Uranus: Predicted origin and composition of its atmosphere, moons and rings

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17 February 1986

URANUS: PREDICTED ORIGIN AND COMPOSITION OF ITS ATMOSPHERE, MOONS AND RINGS A.J.R. P R E N T I C E Department of Mathematics, Monash University, Clayton, Victoria 3168, Australia

Received 23 December 1985; accepted for publication 2 January 1986

Numerical modelling of the gravitational capture and contraction of Uranus" primitive atmosphere with the modern laplacian theory, suggests that the four outer moons consist mostly of frozen methane and have densities -0.53 g/cm 3. Miranda, and any new moons interior to its orbit, contains mostly rock and water ice. with density - 1.35 g/cm 3.

1. Introduction. The January, 1986, encounter of Voyager 2 with Uranus will provide the first direct data on the physical and chemical nature of this planet and its family of moons and rings. Measurements of both the mass and size of the moons will yield the density of these objects, thus providing a handle on their chemical make-up and the early thermal history of the planet. It is the purpose of this letter to present the results of calculations based on the modern laplacian theory [MLT] for the formation of the solar system [ 1 - 4 ] which yield a set of predicted densities for the uranian satellites. According to the MLT, the planetary system and, in turn, the regular satellite systems of Jupiter, Saturn and Uranus condensed from a system of orbiting gas rings that were shed by a gravitationally contracting parent envelope. This envelope, of mass Men v and axial moment-of-inertia coefficient f, disposes of its excess spin angular momentum by shedding mass at its equator in discrete amounts m n (n = O, 1, 2 .... ) at distinct orbital radii R n satisfying the equations R n / R n + 1 = (1 + m n + l / M e n v f ) 2 .

(1)

The abrupt nature of the shedding process is linked to the presence of a proposed supersonic turbulent stress which is some 20 times larger than the gas pressure below the photosurface of the cloud. The turbulent stress has the form (pt u2) = (JpGM(r)/ r, where p is the gas density, M(r) the mass interior 0.375-9601/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

to radius r, G is the gravitation constant and/~ ~ 0.1 is assumed to be constant throughout the cloud [5]. It is created by rising convective elements which originate in the deep interior of the cloud and are driven upwards to supersonic speeds by positive buoyancy forces. The existence of such a large nonthermal stress is supported by observations of a high flux o f mechanical energy in the atmospheres of T Tauri stars [6,7]. The temperatures T n of the gas rings at the moment of detachment from the parent cloud vary with orbital radius R n as Tn = T o ( R o / R n ) .

(2)

The inner rings are thus shed at higher temperatures than the outer ones. On this basis we expect to see a well-defined compositional gradient present in both the planetary and regular satellite systems, with the inner planets/moons being progressively rockier and hence denser than the outer ones [8]. Such a gradient certainly exists in the planetary system and the galilean system of Jupiter and, to a lesser extent, in the saturnian system [9]. A single choice for the parameter/3, viz. 0.107, allows us to simultaneously account for the chemistry of all three systems [4]. 2. The Uranus problem. Two fundamental facets of Uranus prevent the immediate application of the MLT to modelling the formation of the satellite system. First, theoretical studies of the planet's structure

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of mass ~103 M e w h i c h was shed by the proto-solar cloud at Uranus' orbit. Such a ring contains ~ 1 8 M e o~f condensable rock and H 2 0 , NH 3 and CH4-6H20 ices, consistent with the estimated mass ~13 M e o f Uranus' core [ 10]. The random-walk nature of the final stages of the gravitational accumulation of those condensates, however, leads to a core orbit having a non-zero eccentricity and inclination [12]. At the same time, convective spreading and thermal evaporation of the gas ring, which takes place when the protosun passes through its most luminous phase, leads to a massive reduction in the mass of gas which can later be captured by the proto-uranian core [1]. As the specific mean angular momentum (h)gas of this gas is 3% larger than that of the core, its capture by the core during the latter's approach to the orbit intersection will l esult in an atmospheric spin axis which is nearly perpendicular to the orbital axis (see fig. 1). It is possible that the uranian core has a prograde spin.

[10,1 I] indicate the presence of a solid core which make up the bulk of its 14.5 M e m a s s ( M e = earth mass). That is, the H 2 - H e gaseous envelope, whose gravitational contraction we propose led to the formation of the satellite system, makes up only a very small component ( ~ 1 - 2 Me) of the planet's mass. It is very difficult, therefore, to see how the envelope could contract homologously in the presence of such a massive core, shedding gas rings at the observed nearly geometrically spaced satellite orbits. Second, the spin axis of Uranus differs "~98 ° from the orbit axis. According to the MLT, however, the planetary spins should be all nearly prograde with the orbital motion since the mean specific angular momentum of the gas is some 3% larger than that on the circular keplerian orbit of the ring, where the accumulation of solid material takes place [2]. Fortunately, the spin-axis problem is not as severe as it first seems since the orbital planes of the uranian satellites are nearly co-incident with the equator of the planet. That is, a co-genetic origin for the atmosphere and satellites is still indicated, in line with a basic premise of the MLT. The solution to the spin problem probably lies in a non-zero orbital inclination which may have existed between the orbit of the core and the mean orbital plane of the gas ring

3. Turbulent stress and viscosity in the presence o f a core. In the absence of a solid core, the turbulent

stress ~otut2) arising from the motions of buoyantlyaccelerated convective elements assumes the form given earlier, with ~ = 2kft(I 1 - 41)/3 [5]. Here ft is

i

hcore=ho

g a

PROTO-URANIAN J CORE J mass - 1 3M~

CAPTURED ! ATMOSPHERE mass - 1 .SM~

Fig. 1. Schematic illustration of the capture of the primitive Hz-He atmosphere of Uranus by a core which occupies a circular orbit that is inclined to the mean orbit of the gas ring. The magnitude (h)gas of the mean orbital angular momentum per unit mass of the ring shed by the proto-sun of mass M, at radius Ro, exceeds that of the core, namely heore = h o ---Gn/'~-~~ , by 3%. As a consequence, when flae gas is focussed onto the mean radius R o by the core, as the latter approaches the orbit intersection, it is sped up in the orbit. This results in a spin axis for the captured atmosphere which is nearly perpendicular to the orbital axis of the planet. 212

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the mass fraction of moving material, ~k = Pe/P is the ratio of eddy mass density to the mean gas density and k = (k)/r is the ratio of the mean acceleration length k to the local atmospheric depth r. Provided that h = r [13], then both k and/3 are constant throughout the cloud. Consider next the situation where a solid core of radius r 0 is present. Here the atmospheric depth is r - r 0, so ( ~ = k(r - ro) and (pt v2) = ~pGM(r) 1 - ro/r

--7----

(3)

In the MLT, turbulent viscosity ~t = (PtVt ?0/3 is responsible for maintaining a uniform angular velocity throughout the cloud and so ensuring the efficient exchange of angular m o m e n t u m between the inner and outer layers during gravitational contraction. Since ut cc x l / 2 , we have [2]

rlt = KtP [Gg(r)r] 1/ 2 [/3(1 - ro/r)] 3 / 2 ,

(4)

where K t 1 - 2ftl/2(ll - ~Ol). Thus r~t declines sharply near the core boundary. This means that the inner regions of the cloud are rotationally decoupled from the outer ones. To illustrate this point, we note that during quasi-static contraction of a uniform sphere of density p and radius R e which maintains a uniform angular velocity We, equal to the Kepler value (GM/ R3e) 1/2 at the equator, each interior sphere of radius r experiences an azimuthal inertial torque H(r) = r2M(r)coeRe/SRe . Here/~e is the radial velocity of the equator. This torque sets up a differential velocity gradient ra(vea/r)/ar which, in turn, is arrested by viscous forces. Assuming for simplicity that ra(vefl r)/ar = 7 sin 0 t~e/Re, where 7 :> 0 is a constant and (0, ~) are the spherical polar angles, the total viscous torque acting on a sphere of radius r is 27r/tM(r)/~e/ oR e. Balancing the inertial and viscous torques, a nearly uniform angular velocity can be maintained only as long as I)t > 0.1 PCOer2/'y or, using eq. (4), if

~(1 - ro/r ) > (10~/Kt)-2/3 = / 3 ' . Thus it is only those points lying exterior to radius r' = ro/(1 -/3'//3) which can rid their excess angular m o m e n t u m efficiently by the process of mass extrusion to the equator. Eq. (1) now needs to be replaced by

Rn/Rn+ 1 = (j~n+l/~n) 2 (1 + mn+ l /Menvfn+ l ) 2 ,

(5)

17 February 1986

where ~n denotes the moment-of-inertia factor of the material external to the co-rotation radius r n in the cloud of equatorial radius R n.

4. Numerical modelling. Numerical modelling of the gravitational contraction of the proto-uranian envelope was initiated by incorporating the revised stress formula (3) and co-rotation criterion (5) into the computer program that had been developed for core-free clouds [ 1 - 4 ] . A new proto-solar model was evolved with an improved equation for the gas which includes the heavy element mass fraction Z = 0.023. The H and He fractions are 0.7807 and 0.1963 [4]. Choosing the turbulence parameter/3 = 0.107 we obtain a temperature distribution amongst the set of gas rings shed by the proto-solar cloud, which very accurately reproduces the observed chemistry of the planetary system, especially the terrestrial planets [ 14]. The earth is a very important temperature reference. If/3 > 0.108 the earth condenses within the stability field of the hydrated silicate tremolite ( T t r e m ~ 530 K) and acquirez Zoo much water. If/3 < 0.106, T ~ > 580 K and no H 2 0 is incorporated at all. Thus we can safely say/3 = 0.107 -+ 0.001. Next, a model of the proto-saturnian cloud was evolved possessing a solid core of mass 13 M~, equal to the value proposed for Uranus and consistent with models of Saturn's structure [15] and formation [1,16]. The density of this core, which consists of rock, H 2 0 and NH3, is taken to be 6 g/cm 3 [17]. Having set/3 = 0.107, the temperature distribution of the gas rings depends uniquely on the ratio mo/ Men v of the initial gas ring to envelope mass. Mimas is the important temperature marker in the saturnian system [4]. Choosing mo/M¢n v = 7.26 × 10 - 4 , a Mimas density of 1.42 g/cm ~ which coincides with the Voyager-determined central value [9], is obtained. The co-rotation radius r'n is adjusted during the evolution so that the cloud sheds gas rings at the observed mean satellite orbital radii ratio (Rn/Rn+l)Sa t = 1.298. The overall low mass distribution in the Saturn system can be accounted for in terms of a depletion of rock, H 2 0 and NH 3 relative to solar values by 50%. This depletion probably arose as a result of core formation in the protosolar gas ring, prior to the capture of Saturn's envelope. Finally, the evolution of the proto-uranian envelope, 213

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whose mass is taken as ~1.5 Me in line with the assumed core mass of 13 Me, can be modelled. The core density is taken to be 3 g/cm 3 [10,11]. Again it is assumed that the envelope is depleted by 50% in rock, H20 and NH 3 due to core formation, but that CH 4 is highly enhanced as a result of decomposition of the unstable CH 4.6H20 ice that was originally incorporated in the core material. To account for the observed atmospheric mixing ratio [CH4 ] / [H 2 ] = 0.045 + 0.025 [18,19] requires 30% of the core's supply of CH 4 to be outgassed, which is not unreasonable. This implies a CH4 mass fraction of 22% in the atmosphere. We set/3 = 0.107, mo/Men v = 7.26 X 10 -4 and adjust r n so that gas rings are shed at the observed mean satellite orbital radii ratio, viz. 1.456.

the five known satellites are indicated. Several additional gas rings are shed at orbits interior to that of Miranda. The contraction is assumed to be uniform in the interval 3 < , R e / R U <. 30 (R U = Uranus' equatorial radius = 26 200 km), with/3 =/30 = 0.107. Below orbit radius 3 RU,/3 is progressively reduced in a manner such that Te approaches a final value of 60 K. All ring shedding ceases at R e ~ 2 R U. Table 1 details the temperature T n and pressure Pn on the mean orbit of each gas ring, along with the compositional mass fractions of the major condensing species. TCH 4 is the local CH4 condensation temperature. The last two columns show the uncompressed satellite density and total mass of condensate in each ring. Fig. 2 and table 1 indicate that two main classes of moons should exist in the Uranus system. Class I objects, which include Ariel, Umbriel, Titania and Oberon, are large, low-density objects consisting chiefly of frozen CH 4. They contain very little rock, and H20 and NH 3 ices simply because the Uranian envelope is so heavily enriched with methane. Class II

t

5. Results and discussion. Fig. 2 shows a plot of temperature Te versus orbital radius R e at the equator of the contracting envelope. The heavy line gives the temperatures T n of the gas rings at their moment of detachment from the parent cloud. The positions of

150

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i

I

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q

I

I

i

I

I

I

I

m

n

LU I- 50

A R I ~ - - "

-

mcore=13Me,

Pcore=3 g/cm z

...

TITANIA OBERON

[CH,]/[H=]=0.045, ~o=0.107 ol .... 1'5

i

I 3

I

L 5

I

I 7

L

~ I 10

EQUATORIAL RADIUS

I 15

1 20

t 30

(R u)

Fig. 2. Temperature at the equator of the proto-uranian envelope, plotted as a function of equatorial size. The small circles mark the positions of the system of orbiting gas rings shed by the contracting envelope. It is assumed that a ring is shed at each of the present satellite orbits. The broken curves define the condensation temperatures of the major volatiles, computed for the central pressure of each gas ring just after detachment from central cloud.

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0 ~ 0 0 0 0 0 X X X X X X X

0

g-

U

o

0 0 0 0 0 0 0

0 ~ 0 0 0 0 0

777T77~ ~E~EEEE X X X X X X X

17 February 1986

objects, which include Miranda and any moons interior to its orbit are much smaller in size and some 2~A times denser than the class I objects. They condensed above the CH 4 condensation temperature and had to make do with the scant supplies of rock, H20 and NH 3 ice. CH 4 is present only as clathrate. On the basis of the estimated condensate mass in each gas ring, class I objects have radii of order 875 km and class II objects radii ~200 km. These values agree well with the observed estimates [20,21 ]. Several final remarks: (i) In computing Tn, the background radiation temperature T b ~ 43 K of the early sun, whose luminosity is estimated to be ~0.4 times the present value [22], was ignored. Including T b leads to the set of modified ring temperatures/~n shown in table 1. It does not alter the conclusions regarding the satellite compositions. (ii) Compression due to self-gravity is unimportant in all of the Uranian satellites. It amounts to a density change of 0.003 g/cm 3 in class I bodies of radius 800 km and 0.001 g/cm 3 in class II bodies of radius 200 km. (iii) Because class I objects contain so little rock and class II ones are so small, all of the uranian moons should have remained geologically inactive since the time of their formation. That is, the surfaces should be ancient, heavily cratered and stress free. They should also be heavily darkened, or tarry, in appearance because of solar wind and cosmic ray bombardment of the CH 4 ice [23,24]. (iv) Choosing a smaller atmospheric mixing ratio [CH4]/[H2] = 0.02 leads to a set of uncompressed densities 0.535, 0.535, 0.535, 0.536 and 1.344 g/cm 3 for Oberon through Miranda. These are essentially the same as before. (v) The rings of Uranus probably formed from gas shed by the envelope in its final stage of contraction, below radius 2 R U. As such, fig. 2 indicates that they should have originally consisted mostly of CH 4 ice. 1 thank Drs. J.D. Anderson and G.L. Tyler for support and encouragement during completion of this work.

g

N

~

~

"N

Note added. The very low density for the four outer satellites reported here depends crucially on the assumption that the proposed large-scale outgassing

. . . .

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o f CH 4 from the uranian core occurred prior to the phase of satellite formation. If this assumption is incorrect, then the masses of the uranian satellites can be reconciled with the MLT only if the captured atmosphere contains nearly 40% by mass o f elements heavier than He. Such an enhancement o f heavy elements relative to H 2 and He can be archieved naturally by the gravitational settling o f solid grains onto the circular axis o f the gas ring shed by the protosun at Uranus' orbit, where the capture of the atmosphere takes place. Repeating the calculations leads to the set o f condensation temperatures T n = 94, 73, 59, 47 and 44 K for the five main moons, with only Titania and Oberon condensing below the CH 4 ice-point. Thus all the moons have class II composition except for Titania and Oberon, which contain an additional 0.5%solid CH 4. The satellite densities now range between 1.2 and 1.5 g/cm 3, depending on the degree of self-compression and fractionation o f CH 4 clathrate. A most likely set of densities is 1.35, 1.4, 1.4, 1.3 and 1.3 g/cm 3 , respectively, counting outward from Miranda, with Titania and Oberon probably being the most fractionated and hence least dense o f the moons. References

[ 1] A.J.R. Prentice, in: The origin of the solar system, ed. S.F. Dermott (Wiley, New York, 1978) pp. 111 - 161.

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[2] A.J.R. Prentice, Moon Planet. 19 (1978) 341. [3] A.J.R. Prentice and D. ter Haar, Nature 280 (1979) 300; Moon Planet. 21 (1979) 43. [4] A.J.R. Prentice, Earth Moon Planet. 30 (1984) 209. [5] A.J.R. Prentice, Astron. Astrophys. 27 (1973) 237. [6] G.H. Herbig, Adv. Astron. Astrophys. 1 (1962) 47. [7] M. Cohen, Nature 291 (1981) 611. [8] A.J.R. Prentice, Phys. Lett. 80A (1980) 205. [9] G.L. Tyler, V.R. Eshleman, J.D. Anderson, G.S. Levy, G.F. Lindal, G.E. Wood and T.A. Croft, Science 215 (1982) 553. [10] W.B. Hubbard and J.J. MacFarlane, J. Geophys. Res. 85 (1980) 225. [11] M. Podolak and R.T. Reynolds, Icarus 46 (1981) 40. [ 12] K. Hourigan, Ph.D. thesis (Monash University, Clayton, Victoria, 1981). [13] D. ter Haar, Astrophys. J. 111 (1950) 179. [14] J.S. Lewis, Science 186 (1974) 440. [15] R. Smoluchowski, Moon Planet. 28 (1983) 137. [16] H. Mizuno, Prog. Theor. Phys. 64 (1980) 544. [17] W.B. Hubbard, J.J. MacFarlane, J.D. Anderson, G.W. Null and E.D. Biller, J. Geophys. Res. 85 (1980) 5909. [18] L. Wallace, Icarus 43 (1980) 231. [19] K.H. Baines, Icarus 56 (1983) 543. [20] R.H. Brown, D.P. Cruikshank and D. Morrison, Nature 300 (1982) 423. [21] R.H. Brown and R.N. Clark, Icarus 58 (1984) 288. [22] A.J.R. Prentice, Astron. Astrophys. 50 (1976) 59. [23] A.F. Cheng, in: Uranus and Neptune, ed. J.T. Bergstralh (NASA Conf. Publ. No. 2330, Washington, DC, 1984) pp. 541-556. [24] L. Calcagno, G. Foti, L. Torrisi and G. Strazzulla, Icarus 63 (1985) 31.