5L=9G ,<9 +7<K5FNG7<=@8 A9H<C8 +7<K5FNG7<=@8

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(21) 5B8 =BJC@J9G 58>IGH=B; H<9 K9=;<HG C: 7C@@97H=CBG C: CF6=HG =B CF89F HC A5H7< H<9 C6G9FJ5H=CBG ,<9 +7<K5FNG7<=@8TG A9H<C8 K5G CF=;=B5@@M =BH9B898 HC F9DFC8I79 H<9 89BG=HM 8=GHF=6IH=CB C: HF=5L=5@ ;5@5L=9G B9J 9FH<9@9GG =HG IG9 K5G @5H9F CB 9LH9B898 HC CH<9F 89BG=HM 8=GHF=6IH=CBG 9 ; *=7<GHCB9

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(23) *=7<GHCB9 ,F9A5=B9

(24) .5@@IF= 9H 5@ +7<K5FNG7<=@8

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(28) 5P :B9F 9H 5@ +CA9 C: H<9G9 FGH =A D@9A9BH5H=CBG C: H<9 +7<K5FNG7<=@8TG A9H<C8 K9F9 89G=;B98 HC KCF? CB ;5@5L=9G K=H< GD<9F=75@ GMAA9HFM K<=7< <5J9 :CIF =BH9;F5@G C: ACH=CB K9@@ 89B98 5B8 H<9 H<F99 7CADCB9BHG C: $ &9J9FH<9@9GG H<9 5DD@=75H=CBG C: GI7< AC89@G =B F95@ ;5@5L=9G 5F9 @=A=H98 +I6G9EI9BH F9GI@HG G<CK98 H<5H 9J9B =B 5L=GMAA9HF=7 ;5@5L=9G K<9F9 HKC =BH9;F5@G C: ACH=CB 5F9 ?BCKB 5B8 $N 5 H<=F8 =BH9;F5@ C: ACH=CB K5G =ADCFH5BH =B CF89F HC F9DFC8I79 F95@=GH=7 89BG=HM 8=GHF=6IH=CBG .9FC@A9 9H 5@ (5FH=7I @5F@M F9@9J5BH =B H<9 89J9@CDA9BH C: +7<K5FNG7<=@8 AC89@=B; :CF 5L=GMAA9HF=7 ;5@5L =9G <5G 699B H<9 KCF? 6M H<9 +-*'& ;FCID 57CB 9H 5@

(29) ,<9M 5B5@=N98 H<9 ?=B9A5H=7G C: 5 G5AD@9 C: 5B8 + ;5@5L=9G :CF K<=7< H<9M =AD@9A9BH98 5 9LH9B898 +7<K5FNG7<=@8 7C89 K<=7< 5@@CK98 H<9A HC GHI8M H<9 D<5G9GD579 8=GHF=6IH=CB C: H<9 ;5@5L=9G #F5>BCJ=O7 9H 5@ 5B8 H<9 79BHF5@ 6@57? <C@9 K<9B DF9G9BH J5B 89 CG7< ,<9=F =AD@9A9BH5H=CB C: H<9 +7<K5FNG7<=@8 H97<B=EI9 9 ; 5DD9@@5F= 9H 5@ K5G 65G98 CB 5 D5F5A9HF=G5H=CB C: H<9 GIF:579 6F=;<HB9GG 8=GHF=6IH=CB IG=B; 5 ;5IGG=5B @=B95F 7CA6=B5H=CB !B 5@@ H<9 75G9G 89G7F=698 H<9 +7<K5FNG7<=@8 A9H<C8 F9 EI=F9G 5 7CAD@9H9 CF6=H @=6F5FM F9DF9G9BH5H=J9 :FCA H<9 F9GD97H=J9 89BG=HM 8=GHF=6IH=CB =B CF89F HC F957< 7CAD@9H9B9GG GI7< @=6F5FM AIGH =B7@I89 H<9 F9;I@5F 5B8 =FF9;I@5F 7CADCB9BHG C: H<9 D<5G9GD579 5F 6I@;9 AC89@G :CF H<9 %=@?M /5M GI7< 5G H<CG9 6M 2<5C

(30) 6 5P :B9F 5B8 CIFG 7CA6=B9 6CH< ?=B8G C: CF6=HG 5B8 =B 588=H=CB 5 HF=5L=5@ DCH9BH=5@ FCH5H=B; 5H 5 65F D5HH9FB GD998 Ωb (F9J=CIG 8MB5A=75@ AC89@G C: H<9 5@57H=7 6I@;9 K9F9 65G98 CB D<CHCA9HFM 5B8 F5 8=5@ J9@C7=HM 85H5 5@CB9 !B H<=G D5D9F K9 DF9G9BH 5B 9::CFH HC 7CBGHFI7H 5 +7<K5FNG7<=@8 AC89@ :CF H<9 %=@?M /5M K<=7< =B7CFDCF5H9G DFCD9F ACH=CB 85H5 5G K9@@ /9 9LD97H GI7< 5 AC89@ HC 69 ACF9 FC6IGH@M 7CBGHF5=B98 6M H<9 85H5 5G =H =G 65G98 CB 5 <=;<9F 8=A9BG=CB5@ 85H5 G9H CF H<=G KCF? K9 IG9 CIF B9K A95GIF9A9BHG C: ∼ 45000 GH9@@5F DFCD9F ACH=CBG 5B8 D<CHCA9HFM :CF H<F99 9@8G 5@CB; H<9 ;5@57H=7 A=BCF 5L=G ,<9 GHFI7HIF9 C: H<=G D5D9F =G 5G :C@@CKG !B G97H=CB K9 9LD@5=B H<9 AC89@=B; H97<B=EI9 65G98 CB H<9 +7<K5FNG7<=@8 A9H<C8 +97H=CB 9LD@5=BG H<9 8=::9F9BH 89DFC +=B79 H<9 HKC6C8M F9@5L5H=CB H=A9 =G trelax ∼ N tcross /(8ln(N )) K<9F9 & =G H<9 BIA69F C: GH5FG 5B8 tcross H<9 7FCGG=B; H=A9 K9 75B G5:9@M 5GGIA9 H<5H CIF GMGH9A =G GH56@9 :CF 5 @CB; H=A9 7CAD5F98 HC H<9 H=A9 :CF K<=7< CIF AC89@ =G 5GGIA98 HC 69 7C@@=G=CB@9GG ,<9F9:CF9 H<9 AC89@ C: H<9 ;5@57H=7 65F 6I@;9 5GGIA9G H<5H CF6=HG 5F9 7CAD@9H9@M 897CID@98 :FCA 957< CH<9F 5B8 H<IG H<9M 75B 69 7CBG=89F98 5G 6I=@8 =B; 6@C7?G C: H<9 8=GHF=6IH=CB C: C6G9FJ56@9G

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(32) . =A9BG=CB5@ MB5A=7G C: H<9 5@57H=7 I@;9. >97H=CB H97<B=EI9G 5DD@=98 HC H<9 ' !* =A5;9G =B CF89F HC C6H5=B 5 6I@;9 65F 89BG=HM 8=GHF=6IH=CB +97H=CB 89H5=@G H<9 GHF5H9;M IG98 H<9 =BDIH =B:CFA5H=CB DFC J=898 HC H<9 AC89@ 5B8 DF9@=A=B5FM 75@7I@5H=CBG HC =ADFCJ9 H<9 CJ9F5@@ FC6IGHB9GG +97H=CB G<CKG H<9 F9GI@HG :CF 5 G9H C: AC89@G 5B8 GCA9 J5F=5H=CBG HC H<9 HH=B; DFC79 8IF9 +97H=CB GIAA5F=N9G CIF 7CB7@IG=CBG 5B8 89G7F=69G DCGG=6@9 :IFH<9F KCF?. 4.2.1. % '$!& , &!)- Formulation. !B H<=G G97H=CB K9 9LD@5=B H<9 :CFAI@5H=CB 5B8 6I=@8=B; C: 5 GH958MGH5H9 AC89@ K<=7< HG 5 G9H C: C6G9FJ56@9G 6M A95BG C: H<9 +7<K5FNG7<=@8 H97<B=EI9 MB5A=75@ GMGH9AG G5H=G:M H<9 7C@@=G=CB@9GG C@HNA5BB 9EI5H=CB K<=7< A95BG H<5H H<9M 75B 69 89G7F=698 6M 5 8=GHF=6IH=CB :IB7H=CB H<5H 89D9B8G CB G=L D<5G9GD579 7CCF 8=B5H9G H<F99 DCG=H=CB 5B8 H<F99 J9@C7=HM 7CADCB9BHG ,<=G =G 5 8=F97H 7CBG9EI9B79 C: H<9 @CB; 6C8M F9@5L5H=CB H=A9 =B GI7< GMGH9AG ,<9 C@HNA5BB 9EI5H=CB 89G7F=69G H<9 ACH=CB C: GH5FG =B 5 ;F5J=H5H=CB5@ DCH9BH=5@ Φ H H<9 G5A9 H=A9 K9 F9EI=F9 H<5H H<=G DC H9BH=5@ 69 ;9B9F5H98 6M H<9 GH5FG H<9AG9@J9G GC K9 AIGH 5@GC G5H=G:M (C=GGCBTG 9EI5H=CB =B CF89F HC <5J9 5 G9@:7CBG=GH9BH GC@IH=CB IFH<9FACF9 H<=G 8=GHF=6IH=CB :IB7H=CB 75B 69 KF=HH9B 5G 5 :IB7H=CB C: 5H ACGH H<F99 isolating integrals 5 F9GI@H ?BCKB 5G H<9 GHFCB; "95BG H<9CF9A 57< G9H C: J5@I9G C: H<9G9 =GC@5H=B; =BH9;F5@G 7CFF9GDCB8G HC 5 8=::9F9BH CF6=H ,<9 =895 C: IG=B; CF6=HG F9;I@5F 5B8 =FF9;I@5F 5G 6I=@8=B; 6@C7?G C: F95@ ;5@5L=9G =G IG98 9LD@=7=H@M =B H<9 +7<K5FNG7<=@8 H97<B=EI9 +7<K5FNG7<=@8

(33) 9F9 K9 K=G< HC AC89@ H<9 65F C: H<9 5@5LM 5G 5 D5HH9FBFCH5H=B; HF=5L=5@ C6>97H ,<9 DFC6@9A =G H<5H H<9G9 H<F99 =BH9;F5@G 75BBCH 69 KF=HH9B 8CKB 5B5@MH=75@@M =B :57H GCA9 CF6=HG 5F9 7<5CH=7 5B8 CB@M C69M CB9 CF HKC GI7< =BH9;F5@G /=H< GCA9 BCH56@9 9L79DH=CBG 9 ; GH5H=7 +HP57?9@ DCH9BH=5@G =BB9M ,F9A5=B9

(34) D

(35) HF=5L=5@ CF D5HH9FBFCH5H=B; GMGH9AG HMD=75@@M CB@M <5J9 CB9 5B5@MH=7 =BH9;F5@ EJ H<9 =BH9;F5@ C: "57C6= K<=7< =G 1 EJ = E − Ω2bar R2 , 2.

(36) . K=H< Ωbar H<9 65F D5HH9FB GD998 5B8 R H<9 F58=IG =B H<9 D@5B9 C: H<9 65F FCH5H=CB BIA9F=75@ 5DDFC57< HC H<9 DFC6@9A =G H<IG DF57H=75@@M IB5JC=856@9 !B 9GG9B79 H<9 DFC6@9A 75B 69 :CFAI@5H98 5G :C@@CKG =J9B 5 DCH9BH=5@ K9 HFM HC B8 5 G9H C: CF6=HG K<=7< F9DFC8I79 5 ;=J9B G9H C: C6G9FJ56@9G oi K<9F9 i = 1, ..., nobs CF 5 6FC58 7@5GG C: C6G9FJ56@9G H<=G 75B 69 KF=HH9B 5G :C@@CKG oi =. . ξi,j wj ,. . j=1,N. K<9F9 & =G H<9 BIA69F C: CF6=HG 5B8 ξi,j =G C6G9FJ56@9 i 8I9 HC CF6=H j 5B8 wj =G H<9 K9=;<H C: H<5H D5FH=7I@5F CF6=H (CGG=6@9 C6G9FJ56@9G H<5H 75B 69 F9DFC8I798 =B H<=G K5M 5F9 H<F998=A9BG=CB5@ CF DFC>97H98 89BG=H=9G @IA=BCG=HMK9=;<H98 A95B J9@C7=H=9G 9H7.

(37) +7<K5FNG7<=@8 AC89@ :CF H<F99 A=BCF5L=G 9@8G. . /9 75B F9KF=H9 9E :CF 5 G9H C: CF6=HG 5G . . (oi −. i=1,nobs. ξi,j wj )2 = 0.. . j=1,N. CA6=B98 K=H< H<9 D<MG=75@ 7CBGHF5=BH H<5H 5@@ CF6=H K9=;<HG AIGH 69 BCBB9;5H=J9 wj ≥ 0. ∀j. . H<=G =G H<9 7@5GG=75@ :CFAI@5H=CB C: H<9 +7<K5FNG7<=@8 A9H<C8 K<=7< 75B 69 J=9K98 5G H<9 G95F7< :CF 5 :95G=6@9 G9H =B @=B95F DFC;F5AA=B; 5 DFC6@9A H<5H IGI5@@M <5G 9=H<9F 5 K=89 F5B;9 C: GC@IH=CBG CF BCB9 M A=B=A=N=B; 5 7CGH :IB7H=CB CJ9F H<9 :95G=6@9 GC @IH=CBG H<9 A9H<C8 75B 69 IG98 HC 7CBGHFI7H :CF 9L5AD@9 A5L=A5@@M FCH5H=B; ;5@5L=9G 9 ; +H5H@9F

(38) CF H<9 DF9G9BH GHI8M K9 <5J9 A95GIF9A9BHG C: =B8=J=8I5@ GH9@@5F DFCD9F ACH=CBG =B 5 BIA69F C: 6I@;9 9@8G 5B8 CIF :C7IG =G CB F9DFC8I7=B; H<=G C6G9FJ98 8=GHF=6IH=CB K=H< 5 8MB5A=75@ AC89@ !B D5FH=7I@5F CIF AC89@ HF=9G HC F9DFC8I79 H<9 :I@@ 89BG=HM 8=GHF=6IH=CB C: H<9 6I@;9 65F ρ(x, y, z) 5B8 H<9 8=GHF=6IH=CB C: HKC DFCD9F ACH=CBG μl μb 5B8 5 D<CHCA9HF=7 D5F5@@5L M∗ =B CIF C6G9FJ98 9@8G ,<9 89H5=@G C: H<9G9 85H5 K<=7< 5F9 D5FH C: H<9 =BDIH =B:CFA5H=CB C: H<9 AC89@ K=@@ 69 9LD@5=B98 =B 89H5=@ =B § ,<9 AC89@ DF9G9BH98 =B H<=G D5D9F HG :CIF ;F=8G G=AI@H5B9CIG@M ,<9 FGH ;F=8 =G H<9 =BDIH 89BG=HM 8=GHF=6IH=CB C6H5=B98 :FCA 5 89DFC>97H=CB C: H<9 ' !* =A5;9 5B8 H<9 F9A5=B=B; H<F99 7CFF9GDCB8 HC H<9 8=GHF=6IH=CB =B DFCD9F ACH=CB 5B8 D<CHCA9HF=7 D5F5@@5L =B 957< C: CIF 9@8G 57< ;F=8 7CBG=GHG C: 32 × 32 × 32 = 32768 79@@G K<=7< ;=J9G 5 F95GCB56@9 89B=H=CB C: H<9 65F GHFI7HIF9 CBG9EI9BH@M K9 75B F989B9 9E =B 5 ACF9 F95@=GH=7 G<5D9 5G χ2 =. . . f =1,4 i=1,32768. (of,i −. . ξf,i,j wj )2 ,. . j=1,N. K<9F9 f =G <9F9 H<9 BIA69F C: H<9 ;F=8 69=B; HH98 ,<=G :CFAI@5H=CB C: H<9 DFC6 @9A 5G 5 @95GHGEI5F9G A=B=A=N5H=CB K=H< H<9 BCBB9;5H=J=HM 7CBGHF5=BH CB H<9 K9=;<HG =G ?BCKB 5G quadratic programming 5B8 75B 69 GC@J98 IG=B; 5 J5F=5H=CB C: H<9 G=AD@9L 5@;CF=H<A 89J9@CD98 :CF @=B95F DFC;F5AA=B; !H K5G =BHFC8I798 =BHC ;5@57H=7 8MB5A =7G 6M 9>CB;<9

(39) !B 7CBHF5GH HC @=B95F DFC;F5AA=B; H<=G :CFAI@5H=CB C: H<9 DFC6@9A K=@@ ;9B9F5@@M ;=J9 5 G=B;@9 CDH=A5@ G9H C: CF6=H K9=;<HG. . (*'",!& , -$ &+!,1 !+,*!-,!'& *'% , ' %( '& !&. ,<9 B95F!* =A5;9G 6M H<9 '!* 9LD9F=A9BH <5J9 5@F958M 699B H<9 GCIF79 C: G9J9F5@ 89DFC>97H=CB AC89@G 6M 8=::9F9BH A9H<C8G K9? 9H 5@

(40) K5G H<9 FGH HC DFC8I79 5 89BG=HM 8=GHF=6IH=CB :FCA H<9G9 =A5;9G 6M IG=B; G9J9F5@ 5B5@MH=75@ HF=5L=5@ AC89@G HC 9LD@CF9 7<5F57H9F=GH=7G C: H<9 65F %CF9 F979BH@M GHI8=9G 5G H<CG9 D9F :CFA98 6M =GG5BHN 9F<5F8 IG=B; 5 ACF9 9@56CF5H98 BCBD5F5A9HF=7 A5L= AIA D9B5@=N98 5@;CF=H<A <5J9 :CIB8 5 65F K=H< A5GG 5B8 G=N9 G@=;<H@M 8=::9F9BH HC.

(41) . =A9BG=CB5@ MB5A=7G C: H<9 5@57H=7 I@;9. z x. y. b. Sun Figure 4.1 =;<H :C@8 GMAA9HFM 5GGIA98 :CF CIF +7<K5FNG7<=@8 AC89@ 'F6=HG <5J9 699B 75@7I@5H98 =B H<9 65F :F5A9 K<=7< =G =B 5B 5B;@9 φb K=H< H<9 @=B9 7CBB97H=B; H<9 GIB K=H< H<9 ;5@57H=7 79BH9F. H<CG9 DF9J=CIG@M 5GGIA98 ,<CG9 89H9FA=B5H=CBG <5J9 699B ACF9 F979BH@M 7<5B;98 6M 9B>5A=B 9H 5@ TG KCF? K<=7< GI;;9GHG 5B 9J9B @5F;9F 65F 577CF8=B; HC H<9=F GH5F 7CIBHG :FCA +D=HN9F 85H5 5G K9 <5J9 5@F958M A9BH=CB98 =B §

(42) ,<IG H<9 DFC6@9A C: H<9 F95@ 8=A9BG=CBG C: H<9 65F =G GH=@@ 5B CD9B 8=G7IGG=CB 'IF +7<K5FNG7<=@8 AC89@ 5G 9LD@5=B98 69:CF9 HG :CIF ;F=8G G=AI@H5B9CIG@M K<9F9 H<9 FGH CB9 5B8 H<9 CB9 K<=7< <5G H<9 <=;<9GH K9=;<H 5GG=;B98 7CFF9GDCB8G HC H<9 JC@IA9 89BG=HM 8=GHF=6IH=CB C: H<9 6I@;9 !B CF89F HC 57EI=F9 5 F9DF9G9BH5H=J9 89BG=HM 8=GHF=6IH=CB :CF H<9 ;5@57H=7 6I@;9 K9 <5J9 D9F:CFA98 CIF CKB 89DFC>97H=CB :FCA H<9 '!* =A5;9G. 4.3.1. Deprojection by analytical bars. 'IF 89DFC>97H=CBG 6M 5B5@MH=75@ 65FG 7CBG=GH98 =B 957< 75G9 =B H<F99 7CADCB9BHG 5 65F 5 8=G? 5B8 5 79BHF5@ 7IGD ,KC 8=G?G <5J9 699B HF=98 5 8CI6@9 9LDCB9BH=5@ 5B8 5 %=M5ACHC&5;5= %& 8=G? %=M5ACHC 5B8 &5;5=

(43) ,<9 %& 8=G? <5G H<9 58J5B H5;9 H<5H =HG DCH9BH=5@ 75B 69 75@7I@5H98 5B5@MH=75@@M ,<IG 8=G? 5B8 7IGD 5F9 ;=J9B 6M H<9 :C@@CK=B; :CFA F9GD97H=J9@M . . e−|z|/z0 e−|z|/z1 ρd (R, z) = +α Rd e−R/Rd z0 z1 ρc (x, y, z) = η rc−1.85 exp(−rc ). .

(44) +7<K5FNG7<=@8 AC89@ :CF H<F99 A=BCF5L=G 9@8G. . /<9F9 Rd = D7 z0

(45) D7 z1 D7 5B8 α 5F9 H<9 8=G? 7CBGH5BHG !B H<9 75G9 C: H<9 7IGD K<=7< 7CFF9GDCB8G HC 5B C6@5H9 GD<9FC=85@ BI7@9IG K=H< 5 GH99D DCK9F @5K 5B8 5B 9LDCB9BH=5@ CIH9F DFC@9 2<5C

(46) 6 rc 7CFF9GDCB8G HC . rc =. q(R02 + x2 + y 2) + z 2 c2. . . /<9F9 q K5G G9H HC 0.6 H H<9 G5A9 H=A9 H<9 8=G? 75B 69 KF=HH9B 5G . ρM N (R, z) =. κ2 β 4π. . ×. √ √ λR2 + (λ + 3 z 2 + κ2 )(λ + z 2 + κ2 )2 . 5/2 √ R2 + (λ + z 2 + κ2 )2 (z 2 + κ2 )3/2. . . /<9F9 a b 5B8 κ 5F9 7CBGH5BHG K<=7< 7CBHFC@ H<9 G<5D9 C: H<9 8=G? !B H<9 75G9 C: H<9 65F K9 <5J9 7<CG9B HC HFM G9J9F5@ 5B5@MH=75@ AC89@G HC 89DFC>97H H<9 GH9@@5F 89BG=HM *5HH9B6IFM 9H 5@ /9 <5J9 =B=H=5@@M IG98 ρG1 = ρb0 exp(−rG1 /2) ρG2 = ρb0 exp(−rG2 /2)..

(47)

(48)

(49) . /<9F9 rG1 5B8 rG2 5F9 rG1 (x, y, z) =. 2. x a. ⎡. rG2 (x, y, z) = ⎣. y + b. 2. x a. 2. +. 2 1/m z. 2 2. y + b. c. ⎤ 4 1/m z ⎦ +. c.

(50)

(51) . ,<9 7CCF8=B5H9 GMGH9A <5G =HG CF=;=B =B H<9 79BH9F C: H<9 ;5@5LM K=H< H<9 xy D@5B9 H<9 A=8D@5B9 C: H<9 ;5@5LM 5B8 H<9 z 5L=G D5F5@@9@ HC H<9 8=F97H=CB C: H<9 ;5@57H=7 DC@9G ,<9 x 8=F97H=CB 89B9G H<9 G9A=A5>CF 5L=G C: H<9 65F !B 588=H=CB CIF 65F :IB7H=CB =G FCH5H98 =B 957< 89DFC>97H=CB =B 5B 5B;@9 φb HC A=A=7 H<9 C6G9FJ98 65F =B7@=B5H=CB ,<9 7CBGH5BHG 5 6 7 5F9 H<9 G75@9 @9B;H< C: LM 5B8 N F9GD97H=J9@M 'IF GHF5H9;M 7CBG=GH98 =B H<9 A=B=A=N5H=CB C: 5 χ2 K<=7< 7CAD5F9G H<9 89DFC>97H=CB K=H< H<9 ' !* =A5;9 ,<=G A=B=A=N5H=CB ;C9G :FCA ID HC

(52) :F99 D5F5A9H9FG 89D9B8=B; C: H<9 7CA6=B5H=CB C: 65F 7IGD 5B8 8=G? DCK9F:I@ A=B=A=N5H=CB 5@;C F=H<A K5G IG98 =B CF89F HC C6H5=B 5 F9@=56@9 G9H C: J5@I9G %!&-!,

(53) ,56@9

(54) GIAA5F=N9G H<9 :F99 D5F5A9H9FG =B 957< 9EI5H=CB 5B8 ;IF9 G<CKG GCA9 C: H<9 8=::9F9BH 89DFC>97H=CBG DFC8I798 +9J9F5@ 65F HMD9G K9F9 DFC698 8IF=B; CIF DF9@=A=B5FM AC89@G =B H<9 B5@ GH5;9 K9 7CBGHF5=B98 CIF 5B5@MG=G HC CB9 D5FH=7I@5F

(55). 5J5=@56@9 5H <HHD. G95@ K96 79FB 7< G95@ KCF?D57?5;9G A5H<@=6G A=BI=H <CA9 <HA@.

(56) . =A9BG=CB5@ MB5A=7G C: H<9 5@57H=7 I@;9. Figure 4.2 +CA9 C: H<9 89DFC>97H=CB D9F:CFA98 :CF H<9 '!* =A5;9G Top left 7CBHCIF =A5;9 C: H<9 '!* =A5;9 5H μm GC@=8 GID9F=ADCG98 CJ9F =HG F9GD97H=J9 89DFC>97 H=CB 85G<98 @=B9G IG=B; 5 AC89@ K=H< 5 79BHF5@ 7IGD 5 8CI6@9 9LDCB9BH=5@ 8=G? 5B8 5

(57) 65F ,<9 89DFC>97H=CB K5G D9F:CFA98 5GGIA=B; 5 φ b = 20◦ Top right, 5 J=9K :FCA H<9 ;5@57H=7 BCFH< DC@9 C: H<9 65F 89DFC>97H98 =B H<9 @9:H D5B9@ Middle row, G5A9 5G HCD FCK =B7@I8=B; 5 7IGD 5 %=M5ACHC&5;5= 8=G? 5B8 5

(58) 65F 89DFC>97H98 5H φb = 20◦ Bottom row, +5A9 CF89F 5G IDD9F FCKG H<9 AC89@ =B7@I89G 5 %=M5ACHC&5;5= 8=G? 5 65F 5B8 5 7IGD ,<9 D5F5A9H9FG C: H<9 65FG G<CK98 =B H<9 A=88@9 5B8 6CHHCA FCK 75B 69 G99B =B ,56@9 .

(59) +7<K5FNG7<=@8 AC89@ :CF H<F99 A=BCF5L=G 9@8G.

(60). Table 4.1 9DFC>97H=CB :F99 D5F5A9H9FG. CADCB9BH F99 D5F5A9H9FG 5F

(61) ρb0 , a, b, c, m 5F ρb0 , a, b, c, m =G?CI6@9 9LDCB9BH=5@ γ =G?%& β, λ, κ IGD c, η. AC89@ 8I9 HC =HG GID9F=CF D9F:CFA5B79 8IF=B; H<9 89DFC>97H=CB GH5;9 ,<9G9 89DFC>97 H=CBG =B7@I898 G9J9F5@ C: H<9 ACGH 7CAACB@M IG98 89BG=HM DFC@9G :CF H<9 ;5@57H=7 65F 5B8 8=G? 9 ; *5HH9B6IFM 9H 5@ K<9F9 957< 9EI5H=CB <5G 699B G@=;<H@M AC8= 98 HC =B7@I89 GCA9 :F99 D5F5A9H9FG AC89@G

(62) 5B8 F9GD97H=J9@M @95F 8=::9F 9B79G <5J9 699B :CIB8 =B H<9 F9GI@HG C: H<9 89DFC>97H=CBG 577CF8=B; HC H<9 7CADCB9BHG 7<CG9B 957< H=A9 CI6@9 9LDCB9BH=5@ 8=G? HMD=75@@M M=9@8 <=;<9F χ2 7CBJ9FG9@M

(63) 5B8 65FG <5J9 DFC8I798 G=A=@5F 65FG =B G=N9 5B8 JC@IA9 89BG=HM DFC@9 K=H< 5 G@=;<H@M GA5@@9F χ2 :CF H<9 :CFA9F 5G H56@9 5B8 ;IF9 G<CKG @GC H<9 65FG =B CIF 89DFC>97H=CBG H9B8 HC <5J9 GA5@@9F 5L=G F5H=C H<5B GCA9 :CIB8 69:CF9 9 ; =BB9M 9F<5F8 +D9F;9@

(64) B9J9FH<9@9GG GA5@@ 65FG <5J9 699B 5@GC :CIB8 F979BH@M 6M *5HH9B6IFM 9H 5@ IG=B; '$!! F98 7@IAD ;=5BHG GH5FG 88=H=CB5@@M CIF 65F 7CAD9BG5H9G =HG F9@5H=J9 GA5@@9F 5L=G F5H=C 6M <5J=B; GA5@@9F J5@I9G C: H<9 9LDCB9BH m =B 9E

(65) 5B8 9E

(66) ,<9G9 J5@I9G A5?9 H<9 65F 89BG=HM DFC@9 8975M ACF9 G@CK@M 9DFC>97H=CBG K=H<=B H<9 F5B;9 C: φbar ∼ 20◦ − 40◦ <5J9 G<CKB 5 F95GCB56@9 7CBG=G H9B7M 'IF 65FG G<CK 5 F5H=C C: G9A=A5>CF 5B8 G9A=A=BCF 65F 5L=G G75@9 @9B;H<G =B H<9 ;5@57H=7 D@5B9 a, b 5B8 J9FH=75@ 65F G75@9 @9B;H< c K<=7< =G a : b : c = 10 : 3 : 3 ,<IG 8I9 HC H<9 7CBG=GH9B7M C: H<9 89DFC>97H=CBG K9 <5J9 7<CG9B CB9 C: CIF 65FG HC ;9B9F5H9 5 7CAD@9H9 ;F=8 C: AC89@G ,<=G 65F K<=7< =B7@I89G H<9 C6@5H9 79BHF5@ 7IGD H<9 %=M5ACHC &5;5= 8=G? 5B8 H<9

(67) 65F K=@@ 69 F9:9FF98 5G

(68) AC89@ <9B79:CFH<. 4.3.2. Potential formulation. 5J=B; HH98 H<9 89BG=HM 8=GHF=6IH=CB C: H<9 6I@;9 65F B9LH K9 F9EI=F9 5 DCH9BH=5@ K<=7< =G 7CBG=GH9BH K=H< GI7< 89BG=HM =B CF89F HC =BH9;F5H9 7CBG=GH9BH CF6=HG /9 <5J9 8CB9 H<=G 6M DFC8I7=B; 5 AI@H=DC@5F 9LD5BG=CB C: H<9 89BG=HM B;@A5=9F 9F<5F8

(69) 77CF8=B;@M K9 75B KF=H9 CIF 8=A9BG=CB5@ DCH9BH=5@G 5G Φ(r, θ, ϕ) = −Gρ0. . Φlm (r) Plm (cos(θ)) cos(mϕ).

(70) . l,m. /9 CB@M =B7@I89 H9FAG K=H< 9J9B l 5B8 m HC 9B:CF79 F997H=CB GMAA9HFM 'IF DCH9B H=5@G 5F9 89F=J98 :FCA D<CHCA9HF=7 AC89@G :CF H<9 5@5LM 5B8 GC =BJC@J9 5B IB?BCKB A5GGHC@=;<H F5H=C K<=7< 75B 69 58>IGH98 H<FCI;< H<9 D5F5A9H9F ρ0 /9 BCFA5@=N9 =H 6M :CF7=B; H<9 7=F7I@5F GD998 C: H<9 5@5LM HC A5H7< H<9 C6G9FJ98 200 km s−1 5H 8 kpc #I=.

(71) . =A9BG=CB5@ MB5A=7G C: H<9 5@57H=7 I@;9. >?9B ,F9A5=B9

(72) H<IG H<9 A5GG C: CIF 6I@;9 AC89@ 7CFF9GDCB8G HC ∼ ×1010 % :CF 5 ρ0 = % /I2.2μm 'IF 897CADCG=H=CB C: H<9 DCH9BH=5@ =B AI@H=DC@9 7CADCB9BHG :C@@CKG H<9 DFC798IF9 IG98 6M B;@A5=9F 9F<5F8

(73) :CF H<9=F 8=A9BG=CB5@ <M8FC8MB5A=75@ AC89@ C: H<9 %=@?M /5M !B CF89F HC C6H5=B H<9 AI@H=DC@9 7CADCB9BHG C: H<9 DCH9BH=5@ K9 GH5FH 75@7I@5H=B; H<9 AI@H=DC@9 7CADCB9BHG C: H<9 89BG=HM 8=GHF=6IH=CB ρlm H<9G9 5F9 ρlm (r) =. π 0. dθ sin(θ) Plm (cos(θ)). 2π 0. dϕ cos(mϕ) g(r).

(74) . /<9F9 g =G H<9 89DFC>97H98 89BG=HM 8=GHF=6IH=CB C: H<9 ;5@57H=7 6I@;9 C6H5=B98 69 :CF9 K<=7< =G F9@5H98 K=H< H<9 89BG=HM 6M ρ(r, θ, ϕ) = ρ0 g(r, θ, ϕ),.

(75) . !BH9;F5H=CBG =B 9E

(76) K9F9 D9F:CFA98 K=H< H<9 *CA69F; A9H<C8 :CF AI@H=DC@9 7CADCB9BHG G5H=G:M=B; 9=;<H:C@8 GMAA9HFM 5G A9BH=CB98 69:CF9 H<=G K5G 8CB9 HMD= 75@@M IBH=@ l = 6 K<=7< A95BG

(77) AI@H=DC@9 H9FAG ,<9 ρlm (r) H56@9G K9F9 75@7I@5H98 :CF @C;5F=H<A=75@@M 9EI=8=GH5BH F58== 69HK99B

(78) D7 5B8

(79) ?D7 -G=B; H<CG9 ρlm (r) H56@9G HKC =BH9;F5@G K<=7< K=@@ 69 IG98 @5H9F K9F9 75@7I@5H98 Iint (r) = Iext (r) =. r 0. ρlm (s) sl+2 ds,. ∞ r. ρlm (s) s−(l−1) ds.

(80)

(81) . FA98 K=H< CIF 5IL=@=5FM =BH9;F5@G K9 75B H<9B 75@7I@5H9 H<9 DCH9BH=5@ AI@H=DC@9G Φlm Φlm (r) = (2 − δm0 ). (l − m)! −(l+1) [r Iint (r) + r l Iext (r)], (l + m)!.

(82) . 5B8 H<9=F FGH 89F=J5H=J9G Φlm (r) = (2 − δm0 ). (l − m)! [−(l + 1)r −(l+2) Iint (r) + l r l−1 Iext (r)]. (l + m)!. . /<9F9 H<9 D5FH=5@ 89F=J5H=J9G C: 9E

(83) 5F9 dΦ = −Gρ0 Φlm (r) Plm (cos(θ)) cos(mϕ), dr l,m.

(84) . dΦ R = Gρ0 Φlm (r) Plm (cos(θ)) cos(mϕ), dθ r l,m. . dΦ = Gρ0 mΦlm (r) Plm (cos(θ)) sin(mϕ), dϕ l,m. . 5B8 K<=7< 75B 69 IG98 =B HIFB HC 75@7I@5H9 H<9 5FH9G=5B 5779@9F5H=CBG H<FCI;< a = −∇Φ ax = −. dΦ dΦ dΦ (x/r) + (y/R2 ) − [xz/(r 2 R)], dr dϕ dθ. .

(85) +7<K5FNG7<=@8 AC89@ :CF H<F99 A=BCF5L=G 9@8G. . Figure 4.3 Top, F9@5H=J9 7CB HF=6IH=CB C: AI@H=DC@9 7CA DCB9BHG Φlm =B DCH9BH=5@

(86)

(87) AI@H=DC@9 7CADCB9BHG <5J9 699B IG98 HC 897CADCG9 H<9 CF=;=B5@ DCH9BH=5@ Bottom, F9@ 5H=J9 7CBHF=6IH=CB C: 89F=J5 H=J9G C: H<9 AI@H=DC@9 9LD5B G=CB dΦlm /dr C: H<9 DCH9BH=5@ 89F=J5H=J9G C: AI@H=DC@9 9LD5B G=CB 5F9 B979GG5FM HC H<9 75@7I @5H=CB C: :CF79G =B FU . dΦ dΦ dΦ (y/r) − (x/R2 ) − [yz/(r 2 R)], dr dϕ dθ dΦ dΦ az = − (z/r) + (R/r 2 ). dr dθ ,<IG CIF :CF79G =B 957< 75G9 K=@@ 69 DFC8I798 :FCA 5 IB=EI9 H56@9 7CBH5=B=B; H<9 Φlm 5B8 Φlm ay = −. 4.3.3. Orbit library. /9 <5J9 5@F958M A9BH=CB98 H<5H CF6=HG 75B 69 7CBG=89F98 5G 6I=@8=B; 6@C7?G C: H<9 ;5@5LM &9J9FH<9@9GG H<9G9 6I=@8=B; 6@C7?G 75B 7CA9 =B HKC 5JCFG F9;I@5F 5B8 =FF9; I@5F *9;I@5F CF6=HG F9GD97H H<F99 =GC@5H=B; =BH9;F5@G C: ACH=CB 5B8 H<9F9:CF9 <5J9 RK9@@.

(88) . =A9BG=CB5@ MB5A=7G C: H<9 5@57H=7 I@;9. 69<5J98S HF5>97HCF=9G 5B8 5F9 F9@5H=J9@M G=AD@9 HC 7CADIH9 K<9F95G =FF9;I@5F CF6=HG ACGH@M F9GD97H CB@M EJ 5G 5B =BH9;F5@ 5B8 H<9F9:CF9 K5B89F 5FCIB8 =B 5 8=A9BG=CB5@ D5FH C: 957< 7CBGH5BHEJ G@=79 C: D<5G9 GD579 5JC=8=B; H<9 F9;I@5F CF6=HG 'B9 K5M HC =B7@I89 =FF9;I@5F CF6=HG =B H<9 AC89@G =G 6M IG=B; 5 GD97=5@ ?=B8 C: CF6=HG 7F95H98 HC A=A=7 H<9 DF9G9B79 C: H<9G9 =FF9;I@5F 7CADCB9BHG C: H<9 D<5G9GD579 7CA DCG=H9 CF6=HG CADCG=H9 CF R7C@@97H=J9S CF6=HG %9FF=H F=8A5B

(89) 2<5C

(90) 6 5F9 BCH CF6=HG =B H<9 IGI5@ G9BG9 6IH F5H<9F 7C@@97H=CBG C: A5BM CF6=HG K=H< H<9 G5A9 =BH9;F5@ C: ACH=CB ,<9F9:CF9 =B H<9 75G9 C: HF=5L=5@ AC89@G GI7< 5G 65FG 5 7CADCG=H9 CF6=H K=@@ =B7@I89 5@@ CF6=HG K=H< H<9 G5A9 9B9F;M EJ 5B8 K=@@ 69 =B89D9B89BH C: CH<9F =BH9;F5@G C: ACH=CB 9B79 5 AC89@ =B7@I8=B; F9;I@5F 5B8 7CADCG=H9 CF6=HG K=@@ <5J9 5 8=GHF=6IH=CB :IB7H=CB K<=7< =G H<9 GIA C: H<9 8=GHF=6IH=CB :IB7H=CBG C: F9;I@5F CF 6=HG K=H< H<F99 =BH9;F5@ C: ACH=CB 5B8 7CADCG=H9 CB9 =BH9;F5@ C: ACH=CB 7CADCB9BHG 9LD@=7=H@M ,<5H =G f = f1 + f3 = f (EJ ) + f (I). /<9F9 EJ 5B8 I F9DF9G9BH H<9 CF6=HG K=H< CB9 5B8 H<F99 =BH9;F5@G C: ACH=CB F9GD97 H=J9@M +=B79 H<9 8=GHF=6IH=CB :IB7H=CB =G H=A9=B89D9B89BH H<9 G5A9 AIGH 5DD@M :CF 6CH< 7CADCB9BHG 7CADCG=H9 5B8 F9;I@5F ,<IG K9 <5J9 H<5H 6CH< 7CADCB9BHG G5H=G:M 7C@@=G=CB@9GG C@HNA5BB 9EI5H=CB 5B8 5F9 =B89D9B89BHG C: H=A9 ,<9F9:CF9 6CH< 75B 69 7CBG=89F98 5G J5@=8 =B89D9B89BH 6I=@8=B; 6@C7?G C: H<9 8=GHF=6IH=CB :IB7H=CB &9J9FH<9@9GG H<9F9 =G 5 F=G? 5GGC7=5H98 K=H< =B7@I8=B; 7CADCG=H9 CF6=HG =B 5 @=6F5FM +=B79 7CADCG=H9 CF6=HG 5F9 9BH=H=9G K<=7< F9DF9G9BH A5BM CF6=HG G=AI@H5B9CIG@M H<9M 75B 95G=@M CJ9FDCDI@5H9 CIF D<5G9GD579 6M G9@97H=B; A5BM HMD9G C: CF6=HG K<9B 5G G=;B=B; 5 D5FH=7I@5F K9=;<H HC >IGH CB9 7CADCG=H9 CF6=H I9 HC H<9 :57H H<5H 7CADCG=H9 CF6=HG >IGH 89D9B8 CB CB9 =BH9;F5@ C: ACH=CB EJ 5 G=B;@9 7CADCG=H9 CF6=H KCI@8 @@ 7CAD@9H9@M 5 DCFH=CB C: H<9 D<5G9GD579 =B7@I8=B; 5@@ F9;I@5F CF6=HG 5H H<5H 9B9F;M ,C 7CAD9BG5H9 H<=G KCI@8 F9EI=F9 GCA9 F9;I@5F CF6=HG HC <5J9 B9;5H=J9 K9=;<HG /<=@9 H5?=B; 75F9 HC C6H5=B 5 D<5G9GD579 89BG=HM DCG=H=J9 9J9FMK<9F9 /9 <5J9 897=898 HC 5JC=8 H<=G 7CAD@=75H=CB 6M G=AD@M 6I=@8=B; CF6=H @=6F5F=9G H<5H =B7@I89 5 7CBG=89F56@9 BIA69F C: CF6=HG F9;I@5F 5B8 =FF9;I@5F 'IF AC89@G 7CBH5=B 69HK99B ∼ 700 5B8 ∼ 4000 CF6=HG 89D9B8=B; CB D5FH=7I@5F 7CB8=H=CBG C: H<9 65F AC89@98 Orbit families 5A=@=9G C: F9;I@5F CF6=HG =B 5 65F DCH9BH=5@ 5F9 K9@@ 9GH56@=G<98 ,<9G9 CF6=H :5A= @=9G 5F9 5@@ F9@5H98 HC F9GCB5B79G 5B8 75B 69 9J9B 7@5GG=98 H<FCI;< H<9A +9@@KCC8 /=@?=BGCB

(91) %5=B F9;I@5F CF6=H :5A=@=9G 5F9 =BHF=BG=75@@M F9@5H98 HC F9GCB5B79G 5G 7CFCH5H=CB =BB9F $=B86@5B8 !$* CIH9F $=B86@5B8 '$* CF =BB9F I@HF5 <5FACB=7 F9GCB5B79G 9B9F5@@M GD95?=B; CF6=HG =B 65F DCH9BH=5@G H9B8 HC 7<5B;9 CF=9BH5H=CB 57FCGG 9J9FM CB9 C: H<9 F9GCB5B79G ,<9 657?6CB9 C: H<9 65F GHFI7HIF9 =G H<9 x1 :5A=@M C: CF6=HG H<9M 7CAACB@M 5DD95F =B G=AI@5H=CBG :CF 65FG 5B8 9J9B H<CI;< H<9M 5F9 @9GG DFCA=B9BH =B 8=A9BG=CB5@ 65F G=AI@5H=CBG K<9F9 J9FH=75@ =BGH56=@=H=9G 5F=G9 J9FH=75@ F9GCB5B79G 5F9 :CIB8 5G C:H9B 5G <CF=NCBH5@ H<9M GH=@@ DF9G9FJ9 5 J=H5@ FC@9 =B 8=A9BG=CB5@ 65FG ,<9 x1 :5A=@M C: CF6=HG =G DFC;F589 5B8 BCFA5@@M 9@CB;5H98 5@CB;.

(92) +7<K5FNG7<=@8 AC89@ :CF H<F99 A=BCF5L=G 9@8G. . Figure 4.4 (9F:CFA5B79 C: DCH9BH=5@ AI@H=DC@9 9LD5BG=CB !B 957< D@CH 7CBHCIF GC@=8 @=B9G 7CFF9GDCB8 HC 5B 9@CB;5H98 5B5@MH=75@ DCH9BH=5@ 85G<98 @=B9G 5F9 :FCA =HG F9GD97H=J9 AI@H=DC@9 9LD5BG=CB =B7@I8=B; H9FAG :CF 0 ≤ l ≤ 6. H<9 65F 5H @5F;9 F58=IG 5B8 D9FD9B8=7I@5F 5H GA5@@ F58=IG !B 588=H=CB H<9 x1 :5A =@M C: CF6=HG <5G 699B 5GGC7=5H98 6M G9J9F5@ 5IH<CFG K=H< H<9 6CLM 5B8 D95BIH G<5D9G C6G9FJ98 =B A5BM ;5@5L=9G G99 IF95I 9H 5@ 5B8 F9:9F9B79G H<9F9=B %CF9 GD97=: =75@@M H<9 x1 HKC A5=B 6F5B7<9G =B H<9

(93) J9FH=75@ F9GCB5B79 :CFA 5 R0S G<5D9 J=9K98 G=89CB H<5H =G HF5BG:CFA98 =B 5 6CLM 6I@;9 K<9B DFC>97H98 =B 5 8=::9F9BH J=9K 5B;@9 +=B79 CIF 6I@;9 G99AG HC <5J9 5 6CLM GHFI7HIF9 5G G99B =B H<9 '!* =A5;9G 5 G=;B=75BH BIA69F C: x1 CF6=HG G<CI@8 69 7CBH5=B98 5B8 G9@97H98 =B H<9 69GH GC@I H=CB C: CIF @=6F5FM ,<=G <5G 699B :CIB8 =B DF9J=CIG AC89@G 2<5C

(94) 6 :CIB8 5 C: H<9 @5HH9F CF6=HG =B H<9 65FG 7CBGHFI7H98 6M <=G AC89@ +=A=@5F@M R65B5B5S CF6=HG G<CI@8 5@GC D@5M 5B =ADCFH5BH FC@9 =B H<9 65F GHFI7HIF9 @CB;D9F=C8 65B5B5 CF6=HG C:H9B 5DD95F 5FCIB8 $5;F5B;9 DC=BHG L4,5 G99 ; CBJ9FG9@M G=B79 H<9 x2 5B8 H<9 x3 :5A=@=9G C: CF6=HG 5F9 D9FD9B8=7I@5F HC H<9 65F H<9M <5J9 HMD=75@@M 699B 7CBG=89F98 C: @=HH@9 =ADCFH5B79 =B 65FG ,<9 x4 F9HFC;F589 :5A=@M C: CF6=HG G<CI@8 5@GC 69 DF9G9BH =B CIF @=6F5F=9G 5B8 GC@IH=CBG 5G =H <5G 699B C6G9FJ98 =B DF9J=CIG AC89@G ,<9 x4 :5A =@M G99AG J9FM FC6IGH 5B8 @=HH@9 5::97H98 6M F9GCB5B79G 9L=GH=B; :FCA 7@CG9 HC H<9 79BH9F H=@@ H<9 8=G? ACF9 89H5=@98 577CIBH CB 65F CF6=HG 75B 69 :CIB8 =B +?C?CG 9H 5@ 5B8 (5HG=G 9H 5@ /9 K=@@ 8=G7IGG H<9 CF6=H :5A=@=9G =BJC@J98 =B CIF GC@IH=CBG 5B8 H<9 =AD@=75H=CBG :CF H<9 AC89@G @5H9F =B H<=G 7<5DH9F Orbit integration scheme ,<9 =B=H=5@ 7CB8=H=CBG C: CIF CF6=H @=6F5FM 5F9 G=A=@5F HC H<CG9 =B DF9J=CIG AC89@G +7<K5FNG7<=@8

(95) (:9BB=;9F

(96) 2<5C

(97) 6 'F6=HG 5F9 @5IB7<98 D9FD9B8=7I @5F@M HC H<9 MN D@5B9 5@CB; H<9 65F =B 6CH< 8=F97H=CBG !B 588=H=CB 5 8MB5A=75@ H=A9GH9D <5G 699B 75@7I@5H98 =B 957< CF6=H 7CBG=89F=B; H<9 =B=H=5@ H5B;9BH=5@ :CF79 ,<9 =B=H=5@ H=A9GH9D <5G 699B G9H HC dt = 0.005 T K<9F9 T = (Fr /r)1/2 H<9 FCH5H=CB5@ D9F=C8 75@ 7I@5H98 :FCA H<9 =B=H=5@ :CF79 ,<9B H<9 CF6=H =BH9;F5H=CB <5G 699B G9H HC 7CAD@9H9 H<9 9EI=J5@9BH C:

(98) FCH5H=CB5@ D9F=C8G IG=B; 5B 585DH=J9 GH9DG=N9 7CBHFC@ 5@;CF=H<A K=H< 5 F9EI=F98 577IF57M C: ∼ 10−7 D7 ,MD=75@@M H<9 @5HH9F DF97=G=CB M=9@8G 7<5B;9G =B H<9 =BH9;F5@ C: ACH=CB @CK9F H<5B ∼ 0.01 EJ % 5H H<9 9B8 C: H<9 =BH9;F5H=CB ,<9 B5@ CF6=HG <5J9 699B =BH9FDC@5H98 K=H< 5 :CIFH< CF89F DC@MBCA=5@ HC F9;I@5F H=A9GH9DG HC 7CAD@9H9 2 × 106 GH9DG D9F CF6=H.

(99) . =A9BG=CB5@ MB5A=7G C: H<9 5@57H=7 I@;9. H 957< =BH9;F5H=CB GH9D K9 <5J9 F97CF898 H<9 89BG=HM ;F=8 79@@ IFH<9FACF9 K<9B H<9 DFC>97H98 DCG=H=CB :5@@G K=H<=B 1.2 C: CB9 C: CIF +, 9@8G K9 F97CF8 DFCD9F AC H=CBG F58=5@ J9@C7=H=9G 5B8 @=B9 C: G=;<H 8=GH5B79G J9B H<CI;< F58=5@ J9@C7=H=9G <5J9 699B F97CF898 H<9G9 <5J9 699B IG98 HC 7CAD5F9 HC 9L=GH=B; 85H5 =B H<9 G5A9 9@8 F5H<9F H<5B C6H5=B=B; 588=H=CB5@ 7CBGHF5=BHG HC CIF AC89@G !B CF89F HC :C7IG CIF 5HH9B H=CB =B H<9 6I@;9 GHFI7HIF9 K9 <5J9 8=G75F898 CF6=HG ;C=B; 69MCB8 ?D7 :FCA H<9 ;5@57 H=7 79BH9F +=A=@5F@M CF6=HG K=H< <=;< 9B9F;=9G 5B8 H<9F9:CF9 BCH GI:7=9BH@M 6CIB8 K9F9 BCH IG98 'B79 H<9 ;F=8 89BG=H=9G <5J9 699B 7CADIH98 :CF 5@@ CF6=HG =B 5 @=6F5FM CIF H5G? =G HC B8 GI=H56@9 K9=;<HG wj ≥ 0 H<5H F9DFC8I79 H<9 C6G9FJ5H=CBG 5G 577IF5H9@M 5G DCGG=6@9 9E !B CF89F HC 9BGIF9 H<5H H<9 89BG=HM C: H<9 AC89@ A5H7<9G H<9 DCH9BH=5@ 5G K9@@ 5G DCGG=6@9 K9 ;=J9 H<9 89BG=HM ;F=8 5 K9=;<H H<5H =G 9EI5@ HC H<9 GIA C: H<9 H<F99 ?=B9A5H=7 ;F=8G ,C :IFH<9F =B7F95G9 H<9 AC89@@=B; K9 =ADCG9 9=;<H:C@8 GD5H=5@ GMAA9HFM CB H<9 79@@ 89BG=H=9G C: 957< CF6=H H<=G =G >IGH=98 6M H<9 GMAA9HFM C: CIF 65F AC89@. . + /*2+ !$ % '$+. +=B79 CIF ;C5@ =G H<9 89H9FA=B5H=CB C: F9@9J5BH 65F D5F5A9H9FG GD97=75@@M φbar H<9 5B;@9 69HK99B H<9 GIB @=B9C:G=;<H HC H<9 ;5@57H=7 79BH9F 5B8 H<9 A5>CF G9A=5L=G C: H<9 65F 5B8 Ωb H<9 65F D5HH9FB GD998 C: H<9 65F K9 <5J9 89J=G98 5 GHF5H9;M K<=7< 7CBG=GHG =B FIBB=B; 5 ;F=8 C: @=6F5F=9G K=H<=B H<9 F5B;9 =B K<=7< K9 KCI@8 9LD97H CIF HKC D5F5A9H9FG φbar 5B8 Ωb HC :5@@ 'IF ;F=8 GD5BG φbar = 0◦ IBH=@ 40◦ =B GH9DG C: 10◦. Figure 4.5 ::97H=J9 DCH9BH=5@ :CF AC89@

(100) =B 5 65F 5@CB; H<9 x 5L=G FCH5H=B; 5H Ω = 80 km s−1 kpc−1 G<CK=B; H<9 :CIF IGI5@ $5;F5B;=5B DC=BHG =B H<9 7CFCH5H=CB F9;=CB.

(101) +7<K5FNG7<=@8 AC89@ :CF H<F99 A=BCF5L=G 9@8G. . Table 4.2 9DFC>97H98 5B8 7<CG9B D5F5A9H9FG :CF AC89@G. (5F5A9H9F a D7 b D7 c D7 m ρb0 % /pc3 β % × 108 λ D7 κ D7 η % /pc3 χ2.

(102) ± ±

(103) ±

(104) ±

(105) ±

(106) ±

(107)

(108)

(109) ± ± ±

(110)

(111) .

(112) ± ±

(113) ±

(114) ± ±

(115) ± . 5B8 Ωb = 0 ID HC 80 km s−1 kpc−1 =B GH9DG C: 20 km s−1 kpc−1 ,<9 ;F=8 7CBG=GH=B; C: 7CAD@9H9 @=6F5F=9G <5G 699B FIB :CF CIF 69GH 89DFC>97H=CB §

(116) K<=7< =B7@I89G 5

(117) 65F D@IG 5 7IGD 5B8 5 %=M5ACHC&5;5= 8=G? CF 8=::9F9BH Ωb H<9 @=6F5F=9G 7CBH5=B 8=::9F9BH BIA69FG C: CF6=HG +@CK 65FG H9B8 HC 7CBH5=B A5BM 7<5CH=7 CF6=HG K<9F95G H<9 F9GCB5B79G C: :5GH 65FG ?=7? A5BM CF6=HG CIH C: H<9 ;F=8 !B CF89F HC DFCJ=89 5 :5=F 7CAD5F=GCB 69HK99B H<9 8=::9F9BH 65F D5F5A9H9FG K9 F5B8CA@M GI6G5AD@9 957< @=6F5FM @=A=H=B; H<9 BIA69F C: CF6=HG K9 H HC

(118) /9 <5J9 7<97?98 H<5H 8=::9F9BH F5B8CA GI6G5AD@9G @958 HC 7CBG=GH9BH F9GI@HG. 4.4.1. Proper motions and Photometric parallax. !B 588=H=CB HC H<9 DFCD9F ACH=CBG 957< ?=B9A5H=7 ;F=8 =B7@I89G 5 F97CF8 C: H<9 8=G H5B79G C: 957< GH5F 5@CB; H<9 @=B9 C: G=;<H ,<=G F97CF8 <5G 699B C6H5=B98 6M IG=B; 5 D<CHCA9HF=7 D5F5@@5L :CF 5 GI6G5AD@9 C: HIFBC:: 5B8 A5=B G9EI9B79 GH5FG =B 957< 9@8 ,<9 G@CD9 C: H<9 % <5G 699B F9ACJ98 =B 957< D@CH 6M A95BG C: H<9 EI5BH=HM % K<=7< 7CFF9GDCB8G HC 5 7FI89 8=GH5B79 AC8I@IG ,<9 @5HH9F EI5BH=HM =G M∗ = m814 − ζ × (m814 − m555 ) /<9F9 ζ :CF Sagittarius-I 5B8 Baade’s Window 9@8G =G 5B8 :CF NEAR NGC 6558 9@8 K<=7< IG9G H<9 / @H9F =BGH958 C: / =G ,<=G F9@5H=J9 8=GH5B79 =B8=75HCF M∗ =G 7CFFC6CF5H98 6M H<9 GACCH< ;F58=9BH C: FCH5H=CB C6G9FJ98 :CF CIF H<F99 79BHF5@ 9@8G =B #* 5B8 # 5B8 =G 5@GC 7CBG=GH9BH K=H< F58=5@ J9@C7=HM GIFJ9MG K<=7< <5J9 F9J95@98 H<9 FCH5H=CB D5HH9FB :CF 6I@;9 DCDI@5H=CB G99 %=BB=H= 2C775@= 5B8 F9:9F9B79G H<9F9=B &9J9FH<9@9GG GI7< FCH5H=CB =B DF57H=79 A=;<H 69 6@IFF98 6M H<9 G9@97H=CB C: GH5FG 5 DIF9 6I@;9 DCDI@5H=CB =G 8=:7I@H HC 57<=9J9 G=B79 =HG DCDI@5H=CB F5D=8@M 897F95G9G 5H H<9 G5A9 H=A9 8=G? 5B8 <5@C =B7F95G9 5G CB9 ACJ9G C:: 5L=G 'IF G5AD@9G =B H<9 H<F99 A=BCF 5L=G 9@8G 7@95F@M G<CK H<9 FCH5H=CB C: 6I@;9 8CA=B5H98 DCDI@5H=CBG.

(119) . =A9BG=CB5@ MB5A=7G C: H<9 5@57H=7 I@;9. !B CF89F HC 7CAD5F9 H<9 CF6=H ;F=8G K=H< H<9 ?=B9A5H=7 85H5 K9 B998 HC IB89FGH5B8 H<9 C6G9FJ5H=CB5@ 9FFCFG 5B8 7CBJC@J9 H<9 AC89@G K=H< H<9A CH< H<9 DFCD9F ACH=CBG 5B8 H<9 @=B9 C: G=;<H 8=GH5B79G <5J9 9FFCFG CF DFCD9F ACH=CBG K9 <5J9 5GGIA98 5 HMD=75@ 9FFCF C: A5G MF H<=G J5@I9 =G 7CBG=GH9BH K=H< H<9 C6G9FJ98 DFCD9F ACH=CB 9FFCFG =B CIF H<F99 A=BCF5L=G 9@8G CF H<9 8=GH5B79 AC8I@IG M∗ CB H<9 CH<9F <5B8 K9 <5J9 75@7I@5H98 H<9 C6G9FJ98 9FFCF 6M 9@8 5G =H =G =@@IGHF5H98 =B ;IF9 CF 957< 9@8 K9 <5J9 7CAD5F98 H<9 8=GHF=6IH=CB C: GH5FG 5@CB; H<9 @=B9 C: G=;<H K=H< H<9 8=GHF=6IH=CB 89F=J98 :FCA H<9 65F AC89@ 7<CG9B G 9LD97H98 H<9 8=GHF=6IH=CB :FCA H<9 AC89@ =G 5@K5MG K9@@ 89B98 K=H< 5 G<5FD9F D95? ,<IG K9 <5J9 IG98 HKC :F99 D5F5A9H9FG =B 5 $9J9B69F;%5FEI5F8H @95GHGEI5F9G A=B=A=N5H=CB HC B8 H<9 69GH H D5F5A9H9FG K<=7< 7CBJC@J9 H<9 AC89@ 8=GHF=6IH=CB %C89@

(120) M=9@8G 5 σM ∗ ∼ 0.3 (mag) :CF 5589TG /=B8CK 5B8 +5;=HH5F=IG 5B8 ∼ 0.6 (mag) :CF &95F & ,<=G GACCH<=B; <5G 699B D9F:CFA98 IG=B; 5 5IGG=5B ?9FB9@ =B H<9 DFCD9F ACH=CB D<CHCA9HF=7 D5F5@@5L ;F=8G :CF H<9 73 − 1 = 342 58>579BH D=L9@G. . * +-$,+ ' , + /*+ !$ , &!)-. =;IF9 G<CKG H<9 χ2 7CBHCIF D@CH C: H<9 @=6F5F=9G FIB :CF H<9

(121) AC89@ +9J9F5@ :57HG 75B 69 DC=BH98 CIH =AA98=5H9@M :FCA H<=G D@CH =FGH@M 5 7@95F A=B=AIA 75B 69 C6G9FJ98 :CF H<=G AC89@ 5H φbar = 0◦ 5B8 Ωb = 60 km s−1 kpc−1 H<=G A=B=AIA G99AG HC 69 D5FH C: 5 NCB9 K=H< @CK9F χ2 K<=7< FCI;<@M GD5BG :FCA Ωb = 40 HC 60 km s−1 kpc−1 G97CB85FM A=B=AIA =B H<=G NCB9 75B 69 C6G9FJ98 :CF H<9 AC89@G 5H φbar = 40◦ 5B8 Ωb = 40 HC 60 km s−1 kpc−1 K<=7< DFC8I79G 5 χ2 5@ACGH =89BH=75@ +97CB8@M AC89@G K=H< G@CK 65FG G99A HC 69 FI@98 CIH 6M H<9 D@CH 65FG 5G :5GH 5G Ωb = 80 km s−1 kpc−1 75B G=A=@5F@M 69 9L7@I898 /9 =BH9FDF9H H<9 F9GI@HG K<=7< 5F9 65G98 CB@M =B DFCD9F ACH=CBG 5G 5 J5@@9M C: GC@IH=CBG H<5H FIBG :FCA σbar = 0◦ , Ωb = 60 HC σbar = 40◦ , Ωb = 50 &9LH K9 =BJ9GH=;5H9 K<9H<9F H<9 F58=5@ J9@C7=H=9G 75B DFCJ=89 9LHF5 7CBGHF5=BHG H<5H 75B 69 IG98 HC 6F95? H<9 5DD5F9BH 89;9B9F57M C: AC89@G *58=5@ J9@C7=H=9G 5@@CK IG HC A95GIF9 J9FH9L 89J=5H=CB 5B8 J9@C7=HM 8=GD9FG=CBG σr 5B8 σrl .9FH9L 89J=5H=CB G=AD@M 7CFF9GDCB8G HC H<9 5B;@9 69HK99B H<9 A5>CF 5L=G C: H<9 J9@C7=HM 9@@=DGC=8 5B8 H<9 @=B9 C: G=;<H 2<5C +D9F;9@ *=7<

(122) !B 5B 5L=GMAA9HF=7 DCH9BH=5@ H<9 J9@C7=HM 9@@=DGC=8 AIGH 69 5@=;B98 K=H< H<9 5L=G H<IG H<9 J9FH9L 5B;@9 lv =G 0◦ CF 90◦ 89D9B8=B; C: H<9 5L=G C: H<9 5@=;BA9BH BM 89J=5H=CB C: H<=G G=HI5H=CB B979GG5F=@M =AD@=9G HF=5L=@=HM C: H<9 DCH9BH=5@ 5B8 =B HIFB C: H<9 89BG=HM 8=GHF=6IH=CB K<=7< ;9B9F5H9G =H 9B79 J9FH9L 89J=5H=CB =G 5B =B8=75HCF C: HF=5L=@=HM K<=7< <5G 699B F9D95H98@M 89H97H98 :CF GH5FG C: H<9 ;5@57H=7 6I@;9 G G<CKB =B 2<5C 9H 5@

(123) 5B8 § 5 G=;B=75BH J9FH9L 89J=5H=CB 5DD95FG :CF G5AD@9G C: # 5B8 % ;=5BHG =B 5589TG /=B8CK %CF9 F979BH@M K9 5B5@MN98 ∼ 2000 F58=5@ J9@C7=H=9G :CF GH5FG =B CIF H<F99 A=BCF 5L=G 9@8G K<=7< 7CFF9GDCB8 HC H<9 GD97HFCG7CD=7 7CIBH9FD5FH C: H<9 DFCD9F ACH=CBG 5@F958M C6H5=B98 =B #* 5B8 # § G=;B=75BH J9FH9L 89J=5H=CB =B H<9G9 H<F99 9@8G <5G 699B :CIB8 K=H< J9FH9L 5B;@9G C: −41◦ −40◦ 5B8 −17◦ :CF 5589TG /=B8CK +5;=HH5F=IG! 5B8 &95F & F9GD97H=J9@M /9 KCI@8 9LD97H CIF 69GH H AC89@ HC F9DFC8I79 H<CG9 G=;B5HIF9G C: HF=5L=@=HM =;IF9 G<CKG H<9 DF98=7H98 J9@C7=HM 9@@=DGC=8G 5B8 J9FH9L 5B;@9 lv :CF H<9 .

(124) +7<K5FNG7<=@8 AC89@ :CF H<F99 A=BCF5L=G 9@8G. . Figure 4.6 5@7I@5H=CB C: H<9 σM ∗ =B CIF H<F99 9@8G /9 <5J9 7CAD5F98 H<9 8=GHF=6IH=CB C: GH5FG 5@CB; H<9 @=B9 C: G=;<H K=H< H<9 DF98=7H=CB A589 :CF 957< AC89@ Top row left, H<9 G9@97 H=CB C: GH5FG =B 5589TG /=B8CK K=H< D<CHCA9HF=7 D5F5@@5L 89H9FA=B98 K<9F9 5 GI6G5AD@9 =B 6@57? <5G 699B IG98 HC 6I=@8 H<9 <=GHC;F5A 5@CB; H<9 @=B9 C: G=;<H =B CF89F HC F957< 7CAD@9H9 B9GG (Top right) <=GHC;F5A :CF K<=7< K9 <5J9 :CIB8 H<9 69GH H D5F5A9H9FG 85G<98 HC A5H7< H<9 F95@ GH9@@5F 8=GHF=6IH=CBG<=GHC;F5A K=H< H<9 =B=H=5@ DF98=7H=CB C:

(125) AC89@GC@=8 @=B9 Middle and bottom row, G5A9 5G HCD FCK 6IH :CF 9@8G +5;=HH5F=IG! 5B8 &95F & F9GD97H=J9@M. AC89@G K=H< H<9 @CK9GH χ2 C: H<9

(126) ;F=8 !B 588=H=CB H56@9 G<CKG H<9 χ2 5B8 8=GD9FG=CBG =B r :CF H<9 AC89@G +=A=@5F J9@C7=HM 9@@=DGC=8G HC H<CG9 =B ;IF9 <5J9 699B C6H5=B98 =B 7<5DH9F :CF 5 GI6G5AD@9 C: HIFBC:: 5B8 A5=B G9EI9B79 6I@;9 GH5FG =B CIF H<F99 A=BCF 5L=G 9@8G =;IF9

(127) ,<9G9 RF95@ 85H5S J9@C7=HM 9@@=DGC=8G :FCA 7<5DH9F 75B 69 8=F97H@M 7CAD5F98 K=H< H<CG9 C6H5=B98 :FCA H<9 AC89@G !B ;9B9F5@ K9 :CIB8 H<5H 5 65F =B7@=B5H=CB =G B99898 =B CF89F HC <5J9 5 J9FH9L 89J=5H=CB J9@C7=HM 9@@=DGC=8G =B AC89@G 5H φbar = 0◦ DC=BH =B H<9 CDDCG=H9 8=F97H=CB C: H<9 F95@ 85H5 J9@C7=HM 9@@=DGC=8G H<5H :57H =AA98=5H9@M 5@@CKG IG HC FI@9 CIH H<9 69GH χ2 AC89@ +=A=@5F@M H<CI;< ACF9 K95?@M AC89@G 5H (φbar , Ωb ) = (10◦ , 60) 5B8 (φbar , Ωb ) = (40◦ , 60) 75B 69 8=G75F898 8I9 HC H<9=F GHFCB; 8=G5;F99A9BH =B 5H @95GH CB9 C: H<9 J9FH9L 5B;@9 75@7I@5H98 =B 957< 9@8 !B CF89F HC EI5BH=:M H<CG9 8=::9F9B79G K<=7< K9 <5J9 >IGH 89G7F=698 K9 7CAD5F98 H<9 8=GD9FG=CBG σr 5B8 σrl DF98=7H98 :CF H<9 ;F=8 C: AC89@G K=H< H<CG9 :CIB8.

(128) . =A9BG=CB5@ MB5A=7G C: H<9 5@57H=7 I@;9. Figure 4.7 CBHCIF D@CH :CF H<9 ;F=8 C: AC89@G IG=B; H<9

(129) 65F 5F? NCB9G 7CFF9GDCB8 HC H<9 AC89@G K=H< GA5@@ χ21 . =B H<9 7<5DH9F =;IF9 G<CKG H<9 χ2 (σr ) 5B8 χ2 (σrl ) K9 C6G9FJ9 H<5H χ2 (σr ) =B ;9B9F5@ :5JCIFG 65FG K=H< <=;< =B7@=B5H=CB 5B8 F5D=8 FCH5H=CBG K<9F95G σrl CB H<9 CH<9F <5B8 FCI;<@M F9DFC8I79G H<9 A=B=AIAG C6G9FJ98 =B ;IF9 D@IG 5 B9K A=B=AIA 5H (φbar , Ωb ) = (20◦ , 40) /<9B 7CA6=B98 K=H< 9EI5@ K9=;<HG H<9M G<CK 5 F9A5F?56@9 5;F99A9BH K=H< ;IF9 H<9 G5A9 HKC 7@95F A=B=AIAG 5H (φbar , Ωb ) = (0◦ , 60) 5B8 (φbar , Ωb ) = (40◦ , 40) 5DD95F <CK9J9F H<9 A5=B A=B=AIA =B H<=G σr + σrl ;F=8 <5G G<=:H98 HC H<9 AC89@ 5H (φbar , Ωb ) = (40◦ , 40) 5G K9 KCI@8 9LD97H 577CF8=B;@M HC H<9 J9@C7=HM 9@@=DGC=8G ,<IG H<9 69GH 75B8=85H9G :CF H<9

(130) 65F 577CF8=B; HC CIF ;F=8 G99AG HC 69 H<9 AC89@ 5H (φbar , Ωb ) = (40◦ , 40) !BH9F9GH=B;@M H<=G 65F =B7@=B5H=CB <5G 699B DC=BH98 CIH DF9 J=CIG@M IB89F 8=::9F9BH H97<B=EI9G 5G H<9 7CFF97H =B7@=B5H=CB C: H<9 65F 5G K9 <5J9 5@F958M A9BH=CB98 ,<9 @=GH C: CF6=HG A=B=A=N98 =B H<=G G97H=CB 8C BCH <5J9 5BM DF9G 9@97H=CB 7F=H9F=CB &9J9FH<9@9GG =H A=;<H 69 =BH9F9GH=B; HC 9LD@CF9 =: F9GHF=7H=B; CIF @=GH C: CF6=HG =B CF89F HC :5JCIF F9;I@5F CF6=HG 5G DF9J=CIG KCF? <5J9 8CB9 G=;B=75BH@M 7<5B;9G H<9 F9GI@HG.

(131) +7<K5FNG7<=@8 AC89@ :CF H<F99 A=BCF5L=G 9@8G.

(132). Figure 4.8 ,F5BGJ9FG5@ DFCD9F ACH=CB μl JG F58=5@ J9@C7=HM V r :CF H<9 H<F99 A=BCF 5L=G 9@8G C6H5=B98 :FCA H<9 @CK9GH χ2 AC89@G =B

(133) ;F=8 HKC8=A9BG=CB5@ 5IGG=5B H <5G 699B D9F:CFA98 =B 957< D@CH.

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(173) . 5589TG /=B8CK lv σr σrl (deg)

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(207). +5;=HH5F=IG! lv σr σrl (deg)

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(233) . &95F & lv σr σrl (deg)

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(268) . &CH9 5F D5HH9FB GD998 Ωb =B km s−1 kpc−1 J9@C7=HM 8=GD9FG=CBG σr =B (km s−1 ) 5B8 σrl =B (km2 s−2 ). φbar Ωb (deg).

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(273) . χ21. Table 4.3 .9FH9L 5B;@9 5B8 J9@C7=HM 8=GD9FG=CBG =B r :CF

(274) AC89@G ;F=8. =A9BG=CB5@ MB5A=7G C: H<9 5@57H=7 I@;9.

(275) +7<K5FNG7<=@8 AC89@ :CF H<F99 A=BCF5L=G 9@8G. . Figure 4.9 χ2 7CBHCIF D@CHG C:

(276) AC89@G K<=7< 7CAD5F9 H<9 σ r 5B8 σrl DF98=7H=CBG :FCA H<9 AC89@G 5B8 H<CG9 C6H5=B98 =B 7<5DH9F :FCA 5 GI6G5AD@9 C: HIFBC:: 5B8 A5=B G9EI9B79 6I@;9 GH5FG. Orbit classiﬁcation 7CBG=89F56@9 :F57H=CB C: 957< @=6F5FM AIGH 7CFF9GDCB8 HC =FF9;I@5F CF GHC7<5GH=7 CF 6=HG !B CIF @=6F5F=9G H<=G :F57H=CB =B7F95G9G 5G H<9 65F 697CA9G G@CK9F 5B8 IB89F H<9 =BI9B79 C: H<9 79BHF5@ A5GG 7CB79BHF5H=CB % B CF6=H A=;<H 69 7CBG=89F98 5G TF9;I@5FT K<9B =HG HF5>97HCFM 8C9G BCH M=9@8 5 G=;B=75BH 7<5B;9 =B =HG A5GG 89BG=HM 8=GHF=6IH=CB CJ9F 5 D9F=C8 C: H=A9 @CB;9F H<5H H<9 CB9 K9 9LD97H CIF AC89@ HC 69 GH5 H=CB5FM ,<=G 7CB79DH <5G 699B H<9 65G9 C: H<9 7@5GG=75H=CB 6M $=5DIBCJ 9LDCB9BHG =B H<=G H97<B=EI9 9 ; +7<97?

(277) H<9 8=GH5B79 69HK99B D5=FG C: 7@CG9 CF6=HG =G A95GIF98 HFCI;< H<9 =BH9;F5H=CB =B GHC7<5GH=7 CF6=HG H<9 @5HH9F 8=GH5B79 =G 9LD97H98 HC 8=J9F;9 9LDCB9BH=5@@M !B CF89F HC DFCJ=89 5 G=A=@5F 7@5GG=75H=CB K9 <5J9 89J=G98 5 AI7< G=AD@9F 5D DFC57< K<=7< =G IG9:I@ :CF CF6=HG @5IB7<98 BCH =B D5=FG 'IF ;C5@ =G BCH HC CJ9F@C58 CIF A=B=A=N5H=CB A5HF=L K=H< GHC7<5GH=7 CF6=HG K<=7< =B ;9B9F5@ @@ 5 G=;B=75BH BIA69F C: 79@@G =B 7CAD5F=GCB K=H< F9;I@5F CF6=HG 'B H<9 CH<9F <5B8 H<9 :CFA9F CF6=HG A=;<H 69 F98IB85BH G=B79 A5BM C: H<9A H9B8 HC @@ 5 G=A=@5F :F57H=CB C: H<9 D<5G9 GD579 K<=7< ;=J9G H<9A 5 7<5F57H9F=GH=7 5DD95F5B79 /9 D9F:CFA98 5 G=AD@9 =89BH=75H=CB C: H<CG9 CF6=HG 6M 7CAD5F=B; H<9 F9@5H=J9 BIA69F C: 79@@G C77ID=98 =B 957< C: H<9 ;F=8G G5AD@98 !B CIF H9GHG H<=G J9FM G=AD@9 5DDFC57< <5G DFCJ9B HC 69 9::97H=J9 HC G9D5F5H9 H<9 F9;I@5F CF6=HG :FCA H<9 =FF9;I@5FG H<5H @@ 5 7CBG=89F56@9 DCFH=CB C: H<9 D<5G9GD579 CBG9 EI9BH@M K9 <5J9 F975@7I@5H98 H<9 &&$+ 69GH H 7CBHFC@@=B; H<9 F9@5H=J9 F5H=C C: F9;I@5F 5B8 =FF9;I@5F CF6=HG =;IF9

(278) G<CKG H<9 7CBHCIF D@CH :CF H<9 ;F=8 C:

(279) AC89@G 70% C: H<9 =BDIH CF6=H @=GH K9F9 G9@97H98 =B H<=G G97CB8 A=B=A=N5H=CB HC 69 CIF GC75@@98 RF9;I@5FS =B CF89F HC :5JCIF 5 G9@97H=CB K<9F9 H<9 @5HH9F HMD9 DF9J5=@G ,<9 B9K A=B= A=N5H=CB ;F=8 75@7I@5H98 =B7@I8=B; CIF CF6=H 7@5GG=75H=CB 89@=J9FG χ2 J5@I9G 7CBG=GH9BH K=H< H<9 ;F=8 C: BCB7@5GG=98 CF6=HG.

(280) . =A9BG=CB5@ MB5A=7G C: H<9 5@57H=7 I@;9. Figure 4.10 CBHCIF D@CH :CF H<9 ;F=8 C: AC89@G IG=B; H<9

(281) 65F G=AD@9 7@5GG=75H=CB <5G 699B D9F:CFA98 HC 7CBHFC@ H<9 F9@5H=J9 F5H=C C: F9;I@5F 5B8 =FF9;I@5F CF6=HG !B 957< @=6F5FM 70% C: H<9 CF6=HG G9@97H98 5G =BDIH C: H<9 &&$+ 5F9 F9;I@5F. G =B H<9 DF9J=CIG A=B=A=N5H=CB ;F=8 K9 5@GC 75@7I@5H98 5B8 7CAD5F98 H<9 DF98=7 H=CB C: H<9 AC89@G =B σr 5B8 σrl K=H< H<9 J5@I9G C6H5=B98 =B H<9 7<5DH9F ,<9 F9GI@HG 5F9 GIAA5F=N98 =B H56@9 5B8 ;IF9

(282)

(283) χ2 σr G<CKG 5 G=A=@5F 69<5J=CIF H<5B =B H<9 BCBG9@97H98 CF6=H @=6F5FM A=B=A=N5H=CB ;F=8 CBJ9FG9@M χ2 σrl G<CKG 5 G@=;<H@M 8=::9F9BH 8=GHF=6IH=CB K<9F9 =H <5G BCH DF9G9FJ98 CB9 C: H<9 H<F99 A=B=AIA C6G9FJ98 5H (φbar , Ωb ) = (40◦ , 40) =B 588=H=CB <5G :5JCIF98 G@CK9F 65FG K=H< GA5@@ =B7@=B5H=CBG (φbar , Ωb ) = (10◦ − 20◦ , 20) G 5 F9GI@H C: H<=G GA5@@ 8=::9F9B79G H<9 7CA6=B98 69GH H AC89@ K=H< 9EI5@ K9=;<HG :CF H<9 χ2 σr + σrl =G =B H<=G 75G9 5H (φbar , Ωb ) = (30◦ , 40) ,<9 J5@I9G 89H9FA=B98 6M H<9 69GH H AC89@ =B H<9

(284) ;F=8 5F9 BCH 5 GIFDF=G9 /9 <5J9 5@F958M =B8=75H98 <CK H<9 @5GH 89H9FA=B5H=CBG 65G98 CB GH5F 7CIBHG =B % 5B8 # ;=5BHG IG=B; +D=HN9F 7CIBHG 6M 9B>5A=B 9H 5@ <5J9 :CIB8 =HG 69GH H :CF 5 @CB; 65F 5H φbar = 44◦ !H =G F9A5F?56@9 <CK CIF 69GH AC89@ =B GD=H9 C: H<9 8=::9F9B79G =B H<9 65F 8=A9BG=CBG 5;F99G =;IF9G

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(286) 5B8

(287) G<CK H<9 69GH H AC89@ 5H (φbar , Ωb ) = (40◦ , 40) CB K<=7< K9 K=@@ :C7IG =B CIF GI6G9EI9BH 5B5@MG=G !B CIF AC89@G H<9 D9F79BH5;9 C: GHC7<5GH=7 CF6=HG GHFCB;@M 89D9B898 CB H<9 65F D5HH9FB GD998 5G 9LD97H98 @=6F5F=9G :CF F5D=8 65FG 5H 80 km s−1 kpc−1 <5J9 ∼ 80% C: F9;I@5F CF6=HG K<9F95G G@CK 65FG 5H 0−20 km s−1 kpc−1 CB@M <5J9 ∼ 15% ,<=G CJ9F56IB85B79 C: GHC7<5GH=7 CF6=HG K<=7< CJ9F @CB; =BH9;F5H=CB H=A9G <5J9 =B A5BM 75G9G 5 G=A=@5F 5DD95F5B79 <5G DFC8I798 BCH=7956@9 9::97HG =B H<9 69GH H =B 957< AC89@ !B H<9 75G9 C: CIF 69GH AC89@ 5H (φbar , Ωb ) = (40◦ , 40) 9J9B H<CI;< H<9 H =G 8CA=B5H98 =B BIA69F C: CF6=HG 6M F9;I@5FG 75% C: H<9 CF6=HG K=H< K9=;<HG> 0 5F9 F9;I@5F H<9 GHC7<5GH=7 7CADCB9BH 577CIBHG 6M ∼ 80% C: H<9 A5GG ,<IG H<9 9@CB ;5H98 65F G<5D9 =G DFCJ=898 A5=B@M 6M H<9 GACCH< GHC7<5GH=7 7CADCB9BH F5H<9F H<5B x1 F9;I@5F :5A=@M C: CF6=HG 5G K9 7CI@8 <5J9 9LD97H98 577CF8=B; HC DF9J=CIG AC89@G ,<9 A5=B FC@9 H<5H F9;I@5F CF6=HG G99A HC D@5M =G HC ;=J9 5 ACF9 F95@=GH=7 G<5D9 HC H<9.

(288) +7<K5FNG7<=@8 AC89@ :CF H<F99 A=BCF5L=G 9@8G. . Figure 4.11 +5A9 5G ;IF9 6IH IG=B; CIF CF6=H 7@5GG=75H=CB HC :CF79 C: H<9 CF6=HG HC 69 F9;I@5F. 69GH H 577CIBH=B; :CF H<9 89H5=@G =B H<9 H =;IF9

(289) G<CKG H<9 7IAI@5H=J9 :F57H=CB C: =FF9;I@5F 5B8 F9;I@5F CF6=HG 5G 5 :IB7H=CB C: H<9 "57C6= 9B9F;M EJ :CF H<9 AC89@ 5H (φbar , Ωb ) = (40◦ , 40) "57C6= 9B9F;M EJ A95GIF9G FCI;<@M H<9 F58=5@ 9LH9BH C: 5B CF6=H J9B H<CI;< F9;I@5F CF6=HG G99A HC 8CA=B5H9 5H GA5@@ F58=IG K9 ?BCK H<5H CIF G=AD@9 DFC798IF9 HC 7@5GG=:M CF6=HG G<CI@8 69 =B9::97H=J9 HC F97C;B=N9 GHC7<5GH=7 CF6=HG 7@CG9 HC H<9 79BH9F H<9F9:CF9 K9 KCI@8 9LD97H H<CG9 F9;I@5F CF6=HG HC 69 CJ9F9GH=A5H98 H<9F9 CF @5F;9F F58== 6CH< F9;I@5F 5B8 =FF9;I@5F :F57H=CBG GH958=@M =B7F95G9 BCH G<CK=B; 5 D5FH=7I@5F F58=IG =B K<=7< F9;I@5F CF GHC7<5GH=7 CF6=HG 7@95F@M DF9J5=@ CJ9F H<9 CH<9F HMD9 CAD5F5H=J9@M H<9 65F D5HH9FB GD998 :5JCIF98 6M CIF 69GH χ2 =G G@CK9F H<5B H<CG9 =B DF9J=CIG 65F AC89@G =BB9M 9H 5@

(290)

(291) 89H9FA=B98 5 65F FCH5H=CB C: 81km s−1 kpc−1 K<=@9 2<5C

(292) 6 5B8 5P :B9F IG98 G@CK9F 65FG 5H 60km s−1 kpc−1 H<IG CIF AC89@ DF98=7HG 5 65F 5H @95GH 20km s−1 kpc−1 G@CK9F J=89B79 C: 5 G@CK9F FCH5H=CB H<5B 60km s−1 kpc−1 <5G 5@F958M 699B GI;;9GH98 =B DF9J=CIG AC89@G 2<5CTG AC89@ 9B7CIB H9F98 DFC6@9AG HC H C6G9FJ5H=CBG 5H @5F;9 F58=IG <=G AC89@ DF98=7H98 HCC AI7< FCH5 H=CB 5H @5F;9 F58== K<=7< 7CI@8 69 GC@J98 6M 5 G@CK9F 65F >IGH 5G CIF ;F=8 C: AC89@G DF98=7HG 'IF 69GH H F9GI@HG DIH :CFK5F8 HKC DCGG=6=@=H=9G H<9 FGH =G F9@5H98 HC H<9 65F AC89@ =B=H=5@@M 5GGIA98 CIF 89DFC>97H98 65F =G GA5@@ 5B8 7CAD5F5H=J9@M 7CB79BHF5H9G ACGH C: =HG A5GG K=H<=B H<9 ?D7 F58=IG 9B79 CB9 75B 9LD97H =B 58J5B79 H<5H 5 7CB G=89F56@9 :F57H=CB C: H<9 CF6=HG =B H<9 @=6F5FM 5F9 GHC7<5GH=7 K<9B H<9 65F =G GI:7=9BH@M G@CK H<=G @=A=HG =B HIFB H<9 BIA69F C: F9;I@5F CF6=HG ,<9 @5HH9F DFC6@9A =G 7CID@98 K=H< H<9 :57H H<5H H<9 79BHF5@ F9;=CBG C: H<9 ' !* =A5;9G 5F9 <95J=@M C6G7IF98 5B8 <5F8@M 75B 69 7CBG=89F98 HC G5H=G:M 5 @=B95F 9LH=B7H=CB F9@5H=CB K<=7< A=;<H 57 7CIBH :CF H<9 CJ9F9GH=A5H=CB C: H<9 A5GG =B H<9 IBF9GC@J98 79BHF5@ 7IGD C: H<9 6I@;9 !B 588=H=CB CIF AC89@ <5G BCH 9LD@=7=H@M 5JC=898 F9HFC;F589 CF6=HG 5B8 A5H<9A5H=75@ =B79FH5BH=9G 65G98 CB CIF A5BM :F99 D5F5A9H9FG IG98 8IF=B; H<9 89DFC>97H=CB 75B 5@GC.

(293) . =A9BG=CB5@ MB5A=7G C: H<9 5@57H=7 I@;9. 69 5 GCIF79 C: F=G? &9J9FH<9@9GG =: CB9 5GGIA9G H<9 :95G=6=@=HM C: H<9 65F =BDIH AC89@ 5B8 7CBG=89F=B; H<9 F9A5F?56@9 5;F99A9BH K=H< DF9J=CIG KCF? F9GD97H HC 65F D5F5A9 H9FG K9 A=;<H K5BH HC 5GGIA9 H<5H H<9G9 F9GI@HG GI;;9GH 5B 57HI5@ D5FH=7I@5F :CFA5H=CB G79B5F=C :CF H<9 6I@;9 65F !: H<9 65F <5G ;FCKB G97I@5F@M :FCA H<9 H<=B 8=G? 5:H9F 5 :9K 8MB5A=75@ FCH5H=CB D9F=C8G 5 7CBG=89F56@9 DCDI@5H=CB KCI@8 69B8 =HG HF5>97HCF=9G HC @@ H<9 D<5G9GD579 =B CF6=H :5A=@=9G K<=7< GD9BH 5 G=;B=75BH D9F=C8 C: H=A9 5K5M C: H<9 D@5B9 *9;I@5F CF6=HG :5F :FCA F9GCB5B79G 75B GIFJ=J9 58=565H=7 7<5B;9G =B H<9 DCH9BH=5@ H<9 F9HFC;F589 CF6=HG CB H<9 CH<9F <5B8 5F9 5 F9GI@H C: H<9 G75HH9F=B; =B F9G CB5B79G 5B8 DCH9BH=5@ I7HI5H=CBG ,<=G DFC79GG 75B 69 9B<5B798 6M H<9 79BHF5@ A5GG 7CB79BHF5H=CB 5B8 H<9 ;5G =BCK 5G5B 9H 5@

(294) K<=7< <5G 699B DC=BH98 CIH <5G 5 7FI7=5@ :57HCF =B H<9 G9@:89GHFI7H=CB C: 65FG 5G =BCK =G 8F=J9B =BG=89 H<9 7CFCH5 H=CB F58=IG 6M H<9 65F HCFEI9G 5B8 F97=DFC75@@M HCFEI9G H<9 65F CA69G ,<=G 5B;I@5F ACA9BHIA @CGG A5?9G H<9 65F ;FCK @5F;9F 5B8 G@CK9F IBH=@ =H B5@@M 89GHFCMG =HG9@: +=B79 H<5H 5 7CBG=89F56@9 :F57H=CB C: H<9 CF6=HG :CIB8 =B H<9 65F 5F9 GHC7<5GH=7 G99AG HC =B8=75H9 H<5H H<9 65F =G =B 5B 58J5B79 9JC@IH=CB5FM GH5;9 'B H<9 CH<9F <5B8 H<9 :57H H<5H GHC7<5GH=7 CF6=HG 8CA=B5H9 H<9 A5GG C: H<9 65F GI;;9GHG 5 J=C@9BH :CFA5 H=CB G79B5F=C K<9F9 A5BM F9;I@5F CF6=HG <5J9 BCH GIFJ=J98 8I9 HC F5D=8 7<5B;9G =B H<9 D<5G9GD579. . + -%%*1 & -,-* / '*#. /9 <5J9 DF9G9BH98 =B H<=G D5D9F H<9 F9GI@HG C: 5 G9H C: +7<K5FNG7<=@8 AC89@G K<=7< 5HH9ADH HC F9DFC8I79 H<9 89BG=HM DFCD9F ACH=CB 5B8 D<CHCA9HF=7 D5F5@@5L 8=GHF=6I H=CB :CF H<9 ;5@57H=7 6I@;9 'IF 85H5 7CADF=G9G ∼ 45000 GH5FG K=H< DFCD9F ACH=CBG @M=B; =B H<F99 @CK :CF9;FCIB8 9LH=B7H=CB K=B8CKG 5@CB; H<9 ;5@57H=7 A=BCF 5L=G 5B8 5B =BDIH 89BG=HM 8=GHF=6IH=CB C: H<9 65F 6I@;9 K<=7< 7CA9G :FCA 5 89DFC>97H=CB C: H<9 8IGH9LH=B7H=CB 7CFF97H98 '!* =A5;9G ,<9 H5F;9H <5G 699B HH98 :CF 5 ;F=8 C: AC89@G K<=7< =B7@I89G AC89@G K=H< 8=::9F9BH 65F =B7@=B5H=CBG 5B8 65F D5HH9FB GD998G ,<9 F9GI@HG G<CK HKC A=B=A5 5B8 9L7@I89 F5D=8 65FG CF 65FG K=H<CIH FCH5H=CB !B CF89F HC 7<CCG9 69HK99B CB9 C: H<9 HKC A=B=A5 5H (Ωb , φbar ) = (60km s−1 kpc−1 , 0◦ ) 5B8 (Ωb , φbar ) = (40km s−1 kpc−1 , 40◦ ) K9 <5J9 H9GH98 H<9 DF98=7H=CBG C: H<9 AC89@G :CF H<9 F58=5@ 7CADCB9BHG C: H<9 8=GD9FG=CB H9BGCF 7CAD5F=B; H<9A K=H< ∼ 2000 F58=5@ J9 @C7=H=9G =B H<9 G5A9 9@8G ,<9 F9GI@HG C: H<=G G97CB8 G9H C: 7CBGHF5=BHG G<CKG H<5H 5 65F =B7@=B5H=CB =G B99898 =B CF89F HC C6H5=B 5 J9FH9L 89J=5H=CB ,<IG K9 <5J9 7<CG9B (Ωb , φbar ) = (40km s−1 kpc−1 , 40◦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Ωb , φbar ) = (40km s−1 kpc−1 , 30◦) !H =G F9A5F?56@9 H<5H CIF AC89@G G<CK 5 69GH H K<=7< G99AG HC 5;F99 K=H< H<9 @5GH 89H9FA=B5H=CBG C: 65F D5F5A9H9FG 7.

(295) +7<K5FNG7<=@8 AC89@ :CF H<F99 A=BCF5L=G 9@8G. . 7CF8=B; HC CIF F9GI@HG H<9 ;5@57H=7 65F DF9G9BHG 5 <=;< =B7@=B5H=CB 30◦ − 40◦ =B H<9 FGH ;5@57H=7 EI58F5BH 5B8 5 AC89F5H9 D5HH9FB GD998 C: 40 km s−1 kpc−1 88=H=CB5@ 5B5@M G=G HC CIF 69GH AC89@ G<CK H<5H H<=G 65F =G <=;<@M 8CA=B5H98 6M GHC7<5GH=7 CF6=HG 9J9B K<9B 75% C: H<9 CF6=HG G9@97H98 :CF H<9 69GH H 7CFF9GDCB8 HC F9;I@5F CF6=HG H<9M >IGH DFCJ=89 ∼ 20% C: H<9 HCH5@ A5GG C: H<9 65F 9B79 H<9 A5=B GHFI7HIF9 C: H<=G AC89@ =G ;=J9B 6M 9@CB;5H98 GHC7<5GH=7 CF6=HG K<=7< GACCH<@M @@ H<9 D<5G9GD579 K<=@9 ACF9 89H5=@98 :95HIF9G 5F9 F9DFC8I798 6M H<9 F9;I@5F 7CADCB9BH C: H<9 65F /9 69@=9J9 H<5H H<9G9 F9GI@HG GI;;9GH 5 65F :CFA5H=CB G79B5F=C =B K<=7< H<9 65F <5G 69=B; :CFA98 :FCA H<9 8=G? 5B8 =G =B 5B 58J5B798 9JC@IH=CB5FM GH5;9 =H <5G 699B G@CK98 8CKB 5B8 A589 ACF9 A5GG=J9 H<FCI;< 5B;I@5F ACA9BHIA 9L7<5B;9 5B8 ;5G 577F9H=CB H H<9 G5A9 H=A9 H<9 79BHF5@ A5GG 7CB79BHF5H=CB D@5M 5B =ADCFH5BH FC@9 =B 8=GFIDH=B; H<9 CF6=HG 7@CG9 HC H<9 79BH9F 5B8 =B7F95G=B; H<9 :F9EI9B7M C: GHC7<5GH=7 CF6=HG. Figure 4.12 CBHCIF @@98 D@CH K=H< H<9 H5F;9H HH98 =B 5@@ H<9 @=6F5F=9G ,<=G H5F;9H <5G 699B G75@98 HC H H<9 AC89@G =B 957< 75G9.

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(333) . 5589TG /=B8CK lv σr σrl (deg)

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