THE UNIVERS ITY OF MICHIGAN COLLEGE OF ENGINEERING Department of Electrical Engineering Space Physics Research Laboratory Scientific Report No. KS-1 THE LUNAR ATMOSPHERE F. Hinton D. R. Taeusch ORA Project 03555 under contract with: NATIONAL AERONAUTICS AND SPACE ADMINISTRATION GODDARD SPACE FLIGHT CENTER CONTRACT NO. NASw-133 GREENBELT, MARYLAND administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR October 1962

TABLE OF CONTENTS Page LIST OF TABLES v LIST OF FIGURES vii NOMENCLATURE ix CONSTANTS xiii SECTION I. INTRODUCTION 1 SECTION II. A SURVEY OF EXPERIMENTAL AND OBSERVATIONAL RESULTS 3 A. The Lunar Atmosphere 3 B. The Solar Wind and the Interplanetary Material 3 C. The Selenomagnetic Field 5 D. The Lunar Surface Temperature5 SECTION III. SOME GENERAL IDEAS CONCERNING THE LUNAR ATMOSPHERE 9 A. The Molecular Distribution Function and Liouville's Theorem 9 B. The Migration of Molecules Over the Lunar Surface 12 C. The Barometric Law 19 1. Gravitational Condensation 20 2. The Scale Volume 22 3. The Assumption of No Collisions 23 D. Criticism of the Use of the Barometric Law for the Lunar Atmosphere 24 E. The Jeans Evaporation Flux 28 F. Sources for the Lunar Atmosphere 29 G. The Effect of the Solar Wind on the Lunar Atmosphere 32 1. Assumptions Concerning the Solar Wind 32 2, Elastic Scattering 35 35 Charge Exchange 39 H. Photoionization and Photodissociation in the Lunar Atmosphere 40 SECTION IV. A SIMPLE MODEL OF THE LUNAR ATMOSPHERE 43 A. Hydrogen and Helium 44 B. Water Vapor 49 C. Argon 49 D. Krypton 50 E. Xenon 50 iii

TABLE OF CONTENTS (Concluded) Page F. The Variation With the Strength of the Solar Wind 50 G. The Lunar Surface Charge and Charged Particles in the Lunar Atmosphere 51 SECTION V. DISCUSSION OF THE PROPOSED MEASUREMENT 57 A. Instrumentation 57 B. Calibration 58 C. Measurement 58 D. Contamination 59 Eo Solar Flux 61 REFERENCES 65 iv

LIST OF TABLES Table Page 1 Atmospheric Particle Range, Time of Flight and Scale Height (T = 300~K) 17 2 Assumed Source Strengths 31 3 The Assumed Minimal Model of the Solar Wind 34 4 Cross Sections for Elastic Collision and Charge Exchange With 1300 ev Protons 39 5 Numerical Values for Eq. (4-5) 47 6 Ion Scale Height, Scale Volume and Escape Cross Sections 55 7 Retro Rocket Gases 64

LIST OF FIGURES Figure Page 1. Assumed equatorial moon surface temperature (Tm) and instrument package temperature (Tp) vs. suns angular position. 0~ at night to day terminator. 7 2. Particle trajectory coordinates. 13 3. Angular range vs. temperature. 18 4. Scale height vs. sun angle; dimensioned from center of moon, 21 5. Collision coordinate system. 35 6. Collision coordinate system. 35 7. Lunar model atmosphere. Surface number density vs. solar particle flux. 56 8. Ratio of gauge number density to atmosphere number density. 60 vii

NOMENCLATURE A atomic weight B magnetic field in gauss C root-mean-square thermal speed of particle E kinetic energy of a particle Ef kinetic energy of incident particle after collision Eflow kinetic energy of a particle at the mean velocity of the flow in the solar wind ET total energy of a particle, kinetic plus potential G universal gravitational constant H Hamiltonian function for a particle TlI2,I 3 escape and production mechanism integrals J solar wind flux in particles/cm2/sec Jj Jeans evaporation flux Jtp proton flux in the solar wind Js source strength in particles/cm2/sec M mass of moon N total number of particles in the lunar atmosphere Nc total number of particles in a square centimeter column Ni total number of ions in the lunar atmosphere N(t) number of atoms produced per cm2 of lunar surface in a time t by radioactive decay processes P magnitude of angular momentum of a particle ix

NOMENCLATURE (Continued) S volume production rate of particles T temperature; QK Tg temperature of gauge Tm temperature of moon V scale volume Z charge of an ion in units of the electron charge a(t) activity in disintegrations/sec/cm2 b impact parameter c speed of particles mean speed of particles e electron charge in esu f molecular distribution function g acceleration of gravity at the moon's surface h scale height for an isothermal atmosphere k Boltzmann's constant m mass of a particle in grams me mass of electron mP mass of proton ml mass of incident particle in a collision m2 mass of target particle in a collision n number density of particles in particles/cm3 ng number density of particles inside gauge

NOMENCLATURE (Continued) no number density of particles in the lunar atmosphere near the surface nHi number density of positive ions near the surface n>, number density of particles in interplanetary space nsg number density of particles inside gauge which originated in the solar wind Pi momentum conjugate to the ith position coordinate qi ith position coordinate of a particle r distance from the center of the moon ro radius of the moon tc "self-collision time" for particles in a plasma teq time of equipartition for particles in a plasma tf time of flight of a particle t1/2 half-life for radioactive decay u velocity of particles <U>.avg mean velocity of particles vesc escape velocity at the lunar surface Vorb orbital velocity at the lunar surface angle between the velocity vector of a particle and the vector from the center of the moon, or the vertical direction the fraction of the particles in the lunar atmosphere which are exposed to the sun 7 the geometric probability for escape of a particle from the atmosphere d2 "volume" element in phase space xi

NOMENCLATURE (Concluded) angle between the centerline of the orifice and the sun angle of scattering in center-of-mass coordinates angle of scattering of incident particle A mean-free-path of particle e9o mean path for deflection by 90~ ira2 gas kinetic cross-section of a particle, calculated from measured transport properties p range of a particle on the moons surface ZCoo photoionization or photodissociation probability per sec a cx) measured charge exchange cross section a (l) measured elastic collision cross section with a charged particle a(cx) charge exchange "escape cross section" esc'eI elastic "escape cross section" ia'e elastic "escape cross section" for a positive ion a(e) differential scattering cross section T mean-free-time of a particle angle of scattering of target particle in a collision with an incident particle angular range of a particle on the moons surface 4r angle used in describing the trajectory of a particle, measured from the apogee xii

CONSTANTS Earth: Radius: 6,371 km Mass: 5.975 x 1024 kg Mean Distance from Sun: 1,495 x 108 km Mean Velocity in Orbit Around Sun: 29.76 km/sec Moon: Radius: 1,739.3 km Mass: 75343 x 1022 kg Mean Distance from Earth: 53844 x 105 km Mean Velocity in Orbit Around Earth: 1.03 km/sec Surface Gravity: 161.9 cm/sec2 Escape Velocity at Surface: 2.37 km/sec Orbital Velocity at Surface: 1.68 km/sec Universal Gravitation Constant: G = 6.670 x 10-8 dyne cm2/gram2 One Electron Volt = 1.602 x 10-12 erg = k(ll606~K) Boltzmann s Constant: k = 15380 x 10-16 erg/~K Avagodro's Number: 6.025 x 1023 mole-1 Proton Mass: mp = 1.660 x 1024 grams Electron Mass: me = 9.108 x 10-28 grams (mp/me = 1836) Electron Charge: 4.803 x 10-10 esu. xiii

SECTION I INTRODUCTION At present, this laboratory, in cooperation with the Goddard Space Flight Center, Aeronomy and Meteorology Division, is under contract with NASA to design, construct, test and calibrate an instrumented package capable of measuring certain atmospheric parameters on the moonts surface for the purpose of determining the lunar atmospheric structure. This package is to be flown on one or more of the Surveyor soft-landing lunar vehicles to be launched in 1965. This report is the result of a study conducted to obtain information about known facts and theories concerning the lunar atmosphere and to postulate a theory of its structure. The problem that presents itself here is that the moon's surface atmosphere is almost certainly an exosphere, whose characteristics are determined not only by the moon's presence and gaseous contribution itself but also by the effect of the solar wind on the surface materials and gaseous envelope. Since the solar wind itself is not well understood, theoretical determination of the structure of a lunar atmosphere becomes complex, and at best only order-of-magnitude effects of the various known inputs can be determined by theoretical means. The over-all effect of all inputs on the lunar atmosphere has been a subject of much debate and will be known only when we can measure experimentally the parameters, such as neutral number densities of the various constituents, ion number density and particle temperatures Section II of this report delineates the limited experimental and observational results to date of measurements of the physical properties of the lunar atmosphere, "solar wind," lunar magnetic field and lunar surface temperature. In Section III, the general concepts of the lunar atmosphere are discussed with reference to the mechanisms which affect an exosphere. A simple model of the lunar atmosphere is described in Section IV. Section V describes the experimental measurements to be made by this laboratory, the expected problems and the meaning of the expected results. Since the period between July of 1964 and July of 1965 is expected to be a period of minimum solar activity, we will be concerned here primarily with conditions during "quiet sun, " 1

SECTION II A SURVEY OF EXPERIMENTAL AND OBSERVATIONAL RESULTS A. THE LUNAR ATMOSPHERE For a long time, astronomers attempted to find some evidence of an atmosphere on the moon, but their results showed that the lunar atmosphere is practically non-existent. A summary of the attempts to find evidence of the lunar atmosphere by using refraction phenomena, spectroscopic methods, and the brightness and polarization of the scattered light'has been given by Sytinskaya.l Here, we will only mention the measurements of the French astronomer Dollfus,2 which have been the most sensitive attempt at an optical determination of the density of the lunar atmosphere, Dollfust results do not give a positive determination of the density; rather, an upper limit for the density is set by the sensitivity of his apparatus. In measuring the polarization of the scattered light from the moon, Dollfus showed that the density of the lunar atmosphere is less than 10-9 of the density of the terrestrial atmosphere at sea level, which corresponds to a number density of about 1010 particles/cm3, Similarly, an upper limit for the density of electrons in the lunar atmosphere at the surface was derived by Elsmore.3 On the basis of a refraction of radio waves of 357 meter wavelength estimated from the difference between the observed and calculated times of obscuration for the lunar occultation of the Crab Nebula at Cambridge on Jan. 24, 1956, Elsmore deduced an electron density of about 103/cm3 in excess of that of the surrounding interplanetary medium, He estimates that the density of the atmosphere at the surface of the moon is about 2 x 10-13 of that of the earth's atmosphere at normal temperature and pressure, or about 5 x 106 particles/cm3. (It is interesting to note in this connection that Jan., 1956 appeared approximately halfway between sunspot minimum and sunspot maximum. Thus, the density during quiet sun, or sunspot minimum, would be expected to be somewhat different. This will be discussed in Section IV,) Evidence of an increased electron density in the vicinity of the moon has also been given by Krasovskii,4 On the basis of ion trap measurements performed on Soviet cosmic rockets, he states that the density of ionized particles at a distance of some radii from the earth is of the order of several thousand ion pairs/cm3 or less, and that somewhat higher currents between the electrodes of an ion trap were observed near the moon. B. THE SOLAR WIND AND THE INTERPLANETARY MATERIAL Because the lunar atmosphere is so tenuous we expect that the interplan5

etary medium may have a considerable effect on its structure. It appears that the moon's surface cannot possibly be shielded by its atmosphere from the interplanetary medium or the solar radiation. In addition to the electromagnetic radiation, the sun also emits radiation in the form of charged particles. As shown by Herring and Licht,5 these have a considerable effect on the lunar atmosphere. The following is a summary of the evidence concerning the corpuscular radiation, which has been called by Parker6 the "solar wind." Based on the observation of the acceleration of Type I comet tails, Biermann7 has presented evidence for the presence of an outward streaming component of ionized gas from the sun. He suggests that the solar corpuscular radiation consists of ions and electrons in the same number, about 103 to 105/ cm3, which are emitted from the sun with ordered velocities of about 1000 km/ sec. In a discussion of Biermann's paper, Kiepenheuer8 has derived a lower limit for the particle density of about 6/cm3 assuming a velocity of 1000 km/ sec. In a series of papers6 Parker summarized various geomagnetic effects which are assumed to be associated with discrete solar streams from an active sun. From these considerations, densities for quietest conditions appear to be at least 100 ions/cm3 with a minimum velocity of 500 km/sec. Parker9 asserts that the corpuscular radiation represents an outward streaming of matter from the sun that is hydrodynamic in character: it consists of an electrically neutral plasma in which the mean-free-path for interparticle collisions is small compared with the dimensions of the flow. Using the hydrodynamic equations, Parker shows that the solar corona must expand, and he believes that the interplanetary material and the solar corpuscular radiation are both merely the hydrodynamic continuation of the expanding solar corona. The density of the interplanetary material has also been estimated from the results of optical measurements. In Feb. and March, 1952, Behr and Siedentopf10 observed the brightness and the polarization of the evening zodiacal light. (This was a time of relatively quiet sun, since sunspot minimum occurred in 1954.) After examining the polarized component of the light, they concluded that the electron density in the vicinity of the earth is about 600/cm3, assuming that the scattering of light by free electrons is entirely responsible for the polarized component. However, Blackwell and Ingham11 have concluded from their data that the scattering is almost entirely due to dust and that it is unlikely that the electron density in the vicinity of the earth exceeds 120/cm3. Direct measurements of corpuscular fluxes have been reported by Gringauz.l2 He states that during the last part of the trajectory of Lunik II, positive collector currents in ion traps were recorded which correspond to positive ion fluxes of about 2 x 108 ions/cm2 sec. (Geomagnetic disturbances during this portion of the flight were characterized by a K-index of 5, which indicates a moderate disturbance level.) During the flight of space probe IV launched in the direction of Venus in Feb., 1961, currents were recorded which correspond to a density of positive corpuscular flux of about 109 ions/cm2 sec. (Shortly 4

after these readings were taken, the commencement of a magnetic storm with amplitude of perturbation of the horizontal component of the geomagnetic field of 10o3 gauss = 100ly- was observed, an average magnetic storm which is consistent with a K-index of 5.) Both of the Russian results would be expected to give fluxes greater than those expected during quiet sun. However, these results are smaller than Parker's estimate of the minimum flux for quietest conditions of 5 x 109 ions/cm2 sec corresponding to a density of 100/cm3 and a velocity of 500 km/sec. C. THE SELENOMAGNETIC FIELD Since the solar wind is expected to have a considerable effect on the lunar atmosphere, and the stream of charged particles will be affected by any magnetic field, it is of interest to know the magnitude of the magnetic field of the moon, A magnetic field would shield or markedly change the influence of the solar wind upon the lunar surface and atmosphere. A criterion for shielding has been given by Parker13 as 1 nmu2 B2 n m <u> = -- 2 avg 8t It will be seen in Section III-C that a good estimate for the density of the ordered translational kinetic energy in the solar wind is about 5 x 10-8 ergs/ cm3. The field required by the above formula for shielding is therefore B = 1.12 x 10-3 gauss = 112y. The Russian scientists Dolginov, et al,14 have reported an attempt to measure the magnetic field of the moon. They state that the magnetometer aboard Space Rocket II did not detect a magnetic field up to 50 km from the lunar surface, and that if the moon had a magnetic field whose intensity at the lunar surface exceeded 100y, it would have been detected. Therefore, it appears that the moon is not effectively shielded from the solar wind by its magnetic field. However, it is possible that the lunar magnetic field itself is greater than 100y, but that the solar corpuscular radiation, deflected in that field, creates a current having a magnetic field which tends to cancel the lunar field. This has been discussed by Neugebauer.15 Do THE LUNAR SURFACE TEMPERATURE The structure of the lunar atmosphere will be greatly affected by the surface temperature of the moon. High surface temperatures will increase the escape of molecules from the vicinity of the vicinity of the moon because of the high velocities which they can acquire upon contact with the surface. Low surface temperatures can cause condensation of atmospheric volatiles on the lunar 5

surface if their partial pressures exceed their vapor pressures at that temperature It is therefore of interest to know the variation of temperature on the surface. Measurements of the lunar surface temperature were made by Pettit and Nicholson16 in 1927 and by Pettit17 in 1939. Although these measurements were made over twenty years ago, they are still the most valuable measurements which we have of the lunar surface temperature. During the eclipse of June 14, 1927, Pettit and Nicholson determined the temperature of a small area near the border of the lunar disc. With an improved technique Pettit observed the temperature of a small area near the center of the lunar disc during the eclipse of October 28, 1939. Pettit and Nicholson found the temperature of an area opposite to the sun to be 120~K, with an uncertainty of 15% corresponding to the probable error of 50% in the recorded amount of radiation, The temperature near the center of the lunar disc at full moon, as found by Pettit just before the beginning of the 1939 eclipse, was 3700~K These observations were discussed by Wesselink,l8 who gave a predicted curve of the variation in surface temperature of an area near the center of the lunar disc during a lunation, Given in Fig. 1 is a curve similar to Wesselink!'s except that it has been shifted by about 50~K, so that the maximum temperature is 400~K. This curve has been used in the thermal design of the instrument package as stipulated by the Jet Propulsion Laboratory. It can be seen from this curve that a reasonable average temperature for the sunlit side is about 500~K and a reasonable average temperature for the night side is about 130~K. The curve labeled Tp is the expected variation in the instrument package temperature which is discussed in Section V. 6

0 Q) ~....~- I J. \r\ | -') c iP a) N Cc jo 1o0 o CC) N "3 E fz- g o o~l 4 —-J_ e~ 0 0 0 -4 oet d N3 AftA 7\L

SECTION III SOME GENERAL IDEAS CONCERNING THE LUNAR ATMOSPHERE In this section, we discuss the equations governing the density distribution in an exosphere, and an isothermal atmosphere, the possible sources of gas for the lunar atmosphere, the effect of the solar wind on the lunar atmosphere and the molecular escape mechanisms. In all cases, it will be assumed that there are no collisions between neutral particles, since their mean-free path is long compared with their ballistic trajectory. The so-called base of the exosphere is, therefore, taken to be at the lunar surface. This and other assumptions are discussed in relation to their use in the presentation of the model lunar atmosphere in Section IV. A. THE MOLECULAR DISTRIBUTION FUNCTION AND LIOUVILLE'S THEOREM We could determine the structure of the lunar atmosphere if we knew, at any given time, the positions and states of motion of all of the particles up to a great height above the lunar surface. For a given particle, we know the necessary information if we are given, as functions of time, its three position coordinates, ql, q2, q3, and their three conjugate momenta, Pl, P2, P3. The motion of the particle is described by the solution of the equations of motion - s S= - H s = 1,2,3, (3-1) subject to certain initial conditions, where H is the Hamiltonian function for that particle. If the particle is not charged, and if we can neglect collisions between the particles, the Hamiltonian function in terms of rectangular coordinates with origin at the center of the moon is H(x,y,z,px,py,pz) = 2 (Px2 + py2 + pz2) MG, (3-2) 2~~m 14x2+y2+z2 for N/x2+y2+z2 > ro. Since collisions are to be neglected, the initial conditions for the motion need to be specified only after an encounter with a solar wind particle or 9

after the particle strikes the lunar surface. The motion of the particle may be represented by the path of a point in the six-dimensional space whose coordinates are ql, q2, q3, Pi, P2, P3, the phase space of the particle. Through any given point in phase space, perhaps given by the initial conditions, there passes a uniquely determined path for the representative point of a particle.. Since the determination of the motions of the large number of particles in the atmosphere is practically impossible, we must introduce some statistical postulates. These are discussed by Present.19 Here, we assume the existence of a function f(ql, q2 q3, Pi, P2, P3, t) such that fdql*-ddps is the number of particles at time t whose motions can be represented by points in phase space located in the element dQ with boundaries ql, ql+dql, *' ps, p3+dp3, and having extension dq1.. dp3s The function f is called the molecular distribution function and represents the density of the representative points in phase space. Its use is illustrated by the way in.which we find the spacial density and temperature. The density is given by nl( q,2, q3) = f fdpzdp2dP3, (5-3) where the integration is carried over the limits of the momenta. The socalled kinetic temperature T is defined by 5 kT - 1 C2 (3-4) 2 2mC, where C is the root-mean-square speed:? - c2fdpldp2dp3 (3-5) and fdpldp2dp3 is expressed in terms of c, the particle speed. The question arises as to how one can find f. It can be shown to be the solution of a partial differential equation, as follows: imagine a cloud of representative points in phase space whose density at any point at a given time is f. The cloud will move in phase space, as each representative point traverses its dynamical path, and will pass through a stationary element dS. The representative points crossing the face of d2 normal to the pi direction, which has area dql *** dp3/dpi, have a component of velocity, in phase space, Pi normal to that face. The rate of increase in the number fdql -.* dp3 of representative points in dQ2 due to motion across this face is fi dql -- dp3] dpi J 10

There is a loss fp dqj e dp3] dPi 1Pi+dPi due to motion across the opposite face, and so the net increase is: I f] pi d dql1''e dps = - - (fpi)dq1' dp3. (-6) f~i (f(i)dql dP3 (3-6) Pi Pi+dP dpi api Summing over all the six pairs of faces, the net increase in the number of representative points in dQ is' r0 - [ (fqs)+ Ma O dq dP3 +t 1 3e (fus + TPs (f i = o. (3-7) s=l This is also equal to the rate of increase of f in the fixed element ds, Using the equations of motion, this may be written 3 + t, q + P-s s + 0- (3-8) s=l But this simply says that the substantial derivative of f in phase space vanishes: Df - 0. (3-9) Dt Thus, the density of representative points remains unaltered as the motion progresses, which is the statement of Liouville's theorem. 11

The general solution of the above partial differential equation is f(ql, * *,P3) = F[I1(q1, *,P),,I*6(. * *.,ps) ] (3-10) where F is any arbitrary function and the IIPs are six integrals of the motion, which may be the initial conditions, and they may include constants of the motion such as the Hamiltonian and the angular momentum. To show this, substitute into the equation: 6 Df = DF = FIi (-11) Dt Dt i (3-11) i=l This is identically zero because the integrals of the motion are defined by ti O0. The general solution can be used if we have enough additional information about the function f to determine it uniquely. This is equivalent to specifying the distribution of initial conditions, that is, by giving the function f over the entire phase space at some particular time. If the particles all begin their trajectories at the surface of the moon, then the specification of the function f at the lunar surface at some time will be sufficient. In Sections III-C and III-D this is done for special cases. B. THE MIGRATION OF MOLECULES OVER THE LUNAR SURFACE The picture that we envisage of the lunar atmosphere is that of particles traveling without collisions from the time they leave the surface of the moon until the time they either return to the surface or are lost into interplanetary space. We would like to know just how far a given particle will travel over the moon's surface, on the average, and what the length of time is during its flight. This will give us an idea of how fast the particles are able to migrate from one place on the moon to another. Let us assume that a particle leaves the surface of the moon at an angle aO = 45~ with the vertical direction and with a speed equal to the mean speed 7 of the distribution of velocities characteristic of the temperature T. If we use the Maxwellian distribution formula, this is = 8kT (3-12) V am (The values of c for various gases are listed in Table 1.) We now want to calculate the range as a function of T for various gases. For the lighter gases, such as hydrogen and helium, the range will be comparable in magnitude 12

to the lunar dimensions. In spherical coordinates, the law of motion for a particle near the moon with a total energy ET and an angular momentum with magnitude P is given as follows (see Goldstein20): 1 m2MG m2MG 2 2mET - = m —- - 7 m- M + -- cos I. (3-13) r p2 2p2 In this equation, r is measured from the apogee of the orbit (see Fig. 2). For a particle which leaves the surface r = ro with a speed c at an angle ao with the vertical, the energy and angular momentum constants are given as follows: ET = 2 - MG (3-14) m 2 ro m - Erosin o. (3-15) m/ / 2/ \! Fig. 2. Particle trajectory coordinates. 13

The law of motion can therefore be written - = A - B cos, (3-16) r where 2A (Vorb/_)2 and B = 2 L orb4 + 1 _ orb2 B 2 ) for aO = 45~. Here, vorb - MGr7o is the orbital velocity at the surface; its value is 1.68 km/sec. (This is the velocity for which a circular orbit is possible just outside the lunar surface.) When r = ro, * = 7 ~, where /~ is the angular range in radians. The angular range is therefore given by 1 /vorb\2 1 cos t = 2 _ (3-17) 2 B jVorb 4 +1 ( orb2 The range is then given by i- 1 2 p = ro# = 2roarccos 1 2 vorb _ (3-18) l+ (s assuming U < vesc, the escape velocity. (vesc = X vorb = 2.38 km/sec.) For the heavier atoms, such as argon, the ratio c/vorb may be small compared to 1 (for argon at 300~K, it is equal to 0.24.) The asymptotic form for the above equation for c/vorb < < 1 is found as follows:

p = 2rOarccos 1 44r 1 + - 1 <E ~] L 2 vorb/ _ 2roarccos 1 ( Since cos x ~ 1 - x2, for x < < 1, ___2ro - 2 (5-l9) WP^2ro( ^c 0Kor g Vorb where g is the acceleration of gravity at the lunar surface, g = MG/ro2. This is just the expression which is found by neglecting the curvature of the surface and the variation of the acceleration of gravity, by solving the equation of motion x = (c sin ao)t (3-20) z = (c cos 0o)t - 2 gt2 for the case ao = 45~. The magnitude of the angular momentum constant is given by P = mr2(d*/dt) for any time t. Therefore, the time of flight is given by 15

p/2 k/2 tf = / r2d 2m 2. -/2 P (A-B cos t) 2m B sin 2 2A... 2m1 22 2A,iIAtB an1 { ( -B)( o + arctan Larc tt = ^ --— ^(A 2- B 2) 3 / 2 L ^A^B' ro0 1/2 ( /Vorb\2 1 kP) 2 sin 2+ 2 +. - arcta tan (3-21) Vorb32 )2 L -) sin If 7/vorb is small compared to 1, the time of flight may be shown to have an asymptotic form as follows: r~ t.l~f _ frs 4 1/2 orb - 2 vorb/ ( rb) V orb | Vorb +l 2 sin27L sin + - 23/2 arctan 2 1 + cos Vor sin2 / re o 1 + orb arctan xo 1 vr 12 7 L2 o Since arctan x ^ x, for x < < 1, tf _ N'o ~= /c - (3-22) V2 g Vorb This is just the expression found by neglecting the curvature of the surface and the variation of the acceleration of gravity. 16

For temperatures less than 2670K, the mean velocity for hydrogen is less than the escape velocity, and formulas (3-18) and (3-21) are applicable. (These formulas give an angular range 180~ and an infinite time of flight, for hydrogen at T = 2670K, when the mean velocity is just equal to the escape velocity.) In Fig. 3, we have plotted the range and angular range for hydrogen and helium, as functions of the temperature of the surface where the flight begins, along with the asymptotic forms. For water vapor only the asymptotic form is shown. Typical values of range and time of flight are given in Table 1. (The times of flight have been calculated using the asymptotic formula except for helium and hydrogen; we have used the exact formula for helium.) Figure 3 shows that the range versus temperature curve is much steeper for hydrogen than for the other gases. This means that migration of hydrogen atoms will be much more efficient from the hot side to the cold side than in the opposite direction. Therefore, even though hydrogen is supplied only on the sunlit side, the density at the surface is likely to be fairly uniform over the entire surface of the moon. TABLE 1 ATMOSPHERIC PARTICLE RANGE, TIME OF FLIGHT AND SCALE HEIGHT (T = 3000K) Mean Thermal Range p and Angular Time of Scale rto2 Gas Speed, c Range 7 (a = 45~) (km) Flight, tHiglf.t, V (cm/sec) (T = 3000K) (T = 130~K) tf (sec) h (km) (cm1) H.51x10r" 26352 - H 2.51xl*5 26872 1530** 7.9xlt10 He 1.26x105 16 48 1923 385 5.3x10-9 (43~) (16o) 4 217 93.6 -8 H20 5.94x10 27 9356 519 85.6 2.8x108 (70) (30) Ar 3.98xlO4 979 42. 348 38.6 7xl8 (3;.20) (1.40) Kr 2 4 4.75x 201 240 18.4 1.5x10-7 (1.50) (0.70) Xe 2.19x104 29.8 12.8191 11.7 2.x10-7 (1.) (0.40) *Compare this with the escape velocity, vesc = 2.57 x 105 cm/sec. **Compare this with the lunar radius, ro = 1740 km. 17

180 X 160 H 140 x 0'120/ _ / ----- ASYMPTOTIC FORM 1 00 - EXACT FORM vO J I 80x 60 6 0 o < lx( 40 He He 20 WX~H20 4 o - _X _ _- X- --- - --- 01.. I...I.I. gn I 120 150 200 250 300 350 400 TEMPERATURE ~K Fig. 3. Angular range vs. temperature. 18

C. THE BAROMETRIC LAW If no collisions take place between particles in the atmosphere, the distribution function is a solution of the partial differential equation at i Qs+ 4 L1s +ps s=l as we have shown in Section III-A. The assumption of no collisions will be justified later on in this section, and we will use it for now without justification. Let us assume that the surface of the moon is at the constant temperature T and that the atmosphere is in equilibrium out to large distances, that is, just as much mass, momentum, and energy is being transported outward as inward at any given time and position. Then we can consider the particles to start or end their flights, or both, at the lunar surface. In this case, the function f can be found if it is given at the surface; we assume of course, that f does not depend upon the time explicitly: af/at = 0. Let us assume that f takes the following form at r = ro: fo - f(ro,,,PrpP) = no 2k 2 e p2/2kT, (3-23) where p2 = r 2 + (p2 + m2(vx2 + vy2 + V2), and no is the number density of the particles at the surface. This is simply no multiplied by the Maxwellian velocity distribution formula for a gas in contact with a surface at temperature T. The only function of the kind F(I1,'-' IG) of Eq. (3-10) which takes the above form at r = ro is the following: H(ql, —,p3) 3/2 MG 1- T f(q l, P) = no( /2 Troe. kT (3-24) This is the molecular distribution function, sometimes called the MaxwellBoltzmann distribution, for an atmosphere where the following assumptions are valid: 1. No collisions take place between particles in the atmospheres 19

2. The entire lunar surface is at the same temperature T. 3. The atmosphere is in equilibrium out to large distances. Although' the assumptions valid for an isothermal atmosphere in equilibrium with no collisions are not strictly valid for the lunar atmosphere, as described in the next section, the results for this idealized atmosphere are extremely useful and instructive. If we integrate the Maxwell-Boltzmann distribution function over the entire momentum space, we get the spacial distribution of density in the isothermal atmosphere: n(r,e,) = f d fddpddpg hmMG /I 1 no exp iM kT(r ro) -no exp -Q -- ], (3-25) where kTrO2 kT mMG mg This is called the barometric law, and h is called the scale height. The scale height is a measure of the rate of decrease of density with increasing height. Scale heights for various gases are given for T = 5000K in Table 1. To help in visualizing the shape of the atmospheric distribution of density, we plot, in Fig. 4, the scale height for hydrogen, helium, and water vapor at the lunar equator, as functions of the angle which the sun makes with the zenith. 1. Gravitational Condensation We may now use the barometric law to calculate the effect of gravitational condensation of interplanetary material: the increase in density of the interplanetary material near the moon. If this material were in hydrostatic equilibrium at the temperature T, then the ratio of the density at the lunar surface to the density far away from the moon is given by no ro/h -- = e, (3-26) n, 20

180~ MO~ He / 270 900 11 1 I000 0oo / 2000 He 0~ H SUN Fig. 4. Scale height vs. sun angle; dimensioned from center of moon. 21

where h = kTr02 h - mMG and m is the mean weight of particles in the interplanetary material. If we take the interplanetary material to be fully ionized with a mean molecular weight of about 1/2 and a temperature of about 105lK, then the above ratio is no = 1.0002 noo This is obviously too small to explain Elsmore's result for the increased electron density near the moon. The gravitational condensation of interplanetary gas has been discussed by Firsoff,21 Brandt,22 and Singer.23 2. The Scale Volume If we introduce the height above the lunar surface, z = r-ro, the barometric law may be approximated as follows, for z < < ro: n = no ~ (+ - 1)j no e * (5-27) i_ o ror_~ 1 - z/h If the last expression were valid for all z, we could integrate to get "the total number of particles in a square centimeter' column": 00 - z/h NC = no e Z/hdz = noh. (-28)'Jo Although this number does not exist, since the number density does not really go to zero with increasing height above the surface, it may give a good estimate for the number of particles in the unit column which are associated with the atmosphere, especially if the scale height is small in comparison to the lunar radius: h < < ro. This is further substantiated by the work of Herring and Kyle.24 They have derived the distribution of density in an exosphere without assuming that it is in equilibrium out to large distances. We will give their result in Eq. (3-46); they have shown by integrating this equation numerically that the number of particles in a square centimeter column which began their trajectories at the lunar surface is close to noh for large values of ro/h, although it departs considerably for smaller values of ro/h than about 40. 22

The total number of particles which are associated with the atmosphere is then gotten by multiplying by the surface area of the moon: N = 4rro2hno. (3-29) If h < < ro, this is approximately equal to the density at the surface multiplied by the scale volume V, the volume contained between the surface and a sphere at r = ro+h. We can then write this ficticious number as N = noV, where V, the scale volume is given by V = 3 [(ro+h)3 - ro3]. (3-0) If h < < ro, V _ 4tro2h, and N is given by Eq. (3-29) above. 35 The Assumption of No Collisions Finally, we wish to justify the assumption of no collisions in the lunar atmosphere. The so-called Maxwell mean-free-path formula is 1 = ----, (3-31) 4. V2 ta2no where ta2 is the gas kinetic cross section of the particles. As a criterion for a collisionless atmosphere, let us set the mean-free-path equal to the scale height: 1 kT (3-32) 4 X-2 tra2no mg The upper limit for the density at the surface for a collisionless atmosphere is then given by no = m- - (3-33) 4 V ta2kT The smallest value of no is given by this formula in the case of hydrogen. At T = 3000K, Eq. (3-33) gives no = 2.5 x 106 atoms/cms. Elsmore's estimate for the upper limit of the density of neutral particles is of this order of magnitude. Therefore, the mean-free-path for particles in the lunar atmosphere is likely to be long enough to allow us to neglect collisions. 23

Another way of justifying this is to compare the time of flight with the mean-free-time. Using the Maxwell mean-free-path formula, the mean-freetime is given by 1 T = —.-.-.. o (3-34) 4 1 ta2noc For helium at 300~K, at a density of l06/cm3, the mean-free-time is T = 3762 seco The time of flight is, from Table 1, 1923 seco The mean-free-time is calculated for a flight in a region of constant density, and since it is, even so, about twice as great as the time of flight itself, we see that collisions are unlikely. D. CRITICISM OF THE USE OF THE BAROMETRIC LAW FOR THE LUNAR ATMOSPHERE The following are the reasons why the assumptions valid for an isothermal atmosphere in equilibrium with no collisions are not strictly valid for the lunar atmosphere. 1. Collisions will always take place, even though they are extremely infrequent. These will put some particles into bound orbits around the moon which do not intersect its surface, The number of these orbiting particles is difficult to calculate, and the motions of these particles are not described by giving f at the surface. 2. The temperature of the lunar surface varies from about 400~K at the subsolar point to about 1200K at the coldest parts of the lunar surface. Therefore, the function f cannot be given by the same expression over the entire surface. Also, because of the long range of the atoms of hydrogen and helium, these particles will always be arriving at certain places on the lunar surface with velocities characteristic of the temperature at certain other places on the lunar surface. The distribution function at any given place on the lunar surface is very difficult to find because of this situation. 3. The atmosphere is not in equilibrium out to large distances; this state would require an infinite length of time to establish. Because of this, particles which leave the surface of the moon with velocities exceeding the escape velocity will be lost from the atmosphere, since they suffer no collisions, and these will not be compensated for by particles coming from a large distance, with correspondingly high velocities. The distribution of velocities will therefore be truncated for velocities directed toward the moon and exceeding the escape velocity. (Also, the atmosphere will eventually escape entirely, unless some supply mechanism exists on the lunar surface, because of the escape, or evaporation, of the tail of the velocity distribution. The rate of escape of the lunar atmosphere due to this mechanism is calculated in Section III-E.) 24

Let us now find the distribution of density in the lunar atmosphere assuming only that no collisions take place and that the temperature of the moon is constant over the surface. We will no longer assume that the atmosphere is in equilibrium out to large distances. In this we use Liouville's theorem, which was derived in Section III-A, which states that the density of representative points in phase space, f (the molecular distribution function) is a constant along a dynamical trajectory. It will be convenient to express f as a function of position and velocity: f(r,~,,VrVO,vO), rather than as a function of position and momentum, as we have done previously. The relations which show how the velocities replace the momenta are as follows: Pr = mvr, pg = mrv sin Gv Liouville's theorem can then be expressed by the equation f(rGG vr,V v,v) = fo(roGo,~ovrov9ovo), (3-35) where fo is the function f evaluated at the point on the surface where the trajectory began. The variables in the two functions above are related as follows, using the conservation of energy and angular momentum: 2 2MG 2 2MG r r O _ = v r -o (3-36) vr sin a = vorosin O o Here, v is the magnitude of the velocity, a is the angle between the velocity vector and the radius vector r, and subscripts o indicate the values of these variables at the point on the lunar surface where the trajectory began. Assuming the density at the lunar surface does not depend upon 9 or I., the spacial distribution of density in the atmosphere is given by n(r) = f'o _ f(r,vr,,vQ,v)dvrdvodv. (5-57) 25

Using spherical coordinates, vr = v cos a, v0 = v sin a cos 3, v = v sin a sin P, the integral is 2jrt tr oo n(r) = o o f(r,v,'a,)v2sin c dvdadp. (3-38) 0 0 0 Using Liouville's theorem, this may be written as follows: 2Jp Jt o00 n(r) = o o fo(ro, vooao)v2sin o dvdadp. (3-39) The density can be found if we know the molecular distribution function at the surface, assuming that each particle started its trajectory at the surface. (The density of particles in bound elliptic orbits which do not intersect the surface of the moon cannot be found in this way.) To carry out the integration, we need to transform the volume element dvdadp to one involving vo, Go, and Po. Using the relations in Eq. (3-36) and the fact that B = Po, we find that the functional determinant has the value (vo,) = /vo)2 rO cos Co (3-40) (vo ao(o) v r cos a The integral can then be wrriten as follows: 2t rt oo n(Pr) = sin n(r) = vfo(r o,v o, o,)vo2 a cos aodvodaodPo. (3-41) omin Since the range of integration must be over the part of velocity space compatible with the relations in Eq. (3-36), i.e., only over those orbits which intersect the sphere at ro and r, the lower limit in the integration over vo is the minimum velocity for which a particle may reach the sphere at r if it is emitted from ro at the angle ao. From the relations given in Eq. (3-36), we find oi2 _26 vomin r (3-42) 1 - ( 2sin2o To express the initegrand entirely in terms of vo, %,o and Bo, we again use the relations in Eq. (3-36). We have 26

r~ sin ao sin r in O cos a 2MG 2~ ~cos of v2G2 (_r r) _sin2ao NO Therefore, the integral becomes ( ) tP tF J ~ ~ ~,2 VcOS ao sin aOdvododPpO n(r) = f fo( ro vo o, ) vcs sin d 0 vomin 2MG 1 ro 2a. ( v r, r r (3-44) Now we use the truncated Maxwellian distribution for fo: T0 0 < 0 < < v < v ( m 3/2 -2kT and | no2k-T e 0 < U <,/~J L /2 < a < Tr fO = < es < V < 00 I [jr/2 < a < 3i ( 3-45) (Note that the factors vo2 and sin ca have already been included, in the functional determinants.) The integral can now be written as follows: (r o J esc 3/ 1 n(r) =J2~~) S n (i)3k,e v02cOs aosin aodvodao romin 2MG I 1 2sin o y02 Vo1 + 2 Lr - ) - (r) o (546) This result has been obtained by Opik and Singer25 and by Herring and Kyle.2 The latter authors have evaluated the integral as follows: 27

ro(1 r n(r) - r o\ - 2 -.2 erh(r+r) -=o _T| e"- ro/h(l + erc ~T- r ( - S o e ro (3-47) For values of r/ro close to one, the above equation departs very little from the barometric law, particularly in the case of a heavy gas such as argon. For larger values of r/ro, the above equation gives values of n(r)/no considerably smaller than the barometric law, and the ratio n(r)/no goes to zero as r/ro gets very large. This is a more realistic result than that given by the barometric law, which predicts the constant value e~ ro/h for n(r)/no as r/rO becomes very large. E. THE JEANS EVAPORATION FLUX In Section III-D we mentioned the fact that some high velocity particles will escape from the atmosphere and will not be compensated for by incoming particles. This evaporation of the tail of the velocity distribution was first discussed by Jeans.2 Since the escape velocity at the surface is Vesc = j2MG/ro, the number of particles which leave one cm2 of the surface/sec with velocities exceeding the escape velocity is JJ = JJ/ fo vzdvxdvydvz (3-48) (vz > 0, x + Vy + V 2 > 2M z VX y ro ( m\3kT/2 rf 0 - mc/2kT Jj = noy 3.-)' 2 e cSdc 28

The Jean's evaporation flux is seen to be a sensitive function of the quantity a _ romg h kT kT The smaller the mass of the atoms and the higher their temperature, the more readily they escape by this process. In order for the density of any gas in the lunar atmosphere to be constant over long periods of time, some supply must exist to balance the escape of gases by this and other processes. F. SOURCES FOR THE LUNAR ATMOSPHERE It will be seen in Section IV that an atmosphere cannot exist on the moon without having a continuous gas supply from some source. In this report we will consider the following sources. (a)- The Solar Wind, as a Source of Neutral Hydrogen and Helium - We assume that every positively charged particle from the solar wind which strikes the moon's surface is confined within the crystal latticedf the surface material for a certain length of time and then re-emitted as a neutral atom, such that the velocity distribution of the emitted particles is Maxwellian at the temperature of the surface where they are emitted. (The velocity distribution we are speaking of has only velocities directed away from the surface, so it is not Maxwellian in the strict sense.) We assume that the number of particles does not buildup in the surface layer, so that the incoming flux of charged particles equals the emitted flux of neutrals, and we call this process "diffuse reflection." (Actually, a certain number of the particles will be specularly reflected ard not accommodated to the surface temperature, and possibly not neutralized, but for simplicity we neglect these.) (b) The Supply of H20, CO2, and SO2 From the Crustal Rocks, From Hot Springs, From Lava Flows and From Volcanos, and the Supply of H20 From Trapped Ice. -Assuming that the composition of the moon is similar to the earth's crust and mantle, and that the yearly yield of volatiles is proportional to surface area, Vestine27 predicts equivalent source strengths for these gases on the moon as follows: - H20: 4.5 x 1010 molecules/cm2 sec C02: 1010 molecules/cm2 sec S2: 1.6 x 108 molecules/cm2 sec. 29

It is unlikely that the actual source strength for water vapor is as high as that given above. If appreciable amounts of hydrated silicate were present in the material that formed the moon, it is possible that water could be trapped below the surface as permafrost. Since the temperature at shallow depths is so low that the evaporation rate would be very small, all of the water contained in the moon could not escape readily into the atmosphere. The possibility of permafrost on the moon has been discussed by Gold,28 Watson, Murray, and Brown29 have discussed the possibility of ice being retained on the moon on portions of its surface which are permanently shaded from the sun. If these portions of the surface are at a temperature of about 120~K, the estimated evaporation rate is 10-15 grams/cm2 sec, which is about 3 x 107 molecules/cm2 sec. These authors estimate that the fraction of the lunar surface that is permanently shaded is most likely about 0.005. The equivalent source strength for water vapor based upon the entire lunar surface is therefore about 1.5 x 105 molecules/cm2 sec. Although the estimated source strengths for C02 and SO2 are high enough that these gases may constitute an appreciable portion of the lunar atmosphere, they will be ignored here because of the impossibility of treating them with The simple model of Section IV, where photochemical reactions and chemical combination on the lunar surface are not considered. It must be stressed, however, that our simple model will give only a minimum estimate for the lunar atmospheric density, because we have not considered these gases. (c) Natural Radioactivity of K40 as a Source of Ar.-Assuming that about 0.12% by weight of the moon is composed of potassium, of which about 0.011% is K0, and that each gram of K40 has produced 0.1 gram of Ar by electron capture or positron decay processes, as has apparently been the case for the earth, then Vestine'7 calculates that the number of atoms of Ar produced/cm2 of the lunar surface over a period of 3 x 109 years would be 4.0 x 1022 atoms of Ar/cm2. The half-life of K40 is about 1.4 x 109 years, and the activity is not constant over any period of time comparable with the half-life. If a(t) is the activity in disintegrations/sec, per cm2 of surface, then a(t) = a2 t/t/ (-50) where ao is the original activity and tl/2 is the half-life. The number of atoms of Ar produced/cm2 of surface in a time t is N(t) = a(t)dt = 2 (1 - 2 t/tl/2) (3-51) 50

Therefore, the activity at time t is a(t) = N(ln2) (3-52) tl/2(2 / 1 (3-52) ^ 2t/tl/2 - 1) Using N = 4.0 x 1022/cm2 and t = 2tl/2, we find a = 1.9 x 105/cm2 sec. Assuming that the time of outgassing for the Ar atoms is much smaller than the half-life, the source strength of Ar at the surface is the same as the activity, 1,9 x 105 atoms of Ar/cm2 sec. (d) Production of Krypton and Xenon by Spontaneous Fission of U238, Decay of 1729, and From Primeval Gases Trapped in Rocks and Released by Meteor Bombardment.-These processes have been examined by Edwards and Borst.50 The primeval gases give the largest contribution; in a layer 10 cm deep they give the following values for the total number of atoms: 1051 Kr: lO^^'.toms"" Xe: 7 x 1031 atoms If we assume that these are released at a constant rate over the entire surface by meteor excavation over a period of 3 x 109 years, the equivalent source strengths are: Kr: 2.8 x 10-2 atoms/cm2 sec Xe: 2 x 103 atoms/cm2 sec. TABLE 2 ASSUMED SOURCE STRENGTHS Source Strength Gas (atoms/cm2 sec) Area of Source (atoms/cm2 sec) H 93 J ro2 He 7x10-2J ro2 H20 1.5x105 4h3ro2 Ar 1.9x105 4htro2 Kr 2.8x10l2 4rro2 Xe 2x10-3 4tro2 31

G. THE EFFECT OF THE SOLAR WIND ON THE LUNAR ATMOSPHERE 1, Assumptions Concerning the Solar Wind To calculate the effect of the solar wind on the lunar atmosphere, we must make some specific assumption about the parameters of interest. Since the solar wind flux, J, is known to vary with solar activity, but its magnitude is not well known for any specific level of solar activity, the densities of atmospheric gases predicted by our model in Section IV will be presented as a function of J, for values of J between 107/cm2 sec and 1013/cm2 sec. We must, however, be more specific for assumptions concerning the velocity of the solar wind so that we may predict the elastic scattering and charge exchange cross sections for the various atmospheric gases. For the mean flow velocity of the solar stream, we will use the value <u>avg = 500 k/sec, which is, according to Parker, the minimum velocity for a quiet sun. This assumption is somewhat incompatible with our use of a variable flux; however, we are primarily interested in conditions during quiet sun, and the problem is much less complicated if we make the assumption that <u>avg is constant. To exemplify the use of the equations in Section IV, we will use the value J = 109/cm2 sec. Since this is the highest flux measured by the Russians during an average geomagnetic storm, we would be persuaded to assume an average value of J for quiet solar conditions less than this. However, the majority of the remaining the retical evidence is for higher values of J, even during quiet solar conditions. More recent measurements by U. S. satellites corroborate the Russian measurements, so that a flux of about 109 part/cm2 sec is a reasonable value to assume. The above two assumptions for J and <u>avg force us to use the value n = 20/cm3 for the number of ion-electron pairs/cm3 in the solar wind. Hinterreger31 has shown that the ratio of densities of neutral and ionized particles in the interplanetary space is unlikely to exceed 10-3. Because this is the result of considering only photoionization, we expect this ratio to be even lower when the ionizing effects of the solar wind are taken into consideration. Therefore, we will neglect the presence of any neutral particles in the solar wind and assume that it is completely ionized. The question now arises concerning the temperature of the interplanetary gas, or the solar wind. Pottasch32 has computed the temperature distribution from 1.0043 to 20 solar radii in the solar corona, using the observed distribution of density in this region and the assumption of hydrostatic equilibrium. If we extrapolate his curve to the distance of the earth's orbit, we find the value T S 10o5~K. This is about equal to the temperature which was calculated by Chamberlain553 using an exospheric model of the solar corona, and is somewhat less than the temperature found by Chapman34 based upon heat conduction in a static, spherically symmetric model solar corona. 32

We will assume that the solar wind is simply the hydrodynamic expansion of the solar corona, so that helium is present as well as hydrogen. Assuming that the composition of the corona by weight is 75% hydrogen, 23% helium, and 2% heavy elements with a mean atomic weight of about 32, all completely ionized, the number density of helium ions should be about 6/75 of that of hydrogen ions or about 7% of the total density of positive ions in the stream. Since collision frequencies in the corona close to the sun can be assumed relatively high, we will assume that the helium ions travel at the same mean velocity in the solar stream as the hydrogen ions and the electrons. Although the solar wind is believed to flow radially outward from the sun, we will assume that the flow in the vicinity of the earth is a one-dimensional uniform flow, over a region the size of the moon, in the direction away from the sun, The ordered mass motion of the solar stream at 500 km/sec corresponds to kinetic energies of 1300 electron volts for protons, 5200 ev for helium ions, and 0.7 ev for electrons. At the temperature T = 105K, the thermal energy per particle is about 15 ev for all kinds of particles, assuming that equipartition of energy exists. (This assumption will be discussed in the last part of this section,) We see that the energy of the ordered motion is much greater than the energy of the random motion for protons and helium ions, and is considerably less than the random energy for electrons, The physical situation is very similar to that of a spherical satellite moving with a uniform speed in the earth's ionosphere, with a velocity much greater than the thermal velocities of the positive ions but less than the root-meansquare thermal velocity of the electrons, For the purpose of describing various physical processes, we will assume that the velocity distribution for the particles in the solar wind is Maxwellian and isotropic in a coordinate system moving with the mean velocity of the flow. This assumption may be partially justified by comparing the times of relaxation for the particles and the length of time necessary for the particles to. tkavel from the sun to the orbit of the earth. The "self-collision time" for positive ions in a plasma interacting with others of the same species is given by Spitzer35 as follows: 11. 4 A1/2 T3/2 tc 1= A T. (3-53) nZ42n A where A = e3 T3 2 1.24xl o04 x3 A2a3 e — =~,-//n A is the atomic weight, and Z is the number of unit charges of the positive ions. This is the time in which a particle is gradually deflected by 90~ 33

due to the interactions with other particles, on the average. The self-collision time is a measure both of the time required to reduce substantially any lack of isotropy in the velocity distribution and also of the time required for the distribution of kinetic energies to approach the Maxwellian distribution. If T = 105lK and n = 20/cm3, then tc = 7 x 105 sec, for H. This is of the same order of magnitude as the length of time taken for protons traveling at 500 km/sec in traveling from the sun. The self-collision time for electrons in the plasma is a fraction Ime/mp = 1/43 of that for protons, which is considerably smaller than the time taken in traveling from the sun, Therefore, the particles may be assumed to have velocity distributions which are Maxwellian and isotropic, as stated previously. The temperature may be different, however, for the different kinds of particles. The rate at which equipartition is established between protons and electrons is measured by the "time of equipartition," teq = 43tc = 3 x 107 sec. Thus, if the particle density was as low as 20/cm3 during the entire duration of time in which the stream travels from the sun, it is unlikely that the temperatures of electrons and positive ions would be the same at the earth's orbit if they were not the same in the solar corona. It is possible, however, that equipartition may have become established closer to the sun, where the density is higher. We will assume here that the temperatures of all kinds of particles in the solar wind at the orbit of the earth are the same, T = 105~K. Table 3 summarizes our assumptions concerning the solar wind. The number nEflow is the number density multiplied by the kinetic energy per particle, or the kinetic energy density. The total for all three types of particles is about 5 x 10-8 ergs/cm3, as was mentioned in Section II-C. TABLE 3 THE ASSUMED MINIMAL MODEL OF THE SOLAR WIND Electrons e- Protons H Helium Ions He++ J 109/cm2 sec 0.93xlO9/cm2 sec 7xlO7/cm2 sec <u>avg 500 km/sec 500 km/sec 500 km/sec Eflow 0.7 ev = l.lxlO-12erg 1300 ev = 2.1xlO-9erg 5200 ev = 8.3xlO-9erg n 20/cm3 18.6/cm3 1.4/cm3 nEflow 2.16xlO-11ergs/cm3 3.87xlO-8ergs/ 6x ergs/cm3 1.6xl8ergs/cm3 T O15K 105~K lO5OK 2 kT 15 ev 15 ev 15 ev 34

2. Elastic Scattering The molecules in the lunar atmosphere will be subjected to collisions with the high energy protons. The difference between protons and helium ions will be, ignored here and we will also ignore the effect of the electrons,'since their energy is so low. These collisions may ionize an atmospheric molecule, effect a charge-exchange reaction, or collide elastically, with no change in the internal states of the colliding particles. As shown by Massey and Burhop,36 charge exchange is about an order of magnitude more probable than ionization, so it is the only inelastic collision process considered here. To calculate the probabilities for the processes of charge exchange and elastic collision, we need to know the cross sections. Since we are interested in the probabilities for ejection of atmopsheric molecules by these processes, we will calculate the cross sections for scattering through all angles greater than a minimum angle, so that the atmospheric particle gains at least enough energy in the process to enable it to escape. We will call this the "escape cross section." A consideration of the dynamics of elastic collisions is necessary now. Since the de Broglie wavelength for protons of 1300 ev kinetic energy is very much smaller than the dimensions of the smallest atoms, the classical equations of motion provide a very accurate approximation to the correct quantum mechanical equations of motion, and the treatment of collisions will be done here using classical mechanics. Consider an incident particle of mass ml, velocity v and kinetic energy E, and a particle of mass m2 which will be assumed to be at rest with respect to the moon. We will consider that some interaction takes place between these particles such that the energies and angles of motion of the particles after the collision are as shown in Fig. 5. (Note that we have implicitly used the assumption that the sum of the kinetic energies of the particles before the collision is the same as the sum after the collision, in writing E-Ef for the final velocity of particle 2.) Figure 6 shows the representation of the collision in a coordinate system which always has as its origin the center of ml m, Ef m,v,E ml m ~ Collision coordinate systems. 55

mass of the two-particle system. It can easily be shown (see Goldstein20) that the following relations are true between the scattering angles: s min L= 2 ~ 2' tan = sin (3-54) cos E + m m2 Now let us write the equations expressing conservation of energy and momentum in the following form: 2miE = N/2mEf cos G + 2m(E-Ef) cos / (5-55) 0 = 2mlEf sin 9 - x/2m2(E-Ef) sin. Squaring and adding to eliminate G, we get the energy gained by particle 2 in terms of the angle through which it is scattered: E - Ef = 4E mlm2 cos2~. (3-56) (ml+m2) It will be convenient to express this in terms of the angle of scattering in the center of mass system: E - Ef = 4E mlm2 sin2 (3-57) (m +m2)2 2 1 Now the energy of escape for particle 2 is (E-Ef)esc = m2 v2sc, where vesc is the escape velocity at the moon's surface. Therefore, the minimum angle 8 through which the scattering can take place and still impart to particle 2 the escape energy is given by @,,, (ni+mI+MV2gV sin = ((m+) c (-8) 2 8mlE Now the elastic "escape cross section" is given by (el) esc = 2 c(() sin 0 dE, (3-59) esc where a(e) is the differential scattering cross section, which depends upon the interaction potential. The above integration does not need to be actually 36

carried out if we can express the so-called impact parameter b as a function of B. The impact parameter is the perpendicular distance between the center of force and the incident particle velocity, which is associated with scattering through a particular angle 8. If 8 = Gesc, then the elastic "escape cross section" is given by (el) = tb2(eesc) (-60) esc Now for the light atoms of H and He, the above cross sections are very nearly equal to the gas kinetic trasport cross sections, given by Jeans.26 For heavier atoms, however, eslc may be considerably smaller than the gas kinetic cross section. This is for two reasons: (1) the energy transferred from incident protons is smaller when the mass of the atmospheric particle is larger; (2) the escape energies for heavy particles are higher than for light particles. We will evaluate a(ec) for heavy particles in the following way: The impact parameter b(Oesc) corresponding to the minimum scattering angle for escape is found using the result of the Rutherford scattering of protons by charged particles of charge Zeffe: Zeff e2 b - -f cot (2), (3-61) 2EO where Eo is the incident kinetic energy minus the kinetic energy of a particle of mass (ml+m2) moving with the center of mass (it is the incident energy for the problem in the center of mass coordinates). In terms of the incident energy, x im+m2 The formula which we want is therefore Zeff e2cot ~ b = m (3-62) 2m2 E ml+m2 so that b(esc) Zeff e (ml+m2) 8mlE - b(Oesc^ ---- J -- - - 1 2m2E (ml+m2) vsc Zeff e2 /2m, | ( +m 2 2e m2E vsc3 2 37

Since E >>> 1Vsc b(esc) = Zeff e E (3-63) m2 vesc In this formula, Zeff e is the effective charge of the atmospheric atom for the scattering, i.e., the nuclear charge Ze minus the charge of the electrons which screen the nucleus. If the deflection angle G is small (and so E is also small), the force which deflects the incident proton will be approximately given by the force due to a point charge Zeff e at the center of the atom, where Zeff is defined by (r) _ Zeffe2 (3-64) v(r]f rb and where v(r) is the electrostatic potential energy. For heavy atoms, this potential energy can be calculated, using the Thomas-Fermi atomic model. We have Zeff(b) = (b) = ZX[x(b) ], (3-65) where X(x) is the solution of the problem /2 d2X X3/2 X(O) = 1, X(o) = O (3-66) dx2 and is graphed on page 364 of Ref. 37. Also, x(r) = 2(4/3)32/3 Z1/3, (5-67) where ao = 5.29 x 10-9 cm is the first Bohr radius. Since 1 3/5t\2/3 ao r = () x 38

and b( ) = ZX(x[b( esc) ] e2 b(~esc) = ---------- 2m/E m2vesc we have X(x) = 1 2/3 aom2ves x. (3-68) 2 a om Z4/3e2 2mE This equation has been solved graphically for x in the case of Ar, Kr and Xe. The results are given in Table 4 along with the gas kinetic cross sections for the light molecules which are given by Jeans26 and Loeb38 TABLE 4 CROSS SECTIONS FOR ELASTIC COLLISION AND CHARGE EXCHANGE WITH 1300 ev PROTONS eel) c gel) ael ) - cx) C- cx) Ga esc rsSC ese Gas (cm2) (cm2) lT (cm2) (cm2) H 4.5xl1016* 4.5xlO016 1 2x10-15 2x10l15 He 3.7xlO-16 3.7x10-16 1 1.8xl0-17 1.8x10-17 H20 5.8x10 16 5.8x1016 1 2xlO15** 2xl015 Ar 10.5x10-16 3.2xlO-17 0.030 4 4xl16 1.3x10-17 Kr 13.5x10-16 2.5xl117 0.018 10o6x10-16 1.9x10-17 Xe 18.7x1l-16 2.1x0l"17 0.011 7.9x10-16 8.7x1l-18 *This is an estimate; it is intermediate between the values for He and H2, given by Jeans. **This value was adopted as the same as that for H. 3. Charge Exchange Collisions which result in exchange of charge are not elastic, but the change in internal energy of the colliding particles is only about 2 ev for 39

protons in Ar and less than 1 ev for protons in Kr or Xe (see Massey and Burhop36). This produces a very small effect dynamically when the incident proton energy is 1300 ev. Therefore, we can treat these collisions as approximately elastic. For the light atoms of hydrogen and helium, the charge exchange "escape cross sections" CX: exchange "escape cross sections" esc, are almost the same as the charge exchange cross sections which have been measured by Fite, et al.,39 and Keene. We will take the charge exchange cross section for water vapor to be the same as that measured for hydrogen by Fite, et al, Charge exchange cross sections for Ar, Kr, and Xe have been measured by Hasted.41 Although these would be expected to be larger than the charge exchange "escape cross sections," it is difficult to say what the ratio is exactly. Therefore, we will take the ratio of "escape cross section" to the measured cross section to be the same for both charge exchange and elastic collisions: a(cx) jcx) c(el) ~(cx) esx) (3-69) esc co(el) We have listed the measured values of cross sections along with the values for "escape cross sections" used by us in Table 4, Although we have called these cross sections "escape cross sections" the probability of the particles hitting the moon and failing to be ejected from the lunar atmosphere must be taken into consideration; this is done in Section IV. H. PHOTOIONIZATION AND PHOTODISSOCIATION IN THE LUNAR ATMOSPHERE Although we have neglected ionization of atmospheric molecules due to solar protons, we cannot neglect ionization due to solar ultraviolet radiation. At times of quiet sun when the solar corpuscular radiation flux is quite small, the effects of the solar ultraviolet radiation with wavelengths longer than l00O may become more important than the effects of the solar wind, since the intensity of the radiation in this portion of the spectrum would be expected to be approximately the same during active and quiet solar conditions. Hinterreger31 has measured the solar photon fluxes between 1300 and 60o at an altitude of 200 km and estimated the fluxes in interplanetary space near the earth. On this basis, he gives total ionization rates for several kinds of gases, among them monatomic hydrogen and helium, by summing the products of photoionization cross sections and particle fluxes over the wavelength bands. He obtains the following results: H: -Eoo = 4.5 x 10-7/sec He: Zo0o = 10-7/sec, 40

We have estimated the ionization rate for water vapor by using Hinterreger's flux measurements and averages of the photoionization cross sections measured by Wainfan, et al. 42 in the range from 1000 to 300 and those measured by Astoin, et al.,3 in the range from 300 to 250. Our result is Zcao = 2.45 x 10-/sec. We have estimated the photodissociation rate for water vapor in a similar way. Near the threshold for dissociation, 2400A, Granath4 found the absorption cross section to be less than 10-22 cm2. We have therefore neglected the absorption in the wavelength region between the threshold for dissociation and 1800A. Using the results of Watanabe, et al.,45 for the absorption cross section in the wavelength region between 1800 and 1000X, and the intensity of the solar spectrum given by Detkilers. et al.,46 we find Zoo = 1.62 x 10-5/sec. The total probability for removal of water vapor by either photodissociation or photoionization is therefore ZoOo = 1.86 x 10-5/sec. We have also estimated the ionization rate for argon using Hinterreger's flux measurements and the photoionization cross sections measured by Po Lee and Weissler.47 Our result is Zo0o = 1.95 x 106/sec. Although the ionization thresholds of krypton and xenon occur at longer wavelengths than does that of argon, the photoionization cross sections, which have never been measured, may be greater than that of argon or they may be less. In view of this uncertainty, we will use the same value of photoionization rate for Kr and Xe as we calculated for Ar. 41

SECTION IV A SIMPLE MODEL OF THE LUNAR ATMOSPHERE We now wish to calculate the expected partial densities of various gases on the moon, Their source and escape rates are based on the different processes discussed in the previous sections. For our purposes, we will use an even simpler model of the lunar atmosphere than the prementioned isothermal model. The model is constructed as follows: the entire atmosphere is considered to be contained between the surface of the moon and a surface at height h(T) where h(T) is the scale height for the particular gas considered It was shown in Section III that the total number of particles which are associated with an isothermal atmosphere is given by the density at the surface, no, multiplied by the volume included between the lunar surface and the inner surface of a sphere of radius ro+h, which we called the "scale volume." We will use an average temperature of 300K in computing the total number of particles which are included in our model atmosphere. This number will be calculated as follows: N = n0V where no is the average number of particles/cm3 at the lunar surface, and V is the scale volume for an isothermal atmosphere at 300 ~Ko The change of N with time is due to the following processes. (a) The Net Influx of Particles at the Inner Surface, —Included in this term are the contributions discussed in Section III-F from the diffuse reflection of hydrogen and helium, the supply of water vapor from trapped ice, the production of Ar from the radioactive decay of K4, and the production of Kr and Xe from primeval gases trapped in rocks and released by meteor bombardment, We neglect recombination of atmospheric ions on the lunar surface, (b) The Net Outflux of Particles at the Surface of the Outer Sphere.This term includes only the evaporation escape flux discussed in Section III-E. Although the integration was taken over the surface of the moon, the same total flux escapes through any finite surface surrounding the moon. (c) The Net Rate at Which Particles are Produced Within the Scale Volumeo -In this term we take into consideration the removal processes of photodissociation and photoionization, due to solar ultraviolet radiation, and charge exchange and elastic collision processes due to the solar wind, 453

all of which result either in the ejection of particles from the atmosphere or in the ionization of atmospheric particles. We neglect recombination in the scale volume. The three kinds of processes are related to the time rate of change of the total number of particles by the equation dNt J ds - d SdV inner surface outer sphere scale volume I - I2 + Is, (4-1) where Js is the source strength, 5j is the Jeans escape flux and S is the volume production rate. We have abbreviated the three terms, I,, I2, and Is. A. HYDROGEN AND HELIUM The calculation of the atmospheric partial densities is well illustrated by the cases of hydrogen and helium. For these gases, I1 is equal to the proton flux Jp multiplied by the effective capture cross section of the moon. If we assume that the charged particles in the lunar ionosphere form a space charge region close to any portions of the lunar surface which are charged, so that the charge on the moon is shielded a long distance away, the only forces acting on the particles in the solar wind are due to the gravitational attraction of the moon. Let b be the impact parameter corresponding to particles which just graze the moon's surface. From continuity we have b2 J = Jo _ — ro2 where JO is the solar wind flux far away from the moon. From conservation of energy and angular momentum we find that 0, r o.b ro2.+ ].mMG = 2 2mv2 so that b2 2mMG vesc - = 1 +- M- = 1 +- r02 mvyro v 2 44

where vo is the velocity of the stream a long distance away from the moon. Since Vesc/Voo _ 5 x 10-3, b c ro, and the effective capture cross section of the moon is just itro2. Therefore, I, = Jplro2. (4-2) (We have neglected the contribution to the number of neutral hydrogen atoms in the atmosphere from dissociation of 320., which is negligible in comparison to the solar wind contributions) 12 is equal to the evaporation escape flux multiplied by the total surface area of the moonI_ = 24rca Bynoc - ro/h ro)y I2 4 Iro2 e r (l /h (4-3) The term I3 is negative for neutral particles, Its magnitude is equal to the "scale volume" V multiplied by P, the fraction of the particles in the atmosphere which are exposed to the sun, multiplied by the average volume rate of all processes which eject hydrogen or helium from the atmosphere: 13 = -V[noJo sc) 7 + + noJo + noso] (4-4) The fraction P will be taken to be equal to 1 for hydrogen and helium. As we stated in Section IIIB, we expect the density of hydrogen atoms at the surface to be fairly constant over the surface. As shown in Fig. 4. most of the scale volume, which may be considered to be the volume contained between the surface and the scale height, is exposed to the sun in the case of hydrogen and helium. Therefore, we can take (3 1 for hydrogen, The value of P for helium is expected to be even more nearly one, since the net rate at which helium atoms can migrate to the dark side after being formed on the sunlit side is considerably smaller than the net rate for hydrogen atoms. The three contributions to I3 are due to elastic collision and charge exchange with solar wind protons and to photoionization and photodissociation. The y-factor is the result of averaging the rate of escape of ejected particles over the entire exposed lunar atmosphere, taking into account the geometric probabilities of ejected particles striking the moon and not escaping. The distribution of scattered intensity with scattering angle for elastic collisions and charge exchange which has been measured (see Massey and Burhop36) shows that the vast majority of incident particles are scattered through very small angles. Since the exchange of energy in charge exchange is small com45

pared to the total kinetic energy of the particles, the dynamics of those processes would be expected to be almost the same as for elastic collisions, as mentioned previously. When the incident particles are deflected at very small angles Q, the struck particles are scattered at angles very close to 90~, since sin 0 sin 2~ cos G ml - cos 20 m2 (see Section III-F) and ml/m2 < 1. Most of the particles in the atmosphere are, therefore, scattered nearly at right angles to the incident solar wind direction, so they have a geometric probability of escape close to 1 near the portion of the atmosphere roughly above the subsolar point, and a geometric probability of escape of about 1/2 near the limbs. The geometric probability averaged over the exposed part of the.model atmosphere is therefore between 1 and 1/2. y as a function of the molecular weight would be expected to approach 1 as the molecular weight became very small, and to approach 1/2 as the molecular weight became very large. For hydrogen and helium we will, therefore, use y = 1, and we will use y = 1/2 for the remaining gases. Using the values of C, h, and i-ro2/V taken from Table 1, the values of J and Jp taken from Table 3, the values of oesc taken from Table 4, and the values of ZcTo taken from Section III-G, we can proceed to calculate the time rate of change of the atmospheric density of hydrogen and helium as follows: dN dno dt dt or dno - crO2 n c e2 ro/h( + ) dt V P V- - Pno(J(ec) 7 + Jec7 + oCo). (4-5) Table 5 gives the:numerical values for use in Eq. (4-5). For both hydrogen and helium, the primary escape mechanism is seen to be the evaporation of the tail of the velocity distribution, which does not depend upon assumptions concerning the scale volume, as previously noted. This is expected, since the thermal velocities are comparable to the escape velocity. Given any initial density, the above differential equations govern the time rate of change of the density at the lunar surface according to our model. For J = 109 part/cm2sec, it is seen that the time constant for approach to an equilibrium value is 7.7 x 103 sec, or about two hours, for hydrogen, and 2.5 x 10 sec, or about seven hours, for helium. The equilibrium values are 46

oN cO o\ cO tbr-4 r-1 r-4 r-q r-1 rA O O O O O O r- r r- r- Hr X X X X X X L r - -H \ OJ 0 O CO H \ O\ t - \ O p H O O O O O O O r- r- r- r r- r-H 41X X X X > X pi O Ir 0C \ K CO O CO o C 4 \L KO r X X X X X X o H 1 I I I 0 o 0 0 0 0 a 3 CH *H dl 0 k U) LCU rH rdj U) rO M [t\ CD U) CM C O O D II H Do o -I o- k~ o!) o? ( 0- H - H ri H+' H1r-l r-A r-A r-q pi4-p P 40Wo, r-^n~ ~ ~ 1 x x x x Ho O LC\ F'. 0 0\ 0 0 _ c } p^ 0 +I>- r-| -J- r-l+^ r -p o 0 x x rx r-l0 r-i -0 <Lr\ N H 0 0 0 o o 00 CO) D - -H H H1+ G LCI\ CM CM J zI.. ) I H H C C H H CO 0 O H U 0 -t — dI hI (D co -o C*c) ~ lH I IO ~ s O e a) CE <I r) C ) CP 0 C CH UO O C. O O O M H c - * C 0 z O1 fl kG L\ zj z1- rr 1 I! J Hd 0 J O d LCM CM JC CM CM ^5 6\1 <^ X X II 0 0I 11 M4 A\0 CD?- 0 M H H H e C iCUd) CO C CO C C H |m g Q Q 00 0 47

thus attained rather quickly; they are given for J = 109 part/cm2 sec as follows: H: no = 5.30 x 103 atoms/cm3 He: no = 9.13 x 103 atoms/cm3. Thus, although the number density of helium in the solar wind is likely to be from 1 to 2 orders of magnitude less than that of hydrogen, the evaporation of the helium provides an escape rate of from 1 to 2 orders of magnitude slower than that of hydrogen, so the equilibrium number densities are about the same. The larger scale volume of hydrogen also adds to this effect. Since the evaporation flux is of predominant importance for the escape of these light gases, we may also estimate the average number density at the lunar surface by equating the number of particles escaping/cm2/sec from some particular place on the lunar surface due to the evaporation, to the influx Jp cos 8, where 8 is the angle which the sun makes with the zenith at that place. Therefore, Jpcos E exp foO<m< for_ 0 << 2 kT() 1 + mgro n(e) = T(4-6) 0, for < Q < This formula implies a detailed balancing at every point on the surface, which cannot be the case. Because of the long flight paths of particles emitted from the surface on the day side of the moon, there will always be more particles near the terminator, 0 = t/2, and on the night side than given by the above formula. The value of n given by the above formula is, therefore, too high in the region near 8 = 0, since the only input is the solar proton flux Jpcos 8. The value predicted at 8 = 0 is: n(0o) = 2.17 x 104 atoms/cm3 for H. Applying the same reasoning in the case of helium gives n(0O) = 3.67 x 10 atoms/cm. These values are about four times higher than those gotten with our simple model, where migration of atoms was taken (implicitly) to be efficient 48

enough that the density was uniform over the entire surface. To see which point of view is more justified, let us assume that an atom of hydrogen and of helium is emitted from the lunar surface at an angle of 45~ with the horizontal plane and with the mean speed characteristic of the temperature T. As shown in Fig. 3, the angular ranges of hydrogen for temperatures between 2000K and 400~K, which are typical of the day side, are enough longer than those for temperatures between 120 and 2000 to enable migration to take place fairly readily from the day side to the night side. Therefore, the density of hydrogen would be expected to be fairly uniform over the entire surface, as assumed in our simple model. On the other hand, the curve of angular range versus temperature for helium is not nearly as steep, indicating that the net migration of helium atoms from the day side to the night side is not as efficient, resulting, therefore, in a somewhat higher density on the day side. B. WATER VAPOR The term I, is equal to the equivalent source strength for the entire surface given in Section III-F multiplied by the lunar surface area: Ii = Js4ro2. (4-7).1 and I3 have the same form as for hydrogen and helium. Although the scale height is larger on the sunlit side, so that a slightly larger fraction than 1/2 of the scale volume is exposed to the sun, we expect somewhat higher densities for water vapor and the heavier gases on the dark side. We will, therefore, use the value P = 1/2, in the term 13 and use the value 1/2 for the yfactor for water vapor. The largest contribution is seen in Table 5 to be the ionization and dissociation. The contributions due to the solar wind are somewhat smaller. The time constant is 105 sec, or about 28 hours. The equilibrium value is no = 1.7 x 103/cm3 for J = 109 part/cm2 sec. The corresponding partial pressure of water vapor in regions where the temperature is about 120~K is about 2 x 10"14 mm Hg. Thus, the partial pressure is considerably lower than that needed to stop the net evaporation (according to Vestine27 the equilibrium vapor pressure of water at 120~K is about 5 x 10-12 mm Hg), and we are justified in using the evaporation rate which we used. C. ARGON The term I1 has the same form as for water vapor. We neglect I2, since it is much smaller than 13. The time constant for approach to equilibrium is 106 sec, or about 11.6 days, and the equilibrium value is no = 5.4 x 104/ cm3 at J = 109 part/cm2 sec. 49

D. KRYPTON The time constant is about the same as for argon, about 11.6 days, and the equilibrium value is no = 1.7 x 10-2/cm3. E. XENON The time constant is also about 11.6 days, and the equilibrium value is no = 1.87 x 10-3/cm3. We see that krypton and xenon comprise a very small portion of the lunar atmosphere. F. THE VARIATION WITH THE STRENGTH OF THE SOLAR WIND The composition of the lunar atmosphere will be expected to vary considerably with the strength of the solar wind. To see this, consider the following: in the case of hydrogen and helium, the predominant escape mechanism is the evaporation of the tail of the velocity distribution, while removal by solar corpuscular streams and ultraviolet radiation is negligible in comparison. For water vapor, argon, krypton and xenon, the situation is reversed: the evaporation term is negligible in comparison to the other terms. For hydrogen and helium the source is the solar corpuscular flux itself, while the sources of the other gases lie within the moon or on its surface. Since the time constants for approach to equilibrium of all of these constituents are less than twelve days, the densities, which depend upon the solar wind flux J, would be expected to change in response to a change in J in a correspondingly short time. Figure 7 shows the equilibrium values of density for the constituents whose densities are greatest, Ar, He and H, as a function of J. For values of J much below 109/cm2 sec, the main constituent is Ar, at a density of about 5.5 x 104/cm3. For values of J above 109/cm2 sec the densities of H and He increase with J while the density of Ar decreases, since the effects of the solar wind in sweeping it away begin to predominate over the photoionization. For values of J above about 1011/cm2 sec, the density of H increases more slowly with J, and approaches a limit at about 2.9 x 105/cm3, when the increase of hydrogen accretion from the solar wind with increasing J is just balanced by the sweeping away effect of the solar wind. The density of helium continues to increase with increasing J until J reaches about 10l2/cm2 sec, when it begins to level off and approach the limit 9.6 x 105/ cm3. We have assumed that the flux of ultraviolet radiation is not affected by the level of solar activity, while the corpuscular streams are. Since most of the ionizing and dissociating effects are caused by radiation of wavelengths greater than 100X, this is justified if the short wavelength region whose flux is enhanced during periods of solar activity lies below 100b, as indicated by Shaw.48 50

We see from Fig. 7 that the minimum density of the lunar atmosphere is about 5.5 x 104 particles/cm3, during conditions of very small solar flux. We must emphasize again, that the representation of the partial densities in the lunar atmosphere in Fig. 7 is incomplete, since we have omitted the gases CO2 and S02. Our simple atmospheric model may be thought of, therefore, as a minimal model of the lunar atmosphere. G. THE LUNAR SURFACE CHARGE AND CHARGED PARTICLES IN THE LUNAR ATMOSPHERE Because of the photoionization of atmospheric particles and charge exchange with solar protons, we expect that charged particles will be present in the lunar atmosphere. These charged particles would be able to recombine on the lunar surface if the surface had no repulsive potential. However, there are several ways in which the surface may become charged. On the sunlit side of the moon, the predominant effect is expected to be photoemission of electrons caused by solar ultraviolet radiation. Since this effect tends to charge the surface to a positive potential, we expect the surface to be positively charged over most of the sunlit side of the moon. Therefore, an atmospheric ion cannot strike the surface unless its kinetic energy is greater than the surface potential. Consequently, a negative space charge region will form close to the surface. The potential of the surface is determined by the condition that the number of electrons emitted per unit area and time due to photoemission must equal the difference between the random electron current to a unit area from the space charge region and the solar wind, and the current of ions from the solar wind and those from the atmosphere which have energy greater than the surface potential. Near the limbs of the moon the photoemission of electrons should be smaller than at the subsolar point, and it is possible for the lunar surface to acquire a negative potential because the random current of electrons from the solar wind and atmosphere is larger than that due to positive ions. Therefore, if the moon is a poor conductor of electricity, we expect most of the sunlit side to be positively charged and some of the shaded side to be negatively charged. This is justified by the results of Gibson49 concerning radio: measurements of the properties of the lunar surface. Assuming a dielectric constant K = 5, he finds the conductivity of the top few millimeters of the moonts surface corresponding to the measured attenuation to be 1.16 x 10-2 mho/meter. Therefore we expect that the lunar surface will be able to maintain a substantial potential gradient with reasonably small surface currents The positive surface charge has another effect on the charged particles in the atmosphere in addition to hindering their recombination, If a neutral particle is ionized while it is within the shielding distance of the surface, it will be ejected from the atmosphere if the potential in volts at the point where it is ionized exceeds the escape energy for the particle in electron 51

volts. (Escape energies for particles of atomic weight less than 140 are less than 4 ev.) This has been pointed out by Opik and Singer.50 These authors estimate the screening length for an electron density of about 500/ cm3 to be about 400 cm. Since this is so small compared to the scale heights for the gases which we consider, this effect will be ignored here. We will not attempt to compute the magnitude of the surface potential. If we assume that an atmospheric ion has an energy equivalent to temperatures of 1500~K, which is approximately true for the earth's ionosphere, the surface potential need only be about.2 volts to effectively hinder surface recombination on the sunlit side of the moon. If the surface potential is not at least this high, there will be too few charged particles in the lunar atmosphere to worry about. Therefore, for simplicity, we will assume that no surface recombination or volume recombination will occur on the sunlit side in the moon's atmosphere. Also, in the following computation of ion density, we will assume that the diffusion of ions in the atmosphere to the limbs of the moon, and their eventual recombination there, is negligible compared to the other processes considered which effect ion removal. This, of course, is not strictly true. Our model therefore, predicts the maximum number of ions that can be present in the moonts atmosphere, assuming that the only atomic species present are those considered here. The model we wish to present here for the lunar atmospheric ion density is similar to the model for the lunar neutral particle atmosphere. We state that all the ions associated with the atmosphere are contained in the scale volume Vi, and have a constant number density, noi, which will be equal to the surface number density in the real case of an exponentially decaying exosphere. For this model, we assume an ion temperature of 1500~K. Since the plasma must remain neutral, the scale height for each species is determined by assuming that the effective mass of the ion is half the actual mass to account for the electrical forces caused by the much larger scale heights associated with the electrons. For half the mass and five times the temperature the scale heights for the various ions under consideration are 10 times the scale height of the gases given in Table 1 of Section III-C. The source of the ions is assumed to be that of the number of ions produced by photoionization plus the number of ions experiencing charge exchange with solar protons which do not escape. This term is expressed; by: I1 = DnoV[J((c) - cx)) +o]. (4-8) 52

HEere, 5 is the fraction of the neutral particle scale volume which is not shielded from the sun, We will assume here that for H and He, 3 = 1; and for H20 and Ar, P = 1/2 as before. The two processes considered here which cause a removal of ions from the atmosphere are the evaporation of the tail of the velocity distribution and the ejection of ions due to collisions with the solar protons. The Jean's escape flux is expressed by: I2 = PiTr2noiie /( + h) (4-9) where Pi is the fraction of the ionized scale volume which is effected by this term and subscript i denotes a quantity associated with the ions. The removal of ions by collisions with solar protons is given by: I3 = PinoiJVi i el) (4-10) esc where i (sl) is given from the results of Section III-G to be: a(el) Icb2(es es = b(xesc) where b(8esc) = Zeff e2 2 (4-11) m2Vesc Here, m2 is the mass of the positive ion and m is the mass of the particle which strikes it, which we assume to be a proton. For singly ionized ions, we use Zeff = 1 so that,(el) 1.7xlO-5 cm2 (4-12) i esc (M W.) where M.W. is the molecular weight of the atmospheric ion. Therefore, our equation becomes: dNi dnoi d V dt = I1- 12 I3 53

or dnoi V odn = pno [J(a() (cX)) + Zoo] dt Viesc - Bi - noiie - ~/ i(1 + i)(4-13) vi PinoiJ isael) setting dnoi/dt = 0 for equilibrium, we get: V[J((CX) - a ) + ZEo] flnj - Vi oi = ----. (4-14) no iFr2 - ro/hi( r noi * cie (1 + ) + J e Obtaining the desired values from Table 4 and Table 6, we get, for J = 109: noi/no noi H+ 6.2 x 10-4 3.3 x 10~ He+ 7.2 x 10-6 5.9 x 10-2 H2aO 1.8 x 10-3 3.0 x 100 Ar+ 6.31 x 10-3 3.41 x 102 The total ion density is 347/cm3 for quiet solar conditions. This result argrees well with that of Elsmore,3 which is about 1000/cm3 for somewhat more active solar conditions. 54

C) C C) C) A lf -0 CO CO Ht H H H1 H H n H -rl. HH H H i ri 0 H o 1 I I I 0 0 0 0 U) *, H H H H CQ C OJ \ O -t L- 3O. C Cd O. - r; C CM C M r-q r-A r-{ o ^ Po o Co sq gg S g 6 Q rn -H c. Co Co r \ H H o 1 1 E U O H H \ os o M M\ 0o -:1 r' 11 I s r 0 00 0 + +U 0) 0 H a) ^ ro H r S P 11 X X X X O Q 1 \-i cO N ON O + +O + O r, Cs 55

106,' l I I 11 llI I I i i i I! I1 1 I- I,1I I 1 l ~!lI,'',,,,_,XI 105 X XX~ i0 ~ 10 x *X 7- 0 t~ ~ / X /) < )^ r r y —--— ~ x j- ^^. Z L_ ~~~~~~~~~~~~~~~~~~~~~~~'X~ ^ - ^ / L —~/(Taotl fon density) aAr S HW nH20 M I 0 $ - ~xx~~~~~~~~~~~ Z X 107 108 109 10'~ 10" 1012 1013 SOLAR PARTICLE FLUX J ATCE 0 %A. cmn sec Fig. 7. Lunar modetal atmosphere. Surface number density vs. solar particle fl.Ar 56 032 HI0 I t I I t i l li I i I t i li I I I! l t - I i I, t lI I I I J! 1 ]J i I J 107 I 08 I 09 I 0~ I0t I 02 I01 SOLAR PARTICLE FLUX 5 ATCE cm 2sec

SECTION V DISCUSSION OF THE PROPOSED MEASUREMENT The experimental determination of the lunar atmospheric structure is a task similar to that required for any planetary atmospheric structure determinationo The parameters desired are neutral and charged particle number density, composition, and temperature as a function of altitude, time and latitude. In the case of the terrestrial atmosphere, instruments have been devised to measure all of these quantities to great altitudes. However, the tenuous lunar atmosphere precludes the use of most of the instruments for atmospheric structure measurements on the moon, The most important single deficiency in measurement capability is our inability to measure atmospheric composition on the lunar surface with instruments available today, State-ofthe-art flight hardware does very well, with special handling, if it measures partial pressures on the order of the expected maximum total pressure on the moon's surface, Also, this instrumentation is heavy and temperamental and not easily adaptable to the environment expected on the moon, Therefore, we are limited! to the possibilities of neutral and charged particle number density and temperature measurements for initial experiments. Of these, we feel that the neutral particle number density is the most important single measurement, As described in the previous section of this report, the number of charged particles in the lunar atmosphere is probably less than 1% of the number of neutral particles. Therefore it seems important, for initial flights, to measure the neutral density to better understand the ordersof-magnitude with which we are dealing. Once the neutral number density is known, it is possible to view the problem of composition determination more objectively. A. INSTRUMENTATION Our initial choice of instrumentation is a cold-cathode discharge gauge (Redhead gauge) with its required electronics. This instrumentation is capable of measuring total neutral number densities from approximately 200 cm"3 to 2 x 1010 cm-3. Details of this gauge and the instrument package can be found in Refs. 51 and 52. Basically, the Redhead gauge is an ionization gauge. It utilizes electrons obtained by a cold-cathode discharge in a crossed electric and magnetic field to ionize the gas. Neutral gas particles enter the gauge through an orifice and tube and are ionized. The resulting ions are collected, producing an ion current which is monitored by an electrometer amplifier. Our instru57

mentation is capable of detecting ion currents from approximately 10-6 amps to 1 1 amps B. CALIBRATION As mentioned previously, the Redhead gauge to be used for the: measurement of the lunar atmospheric neutral number density is an ionization gauge whose ion current is approximately linearly proportional to gas density for any particular gas. The proportionality: constant for each particular gas must be experimentally determined, as must the slight variations from linearity for each particular gauge, i.e., the gauge must be calibrated against a known standard. The proportionality constant is called the sensitivity, and is determined primarily by the ionization cross section or ionization probability for a given gas. The performance of this type of gauge is covered in more detail in Ref. 51. Since we do not know:. the composition of the moon's atmosphere, it is impossible to calibrate a gauge for the correct gas or mixture of gases. However, we do know that, of the expected gases in the moon's atmosphere, the gauge will have the lowest sensitivity for helium and hydrogen and the highest sensitivity for argon and the residual volcanic gases, with a maximum spread in sensitivity of approximately one order of magnitude. The gauge will be calibrated for as many of these gases and mixtures as possible, so that data received during early flights will be useful when further knowledge of composition is obtained. It is expected that argon and the volcanic gases will exhibit highest density on the dark side of the moon since -they are being evolved from all parts of the moon equally and will be shielded from the solar wind on the dark side. Helium should exhibit highest density on the sunlit part of the moon since it is being produced there. Therefore, the measurement for one lunation should give some indication of the predominant gas, thereby increasing the accuracy of our interpretation of the measurement. C. MEASUREMENT The measurement to be made will give us neutral number density versus time at the lunar surface for at least one complete lunation. The expected minimum equitorial number density for our theoretical model is described in Section IV. The moon's atmospheric neutral number density is interpreted from our measurement of gauge number density by the thermal transpiration equation: n = id ng n (5-1)

It will be assumed that the gas particles in the gauge will be at the temperature of the gauge (Tg), a measured parameter. If we assume that the moon's atmosphere is predominantly composed of gas molecules of high molecular weight, such as argon, CO0 and SO2, then the gas temperature outside the gauge (Tm) can be assumed to be at the temperature of the local surface. This can be seen by noting that the range of these molecules, as discussed in Section III, is short enough such that the surface temperature does not vary appreciably over the distance. As an example of this, we see in Section III that argon, at 3000K, has a range of 97.9 km (angular range of 3.20"~). Figure 1 gives the variation of surface temperature with angle of rotation with respect to the sun for the lunar equator, and the design package temperature for this condition. As can be seen, an angular range of 35.2~ gives a maximum error of approximately 100K in gas temperature at the point where the surface temperature is 3000~K, Of course, near the terminator the temperature varies much more with distance, but the range will also be smaller due to the lower surface temperature, With the above assumptions, we can plot an expected ratio of gauge number density to atmospheric number density for one lunation (Fig. 8). We divide the measured number density by this ratio for each specific time to obtain the atmospheric number density. All of the above considerations assume that the spacecraft will be on the lunar equator, which will not necessarily be the case. At the time of the experiment, ng, Tm, and Tg will all be measured, and a true correction factor computed. D. COCTAMINATION One problem that confronts us in our measurement is that of the contamination of the moon's atmosphere with gases evolving from the spacecraft, Contamination of the Redhead gauge itself is minimized by evacuating and sealing it on the earth, then breaking the seal after landing on the moon (see Ref. 52). However, the spacecraft itself will be a contributor of gas for some time after being launched from earth, not to mention the retro-rocket contribution of gas upon landing on the moon. The contaminating effect of the outgassing of the spacecraft on our experiment will be minimized by mounting our package on a 15-foot boom, which extends after impact, and orienting the package such that the orifice of our gauge "sees" none of the spacecraft in its hemispherical view. Since the mean-free-path of a gas molecule in the moon's atmosphere is certainly much larger than the dimensions of the spacecraft, we can assume that each molecule will travel in a ballistic trajectory away from the spacecraft. The number of molecules of spacecraft gas having a trajectory which intersects the gauge orifice is assumed negligible, if the gauge is oriented as described above. Also, the total amount of gas which the spacecraft can contribute to the moon's present atmosphere is assumed negligible. 59

0 L ~ a) PA /0 /-sgz- ^ 60( o - wi 0 p60 o ~ o zl br0 co -0 60 6o

The major problem of contamination is afforded by the retro-rocket motor used during descent to the moonts surface. Table 7 gives the composition and amount of these gases. A quick computation assuming an effective molecular weight of 35 for these gases, a scale height of 50 km, a. total weight of gas of 1600 lb yields a surface number density of 6 x 103 cm-3 if the gas is evenly distributed about the moon's surface and escape mechanisms are neglected. Of course, the high initial thermal velocity of these gas molecules will cause a significant percentage of them to escape immediately. The remaining gas will disperse rapidly, becoming a part of the lunar atmosphere. As discussed in Section IV we feel that the minimum atmospheric density on the moon is about 5.5 x 104 cm3, Therefore, soon after landing it is expected that the density of the gases from the retro-rocket will be no more than 10% of the atmospheric density. The prementioned escape mechanisms cause a decay time constant of approximately twelve days or less for gases similar to the retro-rocket gases. Therefore, in the 30-day lifetime of the spacecraft, the amount of retrorocket gas in the atmosphere will decrease by more than one order-of-magni'tude. If surveyors are landed no gore often than every 30 days, the effect of the retro-rocket gases on our measurement will be negligible. E. SOLAR FLUX Protons from the solar wind will be expected to enter the orifice of the gauge, become neutralized on the inside surfaces, and contribute to the number of neutral particles in the gauge. The contribution to the number density inside the gauge due to solar wind particles will be denoted nsg and will now be calculated, using our previous assumptions concerning the solar wind. If <u>avg is the mean velocity of motion of the plasma (500 km/sec), the distribution of the components of velocities normal to the orifice of the gauge, u, in a coordinate system at rest on the moon at some place where Ir is the angle between the centerline of the orifice and the sun is 1m M/2 "i ( -<u~ avg cos )2 f(UJr) = (i e (5-2) where T is the kinetic temperature of the solar wind (105~K). If A is the area of the orifice, the number of particles entering the gauge per second is 00o o m 2 dni P /m 1/2 0 - (u-<u>avg cos ) dtn = -nA ufdu = nA( ) kT ue du in.( e ).kT du 61

where n is the number density in the solar wind (20/cm3). Let = 2VkT (u-<>avg cos ); the integral can then be evaluated as follows: dtJin 2 m 1- K 2 kT avg <U>avg cOs' + <u>-avg cos (1 + erf 2 <u>avg cos; (5-4) Now <u>avg cos U =. <uavg cos = 12.1 cos r, t 2kT <Uavg COs 2 Urms using values taken from Table 2. If we take r = 0~, then we can use the asymptotic form for the error function: e-s 1 + erfs 2 - H- 2 for s 1. (5-5) sJt s Therefore, we have, with Em s = U2kT avg cos 1 a dtn nA <u>avg = AJ(56) where J is the solar wind flux. The number of particles which were contributed by the solar wind which leave the gauge per second:.is given by the well-known effusion equation: dnsg] = n gi (5-7) dtl out nsgA 2n 62

where Tg is the temperature inside the gauge. When the net number of particles entering the gauge from the solar wind is zero, dn dntg out or n (5-8) Tjin or nsg =..Fg J k~g dt in dtOdut Jo ut' If the temperature inside the gauge is 300~K, then nsg = 1.6 x 104/cm3. When T = 90~, 1 = nA kT (5-9) dtjin =m 2 so the equilibrium density of particles inside the gauge due to the solar wind is given by nsg = n (5-10) the ordinary thermal transpiration equation. When T = 105~K and n = 20/cm3, nsg = 365/cm3. In the cases where q is not close to 90~, the contribution of particles inside the gauge due to the solar wind is comparable to that which we expect due to argon, and so the contribution due to the solar wind may be comparable to the total contribution from the neutral particles in the atmosphere. If the flux J is measured independently, we will known nsg during the times when it is large, and corrections to our measurement will be made. 63

TABLE 7 RETRO ROCKET GASES Vernier (160 lb) Main Retro (1450 lb) 0 0 = 0.1 to 0.3 lb/sec m = 30 lb/sec H2 32.4% H20 12.9% H20 19.2 N2 9.2 N2 31.0 CO 23.7 NH3.001 HC1 17.2 CH4.2 C02 3.6 CO 2.7 H2S.1 CO2 14.5 H2 33.3 Burns from 280,000 ft Burns out between to 13 ft 9000-55,000 ft Ignition ~ 280,000 ft 64

REFERENCES 1. Sytinskaya, N. N., "The Problem of the Lunar Atmosphere," The Moon, edo by A. V. Markov, Univ. of Chicago Press, 1962. 2. Dollfus, A., "A New Investigation of an Atmosphere in the Neighborhood of the Moon," Comptes rendus, 234, 2046 (1952). 3. Elsmore, B., "Radio Observations of the Lunar Atmosphere," Phil Mag,, 2, 1040 (1957). 4.: Krasovskii, V I,, "Results of Scientific Investigations Made by Soviet Sputniks and Cosmic Rockets," A.R.S. Journal, 30, 27 (1960). 5. Herring, J. R., and Licht, A. L., "Effect of the Solar Wind on the Lunar Atmosphere," Science, 30, 266 (1959). 6. Parker, E. N., "Interaction of the Solar Wind with the Geomagnetic Field," Phys. Fluids, 1, 171 (1958); "Dynamics of the Interplanetary Gas and Magnetic Fields," Astrophys. J., 128, 664 (1958); "Cosmic Ray Modulation by Solar Wind," Phys. Rev,, 110, 1445 (1958). 7. Biermann, L., "Kometenschweife und solare Korpuscularstrahlung," Zeit. f. Astrophysik, 29, 274 (1951), 80 Kiepenheuer, K. O., "Solar Activity," The Sun, ed. by G. P. Kuiper, Univ. of Chicago Press, 19535 9. Parker, E. N., "The Hydrodynamic Treatment of the Expanding Solar Corona," Astrophys. J,, 132, 175 (1960); "The Hydrodynamic Theory of Solar Corpuscular Radiation and Stellar Winds," Astrophys. J., 152, 821 (1960). 10. Behr, A,, and Siedentopf, He, "Untersuchungen fiber Zodiakallicht und Gegenschein nach lichtelektrischen Messungen auf dem Jungfraujoch," Zeit. f. Astrophysik, 32, 19 (1953). 11. Blackwell, D. E., and Ingham, M. F., "Observations of the Zodiacal Light from a Very High Altitude Station II. Electron Densities in Interplanetary Space," Monthly Notices, Royal Ast. Soc., 122, 129 (1961). 12. Gringauz, K. I., "Some Results of Experiments in Interplanetary Space by Means of Charged Particle Traps on Soviet Space Probes," Space Research II, p. 539, North-Holland Publ. Co., Amsterdam, 1962. 65

REFERENCES (Continued) 135 Parker, E. N., "Cosmic Ray Modulation by Solar Wind," Phys. Rev., 110, 1445 (1958). 14. Dolginov, S. Sh., Erashenko, E. G., Zhuzgov, L. N., and Pushkov, N. V., "Measuring the Magnetic Fields of the Earth and Moon by Means of Sputnik III and Space Rockets I and II," Space Research, p. 863, North-Holland Publ. Co., Amsterdam, 1960. 15. Neugebauer, M., "Question of the Existence of a Lunar Magnetic Field," Phys. Rev. Letters, 4, 6 (1960). 16. Pettit, E., and Nicholson, S. B., "Lunar Radiations and Temperatures," Astrophys. J., 71, 102 (1930). 17. Pettit, E., "Radiation Measurements on the Eclipsed Moon," Astrophys. J., 91, 408 (1940). 18. Wesselink, A. F., "Heat Conductivity and Nature of the Lunar Surface Material," Bulletin of the Astronomical Institutes of the Netherlands, 10, 351 (1948). 19. Present, R. D., Kinetic Theory of Gases, p. 8, McGraw-Hill, N. Y., 1958. 20. Goldstein, H., Classical Mechanics, Addison-Wesley, Reading, Mass., 1959. 21. Firsoff, V. A., "Dissipation of Planetary Atmospheres," Science, 130, 1337 (1959). 22. Brandt, J. C., "Density of the Lunar Atmosphere," Science, 131, 1606, (1960). 23. Singer, S. F., "Atmosphere Near the Moon," Astronautica Acta, 7, 135 (1961). 24. Herring, J., and Kyle, L., "Density in a Planetary Exosphere," J. Geophys. Res., 66, 1980 (1961). 25. Opik, E. J., and Singer, S. F., "Distribution of Density in a Planetary Exosphere," Phys. Fluids, 2, 653 (1959); "Distribution of Density in a Planetary Exosphere," Phys. Fluids, 3, 486 (1960). 26. Jeans, J. H., The Dynamical Theory of Gases, p. 342, Dover, N. Y., 1954. 66

REFERENCES (Continued) 27. Vestine, E. H., RAND Corp. Research Memorandum RM-2106, 1958. 28. Gold, T., "Permafrost on the Moon," paper presented at the 42nd Annual Meeting of the American Geophysical Union, Wash., D. C., April 18-21, 1961. 29. Watson, K., Murray, B. C., and Brown, H., "The Behavior of Volatiles on the Lunar Surface," J. Geophys. Res., 66, 3033 (1961). 30. Edwards, W. F., and Borst, L. B., "Possible Sources of a Lunar Atmosphere," Science, 127, 325 (1958). 51. Hinteregger, H. E., "Interplanetary Ionization by Solar Extreme Ultraviolet Radiation," Astrophys. J., 132, 801 (1960). 32. Pottasch, S. R., "Use of the Equation of Hydrostatic Equilibrium in Determining the Temperature Distribution in the Outer Solar Atmosphere," Astrophys. J., 131, 68 (1960). 33. Chamberlain, J. W., "Interplanetary Gas II, Expansion of a Model Solar Corona," Astrophys. J., 131, 47 (1960). 54. Chapman, S., "Notes on the Solar Corona and the Terrestrial Ionosphere," Smithsonian Contributions to Astrophysics, 2, 1 (1957). 35. Spitzer, L., Physics of Fully Ionized Gases, p. 78, Interscience, N. Y., 1956. 56. Massey, H.S.W., and Burhop, E.H.S., Electronic and Ionic Impact Phenomena, Oxford, the Clarendon Press, 1952. 37. Leighton, R. B., Principles of Modern Physics, McGraw-Hill, N. Y., 1959. 38. Loeb, L. B., The Kinetic Theory of Gases, Dover, N. Y., 1961. 39. Fite, W. L., Brackmann, R. T., and Snow, W. R., "Charge Exchange in Proton-Hydrogen Atom Collisions," Phys. Rev., 11, 1161 (1958). 40. Keene, J. P., "Ionization and Charge Exchange by Fast Ions of Hydrogen and Helium," Phil. Mag., 40, 369 (1949). 41. Hasted, J. B., "The Exchange of Charge Between Ions and Atoms," Proc. Roy. Soc. A., 205, 421 (1951). 67

REFERENCES (Concluded) 42. Wainfan, N., Walker, W. C., and Weissler, G. L., "Photoionization Efficiencies and Cross Sections in 02, N2, C02, A, H20, H2 and CH4," Phys. Rev.. 99, 542 (1955)o 43. Astoin, N., Johannin-Gilles, A., and Vodar, B., "Absorption of Water Vapour in the Extreme Ultraviolet," Comptes rendus, 237, 558 (1953). 44. Granath, L. P., "The Absorption of Ultraviolet Light by Oxygen, Water Vapor and Quartz," Phys. Rev., 34, 1045 (1929). 45. Watanabe, K., and Zelikoff, M., "Absorption Coefficients of Water Vapor in the Vacuum Ultraviolet," J. Opt. Soc. Amer., 43, 753 (1953). 46. Detwiler, C. R., Garrett, D. L., Purcell, J. D., and Tousey, R., "The Intensity Distribution in the Ultraviolet Solar Spectrum," Annales de Jeophysique, 17, 263 (1961). 47. Po Lee and Weissler, G. L., "Absorption Cross Section of Helium and Argon in the Extreme Ultraviolet," Phys. Rev., 99, 540 (1955). 48. Shaw, J. H., "The Radiation Environment of Interplanetary Space," Applied Optics, 1, 87 (1962). 49. Gibson, J. E., "Lunar Surface Characteristics Indicated by the March, 1960, Eclipse and Other Observations," Astrophys. J., 133, 1072 (1961). 50. Opik, E. J., and Singer, S. F,, "Escape of Gases from the Moon," J. Geophys. Res., 65, 3065 (1960)o 51. Redhead, P. A., "The Magnetron Gauge: A Cold-Cathode Vacuum Gauge," Canadian J. Phys., 37, 1260 (1959). 52. Taeusch, D. R., Lunar Atmosphere Experiment, Interim Engineering Report No. 1, Project 03555, Univ. of Mich. Res. Institute, June 1962. For information on photoionization cross sections, see also the following: Weissler, G. L., "Photoionization in Gases and Photoelectric Enission from Solids," Encyclopedia of Physics, Vol. XXI, p. 304, Springer, Berlin, 1956. 68

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