G~OPHYSICAL RESEARCH LETTERS, VOL. 15, NO. 13, PAGES 1455-1458, DECEMBER 1988
BACKGROUND HEATFLOW ON HOTSPOT PLANETS: 10 AND VENUS
David J. Stevenson and Sean C. McNamara
Division of Geological and Planetary Sciences, California Institute of Technology
Abstract. On planets where most of the heat is trans the global heatflow of lo have expressed the view that ported to the surface by igneous activity (extrusive vol the total background heatflow is less (perhaps much less) canism or near-surface intrusions), the surface heat flow than the hotspot component (e.g., Gaskell et a!. [1988]). at localities well away from regions of current igneous The purpose of this paper is to challenge this assump activity need not be even approximately the conductive tion. The potential fallacy in the assumption is easily heatflow through the entire lithosphere but may instead understood: Consider a planet for which the volcanic be dominated by the residual heat leaking out from the activity is large enough that every point on the surface last igneous event in that locality. On lo, it is likely that has been within a volcanically active region many times (~tr) 1 1 2
Copyright 1988 by the American Geophysical Union. As one would intuitively expect, F ex C 112 (like a cooling Paper number 88GL04023. half space) fort ;S tf2 /4~t and F ex t-3 / 2 once the entire 0094-8276/88/88GL-04023$03.00 interior of the flow is losing heat to the surface and below.
1455 1456 Stevenson and MeN amara: Hotspot Planet Heatflow
This exact solution does not quite suit our purpose be cause it is a single event. Consider a model in which resurfacing occurs by emplacing a layer of thickness d at times T, 2T, 3T, · · ·. We carried out a number of nu merical solutions of the diffusion equation under these circumstances. For rP ¢: 4ttT we found, not surprisingly, that little memory is retained of previous flows and the behavior is well described by eq. (4) with t measured rel 0.1 ative to the time of last emplacement. If rP > 4ttT then the flows do not cool internally before the next flow is emplaced. In this limit, no lithosphere can exist. In fact, this is not a meaningful limit for our purposes (it might, however, describe the development of a magma ocean). 2 The most difficult case is the regime where d I ItT is not enormously different from unity. This might seem like a 0.01 special (hence implausible) regime, but it may even be a 0.01 preferred regime if the eruption process proceeds once the overlying layer has cooled enough to allow fracture. In F/F this regime, an approximate solution can be obtained, by Figure 2. Fraction (if>) of total heat flow that comes from allowing for a fraction € of the heat to be retained from the regions where the heat flow exceeds a value F (non- each resurfacing event. This suggests an expression for dimensionalized by the mean heatflow F). Each curve is the heat flux of the form labeled by the value of Pe, the Peclet number (see text). At high Pe, most of the heatflow comes from low heat flow regions. F ~ ..;;;ctkf:l.T [1- (1- E)e-d'/4"'t]. (5)
where the value of € was estimated from numerical sim where u is the resurfacing rate. The requirement that this ulations such as that shown in Figure 1, and is given is large compared to the conductive heat flux through the roughly by lithosphere is u(1- €) drltt > 1 (7)
The requirement that a lithosphere exist is € ~ 112 or udl 11: ;$ 1. It is easy to satisfy this simultaneously with 1/2 equation (7) if d ¢: dr. All these equations omit latent € ~ 1-0.8 ( ; ) heat effects but these are only a 10-20% correction, less important than the other uncertainties in the model. By conservation of energy, the time averaged heat flux at From the poi·nt of view of observations, it is much more any locality is interesting to ask how the heat flow is distributed on the surface, since this may provide an observational con pCvf:l.Tu(1 - F ~ €) straint. Consider, first, the "boxcar" model (6) u= diT 1 P(t) =- O 00 00 if>( F) = f F P(F) dF I f F P(F) dF (10) F Fudn 00 05 10 I 5 20 25 30 35 5 Z (10 cm) and has the value Fmiu IF in this particular case. Half of Figure 1. A sequence of numerically computed tempera the heat output occurs from regions that have heatflows ture profiles at successive times (T, 2T, 3T, · · ·) just prior less than the mean heatflow (2 Fmin )· This has important to a resurfacing event, for the particular case rP = ItT and implications for the (non)detectabihty of the background an equilibrium surface temperature appropriate to lo. In heatflow on Io. In Figure 2, if>(F) is shown for a variety each case, zero on the horizontal axis refers to the ac of models. If the heatfl.ow is non-dimensionalized by its tual surface, so the residual (unescaped) heat is trapped mean value then the only remaining variable is a Peclet deeper down after each new resurfacing. number Stevenson and McNamara: Hotspot Planet Heatflow 1457 d2 ud Pe=-=- (11) 150 2~~:r 2~~: which can be best interpreted as the fractional time of high heatflow (i.e., the ratio of thermal diffusion time for the resurfacing layer to the recurrence time of vol~ canic events at any given locality). The factor of two in the definition improves this convenient interpretation. The simple example above (eq. (10)) corresponds to the ~aximal value, Pe ~ 1. As Pe decreases, a larger frac tiOn of the heat output occurs from regions with higher heatflows, but there is always a substantial low heatflow component even at low Pe. --·- -·-...... Application to Io ·---....._. 120~~~~~----~~--~------~ 15 20 30 The anomalous infrared emission properties of Io have A (!l) long been known but were not correctly interpreted un til after the discovery of volcanism during the Voyager 1 Figure 3. Brightness temperature 11, as a function of flyby. The analysis of both ground-based and Voyager wavelength for three sets of models: --- (Pe = 1), data has led to the unequivocal identification of an inter• - ·- (Pe = 0.1),- -- (Pe = O.Dl). In each case, 2 1 the lower curve corresponds to a mean heatflow of 1500 nal heatflow of at least 1400 erg cm- s- and possibly 2 1 more [Matson et al., 1981; Johnson et al., 1984; Nash et erg cm- s- and the upper curve corresponds to 3000 al., 1986; Johnson et al., 1988]. The identification is un erg em - 2 s-1 . The Io data are triangles at 8.4, 10.6, equivocal not so much because of the magnitude of this and 21 p, (Matson et al. [1981]); the curves are not meant heatflow (which is only about 10% of the absorbed in to fit the data (but should lie below them). solation) but because it is associated with temperatures (ranging from 200 K upward) that are high compared with typical surface temperatures. In other words, the dally when allowance is made for near surface intrusive "spectral contrast" (the steep rise of brightness temper events. Certainly, Pe is at least as large as the fractional ature with decreasing wavelength) is an essential part of surface of Io that is currently active, assuming that Io identifying this heatflow. In the context of the models is in a typical state at present. (This is a debatable as presented above, we must ask: What is the spectral con sumption, but probably not wrong by more than a factor trast of the background heatflow? The brightness tem 2 of a few; see below.) This suggests Pe ;:::; w- • How perature at each wavelength can be evaluated from ever, a substantially larger value would seem plausible. 4 00 For example, a volcanic province that erupts every 10 years and lays down three hundred meters of material at B(.X, Tb) = Jf(T) B(.X, T) dT (12) each eruption would have Pe ...., 0.3, assuming K ~- w-2 0 cm2 s-1 • If the actual eruption (i.e., period of activity) takes ...., 102 years and all parts of the surface of Io are where B( .X, T) is the blackbody emission (Planck func equally suitable for eruptions then the "active fraction" tion) at wavefength .X, 11, is the brightness temperature, of the surface would then be ...., w- 2 , much less than the and f(T) is the probability that a surface element of Io value of Pe. Figure 3 then implies that the spectral con lies in the temperature range between T, T + dT: trast is low and the globally averaged background heat flow could be large; comparable or even larger than the f(T) = - P(t) dF /dT (13) hotspot component. dF/dt Recent analysis of the coupled thermal and orbital evo lution of Io are consistent with this possibility. If the total 2 1 with the heatflux F given by equation (5). We also ex average heat output were less than...., 2000 er~ cm- s- press then it could be argued (e.g., Schubert et al. [1986]) that (14) Io is in steady state, with its orbit steadily expandmg at a rate given by Jupiter's tidal Q and its orbital eccentric ity determined by the ratio of Io's Q to that of Jupiter where u is the Stefan-Boltzmann constant and T0 is the insolation temperature. Our purpose here is not to model [Yoder and Peale, 1981]. However, data on the expan Io specifically (since that requires detailed assumptions sion of Io's orbit suggest a much slower expansion than about surface properties) but to show how the spectral expected for the steady state model [Lieske, 1987; Green contrast varies with total heat flow and Peclet number. berg, 1987] and accordingly allow for a higher heatflow, similar to {but not necessarily exactly compatible with) This is illustrated in Figure 3, for the choice T0 = 120 K and two background mean heatflows ( 1500 and 3000 erg the non-steady state model of Ojakangas and Stevenson ]1986]. The higher heatflow arises if the eccentricity of cm-2 s-1 ). As these calculations show, a large back lo's orbit is currently decreasing. A global heat output of ground heatflow is possible without substantial spectral 2 1 contrast, provided Pe is quite large. These heatflows cor 3000-4000 erg cm- s- is possible. These arguments respond to mean resurfacing rates of up to a few cm/yr are only suggestive and the true test is observational. [Johnson and Soderblom, 1982; Carr, 1986] and imply that the interior of Io is only partially molten [Webb and Application to Venus Stevenson, 1987]. It is difficult to make independent estimates of plausi The current uncertain state of the interpretation of ble values for Pe, the ratio of cooling time for a resurfac Venus geology [Basilevsky and Head, 1988] renders any ing event to the time between resurfacing events, espe- statements about Venus conditional at best. If the total 1458 Stevenson and McNamara: Hotspot Planet. Tlcatflow magmatic activity of Venus is not much different from lithospheric heat transport on Venus from impact Earth, as several have argued [Solomon and Head, 1982; crater densities, Geophys. Res. Lett. 14, 530-541, Morgan and Phillips, 1983; Phillips and Malin, 1984], 1987. then the considerations developed above have marginal Johnson, T.V., D. Morrison, D. L. Matson, G. J. Veeder, relevance. Specifically, consider a recurrence interval of R. H. Brown, and R. M. Nelson, lo volcanic hotspots: 108 years for magmatic activity, with 10 km of material Stability and longitudinal distribution, Science 226, (basaltic crust or gabbroic intrusion) "laid down." This 134-137, 1984. corresponds to Pe ~ 0.03 and .$ 10% enhancement of the Johnson, T.V., G. J. Veeder, D. L. Matson, R. H. Brown, background heat flow. On the basis of crater counts [Bar R. M. Nelson, and D. Morrison, lo: Evidence for sili sukov et al., 1986; Grimm and Solomon, 1987], this recur cate volcanism in 1986, Submitted to Science, 1988. rence interval is about the smallest conceivable. However, Johnson, T.V. and L.A. Soderblom, Volcanic eruptions it must be stressed that intrusive activity can come close on lo: Implications for surface evolution and mass loss, to achieving much of the same effect (in terms of heat In Satellites of Jupiter, D. Morrison, (ed.), pp. 634- flow) as surface lava flows, yet have a less direct impact on 646, Univ. of Arizona Press, Tucson, 1982. geomorphology, so the situation is not fully determined Landau, L. D. and E. M. Lifshitz, Fluid Mechanics, from the radar data. Turcotte [1988] has suggested that Addison-Wesley, Reading, Mass., p. 199, 1959. the magmatic activity could be two orders of magnitude Lieske, J. H., Galilean satellite evolution: Observational higher on Venus than on Earth. Under these circum evidence for secular changes in mean motions, Astron. stances, the "background" heatflow will have comparable Astrophys. 176, 146-158, 1987. contributions from conduction through the lithosphere Matson, D. L., G. A. Ransford, and T.V. Johnson, Heat and cooling of previous igneous events, if the lithosphere flow from lo (Jl), J. Geophys. Res. 86, 1664-1672, thickness dT ~ 100 km. 1981. Morgen, P. and R.J. Phillips, Hot spot heat transfer: Concluding Comments Its application to Venus and implications to Venus and Earth, J. Geophys. Res. 88, 8305-8317, 1983. The main message of this paper is a rather obvious one, Nash, D. B., M. H. Carr, J. Gradie, D. M. Hunten, but one that seems to have been insufficiently appreci and C. F. Yoder, lo, In Satellites, J. Burns and M. ated: On planets where volcanism dominates the heat Matthews, (eds.), pp. 629-688, Univ. of Arizona Press, flow, there is no simple relationship between lithospheric Tucson, 1986. thickness and heatflow, either locally or globally, even O'Reilly, T. C. and G. F. Davies, Magma heat trans away from regions of current volcanic activity. There is port on lo: A mechanism allowing a thick lithosphere, no basis, either observationally or theoretically, to advo Geophys. Res. Lett. 8, 313-316, 1981. cate the minimalist view that the Io heatflow is as low Ojakangas, G. W. and D. J. Stevenson, Episodic volcan as the hotspot component alone would suggest. Our de ism of tidally heated satellites with application to Io, veloping views of lo and (to a lesser extent) Venus will Icarus 66, 341-358, 1986. require a better understanding of how volcamsm operates Phillips, R.J. and M.C. Malin, Tectonics of Venus, Ann. on these bodies. Rev. Earth Planet. Sci. 12, 411-443, 1984. Schaber, G. G., The geology of Io, In Satellites of Acknowledgements. Discussions with T. V. Johnson Jupiter, D. Morrison, (ed.), pp. 556-597, Univ. of Ari and D. L. Matson, and comments from a reviewer were zona Press, Tucson, 1982. helpful. Contribution number 4657 from the Division of Schubert, G., T. Spohn, and R. T. Reynolds, Thermal Geological and Planetary Sciences, California Institute histories, compositions and internal structures of the of Technology, Pasadena, California 91125. This work is moons of the solar system, In Satellites, J. Burns and supported by the NASA Planetary Geophysics program, M. Matthews, (eds.), pp. 224-292, Univ. of Arizona grant NAGW-185. Press, Tucson, 1986. Solomon, S.C. and J.W. Head, Mechanisms for litho References spheric heat transfer on Venus: Implications for tec tonic style and volcanism, J. Geophys. Res. 87, 9236- Barsukov, U.L., et al., The geology and geomorphology 9246, 1982. of the Venus surface as revealed from the radar images Turcotte, D. L., A heat-pipe mechanism for volcanism obtained by Veneras 15 and 16, J. Geophys. Res. 91, and tectonics on Venus. LPSC XIX Abstracts 1207- D378-D398, 1986. 1208, and submitted to J. Geophys. Res., (1988). Basilevsky, A. T. and J. W. Head III, The geology Webb, E. K. and D. J. Stevenson, Subsidence of topog~ of Venus, Ann. Rev. Earth Planet. Sci. 16, 295-317, raphy on Io, Icarus 10, 348-353, 1987. 1988. Yoder, C. F. and S. J. Peale, The tides of lo, Icarus 47, Carr, M. H., Silicate volcanism on Io, J. Geophys. Res. 1-35, 1981. 91, 3521-3532, 1986. Gaskell, R. W., S. P. Synnott, A. S. McEwen, and G. G. Schaber, Large scale topography of Io: Implications S. C. McNamara and D. J. Stevenson, Division of for internal structure and heat transfer, Geophys. Res. Geological and Planetary Sciences, California Institute Lett. 15, 581-584, 1988. of Technology, Pasadena, California 91125. Greenberg, R. Time-varying orbits and tidal heating of the Galilean satellites. International Workshop on Time-Variable Phenomena in the Jovian System, (Received June 20, 1988; Flagstaff, Arizona, 1987. revised October 24, 1988; Grimm, R.E. and S.C. Solomon, Limits on modes of accepted October 26, 1988.)