Download Energy Loss of a Charged Particle Traversing Ionized Gas and

139
Progress ·of Theoretical Physics Vol. 16, No. 2, August 1956
Energy Loss of a Charged Particle Traversing Ionized Gas
and Injection Energies of Cosmic Rays
Satio HAYAKAWA* and Kazuo KITAO**
*
Research Institute for Fundamental Physics, Kyoto Uni'l'ersity, Kyoto
**Department of Physics, Osaka City Uni'l'ersity, Osaka
(Received April 16, 1956)
The energy loss of a charged particle traversing an electron gas is calculated by taking the direct
collisions with free electrons and the plasma excitation into account. The contributions from ions and
neutral atoms are added and the resultant energy loss is given as a function of the degree of
ionization. The energy loss is found to increase with the degree of ionization, to which excess loss is
approximately proportional. The critical injection energy of cosmic rays, based on the Fermi mechanism
of acceleration, is given in Table 1. Its value is al:out fifty percent larger for the degree of ionization
of ten percent than for a neutral medium and nearly four times larger for a half ionized medium. In
the case of the solar production of cosmic rays the critical injection energy is below one MeV per
nucleon; so that the secondary particles generated by nuclear reactions could be accelerated to cosmic
ray energies. The mechanism of acceleration in a coronal region is proposed in the Appendix to explain
the time delay of the cosmic ray increase compared with the solar eruption as well as the energy
spectrum of solar cosmic rays.
§ 1. Introduction
In some astrophysical problems, it is important to know the energy loss of a charged
particle in a highly ionized gas. This fact has been emphasized in a number of occasions,
for example, by Morrison, Olbert and Rossi1 J in connection with the injection of cosmic
rays and by Greenstein2J in discussing the role of electrons in the stellar atmosphere.
These authors had to use the formula of the ionization loss for a neutral medium or its
approximate modification, because no reliable knowledge of that for an ionized medium
was available. Recently, however, Neufeld and Ritchie3> have worked out this problem
by paying attention to the excitation of plasma waves. lndepently of them the present
authors4 > have also treated the same problem in a simple way. In this paper we present
our method of derivation as well as its numerical results which may serve to astrophysical
problems, and discuss the bearing of our formula in the injection of cosmic rays. 5J
In deriving the formula of the energy loss in an electron gas, we use the method
of impact parameter. For small impact parameters the energy loss is regarded as due to
the direct collision with electrons, while for large impact parameters as due to the excitation of plasma waves. The two regions are divided bv a critical impact parameter which
is of the order of the Debye length. Adding the contributions from the respective regions,
the critical impact parameter is eliminated from the final formula.
140
S. Hayakawa and K. Kitao
The energy loss due to free electrons thus derived is added to the contribution from
ions and neutral atoms. Then the total energy loss is expressed in terms of the degree
of ionization, a. The fractional excess energy loss is found to be of the order of ajZ,
where Z is the atomic number of the medium concerned, and to be nearly independent
of energy in the relativistic region, while it increases slowly with decreasing energy in
the non-relativistic region.
In order that our result is applicable to astrophysical problems, numerical values are
presented for the cases of the interstellar space and of the solar corona, in which gases
are mostly composed of hydrogen with the degrees of ionization of ten percent and of
hundred percent, respectively. In the case of the interstellar gas, we also show the dependence of the energy loss upon the degree of ionization, expecting the application to
the galactic halo, in which the degree of ionization is supposed to be higher. The effect
of ionization of a gas is found considerable for the solar corona.
The energy loss of a
proton is so large that it can not penetrate the corona, if it would start from the bottom
of the corona with the velocity of auroral particles.
The large ionization loss sets the
lower limit of injection energies of cosmic rays from the sun as below 1 MeV, provided
that the Fermi mechanism of acceleration is operative.
§ 2.
Derivation of formula
Let the density of atoms with atomic number Z be n 0 • For the degree of ionization,
a, the density of neutral atoms is ( 1 -a) n0 and their contribution to the energy loss is
given by
-JWjdzol"=27rZ21 e4/mt!{f ·Z(l-a)n0
X~ t:-[ln
J
3
4m2c4[f1 r2
fl1-{fE;;;jJ1lwf
(2·1)
r=
where
(1- {f) _,, 2 and other notations are the same as used in our preceding paper6>~
There are ions of density an 0 which are assumed to be singly ionized.
The extension
to the case of higher degree of ionization is straightforward. Then the contribution of
those ions is
-dW/dZol 1=27rZ/e4/mt!{f · (Z-1) an0
X~ fj[ln
J
4m2c4f14F2
fJ1- {fE,;;jJ1lwf
(2·2)
The summation over j in (2 · 2) is a little different from that in (2 ·1), but the difference
is insignificant for large
The ionization produces free electrons of density an 0 • Their contribution is what
we want to calculate. For small values of impact parameter, T< T0, where T0 is of the
order of the Debye length, the energy loss is considered as due to the direct collision
with such free electrons. This gives rise to the energy loss :
z.
(2·3)
Energy Loss Of a Charged Particle Tra'Persing loniz:ed Gas
141
For large values of impact parameter, r>r0 , on the other hand, the energy loss is caused
by the excitation of plasma waves. The energy loss can then be calculated, if the dielectric
constant of the electron plasma is known. This is
(2·4)
where JJ is the collision frequency of an electron in the medium and wP the plasma frequency given by
wP2 =4 1r
a n0e'l/m=aw02 •
(2·5)
With use of (2 · 4) we obtain the energy loss for the region concerned as
-JW/dz:ol)ro=21rZte4 /m~f?l· an0
X[1n 2 4~[?l
f?letwp]·
T ll- f?lEwplr/cop2
(2·6)
Adding (2 · 3) and (2 · 6), the energy loss due to free electrons is obtained as
-dW/dz:0 l' =
21rZ 1 2e4 /m~f?l· an 0
(2·7)
It must be noticed that r0 does not appear in the final formula (2 · 7).
(2 · 7) can be compared with formula (87) obtained by Neufeld and Ritchie3>, who
used a more complicated method, as shown in what follows. In practical cases both the
density of atoms is not too large and the degree of ionization is not too small, so that
there holds
wP2=
Hence
Ewp
and
w,
E1
al/Jt'> Jl-.
can be assumed as zero.
Thus we have
-JWjdz:ol"=21rZtifmcf?l· an 0 ln( 4m2c4{14F2//Iho,2 ).
(2. 7')
This is nothing but their formula (87), if one notices bmm=fi/mc{JT there.
Adopting (2 · 7') instead of (2 · 7) for practical purpose, the dependence on a is
expressed as -a ln a. In a similar way we write down the a dependence of the total
energy loss, the sum of (2 ·1), (2 · 2), and (2 · 7'), as
JW
a (C-B-lna) ] ,
-= A[ B+dz:o
z
(2·8)
where
(2 ·9)
(2·9)
142
S. Hayakawa and K. Kitao
(2 · 8) indicates that the maximum of the energy loss appears for C-B-1 < 0 at
(2 ·10)
ama_.,=exp(C-B-1).
Introducing
w,
the average of
wJ
over oscillators, we can approximate
(2 ·11)
As w2 ~ liJ02in usual cases, (2 · 11) is always positive, so that the energy loss increases with
a in a monotonic way. The energy dependence of the excess energy loss in the ionized
medium is due mainly to 11- j9~E;;; I in (2 · 11). This tends toi ncrease with decreasing energy;
the difference of the energy loss between in ionized and neutral media is larger for lower
energies.
Since the logarythm in (2 · 11) is as large as B, the fractional excess energy
loss is roughly estimated as ajZ.
§ 3. Numerical results for hydrogen gas
In most celestial cases the gas is mainly composed of hydrogen.
Then (2 · 8) with
Z = 1 is of practical importance and is expressed as
JW
d~o
(3 ·1)
Numerical examples of (3 ·1) are illustrated for the interstellar gas and the solar corona.
In the galactic plane the interstellar gas is composed mainly of hydrogen with the
degree of ionization of about a=1/10. The average density of hydrogen is supposed to
be approximately n0 =1/2 cm- 3• This results in w 0 ~4X 10~ sec-\ far smaller than iii
~2 ·1 X 1016 sec-1 • Owing to the small density E;;; in (3 ·1) can be set equal to unity.
On account of that a may change from place to place, we express (3 · 1) with a
as a parameter. In a medium of density p g cm- 3, the energy loss is given by
_ _!__dW = 0.153ZI2[{21·3+2ln
p d~0
P
p
P}
1-P
+a {ln(1-P) +P-lna+53 · 9} J MeV/g cm- 2•
(3 ·2)
For Z 1 =1 and a=0.1 and 0.5, (3 ·2) is plotted against the kinetic energy of a
proton in Fig. 1. For comparison the energy loss for a= 0 is also shown in the same figure.
In comparison with the latter case the energy loss for a=0.1 increases nearly by twenty
percent. The excess loss is nearly independent of energy in a relativistic region, while it
increases weakly with decreasing energy in a non-relativistic region. This can be understood, on account of that' the plasma frequency is low, while the atomic frequency is so
high that a part of the electric ·field caused by a charged particle becomes ineffective, as
its velocity decreases.
In order to show the dependence on the degree of ionization; the a dependence at
Energy Loss
of a Charged Particle Traversing Ioni<ed Gas
143
17
2
0 ~ 5 6 7 8 910'
3
4 5 6 78910
3
4 5 6 7 8 910'
2
3
4 5 6 7 8 910'
Kinetic Energy of Proton (MeV)
f~ ~~
IQ
:"'~ ~~
~~ f'\o
2
~~ ~
~['.._ ~ ~
~~ ~~
..
" ~r--.~.~
~ ::--~
.
9
B
7
6
5
""
4
3
2
...
....
~
8
7
6
5
4
3
2
i
3
4
5 6 7 8 910
3
4
5 6 7 8 910'
Kinetic Energy of Proton (MeV)
Fig. 1. Energy loss (MeV/g cm-~J vs the kinetic energy of proton (MeV)
Curves A, B and C correspond to the hydrogen gas of density 1/2 cm-3 with
a=O, 0.1 and 0.5, respectively. Curve D corresponds to the solar corona with
a=l.
Fig. la for kinetic energy above 400 MeV; Fig. lh below 1 GeV.
S. Hayakaw:a and K. Kitao
144
W=2.18 GeV*> for protons, at which the energy loss is minimum in a neutral medium, is
shown as
(3 ·3)
-1/p· dWjdzolmtn=0.168[25.0 +a (52· 4-ln a) ]MeV/g cm- 2•
(3 ·3) is plotted in Fig. 2, together with its ratio to the case of a=O .
5 .$ r-
1
j:
.Or
4 .Sf-
v
4 .o·
3 .5
3
.or.Sf-
•ort .6
....-..,..- ---- --- - "'
~
1.0.
ICI-"
~~-
2
v
v
3
2
4 5 Q 7 8 9 JO-.
Degree of Ionization (a)
-!
/
//_ 1'
~....
/"'
5
I
3
4
6
5 6 7 89
4
1
Fig. 2. Energy loss vs degree of ionization at W=2.18 GeV for proton
in the hydrogen gas of density 1/2 an-s.
Solid curve indicates the energy loss (MeV/g an-2 represented in the right
ordinate) and dashed curve the ratio to the neutral medium (in the left
ordinate).
In the solar corona the degree of iqnization is supposed to be unity. Hence the
energy loss is obtained by putting a= 1 in ( 3 · 1) . The numerical result is shown also
in Fig. 1. The density of electrons is of the order of 108 em-s which is still small
enough to neglect the density effect. The electron temperature is so high as 106 °K that
the motion of electrons plays a significant role in the energy loss of slow particles. Since
the average velocity of electrons in the corona is about 3 X 108 em seC\ however, the
effect of the electron motion may be neglected for protons of energies above 1 MeV. In
a MeV region the energy loss is approximated as
-1/p·dWjdz:o=144/W·[ln W+22·.3]MeV/gcm- 2
(3 ·4)
where W is measured in MeV.
*) By W we mean the kinetic energy, while E is used for the total energy. Wand E with the same
suffix, as employed in § 4 and Appendix, are related as, say, E0 =Mc2+W0 , where Mt? is the rest energy
of a proton.
Energy Loss of a Charged Particle Traversing Ionized
145
Neglecting the electron motion which is of little practical effect at energies concerned,
the range of protons can be estimated in the following way. We assume that (3 ·4)
holds down to the energy at which the velocity of a proton is equal to the thermal
velocity of electrons. The range for a proton with energy W,. in MeV to loose its energy
down to this energy is approximately given by
(3 ·5)
The range for a proton below this energy is negligible for practical purpose. As the depth
of the corona is of the order of 10-5 g em - 2, the proton of energy below 200 Ke V can not
penetrate the corona. If the spiral motion of protons in a magnetic field is taken into
account, the minimum energy of penetration will be larger. The protons of currently
believed velocity, about 108 em sec-t, as the cause of magnetic storms, can not at all
penetrate the corona. If such particles were injected from the sun, they would have to
be accelerated near the top of the corona.
§ 4. Injection energy of cosmic rays
According to the Fermi mechanism of acceleration of cosmic rays, cosmic ray particles
are injected into the interstellar space, in which wandering magnetic clouds take part in
the acceleration of the particles. In order that the acceleration is effective, the particles
have to be injected with energies larger than a critical injection energy, the energy at
which the gain and loss of energy of a particle are equal. This is given by Morrison et
aU> on the assumption that the cosmic radiation is stored in our galaxy of disk shape.
This assumption seems, however, to be in contradiction with recent evidences in cosmic
rays as well as in astronomy. As described by one of the present authors (S. H.r>, it
seems better to regard the stored region as spherical, containing the galactic halo, in which
the average thickness of matter for cosmic rays to traverse is of the order of x~1 g cm- 2 *>.
Then the critical injection energy is, as in (3 · 5) of reference 7, given by
(4·1)
where factor 1.5 comes from the exponent of the power energy spectrum of cosmic rays.
j9(W.) is the energy loss per g cm- 2 for a particle of energy We.
The value of Wr
has been computed by employing the energy loss in a neutral medium1>5m. Now we are
able to get a more realistic estimate of W., based on our calculation.
The critical injection energies per nucleon are given in Table 1 for a=0.1 and 0.5
and for some typical cosmic ray nuclei, assuming x= 1 g em - 2• These values of Wr are,
of course, larger than in the case of the neutral medium. The ionization loss for a= 0.5
1s appreciable, so that the energy spectrum at low energy would be affected considerably,
*) The thickness may be different for protons, because a proton has a higher momentum than heavy
nuclei for the same rigidity. On account of that the mechanism of escape from the galaxy is not solved yet,
however, we use the same value of x for all nuclei.
146
S. Hayakawa and K. Kitao
if the mean path length traversed were larger by one order.
Table 1.
Critical injection energy (We/A in MeV)
Zt
1
2
4
7
14
26
A
1
4
9
14
28
56
tl<
0
0.95
0.95
2.2
4.8
11
0.1
1.6
1.6
3.5
7.2
16
29
0.5
4.5
4.5
8.5
32
65
16
20
Actually -the average energy of cosmic ray particles injected into the interstellar space
is supposed to be much higher than the critical injection energy W..
As discussed in
our previous paper7>, cosmic rays suffer acceleration in the vicinity of local sources, such
as Crab nebula. Also from the sun particles of energies above 1 GeV are generated.
Such high energies may also be explained in terms of the Fermi mechanism that is
operative near local sources, where physical conditions are significantly different from those
in the general interstellar space. For example, the density of matter, the degree of ionization, the velocity of magnetic clouds and the strength of magnetic fields may be much
higher in the vieinity of a local source than in the general interstellar space. Hence
the acceleration in the former region is considered as more efficient than in the latter.
And it should also be subjected to the critical injection energy that will intimately be
related to an instantaneous process of accceleration, such as the betatron mechanism. Thus
we presume that the acceleration of cosmic rays can be divided into three steps, the instantaneous injection from local sources, efficient acceleration near the local sources and the
gradual acceleration in the general interstellar space. As we have been concerned with
the last step in the preceding paragraphs, the second step will be discussed in the
Appendix.
Based on (A·5) with (A·6) the critical injection energy in the solar corona is approximately given by
(4·2)
Here p is the density of matter in the corona and is supposed to be of the order of 10-16
g cm- 3• As the corona is completely ionized, a is taken as unity in (3 ·1). The values
of W 1 per nucleon are found to be below 1 MeV, at which our formula may hardly be
applicable, because of the influence of rapid electron motion.
It seems to us worth remarking that the critical injection energy for protons is so low
that a certain fraction of protons from d- d reactions may contribute to cosmic rays. These
reactions possibly supply a sufficient amount of energy to the disturbance occurring on the solar
surface, as suggested by Greenstein. 2> Furthermore, this might cause such a stream of
high velocity,--..1010 em sec-\ that could cause the solar radio emission of Type III observed by Wild et al. 8> In this connection it will be very interesting to study the intercorrelation between the energy supply and the particle acceleration which will take place in
147
Energy Loss of a Charged Particle Traversing Ionized Gas
the stellar atmosphere.
Appendix
An acceleration mechanism of solar cosmic rays
Let the dimension of a region be R, in which the acceleration in the second step is
operative. Inside this region there are numerous magnetic clouds which move with the
average velocity V. The average strength of magnetic fields in the clouds is represented
by H Owing to the magnetic field the particles injected undergo the r~m walk with
transport mean free path l, which is as large as the average dimension of the magnetic
clouds. H and l .determine a rigidity, above which the rigidity spectrum becomes steep.
As we are mainly concerned with protons and electrons, the rigidity is equal to the momentum divided by unit charge. Thus in the unit of eV/c the corresponding momentum
is given by
P1 ~.300
H ljc,
(A·l)
where H and l are measured in gauss and em respectively. The particles with momenta
below P1 suffer collisions with the magnetic clouds before they escape out of the region.
The number of collisions is roughly estimated as
(A·2)
Hence the average lifetime of particles for escape is
(A·3)
where v is the velocity of particles and is nearly equal to the light velocity c. A collision with a magnetic cloud results in the rate of energy gain by 2 (Vjv) 2, so that the
exponent of the power energy spectrum turns out to be
r::::::..1jN2 (Vjv) 2:::::::.. ( / / R) 2/2 (Vjv) 2•
(A·4)
In the medium of density p the average thickness of traversed matter is Nlp::::::..tvp; It
is almost always satisfied that this quantity is much smaller than the mean free path for
nuclear collisions. The critical injection energy, W" in this case is given by
E1 =rf1 (W,) Nlp::::::..rf1 (w;) tvp,
(A-5)
analogous to {4 · 1) .
Now we apply the above considerations to the solar production of cosmic rays. Among
various quantities introduced above, Pu t and
are directly related to observations. The
energy spectrum of the solar cosmic rays becomes steep above several GeV.
Below this
the energy spectrum is not so steep and
is estimated as nearly unity.
The lifetime
for escape may be estimated on the time interval between the solar eruption and the
cosmic ray increase as about ten minutes. 9> Therefore we take the numerical values of P1,
r
r
tandras
(A·6)
148
S. Hayakawa and K. Kitao
These values indicate only the order of magnitude and are subject to ambiguities of factor
two or so. In addition to these values, we assume R as large as the radius of the coronal
region, namely
R=l011 cm.
(A·7)
Four quantities given in (A· 6) and (A· 7) are sufficient to determine others, if v is
assumed as equal to c. Thus we have
(A·8)
These values seem to be reasonable in view of the following facts : l is as large as
the thickness of a shock front generated by a stream of velocity V; the magnetic field
strength on the solar surface is about one gauss and the equipartition between magnetic and
kinetic energies predicts H of a fraction of gauss ; the velocity of particle streams associated
with solar disturbances is believed as about 108 em sec-1• However, there arises a question
on the behaviour of electrons. The magnetic field strength given in (A· 8) allows to
accelerate the electrons by the Fermi mechanism up to about several GeV, above which
the energy loss due to the synchrotron radiation forbids the acceleration. The spectrum
of radio waves generated by these electrons may have a maximum intensity at frequency
of about 1013 eye/sec which is larger than the observed frequencies of solar disturbed radio
waves, around lOR eye/sec. However, the mechanism of the solar radio burst is due likely
to the plasma oscillation taking place in the corona6>Io> and the synchrotron radiation may
be relatively unimportant as its cause.
References
1)
2)
P. Morrison, S. Olbert and B. Rossi, Phys. Rev. 94 (1954), 191.
L. Greenstein, Conference on Radio Aarronomy at Manchester (1955); The authors are grateful to
Professor Hatanaka who let them know the report of Greenstein.
3) J. Neufeld and R. H. Ritchie, Phys. Rev. 98 (1955), 1632.
4) S. Hayakawa and K. Kitao, Lecture at the Physical Society of Japan (1954).
5) Our fundamental idea is based on E. Fermi, Phys. Rev. 75 (1949), 1169.
6) S. Hayakawa and K. Kitao, Prog. Theor. Phys. 16 (1956), 131.
7) S. Hayakawa, Prog. Theor. Phys. 15 (1956), 111.
8) J. P. Wild, J. D. Murray and W. C. Rowe, Aust. J. Phys. 7 (1954), 439.
9) H. Elliot, Progress in Cosmic Ray Physics 1 (1952), 455.
10) S. Hayakawa and N. Hokkyo, Prog. Theor. Phys. 15 (1956), 193.
*)
protons.
J.
The value of l is regarded as fixed.
So the P 1 per nucleon for heavy nuclei is a half of that for