Prog. Aerospace Sei. 1975. Vo}. 16, No. 3. pp. 273-297. Pergamon Press. Printed in Great Britain

ON IONOSPHERIC AERODYNAMICS*

V. C. LIU The University of Michigan, Ann Arbor

Summary--This paper presents theoretical methods to determine the gas dynamic and the electrostatic effects due to the interaction caused by a rapidly moving body in the ionosphere. The principles of the methods are derived from the kinetic theory of collision-free plasma. It is shown that the collective behavior of the collision-free plasma makes it possible to use the fluid approach to treat the problems of ionospheric aerodynamics. Various solutions to the system of fluid and field equations that have direct bearing on the ionospheric aerodynamics are presented and discussed. Physical significances of the mathematical results are stressed. Some outstanding unsolved problems in ionospheric aerodynamics are elaborated and discussed.

LIST OF SYMBOLS AND UNITS (EM) l. INTRODUCTION All symbols are defined in the text. Only those that are repeatedly used are listed here. In the description of The advent of space flight which entails the electromagnetic fields Gaussian cgs units are adopted interaction between a rapidly moving, electrically throughout. These symbols are compiled in a separate charged body, e.g. a spacecraft, and a tenuous group with their corresponding quantities in rationalized ionized gas, e.g. the terrestrial ionosphere or the inks units with conversion factors listed in tandem. interplanetary gas, brings the aerodynamicists to GROUP A (electromagnetic) bear with a new aspect of aerodynamics--the ionospheric aerodynamics.t This interdisciplinary Gaussian Quantity mks science which combines gas dynamics with elec- B magnetic induction (47r/p.o)'r-B H magnetic field (4 ~-p.o)'nH tromagnetic fields poses some distinctive traits of c speed of light (~ot~o)-''~ its own even though it shares the basic principles of E electric field (4rr¢o)'r~E plasma physics with studies on electrical discharges 4, electric potential (4,n.eo)'1:~ and nuclear fusion. The research on electrical dis- e charge on proton e/(47reo) '/~ p. charge density p,/(4~'eo) ''2 charges in gases dated back several decades to the J current density J/(4~eo) 'n works of J. J. Thomson and then I. Langmuir and dielectric constant many other prominent physicists. It was Langmuir (Co, in free space) who coined the word "plasma" to mean that part of p. permeability ~//~o a gaseous discharge which contains almost equal (t~o, in free space) D displacement (4~r/~o)'2D charge densities of free electrons and positive ions. Intensive studies on plasma began with the realiza- ~r electrical conductivity a'l(4~rEo) tion of possible controlled release of nuclear fusion tap plasma frequency energy in the earlier 1950s J" Important contribu- lq~ Larmor frequency for particle tions to the basic understanding of plasma of type t~ phenomena have also been made by astronomers ho Debye shielding length and geophysicists in their studies of the solar e, charge on particle of type a activities <2~ and of the atmospheric electricity. °> A GROUP B (mechanical) new incentive to the study of space plasma was x position v particle velocity added in the late 1950s when the solar wind in the va thermal speed of particles interplanetary space was discoveredJ 4~ of type a The speed of a spacecraft, of course, varies but /o(X, v, t) distribution function of its typical value is mesothermal which means it lies particles of type a intermediate between the thermal speeds of am- T temperature na number density of particles bient ions and electrons; specifically, it satisfies the of type a following conditions: (KTj/m~) 'rz "~. V~ ~ (KT, Im,) '12. ma mass of particle of type a The typical values of these thermal speeds in the l mean free path upper ionosphere of the earth are 1 and 100 km/sec v collision frequency respectively while a representative speed of an K Boltzmann constant h Planck constant artificial earth satellite is of the order of 10 km/sec. p scalar pressure The fact that the motion is hyperthermal with pressure tensor respect to the ambient ions and almost stationary to V flow velocity the swiftly moving thermal electrons tends to cause q thermal flux separation in the flow field. This, however, is re- p mass density

*Prepared through the support of NASA Grant t Also known as electrogasdynamics or simply ab- NGR23-005-094. breviated as EGD. 274 V.C. Liu

strained by the natural tendency of the plasma to microscopic velocity distribution of the particles preserve electrical neutrality, i.e. to have a minimal if will be lost through the preliminary averaging pro- not zero charge separation. These contrasting influ- cess over the velocity space from which the mac- ences on a mesothermally moving plasma force it to roscopic equations of flow and field have been develop local charge separation that confines to a formulated. A case in point here is the Landau distance of the order of a Debye shielding length damping of collison-free plasma oscillations which (AD) in the absence of an external magnetic field)~ disappears when the fluid approach is used. The It is of interest to investigate further the nature of justification for the use of a fluid approach can be the disturbances of particles and felds caused by a strengthened by citing the successful continuum charged body in rapid motion. It is commonly theory of bow shock resulting from the impinging known by an aerodynamicist that there is fluid of solar wind on the terrestrial magnetosphere. On disturbance in the vicinity of a moving body. If the the other hand, the elucidation of the collision-free disturbed fluid is of the charged species, electric shock transition process which is obviously a prob- current can be induced whenever a differential lem of kinetics, is beyond the reach of the fluid motion of the oppositely charged particles prevails: approach. in other words, there will be induced field of charge It is the intent of the present review to discuss separation present. An electric field will influence both methods of approach particularly the connec- the motions, hence the distributions of the charged tion between them. Several applications of these particles which, in turn, further affect the field. This methods to the ionospheric problems of interest inherent coupling effect of particles and field almost will be presented. The topic of plasma diagnostic invariably plays a primary role in the problems of probes which is a very important subject of the field ionospheric aerodynamics. We shall see that this is excluded here because there have been several particle-field interaction is governed by a system of recent reviews on plasma probes. '~'7' coupled nonlinear equations. It is in the unravelling The presentation includes primarily theoretical of this nonlinear coupling effect that various studies of the problems of interest. Efforts are schemes of mathematical approximations have made to elucidate the physical essence of the been introduced in the contemporary studies of phenomena using the simplest mathematical tools ionospheric aerodynamics. Unfortunately, some of we are acquainted with. The description of experi- those approximations are not justifiable for the mental results is kept to a minimumt partly because problems at hand and the consequences are particu- of the scarcity of laboratory experiments that really larly serious for those cases where the nonlinear meet the requirement of a collision-free plasma. coupling of particle and field is inherent with the The in situ observations from space probes are problem, such as the near wakeJ 5~ A contemporary indeed plentiful. Unfortunately, these measure- belief that a bad approximation can be compen- ments of aerodynamically related quantities for sated by a numerical iteration of a nonlinear system purposes other than the interest of aerodynamics is not necessarily valid because the iteration pro- are inadequate to be used for elucidating the cess in question may not always converge. Thus a physics of fluids in question. In fact, the exposition self-consistent solution of particle and field dis- of the need to appreciate the aerodynamic signifi- tributions may not be obtained as expectedJ s~ cances in the space research and teaching is the Another aspect of the ionospheric aerodynamics primary motivation of the present paper. that could be of eminent interest to the aerodynami- cists concerns with the method of formulation of 2. NATURE OF THE EARTH'S IONOSPHERE the problems, to wit, the particle versus the fluid view of the problems. It is to be reminded that the The atmosphere of a planet is characterized by flow medium, namely the ionosphere, is very tenu- several parameters: the temperature, the density, ous; the mean free paths of the ambient particles, the chemical composition and the degree of ioniza- based on some binary collision cross sections, are tion. These properties vary with increasing altitude much larger than the size of the body in motion. above the planetary surface. The density of an Had the particles of flow been neutral species, the atmosphere decreases continuously with increasing particle approach, i.e. to use the Boltzmann equa- altitude and is related to the temperature of the tion of kinetics, will be naturally adoptedJ 6~ With atmosphere by the hydrostatic equation and the charged particles, the long-range nature of the equation of state of the atmospheric gas in ques- inter-particle forces leads to their collective be- tion. It is well known from statistical mechanics of havior in a collision-free plasma hence makes it gases that a mixture of gases with different molecu- much more amenable to the fluid approach* which lar masses, such as the atmosphere when left undis- is comparatively simpler. It must, however, be turbed to itself for a sufficient length of time, would noted that with the use of a fluid approach, those diffuse through each other and attain an isothermal plasma phenomena which are associated with the t Additional references on the contemporary studies of plasma, both theoretical and experimental, are available in * Also known as the continuum approach. ref. 5. On Ionospheric Aerodynamics 275 equilibrium in which each constituent gas is distri- ionization by recombination of positively and nega- buted exponentially according to Dalton's law, tively charged species in the ionosphere. The elec- whereby the lighter gases will predominate in the tron concentrations in the ionospheric layers are higher altitude and the heavier ones in the lowerfl ~ essentially controlled by the balance in production The atmosphere near the ground level is constantly and loss of ionization. As an example, it is known disturbed by the atmospheric circulation and tur- that F~ and F2 layers are produced by the same bulent mixing. This keeps the atmosphere photon radiation; the electron density in the F2 homogeneous in composition until about 105 km layer is, however, greater even though the radiation above the Earth, according to the sounding rocket is relatively weakly absorbed. The compensating observations, where the diffusive separation of the factor is that the recombination of electrons and constituent gases commences. It has been found ions in the F2 layer is much slower. This steady that at this altitude, called the turbopause, where state theory holds except at night. In the lower the molecular diffusion begins to dominate over the ionosphere the balance between production and turbulent diffusion, the atmosphere starts to distri- loss is complicated, particularly below the tur- bute according to Dalton's law. At an altitude of bopause where the small-scale eddies and wave about 200 km atomic oxygen, produced by photo- motions keep the atmosphere well mixed. On the chemical dissociation of the molecular oxygen, other hand, at altitudes higher than 250 km, trans- takes over from the molecular nitrogen as the main port processes including effects of wind and electric constituent, only to be superseded by helium and field become important. At much higher altitude, the then hydrogen. The Earth indeed has an outer Earth's magnetic field plays a conspicuous role in hydrogen atmosphere, but not below 1000km or determining the structure of the ionosphere. Elec- SO. ~9~ trons and protons enter the atmosphere from above From experiments on radio transmission, it was spiralling down magnetic lines of force and give rise found that ionized layers, conducting electric cur- to auroral emissions and ionization. The same lines rents and reflecting radio waves, must exist in the of force extend outwards to form part of the upper atmosphere of the Earth. These layers are boundary of the magnetosphere, m'~-'' called the ionosphere which contains free electrons Typical electron density vs. altitude at mid- and ions. Accurate soundings of these layers has latitude is shown in Fig. 1 for the purpose of been made possible by means of radio-echo techni- illustration. It is based on the midlatitude ionos- ques. In fact, the structure of the Earth's lower pheric measurements "°~ and intended to demon- ionosphere had been indirectly measured long before strate that the D-layer almost disappears at night, the space age. None the less, extensive direct and the Ft and F,. layers come close to merging. measurements of the ionospheric parameters, the Curve "I" corresponds to minimum of sunspot density and temperature of electrons and ions, from cycles while "II", maximum of sunspot cycles. the use of sounding rockets and spacecrafts con- solidate our knowledge of the earth's ionospherefl ~ I000 The ionospheric layers are designated by letters D, E and F, subdivided when necessary. The E boundary between the D and E regions is generally taken to be at 90 km altitude and that between the E and F regions at 150km. The upper ionosphere "~ 50C merges without clear-cut distinction into the mag- netosphere, a tenuous region extending out to dis- tances of several earth radii. The magnetosphere contains ionized gases and energetic charged parti- cles trapped in the earth's magnetic field: besides I I I I the MeV particles of the "Van Allen" radiation 103 104 i0 -~ 106 107 belts, there are the lower energetic particles that Electron density, elect rons/cm 3 cause auroral displays when they are dumped into the upper atmosphere near the poles. "°~ FIG. 1. Ionospheric profile of the Earth at mid-latitude. T--minimum of sunspot cycle. It has long been established that the solar ioniz- 'IF--maximum of sunspot cycle. ing radiation is primarily responsible for the pro- duction of the ionosphere. Measurements by 3. KINETIC PROPERTIES OF IONIZED GASES rocket-borne detectors have convincingly shown that the ionizing UV- and X-radiations from the sun Although an ionized gas is a gaseous mixture, it are sufficiently strong and are absorbed at the behaves kinetically very different from that of the appropriate altitudes to produce the D, E and F, neutral molecular species. Many of these properties layers. At times, and particularly in high latitudes, peculiar to the ionized gases have important bear- the energetic particles bombard the atmosphere ings on the understanding of ionospheric from above and produce ionization in the ionos- aerodynamics. We shall discuss them in the phere. Various processes exist to annihilate the following. 276 v.C. LIu

3.1. Electrical Neutrality and Shielding Effect field occurs at some finite-distance (hD) which in- creases with temperature (T) and decreases as the An ionized gas may be composed of several number density (n~) increases. If we imagine that components: positively and negatively charged all the positively charged ions in the gas behave as ions, free electrons and also neutral atoms and the above-mentioned point charge, each of them molecules. Each of these component species can be must have a shielded field of its own. A quiescent characterized by its concentration and temperature. ionized gas in equilibrium thus has a lumpy (grainy) In the simplest case all the ions are atomic species structure where hD defines the size of the grained- with single positive charges, and neutral molecules ness. The above disclosure has an important bear- are fully dissociated into neutral atoms. The ionized ing on the dynamics of ionized gases: the fact that gas thus contains only three components: free the potential between charged particles is the electrons, ions and neutral atoms; it contains no shielded Coulomb rather than Coulomb field means surplus charge of either kind, that is to say, the total that the effective range for the binary collisions is amount of positive charge contributed by the ions significantly shorter, hence the Boltzmann collision will be closely equal to the total amount of negative integral for the charged particles is no longer di- charges contributed by the electrons, hence the vergent as with Coulomb interactions. "3~ state of the gas is electrically neutral. To preserve The parameter ND =--n~A~ is a measure of the this electrical neutrality-status has been a unique maximum number of charges that bunch together. characteristic of an ionized gas. It is of interest to It is basic to the concept of plasma that the see how the charged particles react when a local concentration of the charged particles in an ionized deviation from charge neutrality is forced upon the gas is sufficiently large for their motions to be gas. dominated by their cooperative electromagnetic For simplicity, consider a fully ionized gas con- interactions. To fulfill the above-mentioned defini- taining free electrons and singly charged ions only. tion of a plasma we may specify the conditions as Let an isolated positive point charge (Ze) be intro- follows: duced to the gas. In equilibrium, the spatial dis- tribution of the electrons is given by the Boltzmann n~h ~ ~> 1, AD "~ L (3.3) distribution:* n, = n~ exp (ecb/KT) where ~b(x) is where L is a length characterizing the dimensions the field potential due to charge imbalance. Ignoring of the plasma. These conditions are also sup- ion motiont and letting the site of the charge plemented by that of charge neutrality in the defini- particle be the origin of a spherical coordinate, (Ze) tion of a plasma herein discussed. the Poisson equation becomes ~r1 0 (r2~) :-4~[en~-en~exp (~)- ZeS'x'J 3.2. Multi-mode Oscillations and Waves 4 zrn~e2 With the induced electrostatic and magnetic = KT +47reZS(x) (3.1) fields acting as restoring forces, various types of oscillations and waves may be initiated by distur- at distances such that ec~(r)/KT ,~ 1 where 8(x) is bances in a plasma. Furthermore, on account of the the Dirc delta function. The solution to eqn. (3.1) is particle bunching discussed in Section 3.1, the induced fields can provide a means of coupling ~b(r) = ---Zeexp(r - ~) (3.2) between the particles so that they can participate in, and influence, coherent wave propagation even where ;to = (KT/4~rn~e2)m is known as the Debye in a collision-free plasma. The nature of the restor- shielding length (or simply Debye length), being ing forces depends on the types of disturbances, first derived by Debye in his electrolyte theory. e.g. the effect of charge separations is restored by Notice that the potential (3.2) is a shielded electrostatic forces and the displacement of plasma Coulomb potential field where ;to is the effective as a whole can be restored by the "elasticity" of the cut-off distance; it gives the magnitude of the magnetic lines of force which are displaced with sheath thickness within which charge neutrality the conducting plasma. In addition, a plasma, just may not be valid. The above analysis shows that the like any other gas, exhibits elasticity towards pres- response of the charged particles to the extraneous sure changes. Owing to the presence of these charge (Ze) is to create a potential field to shield the possible restoring actions, the initial disturbance of influence of its charge imbalance. The physical whatever cause may lead to the appearance of interpretation of the shielding effect is that the elec- various oscillations and waves. We shall draw tric field tends to polarize the gas while the thermal attention here only to some of the basic modes of motion of the charged particles tends to weaken the plasma oscillations. polarization. As a consequence, shielding of the It is convenient to treat the response of the plasma to an electromagnetic field by the use of a * Also called the Maxwell-Boltzmann distribution. dielectric tensor (e). In so doing, we can separate t If ion distribution is also taken to be Boltzmann, the the dynamical part of the problem from the part Debye shielding length becomes 1/~/2 times smaller. concerned with the electromagnetic field equations. On Ionospheric Aerodynamics 277

The Maxwell equations for a disturbed elec- ie v(to) = - E(to). (3.12) tromagnetic field can be written as follows: meto

VxB=lO_EE+47rJ_ 1 0D From eqn. (3.6), after Fourier transform and c 8t c Ot' (3.4) substitutions of eqns. (3.11) and (3.12), we obtain 10B V x E .... (3.5) the conductivity tensor in (k, to)-space. c or" - ine 2 - The oscillatory motions of the plasma particles tr= I (3.13) meCto depend on the fields to which the particles are subjected; the only fields that need considera- where I is the unit dyad. From (3.7), after Fourier tion in eqns. (3.4) and (3.5) are the small perturba- transform and some algebraic manipulations, we tions. Any static, external fields present will obtain only affect the dielectric properties of the plasma through the equation of motion. To complete the t= 1- I (3.14) set of the above equations we must have Ohm's law to connect the current with the electric field: where = (4wne2~ m J = o" • E (3.6) top ,~-, (3.15) where ~ is the conductivity tensor, or equivalently denotes the electron plasma frequency. we may introduce the dielectric tensor to connect The wave equation (3.8), after substitution of the displacement vector (D) with the electric field: (3.14), becomes D = ~ • E. (3.7) k x (k x E) +-C-r 1- E=0 (3.16) If the system in question is homogeneous spa- tially and all perturbed quantities have time depen- which can be rewritten after the field E is separated dence proportional to exp (-itot), we may intro- into longitudinal and transverse components, with duce Fourier transforms of variables in the above respect to the wave vector k, i.e. E = E, + E~ in the equations and considering variation of the form following form: exp [i(k. x-tot)], we may replace the differential operators (O/Ot) by - iw and V by - ik to obtain the kX[kx(Eii+Ei)]+~. 1- (ELI+E~)=0. Fourier transformed system of the electromagnetic field equations which, after the elimination of (3.17) variable B, becomes 2 Notice that k × Ell = 0 and k x E = 0 by definition of kX(kxE)+~-rE "E=0. (3.8) EIj and E~ respectively, so that C (to2_ to2p)E,+(to2 - to'p- c'-k2)E~ = 0. (3.18) Equation (3.8) describes the basic behavior of plane waves in a plasma in the (k, to)-space. In the case of the longitudinal mode of oscilla- tion, we have E = 0 and Eu ~ 0 and the dispersion 3.2.1. Collision-.free plasma high-frequency oscilla- relation leads to tions in a weak magnetic field to: = to2 (3.19) If the range of oscillation frequencies is suffi- which denotes the plasma electron oscillations; on ciently high, we can discuss the electrostatic and the other hand, when the oscillation mode is trans- electromagnetic waves with the assumption that the verse, we have E~ = 0 and El # O. Thus the disper- ions play no essential role other than to provide a sion relation becomes background which assures the existence of an equilibrium, quasi-neutral state. We also ignore the toz = to~ + cZk 2 (3.20) particle collisions and external magnetic field in the which refers to the electromagnetic waves. present cold plasma approximation. In the electrostatic mode of wave propagation in The electric current is a plasma, negligible magnetic field is generated lie because Ell is parallel to the wave vector implying J = ---v (3.9) C that the curl of the electric field vanishes. It is where the electron velocity can be determined from further noted that EjL is nonzero only if to" = to'-p for its equation of motion in a homogeneous medium any k; hence all waves in this mode have the same frequency. Another noteworthy aspect of the pres- av m~-~- = - eE. (3.10) ent electrostatic waves is the consequence of the cold plasma approximation herein used. It is valid if Equations (3.9) and (3.10), after Fourier trans- the phase velocity of the waves to /lkl "> o,, the form, become electron thermal velocity. If an electron moves ine" with a velocity comparable to the phase velocity, it J(oj) = E(to), (3.11) mec to sees a d.c. field and hence can resonate with the 278 v.C. Liu

wave, either gaining or losing energy, even for a It should be noted that the magnetic field should weak field. Consequently, electron thermal effects have no appreciable effect on the electromagnetic modify the wave dispersion giving rise to a change wave field in a plasma if the frequency of oscilla- of phase velocity and to a damping known as tion is much greater than the Larmor frequency of Landau damping in a collision-free plasma, m~ the plasma electrons. On the other hand, if it is The inclusion of particle collision effect in the much lower than the Larmor frequency of the wave analysis will show that collisions tend to plasma ions, assuming for simplicity that the direc- damp out the plasma electron oscillations in the tion of propagation is parallel to the direction of the time of the order 2Iv where v is the collision constant external magnetic field, the lines of force, frequency. The electromagnetic waves are less frozen into the medium, assume a wave-like form sensitive to the collision effect. The reason is that and bring the plasma into oscillatory motion with its wave energy is carried partly by the plasma them. A large proportion of the electromagnetic particles and partly by free space. At high frequen- energy associated with the wave is thus converted cies, it is carried mostly by the space and thus into the kinetic energy of oscillation of the medium. particle collisions are unimportant. The situation is somewhat similar to the propaga- The effect of an external magnetic field on the tion of elastic waves along a string. If the mass per electrostatic and the electromagnetic waves is very unit length is higher at some point of the string than complicated. Interested readers should consult spe- elsewhere, the velocity of the wave is reduced cial treatise. "4~ accordingly. In the propagation of electromagnetic waves in the direction of a constant magnetic field 3.2.2. Low-frequency plasma oscillations in plasma, the lines of force behave like elastic When oscillations of low frequencies are excited strings along which oscillations are transmitted. in a plasma, both ions and electrons will take part in These strings are loaded by the plasma, and there- action. If the frequency of oscillation is sufficiently fore the velocity of the wave is reduced. "6~ low, the thermal motions of the electrons are able to Our discussion on the plasma wave properties set themselves in equilibrium in the electric field has been formal in the sense that some of the created by the motion of the ions. A similar proce- dispersion relations of the waves were formulated dure as in the high-frequency case discussed earlier from the basic governing equations of particles and can be set up with, however, much heavier algebra field without questioning the physical basis for the to determine the dispersion relation for the low- existence of wave motions, which are fluid proper- frequency plasma waves. "~ We shall not consider ties, in a collisionless plasma. This subtle implica- in detail the mechanism of ion oscillations except to tion is based on the long-range nature of Coulomb note that their propagation through a plasma is forces between charged particles whose interac- analogous to the propagation of acoustic waves tions is not so much in the nature of a collision as it through a gas of neutral particles. If there is thermal is a reflection of the bunching effect of the so-called equilibrium between the electrons and ions in the self-consistent field due to the encounterings at plasma, i.e. if T, = T~, the ion acoustic wave propa- large distances. This collective behavior of charged gates with a velocity of the order of thermal particles is one of the fundamental properties of a velocity of the ions. In a bi-thermal plasma where plasma medium and will be more thoroughly dis- Te-> T~, the velocity of propagation of the ion cussed in the following. acoustic wave is found to be of the order of (KT,/m~)~2; in other words, the ion wave travels 3.3. Collective Behavior of Collision-free Plasma with the velocity which the ions would have if their temperature were equal to the electron tempera- In the earlier studies on rarefied plasma, it had ture. It has been shown in theoretical analysis that been found that there were many anomalous ion acoustic waves can propagate freely in a plasma phenomena which could not be explained on the which has ions much colder than electrons, and are basis of the classical Boltzmann kinetics, e.g. thick- rapidly attenuated in an isothermal plasma. So far, ness of a plasma shock front is appreciably smaller ourdiscussion on the plasma low-frequency oscilla- than the mean free path of the medium; the plasma tions have been limited to the longitudinal modes. diffusion flux across magnetic field is abnormally Among the transverse modes of plasma oscilla- high--a phenomenon known as the Bohm diffusion tions that have frequency much lower than the after its discoverer. Larmor frequencies of the ions the Alfven wave Following our earlier discussion of the Debye which is an electromagnetic wave propagating shielding effect on the Coulomb field of a charged along the magnetic lines of force in a plasma is of particle, we note that the statistical effects of special interest in the ionosphere studies. In the charged-particle interactions can be conveniently idealized, dissipationless plasma, the nature of AlL classified into two groups: (i) with particle separa- yen waves can be elucidated with the concept of tion r < AD their random interactions can be treated frozen fields which means the magnetic field lines in the manner of the Brownian motions, i.e. expres- stick to the plasma fluid and move with the local sed in the form of the Fokker-Planck "collision" in fluid. the theory of Brownian motion;m~ (ii) with r > AD On Ionospheric Aerodynamics 279 the particles have finite correlations, i.e. behave study must be done with caution. The adsorption of collectively, the plasma hence having a configura- gas by a real surface is another factor that adds to tion somewhat like a lattice that has a spacing equal the uncertainty of the surface condition. Physically to a wavelength of propagating plasma oscillations. adsorbed gas layers attached to a solid surface may Thus )tD can be identified with the shortest possible be removed by moderate heating. Adsorption be- wavelength of a longitudinal plasma wave. The havior, from the kinetic point of view, may be collective behavior of plasma particles due to dis- conveniently described by the sticking probability tant interactions is reflected in the self-consistent for molecules of the gas striking the surface. Stick- field effect. This finite correlation of particle mo- ing probability lies in the range 0.1 to 1 for the first tions accounts for the quasi-continuum nature of a monolayers and falls sharply as more layers are collision-free plasma that supports the wave mo- adsorbed. The amount of adsorption quickly tions and even shock fronts. "a~ reaches an equilibrium value which depends upon The resonant interactions between propagating the partial pressure of the adsorbate and the sur- plasma waves and those particles which have vel- face temperature, e.g. with a partial pressure of ocities close to that of the wave could provide the l0 -7 mm Hg of oxygen a clean metallic surface will mechanism of damping or growth of the wave acquire an oxide monolayer in about a minute. depending on the particle velocity distribution as Fortunately, in an ionospheric aerodynamic first discussed by Landau. The case of wave- problem where the surface potential is never larger growth which is the cause of a micro-instability of than a few volts, it is a close approximation to the system can lead to the Bohm diffusion consider all incident ions neutralized by picking up phenomenon mentioned earlier and the plasma a negative charge of equal amount at impact; turbulence, to cite a few. Since plasma in a magne- electrons, absorbed. This assumption is used tic field contains a large number of natural resonant throughout the present study. frequencies, considering the collective oscillations of both ions and electrons, their Larmor frequen- 4.1. Equilibrium Surface Potential without cies and several combinations of these, it is one of Photoemission (zo.2~ the most challenging jobs in plasma research to determine various modes of micro-instabilities, the We shall illustrate the attainment of equilibrium wave-particle interactions, particularly the non- surface potential (~b,) by using a spherical model of linear ones if the wave amplitude grows sufficiently radius R in a quiescent plasma in equilibrium at a large. One particular mode worth noticing is the temperature T. To a stationary sphere of zero initial drift wave which can occur spontaneously by an surface potential, the incident flux of thermal elec- instability driven by density and temperature gra- trons is larger than that of thermal ions, assuming dients. "9~ It could be primarily responsible for the singly charged. The negative surface potential thus collision-free shock transition process, a problem acquired on account of the net amount of negative yet to be solved. charges accreted by the sphere tends to repel the electrons and attract the ions. An equilibrium sur- face potential (~bs) will be reached finally when the 4. PARTICLE-SURFACE INTERACTION thermal electron influx balances the thermal ion When a charged particle impinges on a solid influx. We have surface, one of several alternatives may happen, 8KT ~12 e.g. it may be neutralized electrically and reflected as a neutral particle, thus lost to the population of the charged species; it may reflect specularly and where A,, A~ denote the effective collection areas retains its charge, etc. An incident particle of high of the ambient thermal electrons and ions respec- energy may cause secondary emission from the tivelyfl°~ An estimation of the equilibrium surface surface. This charge accretion by the surface is one potential of a spherical body moving at mesother- of the inputs that determine the resultant surface mal speeds, namely (KTJlmt)'I2~ V=~(KT, Im,) '12 potential of the body in question. '~ was madefl ') It goes without saying that other The study of particle-surface interaction has sources of charge inputs such as the photoemission been made difficult by the uncertainties of the of electrons from the surface due to solar radiation, actual surface condition which is one of the factors the intensive charge flux input ff the body is in the influencing the nature of particle accommodation. radiation belt of the earth, etc., would change the An ideal surface may be defined as one with which equilibrium surface potential. the bulk properties of the material persists to the geometric surface. On the other hand, with a real 4.2. Equilibrium Surface Potential with surface which is generally finished by mechanical Photoemission (~o.2u grinding, polishing, etc. there exists a surface layer, about lO/xm thick, which has usually The photoemission effect upon the surface po- been violently distorted. Hence the use of ideal tential of a spacecraft in deep space can be strong solid surface models for particle accommodation enough that the equilibrium potential takes a posi- 280 V.C. LIU

tive value (~, > 0) relative to the ambient plasma. intended here to formulate the governing equations Under such conditions, the thermal ion flux be- of the system from both the kinetic and the con- comes negligible. We consider now the extreme tinuum points of view emphasizing particularly the case with a sphere of radius R that has the thermal connection between them. It is expected that in so electron flux in balance with the photoemission doing we can appreciate how certain plasma be- current Ip,(th,, ~b,.) which depends on the surface havior peculiar to the microscopic velocity dis- potential (~b,) and the photoelectric work function tribution have been averaged out and lost to the (~.) system at the level of macroscopic description of [8KT'~ 'n ( ecb,) Ip, the flow. Nevertheless, the asset of simplicity and en, ~,-~]-m) 1 - KTJ = ~R-~" (4.2) directness of the continuum approach is too famil- iar to the aerodynamicists to be enumerated here. Equation (4.2) can be solved by a graphical method provided the photoelectric work function (4~,.) of the body material is given. It has been estimated 5.1. Kinetic Equations that a positive surface potential larger than a few The ionospheric plasma is considered to be com- volts is unlikely unless the spacecraft is in the posed only of singly charged and neutral particles radiation belt where extremely intensive radiation which interact according to the laws of classical flux is expected. mechanics. Inelastic collisions, e.g. charge- exchange collision which is possibly important in 5. KINETIC AND CONTINUUM THEORIES OF comet gas dynamics, are excluded herein. IONOSPHERIC FLOWS

The mathematical representation of a gaseous 5.1.1. Distribution function in the phase space motion can be made at various levels of approxima- (x, v) tions. Considering a gaseous motion as a collected The statistical mechanical state of a plasma can motion of a large number of particles each of which be described by separate distribution functions may interact with the others by means of the each of which denotes a particular species of the inter-particle forces, a microscopic approach to the mixture, e.g. fo(x, v, t) dx dv represents the proba- problem starts with the application of the conserva- ble number of particle species a in the six- tion laws of mechanics to the particle motions and dimensional phase space dx dv at time t where a is the use of mathematics of statistics to determine used to designate ions (a = i), electrons (a = e) or mean values of the particle motions that are mac- neutral (a = n). The kinetic equation known as the roscopic observables of the flow field. This is Boltzmann equation for the distribution function known as the kinetic approach to the problems of fi(x,v, t) governing the statistical distribution of flows. On the other hand, a macroscopic approach species a that stream in the phase space (x, v) can starts with the application of the laws of mechanics be written"3~ to the motion of a representative fluid element whose macroscopic behavior can be construed as ~+v ~+a.~ ~, (5.1) giving the average behavior of particles it contains. The latter approach is called the continuum theory of flow or fluid theory because it focuses attention where a(x, v, t) is the particle acceleration due to on a fluid element as the starting point. It appears the Lorentz force eo(E+vxB/c) on a particle that the formal difference between the two- charge eo and possibly the gravitational force if it is mentioned microscopic and macroscopic ap- significant. The electrostatic field E is the self- proaches is merely a matter of procedure. There is, consistent electrostatic field which takes account of however, much subtlety in the choice between the distant (collective) part of the Coulomb interac- them for the solution of a flow problem. tion while the collision term (~fo/6t)c takes into If our interest in the upper atmospheric flows is account the near-neighbor interaction of the parti- restricted to the neutral particle species for which cles. This ad hoc division of the many-body in- the mean free paths are many order of magnitude teraction effect into the long-ranged collective in- larger than a characteristic length of the moving teraction and the short-ranged discrete collisions body, a free molecular flow would be expected; the and the subsequent assumption of the dominance kinetic approach is the logical choice because there of the former contribution must be considered as is no meaningful representative fluid element in one of the fundamental assumptions of the contem- such a collision-free flow. On the other hand, with porary plasma physics. an ionospheric flow where the charged species The self-consistency of the electromagnetic field dominate, a quasi-continuum flow prevails even for is built into the equation when the field is required a collision-free plasma because of their collective to satisfy the Maxwell equation of the elec- behavior discussed in Section 3.3. Consequently, tromagnetic fields (3.4), (3.5) supplemented by fluid approach to an ionospheric flow becomes 0 possible provided certain conditions are met. ~8~ It is a-x ' E = 4~rpc (5.2) On Ionospheric Aerodynamics 281 and assuming that a time interval At exists, which is 0 long enough for a particle to suffer a large number --. B = 0 (5,3) Ox of collisions, but short enough so that its velocity does not change much. The discrete collision effect where pc is the net charge density. The current J in is eqn. (3.4) is (Sf.) [f,(x,v,t)-fa(x,v,t-At)] -~- = lira (5.7) J=Jo,,,+~e~fvf~(x,v,t)dv (5.4) c ~,-0 At where J .... is an externally produced current while To evaluate (SfQ/St)c (5.7), we expand the functions the second term on the right-hand side of eqn. (5.4) under the integral in eqn. (5.6) into Taylor series, represents the current produced by the plasma itself. The summation is over various species that /~(x, v, t)= f d(hv) compose the plasma. A similar equation relates the net charge density to the distribution functions: × {so(x,v, r +:o l ot LOV ovj +~ AvAv: [~"' a--G~,02f0 + 2 af° aq'+ a% , pc = ~ eo S.f~dv. (5.5) av av The determination of the term (Sf~/Ot)c in eqn. which, after the use of the normalization condition (5.1) that represents the effect of nearby collisions for ~, has been of considerable interest in the contempor- ary plasma studies. It is noted that in the classical kinetic equation (5.1) of Boltzmann, the collision f tp d(Av)= 1 (5.8) term has been formulated on the basis of the binary assumption which says that encounters in which becomes more than two molecules take part are negligible in 3 1 2 frequency and effect compared with binary encoun- - --- ((Av).fo) + z a_~__~:((AvAv).fo) L OVOV ters. The binary assumption has worked out well (5.9) for the dilute gases of neutral particles having small fields.* In the analysis of collisions between where charged particles difficulties arise that stem from the long-range nature of the Coulomb force. A (Av) = f $(v, Av)Av d(Av), given particle is subject to the influence of a large number of surrounding particles, and all of these (AvAv) = f $(v, Av)AvAv d(Av). particle interactions must be taken into account. Thus, instead of dealing with the binary collisions, The expression (5.9) can be used after ~ is pre- we must cope with a many-body problem, for scribed for an assumed model of the scattering which an effective rigorous analysis is not yet process; its first term gives rise to a slowing-down available. A working hypothesis has been intro- effect and is called the coefficient of dynamic duced to approximate this many-body interac- friction; its second term has the effect of spreading tion. "3) Other than the collective effect of distant out an initial particle beam thus named the coeffi- interactions and cumulative effect of interactions cient of diffusion. between a given particle and its neighbors within a A result essentially equivalent to expression (5.9) distance of Debye length (AD) can be effectively was obtained by Landau "2) who started with the treated as equivalent to diffusion in velocity space Boltzmann binary collision integral and postulated (Fokker-Planck Process"3)). that owing to the long-range nature of electrostatic Consider the plasma particles undergoing ran- forces, the number of collisions resulting in a small dom motions, akin to the Brownian motions, "7) as a deflection 0 are much more numerous than those of result of a large number of small deflections. It is large 0, hence change of particle velocity Av will be appropriate to describe such a stochastic process small for most collisions so that it is reasonable to by the transition probability qJ(v, Av) that a particle expand the integrand of Boltzmann's collision in- with velocity v acquires a velocity increment Av in tegral in powers of Av and keep the first and second time At. For the Markov Process "7) ~b does not powers. A justification for Landau's procedure, depend explicity on time and the distribution then which caused some earlier controversies, can be evolves according to the relation: made on the ground that it makes no difference to the final result whether the interactions are simul- fo(x,v,t) taneous (as a many-body problem) or successive f f,(x, v - Av, t - At)~b (v - Av, Av) d(Av) (5.6) (as the summed binary effects). d The collision effect between charged and neutral *Referring to the short-range inter-molecular force particles can be adequately represented by the fields. classical Boltzmann binary collision integral with 282 V.C. LIU

the Maxwellian molecular model (interaction force sight because of its relative simplicity in formula- law ~ r -~) as, for example, was done by Allis.~' tion. In the upper ionosphere where the density is very low and the atmosphere is strongly ionized, not 5.1.3. Particle orbit theory of collision-free plasma only the effect of the neutral atoms but also the In view of the equivalence between the Jeans complete discrete collision (Sfa/St)~ are negligible theorem approach and the consideration of the compared with the collective effect of the distant particle trajectories to treat the collision-free charged particles, eqn. (5.1) can be simplified to the plasma as shown in Section 5.1.2. it should be collision-free Boltzmann equation with self- possible to replace the Vlasov equation by the consistent field, often called the Vlasov equation: equations of motion (5.11). As to how effective the latter approach is depends on the method used to -~-Ofo + v ' 00--~+ a - ~-~of, = 0. (5.10) handle the results of particle trajectories thus ob- tained in order to formulate the space charge 5.1.2. Formal solution of Vlasov equation density (pc) needed in eqn. (5.2). The job can The distribution function f(x, v, t) satisfying eqn. become unwieldy considering the number of essen- (5.10) and a given initial state [o(x,v, 0) can be tial parameters for the determination of the as- determined formally by the method of characteris- sorted trajectories. It is for the purpose of efficient tics. Notice that the subsidiary (characteristics) mathematical classification of the particle trajec- equations to eqn. (5.10) in Cartesian coordinates can tories that an alternative approach has been intro- be written duced ~24~ which uses the Hamilton-Jacobian equa- tion in classical mechanics to describe particle dt_ dx, dx~._ dx~ dv, dv: dr3 (5.11) motion in lieu of the Newtonian equations (5.11). 1 v~ v: 03 a~ a,. a3 However, since the wave function of the which define the particle trajectories of the Schroedinger equation which is the counterpart of collision-free plasma. The general solution to eqn. the Hamilton-Jacobian equation is more accessible (5.11) can be prescribed as follows: for describing particle density, it is initially used even though the quantum mechanical effect is x, = x~(I,, 12..... 16, t); v~ = v,(I,, I2 ..... /6, t) negligible. The transformation of the Schroedinger (/3 = 1,2,3) (5.12) equation to the Hamilton-Jacobian equation can be where L, I, ..... I, are the six constants of integra- made by the introduction of the W.K.B.J. approxi- tion. The integrals of motion are mation. ~m This approach is found particularly effec- tive to treat the interaction problems of mesother- Ia=I~(x,v,t) (/3=1,2 ..... 6). (5.13) mal collision-free plasmas in the absence of magne- According to the theory of partial differential tic fields~ where the electrostatic approximation is equations, the general solution to eqn. (5.10) is an appropriate. arbitrary function of the integrals of motion (5.13), The Schroedinger equation for an ion of mass m~ namely and charge in steady motion is f = F(L, I2..... /6). (5.14) h'V2~(x) + 87r"m,[E, - e~b(x)]~ = 0 (5.15) The particular solution to the given initial value where h denotes the Planck constant; qb(x) the field problem is obtainable by requiring that potential; E, the total energy of the particle in F(L, I: ..... /6) satisfies the given initial distribu- question. The wave function ~(x) has meaning such tion fo(X, v, 0). that Iq, I2 dx is the probability of finding a particle in An application of the above result to the study of volume element dx. Thus, if there are n~ nonin- stellar dynamics was made by Jeans. The functional teracting identical particles per unit volume, n=rq'l 2 relation (5.14) is known as Jeans theorem in stellar will represent the particle density at x. Equation dynamics. Extensive applications of the theorem (5.15) can be used to describe the distribution of (5.14) to specify ion trajectories in an electrostatic monoenergetic ions in the problems of mesother- field of sheath and wakes in ionospheric gas mal flows, t24' dynamics were also made. ~5~The advantage of using A short-wavelength approximation, known as the the theorem (5.14) to the study of collision-free W.K.B.J. method, t2~) can be introduced to eqn. plasma over the calculation of assorted ion trajec- (5.15), we let tories with assumed parameters of field and initial conditions is computational. A judicious choice of = A exp [iW(x)] (5.16) F(I,, 12..... /6) with sufficient number of integrals and substitute it in eqn. (5.15) (Ia) not only saves much computations for indi- vidual trajectories but makes the numerical h2[(VW) 2- iV2W] = 8~r2m~(E~ - eck ) (5.17) iterations* much illuminating to gain physical in- which, under short-wavelength approximation, re- duces to * Which is almost always needed when the system is nonlinear such as the near-wake problem,c~ h2(VW) 2 = 87&m,(E, - ech). (5.18) On Ionospheric Aerodynamics 283

Equation (5.18) is the Hamilton-Jacobian equation for number density, flow velocity, pressure tensor in classical mechanics. W is proportional to the or thermal flux .... respectively, as given in flow action. It can be determined as the formal solution quantities (5.20). to eqn. (5.18) Multiplying both sides of eqn. (5.1) by Q(v) and integrating over the velocities of the particles of w(x) = f 27r (2m')'t"h {E~ - eth [x(S)]} '/2 dS (5.19) type a, we obtain "~ where S is a parameter in terms of which the ion 0(n,(Q))0t +O0x " (no (vQ))- n,a. ~ orbits are described. It is noted that the mathematical advantage of aIo using the Schroedinger equation for the present purpose stems from the fact that it is linear for a where an angular parenthesis denotes the average given ~(x). This makes it very convenient to un- over velocity space of the quantity enclosed. The ravel the nonlinearly coupled system of the term on the right-hand side of eqn. (5.21) needs particle-field interactions under electrostatic ap- further clarification. It is recalled that (af,/~t)c proximation which will be shown later. represents the rate of change of f, due to close clllisions, a comparison of eqns. (5.1) and (5.21) 5.2. Continuum Equations suggests the moment of the collision term might represent the rate of change of no(Q) due to The kinetic flow analysis (Section 5.1) which collisions. involves the use of distribution functions in a It is of interest to observe the analogous relation- six-dimensional phase space (x, v) is not only sad- ship between eqns. (5.1) and (5.21) to that between dled with considerable mathematical difficulties but the equations of boundary layers, in fluid also provides too much detailed information on dynamics, '26' due to Prandtl and yon Karman.* plasma particle behaviors which are not needed in Recall that Prandtl's equation satisfies pointwise determining the flow observables: the boundary-layer flow field while yon Karman's equation, being a moment of PrandtFs equation, number density: no(X,t)= f fodv, satisfies the momentum conservation only on the average across the boundary layer. The relaxed flow velocity: Vo(×, t) =~ vLdv, condition imposed in the latter approach to the boundary layers makes the solution untenable for a unique pointwise description of the flow field. This pressure tensor: (x, t ) = mo f (v - Vo) (5.20) loss of accuracy in flow-field description is compen- x (v - Vo)/~ dv, sated by its mathematical simplicity and versatility in applications. The comparison of the cited alter- thermal flux: q(x, t) = (v - vo) native approaches to the boundary-layer problems × (v - V. )2f, d v, in fluid dynamics is very much in parallel with that of the two alternative approaches (5.1) and (5.21) to etc. gas-kinetic problems. While the kinetics eqn. (5.1) It would be desirable to develop governing equa- provides a pointwise description in the phase-space tions of flows that have the flow quantities (5.20) as (x,v), the continuum eqn. (5.21) obtained as the their unknown functions. This is the primary moti- velocity moments of eqn. (5.1) satisfies the trans- vation for introducing the moment form of the port of Q(v) on the average in the velocity-space. kinetic equation (5.1). Notice that the flow quan- The physical basis for the use of eqn. (5.21) is that tities (5.20) are the zeroth, the first, the second and the time characteristic of the randomizing collision the contracted third moment respectively of the effect on the relaxation of an arbitrary velocity distribution function. distribution to its locally near-equilibrium state is 5.2.1. Transport equation very small compared with the time constant of the hydrodynamic flow. Under such a condition the In spite of the importance of the distribution flow quantities (5.20) would be relatively insensitive function which serves as a universal joint to all the to the microscopic velocity distributions. It thus flow quantities (5.20) of interest, it is sometimes points out the important role played by the molecu- found necessary to circumvent the mathematical lar collision effects. difficulties by forming moments of the terms of its There is another thorny question pertaining to governing eqn. (5.1). Physically speaking, it is in- the moment approach (5.21), namely the closure tended to transform eqn. (5.1) in such a way that it problem of the moment equations. The formal may be possible to calculate the average value of development of the moment equations by assigning any quantity Q(v) without knowing much about the distribution function itself. Notice that Q(v) desig- nates * More properly, known as the yon Karman's momen- 1,v, m(v-V°)(v-V,,) or ½m°(v-V°)(v-V,) 2 .... tum integral. 284 V.C. Ltu successive order of velocity moments leads to a set Equations (5.22) and (5.23), together with Max- of equations which contains more unknowns than well equations of the electromagnetic field, form a the number of available equations. For instance, an closed set which has been used extensively particu- (n + l)th moment is, in general, generated in an nth larly in the study of wave propagation in the moment equation. Various means of closure on the ionosphere, aS' When the cold plasma equations are basis of physical considerations are introduced to used, it is important that the speed of plasma obtain a closed set of moment equations as will be disturbances of interest must be large compared shown later. with the thermal velocities of the particles. (b) Locally Maxwellian hypothesis• If the colli- 5.2.2. Conservation equations of a two-component sions are so frequent that the microscopic velocity "plasma distribution is sufficiently isotropic such that to the A two-component plasma is regarded as two first approximation a local Maxwellian distribution interacting fluids, the electron gas and ion gas. They is maintained interact through collisions as well as the elec- [ mo ]3,,. f~(x,v,t)=no(x,t) 2~rKT~x, t) tromagnetic field to which both fluids contribute and are attached ×exp{ ma[v-V.(x,t)]]2s

* Tantamount to zero bulk viscosity, t Note that all the collision terms vanish. On Ionospheric Aerodynamics 285

lapses into Poisson equation less representations (with superscript*) are as follows: V2q~ = - 47rp~. (5.27) Equation (5.27) together with Vlasov equation com- t= (R/rOt*, x = Rx*, ~ = (KT/e)4~*, v = v~v*, f, = plete a mathematical description of a collision-free (n~/v])f*, At = (d/v~)(At)*. plasma with only electrostatic interparticle forces In terms of the above dimensionless quantities, included. This is an important set of equations eqns. (5.1), (5.7) become governing many of the problems of interest in ionospheric aerodynamics. Of~' 4-v*. af~ ad~*. Of,, cgt* 3x* 3x* Ov* _ R { O [(Av*) £] 1 3 z 6. SIMILARITY PARAMETERS X/2x 7rl _- Ov---~ • L(At), a, j 4 2 Ov*0v* OF IONOSPHERIC MESOTHERMAL FLOWS [!Av*Av*) In the studies of many mesothermal aerodynamic 'L (At)* f*]} (6.1) problems of the ionosphere where the plasma in- where the characteristic mean free path I= teraction near the moving body is of interest, the (X/2 zr d2n~)-'n eqn. (5.27) becomes electrostatic approximation of the equations of flows (Section 5.2.2(c)) is valid. The magnetic field h 2 effect is small because the Larmor radii of ions are (6.2) usually much larger than the characteristic size of the body. Under the mesothermal flow condition, where ;to is the Debye shielding length. the thermal electrons maintain a quasi-equilibrium It is observed from eqns. (6.1) and (6.2) that the state in spite of the moving body, hence the gas dynamics and the electrostatic similarity be- magnetic-field effect on the plasma interaction tween different model systems, depends on two through the Larmor electron gyrations is not impor- similarity parameters: Knudsen number (I/R) and a tant. The physical picture of the steady state shielding parameter (AD/R). Let us investigate plasma interaction due to a mesothermally moving further the physical significance of the dimension- and negatively charged body of characteristic size less equations (6.1) and (6.2). From eqn. (6.1) we R in the ionosphere can be viewed as follows. The observe that when the Knudsen number is very ions, which are significantly disturbed from their small, say IIR ,~ 1, the collision term dominates equilibrium distribution, move among electrons over the streaming-term. This implies that the flow which are essentially in a Maxwell-Boltzmann dis- field in question is collision-controlled, hence the tribution: n, = n~exp [e~b/KT] where the field po- particle velocity distribution is not expected to tential ~(x) is due to charges both on the body and deviate much from the locally Maxwellian distribu- of the space. It is of interest to determine the tion on which corresponds to continuum flow of an particle and field distributions in the disturbed zone inviscid fluid (see Section 5.2.2(b)). On the other of particle-field interaction. hand, when the Knudsen number (I/R) is extremely large, we have the collision-free plasma flows with 6.1. Dimensionless Transformations of the Govern- self-consistent fields due to particle distant encoun- ing Equations ters, known as their collective behavior (Section 3.3). This is of primary interest to the study of The equations governing the above-mentioned ionospheric aerodynamics. plasma interaction consist, as usual, of two groups: It is observed that eqn. (6.2) has several distinc- the particle equations are the kinetic equation (5.1) tive features: (i) its coefficient of the most high for the ions and the Maxwell-Boltzmann distribu- differentiated term (AD/R)2~ 1 for aerodynamics of tion for the electrons; the field equations of Max- finite bodies; (ii) the other coefficients are of order well is reduced to a single equation (Poisson equa- unity. The electrostatic field potential ~b(x) must tion) (5.27) under the present electrostatic approxi- thus behave akin to the boundary layers in applied mation. mechanicsfl °~ The field potential ~(x) would have In order to gain some physical insight into this small gradient* except in a narrow region very coupled particle-field set of equations, we rewrite close to the body, known as the "boundary layer" them in terms of dimensionless quantities. In defin- in aerodynamics. It will be called the plasma sheath ing these quantities, the following reference mag- in the ionospheric flows where the stream impinges nitudes as standard of comparison are introduced: on the body. The field configuration in the near the ion thermal speed vl = (KTdmi) ~n as characteris- wake behind a moving body is much more com- tic velocity; the body size R as the macroscopic plex. c5~ In any event for a region far away from the characteristic length; effective range of particle body, the term on the left-hand side of eqn. (6.2) collisions d (defined as a transverse distance from a scattering center that a scattered particle is deflected by an angle 0 ~ 90 °) as the microscopic characteris- *The gradient can be moderately large in the near tic length; ambient ion density, n.. The dimension- wake, see the discussion that follows. 286 V.C. Lxu

plays an insignificant role, we have a quasi-neutral 7.1. Plasma Sheath flow for which n~/n~ = n,/n~ = exp ~b*. On account of the boundary-layer nature of the field equation (6.2), it is expected that adjacent to 6.2. Asymptotic Solutions the surface where the plasma stream impinges on the body, a thin sheath exists in which the variation Following the discussion of Section 6.1, we have, of 4~(x) across the sheath is rapid. In view of the under the condition ,~D/R "~ I, the quasi-neutral extreme disparity in field potential gradient across flows i.e. with n,(x)=n~(x) outside of the sheath and along the sheath, we shall use a simplified and the near wake. This corresponds to the zeroth quasi- one-dimensional sheath analysis.~u~ order approximation to eqn. (6.2). It designates Consider a collision-free mesothermal plasma flows both upstream beyond the frontal sheath stream, which is composed of fully ionized and and the far wake downstream. We have from eqn. singly charged particles, impinging on a negatively (6.2) charged plate at a small angle /3 (Fig. 2). It is assumed that the incident electrons are absorbed; n,(x) - n,(x) = n~ exp (ec~lKT) (6.3) ions neutralized and re-emitted as neutrals. The plate is fixed to the yz-plane as shown in Fig. 2 and the field potential is assumed to be infinitely large, hence no edge effect will be considered, and the field potential (6.4) depends on x only, 4~ = 4~(x). Furthermore, it is assumed that the absolute value is large enough that Gurevich°" suggested that the wake behind a only the electrons at the rare high energy tail of its spherical satellite be divided into two regions in one distribution can reach the plate and be lost. Thus of which the ion density is not very small such that the Maxwell-Boltzmann equilibrium distribution (R/ho)2n,(x)/n~> 1; in the other the ion density is for electrons is approximately valid so small that (R/,kD)2n~ (x)/n~ < I. These refer to the he(X) = n~ exp [e~b(x)] regions far from and close to the body respectively. L KTe / (7.1) The boundary separating them is so defined by the condition where n~ denotes the electron density at the outer edge of the sheath where n~ = n,~ = n~. ni(x) R) 1 (6.5) × V The field potential in the near-region can be approximated by the use of

V24~ = 0 (6.6) which is equivalent to assuming that the effect of space charge density is negligible compared with that of the surface charge. FIG. 2. Plasma sheath at an inclined plane (cb, < 0).

The field potential obeys the Poisson equation 7. PLASMA SHEATH AND NEAR WAKE d~4)2 = - 4 7re( m - nc ) = - 4 7re [ n~ - n~ exp (~T ) In view of the discussion on the nature of eqns. (6.1) and (6.2) which govern the distributions of (7.2) particles and field near a moving body (R), non- and boundary conditions: linear analysis for the sheath region in front of the ~b = 4), x = O, body and the near wake behind it are mandatory (11.3) unless the body size is small (R ,~ ;to). This stems 4,=0, x=~, from the fact that the nonlinear relation of the which implies that the sheath potential decays particle-field coupling is inherent with the problem. asymptotically to zero at ambient. The ion density We shall treat these aspects of the ionospheric n~(x) in eqn. (7.2) can be determined by means of aerodynamics first in order to gain a glimpse of the either the Vlasov equation (5.8) or the Schroedinger nature of plasma interaction. Our contemporary equation (5.13). The latter is chosen for mathemati- apprehensions about these problems are not at par. cal simplicity. While the solution to the sheath problem has almost a2~ 0"g' 8rrzm, been satisfactorily resolved, the theory of near -~-rx +~Ty,_ +~[E,-erb(x)]~=O (7.4) wake is very primitive, nowhere near to being a satisfactory solution. where E~ denotes the energy of an impinging ion On Ionospheric Aerodynamics 287 relative to the reference frame fixed to the plate. values occur in a minute distance from the plate The stream is approximately monoenergetic (E~) which may be defined as the thickness of the since V-~ >> v~. sheath. This unique "boundary-layer" behavior of the sheath makes it possible for the use of singular Let perturbation analysis,m~ The fact that the variations of field variables, 4>, • (x, y) = qt,(x)~,.(y) (7.5) ni and n, are infinitely stronger across than along be a trial for the separation of variables in qt(x, y). the sheath makes it feasible to use the present Equation (7.4) is resolvable into two component inclined-plate solution to approximate the frontal equations: sheath of a three-dimensional body which can be sliced into convex ring sections facing a mesother- dxP't 87r"md dx'- F----ffr-[Ei - ecb(x)]~' = A 2xtt'' mal stream at different inclination angles (/3). This approach is analogous to the modified Newtonian d "-xlt: dy " - A 2xtt2 theory of sphere drag at hypersonic speeds using the oblique shock relation/TM where A'- is a constant. Through the W.K.B.J. transformation (5.16), the solutions to the above equations can be readily obtained which, after the 7.2. Near Wake introduction of the upstream boundary conditions (7.3), gives When a negatively charged circular disk of radius R moves at a mesothermal speed along the direc- I'I'l: = leqb ) tl2 n,/n~. (7.6) tion of its axis in a collision-free plasma, a momen- tary void develops immediately behind the disk. 1 Ei cos"/3 Both the ambient electrons and ions rush to popu- It is of interest to compare the present result (7.6) late the void region. Since the electrons have a with that obtained by Ginzburg(32~ who studied a greater mobility they leave the ions behind giving special case (/3 =0) of the problem using hyd- rise to an electric field which decelerates them very rodynamic equations suitable for continuum flows. rapidly while slightly accelerating the heavy ions. It is found that if the contribution due to pressure- As a result, the velocities of electrons and ions term in reference 32 is neglected for a hyperthermal filling the void tend to become equal. The front of flow as with the present case, the result thus the plasma motion thus has a local negative field obtained would agree with eqn. (7.6) when/3 is put potential because of the excess electrons therein. to zero. This apparently is responsible for the presence of a The solution to eqn. (7.2) after the substitution V-shaped potential valley in the near wake/5' For a of n~ (7.6) can be written: moving blunt body having a width R across the stream and with the above depicted nonlinear cou- X = (87rn~)-w'{2Ei cos-"/3 (~,/(I - ecklE, cos"/3) pling between particles and field is an inherent trait s of the near wake. - 1)- KT,.[1 -exp(ecMKT,)]} -~12. (7.7) Inasmuch as a satisfactory theory of near wake is The sheath potential ~b(x), given in solution (7.7) not yet available, it appears more important now to is shown in Fig. 3. It is observed from Fig. 3, along expose the intricacy of the problem and make some with the number densities m (7.6) and n, (7.1), that suggestion that might serve as a guide to a prospec- they all show the rapid variation nearest to the tive solution. In order to make the discussion plate, in fact more than 95% of the drops in their definitive, we introduce an idealized model of near wake. Consider a sphere of radius R and with a negative surface potential ~b, <0 moving in a collision-free fully-ionized plasma of singly charged particles. It is assumed that the ambient plasma may be bi-thermal, i.e. having different temperatures (T,, T~) for the electrons and ions k Te respectiyely. The sphere moves at a steady mesothermal speed, namely (KT~/m~)m~ V~ (KT,[m,) ~n'. It is further postulated that upon colli- sion with the body an electron is absorbed; an ion, neutralized and re-emitted as a neutral. The body size (R) is much larger than the Debye shielding x_ length ()to). Magnetic field is absent. xD The equations governing the distributions of par- FIG. 3. Field potential of sheath. ticles and field in the near wake consist of the T: V®c°s ~= 2. qI': V®c°s ¢=4. Vlasov equation (5.10) for ions, Maxwell-Boltz- 1)l 1)1 mann distribution for electrons (7.1) and the Poisson 288 V.C. LIU

equation (5.27) in dimensionless quantities. starts with neutral particle approximation. The mathematical question of convergency in the num- af,. 1 0q~ o)~_ v ...... 0, (7.8) erical iteration of the system of eqns. (7.8)-(7.11) Ox 2 dx Ov has not been satisfactorily resolved. Suffice to say n, = exp q5, (7.9)* that there is no guarantee that such an iteration will (~ ) 2V2,~ = n, - n, = exp ¢ - f f, dv (7.10) always converge. It seems obvious that a self- consistent solution of the system must emerge only where the superscripts * have been dropped. The after a convergent iteration. set of eqns. (7.8), (7.9) and (7.10) provides a self- It is noted that the difficulty of treating the consistent system together with the boundary con- system of coupled eqns. (7.8)-(7.11) is computa- ditions: tional. For instance, if the computer capacity is of no problem, one could start with a guessed initial ¢(R) = ~b,, ~b(~)= 0, cb(x) to perform the trajectory study and then the /;(R, Vr >0)= 0, iteration. It could serve as a test for the con- ~(~, v)= ~){ 1 ~3/2exp [--~(V V,)"]. (7.11) vergency of the iteration scheme. In so doing, however, the number of parameters needed to A common practice in the contemporary studies specify an ion is considerably increased as com- of near wakes is to treat the ions as if they were pared with the neutral particle calculations. At- neutrals as far as the initial determination of ion tempts have been made to simplify the trajectory density n~(x) in the wake is concerned. Assorted specifications by the use of Jeans theorem (5.14) to free "ion" trajectories, starting from the free prescribe the ion distribution in terms of three stream, are calculated. The amount of computa- integralst of the ion motion~5'36~ in order to reduce tions thus involved depends on whether the thermal the number of parameters for specifying an ion random velocities of the free stream "ions" are trajectory. On the other hand, it is deprived of the taken into account in the parametric trajectory flexibility in the method of parametric trajectory studies. This is important because it is the transver- calculations. The primary difficulties involved in sal thermal motions of the free stream particles that the application of Jeans theorem to the near wake are primarily responsible for the filling in the void. study are as follows. (1) Two of the integrals, the Since they are perpendicular to the direction of the total energy and the axial angular momentum of a free stream, the transversal thermal velocities particle, are well known; the third one is not yet though much smaller than the longitudinal free available in the general case if at all. There has been stream velocity (V,) must be retained in the trajec- some scheme suggested to be used for the purpose tory calculations. This neutral particle assumption of short-range numerical integrationsY~ A special is tantamount to the neglect of the second term on variational method °6~ has been proposed to general- the left-hand side of eqn. (7.8). In so doing, the ize a third integral which is valid for a restricted particle and field equations are thus decoupled. The axisymmetric fieldY7~ (2) It is difficult to find the population of ions in the wake can be determined function F(L, I:, 13) tailored to satisfy the boundary by the use of the trajectory results. Once the ion conditions (7.11). It goes without saying that the density n~(x) is known, it is a simple matter to use efforts of applying Jeans theorem to the study of the Poisson equation (7.10) to evaluate a first near wake could be very rewarding. approximation to the field potential d~(x). A physical interpretation of the use of Jeans In some studies, efforts apparently have been theorem in lieu of individual trajectory specifica- made to iterate for the higher-order approximations tion can be made. While the prospective path of a to the ion density and the field potential by using given ion may be specified in terms of the parame- eqns. (7.8)-(7.11) in order to obtain a set of self- ters for its initial position, velocity and the field consistent solutions of particle and field distribu- according to the Newtonian equation of motion, it tions. No published results on such efforts for the may also be determined in terms of its integrals of near wakes with R ~>AD seem available. This motion (5.14). shruld not be confused with the works for the studies of far wakes or of near wakes with R ,~ AD 8. QUASI-NEUTRAL FLOWS AND FAR WAKE which can be done on the basis of entirely different sets of approximations as will be seen later. The condition of quasi-neutral, i.e. n,(x)= n,(x), The effectiveness of the above-mentioned prevails in the plasma flows beyond the sheath and neutral particle approximation depends on the con- the near wake. Under this simplified condition, it is ditions of the wake in question. With R -> )to the of interest to investigate the nonlinear interaction formation of a potential valley in the near wake, as between the gas dynamical and the electrostatic depicted earlier, seems inevitable. It might be diffi- phenomena for the mesothermal collision-free cult to recover this result in a computation that flows.

* The use of MaxwelI-Boltzmann distribution for field potential with valleys has been cautioned by Lam. °~ t Also called adiabatic invariants. On Ionospheric Aerodynamics 289 8.1. Isothermal Compressible Flow Analogy ( t+vo We restrict the discussion to a negatively charged body of characteristic size R which is much larger av,, 0xv (a ) • ax ~ o\-ff-~x.Vo (a=i,e). (8•7) than ;to. The electron temperature (Te) is assumed to be much larger than the ion temperature (Z)--a Recall that d/0x×V~ and a/0x×V, are initially bi-thermal state commonly prevails for the upper zero in the oncoming stream and therefore remain ionosphere particularly during the daytime. so in the flow field• The fact that the flows are We shall use the continuum equations and show irrotational proves to be important in the analysis that both the ion and the electron flow fields are later because a velocity potential exists for an irrotational and mathematically uncoupled. The irrotational flow• governing equation for the uncoupled ion flow velocity potential has been shown ~3~ to be identical 0qb~ Oqb, to that of an isothermal irrotational compressible V~=- Ox' V,= ax" (8.8) flow in classical gas dynamics. Consider now the Poisson equation (6.2) which The continuity equations (5.22) for the steady can be written: flows of ions and electrons are O (--~)'V*:d~* = n* - n, (8.9) ax" (n,V,) = O, (8.1)

a where the superscript * denotes a dimensionless ax' (n~V~)= 0. (8•2) quantity (Section 6.1), and ;tD = (KT,/47rn~e2) ''. In the limiting case of small value for the parameter The electron momentum equation (5.23) can be ;tD/R, we have written: n ~, = n* + 0[(AD/R)"] = n* (8•10) OV~ 4-V~ OV, e Oq5 1 dp, which is the precise quasi-neutral condition and is at " Ox me 8x ntm~ 8x (8.3) valid where ever 7*:&* is of order unity or less• where p, is the scalar electron pressure. The equations (8.1), (8.2), (8.4), (8.6), (8.8) and It is noted that in the absence of collisions, the (8.10) can be combined for the elimination of all the electron gas is essentially isothermal. This and the flow variables except qb~ and qb,. Recognizing the equation of state for a perfect gas, p, = n,KT,, can conditions m,/m~ ,~ 1 and V,/v,,~2z I (mesothermal be introduced to eqn. (8.3) which becomes integra- flows), we obtain finally ble along a streamline {e~ I~(~ )} V:O~ = ~mN~_. Vq~ • V (2Vqb, Vqb~), (8.11) n, = n~ exp 2 - 1 (8.4) m~V2~ -" • VOw). where V~ is the free stream velocity and v, = V'(1)e = -~-7-. vq), • V (~ V*, (8.12) (KT,/m,) ~n is the electron thermal speed. In the limit of Vdv, ~1, eqn. (8•4) is reduced to the Where mN:~/(KT,) represents the ratio of the di- Maxwell-Boltzmann distribution (7.9). rected kinetic energy of the free stream ions to the The ion mome0tum equation (5.23) can be thermal energy of the electrons it can be interpreted written: as the square of a flow roach number on the basis of ion-acoustic speed which is equal to (KT,/m,) 'n OV,+v~. OV~ e Oq5 (Section 3•2). Equation (8.11) appears like the po- at ax m, Ox" (_8.5) tential equation for an isothermal compressible Equation (8.5) is also integrable along a streamline flow in classical gas dynamics,a7~ The classical and gives the ion energy equation solutions developed for the compressible flows of neutral particles can be adopted with caution for ~.m~ ~+ e~b = Const. (8.6) the ion flows in view of the mathematical analogy which is valid as long as we can identify ion shown herein. Once the ion flow field is determined, trajectories with macroscopic ion streamlines•* the electron flow field can be obtained from eqn. This condition prevails under the mesothermal con- (8.12) which is then a linear equation• Applications dition because the oncoming ion trajectories look to mesothermal collision-free flows about circular essentially like a parallel beam. cones and other configurations are available)3~'3.~ It is important to note ~m that both the ion and the electron flow fields are irrotational since taking curl 8.2• Far-wake Flows of both eqns. (8.3) and (8.5) we obtain Another class of quasi-neutral flows that are of aerodynamic interest is that which is far behind a * For a more rigorous justification see the discussion in mesothermally moving body. We call it herein the reference 5. far-wake flows. In contrast to the near-wake flows 290 V.C. LIu

which are governed by a set of nonlinear equations, there will be a considerable elongation of the wake the gas dynamical as well as electrostatic distur- if the electron collisions with ions and neutral are bances are small enough such that perturbation neglected. Otherwise, the electrons may diffuse a analysis is applicable, hence a method of linear distance of the order of Larmor electron radius analysis can be used effectively. The structure of across the field lines as a result of a collision. When the disturbances undergoes sharp variations in the body moves oblique to the magnetic field lines, the near-wake region which extends a distance electrons and ions tend to spread into the void due of only a few body diameters. Thereafter the to their thermal motions. disturbed flow becomes essentially quasi-neutral and persists for a relatively long period of time, 9. SLENDER-BODY THEORY during which collision-free Landau damping pro- cess acts to nullify the disturbances. The body- The ionospheric aerodynamic problems treated plasma interaction can also be viewed in a different so far are of the type where the characteristic light: the rapid motion of the charged body can be dimension of the moving body is much larger than considered as a generator of low-frequency collec- the Debye shielding length (,~D). On account of the tive oscillations of the positive ions--the ion acous- distinctive differences in flow and field characteris- tic waves. tics in front and behind the moving body, they have The unique feature of a great extension of the been treated as separate entities. It is intended now collision-free plasma disturbances behind a moving to develop a unified theory of plasma interaction for body is technically important. For instance, it con- a slender body, for which R ,~ Am that moves at stitutes an excellent target for radio-wave tracking* mesothermal speed in a collision-free fully ionized of the moving object when the body itself does not plasma where magnetic field effects are unimpor- provide large enough scattering cross-section for tant. wave-echos. The plasma tail which has perturba- The dynamical consequences of the motion of a tion of the electron density that leads to the varia- charged body in a plasma can be considered from tion of the plasma dielectric constant which is two different points of view: responsible for the scattering effect. Although the (i) Waves and energy. A charged body travelling dielectric constant of the trail is lower than that through a tenuous plasma may lose energy to the of a metallic body it compensates in total scattering collective motions of oscillations by the plasma effect many-fold over by its extremely larger size. particles in addition to the energy loss by means of The linear theory of mesothermal far wakes in a aerodynamic drag. When the body speed is collision-free plasma without magnetic field has mesothermal, the important mode of waves thus been reviewed recently~5~ and will not be repeated excited must be the ion acoustic type (Section 3.2). herein. It is, however, pertinent to discuss the (ii) Disturbed particle and field distribution. The magnetic field effect on the electron motion in the motion of the body makes it necessary for the wake hence the radio-wave scattering by the wake ambient plasma in the neighborhood to adjust to of moving body in the ionosphere. The magnetic new distribution together with a new local field. The field effect on the plasma interaction in the case of a self-consistent particle and field distributions under far wake is no longer negligible because the length the influence of a prescribed motion of the body are of the plasma tail could be much larger than the the electrogasdynamic effects of interest. We shall Larmor radii of the ions. confine our discussions to the latter viewpoint. It is expected that the geomagnetic field could critically affect the length of the wake depending on 9.1. Linearized Equations of Particles and Field the orientation between the magnetic line of force and the axis of the wake. If the body is moving The ion distribution, under the influence of a along the magnetic field lines ambient electrons slender body having minute negative charge and which are moving into the void with thermal speed moving at mesothermal speeds, is described by the will be turned around by the field before they Vlasov equation (5.10)" penetrate a distance of the order of its Larmor radius. The ions, having larger Larmor radii, are f~+v. 0J~ e O~ Of,=O (9.1) much less influenced by the magnetic field. As it //t 8x m~ 8x 8v was depicted earlier, the ions cannot separate from the electron distribution, as usual, complies with the electrons by more than a distance of the order the Maxwell-Boltzmann law: of Debye shielding length ()tD). The net effect on the rne 3/2 l electrons of dragging by the ions and deterring by the magnetic field is to fill in the void with a drift motion of the electrons around the circumference (9.2) of the cylindrical void which is inconsequential as and the field potential ~b(x) obeys the Poisson far as void-filling is concerned. The net effect is that equation:

* Technically known as the RADAR technique. V% =-4~e(f f, dv- f f, dv). (9.3) On Ionospheric Aerodynamics 291

If the body is sufficiently thin and the charge on where the body sufficiently small, the perturbed ion dis- B.k tribution can be represented as o¢(k, to) =---~ f(~ "~k)Z~- _--z-.- d~' (9.16) /~(x, v, t) = f~(v) + f'(x, v, t) (9.4) Under the mesothermal condition and negligible where L is the Maxwell distribution for the undis- Landau damping, the integral (9.16) can be simp- turbed plasma lifted~9~ I m, "~"~ (_ m o~ f==n=~2~-~KT~) exp\ 2KT~] (9.5) 5~=1+0(~) (9.17) f', the perturbation. The electron density n,(x) and eqn. (9.15) becomes evaluated from eqn. (9.2) becomes t = n=mi f n KTI 3 (B • k) e "k ..... ~dk do). (9.18) n,=n~exp(-~--~)=n~e4~ (1 + KeT~T+" .') (9.6) Let U(x, t) = o/at(oG/ax), the eqns. (9.18), (9.12) where le~/KT, I ~ 1. The substitution of eqns. (9.5) and (9.9) can be simplified to the following and (9.6) in eqns. (9.1) and (9.3) yields form: ~'°.4,

Of' 4- v .....Of t - e Odp Of= (9.7) Ot Ox m~ Ox Ov' -ff~-= ~ 4~ (9.19) and (V 2- A ~")~b = -4~re( f' dv (9.8) 4~ren~m~ where AD = (KT,/4rrn~eZ) Ip-. (V2 - A ~,2)4> KT~ V2G" (9.20)

The boundary and initial conditions complying 9.2. Fourier Transform Analysis with given physical situations that are imposed on the functions ~b and G can be formulated as To solve for f' we follow the standard proce- follows: at the surface of the body we may pre- dure ~39'4°~with the introduction of a direct velocity U scribe ~b = ~b,, a constant value for the surface of a particle caused by the field potential ~(x) as potential in the case of a conducting body.* Far represented by upstream of the body,

= v - U(x, t). (9.10) f=y-*lo(x-V~t) onS (9.21) Equation (9.7) becomes where V~ is the body velocity which is along the Oft+ ~ . Of t O ( Of~) x-axis; x and y measure along the longitudinal and 0--t- -~x = ~ .-~x U.-~- (9.11) transversal dimensions of the slender body respec- and eqn. (9.8) becomes tively. Differentiating eqn. (9.21) and neglecting the highest-order term, we have the remaining terms: (V 2 - = -4Trey ft dv. (9.12) 02G OyOt =- V.rl'o on S. (9.22) The method of Fourier transform can be used to solve the system of linear equations (9.11) and Following the usual practice in thin airfoil theory, (9.12). Let the condition (9.22) may be evaluated at y = 0 instead of at the actual boundary of the body. f'(x, ~, t) = f A(k, co, ~ ) e ',~ ) dk doJ J Some physical insight into eqns. (9.19) and (9.20) (9.13) may be gained by introducing an approximation on ~b, which is appropriate for the slender bodies and (R ,~ Xo) U(x, t) = f B(k, to) e ~a..... ~dk do). (9.14)

The ion density perturbation is Under the condition (9.23), eqns. (9.19) and (9.20) n~=n,-n== f f' dv

= m----Lf e "~ ..... '3~(k, w) dk do) (9.15) * In the case of a dielectric body, either th or its normal KT, 3 derivative may be prescribed. 292 V.C. Liu become From the aerodynamic point of view, the meteor phenomenon is a conglomeration of problems of KT,. 1~ V2G (9.24) aerodynamic heat transfer, ablation, plasma e v~ dynamics in addition to radar tracking. It has O2 G practically all the essentials of the missile re-entry (KT') 2 v:c = o. (9.25) - ,,,---7/ problems and possibly at a higher intensity because meteors move at higher speeds into the Earth's For coordinates fixed to the body, eqn. (9.25) atmosphere. This has been one of the motivations becomes the usual wave equation with the speed of in the earlier studies of meteor aerodynamics. The propagation (KT,/m~) ~n equal to the speed of the ion specific aspects of meteor aerodynamics that fit in acoustic wave. Here once again we see the fluid- the present study are: (i) the problem of the initial nature behavior of the collision-free plasma due to radius of ionized meteor trails, i.e. the kinetic their collective action through a self-consistent process of interaction between the evaporated field. atoms and the ambient atoms; (ii) the ambipolar diffusion of the trail in the presence of an external I0. AERODYNAMICS OF METEORS magnetic field. The former is germane to the meteor detectability via electromagnetic wave scatter° The discussion on aerodynamic problems, so far, ing:'4:3 it also reveals that the evaporated meteor has been limited to the cases where the physical atoms undergo stages of free molecular, transi- conditions of the moving body remain unchanged. tional and continuum flows--a problem of funda- In reality, there are many situations, e.g. a missile mental importance to rarefied gas dynamics. The re-entering into the atmosphere, where the body latter constitutes an interesting application of the surface is caused to transform with considerable theories developed in Section 5.2.1. aerodynamic consequences. The remaining space A meteor aerodynamic problem is often made of the present review will be used for a glimpse at a extremely complicated by the nonlinear coupling class of aerodynamic problems for which the mov- between the various after-effects of meteor ing body of interest is allowed to change as a aerodynamics, e.g. ionization, recombination, radi- feedback to the dynamics of flow. The meteor ation, etc. The essentials of meteor phenomenon phenomenon will be taken up first and then the can be effectively treated by dividing it into two comet for which the kinetic processes of interac- parts: the gas kinetics of flow of the evaporated tion between the cometary atmosphere and the atoms and the microscopic processes of inelastic solar wind are not yet completely understood. atomic collisions. It is the first part that we shall be concerned herewith. An idealized and viable model 10.1. Statement of the Problem of the meteor trails has been proposed t43~ of which the meteor is represented by a point source of A meteor is an extraterrestrial body which moves evaporated meteor atoms whose intrusion into the at an extremely high speed, e.g. 30 km/sec, into the ambient atmosphere undergoes stages of free Earth's atmosphere. Solid bodies entering the at- molecular, transitional and diffusional flows. In the mosphere at such high speeds will be heated to initial period prior to their collisions with the am- incandescence by friction with the air and in some bient air molecules, the atom convection can be cases they are melted or vaporized. When this considered as a free molecular flow which is related happens we see a momentary streak of light in the to the determination of the initial radius of a trail; upper atmosphere which is part of a meteor after several collisions the distribution of the phenomenon. Atoms evaporating from the surface evaporated meteor atoms become essentially ther- of a meteoroid possess energies as high as hundreds realized and is close to equilibrium with the ambient of electron volts. The vaporizing atoms are ionized air--a diffusional flow which is often called a principally in the first collisions with atoms of the convection-diffusion process. The motion of the ambient air. The ions and electrons produced ion species, produced by the meteor atomic colli- thereby form a cloud of quasi-neutral plasma sions, obey the law of ambipolar diffusion which is whose concentration at the time of plasma forma- further complicated by the presence of a magnetic tion is usually higher than the concentration of the field. ionospheric plasma at those altitudes. The meteor wake formed in the process dissipates by ambipolar 10.2. Formulation of Aerodynamics of the Meteor and possibly turbulent diffusion. Radar techniques Trail which are adopted to observe echoes from the ionized meteor trails have been useful tools for the Consider a stationary point source of evaporated study of the upper atmosphere. Various scattering meteor atoms at the origin (x = 0). The distribution theories have been developed that give the amp- of meteor atoms having velocity v at time t and litudes and durations of the radar echoes in terms position x is denoted by f(x, v, t). They interact of the ionization densities in the trails among other with the ambient atmospheric molecules (or atoms) factors. whose distribution is denoted by F(v~) which is On Ionospheric Aerodynamics 293 assumed stationary. The transfer of the dynamic the ambient moleculesJ'3~ In addition the solution characteristics of the meteor atoms Q(v) in the also has terms denoting residual disturbances phase space (x, v) can be represented by a moment which persist at all points traversed by the wave equation of Boltzmann as given in Section 5.2.1. front. ~4~ The telegrapher equation thus lies between the simple wave equation whose solutions have a O(nm(Q)) ~_ 0_.0_ . (n,,(vQ)) wave front but no residual disturbances and the Ot Ox classical diffusion equation whose solutions have = f f f [O(v')-O(v)]F(v,)f(v) residual disturbances but no wave front. The differ- ence between them becomes indistinguishable after xlv~-vlGdg) dvdv~ (10.1) a few collisions from the initial instant at t = 0. The where the collision-term (Q(Sf/6t)c) has been pre- stationary point source solution to eqn. (10.4) has scribed in terms of Boltzmann's binary collision been obtained~'3~ which gives the distribution of integral~23'~3~ which is valid for the collisions be- meteor atoms evaporated from a stationary point tween meteor atoms and the atmospheric molecules meteorite. The corresponding distribution for a or atoms. uniformly moving meteorite can be determined by a The incomplete understanding of the initial vel- simple application of the Galilean transformation to ocity distribution at the source and the interest to the stationary source solution as has been done in obtain a uniform representation including free the heat conducting problem for a uniformly mov- molecules to diffusional flow prompt us to use the ing sourceJ 45~ transport equation (5.21) instead of the Boltzmann The present result of the initial expansion of the equation (5.1). With the choice of Q = I and Q = v, cloud of evaporated meteor atoms can be used to eqn. (10.1) can be simplified into the following make a rational determination of the "initial radius" forms: ~23,431 of the ionized meteor trails which were defined on the basis of a variable mean free path approach in 0nm + ___O Ot 0x '(nV"')=n"(vl-va) (10.2) the meteor publications, which is an unsatisfactory way of treating nonequilibrium kinetic processes. and Considering the high speed with which the evapo- O(mn,.V,,,)+O = -mn.,vV,,, (10.3) rated meteor atoms leave the meteor surface, cor- responding to a kinetic energy of several electron respectively where v, denotes the frequency of volts for some meteors, the present theory thus ionization of meteor atoms: vA that of attachment, provides an elucidation of an earlier observation which found the initial spreading of a meteor trail P,., the pressure tensor that has diagonal elements* "explosively" fast, not at all diffusionlike."6~ only. The elimination of V,, between eqns. (10.2) and It is also of interest to use the present theory to (10.3) yields explain the double trails observed for some fast meteorsS ~ It was observed that a meteor at an 102n,,, 1( v,- va) Onm VI-- IJA V"n°, = ~--~-~- + ~ 1- nm altitude of 100-110 km has a broad trail that shows v Ot D an outer trail with a diameter about 20cm that (10.4) surrounds a bright inner trail of much smaller width where A 2= KT/m, D = KT/mv with T, assumed with sharp borders between these concentric trails constant, denoting an effective kinetic temperature at the initial instant of the trail's formation. Now of the evaporated meteor atoms of mass m. The consider the atomic streams emitted from the sur- quantity D approximately equals the diffusion face of a meteorite; two distinctive groups must coefficient of the meteor atoms in the ambient air. exist: (i) the evaporated meteor atoms with a cer- The quantity A represents the propagation speed of tain kinetic energy of escape from the meteor a small pressure impulse of meteor atoms assum- surface, (ii) the atmospheric molecules which are ing an isothermal process during the propagation. reflected specularly from the meteor surface with velocities of the order of meteor speed. Besides the 10.3. Nature of the Solution of Meteor Trail Equa- difference in their velocities, these streams differ, tions of course, in composition and ionization collisions as well. Therefore there exist two separate wave Notice that eqn. (10.4) is a so-called telegrapher fronts with different speeds of propagation into the equation~'~ where A denotes the dissipationless ambient air that account for the observed double propagation speed of a telegraph signal; D, the trails according to the present theory of meteor coefficient of diffusion of the signal during propaga- trails. tion. The solution to eqn. (10.4) will have a well- defined wave front representing an average be- 10.4. Ambipolar Diffusion of Ionized Meteor Trails havior of meteor atoms prior to their collisions with The ionization collisions between the evaporated * This assumption implies that the flow of the meteor meteor atoms and the ambient air are the primary atoms is frictionless (see Section 5.2.2(b)). source of ionization production in the meteor trails. 294 V.C. LIU

Since the electron density of a meteor trail deter- where v~, v, and the ion and electron collision mines its detectability during a radar observation, it frequencies and fL, ~, are the Larmor gyrofre- is of interest to study the diffusion process of the quencies respectively. At meteor altitudes, D~l will positively charged ions and the free electrons in a be close to D~ but D~ is greatly reduced at altitude, meteor trail particularly in the presence of an say, above 95kin. It is reasonable to treat the external magnetic field. ambipolar diffusion of a meteor trail above 95 km as In the absence of a magnetic field, it is noted that an isotropic diffusion with the longitudinalcoefficient the large difference in the diffusion coefficients for equal to DA and the transverse coefficient D. ions and electrons will lead to their separation. The On _ [32n+ 32n\ 02n separation of the charged particles will produce a--~ = u~ ~ ~-~x ~. ~y~ } + D,, ~y , (10.9) electrostatic forces tending to resist the separa- tion. As a result, the electron diffusion rate is with the magnetic field in the y-direction and decreased while that for the ions is increased and K, ID~ + K~zD, l both particles diffuse at the same rate given by the Dl (10.10) K,~ + K;~ coefficient of ambipolar diffusion Da which is a function of the diffusion coefficients and mobilities The solution to eqn. (10.9) is of ions (D~, K,) and electrons (D,, K,). /x'+z" y" \] It has been showff ~ that in the absence of a . = 47rDI(DAQ ):'t exp +4- at)j. (lOll) magnetic field, the ambipolar diffusion of a weakly ionized gas obeys the classical diffusion equation: 11. COMETARY GAS DYNAMICS ~-On = Da V2 n (10.5) From the gas dynamics of meteors to that of where quasi-neutrality is assumed, i.e. n = n~ = n, comets, another degree of complexity is added. with singly-charged particles and While the cause and nature of the meteor atom evaporation and dispersion as a result of K,D~ + KiD, aerodynamic heating are well known, the composi- DA = K, + K; (10.6) tion of the cometary atmosphere and its micros- The use of Einstein's relation between the mobility copic interaction with the oncoming solar wind are and diffusion coefficient of the particles, namely yet to be settled. None the less, macroscopic K,/D, = e/(KT,) and K~/D~ = e/(KT~) where T; and theories of cometary gas dynamics have already T, are the ion and the electron temperatures re- served important causes in the contemporary as- spectively, helps to simplify eqn. (10.6) for Da. This trophysics. simplification is found particularly effective consid- ering the large mass ratio of ions and electrons thus 11.1. Astrophysical Significance of the Cometary (Te/D,)(DdTO~ 1 we have DA =D~(I + T,/T~) and Phenomena with an isothermal plasma T~ = T,, DA = 2D, If we can represent a meteor trail by a line source The primordial origin of comets in the solar of strength Q which diffuses into a neutral atmos- system is still being debated among astrophysi- phere, the solution to eqn. (10.5) becomes cists. Suffice to say that comets are astronomical objects that may contain frozen gases like ammonia and probably water. They appear to contain large n(r,t)=4~at exp[-4-~at] (10.7) amounts of hydrogen, trapped in these molecules, where t denotes the time lapse after the passage of or larger molecules that can break up into NH~, OH, the meteor; r, the radius from the trail axis referring CO2 and CH on exposure to solar radiation. There to a coordinate system fixed to the meteor in may be large amounts of frozen hydrogen gas question. It is noted that in the above diffusion present as well. Some of the comets at large analysis, the initial radius effect which accounts for distances from the sun may represent deep-frozen the free molecular and transitional regions of the samples of matter preserved from the early solar trail has been ignored. system and therefore are interesting objects to In the presence of a magnetic field, charged study if the history of the solar system is to be particles tend to gyrate around the magnetic field reconstructed. with their respective Larmor frequencies until The continual heating by solar irradiation can these are interrupted by particle collisions. As a evaporate most of the short-period comet gases. result, a magnetic field reduces the ion and electron The comet nucleus itself is too small to hold on to diffusion coefficients transverse to the field to D~ these gases through its gravitation and soon the and D,~, respectively, leaving the longitudinal entire comet disintegrates. If it has a solid core, coefficients unchanged. These transverse diffusion only that core would remain and become an asteroid or a meteorite in the solar system. Some comets coefficients can be approximated as follows:TM have elliptic orbits about the sun; their periods may 2 D,~=~D, (a=i,e) (10.8) range from a few years to many hundreds of years. Other comets have nearly parabolic orbits and must On Ionospheric Aerodynamics 295 be reaching the sun from the far reaches of the relevant to the study of the contemporary cometary solar system. gas dynamics. A typical comet has a fuzzy head, called the To simplify matters for the sake of discussion, coma, surrounding a bright nucleus and a long tail we consider a cometary atmosphere with sufficient when its orbit is under closer influence by the sun. ionized components such that its conductivity is The intense interest in the studies of comets stems high enough to induce an electric current that also from the fact that comets can be considered as interacts with the solar magneto-plasma. Under natural probes of the interplanetary plasma or solar such conditions, a bow hydromagnetic shock wave wind; hence potentially may supply astrophysical can be expected for the interaction between the information which can be extracted from ground or oncoming solar wind and the comet. With an as- in situ observations. The gasdynamic interaction of sumed shock discontinuity in the flow field, solar wind with a comet which has its own atmos- the gas dynamic problems becomes a simple phere without its own magnetic field is somewhat hypersonic flow over a blunt body c27~ with, of like that with a non-magnet planet, e.g. Mars or course, proper consideration of the hydromagnetic Venus. In fact, it was Biermann's observation~'~ of aspect of the flow. The kernel of the difficulties with the comet tails that led him to predict a continuum such an approach is the irreconcilable time scales: streaming of plasma from the sun, now known as the observed time scales of the variation in comet- the solar wind.~4~ ary structure and of the likely ionization processes either by charge transfer or by absorption of solar ultraviolet light are orders of magnitude apart. 11.2. Nature of Cometary Gas Dynamics It is illuminating to follow Biermann et al. "8~ who investigate the macroscopic feature of the plasma On the basis of observational data, a gross de- flow on the sunward side of a comet by means of a scription of comets can be made. The coma has stationary quasi-hydrodynamic model. The hyd- essentially a spherical volume centered on the rodynamic equations are modified by adding source cometary nucleus from which neutral molecules terms to take into account the influence of the appear to be moving away with velocities of about added plasma of cometary origin. 0.5 km/sec. Charged particles are produced from The inviscid hydrodynamic equations modified the neutrals either by absorption of solar radiation by source terms are as follows: or by their interaction with the solar wind. The question pertaining to the mechanism of this in- an+ a ~- ~x • (nV) = A, teraction is still unsettled. The coma can be de- tected out to distances of 10~ to 10 6 km from the Op a nucleus. Another basic structural element of a ~+~ .(pV)=B, comet is the tail which can be as long as 108 kin. There are two distinctive types of tail structures. °(PV)~v. (pVl+pV 7x.V +-~x=C, The Type I tails which point straight away from the at sun with filamentary structure in the form of tail rays are composed of ionized gases. The Type II 3-t PT + +-~x "V p + p =D, tails, which are composed mainly of dust, are strongly curved, broad and apparently without ex- where n, p, p represent the number density (ions tensive fine features. The Type II tails are generally plus electrons) the mass density and the pressure shorter than the Type I tails. tensor of the plasma (including the magnetic stress It should be noted that the construction of a tensor) which has only isotropic diagonal compo- theory of cometary gas dynamics is still plagued nents hence have been replaced by a scalar, V its with uncertainties about some basic physical con- mass velocity vector and 3' denotes the ratio of cepts on which it is built. For instance, the hyd- specific heat (3' = 2 to include the magnetic pres- romagnetic interaction between the solar (magneto- sure H:/8~'). The source terms A-D describe the plasma) wind and the cometary atmosphere rests local gains which the plasma undergoes in the on the assumption that the latter becomes ionized corresponding quantities due to the (charge) ex- sufficiently fast somehow in order to be coupled to change of particles with the neutral cometary gas. the former by means of a convected or fluctuating Simple order of magnitude analysis shows that in a field; yet an ionization process suitable for such steady supersonic flow the source term B for the results has not been established. Besides, the study mass conservation equation has the predominant of cometary gas dynamic interaction depends criti- source effect on the system. Using a simplified cally on the rate of ionization of the cometary gas in one-dimensional model, it can be shown "~ that question. In view of this dilemma, it serves no under their assumed ionization rate for the comet- important purpose by performing detailed exercise ary molecules hence the mass accretion rate for the of cometary gas dynamic calculations. Instead, we plasma flow, a shock transition from supersonic to shall dwell on the criterion for the establishment of subsonic flow is necessary. Biermann et al. ~"~ pro- hydromagnetic interaction between the solar wind ceeded to give an elaborate flow-field analysis for and the ionized cometary atmosphere which is the interaction between the solar wind and a comet. 296 V.C. Liu

The tentative solution of Biermann et al. 148~ for buch der Physik, Bd. 22 (ed. S. FLUGGE), Springer- comets and their extension to Mars and Venus must Verlag, Berlin, 1956. 4. E. N. PARKER, blterplanetary Dynamic Processes, be subjected to experimental verification on the Wiley, New York, 1963. basis of in situ observations by space probes. Some 5. V. C. LIu, Ionospheric gas dynamics of satellite and features of the comet, Mars or Venus that have diagnostic probes, Space Sci. Rev. 9, 423 (1969). bearings on the consequence of the theory are 6. V. C. LIU, S. C. PANG and H. JEW, Sphere drag in flows of almost-free molecules. Phys. Fluids, g, 788 experimentally verifiable, e.g. the extent of the (1965). upstream disturbance must be limited by the shock, 7. F. F. CHEN, Electric probes. In: Plasma Diagnostic existence of high energy spikes of electrons or ions Techniques (ed. R. H. HUDDLESTONE and S. L. as a result of a strong shock, etc. LEONARD), Academic Press, New York, 1965. The causes of ionization of the cometary gas are 8. S. K. MITRA, The Upper Atmosphere, The Asiatic Society, Calcutta, 1952. still not clear since all the mechanisms considered 9. M. NICOLET, The properties and consitution of the are too weak to provide the observed degree of upper atmosphere. In: Physics of the Upper Atmos- ionization. It was proposed c49~that the motion of the phere (ed. J. A. RATCLIFFE), Academic Press, New solar magneto-plasma relative to the neutral comet- York, 1960. ary atmosphere obeys the critical velocity 10. J. A. RATCLIFFE and K. WEEKS, The ionosphere. V (~0) Ibid. hypothesis of Alf en which implies that when the 11. H. FRIEDMAN, The Sun's ionizing radiation. Ibid. relative velocity between a neutral gas and a mag- 12. J. A. RATCL[FFE, An Introduction to the Ionosphere netized plasma increases to a value of (2e~/mn)~, and Magnetosphere, Cambridge, 1972. then ionization of the gas will increase abruptly. 13. J. M. BURGERS, Flow Equations for Composite Gases, Academic Press, New York, 1969. Here 4~ and mn are the ionization potential and 14. T. H. STIX, The Theory of Plasma Waves, McGraw- mass of the neutral atom respectively. Considera- Hill, New York, 1962. ble discussions pertaining to Alfven's hypothesis 15. D. V. SHAFRANOV, Electromagnetic waves in are available in the literature. ~-~ plasma. In: Reviews of Plasma Physics, 3 (ed. M. A. So far our discussion has been limited to the LEONTOVICH), Consultant, Bureau, New York, 1967. 16. H. ALFVEN and C.-G. FALTHAMMAR, Cosmical sunward side of the cometary gas dynamics. To Electrodynamics, Clarendon Press, Oxford, 1963. study the dynamics of the tails, let us start with the 17. S. CHANDRASEKHAR, Stochastic problems in acceleration observed in the comet tails. The Type I physics and astronomy. Rev. Mod. Phys. 15, 1 (1943). tails normally have high accelerations of the order 18. V. C. LIu, Interplanetary gas dynamics. In: Advances in Applied Mechanics, 12, Academic Press, New of l02 or 103 times solar gravity, while the dust tails York, 1972. (Type II) have accelerations of the order of solar 19. R. J. HUNG and V. C. LIu, Effect of the temperature gravity (which is equal to 0.6 cm/sec: at l astronom- gradient on the drift instabilities in a high-/3 collision- ical unit from the sun). The accelerations in dust less plasma. Phys. Fluids, 14, 2061 (1971). tails can be explained by consideration of the solar 20. P. G. KURT and V. I. MOROZ, The potential of a metal sphere in interplanetary space. Planet. Space radiation pressure, the solar gravity and the Sci. 9, 259 (1962). aerodynamic drag of the free expanding dust parti- 21. G. L. BRUNDIN, Effect of charged particles on the cles in a free expanding cometary gas. 's2' However, motion of an earth satellite. AIAA J. 1, 2529 0963). the accelerations in ionized tails are far too high to 22. L. D. LANDAU,The transport equation in the case of be explained by this mechanism. It was this ob- Coulomb interactions. In: Collected Papers (ed. D. TER HAAR), Pergamon Press, London, 1965. served anomalous high outward acceleration of the 23. W. P. ALLIS, Motion of ions and electrons. In: comet tails that prompted Biermann to propose the Handbuch der Physik Bd. 21 (ed. S. FLUGGE), existence of some general streaming of plasma Springer-Verlag, Berlin, 1956. particles outward from the Sun. Biermann has 24. V. C. LIo, A wave mechanical approach to the plasma interaction problems. Nature, 208,883 (1965). pointed out that the ionization and excitation of the 25. P. M. MORSE and H. FESHBACH, Method of molecular ions can be explained only by some kind Theoretical Physics, McGraw-Hill, New York, of particle flux, presumably the same solar stream- 1953. ing responsible for the acceleration. 26. S. GOLDSTEIN, Modern Development in Fluid Dynamics, Clarendon Press, Oxford, 1938. .The basic filamentary structure of the Type II 27. H. W. LIEPMANN and A. ROSHKO, Elements of comet tails appears to be compelling evidence for Gasdynamics, Wiley, New York, 1957. magnetic tails in the ionized gas comet tails. Pursu- 28. J. A. RATCLIFFE, Magneto-ionic Theory and its ing along this line of argument, one may suggest Applications to the Ionosphere, Cambridge, 1959. various modes of plasma instabilities to explain the 29. G. F. CHEW, M. C. GOLDBERGER and F. E. LOW, The Boltzmann equation and the one-fluid hyd- fine structure of the tails. romagnetic equations in the absence of particle colli- sions. Proc. Roy. Soc., A 236, 112 (1956). 30. G. F. CARRIER, Boundary layer problems in applied REFERENCES mechanics. In: Advances in Applied Mechanics, 3, Academic Press, 1953. 1. L. A. ARTSIMOVICH, Controlled Thermonuclear 31. A. V. GUREVICH, Perturbations in the ionosphere Reactions, Gordon & Breach, New York, 1964. caused by a travelling body. Planet. Space Sci. 9, 32 2. G. P. KUIPER, The Sun, University of Chicago, (1962). Chicago, 1962. 32. M. A. GINZBURG, Double electrical layer on the 3. B. F. SCHONLAND, The lighting discharge. In: Hand- surface of a satellite. ARS J. (Suppl.), 32, 1794 (1962). On Ionospheric Aerodynamics 297

33. S. H. LAM, Unified theory for the Langmuir probes in Soc. 121, 183 (1960). a collisionless plasma. Phys. Fluids, 8, 73 (1965). 43. V. C. LIU, Fluid Mechanics of meteor trails. Phys. 34. V. C. LIU, On the drag of a sphere at extremely high Fluids, 13, 62 (1970). speeds. J. Appl. Phys. 29, 194 (1958). 44. J. A. STRA'VrON, Electromagnetic Theory, McGraw- 35. S. H. LAM, On the capabilities of the cold-ion approx- Hill, New York, 1941. imation in collisionless plasma flows. In: Rarefied Gas 45. H. S. CARSLAW and J. C. JAEGER, Conduction of Dynamics (ed. G. L. BRUNDIN), Academic Press, Heat in Solids, Clarendon Press, Oxford, 1959. New York, 1967. 46. I. HALLIDAY, Meteor wakes and their spectra. As- 36. V. C. LIu and H. JEW, On the field anomaly of near trophys. J. 127, 245 (1958). wakes in a collisionless plasma. Phys. Lett. 43A, 159 47. B. A. MIRTOV, The mechanism of formation of a (1973). meteor trail. Soviet Astron. A. J. 4, 485 (1960). 37. D. LYNDEN-BELL, Stellar dynamics. Potentials with 48. L. BIERMANN, B. BROSOWSKI and H. U. SCHMIDT, isolating integrals. M.N. Roy. Astron. Soc. 124, 95 The interaction of the solar wind and a comet. Solar (1962). Phys. 1, 254 (1967). 38. S. H. LAM and M. GREENBLATT, Flow of a collision- 49. W. F. HUEBNER, Relation between plasma physics less plasma over a cone. AIAA 3. 3, 1950 (1965). and astrophysics. Comment about comet tails. Rev. 39. L. KRAUS and K. M. WATSON, Plasma motions Mod. Phys. 33, 498 (1961). induced by satellites in the ionosphere. Phys. Fluids, 50. H. ALFVEN, Collision between a nonionized gas and 1,480 (1958). a magnetized plasma. Rev. Mod. Phys. 32, 710 (1960). 40. H. YOSHIHARA, Motion of thin bodies in a highly 51. J. C. SHERMAN, Review of the critical velocity of rarefied plasma. Phys. Fluids, 4, 100 (1961). gas-plasma interaction. Astrophys. Space Sci. 24, 487 41. S. RAND, Damping of the satellite wake in the ionos- (1973). phere. Phys. Fluids, 3, 588 (1960). 52. M. L. FINSON and R. F. PROBSTEIN, A theory of 42. J. S. GREENHOW and J. E. HALL, The importance of dust comets. I. Model and equations. Astrophys. J. initial trial radius on the apparent height and number 154, 327 (1968). distribution of meteor echoes. M.N. Royal Astron.