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Evolution of galactic density waves at the unstable stage
inese Astronomy
© P e r g a m o n Press, Printed in Great Britain
2 (1978) 139-146
01~/78/0601-01~$07.50/0
Acta Astr. Sinica 18 (1977) 105-112
EVOLUTION OF GALACTIC DENSITY WAVES AT THE UNSTABLE STAGE
Yue Zeng-yuan (Received
1976 September 23)
ABSTRACT An evolution law of galactic density waves is derived from the equations of stellar dynamics and compared with the results [4] obtained earlier by the author using fluid mechanics. The comparison shows that the results for the unstable stage obtained in the two approaches are close to each other.
1. INTRODUCTION Galactic spiral structure has been discussed by Lin and coworkers [I-3] on a quasistationary model (the QSSS hypothesis), using both fluid dynamical and stellar dynamical methods.
The former method is rigorous, while the latter is approximate.
It was shown that,
in the quasi-stationary model, the results from the two approaches are close to each other, except for the regions around
Ivl = 1 (the inner and outer Lindblad resonances).
According to the world view of dialectical materialism, density waves, in common with other things in nature, must pass through evolutionary stages.
Before reaching the quasi-stationary
stage, they must have passed through an unstable stage when the wave amplitude was on the increase.
Starting from an evolutionary model of fliud dynamics, the present author [4] has
found that the main property during the unstable stage is that of a "quasi-material arm". Because of this property,
the form of the wave will undergo winding or curling, forming a
retarding wave, and the wave number will increase. a quasi-stationary state will automatically result.
When the wave number reaches certain size, After supposing that Q < I around the
co-rotation circle, the following physical picture was obtained: density waves are produced in the unstable region about the co-rotation circle, gradually winding up into retarding, short waves; after becoming quasi-stationary, with the group velocity.
they move out on both sides of the co-rotation circle
Because density waves can be continually formed in the unstable
region, the maintenance over a long time is no longer a serious problem. In this paper, I shall use the stellar dynamical approach to discuss the evolution of density waves during the unstable stage and shall make a comparison with the results obtained on the fluid dynamical approach.
140
Galactic Density Waves
2. EQUATIONS OF STELLAR DYNAMICS AND AN EVOLUTIONARY MODEL The stellar dynamical equation is simply the collision-free Boltzmann equation, which, fol a flat star system with differential rotation, has the form, [4],
(2) and $ is the two-dimensionaldistribution function (i,- +(ij,f3,Csr, Ce, t>
(3)
all other symbols have the same meanings as in [3]. Let &,(cj,Cm, CO)
be the constant, axi-symmetric distribution function for the basic
state, Vo(6) the corresponding potential. Take G(G) such that (4) Let
9 = 40 + 9, = $b,(l -I- c/5*) = e-Qo(l -I- +*), V = Vo(L;)
+ v&3,
0, t)
(5) (6)
then the relative perturbation distribution function $* and the perturbing potential vl satisfy the linearised equation
at =
am,
F a3+ a,%
(7)
and the Poisson equation &+'"+-&++= ( 6 a;,
4% G a*1 6(z)
)
(8)
in which aJcIis the surface perturbing density f'*,= me and a61 and aeI satisfy
JJ
cCtrdCtrdCs= m*
oc$,= - av -1,
a65
JJ
&,+* dCsdCe
081 F----I i av
63 a3
We seek particular solutions of JI*and VI of the
in which
O.and A vary slowly with Liand t,$,
t)p’mOj, fhum A,
(10)
form
&* = &(&, t, Ca, C&‘[Y(“*‘)-mel
v, -_ A(&,
(9)
(11) (12)
0 are complex functions, and m is a positive
number (representing the number of spiral arms).
Introduce complex wave number and complex
Galactic Density Waves
frequency
I( = l(R + if(, ~ ..0~ =
141
O~..___,~R+ iO~_.__A~
O #o
0 #~
O Co"
(13) O~R + i 0~, Ot Ot "
w = w~ + iwt =- O__~_~= Ot We shall use ~
and ~
The relative change in ~ over 1/4 period.
(14)
to denote respectively any quantity that varies slowly with ~ and t. is then much smaller than I over I/4 wavelength, and so is that of
Mathematically expressed, these conditions are
O~R 'O~l-/-/-Ind~" << -~-- =
oiinl<
-~,
Ikk],
(15)
= I~[.
(16)
We now seek the following type of solution: even at the unstable stage, the spatial variation of the physical quantities already assume a spiral form.
In other words, whether in the
quasi-statlonary or in the unstable stage, the actual amplitudes @espectively [A[e-~)
]~[e - ~
and
of the perturbing quantities ~* and Y 1 vary slowly with ~; while on the other hand,
since amplitudes may increase appreciably over I/4 of the period, we do not regard e-~ as slowly varying with t.
Surmnarising, the slowly varying quantities at the different stages are
slowly varying with 5,
quasi-stationary or unstable stage
, - ~ -'~ {4', A: os, k; e-~'}
(17)
~" {q6, A; w, k; e'~t} ! {4,, -~; o0, k}
slowly 9arying with t~
quasi-stationary stage unstable stage
(18)
Hence we have
]kt] << IkR[ I~;I~ << I~R] [ "- I~RI
quasi-stationary or unstable stage
(19)
quasi-stationary stage unstable stage
(20)
The forms (11), (12), plus the conditions (15)-(20) define the type of evolution model we wish to find. Substituting (11) in (7), and using the approximation of fast varying phases (of which (15) -(20) are the concrete expressions), and using the fact that for the majority of stars,
Ic~,l, Icel < < , ~
(21)
we obtain
04' o4, ~-A-~-= - - ~'-A-f-_ + i ( v + ag ) 4~
%
iaA U~
OQo, O~
(22)
in which
V
~---
(~0
-
-
m~)lx,
a-----(]{R~)(2~/~2),
(23) (24)
142
Galactic Density Wave
(25)
U, ~- ( ~ ) ( 2 ~ / x ) ,
It is to be noted that
---- C e / U l ,
(26)
~--- C o / ~ .
(27)
(22) is of the same form as equation
(C15) in [3] (which contains
incidentally a misprint of 2 in the denominator on the right), the only difference is, here, and v are complex functions of (~, t). If we take a basic state which satisfies a generalised
distribution
Qo(r', ~),
Qo = where
Schwarzschild
(28)
r2~_ ~ 2 + ~ 2
(29)
then the solution of (22) is, [3],
2.4 O ' 0 ( r ' ) { 1 - - q(a~, aT, v ) } , U]
4'
where
q(ff~, aT, ~)
~= sinu~
- -
2~
Hence the perturbing distribution 4', ==
I I"
*
+co|S)]
--*[vS--a~iln,~--a*l(!
_
(30)
e
dS.
(31)
function is
d~og'* = ~o dPe'[¢~''n-"az =
2z/ed~<'")-'a] U~ ~'°(ra)(1 -- q) = - - ~ ( r ' ) ( 1
If we further take the basic state to have the original Schwarzschild
-- q)
(32)
distribution
PO
~bo = P 0 ( ~ ) e - - ~ -(~'+n') where
1% ~ U~/,
(33)
t h e n we have q~o(r,)( 1 __ q ) .
V,
(34)
and h e n c e a., =
v.,
v,
H*0(]
-
q)eC.aCo
=
_
v,
<(1 _
q)>,
S;*o d C , d C o
(35)
in which o,0 is the basic surface density and the angular brackets denote averages over the velocity space according to the basic state distribution function.
_The solution of the
Poisson's equation to the first order in the fast varying phase approximation,
~**
is [1],
2~G
(36)
From (35) and (36), we have
I~RI =
~*o < ( x - q ) > .
2uG Hence, < ( l - q ) > generalised that is,
must be r e a l .
(37)
Under t h e p r e m i s e t h a t t h e b a s i c s t a t e
Schwarzschild distribution,
it
v =
is not difficult
vR =
~ol ~
O,
(~o,~ - co ~
m_O)/~, wR,
satisfies
the
t o p r o v e t h a t when ( i ) v i s r e a l ,
(38) (39)
Galactic Density Wave
143
or (ii) ~ is purely imaginary,
v=iv~,=iwJ~,
VR~ O, <(l-q)>
is real.
(40)
(41)
WR-~" m ~0
The case of (i) was proved in [3]. sinv~
For case (ii), we note that, here,
sh vl~
is real, hence it is only necessary to prove that
II~ Ii e-'[vs-a~""s-~(t+c°'s)]~l,o(~2+ ~72)dSd~ d~? is real.
Now, its imaginary part is equal to
Ifl
I:_.
-'-
s+
+
(42)
but the last integrand is an odd function, and the first, an even function of (~, ~), hence the whole integral is zero.
Therefore, when
v = Z~l, < ( l - q ) >
is indeed real.
As was pointed out in [4], case (i) ~orresponds to the quasi-stationary stage, and case (ii), the unstable stage. When ~0 satisfies the Schwarzschild distribution (33), we have
<(l--q)>
1--
=
v~
. 1
T~I
sin v=
--
" e_,,s,
e~Cl+¢o,s~dS ~o
"
-.
fl 1)R~"~~I -,,c°svRS" c-x(l+¢°'s) dS
(43)
sinvR~:
--
,
.
• vt~
1
sh vl~:
e~tS_X(t+co, S) dS
2~ d-,,
(v=iv,)
(44)
where
~
"
(45)
When ~ = ~R' the substitution of (43) in (37) gives the usual dispersion relation in the quasl-stationary model.
When ~ = ~Vl, if we denote
~,,(x)
=
~I_e ",s-xc'*'°'s'as, I
"
~vl(X) "~" 1 "t- V~(l X
VI"
GaI(X)},
(46)
(47)
shvt~
t
then we have <(i
-
q)>
=
"'" ~,,(x)
i -
sh I)I~
=
x
1 +,,i
(48)
P,,(x) (49)
Inserting (37), we have
(50)
144
Galactic Density Waves
N2
where
]~o "~- - 2~rGO',o
is the critical wave number for a cold disk introduced by Toomre [5], and xo =
~
(51) Toomre proved that 2 2 ~o< C~>~,. x~
-- 0.2857,
(52) where
mla
is the minimum value of < C ~ >
that can suppress instability against
axi-symmetric perturbations of various wavelengths. xo =
Hence
0 . 2 8 5 7 Q2,
(53)
where 2 2 Q2=/(C.> ....
(54)
The second expression in (41) shows that, for the unstable stage, the pattern speed ~(
= ~R/m) is equal to the matter speed ~.
(This is approximately, but not strictly, equal
to the average angular speed of all the stars.
According to its definition in [4], it is
that velocity which all the stars would have if they all moved with the same velocity under the basic mass distribution).
This is the "Quasl-material arm property" during the
unstable stage discussed in detail in [4].
Hence we have the evolution law of the wave
number for the unstable stage [4]: O~R =
m~'(~),
Ot
(55)
and should the wave be of the retarding type,
Ot
(56)
On the other hand, (50) gives the relation between the rate of increase of wave amplitude and the wave number.
Thus. the second expression of (41) and the expression (50) completely
describe the entire evolution during the unstable stage and can be regarded as the basic relations for the unstable stage. 1.0 ~
z,, ~ 0 v
,~ 0.5
i
t
2
i
$
Fig. 1 The calculated results of the function
~,,(x)
are given in TABLE I and shown in Fig. 1
Galactic
TABLE I
3.0
0.I 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.5342 0.5352 0.5379 0. 5423 0.5484
0.3457 0.3468 0.3499 0.3549 0.36J8
0.2523
These numerical increase
4.0
0.5559 0.5647 0.5747
0.3704 0.3805 0.3920
0.5855 0.5971 0.6092
0.4045 0.4180 0.4322
0.2758 0.2855 0.2965 0.3087 0.3218 0.3357
0.2533 0.2562 0.2610 0.2676
145
~, ,(x)
The F u n c t i o n
2.0
1.0 0
Density Waves
5.0
6.0
7.0
8.0
0~1982 0.1991
0.1633 0.1641
0.1389 0.1396
0.2018 0.2062 0.2122 0.2198
0.1665 0.1705 0.1760 0.1829
0.1419 0.1455 0.1505 0.1569
O. 1209 0.1216 O. 1236 0.1270 0.1316
0.1071 0.1077 0.1096
0.2287
0.1911 0.2005 0.2109 0.2222 0.2343
0.1644 0.1730 0.1826 0.1931 0.2043
0.1374 O. 1444
0.1224 0.1288 0.1362 0.1445 0.1536 0.1633
0.2389 0.2502 0.2625 0.2755
0.1524 0.1613 0.1710 0.1814
results can now be used to give the relation between the amplitude
rate and the wave number.
We introduce
the relative wave number
~- ~R/ko, and show in Fig. 2 the relation between ~ z 0 = 0.2312,
0.1127 0.1170
0.1828,
0.1400).
In Fig. 3,
(57)
and E for Q = 0.9, 0.8 and 0.7 (corresponding
we show the relation for the same three values of
Q=0.7 (xQ -~0.1400) !
1
Fig. 2
The ( ~
2
3
~ ~) relation
4
5
8
in the stellar dynamical
(arrows mark the direction
approach.
of evolution)
2
xo = O. 1400
1
Fig. 3
The (~l2
x0, derived from the elementary
2
to
3
4
5
6
ff
~ E) relation in the fluid dynamical
approach.
theory
A comparison of these
(the fluid dynamical model).
146
Galactic Density Waves
two figures shows that the fluid dynamical
model is a satisfactory approximation to the
stellar dynamical theory. Since the duration T 1 of the unstable stage and total proportional increase n during the unstable stage are calculable from the v~ (~;x 0) curves, for which the two approaches give entirely similar results, we can expect that the results for TI(~ , z0) and
n(T1~ z0) in the
two theories will also be close to each other. Fig. 4 shows the boundary of the unstable region according to fluid dynamics and stellar dynamics, and the direction of evolution of density waves within the unstable region.
X 0 '
stable region 0.3
- - - 0.2837 - 0.23 ,~
0.2 / 0.1
Stellar dynam~ cs
unsEaDle !r
0.5 Fig. 4
-,,\\\~ i
/7"/_ .= . " \ \ \ ~ fluid dynamics unstable rlgion
1.0
A,',~
Boundary of the unstable region and the direction of evolution within the region.
The author wishes to express the warmest thanks to Comrade Zhang Cong-jin for enthusiastic help with the numerical calculation.
REFERENCES [1] [2] [3] [4] [5]
Lin, C.C. & Shu, F.H., A8trophys. J., 140 (1964), 646. Lin, C.C. SIAM J. Appl. Math., 14 (196--6~, 876. Lin, C.C., in "Galactic Astronomy" (ed. Hong-Yee & Amador Muriel), 2, 1 - 93. Yue Zeng-yuan, ghongguo Ke~:ue (Chinese language edition of Scientia Sinica) 1976 6, 624. Toomre, A., Astmophy8. J., 139 (1964), 1217.
© P e r g a m o n Press, Printed in Great Britain
2 (1978) 139-146
01~/78/0601-01~$07.50/0
Acta Astr. Sinica 18 (1977) 105-112
EVOLUTION OF GALACTIC DENSITY WAVES AT THE UNSTABLE STAGE
Yue Zeng-yuan (Received
1976 September 23)
ABSTRACT An evolution law of galactic density waves is derived from the equations of stellar dynamics and compared with the results [4] obtained earlier by the author using fluid mechanics. The comparison shows that the results for the unstable stage obtained in the two approaches are close to each other.
1. INTRODUCTION Galactic spiral structure has been discussed by Lin and coworkers [I-3] on a quasistationary model (the QSSS hypothesis), using both fluid dynamical and stellar dynamical methods.
The former method is rigorous, while the latter is approximate.
It was shown that,
in the quasi-stationary model, the results from the two approaches are close to each other, except for the regions around
Ivl = 1 (the inner and outer Lindblad resonances).
According to the world view of dialectical materialism, density waves, in common with other things in nature, must pass through evolutionary stages.
Before reaching the quasi-stationary
stage, they must have passed through an unstable stage when the wave amplitude was on the increase.
Starting from an evolutionary model of fliud dynamics, the present author [4] has
found that the main property during the unstable stage is that of a "quasi-material arm". Because of this property,
the form of the wave will undergo winding or curling, forming a
retarding wave, and the wave number will increase. a quasi-stationary state will automatically result.
When the wave number reaches certain size, After supposing that Q < I around the
co-rotation circle, the following physical picture was obtained: density waves are produced in the unstable region about the co-rotation circle, gradually winding up into retarding, short waves; after becoming quasi-stationary, with the group velocity.
they move out on both sides of the co-rotation circle
Because density waves can be continually formed in the unstable
region, the maintenance over a long time is no longer a serious problem. In this paper, I shall use the stellar dynamical approach to discuss the evolution of density waves during the unstable stage and shall make a comparison with the results obtained on the fluid dynamical approach.
140
Galactic Density Waves
2. EQUATIONS OF STELLAR DYNAMICS AND AN EVOLUTIONARY MODEL The stellar dynamical equation is simply the collision-free Boltzmann equation, which, fol a flat star system with differential rotation, has the form, [4],
(2) and $ is the two-dimensionaldistribution function (i,- +(ij,f3,Csr, Ce, t>
(3)
all other symbols have the same meanings as in [3]. Let &,(cj,Cm, CO)
be the constant, axi-symmetric distribution function for the basic
state, Vo(6) the corresponding potential. Take G(G) such that (4) Let
9 = 40 + 9, = $b,(l -I- c/5*) = e-Qo(l -I- +*), V = Vo(L;)
+ v&3,
0, t)
(5) (6)
then the relative perturbation distribution function $* and the perturbing potential vl satisfy the linearised equation
at =
am,
F a3+ a,%
(7)
and the Poisson equation &+'"+-&++= ( 6 a;,
4% G a*1 6(z)
)
(8)
in which aJcIis the surface perturbing density f'*,= me and a61 and aeI satisfy
JJ
cCtrdCtrdCs= m*
oc$,= - av -1,
a65
JJ
&,+* dCsdCe
081 F----I i av
63 a3
We seek particular solutions of JI*and VI of the
in which
O.and A vary slowly with Liand t,$,
t)p’mOj, fhum A,
(10)
form
&* = &(&, t, Ca, C&‘[Y(“*‘)-mel
v, -_ A(&,
(9)
(11) (12)
0 are complex functions, and m is a positive
number (representing the number of spiral arms).
Introduce complex wave number and complex
Galactic Density Waves
frequency
I( = l(R + if(, ~ ..0~ =
141
O~..___,~R+ iO~_.__A~
O #o
0 #~
O Co"
(13) O~R + i 0~, Ot Ot "
w = w~ + iwt =- O__~_~= Ot We shall use ~
and ~
The relative change in ~ over 1/4 period.
(14)
to denote respectively any quantity that varies slowly with ~ and t. is then much smaller than I over I/4 wavelength, and so is that of
Mathematically expressed, these conditions are
O~R 'O~l-/-/-Ind~" << -~-- =
oiinl<
-~,
Ikk],
(15)
= I~[.
(16)
We now seek the following type of solution: even at the unstable stage, the spatial variation of the physical quantities already assume a spiral form.
In other words, whether in the
quasi-statlonary or in the unstable stage, the actual amplitudes @espectively [A[e-~)
]~[e - ~
and
of the perturbing quantities ~* and Y 1 vary slowly with ~; while on the other hand,
since amplitudes may increase appreciably over I/4 of the period, we do not regard e-~ as slowly varying with t.
Surmnarising, the slowly varying quantities at the different stages are
slowly varying with 5,
quasi-stationary or unstable stage
, - ~ -'~ {4', A: os, k; e-~'}
(17)
~" {q6, A; w, k; e'~t} ! {4,, -~; o0, k}
slowly 9arying with t~
quasi-stationary stage unstable stage
(18)
Hence we have
]kt] << IkR[ I~;I~ << I~R] [ "- I~RI
quasi-stationary or unstable stage
(19)
quasi-stationary stage unstable stage
(20)
The forms (11), (12), plus the conditions (15)-(20) define the type of evolution model we wish to find. Substituting (11) in (7), and using the approximation of fast varying phases (of which (15) -(20) are the concrete expressions), and using the fact that for the majority of stars,
Ic~,l, Icel < < , ~
(21)
we obtain
04' o4, ~-A-~-= - - ~'-A-f-_ + i ( v + ag ) 4~
%
iaA U~
OQo, O~
(22)
in which
V
~---
(~0
-
-
m~)lx,
a-----(]{R~)(2~/~2),
(23) (24)
142
Galactic Density Wave
(25)
U, ~- ( ~ ) ( 2 ~ / x ) ,
It is to be noted that
---- C e / U l ,
(26)
~--- C o / ~ .
(27)
(22) is of the same form as equation
(C15) in [3] (which contains
incidentally a misprint of 2 in the denominator on the right), the only difference is, here, and v are complex functions of (~, t). If we take a basic state which satisfies a generalised
distribution
Qo(r', ~),
Qo = where
Schwarzschild
(28)
r2~_ ~ 2 + ~ 2
(29)
then the solution of (22) is, [3],
2.4 O ' 0 ( r ' ) { 1 - - q(a~, aT, v ) } , U]
4'
where
q(ff~, aT, ~)
~= sinu~
- -
2~
Hence the perturbing distribution 4', ==
I I"
*
+co|S)]
--*[vS--a~iln,~--a*l(!
_
(30)
e
dS.
(31)
function is
d~og'* = ~o dPe'[¢~''n-"az =
2z/ed~<'")-'a] U~ ~'°(ra)(1 -- q) = - - ~ ( r ' ) ( 1
If we further take the basic state to have the original Schwarzschild
-- q)
(32)
distribution
PO
~bo = P 0 ( ~ ) e - - ~ -(~'+n') where
1% ~ U~/
(33)
t h e n we have q~o(r,)( 1 __ q ) .
V,
(34)
and h e n c e a., =
v.,
v,
H*0(]
-
q)eC.aCo
=
_
v,
<(1 _
q)>,
S;*o d C , d C o
(35)
in which o,0 is the basic surface density and the angular brackets denote averages over the velocity space according to the basic state distribution function.
_The solution of the
Poisson's equation to the first order in the fast varying phase approximation,
~**
is [1],
2~G
(36)
From (35) and (36), we have
I~RI =
~*o < ( x - q ) > .
2uG Hence, < ( l - q ) > generalised that is,
must be r e a l .
(37)
Under t h e p r e m i s e t h a t t h e b a s i c s t a t e
Schwarzschild distribution,
it
v =
is not difficult
vR =
~ol ~
O,
(~o,~ - co ~
m_O)/~, wR,
satisfies
the
t o p r o v e t h a t when ( i ) v i s r e a l ,
(38) (39)
Galactic Density Wave
143
or (ii) ~ is purely imaginary,
v=iv~,=iwJ~,
VR~ O, <(l-q)>
is real.
(40)
(41)
WR-~" m ~0
The case of (i) was proved in [3]. sinv~
For case (ii), we note that, here,
sh vl~
is real, hence it is only necessary to prove that
II~ Ii e-'[vs-a~""s-~(t+c°'s)]~l,o(~2+ ~72)dSd~ d~? is real.
Now, its imaginary part is equal to
Ifl
I:_.
-'-
s+
+
(42)
but the last integrand is an odd function, and the first, an even function of (~, ~), hence the whole integral is zero.
Therefore, when
v = Z~l, < ( l - q ) >
is indeed real.
As was pointed out in [4], case (i) ~orresponds to the quasi-stationary stage, and case (ii), the unstable stage. When ~0 satisfies the Schwarzschild distribution (33), we have
<(l--q)>
1--
=
v~
. 1
T~I
sin v=
--
" e_,,s,
e~Cl+¢o,s~dS ~o
"
-.
fl 1)R~"~~I -,,c°svRS" c-x(l+¢°'s) dS
(43)
sinvR~:
--
,
.
• vt~
1
sh vl~:
e~tS_X(t+co, S) dS
2~ d-,,
(v=iv,)
(44)
where
~
"
(45)
When ~ = ~R' the substitution of (43) in (37) gives the usual dispersion relation in the quasl-stationary model.
When ~ = ~Vl, if we denote
~,,(x)
=
~I_e ",s-xc'*'°'s'as, I
"
~vl(X) "~" 1 "t- V~(l X
VI"
GaI(X)},
(46)
(47)
shvt~
t
then we have <(i
-
q)>
=
"'" ~,,(x)
i -
sh I)I~
=
x
1 +,,i
(48)
P,,(x) (49)
Inserting (37), we have
(50)
144
Galactic Density Waves
N2
where
]~o "~- - 2~rGO',o
is the critical wave number for a cold disk introduced by Toomre [5], and xo =
~
(51) Toomre proved that 2 2 ~o< C~>~,. x~
-- 0.2857,
(52) where
is the minimum value of < C ~ >
that can suppress instability against
axi-symmetric perturbations of various wavelengths. xo =
Hence
0 . 2 8 5 7 Q2,
(53)
where 2 2 Q2=
(54)
The second expression in (41) shows that, for the unstable stage, the pattern speed ~(
= ~R/m) is equal to the matter speed ~.
(This is approximately, but not strictly, equal
to the average angular speed of all the stars.
According to its definition in [4], it is
that velocity which all the stars would have if they all moved with the same velocity under the basic mass distribution).
This is the "Quasl-material arm property" during the
unstable stage discussed in detail in [4].
Hence we have the evolution law of the wave
number for the unstable stage [4]: O~R =
m~'(~),
Ot
(55)
and should the wave be of the retarding type,
Ot
(56)
On the other hand, (50) gives the relation between the rate of increase of wave amplitude and the wave number.
Thus. the second expression of (41) and the expression (50) completely
describe the entire evolution during the unstable stage and can be regarded as the basic relations for the unstable stage. 1.0 ~
z,, ~ 0 v
,~ 0.5
i
t
2
i
$
Fig. 1 The calculated results of the function
~,,(x)
are given in TABLE I and shown in Fig. 1
Galactic
TABLE I
3.0
0.I 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.5342 0.5352 0.5379 0. 5423 0.5484
0.3457 0.3468 0.3499 0.3549 0.36J8
0.2523
These numerical increase
4.0
0.5559 0.5647 0.5747
0.3704 0.3805 0.3920
0.5855 0.5971 0.6092
0.4045 0.4180 0.4322
0.2758 0.2855 0.2965 0.3087 0.3218 0.3357
0.2533 0.2562 0.2610 0.2676
145
~, ,(x)
The F u n c t i o n
2.0
1.0 0
Density Waves
5.0
6.0
7.0
8.0
0~1982 0.1991
0.1633 0.1641
0.1389 0.1396
0.2018 0.2062 0.2122 0.2198
0.1665 0.1705 0.1760 0.1829
0.1419 0.1455 0.1505 0.1569
O. 1209 0.1216 O. 1236 0.1270 0.1316
0.1071 0.1077 0.1096
0.2287
0.1911 0.2005 0.2109 0.2222 0.2343
0.1644 0.1730 0.1826 0.1931 0.2043
0.1374 O. 1444
0.1224 0.1288 0.1362 0.1445 0.1536 0.1633
0.2389 0.2502 0.2625 0.2755
0.1524 0.1613 0.1710 0.1814
results can now be used to give the relation between the amplitude
rate and the wave number.
We introduce
the relative wave number
~- ~R/ko, and show in Fig. 2 the relation between ~ z 0 = 0.2312,
0.1127 0.1170
0.1828,
0.1400).
In Fig. 3,
(57)
and E for Q = 0.9, 0.8 and 0.7 (corresponding
we show the relation for the same three values of
Q=0.7 (xQ -~0.1400) !
1
Fig. 2
The ( ~
2
3
~ ~) relation
4
5
8
in the stellar dynamical
(arrows mark the direction
approach.
of evolution)
2
xo = O. 1400
1
Fig. 3
The (~l2
x0, derived from the elementary
2
to
3
4
5
6
ff
~ E) relation in the fluid dynamical
approach.
theory
A comparison of these
(the fluid dynamical model).
146
Galactic Density Waves
two figures shows that the fluid dynamical
model is a satisfactory approximation to the
stellar dynamical theory. Since the duration T 1 of the unstable stage and total proportional increase n during the unstable stage are calculable from the v~ (~;x 0) curves, for which the two approaches give entirely similar results, we can expect that the results for TI(~ , z0) and
n(T1~ z0) in the
two theories will also be close to each other. Fig. 4 shows the boundary of the unstable region according to fluid dynamics and stellar dynamics, and the direction of evolution of density waves within the unstable region.
X 0 '
stable region 0.3
- - - 0.2837 - 0.23 ,~
0.2 / 0.1
Stellar dynam~ cs
unsEaDle !r
0.5 Fig. 4
-,,\\\~ i
/7"/_ .= . " \ \ \ ~ fluid dynamics unstable rlgion
1.0
A,',~
Boundary of the unstable region and the direction of evolution within the region.
The author wishes to express the warmest thanks to Comrade Zhang Cong-jin for enthusiastic help with the numerical calculation.
REFERENCES [1] [2] [3] [4] [5]
Lin, C.C. & Shu, F.H., A8trophys. J., 140 (1964), 646. Lin, C.C. SIAM J. Appl. Math., 14 (196--6~, 876. Lin, C.C., in "Galactic Astronomy" (ed. Hong-Yee & Amador Muriel), 2, 1 - 93. Yue Zeng-yuan, ghongguo Ke~:ue (Chinese language edition of Scientia Sinica) 1976 6, 624. Toomre, A., Astmophy8. J., 139 (1964), 1217.