Shkarofsky: Generalized 4 ppleton-Ilartree Equation

VI. CoNCLUSIONS

The total noise power available at a receiving antennaon a hypervelocity space vehicle has been determined bytreatinig the enigulfinig plasma sheath as a uniform slabof plasma. The nioise emission from the plasma cani thenbe evaltuatedl in terms of a bouiidary-value problemwhich for this simple model cani be completely solved.The effect of high temperatures of the vehicle surfaceajid of noise sources externial to the vehicle and plasmasheath are inclu(ledl in the analysis.

Wheni the plasma sheath is isotropic, the main noisepower is due to emissioni from the vehicle surface if theRE frequency is much lower thain the plasma frequenicy;it is due to sources outside both the vehicle anid plasmafor RF freqLuenicies much greater than the plasma fre-quency antI results from direct emissioni from the plasma

for RF frequencies in the neighborhood of the plasmafrequency. The noise power from the plasma itself is themost significant and exhibits a sharp maximum at ani REfrequenicy just above the plasma frequenicy. This eni-hances the suggestion8 of the value of noise emissionmeasurements as a diagnostic tool for the explorationiof the hypersoniic enivironiment of a space or re-entryvehicle.The effects of anl amisotropic plasma sheath (due to

anl auxiliarv nmagnetic field) have also beeni inivestigatedfor the two cases of the magnetic field niormal to thesheath and parallel to the sheath. Ihe effect of theaniisotropy is to alter completely the spectral charac-teristics of the available nioise power at the receiving an-tenna. Several regionis of both weak anid initenise nioiseenmissiol) are now possible, (lepenclinig upon magnieticfiekl strength and orientation.

Generalized Appleton-Hartree Equation for AnyDegree of Ionization and Application

to the Ionosphere*I. P. SHKAROFSKYt, MEMBER, IRE

Summary-A generalized Appleton-Hartree equation is derived,applicable to any variation of electron collision frequency with elec-tron speed and any degree of ionization. Regions of the parameterswhere simplification occurs are given. Results are shown for theionosphere (60 to 320 km) and compared with experimental data.Good agreement is obtained for the D layer'Experimental data forthe E layer can also be explained by assuming an electron tempera-ture several times the gas temperature. This is consistent withrocket data and a measurement by Sputnik III. It is shown thatthe classic Appleton-Hartree equation should be applicable withno corrections necessary for the F2 layer, provided the collision fre-quency is averaged appropriately.

1. INTRODUCTION

Mlost inivestigationis on the propagation of an elec-tromagnetic wave in a mnagnietized plasma are based onthe concept of a constanit value for the electron colli-sion frequency, independent of electron speed. The clas-sic Appleton-Hartree equationi, used for propagation in

* Received bv the IRE, April 6, 1961. The work reported inthis paper was done under Contract GX9-998054-0001-70-Q82 toRCA, Missile and Surface Radar Div., Moorestown, N. J.

t RCA Victor Co., Ltd., Research Labs., Montreal, Canada.

the ionosphere, has this inherent assumption. Becausethe electron elastic collision frequency with the nitro-geni molecules in the atmosphere varies as the square ofthe electron speed,' a large discrepancy between classictheory and ionospheric experiments can be expected. Anaccurate analysis miiust include these effects. Severalcorrections to inclutde this speed variation have ap-peared in the literature.2-4 These references only ac-count for the neutral particle effects, under the assuiip-

I A. V. Phelps and J. L. Pack, "Electron collision frequencies innitrogen and in the lower ionosphere," Phys. Rev. Lett., vol. 3, pp.340-342, October, 1959; "Drift velocities of slow electrons in helitm,neon argon, hydrogen, and nitrogen," Phys. Rev., vol. 121, pp. 798-806, February, 1961.

2 R. Jan(cel and T. Kahn, "Theorie du couplage des oindes electro-magnetiques ordinaire et extraordinarie dans un plasma inihomogneieet anisotrope et conditions de reflexion. Applications a l'ionio-sphere," J. Phys. Radium, vol. 16, pp. 136-145, February, 1955;"Th6orie non Maxwellienne des plasmas homogenes et anisotropes,"Nuovo Cim., vol. 12, pp. 573-612; November, 1954.

3 H. K. Sen and A. A. Wyller, "On the generalization of theAppleton-Hartree magnetoionic formulas," J. Geophys. Res., vol.65, pp. 3931-3950; December, 1960.

4A. V. Phelps, "Propagation constaiits for electromagnietic wavesin weakly ionized dry air," J. Appl. Phys., vol. 31, pp. 1723-1729;October, 1960.

1961 1857

PROCEEDINGS OF THE IRE

tion that the gas is very slightly ionized. This may be agood approximation to conditions in the D layer, butnot in the F layers of the ionosphere. In the following, ageneralized Appleton-Hartree equation, valid for anyvariation of the collision frequency with electron speedand any degree of ionization, is derived. The results arethen applied to the ioinosphere. Various domains of theparameters where simplificationi may result are investi-gated.A method for calculating the electronic conductivity

for any degree of ionization has been developed.5 Inthis method, the usual series expansion of Laguerrepolynomials is substituted into the Fokker-Planckequation for Coulomb (electron-ion and electron-elec-tron) collisions,6 and into the Boltzmann equation forelectron collisions with neutral particles. A Maxwelliandistribution in electroni velocity is assumed. For Cou-lomb collisions, the expressions are the samne as thosederived by Landshoff.7 Collisions of electronis withneutral particles and ionis, ac electric fields, as well asstatic magnetic fields, are treated in this analysis. Inthe limit of a completely ionized gas, the results agreewith those of Spitzer and Harm.8 For a slightly ionizedgas, no expansion is necessary and the results are identi-cal with those of Allis.9 The conductivity can be calcu-lated by usinlg the Dinlgle tabulations,10 if the electroncollision frequenicy varies as an integral power (between-3 and 3) of velocity. This method, to be reviewed inSection II, is utilized here to yield the proper averages(or multiplyinig factors) for the collisioin frequency andmultiplying factors for the angular frequeincy.

II. BASIS OF ANALYSIS

In this analysis, existing gradients and fluctuations inthe dielectric coefficient, ion motion and thermal motionof the electrons which give rise to plasma waves, are allneglected. The limit for the convergence of the theory,if thermal mnotions are neglected, is that the ratio of thethermal velocity of the electrons VkT/m to the phasevelocity of the wave c/n is small. That is, n2kT/mc2K<1,where n is the refractive index and c is the velocity oflight.The electronic conductivity matrix (d) for any de-

gree of ionization and any variation of electron-neutralparticle collision frequency vm with electron velocity v

5 I. P. Shkarofsky, "Values of the transport coefficients il aplasma for any degree of ionization based on a Maxwellian distribu-tion," Canad. J. Phys., vol. 39; November, 1961.

6 M. P. Bachynski, I. P. Shkarofsky and T. W. Johnston, "Plas-mas and the Electromagnetic Field," McGraw-Hill Book Co., Inc.,New York, N. Y.; in press.

7R. Landshoff, "Transport phenomena in a completely ionizedgas in presence of a magnetic field," Phys. Rev., vol. 76, pp. 904-909;October 1, 1949.

8 L. Spitzer and R. Harm, "Transport phenomena in a completelyionized gas," Phys. Rev., vol. 89, pp. 977-981; March 1, 1953.

9 W. P. Allis, "Motions of ions and electrons," in "Handbuch derPhysik," S. Fluigge, Ed., Springer-Verlag, Berlin, Ger., vol. 21; 1956.

10 R. B. Dingle and D. Arndt, "The integrals 2[p(x) and Zp(x)and their tabulation," Appl. Sci. Res., vol. 6-B, no. 3, pp. 144-154,1957; "The integrals (Sp(x) and )p(x) and their tabulation," pp.155-164; "The integrals (p(x) and 7p(x) and their tabulation,"no. 4, pp. 245-252; 1957.

can be represented as5

/ b + c j(b-c) 0

(d) = |-j(b - c) b + c 0 },

O 0 d/

with

Ne2 1d=--

m (v9)go + jwho

Ne2 12c =--

m (vP)g± +j(c + Ob)h+

Ne2 12b = g±(

m (vg)g_ + j( - cob)h-

(1)

(2)

(3)

(4)

whereN,e,m are respectively electroni denisity, charge, and

mass,

w is the RF angular frequency,cb is the cyclotron angular frequenicy, Wb =| eB/m |

where B is the imiagnitude of the dc mnagneticfield. The niagnetic field is taken parallel to thez-axis of a Cartesiani or cylindrical system ofcoordinates.

Also

(v,) is the averaged electron collisioni frequencywith neutrals plus ioIIs anld will be defined be-low.

go,+,- and ho,+, are correction factors to account forthe variation with velocity, of the electron-nieutral particle collision frequency aind forelectron-ion and electron-electroni effects.

'I'he averaged collisioni frequency is defined so as tomake g and h equal to 1 at high angular frequencies.'I'his requires5 that

4ir J o fo0(v11)- ,) - Vv(dz(5

where

fJo is the distribution function of electron velocity,v is the electron velocity,

vP is the unaveraged total electron collisioni fre-quenicy-a funiction of electron speed.

Let the contributions to P, be Pm fromn ineutrals, whosevariation with velocity depenids oni the constituentgases, and Pei from ions, known6 to be of the form

N+ YPei =

v3

where

V Ze= n,Y -r47reom I A

3(4rEok T)3122Ze3(7rN) 1/2

1858 December

(5)

(6)

Shkarofsky: Generalized A ppleton-Hartree Equation

where

k is Boltzmanin's constant,1 is the electron temperature,

N+ is the ionl density,Z is the ion charge number, andE0 is the permittivity of free space.

For the temperatures considered here, N4 =N and Z = 1.Thus

Vg = vm(v) + Vei(V) (7)Suppose that one can express vm as a power index of

velocity, not necessarily anl integer, viz.,

Vxr = CV'. (8)

In nmost cases, this can be done over the velocity rangewhere the largest contribution to the integral in (5) oc-curs. Also, if the imiagnitude of the electric field is suffi-cienitly simall, one call let fo° be a Maxwellian distribu-tioIl

J lO mN(2;T)312Env2/2 Tfo°0- N fT -^4 2T (9)

With these assumuptionls, substitutioni of (7) into (5)yields

(V0g) = (im) + (ei), (10)

gas to approximately unity, at (vrn)/(vei) between 1 and10, and then g and h deviate miore from unity as(Vm)/(Pei) decreases, unitil thev approach the limnits forthe completely ionized gas. The explanationi for thisvariation is the fact that vg, equal to (vm+vei), is ap-proximately constant5' 6 when (i'm) (iei) andi wheii P.varies as a positive power of speed. In other words, sinicevei decreases with electron speed as 1/v3 anid VP, increasesas vr, there exists a ratio, (i'm)/(iei), for which the sumof the decreasing and increasing velocity function re-sults in a more or less constant P., with respect to veloc-ity, and when this occurs, g and h are approximately 1.Consequently, g and h shift towards 1 anid theni deviatefrom 1 as the degree of ionization in the gas inicreases.

III. GENERALIZED EQUATION FOR

THE REFRACTIVE INDEX

From Maxwell's curl equation, we can write

AxH = [o + j.wEo]* E - j E (13)

where

H is the magnetic field vector, and(d and K are respectively the conductivity and dielec-

tric coefficient matrices.

If the elemnents of the dielectric tensor are denoted by

4(2Xr)12 Ze2 \2/kT 1/2(Pei) = - --N+ 47eoT

In A (11)(Vc) = -jE21

(

then one can write

(pm)= T(5 2 (2kTjr/2 E22 = 11 = 1- -1(12)

E12 = E21 = -t +

where r is a Gamma function.The (go, ho), (g±, h+) and (g_, h_) factors are respec-

tively functions of w/(v,), (W+&b)/(VI) and (co-b)/(Vg).They are also functions of (vim)/('vei), of Z, and depend on

the variation of vim with velocity. These functions are

tabulated in a recenit article by the author5 for Z= 1,power law variations of r = -3, -2 3, variousratios of (iv,)/(I(ev) and continuous (CwOb)/(iV,) values.In a slightly ionized gas, when (im)/(I'ei) becomes quitelarge, the values of g and h can be obtained from simpleanalysis, using the tabulations of the Dingle'0 functions.The g and h functions are nmonotonic; g increases from

an asymptotic value less than unity at small values of(W±W) /(fi,) to unity at large (W±_Wb)/(V',) values, andh decreases from a limilitinig value greater than unity tounity. If the power index r in (8) is less than or equal tozero, g and h also vary monotonically as (vim)/vi'ej) in-creases. When r is equal to 0, g and h are identicallyunity at large ratios of (vi')/(i'ei), that is for the slightlyionized gas. However, when r is greater than or equal to1, g and h vary from the values for the slightly ionized

-33 = - 'Y.

(16)

(17)

Note that the 4, i.', y coefficients are equal to (j/lwo)times the b, c and d coefficients represented in (2) to(4). For any (legree of ionization and anly variationi ofPm with velocity, i, ', y are related to the g and h funic-tions by

-wp2

A=

ey =

jiw x/2

(Pv0)g_ +J(w-ob)hl /_-jg z-lzhyicc x/2

(V5,)g+ + j(w + Wb)h± I+ -gjgz + 1,_y

jw

(vg)go + jwho ho -jgoz

(18)

(19)

(20)

where

cp= (Ne2/mEo)112 is the plasma frequency,(v,) is the average collision frequencv of electrons

with

and

jE12 0

E22 0

0 E133

(14)

(15)

1961 1859

PROCEEDINGS OF THE IRE

In terms of the (, 41 and y coefficients, this is

4~V 4~k - y!-tSa_ _tA sin2 0a a(y- 1)

a + 4 s-in 2 {[0-'-40 +a 2 12 } 1/22a (y -1) s 2a (y -1) J \ a J

with neutrals plus ions and is defined in (5) and(10), and1'

X - Up2/W2 y -= Ub/ Z = (VP)/l. (21)

The following equation results2,6 for n, the mag-nitude of the refractive index, from Maxwell's equa-tions, assuming a plane-wave solution of the formexp( -j(co/c)n * r).

where a = +. This expansion follows after consider-able algebraic manipulation from (28) and after2(A-B+C), (2A-B) and (B2-4AC)y have been di-vided by 2a(y-1).

Let us further manipulate the equations with thevalues of 4, i1 and y given in (18) to (20). Again, afterconsiderable algebra, one finds'2

X+ 0x(zol+je3_ _n__

1~~~~~~~~~~~~~~~~~~t-- --_zn2= 1 -y2g3+z205+jO4z+jx(zeljZ1 )\ [(2- y2g3+Z296+je4z-jx(z4 +j ) \2

* *(Z(1 O3 21/2j( 2(ho-x-jgoz) - sin 0± 2(ho-x-jgoz) r yCOSVJ

n4[Ell sin2 + E33 COS2 ]- n2[(6112 - E122) sin2

+ E11C33(t + COS2 0)] + E33(E112 - E122) = 0 (22)

where

tan2 = (n.2 + n,2)/n,2. (23)

That is, is the angle between the direction of propa-

gation and the dc magnetic field, assumed along the z

axis. Eq. (22) can be written as

An4-Bn2+ C = 0 (24)

where, according to (22) and (15) to (17), the coefficientsare given by

A E llSin' + C33 cOs2'

- 1 - Q +tb) sin2 'ycos26 (25)

B = (,l2 -E122) sin2 0 + ElIE33(l + COS2 6) (26)

= 2 -t(2 + sin2 60) -14(2 + sin2 6)

- y(1 - - if)(2 - sin2 6) + 40Vsin20

C = E33(6112--122)

- 12(t+ )+44+-y[1-2(t+0+40+]. (27)

The solution of the quadratic equation for the square

of the refractive index, is usually, in ionospherictheory, written with the square root in the denomina-tor. That is,

B VB2 _4ACn2_

2A

2(A-B + C)-A -B-x- C(28)

2A -B + \/B2-4AC

11 In ionospheric theory, the usual notation is capital X, Y, Zinstead of the lower-case letters adopted here.

where

x = wp2/W2 y (b/W

201 = 2go - g- + g+ 2)2 = g -g

z = (vg)lw

(31)(32)

203= 22ho - h+ - y(h+- h-)

204 = go(h_ + hk) + ho(g_ + g+) - 2g_h± - 2g1h-

+ y(-gohl + gohl - 2g-h1+ + 2g+h_)205 = gog-+ gog+ - 2g-g±

(33)

(34)

(35)

29, = hk + h_ + y(h+±-h) 222 = g+ + g_ (36)(37)

2y2,43 = ho(h_ + h+) - 2h1hJ + yho(h+- h-)

+ 2h1hzy2

2Yi44= h+ - h_ + y(h+ + h_)

(38)

(39)

and where

go, ho are the values of g and h for argument (0/(vq),g±, h+ for argument (W+WOb)/(V, andg-, h. for argument | X-Cob / (P) .

In (31) to (39), two sets of functions, 50 and 44 func-tions, are defined. The 0 ones are identically zero, andthe S's are 1, when the g and h parameters are all equalto 1 (which occurs for constant collision frequency in a

slightly ionized gas).The complicated equation can also be written as

n2 = I x(l + iS)

S1 - jZS2 -yLR(40)

12 Since the g and h parameters are defined to be corrections inthe denominators of (, Vk and -y, these correction factors combine read-ily in the following evaluation, and this fact justifies the originaldefinition.

2= 1 - (29)

(30)

1860 December

Shkarofsky: Generalized A ppleton-Hartree Equation1

where

-Y j12 /Z( )1 + Jo(3)S= -go

O is arbitrary). Obviously, ini this caIse go=-g+=g_=1,hn=h+=h =1, and consequiently

4Sl4 = 8(t + v/)

YL=Y COS 0 YT = y sin 0 _i - " i - - - mktI'

R = V-j;XS/ 2YL T | (g4-jz( 2/y)2+ (V +jxS/2YL)21112 V = YT2/[2yL(1 - X -jz)] R = V + Vl + V2. (46)

= YT{2L3 - (Z205 + j(4Zz)/y2]2yL(ho - X - jgOZ)

2= I -

(41) The equation for the square of the refralctive index be-collmes the Appletoin-Hartree equationi, ilamiiely:

x--T --(47)

1 j- 2(1 _ ±z [-[4( + YVLA21- x - jz) L4G - x jz)2 J

IV. PARTICULAR CASES

Let us now consider particular cases, when simplifi-cation results.

1) Propagation alonig the miiagnetic field, i.e., 0=0:One obtains two transverse cvclotroin waves, accord-

ing to the ± sign in the denlominator, given byx

n2)-- = I --2 =El 1- C12 (42)k+ -jg z±+ J±y

andx

n2 I1-- =1-20 ='ell +E12. (43)h_-jg-z-h-y

The wave with h_ and g_ corresponds to the electroncyclotron wave (sometimnes called the extraordinarywave for 6=0), and displays a resonance at y = 1 orw = Wb. The wave with h+ and g+ corresponds to the ioncyclotron wave (sometimes called the ordinary wave for6=0). The terminology of "ioni cyclotroln wave" isadopted, because this wave would also display a reso-nance at the ion cyclotron frequenicy (at w=eB/M+)if ion motions were included in the analysis.6

2) Propagationi perpendicular to the magnietic field,i.e., 6=90:One obtains the ordinary and extraordinary waves.

The ordinary wave results from choosing the minus signin the denominator,

4) High collision frequenicies, wheni ((P,))>>wCb and( (V!8 )) >>w:

In a recent article by the autilors it is showni thlatho = h+ = h- = h(0) and go-g- g±g(0), where h(0) andg(0) are the values of the fuLnctionis for zero argumllenit.Let us define an effective normalized collisioni frequenicvfor this case as

Zi = zg(O)//z(0) = (YV)g(O)jwfr(0)

and a reduced plasmla frequenicy x, as

xI = x/h(0) = Wp2/w2/1(0).

(48)

(49)

The simplification in the 0 anid Y funictions is given inTable I, Column 2. The refractive index satisfies theAppleton-Hartree equation (47), with xi and z, definiedin (48) and (49), replacinig x and Z. TIhe correction fac-tors, g(0) and h(0), depend oni the degree of ionizationand oni the variation with electroni speed of i'm, the elec-troll collision frequenicy with neutrals. For example,5 invery slightly ionized niitrogen, g(0) =0.6 and h(0) = 3.0,since in nitrogeni, v,M V2.

5) High RF angular frequencies, wheln

a) w »> ((v!)) and b) Wc>> b

or

a) w>>((v,0)) anid b) Wb>> I

xn2-1- = 1- y= -33

ho- jgoz(44)

and the extraordinary wave results fronm the plus sign inthe deniominator,

(1- 24(1 - 24) (El - E12)(E11 + Eu12)n2= (45)

3) Cojnstant collisionl frequency and a slightly ionizedgas:The classic Appleton-Hartree equation, usually ap-

plied to ionospheric xwork, results. Note the simplifica-tion for this case niaimely that the angular term in thenumerator is zero and so are several factors in the de-nominator. (Here, and in the following cases, the angle

01'

a) (v,6)0 and b) COa4b.As discussed in Section II, all the g and h functioins

approach 1, and hence, the usual Appleton-Hartreeequationi (47) results, with z= (^,,)/A. Note, that the col-lisioin frequency to be applied is that defined in (5), andresults quoted in the literature m11ust be checked to de-term-line whether they refer to the samie average.

6) High imiagnietic fields, satisfying

a) COb >> ((Vg)) b) wb >>C.

In this case, h+ = h-=g+ = g-= 1, but go and h0 dependo01 w01(v,). If onie also has >>((v,)), case 5) results. If on1the other hand, co<<((v,)), then go-g(O) and ho = h(0).

18611961

A = n cf = .£ =7) = Ui /t.1

862 PROCEEDINGS O0/ THE IRE Deccmb

T'ABBLE I

SIMPLIFICATION OF THE ( AND g FUNCTIONS IN THE GENERAL/L1D) AITPLETON-HARTREE EQUATION FOR PARIICULAR CASES (SEE TEXT)

a) Case 3v7, constant aIld nieuitral

dominiatedb) Case 5

a»( (vs ))and c@>Noo

c) Case 5»>W((v))

and wK<<Wbd) Case 5

and wAgse) Case 8

VrJ = Vm8 +Veic'onstanIt

0

0

y7,22YL( I-X-jz)

Function

go

g+hohh+(-)4

I 5

59I

1(,42

S

g4

iV

Case 4

( (v))>>wlaiid((vPg)»>>

g(o)g(o)g(O)

h(O)h(030

0

()

0

0

,(O,g(o)

fh(0)12

h(O)0 .

[h(0)]2y7,22yL(h(O) -x -jg(O)z)

Case 6b»>>&Janid

1b>>( (^g)). ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i_

g,, i-go

IIt

go I0

ho-1

g(, ho-2

go+1oII

Case 75, = Iannd

((V, )<<W = Cb

1g(O)

h(O,'

[1 -g(0)J/2

03f1 g(1O)J 2

3[l -g(o)j/2

1 -g(o)j/2

[I+g(0)112I-(-)/y2--1I+(hncey->

siiice y>> I

_I

/ z(go-I ) +j(h( -1I_ JgoZ ) Sill2

ho-x-jgo,

fh0-1 -jz(go-1)1(1 -jz) sin52

2(ho-x -jg,)

(2

4,[1 -g(O)f sill, 0

2(1 -.x J-)

-jzfl -g(O)j(3 -j.'\2(1 x j . Siz12

2(1 -x-jz)

The general equation for the refractive inidex re-duces to

2= 1-.x.-- x [-jz(go-1)+(ho-1) ] sin' 0/(ho-x-jg, -)

1 -jz-j(Y.+---z+x-[ko1IZ(go1)I sin2 0+[-(ga-J)] 2

sin4 0±y2 Cos21/2

2(ho-x-jgoz) 22(koo-x-jgoz)

_ (501)

7) Cvclotron resonance (y=1) with wco=co>>((g)):We now have g_-g(O), h-=h(O) and go=g+==1,

ho = h+=1, y = 1. The modified Appleton-Hartree equa-tion is (see Column 4 of Table I)

x+jxz(l-g(O)) sin'2 /t2(1-jz-x)J- (1+g(0)) (jz(1-g(O))(3-jz+x)-2 sin'0±a (jzj-g(O))(3-jz-x)-2\2 I

(I-g(O)) \2 1c2o1-jz __+ kt 4(1-x-jz) / L\-4(1 --j - )smn4 0+ l\1 -J4 x2

Note that the results, exactly at resonance, are inde-pendent of h(O).

8) Constant total collision frequency:We consider the situation when over a reasonable

velocitv range, v0, given by (7),

47rA+ / ze2 \2v1 = 'm(V) + vei(v) = VV(1)+)-3In A, (52)

V3 47rf0M/

is constant with respect to velocity. If vl'r varies as V2,

then P, is nearly constant5 when (Pm)/(va) is between2.25 and 3.7, since in this range 0.92 <g < 1 and1<h<1.038. If Pm varies as v, then 0.96<g<1 and1<h<1.035, when (Pm)/('ei) is between 3 and 15. As-suming that the error in taking g and h equal to 1 issmall, the relationship for n reduces to the Appleton-Hartree equation (47), with z denoting v,/&.

I r

Function

go, +

ho, +

01,23.4. 5

41,2,3,4

s

v

2= 1 - (5I)

Shkarofsky: Generalized l ppleton-I-lartree Equation

V. Appi.1( ATION To THE IONOSPHERE

In general, oine can plot fromii the calculated g and hcurves, the (i anid 4 funictioIns vs (0)/Cw for variousWb/w anid (vi'm/('ei). The investigation that follows is forvm proportional to tl. 'IThis applies up to about- 120 kin inthe ioniosphere, where electron collisions with the nitro-gen species are predomuinant over collisionis witlh theoxygenl atonms. The usual assumption that the atmiios-phere is dry (conitains no water vapor) is niiade throtugh-out this discussioni. Typical graphs are showni in Figs.1-9 (pp. 1864-1868) for degrees of ionization correspond-inig to 1) a weakly ionized gas (Ki'm)/(i'ei+ x), 2) equLalfrequencies of electron collisions with neutrals and ionls~(i(vn)/ 1), and 3) anl ion-dominated l)hlsma((V,,,)i(V,W)0). Values of Wb/wequal to 104 or 101, 102, 10,1 or 0.999, 10-', 10-2, 10-4 are used.Note the following characteristics in the figures:1) For large (v,)/w, the () curves approach zero and

the g curves approach the values given in Columnli 2 ofTable I. Also, the lower the value of WbIco, the quickcerthe curve attains its limilit.

2) For negligible (v,,)/co and W0£Wb, the (0) anld 4curves are respectively equal to 0 anid 1, which shouldbe so according to Column I of Table 1. At cvclotronresonance, the curves assumiie the values tabulated inthe last columnll of Table l. Note that if Wb /-0.999,deviations towards zero or 1 occur for extremely small(Vg )/C0.

3) In the limit of Wb>>((iPq))>>N, which, for examiiple,occurs when Wb/W = 104 and 10< ((v,)/w) <103, somiecurves flatten out anid tend towards the values given inColumln 3 of Table 1. If co/O is lower than 10), theNynever actually attain these limllits.

4) The largest deviations are presenlt in the curvesfor the slightly ioniized gas, and the least in the partiallyionized gas, (iv)(vet). As the ionizationi increases, the4 curves shift towards 1 anid the (0) curves towards zero,and then they deviate more wheni (ivnt)(v,j becomiiesless than 3. Finally, the limiiiting valuies for a coIml-pletely ioniized gas are attained.The graphs show that the approaclh to the limiiitiiig

values i's slow in milost cases, and henice the tuse of equiva-lent collision frequencies in the Appletoni-Hartree equa-tion milust be justified by inivestigatinig curves such asillustrated here.The curves in Figs. 1-9 are nlot sufficienitly accurate to

calculate the refractive index. As anl example of the dif-ficulty, conisider the miiagnlitude of the (-y2 3+Z2 ()b)combination in the generalized Appletoni-Hartree equa-tion (30). In imaany cases, one caninot obtain reasonablevalues, simlply because of the inaccuraey in the 93 anid(0\ funictionis. One canl conclude that the 0 and 4 func-tionls are only useful for illustratinig the physical factthat large deviatiolns fromii zero and uniity canl occur,and for providing regions where their replacemlenit bylimiting constants canl be justified.

Values for the electron collision frequenicy with nitro-gen miiolecules will niow be given, anid applied to the Dlayer of the ionosphere, or in genieral to heights below85 kmii. Basically, the formiiula used for nitrogen is ideni-tical to that given by Huxley.'3'4 The cross section forniitrogen1, QN., is1

QN = 3.29 X 10- 23 cm2 (v in cm sec)ani(l 0.8 timiies this value for air.

Q, ir = 2.63 X 10-23V Cm2.

(53)

(54)If Nm, is the niumiiber denisity of air mi-olecuiles, the colli-sionl fre(qLuenlcy is

pill - V,1IZQ = 2.63 X 10 23v2(p/po)Lo, (55)

w-lherep is the denisity,po is the density at stanidard temlperature atnd pres-

suire, an(lLo is Loschm1idlth 's n umberLo=p,/k7'o0 2.687 X 10I9 molecules /atm.Cn3.

Hence for air

Vitt= 7.073 X 10-4(p/pO)V2(7v iln cn/c sec)

= 7.073(p p/o)v2(V in m,/sec). (56)The high-frequencyv average is calculated from (12) to beequal to

(i',,m) = 5.359Tp/po X 10X= 1.9944N,AT X 10-17(N,,in 1/r3). (57)

TIhree dlifferent averige collisionl freqluenicies are cur-rently in use in the literature, inamelNy (zv,l), P'n and Vp, (le-fined, for a variationi of P,, x v

respectively by (i ).. (1 /z,'2)d7dv(z 3V,,1/3)=5v.13

i'm V'm

anid V'-0.4(r',,). A bar (leniotes the following averagefor a scalar quantity f

fr 4(1(/NW) fJ ~fo0)4ir27"./

13 L. G. H. Hnxley, "A discussion of the iilotion in nitrogen offree electrons with small energies with reference to the ionosphere,"J. Atliii. and Terr. Phys., vol. 16, pp. 46-58; October, 1959.

14 1. P. Shkarofsky, et al., "Collision freqtuenicy associated withhigh teim3perature air and scattering cross-sections of the constitu-ents," in "'Electromagnetic Elffects of Re-enitry,' Piergamon Press,New York, N. Y.; 1961.

15 Phelps in a recenit atrticle4 uses values for initrogeni which are15 per cent larger thani these. In a footniote, he quiotes in3easture-mients, which should read, "are 10-12 per cenit smaller" thani givenin his paper, so that his new values are 3-5 per cent larger thani giveiihere. For conisistency, the anialysis here is based exclusively oni thecross sectioiis plotted in Shkarofsky.14 Also, for simiplicity, the con-tributioii fronm oxygeni is neglected, since the additionali. error intro-duced is less thani 15 per cenit.

1961 1863

PROCEEDINGS OF THE IRE

,," >w4

I~

1 92

Fig. 1-The function 03. The numbers beside the curves are values Fig. 2-The function 02. The numbers beside the curves are valuesOf Wb/W. Of cob/c,.The seti/ corresponds to (VP)/(VPi) = °O The set corresponds to (v,)/(v,-'oo

* * -(v-)/(v.)- - _(-m)/(va)=O

1864 December

I

<"R>w

T I

1961

I

Shkarofsky: Generalized A ppleton-Hartree Equation

-o,3

1865

-4p B

Fig. 3-The functioni 03. rhe numbers beside the curves are values Fig. 4-The functioni 34. The nllllbers beside the curves are valuesof wOb/W. of (c.b/IW-The set correspouds to (v. )/ (Vei ) = .o The set corresponds to (v,. )/ (v,i) = XTake 03a= 0 for (v.)/(vi,)=I -(-v-V (V.i)=I_ - - (V,.N)/1(v )=0 - - - (VN(V,i )

PROCEEDINGS OF THE IRE

t

*5 >f,Fig. 5-The function (-)6. The numbers beside the curves are values Fig. 6-The function 41. The numbers beside the curves are values

of wb/&J. of wb/w.The set corresponds to (vP, /(va,) = X The set corresponds to (vP.)/(vi)= X

Take 9, =1 for (v,p)/(v)= I_ _ - (v.,,)/ (Vi) = ° - - - (V.)(P'f)=O

1866 December

Shkarofsky: Generalized A ppleton-Hartree Equation

<a> .-I_

I de

0 2 4 6 8 10 12 14

lof0I2

Fig. 7-The function 92. The numbers beside the curves are values Fig. 8-The functioni g3. TIhe numbers beside the curves are valuesof COb/CO. of WI,/W.The set corresponds to (., )/(vPi) = - The set corresponds to (v'...)/(v.i)= -

( -V(vel)/<)=1 Take 93=1 for (.VN)/(V)=I-- (V.NI) (VCX) ° (Vl... (V¢E) O0

18671961

PROCEEDINGS OF THE IRE

cules by Huxley,'" by Phelps, et al.,'7 and by Phelps andPack,' using different respective methods, all practicallycoincide numerically.An attempt is now made to plot for the ionosphere

the quantities (v,,) and (vm)/(P,i) vs altitude up to 340km, which is above the F2 layer. One is immediatelyfaced with three unknowns.

1) The electron density profile depends on local time,season, solar activity and latitude.

2) Although nitrogen is the basic constituent of airbelow 90 km, electron collisions with oxygen atoms be-come important at higher altitudes and comparable tothat with nitrogen molecules in the F2 layer. The exactrelative proportion of nitrogen molecules and oxygenatoms vs height is unknown.

3) The electron temperature is unknown; it may bequite different, by as much as a factor of 5, than theteniperature of the neutral particles. Furthermore, thevelocity distribution of the electrons may also be non-Maxwellian.For discussion, one can assume an average daytime

profile'6"18'19 of electron density, shown in Fig. 10. Aplasma frequency scale is also shown. We will also adoptthe 1959 ARDC model of the atmosphere,20 which pro-vides values for the molecular weight, as well as for thegas temperature Te and neutral particle density NM.Electron collisions with neutral oxygen molecules areneglected as being small compared with collisions withnitrogen molecules.

Below 90 km, effects due to oxygen atoms20 can alsobe neglected, and (57) applies for (Pm) in air.Above 180 km, oxygen is assumed to be completely

dissociated, and air can be considered to consist onlyof N2 and 0. Then the relative number densities aregiven in terms of the molecular weight of air

T.1UwIV

0i 05 10 150 20 5 50 35

4

Fig. 9-The function 94. The numbers beside the curves are valuesof (lb/CO.The set corresponds to (V.)/(vi)=Take 94 =1 for (vP.)/(v,)= 1

---(P.)/4'.0 0

Many discrepancies between quoted values, such asthose of Huxley,"3 who uses the second average, andthose of Phelps and Pack'4 and Seddon,"6 who adoptthe third average, vanish, if this point is remembered.Here, the first average (P-) is adopted. As a result, thevalues of the collision frequency (P-) here are 2.5 timeslarger than those plotted by Phelps, Pack and Seddon.Another point to note'4 is that the measurements of thecollision cross section of electrons with nitrogen mole-

16 J. C. Seddon, 'Summary of Rocket and Sate lite ObservationsRelated to the Ionosphere," Natl. Aeronautics and Space Admin.,Goddard Space Flight Ctr., Washington, D. C., NASA Note D-667;1961.

M = (28.016NN2 + 16N0)/N, (58)by

NN, = NM(M - 16)/12.016 and NO = NM - N2 (59)

where NN, is the number density of nitrogen, whosemolecular weight is 28.016, NO is the number density ofoxygen atoms (molecular weight= 16), and NM is thenumber density of air (molecular weight=M). Fromthe tabulated data20 for NM and M, the quantities NN,and NO can be obtained. Let (PN,) and (vo) be the aver-aged collision frequencies for N2 and 0. Then we know(see Section II) that the averages can be added to yield

17 A. V. Phelps, et al., 'Microwave determination of the probabil-ity of collision of slow electrons in gases," Phys. Rev., vol. 84, pp.559-562; November 1, 1951.

18 See "The Ionosphere"-IGY issue of PRoc. IRE, vol. 47, pp.167, 169, 273, 281, 292; February, 1959.

19,1. A. Ratcliffe, Ed., 'Physics of the Upper Atmosphere,"Academic Press, New York, N. Y., pp. 409-411, 442-445; 1960.

21 R. A. Minzner, et al., 'The ARDC Model Atmosphere, 1959,"A.F. Cambridge Research Ctr., Cambridge, Mass., Rep. AFCRC-TR-59-267; August, 1959.

1868 December

Shkarofsky: Generalized Appleton-Hartree Equation

108

to7

z

A

VK)

N5 _CVm>_ T r

Z5 _ _ ____ <e> ____ ___ ____2 0

~log ____:104

5 _ __.,2 O_ T_ -T-C;ICo235

_ _ _ T=3T __

1040 50 100 150 200 250 300 340

- -HEIGHT IN <KM

Fig. 10-The ratio (vP)/()v,j) and the total collision frequency (v,%,vs height for an assumed average daytime profile of electrondensity (N) or plasma frequency (c,w,), if the electron and neutralparticle temperatures are equal. The range in value of the cyclo-tron frequency (W,)O is also shown. The arrows indicate possibledeviations if the electron temperature is larger than the neutralparticle temperature by the amount shown.

the average collision frequency with all neutrals, (Pm),viz.,

(Vm) - (VN,) + (VO) (60)

The value of (PN,) is (1/0.8) times that in (57), since(PM) was assumed there to be 80 per cent that of (PN,).Thus,

(VN2) = 2.492NN2T X 10-17 (61)This equation applies if the electron temperature is lessthan about 1500°K, which would be true20 up to 400 km,if the electron and gas temperatures are equal (TcTJ).For higher temperatures, (VN2) does not vary as T,sinCe'4 I'N, is not proportional to V2. It is known" thatthe cross section of electron collisions with oxygenatoms, based on the analyses of Klein and Brueck-ner, is nearly constant with speed and equal to 10-19 M2.The analyses of Mitra, Yamanouchi, Robinsoni, Ham-merling, et al., and Lamb and Lin, which predict largecross sections at low electron energies, have beenshown'4 to be questionable. A low cross section was alsofound experimnentally by Lin and Kivel.22 We therefore

21 M. M. Klein and K. A. Brueckner, "Interaction of slow elec-trons with atomic oxygen and atomic nitrogen," Phys. Rev., vol. 111,pp. 1115-1120; August, 1958.

22S. C. Lin and B. Kivel, "Slow-electron scattering by atomicoxygen," Phys. Rev., vol. 114, pp. 1026-1027; May 15, 1959.

write

Vo--Nov X 10-19

and averaging according to (12) yields

(vo) = 0.8282 X 10-5'T"12N0.

(62)

(63)

Adding, (60) becomes (for T< 1500'K)

(nm)= 0.8282 X 10-"NoT"l2 + 2.493 X 10-I7NN,T. (64)

For reference, the value of vm before averaging is, ac-cording to (55) and (62), given

vm- Nov X 10-19 + NN,v23.29X 10-25(V in m/s). (65)

Between 90 km and 180 km, the relative proportionsof N2, 02 and 0 are quite uncertain.2' However, it is easyto shiow, from the values20 of the molecular weight of airas a function of height and from (59), that up to 180 km,the number density of atomic oxygen is certainly notgreater than 10 per cent that of nitrogen. Further evi-dence that this is so is given by Johnson.24 Assuminigthat for lower altitudes NN, . 10 No, one cain neglectthe contribution of atomic oxygen given by (63) to theaveraged collision frequency and extend the range of(57) up to 180 km, as a first approximation. The error isless than 12.5 per cent.Most of the uncertainty in the calculationi arises from

a lack of knowledge of the electron temperature andelectron distribution function. The results, to be showni,are based on the ideal assumptions of a Maxwellian dis-tribution and equal electron and gas temperatures.Some calculations based on higher electron tempera-tures are also shown. The electron and ion densities areassumed equal, and negative ions are neglected. Ioniswith charge numbers greater than one are assumed tobe insignificant.

In Fig. 10, (V,), the sum of electron collisions with allneutrals plus ions ((v.)+(vei)) is also plotted where (nm)is given by (57) up to 180 km and by (64) above 180 km,and where (i'ei) is given by (11). In MKS units,

(Pe) = 8.375 NIlog A X 10-6/T3/2

andA = 1.24 X 107(T3/N) "/2. (66)

The ratio ('m)/I(vei) is also given vs height. Both are basedon the assumption that the electron and gas tempera-tures are equal. No attempt is made to calculate theconductivity functions g and h, mainly because of theuncertainties in the atomic oxygen concentration andelectron temperature, and also because it is shownbelow that there are regions where g and h are 1.

Several calculations obtained by assuming a higherelectron temperature are shown by arrows in Fig. 10.

23 Ratcliffe, op. cit., p. 34.24 C. Y. Johnson, "Aeronomic Parameters from Mass Spectrome-

try," presented at the Symposium on Aeronomy, Copenhagen, Den-mark; July, 1960.

1961 1869

PROCEEDINGS OF THE IRE

One point is based on measurements with Sputnik III.Krassovsky2'5 reports an effective electron temperatureof about 7000°K at 242 km, whereas the ARDC value20for the air tenmperature is 1415'K. As a result of T 5 TQ,the value of (vm)/KVe-i) is raised and (v,) is decreased atthis height. The decrease in (v,) is due to the smallervalue of (v,j) which decreases faster with temperaturethan (vm) increases. At heights below 200 km, the ratio(vm)/(vei) would also be larger, but (v,) would be in-creased rather than decreased. This is because (vm) isnow the main term in (v,) and it increases with tempera-ture. At 200 km, there is only a small change in (v,),first a (lecrease as T-÷3 Tq and then an increase asT--57,. This result is obtained by plotting vPmP-N2+vovs velocity up to 1 ev, using the collision cross-sectionidata in Shkarofsky, et al.,'4 and using the values NN2-7.220 X 1015 and No = 1. 186 X 101" deduced from Minz-ner20 for 200 kimn. Note that the exact cross-section datafor nitrogeni is required for these higher temperaturesand that PN, does not vary as v2. One finds approxi-mately that IJm obeys the following power law,

Vm = 1.265 X 10-IvI.'224 (67)

so that

(i'm) = 1.965T6612 (68)

The sum of (vm) anid (Oei) varies little with temperaturefor 200 km.The magnetic field of the earth varies between 0.365

and 0.575 gauss, depending on latitude and height. Thecyclotron frequency Cb= eB/m is then between 6.42 X 106and 1.01 X 107, and this is indicated in Fig. 10. We thussee that the largest corrections in the conductivity occur

in the D layer (at about 70 km), since b-b(VP). Above110 km, in the E and F layers, Wb.(102(Kv)), evein ifT-5T, or (v,) is five times that shown at 110 km. Also,the plasma frequency cop= (Ne2/mEO)"12 is indicated inthe figure, and obviously, above 110 km w, is muchgreater than (v,9). Thus both x =c,2/Co2 and y= Wbl/ are

much greater than z = (v0)/w. Let us look at Table I. Atcyclotron resonance, the last column (case 7) applies.Otherwise, depending on whether fWb>>W or CbK<<Kw, thethird (case 6) or first column (case 5) can be used. Notethat provided CLb iS not too niear to o, all the g futictioniscan be takeni as one; the (0 functions, however, do notbehave as nicely. In the genieral equationi (30), one can

neglect (k)5 Si nlCe y2 g3>>z2(_),.In the region where >b.(6X 103(v,)), which occurs in

the ionosphere above 150 kml- if T= T0, and above 190kmn if T=57I',,, inivestigationi of the .f and curves, Figs.1-9, indicates that it is safe to use the ordinary Apple-ton-Hartree equation with (P,) as the collisioni fre-quency, provided the RF angular frequency co satisfiesC _b/W< 0.9 or 100 < (wOb/w)< 2. The limit Cob.6 X O13(io),

25 V. I. Krassovsky, "Exploration of the upper atmosphere withthe help of the third soviet sputnik," PROC. IRE, vol. 47, pp. 289-296; February, 1959.

nmay in fact be less, since as (67) indicates i'm varies as apower much less than 2, whereas the 1 and 0 functionsare for v c v2. Thus in the F2 layer, except at cyclotronresoniance, it should always be possible to let the g andh correction fuinctioins be 1, and the ordinary Appleton-Hartree equation should be applicable.The regioin where v, is coinstant with respect to speed

depeinds on (v',)/(v' ). This ratio is quite uncertain be-cause T is un-knowni. Assuming T- T0, the regionstarts in the E layer and extends to 150 km. If T is fivetimes greater than T,,, theni i, is constant in the F2layer. (See arrow in Fig. 10 anid limits for constant v,,if pi' a v2 or v.. v, given in Section IV, case 8.) This factprovides aniother justification for letting g anid h be 1 inthe F2 layer. In any case, there exists a region where niocorrection need be applied to the Appletoni-Hartreeequation, as discussed in case 8. It also seems from thesingle poinit of Krassovsky2' that onie has to go to alti-tudes above the F2 layer before electron-ion collisioniswill dominate neutral collisions, siince, above 340 kim,the neutral particle density decreased miiore rapidly thainthe electron density.26

VI. COMPARISON WITH EXPERIMENTS

The values of (v,) are compared in Fig. 11 with ex-perimental data of Kane,27 Schlapp,28 Whitehead29 andAtaev.Y' Kane's data, plotted here, are those reevalu-ated"'3 with the proper v2 relation for the collision fre-quency. Recall that the (v,) defined here is 2.5 timesvp, the collision frequency used by Phelps.' The valuesof (v,) agree very favorably with Kanie's experimentalresults for the D layer. Ratcliffe" also finds good agree-ment with other experimental data. However, above 90kni, Ratcliffe, Schlapp and Whitehead iiote that thetheoretical valties for (i') are appreciably smaller thanithose obtained experimentally. Ratcliffe and Schlappsuggest that this miiay be due to atomiiic oxygen, whichthey neglect theoretically. In these calculations, how-ever, it is found that the influence of atomic oxygen issmall as far as values of (z,) are conicerned, provided thethree assumlptions in the analysis are true, niamnely:

1) atomic oxygen is less thanl 1/10 the niitrogen con-centration below 180 kni,

26 Y. L. Albert, "State of the outer ionosphere," Priroda, vol. 6,pp. 85-87; 1958. Translated by E. R. Hope, DRB Canada, nio.T304R; September, 1958.

27 J. A. Kane, "Arctic measurements of electroni collision frequeln-cies in the D-region of the ionosphere," J. Geophys. Res., vol. 64,pp. 133-139; February, 1960.

28 D. M. Schlapp, "Some measurements of collision frequency inthe E-region of the ioniosphere," J. Atmos. Terr. Phys., vol. 16, pp.340-343; November, 1959.

29 J. D. Whitehead, "The absorption of short radio waves in theD-, E- and F-regions of the ionosphere," J. Atmos. Terr. Phys., vol.16, pp. 283-290; November, 1959.

30 0. M. Ataev, "The determination of the number of collisionsin the ionosphere," Radio Engineering and Electronics, (Translationof Radiotekhnika i elektronika by Pergamon Press, New York, N. Y.),vol. 4, no. 9, pp. 37-45; 1959.

31 J. A. Kane, "Re-evaluation of ionospheric electron densitiesand collision frequencies derived from rocket measurements of refrac-tive index and attenuation," Natl. Aeronautics and Space Admin.,Goddard Space Flight Ctr., Washington, D. C., NASA TN- D-503;November, 1960.

1870 December

Shkarofsky: Generalized Appleton-Hartree Equation

100 150 200

,HEIGHT IN KM

Fig. 11-Comparison of experimental data with the theoretical totalcollision frequency (v) vs height, replotted from Fig. 10, whenT= T, (solid curve) and T is greater than T, (dashed curve).The experimental points indicated by vertical lines are those ofKane, by arrows are those of Ataev, by crosses are those ofSchlapp and by squares are those of Whitehead.

2) the collision cross section of atomic oxygen is notmuch greater than that given by Klein andBrueckner2",

3) the electron and gas temperatures are equal.

Suppose that the electron temperature is greater thanthe gas temperature above 110 km. Note that at 110km the gas temperature starts to inicrease rapidly fromabout 287°K and reaches 1031°K at 150 km, afterwhich it increases at a slower rate to 1404°K at 200 km.Assume that the electron temperature is much greaterat the high temperatures. In Fig. 12, a possible variationis shown, which includes the electron temperatures of7000°K at 242 kin and 15,000°K at 795 kni, givein byKrassovsky.25 The latter point has been reevaluated byWhipple32 to be near 9000°K. If, in the E layer, the elec-tron temperature is about three times the gas temiipera-ture (see Fig. 11), the theoretical electron collision fre-quencies, based on the higher temperature, agree withthe experiinental values of Schlapp and Ataev in the Elayer. Boggess, et al., report rocket data33 of electronitemperatures in the E layer, which indeed indicate a

32 E. C. Whipple, Jr., "Ion-trap results in 'Exploration of theupper atmosphere with the help of the third Soviet Sputnik,' " PROC.IRE (Correspondence), vol. 47, pp. 2023-2025; November, 1959.

33 R. L. Boggess, et al., "Langmuir probe measurements in theionosphere," J. Geophys. Res., vol. 64, pp. 1627-1630; October,1959.

Lli

0 100 200 300 400 500 600 700 80oi-HEIGHT IN KM

Fig. 12-The temperature of air TF vs height according to the 1959ARDC model, and the assumed electron temperature T (dashedcurve) vs height. The points at 242 and 795 km are those fromKrassovsky. The arrow at 795 km indicates Whipple's interpre-tation of the Russian measurement, initially deduced to be15,000°K. The crosses indicate values given by Boggess et al.The point 0D at 120 km is deduced from Schlapp's and Ataev'svalues of (v,), shown in Fig. 11.

high electron temperature (see Fig. 12), but also a sur-prising minimum at 175 km. Recent measurements byBoggess34 indicate that the electron temperature rises toa maximum of 2800°K and becomes isothermal near theF2 layer (no decline in temperature observed). White-head's result for 190 km still remains inexplicable, since,as discussed above, a large change in electron tempera-ture produces a small change in collision frequency at200 km. Theoretically, the new curve for the collisionfrequency, shown dashed in Fig. 11, is above the originalcurve for heights less than 200 km and below the originalcurve for heights greater than 200 km. Ataev's resultfor the F2 layer seems to favor the originial curve basedon T= T. However, ineasurements of collision fre-quency in the F2 layer are widely scattered30 in valueand inconclusive."9 More accurate and systematicmeasurements, possibly sounding by satellites fromabove the ionosphere, will check the hypothesis thatT> T, in the F2 layer, since the hypothesis predictssmaller collision frequency values than given by lettingT- T9. This effect is reversed in the E layer wherehigher temperatures give higher collision frequencies,and, indeed, Schlapp and Ataev find the collision fre-quency to be higher than expected. One may also sug-gest that if the hypothesis proves to be correct a meas-urement of electron collision frequency in the ionospheremnay be a sensitive method for determining the electrontemperature, especially in the E layer.5

ACKNOWLEDGEMENT

It is a pleasure to thank Dr. T. W. Johnston and Dr.M. P. Bachynski for stimulating and fruitful discussions.

3 R. L. Boggess, private communication; March 13, 1961.35 A similar suggestion has been made by M. Nicolet, "Collision

frequency of electrons in the terrestrial atmosphere," Phys. Fluids,vol. 2, pp. 95-99; March-April, 1959, in connection with the electron-ion collision frequency in the F2 layer.

1961 1871

y40