Mass-radius relationships in icy satellites after voyager

Mass-radius relationships in icy satellites after voyager

ICARUS52, 40-53 (1982) Mass-Radius in Icy after Voyager MARK J. LUPO’ Department of Earth and Planetary Sciences, Massachusetts Massachusetts 02...

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ICARUS52, 40-53 (1982)

Mass-Radius

in Icy

after Voyager

MARK J. LUPO’ Department

of Earth and Planetary

Sciences, Massachusetts Massachusetts 02139

Institute

ofTechnology.

Cambridge,

Received July 22, 1981; revised April 28, 1982 Using improved data for the masses and radii of the satellites of Jupiter and Saturn, models accounting for self-compression effects are presented for the interiors of Europa, Ganymede, Callisto, Rhea, and Titan. For the differentiated models, two different possible scenarios for heat transport are treated, as well as two different compositions for the silicate component. Undifferentiated models are also treated. In each case, the models of Ganymede, Callisto, and Titan show noticeable similarities. It is found that estimates of the ice-rock ratio are dependent upon the assumptions made about the heat transport mechanisms, the rock composition, and on the distribution of rock and ice in the satellite’s interior.

In the earlier work, the objects were assumed to be composed of Hz0 and ordinary chondritic material, which was referred to simply as “rock.” It was further assumed that the melting and differentiation was total and complete, so that the silicates would be found exclusively in the center of the bodies and the Hz0 component would be pure. Second, it was assumed that the satellites’ rocky component was producing heat at a rate consistent with that of ordinary chondritic material that is 4.5 by old. Third, it was assumed that there were no inversions in the temperature and pressure profile, that the body was in a thermal steady state, and that the satellite was spherical. Diurnal and latitude effects were ignored and the surface temperature was taken everywhere as the global mean average value. In this work, the above assumptions will also be made. Objects whose ice : rock ratio is taken as solar, i.e., 60% ice and 40% rock (Lewis, 1972), and whose heat produced by the long-lived radioactive nuclides in the core is removed by conduction will have internal structure as shown in Fig. 1. Figure 1 supersedes Fig. 5 in Lupo and Lewis (1979). We now model the temperature-pressure profile in the ice I stability field as much steeper, so that the profile intersects the ice

A modeling technique was developed by which the bulk density, density and temperature profile, and rotational moment of intertia of an ice-rich body could be computed as a function of the radius, heliocentric distance, and silicate composition (Lupo and Lewis, 1979). Measurements of mass, radius, and, if possible, the dimensionless moment of inertia by spacecraft can be interpreted directly by the application of this model to yield substantial information about the internal structure and the ice : rock ratio of the body. The modeling technique was applied to the mass and radius data of numerous satellites in the Jovian and Saturnian systems. Since that time, Voyager flybys of both satellite systems have significantly sharpened the mass and radius data for these satellites (Table I). Further, the computer program itself has been improved. For the case of a conducting model, referred to in Lupo and Lewis (1979) as an “adiabatic model,” a minor error in the routine had underestimated the slope of the temperature-pressure curve in the ice I region. This paper will therefore supplement the earlier work, and in some aspects supersede it. ’ Present address: Chevron Oil Field Company, La Habra, Calif. 90631.

Research 40

0019-1035/82/100040-14$02.00/O Copyright 0 1982 by Academic Press, Inc. All rights of reproduction

in any form reserved.

MASS VS RADIUS IN ICY SATELLITES TABLE I VOYAGER DATA Radius (km)

4.79 + 0.05’ 14.81 f 0.06a 10.75 ?Y0.034 0.249 * 0.015* 13.457 * O.tKtSb

Europa Ganymede Callisto Rhea Titan 0 Data b Data c Data d Data

Mass (1025 g)

drawn drawn drawn drawn

from from from from

1565 2638 2410 765 2570

+ ? + 2 f

15’ lad Iod 10b ?b

Null (1976). Tyler et al. (1981). Smith et al. (1981). Davies et a/. (1979).

I-liquid phase boundary in the smaller bodies. The profile is nearly flat as it cuts through the liquid water stability field, with a value of only about 2.O”K kbar-r. This is due to convection. The T-P profile then intersects the melting curve in the ice III region (or, in the case of the actual satellites, which possess more rock, the ice V region), and follows the melting curve thereafter.

41

These models will be termed here “conductive.” If large-scale solid-state convection is taking place, as suggested by Reynolds and Cassen (1979), the temperature-pressure curve is flatter. As in the earlier work, the temperature-pressure curves in these bodies will be approximated as isothermal. The internal structure and local composition under this assumption was presented in Fig. 6 of Lupo and Lewis (1979) as a function of the object’s radius. Since present available observational and laboratory data are insufficient to determine the degree to which solid-state convection is taking place in the icy portions of the interiors of large icy satellites, both cases were used to model Europa, Ganymede, Callisto, Rhea, and Titan. Even if such data were available, the presence of small amounts of solutes in the Hz0 regions of the satellites could profoundly affect the viscosity of ice. Table II lists the properties that certain selected models had, selection

Radius (km)

FIG. 1. The local composition of an object which is 60% Hz0 by mass and 40% silicates as a function of distance from its center given its radius. These models have surface temperatures of 103°K. They represent the extreme case of the absence of solid-state convection. Objects smaller than 1200 km are probably not fully differentiated. This figure supersedes Fig. 5 of Lupo and Lewis (1979).

42

MARK J. LUPO TABLE

II

MODELSOFTHESATELLITES-CHONDRITICROCKCOMPONENT Percentage

TS

M

R

P

P

R rock IlMR2

X

rock

Convecting Europa Ganymede Callisto Titan Rhea

103 103 103 93 77

4.796 14.83 10.757 13.458 0.248

1,562 2,637 2,409 2,568 765

3.004 1.930 1.837 I.897 1.321

41,700 81,000 62,500 74,500 3,800

1,398 1,645 1,470 1,581 379

0.346 0.309 0.310 0.311 0.318

0.059 0.260 0.265 0.267 0.320

89.97 49.44 47.89 48.12 33.81

1,402 1,689 1,501 1,633 404

0.345 0.305 0.305 0.305 0.313

0.054 0.230 0.243 0.233 0.265

90.78 53.50 51.00 52.92 40.50

0.390 0.378 0.379 0.379 0.387

-

85.00 43.00 41.80 42.30 34.00

Conducting Europa Ganymede Callisto Titan Rhea

99 105 101 90 74

4.793 14.809 10.746 13.467 0.251

1,563 2,638 2,410 2,566 763

2.991 1.926 1.833 1.903 1.347

Europa Ganymede Callisto Titan Rhea

100 104 103 94 77

4.763 14.810 10.772 13.461 0.249

1,566 2,639 2,410 2,569 764

2.961 1.924 1.837 1.895 1.335

41,700 82,000 63,300 76,000 4,000

Homogeneous 32,000 42,000 31,100 38,600 1,590

-

Nofe. T, denotes surface temperature. M is mass in 10zS grams. R is radius in per cubic centimeter. P is the central pressure in bars. Rrock is the radius at which MR* is the dimensionless moment of intertia. X is the pressure at the rock-ice pressure. Percentage rock is by mass. The order of the listing of the satellites, and Callisto, is deliberate.

being based upon the matchup with the observational data from Table I. It should be noted that these two thermal models are extremes; the actual temperature-pressure curves should lie between these. So far, it has been assumed that the rocky component is ordinary chondritic rock, whose density, p (in g cm-3), is a function of pressure, P (in bars). The following equation of state was used in Lupo and Lewis (1979): p = 3.661 + 6.1 x 1O-6 P.

(1)

When density is plotted against radius for bodies which are 100% “rock” by mass (Fig. 2), we see a discrepancy between the density of 10 and the density of an object of “rock” of the same radius. It was admitted in the earlier work that the principal source

kilometers. p is density in grams the rock-ice interface occurs. I/ interface divided by the central placing Titan next to Ganymede

of error in precise modeling of the satellites is our ignorance of the composition and density of the rocky component. It is not unreasonable to consider 10 a representative of Galilean satellite rock. An equation of state, p = 3.361 + 6.1 x 1O-6 P,

(2)

is assumed for a substance which will be referred to in this paper as “Iorock.” It is assumed to have the same compressibility as ordinary chondritic rock and an uncompressed density not atypical of type 3 and 4 carbonaceous chondrites, although no assumptions will be made here regarding its bulk composition. It will be pointed out, however, that to oxidize the metallic iron to magnetite of what was called “rock” [see Table III of Lupo and Lewis (1979)] merely

MASS VS RADIUS IN ICY SATELLITES

43

TABLE III MODELS OF THE

T,

M

R

SATELLITES-I•

ROCK

P

P

SILICATECOMPONENT Percentage rock

R rock

IIMR2

X

1,456 1,727 1,544 1,660 401

0.362 0.315 0.315 0.315 0.319

0.040 0.252 0.256 0.258 0.318

93.38 52.44 50.90 51.19 36.60

1,460 1,770 1,568 1,700 421

0.363 0.311 0.310 0.311 0.316

0.037 0.224 0.240 0.232 0.270

94.02 56.45 53.36 54.81 41.76

Convecting Europa Ganymede Callisto Titan Rhea

103 103 103 93 77

4.785 14.841 10.761 13.445 0.249

1,564 2,638 2,411 2,570 763

2.986 1.930 1.833 1.891 1.338

37,300 74,400 57,400 68,400 3,580

Conducting Europa Ganymede Callisto Titan Rhea

105 101 102 96 74

4.791 14.820 10.746 13.461 0.252

1,562 2,638 2,412 2,572 765

2.990 1.927 1.828 1.889 1.346

37,350 75,250 58,000 69,300 3,700

Homogeneous Europa Ganymede Cahisto Titan Rhea

101 104 102 93 77

4.789 14.821 10.749 13.457 0.250

1,565 2,638 2,410 2,569 765

2.983 1.927 1.833 1.895 1.331

31,880 41,900 31,800 38,430 1,582

-

0.393 0.379 0.379 0.379 0.387

-

90.00 45.60 43.80 44.70 35.00

Nofe. T, denotes surface temperature. M is mass in 10z5grams. R is radius in kilometers. p is density in grams per cubic centimeter. P is the central pressure in bars. R mk is the radius at which the rock-ice interface occurs. I/ MR* is the dimensionless moment of inertia. X is the pressure at the rock-ice interface divided by the central pressure. Percentage rock is by mass. The order of the listing of the satellites, placing Titan next to Ganymede and Callisto, is deliberate.

reduces the uncompressed bulk density of the rock to 3.592 g cm-3, which is denser than 10. The bulk composition of 10, therefore, is not simply ordinary chondritic rock whose iron is in the form of magnetite, and speculation regarding the composition of Iorock is beyond the scope of this paper. It suffices to point out that when the density of objects made of rock [with Eq. (2) as the equation of state] is plotted against radius, there is a good match with 10. Figure 3 shows the local composition of objects 60% H20 by mass and 40% Iorock as a function of distance from the center, given the body’s radius. If the rocky component of the three icy Galilean satellites, as well as of Titan and Rhea, is taken to be Iorock, Table III is the result, with a model satellite listed for each of the two theorized heat transport mechanisms.

That the large icy satellites are differentiated is not unanimously accepted. Arguments have been presented (Schubert et al., 1981) that subsolidus convection in ice-rich bodies could remove the radiogenic heat from the primordial satellite preventing the melting of the ice and thus the separation of the rock and ice. Since the possibility that these satellites are undifferentiated cannot be overlooked, Tables II and III include sets of models for which it was assumed that the bodies were homogeneous mixtures of rock and ice. The subject of homogeneous ice-rock spheres is of particular interest, since it is not likely that Saturn’s smaller satellites have differentiated (Ellsworth and Schubert, 1982). Table IV shows the densities of such bodies under self-compression. Figure 4 shows the local composition of homoge-

44

MARK J. LUPO

7 6

3.6 -

c” > .= E ; 3.5 Y 2 3.4 -

“lo

Rock” Moon

.

3.3 -

3.2

I 500

I

I

1000

1500

2ooo

Radius (km)

FIG. 2. The bulk density of spheres of rock under self-compression as a function of radius. The lower plot is for “Iorock,” whose density equation of state is given in the text as Eq. (2). Its agreement with the current mass-radius data for IO is much better. For IO, the mass estimate of Null (1976) is used and the radius is from Smith et al. (1979).

1000

1500

2500

3MK)

Radius (km)

FIG. 3. The local composition of an object which is 60% Hz0 by mass and 40% “Iorock.” caption of Fig. 1 for other details.

See the

MASS VS RADIUS IN ICY SATELLITES

45

TABLE IV HOMOGENEOUS SPHERES

P

M

R

P

IIMR2

30 100 200 300 500 750 1,000 1,500 2,000 3,000 5,000 7,500 10,000 12,500 15,000 17,500 20,000 25,000 30,000 35,000 40,000 45,000 50,000

0.0010 0.0053 0.0144 0.0260 0.0548 0.0987 0.1346 0.2153 0.3137 0.5500 1.1347 1.9971 2.8129 3.7388 4.7309 5.7617 6.6101 8.4794 10.566 12.780 15.115 17.536 20.024

120 213 297 361 461 563 622 719 807 963 1,212 1,454 1,618 1,768 1,902 2,021 2,107 2,269 2,424 2,565 2,698 2,821 2,935

1.314 1.315 1.316 1.317 1.319 1.321 1.335 1.383 1.425 I .470 1.522 I.551 1.585 1.615 1.641 1.666

0.398 0.399 0.399 0.400 0.399 0.399 0.396 0.389 0.386 0.387 0.389 0.390 0.388 0.387 0.386 0.386 0.385 0.382 0.380 0.379 0.379 0.379 0.379

P

M

R

I-J

IIMR2

0.0009 0.0052 0.0140 0.0252 0.0534 0.0963 0.1311 0.2080 0.3040 0.5328 I,1004 1.9379 2.7275 3.6144 4.5747 5.5139 6.3835 8.1811 10.183 12.322 14.574 16.902 19.304 21.754

118 210 293 356 457 556 614 707 795 948 1,194 1,432 1,593 1,739 1,871 1,989 2,072 2,230 2,381 2,520 2,650 2,771 2,883 2,989

1.331 1.332 1.333 1.334 1.336 1.338 1.352 1.405 I.444 1.493 1.543 I .576 I.611 1.641 1.667 1.691 1.713 1.761 I.801 1.838 1.870 1.897 1.923 1.945

0.398 0.399 0.399 0.400 0.399 0.399 0.396 0.388 0.386 0.386 0.389 0.390 0.389 0.387 0.386 0.386 0.384 0.382 0.380 0.379 0.379 0.378 0.379 0.378

B

A

1.687 1.733 1.771 1.808 1.837 1.865 1.891

30 100 200 300 500 750 1,000

1,500 2,000 3,000 5,000 7,500 10,000 12,500 15,000 17,500 20,000 25,000 30,000 35,000 40,000 45,000 50,000 55,000

Note. The result of computing the densities of homogeneous spheres under self-compression that are exactly 40% rock by mass and 60% H,O. The thermal model is described in the text and the surface temperature is 77°K. typical of a Satumian moon. M is the mass in 1O25grams, P is the central pressure in bars, R is the radius in kilometers, p is the density in grams per cubic centimeter, and IIMR2 is the dimensionless moment of inertia. A is for “Iorock” as the silicate component. B is for ordinary chondritic rock as the silicate component. Results at 103” and 55”K, typical temperatures for Jovian and Uranian moons, respectively, were not published, but they fall within a percent of these. except the smaller bodies where radii may differ by 4%.

neous bodies as a function of distance from the center given the body’s radius. The thermal profile used for all homogeneous models was a hybrid of the two used above. I assumed that solid-state convection was taking place, but that the motion of the ice-rock mixture was hindered below a certain critical temperature which varied as a function of pressure. Conduction is a slower heat transport mechanism, so the temperature would increase to the critical temperature and convection would prevent this temperature from being exceeded at that given pressure. Thus the pressuretemperature profile would be a regular

conduction curve until it intersected the critical temperature curve, which it would follow from there on. The critical temperature was taken as 60% of the melting temperature . In cases when the conduction curve intersected the critical curve in the ice I stability field, where the slope of the melting curve is negative, the temperature was held constant (the slope of the adiabat is small, about 1” per kilobar) as pressure increased until the critical curve exceeded the temperature of intersection. This is to avoid a temperature inversion. The thermal conductivity of the ice-rock

46

MARK J. LUPO

0

500

1000

1500 2ooa Radius(km)

2500

3ooo

FIG. 4. The local ice phase composition of an object which is 60% Hz0 by mass and 40% ordinary chondritic rock as a function of distance from its center, given its radius. These models have a surface temperature of 77°K and their rocky component is distributed homogeneously.

mixture is taken as the thermal conductivity of ice [see Table IV of Lupo and Lewis (1979)] because the conductivity of ice is greater than that of ordinary rock at the temperatures in which conductivity was the sole heat transport mechanism. Six models are presented for each of the satellites treated (Tables V to IX). The choice of thermal model, rock type, and rock distribution is left to the reader, and those who disagree with my opinion on each issue can still derive benefit from this modeling technique. Assuming that the silicate was ordinary chondritic and the rock distribution was homogeneous, models of other satellites were generated. These are given in Table X. It was pointed out by Smith et al. (1981) that the densities of Titan, Ganymede, and Callisto are similar. In each of the six situations, we see a remarkable similarity between the three largest satellites in several other categories as well. These categories are: percentage rock by mass; the dimensionless moment of inertia ( Z/MR2); and the parameter X which is the pressure in which

the rock-ice interface occurs, divided by the central pressure. This would tend to set these objects apart as a class. When selfcompression is accounted for, Mimas and Dione are shown to have a composition compatible with this class in spite of their lower densities. The other icy satellites of Saturn, whose bulk properties suggest rock compositions of no more than 40% by mass, appear to be another class-if indeed a diverse one. It is surprising that the dichotomy would not be more strongly based on parent planet. It is possible that the solar ice : rock ratio is indeed 3 : 2 and that the democratic loss of rock and ice on the smaller bodies maintains this ratio. At the same time, differentiation in the larger bodies has exposed them to selective loss of ice, since the rock would be in the center of the body and more difficult to remove. This suggestion is supported by the trend within the class of the three largest satellites of an increasing rock abundance with mass. This explanation is contradicted by Dione’s high mass and by the higher of the two accepted mass esti-

1.185 0.948 0.946 0.932

1.176 0.947 0.933 0.932

1,398 1,493 1,503 1,562

1,456 1,501 1,561 1,564

1,491 764 30 0

2,430 847 733 0

P

Rock-ice II Ice II-ice I Surface

Rock-ice 11 Ice II-ice I Surface

+

1,460 1,485 1,564

1,402 1,477 1,507 1,563

Z

0.937 0.932

0.938 0.939 0.932

P

1,345 1,044 0

2,217 1,136 692 0

P

262 262 105

263 261 175 99

T

Melt

100.0 0.0 0.0

100.0 0.0 0.0 0.0

Rock

Conductive

Rock liquid Liquid-ice I Surface

Rock-liquid Liquid-ice I Surface

4

105 1,025 1,365 1,405 1,505 1,545 1,565

105 1,015 1,355 1,405 1,505 1,545 1,565

Z

3.203 3.047 2.887 2.875 2.844 2.681 2.666

3.236 3.045 2.839 2.823 2.788 2.562 2.543

P

3 1,745 17,428 6,967 5,618 2,123 844 0

31,862 17,540 7,150 5,500 2,081 834 0

P

256 199 166 162 151 118 101

256 199 167 161 151 117 100

T

Homogeneous

(Ice VIII) Ice VIII-ice VI Ice VI-ice II Ice II-ice I Surface

(Ice VIII) Ice VIII-ice VI Ice VI-ice II Ice II-ice I Surface

4

Note. For these models, Z is the distance from the center of the sphere in kilometers, and p is the local density in grams per cubic centimeter. In the case of a phase change, this would be the density of the phase nearer to the surface. P is the local pressure in bars, Tis the local temperature in degrees Kelvin, and “Melt” signifies the melt percentage by mass. In regions where liquid coexists with ice, the density is given as the density of the solid-liquid slush in that shell. The column marked 4 is where any phase changes are recorded.

P

Z

Isothermal

TABLE V EUROPAMODELS

1.636 1.620 1.417 1.388 1.363 1.240 1.225 1.208 1.192 1.177 0.946 0.934 0.932

1,727 1,832 1,922 2,032 2,132 2,142 2,232 2,332 2,432 2,532 2,582 2,632 2,638

4

Surface

V.

Ice II-ice -

-

Ice VI-ice -

I

II

VI

VIII

Ice VIII-ice -

Rock-ice -

Rock-ice VIII Ice VIII-ice VI Ice VI-ice 11 Ice II-ice I Surface

for Table

18,711 15,803 13,401 10,934 8,779 8,567 6,881 5,059 3,292 1,572 728 72 0

21,038 18,008 15,253 13,385 12,708 10,470 8,748 6,496 4,682 2,923 1,209 872 0

P

See caption

1.649 1.632 1.617 1.416 1.408 1.382 1.241 1.221 1.205 1.189 1.174 0.948 0.932

1,645 1,750 1,850 1,920 1,950 2,050 2,130 2,250 2,350 2,450 2,550 2,570 2,637

Note.

P

Z

Isothermal

1,770 1,875 1,975 2,075 2,175 2,255 2,275 2,375 2,475 2,515 2,575 2,638

1,689 1,794 1,894 1,994 2,094 2,194 2,254 2,294 2,384 2,494 2,514 2,594 2,638

z

1.314 1.257 I .254 1.139 I .097 0.946 0.943 0.932

1.329

1.368 1.347

-

1.251 1.136 1.087 0.946 0.941 0.932

1.257

1.388 1.364 1.343 1.326 1.311

P

16,796 14,285 12,028 9,873 7,801 6,191 5,811 3,931 2,282 1,717 836 0

18,834 16,190 13,834 I 1,602 9,469 7,413 6,210 5,451 3,767 1,968 1,724 574 0

P

56.8 46.9 38.2 30.2 22.7 15.7 11.3 8.8 100.0 100.0 0.0 0.0 0.0

328 315 303 292 281 273 271 260 256 255 153 101

49.1 39.9 31.7 24.1 16.9 11.3 10.0 100.0 100.0 0.0 0.0 0.0

Iorock

338 325 312 301 290 279 273 268 258 254 255 137 105

+

VI

1

Ice V-liquid Liquid-ice I Surface

Rock-ice VI Ice VI-ice V

Liquid-ice Surface

Ice VI-ice V Ice V-liquid

Rock-ice -

MODELS

VI

Melt

Rock

T

Conductive

GANYMEDE

TABLE

10s 1,005 1,885 2,005 2,105 2,205 2,305 2,325 2,405 2,505 2,585 2,605 2,638

105 1,005 1,885 2,005 2,105 2,205 2,305 2,325 2,405 2,505 2,585 2,605 2.639

Z

2.284 2.243 1.982 1.946 I.916 1.886 1.857 I.726 1.707 I.683 1.411 1.404 1.390

2.292 2.250 1.980 1.942 1.912 1.881 1.851 1.715 1.696 1.672 1.393 1.386 I .372

P

41,831 34,778 17,546 14,781 12,417 IO,01 I 7,565 7,072 5,231 2,903 1,136 649 0

41,931 34,830 17,491 14,724 12,361 9,951 7,515 7,023 5,194 2,881 1,127 648 0

P

136 122 104

168 167 160 152

300 269 199 190 183 176

301 270 199 190 183 175 168 166 160 152 135 122 104

T

Homogeneous

I

II

VIII VI

Ice II-ice Surface

-

Ice VI-ice

1

II

(Ice VII) Ice VII-ice VIII Ice VIII-ice VI -

Surface

Ice II-ice -

Ice VI-ice -

-

(Ice VII) Ice VII-ice Ice VIII-ice

cb

1.206 1.192 1.178 0.948 0.941 0.932

1.614 1.418 1.404 1.381 1.240 1.224 1.210 1.196 1.182 0.947 0.932

2,075 2,175 2,275 2,335 2,375 2,409

1,544 1,589 1,649 1,749 1,839 1,949 2,049 2,149 2,249 2,339 2.411

for Table V

Rock-ice VIII Ice VIII-ice VI Ice VI-ice II Ice II-ice I Surface

14,645 13,551 12,315 10,323 8,611 6,794 5,199 3,653 2,147 819 0

I

Ice II-ice Surface

Rock-ice VIII Ice VIII-ice VI Ice VI-ice II -

4

4,764 3,227 1,727 843 386 0

16,513 13,894 13,410 11,768 9,797 8,659 7,991 6,349

P

See caption

1.624 1.609 1.417 1.397 1.375 I.241 1.235 1.220

1,470 1,575 1,595 1,675 1,775 1,835 1,875 1,975

Note.

P

z

Isothermal

1,568 1,673 1,773 1,873 1,973 2,073 2,123 2,173 2,266 2,412

2,126 2,206 2,258 2,288 2,388 2,410

1,501 1,606 1,706 1,806 1,906 1,976 2,006 2,106

z

1.345 1.329 1.315 1.257 1.244 1.135 1.117 0.947 0.932

-

1.134 1.104 0.948 0.947 0.937 0.932

1.357 1.339 1.324 1,311 1.257 1.253 1.240

P

13,891 11,749 9,822 7,982 6,206 4,550 3,737 3,014 1,768 0

3,662 2,509 1,845 1,431 253 0

15,346 13,098 11,096 9,195 7,373 6,135 5,634 3,986

P

100.0 100.0 0.0 0.0 0.0 0.0

313 302 292 282 273 263 258 257 254 102

43.8 35.6 28.4 21.7 15.5 11.1 9.4 3.8

Melt

38.4 30.7 23.9 17.6 11.3 5.8 100.0 100.0 0.0 0.0

Iorock

257 255 254 203 111 101

320 309 298 288 279 273 269 259

VII MODELS

Rock

T

Conductive

CALLBTO

TABLE

Rock-ice VI Ice VI-ice V Ice V-liquid Surface

Ice V-liquid Liquid-ice 1 Surface

Rock-ice VI Ice VI-ice V -

4

2.197 2.184 2.166 2.142 1.955 1.927 1.903 1.880 1.856 1.699 1.685 1.666 1.385 1.364

1,905 2,035 2,105 2,205 2,345 2,410

1.860 1.697 1.684 1.663 1.376 1.354

2.214 2.200 2.182 2.157 1.962 1.933 1.909 1.884

P

105 605 90.5 1,205 1,485 1,605 1,705 1,805

1,905 2,035 2,105 2,205 2,345 2,410

105 605 905 1,205 1,485 1,605 1,705 1,805

z

9,751 7,129 5,792 3,846 1,163 0

31,736 29,417 26,449 22,349 17,557 15,441 13,604 11,706

9,778 7,139 5,798 3,847 1,160 0

32,035 29,682 26,671 22,513 17,653 15,517 13,662 11,748

P

175 167 162 155 136 102

256 246 232 214 199 192 187 181

175 167 162 155 134 103

257 247 233 215 199 193 187 181

T

Homogeneous

VI

Ice VI-ice II Ice II-ice I Surface

(Ice VIII) Ice VIII-ice -

VI

Ice VI-ice II Ice II-ice I Surface

(Ice VIII) Ice VIII-ice -

4

1.644 1.628 1.613 1.414 1.401 1.376 1.243 1.235 1.219 1.203 1.188 1.174 0.946 0.933

1.631 1.616 1.414 1.406 1.381 1.243 1.238 1.222 1.207 1.192 1.177 0.946 0.934 0.933

1,581 1,686 1,786 1,836 1,886 1,986 2,036 2,086 2,186 2,286 2,386 2,486 2,516 2,568

1,660 1,765 1,835 1,865 1,965 2,035 2,065 2,165 2,265 2,365 2,465 2,515 2,565 2,570

17,641 14,869 13,083 12,438 10,304 8,864 8,323 6,532 4,801 3,122 1,484 678 52 0

19,858 16,973 14,350 13,075 11,992 9,871 8,844 7,937 6,156 4,432 2,758 1,125 642 0

P

Note. The surface

P

z

+

1.377 1.355 1.336 1.321 1.307 1.257 1.247 1.158 1.114 0.949 0.949 0.940 0.933

1.362 1.343 1.326 1.312 1.257 1.253 1.239 1.158 1.122 1.096 0.948 0.944 0.933

1,633 1,738 1,838 1,938 2,038 2,138 2,168 2,238 2,328 2,358 2,418 2,438 2,538 2,566

1,700 1,805 1,905 2,005 2,105 2,175 2,205 2,305 2,335 2,345 2,405 2,435 2,505 2,572

52.5 43.1 34.9 27.3 20.2 13.5 11.3 7.1 3.2 100.0 0.0 0.0 0.0 0.0

324 312 301 290 280 273 269 259 256 256 254 254 146 96

46.4 37.7 30.0 22.8 16.0 11.2 9.4 3.4 3.1 100.0 100.0 0.0 0.0 0.0

Iorock

332 319 308 297 286 276 273 265 256 254 252 213 105 90

VI

models.

Ice VI-ice V Ice V-ice III Ice III-liquid Liquid-ice 1 Surface

Rock-ice -

Surface

Rock-ice VI Ice VI-ice V Ice V-ice III Ice Ill-liquid Liquid-ice I

4

but it does not affect these

16,074 13,688 11,542 9,494 7,522 6,179 5,635 3,843 3,314 3,155 2,220 1,830 856 0

Melt

Rock

T

Conductive

17,690 15,171 12,925 10,797 8,760 6,796 6,219 4,941 3,331 2,842 1,979 1,654 346 0

P

into account,

P

z

of Titan was not taken

Rock-ice VIII Ice VIII-ice VI Ice VI-ice II Ice II-ice I Surface

Rock-ice VIII Ice VIII-ice VI Ice VI-ice II Ice II-ice I Surface

pressure

Isothermal

VIII

TITAN MODELS

TABLE

P

38,363 35,030 30,086 23,113 17,580 14,595 12,395 10,146 7,157 5,643 3,449 1,235 1,001 0

38,533 35,169 30,181 23,148 17,570 14,574 12,368 10,115 7,125 5,616 3,430 1,226 882 0

P

for Table

2.250 2.230 2.201 2.160 1.969 1.930 1.902 1.874 1.713 1.698 1.675 1.653 I.397 1.378

2.260 2.240 2.211 2.169 1.969 1.929 1.900 1.872 1.705 1.690 1.667 1.645 1.380 1.362

See caption

105 705 1,105 1,505 1,765 1,905 2,005 2,105 2,235 2,305 2,405 2,505 2,520 2,569

105 705 1,105 1,505 1,765 1,905 2,005 2,105 2,235 2,305 2,405 2,505 2,525 2,569

z

V.

285 270 249 218 199 190 183 176 167 162 154 141 118 93

286 271 249 218 199 190 183 176 167 162 154 140 115 94

T

Homogeneous

Ice II-ice Surface

-

Ice VI-ice -

Ice VIII-ice -

(Ice VII) Ice VII-ice -

Ice II-ice Surface

Ice VI-ice -

Ice VIII-ice -

(Ice VII) Ice VII-ice -

4

I

I

II

VI

VIII

II

VI

VIII

8

E

z LI

$

1,212 792 611 475 210 0

1,134 719 610 417 150 0

1.176 1.172 0.947 0.944 0.940 0.934

1.175 1.171 0.947 0.943 0.938 0.934

319 484 534 584 684 165

401 506

Rock-ice II Ice II-ice 1 Surface

Rock-ice I1 Ice II-ice I Surface

4J

P

0.945 0.942 0.938 0.934

0.945 0.942 0.937 0.934

Z

404 509 609 709 763

421 526 626 726 765

998 654 370 102 0

1,053 698 410 141 0

P

147 108 89 77 74

0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0

Melt

lorock

152 111 91 78 74

Rock

T

Conductive

Rock-ice Surface

Rock-ice Surface

+

I

1

IO5 205 305 405 505 605 705 765

105 205 305 405 507 764

Z

1.518 1.518 1.518 1.517 1.266 1.262 1.256 1.250

1.524 1.524 1.524 1.524 1.267 1.251

P

1,552 1,458 1,300 1,078 806 515 202 0

1,559 1,465 1,306 1,082 801 0

P

176 165 148 124 95 88 81 77

176 166 149 124 94 77

T

Homogeneous

(Ice II) Ice II-ice I Surface

(Ice II) (Ice B-ice 1) Surface

4

Note. For these models, Z is the distance from the center of the sphere in kilometers, and p is the local density in grams per cubic centimeter. In the case of a phase change, this would be the density of the phase nearer to the surface. P is the local pressure in bars, T is the local temperature in degrees Kelvin, and “Melt” signifies the melt percentage by mass. In regions where liquid coexists with ice, the density is given as the density of the solid-liquid slush in that shell. The column marked + is where any phase changes are recorded.

536 606 706 763

P

P

Z

Isothermal

RHEAMODELS

TABLE IX

9

52

MARK J. LUPO TABLE

TS

Mimas Enceladus Tethys Dione Iapetus

77 77 77 17 77

X

M

R

P

0.4538 0.8452 7.5378 10.585 18.822

196 250 530 561 730

1.291 1.209 1.431 1.155

1.439

P

IIMR2

101 135 555 890 1078

0.399 0.399 0.399 0.399 0.392

Percentage rock 47. 37. 30. 46. 22.

Note. T, denotes surface temperature. M is mass in 1O23grams. R is radius in kilometers. p is density in grams per cubic centimeter. P is the central pressure in bars. Z/MRz is the dimensionless moment of inertia. Percentage rock is by mass. All the models have only mild temperature gradients. The central temperature would be 125°K for Iapetus, 103°K for Dione, and less for others. The ice phase present would be ice I in all the models. The innermost 105 km of Dione and the innermost 365 km of Iapetus would contain ice II.

mates for Mimas. This idea would be further detracted from by the discovery that Saturn’s smaller satellites were differentiated-were such a discovery made-or by the discovery of a large indigenous silicate component in the crust of Call&to, which is widely hypothesized. Since methane has been confirmed by Voyager as a constituent of Titan’s atmosphere (Stone and Miner, 1981), it is conceivable that the gas is buffered by solid methane on the surface. It should be pointed out that a significant quantity of methane in Titan’s bulk composition would greatly enhance the satellite’s rock abundance, which is high with respect to its neighbors as it is. The density of methane ice is 0.5 g cmd3. Further, the proximity of Titan’s surface conditions to the triple point of methane and its low abundance in the atmosphere with respect to nitrogen do not favor a large amount of methane in Titan’s bulk composition. Clathrated methane, however, is denser than water ice and a significant quantity of this ice would place Titan’s rock abundance at a lower value and more in line with that of Rhea and other moons of Saturn. This paper was intended to provide theorists with a simple rock-ice model of each satellite without inflicting any preferences of rock type, rock distribution, and thermal assumptions on the reader. It was also intended as a tool to aid in the determination

of composition and heat transport mechanism for each satellite should additional information, such as the dimensionless moment of interia, be acquired. The data presented here demonstrate that estimations of the rock abundance and the central pressure of an icy satellite depend upon assumptions concerning the heat transport mechanism, the silicate type, and to a large degree upon whether or not the body has been differentiated. In the ice-rich satellites, where Z/MR*was -0.31 for the differentiated case and -0.38 for the homogeneous case, the crudest determination of the moment of inertia should be helpful in determining the distribution of the silicates in these satellites. We have yet to consider the effects of the departures from spherical symmetry due to planetary tidal forces and latitude-dependent subsurface temperatures. The wait for a reliable value of Z/MR*for any of these bodies is likely to be a long one.

ACKNOWLEDGMENTS I would like to thank Professor John S. Lewis for helpful discussion and for paying for some of the computer time through his grant from the National Aeronautics and Space Administration through the Planetary Atmospheres Program under Grant NGL 22-009-521. Helpful comments by Dr. Gerald Schubert were appreciated. I would also like to thank the Chevron Oil Field Research Company for the computer time needed in the revision of this manuscript.

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