Helium ions in the upper atmosphere

Planet.

Space Sci.. 1962, Vol. 9, pp. 599 to 605.

HELIUM Department

Pergamon

Press Ltd.

Printed

in Northern

Ireland

IONS IN THE UPPER ATMOSPHERE

D. R. BATES and T. N. L. PATTERSON* of Applied Mathematics, The Queen’s University, Belfast, N. Ireland 27 June 1962)

(Received

Abstract-Reasons are given for believing that ion-atom interchange between He+ and Nz is strongly endothermic but that ion-atom interchange between He+ and 0, is exothermic. It is concluded that the former ion-atom interchange process does not occur to an appreciable extent but that the latter may be responsible for the removal of the He+ being produced by photoionization. Calculations are carried out on the steady state. In harmony with the prediction of Nicolet He+ is found to be abundant at great altitudes even if a high rate coefficient is assigned to the collision process leading to its removal. Attention is drawn to the possibility that radiative dissociation of excited HeOf is important in connection with the problem of the escape of helium into interplanetary space. 1. INTRODUCTION

Nicoletu) has recently shown that the slow decrease of atmospheric density with altitude between 750 km and 1500 km indicates that helium is a major constituent in the region concerned. Noting that the rate of production of He+ ions at altitudes above 500 km by the action of solar radiation is about one hundredth the corresponding rate of production of Of ions, he suggested that the number density n(He+) is about lo3 cm-3 at the level where the number density n(O+) is lo5 cm-3 (which he took to be at 500 km). Recalling that n(He+) is initially an increasing function of the altitude, he variously estimated that it may rise to 2.5 x 103 cm-3 or lo4 cm-3 at 1000 km. Experimental evidence supporting Nicolet’s striking prediction that n(He+) is very much higher than was assumed in the past has been reported by Bourdeau, Whipple, Donley and Bauerc2). This paper is concerned with the steady state of an He+ ion layer. 2. MODEL

ATMOSPHERE

The calculations to be described are not sufficiently precise to justify the use of refined models of the upper thermosphere and we were content to take the number densities of the systems of interest to be as given by the following simple expressions: n(z 1N2) n(z 10,) n(z 10) n(z 1He)

= = = =

7.3 3.9 3.1 2.3

x

lo6 lo5 X 10’ x lo6 x

exp exp exp exp

[-2.0 [-2.3 [-1.1 [-0.3

x

10-2(z 10-2(.z X 10-2(2 x 10P2(z x

- 500)] cm-3, - 500)] cm-3, - 500)] cm-3, - 500)] cm--3,

‘(1)

and n(2 10+) = 1.0 X lo5 exp [-0.5 z being the altitude lo-l5 g cm-3.

in kilometres. 3. RATES

OF

X

10P2(2 -

The corresponding GAIN

AND

LOSS

500)] cm-3,

mass density

at 500 km is 1.2 x

OF He+ IONS

According to Hinteregger (3) the rate of photoionization of a helium atom at the top of the atmosphere is 1 x 10P7 set-r during the sunlit period. Accepting this and allowing for the dark period and for the absorption of the solar radiation by the main atmospheric * Now at the Graduate Research Centre of the Southwest, Dallas, 5, Texas. 1

599

600

D. R. BATES and T. N. L. PATTERSON

constituents it is found that the average value of the rate at altitude I is approximately ^/ = 5 X lop8 exp f-S.4

X 10-%(~ 1N2)- l-1

x

10-lotz(z 1 0)] set-1.

(2) Nicolet has tentatively suggested that the He+ ions may, in some unspecified way, escape and indeed that this escape may even control the amount of helium in the atmosphere. We shall here confine ourselves to the possibility that the He+ ions diffuse downwards and are destroyed by the collision processes. Hanson’@ has suggested that the most likely mechanism is ion-atom interchange. He+ + N2 --)rHeN+ + N or

(3)

Hef+O,-+HeO++O

(4) followed by dissociative recombination. Since molecular nitrogen is more abundant than molecular oxygen Hanson favoured (3) rather than (4) as being the ion-atom interchange process operative. In order that the rate coefficient of either process should be su~~iently high, the energy released, which will be denoted by AE with the equation number as an identifying subscript, must be positive or, if negative, must only be of the order of thermal energy. If D(XY) represents the dissociation energy of the molecule XY then and

AE, = B(HeN+) - D(N,)

(5)

AE4 = 1)(HeO+) - D(0,).

(6)

It is now convenient to introduce 6, = D(HeN+) - B(HN)

(7)

8, = D(HeO+) - @HO),

(8)

so that (5) may be written

AE, = D(HN) - I)(N,) + 6, = -6.1 eV + S,,

(9)

and (6) may be written

AE, = D(H0) - D(0,) -t_ d, = -0.7 ev + 6,.

(10)

and

In view of the isoelectronic nature of HeN*and HN and of HeOC and HO, and in view of the pol~ization potential arising in the case of an ion it is apparent that 6, and 6, are rather small and positive. Hence process (3) is almost certainly so endothermic that it may be neglected but process (4) is probably exothermic. Use of the formula of Gioumousis and Stevensonc5) yields that the rate coefficient describing close encounters between He+ ions and 0, molecules is about 1 X I Opscm3 se&. Beiause of the simple nature of the reactants the rate coefficient K for (4) would be expected to be quite close to this upper limit to its value. However, the factors on which ion-atom interchange depends are still not fully understood and it is possible that K is in fact much smaller. Dissociative recombination is not responsible for the destruction of the HeO+ ions (cf. @). Little can be said about charge transfer to N, and 0, molecules because of lack of information on the relevant attractive and repulsive potential energy surfaces. A high rate coefficient is unlikely unless the change in electronic energy along the primary reaction path is small. The appearance potential of N+ ions produced by electron impact suggests that dissociative charge transfer collisions between He+ ions and N2 molecules are in close energy balance. Since 0 atoms are relatively abundant in the region concerned special interest is attached

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to charge transfer to them. The energy released is so great that most of the charge transfer collisions occurring must be radiative Hef+O+He+Of+hv. Quanta1 calculations

(11)

on a process of this type, He2f + H -+ He+ + Hf + hv

have been carried out by Arthurs lo-l3 cm3 se& (almost independent less by at least a power of ten.

(12)

and Hyslop w who obtained a rate coefficient of 1.7 x The rate coefficient of (11) is probably of temperature). 4. STEADY

STATE

4.1 In the steady state we have that dF(z

1He+) dz

= m(O,)n(z 1He+) -

yn(z 1He)

(13)

where F((z 1He+) is the downward flux of He+ ions, K is the rate coefficient of the ion-atom interchange process (4) which is provisionally assumed to be responsible for the loss* and y, given by (2), is the average rate of photoionization. We also have dn(z I He+) F(z I He+) + an(z I He+) (14) dz

where D is the diffusion

coefficient

D

=

of He+ ions and a = i

log n,(z I He+)

(15)

n,(z 1He+) being the number density in electrostatic equilibrium. and (15) are in c.g.s. units. The boundary conditions are F(z I He+) -+ 0

as

z-0

andas

All quantities

in (13) (14)

z”+co.

(16)

Using estimates given in a review by Dalgarno (‘) it is judged that to a sufficient approximation l/D = [5-O x 10-20n(z 1N2) + 3.1 x 10-20n(z 1 0) + 1.6 x IO-%(z 1 0+)] (17) cm2 sec. (1) requires a to be for diffusion along the geomagnetic lines. Consistency with distributions 3 x 1O-s cm-l in the important region where Of is the most abundant ion. Solutions to equations (13) and (14) which satisfy boundary conditions (16) were obtained with the aid of a digital computer, n(800 km 1He+) being taken as known and K being treated as an eigenvalue. Figure 1 displays a selection of the solutions for z between 300 km and 600 km. These make it difficult to avoid the conclusion that He+ ions are quite abundant as suggested by Nicoleto). Figure 2 shows how n(600 km; T 1He+) the calculated number density at 600 km, depends The dependence on -r the lifetime (in sets) of an He+ ion towards destruction in a collision. * If instead dissociative charge transfer to N, is responsible in (13) by K’~(N,) where K’ is the relevant rate coefficient.

it is only necessary

to replace in

602

D. R. BATES and T. N. L. PATTERSON

2

3

4

log n (zl

He+)

FIG. 1. CALCULATEDSTEADYSTATEDISTRIBUTION~FH~+IONSINTHEUPPERPARTOFTHETHERMOSPHERE. THE NUMBERONEACHCURVEISTHEASSUMEDVALUEOFTHERATECOEFFICIENTOFTHE ION-ATOMINTERCHANOEPROCESS (4) BETWEEN He+ IONS AND 0, MOLECULESINCM~SEC-I.

FIG. 2. DEPENDENCE OF THE STEADY STATE NUMBER DENSITY OF He+ IONS AT 600 KM ON THE MEAN LIFE TIME OF AN ION TOWARDS DESTRUCTION BY C~LLI~IONAL PR~CESSES(THE VALUE OF n(600 KM (O+)IS TAKEN TO BE 6 X lo*CM-'AS IN (1)OF TEXT).

may be expressed

approximately

as

n(600 km; provided

T exceeds about

7 1 He+) = 77* cm-3

lo3 sec. If the main destruction T = l/[~n(600

km 1O,)] .

process

(18) operative

is (4) then (19)

As already noted an upper limit to K is provided by the rate coefficient for close encounters between He+ ions and 0, molecules which is about 1 x 1O-g cm3 se&. Hence, even if the temperature of the thermosphere were high enough and the degree of dissociation of molecular oxygen were low enough to make n(600 km 1 0,) 20 times as great as the value given by (l), 7 would be at least 2 x lo2 sec. It is unlikely that He+-N, charge transfer collisions could make T much shorter than this. Inspection of( 13) and(l4)reveals that if Kn(.z /0,)/D is held fixed n(z ) He+) is proportional to an(z 1He)/D; and therefore if K&Z I 0,) is held fixed n(z I He+) is approximately proportional to an(z I He)/@ p rovided a relation of the form of (18) is satisfied. The uncertainties n the adopted an(z 1 He) and D should not affect the results to a serious extent.

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4.2 The steady state distribution of He+ ions in the region above that covered by Fig. 1 is almost the same as the electrostatic equilibrium distribution.* This is given by n&z 1He+) = P(z)Y(z 1He+)

(20)

P(Z) = n(600 km 1e)+j[p(z 1H*) + Y(Z1He+) 5 Y(Z1 O+)lk

01)

with and Y(Z[ X+) = n(600 km 1X+) exp [--m,V/K!J

(22)

where wf, V is the sum of the gravitational and centrifugal potential energies of an ion X+ at altitude z, the zero of potential being taken to be at 600km, and where the other symbols have their usual significance@~9Jo). From (20) and the corresponding formulae for n(zS [ H+) and n&z IO+) it may be seen that there is a layer in which He+ ions are more abundant than H+ or 0’ ions provided n(600 km 1 He+)/n(600 km 1H+) > [?2(600km 1O+)/n(600 km I He+)]’ .

(23)

Assuming that H++O+H+O’

(24)

is sufftciently rapid to ensure that n(600 km ] O+)/n(600 km 1H+) = ~~(600 km I O)/n(600 km I H), 0 = s/9

(25)

it is apparent that condition (23) is equivalent to n(600 km I He+)/n(GOOkm I O+) > r,

(261

where r, =

[~(600 km 1H)~~~(600 km 1O)]4’5.

(27)

Table 1 gives the values of rc for the model atmospheres described by Bates and Patterson (which are characterized by the temperature T( co) of the exosphere) with ~(100 km I H) taken to be 6 x IO6cm-3 (cf. Bates and PattersorW ). The natural inference from the combination TABLE 1

T( co) “K r, of (27)

1000 2.2 x 10-Z

1250 4.2 x lo+

1500 1.0 x 10-S

2000 2.1 x 10-4

of this table and Fig. 2 is that for much of the time there is indeed a layer in which He+ is the most plentiful ionic species. The distribution of the different ions depend on the temperature and on the reIative abundances. To illustrate the position Fig. 3 depicts the distributions for the case of a low temperature and low n(600 km I He+)/n(600 km I H+) ratio, for the case of a high temperature and high $600 km f He+)/tr(600 km 1H+) ratio, and for an intermediate case.? In connection with the dispersion of whistlers (cf. Patterson 02)) it is of interest to note that the relevant formula analogous to(20)shows that the abundance of H+ions at great altitudes is unaffected by the presence of a He+ ion layer lower in the exosphere. *The asymptotic fall-off with altitude of this dis~ibution appears to be too slow (cf. Bates and PattersorFJ). This is not properly understood. i’ Representative distributions have also been given by Nicolet’l) and by HansorF.

D. R. BATES and T. N. L. PATTERSON H’ T(m)-IOOO”K

Tko)=ZOOO'

T(m)=l500"1

0

lwn(zlW

3

2

log

”

4

(2 I X)

FIG.~. CALCULATED STEADY STATEDISTRIBUTIONS OF H+,He+ AND Of IONSINTHEEXOSPHERE. (Thevalue of n(500km 1H+) istakento beas given by an equation correspondingto(25)with n(500km 1O+) and n(500 km 1H) as indicated after equation (27). The value of n(500 km 1He+) is taken from the 1.2, x lo-% cm* set-l curve of Fig. 1.)

5. ESCAPE

OF HELIUM

Any helium oxide ions formed by (4) are electronically excited and undergo a radiative transition to a repulsive state breaking up into a He atom and an Of ion: thus HeO+ + [He+ + O+] + hv.

(28)

It is possible that the helium atoms are liberated with kinetic energy in excess of the 2.4 eV that is required for escape from the exosphere into interplanetary space. Should ~n(500 km 1 0,) be 5 x 10e4 see-l the rate of production of energetic helium atoms in the exosphere might be about 2 x lo6 cme2 set-l which is of the same order of magnitude as the escape flux of helium atoms is commonly estimated to be. The hypothesis that (4) is the main process responsible for the removal of He+ ions thus gives rise to an attractive possibility. Acknowledgements-The research reported has been sponsored by Cambridge Research Laboratories, OAR, through the European Office, Aerospace Research, United States Air Force, under Grant No. AF-EOARDC 61-16.

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REFERENCES 1. M. NICOLET, Space Research II, Proceedings of the Second International Space Science Symposium, Florence (eds. H. C. van de Hulst, C. de Jager and A. F. Moore) p. 896, North Holland Pnbl. Co., Amsterdam (1961); J. Geophys. Res. 66, 2263 (1961); Proceedings of the International Conference on Cosmic Rays and the Earth Storm, Kyoto 1961, I. Earth Storm, J. Phys. Sot. Japan 17, Supplement Al 314 (1962). 2. R. E. BOURDEAU,E. C. WHIPPLE, J. L. DONLEY and S. J. BAUER,J. Geophys. Res. 67,467 (1962). 3. H. E. HINTEREGGER,Astrophys. J. 132, 801 (1960). 4. W. B. HANSON,J. Geophys. Res. 67, 183 (1962). 5. G. GIOUMOUSISand D. P. STEVENSON, J. Chem. Phys., 29,294 (1958). 6. A. M. ARTHURSand J. HYSLOP, Proc. Phys. Sot. A, 70,849 (1957). 7. A. DALGARNO, Ann. Gdophys. 17, 16 (1961). 8. J. W. DIJNGEY,The Physics of the Ionosphere, Report of the 1954 Cambridge Conference, p. 229, Phys. Sot. London (1955). 9. P. MANGE, J. Geophys. Res. 65, 3833 (1960). 10. D. R. BATESand T. N. L. PATTERSON,Planet. Space Sci. 5,257 (1961). 11. D. R. BATESand T. N. L. PATTERSON,Planet. Space Sci. 5,328 (1961). 12. T. N. L. PATTERSON,Planet. Space Sci. 8,71 (1961).