Self-gravitující disk s nulovou disperzí rychlostí
dZ       1    čREvr dEv9
-+ - {-+ -} = 0
dt     r čr ae
č\R d\R        ve  d\R     ve2      d O
- +Vr- +----=--
St ČR R   50        R SR
d Ve č Ve        v«   íve      vRve       1 5$
- +vr-+--+ -=---
dt ČR        R  čQ        R R ČQ
d2<D    1 dd>     1 ^<D č2®
-+--+--+-= 4 7r G Z(R, 0, t) S(z)
ČR2   R dR     R2dQ 6z2
Počáteční rovnovážný stav: Lineární porucha:
2 = 2o (R) E = Se (R) + Ei (R,e,t)
O = 0>o(R,z) O = O0(R,z) + 0>i(R,9,z,t)
vR= 0 vr= vR1(R,0,t)
ve = V(R) = R £1(R) > 0 ve = V(R) + v9i(R,8,t)
1-3
Linearizace rovnic s poruchou
-+ Í1-+--+--= 0
dt č Q     R ČR R ÔQ
Ô Vri d Vri Ô Oi
-+ íl--2 íl Vei =--
dt ôq ôr
dvei dvei 1 d<t>i -+Í1-+ 2 B vR, =---
et      ae r ôq
-+--+--+-= 4 7T G Lx ô(z)
čr2   r ôr    r2dB ôz2
9R» K
Elementární řešení (vlnové módy):
Hustotní vlna spirálního tvaru
U = £*(R) exp [i(cot - m9)] = £'(R) exp [i(o»t - m8 + F(R))] E*(R) komplexní:      £*(R) = E (R) exp [ i F(R) ]
F(R): „fázový faktor" (phase factor),
„tvarová funkce" (shape function)
podobně pro <Di, vRi a vei:
Ox=0* (R) exp[i(ot-m8)] Vri= Vri*(R) exp[i(cot - m9)j Vei= vei* (R) exp[i(ot - m9)]
Frekvence vlny:
co = Or + ÍCOi
oscilující módy:    | coR | » | g>i
nestabilní módy: - g>i » | coR |
tlumené: rostoucí: neutrální:
Oi>0
o>i < 0 (overstability) 0)i= O
SP it AIM* STtV*Tt/ltA
otct'Uet
Tr»* reifet W -   C^Vtourf]
logaritmická'
dg* -
2-9
• Ä A*/At*// Vt 4/0*4 '>M4!4
><?, A ft, ť)
* u»»a lmí ' hm eye e/sio
" TRA ILM*"       »IMM* *
1-44
Figure 6-5. Leading and trailing
observer's view
side view
lo the sun
<g)= receding     Q = approaching
Figure 6-6. The appearance of leading and trailing arms. Galaxy A has leading arms, while galaxy B has trailing arms, but both exhibit the same pattern on the sky and the same radial velocity field.
To Ihe sun
Figure 6-7. Distinguishing near and far sides of a disk galaxy. The dots represent objects such as novae or globular clusters. There is an obscuring dust layer in the central plane of the disk which is shown as a line in the side view at left. In tne observer's view, at right, objects behind the dust layer are fainter and are shown as open circles.
t
»hu
* TMA SPttALHfGH *A*f£řJ
Ě
?í . *m, íftí - i.l i/j &t>\I. t „Tise*
z y /   *(+-iWHS?/
* /0ex\4.ct *****
^  y o. 06 s o, or
2f AT&4i o.c
F * @. Ote C
j
«--
A
M/
"Čŕ,
r.
I
» 5
-Oř
Mr4T
1—r—I—r
III'
Ik|/kcm
i i i i i i i i i i i i i i i i t i i i i ill i i i i i
-3      -2 -1 leading
0
k/kertl
1 2 trailing
Figure 6-14. (a) The dispersion relation for tightly wound disturbances in gaseous (eq. (6-40), dashed ylines] and stellar [eq. (6-*46), solid lines] disks. The curves shown are (bottom to top) Q — 1, 1.5, and 2.  Since only \a\ and \k\ are shown, there is no distinction between leading and trailing waves, or waves inside and outside corotation. (b) Dispersion relation in the form of wavenumber versus radius for an m = 2 tightly wound wave in a stellar Mestel disk with Q = 1.5. The radial scale is in units of the corotation radius tcr, and only waves inside corotation are shown. The inner Lindblad resonance is at r = 0.293rcR.  The direction of the group velocity is shown by-arrows.
toomiu: 1
0.8
. 0.6
z04
02
	i i Q = 2
	
	
\ ^^13—-^^	
—      \      b ,s Short \               / long	X
\Q=y i	1 1 1
0.4 0.8 1.2
Wavelength, ^/^crjt
16
Fifiure 4 The Lin-Shu-Kalnajs dispersion relation for axisymmctrie density waves of 1 frequency to = vk and modest radial wavelength A in a thin, rotating disk of stars endo with Q times the minimum random motions required by Equation (3).
Lindblad resonance
Corotation
X/X,
1.5
1.0
0.5
1 IV ■ I	1-1-1-1-           i 1
Vx i vy	{
	I \
• t t i i	w
Q = 2 I i i » i i	\ \ \ \Lonq waves \\. ■ Q=1.2 \
i	1 \ / \ Q=1 >
- /	
i i	^--^Short waves
1 *' ^-— ; o*"^ 1           ! 1_	.j_i__i-1-1-1-
1.0
0.5
0
Fic. 6 — Dispersion relation ,\( v ) for demity waves (Lin and Shu). The arrows show tin- direction of propagation of the waves. Q measures die velocity dispersion.