Aerodynamics. - Same problems ot the motion ot interstellar gas clouds.
I. By J. M. BURGERS. (Mededeling No. 48 uit het Laboratorium
voor Aero- en Hydrodynamica der Technische Hogeschool te Delft.)
(Communicated at the meeting of May 25. 1916.)
1. Introduction. The following considerations have arisen from
discussions with prof. dr. J. H. OORT at Leiden . who suggested most of
the problems and to whose interest and advice I am greatly indebted in
preparing these not es 1). They must be taken. nevertheless. as a very
preliminary attempt towards the application of aerodynamics in analysing
some of the intricate riddles that are presented by the results of astronomical observations on the interstellar material within our Galaxy. Apart
from the diffieulties inherent in aerodynamics itself or arising when it is
applied to matter in a state of such extreme dilution . astronomical observation. notwithstanding the powerful methods of photography and spectroscopy and the beauty of all what has already been revealed. still gives us
only very scanty information about the structure and the motion of the
cosmic clou ds. In attempting to make a few calculations many guesses must
be introduced. as otherwise it is impossible to construct simple pictures
whieh can serve as starting points for a provisional theory. The reader
therefore must take for granted that the cases treated in the following pages
are no more than examples. which it is hoped may elucidate a few features.
and perhaps point to a way for further work.
In the calculations the interstellar gas is considered as an ideal gas. to
whieh the ordinary equations of aerodynamies can be applied. Although
various elements and even certain chemical combinations are present (Ca.
Na. K. Ca+. Ti+. Fe+. CH. CH+. CN). the bulk (about 99.9 % of the
number of atoms) is formed by hydrogen . We will assume hydrogen atoms
to be the only constituent. and take the gas law in the form: ple
R Tim.
with p
pressure. e
density. T
absolute temperature. R
molecular
gas constant = 8.315.10 7 (cm-gram-second units will be applied throughout). and m
molecular weight
1 for ordinary monatomic hydrogen .
The ratio of the specifie heats is taken as k
5/3'
Dust particles (" smoke") are present in the interstellar gas. and it is
mainly due to their presence that clouds become visible. either as dark
=
=
=
=
=
=
=
=
1) GORT also kindly put at my disposal the manuscript of the George Darwin lecture
"Some phenomena connected with interste/lar matter". delivered by him before the Royal
Astronomical Society in London on May 10. 1916. and the results of his discussions with
prof. dr. H. A. K'RAMERS at Leiden. Some data used in the present communication have
been taken from this manuscript. which is to be published in the Monthly Notices Roy.
Astron. Society 1916.
590
masses , screening oH the light of the stars, or as luminous patches when
stars of suHicient luminosity are near. (Light emitted by the gas itself is
visible in some cases, probably in consequence of an important rise in
tempera tu re produced by collisional eHects as will be considered below. )
No account, however, has been taken of the dust particles in the calculat~
ions. They will give a small contribution to the density and perhaps might
have some influence upon the viscosity of the gas.
The average density of the interstellar material is of the order of
3.10 - 2 4 gr/cm 3 • It seems probable that a considerable fraction of the
material is concentrated in more or less separat,e clouds, with an average
density of the order of 10- 22 gr/cm 3 , an average radius of 20 parsecs
~ 6.10 19 cm (1 parsec
3,083 . 10 1 8 cm) , and consequently a mass of
approximately M = 9. 10 3 7 gr (46000 times the mass of the sun); and
distances apart of the order of 65 parsecs ~ 2 . 10 20 cm (or more). In the
intermediate regions the density then will be much helow 3 . 10- 24 • These
clouds must take part in thegeneral rotation of the Galaxy so as approx~
imately to halance the gravitational attraction towards the galactic centre.
At the sun's -distance from the centre (approx, 30.000 lightyears
3,2 . 10 22 cm) this rotational velocity is 300 to 320 km/sec
3,0 à
3,2 . 10 7 cm/sec. Superposed up on th is general motion the clou-ds have
peculiar motions of the order of 15 to 20 km/sec (1.5 à 2,0. 10 6 cm/sec).
A density of 10"': 22 gr/cm 3 corresponds to n = 60 hydrogen atoms per
cm 3 (mass of a H~atom : 1,66.10- 24 gram). Taking as the collisional
diameter in the sense of the classical kinetic theory of gases the -diameter
of the first BOHR orbit a
1.1 .10- 8 cm, the old formula for the free
2
path gives: Z
1/(n na )
4,38 . 1O- 9 /e, which for e
10- 2 2 leads to
13
1
3
Z~4,4 .10
cm (the distance earth~sun is nearly l,S . 10 cm) .
It will be evident that the possibility of obtaining a Maxwellian velocity
distrihution and the normal relation between pressure, density and tem~
perature can he foun-d only in elements of volume with dimensions large
compared with Z. In the case of the cloud around the star Merope of the
Pleiades the diameter of the illuminated portion is about 500", which at a
distance of 100 parsecs amounts to about 7.5 . 10 17 cm. This cloud shows
stratifications or waves with a thickness of approximately 8" = 1,2. 10 1 6 cm.
In the Cygnus nebulae NGC 6960 and 6992 there are extremely thin strips
or sheets, with an apparent thickness from 1" to 5". Taking the distance
to be 350 parsec
1,08. 10 21 cm, these ,t hicknesses amount from O,S to
2.5 . 10 16 cm. The density is supposed to be somewhat smaller than 10- 2 2
and to correspond to 30 atoms per cm 3 , giving e
5. 10- 2 3 ancl
Z
8,7 . 10 13 cm. In both cases the applicability of the ideal gas laws
to the thinnest observable sheets seems possible.
The temperature of the interstellar gas is assumed to be of the order of
=
=
=
=
=
=
=
=
=
=
=
=
10.000° . The mean molecular velocity then isgiven by: Cm
V 3 R Tjm
6
,= 1.58 . 10 cm/sec. The velocity of sound and velocities of expansion are
of the same order of magnitude. In certain cases (clouds produced by the
591
cxplosion of a nova) velocities of the order of 1000 km/sec = lOs cm/sec
have been observed. In all these cases there is no need for applying
relativity corrections.
As the general gravitational field in the Galaxy regulates the mot ion of
a cloud as a whoIe, it will not have much effect upon expansion or com~
pres sion phenomena. The gravitational field of a cloud itself perhaps may
be of some importance. Taking by way of example a spherical cloud of
mass M
9. 1037 gr, having a mean density "ij
10- 22 , application of
EMDEN' s results for the case of statie "polytropie" equilibrium 2) gives
for the density at the centre eo
6,0. 10- 22 ; for the temperature at the
centre T 0
650°; and for the pressure at the centre Po
3,24. 10- 11 (the
radius comes out as 6 . 10 19 cm as should be). Hence the pressure due to
the gravitational attraction inside the spherical mass in the neighbourhood
of the centre becomes comparable with the pressure 8,3. 10- 11 calculated
for a temperature of 10.000° and a density of 10- 22 • - At the surface of
this spherical cloud the value of 9 is equal to 1,67.10- 9 . In the case of
the expansion problem considered in the next section we will make an
estimate of its influence.
Viscosity, heat conduction and radiation in the first in stance will be left
out of account. A few remarks concerning their possible effects will be
made in connection with some results.
Finally the calculations will rder to motions in one dimension only, so
as to obviate the mathematical difficulties connected with problems of
spherical expansion.
=
=
=
=
=
2. Equatians of motion. - Applicatian ta a case of simple expansian in
a vacuum. - The equations for the one~dimensional mot ion of an ideal gas,
in the absence of gravity, have the form:
Du = ou + u ou
Dt ot
ox
D~ _ oe
Dt - ot
= _ ! op
+ u oe _
ox -
(1 )
e ox
_
ou
(2)
e ox
DT = oT + u oT =-(k-l) T ou
Dt
ot
ox
p=RTe/ m
OX
(3)
(4)
Viscosity, heat conduction and radiation have been neglected; u is the
velocity of the gas, while D/Dt in the usual way indicates the total derivative
with respect to t for an observer moving with a partieular element of volume.
For such an ob server combination of (2) and (3) gives: T/r.l-l = constant,
2) R. EMDEN. Gaskugeln (Leipzig 1907). Tab. 4. p. 79. for the case k = :;/:1. and the
formulac of p. 69. pp. 96/97.
592
expressing POISSON's isentropic relation for an individual element of
volume. This relation holds so long as the differential equations can be
applied; it breaks down, however, in so-called shock waves where iJu/iJx
and iJT/iJx assume such large values th at viscosity and heat conductio-:
become important and lead to an increase of entropy.
As the density in a stellar cloud is much larger than that in the space
around it, we must expect that it will expand, with an accompanying compres sion of the surrounding matter. The simplest case is that of an
originally homogeneous mass, expanding into a vacuum, assuming th at for
t < 0 we had: P
Po' e
eo for x < 0, and p 0, e 0 for x> O.
Introducing the velocity of sound
=
=
C
=
=
=
V k p/e = Vk RT/m
(5)
(counted positive in the same direct ion as u) and making use of the
relation p/[l
constant, the solution can be put into the form 3):
=
x
2
u
C
2
= k + 1 t + k + 1 Co
k-l x
k 1
2
= + t - k + 1 Co
)
(
(6)
\
This solution is limited to a reg ion defined by two values of x/t. The first
one characterizes the front of the expansion wave; here:
(x/t)'
= 2 co/(k ---.: 1) = u; = 0
C
(expansion to zero density);
the second one determines the back of the wave, where
(x/t)"
= -Co; u = O.
=
In the case k
513 the front moves outward with 3 times the velocity of
sound in the original state of the gas, while the back of the wave moves
inward with just this veloçity. For T
10.000 0 , m
1, k
5/ 3 we obtain:
Co
1,18 . lOG cm/sec; U front
3,53 . lOG cm/sec (35,3 km/sec). In a year
3,16 . 10 7 sec the front moves outward over 1.12. 10 14 cm, which at a
distance of 100 parsecs would correspond to 0,075" and thus perhaps might
be just appreciable. Nevertheless it will be evident that a cloud with a
diameter of 7,5. 10 17 cm for many centuries will retain its appearance.
Estimatian of the influence of gravity. - In order to take gravity into
account, a term - g must be added to the right hand side of eq. (1). In
connection with the restriction to one-dimensional mot ion we take g to be a
=
=
=
=
=
=
3) Compare: J. M. BURGERS, Over de eendimensionale voortplanting van drukstoringen
in een ideaal gas, Vers\. Neder\. Akademie v . Wetenschappen, afd. Natuurkunde, 52,
478 (1943).
593
constant. Introducing u, C as dependent variables, eqs. (1) and (2) can be
transformed into the system:
\~
~Î(
(àt +(u+c)àx~ u+ _k-l
2
~~+(U-C)~((u-_2
dx~
( àt
k-l
)--l
c)=-g
C
-
9
(7)
This system can be solved with the aid of its characteristics, which are
determined by (dx jdt)[
u + c; (dx jdt)Il
u-co Passing over the
details and taking for simplicity the same initial state as considered above
(although th is cannot be an equilibrium state) , the solution is found to be :
=
=
x
2
u
C
2
= k + 1 t + k + 1 Co k-lx
k
1t
= +
2
- k+1 +
Co
k
k
+1
gt
(8)
k-l
2(k+l) gt
The front of the wave, determined again by c
the equation:
= 0, moves according
to
(9)
while the back, where c
=-
Co' is to be found at:
x" =-co
=-
t-t ge .
(10)
At the back of the wave u
gt; likewise for all values of x to the left
of those determined by (10) u
gt, which means that here the whole
mass of the gas is moving with the acceleration - 9 just as a solid body
would do .
This latter result is of no particular importance, as it is a consequence of
the condition assumed for t = 0, no account having been taken of the
possibility of acompression (which would require the introduction of a
boundary condition for some negative value of x). It will be evident,
however, that the influence of gravity will be of importance only wh en gt
becomes equal to, say Co. With 9 = 1,67. 10- 9 as found in the example at
the end of section I, this will be the case af ter a lapse of time of 7.10 14
sec
2,2. 10 7 year. Hence no great error is made by neglecting the
influence of gravity in expansion and compression phenomena of clouds
during comparatively short periods.
=-
=
3. Expansion with compression of the surrounding gas of [ow density.
- When for t < 0 we have a gas (I) with a density 10- 2 2 to the left of
x = 0, and to the right a gas (11) with a density of. say, 3 . 10- 24 , both
gases originally being at the same temperature T 0' the expansion of the
denser gas (I) will produce a compression wave in gas (11). which com-
594
pres sion wave, owing to the high velocities involved, will take the form of
a shock wave. The ex pan sion of gas (I) will extend to a pressure equal
to that produced by the passage of the shock wave through gas (II). As
the front velocity (xlt)' of the expansion region is smaller than the gas
velocity u when the ex pan sion does not proceed to zero density, a reg ion
of constant velocity, density and pressure appears between the front of the
expansion region and the boundary separating gas I from gas 11.
Fig. 1.
The propagation of the waves has been represented schematically in
fig . 1. Starting from the right hand side, the shock wave in gas II (original
state: PI' th' Tl; zero velocity) moves with the velocity ~. Behind th is wave
gas II moves with the constant velocity V, its state being defined by P'2'
e2' T 2 • We then have 4 ):
The boundary between the two gases moves with the velocity V and so
does gas I in the region between th is boundary and the front of the
expansion wave, the latter moving with the velocity (xlt)'
17. At this
front eqs. (6) give:
=
2
u= k+l (17
+ co);
k-l
At the same time:
P
= (~! f(k-I) (~2 r/(k-I); e =
=
2
c= k+l YJ- k+l co·
(:! f(k-I) (
~2
f(k-I)
(12)
(13)
At this front we must have u
V; moreover with regard to the boundary
between gas I and gas II the condition P
P2 must be fulfilled.
The equations can be solved numerically. With the data:
=
=
(gas I ) eo= 100.10- 24 ; T o= 10 4 ; Po
83,2.10- 12 ; Co= 1.18.10 6 ;
1
24
(gas 11) el
3.10- ; TI = 10 ; PI = 2.5.10- 12 ; Cl
1.18.106 ,
=
4)
=
See the paper mentioned i:1 footnote 3), p. 477, eqs. (9) and (A).
595
we obtain:
and for the state of gas I in the region of constant velocity u
= V:
p=10,5.1O- 12 ; e=29.1O- 2i ; T=4370o; c=-O,78.10 6 ,
while for gas 11 in the reg ion behind the shock wave:
Owing to the compression the rarer gas takes a much higher temperature
than the expanding denser gas. The region of this high temperature
increases in width at the rate .; - V = 1,02 . 10 6 cm/sec (10,2 km/sec).
With regard to the shock wave front it must be remarked that when
account is taken of viscosity and heat conduction, it is found that the sudden
change of state in reality is a continuous one, although with very steep
gradients. The thickness of the transition layer is of the order ,tI/eV, where
p is the viscosity 5). Assuming the viscosity to be given by ft = -ft- (! Z Cm,
with Z= 4,3. lO-!J/e, we find, for T = 10000°, Cm = 1,58.10 6 cm/sec:
u D,;l 0,0023 6). The thickness of the layer becomes of the order of magnitude of Z. This generally will be inappreciable from the observational point
of view. In the transition region the Maxwellian velocity distribution will
not hold; a kinetic treatment of the phenomena in this reg ion would be
necessary, which, however, will be extremely difficult.
4. Callisian af twa clauds. - The examples given in the preceding
sections will appear artificial, and the reader may ask how the state assumed
for t = 0 can have arisen. This, however, is a reflection of our general lack
of knowledge about the fields of motion in the interstellar gas, which makes
it necessary to start from imagined cases. The examples were meant to
illustrate that when regions of highly different densities occur side by side,
expansion and compression phenomena certainly will be present; their
order of magnitude will be apparent from the results obtained.
Clouds of a much higher density than that of the average surrounding
gas can be produced in star explosions. Such clouds in the first instants
(the first few years or perhaps decades of their existence) will move with
enormous velocities, compared to which the velocity of expansion often
5) Compare e.g.: R. BECKER, Stosswelle und Detonation, Zeitschr. f. Physik 8. p. 339.
1922: G. I. TAYLOR and J. W . MACCOLL, The mechanics of compressible fluids, section
6 (W. F. DURAND, Aerodynamic Theory. Berlin 1935, vol. lIL Div. H, p. 218) .
!l)
The value of ,u also can be calculated with the aid of SUTHERLAND's formula :
l'=lloVT12?3 . (I + C /273)/(1+C/T). Here l'O (at T=273 ° ) is proportio:1al to
=
=
=
=
=
Vmla~. For H~ (m
2; r;
2.730. 10-s cm) llO
0,000085. C
72; for H e (m
4:
0 = 2,174. 10-s cm) .l /o
0,000189, C
80. Estimating for Hl (m
I; 0 = 1.1 . IO- s
cm) ,"0
0,00037, C
76, the formula gives ,ft
0,0029 at T
10000°, which is
=
=
=
=
slightly larg er than the value found above.
=
=
=
596
may appear negligible. In certain cases such clouds perhaps may impinge
upon each other; or they may impinge upon a portion of the interstellar
gas with more than minimum density; also the gradual loss of velocity
experienced in moving through the interstellar gas deserves attention.
Certain problems iIIustrating these possibilities will be treated now.
The simplest case is that of a "head on" colli sion of two clouds. having
both the same density and temperature. with parallel boundary planes. In
each cloud a shock wave will appear. When we introduce a system of
coordinates with respect to which the two clouds have equal and opposite
velocities of absolute magnitude V. the situation is a symmetrical one
(comp. fig. 2). The two shock waves move outwards with the absolute
velocity ~; between them the gases are at rest. Indicating the original state
of each gas by Po. t!a. T 0; and the compressed state between the shock
waves by p'. r/. T'. we have 7):
v +~ = k
p'
!
1V
V
+ c~ + (k +1~2 V2 ~
= Po + 00 V (V + ~); e' = eo (V + ~) I ~
(14)
~
Fig. 2.
When V is large compared with co. we may use the approximations:
~
f'}
t
(k-I) V; p'
f'}
-Ir (k
+ I) eo V2;
I!~
f'}
(k
+ 1) f:!o/(k-I)
(14a)
from which:
RT' ~ t (k- I) V2.
=
=
(I 5)
=
By way of example we take eo
1O - 2~; T o
10000° ; V
10 7 cm/sec
(100 km /sec). With Ic = Gh as before we find,: ~~ V13; p'~ 1.33.10- 8 ;
r;/ ~ 4 I!o; T'~ 400.000° .
In the more general case indicated in fig. 3. where gas I. moving with
the velocity V 1. overtakes gas II (velocity V 2). the solution still can be
obtained by means of straightforward algebra. In the case of large
7) These formulae follow from eqs. (9). (A). (B) of the paper mentioned in footnote :l). if it is observed that relatively to the uncompressed gas the shock wave moves
with the velocity V
and the compressed gas with the velocity V.
+;
597
velocities the following approximations are obtained, where
(supposed to be < 1):
+ 1) ez (VI - V
e' "' ·(k + 1) Ih/(k-l)
p' = p" '" -!r (k
2 )2/(1
RT' '" ~ (k-l) E2(V I- V 2)2/(1 +E)2;
+
E
=V
e'2 /(21
E)2
el' '" (k + 1) e2/(k-l)
( (16)
RT" ~ t (k-l)(V I- V 2)2/(1 +E)2
=
so that T"/T' çSl (!J /rh. - Particular cases, e.g. V';!
0, can be deduced
from these formulae. It will be seen th at the temperature of the rarer gas
increases to a much higher degree than that of the denser gas. The thickness of the zone of high temperature in gas 11 increases at the rate :
~';!-V*çSl~- (k-l) (V 1 - V 2 )/ (1 +E) .
Fig. 3.
[It must be observed that with the very high temperatures and consequent high molecular velocities obtained in collisions between clouds with
velocities of 50 km/sec and more, dissociation (ionization) of the hydrogen
atoms must be taken into account. In all collisions with sufficiently high
relative velocity of the colliding atoms, there will be a 10ss of energy in
consequence of excitation or of the production of ions, which energyafterwards is radiated out in the form of light quanta. There is thus a continuous
10ss of energy in the compressed gas, at a ra te which depends upon the
number of favourable collisions. In those cases where this 10ss of energy
is considerable, the temperature of the gas behind the shock wave will
decrease, and a temperature gradient will be set up , which will influence
the density and the pressure. ] R)
8) In the case of the N ebuIa south of 1; Orionis (IC 434) the impression is gained
o f a dark cloud of greater density penetrating into aluminous cloud of smaller density ,
while bright fringes appear just along the boundary. These fringes are rather sharply
limited on the side of the dark cloud (in particular around the "Dark Bay"). while they
shade oH gradually towards th e side of the smaller density _ Similar features are observed
in other cases. With the "Crab NebuIa" in Taurus a th in luminous layer se ems to
precede at some dista nce the ex panding central luminous mass. The spectrum of this thin
b yer shows many "forbiddeh" lines. which ma kes it probable that excitation here is
due to collisions in a gas of not too small density.
When an cxplanation of these features is asked for . the picture devcloped in the text is
too simple and presents the difficulty that it leads to clear cut front iers limiting the
598
5. Cloud with high velocity impinging upon gas at rest. - We wil!
consider what wil! happen wh en a cloud moving with a very high velocity.
after having passed through a region of extremely low and negligible
density. suddenly impinges upon a gas at rest of sufficient density to make
itself felt. Such a case according to DORT seems to be present with a part
of the cloud produced in the explosion of Nova Persei in 1901. It is
estimated that this cloud moves with a velocity V 1
1200 km/sec
1.2. 10 8 cm/sec.
During the first instants af ter the collision of the cloud with interstellar
gas at rest of appreciabIe density. the solution appropriate to the case is the
same as the one considered in the second part of section 4. with V 2 = O.
In the interstellar gas a shock wave appears. moving with the velocity:
~:2
i (k + 1) V 1 /( 1 + e) t V 1/ ( 1 + el. while the boundary separating
the cloud from the interstellar gas moves with the velocity: V*
V 1/(1 + e).
Hence the region of compressed interstellar gas broadens at the rate:
$" - V*
!(k-l) V1/(1 + e)
t V 1/( 1 + el. The tempera tu re in this
region. calculated without taking account of radiation. would rise to the
value: T"
t(k - 1) (V 1 2/R)/( 1 + e)2
5.77 . 10 7/( 1 + e)2. The
shock wave in the cloud has the forward velocity
=
=
=
=
=
=
=
=
=
=
and thus relatively to the material of the cloud moves backward with the
velocity:
Wh en the cloud has a finite thickness Land is assumed to have constant
regions of compression and of increased temperature. However. it has been mentioned
already that heat conduction and viscosity turn shock waves into continuous transition
regions; at the boundary where the denser and the rarer gas meet with equal pressures
3:1d ma ss velocities (but with different temperatures). also diffusion will play a part.
Now i:1 those cases where on one si de we have a high density combined with low
temperature and on the other side a low density with higher temperature. it is possible
that somewhere in the transition region optimal conditions for the emission of radiation
will be found . This perhaps could give a:1 explanation of the appearance of narrow
bright fringes. In those cases where a shock wave would bring a theoretical increase of
temperature of 100.000° or more. optima I conditions for radiation might be found possibly
not in the reg ion of high temperature. but i:1 a sheet of the transition layer with some
intermediate temperature.
It must be observed nevertheless that as soon as kinetic energy in collisions of atoms
is lost in conseque:1ce of excitation and ionization (Ieading to emission of radiant energy)
the phenomena become much more complicated. It is possible that owing to this loss of
energy no constant condition is obtained behind the shock wave ; in certain cases perhaps
even the shock wave itself may be absent. Before a satisfactory description can be given
of the relations to be expected in such cases and the problem of sharp and diffuse
boundaries can be attacked. a more penetrating investigation therefore will be necessary.
based up on equations in which due account is taken of radiation and diffusion phenomena.
etc.
599
values of p. e. T throughout. th is shock wave will reach the back of the
cloud after the interval 7:
L/( VI - ~ 1)' When tentatively we estimate the
thickness of the Nova Persei cloud at L = 0.5" = 4.1 . 10 15 cm (assuming
a distance of 550 parsecs = 1.7.10 21 cm). then with VI = 1.2.10 8 •
(h/e l
0.01. E
0.1 we find VI -~l
1.45.10 7 and 7:
2.83. lOs
9 years.
sec
At the end of this interval the whole cloud has undergone a compression;
it moves as a whole with the velocity V*
1.09. lOs and has obtained the
temperature T'. lts thickness has been reduced to: L-7:(V l - V*)
(k -1) L/ (k + 1)
1.03.10 15 • that is in the inverse ratio of the
increase in density (e'jed .
At the back of the cloud the shock wave is reflected as an expansion
=
=
=
=
=
=
=
=
=
=
=
=
wave. The front of th is wave has the velocity e"
V kR T" 8.1 . 10 7
relatively to the gas. and thus af ter a further interval of time of approximately 0.4 year reaches the front boundary of the cloud. As soon as the
expansion wave has reached the front of the cloud. the velocity and the
p;:essure will decrease here. so that also the compression produced in the
interstellar gas will become less. Prom now onward the whole phenomenon
assumes a rather complicated form !J). To work it out in details does not
seem promising in view of the uncertainties of the available data. In
particular the assumption of a homogeneous state in the original cloud is
an oversimplification of the case.
In view of this situation OORT put forward the question wh ether ä
stationary solution could be found. describing a case where owing to the
general slowing down of the cloud a "barometric" gradient of the pressure
and the density is set up. OORT supposes that such cases may be found
in the very thin cloud strips. observed in NGC 6960 in Cygnus. An
approximate solution of this type will be constructed in the next section.
(To be continued.)
9) Certain phenomena of this nature have been investigated i!J the paper mentioned in
footnote 3).