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Rotation curves velocities obtained by warm low density
plasma simulating dark haloes and dark matter
Y. Ben-Aryeh
Physics Department, Technion- Israel-Institute of Technology , Haifa, 32000, Israel
e-mail: phr65yb@physics.technion.ac.il
The properties attributed to dark haloes in galaxies can be obtained by using free electrons
model with Boltzmann statistics for low density plasma with temperature above (or equal) 20000
(K) for which Hydrogen plasma is completely ionized but below (or equal) 300000 (K) for which
free electrons model is still valid and quantum and relativistic effects can be neglected. We
perform the analysis for Hydrogen plasma which is the main component of stellar plasmas but we
expect that more complicated plasmas will follow similar behavior. The density (per unit volume)
of electrons is assumed to be smaller than (or equal) one ionized electrons per
3
cm
. For
Hydrogen plasma under these conditions the amount of neutral atom is negligible. Any damping
effects related to bremsstrahlung are proportional to ionized electrons density squared so that
they are also negligible. For small distances from the galaxy center the opposing force to the
centrifugal force on the rotating mass is its attraction with the high density stars. We need,
however, to add the attraction with the low density plasma which becomes the dominant factor
for long distances from the galaxy center. The velocity of the rotating mass can be measured by
its spectrum e.g. Hydrogen spectral lines as it belongs to the high density stars which are
observable (have large electrons density). In the high velocity limit the dynamic of such rotating
mass is affected, however, by its gravitational attraction with the low density plasma (which is
not visible) and the results for rotation curves velocities are obtained.
Keywords: Low density Hydrogen plasma: dark haloes: rotation curves: ionization and
transparency
1
1. Introduction
Rotation curves (RC) describe the rotational velocities of objects in a galaxy as
function of their radial distance from the galaxy center. Various methods for deriving RC
were described in the extensive literature on this subject [1-6] and on dark halo [7-12]. By
studying many galaxies it was found that the stellar RC velocities become nearly constant,
or “flat” with increasing distance away from the galactic center. This result is
counterintuitive since based on Newtonian motion the rotational velocities would decrease
for distances far away from the galactic center. By this argument the flat rotation curves
suggest that each galaxy is surrounded by large amount of dark matter. There are various
models describing dark matter haloes. While such models give a good agreement with
astronomical observations, such as gravitational lensing [13] and with the empirical
gravitational forces, the composition and the physical nature of such dark matter haloes is
not clear. In previous work [14] the dark haloes were related to low density ionized
transparent plasma. The absorption and emission of such plasma is proportional to product
of the electrons and ions extremely low densities (i.e. proportional to the electrons density
squared) so that absorptivity can be neglected. But, as such low density plasma extends
over extremely large volumes gravitational effects become important.
Quite long ago it has been shown that there is a problem of "missing mass" for high
RC velocities [15]. Common approach to solve this problem is to assume that in addition
to the masses of stars there is the mass of dark matter which does not show any
interactions with the EM field. For this purpose it was assumed that there are new non-
baryonic interactions that simulate the gravitational forces but there is not any
experimental evidence to the existence of such interactions. I propose alternative new
model for dark halo where it is obtained by low density plasma which is transparent and
have the properties of dark matter,
As shown in the next section we treat the present plasmas in enough high
temperatures and low densities so that the plasma is completely ionized and spectral lines
of atoms cannot be observed. We find according to Saha equation [16, 17] that the
amount of ionization is inversely proportional to the number of electrons. So at the same
temperature in which the present plasma is completely ionized the stars which have large
number of electrons show rich spectra of Hydrogen and other atoms. RC velocities can be
2
measured by the spectrum of a rotating mass e.g. that of Hydrogen as it belongs to the
high density stars which have large electrons density and therefore are observable. This
rotating mass is attracted to the low density transparent plasma in addition to its attraction
with the high density stars. Another way to detect such completely ionized plasma is to
find stimulated bremsstrahlung effects. But such effects are proportional to the electron
density squared so that such effects are also negligible for low density plasma.
In the following extensive analysis we show how low density plasma can
simulate dark halo effects. We follow here previous work [14] but we extend the analysis
to many more simulations of dark halo effects. We calculate the radius of the dark halo,
the mass of its central part and its tail mass. From these values we derive the potential
produced by the low density plasma so that on the high velocity limit it gives the RC
velocities. We find that our results are in good agreement with experiments.
2. Ionization and transparency of Hydrogen plasma as function of temperature
The Boltzmann distribution can be applied to describe ionization equilibrium
A e A
�
in plasma [17]:
2
2exp
e i e i e B I
a a B
n n g g m k T E
n g h k T
� �
� �
� �
� �
� � � �
. (1)
Here
A
is a neutral atom,
A
is this atom with one ionized electron, with charge
e
,
I
E
is the ionization potential and
B
k
the Boltzmann constant,
,
a i
g g
and
e
g
are the
statistical weights of atoms, ions and electrons;
,
a i
n n
and
e
n
are their number densities
and
e
m
is the electron mass. This equation is known as Saha Equation and is widely used
for calculating ionization degree in thermal plasmas,
The simplest case of pure Hydrogen plasma is amenable to analytic solution,
since there is only one Saha equation to solve. For this case the number of ions is equal
3
to the number of electrons i.e. ,
e I
n n
. Also
1
i
g
(because ionized state is just a
proton) while
a
g
and
e
g
are equal 2. So For Hydrogen plasma Eq. (1) is reduced to:
3/2
2
1 2 13.6
exp
i e B
a i B
n m k T eV
n n h k T
� �
� �
� �
� �
� � � �
. (2)
Substituting numerical values we get [16]:
3/2
21
13.6
2.4 10 exp
i
a i B
n T eV
n n k T
� �
�� �
� �
.
(3)
where
i
a
n
n
is the ratio between the number of ionized electrons
3
( )
i
n m
and the
number
3
( )
a
n m
of neutral hydrogen atoms and by conversion to MKS unit
18
13.6 2.179 10eV x J
�
. For temperature of 20000(K) and ionized electron density
6 3
10 ( )
i
n m
(1 ionized electron per
3
cm
) we have approximately
18
6 10x
ions per
one neutral hydrogen atom! , and for higher temperatures and lower densities this
ratio is increased much further. So under this condition absorption and emission of
radiation related to spectral lines can be neglected and the low density plasma can
be considered as a new kind of dark matter. One should take into account that
although in this calculation the thermal energy is small relative to the ionization
energy the ionization is very close to unity due to the effective statistical weight of
the continuum spectrum which is very high (proportional to
3
3/2
2
2 1
e B
DB
m k T
h
� �
� � � �
� �
� � � �
where
e
m
is the electron-mass,
h
is Planck constant,
DB
is the De Broglie wavelength and the proportionality constant in Eq. (3) is
obtained by using this density of states). One should take into account that at the
same temperature for which low density plasma is transparent the high density stars
4
can have strong absorption and emission of radiation spectral lines. This fact is
related to the
1 /
i
n
dependence in Eq. (3) where for high density stars the density
i
n
becomes very large reducing very much the ratio between the density of ionized
electrons and the neutral atoms. Such high density stars might therefore lead at the
same above range of temperatures to emission and absorption of radiation related to
spectral lines of Hydrogen and other atoms included in the stars atmospheres. The
rotating mass belongs to the high density stars which have large electrons density so
its velocity can be measured by the spectral lines measurements e.g. of Hydrogen.
Low density plasma is described in the present work as a plasma for which
the density of ionized electrons is smaller than (or equal) 1 ionized electrons per
3
cm
. I have shown above by using Saha equations for such plasma that for
temperatures above 20000 (K) the amount of neutral Hydrogen atoms is completely
negligible. Then the Hydrogen plasma is composed of protons and electrons
(without any neutral Hydrogen atoms).
We assume a stationary state for which free electron model with Boltzmann
distribution is approximately valid. Maximal temperature for which the present free
electron model is valid is given under the condition [18-20]:
2
0
1
4
Ze
v
h
(in CGS unit:
2
1
Ze
v
h
) where
e
is the electron charge,
Z
the atomic number,
v
the electron velocity
and
h
Planck constant (divided by
2
). So, the maximal velocity
max
v
for Hydrogen
plasma for which our model is valid is given by:
6
2.3 10 ( / sec)m�
.
2. Dark Halo produced by the low density plasma
In previous work [14] we studied the stability of low-density star plasma where the star
stability is produced by the balance between the gravitational forces and the pressure
produced by perfect gas conditions, and found the impact of such model on the properties
5
of low-density plasmas. While we treated this topic in the previous work we obtained
there only approximate analytical results for the central body of such stellar plasma. By
using now numerical solutions (using Mathematica NDSolve) we get in the present work
explicit numerical solutions. These numerical results help us to get asymptotic equations
for the change of the electrons density
( )
e
n r
and the potential
( )r
as function of the
distance
r
from the star plasma center. I assume a stationary state for which Boltzmann
distribution is valid. Stellar plasma behaves as a perfect gas at densities lower than a
critical value. This critical value is given by the condition that the Coulomb interaction
energy is smaller than the thermal energy. We assume one ionic component with atomic
number
Z
but the analysis can be easily generalized if we have more ionic components.
By using the low-density stellar plasma-stability conditions we obtained [14] the
change of the density of electrons
3
( ) ( )
e
n r m
as function of the distance
( )r m
from the
plasma center
0
( ) ( )
N
e
B
m
n r n Exp r
k T
� �
� �
� �
� �
� �
� �
. (4)
Here
3
0
n m
is electrons density at the galaxy center,
B
k
is the Boltzmann constant,
T K
the absolute temperature in Kelvins,
is the number of nucleons per ionized
electron,
27
1.67 10
N
m kg
�
is the nucleon mass, and
( )r
is the potential.
If there are
nucleons for each ionized electron, then the mass density
r
is
given approximately by
e N
r n r m
(5]
We notice that for fully ionized hydrogen plasma
1
but
will be larger for partial
ionized Hydrogen plasma (according to the amount of ionization) and for other atoms.
For generality purpose we consider the parameter
in our equations but we specify the
results later to completely ionized Hydrogen plasma for which
1
.
6
Using stability conditions for low density plasma the equation for the potential
( )r
is
given as
2
0
2
( ) 2 ( ) 4 4 ( )
N
N
B
m
d r d r G G m n Exp r
dr r dr k T
� �
� �
� �
� �
� �
� �
.
(6)
Here
G
is the gravitational constant and we inserted here the relation
( ) ( )
e N
r n r m
.
We substitute
22
40
00
; 4 1.69 10
N
B
m n n
x r G k T T
�;
(7)
where
is extremely small number (under the present conditions).
Then Eq. (6) is changed to
2
2
( ) 2 ( ) 1
( ) ( ) ;
N N
B
N b b
m m
k Td x d x Exp r Exp b x b
dx x dx m k T b k T
� �
� � � �
�
� �
� � � �
� � � �
� �
.
(8)
One should take into account that
x
is a normalized pure number where for
1x
the
distance
r
from the star center stretches to extremely very long distance equal to:
1 /
where according to Eq. (7)
is a very small number.
Our solutions for the number of electrons and the potential are given as function of
the pure numerical parameter
x
i.e. as
( )
e
n x
, and
( )x
. So, by transforming our
solutions to those of functions of the distance
r
they become functions of astronomical
long distances.
We assume a stationary state for which Boltzmann distribution is approximately
valid .We would like to use the free electron model for completely ionized Hydrogen
plasma for which absorption and emission related to spectral lines can be neglected, and
discuss the possibility to relate such plasma to dark matter. Due to this condition we treat
7
low density stellar plasma with temperatures above
4
2 10 ( )K�
, since only under this
condition Hydrogen plasma will be completely ionized according to Saha equation [16,
17] . I suggest the idea that the present analysis describes some kind of dark matter. I
would like to check this idea by extending the previous treatment of the electromagnetic
(EM) properties of low-density plasma to that of rotation curves velocities..
4. The stability of spherical low density stellar plasma analyzed by the use of
Boltzmann statistics
We extend here the previous analysis for the stability of spherical low density stellar
plasma. Magnetic fields might be important for high density stars but it has been
shown by following Bohr-Van-Leeuwen theorem, [21, 22] that under free electrons
model and classical conditions the averaged magnetic fields vanish so we neglect their
effect.
In the previous work [14] we obtained the differential equation for
0
( )
( )
e
n x
xn
which is given as
2
2
2
2
( ) 2 ( ) 1 ( ) ( ) ;
( )
x x x x x r
x x x x x
� � �
� �
� �
� � �
� �
. (9)
It is quite easy to find that Eq. (9) is satisfied for
0 0
2 2 2
0
( ) 2 2 2
( ) ( ) ; ( )
e
e e
n x
x n x n n r n
n x x r
�
. (10)
But as the solution of Eq. (9) by Eq. (10) does not satisfy the boundary condition
0
0
( 0) 1
e
n x r
xn
this solution is valid only for
1x?
.
8
By numerical calculations (using Mathematica NDSolve) we obtained exact
numerical solutions of Eq. (9). Such solutions give the density of electrons
3
( )
e
n x m
as function of the distance
x
r m
from the star center, and as function of the
density of electrons
3
( )
o
n m
in the star center, which is taken as experimental
parameter. These solutions enable us to get asymptotic results at large distance
r
from
the star center for the electrons potential and for the density of electrons. We find that in
addition to the central part of such stellar plasma we get now a very long tail of the
stellar plasma potential and its density.
An approximate solution is obtained for the region near the center of the star by
using series expansion of the exponential function of Eq. (4). Then, we get
2
2 2 40 2
0
0 0
( ) 1 ( ) 1
( ) ; 1.69 10 ( )
6 6
e e
n x n r n
x Exp x Exp r m
n n T
� � � �
� �
� � � �
� � � �
; ;
.
(11)
By using Mathematica (NDSolve) the general solution for
0
( )
( )
e
n x
xn
of Eq. (9)
(including the boundary condition
( 0 )
e o
n x r n
) as function of
x
was
calculated. The approximation (11) was found to be valid near the stellar plasma
center. The numerical calculations for
10 x�
were found to be equal to the
approximate solutions by Eq. (10). The use of the function:
0
( )
( )
e
n x
xn
is
remarkable. It gives also the mass density
3
( )
e n
kg m n m
�
as function of the
distance
( ) x
r m
from the star center which is proportional to the electron density
9
3
0
( )n m
at its center. One should notice that the density of electrons
( )
e
n x
for
x≫1
is decreasing inversely proportional to
2
x
.
5. The radius of the dark halo the mass of its central part and its tail mass
One should take into account that
x
is a normalized pure number where for
1x
the
distance
r
from the star center stretches to extremely very long distance equal to:
1 /
where according to Eq. (7)
is a very small number. Important aspect in the
interpretation of Eq. (8) and other similar calculations is the scaling of astronomical
distances where the distance
( )R m
from the galaxy center is given by Eq. (7) as
19
20
40 0
1 7.7 10 ( )
1.69 10
x x T
R x m
n
n
T
�
�
(12)
where for completely Hydrogen ionized plasma
1
. We find that the distance is
proportional to the dimensionless parameter
x
, proportional to the square root of the
temperature and inversely proportional to the square root of the ionized electrons density
in the galaxy center. So, for lower density
3
0
( )n m
and larger temperature
T
the distance
R
becomes in the order of dark halo radius.
Using Eq. (11) the total number of electrons
,e cent
N
in the central part of the present
star is calculated by substituting
of Eq. (7) and performing the integral:
3/2
2 2 61 0
, 0 3
0
0
14 3.72 10
6
e cent
T n
N n Exp r r dr n
�
� �
� �
� �
� �
� �
� � � �
�
.
(13)
By multiplying Eq. (13) by
N
m
we get for the central part mass
cent
M
of this galaxy
10
3/2
34 0
2
0
6.21 10
cent
T n
M kg n
� �
� �
� �
� �
.
(14)
The central part mass
cent
M
is proportional to
3/2
T
and inversely proportional to
0
n
Using Eq. (10), the number of electrons in the tail of the low-density star plasma
,e tail
N
is
max
20 max
, 0 2
0
2 8
4
R
e tail
R
n R
N n r dr
r
�
.
(15)
We find here the interesting point that the amount of electrons for spherical shell with
thickness
dr
is constant, so that the divergence of the integral is prevented by assuming a
maximal value
max
max
x
R
. We simplified the integral in Eq. (15) by taking its lower
limit as
0R
. By multiplying Eq. (15) by
N
m
we get for the tail mass
cent
M
of this
galaxy
0 max
8
tail N
n R
M kg m
. (16)
We obtained here a very simple equation of the galaxy tail mass and it is proportional to
maximal radius of the galaxy
max
R
which might be taken as experimental parameter..
6. Rotation Curves
As the low density plasma has spherical symmetry its potential
p
at the point of the
rotating mass can be given as:
11
( )
cent tail
p
GM G M M
R R
. (17)
where G is the gravitational constant, R is the distance from the rotating mass to the
galactic center and
cent
M
,, and
tail
M
are the low density galaxy plasma central mass and
its tail mass obtained, respectively, by Eqs. (14) and (16). The kinetic energy per unit
rotating mass is equal according to the virial theorem to one half of its potential energy.
Then, we get
2
R p
v
(18)
where
R
v
is the rotating mass velocity
The contribution of the plasma tail to the velocity of the rotating mass is
obtained from Eqs, (16) and (17) as:
20
1
8
( )
tail
p N
GM n G
tail v m
R
. (19)
Substituting
2
40 0
1.69 10 n
T
�
from Eq. (7) ino Eq. (19) we get:
2 4
140
8 1.66 10
1.69 10
N
Gm T
v T
�
�
. (20)
We find here the interesting result that the contribution of low density plasma tail
contribute velocity which is proportional to
T
and independent of the distance
R
and
the density of the ionized electrons (within our approximations).
The contribution of the central part of the Hydrogen plasma to the velocity of the
rotating mass is obtained from Eqs, (14) as:
3/2
34
20
22
0
6.21 10
( )
cent
p
GM G T n
cent v R R n
� �
�
�
� �
� �
. (21)
Substituting R from Eq. (12} into Eq. (21) we get:
12
4
2
2
5.38 10 T
vx
�
. (22)
Here again the contribution of the plasma central part to the high limit rotation velocity is
independent of the density
0
but inversely proportional to the normalized distance
x R
.
The total rotational velocity squared in the high velocity limit is obtained by adding
the contributions from Eq. (20) and (22) .Then we get:
4
2 2
2
1
10
1.66 5.38 / T
v v x
, (23)
where for Hydrogen plasma
1
. For example for trmperature
5
3 10T�
for which
the present free electron model is valid we get for Hydrogen plasma:
2
2 2 9
2
1
5 16 / 10 sec
m
v v x � �
� �
� �
.. (24)
Taking into account that
x
is small number (nearly 1-3) we get values of the high
velocity limit which are in order of those of experimental values (see for comparison [2]).
7. Conclusion and summary
Low density plasma can simulate dark matter effects. Although we made the
calculations for completely ionized Hydrogen plasma similar relations are expected for
more general plasmas in which Hydrogen plasma is mixed with other atoms. The present
analysis is based on semi-classical free electrons model which is valid for temperature
above (or equal) 20000 (K) in which the plasma is completely ionized but below (or
equal) 300000 (K) for which quantum and relativistic effects can be neglected, We
analyze dark haloes and high velocity rotation (RC) curves and the way they can be
related to low density plasma. Although we have used a very simple free electrons model
we find that this model predicts high velocity limits which are in the order of the
experimental values. In a recent article [23] it has been suggested that Helium is dark
13
matter transparent medium but according to the present article low density Hydrogen
plasma is also transparent and the measurement of Hydrogen spectral lines is made only
in high density stars which have high densities ( as predicted by Saha equation).
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