The effect that cosmic rays and the Alfven waves they generate have on the structure of the plasma distribution perpendicular to the galactic disk is examined. It is shown that the plasma distribution exhibits two length scales and the predicted values of gas density far from the galactic plane indicate that models involving hydrostatic equilibrium should be replaced by those allowing for a galactic wind

Dougherty, M. K.

Mckenzie, J. F.

Westergaard, N. J.

English

NASA Technical Reports Server

8 3 OG 7.2-18
The Role of Cosmic Rays and Alf#en Waves
in the Structure of the Galactic Halo.
M.K. Dougherty and J.F. McKenzie (University
of Natal, Durban, R.S.A.) and N.J. Westergaard
(Danish Space Research Institute, Lyngby, Denmark)
ABSTRACT
The effect that cosmic rays and the Alf_en waves they generate
have on the structure of the plasma distribution perpendicular
to the galactic disk is examined. It is shown that the plasma
distribution exhibits two length scales and the predicted values
of gas density far from the galactic plane indicate that models
involving hydrostatic equilibrium should be replaced by those
allowing for a galactic wind.
I. Introduction. It is now recognised [1,2,3] that since the energy
densities of the interstellar gas, magnetic field and cosmic rays are
of the same order (~ 1 eV cm-3) in the interstellar medium, the coupling
between these components, which must be taken into account in a self-
consistent fashion, leads to important dynamical effects. Here we
examine the role of cosmic rays and the Alf_en waves they generate on
the distribution of the plasma perpendicular to the galactic plane.
2. General Equations. Hydrostatic equilibrium perpendicular (i.e.
along" _)requires
= ___(Pg + Pm+ Pw + Pc) (I)
in which p = interstellar gas density with pressure pg, Pc = cosmic
ray pressure, Pm = magnetic pressure, Pw = Alf_en wave pressure and
g_ is the _ component of the galactic gravitational acceleration given
by -_(r_) where _(r,_) is an axisymmetric galactic gravitational
G Mipotential such as [3] 2
¢(r,_) Z
= i=l [r2 + (ai + (z2 + bi2)½)2]½ (2)
al, bz; a2, b2 are parameters for the bulge and disk components of
the galaxy, Mi is the corresponding mass parameter, G the gravitation
constant, and r the radial and _ the axial coordinate.
Treating the cosmic rays hydrodynamically,that is, in terms of their
pressure and effective diffusion constantK, the effects of convection
and diffusion are described by the cosmic ray transport equations [:4]
dec : vdPc + Q (3)
d_ d_
Yc K dPc
Fc = (y_ZT i VPc -(Yc-l) _ (4)
Yc = adiabatic index, Fc = cosmic ray enthalpy flux, V = Alf_en wave
velocity assumed _v_o" The wave energy exchange equation which
describes how the w_ve pressure is generated by the cosmic ray pressure
gradient _,5] is
d dPc -L (5)d--_(V2Pw) = -V d-_-
https://ntrs.nasa.gov/search.jsp?R=19850026572 2020-03-20T16:49:06+00:00Z
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0G 7.2-18
In the absence of additional energy losses or gains, Q = L = 0, (3), (4)
and (5) give the total energy flux conservation law as
Yc K dPc
(_C-I) VPc - (_C--_-) d--_- + V2Pw = constant (6)
3. Results for the cases of strong and weak scattering.
(a) Strong scattering (K . o) If the cosmic ray diffusive flux is
small compared with the cosmic ray enthalpy flux the cosmic ray trans-
port equation (3) integrates immediately to yield the adiabatic form
pcv_c = constant (7a)
or Pc _ p_- ( = p2_ , Yc = _A) (7b)
The total energy equation (6) then tells us how the wave pressure Pw is
related to the plasma density, namely
yc-I
Pw = Pwo (P/Po)½ + Pco 2(_c-l)(P/Po )½ [l-(P/Po )--£--] (8)
where the subscript "0" refers to the value of a quantity at some
reference level, eg. at the galactic plane. It is often assumed that the
_.0- magnetic pressure is proportional to the
plasma pressure [3,6] which in turn we
shall assume is related to the density by
some polytropic law of the form
.75. _ P/Po ._go = (P/P°)Yg (9)
Pc
/Pco Under these assumptions the magnetic
.5o. pressure may be simply incorporated into
II the gas pressure in the form of anPw .co//D effective sound speed also given by
I 2 = Yg(Pqo+ Pmo).25 aso Po (I0)
The above "equations of state" for p_
Pw and Pcenable us to integrate the _'
hydrostatic pressure balance equation
c _ 2 3 4 5 immediatelyto yield
, _P
Figure l :
Distributions for r = 5kpc,
aco = 13,6 kms-l,
3.8 kms-l
aso
Ve = 500 kms-I ., where _ = _/b2
85
0G7.2-18
2 -l I,
_(r,o) - _(r,_) - (_I)[I-(P/Po)Yg ] + awo2[(P°/P )z-I]
+ aco2 (Po/p)½ [l- (P/po) Yc___] (ll)(Yc-l )
2 = Yc Pw 2 _ Pwo
aco 2 awo -
Po Po
Thus for a given shape for the galactic gravitational potential (such as
equation (2)), equation (II) determines the plasma distribution as a
function of r and _. Figure 1 shows the distributions of P/Po, Pc/R
and Pw/pc o perpendicular to the galactic disk. It is interesting to c°
note the qualitative agreement between the theory and observations [6,7]
denoted by crosses. The main point to note is that the model predicts
in a natural fashion that the plasma distribution will exhibit two length
scales; a fast decay near the galactic plane (determined by the plasma and
magnetic pressure)followed by a much slower delay determined by the
self-excited Alf#en wave pressure.
(b) Weak Scattering (m large) If we assume that the cosmic ray
enthalpy flux is negligible compared to the diffusive flux in equations
(3) and (4) then the wave energy equation may be written
w): vFo (12)
where Fo is the (constant) diffusive cosmic ray flux.
This equation (with some assumptions for the behaviour of K) along with
hydrostatic pressure balance generalizes the galactic halo model of
Ghosh and Ptuskin (GP) to include Alfgen waves generated by the cosmic
ray pressure gradient. The GP model and this generalization suffer from
the defect that sufficiently far from the galactic plane the density
(and therefore the gas and magnetic pressures) and wave pressures begin
to increase and the cosmic ray pressure goes negative. This defect in
the model arises because it is not consistent to simply assume that the
diffusive flux is constant since in fact overall energy flux must be
conserved (equation (6)). Thus the wave energy flux can only grow at the
expense of the diffusive flux with the implication that the system will
eventually drive itself into the strong scatteringmode discussed above. We
therefore expect that the results for _#0 will be similar to those for
K = 0 with some modifications which will maximize around the point of
inflection of the distribution of Pc with 2.
86
OG 7,2-18
Conclusions
The strong scattering limit predicts two length scales describing the
gas distribution; the larger one which prevails above the plane arises
because the cosmic rays and the waves they generate lift the gas up and
stretch it out far beyond the galactic plane, asymptotically approaching
a constant value p_ ~ Po (Yc-I)2/ (Ve/aco)" (where Ve is the galactic
escape speed ). Although P_/Po << 1 this asymptotic value for the
density i_ nevertheless much greater than typical intergalactic densities
(2.xlO-8x_)with the implication that the hydrostatic model should be
replaced with a galactic wind model.
4. References :
I. Gin_burg, V.L. and Ptushkin, V.S. (1976) Rev. Mod. Phys 48, 161
2. Parker, E.N. (1969) Space Sci. Rev. 9, 651
3. Ghosh, A. and Ptushkin, V.S. (1983) Astrophysics and Space Science,
92, 37.
4. O'C Drury, L. (1983) Reports on Progress in Physics, 46, 973.
5. McKenzie, J.F. and V_Ik, H.J. (1982) Astron. Astrophysics 116, 191
6. Fuchs, B and Thielheim, K.O. (1979) Astrophysical Journal 227, 801.
7. Schmidt, M. (1957) B.A.N. 13, 247.


The role of cosmic rays and Alfven waves in the structure of the galactic halo

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