Theoretical proton velocity distributions in the flow around the magnetosphere

Theoretical proton velocity distributions in the flow around the magnetosphere

Planet. SpaceSci. 1966. Vol. 14. pp. 1207 to 1220. Peqamon Press Ltd. Printed in Nonhem Ireland THEORETICAL PROTON VELOCITY DISTRIBUTIONS THE ...

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Planet.

SpaceSci. 1966. Vol.

14.

pp. 1207 to

1220. Peqamon

Press Ltd.

Printed in Nonhem

Ireland

THEORETICAL PROTON VELOCITY DISTRIBUTIONS THE FLOW AROUND THE MAGNETOSPHERE JOHN

IN

R. SPRRlTRR, ALBERTA Y. ALJiSNE and BARBARA ABRAHAM-SHRAUNER Space Sciences Division, Ames Research Center, NASA, Moffett Field, California (Received

26 June 1966)

Abstract--Relations from kinetic theory are combined with results from fluid calculations to determine a microscopic interpretation of the properties of the solar wind as it flows around the magnetosphere. Proton velocity distributions are presented as a function of the particle speed and direction for several points in space for representative values of the solar wind parameters. The results illustrate how the velocity distribution is highly collimated in the incident stream, is completely isotropic at the magnetosphere nose, and the degree to which it changes back to that Except for a very small of a highly collimated stream along the flanks of the magnetosphere. region in the vicinity of the magnetosphere nose, the number of particles that move in the upstream direction is insign&antly small compared with the number that move in the direction of the bulk velocity. The most probable speed of the protons, as viewed in a frame 6xed in the Earth, is always in the direction of the local bulk velocity, and tends to be relatively constant throughout the flow in spite of substantial changes in temperature and bulk velocity. The paper concludes with a discussion of the relation of these results to the measurements that would be made by various types of instruments commonly used to observe plasma flows in space. INTRODUCI’ION

The conditions in the solar wind as it flows outward from the Sun and around the Earth and its magnetic field display an interesting duality. On the one hand, the gas is highly rarefied, ordinary collisions are infrequent, and it would appear that a microscopic description based on concepts of kinetics theory would be required to represent its conditions adequately. On the other hand, fluid models have been used extensively to describe both the solar wind itself (see, e.g. Parker(l)), and its interaction with the Earth’s magnetic field (see, e.g. Spreiter et [email protected])and Dryer and Faye-Petersen(3J). The use of such models is usually justified by appealing to possible interaction mechanisms involving electric and magnetic fields, and possibly to plasma instabilities, that operate over distances that are not only much smaller than the ordinary mean free path of the particles, but, more importantly, smaller than the scale of the flow feature under study. Although the precise mechanisms involved in establishing the necessary collective action of the rarefied solar plasma are not completely understood, data obtained in space, particularly from Mariner 2 and IMP-l, have served to establish that many of the gross average features of the flow can indeed be described by the continuum theory of fluid flow. When attention is turned from the gross features of the flow to the quantities actually measured on board spacecraft with, say, a plasma probe, the situation alters significantly because the instruments are much smaller than any of the suggested interaction distances. It thus appears necessary to return to a microscopic picture based on kinetic theory to interpret many features of the data, or to carry the theoretical calculations to the point where the results are directly comparable with the observations. A complete treatment of the solar wind and its interaction with the Earth’s magnetic field based on kinetic theory is, however, an insuperable, and probably unnecessarily complex, task at the present time. 1207

1208

J. R. SPREITER, A. Y. ALKSNE and B. ABRAHAM-SHRAUNER

In this paper, existing results calculated by means of fluid models are combined with relations provided by kinetic theory to calculate results that are more nearly comparable with those that would be measured directly by a plasma probe. The basic assumptions are that the fluid models are able to provide satisfactory values for the bulk properties of the flow, that is, the velocity, density and temperature, and that the instantaneous microscopic velocity distribution of the ions is Maxwellian in a frame moving with the local bulk velocity of the flow. The first assumption is supported by numerous comparisons of calculated and observed results previously reported in the literature. The latter is consistent with procedures frequently employed in the reduction and interpretation of data observed in space. RESULTS

OF FLUID MODELS

Numerous observations in space have served to establish that the solar wind is always present and that it flows nearly radially outward from the Sun. In the vicinity of the Earth’s orbit, its velocity fi ranges between about 300 and 800 km/set, number density it of ions between about 1 and 25 cm*, and temperature T between some tens of thousands and a few hundred thousand degrees Kelvin. It is generally considered to be fully ionized plasma consisting of equal numbers of ions and electrons, with hydrogen ions, or protons, the dominant species with perhaps a 10 per cent admixture of helium ions. Embedded in the plasma is a weak magnetic field having an intensity of the order of 5 x 10” G. These values are quite compatible with those calculated using fluid models for the solar wind together with reasonable values for conditions in the solar corona.(l) Such a stream interacts with the Earth’s magnetic field to form a thin current sheath which separates the geomagnetic field and the solar wind. Its shape in the equatorial plane, as given by the approximate solution of Ferraro, w [email protected]) and Spreiter and Briggs,(6) is illustrated in Fig. 1 for the case in which the dipole axis is normal to the direction of the 3.0 - STREAMLINES ---MACH LINES

2.0 -

r/o

I.0 -

OL

FIG.

1. STREAMLINES AND

WAVEPATTERNS

1.0

0 X/D FOR

-1.0

SUPERSONICPLOW

PASTTHE

MAGNETOSPHERE;

Mm =S,y=&

incident solar wind. Although this shape was originally calculated using the classical Chapman-Ferraro theory based on particle concepts, it has been show0 that an approach based on the continuum equations of magnetohydrodynamics leads to virtually the same shape. The distance D used for making the distance scale dimensionless is the distance from the center of the Earth to the magnetosphere nose. It is generally of the order of ten earth radii, and fluctuates in response to slow variations in the incident stream in accordance withtheexpression D = a,H~~/(2~Kp,tV2,)11swhere a, = 6.37 x 10s cm is the radius of the Earth, H,,, = O-312 G is the intensity of the Earth’s magnetic field at the equator, and K is a

1209

MAGNETOSPHERE

constant usually taken as unity in applications based on fluid concepts but is more accurately given by 088 for a gas having a ratio of specific heats y of 8. Of critical importance to the fluid models is the observation that the solar wind approaches the Earth with a velocity considerably in excess of both the speed of sound Q and the AlfvCn speed A. These quantities are defined by a = (ap/ap)l’Z = (rP/P)1’2 = (YRT/PY2

(1)

A = (H2/4.rrp)1’2

(2)

wherep represents the pressure, R = 8.314 x 10’ er$K is the gas constant, ,u is the mean molecular weight = 4 for fully ionized hydrogen plasma, y = (N + 2)/N represents the ratio of specific heats, and Nrepresents the number of degrees of freedom of the gas particles; and H represents the intensity of the magnetic field. With values selected so as to be representative of conditions in the undisturbed solar wind in the vicinity of the Earth’s orbit, that is, T, = lo6 “K, y = $, n, = 10 protons/cma, poo = 1.67 x lo_23 g/en?, H, = 5 x lob G, it follows that a, = 5.3 x 106cm/set = 53 km/set and A, = 3.5 x lo6 cm/set = 35 km/set. Since both of these speeds are much less than P’a, in the undisturbed solar wind, a bow wave forms in front of the magnetopause much as in front of a supersonic airplane or bullet. Because the magnetic field is sufficiently weak that the Alfven Mach number #/lA is much greater than unity, the magnetohydrodynamic flow simplifies to essentially that of gasdynamics. It is thus possible, although still a complicated numerical task, to calculate the location of the bow wave and the properties of the flow in the region between the bow wave and the magnetosphere. The configuration and intensity of the magnetic field can then be calculated as a subsequent step. Results of gasdynamic calculations(2) for the special case of M, = 8 and y = Q that will be used as the basis for illustrating the present analysis are presented in Figs. 1 and 2. In Fig. 1, the locations of the magnetosphere boundary and shock wave are shown together with several solid lines representing streamlines, and broken lines representing characteristic or Mach lines of the flow. The latter correspond to standing compression or expansion waves of infinitesimal amplitude. They cross the streamlines at such angles that the local velocity component normal to the wave is exactly equal to the local sound speed. Mach lines thus exist only where the flow is supersonic. They are absent from the vicinity of the magnetosphere nose because the flow there is subsonic. Contour maps showing lines of constant density, velocity and temperature, each normalized by dividing by the corresponding quantity in the incident stream, are presented in Fig. 2. The results show that the density ratio p/p, along nearly the entire length of the portion of the bow wave shown remains close to the maximum value (y + I)/@ - 1) = 4 for a strong shock wave in a gas with y = 5. The gas undergoes a small additional compression as it approaches the stagnation point at the magnetosphere nose and then expands to less than free-stream density as it flows around the magnetosphere. The velocity remains less than in the free stream, however, throughout the same region. The temperature T/T, is closely related to the velocity ratio through the expression T

-=1+(~-;~~‘(1-~) T,

WC0

(3)

Of particular interest is the tremendous increase in temperature of the solar wind as it passes through the bow shock wave. If, for example, the temperature of the incident solar wind

1210

J. R. SPREITER, A. Y. ALKSNE and B. ABRAHAM-SHRAUNER 3.0 -

2.0 -

r/0 MASNETOSPHERE

MASNETOSPHRE

1.0 -

o,

,,82&#$i, LET

1.0

0 X/D

,

,

-1.0

o;s;

j/E? I .o

0 X/D

, -1.0

FIG. 2. DENSITY,VELOCITY AND TEMPERATURE cmvoms POR SUPERSONICPLOW PAW THE MAGNETCSPHEXB; MC,, = 8, y = f.

is 1OO,OOO”K, the temperature at the magnetosphere nose is indicated to be 2,230,OOO”K. This value is consistent with the temperature of the gas in the solar corona before it is accelerated to the high velocities characteristic of the solar wind in the vicinity of the Earth’s orbit. Equation (3) shows that contours of constant q/e), are also contours of constant T/T,, as illustrated in Fig. 2. Since equation (1) shows that the speed of sound a depends only on the temperature T for a given incident flow, these contours are also contours of constant local Mach number M = Q/a. Numerical values for M/M, can be determined readily since M/M, = (IV/IV~)(T,&!‘)~/~. MICROSCOPIC

JNTERPRETATION

OF FLUID RESULTS

In ordinary gases, near standard conditions, the rapid rate of collisions between particles causes the velocity distribution to be very nearly Maxwellian. In the present application, the collision rate is very low, and it is not obvious, nor necessarily correct, that the velocity distribution is Maxwellian. Nevertheless, there are reasons to believe that the Maxwellian distribution is useful at least as a rough guide, and may even be a reasonably good approximation to the conditions actually encountered in space. The use of the Maxwellian distribution is, moreover, consistent with the use of continuum equations of magnetohydrodynamics and gas dynamics in the fluid models, and with current practice in analyzing and interpreting data from plasma probes on some of the recent spacecraft. According to the Maxwellian velocity distribution, the number density dn of a species of particles of mass m having velocities between w and w + dw in a gas moving with bulk velocity W is (4) where k = Boltzmann’s constant = l-38 x lo-l6 erg/“K, n is the total number density of the species of particles independent of their velocity, and U,u and ware Cartesian components of w. We proceed to apply equation (4) to the ions, and note that m = l-67 x lO-= g if they are hydrogen ions, or protons. The quantity (2kT/m)‘i2 = ct has the dimensions of velocity, and is recognized to be the most probable particle velocity in a reference frame in which the bulk velocity W is zero, that is, in a reference frame moving with the local bulk

1211

MAGNETOSPHERE

velocity. relation

It is proportional

to, and less than, the speed of sound in accordance with the

a = (2kT/m)lla = (p/p)‘l” = a/y112 = O-775a

(5)

derived from equation (1) by replacing the pressure p by the combined pressure 2nkT of the ions and electrons, and the density p by the product mn of the mass and number density of the ions, the contribution of the electrons to the density being negligibly small. W

FIG. 3. SPHERICALAND CARTESJAN COORLXNATESINVELEITYSPACE.

It is convenient to introduce a spherical coordinate system in velocity space related to a rectangular cartesian coordinate system having the w axis alined with the bulk velocity vector w, as illustrated in Fig. 3. Equation (4) can thus be rewritten as

(6) where c = (u2 + v2 + w2)l12represents the particle speed, and dQ = sin 8 de dpl represents an infinitesimal solid angle. Introducing the dimensionless quantities C = c/a,

dC = de/a,

FF = G/a

(7)

leads to the following dimensionless form for equation (6) dn -= n

C2 p2 exp [-CC2 + FP - 2civcos

e)] dC dQ

This expression shows that the fraction dn/n dC dQ of the particles per unit solid angle and unit dimensionless speed depends only on the dimensionless speed C and direction 8 with respect to that of the local bulk velocity of the particles being counted, and the ratio V of the local bulk speed to the most probable speed a in a frame moving with the local bulk velocity. The quantity w is moreover directly related to the local Mach number of the flow since w = @/a = y112fi/a = 1.29M. Plots of dn/n dC dQ as a function of 8 for several representative values of C and w are presented in Fig. 4. These plots display how the velocity distribution changes from being completely isotropic when V is zero, as at the magnetosphere nose, to being highly collimated when w = 10, as in the undisturbed solar wind when the free-stream Mach number M, = (3/5)ln x 10 = 7.75. For fixed C and w, dn/n dC dQ is a maximum or minimum

1212

J. R. SPREITER, A. Y. ALKSNE and B. ABRAHAM-SHRAUNER

-t

b

I

1213

MAGNETOSPHERE

when sin 8 = 0, that is, when 8 = 0 or 0 = 7~, as can be seen by equating to zero the derivative of equation (8) with C and w held constant

(9) That dn/n dC da is a maximum when 0 = 0 and a minimum when 0 = rr for all values of C and w follows from the fact that the sign of the next higher derivative = [(2CWsin tQ2 - 2CWcos 191&a

(10)

is negative for 8 = 0, and positive for 0 = rr. The maximum and miminum values for dn/n dC da follow directly upon substitution of these values for 0 into equation (8). In Fig. 5, the velocity distribution dn/n dC da is illustrated a second way, as a function of the dimensionless particle speed C for fixed Wand 8. The minimum value for dn/n dC da for fixed 8 and Vis zero, and occurs when C is either zero or infinite. The maximum value occurs when C has such a value that

(11) and dn/n dC da # 0. Since C and dn/n dC da are positive quantities, the maximum value occurs when c = ii7 cos

e + (r2 cos2 8 + 4)1/a (12)

‘I

For small V cos 8, dn/n dC dQ is thus a maximum when C = 1 + (r cos (Q/2. For large w cos 8, the maximum value occurs at either C = r cos 0 or C = - l/( I7 cos e), depending on whether P cos 8 is positive or negative. Since there is only one solution for C for given Vand 8, dn/n dC dadecreases monotonically on both sides of the maximum, and approaches zero as C approaches zero and infmity. Combining the results of the two preceding paragraphs, we see that the absolute maximum value for dn/n dC da occurs when C is given by equation (12) with 0 = 0, that is, when c=

c

M

= W+(~2+4)1’2 2

(13)

Curves illustrating the variation of dnfn dC da with 8 for C = CM are included on Fig. 4, and indicated by dashed lines. Similarly, the smallest maximum value for dn/n dC d!A for fixed r occurs when C is given by equation (12) with 8 = 72,that is, when

c=c,=

- W + (P 2

+ 4)1’s (14)

The total number N of particles with direction of motion in small solid angle dB inclined an angle 8 with the direction of the bulk velocity is a quantity of interest in certain applications, such as the discussion of the response of plasma probes with wide energy acceptance

J. R. SPREITER, A. Y. ALKSNE and B. ABRAHAM-SHRAUNER

1214

-z N



-

0

7 I3

1

3

I

I

R

I

I 0

0

(D -

S

0

N

0

1215

MAGNETOSPHERE

windows. It is given by the area under a curve for dn vs. C for fixed r and 13,thus N=

s* c=o

(dn) dC = $v

7

[exp (-P x

sin2 @][l + (P(Wcos 13)]

(1 +2wZc0s2f3)

+ ~(W~ose)exp(-W~)

(15)

I

where 6 ~WCOS

e) =

m(t) =

ej’dt

&2

(16)

s0

is the error function with argument wcos 8. A plot showing the variation of N/n d&2with w for various 8 is presented in Fig. 6. The preceding results can be applied to the flow around the magnetosphere by inserting into equations (4) through (8) the values for density, temperature and velocity calculated 100

FIG. 6. TOTALNUMBW INCLINED

NOP PARTICLESWITHDIREC~ONOFMOTIONINASMALLSOLIDANGLE AN

ANGLE 0 WITH THE

DIRFCTION

OF THE BULK

di2

VJZLOCITY.

using the fluid models. Figure 7 shows the results of a specific application for the case illustrated in Figs. 1 and 2, namely M, = 8 and y = 9. The solar wind is considered to consist entirely of protons and electrons, and to flow in the undisturbed incident stream with a velocity Gj_ of 4 x 10’ cm/set. The temperature T, in the free streamis thus 0% x 106 “IS. Several kinds of results are included on this figure. First of all, there are lines representing the magnetosphere boundary and the bow shock wave, streamlines and contour lines of constant speed and temperature, all of which correspond to those shown on Figs. 1 and 2. The latter are also contour lines of constant local Mach number M and w as indicated on the figure. Small diagrams are presented at several points of intersection of the streamlines and the contour lines that illustrate the velocity distribution of the protons. These diagrams are similar to those of Fig. 4 in that they show on a polar plot the variation of number density with direction of motion for particles of selected speeds. The speeds selected for each of the diagrams include c = cM = UC, so that the curve displaying the absolute maximum value for the number density is always shown. The values for c for the other curves are simple fractions and multiples of c,. The diagrams differ from those of Fig. 4 in that dn/n, dc da is plotted rather than dn/n dC dQ in order to illustrate better the relation between the velocity distributions at different points. They are also oriented on Fig. 7 so

1216

J. R. SPREITER, A. Y. ALKSNE M,=8,

and

B. ABRAHAM-SHRAIJNER

7=5/3 BOW SHOCK WAVE

iia IO.31 Ej,=4x107 cm/set Tm =.90x105’K q

m=.3, ‘Z/l,=

M=2.32 .B3, T/T,

= 7.7.

h

i’

.N

Z&,=,69,

T/T,=l2.1

-NM

/'cU=3.54x107cm/sec

-;)T’\\ I

/

,--T--

f/,/--W=l, M=.775 z/i? ,=.43, T/T,=lB.4

Cm = 4.03 x 107cm/sec

0

\m

A/-----

MAGNETOSPHERE BOUNDARY LEGEND FOR VELOCITY DISTRIBUTION DIAGRAMS

B-

FIG. 7. %LOCITY DISTRIBUTIONS dn/(n co dc d!2) OF PROTONSAT SEVERALSELE43’EDPOI!+~ IN THE FU>WAROUNDTHBMAG~P~PORASOLARWMD~M~D =8; r=!t,fl,, =4X 10’ cm/set.

that 8 equals 0 in the direction of the local flow velocity. These plots, together with those of the corresponding velocity spectra (h/n, dc dO vs. c) shown in Fig. 8, illustrate how the velocity distribution is highly collimated in the incident solar wind, completely isotropic at the stagnation point at the magnetosphere nose, and changes back to that of a highly collimated stream along the flanks of the magnetosphere. Except for a limited region in the immediate vicinity of the nose, only a small fraction of the total number of particles at a given point move in the upstream dire&ion (0 = r). Throughout the entire flow, however, the speed cN corresponding to the maximum number density at each of the stations tends to be relatively constant in spite of substantial changes in the velocity and temperature of the gas. RELATION TO RESULTS OF PLASMA PROBE MEASUUMENTS

Plasma probes currently employed in spacecraft do not measure density directly as a function of vector velocity of the incident particles, but rather the current produced by a flux into the instrument of charged particles of either positive or negative charge. The particles that contribute to the current at any instant, moreover, are not unidirectional in theirmotion, but directed within a range of angles about some nominal viewing direction of the instrument. The width and shape of this angular receiving window, as well as the efficiency with which the effects of the particles are measured, depend on the type and design of the instrument. A circular Faraday cup plasma probe, for example, may receive particles of a given speed with equal effectiveness from all directions that make a iixed angle with the center line of the instrument. Contour lines of equal efficiency would thus appear as circles when plotted in terms of angular direction of approach of the particles. Similarly, contours of

0

FIG.&

J=O

3 c, cm/set

4

(d)

5x107

4x107

0

0

P =.5

c

4x10-8

c, cdsec

2 c. cm/set

f---l

,8=0’

(bl

4x107

8X10-8 r

i = 10.32

c,

cm/set

c, cmhec

VEU)CITYSPE~OFPROTONSATTHEPOMTSINTHEFLOWAROUNDTHEMAGNETOSPHEREFORWI~~CH mm~xsmrnrno~s ARE SHOWN IN FIG. 7; Ma = 8,~ =$!, GOD =4 x lO’cm/sec.

2

2 c, cmkc

,

(0)

(cl

X10’

1218

J. R. SPREITER,

A. Y. ALKSNE

and B. ABRAHAM-SHRAUNER

equal efficiency for a cylindrical curved plate electrostatic analyzer would more nearly resemble narrow rectangles when plotted in the same manner. Furthermore, the particles that penetrate the instrument to the current collector and contribute to the measurements are not of a single speed or energy. Design factors enter in such a way that curved plate electrostatic analyzers tend to have narrow energy acceptance windows, whereas Faraday cup plasma probes usually have broad energy windows. The energy acceptance will, in general, depend on direction as well, and even strongly in some instances. A spherical curved plate electrostatic analyzer would, for example, tend to respond at any instant to particles having a relatively smaIl range of velocities. On the other hand, a Faraday cup instrument would tend to respond to particles from different directions that have nearly the same range of values for the velocity component normal to the face of the instrument. In all cases, the response of the instrument is given by an integral of the particle flux multiplied by an efficiency factor that depends on the direction, speed or energy, and charge-tomass ratio of the particles. Since the efficiency factor is a function of the instrument design and can best be discussed in terms of specific cases, we will not develop those aspects of the discussion further here. Consider, instead, an idealized telescope-like instrument that measures the infinitesimal current dj produced by protons coming into the instrument within an im?nitesimaI range of energies dE and directions da about some nominal values. Consider further that the stream to which the instrument is exposed consists entirely of protons and electrons. The current is then given by dj = ec dn where e refers to the charge on a proton (e = 4.8 x lo-lo e.s.u.), and c and dn are as defined previously herein in connection with equations (4) and (6). Since dE = d(mc”/2) = mc dc where m = 1.67 x lO-% g is the mass of a proton, the current is thus given by dj=i(&)a’zexp[

-m(c2

+ fi2 - 2~19 cos 0)

2kT

] c2dEdQ

(17)

Comparison with equation (6) shows that dj -=ndEdSJ

m dn e ( n dc dR,)

(18)

from which it follows that the properties of dj/n dE da are identical, except for a numerical constant given by the mass-charge ration m/e = 3-5 x 10-15 g/e.s.u. for protons, with those discussed previously herein for dn/n dc dQ. Specific application of the foregoing results requires determination of the angle 8 between the viewing direction of the instrument and the local flow direction. Although this is a straightforward task, the necessary expressions are somewhat lengthy, and are presented here to facilitate applications and comparisons with observational data. Consider the coordinate systems illustrated in Fig. 9 and let ii, and c! be unit vectors parallel to the directions of the local flow velocity and the spin axis, or other reference line fixed in the spacecraft. The Cartesian components of $ may be expressed in terms of the angles A; and il; of the spherical coordinate system as follows: _ W, = cos J$ cos GV= cos 2~ sin $)B= sin 40

(19)

1219

MAGNETOSPHERE

FIG. 9. VIEW OF COORDINATE SYSTEMS.

The corresponding relations for the components of Q can be written similarly by substituting Cc,for + everywhere in equation (19). The appropriate expression for the cosine of the angle j3 between G and CGfollows immediately upon substitution of these relations in the scalar product of 6 and CG cos B = i+ - ul = [email protected]& + $&i& + 0,6,

(20)

Let y designate the angle between the nominal viewing angle of the instrument and the spin axis, and consider the vehicle to have rotated an angle co(t - th) about that axis since the plane of the spin axis and the viewing direction passed through the direction of the velocity vector G. The angle 8 between B and the viewing direction used in equation (6) and following can now be determined by application of the law of cosines, and is given by cos e = cos y cos p + sin y sin /I cos [~(t - Q)] An important special case is that for which the viewing direction is perpendicular spin axis of the spacecraft. Then y = 90”, and cos 8 = sin b cos [w(l - t;)]

(21) to the (22)

If, on the other hand, y is arbitrary, but the coordinate system is oriented so that the x axis points directly into the wind, iGz= - 1, GV= 8, = 0, or equivalently A; = 0, & = 180”, equation (20) reduces to cos /I = -cSz = -cos 2,: cos A;,

(23)

and equation (21) to cos 8 = -cos

y cos Pa cos A: + (1 - cos2 ill: cosa A$‘2 sin y cos [o(t - th)]

(24)

Further simplification occurs, of course, if the viewing direction is parallel (y = 0”), perpendicular (y = go”), or opposite (y = 180”) to C& As a further application of these results, it may be observed upon examination of equation (21) that cos 0 is a maximum, for fixed y and /3, when t = th, decreases monotonically with increasing t until a minimum is reached when co(t - Q,) = 180°, and then increases again until o(t - t;) = 360”. The maximum and minimum values for cos 8 are thus (COS6),,,

= ~0s (Y - B);

(~0s e),,

= cos (Y + PI

(25)

1220

J. R. SPREITER, A. Y. ALKSNE and B. ABRAHAM-SHRAUNER

This result, when combined with equation (18) and the properties of dn/n dC dQ discussed in connection with equation (9), shows that the current measured by the idealized instrument described above, when exposed to a flow characterized by constant values for the temperature and bulk velocity and set to register the effects of particles of a fixed iniinitesimal energy range, is a maximum or minimum when the viewing direction of the instrument is in the plane of the vectors representing the bulk velocity and the spin axis. The maximum current is observed when the instrument looks as nearly as possible into the stream. The minimum current is observed when the spacecraft has rotated half a revolution from that orientation. REFFBENcES 1. E. N. PARKER,Interplanetary Dynamical Processes. Interscience, New York (1963). 2. 3. 4. 5. 6.

J. R. SPREITER, A. L. SUMMJXS and A. Y. ALKSNE,Planet. Space Sci. 14,223 (1966). M. DRYERand R. FAYIQETZRSEN, AIAA J. 4,246 (1966). V. C. A. FERRARO, J.geophys. Res. 57,15 (1952). D. B. BEARD,J. geophys. Res. 65,3559 (1960). J. R. SPREITER and B. R. BRIGGS,J. geophys. Res. 67,37 (1962). P0BH)ltit+OTHOIIIeHUH, BBRTbIe ULl TeOpuu KHHeTuKH, COBMeIJJalOTCK C pe8yJIbTaTaMH [email protected] IKUnKOCTU, gTO6bI OIIpe~eJIuTb MuKpOCKOnHOe TOi'IKOBaliue CBOlfCTB conHe9Horo BeTpa~o Rpenis era ~BumeHuK BOKPYP [email protected] Pacnppefienemis CKOpOCTu UpOTOHOB AaIoTCJ-4B Bu&e (PyHKIJuu CKOpOCTu u HalIpaBJIeHUII YaCTuQ AJIR HeCKO~bKuXTO~eKB~pOCTpaHCTBe,~~KTHKuYHblX8Ha¶eHU~UapaMeTpOBCO~He~HO~O BeTpa. PeaynbTam ~oHaz.rBamT, mo pacnpeAeneme CKOPOCTH, ~h1c0~0-K00p~UHUpOBaHHOeBHaKJIOAHOMTeVeHuu,a6COJlH)THO USOTpOUHO B HOCyMalWuTOC~epbl,aTaKHte yKaBbIBaIOT Ha CTeIIeHb eP0 uEZ.leHeHUFI B 06paTHyIO CTOpOHy, K CTelIeHU BbICOKOKOOp~uHHpOBaHHO~O Te¶eHuH BAOJIb TpaHUq [email protected] 3a UCKJIWieHueM OqeHb He6OJIbIJIOrO paiiOHaB6JlUBU HOCaMal'[email protected],KOJlU¶eCTBO ~BU~y~UXCfl BBepXIlO TeqeEHIO SaCTUIJ BeCbMa He3Ha4UTeJIbHO II0 CpaBHeHHIO C KOJfUPeCTBOM, KOTOpOe ~WKeTCH BHaIIpaBJIeHUU OCHOBHOt MaCCbICKOpOCTU. CaMaS BepOHTHaJZ CKOpOCTb IlpOTOHOB, COrJlaCHO saKpeUJIeHHOi8 B 3eMJIe CKCTeMhI HallpaBJIeHa B CTOpOHy OCHOBHOti MaCCbI CKOpOCTU U CpaBHHTeJlbHO KOOpAuHaT, IIOCTOIIHHa Ha BCeM KpOTFllKeHUU Te¶eHUK, HeCMOTpR Ha FlHaWiTeJIbHbIe HRMeHeHHR B CTaTbH 8aKaHWlBaeTCH o6cymneKueM TemepaType u u OCH~BHO~~ Macce CKOPOCTE. OTAOllleHUt aTUX peFlyJlbTaTOB K UaMepeHURM , KoTopble 6yw~ II~OU~BOAUT~W~ npu nomo~qu paEzHOo6pamibIX IIpU60pOB,KoTopbIe o6bnHo IIpuMeHKmTCK~AJIFI Ha6nIoAeHufi 110~0K0BnnaaMbI~UpoC3pRKcTBe.