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Solar wind flow past nonmagnetic planets—Venus and Mars
Planet. Space Sci. 1970, Vol. 18, pp. 1281 to 1299.
SOLAR
Department
Pergamon Press Printed in Northern Ireland
WIND FLOW PAST NONMAGNETIC PLANETS-VENUS AND MARS
JOHN R. SPRRITER of Applied Mechanics, Stanford University, Stanford, Calif. 94305, U.S.A.
AUDREY L. SUMMERS Space Sciences Division, Ames Research Center, Moffett Field, Calif. 94035, U.S.A. and Department
ARTHUR W. RIZZI of Applied Mechanics, Stanford University, Stanford, Calif. 94305, U.S.A. (Received infinal form 9 December I.969)
Abstract-The hydromagnetic theory of solar wind flow past the Earth has been extended and modified so as to be applicable to nonmagnetic planets, such as Venus and Mars, that have a sutIicient ionosphere to deflect the solar plasma around the planet and its atmosphere. The principal difference in the analysis stems from the fact that the current sheath that bounds the solar wind away from the planet is formed by interaction with the ionosphere rather than with the geomagnetic field as in the case of the Earth. After stating the principal assumptions and equations, the theory is applied to determine the shape of the ionosphere boundary, or ionopause, across which the Newtonian approximation for the pressure of the flowing plasma is balanced by the pressure of a stationary ionosphere. Specific numerical results are given for a wide range of ionospheric parameters representative of those proposed for Venus and Mars. The location of the bow wave and the properties of the flow field are then calculated using magnetohydrodynamic and gasdynamic considerations in a manner similar to that employed for the flow of the solar wind plasma past the Earth’s magnetosphere. Examination of the results reveals a correspondance rule that enables results presently available for the location of the bow wave and the properties of the flow about the Earth’s magnetosphere to be converted rapidly into those for a nonmagnetic planet by a simple relabeling of the s&es. The results are shown to be in general accordance with observations made by Mariner 5 as it flew past Venus, although certain differences near the theoretical location of the ionopause suggest the presence of a thick boundary layer. A similar analysis of the data from Mariners 4,6 and 7 indicates that Mars has a sufficient ionosphere for the theory to be applicable. Further comparison is relatively uninformative beyond revealing no inconsistencies with the theoretical results, however, because Mariner 4 did not approach close enough, and Mariners 6 and 7 were not equipped to make the necessary measurements. INTRODUCTION
Data acquired in recent years in the vicinity of the Earth, the Moon, Mars and Venus have disclosed three essentially different types of interaction with the solar wind. For the Earth, the geomagnetic field prohibits the solar wind from approaching nearer than about 10 Earth radii under ordinary circumstances. The shielded region, the magnetosphere, acts as an obstacle in the supersonic flow of the solar wind, and a bow wave forms a few Earth radii upstream of the magnetosphere boundary, as illustrated in Fig. 1(a). Interaction of the solar wind and the Moon is very different because of the Iack of either a sensible magnetic field or ionosphere to deflect the incident flow. The particles of the solar wind thus proceed unchecked until they contact the lunar surface, where they are effectively removed from the flow. As a consequence, no bow wave forms, and the principal features of the flow field are associated with the closure of the lunar wake, or cavity in the solar wind, that extends downstream from the Moon as illustrated in Fig. l(b). Neither Venus nor Mars has a significant magnetic field, but the electrical conductivity of the tenuous upper ionosphere of these planets is evidently sufficiently high that the solar wind is prevented from flowing directly into the planetary surface or lower absorbing levels of the atmosphere. The solar wind is thus deflected around the ionosphere, and a bow wave is 1
1281
_/---
field
__--
-.
./--i)
-0.
-
c---.~ 4
Bow wave,
Fm. 1. PRINCIPAL FEA’MJRES OF SOLARWIND FLOWPASTTHEEARTH(a), THE MOON (b) AND VENUSOR MARS (c).
Nanalined
7.p
field
Cavify Moon Alined
/Streamline
-----f
/3trramline
SOLAR WIND
FLOW PAST NONMA~~~C
PLANETS
1283
formed upstream of the planet, similar in many ways to that associated with the Earth. Aside from evident differences in the underlying physical processes at the boundary surface between the ionosphere and the solar wind, the principai difference between the flow fields around Venus or Mars and Earth is the size of the cavity. As illustrated in Fig. 1(c), the ionosphere boundary is wrapped much closer around a nonmagnetic planet than the magnetosphere boundary is around the Earth, the nose being at an altitude of about 500 km for Venus compared with about 60,000 km for Earth. The corresponding altitude for Mars is not known as well, but available results combined with the present theory indicate that it is probably between 155 and 175 km. Extensive theoretical studies, specific calculations, and detailed comparisons with observations have shown that the interaction of the solar wind and the Earth may be represented satisfactorily, insofar as gross features of the flow such as density, velocity, temperature, and magnetic field are concerned, by the standard continuum equations of magnetohydrodynamics (see Spreiter et al., 1968 or Spreiter and Alksne, 1969 for recent summaries). Recent investigations (see Spreiter et al., 1970 for an extensive account) have shown that the same is true for the Moon, provided boundary conditions are applied at the lunar surface that correspond to the Moon absorbing, or neutralizing, all particles of the solar wind on impact. The present paper extends the same line of analysis to objects in the solar system that have no significant intrinsic magnetic field, but a sufficiently dense and electrically conducting ionosphere to stop and deflect the solar wind before it is absorbed by the planetary surface or atmosphere. Venus is a known example of such an object. We believe Mars to be another al~ough present data are insu~cient to support this with the same degree of surety as for Venus. The anaIysis commences with a presentation of the basic equations used to represent the variation of pressure with height in the ionosphere, and the dynamical properties of the solar wind. There follows an account of the manner in which these equations may be simplified to obtain a tractable mathematical problem without undue loss of realism or accuracy of representation of principal features of the interaction. Specific numerical results are presented for the shape of the ionosphere boundary and the bow wave for a substantial range of values for the parameters describing conditions in the solar wind and in the planetary ionosphere. It is found that the results for a wide range of ionospheric parameters may be brought into close correspondence with those for solar wind flow past the Earth’s ma~etosphere by application of a simple geometric transfo~ation of the coordinates. This correspondence rufe enables a substantial body of theoretical results for the density, velocity, temperature, magnetic field, and velocity distribution of the protons presently available for solar wind flow past the magnetosphere to be applied to Venus or Mars with only minor modification. FinaIly, the theoretical results are compared with the observations made by Mariner 5 as it flew past Venus, and implications of the points of agreement and disagreement are discussed. The corresponding Mariner 4 data for Mars provide little opportunity for comparison, beyond the simple observation that the absence of planetary effects in the plasma, magnetic field, and energetic particle data is consistent with the indications of the present results that Mariner 4 was outside the theoretical location of the bow wave at all points on its trajectory for which data were received. Mariners 6 and 7 subsequently approached within 2000 km of the Martian surface, but neither spacecraft carried a plasma probe, ma~etometer or energetic particle detectors. As a resuh, the only observation of relevance to the present study is that the
1284
J. R. SPREITER,
A. L. SUMMERS
and A. W. RIZZI
occultation experiments indicated that the peak electron density was 1.5 x lo5 electrons/ cm8 and that it was at an altitude of 130 km (Anonymous, 1969), essentially the same as indicated earlier by the occultation experiment of Mariner 4 (Kliore et al., 1965 and Fjeldbo and Eshleman, 1968). MATHEMATICAL REPRESENTATION OF INTERACTION OF THE SOLAR WINJl AND THE IONOSPHERE OF A NONMAGNETIC PLANET
It is the plan of the present investigation to base the analysis on the equations of magnetohydrodynamics and proceed to the solution through approximations that are well known and tested in previous applications to the interaction of the solar wind and the Earth, with modifications introduced only as required because it is the ionosphere rather than the geomagnetic field through which the presence of the planet is communicated to the solar wind. In broad outline, the elements of the theory are as follows. The incident solar wind is considered to be steady and supersonic, and to flow in accordance with the equations of magnetohydrodynamics for a perfect gas in which all dissipative processes, such as those associated with shock waves and boundary layers, take place in the interior of thin layers idealized as magnetohydrodynamic discontinuity surfaces of zero thickness. The ionosphere, or at least the outer part of it that participates in the interaction with the solar wind, is idealized as spherically symmetric and hydrostatically supported plasma having infinite electrical conductivity. Since the two bodies of plasma are of different origin, and have different properties, they must be considered to be mutually impenetrable in the idealized hydromagnetic representation, and to be separated by a tangential discontinuity surface. This surface is called the ionopause, since it marks the outer boundary of the ionosphere. Once the problem is formulated, the solution proceeds in two steps, just as in the analogous problem for the Earth. First the shape and location of the ionopause is calculated for selected values of the parameters characterizing the solar wind and the ionosphere. Following that, the location of the bow wave and the properties of the flow field are determined. More explicitly, the assumption of hydrostatic support is equivalent to assuming that all motions of the gas within the ionosphere are sufficiently small with respect to the planetary body that equilibrium exists between the pressure gradient and the force of gravity, thus dP -= dr
-t%
(1)
where p and p are the gas pressure and density, r is the radial distance from the center of the planet, and g is the acceleration of gravity. Values for the latter are inversely proportional to r2. Thus g = g,(r,/r)2, where subscript s refers to values at the surface of the planet. Values for g, for Venus, Mars and Earth are about 870, 375 and 982 cm/sec2, and those for rr are about 6.1, 3.4, and 6.4 x lo* cm, respectively. The pressure is assumed to be related to the density by the perfect gas law p = nkT = pRT/M
(2)
in which n is the total number density of particles, Tis the absolute temperature, k = 1.38 x lo-la ergsrK is Boltzmann’s constant, R = 8.31 x 10’ ergs/g “K is the universal gas constant, and M is the mean molecular mass nondimensional&d so that M = 16 for atomic oxygen. M is thus equal to + for fully ionized hydrogen plasma, and 1 and 2 for singly
SOLAR WIND
FLOW PAST NONMAGNETIC
PLANETS
1285
ionized molecular hydrogen and helium. The density may be eliminated from Equation (1) by introduction of Equation (2), and the result integrated to yield
in whichpR is the pressure at the reference radius rn, and H, the local scale height of the atmosphere, is given by J&T
RT mg = Mg
(4)
where m = 1.67 M x 1O-24g is the mean molecular mass. If H is constant, Equation (3) may be integrated to obtain p =pRexp
(-y)
which shows that H represents the height interval in which the pressure decreases by a factor e. If the variation of g with r is disregarded over the range of application of Equation (5), Tis also constant, and the density varies with height in the same way as the pressure, thus P --=-= n nR
&-
P
exp(-y).
PR
In the upper atmosphere, where little mixing would be presumed to occur, diffusive equi~brium may be considered to prevail among the various ionic constituents, at least in an idealized sense. As emphasized recently by Bauer (1969), however, this equilibrium is more complicated in an ionosphere than in a neutral atmosphere for which Dalton’s law of partial pressures is customarily invoked. This is because an electric polarization field, dependent on the mean ionic mass and charged particle temperatures, acts on all the ions and couples the diffusive equilibrium distribution of all the ions. In most circumstances, however, the lightest constituent, ionized atomic hydrogen, would emerge as dominant at great altitudes. At lower and intermediate altitudes, more specific knowledge, such as is now resulting from space experiments conducted by the USSR and USA, is needed to specify the variation of pressure with radius with any degree of precision. Figure 2 shows the variation with distance from the center of Venus of the electron number density, as deduced from the dual frequency radio occultation measurements by the Mariner Stanford Group (1967). For a plasma with singly ionized ions, charge neutrality requires that these curves also describe the ion number density, summed over all constituents. Also included on Fig. 2 are three small inserts showing the sIope of the densityaltitude curve for four possible atmospheric constituents covering a wide range of M at three substantially different temperatures, as calculated using Equations (4) and (6). These measurements indicate that the nose of the ionopause is at a distance of about 6500 km from the center of Venus, or about 450 km above the surface. The experimenters interpret the results between 6800 and 7500 km as almost certainly indicating the dominant constituent is ionized atomic hydrogen plasma with a mean ion-electron temperature between 625 and 1100°K. Between 6800 and 6300 km, they conclude that the atmosphere is most likely composed of ionized helium with a temperature between 620 and 970°K.
1286
J. R.
6ooo
SPREITER,
5
2 I02
A.
L. SUMMERS
5
2 IO3
Electron
5
2
and A.
5
concentration
2 IO5
IO4
W.
RIZZI
5 IO6
,CfK'
F1cz.2, PROFILESOF ELECXRON CONCENTRATIONSIN 'IXENW3HT AND DAY IONOSPHERESOF ~OMTHE D~~~F~Q~ENCY 8~~x0 -TATION ~~~E~~SBY~ MARINERSTANFORD G~oup(1967).
VEMJS, ASDEDUCED
The proper interpretation of these data, and also those from the Mariner 5 Lyman alpha measurements of Barth et al. (1967), has, however, been the subject of considerable discussion. Barth (1968) and Barth et al. (1968) conclude that the entire upper atmosphere of Venus above about 6500 km can be represented best by an atmosphere consistingp~~rily of molecular hydrogen at 650 & 50°K. On the other hand, Wallace (1969) and Donahue (1969) have concluded from their analyses of the Mariner 5 Lyman alpha data that the dominant constituent above 6500 km is deuterium at about 650°K. McElroy and Strobe1 (1969) have compared results from a number of models of the topside night-time ionosphere of Venus with obse~ations of Mariner 5 and concluded that the primary ionized constituent is either helium or molecular hydrogen. In view of these uncertainties, and even greater ones with respect to Mars at the time of this writing, we put aside any further discussion of refinements of the variation of N with altitude and proceed to adopt Equation (5) as sufficient to represent the variation of p with r in the upper atmosphere of either Venus or Mars. Results will be presented for several values for H, however, in order to provide an indication of the variations to be expected between ionospheres of different chemical composition. For Venus, with r, equated to 6500 km and T to 700”K, N is approximately 1500, 760, 380 and 35 km for ionized atomic hydrogen, singly ionized molecular hydrogen, helium, and carbon dioxide. The corresponding values for the nondimensional ratio H/r,, in which r,, the distance from the center of the planet to the nose of the ionopause, is equated to 6500 km are O-23, O-115, O-058 and O-0052, For Mars, the smaller radius and acceleration of gravity and lower ionospheric temperature of about 200°K (Kliore et al., 1965 and Fjeldbo and Eshleman, 1968) lead to values that are about l-2 times larger than for Venus, namely about O-28, O-138, O-069 and 0.0062. The calculations of these values makes use of the result, indicated by the data from Mariners 4, 6 and 7 together with the present theory,
SOLAR WIND
FLOW PAST NONMAGNETIC
PLANFXS
1287
that the nose of the ionopause is at an altitude between 155 and 175 km, and that to therefore scales nearly in proportion to the planetary radius. The dynamical properties of the solar wind flow are considered to be represented by the standard magnetohydr~ynami~ equations for steady flow of a dissipationless perfect gas. The governing differential equations are thus v.pv=o p(v . v)v + Vp == - (lj&)B
div B = 0
cur~(Bxv)=0, (v . V)S = 0,
x curl B
S - S, = c, In ~
OY in which v and S refer to the velocity and entropy of the gas, B represents the magnetic field, and y = c,/c~ = (N + 2)/N in which cP and c, are constants representing the specific heats at constant pressure and volume, and N represents the number of degrees of freedom of the gas particles. For a monatomic gas, such as either of the two leading constituents of the solar wind, atomic hydrogen and helium, iV = 3, and y = +. Implicit in the relations of Equation (7) are two important nondimensional parameters, the Mach number M and the Alfven Mach num~r Pn;, defined as follows:
Conditions on opposite sides of discontinuity surfaces that develop in the ffow because of the omission of dissipative terms in Equation (7) are related by the following conservation laws : [P%l = 0 [plt,v + @ C B2fs7T)li- S,B,/~W] = 0
WGi - B&f = 0,
I&al = 0
[pv,@ + u*/2) + v$~/~w - B,v t B/47r]= 0
(91
in which h = c&I’is the enthalpy, A and I represent unit vectors normal and tangential to the discontinuity surface, and the square brackets are used to indicate the difference between the enclosed quantities on the two sides of the discontinuity, as in [Q] = Qa - Q, where subscripts 1 and 2 refer to conditions on the upstream and downstream sides of the discontinui~. It is well known (F~ed~chs and Kramer, 1958; Landau and Lifshitz, 1960; Jeffrey and Taniuti, 1964; Jeffrey, 1966) that five classes of discontinuities are described by the conservation equations. As in the corresponding application to solar wind flow past the Earth, (Spreiter et al., 1966b, 1968 and Spreiter and Alksne, 1969), two of these, the fast shock wave and the tangential discontinuity, are of concern in the present analysis. The first relates conditions on the two sides of the bow wave, The latter, which has the properties v, = 3, = 0, lvt] # 0, RlJ.0
IPl f 0,
[p + P/857] = 0
(10)
1288
J. R. SPRlEIl-ER, A. L. SUMMERS
and A. W. RIZZI
relates conditions on the two sides of the ionopause under the assumption that the upper ionosphere can be treated as a perfectly conducting fluid effectively bound to the planet and incapable of mixing with the solar wind plasma. As in the corresponding application to the Earth’s magnetosphere, important simplifications may be introduced in these relations In the present application, p tends to be much larger than B2/87r on both sides of the ionopause. Therefore the pressure balance relation of Equation (10) may be reduced to a simple balance between the ionospheric pressure given by Equation (5) and the static pressure of the flowing solar plasma adjacent to the ionopause. With the further introduction of the Newtonian approximation p = psi cos2 y for the pressure on a body in a steady hypersonic flow, we arrive at the following expression for the pressure balance at the ionosphere boundary: pst
cofy = Kpmueo2 co9 y
= pR exp (_r+)
(11)
in which y is the angle between the outward normal to the ionosphere boundary and the flow direction in the undisturbed incident solar wind, and pst = Kpoouco2is the stagnation pressure of the solar wind exerted on the nose of the ionopause. The subscript co indicates that values associated with conditions in the solar wind upstream of the bow wave are to be used, and K is a constant equal to 0.88 for high Mach number flow of a monato~c gas, although usually taken as unity in applications to the Earth’s magnetosphere (Spreiter er al., 1966b). Use of the foregoing approximations permits the coordinates of the ionopause to be calculated without determining any further properties of the flow field. Once the shape of the ionopause has been determined, the density, velocity, and temperature in the surrounding flow may be calculated by solving Equations (7) and (9) with the terms containing B omitted because of their smallness in the characteristically high Alfven Mach number flow of the solar wind. The magnetic field can be calculated subsequently by solving the remaining equations: curl (B x v) = 0, div B = 0 F&v, -
W,l = 0,
P4,l = 0
(12)
with v considered known from the preceding steps, as has been done for the case of the Earth by Alksne (1967). The calculations can be further extended to provide information on the velocity distribution of the solar wind particles, if desired, by employing the procedures applied to the Earth by Spreiter et al. (1966a). LOCATION
OF THE IONOPAUSE
It
is convenient in the calculation of the shape of the ionopause to let rn = ro, the distance from the center of the planet to the nose of the ionopause. Since cosa 1y = I at this point, Equation (11) simplifies to PR
=
PO =
Kf%st),2
(13)
at the ionopause nose, and to cof
y =
e&-rO)lE
(14) elsewhere along the ionosphere boundary. If, for example, the solar wind is considered to approach Venus with a number density of 3 protons/cm8 and #abulk velocity of 5 X IO’ cmlsec, as the data reported by Bridge et al. (1967) show occurred at the time of the Mariner 5 measurements, Equation (13) indicates that p,, = 1.1 x 10” dyn/cm2. Equation (2)
SOLAR WIND
FLOW PAST NON~G~TIG
PLANEiTS
1289
shows that such a value corresponds, assuming 2’ = 7OO”K, to an ion or electron number density n, = n, = n/2 at r = r, of about 5.7 x 104 particlesjcms. This value is somewhat greater than that indicated by the data of Fig. 2 for r = 6500 km, but is not unreasonable, considering the preliminary nature of the Mariner 5 data involved in the estimate, and the uncertainties attendant with all plasma measurements in space. In any case, it may be seen from Fig. 2 that the essential condition that the ion-electron number density required to stop the solar wind be exceeded at some altitude is satisfied by a considerable margin,
FIG. 3. VKEWOF ELEMENTOF I~NOPAUSEAND COORDINATES USEDIN EQUATION(15).
since the peak daytime value of 5 x lo5 electrons/ems is larger than the required value by a factor of about 10. Similarly for Mars, the values 0, = 3.3 x lO’cm/sec, n, = 0.8 protons/cm8, and T = 200°K inferred from the measurements of Mariner 5 (Lazarus et al., 1967; Kliore et al., 1965) indicate that an electron number density of 2.6 x lo* electrons/ cm3 is required at the subsolar point for the ionosphere to stop the solar wind. This value is exceeded by a factor of about 6 by the peak electron density of 1.5 x 105 electrons/cm3 deduced by Kliore et al. (1965) from Mariner 4, and subsequentIy confirmed by similar experiments with Nariners 6 and 7 (Anonymous, 1969). On the basis of the further observations that the altitude of the peak electron density is between 120 and 130 km above the Martian &ace, and that the electron scale height there is between 20 and 25 km, we may estimate from the above relations that the subsolar point of the ionopause is at an altitude between 155 and 175 km. For these reasons, we feel that the applicability of the theory to both Venus and Mars is supported by existing data, even though those for Mars are restricted in quantity and inferential in nature at the present time. To proceed, we must express cos2 y in terms of r and 0, where r(t9) represents the coordinates of the boundary, and 6 is the angle measured at the center of the planet with respect to a line that extends directly upstream. Upon carrying out this step in the way made clear by the illustration in Fig. 3, and substituting the result in Equation (14), we find (r d@cos 8 + dr sin 19)~ cog y ziz 4v = =exp(-r+)=E. (15) dr2 + (r dQ2 ds
0a
1290
J. R
SPREITBR, A. L. SU-W
and A. W. RIZZI
The numerical solution of this differential equation is facilitated by solving for dr/r de to obtain dr rde=
sin2&2dE-E2 2(E -
sins i3)
’
(16)
At the ionopause nose 0 = 0, r = r,, and dr/r d0 =r?0 with either choice of sign. The proper choice of sign is dictated by the following considerations based on the assumption that r increases monotonically from r, to co, and hence that E diminishes monotonically from 1 to 0, as 8 increases from 0 to 180”. Since sin* 8 = 0 at 8 = 0, increases to unity at 0 = 90”, and then returns to 0 at 180”, the denominator of Equation (16) must vanish at one or more values for 8, and dr/r de would be infinite unless the n~erator vanishes simultaneously. If the critical values for 8 and E are designated by the subscript cr, we have that EC, = sins 0,,
(17)
sin 20,, f 2(E,, - Eo,a)l/a= 0.
(18)
and also that
Substitution of the former into the latter to obtain sin 28, j, 2(sin* 8,, - sin* ep
= 2 sin ecr cos 8, + 2 lsin @,,cos e,l = 0
(19)
shows that the minus sign must be used in Equation (16). That the resulting indete~inate form actually leads to a finite value for drjde at 0 = ear may be confirmed by application of L’Hospital’s rule. The shape of the ionopause may therefore ,be determined directly by integration starting from the boundary condition that r = r, at 8 = 0. Results obtained by numerical integration are presented in terms of cylindrical coordinates x = r cos 0 and r’ = r sin 8 for several values for H/r, from 0.01 to 1 in Fig. 4. As noted previously, this range is ample to include all likely possibilities for both Venus and Mars. (Although the displayed results are independent of the altitude of the ionopause nose, a planet silhouette has been added to the plot to indicate the radius r,, of Venus for the ratio r,fr, = 6050/6500 = O-93). The corresponding ratio r&r,, for Mars is not known so definitely, but the estimate given above indicates that it is about 0.95. Also included on Fig. 4 is a dotted line indicting the coordinates of the equatorial trace of the magnetosphere boundary, as determined by Beard (1960) and Spreiter and Briggs (1961, 1962), that has been used extensively in the calculation of solar wind flow past the Earth. These coordinates, as well as those for the location of the associated bow wave for M, = 8, y = 513given by Spreiter et al. (1966b), have been nondimensionalized by dividing by the distance from the center of the Earth to the magnetosphere nose. We shall continue to call this distance r,,, even though it is much larger than for Venus or Mars, and evaluated differently using the expression r, = ra(BeQ2/2rrKp,v,2)11*in which rs = 6.37 x 10s cm is the radius of the Earth, and B,, = 0.312 G is the average intensity of the geomagnetic field at the geomagnetic equator. It may be seen that this curve is very similar to that for the ionosphere boundary for H/r, = O-2. The size, with respect to the planet, of the cavity carved in the solar wind is very different in the two applications, however, since r, is only a few percent greater than the planetary radius for Venus or Mars, whereas it is usually of the order of 10 Earth radii for the magnetosphere.
SOLAR WIND FLOW PAST NONMAGNETIC
1291
PLANETS
6Ho-0
5/
/
/
,
c’
/
/ /
/
,
, I’0 -75
4-
7/r,
3-
/
/
/
‘/ / I ,
N I
..
.. .’ ,
/:
/I
-
lonopouse
---
Bow
wave
~~~~~~_~~ MaQnelDpa~se and bow wave
0
-1.0
-2.0
-3.0
x/r, Fra.4. CALCULATEDLOCATIONOFIONOPAUSE ~WWAVEFOR~*
FOR~ARI~US~~~~,AND
ASSOCIATEDLOCATIONOF
= 8,y -$.
results for the shape of the Earth’s magnetopause and bow wave, nondimensionalized so that the magnetosphere nose is at x[rp - 1, is included for purposes of comparison. The dashed line for the bow wave for H/r0 - 0.2 has been omitted because it is indistinguishable from the dotted line representing the Earth’s bow wave.
The cormpon&ng
LOCATION
OF THE BOW WAVE AND PROPER~
OF THE FLOW FIELD
Upon specification of a value for the free-stream Mach number M, >> 1, the location of the bow wave and the properties of the flow field may be computed for any of the ionopause shapes shown in Fig. 4 by application of highly developed numerical techniques of nonlinear gas dynamics. Several alternatives are available for this, but as in our previous analyses of the Sow past the Earth, we have continued to employ the methods developed at Ames Research Center by Van Dyke (1958), Van Dyke and Gordon (1959), Fuiler (1961), Inouye and Lomax (1962), and Lomax and Inouye (1964). Results for the location of the bow wave for flow with M, = 8 and y = Q are presented in Fig. 4 for each value for H/r, for which the coordinates of the ionopause are ihustrated. Although these results are for a specific M,, it has been shown in our previous
J. R. SPRJXI’EJR, A. L. SUMMERS
1292
H&,=.20
Ma=8
and A. W. RIZZI
y=5/3
x/r0
x/r0 _T _ , + %I
Fm.5.
(Y-I)& 2
= 22~33-21.33(V/V,)2
I)ENSITY, VELOCITY, AND TEMPERATURE FIELDS FOR SUPERSONIC FL.OW PAST THE IONOSPHERE; H/r,, = 0.20, Moo = 8, y = #.
papers relating to the Earth that the shape of the bow wave, and also the associated distributions of p/p,, Y/V,, and B/B, for any given direction for B, are relatively independent of free-stream Mach number for all Iw, greater than about 5. In Fig. 5 are shown contour maps for the density ratio p/p,, velocity ratio v/v,, and temperature ratio T/T, for flow with.HfrO = O-20. Such a value is appropriate, for example, for a Venusian ionosphere composed primarily of atomic hydrogen at about 600°K. In view of the current tendency for some to interpret the Mariner 5 results as indicating that the primary constituent in the Venusian atmosphere is molecular hydrogen, or even heavier ions, we have also made a set of calculations for the density, velocity, and temperature in the flow field for H/r, = 0.1. The results are shown in Fig. 6. Corresponding plots for other quantities of interest, such as the magnetic field strength and direction, or Maxwellian proton velocity distribution, can also be calculated in the same way as has been done previously for the Earth. Such calculations have not been carried through, however. This is partly because they are time consuming to do and require considerable space to display the results for even a ~nimum number of cases, but p~ncipally because an approximate correspondence rule to be described in the next section makes it possible to convert quickly any of the numerous contour plots already available for the properties of the tlow around the Earth’s magnetosphere into that for the flow around an ionosphere having any H/r, between O-01 and 1 by a simple relabeling of the coordinate axes. CORRESPONDENCE NO~G~~C
RULE RELATING SOLAR WIND FLOW PAST A PLANET TO THAT PAST THE EARTH
The close relationship between the shape of the magnetopause and that of the ionopause for H/r0 = 0.20, and the general similarity of the ionopause shapes for all H/r,, suggests the possibility that a correspondence rule relating ionopause shapes for H/r, other than 0.20 to an appropriately scaled magnetopause may be found if the coincidence between
SOLAR WIND FLOW PAST NONMAGNETIC H/r0
= PI
Ma=8
1293
PLANETS
y=5/3
Density ratio
3-
.5 ,25
i/r0
2 .7 '6
2-
‘5
x/r0
x/r0
T _-_,+ TQ Fm.6.
RENSI~,~~~Y,
AND TE~E~~RR PHERE; Ii/r6=
(Y-I)Mtl 2
FIELDS FOR SUPERSONIC FLOW
= 22*33-21~33WVm~* PAST THEIONOS-
O*lO,Mm = 8, y = g.
the Earth and planetary centers be relinquished. Great practical utility would result from the availability of such a correspondence rule, because it would enable a substantial body of results already calculated for solar wind flow past the Earth to be applied with minor change to nonmagnetic planets having a wide range of ionospheric parameters. The results displayed in Fig. 7 show that it is indeed possible to achieve a fit that is probably sufficiently good for most purposes relating to the interpretation of data obtained in space. This figure is basically that given originally by Spreiter et al. (1966b) for the magnetosphere boundary and shock wave for the Earth, and the associated characteristic or Mach line pattern, for M, = 8 and y = $. Superposed on it are the ionopause curves from Fig. 4, each with its nose retained at x/r,, = 1,8 = 0, but with its scale adjusted so that its ordinate coincides with that for the magnetopause curve at 8 = n/2. The center of the nonmagnetic planet is, in general, no longer at the origin of coordinates, but at the points x,/r,, indicated along the axis of symmetry for each value for H/r,,. The coordinates of these points are listed on Fig. 7 for accuracy and convenience in applications of the correspondence rule. To convert the nondimensional shape for the ma~etopause into that for the ionopause for a given H/r,,, we must thus place the center of the planet at x,/r0 and change the labeling of the scales so that x/r,, = 0 at x,/ro, and x/r,, = 1 at the ionopause nose. It follows from the close correspondence between the coordinates of the ionopause and the magnetopause that the coordinates of the bow wave should also display a similar relations~p. It may be seen from Fig. 7 that this is indeed true. A similar degree of correspondence may be anticipated for the coordinates of contour lines for constant values of the flow parameters, such as those for the density, velocity, and temperature shown in Figs. 5 and 6, Although some of the ionopause curves depart significantly from the curve for the
1294
J. R. SPREITER, A. L. SUNMERS
and A. W. RIZZI
2.0 F/r0 I.5-
I,0-
_
lonopouse and bw
. . . . . . ..* . . . .
'5 -
wove
Moqnetopouse
and bow wave
?. bJ.USTRATlON OF DEGREE OF CUXNC~ENCB OF CXTRVESREPRESEl4’lTNQ T?lE IONOFAUSE AM) BOW WAMt FOR VARIOUS f@e OBTAINED BY APFLICATiON OP CORRESFONDENCE RULE. The solid lines have been omitted where they are ~~stjn~s~ble from the dotted lines.
%A
magnetopause somewhat downstream of the phmet, the effects of these differences have no infhtence upstream of the rearwardly inclined characteristic line emanating from the point on the boundary where the differences first become significant. It may be seen from Fig. 7 that this is su~cient~y far do~stream for ionospheres having Hfro between 041 and 1 that nearly all of the results we have presented previously for the properties of the flow field about the magnetosphere can be carried over with uo other change than relabeling the scales to obtain a good approximation for the conditions around Mars and Venus, COMPARISON It
WITH
MARINER
5 DATA FOR VENUS
is appropriate at this point to make a general assessment of the relation between the theoretical results shown in Figs. 4-6 and those observed by Mariner 5 as it flew past Venus. Figure 8 shows the time variation of the bulk velocity v, ion number density rz, and intensity 1231 and direction a and /i of the magnetic field together with a plot of the trajectory giving the position of the spacecraft as a function of time (Bridge et al., 1967). The angles a and 8 are in spherical coordinates in which a is the longitude measurement, in the RT plane, from the R direction, where R is a unit vector in the antisolar direction and T is an orthogonal vector which is parahel to the Sun’s equatorial piane and points in the direction of the planet’s orbital motion; and ,L3 is the latitude angle of the magnetic
-180
o
-240
’
-180 Time
-I20 -60 from encounter ,min
0
60
I
To Sun 2
I 0
-I
X/f”
-2
I
--Venus
-3
I
-4
I
‘lozpo”se / -5
j -6
F1a.8. PLASMA ANDMAGN~C~~CPIELDRATAMEASUREDBY~INER 5 INITSENCOUNTERWITH VENUSON~~~CIY)BER~~~~ (BRIDGE et al.,l967). The indicated trajectory has been projected into a plane by rotation about the Sun-Vent line. Times are in minutes relative to the encounter time E of closest approach to the planet.
a,deg
I80
-180’
1296
J. R. SPREITER,
A. L. SUMMERS
and A. W. RIZZI
field vector, considered positive when northward. For ease of reference, lines indicating the theoretical location of the ionopause and bow wave for H/r, = 0.10 and 0.25, for M, = 8, y = $, and r, = 6500 km, are superposed on the plot of the trajectory. The corresponding location of the bow wave for M, = 5 and H/r,, = 0.25 is also included in order to better assess the consequences of the fact that the values of about 590 km/set and 300,OOO”K reported for the velocity and temperature in the solar wind before and after Mariner 5’s encounter with Venus indicate a value of about 6.5 for M,. The magnetometer and plasma probe experimenters have concluded from the presence of abrupt and easily recognizable changes in their data that Mariner 5 crossed the bow wave at the points labeled 0 and 0. These points are reasonably close to the theoretical locations of the shock wave for H/r,, = O-25 for M, = 5, and somewhat farther from Venus than indicated by the theoretical results for M, = 8, or for H/r, = 0.10. On the other hand, the theoretical results for H/r, = 0.25 indicate that Mariner 5 should have crossed the ionopause, whereas those for H/r, = 0.10 indicate that it should have just skimmed along this surface. Although the data display decreases in n/n, and v/c, that are at least qualitatively similar to those for either H/r,, = 0.10 or 0.25, there is no certain evidence that Mariner 5 actually penetrated the ionopause. In particular, the intensity of the magnetic field remained of the order of that for interplanetary space, and v diminished to only a modest fraction of u, rather than to a value comparable with the speed of the spacecraft relative to Venus. On the other hand, the observations made when the spacecraft was near the theoretical location of the ionopause indicate values for n/n, and v/v, that are only about one-half of those indicated by the theory for the flow exterior to the ionosphere. A possible explanation for these discrepancies is that the tail of the ionosphere might taper inward toward the axis of symmetry rather than extend straight downstream. The theoretical results give no indication of such a trend, but they are of low reliability for this feature of the flow because of the deterioration of the quality of the Newtonian approximation for the pressure of the solar wind as ly approaches 90”, and the substantial departures from a constant scale height of a planetary atmosphere at great heights caused by the diminished gravitational acceleration, and also by probable increase in temperature. In addition, it is virtually certain that the ionosphere in an extended tail would be supported partially by dynamical interaction with the flowing solar wind instead of entirely by the simple hydrostatic means assumed in the present analysis. Although neither theoretical nor observational evidence is as yet sufficient to provide a definitive statement regarding the nature of the ionosphere tail, we are inclined to explain the discrepancy between the theoretical and Mariner 5 results in terms of an extended tail with a relatively thick boundary layer between the planetary ionosphere and the solar wind. The latter may be contrasted with the thin boundary between the solar wind the Earth’s magnetosphere that is becoming increasingly evident as the time resolution of the observations is improved. That the ionopause might be thick well back from the nose, whereas the magnetopause is usually thin, is plausible in view of the fundamentally different nature of the two boundaries. For the Earth, the magnetopause is essentially a boundary between the flowing solar plasma and a relative vacuum, and would be somewhat like the boundary formed by a free streamline in a water flow with embedded air cavity. For Venus or Mars, the ionopause is essentially a boundary between two different bodies of plasma. As such, a significant boundary layer would be anticipated which would provide an increasingly thick transition between the ionosphere and the solar wind with increasing distance from the stagnation point at the
SOUR
WIND FLOW PAST NONMAGNRTIC
PLANETS
1297
nose of the ionopause. It wotdd spread, moreover, both into the outer part of the ionosphere and outward into the surrounding flow, broadening the transition region in which all properties of the plasma including the density, velocity, and magnetic field change from their values for the Aowing solar plasma to those of the planetary ionosphere. At the location of the ionosphere boundary indicated by the present dissipationless theory, the plasma velocity might be expected to be substantially less than indicated by the theory. Since the plasma velocities observed by Mariner 5 in the vicinity of the theoretical location of the ionopause display such a trend, it is tempting to conclude that Mariner 5 entered the boundary layer separating the ionosphere and the flowing plasma, but did not enter the ionosphere proper, A more detailed examination is clearly required before a definite statement can be made, however. C~~~~ON
WITH MARINRR 4, 6 AND 7 DATA FOR MARS
As described in less detail previously, data from the magnetometer (Smith et al., 1965), plasma probe (Lazarus et al., 1967), and energetic particle detectors (Van Allen et al., 1965 and O’GalIagher and Simpson, 1965) on Mariner 4 displayed no effects attributable to the presence of Mars as it flew past that planet along the trajectory shown in Fig. 9, as viewed
Mariner
4 !-I/r, = 25, M, = 8
2
I
0
-I
-2
-3
-4
-5
-6
-7
Fro.9.TR~JE~RYDPEXARINEII~PAST~~,PRO~~DINMAPLANE BY~~A~o~~~ THESIJ%-MARSIJNE,ANDPROSABLB LOCAT~ONSOFTHB~ONOPAUSBANDBOWWAVE. No data were received from the spacecraft as it traversed the dashed portion of the trajectory, because Mariner 4, as viewed from the Ekth, was occulted by Mars. projected onto the Sun-Mars-spacecraft plane. AIthough such a negative finding does not provide much material for comparison, it is at least consistent with the indication of the present theory for the conditions measured at the time of the encounter that Mariner 4 did not cross the bow wave into the region influenced by Mars at any time when signals were being received from the spacecraft. To display this, several curves are shown on Fig, 9 to indicate probabIe locations of the Martian ionopause and bow wave. The latter are drawn for r&r0 = 0.95 and ET/r0= 0.01 as suggested by the observational data for conditions near the location of the peak electron density, and also for H/r,, = O-25, which is approximately the Iargest possible vaIue for Mars. The associated bow waves are for &f, = 8, but would be in virtually the same location for any higher Mach number. Lower Mach numbers are considered unlikeIy for Mars in view of data from the vicinity of Earth 2
1298
J. R. SPRBITBR, A. L. SUMMERS
and A. W. RIZZI
and the prediction of solar wind theory that M, is somewhat greater at Mars than at Earth or Venus. Figure 9 shows the trajectory to be well outside any likely position of the bow wave near the point of closest approach of Mariner 4 to Mars, but suggests the possibility that the bow wave may have been crossed somewhat downstream of that point. Unfortunately for the present purposes, however, it is not possible to determine if Mariner 4 actually crossed the bow wave there because occultation of the spacecraft by Mars, as viewed from the Earth, prevented reception of data for the portion of the trajectory indicated by the dashed line. Mariners 6 and 7 subsequently approached within 2000 km of the Martian surface, but neither spacecraft carried a magnetometer, plasma probe, or energetic particle detectors. As a result, the only observation of relevance to the present study is the confirmation of the previous determination of the ionospheric density by Mariner 4. The data from these experiments, and also those of Mariner 5 for Venus, are of considerable significance to the present theory, however, because they demonstrate that the ionospheres of both Mars and Venus are sufficiently dense to stop the solar wind at the subsolar point and to deflect it around the ionosphere. Since the margin in the product PolVcO2 is only about 6 for Mars and 10 for Venus, considering the ionospheric properties to remain fixed, the known variability of the solar wind suggests that it may be sufficiently enhanced at times to proceed directly through the ionosphere at the subsolar point and into the lower atmosphere where it would be absorbed by collisions. Under such conditions, the interaction would be quite different from that described here, and would probably tend toward that described by Spreiter et al. (1970) for the Moon. Since such conditions can be expected to occur only occasionally, if at all, particularly if the ionospheric density and temperature are enhanced by a strong solar wind flow, we conclude that the theory given here is both plausible and capable of providing a reasonably accurate description of the conditions that prevail most of the time at Venus and Mars. Acknowfedgemenrs-It is a pleasure to recall and to publicly acknowledge a number of stimulating discussions on the topic of this paper with Mr. Ray T. Reynolds, Mrs. Alberta Y. Alksne, and Dr. David S. Colbum of the Space Sciences Division of Ames Research Center, and Dr. Joan Hirshberg of the Applied Mechanics Department of Stanford University. The research at Stanford University was supported by the National Aeronautics and Space Administration, partially under contract NAS2-5355 administered by Ames Research Center, and partially under grant NGR 05-020-330. REFERENCES ANONYMOUS (1969). First findings from the Mariner flybys. Sky and Telescope 38,232-234. ALKSNE, A. Y. (1967). The steady-state magnetic field in the transition region between the magnetosphere and the bow shock. Planet. Space Sci. 15,239-245. BARTH, C. A., PEARCE,J. B., RELLY,K. K., WALLACE,L. and FASTS, W. G. (1967). Ultraviolet emissions observed near Venus from Mariner V. Science 158,1675-1678. BARTH,C. A. (1968). Interpretation of the Mariner 5 Lyman alpha measurements. J. atmos. Sci. 25,564567. BATJER, S. J. (1969). Diffusive equilibrium in the topside ionosphere. Proc. of the IEEE 57, 1114-1118. B~arr-r, C. A., WALXXS, L. and PIXARCE, J. B. (1968). Mariner 5 measurements of Lyman-alpha radiation near Venus. J. eophys. Res. 73,2541-2545. BEARD,D. B. (19 &l). The interaction of the terrestrial magnetic field with the solar corpuscular radiation. J. geophys. Res. 65,3559-3568. BR~X~E,H. S., LAZARUS,A. J., SNYDER,C. W., SMITH, E. J., DAVIS, JR., L., COWMAN,JR., P. J. and JONES,D. B. (1967). Mariner V: Plasma and magnetic fields observed near Venus. Science 158,16691673. I)ONAHUB,T.M. (1969). Deuterium in the upper atmospheres of Venus and Earth. J.geophys. Res. 74,11281137.
SOLAR
WIND
FLOW
PAST NONMAGNETIC
PLANETS
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FJELDBO,GUNNAR and -EMAN, VON R. (1968). The atmosphere of Mars analyzed by integral inversion of the Mariner IV occultation data. Planet. space Sci. 16, 1035-1059. FRIEDRICHS,K. 0. and KRANZER, H. (1958). Non-linear wave motion in magnetohydrodynamics. Institute ofMathematical Sci., New York Univ. Rep. No. MH-8. FULLER,F. B. (l%l). Numerical solutions for supersonic flow of an ideal gas around blunt two-dimensional bodies. NASA Tech. Note No. TN D-791. INOUYE,M. and LOMAX, H. (1962). Comparison of experimental and numerical results for the flow of a perfect gas about blunt-nosed bodies. NASA Tech. Note No. TN D-1426. JEFFREY,A. (1966). Magnetohydrodynamics. Oliver and Boyd. London. JEFFREY,A. and TANIUTI:T. (1964). Non&ear Wave Propagation. Academic Press, New York. KLIORE, ARWDAS, CAIN, DAN L., LEW, GERALD S., ESHLEMAN,VON R., FJELDBO,GUNNAR and DRAKE, Results of the first direct measurement of Mars’ atmosphere FRANK (1965). Occultation experiment: and ionosphere. Science 149,1243-1248. LANDAU, L. D. and L~rsrrrrz, E. M. (1960). Electrodynamics of Continuous Media. Pergamon Press, Oxford. LAZARUS,A. J., BRIDOE,H. S., DAVIS, J. M. and SNYDER,C. W. (1967). Initial results from the Mariner 4 solar plasma experiment. Space Research VZZ,pp. 1296-1305. North-Holland, Amsterdam. LOMAX, H. and INOUYE, M. (1964). Numerical analysis of flow properties about blunt bodies moving at supersonic speeds in an equilibrium gas. NASA Tech. Rep. TR R-284. MARINER STANFORDGROUP (1967). Venus: Ionosphere and atmosphere as measured by dual-frequency radio occultation of Mariner V. Science 158,1678-1683. MCELROY,M. B. and STROBEL,D. F. (1969). Models for the nighttime Venus ionosphere. J.geophys. Res. 74,1118-1127. O’GALLAOHER,J. J. and SIMPSON,J. A. (1965). Search for trapped electrons and a magnetic moment at Mars by Mariner IV. Science 149,1233-1239. SM~, EDWARDJ., DAVIS, JR., LEVERETT,COLEMAN,JR., PAUL J. and JONES,DOUGLASE. (1965). Magnetic measurements near Mars. Science 149, 1241-1242. Rev. Geophys. 7,11-50. SPREITER,J. R. and ALKSNE, A. Y. (1969). Plasma flow around the magnetosphere. SPRE~TER,J. R., AJ_KSNE,A. Y. and ABRAHAM-SHRAUNER, B. (1966a). Theoretical proton velocity distributions in the flow around the magnetosphere. Plartet. Space Sci. 14, 1207-1220. SPREITER,J. R., ALKSNE, A. Y. and SUMMERS,A. L. (1968). External aerodynamics of the magnetosphere. In P/rysr+csof the Magnetosphere (Eds. R. L. Carovillano, J. F. McClay and H. R. Radoski). Reidel, Dordrecht, Holland. SPREITER,J. R. and BRIGGS,B. R. (1961). Theoretical determination of the form of the hollow produced in the solar corpuscular stream by the interaction with the magnetic dipole field of the Earth. NASA Tech. Rep. No. TR R-120. SPREITER,J. R. and BIUGGS,B. R. (1962). Theoretical determination of the form of the boundary of the solar corpuscular stream produced by interaction with the magnetic dipole field of the Earth. J. geophys. Res. 67,37-51. SPREIIER, J. R., MARSH, C. M. and SUMMERS,A. L. (1970). Hydromagnetic aspects of solar wind flow past the Moon. Cosmic EIectrodynamics 1,5-50. SPRE~~XR,J. R., SUMMERS,A. L. and ALPINE, A. Y. (1966b). Hydromagnetic flow around the magnetosphere. Planet. Space Sci. 14,223-253. VAN ALLEN, J. A., FRANK, L. A., K~UMIGIS,S. M. and HILLS, H. K. (1965). Absence of Martian radiation belts and implications thereof. Science 149, 1228-1233. VAN DYKE, M. D. (1958). The supersonic blunt-body problem-review and extension. J. Aerospace Sci. 25,485496. VAN DYKE, M. D. and GORDON,H. D. (1959). Supersonic flow past a family of blunt axisymmetric bodies. NASA Tech. Rep. No. TR R-l. WALLACE,L. (1969). Analysis of the Lyman-alpha observations of Venus made from Mariner 5. J.geophys. Res. 74.115-131.
SOLAR
Department
Pergamon Press Printed in Northern Ireland
WIND FLOW PAST NONMAGNETIC PLANETS-VENUS AND MARS
JOHN R. SPRRITER of Applied Mechanics, Stanford University, Stanford, Calif. 94305, U.S.A.
AUDREY L. SUMMERS Space Sciences Division, Ames Research Center, Moffett Field, Calif. 94035, U.S.A. and Department
ARTHUR W. RIZZI of Applied Mechanics, Stanford University, Stanford, Calif. 94305, U.S.A. (Received infinal form 9 December I.969)
Abstract-The hydromagnetic theory of solar wind flow past the Earth has been extended and modified so as to be applicable to nonmagnetic planets, such as Venus and Mars, that have a sutIicient ionosphere to deflect the solar plasma around the planet and its atmosphere. The principal difference in the analysis stems from the fact that the current sheath that bounds the solar wind away from the planet is formed by interaction with the ionosphere rather than with the geomagnetic field as in the case of the Earth. After stating the principal assumptions and equations, the theory is applied to determine the shape of the ionosphere boundary, or ionopause, across which the Newtonian approximation for the pressure of the flowing plasma is balanced by the pressure of a stationary ionosphere. Specific numerical results are given for a wide range of ionospheric parameters representative of those proposed for Venus and Mars. The location of the bow wave and the properties of the flow field are then calculated using magnetohydrodynamic and gasdynamic considerations in a manner similar to that employed for the flow of the solar wind plasma past the Earth’s magnetosphere. Examination of the results reveals a correspondance rule that enables results presently available for the location of the bow wave and the properties of the flow about the Earth’s magnetosphere to be converted rapidly into those for a nonmagnetic planet by a simple relabeling of the s&es. The results are shown to be in general accordance with observations made by Mariner 5 as it flew past Venus, although certain differences near the theoretical location of the ionopause suggest the presence of a thick boundary layer. A similar analysis of the data from Mariners 4,6 and 7 indicates that Mars has a sufficient ionosphere for the theory to be applicable. Further comparison is relatively uninformative beyond revealing no inconsistencies with the theoretical results, however, because Mariner 4 did not approach close enough, and Mariners 6 and 7 were not equipped to make the necessary measurements. INTRODUCTION
Data acquired in recent years in the vicinity of the Earth, the Moon, Mars and Venus have disclosed three essentially different types of interaction with the solar wind. For the Earth, the geomagnetic field prohibits the solar wind from approaching nearer than about 10 Earth radii under ordinary circumstances. The shielded region, the magnetosphere, acts as an obstacle in the supersonic flow of the solar wind, and a bow wave forms a few Earth radii upstream of the magnetosphere boundary, as illustrated in Fig. 1(a). Interaction of the solar wind and the Moon is very different because of the Iack of either a sensible magnetic field or ionosphere to deflect the incident flow. The particles of the solar wind thus proceed unchecked until they contact the lunar surface, where they are effectively removed from the flow. As a consequence, no bow wave forms, and the principal features of the flow field are associated with the closure of the lunar wake, or cavity in the solar wind, that extends downstream from the Moon as illustrated in Fig. l(b). Neither Venus nor Mars has a significant magnetic field, but the electrical conductivity of the tenuous upper ionosphere of these planets is evidently sufficiently high that the solar wind is prevented from flowing directly into the planetary surface or lower absorbing levels of the atmosphere. The solar wind is thus deflected around the ionosphere, and a bow wave is 1
1281
_/---
field
__--
-.
./--i)
-0.
-
c---.~ 4
Bow wave,
Fm. 1. PRINCIPAL FEA’MJRES OF SOLARWIND FLOWPASTTHEEARTH(a), THE MOON (b) AND VENUSOR MARS (c).
Nanalined
7.p
field
Cavify Moon Alined
/Streamline
-----f
/3trramline
SOLAR WIND
FLOW PAST NONMA~~~C
PLANETS
1283
formed upstream of the planet, similar in many ways to that associated with the Earth. Aside from evident differences in the underlying physical processes at the boundary surface between the ionosphere and the solar wind, the principai difference between the flow fields around Venus or Mars and Earth is the size of the cavity. As illustrated in Fig. 1(c), the ionosphere boundary is wrapped much closer around a nonmagnetic planet than the magnetosphere boundary is around the Earth, the nose being at an altitude of about 500 km for Venus compared with about 60,000 km for Earth. The corresponding altitude for Mars is not known as well, but available results combined with the present theory indicate that it is probably between 155 and 175 km. Extensive theoretical studies, specific calculations, and detailed comparisons with observations have shown that the interaction of the solar wind and the Earth may be represented satisfactorily, insofar as gross features of the flow such as density, velocity, temperature, and magnetic field are concerned, by the standard continuum equations of magnetohydrodynamics (see Spreiter et al., 1968 or Spreiter and Alksne, 1969 for recent summaries). Recent investigations (see Spreiter et al., 1970 for an extensive account) have shown that the same is true for the Moon, provided boundary conditions are applied at the lunar surface that correspond to the Moon absorbing, or neutralizing, all particles of the solar wind on impact. The present paper extends the same line of analysis to objects in the solar system that have no significant intrinsic magnetic field, but a sufficiently dense and electrically conducting ionosphere to stop and deflect the solar wind before it is absorbed by the planetary surface or atmosphere. Venus is a known example of such an object. We believe Mars to be another al~ough present data are insu~cient to support this with the same degree of surety as for Venus. The anaIysis commences with a presentation of the basic equations used to represent the variation of pressure with height in the ionosphere, and the dynamical properties of the solar wind. There follows an account of the manner in which these equations may be simplified to obtain a tractable mathematical problem without undue loss of realism or accuracy of representation of principal features of the interaction. Specific numerical results are presented for the shape of the ionosphere boundary and the bow wave for a substantial range of values for the parameters describing conditions in the solar wind and in the planetary ionosphere. It is found that the results for a wide range of ionospheric parameters may be brought into close correspondence with those for solar wind flow past the Earth’s ma~etosphere by application of a simple geometric transfo~ation of the coordinates. This correspondence rufe enables a substantial body of theoretical results for the density, velocity, temperature, magnetic field, and velocity distribution of the protons presently available for solar wind flow past the magnetosphere to be applied to Venus or Mars with only minor modification. FinaIly, the theoretical results are compared with the observations made by Mariner 5 as it flew past Venus, and implications of the points of agreement and disagreement are discussed. The corresponding Mariner 4 data for Mars provide little opportunity for comparison, beyond the simple observation that the absence of planetary effects in the plasma, magnetic field, and energetic particle data is consistent with the indications of the present results that Mariner 4 was outside the theoretical location of the bow wave at all points on its trajectory for which data were received. Mariners 6 and 7 subsequently approached within 2000 km of the Martian surface, but neither spacecraft carried a plasma probe, ma~etometer or energetic particle detectors. As a resuh, the only observation of relevance to the present study is that the
1284
J. R. SPREITER,
A. L. SUMMERS
and A. W. RIZZI
occultation experiments indicated that the peak electron density was 1.5 x lo5 electrons/ cm8 and that it was at an altitude of 130 km (Anonymous, 1969), essentially the same as indicated earlier by the occultation experiment of Mariner 4 (Kliore et al., 1965 and Fjeldbo and Eshleman, 1968). MATHEMATICAL REPRESENTATION OF INTERACTION OF THE SOLAR WINJl AND THE IONOSPHERE OF A NONMAGNETIC PLANET
It is the plan of the present investigation to base the analysis on the equations of magnetohydrodynamics and proceed to the solution through approximations that are well known and tested in previous applications to the interaction of the solar wind and the Earth, with modifications introduced only as required because it is the ionosphere rather than the geomagnetic field through which the presence of the planet is communicated to the solar wind. In broad outline, the elements of the theory are as follows. The incident solar wind is considered to be steady and supersonic, and to flow in accordance with the equations of magnetohydrodynamics for a perfect gas in which all dissipative processes, such as those associated with shock waves and boundary layers, take place in the interior of thin layers idealized as magnetohydrodynamic discontinuity surfaces of zero thickness. The ionosphere, or at least the outer part of it that participates in the interaction with the solar wind, is idealized as spherically symmetric and hydrostatically supported plasma having infinite electrical conductivity. Since the two bodies of plasma are of different origin, and have different properties, they must be considered to be mutually impenetrable in the idealized hydromagnetic representation, and to be separated by a tangential discontinuity surface. This surface is called the ionopause, since it marks the outer boundary of the ionosphere. Once the problem is formulated, the solution proceeds in two steps, just as in the analogous problem for the Earth. First the shape and location of the ionopause is calculated for selected values of the parameters characterizing the solar wind and the ionosphere. Following that, the location of the bow wave and the properties of the flow field are determined. More explicitly, the assumption of hydrostatic support is equivalent to assuming that all motions of the gas within the ionosphere are sufficiently small with respect to the planetary body that equilibrium exists between the pressure gradient and the force of gravity, thus dP -= dr
-t%
(1)
where p and p are the gas pressure and density, r is the radial distance from the center of the planet, and g is the acceleration of gravity. Values for the latter are inversely proportional to r2. Thus g = g,(r,/r)2, where subscript s refers to values at the surface of the planet. Values for g, for Venus, Mars and Earth are about 870, 375 and 982 cm/sec2, and those for rr are about 6.1, 3.4, and 6.4 x lo* cm, respectively. The pressure is assumed to be related to the density by the perfect gas law p = nkT = pRT/M
(2)
in which n is the total number density of particles, Tis the absolute temperature, k = 1.38 x lo-la ergsrK is Boltzmann’s constant, R = 8.31 x 10’ ergs/g “K is the universal gas constant, and M is the mean molecular mass nondimensional&d so that M = 16 for atomic oxygen. M is thus equal to + for fully ionized hydrogen plasma, and 1 and 2 for singly
SOLAR WIND
FLOW PAST NONMAGNETIC
PLANETS
1285
ionized molecular hydrogen and helium. The density may be eliminated from Equation (1) by introduction of Equation (2), and the result integrated to yield
in whichpR is the pressure at the reference radius rn, and H, the local scale height of the atmosphere, is given by J&T
RT mg = Mg
(4)
where m = 1.67 M x 1O-24g is the mean molecular mass. If H is constant, Equation (3) may be integrated to obtain p =pRexp
(-y)
which shows that H represents the height interval in which the pressure decreases by a factor e. If the variation of g with r is disregarded over the range of application of Equation (5), Tis also constant, and the density varies with height in the same way as the pressure, thus P --=-= n nR
&-
P
exp(-y).
PR
In the upper atmosphere, where little mixing would be presumed to occur, diffusive equi~brium may be considered to prevail among the various ionic constituents, at least in an idealized sense. As emphasized recently by Bauer (1969), however, this equilibrium is more complicated in an ionosphere than in a neutral atmosphere for which Dalton’s law of partial pressures is customarily invoked. This is because an electric polarization field, dependent on the mean ionic mass and charged particle temperatures, acts on all the ions and couples the diffusive equilibrium distribution of all the ions. In most circumstances, however, the lightest constituent, ionized atomic hydrogen, would emerge as dominant at great altitudes. At lower and intermediate altitudes, more specific knowledge, such as is now resulting from space experiments conducted by the USSR and USA, is needed to specify the variation of pressure with radius with any degree of precision. Figure 2 shows the variation with distance from the center of Venus of the electron number density, as deduced from the dual frequency radio occultation measurements by the Mariner Stanford Group (1967). For a plasma with singly ionized ions, charge neutrality requires that these curves also describe the ion number density, summed over all constituents. Also included on Fig. 2 are three small inserts showing the sIope of the densityaltitude curve for four possible atmospheric constituents covering a wide range of M at three substantially different temperatures, as calculated using Equations (4) and (6). These measurements indicate that the nose of the ionopause is at a distance of about 6500 km from the center of Venus, or about 450 km above the surface. The experimenters interpret the results between 6800 and 7500 km as almost certainly indicating the dominant constituent is ionized atomic hydrogen plasma with a mean ion-electron temperature between 625 and 1100°K. Between 6800 and 6300 km, they conclude that the atmosphere is most likely composed of ionized helium with a temperature between 620 and 970°K.
1286
J. R.
6ooo
SPREITER,
5
2 I02
A.
L. SUMMERS
5
2 IO3
Electron
5
2
and A.
5
concentration
2 IO5
IO4
W.
RIZZI
5 IO6
,CfK'
F1cz.2, PROFILESOF ELECXRON CONCENTRATIONSIN 'IXENW3HT AND DAY IONOSPHERESOF ~OMTHE D~~~F~Q~ENCY 8~~x0 -TATION ~~~E~~SBY~ MARINERSTANFORD G~oup(1967).
VEMJS, ASDEDUCED
The proper interpretation of these data, and also those from the Mariner 5 Lyman alpha measurements of Barth et al. (1967), has, however, been the subject of considerable discussion. Barth (1968) and Barth et al. (1968) conclude that the entire upper atmosphere of Venus above about 6500 km can be represented best by an atmosphere consistingp~~rily of molecular hydrogen at 650 & 50°K. On the other hand, Wallace (1969) and Donahue (1969) have concluded from their analyses of the Mariner 5 Lyman alpha data that the dominant constituent above 6500 km is deuterium at about 650°K. McElroy and Strobe1 (1969) have compared results from a number of models of the topside night-time ionosphere of Venus with obse~ations of Mariner 5 and concluded that the primary ionized constituent is either helium or molecular hydrogen. In view of these uncertainties, and even greater ones with respect to Mars at the time of this writing, we put aside any further discussion of refinements of the variation of N with altitude and proceed to adopt Equation (5) as sufficient to represent the variation of p with r in the upper atmosphere of either Venus or Mars. Results will be presented for several values for H, however, in order to provide an indication of the variations to be expected between ionospheres of different chemical composition. For Venus, with r, equated to 6500 km and T to 700”K, N is approximately 1500, 760, 380 and 35 km for ionized atomic hydrogen, singly ionized molecular hydrogen, helium, and carbon dioxide. The corresponding values for the nondimensional ratio H/r,, in which r,, the distance from the center of the planet to the nose of the ionopause, is equated to 6500 km are O-23, O-115, O-058 and O-0052, For Mars, the smaller radius and acceleration of gravity and lower ionospheric temperature of about 200°K (Kliore et al., 1965 and Fjeldbo and Eshleman, 1968) lead to values that are about l-2 times larger than for Venus, namely about O-28, O-138, O-069 and 0.0062. The calculations of these values makes use of the result, indicated by the data from Mariners 4, 6 and 7 together with the present theory,
SOLAR WIND
FLOW PAST NONMAGNETIC
PLANFXS
1287
that the nose of the ionopause is at an altitude between 155 and 175 km, and that to therefore scales nearly in proportion to the planetary radius. The dynamical properties of the solar wind flow are considered to be represented by the standard magnetohydr~ynami~ equations for steady flow of a dissipationless perfect gas. The governing differential equations are thus v.pv=o p(v . v)v + Vp == - (lj&)B
div B = 0
cur~(Bxv)=0, (v . V)S = 0,
x curl B
S - S, = c, In ~
OY in which v and S refer to the velocity and entropy of the gas, B represents the magnetic field, and y = c,/c~ = (N + 2)/N in which cP and c, are constants representing the specific heats at constant pressure and volume, and N represents the number of degrees of freedom of the gas particles. For a monatomic gas, such as either of the two leading constituents of the solar wind, atomic hydrogen and helium, iV = 3, and y = +. Implicit in the relations of Equation (7) are two important nondimensional parameters, the Mach number M and the Alfven Mach num~r Pn;, defined as follows:
Conditions on opposite sides of discontinuity surfaces that develop in the ffow because of the omission of dissipative terms in Equation (7) are related by the following conservation laws : [P%l = 0 [plt,v + @ C B2fs7T)li- S,B,/~W] = 0
WGi - B&f = 0,
I&al = 0
[pv,@ + u*/2) + v$~/~w - B,v t B/47r]= 0
(91
in which h = c&I’is the enthalpy, A and I represent unit vectors normal and tangential to the discontinuity surface, and the square brackets are used to indicate the difference between the enclosed quantities on the two sides of the discontinuity, as in [Q] = Qa - Q, where subscripts 1 and 2 refer to conditions on the upstream and downstream sides of the discontinui~. It is well known (F~ed~chs and Kramer, 1958; Landau and Lifshitz, 1960; Jeffrey and Taniuti, 1964; Jeffrey, 1966) that five classes of discontinuities are described by the conservation equations. As in the corresponding application to solar wind flow past the Earth, (Spreiter et al., 1966b, 1968 and Spreiter and Alksne, 1969), two of these, the fast shock wave and the tangential discontinuity, are of concern in the present analysis. The first relates conditions on the two sides of the bow wave, The latter, which has the properties v, = 3, = 0, lvt] # 0, RlJ.0
IPl f 0,
[p + P/857] = 0
(10)
1288
J. R. SPRlEIl-ER, A. L. SUMMERS
and A. W. RIZZI
relates conditions on the two sides of the ionopause under the assumption that the upper ionosphere can be treated as a perfectly conducting fluid effectively bound to the planet and incapable of mixing with the solar wind plasma. As in the corresponding application to the Earth’s magnetosphere, important simplifications may be introduced in these relations In the present application, p tends to be much larger than B2/87r on both sides of the ionopause. Therefore the pressure balance relation of Equation (10) may be reduced to a simple balance between the ionospheric pressure given by Equation (5) and the static pressure of the flowing solar plasma adjacent to the ionopause. With the further introduction of the Newtonian approximation p = psi cos2 y for the pressure on a body in a steady hypersonic flow, we arrive at the following expression for the pressure balance at the ionosphere boundary: pst
cofy = Kpmueo2 co9 y
= pR exp (_r+)
(11)
in which y is the angle between the outward normal to the ionosphere boundary and the flow direction in the undisturbed incident solar wind, and pst = Kpoouco2is the stagnation pressure of the solar wind exerted on the nose of the ionopause. The subscript co indicates that values associated with conditions in the solar wind upstream of the bow wave are to be used, and K is a constant equal to 0.88 for high Mach number flow of a monato~c gas, although usually taken as unity in applications to the Earth’s magnetosphere (Spreiter er al., 1966b). Use of the foregoing approximations permits the coordinates of the ionopause to be calculated without determining any further properties of the flow field. Once the shape of the ionopause has been determined, the density, velocity, and temperature in the surrounding flow may be calculated by solving Equations (7) and (9) with the terms containing B omitted because of their smallness in the characteristically high Alfven Mach number flow of the solar wind. The magnetic field can be calculated subsequently by solving the remaining equations: curl (B x v) = 0, div B = 0 F&v, -
W,l = 0,
P4,l = 0
(12)
with v considered known from the preceding steps, as has been done for the case of the Earth by Alksne (1967). The calculations can be further extended to provide information on the velocity distribution of the solar wind particles, if desired, by employing the procedures applied to the Earth by Spreiter et al. (1966a). LOCATION
OF THE IONOPAUSE
It
is convenient in the calculation of the shape of the ionopause to let rn = ro, the distance from the center of the planet to the nose of the ionopause. Since cosa 1y = I at this point, Equation (11) simplifies to PR
=
PO =
Kf%st),2
(13)
at the ionopause nose, and to cof
y =
e&-rO)lE
(14) elsewhere along the ionosphere boundary. If, for example, the solar wind is considered to approach Venus with a number density of 3 protons/cm8 and #abulk velocity of 5 X IO’ cmlsec, as the data reported by Bridge et al. (1967) show occurred at the time of the Mariner 5 measurements, Equation (13) indicates that p,, = 1.1 x 10” dyn/cm2. Equation (2)
SOLAR WIND
FLOW PAST NON~G~TIG
PLANEiTS
1289
shows that such a value corresponds, assuming 2’ = 7OO”K, to an ion or electron number density n, = n, = n/2 at r = r, of about 5.7 x 104 particlesjcms. This value is somewhat greater than that indicated by the data of Fig. 2 for r = 6500 km, but is not unreasonable, considering the preliminary nature of the Mariner 5 data involved in the estimate, and the uncertainties attendant with all plasma measurements in space. In any case, it may be seen from Fig. 2 that the essential condition that the ion-electron number density required to stop the solar wind be exceeded at some altitude is satisfied by a considerable margin,
FIG. 3. VKEWOF ELEMENTOF I~NOPAUSEAND COORDINATES USEDIN EQUATION(15).
since the peak daytime value of 5 x lo5 electrons/ems is larger than the required value by a factor of about 10. Similarly for Mars, the values 0, = 3.3 x lO’cm/sec, n, = 0.8 protons/cm8, and T = 200°K inferred from the measurements of Mariner 5 (Lazarus et al., 1967; Kliore et al., 1965) indicate that an electron number density of 2.6 x lo* electrons/ cm3 is required at the subsolar point for the ionosphere to stop the solar wind. This value is exceeded by a factor of about 6 by the peak electron density of 1.5 x 105 electrons/cm3 deduced by Kliore et al. (1965) from Mariner 4, and subsequentIy confirmed by similar experiments with Nariners 6 and 7 (Anonymous, 1969). On the basis of the further observations that the altitude of the peak electron density is between 120 and 130 km above the Martian &ace, and that the electron scale height there is between 20 and 25 km, we may estimate from the above relations that the subsolar point of the ionopause is at an altitude between 155 and 175 km. For these reasons, we feel that the applicability of the theory to both Venus and Mars is supported by existing data, even though those for Mars are restricted in quantity and inferential in nature at the present time. To proceed, we must express cos2 y in terms of r and 0, where r(t9) represents the coordinates of the boundary, and 6 is the angle measured at the center of the planet with respect to a line that extends directly upstream. Upon carrying out this step in the way made clear by the illustration in Fig. 3, and substituting the result in Equation (14), we find (r d@cos 8 + dr sin 19)~ cog y ziz 4v = =exp(-r+)=E. (15) dr2 + (r dQ2 ds
0a
1290
J. R
SPREITBR, A. L. SU-W
and A. W. RIZZI
The numerical solution of this differential equation is facilitated by solving for dr/r de to obtain dr rde=
sin2&2dE-E2 2(E -
sins i3)
’
(16)
At the ionopause nose 0 = 0, r = r,, and dr/r d0 =r?0 with either choice of sign. The proper choice of sign is dictated by the following considerations based on the assumption that r increases monotonically from r, to co, and hence that E diminishes monotonically from 1 to 0, as 8 increases from 0 to 180”. Since sin* 8 = 0 at 8 = 0, increases to unity at 0 = 90”, and then returns to 0 at 180”, the denominator of Equation (16) must vanish at one or more values for 8, and dr/r de would be infinite unless the n~erator vanishes simultaneously. If the critical values for 8 and E are designated by the subscript cr, we have that EC, = sins 0,,
(17)
sin 20,, f 2(E,, - Eo,a)l/a= 0.
(18)
and also that
Substitution of the former into the latter to obtain sin 28, j, 2(sin* 8,, - sin* ep
= 2 sin ecr cos 8, + 2 lsin @,,cos e,l = 0
(19)
shows that the minus sign must be used in Equation (16). That the resulting indete~inate form actually leads to a finite value for drjde at 0 = ear may be confirmed by application of L’Hospital’s rule. The shape of the ionopause may therefore ,be determined directly by integration starting from the boundary condition that r = r, at 8 = 0. Results obtained by numerical integration are presented in terms of cylindrical coordinates x = r cos 0 and r’ = r sin 8 for several values for H/r, from 0.01 to 1 in Fig. 4. As noted previously, this range is ample to include all likely possibilities for both Venus and Mars. (Although the displayed results are independent of the altitude of the ionopause nose, a planet silhouette has been added to the plot to indicate the radius r,, of Venus for the ratio r,fr, = 6050/6500 = O-93). The corresponding ratio r&r,, for Mars is not known so definitely, but the estimate given above indicates that it is about 0.95. Also included on Fig. 4 is a dotted line indicting the coordinates of the equatorial trace of the magnetosphere boundary, as determined by Beard (1960) and Spreiter and Briggs (1961, 1962), that has been used extensively in the calculation of solar wind flow past the Earth. These coordinates, as well as those for the location of the associated bow wave for M, = 8, y = 513given by Spreiter et al. (1966b), have been nondimensionalized by dividing by the distance from the center of the Earth to the magnetosphere nose. We shall continue to call this distance r,,, even though it is much larger than for Venus or Mars, and evaluated differently using the expression r, = ra(BeQ2/2rrKp,v,2)11*in which rs = 6.37 x 10s cm is the radius of the Earth, and B,, = 0.312 G is the average intensity of the geomagnetic field at the geomagnetic equator. It may be seen that this curve is very similar to that for the ionosphere boundary for H/r, = O-2. The size, with respect to the planet, of the cavity carved in the solar wind is very different in the two applications, however, since r, is only a few percent greater than the planetary radius for Venus or Mars, whereas it is usually of the order of 10 Earth radii for the magnetosphere.
SOLAR WIND FLOW PAST NONMAGNETIC
1291
PLANETS
6Ho-0
5/
/
/
,
c’
/
/ /
/
,
, I’0 -75
4-
7/r,
3-
/
/
/
‘/ / I ,
N I
..
.. .’ ,
/:
/I
-
lonopouse
---
Bow
wave
~~~~~~_~~ MaQnelDpa~se and bow wave
0
-1.0
-2.0
-3.0
x/r, Fra.4. CALCULATEDLOCATIONOFIONOPAUSE ~WWAVEFOR~*
FOR~ARI~US~~~~,AND
ASSOCIATEDLOCATIONOF
= 8,y -$.
results for the shape of the Earth’s magnetopause and bow wave, nondimensionalized so that the magnetosphere nose is at x[rp - 1, is included for purposes of comparison. The dashed line for the bow wave for H/r0 - 0.2 has been omitted because it is indistinguishable from the dotted line representing the Earth’s bow wave.
The cormpon&ng
LOCATION
OF THE BOW WAVE AND PROPER~
OF THE FLOW FIELD
Upon specification of a value for the free-stream Mach number M, >> 1, the location of the bow wave and the properties of the flow field may be computed for any of the ionopause shapes shown in Fig. 4 by application of highly developed numerical techniques of nonlinear gas dynamics. Several alternatives are available for this, but as in our previous analyses of the Sow past the Earth, we have continued to employ the methods developed at Ames Research Center by Van Dyke (1958), Van Dyke and Gordon (1959), Fuiler (1961), Inouye and Lomax (1962), and Lomax and Inouye (1964). Results for the location of the bow wave for flow with M, = 8 and y = Q are presented in Fig. 4 for each value for H/r, for which the coordinates of the ionopause are ihustrated. Although these results are for a specific M,, it has been shown in our previous
J. R. SPRJXI’EJR, A. L. SUMMERS
1292
H&,=.20
Ma=8
and A. W. RIZZI
y=5/3
x/r0
x/r0 _T _ , + %I
Fm.5.
(Y-I)& 2
= 22~33-21.33(V/V,)2
I)ENSITY, VELOCITY, AND TEMPERATURE FIELDS FOR SUPERSONIC FL.OW PAST THE IONOSPHERE; H/r,, = 0.20, Moo = 8, y = #.
papers relating to the Earth that the shape of the bow wave, and also the associated distributions of p/p,, Y/V,, and B/B, for any given direction for B, are relatively independent of free-stream Mach number for all Iw, greater than about 5. In Fig. 5 are shown contour maps for the density ratio p/p,, velocity ratio v/v,, and temperature ratio T/T, for flow with.HfrO = O-20. Such a value is appropriate, for example, for a Venusian ionosphere composed primarily of atomic hydrogen at about 600°K. In view of the current tendency for some to interpret the Mariner 5 results as indicating that the primary constituent in the Venusian atmosphere is molecular hydrogen, or even heavier ions, we have also made a set of calculations for the density, velocity, and temperature in the flow field for H/r, = 0.1. The results are shown in Fig. 6. Corresponding plots for other quantities of interest, such as the magnetic field strength and direction, or Maxwellian proton velocity distribution, can also be calculated in the same way as has been done previously for the Earth. Such calculations have not been carried through, however. This is partly because they are time consuming to do and require considerable space to display the results for even a ~nimum number of cases, but p~ncipally because an approximate correspondence rule to be described in the next section makes it possible to convert quickly any of the numerous contour plots already available for the properties of the tlow around the Earth’s magnetosphere into that for the flow around an ionosphere having any H/r, between O-01 and 1 by a simple relabeling of the coordinate axes. CORRESPONDENCE NO~G~~C
RULE RELATING SOLAR WIND FLOW PAST A PLANET TO THAT PAST THE EARTH
The close relationship between the shape of the magnetopause and that of the ionopause for H/r0 = 0.20, and the general similarity of the ionopause shapes for all H/r,, suggests the possibility that a correspondence rule relating ionopause shapes for H/r, other than 0.20 to an appropriately scaled magnetopause may be found if the coincidence between
SOLAR WIND FLOW PAST NONMAGNETIC H/r0
= PI
Ma=8
1293
PLANETS
y=5/3
Density ratio
3-
.5 ,25
i/r0
2 .7 '6
2-
‘5
x/r0
x/r0
T _-_,+ TQ Fm.6.
RENSI~,~~~Y,
AND TE~E~~RR PHERE; Ii/r6=
(Y-I)Mtl 2
FIELDS FOR SUPERSONIC FLOW
= 22*33-21~33WVm~* PAST THEIONOS-
O*lO,Mm = 8, y = g.
the Earth and planetary centers be relinquished. Great practical utility would result from the availability of such a correspondence rule, because it would enable a substantial body of results already calculated for solar wind flow past the Earth to be applied with minor change to nonmagnetic planets having a wide range of ionospheric parameters. The results displayed in Fig. 7 show that it is indeed possible to achieve a fit that is probably sufficiently good for most purposes relating to the interpretation of data obtained in space. This figure is basically that given originally by Spreiter et al. (1966b) for the magnetosphere boundary and shock wave for the Earth, and the associated characteristic or Mach line pattern, for M, = 8 and y = $. Superposed on it are the ionopause curves from Fig. 4, each with its nose retained at x/r,, = 1,8 = 0, but with its scale adjusted so that its ordinate coincides with that for the magnetopause curve at 8 = n/2. The center of the nonmagnetic planet is, in general, no longer at the origin of coordinates, but at the points x,/r,, indicated along the axis of symmetry for each value for H/r,,. The coordinates of these points are listed on Fig. 7 for accuracy and convenience in applications of the correspondence rule. To convert the nondimensional shape for the ma~etopause into that for the ionopause for a given H/r,,, we must thus place the center of the planet at x,/r0 and change the labeling of the scales so that x/r,, = 0 at x,/ro, and x/r,, = 1 at the ionopause nose. It follows from the close correspondence between the coordinates of the ionopause and the magnetopause that the coordinates of the bow wave should also display a similar relations~p. It may be seen from Fig. 7 that this is indeed true. A similar degree of correspondence may be anticipated for the coordinates of contour lines for constant values of the flow parameters, such as those for the density, velocity, and temperature shown in Figs. 5 and 6, Although some of the ionopause curves depart significantly from the curve for the
1294
J. R. SPREITER, A. L. SUNMERS
and A. W. RIZZI
2.0 F/r0 I.5-
I,0-
_
lonopouse and bw
. . . . . . ..* . . . .
'5 -
wove
Moqnetopouse
and bow wave
?. bJ.USTRATlON OF DEGREE OF CUXNC~ENCB OF CXTRVESREPRESEl4’lTNQ T?lE IONOFAUSE AM) BOW WAMt FOR VARIOUS f@e OBTAINED BY APFLICATiON OP CORRESFONDENCE RULE. The solid lines have been omitted where they are ~~stjn~s~ble from the dotted lines.
%A
magnetopause somewhat downstream of the phmet, the effects of these differences have no infhtence upstream of the rearwardly inclined characteristic line emanating from the point on the boundary where the differences first become significant. It may be seen from Fig. 7 that this is su~cient~y far do~stream for ionospheres having Hfro between 041 and 1 that nearly all of the results we have presented previously for the properties of the flow field about the magnetosphere can be carried over with uo other change than relabeling the scales to obtain a good approximation for the conditions around Mars and Venus, COMPARISON It
WITH
MARINER
5 DATA FOR VENUS
is appropriate at this point to make a general assessment of the relation between the theoretical results shown in Figs. 4-6 and those observed by Mariner 5 as it flew past Venus. Figure 8 shows the time variation of the bulk velocity v, ion number density rz, and intensity 1231 and direction a and /i of the magnetic field together with a plot of the trajectory giving the position of the spacecraft as a function of time (Bridge et al., 1967). The angles a and 8 are in spherical coordinates in which a is the longitude measurement, in the RT plane, from the R direction, where R is a unit vector in the antisolar direction and T is an orthogonal vector which is parahel to the Sun’s equatorial piane and points in the direction of the planet’s orbital motion; and ,L3 is the latitude angle of the magnetic
-180
o
-240
’
-180 Time
-I20 -60 from encounter ,min
0
60
I
To Sun 2
I 0
-I
X/f”
-2
I
--Venus
-3
I
-4
I
‘lozpo”se / -5
j -6
F1a.8. PLASMA ANDMAGN~C~~CPIELDRATAMEASUREDBY~INER 5 INITSENCOUNTERWITH VENUSON~~~CIY)BER~~~~ (BRIDGE et al.,l967). The indicated trajectory has been projected into a plane by rotation about the Sun-Vent line. Times are in minutes relative to the encounter time E of closest approach to the planet.
a,deg
I80
-180’
1296
J. R. SPREITER,
A. L. SUMMERS
and A. W. RIZZI
field vector, considered positive when northward. For ease of reference, lines indicating the theoretical location of the ionopause and bow wave for H/r, = 0.10 and 0.25, for M, = 8, y = $, and r, = 6500 km, are superposed on the plot of the trajectory. The corresponding location of the bow wave for M, = 5 and H/r,, = 0.25 is also included in order to better assess the consequences of the fact that the values of about 590 km/set and 300,OOO”K reported for the velocity and temperature in the solar wind before and after Mariner 5’s encounter with Venus indicate a value of about 6.5 for M,. The magnetometer and plasma probe experimenters have concluded from the presence of abrupt and easily recognizable changes in their data that Mariner 5 crossed the bow wave at the points labeled 0 and 0. These points are reasonably close to the theoretical locations of the shock wave for H/r,, = O-25 for M, = 5, and somewhat farther from Venus than indicated by the theoretical results for M, = 8, or for H/r, = 0.10. On the other hand, the theoretical results for H/r, = 0.25 indicate that Mariner 5 should have crossed the ionopause, whereas those for H/r, = 0.10 indicate that it should have just skimmed along this surface. Although the data display decreases in n/n, and v/c, that are at least qualitatively similar to those for either H/r,, = 0.10 or 0.25, there is no certain evidence that Mariner 5 actually penetrated the ionopause. In particular, the intensity of the magnetic field remained of the order of that for interplanetary space, and v diminished to only a modest fraction of u, rather than to a value comparable with the speed of the spacecraft relative to Venus. On the other hand, the observations made when the spacecraft was near the theoretical location of the ionopause indicate values for n/n, and v/v, that are only about one-half of those indicated by the theory for the flow exterior to the ionosphere. A possible explanation for these discrepancies is that the tail of the ionosphere might taper inward toward the axis of symmetry rather than extend straight downstream. The theoretical results give no indication of such a trend, but they are of low reliability for this feature of the flow because of the deterioration of the quality of the Newtonian approximation for the pressure of the solar wind as ly approaches 90”, and the substantial departures from a constant scale height of a planetary atmosphere at great heights caused by the diminished gravitational acceleration, and also by probable increase in temperature. In addition, it is virtually certain that the ionosphere in an extended tail would be supported partially by dynamical interaction with the flowing solar wind instead of entirely by the simple hydrostatic means assumed in the present analysis. Although neither theoretical nor observational evidence is as yet sufficient to provide a definitive statement regarding the nature of the ionosphere tail, we are inclined to explain the discrepancy between the theoretical and Mariner 5 results in terms of an extended tail with a relatively thick boundary layer between the planetary ionosphere and the solar wind. The latter may be contrasted with the thin boundary between the solar wind the Earth’s magnetosphere that is becoming increasingly evident as the time resolution of the observations is improved. That the ionopause might be thick well back from the nose, whereas the magnetopause is usually thin, is plausible in view of the fundamentally different nature of the two boundaries. For the Earth, the magnetopause is essentially a boundary between the flowing solar plasma and a relative vacuum, and would be somewhat like the boundary formed by a free streamline in a water flow with embedded air cavity. For Venus or Mars, the ionopause is essentially a boundary between two different bodies of plasma. As such, a significant boundary layer would be anticipated which would provide an increasingly thick transition between the ionosphere and the solar wind with increasing distance from the stagnation point at the
SOUR
WIND FLOW PAST NONMAGNRTIC
PLANETS
1297
nose of the ionopause. It wotdd spread, moreover, both into the outer part of the ionosphere and outward into the surrounding flow, broadening the transition region in which all properties of the plasma including the density, velocity, and magnetic field change from their values for the Aowing solar plasma to those of the planetary ionosphere. At the location of the ionosphere boundary indicated by the present dissipationless theory, the plasma velocity might be expected to be substantially less than indicated by the theory. Since the plasma velocities observed by Mariner 5 in the vicinity of the theoretical location of the ionopause display such a trend, it is tempting to conclude that Mariner 5 entered the boundary layer separating the ionosphere and the flowing plasma, but did not enter the ionosphere proper, A more detailed examination is clearly required before a definite statement can be made, however. C~~~~ON
WITH MARINRR 4, 6 AND 7 DATA FOR MARS
As described in less detail previously, data from the magnetometer (Smith et al., 1965), plasma probe (Lazarus et al., 1967), and energetic particle detectors (Van Allen et al., 1965 and O’GalIagher and Simpson, 1965) on Mariner 4 displayed no effects attributable to the presence of Mars as it flew past that planet along the trajectory shown in Fig. 9, as viewed
Mariner
4 !-I/r, = 25, M, = 8
2
I
0
-I
-2
-3
-4
-5
-6
-7
Fro.9.TR~JE~RYDPEXARINEII~PAST~~,PRO~~DINMAPLANE BY~~A~o~~~ THESIJ%-MARSIJNE,ANDPROSABLB LOCAT~ONSOFTHB~ONOPAUSBANDBOWWAVE. No data were received from the spacecraft as it traversed the dashed portion of the trajectory, because Mariner 4, as viewed from the Ekth, was occulted by Mars. projected onto the Sun-Mars-spacecraft plane. AIthough such a negative finding does not provide much material for comparison, it is at least consistent with the indication of the present theory for the conditions measured at the time of the encounter that Mariner 4 did not cross the bow wave into the region influenced by Mars at any time when signals were being received from the spacecraft. To display this, several curves are shown on Fig, 9 to indicate probabIe locations of the Martian ionopause and bow wave. The latter are drawn for r&r0 = 0.95 and ET/r0= 0.01 as suggested by the observational data for conditions near the location of the peak electron density, and also for H/r,, = O-25, which is approximately the Iargest possible vaIue for Mars. The associated bow waves are for &f, = 8, but would be in virtually the same location for any higher Mach number. Lower Mach numbers are considered unlikeIy for Mars in view of data from the vicinity of Earth 2
1298
J. R. SPRBITBR, A. L. SUMMERS
and A. W. RIZZI
and the prediction of solar wind theory that M, is somewhat greater at Mars than at Earth or Venus. Figure 9 shows the trajectory to be well outside any likely position of the bow wave near the point of closest approach of Mariner 4 to Mars, but suggests the possibility that the bow wave may have been crossed somewhat downstream of that point. Unfortunately for the present purposes, however, it is not possible to determine if Mariner 4 actually crossed the bow wave there because occultation of the spacecraft by Mars, as viewed from the Earth, prevented reception of data for the portion of the trajectory indicated by the dashed line. Mariners 6 and 7 subsequently approached within 2000 km of the Martian surface, but neither spacecraft carried a magnetometer, plasma probe, or energetic particle detectors. As a result, the only observation of relevance to the present study is the confirmation of the previous determination of the ionospheric density by Mariner 4. The data from these experiments, and also those of Mariner 5 for Venus, are of considerable significance to the present theory, however, because they demonstrate that the ionospheres of both Mars and Venus are sufficiently dense to stop the solar wind at the subsolar point and to deflect it around the ionosphere. Since the margin in the product PolVcO2 is only about 6 for Mars and 10 for Venus, considering the ionospheric properties to remain fixed, the known variability of the solar wind suggests that it may be sufficiently enhanced at times to proceed directly through the ionosphere at the subsolar point and into the lower atmosphere where it would be absorbed by collisions. Under such conditions, the interaction would be quite different from that described here, and would probably tend toward that described by Spreiter et al. (1970) for the Moon. Since such conditions can be expected to occur only occasionally, if at all, particularly if the ionospheric density and temperature are enhanced by a strong solar wind flow, we conclude that the theory given here is both plausible and capable of providing a reasonably accurate description of the conditions that prevail most of the time at Venus and Mars. Acknowfedgemenrs-It is a pleasure to recall and to publicly acknowledge a number of stimulating discussions on the topic of this paper with Mr. Ray T. Reynolds, Mrs. Alberta Y. Alksne, and Dr. David S. Colbum of the Space Sciences Division of Ames Research Center, and Dr. Joan Hirshberg of the Applied Mechanics Department of Stanford University. The research at Stanford University was supported by the National Aeronautics and Space Administration, partially under contract NAS2-5355 administered by Ames Research Center, and partially under grant NGR 05-020-330. REFERENCES ANONYMOUS (1969). First findings from the Mariner flybys. Sky and Telescope 38,232-234. ALKSNE, A. Y. (1967). The steady-state magnetic field in the transition region between the magnetosphere and the bow shock. Planet. Space Sci. 15,239-245. BARTH, C. A., PEARCE,J. B., RELLY,K. K., WALLACE,L. and FASTS, W. G. (1967). Ultraviolet emissions observed near Venus from Mariner V. Science 158,1675-1678. BARTH,C. A. (1968). Interpretation of the Mariner 5 Lyman alpha measurements. J. atmos. Sci. 25,564567. BATJER, S. J. (1969). Diffusive equilibrium in the topside ionosphere. Proc. of the IEEE 57, 1114-1118. B~arr-r, C. A., WALXXS, L. and PIXARCE, J. B. (1968). Mariner 5 measurements of Lyman-alpha radiation near Venus. J. eophys. Res. 73,2541-2545. BEARD,D. B. (19 &l). The interaction of the terrestrial magnetic field with the solar corpuscular radiation. J. geophys. Res. 65,3559-3568. BR~X~E,H. S., LAZARUS,A. J., SNYDER,C. W., SMITH, E. J., DAVIS, JR., L., COWMAN,JR., P. J. and JONES,D. B. (1967). Mariner V: Plasma and magnetic fields observed near Venus. Science 158,16691673. I)ONAHUB,T.M. (1969). Deuterium in the upper atmospheres of Venus and Earth. J.geophys. Res. 74,11281137.
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