Excitation of the oxygen nightglow on the terrestrial planets

A scheme of excitation, quenching, and energy transfer processes in the oxygen nightglow on the Earth, Venus, and Mars has been developed based on the observed nightglow intensities and vertical profiles, measured reaction rate coefficients, and photochemical models of the nighttime atmospheres of the Venus and Mars. The scheme involves improved radiative lifetimes of some band systems, calculated yields of the seven electronic states of O₂ in termolecular association, and rate coefficients of seven processes of electronic quenching of the Herzberg states of O₂, which are evaluated by fitting to the nightglow observations. Electronic quenching of the vibrationally excited Herzberg states by O₂ and N₂ in the Earth's nightglow is a quarter of total collisional removal of the O₂(A, A′) states and a dominant branch for the O₂(c) state. The scheme supports the conclusion by Steadman and Thrush (1994) that the green line is excited by energy transfer from the O₂(A³Σ_u⁺, v≥6) molecules, and the inferred rate coefficient of this transfer is 1.5×10⁻¹¹ cm³ s⁻¹. The O₂ bands at 762 nm and 1.27 μm are excited directly, by quenching of the Herzberg states, and by energy transfer from the O₂(⁵Π_g) state. Quenching of the O₂ band at 762 nm excites the band at 1.27 μm as well. Effective yield of the O₂(a¹Δ_g) state in termolecular association on Venus and Mars is ∼0.7. Quantitative assessments of all these processes have been made. A possible reaction of O₂(c¹Σ_u⁻)+CO is a very minor branch of recombination of CO₂ on Venus and Mars. Night airglow on Mars is calculated for typical conditions of the nighttime atmosphere. The calculated vertical intensity of the O₂ band at 1.27 μm is 13 kR, far below the recently reported detections.     

Altitude variation of ultraviolet oxygen night airglow

Night airglow of oxygen 130.4 and 135.6 nm emissions was measured by a spectrophotometer aborad an S520 sounding rocket, launched at 19:50 JST (10:50 UT) on 14 February, 1982 from Kagoshima, Japan. The altitude variation of the emissions was obtained from 110 to 266 km at zenith angles of 35.5°±4°. The emission intensity around 260 km was about 160R and is roughly compatible with model calculations taking account of O⁺+e ^– radiative recombination as well as O⁺–O^– mutual neutralization. Some excess of about 50R, compared to the model calculation, was observed around 200 km. Possible explanations of the excess are: (i) remnant oxygen ions during the transition period from day to night and (ii) diffuse radiation from the background sky. Model calculations taking account of remnant oxygen ions were also performed by adding an excess electron density to the original density profile. However, it was found that an unreasonably large electron density is required around 200 km (5×10⁵ cm^–3) to produce the observed intensity. It is also probable that some contribution from the background sky is present in the observed intensity.     

Formation of the terrestrial planets

The early phases of formation in the inner solar system were dominated by collisions and short-range dynamical interactions among planetesimals. But the later phases, which account for most of the differences among planets, are unsure because the dynamics are more subtle. Jupiter's influence became more important, leading to drastic clearing out of the asteroid belt and the stunting of Mars's growth. Further in, the effect of Jupiter-- both directly and indirectly, through ejection of mass in the outer solar system-- was probably to speed up the process without greatly affecting the outcome. The great variety in bulk properties of the terrestrial bodies indicate a terminal phase of great collisions, so that the outcome is the result of small-N statistics. Mercury, 65 percent iron, appears to be a residual core from a high-velocity collision. All planets appear to require a late phase of high energy impacts to erode their atmospheres: including the Earth, to remove CO₂ so that its ocean could form by condensation of water.Consistent with this model is that the largest collision, about 0.2 Earth masses, was into the proto-Earth, although the only property that appears to require it is the great lack of iron in the Moon. The other large differences between the Earth and Venus, angular momentum (spin plus satellite) and inert gas abundances, must arise from origin circumstances, but neither require nor forbid the giant impact. Venus's higher ratio of light to heavy inert gases argues for it receiving a large icy impactor, about 10^–6 Earth masses from far out, requiring some improbable dynamics to get a low enough approach velocity. Core formation in both planets probably started rather early during accretion.Some geochemical evidences argue for the Moon coming from the Earth's mantle, but are inconclusive. Large scale melting of the mantle by the giant impact would plausibly have led to stratification. But the "lock-up" at the end of turbulent mantle convection is a trade-off between rates: crystallization of constituents of small density difference versus overall freezing. Also, factors such as differences in melting temperatures and densities, melt compressibilities, and phase transitions may have had homogenizing effects in the subsequent mantle convection.     

Structure of the terrestrial planets

Recent reviews (cf. Runcorn, 1968; or Cook, 1972, 1975) on the structure of the planets omit reference to the phase-change hypothesis for the nature of the terrestrial core, despite that numerous prior predictions of the theory based on this hypothesis have subsequently been borne out as correct. These reviews also ignore the existence of theoretical calculations of the internal structure of Venus which can be computed with high accuracy by use of the terrestrial seismic data. Several examples of numerous mistakes committed in these reviews are pointed out.     

Origin of water in the terrestrial planets

Abstract— I examine the origin of water in the terrestrial planets. Late‐stage delivery of water from asteroidal and cometary sources appears to be ruled out by isotopic and molecular ratio considerations, unless either comets and asteroids currently sampled spectroscopically and by meteorites are unlike those falling to Earth 4.5 Ga ago, or our measurements are not representative of those bodies. However, the terrestrial planets were bathed in a gas of H, He, and O. The dominant gas phase species were H₂, He, H₂ O, and CO. Thus, grains in the accretion disk must have been exposed to and adsorbed H2 and water. Here I conduct a preliminary analysis of the efficacy of nebular gas adsorption as a mechanism by which the terrestrial planets accreted “wet.” A simple model suggests that grains accreted to Earth could have adsorbed 1‐3 Earth oceans of water. The fraction of this water retained during accretion is unknown, but these results suggest that examining the role of adsorption of water vapor onto grains in the accretion disk bears further study.     

Thermal evolutions of the terrestrial planets

The thermal evolution of the Moon, Mercury, Mars, Venus and hypothetical minor planets is calculated theoretically, taking into account conduction, solid-state convection, and differentiation. An assortment of geological, geochemical, and geophysical data is used to constrain both the present day temperatures and thermal histories of the planets' interiors. Such data imply that the planets were heated during or shortly after formation and that all the terrestrial planets started their differentiations early in their history. Initial temperatures and core formation play the most important roles in the early differentiation. The size of the planet is the primary factor in determining its present day thermal state. A planetary body with radius less than 1000 km is unlikely to reach melting given heat source concentrations similar to terrestrial values and in the absence of intensive early heating such as short half-life radioactive heating and inductive heating.Studies of individual planets are constrained by varying amounts of data. Most data exist for the Earth and Moon. The Moon is a differentiated body with a crust, a thick solid mantle and an interior region which may be partially molten. It is presently cooling rapidly and is relatively inactive tectonically.Mercury most likely has a large core. Thermal calculations indicate it may have a 500 km thick solid lithosphere, and the core may be partially molten if it contains some heat sources. If this is not the case, the planet's interior temperatures are everywhere below the melting curve for iron. The thermal evolution is dominated by core separation and the high conductivity of iron which makes up the bulk of Mercury.Mars, intermediate in size among the terrestrial planets, is assumed to have differentiated an Fe–FeS core. Differentiation and formation of an early crust is evident from Mariner and Viking observations. Theoretical models suggest that melting and differentiation of the mantle silicates has occurred at least up until 1 billion years ago. Present day temperature profiles indicate a relatively thick (250 km) lithosphere with a possible asthenosphere below. The core is molten.Venus is characterized as a planet similar to the Earth in many respects. Core formation probably occurred during the first billion years after the formation. Present day temperatures indicate a partially molten upper mantle overlain by a 100 km thick lithosphere and a molten Fe–Ni core. If temperature models are good indicators, we can expect that today, Venus has tectonic processes similar to the Earth's.Paper dedicated to Professor Hannes Alfvén on the occasion of his 70th birthday, 30 May 1978.     

Evolutionary tracks of the terrestrial planets

On the basis of the model proposed by Matsui and Abe, we will show that two major factors — distance from the Sun and the efficiency of retention of accretional energy — control the early evolution of the terrestrial planets. A diagram of accretional energy versus the optical depth of a proto-atmosphere provides a means to follow the evolutionary track of surface temperature of the terrestrial planets and an explanation for why the third planet in our solar system is an aqua-planet.     

Nonuniform cratering of the terrestrial planets

We estimate the impact flux and cratering rate as a function of latitude on the terrestrial planets using a model distribution of planet crossing asteroids and comets [Bottke, W.F., Morbidelli, A., Jedicke, R., Petit, J.-M., Levison, H.F., Michel, P., Metcalfe, T.S., 2002. Icarus 156, 399-433]. After determining the planetary impact probabilities as a function of the relative encounter velocity and encounter inclination, the impact positions are calculated analytically, assuming the projectiles follow hyperbolic paths during the encounter phase. As the source of projectiles is not isotropic, latitudinal variations of the impact flux are predicted: the calculated ratio between the pole and equator is 1.05 for Mercury, 1.00 for Venus, 0.96 for the Earth, 0.90 for the Moon, and 1.14 for Mars over its long-term obliquity variation history. By taking into account the latitudinal dependence of the impact velocity and impact angle, and by using a crater scaling law that depends on the vertical component of the impact velocity, the latitudinal variations of the cratering rate (the number of craters with a given size formed per unit time and unit area) is in general enhanced. With respect to the equator, the polar cratering rate is about 30% larger on Mars and 10% on Mercury, whereas it is 10% less on the Earth and 20% less on the Moon. The cratering rate is found to be uniform on Venus. The relative global impact fluxes on Mercury, Venus, the Earth and Mars are calculated with respect to the Moon, and we find values of 1.9, 1.8, 1.6, and 2.8, respectively. Our results show that the relative shape of the crater size-frequency distribution does not noticeably depend upon latitude for any of the terrestrial bodies in this study. Nevertheless, by neglecting the expected latitudinal variations of the cratering rate, systematic errors of 20-30% in the age of planetary surfaces could exist between equatorial and polar regions when using the crater chronology method.     

Climatic change on the terrestrial planets

In this paper, we review the observational data on climatic change for the terrestrial planets, discuss the basic factors that influence climate, and examine the manner in which these factors may have been responsible for some of the known changes. Emphasis is placed on trying to understand the similarities and differences in both the basic factors and their climatic impacts on Venus, the Earth, and Mars. Climatic changes have occurred on the Earth over a broad spectrum of time scales that range from the elevated temperatures of Pre-Cambrian times (～10⁹ years ago), through the alternating glacial and interglacial epochs of the last few million years, to the small but significant decadal and centurial variations of the recent past. Evidence for climatic change on Mars is given by certain channel features, which suggest an early to intermediate aged epoch of warmer and wetter climate, and by layered polar deposits, which imply more recent periodic climate variations. No evidence for climatic change on Venus exists as yet, but comparison of its present climate state with that of outer terrestrial planets offers important clues on some of the mechanisms affecting climate. The important determinants of climate for a terrestrial planet include the Sun's output, astronomical perturbations of its orbital and axial characteristics, the gaseous and particulate content of its atmosphere, its land surface, volatile reservoirs, and its interior. All these factors appear to have played major roles in causing climatic changes on the terrestrial planets. Despite a lower solar luminosity in the past, the Earth and Mars have had warmer periods in their early history. In both cases, a more reducing atmosphere may have been the responsible agent through an enhanced greenhouse effect. In this paper, we present detailed calculations of the effect of atmospheric pressure and composition on the temperature state of Mars. We find that the higher temperature period is easier to explain with a reducing atmosphere than with the current fully oxidizing one. Both the very high surface temperature and massive atmosphere of Venus may be the result of the solar flux being a factor of two higher at its orbit than at the Earth's orbit. This difference may have led to a runaway greenhouse effect on Venus, i.e., the emplacement of volatiles entirely in the atmosphere rather than mostly in surface reservoirs. But if Venus formed with relatively little or no water, it may have always had an oxidizing atmosphere. In this case, a lower solar luminosity would have led to a moderate surface temperature in Venus' early history. Quasi-periodic variations in orbital eccentricity and axial obliquity may have contributed to the alternation between Pleistocene glacial and interglacial periods in the case of the Earth and to the formation of the layered polar deposits in the case of Mars. In this paper, we postulate that two mechanisms, acting jointly, account for the creation of the laminated terrain of Mars: dust particles serve as nucleation centers for the condensation of water vapor and carbon dioxide. The combined dust-H₂O-CO₂ particle is much larger and so has a much higher terminal velocity than either a dust-H₂O or a plain dust particle. As a result, dust and water ice are preferentially deposited in the polar regions. In addition, we postulate that the obliquity variations are key drivers of the periodic layering because of their impact on both atmospheric pressure and polar surface temperature, which, in turn, influence the amounts of dust and water ice in the atmosphere. But eccentricity and precessional changes probably also play important roles in creating the polar layers. The drifting of continents on the Earth has caused substantial climatic changes on individual continents and may have helped to set the stage for the Pleistocene ice ages through a positioning of the continents near the poles. While continental drift apparently has not occurred on Mars, tectonic distortions of its lithosphere may, in some circumstances, cause an alteration in the mean value of that planet's obliquity, which would significantly impact its climate. Atmospheric aerosols can influemce climate through their radiative effects. In the case of the Earth, volcanic aerosols appear to have contributed to past climatic changes, while consideration needs to be given to the future impact of man-generated aerosols. In the case of Mars, the atmospheric temperature structure and thereby atmospheric dynamics are greatly altered by suspended dust particles. The sulfuric acid clouds of Venus play a major role in its heat balance. Cometary impacts may have added substantial quantities of water vapor and sulfur gases to Venus' atmosphere and thus have indirectly affected its cloud properties. Calculations presented in this paper indicate substantial changes in surface temperature accompany these compositional changes.     

On the rotational temperature of airglow hydroxyl emissions

It is argued that available observational information does not support the contention (recently advanced by Suzuki and Tohmatsu, 1976) that the rotational temperature derived from an airglow hydroxyl emission band depends systematically on the quantum number (ν′) of the vibrational state from which the band originates. In particular, it is shown that bands originating from ν′ = 6, 7 and 8, measured simultaneously, exhibit rotational temperatures which (within the experimental errors) are equal.     

Accretion of the terrestrial planets. II

Some aspects and consequences of the theory of gravitational accretion of the terrestrial planets are examined. The concept of a “closed feeding zone” is somewhat unrealistic, but provides a lower bound on the accretion time. Safronov's relative velocity relation for planetesimals is not entirely consistent with the feeding zone model. A velocity relation which includes an initial velocity component is suggested. The orbital parameters of the planetesimals and the dimensions of the feeding zone are related to their relative velocities. The assumption of an initial velocity does not seriously change the accretion time.Mercury, Venus, and the Earth have accretion times on the order of 10⁸yr. Mars requires well over 10⁹yr to accrete by the same assumptions. Currently available data do not rule out a late formation of Mars, but the lunar cratering history makes it unlikely. If Mars is as old as the Earth, nongravitational forces or a violation of the feeding zone concept is required. One such possibility is the removal of matter from the zone of Mars by Jupiter's influence. The final sweeping up by Mars after this event would result in the scattering of a considerable mass among the other terrestrial planets. The late postaccretional bombardments infrerred for the Moon and Mercury may have had this source.     

Origin of the atmospheres of the terrestrial planets

The monotonic decrease in the atmospheric abundance of ³⁶Ar per gram of planet in the sequence, Venus, Earth, and Mars has been assumed to reflect some conditions in the primitive solar nebula at the time of formation of the planetary atmospheres, having to do either with the composition of the nebula itself or the composition of the trapped gases in small solid bodies in the nebula. Behind such hypotheses lies the assumption that planetary atmospheres steadily gain components. However, not only can gases enter atmospheres; they may also be lost from atmospheres both by adsorption into the planetary interior and by loss into space as a result of collisions with minor and major planetesimals. In this paper a necessarily qualitative discussion is given of the problem of collisions with minor planetesimals, a process called atmospheric cratering or atmospheric erosion, and a discussion is given of atmospheric loss accompanying collision of a planet with a major planetesimal, such as may have produced the Earth's Moon.     

Mid-winter hydroxyl night airglow emission intensities in the northern polar region

Ground-based spectrophotometric measurements of night airglow OH (8-3) band absolute intensities in the polar cap region (78.4°N) during winter solstice are reported. A mean value of 425 ± 40 R is found for the absolute intensity of the OH (8-3) band. Maximum and minimum daily mean values were 770 and 320 R respectively with hourly mean values ranging from 180 to 1020 R. Neither a winter solstice minimum or maximum in the intensity is obvious from the data. No consistent correlation was found between the absolute intensity and geomagnetic and solar activity. A mean transport of O and O₃ into the polar cap region corresponding to a meridional wind speed of at least 20 m s^?1 at 90 km height seems necessary to maintain the observed intensity. A dominant semidiurnal tide component is found in the intensity data, both on a 20-day and a 3-day time scale.     

The formation and habitability of terrestrial planets in the presence of close-in giant planets

‘Hot jupiters,’ giant planets with orbits very close to their parent stars, are thought to form farther away and migrate inward via interactions with a massive gas disk. If a giant planet forms and migrates quickly, the planetesimal population has time to re-generate in the lifetime of the disk and terrestrial planets may form [P.J. Armitage, A reduced efficiency of terrestrial planet formation following giant planet migration, Astrophys. J. 582 (2003) L47-L50]. We present results of simulations of terrestrial planet formation in the presence of hot/warm jupiters, broadly defined as having orbital radii ?0.5 AU. We show that terrestrial planets similar to those in the Solar System can form around stars with hot/warm jupiters, and can have water contents equal to or higher than the Earth's. For small orbital radii of hot jupiters (e.g., 0.15, 0.25 AU) potentially habitable planets can form, but for semi-major axes of 0.5 AU or greater their formation is suppressed. We show that the presence of an outer giant planet such as Jupiter does not enhance the water content of the terrestrial planets, but rather decreases their formation and water delivery timescales. We speculate that asteroid belts may exist interior to the terrestrial planets in systems with close-in giant planets.     

Internal structure of massive terrestrial planets

Planetary formation models predict the existence of massive terrestrial planets and experiments are now being designed that should succeed in discovering them and measuring their masses and radii. We calculate internal structures of planets with one to ten times the mass of the Earth (Super-Earths) to obtain scaling laws for total radius, mantle thickness, core size and average density as a function of mass. We explore different compositions and obtain a scaling law of R∝M0.267-0.272 for Super-Earths. We also study a second family of planets, Super-Mercuries with masses ranging from one mercury-mass to ten mercury-masses with similar composition to the Earth's but with a larger core mass fraction. We explore the effect of surface temperature and core mass fraction on the scaling laws for these planets. The scaling law obtained for the Super-Mercuries is R∝M∼0.3.     

Obliquity evolution of extrasolar terrestrial planets

We have investigated the obliquity evolution of terrestrial planets in habitable zones (at ∼1 AU) in extrasolar planetary systems, due to tidal interactions with their satellite and host star with wide varieties of satellite-to-planet mass ratio (m/Mp) and initial obliquity (γ₀), through numerical calculations and analytical arguments. The obliquity, the angle between planetary spin axis and its orbit normal, of a terrestrial planet is one of the key factors in determining the planetary surface environments. A recent scenario of terrestrial planet accretion implies that giant impacts of Mars-sized or larger bodies determine the planetary spin and form satellites. Since the giant impacts would be isotropic, tilted spins (sinγ₀∼1) are more likely to be produced than straight ones (sinγ₀∼0). The ratio m/Mp is dependent on the impact parameters and impactors' mass. However, most of previous studies on tidal evolution of the planet-satellite systems have focused on a particular case of the Earth-Moon systems in which m/Mp?0.0125 and γ₀∼10° or the two-body planar problem in which γ₀=0° and stellar torque is neglected. We numerically integrated the evolution of planetary spin and a satellite orbit with various m/Mp (from 0.0025 to 0.05) and γ₀ (from 0° to 180°), taking into account the stellar torques and precessional motions of the spin and the orbit. We start with the spin axis that almost coincides with the satellite orbit normal, assuming that the spin and the satellite are formed by one dominant impact. With initially straight spins, the evolution is similar to that of the Earth-Moon system. The satellite monotonically recedes from the planet until synchronous state between the spin period and the satellite orbital period is realized. The obliquity gradually increases initially but it starts decreasing down to zero as approaching the synchronous state. However, we have found that the evolution with initially tiled spins is completely different. The satellite's orbit migrates outward with almost constant obliquity until the orbit reaches the critical radius ∼10-20 planetary radii, but then the migration is reversed to inward one. At the reversal, the obliquity starts oscillation with large amplitude. The oscillation gradually ceases and the obliquity is reduced to ∼0° during the inward migration. The satellite eventually falls onto the planetary surface or it is captured at the synchronous state at several planetary radii. We found that the character change of precession about total angular momentum vector into that about the planetary orbit normal is responsible for the oscillation with large amplitude and the reversal of migration. With the results of numerical integration and analytical arguments, we divided the m/Mp-γ₀ space into the regions of the qualitatively different evolution. The peculiar tidal evolution with initially tiled spins give deep insights into dynamics of extrasolar planet-satellite systems and discussions of surface environments of the planets.     

On the thermal evolution of the terrestrial planets

Physical and chemical constraints for such different planetary objects as the Earth, the Moon and meteorite parent bodies can best be satisfied by thermal history models having high initial temperatures. On the basis of thermal calculations it is suggested that the evolution of the other terrestrial planets (Mars, Venus and Mercury) was also characterized by high initial temperatures. Under these conditions, melting and, consequently, fractionation would set in at an early stage. Because of the resulting redistribution of the long-lived radioactive heat sources and the concentration of these elements in the surface layers, large-scale differentiation could be achieved by partial melting.Paper presented at the Lunar Science Institute Conference on Geophysical and Geochemical Exploration of the Moon and Planets, January 10–12, 1973.     

Cratering of the terrestrial planets by Apollo objects

Abstract— Absolute and relative cratering rates on the terrestrial planets have been calculated using the same asteroidal collision model and Monte Carlo program used for previous studies of the terrestrial meteorite flux, the steady-state number of Apollo-Amor objects, and the orbital distribution of both meteorites and Apollo-Amor objects. The most straightforward result is that projectiles from the asteroid belt appear to provide about one-third the observed present-day production of terrestrial craters larger than 10 km in diameter. When uncertainties in the calculations and observations are included, it cannot be excluded that the entire terrestrial cratering flux is asteroidal. On the other hand, assumption of an additional Apollo-Amor source of extinct comets, in the same quantity permitted by Apollo-Amor observations, provides better agreement with the observed cratering rate. In addition, a significant (e.g., ～30%) terrestrial contribution from active long and short period comets is acceptable within the uncertainties of the assumptions required. The ratios of the cratering rates on the different terrestrial planets are somewhat sensitive to the assumed source. A purely asteroidal source predicts a martian cratering rate per unit area about four times that on Earth, whereas the difference is reduced to about a factor of two for the mixed asteroid-extinct comet source. The opposite effect is found for Mercury. As discussed by previous authors, the predicted lunar cratering rate is significantly higher than that observed. It is not clear whether this is a result of scaling to impacts on a body considerably smaller than Earth, or if it indicates an increase in the cratering flux during the Phanerozoic.     

Primodial retention of carbon by the terrestrial planets

Chemical equilibrium calculations on the stability of pure and dissolved graphite and cohenite (Fe₃C), several other carbides, and several carbonates have been carried out for a system with solar elemental abundances over a very wide range of temperature and pressure. The calculated abundances of condensed carbon compounds are similar to the observed inventories on Earth and Venus, but fully 10 times smaller than the minimum carbon abundance found in ordinary chondrites. The total carbon content of most iron meteorites is compatible with their origin as a cooling FeNiCSP solution which was saturated with dissolved carbon at the solidus, such as would be produced by melting an ordinary chondrite, not by direct condensation from or equilibrium with the primitive solar nebula. It is argued that the carbon content of Mars need not be appreciably greater than that of the Earth. Material with even lower formation temperatures than Mars, such as the primitive material in the asteroid belt, may retain substantially more carbon as disequilibrium polymeric organic matter, possibly by the Fischer-Tropsch mechanism favored by Anders. Carbonates are not found as equilibrium products in a solar-composition system, and are probably secondary alteration products. CaCO₃ might, however, persist in a solar-composition gas at temperatures below 460°K and pressures below 10^?6.6 bar. The most stable condensed carbon compounds are found to be graphite, Fe₃C, and possibly TiC, all in solid solution in the metal phase.     

Magnetism and thermal evolution of the terrestrial planets

Of the terrestrial planets, Earth and probably Mercury possess substantial intrinsic magnetic fields generated by core dynamos, while Venus and Mars apparently lack such fields. Thermal histories are calculated for these planets and are found to admit several possible present states, including those which suggest simple explanations for the observations; whule the cores of Earth and Mercury are continuing to freeze, the cores of Venus and Mars may still be completely liquid. The models assume whole mantle convection, which is parameterized by a simple Nusselt-Rayleigh number relation and dictates the rate at which heat escapes from the core. It is found that completely fluid cores, devoid of intrinsic heat sources, are not likely to sustain thermal convection for the age of the solar system but cool to a subadiabatic, conductive state that can not maintain a dynamo. Planets which nucleate an inner core continue to sustain a dynamo because of the gravitational energy release and chemically driven convection that accompany inner core growth. The absence of a significant inner core can arise in Venus because of its slightly higher temperature and lower central pressure relative to Earth, while a Martian core avoids the onset of freezing if the abundance of sulfur in the core is ?15% by mass. All of the models presented assume that (I) core dynamos are driven by thermal and/or chemical convection; (ii) radiogenic heat production is confined to the mantle; (iii) mantle and core cool from initially hot states which are at the solidus and superliquidus, respectively; and (iv) any inner core excludes the light alloying material (sulfur or oxygen) which then mixes uniformly upward through the outer core. The models include realistic pressure and composition-dependent freezing curves for the core, and material parameters are chosen so that the correct present-day values of heat outflow, upper mantle temperature and viscosity, and inner core radius are obtained for the earth. It is found that Venus and Mars may have once had dynamos maintained by thermal convection alone. Earth may have had a completely fluid core and a dynamo maintained by thermal convection for the first 2 to 3 by, but an inner core nucleates and the dynamo energetics are subsequently dominated by gravitational energy release. Complete freezing of the Mercurian core is prohibited if it contains even a small amount of sulfur, and a dynamo can be maintained by chemical convection in a thin, fluid shell.