Journal of Modern Physics, 2012, 3, 1523-1529
http://dx.doi.org/10.4236/jmp.2012.310188 Published Online October 2012 (http://www.SciRP.org/journal/jmp)
Similarity Criteria, Galactic Scales, and Spectra
Georgy S. Golitsyn
A. M. Obukhov Institute of Atmospheric Physics, Russian Academy of Sciences, Moscow, Russia
Received August 7, 2012; revised September 6, 2012; accepted September 13, 2012
ABSTRACT
An old topic of dimensional analysis in astrophysics is presented and new results, or quantitative explanations of some
observational facts are obtained, in particular, on the base of the supernova, SN, explosions. The presentation starts with
the derivation of two similarity criteria for astrophysical objects constructed out of four measurable quantities: mass, M,
luminosity, Lb, velocity U, size R, and gravitational constant G. The first well known criterium describes the virial prin-
ciple and the other one seems to be new and is based on the Tully-Fisher observational relationship between luminosity
and velocity. The energy generated by SN explosions allows one to estimate well the interstellar turbulent velocities and
magnetic field in our Galaxy, resulting in 3 to 4 microgauss. It is found that for z ≥ 0.6 the observed distant galactic
clusters are far from virial equilibrium and the degree of disequilibrium is increasing with z. It means that to reach such
an equilibrium the cluster age should be of order ten dynamical time scales, see Equation (7). For all considered galaxy
clusters the second similarity criterium was found to be constant with a precision of about ten per cent. Therefore it
could be considered as a general law, though for different classes of objects the numerical coefficient may vary. Some
scales are proposed and two of them are tested for galactic clusters by finding numerical coefficients with accuracies of
about 20 percent or better: e.g. observed luminocities of clusters are

52 32
1b
WL a MR G11.25 0.22a with
for the first eleven objects from the Table for which the virial equilibrium is found with the same accuracy. The
square root of the two criteria ratio

12
321

3
15
UWG explains the Tully-Fisher law and is constant for all
32 available clusters from [1,2] and is equal to 1.8 ± 0.2. This is because

has not global values of total mass and
size.
Keywords: Dimensional Analysis; Galaxy Clusters; Virial Evolution; Tully-Fisher Law; Turbulence and Magnetic
Fields
1. Introduction
Dimensional relationships between the measured quanti-
ties [3-7] are the base of studies of complex phenomena,
certainly for astrophysics. Here we present such an analy-
sis for quantities measured in astrophysics, find similar-
ity criteria they may form, elucidate their meaning and
possible applications, and present certain scales some of
which are new ones, or at least used rarely. This analysis
systemizes our knowledge of this methodology, presents
some new results and may serve at least a methodologi-
cal and educational purpose.
For our Galaxy we estimate turbulent gas velocities
and magnetic fields in interstellar medium basing on the
power supplied by supernova, SN, explosions. At dis-
tances galactic clusters are found to be far from
the virial equilibrium between kinetic and potential ener-
gies and the degree of disequilibrium increases with z.
This is in contrast with what this author was told by some
professionals less than ten years ago that the virial equi-
librium must not to be questioned. This believe prevented
this author from exploring the subject for many years,
through recently another authority in cosmology and as-
trophysics told me that the absence of such an equilib-
rium for early objects is selfevident. How this processes
evolves is shown in this paper. Thanks to the invitation to
present a paper for this Special Issue of the magazine this
text has been written with a methodological goal to re-
new interest for the old subject which has been so pro-
ductive in older times.
0.6z
L
In astrophysics the measured quantities are mass, M,
velocity, V, luminosity, M or b, and sizes, R (in abso-
lute units, if distances to objects are known). Gravity
fields are characterized by the constant G = 6.672 × 10−12
m3·kg−1·s−2. Distances in astrophysics are measured in
parsecs: 1 pc = 3.08 × 1016 m = 3.26 light years. Our four
measured quantities plus the gravity constant G are de-
termined by three independent dimensions: mass, length,
and time. It follows from here that from these five quan-
C
opyright © 2012 SciRes. JMP

G. S. GOLITSYN
1524
tities one can form two dimensionless values, the simi-
larity criteria [3-7]. We choose the following as such
criteria:
12
M
G
RU
 , (1)
35
225
M
G
RW

1
1
,3 5
n
WU n
51
WUG
. (2)
The first value may be naturally called the virial crite-
rium because it is formed by quantites entering the virial
theorem applied to the central gravity field. It is belived
that after sufficient time this theorem is true for an arbi-
trary number of interacting bodies. Then 1, if we
would know with good precision all quantities entering
(1), aside G, which we know with a relatively great pre-
cision. If 1 it would demonstrate inaccuracies in
our knowledge of masses, velocities, orbital and/or tur-
bulent ones, negligence of some factors, e.g. dark matter
or energy, an incompleteness of virialization of suffi-
ciently large size objects, etc. Our Table below will
demonstrate this for clusters of distant galaxies.
The second similarity criterium for the clusters is of
order unity. This demonstrates a close connection among
the quantities entering it, i.e. mass, size and brightness
united by the gravity field. To this author’s knowledge
such a criterium have not yet been introduced. The
long-known Tully-Fisher [8] relationship is a hint of such
a connection:
, (3)
The upper limit for the exponent n is an immediate
consequence from these variables dimensions:



, (4)
where from the velocity scale is
15
UWG

. (5)
Numerical constants entering (4) and (5) should be
determined experimentally and are expected to be of or-
der unity as was noted by P. Bridgman [4] referring to
Albert Einstein. Exponents n lesser than 5 are evidenc
that other similarity criteria act in the determination of
the velocity scale (see [7]). e.g. if we take

15
1
1
UW

 3 WUG we obtain . From Equations
(1), (2) and (4) one may form the nondimensional simi-
larity criterium as
 
12
321
 15
1
cUWG 
c
1
c 
, (4’)
where 1 is a numerical coefficient (see the end of Sec-
tion 5), or 31 . The case that three quantities with
three independent dimensions are forming a nondimen-
sional criterium is a special case revealing a kind of the
phenomenon degeneration. The only other case well
known is the fine structure constant 21137ec
causing some conceptual problems in quantum electro-
dynamics. Such situations are discussed by P. Bridgman
[4] in Chapter 8.
Knowing the size and the velocity scale we determine
the time scale from (1):
12
3
dRR
TUMG




3
, (6)
M
which with the mass density scale R 



equals to
12
d
TG 

. (7)
The last time scale, obvious from dimensions of G and

, can be found in the book by M. Rees [9] where it is
called the dynamical time scale. It is compared there with
the cooling time c of the earlier gas by recombination,
transfer of energy between various atomic levels, etc.
leading to the energy emission and gas cooling. If
cd
T
TT

, then primordial gas cloud is either compressed
to its center, or fragmented into smaller objects, which
may lead to formation of galaxy clusters.
2. Galactic Scales
The observational situation is such that, all four quanti-
ties W, U, M, R are very rarely determined simultane-
ously for the same object. At the same time if we meas-
ure only two values out of four, together with the gravity
constant G we may still form some useful scales.
As an example, let as choose mass M, size R and G.
Then the velocity scale is, as from (1):

12
1
UMGR

, (8)
the time scale is

12
32
2πTRMG


2π
(9)
and is taken for the time of the full revolution of the
object around the gravity center. The brightness, the
power scale is
52 3252
WMGR


24
wMGR
, (10)
the energy density in the unit volume, dimension of
pressure, is

, (11) 
and the power density per unit mass is
32 3252
MGR 


11
410
30
210 kgM 26 410 WW

. (12)
For our Galaxy with stars with the solar mass
 and brightness we
obtain 240 MyT380 msU,

, ,
3

38
1.6 10WW
42
1.6 10ms



, 93 210 Jmw  . First three scales


G. S. GOLITSYN 1525
are rather close to the observed ones. But the last value
requires a comment.
It is known [10] that for our Galaxy the volume den-
sity of cosmic rays energy is close to
3 3
0.6
1
10  0.5 eVcm3 Jm The energy density of the
galactic interstellar magnetic field is 212 8π10H


erg/cm313 110

 3 Jm6 510H 
21
wMUR


33
310
at Gauss. The kin-
etic energy density of turbulent gas motions are of the
same order of magnitude, as well as the energy density of
the star EM emissions and even the density of the relict
emission [10]. The last could be a chance coincidence. At
the same time Equation (11) gives a four magnitudes
higher value. This may be explained by rewriting Equa-
tion (11) using (8) as showing that it is
the energy volume density of the large scale orbital mo-
tions. But we are interested in small scales.
The cosmic rays, CR, and galactic magnetic fields are
generated by SN explosions. SN explode in our Galaxy
two-three times in a century and 1052 - 1053 Jouls are
generated at each explosion [10]. In Watts it corresponds
to W. This and the CR energy volume density
and direct use of the notion of the spectrum are enough to
explain the observed shape of the CR spectrum [11,12].
With the velocity of light c and the Stefan-Bolzmann
constant

one may define the temperature scale
g
T
using the volume energy density w as:
14
3
4π
gwc




T, (13)
where

4π3
13 3
10 Jm

3.35
is the area of a unit sphere. With
we obtain g Tw

K. This is close to
relict temperature 2.7 K. This is a single scale with the
hint of the quantum mechanics because
24
32
23 1
π5.67 10
60
1.3810J K.
k
hc
k



 
82
4
WLK,


33
1.0610J s

 
11
is the Bolzmann constant, is the
Planck constant.
h
M. Rees [9] has introduced the notion of virtual tem-
perature as a measure of kinetic energy in the object of
consideration. It is defined from Equation (1) at

.
Then
2
mMG m
TU
kR k

27
1.66 10  m
v, (14)
where m is the mean atomic weight in the gas cloud. In
its primordial composition protons are about 90 percent
and helium is of order 10 percent. The mass of proton is
kg. Therefore in Equation (14)
p
m

kg.
27
10  
30
210 26 410
1.9
Let us consider what these scales are for a star like our
Sun with mass M kg and luminosity
W. Then the linear scale determined by its mass and lu-
minosity is
25 35
RMWG


13
3.6 10
58
710km 710mR 11 2 210U 1
, (15)
and is equal to m = 240 a.u., astronomic units.
It seems that this was the size of the original cloud from
which the Sun and the Solar system were formed. The
mass of the cloud was several times larger than the mass
of the Sun, therefore the size of cloud was accordingly
larger as Equation (15) is demonstrating. The virial simi-
larity criterium Equation (1) for the Sun at
is . At 1


5
4.510 m/sU we
have

450 km/s

. We see that the vi-
rial criterium to the single star is not applicable. On the other
hand a reasonable estimate is produced by Equation (5):
15
UWG


s
2.3km, which is characteristic for the
convective velocities in the photosphere. One can write
formulas for the scales as Equations (8)-(12) for any oth-
er couple of measured quantities like size R and luminos-
ity W, or mass and luminosity, etc. We see from the above
examples that the purely dimensional analysis is still able
to produce useful results, at least, of educational value.
One may note a very high dependence of luminosity in
Equation (10) on mass and size as

52
MR . It may ex-
plain the high luminosities of compact quasars. If we
know only the luminosity W we may estimate the ratio
M
R for the virialized objects as 25 35 aW G
0.93 0.20a1 which
1

 estimated from eleven first lines of our
Table.
The similarity criterium 3

is found to be unexpect-
edly close to a constant value for all objects of our Table.
The coefficient a varies from 0.50 to 0.65 with the mean
value and dispersion 0.56 0.04a


0.34a
even accounting
for the cluster No. 12 for which . It makes us
think that this cluster is somewhat unusual having a small
mass at a comparatively high luminosity [1].
The addition of ten objects from [2] for which the val-
ue of 3

can be calculated does not change noticeably
the value of 0.56a

. Therefore the similarity criterium

15
UWG  11.8 0.2a
3 is equal to for the ga-
lactic clusters. For other objects it may vary. Certainly
the similarity criterium 3

is of more universal mean-
ing than 1

since it does not depend on the global pa-
rameters of a system like its mass and size determining
the system virial equilibrium between the kinetic and
potential energies.
The paper [2] has some detailed information on 13
nearly clusters with 0.23z

. However only for ten of
them the criteria 1

and 3  can be evaluated using
the larger values for mass, size and temperature, see eq.
(18). One can not directly compare such data with the
parameters from our Table. Using what we have we
compute the virial criterium 1 to be 0.44 0.04

in-
stead of 1. This is mainly caused by the factor
M
R,
evidently by overestimating the size of an object which is

G. S. GOLITSYN
1526
still can be measured but contains only a small fraction
of the total mass. The criterium 3 is close to 1.8 sub-
stantiating the results from the Table.

26
210 W
37
810W

33
310
72 3
310 ms

 
For the Sun at , at a =
0.56 the value For our Galaxy at
W as from stars like Sun and at the mean or-
bital velocities U = 300 km/s we 31.35 . It
looks like that for similar single objects the similarity
erium 3  cou
5750 KW
ry.
e
T
37
11
0
ld va
41
find
crit
3. Turbulence of a Galactic Interstellar Gas
Energy generation for the turbulent gas motions comes
from the SN explosions. The power of this process is
estimated above as W for our Galaxy [10].
During explosions the shock waves tear off the external
shells of SN and spread their matter from random places
in the Galaxy and at random times. Random shock waves
in space accelerate CR particles according to Fermi me-
chanism. We assume the mass of the interstellar gas as M
= 1040 kg. This is two order of magnitude less than the
gravity mass of the Galaxy where over 90 percent is the
dark matter, i.e. the mass of the gas is an order magni-
tude less than the mass of the stars. The SN energy pow-
er for the unit gas mass will be . This
is the energy rate of generation of the matter perturbation
and in the statistically stationary case it will be also the
rate of the energy dissipation.

Armstrong et al. [15] have shown that the spatial en-
ergy spectrum for the electron density fluctuations in the
interstellar gas is proportional to 53 k where 2πkr

is the spatial wave number up to distances
. The velocity fluctuations have the same
structure up to 100 pc [10]. For the velocity fluctuations
A. N. Kolmogorov have proposed in 1941 the structure
function
500 pcr
19
1.5 10m
 
223
r uxbr   

u
Dr ux (15)
where the numerical coefficient b is close to 2 as it was
later obtained from a great number of various measure-
ments (see [14]). With the above estimated

and
18
0m 1031r 0pc we obtain

12 20
ur


uD 
m/s, a value close to the observed velocity fluctuations in
our Galaxy.
4. Magnetic Field in the Galaxy
To describe this field we determine the structure function
using the same arguments as A. M. Obukhov has used in
1949 (see [1]) for the description of the statistical struc-
ture of a passive scalar. But instead of the magnetic field
induction i
H
we shall do it for the modulus of the Al-
ven velocity

4π

1
VH



  
213 23
AA A
DrVxr VxNr 

 

. Then
, (16)
where the rate of magnetic energy generation/dissipation
is defined as
22
dd
d2d 8π
A
VH
Ntt 





. (17)
Since the volume density of the magnetic energy is the
same as the energy of velocity fluctuations, see above,
and both are due to SN explosions, the time scales for
both quantities should be of the same order. Then

23
13231235 432
310ms510 msN 

 
. For


18
100 pc310mr we will have from Equation (16)
2822
10 mcV
60 3
10 mV
A. The mean density of the interstellar gas
at

 is equal to 24 3 10g cmMV 

 
6
3.5 10H
, or
of an order less than one proton on cm−3. Coming from
the square of the Alven velocity to the magnetic field we
obtain, at this density of matter,

 Gauss.
The observed values of the magnetic field in our Galaxy
vary from 3 to 5 microgauss. Its fluctuations in space
should also be described by the Kolmogorov-Obukhov
53
k
7
2.5 10
7
1810 K
14
10 M
spectrum. Such spectra have been obtained for the
magnetohydrodynamic turbulence by Frick and Sokolov
[2].
The last two subsections, just described, relate well the
observed velocity fluctuations and magnetic fields to the
observed rate of SN explosions in our Galaxy.
5. Clusters of Galactics and Their Similarity
Criteria
This author was a reviewer of A. A. Vikhlinin’s D.Sc.
Dissertation “The observational Cosmology and the In-
tergalactic Medium Studies by X-ray Spectra of Galactic
Clusters” [1]. It has a detailed analysis of X-ray and op-
tical data for 21 distant clusters at z from 0.4 to 1.26. A
later paper [2] presents 13 relaxed closer clusters with z ≤
0.23 and necessary references. Table 1 has all the data
from [1] needed for the subsequent analysis. The last
three columns are calculated by us.
In the Table 1 z is the red shift, the difference between
the length of the registered emission and emitted one
related to the length of the emitted wave, temperatures T
in keV from 2.2 to 14 keV, i.e. from to
, bolometric luminosity W from 2 to 260 times
1037 W, the mass is from 0.2 to 8.77 times


44
21022 310 kg, the spectral radius in Mpc, 1 Mpc
m. It was assumed that the gas velocities observed by the
broadening of spectral lines are the thermal ones and the
temperature can be restored from them as
2
3
p
mU
TkA

, (18)

G. S. GOLITSYN
ht © 2012 SciRes. JMP
1527
Table 1. Parameters of the distant clusters and their virial similarity criterium П1 (Equation (1)).
№ z T, (keV) 
37
,10Ww14
,10
M
M,MpcR П1

By
a
T


By
d
T
1 0.394 4.8 9.2 1.24 0.5 0.96 8.12 0.48
2 0.400 3.7 8.9 1.42 0.7 1.02 8.09 0.49
3 0.424 3.6 10.6 1.07 0.5 1.05 7.92 0.61
4 0.426 7.6 27.0 2.89 0.9 0.79 7.91 0.50
5 0.451 14.1 260.4 8.77 0.9 1.23 7.75 0.43
6 0.453 5.8 15.9 1.81 0.7 0.83 7.73 0.65
7 0.460 5.3 16.3 1.57 0.5 1.10 7.68 0.55
8 0.516 5.1 15.7 1.67 0.6 1.02 7.34 0.54
9 0.537 8.1 91.7 3.68 1.0 0.85 7.21 0.78
10 0.541 9.9 113.3 6.43 1.0 1.21 7.19 0.59
11 0.562 4.8 12.5 1.19 0.5 0.92 7.07 0.49
12 0.574 2.7 38.8 0.36 0.5 0.50 7.00 0.88
13 0.583 5.2 10.8 0.95 0.5 0.68 6.95 0.54
14 0.700 7.2 28.7 2.01 0.7 0.74 6.36 0.62
15 0.782 6.3 32,4 1.41 0.7 0.59 5.99 0.74
16 0.805 2.2 2.0 0.21 0.5 0.36 5.89
1.16
17 0.805 4.3 13.2 1.04 0.8 0.56 5.89 1.05
18 0.813 6.6 28.8 1.25 0.7 0.50 5.86 0.79
19 0.823 7.8 70.9 2.58 1.0 0.62 5.81 0.93
20 1.100 3.5 5.9 0.26 0.5 0.28 4.82 1.04
21 1.261 4.7 6.0 0.20 0.5 0.16 4.36 1.19
Copyrig
0.6


27
1.9 10m  2
is the mean molecular weight of plasma with
the space concentration of protons and helium ions, elec-
trons are taken into account, p kg


23
1.38 10 10−27 kg the mean mass of the gas particle, k


J/K, 7
10K keV1.161A
11 42
410 10MM 

.
It is useful to compare the data from the Table with the
corresponding parameters for our Galaxy with R = 15
kpc, U = 300 km/c,  kg. Then
Equation (1) gives 1 close to unity, which is
evidencing that our estimates of the parameters entering
Equation (1) are reasonable.
1.2,
Let us return to our Table 1. In one before the last
column we estimate the object age by dividing the Uni-
verse age of 13.7 By by 1 + z and extracting from the
ratio 1.7 By, the estimate of time by Rees [9] when clus-
ters could form. The last column contains the dynamic
time 12,TG



d Equation (7). This time scale gives
a representation of the nature of the virialization dergee,
i.e. a degree of dynamic equilibrium between kinetic
energy of motions and gravity energy. The ratio of the
two scales ad TT
T gives the cluster age in the time units of .
d
For the first eleven objects the difference of the virial
criterium 1

from unity is rather small and random, as
for our Galaxy. These differences can be related to the
measurements inaccuracies and/or to rounding errors. For
the last ten objects 1

is systematically smaller than
unity with the minimal value of 1 = 0.16 for the most
distant object with

1.26z

. This corresponds to the age
4.36 By at the scale d1.2T

By. Evidently several, up
to ten, units of d are needed for reaching the virializa-
tion. The eleven objects from [2] all have the virial simi-
larity criteria close to unity (spread about 20% or less)
and do not add substantially new information to our Ta-
ble 1. Figure 1 presents the values of criteria 1
T

in
dependence on the ratio of their life time a to the time
of dynamic relaxation . It is evident about 10 dy-
namical time scales of d
T are necessary that for the
virialization of these objects. It is interesting to note that
in a dense gas one or two molecular collision times are
T
d
T

G. S. GOLITSYN
1528
Figure 1. The virial similarity parameter П1 = MG/RU2 for
the 21 galactic clusters from the table versus the ratio of the
cluster age T0 to its dynamic relaxation time Td = (ρG)−1/2, ρ
being the cluster mass density.
needed for the Boltzmann thermal equilibrium, but in
collisionless plasma [16] several dynamic time scales are
needed as well for reaching an equilibrium among vari-
ous degrees of freedom.
6. Conclusions
In astrophysics four quantities can be measured of an
object: mass M, luminocity W, velocity U, size R. There
is also the gravitational constant G, five quantities with
dimensions out of three dimension units: mass, time, and
length. Therefore two non-dimensional similarity criteria
can be formed and various scales. We start to explain
turbulent velocity and magnetic fields. For our Galaxy on
the base of supernova explosions we explain the structure
and intensity of these fields.
As the first similarity criterium we use the well known
virial ratio between potential and kinetic energies:
2
M
GRU
0.6z
z
T
d
T

. For 21 cluster of galaxies from [1] it is found
that this ratio is close to unity for really clusters with
redshift . The 20 per cent scatter may be under-
stood to measurement inaccuracies. The same is for 11
nearly clusters from [2]. The clusters [1] with
have lesser and lesser degree of virial relaxation the
younger they are. A notion of interaction time d is
introduced and it is concluded that of order 10 time
is needed to reach a relaxation for a cluster.
0.6
The second criterium 15 UWG
1.8
is found to be con-
stant at for all 21 clusters from [1] including
not yet relaxed ones and for 11 relaxed clusters from [2].
This unpexected finding needs a model explanation to-
gether with an old Tully-Fisher [8] relation for radiogal-
axies that .
0.02
5
WU
The reasons for this paper to be written are exposed in
the abstract and in the Introduction. I may add two re-
frences for Wesson [17,18] on the application of dimen-
sional analysis to cosmology, which could be a good
starting point in this direction. It should begin, as in any
other case, with an analysis of what the problem is phys-
ically, what is measured and/or should be measured and
how everything is interconnected. Of course, it would not
solve the whole problem of cosmology but may reveal
some important and/or interesting aspects.
7. Acknowledgements
As a student of Moscow University I visited in midfifties
a special lecture course on astrophysics by Prof. A. I.
Lebedinsky. My later work was in environmental physics
in a broad sense, lately there were occasionally a few
specific astrophysical problems, like cosmic rays energy
spectrum, or analogy between earth- and star-quakes.
Sporadic meetings and discussions with I. S. Shklovsky,
Ya. B. Zeldovich, R. A. Syunyaev and some others kept
alive my interest in astrophysics. I am grateful to all
these people and their memories, though occasional talks
with Rashid Syunyaev, Alexey Starobinsky and Alexey
Vikhlinin still continue.
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