A micrometeorite erosion model and the age of Saturn's rings

ICARUS 70, 124--137 (1987)

A Micrometeorite Erosion Model and the Age of Saturn's Rings T. G. N O R T H R O P AND J. E. P. C O N N E R N E Y NASA Goddard Space Flight Center, Greenbelt, Maryland 20771 Received March 20, 1986; revised December 8, 1986

The sharp, about 100-km-wide, transition between Saturn's C and B rings is at the inner stability limit of small (micrometer or less) highly charged debris from micrometeorite bombardment of the main ring bodies. The latter vary from about 1 cm to 5 m in radius. In the C ring this charged debris escapes from the ring plane to Saturn along magnetic field lines because of gravitational pull, thus producing a net mass loss. But in the B ring the debris oscillates stably back and forth through the ring plane until reabsorbed by a large ring body. In this model we assume that what is now the B and C rings was initially formed as one ring with the optical thickness of the present B ring. We estimate the C ring net erosion rate and determine the ring age, assuming that the mass influx is small compared with the erosion flux. The erosion rate has been calculated with the use of presently observed micrometeorite fluxes. We also use the best present estimates of the size distribution and total mass eroded by a micrometeorite of a given size and energy. We find that the ring age is between 4.4 and 67 myr. In either case the age is much younger than the 4.5 byr of the solar system. The sharpness of the transition between the B and C rings indicates that the principal mass loss is carried by particles moving at a few meters per second with respect to the parent bodies from which they were eroded. ©1987 Academic Press, Inc.

1. INTRODUCTION

The origin of Saturn's rings has been a subject of much debate for centuries, a debate that has intensified following the Voyager 1 and 2 Saturn encounters of 1980 and 1981. The classical approach is to regard the rings as a relic of the circumplanetary accretion disk that formed out of the primitive Saturn nebula some 4.5 byr ago. In this cosmogonic view, the rings represent a failed attempt at satellite building, a visible, perhaps highly evolved, primordial accretion disk (Alfu6n and Arrhenius, 1976). An alternative approach is to regard the rings as the relatively youthful remains of a much more recent formation event or process, such as the catastrophic disruption of an inner satellite. In this view, the rings need not have endured in much their present form since the early days of the solar system. Indeed, in this view, the rings are evolving relatively quickly, victims of a

plethora of erosional processes and dispersive effects. The cosmogonic hypothesis is attractive in that it requires only the formation e v e n t - - s o m e 4.5 byr a g o - - t h a t ultimately left the Saturn system as we view it today. In contrast, if the age of the rings is small compared to the age of the solar system, an extraordinary creation event must be postulated. If these events are rare, then seeing a short-lived ring system is improbable. Therefore, we are now either very fortunate, or their production is a c o m m o n occurrence. The most serious objection to the cosmogonic ring theories is the difficulty of reconciling a 4.5-byr ring age with observational evidence for, and theoretical expectations of, a very limited ring survivability. The possibility of rapid mass loss and redistribution of mass in the ring system due to micrometeorite bombardment has been ad124

0019-1035/87 $3.00 Copyright © 1987 by Academic Press, Inc. All rights of reproduction in any form reserved.

AGE OF SATURN'S RINGS vanced (Bandermann and Wolstencroft 1969; Cook and Franklin 1970) as a potential threat to ring survivability. High-velocity impact experimental data (Dohnanyi 1969; Gault and Wedekind 1969) will be used in this paper to demonstrate that this threat is far more serious than originally supposed. Another problem involves the radial spreading of ring material due to particle-particle collisions (diffusion). A narrow ringlet of 1-m-sized objects would spread over a characteristic radial distance of - 1 0 0 km in - 1 5 myr (e.g., Harris 1984; Borderies et al. 1984), relatively quickly on a cosmological time scale. A more serious challenge to the cosmogonic ring age is the rapid radial separation of ring material and ring moons caused by the exchange of angular momentum between them. Borderies et al. (1984) quote an evolutionary age of the A-ring~F-ring shepherds system of order 4 to 20 myr (see also Cuzzi et al. 1984). Northrop and Hill (1982, 1983) and Ip (1983a, 1984) identified electromagnetic mechanisms which, together with micrometeorite erosion, are quite effective in removing ring mass, particularly in the C ring and inner B ring. In this mechanism, mass is removed in the form of high (positive or negative) charge-to-mass ratio particles, ranging in size from ions to submicron grains. Such particles are constrained to move along magnetic field lines under the components of the gravitational and centrifugal forces along the field line direction, and the magnetic mirror force. These particles are not stably confined to the ring plane at all radii. Northrop and Hill (1983) identified the inner edge of the B ring at 1.525Rs (Rs --- 60,330 km) with the radial limit of stability ("marginal stability" limit) of highly charged grains, inside of which they would necessarily be lost, falling into Saturn's atmosphere along magnetic field lines. Connerney (1986a, 1986b) has identified a dark band feature in images of Saturn's Northern Hemisphere at a latitude magnetically linked to the inner edge of the B ring. This and other dark bands appearing in magnetic association with ring features

125

have been attributed to increased water influx to the atmosphere from charged erosion products. The dearth of material in the C ring, at r < 1.525 Rs, was interpreted as a consequence of this electromagnetic erosion mechanism, allowing Northrop and Hill to estimate the age of the rings (10 to 100 myr) within uncertainties in the micrometeorite influx and erosional efficacy. Connerney and Waite (1984) obtained an estimated erosional time constant of - 3 0 myr for Saturn's inner B ring by considering the consequences of an influx of water on the planet's ionospheric electron density. The much reduced ionospheric electron densities that occur at latitudes magnetically linked to the rings (Kaiser et al. 1984) was attributed to the preferential erosion of the inner B ring via Northrop and Hill's mechanism. The estimated erosion flux ( - 2 × 109 water molecules cm -2 sec -I) obtained by Connerney and Waite (1984) depends only on the photochemical model of Saturn's ionosphere, so the -30-myr erosion time constant so obtained is independent of the details of the erosion process (and independent, too, of all other estimates of ring lifetime). We employ in this paper a much more refined and detailed model of the ring erosion than that of Northrop and Hill (1983). The consequences of impact erosion of the rings via micrometeorite bombardment followed by electromagnetic removal of some of the products are calculated using the Morrill et al. (1983b) estimates of the micrometeorite flux. From this we obtain improved estimates of the lifetime of the rings (4.4 to 67 myr). From the radial dependence of the optical thickness at the C - B ring boundary, we identify submicron grains (not plasma) as the principal mass loss carriers. However, mass transport in the form of ions may be important elsewhere in the rings, particularly in the B ring (e.g., Ip 1983a). II. E R O S I O N M O D E L

The gross structure of Saturn's rings is best described with reference to Fig. I, a

126

NORTHROP AND CONNERNEY 1.5245Rs INNER EDGE OF B RING

2.5

=

1,5

70000

80000

90000

I00000

I10000

120000

130000

140000

RADIUS (kin)

FIG. 1. The normal optical thickness of the rings at ultraviolet wavelengths, obtained by the Voyager 2 Photopolarimeter investigation.

plot of the optical thickness of the rings as a function of radial distance from Saturn. These data were obtained by the Photopolarimeter investigation on Voyager 2 (Lane et al. 1982; Esposito et al. 1983). Similar data were obtained by the Radio Science investigation (Tyler et al. 1981 ; 1982). Such a plot does not do justice to the many interesting small-scale features in the rings. Indeed, our erosion model describes one process which we believe is responsible for some gross structure observed in the rings. We ignore other processes that may be important elsewhere, and ignore much structure which is not presently understood. The structure we address here is the low optical thickness of the C ring, which in our model has eroded from an initially optically thick ring like the B ring to its present condition; the sharpness of the B-C ring transition, which can be accounted for by a low (on the order of meters per second) random velocity of the submicron charged grains; and the gross variation of optical thickness, r, with radius in the B ring, characteristically - 1 (at optical wavelength) in the inner B ring (r < 1.63Rs) and ~ ~> 2 at greater radii. This change in ~ at - 1.63Rs has also been attributed to the excavation of mass by impact ionization of ring material and subsequent loss of high charge to mass particles to Saturn's atmosphere (Northrop and Hill 1982; Ip 1984).

Most of the mass throughout the ring system, with the exception of the tenuous E ring, resides in objects of considerable size. Marouf et al. (1983) find that the ring particles are distributed in size according to an inverse 3.3 power in radius from approximately i cm to ! m, i.e., the number of spheres per unit volume with radius in da is A(t) a-33da, where a is the particle radius and A(t) is a coefficient shown here with an explicit time dependence. Beyond - 5 m radius there are considerably fewer particles than expected by extrapolation of the power law relation. If the radius of each particle in a power law distribution of radii decreases at the same rate, as will occur with erosion, the power law will not remain power law. Thus gardening processes must be maintaining the power law during the gradual ring erosion. We have therefore let A depend on time, rather than allowing the form of the distribution to evolve away from a power law. Since electromagnetic forces become appreciable only for particles of radius ~<1 t~m, it has been customary to neglect electromagnetic processes in describing the evolution of the rings. Grun et al. (1984) have reviewed the charging of dust and the electromagnetic consequences. It is becoming increasingly clear that the rings are dynamic entities, continually transferring mass in a process of erosion and reabsorp-

AGE OF SATURN'S RINGS tion/consolidation with time scales as short as l04 to 105 years (e.g., Durisen 1984b). Micrometeorite bombardment excavates neutral water vapor, ionized impact plasma, and a distribution in size of impact ejecta in the form of small chips or grains. Much of this material is eventually reabsorbed by the rings; but during this (brief) reworking or "gardening," small grains are susceptible to charging and immediate removal via the Northrop and Hill (1982, 1983) mechanism, with a loss efficiency approaching 100% inside of the marginal stability radius of !.525 Rs. Thus, even though the ring mass resides most of the time in the form of large meter-sized objects, mass is vulnerable to loss during the exchange from one object to another. The expanding high-density plasma clouds produced by micrometeorite impacts charge existing submicron particles (produced by previous impacts) and the latter are swept away from the rings if they are inside the 1.525Rs marginal stability radius. A similar production and charging process has been described (Goertz and Morrill 1983; Morrill et al. 1983a) in connection with the origin and evolution of spokes in the B ring. We are neglecting ballistic radial transport of the neutral ejecta, although such effects should eventually be combined in a model with the electromagnetic effects of this paper. The present calculation gives young ring ages, comparable with those of Borderies et al., and Connerney and Waite. Therefore we do not believe that the more complicated model including ballistic transport will produce large changes in our age determination. The motion parallel to the magnetic field of highly charged (i.e., large ]ql/m) particles in Saturn's ring plane is governed by the equation (Northrop and Hill 1983, Eq. (2)) dVII = _ M K B ----. dt m B

127

particle, which is assumed to be launched at the prograde Kepler velocity (i.e., with very little velocity with respect to the parent body from which it was eroded by micrometeorite impact); p is the vector from Saturn's rotation axis to the particle; g is the gravitational acceleration, l-I is Saturn's rotational angular velocity; VII is the component of particle velocity along the magnetic field B; m is the particle mass and B the magnetic field magnitude. Saturn's magnetic field is not quite north-south symmetric about the ring plane. Field lines are directed southward and slightly inward at the ring plane, so that none of the first three terms on the right side of (1) vanish there. The first term is acceleration due to the magnetic mirror force and is directed northward, antiparaliel to B. The term proportional to 1)2 is the component of centrifugal force parallel to B and is also directed northward. The third term is the component of gravitational acceleration and is directed southward along B. At the inner edge of the B ring these three terms are, respectively, -3.4, -20.1, and 37.1 cm/sec 2, for a total of 13.6 cm/sec 2 directed southward. In the absence of any other forces, the particle would not be in equilibrium and would move southward along B out of the ring plane. We therefore introduce the equilibrating force Ell, presumably in the form of an electrostatic force, of sufficient magnitude to cancel these terms at the ring plane. It is necessary that Fii be constant with distance along B, at least near the ring plane, in order that it not affect the radial position of marginal stability. The magnitude of the required /711 depends on radius and vanishes at synchronous radius. Some self-adjusting mechanism is needed to do this, local to the field line in question. Equation (1) has the energy integral

VB B

+~.(p~2÷g)

4-Fll

(1)

where M~ is the magnetic moment of the

V~ 2 +

(MmKB_lp2~2 + t~g - SEll ) = constant

(2)

128

NORTHROP AND CONNERNEY

1.8

.o

<.5.

o

I

+ 60 °

i

+ 30 °

0o

30 °

60 °

LATITUDE

FIG. 2. Potential in which high charge-to-mass ratio particles move along a field line. The radii are those at which the field line crosses the ring plane.

where the constant is a constant of the parallel motion along B, ~g is the gravitational potential, and S is the distance along B. The total potential in which the particle moves is the quantity in parentheses. The variation in the potential with respect to latitude in the vicinity of the ring plane is illustrated in Fig. 2 for a few radial positions (r) in the C and B rings. Particles at r > 1.525Rs reside in a potential well; small velocity perturbations perpendicular to the ring plane result in oscillatory motion about the ring plane and presumably reabsorption. Particles at r < 1.525Rs, having no potential well, are not stable in the ring plane and are lost to Saturn's atmosphere. The distance of the potential maximum from the ring plane increases from zero at marginal stability to 600 km at the outer edge of the C to B ring transition (about 150 km outside of the marginal stability radius). This distance is much less than the B ring's radial extent of 25,000 km, so that a selfadjusting Fii that is variable with radius but quite constant near to the ring plane (i.e., within -600 km) is not unreasonable. At each radial position in the ring plane, there is a minimum velocity, Ve(r), directed along the magnetic field, required to escape from the ring plane. This minimum velocity is shown in Fig. 3 as a function of radial distance in the ring plane, for escape toward the north (N) and south (S). Note that due to the asymmetry of Saturn's magnetic

field about the equator (ring~ plane, lesser velocities are required for escape from the ring plane in the southward direction. Thus particles are preferentially lost to the southern hemisphere if the velocity distribution in the ring plane is symmetric about the equator. Note also that Ve ~ 0 as r 1.525Rs, the marginal stability radius. We assume that the charged dust grains have a (small) random velocity characterized by a Maxwellian velocity distribution with thermal velocity, VT. Particles with VII

KEPLER LAUNCHED

PARTICLE~

Vs

100(3

/

M/SEC Ve

10C

i

.0

2.

c A S O RINGP~C __

I

I

RING-

- -

B RING

-IN

- .......

FIG. 3. The velocity, Ve, parallel to the magnetic field that a particle must have to escape from the ring plane to Saturn. Because of the asymmetry of Saturn's magnetic field, the northward escape velocity, VN, is substantially greater than the southward velocity, Vs, throughout much of the ring plane.

AGE OF SATURN'S RINGS > Ve (r) will be able to escape from the ring plane. The velocity distribution of ejecta from micrometeorite impact on basalt is more power law than Maxwellian (Greenberg et al. 1978). But small particles are observed in the B ring and these may become charged by plasma from an impact. If they are Maxwellian prior to charging, then a Maxwellian would be more appropriate than the power law of the immediate ejecta. The fraction F(r) of grains escaping southward along the field line passing through the ring plane at radius r is F(r) = ~

/v~ e -x2 dx.

(3)

It might be thought that F(r) should be only half of (3) because northward moving grains encounter the higher potential barrier in Fig. 3 and are much less likely to have escape velocity in that direction. They do however reflect if they do not escape northward, and return to the ring plane, where they either become reabsorbed or pass through and escape southward if their velocity -> Ve (r) in that direction. We assume here that they all pass through. So in regions of high optical depth (3) may overestimate the loss flux, but not more than by a factor of 2. If we can obtain the (high ]qI/m) mass flux E due to excavation by micrometeorite bombardment, then the ring erosion rate as a function of radial distance is given by OP(r,t) 0----{--

-

F(r) E

(4)

where P is the ring surface mass density. In the following section we develop an estimate for the flux E and a relationship between ring surface mass density and optical thickness. We then obtain an expression for ~. (r,t), the normal ring optical thickness as a function of radial distance and time, which we can compare with Voyager observations of the inner B ring. The width of the transition (-100 km) between the optically thin C ring and the optically thick B ring will determine VT, the characteristic veloc-

129

ity of the mass carriers which are responsible for the removal of ring mass. In the limit Vx = 0, no particles produced at r > re, the marginal stability radius, can escape, and in the absence of diffusion the C-B ring boundary would be a step function in optical thickness vs radial distance. In the opposite extreme of large VT, almost all particles would escape at any radius and the C-B ring division would be broad. The observed width of the C-B transition requires velocities (Vx) of order of meters per second. This characteristic velocity implies that the dominant mass carriers are charged grains and not ions, because ions would have much greater velocities. III. MICROMETEORITE IMPACT PRODUCTS

Morrill et al. (1983b) have given estimates, and extrapolations from observations, of the micrometeorite flux at Saturn as a function of micrometeorite radius. There is an uncertainty of an order of magnitude in the estimated fluxes because of uncertainty in the effectiveness of gravitational focusing by Saturn. The focusing depends on the micrometeorite interplanetary velocity, relative to Saturn, which is not well established. Grun et al. (1980, 1984) summarize the consequence of a micrometeorite impact on dense targets like basalt as follows (these are all approximations): (1) The total mass ejected from the target is about 5 times the square of the incident micrometeorite velocity (in km sec -1) times the micrometeorite mass. Thus there is a mass multiplication factor. The rings are water ice and may actually undergo much greater erosion than basalt (Morrill et al. 1983b). (2) The ejecta have a mass (radius) distribution of the form mass -L8 (radius -3-4) up to a maximum mass (largest fragment) of about 0.5 times the square of the micrometeorite velocity (in km sec -~) times the incident micrometeorite mass. Above this maximum, the mass spectrum cuts off. (3) Target particles with mass less than

130

NORTHROP AND CONNERNEY

a b o u t 103 times the m a x i m u m ejecta mass (i.e., radius less than 10 times the radius of the m a x i m u m ejecta) are catastrophically disrupted by the impact but the fragments still h a v e a m a s s - ' 8 spectrum, In this p a p e r we a s s u m e that ejecta particles larger than s o m e radius amax (typically 0.1/zm) do not b e c o m e sufficiently charged to influence this high charge to mass stability calculation. Thus only a small fraction o f ejecta is lost f r o m the ring, the rest being reabsorbed. The a b o v e can be formulated as follows: let a~, = m i c r o m e t e o r i t e radius (in cm), v~, = m i c r o m e t e o r i t e velocity at the ring plane (in km s e c - ' ; typically v~, = 30 km s e c - ' ) , p~ = m i c r o m e t e o r i t e mass density (in g cm-3), rn. = m i c r o m e t e o r i t e mass (in g). pg ejecta m a s s density (in g cm-3; 0.92 for ice). mE = mass (in g) of largest ejecta, ae = radius (in cm) of largest ejecta, a~ = radius (in crn) o f smallest target particle, below which catastrophic breakup occurs, m~ is its mass (in g), am.x = radius (in cm) of largest particle that b e c o m e s adequately charged, E = fraction of ejected mass with radius ~
Total ejected mass = 5v~rn, =- f l m ,

mu

=

(4rrl3)a3o.

(5) (6)

?

me = 0.5v;m~, =

(2~r/3)vZua3up. = (47r/3)a3 pg aL = \--~pg /

a.

X = \--~L /

lO(4O4,,\ 2Og / a~

= ~vu-~u]

(8)

(9)

mc = 103mL ac = lOaL =

(7)

\ a~. /

y 00o/,

~c /

o

I

/

° 0°/0°

~(aw,O)

_ o 0 o 0 t~.~ 0 o

oO

0

/

FIG. 4, Attenuation of the micrometeorite flux by the ring.

dent on the ring at angle O to the ring normal (Fig. 4) and ~bL(a~,O) da~ be the flux emerging. L e t d~b be the change in 4, caused by absorption of the flux by the ring particles in dl. cos ® 4, , da~ra2A(t) a 3.3

d4) _

cos 4, 7rA \

0_3

(12)

so

t~L (a~,O) = 4, (au,O)e -'~°/c°s o

(13)

where ~-, -- 7rAL (a~-°3 - a2-°3)/0.3 is the optical thickness for micrometeorites incident in the normal direction. It is also the optical thickness to light because light would be blocked out by the spheres to the same extent as the micrometeorites. The fraction of micrometeorites incident on the ring which impact a ring particle is then

1 - f~L (au,O)/~(a~,O) = 1 -- e -'./c°s o

(I0)

L e t P be the m a s s per unit area of the ring residing in particles of size range (aj = i cm, a2 = 500 cm) seen by the 3.6-cm radar:

(11)

P = OgL

4

da -j~ra3A(t) a -3"3 I

IV. RELATIONSHIP BETWEEN RING SURFACE DENSITY AND OPTICAL THICKNESS EROSION RATE L e t 4ffa~,O) da~, be the flux per steradian of m i c r o m e t e o r i t e s with radius in da~ inci-

4 (a 0"7 - a 0"7) = -~rrpgLA 0.7 (14) where pg is the density of the ring particles (taken as ice: pg 0.92 g/cm3). F r o m (10) and the definition of ~-,, =

AGE OF SATURN'S RINGS 4

a 2° ' 7 - -

a °'7

P = ~Pg (a10.3 _ a~0.3) "gn"

(15)

The erosion rate is given by

OP Ot -

F(r)fl fo da~4~rq~(a~)rn~(a~)E(a~,)

f[ ;2 dO sin O cos ®(1 )

,

e-'./c°s°).

(16)

Equate (16) to the time derivative o f (15). With the use of (5) and (11), O~-, . Ot .

7 (a/03 - a2°'3)F(r) ~ 2{ 2pg ]0.2 . 4 . .(a °7. - . a°'7)pg ~v~ Iwt~ .,-52-~ / r'~,"

fo dat'4rtd~(a~')mt'fau)(~-~) °'6 0=/2 dO sin O cos ®(! - e -~./'°S°)

(17)

(where v~, is in km sec-l). The integral over O varies from 0.053 for rn = 0.06 (about the C-ring optical thickness) to 0.29 for z, = 0.53 (in the B ring). The differential equation (17) for r . is easily solved if ~'./cos O in the e x p o n e n t is replaced by rn. This simplification reduces the integral over ® by a factor of 0,55 at r . = 0.06 and 0.71 at r . = 0.53. Thus the erosion rate is reduced by a factor of between 1.4 and 1.8 by this approximation. On the other hand, neglecting reabsorption of charged dust grains produced inside the ring and of grains reflected from the north with velocity >-V~(r) overestimates the erosion flux. F o r present purposes we neglect all three effects and assume they approximately cancel one another. With r , / c o s ® replaced by ~-~, Eq. (17) has the solution % = In [1 + (e ~'.o - l)e -t/rt,)]

(18)

where %o is the optical thickness at zero time, and

1 7 (a[ °'3 - az°'3)F(r) { 2pg ]0.2 2 T(r) - 4 (ao---~7--aO.7)p---~g ~,v~--~J ~v~

fo

\a~l

0"6" (19)

131

To obtain the mass flux, we evaluate the integral over a~ in (19), making use of the Morrill et al. (1983b) mass fluxes. Their fluxes are those striking a unit area of a flat plate from both sides i.e., 2~rtb. The mass fluxes given in Table I are integrals over the indicated decades in micrometeorite radii a~. In evaluating the integral, we have held the flux ~(a~) constant at Morfill's value o v e r that decade in a~,. Morrill et al. give two flux values for each decade o f grain size, corresponding to two different velocities, v0, o f the micrometeorite at infinite distance from the rings. F o r v0 = 3 km/sec, the gravitational focusing by Saturn increases the flux at the ring plane over the interplanetary flux by a factor of 37 and the velocity vu at the inner edge o f the B ring is 29 km/sec. F o r v0 = 10 km/sec (characteristic of a c o m e t a r y origin of the micrometeorites) the flux is increased by a factor of 4.2 and v~ -- 30 km/sec. There are therefore two numbers in Table I for each micrometeorite radius. The second value, in parentheses, corresponds to the lesser flux appropriate to the lesser gravitational focusing. Other authors have estimated the interplanetary micrometeorite flux. Burns et al. (1980) give 3 × 10 -16 g cm -2 sec -1 at Jupiter for the total mass flux. Assume this to hold at Saturn also and multiply it by the Saturnian focusing factors of 4.2 and 37. These fluxes then b e c o m e 1.3 × 10 -15 and 1.1 × 10 -14 g c m -2 s e c - I , which are ] of the next to last column totals in Table I. Ip (1983b) uses an order of magnitude smaller interplanetary flux. A value o f amax = 10 -5 cm has been chosen because a grain of this size would satisfy the large Iql/rn requirement in the present context at the very modest potential o f - 2 . 8 V. (Northrop and Hill (1983) found that a grain must have at least one electronic charge for each 104 atomic mass units to satisfy the high charge to mass requirement.) Only a small fraction (< 10 -3) of the mass e x c a v a t e d by the impact is plasma, but this plasma charges the grains. To come to this potential, the grain needs to

NORTHROP AND CONNERNEY

132

TABLEI EVALUATION OF THE I N T E G ~ L OVER a u IN (19) FOR amax = 10 -5 cm, p,=3gcm 3, p g = 0 . 9 2 g c m 3, v u = 2 9 k m s e c - t au (cm)

aL (cm)

ac (cm)

10 -4

1.1 X 10 -3

!.1 × 10 2

10 3

1.1 X 10 2

1.1 X 10 I

10 2

1.1 X 10 t

1.1

10 I

1.1

11

1

11

110

Total

2rr4am, ~ (g c m 2 s e e 1) 9 × 10 18 (I 7 (0.8 2.5 (0.3 6 (0.7 9 (I

X X × × X x x x ×

10 ~S) 10 -16 10 r6) 10 14 l 0 14) 10 1~ 10 -15) l0 16 l 0 16)

3.3 × 10 -I4 (3.9 x 10 -15)

27r(am, (amax/a~) °6 (g c m -z s e c ')

2.3 × 10 18 (2.5 4.4 (5.0 4.0 (4.8 2.4 (2.8 9.0 (1.0

× × × × × x x x X

10 I';') 10 17 10 -18 ) l 0 16 10 -tT) 10 17 10 18) 10 19 10 -19 )

4.7 x 10 ~6 (5.6 x 10 17)

F r o m Morrill e t al. (1983b).

be exposed to a plasma of only 1.1 eV temperature, quite reasonable for the plasma from a 30 k m / s e c micrometeorite impact. In the last two rows of Table I, ac ~ 1 cm, so that some of the smaller target particles will be catastrophically disrupted. Although the fragment mass distribution still follows the mass -L8 in a catastrophic collision (Grun et al. 1984), the mass multiplication factor is now changed, the total mass of fragments being limited to the target mass. The contribution listed in Table I to the erosion rate from a~ = 10 -1 cm projectiles has been overstated for targets in the 1- to 8-cm range, and understated for those between 8 and 11 cm. F o r example, at a target radius of 1 cm, the ring erosion rate from such projectiles should be reduced by a factor of 32, and for an 1 l-cm target, it should be increased by a factor of 16. Similar considerations apply to au -- 1 cm. We have not corrected for these catastrophic breakup effects because those a~'s do not make the major contribution to the total erosion. Correcting for catastrophic breakup would increase the deduced ages by at most 5%.

V. R I N G A G E

One can immediately estimate the age of the rings, using the erosion mass flux and the ring optical thickness in the C ring and inner B ring. In doing so, we assume that any C-ring mass influx is small compared to our estimated erosion rate. For example, ballistic radial transport of (neutral) ring material from the B ring to the C ring has been neglected in this model. F r o m (19) and the totals in the last column of Table I, one finds that F(r) T(r) = 1.0 × 1014 or 8.6 X 1014 sec, depending on which value of the integral over a , is used. The erosion age of the rings is t in (18) provided r , is taken as the present C-ring optical thickness (0.06 to 0.16) and %° is the B-ring optical thickness at the outer edge of the transition (rno = 0.53). In the C ring, inside of the marginal stability radius (1.525Rs), F (r) = 1, so T(r) is either 1.0 x 1014 o r 8 . 6 x 1014 s e c , d e pending on the assumed micrometeorite influx. These optical thicknesses are from 3.6-cm radar data. In Fig. 5 we show the radial variation of the optical depth in the vicinity of the inner edge of the B ring at 3.6

AGE OF SATURN'S RINGS

133

2.0

1.5

"l'a. 1.0

0.5

0.0 91600

91800

92000 92200 RADIAL DISTANCE (km)

92400

FIG. 5. O p t i c a l t h i c k n e s s o f t h e r i n g s at u l t r a v i o l e t a n d r a d i o (3.6 c m ) w a v e l e n g t h s in t h e v i c i n i t y o f the inner edge of the B ring.

cm and UV wavelengths. In the B ring, the optical thickness at UV wavelengths is greater than that at 3.6 cm because the latter is insensitive to particles up to - 1 cm in size (Cuzzi e t al. 1984) and because of close packing of the larger bodies. In the C ring, however, fine dust is absent, having become charged and pulled in along the field lines by gravity. That is why the ultraviolet and 3.6-cm optical thicknesses agree in the C ring. We use the 3.6-cm optical thickness of 0.53 in the B ring in our mass erosion calculation because this measure contains all but a negligible fraction of the total ring mass. Table II summarizes our estimates of the age of Saturn's rings obtained using both extremes of the micrometeorite erosion rate and present-day C-ring optical thickness, ~'n. We have considered a range of - 0 . 0 6 to 0.16 for T n in estimating the ring age. ~-, falls to a minimum of - 0. 06, on average, near a radial distance of -89,000 km in the C ring. Throughout most of the C ring, the average optical thickness is - 0 . 1 , and just inside the B - C ring boundary ~'n 0.16. However, for small T . the ring age varies linearly with In Zn, and the result depends only weakly on 7n. The extremes in Table II for ring age are 4.4 and 67 myr which are not incompatible with the 30-myr erosion time constant of Connerney and Waite (1984) or the 4 to 20-myr time con-

stant deduced by Borderies et al. (1984), but well less than the 4.5 × 10 9 year age of the solar system. Our deduced ring age is rather insensitive to other model assumptions. For example, the radius amax and micrometeorite density p~, could be changed by a factor of a few times: it would be possible to take amax as large as 2 × 10-5 cm and this would reduce the age, or one could reduce it below 10-5 cm. But amax enters only to the 0.6 power. The incident micrometeorites could be taken as ice (p~, = 0.92) rather than p~, = 3, but p~ enters only to the 0.8 power (because m~ - p~, in (17)). Another effect that has not been accounted for is caused by the orbital velocity of the ring particles, which is - 2 0 km/sec at the inner edge of the B ring. This increases v~, to as much as 49 km/sec on the forward face of a target and decreases it to 9 km/sec for micrometeorites striking the back side. The erosion, being proportional T A B L E 1I RING AGE C-ring T (see)

C-ring 7n

t(sec)

t(years)

High micrometeorite flux

1.0 × 1014

0.06 0.16

2.4 × |014 1.4 × l0 I4

7.6 x 106 4.4 x l06

Low micrometeorite flux

8.6 x 1014

0.06 0.16

2.1 × 1015 1.2 × l013

6.7 × 107 3.8 × l07

134

NORTHROP AND CONNERNEY 1.0

"l'~, 0.5

.......

~

t/T = 0.8 t/T = 1.4 0.0

91600

I

W'~'f

y-

t/T = 0.4

V

V

LIt

,'~/-,

U

. . . . I

91800

I

I

I

92000 RADIAL DISTANCE (krn)

I

92200

I

92400

FIG. 6, Model C-ring optical thickness as a function of radial distance and time ( t / T = 0.4, 0.8, 1.4) compared with the (presently) observed optical thickness at 3.6-cm wavelengths. For each model curve the characteristic random velocity of eroding particles VT = 2.5 m/see,

2 is increased on the front face by a to v,, factor of 2.6 and decreased on the back face by a factor of 10. But the flux striking the front face is increased, and reduced on the back face. A rough estimate of this aberration effect is that the ring ages in Table II should be reduced by about a factor of 2 to account for this. So there are several factors of about 2 in both directions that might affect the estimates in Table II, but it is difficult to see where one could find the two to three orders of magnitude needed to reconcile the ages in Table II with the age of the solar system. VI. RADIAL VARIATION OF RING OPTICAL THICKNESS

To study in detail the radial variation of optical thickness near the inner edge of the B ring, and the evolution in time of the inner edge of the B ring, we need to consider the radial dependence of F(r), the fraction of charged grains escaping the ring plane (Eq. (3)). In the C ring, that is, inside the marginal stability radius, F(r) = 1; all charged grains escape. F(r) varies from 1 to 0 as r increases through the C-B ring boundary and the optical thickness approaches 0.53. The width of this transition is essentially determined by VT, the characteristic random velocity of the charged grains. The radial position of the boundary

is essentially determined by the magnetic field model. Figure 6 illustrates the time evolution of ring optical thickness in the vicinity of the inner edge of the B ring, derived from Eq. (18) with t/T(r) of 0.4, 0.8, and 1.4. This sequence illustrates the erosion history of the rings, ending with the present ring optical thickness at t/T(r) = 1.4. This corresponds to a ring age of either 4.4 or 38 myr, depending on the micrometeorite flux, with ~', = 0.16. The curves in Fig. 6 are obtained using a characteristic random velocity VT = 2.5 m/sec and the zonal harmonic model magnetic field obtained from the Voyager 2 observations (Connerney et al. 1982). A change of more than -0.5 m/sec in Vv results in a conspicuously poorer fit to the data in the vicinity of the transition. The " n o t c h " occurring in the optical thickness midway through the transition is not reproduced by the model. It may be that radial ballastic transport of neutrals is responsible for the notch and other features, such as the buildup of ejecta at ring edges. Durisen (1984a) has demonstrated numerically the appearance of a notch with time due to ballastic transport near a ring edge. The radial position of the C-B ring transition is very sensitive to the magnetic field model used. The Voyager 2 zonal harmonic model used in Fig. 6 allows an excellent fit

AGE OF SATURN'S RINGS to the observed ring optical depth because the theoretical marginal stability radius computed for this model (Northrop and Hill 1983) is 91,967 km, just at the inner edge of the transition region between the C and B rings. With a choice of zn = 0.16 to match the C-ring optical depth just inside the boundary (to the left in Fig. 6), a very good fit to the observations is possible with this model. The Z3 magnetic field model, which was obtained from the combined Voyager 1 and 2 observations (Connerney et al. 1982), has a theoretical marginal stability radius of 91,795 km which is well inside the edge of the C - B ring transition. With this model, the fit to the ring optical depth observations is less satisfactory, assuming again that % = 0.16 (the observed optical thickness just inside the C - B ring transition). The spherical harmonic coefficients (gO, gO, gO) of these two zonal harmonic models differ by only - 1 0 0 nT or less, which represents ~<0.5% of the dipole (gO) term. The 200-kin difference in the computed marginal stability radius demonstrates how sensitive the theoretical C - B ring boundary is to very small changes in the assumed magnetic field model. We have thus far used r, = 0.16 in our detailed fit to the ring optical depth near the C - B ring boundary. If, however, the other extreme rn of 0.06 is chosen as representative of the present C-ring optical thickness (see Fig, 1), the best fit to the C - B ring transition is obtained with the Z3 model magnetic field and a characteristic random velocity VT of --8 m/sec. The value ~-, = 0.06 is found at the inward edge of a linear ramp in optical thickness extending from 90,500 km to 91,970 cm (Fig. 1). There is a similar ramp at the inner edge of the A ring. Both ramps may be caused in some way by the sharp changes in optical thickness, the reason for the sharp changes being different at the inner edges of the A and B rings. We offer no explanation for the ramps. In either case the transition between the C and B ring is well located and indicative of erosion by mass carriers (charged grains)

135

with a characteristic random velocity of a few meters per second. Since ions would acquire velocities of the order of several kilometers per second in impact or photoionization, it is clear that the dominant mass carriers (responsible for the erosion of the C ring, relative to the B ring) are relatively "massive" submicron grains. Submicron grains exposed to an expanding plasma cloud (due to a nearby micrometeorite impact) may well be expected to acquire a (random) velocity of a few meters per second. VII. SUMMARY We have considered an erosional model of Saturn's rings based on theoretical studies of the stability of high charge-to-mass ratio particles in Saturn's ring plane (Northrop and Hill 1982; 1983). Using the best available estimates of the micrometeorite influx and erosion rates, we obtain an estimate of the rate at which mass is broken down into submicron-sized (or smaller) grains. Inside the marginal stability radius, these grains eventually acquire a charge and are lost from the rings, accounting for the erosion of the C ring during the past - 4 to 67 myr, depending on the micrometeorite influx. This estimated age of the rings agrees well with Connerney and Waite's (1984) erosional time constant of 30 myr. The latter estimate results from a consideration of the effect of an influx of water on Saturn's ionospheric electron density and depends only on the photochemical model of Saturn's ionosphere; it is independent of the details of the erosion process. Our estimated age of between 4 and 67 myr also compares favorably with the 4 to 20 myr evolutionary age of the A-ring/F-ring shepherds system (Go|dreich and Tremaine 1982; Borderies et al. 1984) deduced from studies of the transfer of angular momentum between the rings and imbedded moons. All of the above are inconsistent with the 4.5-byr ring lifetime required by the cosmogonic ring hypothesis. Harris (1984) proposes that the present

136

NORTHROP AND CONNERNEY

Saturn's rings and atmosphere. Geophys. Res. Lett. ring system came from the breakup of a Sa13, 773-776. turnian satellite. The satellite was formed outside the Roche limit from the original CONNERNEY, J. E. P., N. F. NESS, M. H. ACOIqA 1982. Zonal harmonic model of Saturn's magnetic accretion disk, migrated inward because of field from Voyager 1 and 2 observations. Nature gas drag, and eventually stopped its radial 298, 44-46. motion when the gas was gone. Then it was CONNERNEY, J. E. P., AND J. H. WAITE 1984. New model of Saturn's ionosphere with an influx of water struck by a meteorite, the resulting fragfrom the rings. Nature 312, 136-138. ments spreading radially and grinding into COOK, A. F., AND F. A. FRANKLIN 1970. The effect of the present size distribution in -105 years. meteoroidal bombardment on Saturn's tings. AsIf this scenario is correct, our work retron. J. 75, 195-205. quires that the breakup by a meteorite in Cuzzl, J. N., J. J. LISSAUER, L. W. ESPOSITO, J. B. HOLBERG, E. A. MAROUF, G. L. TYLER, AND A. fact occurred within the last 67 myr. BO1SCHOT 1984. Saturn's rings: Properties and proWe identify the C - B ring boundary with cesses. In Planetary Rings, pp 73-199. Univ. of Atithe radial stability limit of high charge-tozona Press, Tucson. mass ratio particles, and identify submicron DOHNANYI, J. A. 1969. Collisional model of asteroids and their debris. J. Geophys. Res. 74, 2531-2554. grains (not ions) as the dominant mass loss carriers in the vicinity of the C - B ring DUmSEN, R. H. 1984a. Particle erosion mechanisms and mass redistribution in Saturn's rings. Adv. boundary. We propose that the present-day Space Res. 4, 13. gross structure of Saturn's rings, at least in DUmSEN, R. H. 1984b. Transport effects due to partithe inner B ring and throughout the C ring, cle erosion mechanisms. In Planetary Rings, pp. 416-446. Univ. of Arizona Press, Tucson. is due to the electromagnetic erosion process described herein. We infer that addi- ESPOSITO, L. W., M. O'CALLAGHAN, K. E. SIMMONS, C. W. HORD, R. A. WEST, A. L. LANE, R. B. POMtional structure evidenced in the rings may PHREY, D. L. COFFEEN, AND M. SATO 1983. Voylikewise be influenced by (electromagnetic) ager photopolarimeter stellar occultation of Saturn's mass transport. rings. J. Geophys. Res. 88, 8643-8649.

ACKNOWLEDGMENTS We thank Leonard Tyler and the Voyager Radio Science team, and L. W. Esposito and the Voyager Photopolarimeter team for supplying us with the radio and ultraviolet ring opacities. We also very much appreciate Richard Durisen's care and effort in reviewing this work. His comments have been most helpful. REFERENCES ALFVI~N, H. AND G. ARRHEN1US (1976). Evolution of the Solar System. NASA SP-345. BANDERMANN, L. W., AND R. D. WOLSTENCROFT 1969. The erosion of particles in the rings of Saturn. Bull. Amer. Astron. Soc. 1, 233. BORDERIES, N., P. GOLDRE1CH, AND S. TREMAINE 1984. Unsolved problems in planetary ring dynamics. In Planetary Rings, pp. 713-734. Univ. of Arizona Press, Tucson. BURNS, J. A., M. P. SHOWALTER,J. N. Cuzzl, AND J. B. POLLACK 1980. Physical process in Jupiter's Ring: Clues to its origin by Jove! Icarus 44, 339360. CONNERNEY, J. E. P, 1986a. Saturn: A unique magnetosphere/ionosphere/ring interaction. In Comparative Study of Magnetospheric Systems, pp. 245-251. CNES. CONNERNEY, J. E. P. 1986b. Magnetic connection for

GAULT, D. E., AND J. A. WEDEKIND 1969. The destruction of tektites by micrometeoroid impact. J. Geophys. Res. 74, 6780-6794. GOERTZ, C. K., ANO G. E. MOgFILL 1983. A model for the formation of spokes in Saturn's ring. learns 53, 219-229. GOLDREXCH, P., AND S. TREMAINE 1982. The dynamics of planetary rings. Ann. Rev. Astron. Astrophys. 20, 249-283. GREENBERG, R., J. F. WACKER, W. K. HARTMANN, AND C. R. CHAPMAN 1978. Planetesimals to planets: Numerical simulation of collisional evolution. Icarus 35, 1. GRUN, E., G. E. MORriLL, AND D. A. MENDlS 1984. Dust-magnetosphere interactions. In Planetary Rings, pp. 275-332. Univ. of Arizona Press, Tucson. GRUN, E., G. E. MORFILL, G. SCHWEHM, AND T. V. JOHNSON 1980. A model of the origin of the Jovian ring. Icarus 44, 326-338. HARRIS, A. W. 1984. The origin and evolution of planetary rings. In Planetary Rings, pp. 641-659. Univ. of Arizona Press, Tucson. Ip, W.-H. 1983a. On plasma transport in the vicinity of the rings of Saturn: A siphon flow mechanism. J. Geophys. Res. 88, 819-822. iP, W.-H. 1983b. The collisional interaction of ring particles: A ballistic transport process. Icarus $4, 253-262. IP, W.-H. 1984. On the equatorial confinement of ther-

AGE OF SATURN'S RINGS real plasma generated in the vicinity of the rings of Saturn. J. Geophys. Res. 89, 395-398. KAISER, M. L., M. D. DESCH, AND J. E. P. CONNERNEY 1984. Saturn's ionosphere: Inferred electron densities. J. Geophys. Res. 89, 2371-2376. LANE, A. L., C. W. HORD, R. A. WEST, L. W. ESPOSITO, D. L. COFFEEN, M. SATO, K. E. SIMMONS, R. B. POMPHREY, AND R. B. MORRIS 1982. Photopolarimetry from Voyager 2: Preliminary results on Saturn, Titan and the rings. Science 215, 537-543. MAROUF, E. A., G. L. TYLER, H. A. ZEBKER, R. A. SIMPSON, AND V. R. ESHELMAN 1983. Particle size distributions in Saturn's rings from Voyager 1 radio occultation. Icarus 54, 189-211. MORFILL, G. E., E. GRUN, T. V. JOHNSON, AND C. K. GOERTZ 1983a. On the evolution of Saturn's spokes: Theory. Icarus 53, 230-235. MORFILL, G. E., H. FECHTm, E. GRON, AND C. K.

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GOERTZ 1983b. Some consequences of meteoroid impacts on Saturn's rings. Icarus 55, 439-447. NORTHROP, T. G., ANn J. R. HILL 1982. Stability of negatively charged dust grains in Saturn's ring plane. J. Geophys. Res. 67, 6045-6051. NORTHROP, T. G., AND J. R. HILL 1983. The inner edge of Saturn's B ring. J. Geophys. Res. 88, 61026108. TYLER, G. L., V. R. ESHLEMAN, J. D. ANDERSON, G. S. LEVY, G. F. LINDAL, G. E. WooD, AND T. A. CROFT 1981. Radio science investigations in the Saturn system with Voyager 1: Preliminary results. Science 212, 201-205. TYLER, G. L., V. R. ESHLEMAN, J. D. ANDERSON, G. S. LEVY, G. V. LINDAL, G. E. WOOD, AND T. A. CROFT 1982. Radio science with Voyager 2 at Saturn: Atmosphere and ionosphere and the masses of Mimas, Tethys, and Iapetus. Science 215, 553-558.