Analytical Solutions of Love Numbers for a Hydrostatic Ellipsoidal Incompressible Homogeneous Earth

Tidal forces acting on the Earth cause deformations and mass redistribution inside the planet involving surface motions and variation in the gravity field, which may be observed in geodetic experiments. Because for space geodesy it is now necessary to achieve the mm level in tidal displacements, we take into account the hydrostatic flattening of the Earth in the computation of the elasto-gravitational deformations. Analytical solutions are derived for the semi-diurnal tides on a slightly elliptical homogeneous incompressible elastic model. That simple analytical Earth’s model is not a realistic representation of any real planet, but it is useful to understand the physics of the problem and also to check numerical procedures. We rediscover and discuss the Love’s solutions and obtain new analytical solutions for the tangential displacement. We extend these analytical results to some geodetic responses of the Earth to tidal forces such as the perturbation of the surface gravity field, the tilt and the deviation of the vertical with reference to the Earth’s axis.     

Influence of astronomical factors on the dynamics of lithospheric plates

We suggest the concept of the Earth’s lithosphere as a geocosmic system of mobile lithospheric plates affected by both external astronomical influences (solar radiation, tides) and planetary factors—the Earth’s atmosphere, hydrosphere, and mantle convection. The annual period in seismicity is shown to have a clear cosmic origin related to the seasonal periodicity of solar radiation in the northern and southern hemispheres. The atmosphere can act as a transmitter of the annual periodicity to the lithospheric plates. The formulated concept of the lithosphere has led us to put forward testable hypotheses about the dynamical atmospherelithosphere relationship. These hypotheses form the basis for the next program of research on the Earth’s lithosphere as a geocosmic system.     

On the model structure of the gravity field of Mars

A discussion is presented about the constraints used in constructing a model for the internal structure of Mars. The most important fact is that the Martian chemical model proposed by Wänke and Dreibus (WD) has stood the test of time. This means that the chondritic ratio Fe/Si = 1.71 can be used as a constraint in constructing an interior structure model of the planet. Consideration is given to the constructing of the reference surface of Mars. It is concluded that the effectively hydrostatic-equilibrium model of Mars is well suited for this purpose. The areoid heights and gravity anomalies in the model of Mars are calculated. The results are shown in the figures (maps) and comments made. The results are compared with the similar data for the Earth. Mars deviates much more strongly from the hydrostatic equilibrium than the Earth. It is suggested that the average thickness of the Martian elastic lithosphere should exceed that of the Earth’s continental lithosphere.     

Search for extrasolar planets by polarimetry

Optical polarimetry is suggested as a new method for detecting “hot Jupiter” planets around stars. The polarimetric search method has been tested experimentally; for this purpose, the necessary astronomical observations and their processing have been performed. The results obtained allow us to assert with caution that the suggested method yields positive results and can be of use both in searching for exoplanets and in refining their masses. According to our results, a tangential transit of the planet 51 Peg b may be observed. The angle between the orbital plane of 51 Peg b and the observer’s direction must then be small, sin i ≈ 1, and the mass of 51 Peg b must be close to 0.46 M _J (Jupiter mass).     

Transient mantle layering and the episodic behaviour of Venus due to the ‘basalt barrier’ mechanism

Numerical models of mantle convection that include the ‘basalt barrier’ mechanism are explored for Venus. The ‘basalt barrier’ mechanism is due to the positive buoyancy of subducted basaltic crust between the mantle depths of 660 and 750 km. The inclusion of this mechanism in models of Earth’s evolution has been shown to cause episodic mantle layering early in Earth history and we explore whether it can also operate on Venus. The models presented here include a moderately mobile lithosphere, which is not representative of the current state of Venus, but this allows us to exclude the effects of episodic lithosphere mobility and thus to isolate the effect of the basalt barrier. This is a step in a systematic approach to models with a mostly-static lithosphere. We find the basalt barrier does yield episodically layered mantle convection in some Venus models. The likelihood of episodic layering is increased by Venus high surface temperature and by its less mobile or immobile lithosphere. Surprisingly, secondary differences from Earth, including the lower gravity, density and mantle depth also promote episodic layering. The models suggest that mantle layering and overturns may still be likely to occur in Venus. The breakdown of mantle layering and consequent mantle overturns would lead to dramatic episodes of volcanism, formation of large amounts of crust, and tectonic activity on the planet’s surface, as has been inferred to have happened on Venus around 500 Ma ago from surface morphology and cratering. These results thus suggest that a transient layering of the mantle by the ‘basalt barrier’ mechanism and mantle overturns may be part of the explanation for Venus’s recent resurfacing.     

Giant impacts and the initiation of plate tectonics on terrestrial planets

Earth is the only terrestrial planet with present-day lithosphere recycling through plate tectonics. However, theoretical models of mantle convection based on general considerations find that all the terrestrial planets should be operating in the stagnant lid regime, in which the planets are one-plated and there is no lithosphere recycling. The stagnant lid regime is a consequence of the strong viscosity contrast across the convective layer, and therefore the upper lid (roughly equivalent to the lithosphere) must be sufficiently weakened in order to be mobilized. Here I propose that giant impacts could have provided the upper layer weakening required for surface recycling, and hence for plate tectonics, to initiate on the early Earth. Additionally, giant impacts originated lithosphere thickness and density differences, which might contribute to the initiation of subduction. Impacts are more energetic for Earth than for Mars, which could explain the likely early existence of plate tectonics on the Earth whereas Mars never had lithosphere recycling. On the other hand, convection on Mercury and the Moon might be sluggish or even inexistent, implying a reduced influence of giant impacts on their internal dynamics, whereas there is no record of the earliest geological history of Venus, which obscures any discussion on the influence of giant impacts on their internal dynamics.     

On the possibility of microbiota transfer from Venus to Earth

The possibility of the clouds of Venus providing habitats for extremophilic microorganisms has been discussed for several decades. We show here that the action of the solar wind leads to erosion of parts of the atmosphere laden with aerosols and putative microorganisms, forming a comet-like tail in the antisolar direction. During inferior conjunctions that coincide with transits of the planet Venus this comet-like tail intersects the Earth’s magnetopause and injects aerosol particles. Data from ESA’s Venus Express spacecraft and from SOHO are used to discuss the ingress of bacteria from Venus into the Earth’s atmosphere, which we estimate as ～10¹¹–10¹³ cells for each transit event.     

On the deviation of the lunar center of mass to the East. Two possible mechanisms based on evolution of the orbit and rounding off the shape of the Moon

From the observations of the gravitational field and the figure of the Moon, it is known that its center of mass (briefly COM) does not coincide with the center of figure (COF), and the line “COF/COM” is not directed to the center of the Earth, but deviates from it to the South–East. Here we study the deviation of the lunar COM to the East from the mean direction to Earth.At first, we consider the optical libration of a satellite with synchronous rotation around the planet for an observer at a point on second (empty) orbit focus. It is found that the main axis of inertia of the satellite has asymmetric nonlinear oscillations with amplitude proportional to the square of the orbit eccentricity. Given this effect, a mechanism of tidal secular evolution of the Moon’s orbit is offered that explains up to \(20\%\) of the known displacement of the lunar COM to the East. It is concluded that from the alternative—evolution of the Moon’s orbit with a decrease or increase in eccentricity—only the scenario of evolution with a monotonous increase in orbit eccentricity agrees with the displacement of lunar COM to the East. The precise calculations available confirm that now the eccentricity of the lunar orbit is actually increasing and therefore in the past it was less than its modern value, \(e = 0.0549\).To fully explain the displacement of the Moon’s COM to the East was deduced a second mechanism, which is based on the reliable effect of tidal changes in the shape of the Moon. For this purpose the differential equation which governs the process of displacement of the Moon’s COM to the East with inevitable rounding off its form in the tidal increase process of the distance between the Earth and the Moon is derived. The second mechanism not only explains the Moon’s COM displacement to the East, but it also predicts that the elongation of the lunar figure in the early epoch was significant and could reach the value \(\varepsilon\approx0.31\). Applying the theory of tidal equilibrium figures, we can estimate how close to the Earth the Moon could have formed.     

The internal activity and thermal evolution of Earth-like planets

We model the internal thermal evolution of planets with Earth-like composition and masses ranging from 0.1 to 10 Earth masses over a period of 10 billion years. We also characterize the internal activity of the planets by the velocity of putative tectonic plates, the rate at which mantle material is processed through melting zones, and the time taken to process one mantle mass. The more massive the planet the larger its processing rate (?), which scales approximately as ?∝M0.8-1.0. The processing times for all the planets increase with time as they cool and become less active. As would be expected, the surface heat flow scales with planet mass. All planets have similar declines in mantle temperature except for the largest, in which pressure effects cause a larger decline. The larger planets have higher mantle temperatures over all times. The less massive the planet, the larger the decrease in core temperature with time. The core heat flow is also found to decrease more rapidly for smaller planet masses. Finally, rough predictions are made for the time required to generate an atmosphere from estimates of the time to degas water and carbon dioxide in mantle melting zones. The degassing times depend strongly on the initial temperature of the planet, but for the temperatures used in our model all the planets degas within ∼32 Ma after their formation.     

Motion of a Meteoroid Released from an Asteroid

Evidence of asteroid surface features as regolith grains and larger boulders implies resurfacing possibility due to external forces such as gravitational tidal force during close planet encounters. Motion of a meteoroid released from an asteroid in the gravitational fields of the asteroid and the Earth is modeled. We are interested mainly in a distance between the meteoroid and the asteroid as a function of the time. Applications to Itokawa and some close approaching NEAs are presented.     

Exchange of global mean angular momentum between an atmosphere and its underlying planet

This paper investigates the exchange of global mean angular momentum between an atmosphere and its underlying planet by a simple model. The model parameterizes four processes that are responsible for zonal mean momentum budget in the atmospheric boundary layer for a rotating planet: (i) meridional circulation that redistributes the relative angular momentum, (ii) horizontal diffusion that smoothes the prograde and retrograde winds, (iii) frictional drag that exchanges atmospheric angular momentum with the underlying planet, and (iv) internal redistribution of the zonal mean momentum by wave drag. It is shown that under a steady-state or a long-term average condition, the global relative angular momentum in the boundary layer vanishes unless there exists a preferred frictional drag for either the prograde or the retrograde zonal wind. We further show quantitatively that one cannot have either a predominant steady prograde or retrograde wind in the boundary layer of a planetary atmosphere. The parameter dependencies of the global relative angular momentum and the strength of the atmospheric circulation in the boundary layer are derived explicitly and used to explain the observational differences between the atmospheres of Earth and Venus.     

Geoid and Gravity in Earth Sciences – An Overview

Precise global geoid and gravity anomaly information serves essentially three different kinds of applications in Earth sciences: gravity and geoid anomalies reflect density anomalies in oceanic and continental lithosphere and the mantle; dynamic ocean topography as derived from the combination of satellite altimetry and a global geoid model can be directly transformed into a global map of ocean surface circulation; any redistribution or exchange of mass in Earth system results in temporal gravity and geoid changes. After completion of the dedicated gravity satellite missions GRACE and GOCE a high standard of global gravity determination, both of the static and of the time varying field will be attained. Thus, it is the right time to investigate the future needs for improvements in the various fields of Earth sciences and to define the right strategy for future gravity field satellite missions.     

Numerical simulation of the general circulation of the Cytherean lower atmosphere

The principal features which distinguish the atmosphere on Venus from that of the Earth are the slow rotation of the planet, the large mass of the atmosphere, and the opacity of the atmosphere to long-wave radiation. The slow rotation of the planet gives rise, first of all, to nongeostrophuc dynamics (the atmosphere gas has a tendency to move along the pressure gradient), with the result that the region of the main influx of solar energy is located on one side of the planet, and the region of maximum cooling on the other. These considerations lead to a much simpler scheme of circulation than that in the Earth's atmosphere.The large mass of the atmosphere is the cause of a high thermal and mechanical inertia, which explains why the atmospheric circulation is asymmetrical relative to the solar-antisolar axis. The daily center of circulation is displaced to the second half of the Cytherean solar day, i.e., to the line of zero budget of thermal energy corresponding to a height of the Sun abobe the horizon of about 20°. The notions of cold and warm regions are very relative for Venus. While the horizontal temperature differences on the Earth may reach 100°, a mean horizontal temperature drop as small as 3° in the Cytherean atmosphere may be looked upon as an exceptional phenomenon. This high thermal homogeneity is due to a very large thermal inertia, with cooling at the poles never manifesting itself in the temperature fields obtained.The opacity of the Cytherean atmosphere to long-wave radiation results in vertical heat transfer by turbulence, mesoscale convection, and large-scale currents. This produces adiabatic stratification in the troposphere and a high temperature in the lower layers.These phenomena were studied in a general manner using two- and three-level models. Steps have recently been undertaken to investigate in greater detail the vertical structure of the troposphere on Venus using ten-level models. It appeared that the vertical dynamic structure of the troposphere is very much dependent on the distribution in height of the solar energy influx. In the greenhouse model, the entire atmosphere is affected by circulation. Pronounced velocity maxima are observed in the lower and upper layers. In a model with adsorption of solar radiation in the upper layer, the velocity is small in the lower layers, but it rapidly increases and changes its direction several times in the upper layers. The mean kinetic energy of the atmosphere proves to be two to three times smaller than in the greenhouse model.Attempts have been made in the calculations to find the principal modes of the statistical fluctuations. The results obtained show that atmospheric circulation may be represented by a global mean basic state following the rotation of the planet with deviations from that basic state which are indeterminate disturbances. The mean basic state exhibits a high degree of symmetry relative to the equator. On account of nonlinearity, the disturbances were observed in all the models independently of space and time resolution. This phenomenon appears to reflect the actual properties of the Cytherean atmosphere and has no bearing on the details of the numerical scheme.     

Delivery of Water and Volatiles to the Terrestrial Planets and the Moon

From modeling the evolution of disks of planetesimals under the influence of planets, it has been shown that the mass of water delivered to the Earth from beyond Jupiter’s orbit could be comparable to the mass of terrestrial oceans. A considerable portion of the water could have been delivered to the Earth’s embryo, when its mass was smaller than the current mass of the Earth. While the Earth’s embryo mass was growing to half the current mass of the Earth, the mass of water delivered to the embryo could be near 30% of the total amount of water delivered to the Earth from the feeding zone of Jupiter and Saturn. Water of the terrestrial oceans could be a result of mixing the water from several sources with higher and lower D/H ratios. The mass of water delivered to Venus from beyond Jupiter’s orbit was almost the same as that for the Earth, if normalized to unit mass of the planet. The analogous per-unit mass of water delivered to Mars was two?three times as much as that for the Earth. The mass of water delivered to the Moon from beyond Jupiter’s orbit could be less than that for the Earth by a factor not more than 20.     

Comparative planetology, climatology and biology of Venus, Earth and Mars

Spacecraft studies of the three terrestrial planets with atmospheres have made it possible to make meaningful comparisons that shed light on their common origin and divergent evolutionary paths. Early in their histories, all three apparently had oceans and extensive volcanism; Mars and Earth, at least, had magnetic fields, and Earth, at least, had life. All three currently have climates determined by energy balance relationships involving carbon dioxide, water and aerosols, regulated by solar energy deposition, atmospheric and ocean circulation, composition, and cloud physics and chemistry.This paper addresses the extent to which current knowledge allows us to explain the observed state of each planet, its planetology, climatology and biology, within a common framework. Areas of ignorance and mysteries are explored, and prospects for advances in resolving these with missions within the present planning horizon of the space agencies are considered and assessed.     

Equilibrium rotational stability and figure of Mars

Studies extending over three decades have concluded that the current orientation of the martian rotation pole is unstable. Specifically, the gravitational figure of the planet, after correction for a hydrostatic form, has been interpreted to indicate that the rotation pole should move easily between the present position and a site on the current equator, 90° from the location of the massive Tharsis volcanic province. We demonstrate, using general physical arguments supported by a fluid Love number analysis, that the so-called non-hydrostatic theory is an inaccurate framework for analyzing the rotational stability of planets, such as Mars, that are characterized by long-term elastic strength within the lithosphere. In this case, the appropriate correction to the gravitational figure is the equilibrium rotating form achieved when the elastic lithospheric shell (of some thickness LT) is accounted for. Moreover, the current rotation vector of Mars is shown to be stable when the correct non-equilibrium theory is adopted using values consistent with recent, independent estimates of LT. Finally, we compare observational constraints on the figure of Mars with non-equilibrium predictions based on a large suite of possible Tharsis-driven true polar wander (TPW) scenarios. We conclude, in contrast to recent comparisons of this type based on a non-hydrostatic theory, that the reorientation of the pole associated with the development of Tharsis was likely less than 15° and that the thickness of the elastic lithosphere at the time of Tharsis formation was at least ∼50 km. Larger Tharsis-driven TPW is possible if the present-day gravitational form of the planet at degree 2 has significant contributions from non-Tharsis loads; in this case, the most plausible source would be internal heterogeneities linked to convection.     

AA Dor—An Eclipsing sdOB—Brown Dwarf Binary

AA Dor is an eclipsing, close, post common-envelope binary (PCEB) consisting of a sdOB primary star and an unseen secondary with an extraordinary small mass (M ₂≈ 0.066 M _⊙)—formally a brown dwarf. The brown dwarf may have been a former planet which survived a common envelope (CE) phase and has even gained mass. A recent determination of the components’ masses from results of NLTE spectral analysis and subsequent comparison to evolutionary tracks shows a discrepancy compared to masses derived from radial-velocity and the eclipse curves. Phase-resolved high-resolution and high-SN spectroscopy was carried out in order to investigate this problem. We present results of a NLTE spectral analysis of the primary, an analysis of its orbital parameters, and discuss possible evolutionary scenarios.     

Habitable zones of host stars during the post-MS phase

A star will become brighter and brighter with stellar evolution, and the distance of its habitable zone will become larger and larger. Some planets outside the habitable zone of a host star during the main sequence phase may enter the habitable zone of the host star during other evolutionary phases. A terrestrial planet within the habitable zone of its host star is generally thought to be suitable for the existence of life. Furthermore, a rocky moon around a giant planet may be also suitable for life to survive, provided that the planet–moon system is within the habitable zone of its host star. Using Eggleton’s code and the boundary flux of the habitable zone, we calculate the habitable zone of our Solar system after the main sequence phase. It is found that Mars’ orbit and Jupiter’s orbit will enter the habitable zone of the Solar system during the subgiant branch phase and the red giant branch phase, respectively. And the orbit of Saturn will enter the habitable zone of Solar during the He-burning phase for about 137 million years. Life is unlikely at any time on Saturn, as it is a giant gaseous planet. However, Titan, the rocky moon of Saturn, may be suitable for biological evolution and become another Earth during that time. For low-mass stars, there are similar habitable zones during the He-burning phase as our Solar, because there are similar core masses and luminosities for these stars during that phase.     

Methods of detecting extrasolar planets: I. Imaging

It is proposed that the presence or absence of Jupiter-like planets, and perhaps even Venus-like planets, around nearby stars be studied with a 2-m telescope in Earth orbit. According to the Abbe theory of imaging, the coherence of the light from an unresolvable star can be used to discriminate between planet light and scattered light from the star. Most scattered light is shown to arise from the imperfect figure of the telescope surface, but an analog of a phase contrast trast microscope can be used to control the figure automatically. A number of arrangements are possible for using the interference properties of light to cancel the residual scattered starlight by two to three orders of magnitude without affecting the planet light.     

Obliquity variations of a moonless Earth

We numerically explore the obliquity (axial tilt) variations of a hypothetical moonless Earth. Previous work has shown that the Earth’s Moon stabilizes Earth’s obliquity such that it remains within a narrow range, between 22.1° and 24.5°. Without lunar influence, a frequency map analysis by Laskar et al. (Laskar, J., Joutel, F., Robutel, P. [1993]. Nature 361, 615–617) showed that the obliquity could vary between 0° and 85°. This has left an impression in the astrobiology community that a big moon is necessary to maintain a habitable climate on an Earth-like planet. Using a modified version of the orbital integrator mercury, we calculate the obliquity evolution for moonless Earths with various initial conditions for up to 4 Gyr. We find that while obliquity varies significantly more than that of the actual Earth over 100,000 year timescales, the obliquity remains within a constrained range, typically 20–25° in extent, for timescales of hundreds of millions of years. None of our Solar System integrations in which planetary orbits behave in a typical manner show obliquity accessing more than 65% of the full range allowed by frequency-map analysis. The obliquities of moonless Earths that rotate in the retrograde direction are more stable than those of prograde rotators. The total obliquity range explored for moonless Earths with rotation periods less than 12 h is much less than that for slower-rotating moonless Earths. A large moon thus does not seem to be needed to stabilize the obliquity of an Earth-like planet on timescales relevant to the development of advanced life.