Frictional forces and collision frequencies between moving ion and neutral gases

Journalof Atmosphericand TerrestrialPhysics.1968,Vol. 30,pp. 1965-1985.Pergamon Press. Printed in Northern Ireland

Frictional forces and collision frequencies between moving ion and neutral gases PETER

Laboratory

STUBBE

*

for Space Sciences, NASA-Goddard Greenbelt, Maryland

(Received

16 B’ebruav

1968;

in revised form

Space Flight Center,

19 April

1968)

Abstract-Two different terms sre used in literature for the frictional force between moving ion and neutral gases. It is shown how the corresponding collision frequencies have to be defined in order to keep both terms valid. An expression is derived for the momentum transfer collision frequency yin between ions and neutral particles as a function of temperature and the relative flow velocity lzii - ~~1. The result indicates that for most practical applications vin may be considered as independent of 1vi - v,J. However, for very high velocities, e.g. a convection flow from the magnetosphere into the ionosphere, this dependence may be significant. Numerical values for vin are presented. In the case of collisions between ions and their parent neutral gases, laboratory values are used for the resonant charge exchange cross section obtained for high energies, but a correction is made for the much smaller thermal energies in the temperature range typical for ionospheric conditions in order to take into account the effect of curved particle trajectories due to attractive electrostatic forces. 1. INTRODUCTION

behaviour of the ionospheric plasma and atmospheric neutral gas can be described by the Navier-Stokes equation of hydrodynamics. This equation expresses the acceleration of a fluid in terms of a pressure gradient force, a viscous force and the resultant of all external forces. One of the most important external forces is the frictional force between an ion gas and a neutral gas moving with different velocities. The frictional force can be expressed in terms of the relative velocity between the ion and neutral gas and the number of collisions between ions and neutral particles. Two different expressions are used in literature for the frictional force, both relating the frictional force to the collision frequency between ions and neutral particles. It is necessary, therefore, to define different collision frequencies in order to keep both expressions valid. One of the goals of the present work shall be to give the proper definitions for these collision frequencies. Collision frequencies between ions and neutral particles are usually assumed to be independent of the relative flow velocity. It shall be investigated in this paper to which extent this assumption is correct. The frequency of collisions between ions and their parent neutral particles is greatly influenced by charge exchange processes. Charge exchange cross sections for all processes of atmospheric interest have been measured for energies far above the the thermal range. In this paper an attempt shall be made to extrapolate the charge exchange cross sections to thermal energies by taking into account the influence of THE

DYNAMIC

* On leave of absence from the Max-Planck-Institut 1965

fur Aeronomie, Lindau/Harz,

Germany.

PETER STUBBE

1966

curved particle trajectories due to polarization forces. The effect of polarization forces on collision frequencies between ions and their parent neutral gases has been considered before by DALGARNO (1958) and BANKS (1966a) by adding a polarization contribution to the momentum transfer cross section. In addition to this approach, in the present paper the direct influence of polarization actions on the charge exchange cross section shall be studied which seems to be the more important modification for ionospheric temperatures. 2. EQUATION OF MOTION FOR AN ION GAS equation, applied to an ion gas in the Earth’s atmosphere, reads

The Navier-Stokes as follows : pi ‘2 pi pi vi qi k

= = = = =

= -grad

p)i + v~(Q grad div vi + AvJ + k

(1)

mass density of the ion gas partial pressure of the ion gas ion velocity coefhcient of viscosity resultant external force per unit volume.

The external force k may be written as the sum of a gravity force, an electromagnetic force, a Coriolis force, a centrifugal force and a frictional force (ion drag force) : k = pig + N,e(E + vi x B) + 2p,(vi x a)

+ ~$2 x (r x S?) + kf

(2)

g = acceleration due to gravity Ni = ion number density e = ionic charge E = electric field strength B = magnetic induction Sz = angular velocity of the Earth r = position vector, measured from the Earth’s center kf = external frictional force or ion drag force. For a practical application of equations (1) and (2) to the Earth’s ionosphere the acceleration force, the internal friction force, the Coriolis force, and the centrifugal force may be omitted. 3. FRICTIONAL FORCE AND COLLISION FREQUENCY Two different terms for kf are in use in literature. Some authors (e.g. DOUGHERTY, 1961; KENDALL and PICKERING, 1967 ; KOHL and KING, 1967 ; PRIESTER, ROEMER and VOLLAND, 1967) use the expression kf = -Nivinmi(vi

-

v,) = -ppivi,(vi

-

v,)

(3)

while others (e.g. SCHL~~TER,1950; CHANDRA, 1964) use the expression k, =

-Ni~in~in(~i

-

VJ

(4)

Frictional

,ui,, = mi = m, = Yin = v, =

forces and collision frequencies between moving ion and neutral gases

1967

m,mn/mi + m, = reduced mass ion mass neutral particle mass number of collisions per second of one ion with the neutral particles neutral gas velocity.

Since vin is a fictitious quantity which merely has the dimension of a frequency, it can be defined so that either (3) or (4) is valid. BANKS (1966a) defines a momentum transfer collision frequency as

(5) Q(g)=

277p- cos x)b db =

momentum

transfer cross section

g = relative thermal velocity x = angle of deflection (see Figs. 12 and 13) b = collision parameter (see Fig. 14) = reduced temperature. In order to find out how the frictional force term reads when we use vi,, as defined by equation (5) we refer to the momentum transfer equation provided by statistical plasma mechanics. After BURGERS (1960), the frictional force for Ti = T, in the terminology of this paper is given by k, = -(vi

-

,u,,N,N, (!Y!J ( 2!!&3’2

v,J . :

or, inserting equation

. jomQ(g) . e-_(hJ~fWg5 dg

(6)

(5) :

k, = -Nivinpin(vi - v,). Equation (7) is identical with equation (4). DALGARNO (1961), on the other hand, defines the collision frequency v

IcT -in - m,Di

(7) as

(8)

or, when we insert for the diffusion coefficient Di the expression given by CHAPMAN and CO~TJNCJ(1961) : Yin

By comparison

=

“3”.N,

-

. %

of equation

.

(!I!?)(&)“‘”

(Sa) with equation

kf = -Nivi,mi(vi

/om&(g) e--(J’inIaT)@g5dg.

(&)

(6) we obtain -

v,).

(9)

Equation (9) is identical with equation (3). The numerical values for vin, given by COWLING (1945) and DALGARNO (1961 and 1964), are based on the defining equations (8) or (8a). Hence, when these collision

1968

PETER STUBBE

frequencies are used, Equation (3) must be applied for kf, On the other hand, the collision frequencies presented by BANKS (1966a) and later on in this paper (paragraphs 5 and 6), are calculated by means of equation (5). Therefore, these values have to be taken in conjunction with equation (4). Thereby, the apparent discrepancy between the collision frequencies of DALGARNO (1964) on the one side and of BANKS (19663) on the other side is removed. The frictional force as given by equation (6) leads, when decomposed into the constituents of equation (4), to an expression for the collision frequency which does not depend on the relative flow velocity lvi - v,l. Such a dependency, however, should be expected, since the velocity distribution functions of the ion and neutral gas can be influenced not only by a change in T, but also by a change in lvi - v,l. We will derive here an expression for kf leading to a collision frequency which shows a dependence on lvi - v,l. Our basic assumption is that the velocity distributions of the ion and neutral gas are Maxwellian distributions, displaced by the drift velocities vi and v,, respectively (SMITH, 1964):

e-(W2kT)(cn-vn)’

cd, c, = thermal velocities. The result of our calculations, kf

=

-(vi

_

vn)

.

!f.

N,

d3C

n

which are presented in the appendix, . N,

.fiin

.

(_!$_). (a)“‘.

is:

e-(~i~/2kTdlv~-Vn12

~/~~Q(g)e [l +x.~X2g2+~.~X4g4+~.~X6g6 +...]e-(~~~~2kT~)g2g5dg(10) x =&

IVi - v,l. R

By comparison given by: Yin

=

$

X x

N,

with equation

(&)

(_&_)“’

(4), the momentum

dg_

is

e-_(Pin,2kTB),v~-v,,,a

1 + f * ; xzg2 + ; a; e-_(MWdcPg5

transfer collision frequency

X4g4 +f-;Pg6+... 1 (11)

For vi = v, and Ti = T, equation (11) is identical to equation (5). We will discuss this formula later (paragraph 5). In the following we must distinguish between ions in unlike gases and ions in their parent gases. In the first case, the momentum transfer cross section is determined by electrostatic interactions, while in the second case charge exchange plays a predominant role.

Frictional

forces and collision frequencies between moving ion and neutral gases

1969

4. MOMENTUM TRANSFER CROSS SECTION FOR IONS MOVING IN THEIR PARENT NEUTRAL GAS The momentum

transfer section Q was defined as Q(g) = 27r s,‘il

-

cos ~)b db = 47~jorn cos2 6b db

(12)

x, 6 and b are defined in Figs. 12, 13 and 14. Due to electrostatic interaction forces, which are described later, the integrand is greater than zero for all finite values of b and g, but, for given g and increasing b, it converges fast enough to zero to guarantee the convergence of the above integrals. Y

X’

Y+

lvry L;’ x+

\

Y

\ \

x+

\

i

Fig. la, b. Left side (la): Collision without charge exchange Right side (lb): Collision with charge exchange.

When the ion gas is embedded in its parent neutral gas, the value of Q is much less determined by electrostatic interactions than by charge exchange processes of the type

x++y=x+y+

where X and Y are particles of the same neutral gas species. After HOLSTEIN (1952), the effect of charge exchange processes on the momentum transfer cross section can be described as follows : When the particle trajectories are linear, a critical collision parameter b, with the following features can be defined: For b < b, the charge exchange probability P,, is a rapidly changing function of b, oscillating between 0 and 1 and having an average of l/2. For b > b,, P,, decreases rapidly with increasing b. Therefore, on the average, every second collision with b < b, leads to a charge exchange process. Let us consider a particular impact, in the first instance without charge exchange (Fig. la), then with charge exchange (Fig. lb). In the first case the ion is deflected by x1, in the second case (since X and Y are of the same species) by x2 = rr - x1 (i.e. cos x2 = -cos x1). Hence, on the average, for b < b, the integrand (1 - cos x) - b is equal to b. Therefore, if we neglect the very small contribution of P,, for b > b,, we can describe the net result of charge exchange processes in the following manner: Charge exchange processes between ions and their parent neutral particles influence the momentum transfer so as if all collisions with an impact parameter b smaller than the critical impact parameter b, occur with a deflection angle of 90”. Thus, the momentum transfer cross section can be written as Q(g) = n-b,2 + 477 b;cos2 6 b db s 4

(13)

PETER STUBBE

1970

The charge exchange cross section S,,, which has been measured processes of atmospheric interest, is defined as the integral S,, =

s0

* P,, dq,

(14)

where dq is a differential cross section and Pex is the charge exchange related to dq. Since P,_ is a function of b only, S,, can be written as S,, = 2rr Omitting the small contributions Se, =

s0

for many

“P,, b db.

probability

(144

of the integrand for b > b,, we get

$

?!&. J-

(15)

m cos2 6 b db. s GS,,ln

(16)

b, =

or

3-r

Therefore,

Q(g) and S,, are related by

Q(g) = 2S,, + 4n Laboratory as theoretical energies above relative kinetic

measurements (e.g. STEBBINBS, SMITH and EIFRHARDT, 1964) as well studies (KNOF, M&ON and VANDERSLICE, 1964) showed that for 1 eV, when the trajectories are approximately linear, S,, and the energy E = (,ui,/2)g2 are connected by S,, = (C, -

(32

- log,,

W2,

(17)

where C, and C, are constants. In order to evaluate collision frequencies or diffusion coefficients for ions (e.g. BANKS, 1966b and KNOF et al., 1964), the authors used the unmodified equation (17). However, at thermal energies characteristic of atmospheric conditions, the curvature of the particle trajectories due to long range attractive forces should have a marked influence on S,, and Q. We will therefore make an attempt to correct the charge exchange cross sections for small energies. The corrected values of the charge exchange cross section and the critical collision parameter shall be denoted by S,,’ and b,‘, respectively, to distinguish them from the uncorrected values, given by equations (15) and (17). While the task to relate b,’ to b, for a given relative velocity g can be exactly solved, an exact relationship between S,,’ and b,’ can be found only when all branches of the charge exchange interaction potential are known, but even then it is a very difficult problem. We will assume that S,,’ and b,’ are related in the same manner as S,, and b,:

7i-bI2 Se,’ = +-

or

b,‘=

-2&X

J

lr

’

(1w

This assumption is in agreement with a proposal by HOLSTEIN (1952, p. 835). Equation (15a) would be correct to the same extent as equation (15) if the transformation from b’ to b were strictly linear. The meanings of b’ and b are explained in Fig. 2.

Frictional

forces and collision frequencies between moving ion and neutral gases

Fig. 2. Illustration

1971

of somo quantities used in the text.

When b,’ is known, Q is given by Q(g) = nbc’2 + 477 or, because of equation

(15a),

;

s bc

cos2 8 b’ db’,

P3a)

by

Q(g) = 28,x’ +

477

cos2 6 b’ db’. co I-~ZS,,‘ln

(164

We are left now with the problem to calculate cos2 6 as a function of b’ and g and b,’ as a function of b, and g. WC assume a long range interaction potential between an ion and a neutral part.iclc of the form (e.g. MCDANIEL, 1964) V(r) = -(f)”

[I + (:)“I

for

r 2 (T

for

r
. U(r) = SC.0

(18)

I

The fourth term order is due to dipole interactions, the sixth order term to quadrupole interactions. (T is the sum of the gas kinetic radii of the particles. A can be expressed in terms of the polarizability 7 of the neutral particle.

(1% It would be more realistic to describe the repulsive potential by a term of form C,W, but since no values for C are available, this would only impede calculations without yielding a higher accuracy. The potential given by equation is illustrated in Fig. 3. Using the symbols defined by Fig. 4 and applying the conservation laws energy and angular momentum, we obtain the relations

Pin

‘“4 = Pin gb’

the the (18)

for

(20b)

PETER STUBBE

1972

I

u

r-

Fig. 3. Interaction potential U as a function of the particle distance T.

Fig. 4. Illustration of some quantities used in the text.

w can be expressed as

w= = +2 + r92

+ and 4 stand for the time derivatives identity

dr/dt and d#dt,

+=rp

respectively.

Considering the

. d4

i and c&j/& are given by

(21)

2 By means of equation

=!$[4"(1

_!J _3w-1’2.

(22) we can immediately

obtain

(22) an expression

for 6:

(23)

Frictional forces and collision frequenoiesbetween moving ion and neutml gases

1973

b is the minimum distance between the two particles for given b’ and g. If rr, (PCA = point of closest approach) is defined s,s the greatest nullpoint of +2(~),b is given by (see Fig. 5) b=

for

rPCA

2:

a

for

rPCA

<

(r*

(244

For sufficiently small values of b’, however, there exists no nullpoint of i”(r). In this case, b is simply given by b=CT

(24b)

=

b'bb

i2

Fig. 5. $ (i = radial velocity) as a function of the particle distance I for three different collision parameters b’.

If b is greater than o‘, i.e. b = rpCA,the integrand in equation (23) has a! singularity for r = b. In order to remove this singularity and to make the range of integration finite, we change the variables by using the transformation (MASON and scHAM.r J.958) )

b

sin 6 = - . r

(25)

We thus obtain for 6: -112

1-

b’2 sin2 6 b2 -

The critical collision p&rameter b,’ which has the property that for b’ < b,’ f2(r) has no nullpoint, is of significance for the determination of b,‘. As we read off from Fig. 5, b,’ is given by the condition if b, 2 rpca(b,‘) and by

rr~A(&‘) = 6,

(27%)

b,’ = bo’

(27b)

1974

PETER

STUBBE

if b, < rpcA(b,,‘). Now all formulae necessary for a calculation of Q(g) and Sex’ (g) have been provided. Numerical results are presented in the Section 5. 5. NUMERICAL VALUES FOR MOMENTUM TRANSFER CROSS SECTIONS AND COLLISION FREQUENCIES FOR IONS IN THEIR PARENT NEUTRAL GASES Experimental values of S,, (g) are available for all processes of atmospheric interest. We will base our calculations on the values for S,, and 7 given in Table 1. The values for q have been taken from BANES (1966b). Table 1. Charge exchange cross sections and polarizabilities of neutral particles. Sources: (1) STEBBINGS et al. (1964); (2) FITE et al. (1962); (3) CRAMERend SIMONS(1957); (4) KNOF et al. (1964); (5) AMMEand UTTERBACK (1964) Process

q(10W4 ems)

~$?(10-~ cm)

o++o+o+o+

0.89

5.95-0.63

logI

H++H+H+H+ He+ + He-tHe -+ He+ N++N+N+N+ os+ + 0, + 0, + O$ Ns+ +N,+N, +Ns+

0.67 0.21 1.13 1.60 1.76

7.60-1.06 525-0.74 553-0.46 5.37-0.54 7-36-0.68

log,, log,, log,s logI log,,

Source for S,,

E (eV) E(eV) E(eV) E(eV) E(eV) E (eV)

(1) (2) (3) (4) (5) (5)

Using the formulae derived in Section 4, Q(g) and S,,’ (g) have been calculated numerically with the help of an electronic computer. Since B (defined by equation (18)) is practically unknown, a set of values for B, ranging from O-5 to 2.0 lo-* cm has been used. The result was that the influence of the sixth order potential term is completely negligible, since in no case was the deviation in Q(g) caused by this term more than O-5 per cent. Therefore, all further calculations have been carried out with B = 0, i.e. for a pure fourth order potential law. Results for Q(g) and S,,’ (g), compared with S,, (g), for the processes 0+ + 0 -+ 0 + Of and H+ + H -+ H + H+ are shown in Figs. 6 and 7. The numerical values for Q(g) have been used to calculate the momentum transfer collision frequency vin after equation (11) for the processes listed in Table 1. Figure 8 shows yin for the most important charge exchange collision process in the ionosphere, namely 0+ + 0 + 0 + O+, as a function of the reduced temperature T, for lvi - v,] = 0. In order to indicate the effect of electrostatic interactions, vi,, has also been calculated for the uncorrected momentum transfer cross section Q = 2S,,, which would be the case if no electrostatic interactions would be present. In the temperature range 500°K I T, < 3000°K the collision frequency pin for jvi - v,l = 0, as a function of T,, can be approximated by the following expressions, when the neutral particle number density is measured in the unit cm-3: Y(O+, 0) =

1.86 - 10-g(TR/1000)0~37n(0)

v(H+, H) = 12.03 * 10-g(T,/1000)0~38n(H) y(He+, He) = v(N+, N) = v(O,+, 0,)

=

v(N,+, N,) =

2.92 - 10-g( T,/1000)0J7n(He)

set-1 se+ se+

1.75 - 10-g( TR/1000)0.34n(N) set-l 1.17 * 10-g( TR/1000)0~2Cn(02) se& 2.11 - 10-g(T,/1000)0~38n(N2)

se+

Frictional

forces and collision frequenoies between moving ion and neutral gases

j

I

I ____

0.5c

~_..__

---_-A-

: 1

I

;CUI

Fig. 0. Q, 25,,’

and 25,,

I

&

I

1975

I

/

2000

2500

a8 a function of the relative velocity g for the procass o+ +-o-+0 + o+.

These values can be used to calculate the ambipolar diffusion constant 2?, for an electrically neutral electron-ion gas in the ionosphere, which is given by D _ k(Z7, + ‘,) sin2II avia Clin In the PZ-region, where Of and 0 are the major ionic and n eutral constituents _D, can be written as 0,

= o-55 . 1016

0~~( Ti + T,) sin2 I

em2

sec_l

.

n(O)

In the exosphere, where H+ and H are predominant, D, is 13, = 1.38. 10’6

o.38(T,+ T,) sin2 I (3m2sec_l

.

G-9

The above collision frequencies are greater than those presented by BANES (19663). For 0+ in 0, for example, they are greater by 35 per cent for 500”K, 28 per cent for

PETER STUBBE

1976

kt++tl +

H+H+

6.

0

2000

1000

3OM)

4(

r-----5

0

g-m/xc

Pig.

7. Q, 2Sex’ and ZS,,

as a function of the relaCve velocity H’ + H-H + H’.

g for the process

I

1000

2000

TR+'K as a function of

3000

4

Fig. 8. v(O+, 0)/n(O) the reduced temperature 2’~ for Ivi -v,J = 0. Upper curve for the momentum transfer cross sectionQ after equation (16a), lower cmve for the uncorrected momentum transfer cross so&ion ZS,,.

Frictional forces and collision frequencies between moving ion and neutral gases

1977

lOOO”K, and 22 per cent for 2000°K. Thus, in contradiction to BANKS (1966b), we see that polarization forces affect Yinnot only for very low temperatures, but also

for temperatures of ionospheric interest. vinas a function of lvc -v,l for the processes 0+ + 0 + 0 + Of and H+ + H -+ H + Hf is shown in Figs. 9 and 10. When we consider diffusional flows, electromagnetic drifts or neutral gas winds in the upper atmosphere and ionosphere, Ivi -v,l should not exceed about 300 mjsec

prq Fig. 9. Y(O+, 0)/~(o)

-m/set

as a function of Ivi - v,l for three different temperatures.

Therefore, when applied to these mechanisms, vin may be considered as independent of Ivi - v,I. However, for particle flows from the magnetosphere into the ionosphere, as proposed by AXFORDand HINES (1961), this dependence may be of significance, since the plasma convection velocity is expected to be of the order of kilometers per second. 6. NUMERICAL VALUESOB COLLISIONFREQUENCIES FOR IONS IN UNLIKE

NEUTRALGASES For reasons of completeness, collision frequencies for ions in unlike neutral gases are also presented, although the basic studies in this field have been carried out more than six decades ago by LANGEVIN(1905) and later on by HAS& (1926). LANGEVIN and HAS& give an expression for the ion mobility K which, using the terminology of this paper, can be written as

K= where A(A), 3, being defined as

-44 -

2% dvin r

(28)

1978

PETER STUBBE

1 v,-v,I--+

m/see

Fig. 10. v(H+, H)/n(H) 8s a function of IV, - v,l for four different temperatu~s. is given by HASS~ (1926, Table III). A is not identical with the quantity defined by equation (18). Using equations (l), (2), (4) and taking v, = 0, K and vi,, are found to be related by Yin = 2 * F(A) @I”

N,

(30)

is shown in Fig 11. F(L) = GA(n) Since C, the sum of the gas kinetic particle radii, is not known well enough, the usual method is to take the limiting value P(O), i.e. for vanishing o or infinite 7, instead of the correct value F(A). This leads to the approximation (see also BANKS, 19663) v.tn = 6.94

?f ‘12N ( f4n ) n-

(31)

We must realize, however, that the values obtained from equation (31) can easily be wrong by about 20 per cent for temperatures below 2000°K and by more Especially the temperature dependence of vi,, is than this for higher temperatures. not correctly described by the approximation (31). For 1 < O-6 the collision frequencies decrease with increasing temperature, while for I > 0.6 the opposite is true. For 1 > 1.2 the collision frequencies approximately follow a T1i2-law. Assuming

Frictional forces and collision frequencies between moving ion and neutral gases

1979

Fig. 11. P as a function of A.

T = 500°K and q = 1O-24 cm3, a value of jl 2 1.2 is adopted for G z 4 lO-8 cm. yin after equation (31) is based on the defining equation (5). The defining equation (8a) would yield v#,,

=

6.94. z

*N,.

(314

Although momentum transfer collision frequencies can simply be evaluated from equation (31), we will give some numerical values for the most impo~ant collision processes in the ionosphere: v(O+, 0,) v(O+, N,) v(Of, H) v(H+, 0) v(NO+, 0,) v(NO+, N,) v(NO+, 0) v(O,+, N,) v(O,+, 0)

= = = = = = = = =

1.00 1O-gn(0,) see-r 1.08 1O-gn(N2) se& 2.19 1O-gn(H) see-r 2.52 1O-gn(O) see-r 0.83 1O-s $0,) see-l 0.90 1O-s n(N,) see-l 0.76 1O-s n(O) see-l 0.89 1O-gn(NJ see-l 0.75 1O-gn(O) set+,

As mentioned before, the collision frequencies defined by equations (5) and (Sa) are fictitous quantities, that is, they are quantities having the dimension of a frequency and were introduced merely for reasons of convenience. When two particles

PETERSTUBBE

1980

approach each other so closely that they are separated only by short range repulsive forces or, in other words, when two particles have a direct contact, we will call this a ‘real’ collision. According to this definition, the number of real collisions per second, vR, is given by (STUBBE,1966)

where u0 and w,,are abbreviations standing for the expressions

In terms of ;1, defined by (29), vR can be written as vR = 277

($J,,,Nn [~jol’ie-u”du

+ --$,,i,]

.

(324

For temperatures below about 2000”K, equations (32) and (32a) can be approximated by VR

Wb)

=

By comparision of equations (31), (31a) and (32b) we see that VRlies between vin after (31a) and vin after (31). 8. SUMMARY (a) The relationship between the frictional force and the collision frequency between ion and neutral gases is studied. Proper definitions are given for the collision frequency to fit the expressions used for the frictional force. (b) An expression is derived for the momentum transfer frequency vp, as a function of the relative flow velocity Ivt - v,l. It is shown that for most practical applications vinmay be considered as independent of lvi - v,l , but that for velocities in the km/see range, as may be expected in convection flows from the magnetosphere, this dependence is significant. (c) Numerical values of vin for the most important collision processes in the ionosphere are presented. In the case of ions in their parent neutral gases, the effect of polarization forces on the charge exchange cross section is taken into account. This gives rise to a slight increase of vi, over previously reported values. In the case of collisions between 0+ and 0, for instance, the increase, as compared with the results of BAMES(1966b) and DAL~ARNO(1964) amounts to about 25 per cent for temperatures between 1000°K and 2000°K. (d) The collision frequencies presented here and in other papers (DALCARNO, 1961 and 1964; BANKS, 1966b) are based on experimentally determined values of polarizabilities and charge exchange cross sections. Theoretical manipulations involving some approximative assumptions are necessary to derive vin from these

Frictional forces and collision frequenciesbetween moving ion and neutral gases

1981

data. A much more direct determination of vi,, could be obtained from mobility data, and it would be very desirable to carry out such measurements in the future. want to thank Dr. S. CHANDRAand Dr. J. R. HERUN for very valuable critical comments. The present work was initiated at the Max-Plane-Institut fuer Aeronomie, Institut fuer Ionosphaerenphysik, Lindau/Harz, Germany and completed at the NASA Goddard Space Flight Center, Greenbelt, Maryland, where the author held a postdoctoral resident research associateshipof the NAS-NAE.

Acknowledgments-I

REFERENCES AMMER. C. and UTTERBACK N. G.

1964

AFFORDW. I. and HINES C. 0. BANKSP. BANKSP. BURGERSJ. M.

1961 1966a 1966b 1960

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CHANDRAS. CHAPMANS. and COWLINGT. G.

1964

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COwLINaT. G. CRAMERW. H. and SIMONSJ. H. DALGARNO A. DALGBRNO A. DALGARNOA. DOUGHERTY,J. P. FITE W. L., SMITHA. C. H. and STEBBINGS R. F. HAS& H. R. HOLSTEINT. KENDALLP. C. and PICXERINGW. M. KNOF H., MASONE. A. and VANDERSLICE J. T. KOHL H. and KING J. W. LANGEVINP. MCDANIELE. W.

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29, 1045.

in

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Information

Series, No. R64SD54.

STEBBINGS R. F., SMITHA. C. H. and EKRHARDTH. STUBBEP.

1964

J. Geophys.

1966

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NATO

Res. 69, 2349. A&v.

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Inst.,

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(Edited by J. Frihagen), North-Holland, Amsterdam. (Norway),

April

1965

1982

PETER STUBBE APPENDIX

Derivation of equation (10) The frictional force between two gases is the result of the momentum colliding particles. We introduce the following quantities:

transfer

between

the

C+ c, = ion and neutral particle velocity before the collision ci‘, c,’ = ion and neutral particle velocity after the collision g = ci - C, relative velocity before the collision g’ = ci’ - c,’ relative velocity after the collision Ii = mici ion momentum before the collision Ii’ = mici’ ion momentum after the collision A&, = Ii’ - Ii momentum transfer. Using the conservation

law for momentum, A&

The conservation

law of energy

it can easily be derived

= /+,(g’

-

that

AI,,

is given

by

g).

(37)

which holds for elastic collisions

gives

In order to determine g’ - g we will distinguish between direct collisions (direct contact of the colliding particles) and indirect collisions (no direct contact of the particles, but curved orbits due to long range electrostatic forces). (a) Direct Collisions: Under the assumption that no angular momentum is transferred during the collision, the orbit is symmetrical to the radial unit vector e,. According to Fig. 12, we therefore obtain: g’ - g = lg’ - gl . e, lg’ - gl = 2gcos6 (b) Indirect

Collisions:

According

to Fig.

g’ -

g =

13: -_lg’

-

gl

. e,

lg’ - gl = --2g cos8 In both cmes we get the same result, namely g’ -

g = 2g cos 6 e,

(39)

A&,

= 2pid

(49)

or cos 6 er

We now introduce a Cartesian coordinate system (Fig. 14) and we especially assume that the relative macroscopic velocity vi - v, has the direction of the negative z-axis. In this case we are interested only in the z-component of AIi, since, on the average, all other components cancel out because of the complete symmetry about the z-axis. (AI,,),

= 2,uing cos 8 e,,.

(41)

It can easily be shown that %s = 00s B 00s f3 + cos a sin 8 sin 0. Hence, the z-component of the momentum any particular collision is given by ( AIiJr (AI,,),

and the frictional

= 2~i,g(cos2

from an ion to a neutral

8, cos 0 + 00s a sin 8,cos

force kf are connected dk,

transferred

=

6 sin 0)

particle

for

(42)

by

( AliJZNidvie,

(43)

dvi is the number of collisions of one ion per second impinging on the differential cross section dq = b . db * da (see Fig. 14) and occurring in the velocity range g . . . g + dg and the angular to this definition, dvi is given as the product of the relative range 0 . . . 0 + de. According

Frictional forces and collision frequencies between moving ion and neutral gases

Fig. 12. Relation between g’ -

g and for 6 direct collisions.

Fig. 13. Relation between g’ -

g and 8 for indirect collisions.

1983

velocity g, the cross section dq, and the number of neutral particles dNn per unit volume having a velocity between g and g + dg and being in the angular range between 13and 19+ de, related to the particular ion under consideration. In order to determine dN,, we assume that the ion gas and the neutral gas have Maxwellian velocity distributions, displaced by the velocities vt and v,, respectively. Furthermore we assume, in the first instance, that the ion temperature and the neutral gas temperature are equal.

Wb)

PETER STUBBE

1984

We multiply the distribution functions with each other and introduce a new velocity

After some simple manipulations we get: 312

The integration over IJ can simply be carried out yielding 3/2

e-_(Pin12kT)(vi-v,--g)Z

#g

(45)

Fig. 14. Illustration of some quantities used in the text. When we drop the restriction T = Td = T, we obtain the same result provided that we replace T with the reduced temperature TR defined by TR

(46)

= Pin

Using Fig. 14 and defining

x =g

R

[Vi - v-1

we obtain e-(pin/2kT~)(vi-v,-g)’ = e(Pi,lkTR)glv,-v,lcos

0 =

~-(P~J~BT~)IV~-V~I~ 1 + Xg

cos

13 + ;

dag = gBsin 0 d0 d# dg

. e-(~in/2kT~)~* Xag*

cosa

0 +

s e(‘in/kT~)glV~-v~lcO~ f

X*gs

cos8

0 +

0 . . .

Frictional forces and collision frequencies between moving ion and neutral gases

1985

Inserting these expressions into equation (45), carrying out the integration over #(O 2 # < ZST)and dividing by N+ we get for dN,: 312

. e-WinPkT&i-v,l”

.

. g8 . e-@i,t2kT,)ga

sin5 + Xg sin e cos e f $ X2g2 sin e ~03~ e + , . .

dg de

(47)

dv, is given by dv; = g - b - dzn * db * da.

(43)

Hence, a.fter integrating over GL(OI c( 5 2~): dkt = es4npinN,.gab 00s~ B cos e dx,

db

(49)

6 is a function of the collision parameter b and the relative velocity g. We introduce a velocity dependent collision cross se&ion Q(g) defined by Q(g) = 4~

a,cos2 6 b db s0

(50)

where Q(g) is identical to the momentum transfer cross section

Q&l = 27r *(I - cosX)b db s0

(5Oa)

commonly used in literature since, according to Figs. 12 and 13, the deflection angle x is related to 8 by means of 2 toss 8 = 1 - co9 x. We integrate over 8(0 2 @ 5 ?r) and over b(0 < b < m) in equation (49), insert Q(g) given by (50), consider the relationship Vi -v, ez=

-Ivy

and integrate over g(0 I g < to) to get the flnal result

5