The weak turbulence methods in the problem of galaxy mass distribution function
The analogy between the turbulent Kolmogorov type spectra and the distribution of galaxies on masses is discussed. Due to nonlocality [1] of galaxy mass spectrum formed by merging the additional approximate conservation law affects its intermediate asymptotics power index, which proves to be interve...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2000
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irk-123456789-785512015-03-19T03:02:01Z The weak turbulence methods in the problem of galaxy mass distribution function Kontorovich, V.M. Plasma dynamics and plasma-wall interaction The analogy between the turbulent Kolmogorov type spectra and the distribution of galaxies on masses is discussed. Due to nonlocality [1] of galaxy mass spectrum formed by merging the additional approximate conservation law affects its intermediate asymptotics power index, which proves to be intervening between the constant mass flux and this of the number of massive galaxies. Analytical description of this asymptotics, which essentially uses the Smoluchowsky kinetic equation symmetry transformation, is presented. The result is compared with the recently discovered steepness of galaxy luminosity function on its faint edge. The problem of the main part of barionic mass in the Universe is shortly discussed in this context. 2000 Article The weak turbulence methods in the problem of galaxy mass distribution function / V.M. Kontorovich // Вопросы атомной науки и техники. — 2000. — № 6. — С. 84-87. — Бібліогр.: 18 назв. — англ. 1562-6016 http://dspace.nbuv.gov.ua/handle/123456789/78551 533.9 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Plasma dynamics and plasma-wall interaction Plasma dynamics and plasma-wall interaction |
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Plasma dynamics and plasma-wall interaction Plasma dynamics and plasma-wall interaction Kontorovich, V.M. The weak turbulence methods in the problem of galaxy mass distribution function Вопросы атомной науки и техники |
| description |
The analogy between the turbulent Kolmogorov type spectra and the distribution of galaxies on masses is discussed. Due to nonlocality [1] of galaxy mass spectrum formed by merging the additional approximate conservation law affects its intermediate asymptotics power index, which proves to be intervening between the constant mass flux and this of the number of massive galaxies. Analytical description of this asymptotics, which essentially uses the Smoluchowsky kinetic equation symmetry transformation, is presented. The result is compared with the recently discovered steepness of galaxy luminosity function on its faint edge. The problem of the main part of barionic mass in the Universe is shortly discussed in this context. |
| format |
Article |
| author |
Kontorovich, V.M. |
| author_facet |
Kontorovich, V.M. |
| author_sort |
Kontorovich, V.M. |
| title |
The weak turbulence methods in the problem of galaxy mass distribution function |
| title_short |
The weak turbulence methods in the problem of galaxy mass distribution function |
| title_full |
The weak turbulence methods in the problem of galaxy mass distribution function |
| title_fullStr |
The weak turbulence methods in the problem of galaxy mass distribution function |
| title_full_unstemmed |
The weak turbulence methods in the problem of galaxy mass distribution function |
| title_sort |
weak turbulence methods in the problem of galaxy mass distribution function |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2000 |
| topic_facet |
Plasma dynamics and plasma-wall interaction |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/78551 |
| citation_txt |
The weak turbulence methods in the problem of galaxy mass distribution function / V.M. Kontorovich // Вопросы атомной науки и техники. — 2000. — № 6. — С. 84-87. — Бібліогр.: 18 назв. — англ. |
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Вопросы атомной науки и техники |
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2025-07-06T02:36:45Z |
| last_indexed |
2025-07-06T02:36:45Z |
| _version_ |
1836863356465577984 |
| fulltext |
UDC 533.9
84 Problems of Atomic Science and Technology. 2000. № 6. Series: Plasma Physics (6). p. 84-87
THE WEAK TURBULENCE METHODS
IN THE PROBLEM OF GALAXY MASS DISTRIBUTION FUNCTION
V.M.Kontorovich
Institute of Radio Astronomy of the National Academy of Sciences of Ukraine,
Chervonopraporna Str., 4,Kharkiv, 61002, Ukraine. E-mail: vkont@ira.kharkov.ua
The analogy between the turbulent Kolmogorov type spectra and the distribution of galaxies on masses is
discussed. Due to nonlocality [1] of galaxy mass spectrum formed by merging the additional approximate
conservation law affects its intermediate asymptotics power index, which proves to be intervening between the
constant mass flux and this of the number of massive galaxies. Analytical description of this asymptotics, which
essentially uses the Smoluchowsky kinetic equation symmetry transformation, is presented. The result is compared
with the recently discovered steepness of galaxy luminosity function on its faint edge. The problem of the main part
of barionic mass in the Universe is shortly discussed in this context.
In the expanding Universe the galaxies as we know
run away from each other. In the ideal scheme of such
Hubble's expanding there is no converging of galaxies:
merely the distances between the galaxies grow. Under
such conditions the galaxies evolve independently and
their masses do not essentially change.
In reality the galaxies have not only the Hubble's
recession velocities. They also have some "peculiar"
components which are the result of fluctuations. The
latter are also relevant for formation of galaxies
themselves - the gravitationally coupled systems.
Immediate confirmation of existing of such initial
fluctuations has been received by investigation of the
relic radiation (on the microkelvin's level).
With such fluctuation the Jeans gravitation instability
evolves. In the expanding Universe it transforms from
exponential to power type, but is not removed under
expansion.
In the issue the galaxies gather in groups. Our
Galaxy - the Milky Way - belongs to the Local Group
containing one dozen odd members. The big groups -
clusters - contain themselves groups as a rule. Many
clusters are parts of super clusters, and so on, thus
forming hierarchical structure of the Universe.
In the vicinity of the Milky Way three colossal super
clusters have been detected. The most famous is the
Great Attractor (GA) on which our Local Group falls.
This falling is reflected even in the dipole anisotropy of
relic radiation on the millikelvin's level corresponding to
the velocity of about 600 km/s. The Hubble low (as
straight observation show) fails in the direction of GA:
near GA (remote from us at about 60 Mpc) galaxies
accelerate or decelerate due to their falling on GA. In
similar cases of different scales the galaxy-number
density grows, the peculiar velocity components change
and the collisions and mergers of galaxies take place
(see the review lectures in [2] and the numerous
references there).
As a result of mergers the mass function (MF), i.e.
distribution of galaxies on masses is formed. The answer
to the question: "Where is the main part of barionic
stellar mass of the Universe is concentrated?" depends
very essentially on some of the MF-details.
On the other hand the activity of galaxies and their
nuclei arises owing to merging and such objects as radio
galaxies and quasars appear. The correlation between
the merger process and activity proves this. Probably
even the Black Holes which are responsible for the
nuclear activity may appear in the merging processes.
But this side of the problem goes beyond the scope of
this article (see, for example, [3] and the references
there).
We will be interested only in the galaxy MF and
their analogy with the Kolmogorov spectrum of
turbulence in liquid and with the similar weak turbulent
spectra in plasma.
As we will see the methods being developed in
plasma physics prove to be useful for the analysis of
galaxy mass distribution. The difference is that we have
in the MF case the mass flux on the spectrum instead of
energy flux in the Kolmogorov turbulence case. And the
former is found to be the nonlocal.
The main part of the computations of this paper will
be published in PysicaD, 2000, with more details.
1. So, we will be interested in the mass spectrum
formed while merging, the main attention being paid to
the index of power intermediate asymptotics (IA), which
in the merging model assumes a clear physical sense and
can be obtained by analytical methods. This MF part is
juxtaposed with the recently discovered greater
steepness of the galaxy luminosity function at its faint
edge (see as example [4]). This also may serve as an
argument in favor of the galaxy evolution due to
merging.
Below we will realize what could give us the pure
analytical methods analogous to those in the weak
turbulence theory [5,6]. The idea of nonlocality of the
weak turbulent spectra which was discussed by Balk,
Zakharov and Nazarenko in the plasma physics
problems context [1] and symmetry transformations of
kinetic equations [5,7] will play the most essential role
in this description.
85
2. Before examining the real galaxy interaction we
must recall that SE for MF,
∂
∂
f m t
m
dm dm
( , )
=∫ 1 2 [U f fm12 1 2δ −
− −cycle bicycle ] (1)
where ( )f f m t≡ , , ( )f f m t1 1≡ , , etc, are MFs,
( )δ δm m m m≡ − −1 2 , δ — Dirac’s delta-function,
( )U U m m12 1 2≡ , , which describes the merging,
permits two nontrivial (here we do not touch the case of
U = const) exact solutions describing the MF evolution
from its initial state localized on small masses [8]. In the
second and third terms we have δ (m2-m-m1) and δ (m1-
m2-m) correspondingly.
For the merging probability proportional to the product
of colliding masses the power index of IA is s1=-5/2.
Expressed through the uniformity power u of
coagulation coefficient ( ) ( )( )U am a U mu= it equals
s1= – (u+3)/2 (u=2). As known (see also below) such
index corresponds to the constant mass flux on the
spectrum. In the case of U = c(m1 +m2) the power of IA
is s0 = – 3/2 or s0 = – (u+2)/2 (u=1). The latter
corresponds to the constant flux of the number of
massive objects. Though at first sight such a
conservation law fails with the mergers it is realized in
the form of approximate integral in the case of
predominant interaction of large objects with the small
ones. Such a "nonlocal" situation fits the last solution.
The condition of locality, i.e. that of convergence of the
collision integral in SE for s1 - solution is in the form of
|u2 - u1 |<1 [9], where the indexes u1 and u2 are defined
by the expression for U provided the masses differ
strongly U m mu u∝ 1 2
1 2 ( )m m1 2<< . Obviously, in
the first case ( )u u1 2 1= = the locality criterion is
fulfilled and in the second the marginal case occurs (u1
=0, u2 =1). That is, the galaxy interaction with the most
distinguished scales prevails in the latter case and, thus,
the conservation law of the "number of particles" is
realized.
3. For gravitational galaxy interaction the cross-section
of coagulations is usually taken as product of co-factors,
which describe, respectively, the geometrical cross-
section, gravitational focusing and conditional merging
probability at the frontal collision of galaxies (see
references in 3]):
( ) ( )σ π γ ϕ γ γ= + ≡r v vg
2 2 21 , ,
v G m r m m m r r rg
2
1 2 1 22= = + = +, , .
The homogeneity index differs for "large" and "small"
masses. Further we will focus on this particular region,
regarding a small mass region as contracted to zero.
This scheme can be attributed with a more accurate
formal meaning. On the assumption that ϕ decreases as
square of relative velocity we can take it in the form of
( )ϕ γ= +
−1 1 1
. The resulting cross-section will be a
uniform function in all of the mass-changing interval:
σ π γ= r 2 . By averaging over velocities we come to
the coagulation coefficient U v= < >σ in the form
( )( )U m m m m∝ + +1 2 1 2
β β
where the radius-mass
dependence is chosen as r m∝ β . Below we employ
only the fact that U is the uniform function of masses
with
u u u u= + = =1 01 2β , , ( )β = ÷1 3 1 2
For u>1, as known from general theory of SE 1, the
evolution of MF has an explosive character and a quasi
power asymptotics is established in a wide mass interval
between the region of initial mass localization m m≈ *
and the coagulation front which is turned to the infinity
mass for the finite time [11,12]. Our goal is to find the
power IA of the spatial homogeneous 2 solutions of the
SE (1) with the considered kernel U discussed above.
( )J m dm mIst
m
1 =−∫ ,
( )∂
∂
∂
∂
mf
t
J m
m
+ =1 0 (2)
Here the I st is the right part of SE (1).
4. Both the numerical solution of SE and modelling by
Monte Carlo method show that the power index of IA
α lies between s0 = – (u+2)/2 and s1 = – (u+3)/2 (see
for example [10] Fig. 2b; [13]). In order to understand
what it means consider the symmetry properties of the
collision integral of SE in the case of exact uniformity:
U a Uam am am
u
m m m1 2 1 2
= .
To utilize the similarity of U we must change
simultaneously the scale of all three arguments m m1 2,
and m. But, as one of them (m) is fixed in SE, from the
continuous group transformation only two discrete
transformations remain (except a trivial one): G1 ,
transforming m m1 → , and G2 , transforming
m m2 → . These Zakharov transformations are
1 In this case the initial distribution localized on small masses
within a finite time forms a power "tail" spreading on the
region of formally infinite masses [8]. This the so-called
kinetic phase transition was first discovered and studied in
detail by Stockmayer for the above-mentioned model with U
= cm1 m2 and was utilized for describing polymerization, in
particular, zol–gel transition (in addition to the above-
mentioned see also the references in [10]). In the case of the
gravitating systems we are interested in, the new phase which
emerges at the transition and corresponds to the "infinite"
mass is juxtaposed with cD-galaxies in the center of the
cluster.
2 This surely leads to the loss of a number of distribution
features, including spatial stratification of galaxy clusters, with
a more compact central and less dense periphery, etc. At the
same time the chaotization in the systems considered confirms
the made assumption.
86
considered as some change of variables m1 , m2 with
the fixed mass m conditions:
G m
m
m
m m
m
m
m
G m
m
m
m m
m
m
m
1 1
1
2 2
1
2
2 1
2
2
2
2
1
= → →
= →
→
,
, .
(3)
They form a symmetry group of SE [7]. For these
conformal transformations the integrating paths tending
to infinity in the second and third terms in (1) convert
into the integrating path with the finite mass variation in
the first term of SE. In the issue (using also
x m x→ − symmetry) SE acquires the form of:
( ) { }∂
∂
f m t
t
dxU m x m x
m,
,,= −∫2
0
2
{ } =
( ) ( ) ( ) ( )f m x f x
m
x
f
m
x
m x f m
u
− −
−
+2
( )−
−
−
+m
m x
f
mx
m x
f m
u2
(4)
If in addition a power-law character of the solution is
assumed ( )( )f m ms∝ , then { } in SE is reduced to
[5,14]:
{ } ( ) ( )= − −
−
−
f m x f x
m
x
m
m x
1
ν ν
,
ν = + +2 2u s (5)
In the stationary case we get the exact power solution
f ms∝ 1 , ν = −1, which corresponds to the constant
mass flux P on the spectrum f c P mP
s= 1
1
2 1 . This
can be easily proved by using the definition of mass
flux, thus finding the normalization factor and the flux
sign [9]. The obtained formal solution, however, is
nonlocal: the integrals diverge on small masses, which
thus must contribute mainly.
5. Now consider the MF decreasing steeper than the
power on the largest masses. With this condition the
second term in { } (3) vanishes in the case of essential
contribution of small masses due to nonlocality. In the
issue the approximate power solution arises that
corresponds to conservation of the number of massive
galaxies (if their interaction with the small-mass ones
prevails):
f m s∝ =0 0, ν (6)
Really, the flux of the number of massive galaxies
(below – the galaxy flux) on the mass axis is
( )J m dmIst
m
0 = −∫ , (7)
where through I st we defined the right part of SE (1).
This corresponds rewriting the latter in the form of
approximate conservation law
( )∂
∂
∂
∂
f
t
J m
m
+ =0 0 . (8)
For the power spectrum the galaxy flux
( ) ( ) ( )J m dmm F
m
F
m
0
1
0 0= − = −−∫ ν
ν
ν
ν
ν ,
where ( )F0 ν = (9)
( ) ( )2 1 1
1
11 1
0
1 2
d U f fζ ζ ζ
ζζ ζ
ν
, − − −
−
∫
At ( )ν → +0 (onesiding limit corresponds to
integrability of the expression for ( )j m0 in the origin)
in accordance with (5) F0 0→ and we obtain the
solution with constant galaxy flux J m Q0 ( ) = >0 (cf.
[15]):
Q = = ( )2 1
1
11 1
0
1 2
d U fζ ζ
ζζ ζ, ln− −
−∫ . (10)
The positive sign of the flux corresponds with the
physics of mergers. Using the definition of particle flux
Q we can normalize this distribution too:
f c Q mQ
s= 0
1 2 0 (cf. [16]). With the two concurrent
fluxes it is easy to find the analogous solutions, when
one of the fluxes is smaller than the other. The value of
this ratio depends on the mass: mQ P . Obtain in the
issue the spectrum with the break at P m Qbr ~ 1
which overpasses on its ends to single-flux distributions.
However, our whole case is nonlocal in principle [10],
and even the additional conservation law is connected
with this nonlocality (cf. [1] ).
6. Thus, we have to proceed to the differential
description primarily accounting for the interaction with
multiple dwarf galaxies. With the original SE form (1)
this is difficult to do in view of the equal character of
divergence on small m1 and m2 , as well as on the
infinity masses. After Zakharov transformations we have
only one singular point ( )m1 0= , near-which
expanding gives us the equation:
( ) ( ) ( )∂
∂τ
∂
∂
f M t
A
f m
m
u
m
f m
,
= − +
+
−
2
- ( )f m dx
m
x
f
m
x
u
∫
+
0
2 2
-
( ) ( )f m
m
dx x
f x
x
2
0
∂
∂∫ ; (11)
87
( ) ( )A dx xf x t U m m= = ⋅∫
0
02, τ
The afore-noticed compensation occurs here
automatically. For the pure power distribution obtain the
equation:
− = +
∂
∂τ
∂
∂
ΦΦΦΦ ΦΦΦΦ
ΦΦΦΦA
m
B 2 , (12)
( )ΦΦΦΦ ≡ = = + +∫m f m t B dx x u sµ µ µ, , , 2
0
,
The formal solution of (12)
1 ΦΦΦΦ = +Cm D ( )C B A=
(abstract for a moment from its power form) gives us
two asymptotics. One corresponds to s1 (Cm D>> ):
( )1 1ΦΦΦΦ = ∝Cm f m s ; (13)
the other – to s0 (Cm D<< ):
( )1 0ΦΦΦΦ = ∝D f m s . (14)
Thus we can affirm [13] that if the SE solution is
approximated by the power-type function its index
appears between these limit values
− − < < − −2 2 3 2 2β βs . The resulting index
can be somewhat smaller or larger than -2 and close to
the shechter's index α ≅ − ±171 0 5. . , ([4], field
galaxies), α ≅ − ±2 2 0 3. . , [17] rich clusters galaxies))
on the faint end of LF. If s < −2 , the major part of
mass is concentrated in the smallest galaxies. In the two
flux MF case as we can see above the divergent of the
integral for fQ on the upper limit and the same for fP
on the lower limit results in the main mass concentration
near mbr located by the IA of the whole nonstationary
solution. Some astrophysical appearances of this
question was also discussed in [18].
Note we can not exclude the opposite case when MF is
bent out to the bottom.. Because we can only guarantee
the existence of two power asymptotics but not of their
place in the MF for the assumption we have made of the
pure power form of equation (11) solution.
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