03346-1-T THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING High Altitude Engineering Laboratory Departments of Aerospace Engineering Meteorology and Oceanography Technical Report THEORETICAL MODEL FOR CONVERSION OF OBSERVED NEUTRAL AND ION DENSITIES TO AMBIENT DENSITIES FOR ORBITING GEOPHYSICAL OBSERVATORIES B. B. Hinton, R. J. Leite, and C. J. Mason ORA Project 03346 under contract with: NATIONAL AERONAUTICS AND SPACE ADMINISTRATION GRANT NO. NGR 23-005-383 WASHINGTON, D. C. administered through OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR January 1970

Abstract A spacecraft perturbs the density of ions and neutral molecules and atoms as it moves through the tenuous upper atmosphere of the earth. In this report expressions are derived for the resultant perturbations on measured values of the particle densities for mass spectrometer experiments carried on the OGO spacecraft. Using these equations the ambient values can be recovered from the measured densities for neutral particles, and from the measured fluxes for positive ions. ii

Table of Contents Page Abstract ii List of Figures iv TEXT I Introduction and Purpose 1 II Assumptions and Elementary Considerations 2 III Expansion to Include Multiple Reflection Processes 7 IV Positive Ion Measurements 13 V Conclusions 14 References 14 iii

LIST OF FIGURES Page Figure 1 Incident Particles 3 Figure 2 The Solid Angle Element 4 Figure 3 Number Densities 5 Figure 4 The OGO Spacecraft 6 iv

I. INTRODUCTION AND PURPOSE The neutral and positive ion concentrations obtained by in situ mass spectrometer measurements on an earth-orbiting vehicle must be * adjusted to account for the influence of the vehicle. The following work was initiated to develop the proper transfer functions for converting the OGO-4 Mass Spectrometer measurements of neutral-particle concentrations and positive-ion fluxes into ambient neutral and ionic number densities. The purpose of this report is to present a set of theoretical expressions from which the detailed calculations can proceed. It is planned to present the actual program for carrying out these calculations as a separate report. The near polar orbit of OGO-4 has a perigee of about 400 km and an apogee of 900 km. In this altitude range inter-particle collisions may be neglected since the spacecraft dimensions are orders of magnitude smaller than the mean free path. Except for electrons and the lightest neutrals and ions the spacecraft speed is much greater than the thermal speeds. Therefore, at least for the intermediate and heavier masses, one would expect the densities and fluxes to be significantly greater on surfaces looking into the velocity vector, and significantly less or surfaces looking away from the velocity. However, the density and flux are determined not only by exposure to the incident particles, but also by reflections of particles from one surface to another. The calculation of this transfer from one surface to another as well as the directly incident particle densities and fluxes is considered in this report. *See reference (1) 1

II. ASSUMPTIONS AND ELEMENTARY CONSIDERATIONS The problem is not solvable without certain simplifying assumptions because the interaction of atoms and molecules with surfaces is a poorly understood phenomenon, the surfaces themselves are not well defined, and, in addition, the surface structure may even change with time. Consequently, in order to approximate the real situation by a solvable problem we shall assume the following: (ia) The problem can be approximated for atomic oxygen by a steady state in which the flux of atoms arriving at a surface is equal to the flux of the same species of atoms plus the flux of the recombined diatomic molecules (of the same species of atoms) leaving the surface, the incident molecular oxygen being neglected; (ib) For other species there is assumed to be no chemical change; hence the incoming and outgoing fluxes are assumed to be equal. (ii) The molecules leaving the surface have interacted strongly enough with the surface that they have "forgotten" their direction of arrival and are directed isotropically; or alternativelywe could assume that on the molecular, or atomic scale the surfaces are "rough" and arrive at the same isotropic distribution. (iii) Intermolecular collisions are neglected. In order to arrive at the solution, use will be made of the solutions of two'elementary" problems. The first of these is the calculation of the incident flux of ambient particles on a moving surface and the other is the transfer of particles from one surface to another. In Fig. 1. consider the surface S, which is non-concave; that is, the solid angle subtended by S at any point on the surface is zero. Other surfaces S may subtend non-zero solid angles when viewed from points on S however. The velocity of the surface is V. The Maxwellian velocity distribution of the ambient particles relative to S is f(v = f(V-v) where vm is the Maxwellian velocity of a particle, f3 the three dimensional distribution, and v the relative velocity of the spacecraft with respect to the particles 2

(v +vm = V). The particles approaching a point on S from qSti ~the direction r are those for which v( * n )> 0 where ns is S By., 93- the unit normal at the point, S t/'"' -,, -. " thus 4in, the flux per unit area -|8~ \/A'^~~~ ^at the point on S, is given by L K^ > V(1) where fl is the one-.v.. -/ dimensional Maxwellian dis -.~ v —--.^... vm..-" tribution and N.i, the number Vm in Fig. 1. Incident particle. density of incoming particles, is defined below. co.n =. f. (V rv) vr- ndvd. (1) in inEi A. so -)B, n5>o The integration in solid angle is to be carried out over the entire hemisphere less n' the solid angle shielded from the in-coming flux by other surfaces. The vector r in (1) is a unit vector in the direction of v. Elsewhere in this discussion use will be made of an abbreviated form of (1), as in (4) using 00co~~~~~~~~~~ ~~(2) ( — v A AA <Vn> =, >o f (r (V-rv) vrt n dvd (. no J2o n.o s and the number density N. of incoming particles just outside S, in co (3) Nin=N a T * >o al(a - v) )dvd, where N is the undisturbed ambient density. 3

Thus we have $. = N. <v> (4) in Nin Vro (4) To develop a notation parallel with that to be used for the particle transfer between two surfaces, it is desirable to have a differential form of (1):. d O D in in p n N.J N f (r(V-v) ) vr n dv. r n S> (5) In (5)'in is the flux per unit area onto S per unit solid angle in a direction r. The second elementary problem is the transfer of particles to a surface S. from a second surface Si, at rest with respect to Sj. From assumption (1) the element of flux emitted in the direction?,, (unit vector from a point i on Si to a point pj on Sj) by an element of area dSi into an element of solid angle dQ. is, tSdS (rij.e(nj). w] "v nq dns ni i R\, ^ - -' I irji I // Figure 2. The Solid Angle Element where &i is the efflux per unit area of dS. into the entire hemisphere. The portion of this flux arriving at a unit area of dSj,.ij', may be found by observing that d g j = dS. (nj. r)/rji in Figure 2 and substituting for d in the above equation to obtain, 4

d d.. 2 ds. dJ ij= ii ij)' ji)/2ri 1J (6) Volume=dSi <v> (ni ^iji) - Volume=dSi<v> (ni rij ) A " " -''....._.,. n.i.4 N_< -'.1I \../ Figure 3. Number densities. From Fig. 3 it can be seen that the flux per unit area leaving dSi makes a contribution to the total number density just outside dSi and its magnitude is determined by considering the number of particles contained in an elemental volume <v> dSi ni rij: (dNi)out = i/2 ( (ni. rij) <>) dS. 7) r A The cross sectional area of the volume is dSi n r <v> is the average particle speed in the rij direction and represents the length of the volume, and Ci/2Tr is the number of particles leaving per unit time. Referring to the form of (2), this can be expressed as (dNi)out = i/2r<n> (8) The crossi setonal arao nh ouei S..<v>i h vrg

Similarly, the influx per unit area of dS. from dS. contributes (dN.) = 1 (< ( iX~..)<v>) dS ( 1 (9) i in ij 31 i ij <> n to the number density just outside dS.. J The use of <v> in (8) and (9) is justified by assumption (ii) as well as the fact that D i in (8) refers only to the flux leaving S. and p ij 1~~~~ refers only to flux incident upon S.. j In section III the results obtained above will be applied to the OGO-IV spacecraft pictured schematically in Fig. 4. The numbers refer OPER 12 13 to various surfaces. Quantities 16-{ 1 -14 ^ —— ^ 15-7-n-}^^^ ^,-related to these surfaces will be 9 ^S^.^ ^^^^ ~ labeled with subscripts corre9 sponding to these numbers. The ^3 1.-1^ *:j. mass spectrometer experiments ^'^Main Body are mounted in the OPEP at the Solar Panel surface labeled 1 in the figure. Figure 4. The OGO Spacecraft. Viewed from this surface the shaded areas are shielded and play a secondary role in the problem. Surface 10 cannot be seen from any other surface to be considered; hence, in the absence of intermolecular collisions (assumption (iii) ) this surface plays no role in the problem. Surfaces 11, 12, 13, 14 and 15 are ignored also. Surface 11 can only exchange flux with the portions of 7 and 8 which cannot exchange this flux with 1, 2, 3, 4, or 9. This flux from 11 can contribute only in a multiple -bounce process, which can be neglected since the incident flux on 11 should itself be negligibly small as the surface is always parallel to the velocity V when the spacecraft is functioning properly. * Orbital Plane Experiment Package 6

Similar reasoning leads to the elimination of 12, 13, 14, 15, and 16. Consequently the only surfaces which must be considered are 1, 2, 3, 4, 5, 6, 7, 8, and 9. Furthermore, no flux transfer between certain pairs of these is permitted by the geometry. III. EXPANSION TO INCLUDE MULTIPLE REFLECTION CONTRIBUTIONS The notation to be used in this section is similar to that already introduced, but in the interest of clarity it is explicitly defined below, and useful elementary results are restated. Xi.is the total influx per unit area at at point on the surface S. from all directions. b.. is the total influx per unit area at a point on a surface S. from the entire surface S., except that when i=o, 4. denotes theJinflux per unit area at a point onto S. from all "free'Opace. _.-; is that portion of the flux from a unit area of dS. falling upon a unit area of dS.. For i=o, o- is the flux per unit area falling upon dS. from J "free" space Waving trajectories within the solid arngle d. X.. =- r (n rj) (n- rji)/ rij, from (7) where ni and nj are the unit surface normals at points on Si and Sj and ri' is the vector from the point on Si to the point on Sj. Note also that Xij = Xji. a.ij = Xij.i < rij= (n rji)<v> j where <v> i is the average speed of particles emitted from S.. coj = ( <v ~ o Nin) j (from (2) and (3) 4j =EDij ij = ij d,obJi = jQO d Ojd 7

The remainder of this section is devoted to finding symbolic expressions for (21, c31' Y 41' 51' and C91 which are to be compared with c 01; all of which are evaluated near the location of the mass spectrometer sensors. The procedure consists in expanding the D s, as sums of other > s, and expanding these in turn with the aim of obtaining the ultimate expansion in terms of oj s since these can be related to the ambient density. Each re-expansion generates terms of higher orders in the X's. For the geometry considered one can assume that terms beyond some order may be neglected. Considering the severity of assumption (ii) and the fact that the integrated X's are of the order of 0. 2 or less, it is reasonable to limit consideration to order two or three in X. Consider: 21=f 2 dS2- 2X12dS2=f 02X12dS2+f 12X12 S2+ 32X12dS2 +J 42X12dS2+f I 72X12dS2+f 82X12dS2. It is necessary to continue by expanding 12' (32' (42' 472' and g82. For example, 12=f01' X 2dsl+2f l 2dsl+f 31X l2dsl+3 41Xl2dsl +f47 X12dS + 81 1 ds Substitution of these expansions into (21 and subsequent re-expansion of the terms they contain leads to a sum for 21 which contains terms of increasing orders in the X's. The same is of course true for D31 and t41. If we omit all terms of order three and higher (10), (11), (12), and (13) result: $ 21=O02X12dS2+J Xl2f lXl2dslds2+fXl2f 03x23ds3dS2+fXl204X24dS4dS fXJl2 X07X27dS7 dS2+JffX12J8XdS dS (10) 31= JO3Xl 3dS3+XX 1 3J01X13dSldS3+'X131 05X35dS5dS3+JX1 3 02X2 3dS2dS3; (11) 8

(12) 41 =f04l 4d 4 1 10 X14dSldS4+XlfJ 4 06'46dSS4+>XJ ('02X24dS (12) r 91= 409X91 9ds9+fX1 9f0, 16X1, 16dS16dS9 (13) For a spectrometer with an "enclosed" source or a "closed" pressure gauge, the flux through the entrance aperture due to D 21' D 31' >41' and D91 determines the contribution of the spacecraft surfaces through reflections, to the measured densities. One might consider ( 01- ~oi0+ 1 21+ 31+ 41+ 91)/ (01 to be the relative "error" in ignoring the influence of neighboring surfaces, where <01 is the directly incident flux calculated by (1) and D 91 is the flux one would obtain if the entire hemisphere were free of obstacles to the incoming flux. For certain orientations of the spacecraft, terms of order three in the X's or perhaps even higher, which were discarded in (10), (11), (12), and (13) may actually be larger than some terms retained in these equations. For these cases, certain of the retained terms may be negligible. The four equations will, however, always contain the dominant terms. It will frequently happen that the last two terms in (11) are small. The last two terms in (12) and (13) will be very small in noon-midnight orbits. In sunrise-sunset orbits the second term of (11) is zero, since the OPEP cannot "see" S2. In these orbits either 131 or D41 is zero since the OPEP is exposed to only one solar panel. The entire contribution of (13) is rather small, though perhaps not always negligible. For future reference each of the additive terms in (10) through (13) will be given a symbolic name by extending the bnotation already introduced: 9

fX12o02dS2 4021; JX12JX21101dSldS2 0121; JX12 X23-03dS3dS2 =032;' X12f X24 04dS4dS2 0421; X12fX27%07dS7 S2 = 0721; IX12JX28,08dS8dS2 = 0821; fSX ^X7dS = 1 031; X1303d 3 031S3 X13JX31 01ldSldS3 = 0131; (14) X1 3fX35}05dS3dSS3= 0531; IX13fX23 02dS2dS 3 =0231 Jfx1404dS4 =041; JX14Jx4 1 0 dSdS4 = 0141; JX14fX4066dS 6dS4= 0641; fX14J'X24o02dSS2dS4 = 0241; 1ITQX19% 0dS c6xi 16 QI 91.91 X 90 9dS9 = 091; IX1 QX1,16 0,16d~ 6dS9 01,9. For "open source" instruments the number density contributions of the fluxes must be determined. In order to do this by means of relations similar to (8) and (9), it is necessary to make an assumption about the relation of<v > before a surface collision with<v n >after this collision. n n The simplest assumption to make is that the particles are fully accommodated. That is, <v n >may be calculated from the Maxwellian distribution characteristic of the temperature of the last surface touched by the particles, or in the case of those indicent from "free space", <v > is to be calculated from n o f(V-v) with a temperature corresponding to the ambient atmospheric temperature. Unfortunately, full accommodation is probably a poor assumption. For this reason,formulae will be derived for the number densities which allow the assumption about <vn> to be made later. To take care of this, the notation must be expanded. Let <vn> i.. i be the average speed of particles incident from space onto S., and subsequently transferred to S,..to Sik. Note <vn > o i. is the mean normal speed after interaction with Si not n 0,11...'k Ibefore interaction with Si. 10

Using this notation each of the terms in (14) will contribute twice to the number density just outside a point on S1, once as an incident density as in (8) and once as a reflected density as in (9). Let N021 be the sum of the incident and reflected contributions of c02 1 Then, N+1 N021 021 <v + n and similar relations hold for the other contributions in (14) a complete list of which is given in (15): N 4 1 1 1 1 021:021n> N0121=0 (<V>+);> n 02 n 021 n 012 n 0121 1 1 1 1 N0321 (0321(<v > N0421 0O421( -> 2; n 032 n 0321 n 042 n 0421 N 11 ); 0N721=o 07212(>. N 0821 0821 < 082 <v>082 vN> 072 <vi0721 (n 07821 N 1 + 1.. (15) 0531 0531( N$ N41 041 14 <vn>041 N41 0141 <14 014); No 91 (D 41 1 ~~~~~~~~1 N0140 1 <V-064 <Vrf0641 0241 0241 02 V02 1 N01 ~091 _n>09 n 0<vn> 91 0,16,9,1 01691(v> +19 n 0,16, 9 n 0,16, 91 The above are to compared, along with( - 0) (I + 1 0to n 0l(<f N 21- NN31, N41, N1l, N01 and N01 are defined as in 11

N21= N021 N0121 N0321N 0421 N0721 N082 1; (16) AN N1-N'01+N21+N31+N41+N91 (17) N - Vn 01 V-' For (15) to be correct it is necessary that the<Vn >not be a function of position on the various surfaces. The legitimacy of this should be examined. In the first place it was effectively postulated that< n> 1 N31=N031+N0131+N0531+N02319 i... ik is a function of <vn>o T.... T., and it is evident from (2) that N91=N091+N0, 16, 9, 1. Ky r>o is a function of position on a surface since the limits of integration N01 his compution oaly writn exn ned ) are ios ofsur. e epesitiman o ts suraaalc o i a function of position on a surfacould be deth e limits of integration n o 1 n Since the particles contributing to all the N's in (15) will undergo one, two or three bounces and each interaction tends to t t erase" more and more of the wit dent ity of the incide nt velocity and since the T areb essentially constant on Si and T. is approximately equal to Tj, it is reasonable to assume < vn > to be constant across a given surface which permits the use of (15). This computation could be extended if < vn> could be expressed as an analytic function of<v > and the T.. Then<v >could be determined at each point n o 1 n and integrations similar to those indicated in (14) could be performed explicitly, with<v n> included within the integral, thereby deriving new analytic expressions for the N's. 12

On the surfaces receiving the largest incident flux I oj - those directed approximately into the velocity vector -<vn > will be nearly constant. Other surfaces contribute mostly by higher order processes, hence the statement concerning "erasing" the identity of < Vn >o by subsequent interactions applies to these contributing surfaces. Another aspect of equations (14) should be restated explicitly. The limits of integration are functions of the relative orientations of the OPEP and the solar paddles with respect to each other, and with respect to the main body as well. It is intended that the integrations in a given transfer process extend only over the area of the surface from which the particles are being transferred that can be "seen" from the point to which they are being transferred. Thus the limits depend upon the point to which the transfer is made. This complicated geometric analysis is the subject of a separate report in which the integral equations are set-up in more detail. IV. POSITIVE ION MEASUREMENTS If the incident particles are ambient ions there is no reflection from surfaces since the ions may be assumed to be neutralized on contact. In this case <1 = o01' Equations (1), (2), and (4) are still applicable on the assumption that the electric and magnetic fields around the spacecraft can be neglected when the Debye length is smaller than the characteristic spacecraft dimensions. The neighboring spacecraft surfaces still influence 01 because the spectrometer is shielded from a portion of the incoming flux, DI0' which would reach the instrument if the surfaces were absent. For ~01 the integration in (1) extends over the entire hemisphere. The relative 13

error if the surfaces are neglected is therefore given by ()01- 01) / 01 for ambient ions. One can write an expression for the number density of ambient ions (18) which is of the same form as (16) and (19) gives the relative "error" in neglecting the spacecraft surfaces: Nol 0l~>kt (18) __= it _t ) (<1 ) (19) n 01 V CONCLUSIONS The introduction of a few simplifying assumptions, has permitted a formal solution to be obtained for the influence of the OGO spacecraft upon atmospheric density measurements. Because of the large number of integrations over variable limits, numerical solutions are not easily obtained. Solutions for every important orientation of the solar panels, OPEP, and man body, with respect to the spacecraft velocity, can be expected to consume considerable processing time even when programmed for a digital computer. A program for a Digital Equipment Corporation PDP-8 computer is currently being prepared to perform these calculations. This program and the characteristics of the above solutions will be the subject of a subsequent report. REFERENCES (1) Hinton, B. B., R. D. Kistler, R. J. Leite, and C. J. Mason, IEEE Trans. on Geoscience Electronics GE-7 p. 107 (1969). 14