The increase in random kinetic energy of stars by rapidly fluctuating gravitational fields (collisionless or violent relaxation) in disk galaxy models is investigated for three interaction potentials of the stars corresponding to (1) point stars, (2) rod stars of length 2 kpc, and (3) uniform density spherical stars of radius 2 kpc. To stabilize the galaxy against the large scale bar forming instability, a fixed field corresponding to a central core or halo component of stars was added with the stars containing at most 20 percent of the total mass of the galaxy. Considerable heating occurred for both the point stars and the rod stars, whereas the use of spherical stars resulted in a very low heating rate. The use of spherical stars with the resulting low heating rate will be desirable for the study of large scale galactic stability or density wave propagation, since collective heating effects will no longer mask the phenomena under study

Hohl, F.

English

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NASA TECHNICAL NOTE NASA TN D-7561
-2 -P283-
.C 15( o C cs30
-
---. c
COLLISIONLESS RELAXATION
IN SPIRAL GALAXY MODELS
by Frank Hohl
Langley Research Center
Hampton, Va. 23665
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION * WASHINGTON, D. C. * JUNE 1974
https://ntrs.nasa.gov/search.jsp?R=19740020192 2020-03-23T07:39:09+00:00Z
1. Report No. 2. Government Accession No. 3. Recipient's Catalog No.
NASA TN D-7561
4. Title and Subtitle 5. Report Date
June 1974
COLLISIONLESS RELAXATION IN SPIRAL GALAXY MODELS 6. Performing Organization Code
7. Author(s) 8. Performing Organization Report No.
L-9385
Frank Hohl L-9385
10. Work Unit No.
9. Performing Organization Name and Address 188-41-51-01
NASA Langley Research Center 11. Contract or Grant No.
Hampton, Va. 23665
13. Type of Report and Period Covered
12. Sponsoring Agency Name and Address Technical Note
National Aeronautics and Space Administration 14. Sponsoring Agency Code
Washington, D.C. 20546
15. Supplementary Notes
16. Abstract
The increase in random kinetic energy of stars by rapidly fluctuating gravitational
fields (collisionless or violent relaxation) in disk galaxy models is investigated for three
interaction potentials of the stars corresponding to (a) point stars, (b) rod stars of length
2 kpc, and (c) uniform density spherical stars of radius 2 kpc. To stabilize the galaxy
against the large-scale bar-forming instability, a fixed field corresponding to a central
core or halo component of stars was added with the stars containing at most 20 percent of
the total mass of the galaxy. Considerable heating occurred for both the point stars and
the rod stars, whereas the use of spherical stars resulted in a very low heating rate. The
use of spherical stars with the resulting low heating rate will be desirable for the study
of large-scale galactic stability or density wave propagation, since collective heating effects
will no longer mask the phenomena under study.
17. Key Words (Suggested by Author(s)) 18. Distribution Statement
Spiral galaxy models Unclassified - Unlimited
Computer simulation
STAR Category 30
19. Security Classif. (of this report) 20. Security Classif. (of this page) 21. No. of Pages 22. Price*
Unclassified Unclassified 1 bB 1 $3.00
For sale by the National Technical Information Service, Springfield, Virginia 22151
/
COLLISIONLESS RELAXATION IN SPIRAL GALAXY MODELS
By Frank Hohl
Langley Research Center
SUMMARY
The increase in random kinetic energy of stars by rapidly fluctuating gravitational
fields (collisionless or violent relaxation) in disk galaxy models is investigated for three
interaction potentials of the stars corresponding to (a) point stars, (b) rod stars of length
2 kpc, and (c) uniform-density spherical stars of radius 2 kpc. To stabilize the galaxy
against the large-scale bar-forming instability, a fixed field corresponding to a central
core or halo component of stars was added with the stars containing at most 20 percent of
the total mass of the galaxy. Considerable heating occurred for both the point stars and
the rod stars, whereas the use of spherical stars resulted in a very low heating rate. The
use of spherical stars with the resulting low heating rate will be desirable for the study of
large-scale galactic stability or density wave propagation, since collective heating effects
will no longer mask the phenomena under study.
INTRODUCTION
Recent numerical simulations of the evolution of disk galaxies (ref. 1) show that the
two-body relaxation time in these models is several hundred rotation periods. The large
velocity dispersion acquired by stars in the large-scale N-body simulations (refs. 2 and 3)
is therefore not a result of two-body relaxation or collisional effects. Thus, the increase
in the velocity dispersion must be ascribed to heating by turbulent gravitational fields or
violent relaxation as discussed by Lynden-Bell. (See ref. 4.) Since the time rate of
change in total energy of a star md(V2/2 - )/dt is equal to the time rate of change in
the gravitational field -m(aP/at), the acceleration or heating due to violent relaxation
does not lead to mass segregation. The opposite holds for two-body relaxation or colli-
sional effects which lead to equipartition of energy among stars of different mass. (See
ref. 1.) Disks of stars that are stabilized in accordance with Toomre's (ref. 5) local
criterion for axisymmetric stability are generally found to be unstable to the large-scale
nonaxisymmetric bar-making instability. (See ref. 3.) The heating of stars due to colli-
sionless relaxation can be investigated under more controlled conditions by adding an
axisymmetric gravitational field to suppress the bar instability. In an actual galaxy this
field could be supplied by the central core or halo population of stars. In this paper an
21
investigation is made of the collisionless relaxation of a disk of stars under the influence
of a fixed field. The effects of softening the interaction potential of the stars on the heat-
ing rate are investigated by using the potential corresponding to finite-length mass rods
and uniform-density spheres or spherical stars.
The model used for the calculations is that described earlier (ref. 6) with the fixed
radial field added.
SYMBOLS
f distribution function
G gravitational constant
1 length of rod "stars"
Me  solar mass, 1.989 x 1030 kg
m mass
orQ=
Or,min
r radial coordinate
r s  radius of spherical "stars"
t time
V velocity, km/sec
K epicyclic frequency, km-sec- 1 -kpc- 1
ct surface mass density, Mo-kpc-2
o velocity dispersion, km-sec-1
ar,min = 3.36Gwi/K, km-sec - 1
P gravitational potential
10 angular velocity, km-sec-l-kpc- I
2
Subscripts:
min minimum
r, radial and azimuthal component
RESULTS
The fixed axisymmetric component of the gravitational field used in the first part
of the calculations corresponds to a rotation curve (ref. 7) similar to the Schmidt model
of the galaxy (ref. 8)
4600r 210r km/sec (1)
100 + r 2  0.25 + r 2
for values of r up to 30 kpc.
The 50 000 interacting stars used in the calculation move under the influence of
this field and constitute about 20 percent of the total mass of the system. Three differ-
ent interaction potentials of the stars were used in the calculations:
(1) Point stars
Gm
r
(2) Rod stars of length 2 kpc
02(r) = Gm 2 r) +1+
(3) Uniform-density spherical stars of radius 2 kpc
Gm (3 r2)
3 (r) = 4 (r = 2)
3 (r) (r > 2)
Figure 1 illustrates these three potentials. Note that for r ! 0.5 kpc, the potential is
constant so that the self force on a star is zero. For r > 2 kpc, the interactions are
practically identical. The size of each of the n X n array of cells used in the self-
consistent potential calculation is 0.5 kpc so that the modification of the potential extends
:3
over four cell dimensions. Note that this modification of the potential is essentially an
artificial means to reduce the short wavelength collective instabilities while the long
range or global behavior remains essentially unchanged. For example, such an effect
could be caused in real galaxies if a large fraction of the mass was in large interpene-
trating gas clouds.
The evolution of the disk of 50 000 stars under the influence of the fixed radial field
corresponding to the rotation curve of equation (1) is shown in figure 2. The fixed com-
ponent of the gravitational field corresponds to 80 percent of the total system mass, the
remaining 20 percent of the mass being contained in the stars. Three different systems
are shown corresponding to the interaction potential used for each of the 50 000 stars in
the disk: (a) point stars, (b) rod stars of length 2 kpc, and (c) spherical stars of radius
2 kpc. An initial velocity dispersion corresponding to Toomre's (ref. 5) local criterion
for axisymmetric stability was given to the stars, that is,
ar ar,min = 3.36 Gj(r)
and
K(r)
ag- 2(r) r,min
where G is the gravitational constant, p(r) is the surface mass density of the stars,
and K(r) is the epicyclic frequency including the effects of the fixed field plus stars.
The initial mass density is given by A(r) = Cj(0) 1 _- r where R is the radius of the
disk. The time shown in figure 2 and subsequent figures is in units of the rotation period
at r = 10 kpc. The initial radius of the disk is 20 kpc. As can be seen, in all three
cases a spiral structure develops and remains throughout the simulation period (5 rota-
tions). The spiral structure extends toward the center of the disk to a radius of about
5 kpc. No spiral structure exists within this radius. Also, when the spirals first form,
they rotate with essentially the same angular velocity as the stars, for example,
25 km-sec-l-kpc - 1 at a radius of 10 kpc. During the first two rotations the spiral ele-
ments are not fixed rotating structures. They have a tendency to diffuse or break in one
region and then merge with other nearby components of spiral structure. This is the
case especially in regions of high differential rotation (inner regions). Thus, the spiral
structure is quite dynamic in reforming spiral arm components until after the first three
rotations when there is a tendency to form a two-arm spiral structure. The spiral struc-
ture in this case is formed primarily by a combination of gravitational instability and dif-
ferential rotation.
4
It is of interest to compare the rotation curves n(r) and epicyclic frequency K(r)
for the three interaction potentials used. This comparison is made in figure 3. Also
shown are the curves for 02 + L and 2 - K which are an indication of where spiral pat-2 2
terns can be the result of a density wave as proposed by Lin et al. (See ref. 9.) That is,
for a constant spiral pattern speed Os, a density wave is allowed only between the curves
62 + - and 2 - !. Note that for low spiral pattern speeds (=15 km-sec-l-kpc - 1 ) no2 2
density wave is allowed for small r, and no spiral structure is present for r < 5 kpc
in figure 2. However, the "pattern speed" in figure 2 is only slightly less (at most 10 per-
cent) than the mean rotation speed of the stars. Thus, the spiral structure cannot be con-
sidered a density wave. This result is in contrast to previous simulations without the
background field where the pattern speed was considerably less than the rotation velocity
of the stars (refs. 2 and 3).
To illustrate the heating rates for the three interaction potentials, figure 4 shows
the evolution of
Q = Cr(r,t)
Ur,min(r,t)
the ratio of the radial velocity dispersion to that required by Toomre's criterion. The
results show that for point stars Q quickly increases to a value near 2. For rod stars
the values of Q are generally slightly less, especially in the inner regions of the disk.
However, for the spherical stars, values of Q are only slightly above 1 over most of
the disk. The rise in Q in the region from 15 to 20 kpc is due primarily to the reduc-
tion in the surface mass density as shown in figure 5. This condition effectively reduces
Or,min during the evolution of the system and results in a larger Q. Thus, the use of
finite-size spherical stars effectively softens the interaction to give a slower heating
rate and thus allows a cooler disk. The evolution of the actual radial velocity dispersion
is shown in figure 6. Note that the radial velocity dispersion for the finite-size spherical
stars shows only very little increase, the peak velocity remaining near 40 km /sec. At
the same time or for the point and rod stars increases to around 60 km/sec. More
detailed information can be obtained from the radial velocity distribution of stars in differ-
ent regions of the disk as shown in figure 7. Note the drastic decrease of the low-velocity
stars in the central region (r < 4 kpc) of the disk for the point stars. In contrast for the
spherical stars, the relaxation effects result in only a slight reduction of the low-velocity
stars in that region. Thus, figure 7 illustrates again that for both the point and rod stars,
there is considerable heating, whereas for the spherical stars heating occurs at a much
lower rate. As expected, the same results were obtained for the azimuthal velocity
distribution.
5
These calculations were repeated for a fixed field corresponding to a uniform-
density spherical mass of radius 2 kpc. Therefore, for r > 2 kpc, the fixed field is
Keplerian. Also, the 50 000 stars now contain only 10 percent of the total mass of the
system. Figure 8 shows the evolution of the disk for that system. The evolution of the
radial velocity dispersion is shown in figure 9. Essentially the same result is obtained,
as illustrated in figure 6, with very little increase in ur for the spherical stars and
sizable increases for the point stars. Note that in the region r < 2 kpc, the epicyclic
frequency corresponding to the fixed component of the field is constant and ar remains
at a very small value.
CONCLUDING REMARKS
Since collisional and numerical relaxation effects have been shown to be negligible,
the heating occurring in disk galaxy models is caused by collective or violent relaxation.
The results presented here show that the heating and the associated short-wavelength
unstable modes can be effectively suppressed by using the interaction potential corre-
sponding to uniform-density spherical stars rather than that of point stars. Thus, for
the study of large-scale galactic stability, or large-scale density wave propagation, the
use of an interaction potential corresponding to a uniform-density spherical star will be
desirable since collective heating effects would no longer mask the phenomena under
study.
Langley Research Center,
National Aeronautics and Space Administration,
Hampton, Va., February 12, 1974.
6
REFERENCES
1. Hohl, F.: Relaxation Time in Disk Galaxy Simulations. Astrophys. J., vol. 184, no. 2,
pt. 1, Sept. 1, 1973, pp. 353-359.
2. Miller, R. H.; Prendergast, K. H.; and Quirk, William J.: Numerical Experiments on
Spiral Structure. Astrophys. J., vol. 161, no. 3, pt. 1, Sept. 1970, pp. 903-916.
3. Hohl, Frank: Numerical Experiments With a Disk of Stars. Astrophys. J., vol. 168,
pt. 1, Sept. 15, 1971, pp. 343-359.
4. Lynden-Bell, D.: Statistical Mechanics of Violent Relaxation in Stellar Systems. Mon.
Notic. Roy. Astron. Soc., vol. 136, no. 1, 1967, pp. 101-121.
5. Toomre, Alar: On the Gravitational Stability of a Disk of Stars. Astrophys. J.,
vol. 139, no. 4, May 15, 1964, pp. 1217-1238.
6. Hohl, Frank: Evolution of a Stationary Disk of Stars. J. Comput. Phys., vol. 9,
no. 1, Feb. 1972, pp. 10-25.
7. Hohl, Frank: Dynamical Evolution of Disk Galaxies. NASA TR R-343, 1970.
8. Schmidt, Maarten: Rotation Parameters and Distribution of Mass in the Galaxy.
Galactic Structure, Adriaan Blaauw and Maarten Schmidt, eds., Univ. of Chicago
Press, c.1965, pp. 513-530.
9. Lin, C. C.; Yuan, C.; and Shu, Frank H.: On the Spiral Structure of Disk Galaxies -
III. Comparison With Observations. Astrophys. J., vol. 155, no. 3, pt. 1, Mar. 1969,
pp. 721-746.
7
1.0
.8
.6
.4
.2
0 2 4 6 8 10
r, kpc
Figure 1.- Normalized interaction potential *= m) for point stars1)'
rod stars of length 2 kpc (02)' and uniform-density spherical stars of
radius 2 kpc (0 3)"
8
(C)
t=0 t=l t=2
(a) Point stars.
(b) Mass rods of length 2 kpc.
(c) Spherical stars of radius 2 kpc.
Figure 2.- Evolution of a disk of 50 000 mutually interacting stars under the influence of
a radial gravitational field similar to the Schmidt model of the galaxy. The stars
contain 20 percent of the total mass of the galaxy. The initial velocity dispersions
are such that Toomre's local criterion for axisymmetric stability is satisfied. Time
shown is in units of the rotation period at r = 10 kpc.
9
(C) I : :: -C,.<~ ~ t~'
(a) : 1
(b):
1 I
t=3 t=4 5
Figure 2.- Concluded.
10
POINT STARS
t=0 t 1 t=3 ' t=5
60
40 K/2 Q+K12 +K/2 Q +K2
"- K 
\ 
LQ,\
20
0 .
-K/2 - \ K\ \
20 ----2 -
u 
0
SSPHERICAL STARS, rs =2 kpc
< 80 
t a
t=O t=1 t=3 t=5
60 
K 
K
0\ Q+K2 \ K/2 +K/2 Q+K/2
Q-K2 Q-K/2 Q-K/2
> 
v
0 10 20 300 0 0 300 10 2 300 10 20 30
r, kpc r, kpc r, kpc r, kpc
Figure 3.- Rotation curve, epicyclic frequency, and so forth of the simulated disk galaxies.
11
0 0 0 0 0 0t I t 
5030 0 10 2  3
r, kQer, 2 
+K/ per, /2Fi ur Q+K12 Qncu ve pi ycic K12 Qyan o or K / 2 mu at d i K /2 xi s
40 Q
3 Point stars 1Rod stars, L =2 kpc A Spherical stars, rs=2 kpc
x... . ..... * .. .
t=0 t 1 t=2
3 O
2 A 'A... .. * a . .. * .
t=3 t=4 t=5
0 5 10 15 20 0 5 10 15 20 0 5 10 15 20
r, kpc r, kpc r, kpc
Figure 4.- Evolution of Q U r for the three interaction potentials.
r,min
Note the low heating rate for spherical stars.
8x10 Point stars m Rod stars, L =2 kpc A Spherical stars, r s2 kpc
" . "..A" ""
0 4 A- ' - .--. "". " .-
%*A
t 0 t . t 2.0 =O ". =l "" ^ e
8x107
C:L
0 4 . A,
2 ... . t=5
t = 3 . t = 4 t = 5..
0 5 10 15 20 25 300 5 10 15 20 25 300 5 10 15 20 25 30
r, kpc r, kpc r, kpc
Figure 5.- Evolution of the axisymmetrically averaged surface mass density. There is a considerable reduction
in the mass density in the region from 10 to 15 kpc during the evolution of the disk.
* Point stars a Rod stars, 1=2 kpc ASpherical stars, rs=2 kpe
50 - . .:0.0
go*
40A *.. ANA .A.
SU * A :A30 . I ::
A A
.1=0 " *
70
60- .. *,0 0"0*
.0 •0. .eg.'* O. *00
50 000000 *. 000000. * **a: . AL * && 0&04 0 3 0 AA, A O
5 0 0 a AL 3 .0A L a A A A
40  . * ' AUUU AI a A aA A A
A A
30 00 IA iL
20 - 0 .S
10 A 0
3 A'. L
10 _:A" AA 
A
* t=3 t=4 t=5
0 5 10 15 20 0 5 10 15 20 0 5 10 15 20
r, kpc r, kpc r, kpc
Figure 6.- Time history of the radial velocity dispersion.
1.4x162 Rod stars, = 2 kpc Spherical stars, rs=2 kpc
Point stars t= 0 <r <4kpc t=Or <
1.2 0 r < 4 kp 0 r < 4 kpc
f(V 4 r<8 4 r<8 r
.6. 8 r < 12 8 r < 12 8 <r < 12
12 r < 16 12 < r < 16 12 < r < 16
.4
0
-21.2x10 Spherical stars,r = 2 kpcS
Point stars Rod stars, = 2 kpc t 5 0 < r <4 kpc
t = 5 t = 5 0 < r < 4 kpc
0 r <4 kpc 4 s r < 8(Vr) .6 4 r<8 4<r<8 8<r <12
.4 8 < r < 12 8 5 r < 12 12 < r <16
12 < r < 16 12< r <16
.2
0
-200 -100 0 100 200 -200 -100 0 100 200 -200 -100 0 100 200
Vr, km/sec Vr, km/sec Vr, km/sec
Figure 7.- Evolution of the radial velocity distribution for stars in various regions of the disk.
t= t=l1 t=2
(a) Point stars.
(b) Mass rods of length 2 kpc.
(c) Spherical stars of radius 2 kpc.
Figure 8.- Evolution of a disk of 50 000 stars under the influence of a fixed field
corresponding to a uniform-density spherical mass of radius 2 kpc. The
stars contain 10 percent of the total mass of the galaxy.
16
t=3 t=4 t=5
Figure 8.- Concluded.
17
oPoint stars Rod stars, L =2 kpc ASpherical stars, rs = 2 kpc
20
15 t 
* MCI 0.*.a
S, AA
20 0 A
2 15 
'***-
.AA 0 . AA
* 0 " .**
5 9 
a* 
t
U *.A *x sA
0 
0 A 
=e 3A 
t.
O m A
r, kpc r, kpc r, kpc
0 001 15 0 0 15


Collisionless relaxation in spiral galaxy models

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