On the alignment of interstellar dust

Physica 41 (1969) 100-127

8 North-Holland Publishing Co., Amsterdam

ON THE ALIGNMENT OF INTERSTELLAR

DUST

E. M. PURCELL Harvard University, Cambridge, Massachusetts, U.S.A.

synopsis Two alignment mechanisms have been investigated by means of a Monte Carlo model. In the mechanism suggested by Gold, needles or flakes which have a sufficiently high velocity with respect to the gas around them are aligned in consequence of the anisotropy of the angular momentum acquired by a grain in collision with gas atoms. Spheroids exhibited, in the Monte Carlo runs, the expected alignment. The quantitative dependence of both angular momentum alignment and axis alignment on grain shape and velocity has been established with an accuracy of 5 to 10 percent. The maintenance of grain velocity by radiation pressure and the effect of a magnetic constraint on the trajectory of a charged grain are discussed. With paramagnetic relaxation incorporated, the Monte Carlo model exhibits Davis-Greenstein alignment. The results for different grain shapes, grain temperatures, and magnetic viscosities are compa.red with the theoretical predictions of Jones and Spitzer. Because the lowfrequency magnetic viscosity in any paramagnetic system with dipole-dipole coupling only is independent of the dipole strength and concentration, protons are as effectives as electrons. The viscosity arising from diamagnetic relaxation in graphite is estimated to be much too small for effective alignment of graphite flakes.

1. Introdwtion. polarization of starlight, discovered by Hiltner and by Hall in 1949, shows that the grains for interstellar extinction, substantial fraction of them, must be aligned with respect to something 1). It has generally been assumed that they are aligned with respect to the interstellar paramagnetic grains in the interstellar statistical mechanics of the Davis-Greenstein theoretical basis and have discussed, with respect to their efficacy for grain alignment, a variety of magnetic systems which might be in the substance of the interstellar interstellar the Davis-Greenstein

ON THE ALIGNMENT

of the grain is electrically ditions,

light propagating

isotropic,

OF INTERSTELLAR

and if we exclude special resonance

perpendicular

should suffer least extinction

DUST

if its electric

to the interstellar vector is parallel

magnetic

101

confield

to the field. It

appears that the observations of interstellar polarization are, on the whole, consistent with this prediction and the common assumption that the interstellar magnetic field, at least in the large, is parallel to the galactic plane. In some regions the situation is rather confused. Independent evidence of the configuration of the interstellar fields is perhaps still too spotty to settle the question. In any case, an evaluation of the correlation between polarization observations and galactic magnetic field is outside the scope of this paper. That the grains should be paramagnetic is not a far-fetched supposition. As Davis and Greenstein showed, iron ions present to a few atomic percent might suffice. For reasons that we shall presently review, the mechanical damping attainable is, within certain limits, largely independent of both the density and strength of the atomic magnetic moments. Indeed, under rather reasonable assumptions, the nuclear paramagnetism of pure ice could provide the requisite dissipative coupling to the field. Contentment with the Davis-Greenstein explanation of grain alignment might be nearly universal if it did not call for such a strong magnetic field. The analysis of Spitzer and Jones confirmed, as do results to be reported in this paper, that under the most favorable assumptions paramagnetic particles require for adequate alignment a field no weaker than 10-S gauss. In respect to several aspects of galactic physics, as presently understood, this is regarded as uncomfortably strong. Jones and Spitzer 3) have suggested that a more exotic, though still quite conceivable, grain composition involving isolated, submicroscopic, ferromagnetic domains might permit alignment in considerably weaker fields. As I shall explain below, I believe the “superparamagnetic” system they proposed cannot work as well as they had hoped. However that may be, there is ample reason to look for other mechanisms by which grains might be aligned. Gold 4) described in 1952 a simple, altogether mechanical process which must come into play whenever a grain has a high velocity relative to the surrounding interstellar gas. With respect to the direction of relative velocity, the “streaming direction”, a collision between such a grain and a gas atom contributes dominantly transverse angular momentum to the grain. If on the statistical average a grain has excessive transverse angular momentum, its axis of greatest rotational inertia will tend to be transverse. Consequently an elongated grain will display a net alignment of its long axis parallel to the streaming direction. In Gold’s proposal the streaming velocity was to be provided by a collision between two gas clouds, as the dust from each cloud penetrated the other cloud. The result would be an alignment of prolate particles (“needles”) perpendicular to the cloud-cloud

E. M. PURCELL

102

interface.

Flat particles

(“flakes”)

would be aligned with their planes perpen-

dicular to the interface, so that, as in all other proposed mechanisms I know of, needles and flakes lead to polarization effects of the same sign. A quantitative objection to Gold’s proposal as a general explanation of interstellar polarization was made by Davis 5) on the grounds that the penetration of one cloud by dust from the other is so shallow that only a relatively small fraction of the galactic volume could be occupied by dust thus aligned. On the other hand, there can be no doubt that the Gold mechanism is aligning non-spherical dust grains whenever, for any reason, the dust is streaming through the gas with a relative velocity comparable to the gas atom speed, or greater. The work reported here began as an attempt to survey possible alignment processes in the course of which various effects involving the interaction of radiation with aspherical or anisotropic grains were considered. Among them were effects evoked by the steady differential velocity grains would acquire, relative to the gas, in an anisotropic radiation field. For example, with the optical cross section dependent on grain orientation, and the radiation field most intense, let us say, in the z direction, needles rotating in the x-y plane will be driven to a higher z velocity than needles rotating in the X--Z or y-z planes. Consequently, needles in the first category will be more severely bombarded by gas atoms, and their orientation more quickly altered. The effect is at best quadratic in the velocity, but the grain velocities which can be acquired, under reasonable assumptions, are not small compared to the atomic velocity. As it turns out, this particular process does not yield significant alignment. On the other hand, with a streaming velocity established by any means, the stage is set for the Gold alignment mechanism, of which I became aware through discussion with Professor Gold. The statistical mechanics of irregular grains driven at hypersonic velocity is difficult to analyze quantitatively, even without the possibly important feature of orientation-dependent radiation pressure. However, the system lends itself extraordinarily well to a Monte Carlo treatment, chiefly because the impacts of gas atoms on a grain are widely separated impulsive events. A model was devised that faithfully embodied a number of physical processes. With this model it has been possible to determine quantitatively the degree of alignment of spheroidal grains by streaming mechanisms under various assumed conditions. It proved possible, also, to incorporate the Davis-Greenstein mechanism in the model. This has been worth-while for two reasons. The analysis of Jones and Spitzer provides a rigorous prediction for the expected alignment of the angular momentum vectors of spherical particles, but for that case only. For spheroidal particles, and in particular for the average alignment of the figure axis rather than the angular momentum of spheroidal particles,

ON THE

ALIGNMENT

upon which polarization

depends,

OF INTERSTELLAR

DUST

103

Jones and Spitzer could make only an

estimate from a reasonable but not at all rigorous argument. Their results for the sphere can serve as a check on most features of the Monte Carlo model, while the Monte Carlo machinery, in turn, can provide results for spheroids as reliable and accurate as its results for the sphere. The contribution of this paper will be somewhat miscellaneous. The Monte Carlo results for alignment

by streaming,

the mechanism

of Gold,

will be useful, it is hoped, in estimating the importance of this mechanism in various contexts. The possibility that anisotropic radiation pressure drives grains to a speed sufficient for appreciable alignment will be briefly discussed. It will be pointed out, in this connection, that one cannot neglect the constraint imposed by the magnetic field on the grains, which one must presume to be charged, with the curious consequence that the particles in question, though not aligned by the magnetic field may nevertheless end up aligned with resfiect to the field. The Monte Carlo results for the Davis-Greenstein process, in addition to their quantitative significance, serve to emphasize the critical importance of the grain temperature, a feature pointed out by Jones and Spitzer but perhaps not yet widely appreciated. The often mentioned possibility of the magnetic alignment of graphite particles will be briefly discussed.

2. Alignment by streaming: the Monte Carlo model. In a typical hydrogen cloud a grain 0.5 x IO-4 cm in diameter is struck by an atom of the gas two or three times per hour. The grain is rotating, if its rotational energy is in equipartition with the gas, at an angular velocity of the order of 3 x 104 rad/s. Ordinary fluid dynamics is of course totally irrelevant to the motion of a dust grain through a gas in which the atomic mean free path is of the order of an astronomical unit. However, the calculation of the drag coefficient for low grain velocity is simple. For a spherical particle it yields a retarding force given by Fdrag

=

$T U2pg&46.

(1)

Here a is the radius of the sphere, zl its velocity relative to the gas, pgas the density of the gas, and 6 the mean speed of the gas atoms. The constant in eq. (1) is correct for either of two assumptions: (a) atoms are reflected specularly from the sphere, or (b) every arriving atom sticks and subsequently re-evaporates from an uncorrelated location on the sphere. It follows from eq. (1) that the damping time for linear motion of the grain through the gas, denoted by Tdt is given by the simple relation. Tdt =

(PdParas)

’ (a/c),

(2)

where pgra is the density of the material of which the grain is composed, presumably something like 1.O g/ems. Putting in typical values,

104

E.

M. PURCELL

10-23 g / cm3, a’ = 0.25 x 10-b cm, and d = 105 cm/s, we find that Tdt = 2.5 X 1013 s. In this time, incidentally, the grain will have been

pgas =

bombarded by atoms whose total mass is just i of the mass of the grain. We can similarly compute a time Tdr for relaxation of grain rotation by gas atom impacts. The result, still for u Q fi and a spherical grain, is Tdr =

+(/h&P&(@)

=

+dt.

(3)

This applies only to the “sticking” model; specular reflection from a sphere cannot affect its rotation. We shall return later to the question of the hypersonic drag. The important point here is that the damping time both for translational and for rotational motion of the grain, is typically 106 years, and includes a number of impacts about equal to the ratio of grain mass to atom mass. On the other hand, it is readily shown that the temperature of the grain will be established through radiation exchange in a very much shorter time (indeed, in a few hours). Thus the temperature of the grain is well defined and quite independent of the kinetic temperature of the gas, or the speed u of the grain - and this is true even in the hypersonic case. The way is opened to a practical Monte Carlo model by the observation that the dynamical behavior of a grain bombarded by massive atoms and that of a grain bombarded by light atoms cannot be very different, except in time scale, providing only that the massive atoms are themselves light compared to the grain. If, for example, the grain has only 50 times the mass of the atom, its rotational damping time, which will be the portion of its history needed to yield about one independent sample of some function of orientation angles, will include only 40 or so atomic impacts, instead of 1010. But still, it will be very unlikely that one impact will cause more than a small change in the motion of the grain. It now becomes feasible to calculate precisely the mechanical consequences of each impact, to follow the intervening free rotation of the grain, and to accumulate enough history for reasonable statistical accuracy in the averages of the quantities of interest. As a rough guide, we should expect that an average over a history covering a thousand damping times, equivalent to an ensemble average over a thousand grains, would give results reliable to a few percent. That would include about 40,000 impacts in the example just given. In any other context, one might refer to such a strategy as the substitution of a “coarse grained” model of the physical system, but here the term would be peculiarly confusing; it is a “coarse atom” that we shall employ. The validity of the method can be established a fiosteriori by comparing the results of Monte Carlo runs in which only the relative mass of the atom is different (see table I). Reduced to the appropriate dimensionless form, the independent parameters of the Monte Carlo model are: (a) The ratio of atom mass to grain mass.

ON THE

ALIGNMENT

OF INTERSTELLAR

(b) The eccentricity

of the spheroidal

of inertia is calculated

on the assumption

DUST

grain. (The corresponding

105

spheroid

that the grain has uniform density.)

(c) The magnitude of the assumed steady uni-directional force on the grain. (d) The dependence of the force on the instantaneous grain orientation, if it is desired to include such an effect. (e) The ratio of the temperature of the grain to the temperature of the gas. (This enters here only in the re-evaporation of atoms that have struck the grain, but it will play a more important role later in the Davis-Greenstein process.) Velocities are expressed in units of the rms velocity of atoms in the gas, force in units of the gas pressure on unit area at rest, unit distance being the semi-major axis of the grain.

Fig. 1. Geometry of the Monte Carlo model.

The operations carried out by the Monte Carlo program may be described briefly as follows. Let the steady force act in the z direction, and assume that the grain at a given time has some instantaneous velocity u2 in that direction. Imagine a cube of edge length 2 surrounding the grain, as in fig. 1, and moving in translation with it. Within the cube the grain is rotating freely in a manner determined after the last collision. The symmetry axis A and the angular velocity vector 9 are precessing around the angular momentum J at a rate very high compared to the average collision rate. For each successive “time interval” a suitable random number routine decides whether an atom

106

E. M. PURCELL

of the gas has entered the cube, and if so, where. These atoms must be drawn from a velocity distribution, relative to the cube, which is anisotropic because of the grain velocity uz. The atomic velocity distribution is isotropic in the frame of reference of the gas and is a “pseudo-Maxwellian” distribution designed to be closely equivalent to a Maxwellian distribution for most transport processes, but easier to handle in the computation *. An entry into the cube having occurred, the trajectory of the atom is projected to see if it hits the rotating spheroid. If it does, it is assumed to stick at the point of impact, the resulting angular and linear momentum increments are computed, leading to a new velocity uuz,a new angular momentum vector, and a new angular velocity vector for the grain. The only approximation here, other than purely computational approximations, is the assumption that the momentum transfer is impulsive. It is necessary to take the rotation of the grain into account while projecting the trajectory. The resulting bias of the impact probability, though slight, is essential if the system is to achieve the correct equilibrium distributions **. The program also generates “evaporation” events at an average rate adjusted to equal the average condensation rate. An evaporation event is equivalent, except for the sign of the impulse, to a condensation event in a gas whose velocity distribution is isotropic in grain-centered coordinates and whose temperature is that of the grain. The departure of an atom is assumed to be uncorrelated with its arrival, whether because of migration of the atom, because it is a different atom, or merely because the grain has turned into an uncorrelated orientation in the interval between arrival and departure of a specific atom, Only a modest change in the program would be needed to specify specular reflection instead of sticking with subsequent evaporation, but this has not yet been tried. During a run with specified parameters, the program keeps track of cumulative averages of the z velocity of the grain Uz, the translational kinetic l7 the rotational kinetic energy of energy of the grain in the z coordinate +,, the grain, &, and two alignment measures defined as follows: Let # be the instantaneous angle between the z direction and the symmetry axis A of the grain (major axis of a prolate grain, minor axis of an oblate grain). Let 0 be the angle between the z axis and the angular momentum impacts 0 is constant, of course). We define __QA=3(3coss$1) * The distribution

in velocity,

and has the property

that both 5_lvrmsand F/u3~‘m8agree closely with the corresponding

ratios for a Maxwell

distribution. ** In the “coarse

is composed

atom”

Monte

ness of the momentum

transfer

fluctuation

nonsense

consequences.

can make

of two S-functions

vector J. (Between

Carlo program,

in a typical

collision

approximations

based on the small-

must be used with caution.

of such an approcimation,

with

possibly

A rare

disatrous

ON THE

ALIGNMENT

OF INTERSTELLAR

DUST

107

and QJ = +(3 toss 8 -

1)

(5)

as measures respectively of the average grain axis alignment, and the average angular momentum alignment. The bar denotes a time average over the whole history of the grain. This always spans so large a number of relaxation times that the starting conditions are inconsequential. The statistical uncertainty in the cumulative averages QJ and QA is estimated by examining the fluctuations in these quantities during the course of the run. A run thus yields, along with QA and QJ, a “predicted” standard deviation for each. The routine for this was tested by comparing the actual standard deviation in 12 runs with identical parameters against the prediction generated in each run. The required computing time is best indicated by one or two examples. In a certain run an oblate spheroid of axial ratio 1 : 5 was driven at a terminal velocity about 1.35 times the rms atomic speed. The ratio of atom mass to to grain mass had been set at 0.025. In a run of 7 minutes on an IBM 7094 about 14000 impact events and 14000 evaporation events were included, yielding Q J = -0.19 with standard deviation 0.01 and QA = -0.042 with standard deviation 0.006. In a comparable run of 7 minutes on a prolate spheroid of axial ratio 5 : 1, with the same mass ratio 0.025, only about 4500 impacts and 4500 evaporations were processed. The terminal velocity was 1.5, and QJ = -0.18 with predicted standard deviation 0.016, while QA = +0.077 with predicted standard deviation 0.007. In this case much of the computing time was spent tracing trajectories which missed the slender grain. The program is poorly adapted to thin needles, an efficient program for which would require a different structure. The “flake” limit presents no difficulties, and indeed is reasonably well reprseneted even by the 1 : 5 oblate spheroid, which does not differ significantly, either in the surface exposed to bombardment or in the eccentricity of its spheroid of inertia, from a disk of zero thickness. Perhaps the most stringent test that can be applied to a program of this kind is to run it with the grain temperature set equal to the gas temperature and the steady force turned off, and see if the rotational energy and the translational energy of the grain settle down near the values that correspond to equipartition. The present program tends to produce energy values just a few percent low. I suspect the bias arises from the neglect of the x and y velocities of the grain in the determination of impact, a simplification of the calculation which should have little effect on the averages we are interested in. 3. Alignment by streaming: Monte Carlo results. yield directly the relation of drag to grain velocity

The Monte Carlo runs for high grain velocity

E. M. PURCELL

108

as well as low. On the other hand, for a spherical grain moving with speed zc through a gas in which the atoms all have the same speed v and stick on impact, the drag is easy to calculate exactly. It is given by

F=4rra2Po+(l + $), (6)

379

Fdxa2Pa$;+z-;+g-&

(

>

,

a42 v.

Here a is the grain radius and PO is the gas pressure, +ppgssv2. This could be integrated over a Maxwellian distribution of v but it is more instructive to apply it to the pseudo-Maxwellian distribution used in our Monte Carlo model. If we do so, we obtain the solid curve in fig. 2, where the drag is expressed in units of the gas pressure PO times the surface area of the

THEORETICAL

0

MORTE CARLO SPHERES

() MONTE CARLO PROLATE 0

MONTE CARLO OBLATE

SPHEROIDS SPHEROIDS

1.0

0.5 GRAIN

VELOCITY

Fig. 2. The relation between drag and speed for a grain moving through the gas with average speed 26which is expressed in units of the rms speed of the gas atoms. The curve is the relation of eq. (6)) applicable to any convex body with orientation isotropic on the average. The points are obtained from Monte Carlo runs, ZJbeing the average grain velocity observed in a run with a specified driving force.

ON THE ALIGNMENT

OF INTERSTELLAR

DUST

109

sphere 47~9. The grain velocity is measured in units of the rms velocity of the gas atoms. The circles are the results of Monte Carlo runs on spheres, in which, of course, the drag was specified and the grain velocity was “observed”. If there were no alignment, eq. (6) ought to hold for any spheroid, in fact for any convex body if we replace 47~22 by the superficial area of the body. For in this gas of totally independent atoms, the force on any patch of surface of a convex body at a given velocity relative to the gas, depends only on the orientation of the patch, which will be isotropic on the time average in the absence of alignment. On the same graph a number of Monte Carlo points for both prolate and oblate spheroids have been plotted, with the drag normalized, as for the spheres, by dividing by the superficial area of the spheroid times the gas pressure. The points fall quite close to the curve, but somewhat below it at high velocity. This is to be expected, for the alignment of both prolate and

Fig. 3. Alignment of angular momentum and figure axis, for grains of various shapes driven at various speeds.

E. M. PURCELL

110

oblate spheroids is such as to reduce the time-average which the high velocity drag mainly depends.

frontal

area, upon

The agreement of the Monte Carlo grain velocity averages with the derived drag-speed relation involves no adjustable factors and thus provides a further check on the Monte Carlo program. The results of the Monte Carlo runs are most easily surveyed in a plot of the observed alignments against the grain velocity, expressed in units of the rms speed of the gas atoms. The force needed to establish any particular grain velocity can always be inferred with sufficient accuracy from the drag-speed curve discussed above. Of course, it is the axis alignment QA that really concerns us but it is instructive to show the angular momentum alignment, also. Both are plotted, for a selection of representative cases, in fig. 3. In every case QJ is negative, corresponding, as we expected, to a preference for transverse angular momentum. Therefore -QJ has been plotted in fig. 3. QA is positive for prolate spheroids, negative for oblate spheroids. We show results for prolate spheroids of axial ratio 5 : 1 and 2.5 : 1, and for oblate spheroids of axial ratio 1 : 5. The vertical error bar is twice the predicted standard deviation computed for that Q value in the Monte Carlo run that produced it. The grain temperature was set at 0.2 in all these examples, except for the points labelled W and C which will be discussed separately. In general the axis alignment is smaller than the angular momentum alignment by a factor of Q to 4. Clearly, grain velocities not much less than 1.0 are needed to produce any significant alignment. For a given grain velocity ~4 the angular momentum alignment Q J appears to be nearly independent of the shape of the grain. A single value of Q J for a sphere has been included in fig. 3. Many runs with spheres have been made to check the program in one way or another, and the J alignment always behaves much like that of the other spheroids. In the velocity range above 1.5 there does seem to be a slight but systematic increase of Q J with decreasing prolateness. The axis alignment QA, on the other hand, shows a marked shape dependence in the contrary sense ; the more prolate the grain, the better the alignment. That was to be expected for it is the eccentricity of the ellipsoid of inertia which is responsible for the correlation between the direction of the figure axis and the direction of the angular momentum. In the limit of the needle, or infinitely slender grain, a simple relation connects QJ and QA. In that case the needle must always be perpendicular to the angular momentum and revolving around it, and it is easy to show that, whatever the angular momentum distribution may be,

QA = -~QJ. The results of two runs with 10 : 1 prolate spheroids are plotted

(7) in fig. 3.

ON THE

ALIGNMENT

OF INTERSTELLAR

DUST

111

In each case the relation of eq. (7) is very well satisfied, which suggests that we have reached an adequate representation of the limiting case of the needle. The influence of grain temperature is demonstrated in the points marked W and C, which are results for the 5 : 1 grain with the grain temperature set respectively at 1.0, the gas temperature, for the point W (“warmer”), and 0.111 for the point C (“colder”). The warmer the grain, the more the alignment

is disturbed

by the recoil momentum

from the evaporating

atoms.

The effect of an orientation-dependent force was explored in several runs. The dependence was that which would arise from the orientation dependence of the optical cross section of a grain driven by radiation pressure, the parameter specifying it being the ratio of optical cross section for light polarized parallel to the instantaneous grain axis to that for the perpendicular polarization. The velocity increment between collisions was made to depend on the orientation of the grain axis A, with due allowance for its precession around J. There was no significant effect on the time-average alignment in the cases investigated. This can perhaps be well enough explained by noting that in the history of a grain a period during which an alignment (causing greater force) prevails is associated, not with higher velocity, but with positive acceleration. There is in this sense a phase shift between cross section variations and velocity variations which can reduce their joint influence on the long-time-average alignment to an effect of higher order. However, neither my understanding of this interaction, nor the Monte Carlo exploration of it is satisfactorily complete. Some high velocity cases, in particular, need to be looked at carefully. 4. Alignment of alignment of would be needed several authors,

by stream&g: some astrofihysical im$ications. The degree grains of some specified electromagnetic asymmetry which to explain the observed polarization has been discussed by notably by Professor Greenberg in a number of papers, in

his review article cited in ref. 1, and in his paper at this conference. From these studies it appears that one can hardly make out with grain alignment, measured by the quantity I have called QA, smaller than a few percent. To recall the quantitative relation between QA and the maximum polarization attainable, suppose it is the z direction with respect to which grains are aligned and consider light traveling in the x direction. Let (~11be the optical cross section of the grain for light with electric vector parallel to the symmetry axis of the spheroidal grain, and uI the cross section when the electric vector is perpendicular to that axis. Let a2 denote the average cross section per grain for the light in the x direction with its electric vector in the z direction and or, the average cross section for light polarized in the y direction. If the grain axes were all perfectly aligned parallel to .z, the case characterized by QA = + 1.O, we should have a2 = aI1and by = ol. For any

112

E. M. PURCELL

distribution of grain axes with rotational symmetry around z we have (~2= a,+(1

+ ~QA)+ a,.$(2- ~QA),

q, = q.+(l - QA) +

a;&(2 + QA).

(8)

A suitable measure of the efficacy of polarization is the ratio (u~-u~)/(c~+c+,) which, from eq. (8) is

Of course, the amount of polarization actually introduced by passage through the medium cannot be discussed without bringing in the amount of extinction. The extreme values of QA are + 1, for polar alignment, and -&, for equatorial alignment. The greatest disparities in possible 0 values are represented by needles and flakes of zero thickness. It is interesting to consider the possible combinations of these extremes. For needles we have u, = 0 and

~QA

02 -

uy

cz +

‘TY =

~+QA'

(10)

while flakes, with ~11= 0, give G -

ul/

a2 +

94

=

-~QA

(11)

4-QA'

The streaming mechanism, as we have seen, leads toward polar alignment for needles and equatorial alignment for flakes, so that, if the alignment were in each case perfect, we should have 02 -

uy

UZ+ au

= 1 (needles), = g

(12)

(flakes),

Magnetic alignment by the Davis-Greenstein mechanism, on the other hand, leads to equatorial alignment of needles and polar alignment of flakes, which if perfect in each case would give 02 - or/ = - 1 uz + ‘Ty

= - 1

(needles),

(13)

(flakes).

Equation (12) suggests that if the alignment is of the kind produced by streaming, flakes may be under an a @ori handicap as agents of polarization. Of course we are actually concerned with relatively small QA values. But we notice in eqs. (10) and (11) that (uZ - ~~/)/(a~+ uy) reduces for

ON THE

ALIGNMENT

OF INTERSTELLAR

DUST

113

small QA to $QA for needles, and to -~QA for flakes. This should be kept in mind in comparing

the Monte Carlo results for prolate and oblate grains.

There remains the question whether, in the interstellar medium, there are significant regions where the dust may be expected to acquire the requisite velocity with respect to the gas. A velocity ratio u/v of the order of 1 seems to be needed. If it is to be steadily maintained against the drag, velocity 1 requires, according to fig. 2, a driving force a bit larger than PO times the surface area of the grain, a conclusion which is practically independent of the size and shape of the grain. Whether such a force can be provided by anisotropic radiation is a question which involves only the ratio of anisotropic radiation pressure to gas pressure and the ratio of the optical cross section of the grain to its surface area. In this subject the ratio of optical cross section to geometrical cross section is a frequently occurring ratio, usually denoted by Q. Taking the superficial area of the grain as 4 times its projected area, our criterion reduces simply to QPr&d 2 4Pgas.

(14)

The gas pressure in a hydrogen cloud with density 1O-2s g/cm3 and temperaour criterion ture 100 K is about IO-13 dyn/cm 2. In that environment would call for a radiation pressure 4 x lo-13/Q. We must expect Q to be smaller than 1, so we are asking for an anisotropic radiation density at least of the same order of magnitude as the average total radiation density, which is around lo-l2 erg/ems. We can hardly expect to find this condition through large regions within the Galaxy. An obvious source of anisotropic radiation pressure on a very large scale is the Galaxy itself, as seen from regions just above or below the disk. The figure given by Allen 6) for the brightness of the disk seen from outside, 5.2 mag/degs, implies a flux of 4 x 10-s erg/s *cm2 from each side of the disk, hence a pressure amounting to 1.3 x lo-13 dyn/cms on a black surface lying above and parallel to the disk. That still falls a bit short, for a gas of the density assumed. Of course the density of anisotropic radiation within the Galaxy is so irregular that an “average” or “typical” value cannot be very meaningful. A figure that can at least be computed in a straightforward way is the rms unbalanced force on black spheres randomly located within a stellar population that is described by the accepted luminosity function, but whose members are themselves uncorrelated with one another in location. Using the luminosity function tabulated by Allen 7), I find that this force on a black sphere of cross section 1 cm2 is F

N

1.o x 10-14

rms JRmin

dyny

(15)

E. M. PURCELL

114

where Rmin is in pc and is the radius of the region around the test body from which stars must be excluded to keep the mean square force from diverging. For the process we are concerned with, the characteristic distance is about 1 pc, which is the distance the grain travels in one damping time if u = 1. Rmin must therefore be taken at least that large, which leaves us again with a force about one order of magnitude too small to be interesting. Of course, he assumption that the stars are randomly distributed any clustering will tend to enhance the rms force.

is most unrealistic;

If Q, the ratio of optical cross section to projected area, were substantially larger than unity, it would change the outlook. The best “grain” for the purpose might be a thin conducting filament, half a micron long, say, and a tenth of that in diameter. A good metallic conductor of that shape and size, at 20 K, would be a reasonably well matched dipole antenna with an absorption cross section several times its projected physical area. Wherever there is enough radiation pressure to drive the grains through the gas and bring the Gold alignment mechanism into play, the trajectories of the grains may be constrained by the magnetic field. We must expect a grain to carry some electrical charge 4. Let pl be the potential in volts of a spherical grain of radius a and density pgra. In a magnetic field of strength B, in gauss, the cyclotron frequency for such a particle is

qB Combining

B

w

wcyc=mc=-’300

(16)

($c) ppraa3c .

this with eq. (2) for the translational

BP

OCy,$-dt= 2.7 X lo--l* * ___.

avp gas

damping time gives (17)

Even if pl is as small as 10-s V, surely a conservative estimate, and if B is as small as 10-s gauss, eq. (17) predicts, for the conditions we have been = 10. The grains will spiral around the field diassuming, that UIcycTdt rection. Only a steady force component parallel to B can bring the grain up to the terminal velocity predicted by the drag-speed relation of fig. 2. Thus grains that are aligned as a result of velocity acquired from radiation pressure will generally exhibit an alignment that is correlated with the direction of the local magnetic field, but in a sense opposite to that associated with the Davis-Greenstein process. A high relative velocity of the dust in the gas is bound to occur, as Gold pointed out in his original proposal, when clouds collide at speeds typical of the macroscopic motions in the interstellar medium. More broadly speaking, wherever the medium can be described as turbulent on a scale of 1 pc, with velocity gradients, on that scale, of the order of lo-l3 s-l, we may expect grain alignment by the Gold mechanism. Here too the magnetic constraint must play a role. Presumably, since the gas and the field are

ON THE

themselves

ALIGNMENT

OF INTERSTELLAR

linked, it will favor the occurrence

DUST

of relative

motion parallel

115

to

the field lines. 5. Magnetic

alipmertt;

the Davis-Greenstein

mechanism.

From one point

of view, Gold’s streaming mechanism and the magnetic alignment mechanism of Davis and Greenstein have much in common. The essential step in each is the establishment of anisotropy, in particular, axial alignment, in the distribution of the angular momentum vectors of an ensemble of grains. For a spheroidal grain, thanks to the eccentricity of its ellipsoid of inertia, an alignment of angular momentum will be associated statistically with an alignment of the symmetry axis of the grain. Both alignment mechanisms work on the transverse angular momentum components, transverse to the grain velocity in one case, transverse to the magnetic field in the other. In Gold’s mechanism, transverse angular momentum is preferentially augmented by collisions; in the Davis-Greenstein mechanism, transverse angular momentum is preferentially redticed by a dissipative coupling of the magnetic field to grain rotation around a transverse axis. The dissipative coupling is provided by the imaginary part of the complex magnetic susceptibility of the presumed paramagnetic grain, that is to say, by the time lag in the establishment of magnetization in a changing field. Rotation of a grain at some angular rate or in a constant magnetic field B, is equivalent, in coordinates fixed to the grain, to the application of a rotating magnetic field, and thus to the application of orthogonal alternating magnetic fields of angular frequency CC)~ and amplitude B. Quite generally, the lowfrequency behavior of the magnetic susceptibility, x = x’ - ix”, of any paramagnetic substance, in the absence of a constant magnetic field, is described qualitatively by the graph in fig. 4. The volume susceptibility at zero frequency is ~0, and for a simple paramagnetic crystal consisting of N orientable magnetic moments per ems, of strength ,u, it would be given by

w-

%

Fig. 4. Typical frequency dependence of the complex magnetic susceptibility of a paramagnetic substance in zero applied constant field.

116

E. M. PURCELL

~0 = N,G/3kT.

(18)

In the neighborhood of o = 0, x”( w ) must be proportional to o. This is a feature quite independent of the details of the system, it follows directly from the Kramers-Kronig relations and the fact that x’ --f ~0 > 0 at o = 0. With increasing o dispersion sets in. The curves in fig. 4 are typical of a simple system with a single dispersion region, or expressed differently, a single relaxation time. If we denote by Coda frequency characteristic of this region, the following approximate relation will hold for low frequencies (19) The frequency C0dis closely related to the width of the paramagnetic resonance which would be observed in a strong constant field. It arises from interactions involving the magnetic spins in the solid. One interaction is necessarily present, the magnetic coupling of each dipole p to neigboring dipoles. The energy involved per dipole pair is of order Nps, because the dipole field falls off as r-3 and the distance between dipoles is roughly N-*. If this is the dominant interaction, the absorption-dispersion frequency Wd will be determined, approximately, by AC()d N N/G.

(20)

For electron spins ~2 N 1O-40erg/ems. Taking N = 1021cm-s as a spin density which might be representative of a crystal containing an atomic percent of a magnetic ion, we find from eq. (20), Wd M 10s s-l. This is very much larger than the typical grain rotation rate, so it is clear that we are indeed concerned with the behavior of x” in the lowfrequency limit. If eq. (18) and eq. (20) both hold, eq. (19) reduces to x”(w) 11 Wo/3kT.

(21)

Equation (21) has a look of rather fundamental generality, some implications of which will be discussed in a later section. For our immediate purpose we shall simply write x”, for any system, as x” = Ko.

(22)

The proportionality factor K completely characterizes the coupling of the rotating grain to the magnetic field. Rotation at angular velocity J2 in magnetic field B evokes a torque L, given by L = KVB

x (B x Jz),

(23)

where V is the volume of the grain. A sphere of density pBraand radius a, rotating about an axis transverse to B will have its rotation slowed with a characteristic damping time Tdm.

ON THE

ALIGNMENT

OF INTERSTELLAR

DUST

117

The dissipative torque constant being BsKV and the rotational inertia being I = %Vpm8us,the damping time is

If K is determined by eq. (21) this implies that 7(jm=---.

2 a2kT 5

(25)

#iB2

With a N 3 x 10-s cm, T = 30 K, and B = 10-s gauss, we have T&, % 4 x lO”s, remarkably close to our earlier estimate of Tdr, the characteristic time for the damping of rotation by gas friction. We are merely reviewing here points made in the original work of Davis and Greenstein, who showed that the consequences of a competition between gas atom bombardment and magnetic damping depend essentially on the ratio of the two damping times, which will be denoted by 6

6dfL=__* Tdm

2KB2 a+

gas

(26)

The temperature of the grain is important too, for dissipation is inseparable from fluctuation. The rotation of the grain is now coupled, by way of the magnetic field B, to the internal degrees of freedom of the grain, the temperature of which Tgra, is firmly fixed by a totally unrelated radiation balance. Even in the absence of any disturbance by gas atoms, this coupling would cause an initially stationary grain to rotate, by torques arising from fluctuations around zero in the total magnetic moment of the grain. Each of the two rotational degrees of freedom thus energized (rotation around B is not involved) would acquire ijkT,, on the average. Bombardment by gas atoms, on the other hand, tends to bring the rotational degrees of freedom into equilibrium with the gas at Tgas. Whether the interaction with the magnetic field tends to reduce the transverse rotation of the grain or to &crease it, thus depends on whether the grain is colder or warmer than the gas. As Jones and Spitzer have emphasized, the Davis-Greenstein process is driven by the difference between Tgra and Tgas. If Tgra = T,, we must expect no alignment while if T grs, > Tgas the alignment will be the reverse of that which we associated with the Davis and Greenstein explanation. Our Monte Carlo model, if it is to represent all of this faithfully, must include both the damping torque specified by eq. (23) and a randomly fluctuating torque which properly represents the thermal fluctuations in magnetic moment through which the temperature of the lattice manifests itself. By computing the average of the damping torque over the grain precession and

118

E. M. PURCELL

incrementing the angular momentum at sufficiently frequent intervals, the damping routine can be simplified without serious error. The fluctuations in magnetization can be represented by sufficiently frequent impulsive additions of angular momentum, of random sign, perpendicular to B. Gas atom impacts are generated by the routine used in the streaming model. The external force that drove the grain is ordinarily turned off. However there is no difficulty in exposing the grain to the simultaneous action of the streaming process and the magnetic process, and a few such runs have been made. There are two parameters to be specified for a magnetic alignment run, in addition to the eccentricity of the spheroidal grain and the ratio of atom mass to grain mass. These new parameters are simply the ratios Tgra/Tgas and 6, defined in eq. (26). For grains of a given shape, all possible examples of Davis-Greenstein alignment, regardless of the details of the magnetic system, are embraced in this two-parameter family of cases. As before, a critical test of the program is the requirement that the average rotational energy levels off near the right value. In particular, if we set Tgra/Tgas = 1, the average rotational energy of the grain ought to approach the average kinetic energy of a gas atom, and this ought to hold for aay value of 6. 6. Magrtetic alignmelzt; Morcte Carlo rewlts. Jones and Spitzer were able to analyze exactly the process of magnetic alignment of the angular momentum of spherical particles. They used the Fokker-Planck equation to derive an explicit formula for the angular momentum distribution function. Although the axis alignment of spheroids is our ultimate concern, as it was theirs, their solution for the sphere provides an excellent test of the Monte Carlo design. To apply it we must integrate their distribution function, eq. (15) of ref. 3, to obtain a formula for our alignment measure QJ. The parameter which determines the distribution is the ratio Tav/Tgas, where T

tTgas+ dT,ra)

=q

&”

(I+4

I

With the abbreviation QJ=

(27)

* 72 = Tav/Tgas, the result obtained

,&

’

2(1 - r2)

for

is

QJ

q2< 1,

cos-l q , 1

(28) QJ=

’

2(1 -

2 + r2)

92

-

J&

ln(r + Jll’

-

‘)I’ r2>

’

The smooth curve in fig. 5 is a plot of eq. (28). Each point with an error bar is the result of a Monte Carlo run for a spherical grain, plotted at the value

ON THE ALIGNMENT

-0.3 0.2

0.3

0.4

OF INTERSTELLAR

DUST

119

0.6

0.8 1.0 1.5 2.0 3.0 4.0 TW‘Tsas Fig. 5. Monte Carlo results for the angular momentum alignment of spheres by magnetic relaxation, compared to the theoretical prediction of Jones and Spitzer. The abscissa is Tav/Tgas on a logarithmic scale, where T,,, as defined by Jones and Spiker, is ( Tges + ST,,,) /( 1 + 8). In these Monte Carlo runs the evaporation temperature was set at Tgas rather than Tgra.

of Tav/Tgas computed according to eq. (26) from the values of 6 and Tgra/Tgas specified in that run. In these runs the temperature of the evaporating atoms, which is assignable in the Monte Carlo program, was set equal to Tgas. This is not realistic, but it corresponds to a probably necessary assumption in the analysis of Jones and Spitzer, that all the gas with which the grain is involved is properly characterized by a single temperature. Had the evaporation temperature been assigned as Tgra, which will be the rule in most of the other runs to be reported, the alignment would have been considerably reduced in every case. There may be a small systematic deviation from the prediction of eq. (28) but the agreement is gratifying. There are no adjustable parameters, and the route to eq. (28) from the Fokker-Planck equation is about as different as anything could be from the gyrations of the “grain” in the computer memory* As a further test of the Monte Carlo design we can examine the behavior of spheroids with grain and gas temperatures equal. Table II shows results for two such cases, with 6 = 1. The alignment is zero to within the predicted standard deviation. The average rotational energy of the grain, as well as

120

its average translational equipartition value.

E. M. PURCELL

kinetic

energy,

are satisfactorily

close

to the

The importance of the grain temperature is clear from the results displayed in table III. A prolate grain was run at three different temperatures, respectively 8, Q and 3 times the gas temperature. Notice that warming the grain from iTgas to iTgas severely reduced the alignment. In these runs the evaporating atoms had speeds appropriate to a gas at Tgra, as they would in real life. To show the importance of this, two of the cases were run with the evaporation temperature specified as Tgas. The alignment is substantially greater, as one would expect. The accomodation of the atom to the grain temperature in the interval between its arrival and its departure (events which would ordinarily contribute about equally to the rotational motion of the grain) in effect reduces the contrast between the grain temperature and the gas temperature. The effective Tgas must be something like Q(Tpas + Tsra). The influence of the coupling parameter 6, which is the ratio of magnetic drag to gas drag for a sphere rotating about a transverse axis, can be seen in fig. 6. A 5 : 1 prolate spheroid was run in twelve different cases comprising six 6 values and two grain temperatures.

Fig. 6. The dependence of alignment on the relative strength of the magnetic coupling, for a 5 : 1 prolate grain, at two different temperatures. A logarithmic scale is used for the values of the parameter 6. Solid bars: Tgra/Tgss = 0.111; hatched bars: Tgra/Tgas = 0.333.

ON THE

Following

ALIGNMENT

OF INTERSTELLAR

the work of Davis and Greenstein,

121

DUST

and Jones

and Spitzer,

one

expects the alignments to be proportional to 8, for small 6, but not to increase much beyond 6 = 1. The limit 6 --f 00 would correspond to removing the gas. As it is only through the gas that the rotation around B is controlled, this would leave the alignment an open question. In the Monte Carlo program, the z component of angular momentum would persist throughout the run at its arbitrary assigned initial value, and QJ would depend on that value. Alignment results for five different shapes are presented in table IV. In these runs, 6 was set at 1 and the temperature ratio Tgra/Tgas at 0.192, approximately the geometric mean of the two temperatures represented in fig. 6. The extreme oblate case represents a flake, for all practical purposes. Whether the 10 : 1 prolate spheroid is close enough to a needle to represent that limit is a much more delicate question. It was suggested in section 3 that for the streaming process the 10 : 1 prolate spheroid behaved much as would a needle of infinitesimal thickness. Magnetic alignment, however, involves one torque constant that is shape dependent and one that is not. For a thin cylinder of radius a and length b > a, the time for damping by gas friction of rotation about an axis transverse to the cylinder is 4pgraa Tdr =

(29)

-9

3p gasv

whereas the time for magnetic axis is perpendicular to B, is

damping

of this rotation,

if the rotational

The ratio is _Tdr = Tdm

16KPa vb2pgas

*

On the other hand, if we compare the frictional and magnetic times for rotation about the cylinder axis, we find Tdr -=p,

Tdm

(31) damping

KB2 qgas

differing from eq. (31) by the factor b2/16a2. One is on shaky ground in trying to infer the alignment in the limiting case from an extrapolation at some assumed constant 8. A rigorous theoretical treatment of that limiting case, which is likely to be the one of practical importance anyway, would be a valuable contribution. The results presented in table V provide a test of the formula proposed by Jones and Spitzer for the alignment of spheroids. Four cases were run,

122

E. M. PURCELL

combinations of contrasting temperatures and contrasting shapes. In each case the evaporation temperature was assigned as the gas temperature and the coupling parameter 6 was set at 1. Under the heading QA(J. and S.) are given the QA values predicted for that case by eq. (19) of ref. 3. No great accuracy had been claimed for that formula but, even so, the conspicuous lack of agreement is puzzling. The formula predicts a slightly lower QA, in absolute value, for the cold grain than for the hot grain, although the temperature ratio is considerably greater in the former case, while in the Monte Carlo runs the cold grains acquired about twice as much alignment as the hot grains. 7. Field strength and grain comfiosition. Owing to the inefficiency of a microscopic particle as a radiator of long-wave energy, the temperature of an interstellar grain is elevated above that of a large black body exposed to the same dilute starlight. From the first study of this question by Van de Hulst a) as well as the most recent investigation of Greenberg 9) it appears that the grain temperature can hardly be lower than 10 K. It follows that Tgra/Tgas can hardly be less, in a typical hydrogen cloud, than 0.1, and may well be somewhat greater. The Monte Carlo results demonstrate that with such a temperature substantial alignment is achieved only if the ratio 6 is not much less than 1. If that is required, we see from eq. (26) that the quantity KBs is pretty well fixed. With a = 3 x 10-S cm, z, = 1.5 x 105 cm/s, and pgas = lo-23 g/cma, the condition 6 = 1 implies

KB2 > i$avpgas M 2 x lo-23 erg*s/cm3.

(33)

Equation (21) provides a formula for K which, though approximate, involves only universal constants and the grain temperature. If it holds, we have

K M-.

7% (34)

3kT,r, With 10 K as the lowest reasonable give us a lower bound on B:

B 2 1O-5 gauss.

grain temperature,

eqs. (33) and (34)

(35)

In this argument some numerical factors have been set equal to 1 without discussion, notably in eq. (20). However, the more detailed analysis by Jones and Spitzer led to practically the same value for the factor which relates K to Tgra. For trivalent iron ions their constant differs from !$/k by less than 10%. The universality of the result reflects the dominant role of the dipoledipole interaction in the system in question. The same formula would apply to ice, which is rendered weakly paramagnetic by the magnetic moments

ON THE

ALIGNMENT

of the protons in hydrogen,

OF INTERSTELLAR

providing

DUST

123

only that it is still the lowfrequency

behavior of x” that is relevant. To settle that question we can compare od given by eq. (20) with the grain rotation rate. For ice, the density of protons is N = 6 x 1022 cm-s and ,us = 8 x lo-46 erg ecms, from which we get 0Q = 5 x 104 s-1. That is more than twice the rms rotational speed of a grain of 0.3 x 10-5 cm radius at 100 K. Actually, cr)d in ice is even a bit larger, as can be ascertained from a more careful calculation taking the lattice geometry into account, or directly from the observed line width in nuclear magnetic resonance in ice. We conclude that there will be no significant difference, in capability of alignment, between pure ice grains and grains containing magnetic atoms such as iron, for grains of this size or larger. Putting iron into ice does not increase K, for the nuclear and electronic contributions are not independent of one another. The interaction between the moments of the protons and the unpaired electrons of the paramagnetic ions suppresses the contribution of the protons to K by increasing their dispersion width (reducing the relaxation time T2, in the terminology of magnetic resonance theory). As for the electronic paramagnetism the same interaction, usually called a hyperfine interaction, contributes a width which, unlike the effect of the electronic dipole-dipole coupling, is independent of the concentration of paramagnetic ions. As Jones and Spitzer have pointed out, this hyperfine interaction, or any other concentration independent interaction, sets a lower limit on the concentration of paramagnetic ions needed to achieve the full K value promised by eq. (34). They estimate that a concentration of 1 atomic percent is close to this limit. Because the dipole-dipole interaction is always present, eq. (34) must be regarded as setting an @per limit on the K value that any paramagnetic system can exhibit. As far as I can see, the limit applies with undiminished severity to the “super-paramagnetic” complex suggested by Jones and Spitzer as a substance which could attain much higher K values and so allow alignment in much weaker fields. They propose a matrix of ice containing isolated ferromagnetic precipitates as inclusions, each so small as to act like a single-domain particle. The spins in one of these particles are tightly coupled and orient as a unit, in effect a giant dipole with the moment NIP, where Ni is the number of magnetic ions of moment ,u in the particle. Suppose the grain contains N, such particles. If the particles behave like free spins, and Jones and Spitzer cite experimental evidence that favors the possibility, the susceptibility of the composite grain would be proportional to Ni(Ni+)2 rather than to NiNp,G as it would be if the NiN, magnetic ions were distributed uniformly through the lattice. This would enormously enhance the static susceptibility. Jones and Spitzer suggest that x” may also be very large. But eq. (34) is no respecter of giant dipoles: the dipole moment doesn’t even appear in it ! The explanation appears to be that Jones and Spitzer

124

E. M. PURCELL

have not taken account of the dipole-dipole interaction between the different particles, the “super-spins”. Clearly the dipoles must be magnetically coupled, and in fact the uniting of the clustered spins has enhanced the dipole-dipole energy by precisely the factor needed to offset the increase in ~0. I can see no reason to doubt that the interaction imposes an upper limit on Ts in the superparamagnetic system as it does in the ordinary system. If this view is correct, superparamagnetic relaxation does not offer a way to circumvent the limit expressed by eq. (34) *. Given a B sufficient to make 6 equal to 1, a much stronger field will not cause much greater alignment. This is plain, for angular momentum alignment, from the rigorous result of Jones and Spitzer for spheres. The Monte Carlo results, as far as they go, confirm one’s expectation that axis alignment ought to behave the same way. The decisive parameters for determining the amount of alignment are the shape of the grain and the temperature ratio Tgra/Tgas. To judge from the results shown in fig. 6, for a grain temperature of 30 K, and even the relatively slender 5 : 1 spheroid, we could not expect axis alignment as great as 0.1 at any field strength. Much has been written about graphite grains since Cayrel and Schatzman 10) p ro p ose d in 1954 that they might form an important component of the interstellar dust. I am not concerned here with the possible role of graphite grains in interstellar processes such as the catalysis of reactions, or even with their contribution to interstellar extinction, but solely with the question whether graphite grains can be magnetically oriented at reasonable field strengths. The remarkable crystal structure of graphite manifests itself in a striking anisotropy of several physical properties, including the electrical conductivity and the diamagnetic susceptibility. If one is looking for a polarizer, such extraordinary properties are tantalizing. Cayrel and Schatzman were able, in a truly elegant laboratory experiment, to orient colloidal graphite flakes by means of an intense magnetic field and to measure directly the resulting extinction and polarization of visible light passing through the suspension of flakes. The orientation was effected by the torque resulting from the anisotropy in the static susceptibility of graphite, with the fluid in which the flakes were suspended providing the damping. In interstellar space a different mechanism must be invoked. Cayrel and Schatzman pointed out that diamagnetic relaxation must occur in graphite, and that the dispersion which must appear in the frequency dependence of the anomalous * Note added in proof: I am now persuaded, after discussion with Professor R. V. Jones, that the criticism expressed above is not valid and that the realization of a very high K value in a composite super-paramagnetic system remains a possibility. An essential feature of that system is the barrier to spin rotation imposed by local anisotropy, which can introduce a spin-lattice, rather than a spin-spin relaxation, at a favorable rate.

ON THE

ALIGNMENT

OF INTERSTELLAR

DUST

125

susceptibility must be connected with interruptions of the motion of the quasi-free electrons responsible for the large susceptibility. In principle, orientation by the Davis-Greenstein mechanism should then be possible. There remained the question of the value of K, upon which depends the magnitude of the field required for orientation. This could be estimated, as Cayrel and Schatzman quite correctly observed, from the average collision rate of the electrons, or its reciprocal the mean free time, and a relation equivalent to eq. (19). Unfortunately, in estimating the average rate, they used a velocity of 2 x 105 cm/s, obtained by applying the quantum relation to particles of volume density 101s cm-s. But the thermal velocity of a classical electron at 10 K is already ten times as large as that, so the figure cannot be relevant. This may explain why these authors arrived at the tentative conclusion that graphite flakes could be oriented in fields as low as 10-s gauss. More recently Wickramasinghe 11) has estimated that an even lower field should suffice, but he adopted an electron @in resonance line width for Od and assumed a static susceptibility inversely proportional to the temperature. Neither assumption is valid for diamagnetism. Access to the relevant relaxation time for diamagnetism in graphite is afforded by the cyclotron resonance, a phenomenon which has been studied in graphite, both experimentally and theoretically, by several workers. It is essentially the line width of the cyclotron resonance that we should use for estimating the Od that will be associated with the dispersion in zero field. The situation is complicated by the anisotropy of the effective mass, but according to an analysis by Lax and Zeiger is) of the experimental results, even the longest relaxation time associated with the phenomenon is less than 10-s s. Let us take this as an upper limit on the relaxation time, corresponding to a lower limit on Wdof 109 s-1. The unusually large diamagnetic susceptibility of graphite for a field normal to the basal plane is 2 x 10-s. From eq. (19) we obtain a limit on x” : f

< 2 x 10-140.

(36)

The paramagnetic particles we have been discussing, on the other hand, according to the “universal” formula for K, eq. (34), may have at 10 K a value of x” as large as x” M 2.5 x lo-13w,

(37)

which is more than 10 times larger. Thus even on the most optimistic estimate, it will require a magnetic field several times as strong to orient graphite, as to orient any paramagnetic grain, including one of pure ice. This does not bar graphite from orientation by Paramagnetic relaxation if it should contain paramagnetic centers. In that case the limit expressed in eq. (34) applies. If there are any graphite flakes with ice mantles being magnetically oriented in interstellar space, it

E. M. PURCELL

126

is the ice, in all likelihood, rather than the graphite, which is responsible for the orientation. Acknowledgement. This work was started when I was a summer visitor at the Joint Institute for Laboratory Astrophysics in Boulder, Colorado, in 1966. The hospitality of J.I.L.A. is gratefully acknowledged. The course of the investigation has been decisively influenced by stimulating and enlightening conversations with T. Gold. TABLE I A comparison of Monte Carlo results for different values of the ratio atom mass/grain mass. The grain is a prolate spheroid of axial ratio 2.5 :l, and the driving force, expressed in units of gas pressure on unit area, is the same in all runs. The total number of impacts and evaporation events included in a run is listed under Events. The grain temperature was 0.2 in units of the gas temperature. The figure following j-, for QJ and QA is the predicted standard deviation. Mass ratio

Events

I&

0.0125 0.025 0.050

36375 36277 18394

1.40 1.43 1.43

Computing time (min)

QJ -0.186 -0.192 -0.183

& 0.015 & 0.009 f 0.010

0.049 f 0.005 0.056 f 0.004 0.054 & 0.004

13.5 13.7 6.9

TABLE II Results for equal grain and gas temperatures. The average rotational kinetic energy & and the average translational kinetic energy Et are normalized with respect to the average kinetic energy of a gas atom. TgralTgas = 1 and 6 = 1 for all runs.

QJ

QA J% &

4 : 1 prolate

1:4 oblate

+0.007 & 0.020 -0.002 & 0.008 0.98 1.07

+0.008 + 0.016 +0.001 f 0.005 0.93 1.00

TABLE III Effect of grain temperature on alignment, for a prolate grain of axial ratio 5:1, with&= 1.

TgrslTgas = + QJ

QA QJ OA

TdTms

= Q

TgraiTgas = 3

+0.257 & 0.018 +0.078 & 0.017 -0.139 -0.119 f 0.008 -0.034 f 0.007 +0.041 If evaporation occurs at Tgss we find instead: +0.316 f. 0.020 -0.211 -0.149 f 0.009 +0.068

f f

0.014 0.004

f 0.012 & 0.004

ON THE ALIGNMENT

OF INTERSTELLAR

TABLE

127

DUST

IV

The dependence of alignment on shape. In all cases S = 1 and Tsra/Tsas = 0.192, with evaporation at Tgra. Shape

QA

QJ

prolate 10 : 1 prolate 5 : 1 prolate 2.5: 1 oblate 1 :2.5 oblate 1 : 10

0.267 0.186 0.176 0.138 0.195

& f f f f

TABLE

0.024 0.021 0.014 0.011 0.014

-0.130 -0.086 -0.065 0.032 0.057

& f f f f

0.012 0.009 0.006 0.005 0.008

V

Some Monte Carlo results for axis alignment of spheroids compared with the predictions of the rough analysis of Jones and Spitzer. In each case 6 = 1 and evaporation occurs at the gas temperature. Shape prolate prolate oblate oblate

5: 1 5: 1 1:5 1:5

TgralTzss 0.111 3.0 0.111 3.0

QA W. C.)

-0.149 f0.068 +0.061 -0.023

& f f f

0.009 0.004 0.007 0.003

QA (J. and S.) -0.067 -l-O.072 +0.016 -0.017

.

REFERENCES 1) For a bibliography of this subject and its numerous ramifications one may consult the review paper by Dieter and Goss, Recent work on the interstellar medium, Rev. mod. Phys. 38 (1966) 256, or the even more recent study by J. Mayo Greenberg, Interstellar grains, Ch. 6, in Nebulae and interstellar matter, edited by Middlehurst and Allen (Vol. VIII of stars and stellar systems), University of Chicago Press (1968). The completeness and timeliness of Greenberg’s review will be used as an excuse to omit all but the most specific references from this paper. 2) Davis Jr., L. and Greenstein, J. L., Astrophys. J. 114 (1951) 206. 3) Jones, R. V. and Spitzer Jr., L., Astrophys. J. 147 (1967) 943. 4) Gold, T., Nature 169 (1952) 322. 5) Davis Jr., L., Vistas in Astronomy, ed. A. Beer (Pergamon Press) 1 (1955) 336. 6) Allen, C. W., Astrophysical quantities, 2nd ed., University of London, Athlone Press (London, 1963) 269. 7) Allen, C. W., lot. cit. 238. 8) Van de Hulst, H. C., Rech. A&on. Obs. Utrecht 11 part 2 (1949). 9) Greenberg, J. M., Zoc. cit., 250 ff. IO) Cayrel, R. and Schatzman, E., Ann. Astrophys. 17 (1954) 555. 11) Wickramasinghe, N. C., Monthly Not. Roy. Astr. Sot. 125 (1962) 87. 12) Lax, B. and Zeiger, H. J., Phys. Rev. 105 (1957) 1466.