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Hydrogen ion velocity distributions in the ionosphere
Planet.
Space Sci. 1968.
Vol. 16. pp. 759 to 773.
HYDROGEN
Department
Pergamon
Press.
Printed
in Northern
Ireland
ION VELOCITY DISTRIBUTIONS IN THE IONOSPHERE
PETER M. BANKS of Applied Electrophysics, University of California, San Diego, California and Institute for Pure and Applied Physical Sciences (Received 16 January 1968)
Abstract-The detailed velocity dependence of the hydrogen ion distribution function has been obtained through numerical solutions to the steady state Boltzmann equation for conditions of ionospheric interest. The processes of electron-H +, H+-O+, and charge exchange collisions betwe& atomic oxygen andhydrogen ions and atoms have been included-& the col&ion model. The results indicate that the Hf velocitv distribution is somewhat distorted from the Maxwellian and leads to a H+ kinetic temperat& up to 350°K hotter than that of the ambient 0+ gas. The general trend of the Hf distribution function shows there are fewer particles in the low velocity regions and an enhancement in the tail of the distribution as compared with a Maxwellian distribution having the same kinetic temperature. The application of these results to ionospheric measurements is discussed.
1. INTRODUCTION Theoretical and experimental studies of the ionosphere (14) have shown the existence of a general state of thermal nonequilibrium between electrons, ions, and the neutral atmospheric gases. This condition arises primarily from the excess kinetic energy given to photoelectrons arising from the ionization of the neutral gases by solar ultraviolet radiation. As a consequence of various modes of collisional energy transfer between the charged and neutral constituents, a hierarchy of mean thermal energies is established such that (Welectron 2 where the mean thermal
(Wion
(Qeutral,
(1)
energy is given by (U), = 1/2m(j)Jutf(
and m is the partial
2
mass, v is the velocity,
j 16) d3u
and f is the one particle
(2) velocity
distribution
f(_jl zj)d3v = 1) . The letter j refers to each respective particle species. ) U In earlier theoretical studies of the thermal behavior of the combined ion, electron, and neutral gases, (1*2*5)it was assumed that the ion gases (Of, He+, H+) had a common ‘ion’ temperature with Maxwellian velocity distributions. With this assumption it was shown that at low altitudes, where the neutral to electron number density ratio, n(n)/n(e), is very large (z =G250 km), the heating of the ion gases by the electron gas is unable to raise (U)i,, above GVneutral and T ion _N T,, (T, is the neutral gas temperature). At high altitudes (z 2 600 km) cooling of the ions by collisions with the neutral gas atoms is unimportant and, in the absence of ion thermal conduction, (U)i,, N (U)electron and T,,, ‘u T, (T, is the electron gas temperature). For intermediate density ratios the inequalities of Equation (1) apply. Such an analysis neglects, however, the possibility that the three function
159
P. M. BANKS
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ion species may in fact have different temperatures or have significant distortions from the ~axwelIian form. The heat balance of the individual ionospheric ion gases depends upon the different rates of electron-ion and ion-neutral energy transfer which, in turn, depend directly upon the individual particle velocity distributions. Recent theoretical studies(6*7)have indicated that in the middle ionosphere, where H+ and He+ ions are minority constituents, the unequal rates of electron to ion energy transfer (compared with 0+) should result in temperature separations of up to 250°K between the different ion species. Implicit in these studies, however, has been the assumption that the velocity distributions of the ion species remain Maxwellian with separate temperatures under the strongly distorting influences of Coulomb and charge exchange collisions. Thus, as pointed out previously,@) the theoretical results showing significant differences between the Maxwellian ion temperatures can be interpreted either in terms of temperature differences in the sense of Maxwellian distributions, or as an indication that distortions can be expected in the H+ and He+ velocity distributions (since H+ and He+ in the F-region are considered to be minor constituents, their effect upon the 0* distribution is negligible). In either case the mean kinetic energies of the ion gas species are not the same. The problem of estimating the O+ velocity distribution under ionospheric conditions of heating by the electron gas and cooling by resonance charge exchange with neutral atomic oxygen has been considered recently. ~3) By taking into account electron-ion, ion-ion, and ion-neutral collisions, steady state solutions to Boltzmann’s equation were obtained, giving the 0+ velocity distribution. It was found for the described model there existed no significant distortion of the Of distribution function from the Maxwellian for temperatures and densities of ionospheric interest. Thus, the steady state atomic oxygen ion temperature can be accurately calculated using standard rates of electron-ion and ion-neutral energy transfer based upon Maxwellian velocity distributions.(s) In this paper the results of a study of the H+ velocity distribution are presented. It is shown due to the combined effects of the electron-H+ Coulomb interaction and the process of accidentally resonant charge exchange between atomic hydrogen and oxygen ions and neutral particles, the H+ velocity distribution is somewhat distorted from Maxwellian and leads to a H+ temperature up to 350°K higher than that of the 0’ gas. The distortion of the distribution function, coupled with the increase in H+ temperature (the temperature of a Ilon-Maxwellian gas is given by Equation (19)), acts to reduce the steady state II+ number density below that value appropriate for a H+ gas having a Maxwellian velocity distribution at the O+ gas temperature. 2. MODELS OF THE COLLISION
PROCESSES
To
model the ionospheric environment we consider a partially ionized plasma consisting of electrons, O+ and H+, and neutral atomic oxygen and hydrogen. It is assumed that the ~~e~~~ O+, and neutral gases have isotropic Maxwellian velocity distributions, f(j 1u),
f(j
1V) =
($$$,)“‘” exp [ - $$J ,
(3)
where k is Boltzmann’s constant and T(j) is thej-th constituent temperature. The electron and neutral gas temperatures are regarded as given quantities; no attempt is made to solve the overall ionospheric charged particle energy balance. The oxygen ion temperature
HYDROGEN
ION VELOCITY DISTRIBUTIONS
761
IN THE IONOSPHERE
is not a free parameter but is determined by equating the rates of electron-O+ and 0+-O energy transfer using the values
au(o+) at
e-o+
= 4.8 x 10-7n(e)n(O+)T;3’2 [T, - T(O+)] eV/cm3 per set
au(o+) at
0+-o
= 2.1 x 10-%r(O+)n(O)(T(O+)
+ T,)“s[T(o+)
- T,]
(4b)
eV/cm3 per set taken from the previous work. tg) Here T, is the neutral gas temperature, taken to be the same for atomic oxygen and hydrogen. The Of density remains as an arbitrary parameter. For conditions considered here H+ is a minority constituent and n(e) N n(O+) to within 10 per cent. Hydrogen ions are created and lost through charge exchange involving atomic hydrogen and oxygen ions and atoms according to the reaction H+ + O@‘,) % 0+(*S3/J + H(ls)
[J = 0, 1,2]
(5)
which, from recent measurements, (lo) has a cross section of about 30 x IO-l6 cm2 at ionospheric particle energies. The H+ ions created through the charge exchange reaction (5) undergo elastic Coulomb collisions with the ambient electrons and 0 f, distorting the Hf particle distribution function away from the initial neutral atomic hydrogen velocity distribution which follows from the charge exchange process (0+ + H). Due to the relatively low densities of atomic hydrogen ions and neutrals compared with the 0+ and 0 densities, H+-H+ and H+-H collisions are relatively infrequent and are neglected in this analysis. A diagram of the different types of collisions included in Boltzmann’s equation (described in detail below) is shown in Fig. 1. The heavy lines connecting different constituents indicate the use of Maxwellian distribution functions. The distribution function of H+ must be determined from the characteristics of e-H+, H+-O+, H+-0, and 0-t-H collisions. 3. APPLICATION
OF BOLTZMANN’S
EQUATION
With no external forces acting, Boltzmann’s equation for the gas of H+ ions in a spatially uniform, partially ionized plasma reduces to a!(H+
at
+ am+ / v) Iv> = afw+ ) 0) at at H+--e
H+-0+
+ a/m+ Iv> charge at exchange
,
(6)
where the adopted convention is that the first term in the brackets refers to the gas species and the second term refers to the velocity dependence. The different collisions included in this model which act to alter the Hf velocity distribution are indicated following each term and correspond to the processes shown in Fig. 1. Ion-ion (H+-O+) and ion-electron (H+-e) collisions are represented by FokkerPlanck collision terms valid for isotropic velocity distributions:(rl) 1 a!(H+ 1u> = Acuj a2f(H+ I u) + Bcuj VW+ 10) is au H+-j I’(H+) at
+ C(@f(H+ 14,
(7)
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P. M.
BANKS
FIG. 1. HYDROGEN ION COLLISIONPROCESSES.THE PRESENT MODEL INCLUDESELECIILON-H+, H+-0+, H+-0, AND 0+-H COLLISIONS. The electrons, O+ ions and neutral gases (0, H) are considered to have isotropic Maxwellian velocity distributions with temperatures T,, T(O+), and T,, where r, > T(O+) > T,,. The oxygen ion temperature is found by equating the rates of electron-O+ and 0+-O energy transfer, shown here by the heavy connecting lines. The H+ velocity distribution function is determined by H+ Coulomb collisions with electrons and 0+ and by accidentally resonant charge exchange with oxygen and hydrogen ions and atoms.
where r(H+) = A(v) = $.(j) B(v) = 47rn(_j)[lffjlm)
43re4In A m2(II’)
[Sd)i(jiw)~dw r$
- $)
@a) @b)
+~mfCilw)aldiuj g d0~ + 5 [..Nl
C(U) = 47r?z(j)
c-))F do]
PC)
(84
Here j refers to the j-th constituent, either electrons or Of, e is the particle charge, In A is related to the Debye shielding factor, m is the particle mass, n is the particle density and a is the H+ velocity. To reduce (7) and subsequent equations to a dimensionless form we adopt the notation c2(j) = 2kT(j)lm(j)
(9)
and choose u = u/c(H+) as a dimensionless H+ velocity, The H+ temperature needed to determine c(H+) from Equation (9) has been taken to be TB(H+), the H+ temperature found by solving the energy baIance for the collision processes shown in Fig. 1 assuming an H+ Maxwellian distribution (see Section IV). With these transformations, the electron and Of ion velocity distributions become f(e 1u>= &i2c-0(e} exp [-z2c2(H+)/c2(e)] (loa) f(O+ ( u) = T-~‘~c-~(O+) exp [-u2c~(H+)[c2(O+)J.
(lob)
HYDROGEN
ION VELOCITY DISTRIBUTIONS
IN THE IONOSPHERE
763
Likewise, the H+ distribution function can be re-expressed as F(H+ 1u) = ;;c$+)ftH+ where nilI distributions)
(11)
14,
is the H+ density calculated from the relation
9 n(H)
nnl(H+) = -S n(O) -
(based on Maxwellian
(12)
n(O+>[~,D’dH+P
given by Equation (23). The use of n&H+) and TaZ(H+) normalizes the unknown hydrogen ion velocity distribution such that iff(H+ 1v) were Maxwellian at a temperature Tnl(H+), then F(Hf 1u) = exp (-u2). The subscript M is used here and later to indicate the choice of a Maxwellian H+ distribution. With these changes the HGlectron and H+-Of collision terms become 1 aF(H+ 1u) I’(H+) at = ~- u exp (-u”c”(H+)/c”(j))]
‘exp (-w2c2(H+)/c2(j))
a2Fg2’ ’ ” + dTcF$i&j,.
[(u
dm
+ 2 3)
i 3)
(13) X exP (-u”c”(H+)/c”(j)>
x =(H+
au
+ r$ 1u> I
- 12 ?!?)‘) l c”(H+) u2 4 m(H+) n(j) - cxp l/~ 4j) c”(j)
exp
(-w"c"(H+)/c2(j))
(-u2c2(H+)/c2(j))
F(H+
dw]
1u),
where the letterj refers to electrons or atomic oxygen ions. To take account of charge exchange collisions between atomic hydrogen and oxygen and their ions the source and loss factors corresponding to Equation (5) must be evaluated in terms off(H+ I v). In symmetrical resonance charge exchange between an ion and its parent atom the electron transfer process leaves the individual particle velocities unchanged, essentially interchanging the velocity distribution function labels for the ion and neutral particles. With respect to accidentally resonant charge exchange represented by (5) it is assumed that a similar process is effective; that is, in charge exchange collisions the individual particle velocities are essentially unchanged, leading to the relation f(H I 5) ~f(Hf I 5). Such an assumption does not appear to be contrary to current experimental results for relative particle energies above 5 eV, but it ignores the trajectory curvature brought about by the ion-neutral polarization interaction. Inclusion of this effect would not, however, greatly alter the present results for particle energies above O-1eV, since the largest energy defect of reaction (5) is only 0.019 eV (for J = 2). Thus, even if Hf ions were created and lost through a quasi-orbiting process, the energy changes of the hydrogen atoms and ions would be small.
P. M. BANKS
764
The Boltzmann collision term corresponding to (5) and the above assumptions can be derived as
afW+ at 1v) = n(O+)n(H)f(H 1u) 1 dSw 18 - $1 f(O+ 1 co)oE(O+, H 1!A) dCl -
n(O)fW+ 1v)Jd3w lb - i;l f(0 1o)oE(H+, 0 1a) da,
(14)
wheref(Q ] ~1,fW+ 1w> andf(H 1v1 are the isotropic velocity distribution functions for atomic oxygen, neutral and ionized, and neutral atomic hydrogen, aE is the differential charge exchange cross section for the indicated ion process, and w is a particle velocity. To reduce (14) further it is assumed that charge exchange is isotropic and that the charge exchange cross section qE = Jbz d&J, is a constant, independent of relative velocity. Thus, with (3) and (14),
afH-‘-14 at
I =
charge TqE("+, exchange
W)f(H1V)c4(o+) (f(o+ 1v) +
;
(2-&&+ 1)
X S’AO+ 1w> dw) - rq,(I-I+, 0>c4(O>~-+ I 0) (f(O I v) + ; (2 -&) 0
X /j@
+ I)
/4dm}
- (151
In dimensionless form (15) reduces to
where
X
[exP (-~2c2WYc2(W) E(u) =
+ f (2~’ $$$
+ 1) l
exp
$--rq&O+>H~~(O)c(O)[e~p (-~2c2(H~)~c2(0)) X
u
f0
(-~%2(H+)p(o+))
+ i(2g2 ‘3
exp (-m2cS(H+)/c2(0))
do
&,]
i 1)
1
(17b)
and where the relation(12) qE(H*, 0) = #qE(O+, H), has been adopted. Equation (16) is interesting in several aspects. First, it is clear that for T(O+) > T, (atomic oxygen and hydrogen have a common neutral temperature) in the absence of other collision processes, the steady state distribution (W(H+ 1u)/i?t = 0) given by F(Hf 1u) = ~(~)/~(~) will be substantially non-Maxwellian due to the velocity dependence of the bracketed terms of Equation (17) weighting the Maxwellian dis~ibution function for atomic hydrogen. In the tail of the H+ distribution (~2c~(H~)~e2~0,O+) > l), however, these terms cancel, giving F(H+ I u) CCexp [-~2c2(H~~~c2(~~ CCF(H I u), showing that in
HYDROGEN
10N VELOCITY DISTRIBUTIONS
IN THE IONOSPHERE
the tail of the H+ distribution charge exchange results in the Hf having an identical velocity dependence. Secondly, the steady state depends not only upon the thermal characteristics of the ion and c(O+), c(O), and c(H)), but also upon the densities of the reacting and n(H)) and the specific velocity dependence of F(H+ 1u). 4. MOMENTS
OF THE CHARGE EXCHANGE
765
and neutral hydrogen condition for F(H+ 1u) neutral gases (through particles (n(O+), n(O),
COLLISION
TERM
The velocity moments of the ion-neutral charge exchange collision term yield information regarding the H+ number density, n(H+), and temperature, T(H+), once F(H+ 1u) is known. Thus, using Equation (16), n(H+) =
4n,?l(H+) m z?I;(H+ 1u) du s0 1/n
and, with the relation (U) = g kT(j), T(H+) = -
4
m nM(H+)m(H+)c”(H+)
32/n-k
s0
u41;(H+ 1u) du,
(19)
where it is emphasized that n&H+) and c(H+) are arbitrary initial quantities adopted to reducef(H+ 1u) and o to dimensionless form. For a Maxwellian Hf distribution, Equation (16) can be evaluated to give the concentration and energy moment equations for ion-neutral collisions. Thus, the Maxwellian concentration equation is &(Hf) at
2 = 2/Y~E(O+, H) n(H)n(O+) [c2(O+) + c~(H)]“~
- ; n(0)n(H+)[c2(0)
+ c~(H+)]~‘~ (20)
For steady state conditions, &z(W)/& = 0, Equation (20) gives
n(H+) 9 _n(Of) 8
r+(k)+31 1
c2(O+) + c2(H) 1’2
~~(0) + c2(H+)
(21)
For thermal equilibrium c(O+) = c(0) and c(H+) = c(H), this reduces to n-=-_ (H+) n(O+)
9 n(H) 8 n(0)
which agrees with the result of Hanson and Ortenburger.02)
(22)
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For normal ionospheric temperatures c2(H) > c2(O+) and c2(H+) > ~~(0) due to the mass factors given in Equation (9). Thus, Equation (21) can be reduced to the form n(H+) -z-p n(O+)
1
9 n(H) r, 1’2 8 n(O) [ T(H+)
(23)
as found previously. (6) Current theoretical models of H+ density in the ionosphere have not included the temperature factor. The energy balance moment of (16), assuming the H+ to be Maxwellian is aU(H+) 2 n(H)n(O+) - ar = 2/rr q&O+, H~~(H+~ (c2(H) + c~(O+))~‘~( c4(H) + ; ~z(O~)~z(H)) 8 n(O)n(H+) (c”(H+) -f- ; c2(0)c2(H+))l - 5 (c2(H+) + ~~(0))~‘~
,
(24)
where U(H+) is defined by Equation (2). Again, taking c2(H) > c2(O+) and c2(H+) > G(O), we have
aU(H+)
-
= - -&
at
qE(O+, H)n(H-‘)n(O)c(H[+) $(H+) _ ? tz(tI) !!fo+) ‘3(H)
8 n(0) n(H+) c(H+)
1(25)
which is identical with the corrected Equation (1 lb) of reference 6. In a steady state condition, when &(H+)/at = 0, Equations (25) and (23) can be combined to give
alJ(H+) -;h
-
at
+&do+,W
l/2
~1/2(H~)~(H+)~(0)~ [T(H+) - T,].
(26)
5. METHOD OF SOLUTION
The time dependent, second order, linear partial differential equation for the H+ velocity distribution, which includes e-H f, H+-O+, H+-0, and 0+-H collisions, can be written in the form aF(H+
at
1u)
=IT
Atu)
a2FW-14 +
au2
Bcu)
aJTH+14 + IC(U) - ww(~+ au
I 4 + m4
(27)
where the velocity dependent coefficients have been given in Equations (13) and (1’7). Since H+-H+ and H+-H collisions are neglected in this model (the minority constituent approximation) the magnitude of the calculated H+ distribution function, but not its velocity dependence, depends linearly upon the ratio n(H)/n(O); scaling of the H+ number density to arbitrary Gil ratios may be done once F(H+ 1u> is known for a particular ratio. Hence, the independent parameters for the adopted collision model are T,, T,, n(O+), and n(0). Steady state solutions (%‘(H+ 1u)/& = 0) for Equation (27) were obtained by time relaxation from an initial, arbitrary H+ distribution using a Crank-Nicholson implicit integration scheme (see, for example, Diaz d3)). To improve the numerical stability and accuracy of the implicit method the substitution F(H+ [ u) = exp (--Q(u)) was made. Although this change of variable introduces a nonlinear first derivative in (27), it has been found that the implicit method is stable only for functions which have changes in magnitude less than about 104: 1. The indicated transformation permits the computation of values for F(H+ ] u) over a lo*: 1 range in magnitude.
HYDROGEN
ION VELOCITY
Expansion and linearization triple diagonal expression
DIST~BUT~O~S
of Equation
IN THE IONOSPHE~
767
(27) using central differences leads to the
u.J@jk++ll + pjk@;*l + y$D;i; 3
= s;,
(28)
where the superscripts k refer to linear time elements and the subscripts j refer to linear mesh elements in the dimensionless u-space. The coefficients CC,~ through Bjk are known from the coefficients A(z+) through E(z+) and the H-t- distribution function, F(H+ z+), at the k-th time step. By solving Equation (28) repeatedly (discussed below) for d)j!,+I the steady state H+ distribution can be obtained. The u-space mesh recurrence relations necessary to solve (28) depend upon the specific form of the boundary conditions applicable to (27). At the origin (u = 0) we take aF(H+ 1a)/& lU-_o= 0 which, in terms of (28), gives G?tfl = @ffl. For large velocities (u + co) the usual boundary conditions upon F(H+ 1u) are that the distribution function and its derivatives should vanish. For this analysis it was found that a suitable approximation to this condition could be obtained by extrapolating the known values @,” from j = N - 6 through j = N (N corresponds to the maximum meshj-value) to approximate O%yI. The solutions based upon this bootstrap technique proved to be more stable numerically than those based upon the boundary condition Q$yI = 0 for all k. The solution of Equation (28) for the unknown @Ftl (j= 1 to N) was based upon mesh recurrence relations. Starting from an initial .F(Hf 1u) at time step k = 0, the quantities
were calculated for j = 1 through iV. The unknown functions Qjk were then found using the relation @+I 3 - r;.k@;$ 3 3 = K.k starting with the value found for @:,I and progressing backwards j = N to 1. This basic process was repeated for successive time steps until F(Hf 1ui) computed from the @$+I satisfied Equation (27) with a#‘(H+ 1z+)f& = 0 to better than one part in IO’. 6. RESULTS
Numerical solutions of Equation (27) for the steady state H+ distribution function have been obtained for wide ranges of temperatures and constituent densities. Since these results are generally similar in nature, only one typical distribution function is discussed here in detail. Reference to distributions characteristic of different conditions is made by means of two velocity moment difference coefficients discussed later. The typical H+ distribution function to be discussed is shown by the full line (curve 3) of Fig. 2. Here the parameters T, = 3000”K, T, = lOOO”K, n(O+) = 1 x IO5 per cm%, n(0) = 1 x 10’ per ems, and n(H) = lo5 per cm3 have been used. The steady state oxygen ion temperature was calculated from Equations (4) as T(O+) = 1478%, while the H+ temperature, calculated with the prior assumption of a Maxwellian H+ distribution, was T&H+) = 1629”K, giving c(H+) = 5.2 x lo5 cmlsec. (The subscript M used with the H+ temperature and density indicates the use of a Maxwelljan velocity distribution.)
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0
CALCULATED H+ DISTRIBUTION,F~H+/UI
T,; 3000'K T~O'l=147B°K T =lOOO'K "(0'1 1x105cm? q
n(O) = I"1 IO'clc n(H) = Ix105cm-3 n(H I:8.72~10'cm-'
DIMENSIONLESS H' VELOCITY,u FIG. 2. Hf DISTRIBUTION FUNCTIONS. CURVE 1 REPRESENTS THE NORMALIZED (SEE TEXT) DISTRIBUTION FUNCTION FORA MAXWELLIAN H+ GAS HAVINGTHE OXYGEN ION TEMPERATURE
(1478°K). Curve 3 is the actual H+ distribution function obtained numerically from Boltzmann’s with the collision processes shown in Fig. 1 using the indicated temperatures and The H+ concentration was calculated to be 8.72 x lo* per cm3, while the H+ kinetic ture was found to be 1774”K, 296°K higher than the Of temperature. To assess the which the actual distribution deviates from Maxwellian curve 2 has been plotted as wellian distribution having the same kinetic temperature as the actual distribution, for H+ density changes.
equation densities. temperadegree to the Maxcorrected
The size of the Maxwellian H+-0+ temperature separation (151’K) agrees with previous results.(6*7) For purposes of comparison two additional Maxwellian H+ distribution functions corresponding to different gas temperatures are also shown in Fig. 2. The broken line (curve 1) represents the normalized Hf distribution function which is appropriate for a Maxwellian distribution at the atomic oxygen ion temperature (7’,(H+) = T(O+) = 1478°K). In comparison with the Maxwellian at the Of temperature, the actual distribution function is substantially reduced in magnitude near the origin (u = 0) and the region u ‘U 1, while the tail of the distribution is considerably enhanced. Using Equation (19) the kinetic temperature of the Hf gas corresponding to the actual distribution, F(Hf [ u) , was found to be T(H+) = 1774°K; that is, 296” hotter than the oxygen ion gas. To assess the degree of departure of F(H+ 1u) from Maxwellian the dotted line (curve 2) has been plotted assuming the H+ to have a Maxwellian distribution with the same kinetic temperature as the actual distribution (curve 3). Comparison of curves 2 and 3 indicates that the extent of distortion from the Maxwellian is not large in the velocity regions of
HYDROGEN
ION VELOCITY
DISTRIBUTIONS
IN THE IONOSPHERE
769
u < 2.5. A specific measure of the distribution
function distortion from Maxwellian is discussed later. Since the distribution functions discussed here are isotropic in velocity space, the number of H+ ions in the interval 5, v’+ du’ is given by 47ru2f(H+ 1v) or, in terms of F(H+ 1u), 4/d5-u2F(H+ 1u). The speed distributions for curves 1 and 3 of Fig. 2 are shown in Fig. 3 and indicate the extent to which the actual H+ distribution differs from that
DIMENSIONLESS ION VELOCITY,u FIG. 3. HYDROGENIONSPEEDDISTRIBUTIONS.CUR~BS~ AND3 OFFIG.~HAVEBEENCONVERTED TOSPEEDDISTRIBUTIONSTOSHOWTHEDIFFERENCESBETWEENAMAXWELLIANH+DI~IBUTION ATTHEOXYGENIONTEMPERATUREANDTHEACTUALDISTFUBUTION. The depletion of H+ population at low velocities and enhancement at high velocities can be of
direct importance
to experiments which attempt to deduce Hf density and temperature only selected regions of velocity space.
from
which would be expected for a Maxwellian H+ distribution at the oxygen ion temperature. Associated with the departure of F(Hf 1u) from the Maxwellian distribution expected for T&H+) = T(O+) is a change in the steady state concentration of H+. For the parameters given above the predicted H+ density from Equation (22), which ignores the temperature factor, would be n&H+) = 1-l x IO3 per cm3, while using Equation (23) with TM(H+) = 1477”K, the oxygen ion temperature, gives n&H+) = 9.25 x lo2 per cm3. The direct calculation of n(H+) from F(H+ 1u) using Equation (19) gives the actual value n(H+) = 8.72 x lo2 per cm 3; that is, a reduction of about 30 per cent below the Maxwellian density omitting the temperature factor. However, taking T(H+) = 1774°K from the actual distribution using Equation (23), a density of n(H+) = 8.46 x IO2per cm3 is found. In practical terms the use of T(O+) in place of T(H+) in Equation (23) is sufficient in most cases to determine n(H+) for the atmospheric regions where the chosen model applies, since the ratio n(H)/n(O) is not well known. The shaded area in Fig. 3 gives a measure of the number of hydrogen ions which would
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have to be shifted in velocity space to match one distribution function with the other. defining a movement factor, &f, according to
M=
s
3y
dsu IE;(H+ 1u) - F&H+ f tt)/ (30) d3u);(H+ 1u>
’
s
where FLTx(H+f u) is the H+ Maxwe~ian distribution function at the oxygen ion tem~e~ture, the fractional number of dispfaced H+ ions can be obtained. For the distribu~ons shown in Fig. 3, M = 20.8 per cent. To measure the extent to which .F(H+ f tl) departs from Orwellian a distortion coefhGent, B, can be defined in terms of I;b(H+ f $, the M~wellian dis~ibution having the same kinetic temperature as F(W 1u): d”u /S’(H+ 1ix) - F’(H+ 1u)] D=
s
(31) d%P;(H+ 1u)
’
I
For the above distributions, D = 6.2 per cent, showing that the proportionate number of H+ participating in the non-~axwellia~ part of @Hi- 1EC)is somewhat smaller than those displaced in the transition FX(Hf 1~1)-+ F(H+ 1u)_ The distortions of B’(H+ 1u> from Maxwellian due to the different types of collisions can be evaluated from the individual terms of Equations (6) and (27). Charge exchange is found to enhance the particle population in the velocity region near the mean thermal speed of neutral atomic hydrogen (u CJ 0~85) and to reduce the number of II+ ions in the high velocity tail region (u > 15). Electron -H+ collisions, on the other hand, are very effective in raising the tail population and are responsible for the net tail population being somewhat larger than for a Maxwellian. Ion-ion collisions between H+ and Of tend to bring the H* distribution closer to the Maxwellian form by depleting both the high and low velocity regions while enhancing the H+ population near u rz 1. The smoothing influence of H+-O+ collisions is similar to recent results ~1 showing that Coulomb self-collisions between atomic oxygen ions are not necessary to maintain a near-~axwellian O+ distribution function for conditions of thermal nonequilib~um in a gas mixture of electrons, Of, and atomic oxygen. By varying the temperature and density factors the degree of distortion of F(H+ 1u) can be altered. For given electron and neutral temperatures when ~~0~)~~~~) is large, electron-H+ coupling is strong and F(H+ 1u) b ecomes Maxwellian (iM, D = 0) at the electron gas (and ion) temperature. Likewise, for small density ratios F(H+ 1u) is Maxwellian at the neutral gas (and ion) temperature. For intermediate density ratios the oxygen and hydrogen ion distributions will have different kinetic temperatures (energies) and a distortion of the H+ distribution occurs. The results of many calculations of F(H.+ 1 u) with different densities and temperatures are shown in Fig. 4 in terms of the movement factor, n/r, and the displacement factor, f). The horizontal scale is presented using the oxygen ion temperature fractional separation, X, given by X = 7X0+) - T, T, -
T,
’
HYDROGEN
ION VELOCITY DISTRIBUTIONS
0
0.2
IN THE IONOSPHERE
0,4
06
771
I.0
X
FIG. 4. HYDROGEN ION DISIRIBLJTION FUNCDON COEFFICIENTS. THE MOVEMENT COEFFICIENTIS A MEASURE OF THB PERCENTAGE OF H+ IONS WHICH WOULD BE MOVED IN VELoCITY SPACE IN THE TRANSITION FROM A MAXWELLIAN H+ DISTRIBUTION AT THE OXYGEN ION TEMPERATURE TO THE ACI’UAL H+ SPEED DISTRIBUTION. The distortion coefficient is a measure of the percentage of H+ ions which would be moved in velocity space in the transition from the actual distribution to a Maxwellian H+ distribution having the same kinetic temperature.
where T(O+) is the 0+ temperature calculated from Equations (4). The largest movement (M) and displacement (D) coefficients are found to occur near X = 0.20 and depend upon the magnitude of the electron-neutral temperature separation. The results of a number of Hf distribution function calculations are tabulated in Table 1 in terms of the H+-Of temperature separation, AT. The electron and neutral gas temperatures were chosen as indicated, while the Of temperature was calculated using Equations (4) with different ratios n(O+)/n(O). The H+-0+ temperature separations found here are up to 60 per cent greater than those calculated previously assuming Maxwellian distributions.@) TABLE 1.
H+-0+
TEMPERATURE SEPARATIONS (“K)
T,
T*
T(O+)
AT
2000
1000
3ooo
loo0
4ooo
loo0
1233 1370 1532 1281 1477 1747 2162 1287 1505 1828 2860
170 268 79 257 296 305 253 290 364 419 197
AT = T(H+) - (T(O+)
172
P. M. BANKS 7. IONOSP~~C
~PLICATIONS
In terms of the middle ionospheric regions (250-500 km) the preceding results support the conclusions of previous studies of ion temperature coupling between Maxwellian distributions.(6*7) The H+ mean thermal energy lies significantly above the Of thermal energy as a consequence of the relatively larger (compared with 0+) rate of electron-H+ energy transfer. The direct calculation of F(H+ 1u) has also shown that the kinetic temperature separation can be at least 20 per cent larger than the 0+ temperature and is generally larger than that predicted using the prior assumption of a Maxwellian H+ distribution. This latter situation arises as a result of the relatively greater tail population of F(Hf 1a) than would be predicted for a H+ Maxwellian distribution at the temperature found by equating electron-H f, Hf-O+ and 0+-H energy exchange. Although the H+ distribution [F(H+ 1u)] is somewhat non-Maxwellian with an overpopulated tail (U ;t l-5) and low velocity region (U < l), the largest single change in F(H+ 1u) from conditions of thermal equilibrium between H+ and Of lies in the velocity space displacement of H+ ions from the Maxwellian distribution having TM(H+) = T(O+) to the actual near-Maxwellian F(H+ 1u) at a higher temperature. Since the H+ population for ti G 1 is somewhat reduced (approximately 20 per cent for the example of Section V) below that value for a Maxwellian with TM(H+) = T(O+) experiments which depend upon measurements of the low velocity segments of ,F(H+ 1u) to predict H+ density may underestimate the true H+ concentration. Further study of this problem relating to the interpretation of Thomson scatter composition data is in progress. In a similar manner, analysis of experimental observations which relate to the velocity regions u > I.5 can lead to inconsistencies in the determined H+ density and temperature as a result of the increasingly ove~opulated (relative to a Maxwellian) distribution function, &(H+ 1u). The present results indicate the concentration of H+ in the ionospheric regions where only chemical effects are present, will be reduced as much as 30 per cent from the values found ignoring the temperature factors of Equation (23). For most applications, however, the calculation of F(H+ 1u) and n(H+) can be found by assuming T(H+) = T(0-t) in Equation (23). The collision model adopted here is valid for the atmospheric regions where H+ collisions with electrons, O+, and 0 are dominant. For the high atmospheric regions where ~~/~(O) and ~(H~)~~(Oi) become larger than about 10-l, H+-H+ and H+-H collisions become important and the present model does not apply. However, following the results of the previous analysis for the velocity distributions of Of ions(*) there is no reason to expect that the inclusion of these additional collision effects would distort the H+ distribution more significantly than found in the present model. Acknowledgements-I wish to thank Dr. G. Lewak for his helpful comments and discussion during the course of this work. This research was supported in part by the National Aeronautics and Space Adminis~ation under grant NGR~5~~75 and by the Advanced Research Projects Agency (Project DEFE~ER) monitored by the U.S. Army Research Office, Durham, under contract DA-3I-124-ARO-D257. REFERENCES 1. HANSON,W. B. and JOJu%ON, F. S., 1961. M&m. Sot. R. Sci. Lisge, Series 5,4,390. 2. DALGARNO,A., MCELROY, M. B. and MOFFE~~, R. J., 1963. Planet. Space Sci. 11,463. 3. NAGY, A. F., BRACE, L. H., CARIGNAN,C. R. and KANAL, M., 1963. J. geophys. Res. 68,640l. 4. EVANS, J. V., 1965. Planet. Space Sci. 13, 1031. 5. GEISLER,J. E. and Bowrrr~~, S. A., 1965. J. atmos. terr. Phys. 27,457.
HYDROGEN
ION VELOCITY
DISTRIBUTIONS
IN THE
IONOSPHERE
773
6. BANKS, P. M., 1967.
7. WALKER, J. C. 8. BANKS, P. M. Physics Fluids, 9. BANKS, P. M.,
Planet. Space Sci. 15,77. G. and DALGARNO,A., 1967. Planet. Space Sci. 15,200. and LEWAK, G. J.,
1967.
Ion velocity Distributions
in a Partially Ionized Plasma.
1968.
1966. Planet. Space Sci. 14,110X 10. STEBBINGS,R. F. and RUTHERFORD,J. A., 1968. Low Energy Collisions Between O+ and H. J. geophys. Res. Submitted for publication. 11. MONTGOMERY,D. C. and TIDMAN, D. A., 1964. Plasma Kinetic Theory, McGraw-Hill, New York. 12. ~NSON, W. B. and ORTENBURGER,I., 1961. J.geophys. Res. 66,142s. 13. DIAZ, J. B., 1958. in Handbook of Automation, Computation, and Control (Eds. F. M. GRABBE, S. RAMO and D. F. WOOLDRIDGE)Wiley, New York.
Space Sci. 1968.
Vol. 16. pp. 759 to 773.
HYDROGEN
Department
Pergamon
Press.
Printed
in Northern
Ireland
ION VELOCITY DISTRIBUTIONS IN THE IONOSPHERE
PETER M. BANKS of Applied Electrophysics, University of California, San Diego, California and Institute for Pure and Applied Physical Sciences (Received 16 January 1968)
Abstract-The detailed velocity dependence of the hydrogen ion distribution function has been obtained through numerical solutions to the steady state Boltzmann equation for conditions of ionospheric interest. The processes of electron-H +, H+-O+, and charge exchange collisions betwe& atomic oxygen andhydrogen ions and atoms have been included-& the col&ion model. The results indicate that the Hf velocitv distribution is somewhat distorted from the Maxwellian and leads to a H+ kinetic temperat& up to 350°K hotter than that of the ambient 0+ gas. The general trend of the Hf distribution function shows there are fewer particles in the low velocity regions and an enhancement in the tail of the distribution as compared with a Maxwellian distribution having the same kinetic temperature. The application of these results to ionospheric measurements is discussed.
1. INTRODUCTION Theoretical and experimental studies of the ionosphere (14) have shown the existence of a general state of thermal nonequilibrium between electrons, ions, and the neutral atmospheric gases. This condition arises primarily from the excess kinetic energy given to photoelectrons arising from the ionization of the neutral gases by solar ultraviolet radiation. As a consequence of various modes of collisional energy transfer between the charged and neutral constituents, a hierarchy of mean thermal energies is established such that (Welectron 2 where the mean thermal
(Wion
(Qeutral,
(1)
energy is given by (U), = 1/2m(j)Jutf(
and m is the partial
2
mass, v is the velocity,
j 16) d3u
and f is the one particle
(2) velocity
distribution
f(_jl zj)d3v = 1) . The letter j refers to each respective particle species. ) U In earlier theoretical studies of the thermal behavior of the combined ion, electron, and neutral gases, (1*2*5)it was assumed that the ion gases (Of, He+, H+) had a common ‘ion’ temperature with Maxwellian velocity distributions. With this assumption it was shown that at low altitudes, where the neutral to electron number density ratio, n(n)/n(e), is very large (z =G250 km), the heating of the ion gases by the electron gas is unable to raise (U)i,, above GVneutral and T ion _N T,, (T, is the neutral gas temperature). At high altitudes (z 2 600 km) cooling of the ions by collisions with the neutral gas atoms is unimportant and, in the absence of ion thermal conduction, (U)i,, N (U)electron and T,,, ‘u T, (T, is the electron gas temperature). For intermediate density ratios the inequalities of Equation (1) apply. Such an analysis neglects, however, the possibility that the three function
159
P. M. BANKS
760
ion species may in fact have different temperatures or have significant distortions from the ~axwelIian form. The heat balance of the individual ionospheric ion gases depends upon the different rates of electron-ion and ion-neutral energy transfer which, in turn, depend directly upon the individual particle velocity distributions. Recent theoretical studies(6*7)have indicated that in the middle ionosphere, where H+ and He+ ions are minority constituents, the unequal rates of electron to ion energy transfer (compared with 0+) should result in temperature separations of up to 250°K between the different ion species. Implicit in these studies, however, has been the assumption that the velocity distributions of the ion species remain Maxwellian with separate temperatures under the strongly distorting influences of Coulomb and charge exchange collisions. Thus, as pointed out previously,@) the theoretical results showing significant differences between the Maxwellian ion temperatures can be interpreted either in terms of temperature differences in the sense of Maxwellian distributions, or as an indication that distortions can be expected in the H+ and He+ velocity distributions (since H+ and He+ in the F-region are considered to be minor constituents, their effect upon the 0* distribution is negligible). In either case the mean kinetic energies of the ion gas species are not the same. The problem of estimating the O+ velocity distribution under ionospheric conditions of heating by the electron gas and cooling by resonance charge exchange with neutral atomic oxygen has been considered recently. ~3) By taking into account electron-ion, ion-ion, and ion-neutral collisions, steady state solutions to Boltzmann’s equation were obtained, giving the 0+ velocity distribution. It was found for the described model there existed no significant distortion of the Of distribution function from the Maxwellian for temperatures and densities of ionospheric interest. Thus, the steady state atomic oxygen ion temperature can be accurately calculated using standard rates of electron-ion and ion-neutral energy transfer based upon Maxwellian velocity distributions.(s) In this paper the results of a study of the H+ velocity distribution are presented. It is shown due to the combined effects of the electron-H+ Coulomb interaction and the process of accidentally resonant charge exchange between atomic hydrogen and oxygen ions and neutral particles, the H+ velocity distribution is somewhat distorted from Maxwellian and leads to a H+ temperature up to 350°K higher than that of the 0’ gas. The distortion of the distribution function, coupled with the increase in H+ temperature (the temperature of a Ilon-Maxwellian gas is given by Equation (19)), acts to reduce the steady state II+ number density below that value appropriate for a H+ gas having a Maxwellian velocity distribution at the O+ gas temperature. 2. MODELS OF THE COLLISION
PROCESSES
To
model the ionospheric environment we consider a partially ionized plasma consisting of electrons, O+ and H+, and neutral atomic oxygen and hydrogen. It is assumed that the ~~e~~~ O+, and neutral gases have isotropic Maxwellian velocity distributions, f(j 1u),
f(j
1V) =
($$$,)“‘” exp [ - $$J ,
(3)
where k is Boltzmann’s constant and T(j) is thej-th constituent temperature. The electron and neutral gas temperatures are regarded as given quantities; no attempt is made to solve the overall ionospheric charged particle energy balance. The oxygen ion temperature
HYDROGEN
ION VELOCITY DISTRIBUTIONS
761
IN THE IONOSPHERE
is not a free parameter but is determined by equating the rates of electron-O+ and 0+-O energy transfer using the values
au(o+) at
e-o+
= 4.8 x 10-7n(e)n(O+)T;3’2 [T, - T(O+)] eV/cm3 per set
au(o+) at
0+-o
= 2.1 x 10-%r(O+)n(O)(T(O+)
+ T,)“s[T(o+)
- T,]
(4b)
eV/cm3 per set taken from the previous work. tg) Here T, is the neutral gas temperature, taken to be the same for atomic oxygen and hydrogen. The Of density remains as an arbitrary parameter. For conditions considered here H+ is a minority constituent and n(e) N n(O+) to within 10 per cent. Hydrogen ions are created and lost through charge exchange involving atomic hydrogen and oxygen ions and atoms according to the reaction H+ + O@‘,) % 0+(*S3/J + H(ls)
[J = 0, 1,2]
(5)
which, from recent measurements, (lo) has a cross section of about 30 x IO-l6 cm2 at ionospheric particle energies. The H+ ions created through the charge exchange reaction (5) undergo elastic Coulomb collisions with the ambient electrons and 0 f, distorting the Hf particle distribution function away from the initial neutral atomic hydrogen velocity distribution which follows from the charge exchange process (0+ + H). Due to the relatively low densities of atomic hydrogen ions and neutrals compared with the 0+ and 0 densities, H+-H+ and H+-H collisions are relatively infrequent and are neglected in this analysis. A diagram of the different types of collisions included in Boltzmann’s equation (described in detail below) is shown in Fig. 1. The heavy lines connecting different constituents indicate the use of Maxwellian distribution functions. The distribution function of H+ must be determined from the characteristics of e-H+, H+-O+, H+-0, and 0-t-H collisions. 3. APPLICATION
OF BOLTZMANN’S
EQUATION
With no external forces acting, Boltzmann’s equation for the gas of H+ ions in a spatially uniform, partially ionized plasma reduces to a!(H+
at
+ am+ / v) Iv> = afw+ ) 0) at at H+--e
H+-0+
+ a/m+ Iv> charge at exchange
,
(6)
where the adopted convention is that the first term in the brackets refers to the gas species and the second term refers to the velocity dependence. The different collisions included in this model which act to alter the Hf velocity distribution are indicated following each term and correspond to the processes shown in Fig. 1. Ion-ion (H+-O+) and ion-electron (H+-e) collisions are represented by FokkerPlanck collision terms valid for isotropic velocity distributions:(rl) 1 a!(H+ 1u> = Acuj a2f(H+ I u) + Bcuj VW+ 10) is au H+-j I’(H+) at
+ C(@f(H+ 14,
(7)
762
P. M.
BANKS
FIG. 1. HYDROGEN ION COLLISIONPROCESSES.THE PRESENT MODEL INCLUDESELECIILON-H+, H+-0+, H+-0, AND 0+-H COLLISIONS. The electrons, O+ ions and neutral gases (0, H) are considered to have isotropic Maxwellian velocity distributions with temperatures T,, T(O+), and T,, where r, > T(O+) > T,,. The oxygen ion temperature is found by equating the rates of electron-O+ and 0+-O energy transfer, shown here by the heavy connecting lines. The H+ velocity distribution function is determined by H+ Coulomb collisions with electrons and 0+ and by accidentally resonant charge exchange with oxygen and hydrogen ions and atoms.
where r(H+) = A(v) = $.(j) B(v) = 47rn(_j)[lffjlm)
43re4In A m2(II’)
[Sd)i(jiw)~dw r$
- $)
@a) @b)
+~mfCilw)aldiuj g d0~ + 5 [..Nl
C(U) = 47r?z(j)
c-))F do]
PC)
(84
Here j refers to the j-th constituent, either electrons or Of, e is the particle charge, In A is related to the Debye shielding factor, m is the particle mass, n is the particle density and a is the H+ velocity. To reduce (7) and subsequent equations to a dimensionless form we adopt the notation c2(j) = 2kT(j)lm(j)
(9)
and choose u = u/c(H+) as a dimensionless H+ velocity, The H+ temperature needed to determine c(H+) from Equation (9) has been taken to be TB(H+), the H+ temperature found by solving the energy baIance for the collision processes shown in Fig. 1 assuming an H+ Maxwellian distribution (see Section IV). With these transformations, the electron and Of ion velocity distributions become f(e 1u>= &i2c-0(e} exp [-z2c2(H+)/c2(e)] (loa) f(O+ ( u) = T-~‘~c-~(O+) exp [-u2c~(H+)[c2(O+)J.
(lob)
HYDROGEN
ION VELOCITY DISTRIBUTIONS
IN THE IONOSPHERE
763
Likewise, the H+ distribution function can be re-expressed as F(H+ 1u) = ;;c$+)ftH+ where nilI distributions)
(11)
14,
is the H+ density calculated from the relation
9 n(H)
nnl(H+) = -S n(O) -
(based on Maxwellian
(12)
n(O+>[~,D’dH+P
given by Equation (23). The use of n&H+) and TaZ(H+) normalizes the unknown hydrogen ion velocity distribution such that iff(H+ 1v) were Maxwellian at a temperature Tnl(H+), then F(Hf 1u) = exp (-u2). The subscript M is used here and later to indicate the choice of a Maxwellian H+ distribution. With these changes the HGlectron and H+-Of collision terms become 1 aF(H+ 1u) I’(H+) at = ~- u exp (-u”c”(H+)/c”(j))]
‘exp (-w2c2(H+)/c2(j))
a2Fg2’ ’ ” + dTcF$i&j,.
[(u
dm
+ 2 3)
i 3)
(13) X exP (-u”c”(H+)/c”(j)>
x =(H+
au
+ r$ 1u> I
- 12 ?!?)‘) l c”(H+) u2 4 m(H+) n(j) - cxp l/~ 4j) c”(j)
exp
(-w"c"(H+)/c2(j))
(-u2c2(H+)/c2(j))
F(H+
dw]
1u),
where the letterj refers to electrons or atomic oxygen ions. To take account of charge exchange collisions between atomic hydrogen and oxygen and their ions the source and loss factors corresponding to Equation (5) must be evaluated in terms off(H+ I v). In symmetrical resonance charge exchange between an ion and its parent atom the electron transfer process leaves the individual particle velocities unchanged, essentially interchanging the velocity distribution function labels for the ion and neutral particles. With respect to accidentally resonant charge exchange represented by (5) it is assumed that a similar process is effective; that is, in charge exchange collisions the individual particle velocities are essentially unchanged, leading to the relation f(H I 5) ~f(Hf I 5). Such an assumption does not appear to be contrary to current experimental results for relative particle energies above 5 eV, but it ignores the trajectory curvature brought about by the ion-neutral polarization interaction. Inclusion of this effect would not, however, greatly alter the present results for particle energies above O-1eV, since the largest energy defect of reaction (5) is only 0.019 eV (for J = 2). Thus, even if Hf ions were created and lost through a quasi-orbiting process, the energy changes of the hydrogen atoms and ions would be small.
P. M. BANKS
764
The Boltzmann collision term corresponding to (5) and the above assumptions can be derived as
afW+ at 1v) = n(O+)n(H)f(H 1u) 1 dSw 18 - $1 f(O+ 1 co)oE(O+, H 1!A) dCl -
n(O)fW+ 1v)Jd3w lb - i;l f(0 1o)oE(H+, 0 1a) da,
(14)
wheref(Q ] ~1,fW+ 1w> andf(H 1v1 are the isotropic velocity distribution functions for atomic oxygen, neutral and ionized, and neutral atomic hydrogen, aE is the differential charge exchange cross section for the indicated ion process, and w is a particle velocity. To reduce (14) further it is assumed that charge exchange is isotropic and that the charge exchange cross section qE = Jbz d&J, is a constant, independent of relative velocity. Thus, with (3) and (14),
afH-‘-14 at
I =
charge TqE("+, exchange
W)f(H1V)c4(o+) (f(o+ 1v) +
;
(2-&&+ 1)
X S’AO+ 1w> dw) - rq,(I-I+, 0>c4(O>~-+ I 0) (f(O I v) + ; (2 -&) 0
X /j@
+ I)
/4dm}
- (151
In dimensionless form (15) reduces to
where
X
[exP (-~2c2WYc2(W) E(u) =
+ f (2~’ $$$
+ 1) l
exp
$--rq&O+>H~~(O)c(O)[e~p (-~2c2(H~)~c2(0)) X
u
f0
(-~%2(H+)p(o+))
+ i(2g2 ‘3
exp (-m2cS(H+)/c2(0))
do
&,]
i 1)
1
(17b)
and where the relation(12) qE(H*, 0) = #qE(O+, H), has been adopted. Equation (16) is interesting in several aspects. First, it is clear that for T(O+) > T, (atomic oxygen and hydrogen have a common neutral temperature) in the absence of other collision processes, the steady state distribution (W(H+ 1u)/i?t = 0) given by F(Hf 1u) = ~(~)/~(~) will be substantially non-Maxwellian due to the velocity dependence of the bracketed terms of Equation (17) weighting the Maxwellian dis~ibution function for atomic hydrogen. In the tail of the H+ distribution (~2c~(H~)~e2~0,O+) > l), however, these terms cancel, giving F(H+ I u) CCexp [-~2c2(H~~~c2(~~ CCF(H I u), showing that in
HYDROGEN
10N VELOCITY DISTRIBUTIONS
IN THE IONOSPHERE
the tail of the H+ distribution charge exchange results in the Hf having an identical velocity dependence. Secondly, the steady state depends not only upon the thermal characteristics of the ion and c(O+), c(O), and c(H)), but also upon the densities of the reacting and n(H)) and the specific velocity dependence of F(H+ 1u). 4. MOMENTS
OF THE CHARGE EXCHANGE
765
and neutral hydrogen condition for F(H+ 1u) neutral gases (through particles (n(O+), n(O),
COLLISION
TERM
The velocity moments of the ion-neutral charge exchange collision term yield information regarding the H+ number density, n(H+), and temperature, T(H+), once F(H+ 1u) is known. Thus, using Equation (16), n(H+) =
4n,?l(H+) m z?I;(H+ 1u) du s0 1/n
and, with the relation (U) = g kT(j), T(H+) = -
4
m nM(H+)m(H+)c”(H+)
32/n-k
s0
u41;(H+ 1u) du,
(19)
where it is emphasized that n&H+) and c(H+) are arbitrary initial quantities adopted to reducef(H+ 1u) and o to dimensionless form. For a Maxwellian Hf distribution, Equation (16) can be evaluated to give the concentration and energy moment equations for ion-neutral collisions. Thus, the Maxwellian concentration equation is &(Hf) at
2 = 2/Y~E(O+, H) n(H)n(O+) [c2(O+) + c~(H)]“~
- ; n(0)n(H+)[c2(0)
+ c~(H+)]~‘~ (20)
For steady state conditions, &z(W)/& = 0, Equation (20) gives
n(H+) 9 _n(Of) 8
r+(k)+31 1
c2(O+) + c2(H) 1’2
~~(0) + c2(H+)
(21)
For thermal equilibrium c(O+) = c(0) and c(H+) = c(H), this reduces to n-=-_ (H+) n(O+)
9 n(H) 8 n(0)
which agrees with the result of Hanson and Ortenburger.02)
(22)
P. M. BANKS
766
For normal ionospheric temperatures c2(H) > c2(O+) and c2(H+) > ~~(0) due to the mass factors given in Equation (9). Thus, Equation (21) can be reduced to the form n(H+) -z-p n(O+)
1
9 n(H) r, 1’2 8 n(O) [ T(H+)
(23)
as found previously. (6) Current theoretical models of H+ density in the ionosphere have not included the temperature factor. The energy balance moment of (16), assuming the H+ to be Maxwellian is aU(H+) 2 n(H)n(O+) - ar = 2/rr q&O+, H~~(H+~ (c2(H) + c~(O+))~‘~( c4(H) + ; ~z(O~)~z(H)) 8 n(O)n(H+) (c”(H+) -f- ; c2(0)c2(H+))l - 5 (c2(H+) + ~~(0))~‘~
,
(24)
where U(H+) is defined by Equation (2). Again, taking c2(H) > c2(O+) and c2(H+) > G(O), we have
aU(H+)
-
= - -&
at
qE(O+, H)n(H-‘)n(O)c(H[+) $(H+) _ ? tz(tI) !!fo+) ‘3(H)
8 n(0) n(H+) c(H+)
1(25)
which is identical with the corrected Equation (1 lb) of reference 6. In a steady state condition, when &(H+)/at = 0, Equations (25) and (23) can be combined to give
alJ(H+) -;h
-
at
+&do+,W
l/2
~1/2(H~)~(H+)~(0)~ [T(H+) - T,].
(26)
5. METHOD OF SOLUTION
The time dependent, second order, linear partial differential equation for the H+ velocity distribution, which includes e-H f, H+-O+, H+-0, and 0+-H collisions, can be written in the form aF(H+
at
1u)
=IT
Atu)
a2FW-14 +
au2
Bcu)
aJTH+14 + IC(U) - ww(~+ au
I 4 + m4
(27)
where the velocity dependent coefficients have been given in Equations (13) and (1’7). Since H+-H+ and H+-H collisions are neglected in this model (the minority constituent approximation) the magnitude of the calculated H+ distribution function, but not its velocity dependence, depends linearly upon the ratio n(H)/n(O); scaling of the H+ number density to arbitrary Gil ratios may be done once F(H+ 1u> is known for a particular ratio. Hence, the independent parameters for the adopted collision model are T,, T,, n(O+), and n(0). Steady state solutions (%‘(H+ 1u)/& = 0) for Equation (27) were obtained by time relaxation from an initial, arbitrary H+ distribution using a Crank-Nicholson implicit integration scheme (see, for example, Diaz d3)). To improve the numerical stability and accuracy of the implicit method the substitution F(H+ [ u) = exp (--Q(u)) was made. Although this change of variable introduces a nonlinear first derivative in (27), it has been found that the implicit method is stable only for functions which have changes in magnitude less than about 104: 1. The indicated transformation permits the computation of values for F(H+ ] u) over a lo*: 1 range in magnitude.
HYDROGEN
ION VELOCITY
Expansion and linearization triple diagonal expression
DIST~BUT~O~S
of Equation
IN THE IONOSPHE~
767
(27) using central differences leads to the
u.J@jk++ll + pjk@;*l + y$D;i; 3
= s;,
(28)
where the superscripts k refer to linear time elements and the subscripts j refer to linear mesh elements in the dimensionless u-space. The coefficients CC,~ through Bjk are known from the coefficients A(z+) through E(z+) and the H-t- distribution function, F(H+ z+), at the k-th time step. By solving Equation (28) repeatedly (discussed below) for d)j!,+I the steady state H+ distribution can be obtained. The u-space mesh recurrence relations necessary to solve (28) depend upon the specific form of the boundary conditions applicable to (27). At the origin (u = 0) we take aF(H+ 1a)/& lU-_o= 0 which, in terms of (28), gives G?tfl = @ffl. For large velocities (u + co) the usual boundary conditions upon F(H+ 1u) are that the distribution function and its derivatives should vanish. For this analysis it was found that a suitable approximation to this condition could be obtained by extrapolating the known values @,” from j = N - 6 through j = N (N corresponds to the maximum meshj-value) to approximate O%yI. The solutions based upon this bootstrap technique proved to be more stable numerically than those based upon the boundary condition Q$yI = 0 for all k. The solution of Equation (28) for the unknown @Ftl (j= 1 to N) was based upon mesh recurrence relations. Starting from an initial .F(Hf 1u) at time step k = 0, the quantities
were calculated for j = 1 through iV. The unknown functions Qjk were then found using the relation @+I 3 - r;.k@;$ 3 3 = K.k starting with the value found for @:,I and progressing backwards j = N to 1. This basic process was repeated for successive time steps until F(Hf 1ui) computed from the @$+I satisfied Equation (27) with a#‘(H+ 1z+)f& = 0 to better than one part in IO’. 6. RESULTS
Numerical solutions of Equation (27) for the steady state H+ distribution function have been obtained for wide ranges of temperatures and constituent densities. Since these results are generally similar in nature, only one typical distribution function is discussed here in detail. Reference to distributions characteristic of different conditions is made by means of two velocity moment difference coefficients discussed later. The typical H+ distribution function to be discussed is shown by the full line (curve 3) of Fig. 2. Here the parameters T, = 3000”K, T, = lOOO”K, n(O+) = 1 x IO5 per cm%, n(0) = 1 x 10’ per ems, and n(H) = lo5 per cm3 have been used. The steady state oxygen ion temperature was calculated from Equations (4) as T(O+) = 1478%, while the H+ temperature, calculated with the prior assumption of a Maxwellian H+ distribution, was T&H+) = 1629”K, giving c(H+) = 5.2 x lo5 cmlsec. (The subscript M used with the H+ temperature and density indicates the use of a Maxwelljan velocity distribution.)
P. M. BANKS
768
0
CALCULATED H+ DISTRIBUTION,F~H+/UI
T,; 3000'K T~O'l=147B°K T =lOOO'K "(0'1 1x105cm? q
n(O) = I"1 IO'clc n(H) = Ix105cm-3 n(H I:8.72~10'cm-'
DIMENSIONLESS H' VELOCITY,u FIG. 2. Hf DISTRIBUTION FUNCTIONS. CURVE 1 REPRESENTS THE NORMALIZED (SEE TEXT) DISTRIBUTION FUNCTION FORA MAXWELLIAN H+ GAS HAVINGTHE OXYGEN ION TEMPERATURE
(1478°K). Curve 3 is the actual H+ distribution function obtained numerically from Boltzmann’s with the collision processes shown in Fig. 1 using the indicated temperatures and The H+ concentration was calculated to be 8.72 x lo* per cm3, while the H+ kinetic ture was found to be 1774”K, 296°K higher than the Of temperature. To assess the which the actual distribution deviates from Maxwellian curve 2 has been plotted as wellian distribution having the same kinetic temperature as the actual distribution, for H+ density changes.
equation densities. temperadegree to the Maxcorrected
The size of the Maxwellian H+-0+ temperature separation (151’K) agrees with previous results.(6*7) For purposes of comparison two additional Maxwellian H+ distribution functions corresponding to different gas temperatures are also shown in Fig. 2. The broken line (curve 1) represents the normalized Hf distribution function which is appropriate for a Maxwellian distribution at the atomic oxygen ion temperature (7’,(H+) = T(O+) = 1478°K). In comparison with the Maxwellian at the Of temperature, the actual distribution function is substantially reduced in magnitude near the origin (u = 0) and the region u ‘U 1, while the tail of the distribution is considerably enhanced. Using Equation (19) the kinetic temperature of the Hf gas corresponding to the actual distribution, F(Hf [ u) , was found to be T(H+) = 1774°K; that is, 296” hotter than the oxygen ion gas. To assess the degree of departure of F(H+ 1u) from Maxwellian the dotted line (curve 2) has been plotted assuming the H+ to have a Maxwellian distribution with the same kinetic temperature as the actual distribution (curve 3). Comparison of curves 2 and 3 indicates that the extent of distortion from the Maxwellian is not large in the velocity regions of
HYDROGEN
ION VELOCITY
DISTRIBUTIONS
IN THE IONOSPHERE
769
u < 2.5. A specific measure of the distribution
function distortion from Maxwellian is discussed later. Since the distribution functions discussed here are isotropic in velocity space, the number of H+ ions in the interval 5, v’+ du’ is given by 47ru2f(H+ 1v) or, in terms of F(H+ 1u), 4/d5-u2F(H+ 1u). The speed distributions for curves 1 and 3 of Fig. 2 are shown in Fig. 3 and indicate the extent to which the actual H+ distribution differs from that
DIMENSIONLESS ION VELOCITY,u FIG. 3. HYDROGENIONSPEEDDISTRIBUTIONS.CUR~BS~ AND3 OFFIG.~HAVEBEENCONVERTED TOSPEEDDISTRIBUTIONSTOSHOWTHEDIFFERENCESBETWEENAMAXWELLIANH+DI~IBUTION ATTHEOXYGENIONTEMPERATUREANDTHEACTUALDISTFUBUTION. The depletion of H+ population at low velocities and enhancement at high velocities can be of
direct importance
to experiments which attempt to deduce Hf density and temperature only selected regions of velocity space.
from
which would be expected for a Maxwellian H+ distribution at the oxygen ion temperature. Associated with the departure of F(Hf 1u) from the Maxwellian distribution expected for T&H+) = T(O+) is a change in the steady state concentration of H+. For the parameters given above the predicted H+ density from Equation (22), which ignores the temperature factor, would be n&H+) = 1-l x IO3 per cm3, while using Equation (23) with TM(H+) = 1477”K, the oxygen ion temperature, gives n&H+) = 9.25 x lo2 per cm3. The direct calculation of n(H+) from F(H+ 1u) using Equation (19) gives the actual value n(H+) = 8.72 x lo2 per cm 3; that is, a reduction of about 30 per cent below the Maxwellian density omitting the temperature factor. However, taking T(H+) = 1774°K from the actual distribution using Equation (23), a density of n(H+) = 8.46 x IO2per cm3 is found. In practical terms the use of T(O+) in place of T(H+) in Equation (23) is sufficient in most cases to determine n(H+) for the atmospheric regions where the chosen model applies, since the ratio n(H)/n(O) is not well known. The shaded area in Fig. 3 gives a measure of the number of hydrogen ions which would
P. M, BANKS
770
have to be shifted in velocity space to match one distribution function with the other. defining a movement factor, &f, according to
M=
s
3y
dsu IE;(H+ 1u) - F&H+ f tt)/ (30) d3u);(H+ 1u>
’
s
where FLTx(H+f u) is the H+ Maxwe~ian distribution function at the oxygen ion tem~e~ture, the fractional number of dispfaced H+ ions can be obtained. For the distribu~ons shown in Fig. 3, M = 20.8 per cent. To measure the extent to which .F(H+ f tl) departs from Orwellian a distortion coefhGent, B, can be defined in terms of I;b(H+ f $, the M~wellian dis~ibution having the same kinetic temperature as F(W 1u): d”u /S’(H+ 1ix) - F’(H+ 1u)] D=
s
(31) d%P;(H+ 1u)
’
I
For the above distributions, D = 6.2 per cent, showing that the proportionate number of H+ participating in the non-~axwellia~ part of @Hi- 1EC)is somewhat smaller than those displaced in the transition FX(Hf 1~1)-+ F(H+ 1u)_ The distortions of B’(H+ 1u> from Maxwellian due to the different types of collisions can be evaluated from the individual terms of Equations (6) and (27). Charge exchange is found to enhance the particle population in the velocity region near the mean thermal speed of neutral atomic hydrogen (u CJ 0~85) and to reduce the number of II+ ions in the high velocity tail region (u > 15). Electron -H+ collisions, on the other hand, are very effective in raising the tail population and are responsible for the net tail population being somewhat larger than for a Maxwellian. Ion-ion collisions between H+ and Of tend to bring the H* distribution closer to the Maxwellian form by depleting both the high and low velocity regions while enhancing the H+ population near u rz 1. The smoothing influence of H+-O+ collisions is similar to recent results ~1 showing that Coulomb self-collisions between atomic oxygen ions are not necessary to maintain a near-~axwellian O+ distribution function for conditions of thermal nonequilib~um in a gas mixture of electrons, Of, and atomic oxygen. By varying the temperature and density factors the degree of distortion of F(H+ 1u) can be altered. For given electron and neutral temperatures when ~~0~)~~~~) is large, electron-H+ coupling is strong and F(H+ 1u) b ecomes Maxwellian (iM, D = 0) at the electron gas (and ion) temperature. Likewise, for small density ratios F(H+ 1u) is Maxwellian at the neutral gas (and ion) temperature. For intermediate density ratios the oxygen and hydrogen ion distributions will have different kinetic temperatures (energies) and a distortion of the H+ distribution occurs. The results of many calculations of F(H.+ 1 u) with different densities and temperatures are shown in Fig. 4 in terms of the movement factor, n/r, and the displacement factor, f). The horizontal scale is presented using the oxygen ion temperature fractional separation, X, given by X = 7X0+) - T, T, -
T,
’
HYDROGEN
ION VELOCITY DISTRIBUTIONS
0
0.2
IN THE IONOSPHERE
0,4
06
771
I.0
X
FIG. 4. HYDROGEN ION DISIRIBLJTION FUNCDON COEFFICIENTS. THE MOVEMENT COEFFICIENTIS A MEASURE OF THB PERCENTAGE OF H+ IONS WHICH WOULD BE MOVED IN VELoCITY SPACE IN THE TRANSITION FROM A MAXWELLIAN H+ DISTRIBUTION AT THE OXYGEN ION TEMPERATURE TO THE ACI’UAL H+ SPEED DISTRIBUTION. The distortion coefficient is a measure of the percentage of H+ ions which would be moved in velocity space in the transition from the actual distribution to a Maxwellian H+ distribution having the same kinetic temperature.
where T(O+) is the 0+ temperature calculated from Equations (4). The largest movement (M) and displacement (D) coefficients are found to occur near X = 0.20 and depend upon the magnitude of the electron-neutral temperature separation. The results of a number of Hf distribution function calculations are tabulated in Table 1 in terms of the H+-Of temperature separation, AT. The electron and neutral gas temperatures were chosen as indicated, while the Of temperature was calculated using Equations (4) with different ratios n(O+)/n(O). The H+-0+ temperature separations found here are up to 60 per cent greater than those calculated previously assuming Maxwellian distributions.@) TABLE 1.
H+-0+
TEMPERATURE SEPARATIONS (“K)
T,
T*
T(O+)
AT
2000
1000
3ooo
loo0
4ooo
loo0
1233 1370 1532 1281 1477 1747 2162 1287 1505 1828 2860
170 268 79 257 296 305 253 290 364 419 197
AT = T(H+) - (T(O+)
172
P. M. BANKS 7. IONOSP~~C
~PLICATIONS
In terms of the middle ionospheric regions (250-500 km) the preceding results support the conclusions of previous studies of ion temperature coupling between Maxwellian distributions.(6*7) The H+ mean thermal energy lies significantly above the Of thermal energy as a consequence of the relatively larger (compared with 0+) rate of electron-H+ energy transfer. The direct calculation of F(H+ 1u) has also shown that the kinetic temperature separation can be at least 20 per cent larger than the 0+ temperature and is generally larger than that predicted using the prior assumption of a Maxwellian H+ distribution. This latter situation arises as a result of the relatively greater tail population of F(Hf 1a) than would be predicted for a H+ Maxwellian distribution at the temperature found by equating electron-H f, Hf-O+ and 0+-H energy exchange. Although the H+ distribution [F(H+ 1u)] is somewhat non-Maxwellian with an overpopulated tail (U ;t l-5) and low velocity region (U < l), the largest single change in F(H+ 1u) from conditions of thermal equilibrium between H+ and Of lies in the velocity space displacement of H+ ions from the Maxwellian distribution having TM(H+) = T(O+) to the actual near-Maxwellian F(H+ 1u) at a higher temperature. Since the H+ population for ti G 1 is somewhat reduced (approximately 20 per cent for the example of Section V) below that value for a Maxwellian with TM(H+) = T(O+) experiments which depend upon measurements of the low velocity segments of ,F(H+ 1u) to predict H+ density may underestimate the true H+ concentration. Further study of this problem relating to the interpretation of Thomson scatter composition data is in progress. In a similar manner, analysis of experimental observations which relate to the velocity regions u > I.5 can lead to inconsistencies in the determined H+ density and temperature as a result of the increasingly ove~opulated (relative to a Maxwellian) distribution function, &(H+ 1u). The present results indicate the concentration of H+ in the ionospheric regions where only chemical effects are present, will be reduced as much as 30 per cent from the values found ignoring the temperature factors of Equation (23). For most applications, however, the calculation of F(H+ 1u) and n(H+) can be found by assuming T(H+) = T(0-t) in Equation (23). The collision model adopted here is valid for the atmospheric regions where H+ collisions with electrons, O+, and 0 are dominant. For the high atmospheric regions where ~~/~(O) and ~(H~)~~(Oi) become larger than about 10-l, H+-H+ and H+-H collisions become important and the present model does not apply. However, following the results of the previous analysis for the velocity distributions of Of ions(*) there is no reason to expect that the inclusion of these additional collision effects would distort the H+ distribution more significantly than found in the present model. Acknowledgements-I wish to thank Dr. G. Lewak for his helpful comments and discussion during the course of this work. This research was supported in part by the National Aeronautics and Space Adminis~ation under grant NGR~5~~75 and by the Advanced Research Projects Agency (Project DEFE~ER) monitored by the U.S. Army Research Office, Durham, under contract DA-3I-124-ARO-D257. REFERENCES 1. HANSON,W. B. and JOJu%ON, F. S., 1961. M&m. Sot. R. Sci. Lisge, Series 5,4,390. 2. DALGARNO,A., MCELROY, M. B. and MOFFE~~, R. J., 1963. Planet. Space Sci. 11,463. 3. NAGY, A. F., BRACE, L. H., CARIGNAN,C. R. and KANAL, M., 1963. J. geophys. Res. 68,640l. 4. EVANS, J. V., 1965. Planet. Space Sci. 13, 1031. 5. GEISLER,J. E. and Bowrrr~~, S. A., 1965. J. atmos. terr. Phys. 27,457.
HYDROGEN
ION VELOCITY
DISTRIBUTIONS
IN THE
IONOSPHERE
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6. BANKS, P. M., 1967.
7. WALKER, J. C. 8. BANKS, P. M. Physics Fluids, 9. BANKS, P. M.,
Planet. Space Sci. 15,77. G. and DALGARNO,A., 1967. Planet. Space Sci. 15,200. and LEWAK, G. J.,
1967.
Ion velocity Distributions
in a Partially Ionized Plasma.
1968.
1966. Planet. Space Sci. 14,110X 10. STEBBINGS,R. F. and RUTHERFORD,J. A., 1968. Low Energy Collisions Between O+ and H. J. geophys. Res. Submitted for publication. 11. MONTGOMERY,D. C. and TIDMAN, D. A., 1964. Plasma Kinetic Theory, McGraw-Hill, New York. 12. ~NSON, W. B. and ORTENBURGER,I., 1961. J.geophys. Res. 66,142s. 13. DIAZ, J. B., 1958. in Handbook of Automation, Computation, and Control (Eds. F. M. GRABBE, S. RAMO and D. F. WOOLDRIDGE)Wiley, New York.