Similarity Criteria, Galactic Scales, and Spectra

doi:10.4236/jmp.2012.310188

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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

Email: [email protected]

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

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



321



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

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-

G. S. GOLITSYN

1524

tities one can form two dimensionless values, the simi-

larity criteria [3-7]. We choose the following as such

criteria:

 , (1)

225



1

1

,3 5

WU n

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

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





 3

WUG we obtain . From Equations

(1), (2) and (4) one may form the nondimensional simi-

larity criterium as

 

321

 15

cUWG



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):

dRR

TUMG









, (6)

which with the mass density scale R









equals to





. (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



, 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):



UMGR



, (8)

the time scale is



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





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







410

210 kgM 26

410 WW



. (12)

For our Galaxy with stars with the solar mass

 and brightness we

obtain 240 MyT380 msU,



, ,



1.6 10WW

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

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



wMUR





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

using the volume energy density w as:

4π

gwc









T, (13)

where



4π3

13 3

10 Jm



3.35

is the area of a unit sphere. With

we obtain g



K. This is close to

relict temperature 2.7 K. This is a single scale with the

hint of the quantum mechanics because

23 1

π5.67 10

1.3810J K.







 

WLK,





1.0610J s



 

is the Bolzmann constant, is the

Planck constant.

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

mMG m

kR k



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)



kg.

10 



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





3.6 10

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





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):

UWG



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



MR . It may ex-

plain the high luminosities of compact quasars. If we

know only the luminosity W we may estimate the ratio

R for the virialized objects as 25 35

aW G

0.93 0.20a1 which



 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



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

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.



210 W

810W



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.

T

37

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

1.5 10m

 

223

r uxbr



 



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

0m 1031r 0pc we obtain



12 20

ur







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

we shall do it for the modulus of the Al-

ven velocity



4π









  

213 23

AA A

DrVxr VxNr





 



. Then

, (16)

where the rate of magnetic energy generation/dissipation

is defined as

d2d 8π

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



13231235 432

310ms510 msN





 

. For





100 pc310mr we will have from Equation (16)

2822

10 mcV

60 3

10 mV

A. The mean density of the interstellar gas



 is equal to 24 3

10g cmMV





 

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

k

2.5 10

1810 K

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





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

TkA



, (18)

G. S. GOLITSYN

1527

Table 1. Parameters of the distant clusters and their virial similarity criterium П1 (Equation (1)).

№ z T, (keV)



,10Ww14

,10

M,MpcR П1







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





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





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

T gives the cluster age in the time units

of .

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



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

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:

GRU

0.6z

z



. 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

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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