FormalPara Article Highlights

  • Methods to explore the interiors of terrestrial planets and satellites using space missions are described and compared

  • All planets and most satellites are differentiated into iron-rich cores, rocky mantles and crusts and—for the satellites—ice-water layers

  • Mercury, Venus, Moon, Mars, Ganymede and Enceladus are discussed in some detail as examples

1 Introduction

Since the days of Copernicus (1473–1543), Brahe (1546–1601) and Kepler (1571–1630), we know that the Earth is the third planet out from the sun, orbiting the latter at a distance that is—as we learned to understand—within the sun’s habitable zone. While the Earth is the main focus of our interest as our home planet and of this particular collection of articles (Cazenave et al. 2022), we consider some of the other planets out of curiosity and because comparing similar and not so similar objects is a trusted way in science of deepening our understanding and testing hypotheses.

Of particular interest for scientists studying the interior of the Earth are the interiors of other rocky planets and moons of comparable size (Fig. 1). While the Earth is the biggest and most massive of these bodies in our solar system, other planetary systems host what has been termed super-Earths, planets with radii of up to twice the Earth’s and masses of up to ten times the latter. The pressure in super-Earths ranges up to terapascals and is far beyond what is accessible to date in high pressure experiments, at least in static experiments.

Fig. 1
figure 1

Schematic view of the interior structure and magnetic field evolution of Mercury, Venus, Earth, Moon, Mars and Ganymede. Upper panel: The white surface shows the silicate mantle, the light grey surface the fluid core and the dark grey surface the solid inner core and the white dotted area shows a water/ice mantle. Furthermore, the percentages of the crust/mantle volume ratio are shown. Apart from the Earth, the sizes of the different layers (core, solid inner core, silicate mantle, basaltic crust, or ice mantle) are determined with different precision for the bodies; for more detail, see text. Lower panel: The dark grey surface shows remnant magnetized crust. In case of the Moon dark grey shows a strong field and light grey a weaker field indicating the strong decrease of the magnetic field strength with time. The small hatched regions at the right hand side of the diagram for Earth, Mercury and Ganymede indicate the magnetic fields today. It is important to note that the lack of observed remnant magnetized crust is no clear sign that also an internally generated magnetic field was lacking at that time. This applies in particular to the early evolution of the planets and Venus and Ganymede. It is difficult for crustal rock on Venus to get remnant magnetization due to the high surface temperature close to or above the Curie temperature of magnetic minerals, and the silicate crust of the Ganymede is located under a thick layer of ice and water

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Amongst the bodies of interest in the solar system is Venus as the most similar to the Earth in terms of radius and mass, the former being just a few percent smaller, Mars about half the size and roughly 10 per cent in mass but with a climate that resembles that of Earth’s more than that of the other terrestrial planets, and Mercury, the smallest and least massive but the one with the highest density if evaluated at standard conditions of pressure and density (105 Pa and 273 K) and a present-day intrinsic magnetic field (Fig. 1). Geophysical data are comparatively scarce for these planets, but missions have allowed measurements of the gravity and magnetic fields, and imaging and spectroscopy have allowed geological map** and analysis of the compositions of surface rock and of the atmospheres. Information on planetary composition comes in addition from samples from Mars that we have in our meteorite collection, but also from in-situ measurements from lander and rover missions on Mars (e.g., Abbey et al. 2019) and Venus (e.g., Surkov et al.1984). Most recently, Mars has been the target of a dedicated geophysical observatory mission, InSight (Banerdt et al. 2020), that placed an extremely sensitive seismometer with broadband and short period sensors on the surface of the planet. It also enabled highly accurate measurements of the rotation axis, of the local—small scale—magnetic field and its time variation and of atmosphere pressure, temperature and wind speed (Banfield et al. 2019; Folkner et al. 2018; Lognonne et al. 2019) and of the thermal and mechanical properties of the Martian soil (Grott et al 2021; Spohn et al. 2021a) and interior (Lognonne et al. 2020; Giardini et al. 2020; Knappmeyer-Endrun et al. 2021; Khan et al. 2021; Stähler et al. 2021. Unfortunately, attempts to emplace thermal sensors at the required depth of 3–5 m were not successful and the goal of measuring for the first time the heat flow from another terrestrial planet had to be abandoned. However, the measurements of the other instruments were not affected by this and operated nominally.

Of the planetary satellites, the Moon is of particular interest not the least because it has been explored by humans during the Apollo program (1969–1972), when not only were the gravity and magnetic fields mapped, but seismic data was also recorded and the heat flow was measured (e.g., Taylor 2016; Jolliff et al. 2006 for lunar mission research reviews). In addition, the lunar rotation axis has been accurately measured over extended periods of time, the composition of surface rocks has been mapped and roughly 400 kg of lunar rock have been collected and brought to Earth. The origin of the Moon is closely linked to the early history of Earth (e.g., Canup et al. 2021) and the surface of the Moon, being devoid of traces of erosion apart from the effects of impacts is recording data pertinent to the long-eroded record of the early history of our own planet.

Other planetary satellites of particular interest to geophysics and covered in this article include Ganymede, the biggest of the Galilean satellites, covered with a thick ice mantle, with traces of lateral tectonic movements and a self-sustained magnetic field, and Enceladus, measuring only 500 km in diameter with geysers emanating from the surface near the south pole. Enceladus has been speculated to potentially harbor primitive life and together with Mars, Titan, and Europa belongs to the bodies in the solar system that are regarded as those with the greatest potential for detecting extraterrestrial life.

The evolution of these bodies depends essentially on the functioning of their heat engines. The latter are mostly driven by heat deposited during planet formation and generated by the decay of radioactive elements and, in some cases, by tidal dissipation. In the early evolution large catastrophic impacts likely played an important role as has been discussed for example for the Moon (compare Sect. 3.2). Convection is expected to take place in their rocky mantles, iron-rich cores, and outer ice layers, as has been recognized early on by, e.g., Schubert (1979). In the core, dynamo mechanisms can generate magnetic fields as we will discuss further below in Sect. 2.3. In the rock and ice mantle, convection and the dominant tectonic style are significantly influenced by their composition, the internal structure, and the surface conditions. Among the tectonic styles, a distinction is made between the plate tectonic regime of the Earth and the stagnant lid regime of the other planetary bodies like Mercury, Mars and the Moon as we will discuss in Sect. 3. Mixed tectonic styles may have existed, such as episodes of plate tectonics and stagnant lid convection, that are being discussed for Venus (Sect. 3.2). In the case of icy moons, the thermal evolution is additionally shaped by the outer ice layer, which may have a water ocean either trapped between ice layers (Ganymede, Sect. 3.5) or in direct contact with the silicate mantle (Enceladus, Sect. 3.6).

This article reviews the current state of knowledge about the interior of rocky planetary bodies, including constraints on their interior composition, internal structure, magnetic field generation and their thermo-chemical evolution. We describe the data obtained from planetary missions and ground-based observations and the methods used to analyze these data. We conclude with what understanding we have gained so far of Mercury, Venus, Mars, Ganymede and Enceladus with the help of mission data, modeling and laboratory experiments, and put these results in context with what we know about Earth.

2 Exploring the Deep Interior of Planets

2.1 Composition

Exploring the deep interior of planets commonly relies on the use of indirect methods, because direct access is severely limited. Even on Earth, the deepest hole ever drilled (the Kola Superdeep Borehole, e.g., Pavlenkova 1992) reached a depth of only ~ 12 km before the temperature at the bottom of the borehole exceeded the technical limits of the drilling equipment. This depth corresponds to less than 0.2 per cent of the distance to the center of our planet. Drilling holes in other planets and moons has been even more limited to date: the deepest hole in the Moon (drilled during the Apollo 17 mission in 1972) reached less than 3 m (e.g., Alton and Waltz 1980) and the deepest hole in Mars (dug during the InSight mission) reached ~ 40 cm depth (Grott et al. 2021)–both representing almost negligible distances compared to the bodies overall sizes. No samples of the iron-rich metal cores of the Earth or planets and moons have ever been obtained, and are unlikely to ever be obtainable in the foreseeable future. Core fragments of smaller bodies have been retrieved in the form of iron meteorites, but these represent remnants of planetesimals that were much smaller than the terrestrial planets, and that were broken up and destroyed by violent impacts early in the history of the solar system. NASA plans to fly the Psyche mission to M-type asteroid 16 Psyche, possibly an exposed core of a proto-planet (Elkins-Tanton et al. 2014).

In the absence of deep drilling, surface samples can provide clues about the interior composition and internal structure of rocky planets and moons. Volcanic samples that can be found at the surface form by partial melting of source rocks in the interior. Their chemical compositions reflect, albeit in a non-linear and often non-unique fashion, the composition of the source and pressure/temperature conditions of melting. In addition, volcanic deposits can contain xenoliths, solid rock samples incorporated into magma during magma transport from interior source to the surface. These provide a valuable window into the composition and mineralogy of the interior underneath a volcano. On Earth, some volcanism-related samples retrieved at Earth’s surface originated at up to several 100 km depth (far exceeding the bottom of the deepest drill hole), with some geochemical evidence as reviewed by Dehant et al. (2022), suggesting that a small part of some volcanic rocks may have originated near the core-mantle boundary (~ 2900 km depth).

Tectonic processes including subduction and orogenesis can lead to substantial vertical transport of rock masses, and if accompanied by significant erosion can exhume rocks initially formed tens of kilometers deep all the way to the surface. For example, some of the deepest-formed rocks presently found at the surface in Europe are located in the Swiss Alps, brought to the surface during the Alpine orogeny (e.g., Dobrzhinetskaya et al. 1996). Tectonic emplacement of interior rocks onto the surface is limited in the solar system, as plate tectonics appears to be limited to Earth at present. Strong vertical movements bringing deep interior rocks to the surface can also be induced by impacts–a process that has left clear marks on the surfaces of almost all rocky bodies in the solar system. Large impacts can directly expose subsurface material by evaporating and ejecting overlying rocks or ice. Rebounding of the subsurface after larger impacts leads to the formation of a central peak in the crater, with the material exposed in this peak originating from deep below the crater floor (e.g., Flahaut et al. 2012). The precise depth from which central peak material originated depends, amongst other factors, on the size of the impact (and hence on the size of the crater).

Surface rock samples have been collected, and returned to Earth, only from the Moon (thanks to the Apollo, Luna and Chang’e missions), and asteroids Itokawa and Ryugu (thanks to the two Hayabusa missions) (Kawaguchi et al. 2008), with samples from the surface of asteroid Bennu currently on their way back to our planet (OSIRIS-Rex mission) (Lauretta et al. 2017). Sample return from Mars is being planned for this decade. The NASA Perserverance and the ESA Pasteur rovers will be collecting samples that are planned to be returned to Earth by a dedicated mission (e.g., Kminek et al. 2021). An extensive review of past and outlook to future sample return missions is provided in Anand et al. (2021). On Venus, Mars, and the Moon, lander missions have obtained additional compositional data in situ on the surface. The accuracy and precision of these data is necessarily limited compared to that achievable for returned samples.

For Mercury, the Moon, Mars and outer planetary bodies, surface compositional data is available from remote sensing missions (again with limitations on accuracy, precision, and spatial resolution). A major advantage of remote sensing and lander missions without a sample return component is that they provide more complete global coverage of surface mineralogy and composition compared to the very limited surface areas from which samples have currently been obtained. On top of this, remote sensing missions can provide mineralogical and compositional information for areas that cannot be reached by lander or rover missions, such as steep mountain slopes. As an example, the mineralogy and chemical composition of the central peaks of lunar craters can be mapped from space, and remote sensing analyses of the composition of central peaks in lunar craters of different sizes has been used to map the vertical structure of the lunar crust, as well as its lateral homogeneity (e.g., Martinot et al. 2018, 2020)–something that cannot be done on the basis of sample return or in situ data.

Naturally, combinations of remote sensing, landing/roving, and sample return missions provide most information. In this case, accuracy and precision of space-based remote sensing and/or lander/rover in situ observations can be improved by the data from returned samples, as they enable so-called ground truthing. At the moment this integrated approach is limited to the Moon. A combination of remote sensing and sample return missions is especially important for bodies with a significant atmosphere such as Mars, as the atmosphere complicates remote sensing observations of the surface.

Surface samples from some bodies across the solar system are available on Earth in the form of meteorites. Some of these are remnants of primitive bodies formed very soon after the formation of the solar system (iron meteorites, stony iron meteorites, chondrites); others (the achondrites) were ejected from larger, differentiated bodies including the Moon, Mars, and asteroids. The former provide a cross-section of the nature of potential building blocks of rocky planets and form the basis for the development of models for the bulk chemical composition of Mercury, Venus, Earth, and Mars. The latter provide further insight into the composition of the outer crust and mantle of the Moon, Mars and large asteroids such as Vesta, even though it is often impossible to link these meteorites to a specific location on the surface of the parent body.

Tiny aliquots of returned samples as well as meteorites can be dissected in laboratories on Earth and analyzed for their mineralogy, major and trace elements, stable isotopic, and radiogenic isotopic compositions. These analyses provide constraints on the composition of the silicate reservoirs of planets (planetary crusts and mantles). Sample age information is important as it can yield constraints on models of the dynamics of a planetary interior as a function of time. To link such compositional and age information with geodetic and geophysical data on a planet’s interior structure, experiments in which the high-pressure, high-temperature interior conditions in planets are reproduced are required. Such experiments are important for two reasons. First, they can be used to constrain the chemical composition as well as the formation conditions of planetary mantles and cores, providing boundary conditions for thermochemical models of the evolution of planetary interiors from their origin until the present day. Second, they can help determine the mineralogy and physical state (solid versus molten), as well as the physical properties, of planetary crusts, mantles, and cores as a function of depth, providing independent tests of geophysical models of the present-day geodetic and geophysical properties (such as physical state, density and seismic wave propagation speeds) of planetary interiors.

An overview of the techniques used to recreate the high-pressure and high-temperature conditions in planetary interiors in the laboratory is provided in Dehant et al. (2022). In short, small (on the order of cubic millimeter-sized) synthetic or natural samples consisting of powdered silicate glass and/or iron rich metal of the appropriate composition can be compressed and simultaneously heated in a high-pressure apparatus. Figure 2 shows some examples of the types of experiments that can be performed to assist planetary interior studies. Experiments can be focused on determining the temperatures required to melt a planet’s or moon’s mantle at different depths (Fig. 2a). This information is required to constrain the source depth and temperature at which volcanic rocks found on the surface were formed. Other experiments focus on chemical equilibration between model cores and mantles of planets and moons (Fig. 2b). In such experiments, molten iron-rich metal is chemically equilibrated with molten rock to mimic the conditions of core formation early in a body’s history, and to determine how different elements distribute themselves between metal and silicate during this process. Such information is vital to ascertain under which conditions core formation occurred, and can help constrain the chemical composition of the core of a body on the basis of the chemical composition of its silicate shell. A third type of experiment is focused on determining which minerals are stable at different depths within a planetary mantle or core, and on determining key physical properties of these minerals such as their density or rheological behavior (e.g., Fig. 2c). Complementing and extending such laboratory techniques are computer simulations. Ab initio molecular dynamics approaches are increasingly used to calculate physical and chemical properties of minerals and melts at conditions that cannot be reached in the laboratory, with the added advantage of providing atomic-scale views of processes taking place at these conditions.

Fig. 2
figure 2

a Back-scatter electron (BSE) microscope image of a polished section through the run product of a high-pressure experiment mimicking the interior of the Moon (after Lin et al. 2015) and chapters therein]. Seismology uses the propagation of seismic or acoustic waves generated mostly by quakes through the interior of the Earth and other rocky planetary bodies. The velocities, polarizations and amplitudes as well as the direction of these waves depend on the elastic properties of the interior, in particular on its mineralogy and composition as well as on temperature and pressure. By comparing observed propagation properties with laboratory measurements at high pressure and high temperature conditions (Sect. 2.1), seismological data can provide strong constraints on possible interior layering, composition and mineralogy. Seismic waves travel as compressional and shear waves and are reflected and refracted at discontinuities. They are subject to dispersion and attenuation and compressional waves may induce shear waves and vice versa. In addition to body waves traveling through the interior, seismic energy may propagate as surface waves. Because the velocities of these waves differ, with compressional waves being the fastest followed by shear and surface waves, differences in arrival times of phases can be used. On Earth, strong seismic events are monitored by seismic observatories all over the world while smaller events are recorded by local arrays of stations, sometimes arranged as antennas. Because the origin of the seismic event is usually not known a priori at least three recording stations are necessary to uniquely determine the location of the source from one single phase, e.g., compressional waves.

Seismology is an iterative method that improves prior knowledge with every additional event recorded and evaluated. Seismology started in the late nineteenth century to explore the variation of the seismic velocities with depth leading to the discovery of the major zones such as the core, the upper and lower mantles and the crust and then to 1-D reference models such as the much used IASPEI-91-model (Kennet and Engdahl 1991) or the older PREM model (Dziewonski and Anderson 1981). The huge amount of data that has been accumulated through more than a century of seismic observation has led to detailed knowledge of the Earth’s interior with seismic tomography (e.g., Thurber and Ritsema 2015) allowing exploration of the lateral variation of the velocities and the map** of undulations of interfaces as well as of tectonic features such as subducting slabs.

The Apollo 11 mission in July 1969 not only landed the first human being on the Moon but the Apollo astronauts succeeded in installing the first seismometer on the surface of an extraterrestrial body, followed by three more successful installations by Apollo missions (e.g., Lognonne and Johnson 2015). Two seismometers were aboard flown on Viking 1 and 2 sent to Mars. The Viking seismometers were less successful than expected, largely because they could not be installed properly on the surface of the planet and protected against the Martian winds but remained mounted on the lander platform, whereas the Apollo seismometers yielded a wealth of data on the interior of the Moon. During December 1972 to September 1977, more than twelve thousand events were recorded on the Moon and catalogued with the long-period sensors, including shallow and deep lunar quakes as well as meteorite and man-made impacts (e.g., Toksöz et al. 1974; Nakamura 1983). The moonquakes are typically very small-magnitude events with the largest shallow moonquake having a body-wave magnitude of about 5, whereas the deep moonquakes have magnitudes less than 3 (Goins et al. 1981 Although the seismic signals from the Moon have a high degree of scatter, the seismological surveys allowed the crustal thickness to be determined at the landing site and provided clues to the mantle structure (e.g., Tokzöz et al. 1974; Goins et al. 1981; Nakamura 1983; Khan and Mosegaard 2002; Lognonné et al. 2003; Gagnepain-Beyneix et al. 2006; Khan et al. 2013) and to the core size (Weber et al. 2011; Garcia et al. 2011). See also Garcia et al. (2019a, b) and Nunn et al. (2020) for recent reviews on Apollo lunar seismic data and the interior structure models.

Most recently, in January 2019, the InSight mission (Banerdt et al. 2020) installed a highly sophisticated seismometer on Mars with ultra-sensitive short period and broad band three component sensors (Lognonne et al. 2019; Lognonne et al. 2020). As a single station, the InSight seismometer cannot uniquely locate seismic events. Instead, the team has developed a method that will allow the location of sufficiently strong and not too distant quakes from the delays between body wave phases and surface waves (Panning et al. 2015). Unfortunately, to date surface waves of a Martian quake have not been observed. Instead, the team has used the differences between body wave phases and their polarizations of three sufficiently strong events (magnitude 3.5) together with a map** of tectonic features to locate these events in an area on Mars known as Cerberus Fossae (Giardini et al. 2020). These signals have been inverted to determine the crust thickness (Knappmeyer-Endrun et al. 2021), mantle properties (Lognonné et al 2020; Khan et al. 2021) and core radius (Stähler et al. 2021). Three new magnitude 4.1 and 4.2 quakes were recently reported (https://www.jpl.nasa.gov/news/nasas-insight-finds-three-big-marsquakes-thanks-to-solar-panel-dusting) and await interpretation.

In addition, the seismometer has recorded the hammering of the Heat Flow and Physical Properties Package (HP3) mole (Spohn et al. 2021a) and used it to constrain the elastic properties of the top decimeters of the Martian soil (Brinkmann et al 2019). Moreover, the noise generated by winds and its propagation in the soil is a subject of intense study.

2.3 Magnetic Fields

Spacecraft and ground-based observations have demonstrated that the terrestrial bodies in our solar system display a large range of magnetic field behaviors. Internally generated fields are observed at Mercury, Earth, and Ganymede. Remanent crustal fields, demonstrating that dynamo action occurred sometime in the past, have been detected at Mercury, Earth, Moon and Mars. Magnetized meteorites also constrain the dynamos of Mars, the asteroid Vesta, and other asteroids and planetesimals early in the solar system. An obvious missing body in the list above is Venus, where no crustal or internal magnetic fields have (yet) been detected (O’Rourke et al. 2018). For a recent review of planetary magnetic fields, see Christensen (2019).

The properties of Earth’s magnetic field have been elucidated through a combination of ground-based observatory data, satellite magnetometer data, ship logs of compass readings, and paleo- and archeo- magnetic records. The Earth’s magnetic field data set is plentiful compared to other planetary magnetic data. In particular, its combination of ground-based and satellite observations at the same time and location as well as the long temporal record of observations has allowed for detailed study of the dynamics and structure of Earth’s core.

Data from spacecraft magnetometers typically consist of vector magnetic field measurements along the track of a spacecraft orbiting a planetary body. For the terrestrial bodies and satellites in the solar system (other than Earth), missions that provided orbital magnetic data from intrinsic magnetic fields include Mariner 10 and MESSENGER at Mercury (Ness et al. 1976; Anderson et al. 2012), Lunar Prospector at the Moon (Richmond and Hood 2008), Mars Global Surveyor and Maven at Mars (Acuna et al. 1999; Langlais et al. 2019, Fig. 3), and Galileo at Ganymede (Kivelson et al. 1996). These missions provided spatial coverage that can constrain the field to a maximum length scale based on the orbit coverage and altitude of the spacecraft. Electron reflectometry has also been used to indirectly measure planetary magnetic field intensity at the Moon and Mars (Mitchell et al. 2007; Mitchell et al. 2008) and this data set complements magnetometer data for global coverage of a planetary field. On the local scale, paleomagnetic fields observed in collected Apollo samples from the Moon (Tikoo et al. 2014; Weiss and Tikoo 2014) and meteorites from other parent bodies (Weiss et al. 2010) adds information regarding the intensity and timing of the parent bodies’ magnetizing fields. For Mars, the InSight mission also provided data on the local magnetic field (Johnson et al. 2020).

Fig. 3
figure 3

Maps of the Martian magnetic field at the surface, as predicted by the spherical harmonic model up to n, m = 134. a B, b Br, c Bθ, d Bϕ, Figure from Langlais et al. (2019)

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Furthermore, the time-varying field resulting from the interaction between the solar wind and the planet's environment can be measured. This transient magnetic field induces electric currents in the planet's crust and mantle, which in turn are responsible for an internally induced magnetic field superimposed on the former. Electromagnetic induction is thus a process that provides access to the electrical properties of the planet by observing the transient magnetic field. These depend on the composition, especially the water content and/or the abundance of hydrated minerals, and the temperature. For the Moon, electromagnetic sounding, largely performed during the Apollo program, provided constraints on core size, mantle composition, and interior temperature (Hood et al 1999; Khan et al. 2006). For Mars, the InSight magnetometer, including joint observations with MAVEN, offers the opportunity to study the thermal structure and volatile content of the planet's interior (Johnson et al. 2020). Magnetic induction has also been used to identify subsurface liquid saltwater oceans on the Galilean satellites Europa, Ganymede and Callisto (e.g., Kivelson et al. 2004) and a subsurface magma ocean on Io (Khurana et al. 2011).

Vector magnetic data can be used to create spherical harmonic observational models of the planetary magnetic field. After correction for spacecraft magnetic fields and other known external sources not attributed to the planet’s crustal or internal fields, the magnetic fields are assumed to be measured in an insulating region and hence can be represented as the gradient of a scalar potential: \(\vec{B} = - \nabla V \) where the potential is expanded in spherical harmonics:

$$ V = R_{P} \mathop \sum \limits_{l = 1}^{L} \mathop \sum \limits_{m = 0}^{l} \left( {g_{l}^{m} \cos m\varphi + h_{l}^{m} \sin m\varphi } \right)\left( {\frac{{R_{P} }}{r}} \right)^{l + 1} P_{l}^{m} \left( {\cos \theta } \right) $$
(1)

where \(R_{P}\) is the planetary radius, \(\left( {l,m} \right)\) are spherical harmonic degree and order, respectively, \(\left( {r, \theta ,\varphi } \right)\) are spherical coordinates, and \(P_{l}^{m} \left( {\cos \theta } \right)\) are associated Legendre polynomials. The Gauss coefficients, \(\left( {g_{l}^{m} ,h_{l}^{m} } \right)\) describe the contributions to the field at different spatial wavelengths. Typically, inverse problem methodology is used to determine the best-fit Gauss coefficients for the magnetic field data. Once the Gauss coefficients are determined, the potential, and hence the magnetic field, can be characterized at any radius in an insulating region. Because downward continuation of the field from satellite altitude to the surface or interior of a planet is particularly sensitive to small-scale noise in the data, inversion methods typically employ various regularization techniques to ensure the field is not dominated by unresolved or noisy data when downward continued.

Equation (1) demonstrates that small scale fields (represented by higher l and m Gauss coefficients) are more challenging to observe from spacecraft altitude and there is a minimum wavelength at which a field can be resolved depending on the altitude of the spacecraft. This makes it challenging to determine the total intensity of the field at either the surface or core-mantle boundary of the planet if small-scale fields have significant contributions. This was recently demonstrated at Mars when the InSight mission magnetometer data determined that the surface field strength at the InSight landing site was ten times stronger than expected from downward continuation of spacecraft field models (Johnson et al. 2020). This can be somewhat alleviated if there is an expected distribution of field intensity with scale. For example, arguments based on the idea that a dynamo-generated field should produce a white spectrum in the core have been used to extrapolate the field intensity at larger wavelengths to smaller wavelengths to determine the field strength in the Earth’s core.

If spacecraft data are available at different altitudes at the same latitude and longitude, then the depth of magnetization can also be constrained based on Eq. (1). This can be useful to determine the source of magnetic fields (e.g., whether from a core dynamo or crustal remanence) and the depth of magnetized crustal layers. Crustal magnetic fields are used to constrain properties of previously-active dynamos. For example, if the magnetic mineralogy, thickness, and depth of crustal units are known, then the intensity of the magnetizing dynamo-generated field can be determined. In addition, if the age of the magnetized rocks is known, then the timing of the dynamo can be constrained.

Finer details of dynamo-generated fields may also be related to properties of planetary cores. For example, dynamo simulations demonstrate that the presence of stably-stratified layers, thermal perturbations at the core-mantle boundary, and the geometry of the convective shell provide possible explanations for observed features in planetary magnetic fields, as described for the Earth in Dehant et al. (2022).

Information about a present-day and especially an early magnetic field can be used to constrain the structure and thermal evolution of a planet. Dynamos are usually generated in convecting, electrically conducting liquid cores—there are, however, exceptions, such as magnetic field generation in the oceans of icy moons or possibly even in early basal magma oceans. In the dynamo process, mechanical energy is converted into electromagnetic energy by induction. Necessary conditions are not only that the electrically conducting fluid convects, but the convection must have a certain strength, which is measured by the magnetic Reynolds number, the ratio of the characteristic velocity times a length scale to the magnetic diffusivity. An active planetary dynamo needs to have a Reynolds number larger than ~ 10.

The fluid motions required for dynamo action must have a power source. An important mechanism for dynamo generation in the iron-rich cores of terrestrial bodies is compositional convection that can occur when a buoyant chemical component is released during the freezing of a fluid core with a non-eutectic composition (Braginsky 1964). In the case of Earth, core crystallization started in the center of the core and compositional convection due to inner core growth drives the dynamo. In the small planets, iron may start crystallizing at the core–mantle boundary (CMB) rather than in the center, and iron snow may form (e.g., Hauck et al. 2006). As a consequence, a dynamo that is tied to crystallization may work differently in these planets than it would in Earth. In the iron-snow regime, Fe-rich crystals crystallize, sink and re-melt deeper in the core. The crystallization layer (iron-snow) grows with time until it encompasses the entire core. At the time it reaches the center, an inner solid core begins to grow. A magnetic field is generated under the iron-snow layer (Christensen 2015), which is stable against large-scale convection except for the sinking crystals. As a consequence, the dynamo is less long-lived than in the case of the Earth-like crystallization scenario. Such a process of dynamo generation has been suggested for the magnetic fields of Ganymede and Mercury (e.g., Breuer et al. 2015).

There are not only chemical or compositional dynamos but also those which are only driven by thermal buoyancy, i.e. a thermal dynamo for an entirely fluid core, if a sufficiently large super-adiabatic temperature difference exists between core and mantle. The mantle must remove heat from the core at a rate exceeding the heat flux conducted along the adiabat of the core to become convective. If the mantle removes heat at a rate below this critical heat flux, the core becomes thermally stably stratified and cools conductively, and dynamo action driven by thermal convection is not possible. To study the magnetic field evolution of a planet, therefore, coupling with the mantle is necessary. On the one hand to determine the core-mantle heat flux but also the change of the core temperature with time. The latter determines when the core starts to freeze out and a chemical dynamo can exist. Thermal dynamos are to be expected in the early evolution of the planetary bodies, when the core is still completely liquid and can be cooled effectively. An early thermal dynamo is assumed for the Earth, but also for Mars, Mercury and the Moon. Today's magnetic fields are likely to be generated by chemical dynamos.

Mechanical driving has also been suggested as a potential source for motions in certain planetary cores that could create dynamo action. As examples, libration, precession and impact torques have all been proposed to create dynamo-capable flows due to coupling between the solid mantle and fluid core in the Moon (Dwyer et al. 2011; Le Bars et al. 2011; Stys and Dumberry 2020).

2.4 Thermochemical Evolution and Heat Flow

Modeling dynamical processes in terrestrial planetary mantles on local/regional or global scales is a common approach to learn about the interior evolution of terrestrial planets or, in turn, to study how the interior influences planetary evolution. In particular, modeling on global scales allows us to study the thermochemical evolution and heat transport mechanism of a planet or satellite since its formation.

Mantle dynamics in 2D or 3D domains is calculated by solving a set of differential equations for the conservation of mass, momentum, and energy including an appropriate constitutive equation. Typically, a fluid with Newtonian rheology and infinite Prandtl number is assumed with either the Boussinesq approximation (BA), extended Boussinesq approximation (EBA) (e.g., King et al. 2010) or the anelastic approach for a compressible fluid (e.g., Steinbach et al. (1989) for a comparison of these approaches). As for the Boussinesq approximation, the EBA assumes the density to be constant everywhere except in the buoyancy term of the momentum equation but considers additionally the thermal effects of viscous dissipation and adiabatic compression/decompression. In compressible models an equation of state is solved to consider density variations with pressure. Models can be expanded to further consider the influence of mantle phase transitions on the mantle dynamics. This is particularly important for terrestrial planets as large or larger than Mars but also for icy satellites, in which various exothermal and endothermal phase transitions can occur in the ice mantle. The convection models can further include compositional buoyancy and an equation for conservation of composition to account for compositional changes due to mantle melting and crust formation (e.g., Plesa and Breuer 2014) or for compositional layers formed as a consequence of magma ocean crystallization or of crustal recycling.

Two- or even three-dimensional thermal evolution calculations are very time-consuming on present-day computers, in particular if a large parameter range needs to be investigated. Because of the complexity of these models a semi-empirical approach is often used for which the convective heat transfer rate is parameterized. Parameterizations can be derived using simple theories, e.g., boundary layer theory, that give the heat transport rate as a function of the convective strength, the latter usually parameterized by the Rayleigh number,

$$ Ra = \frac{{\alpha g\Delta TD^{3} }}{\kappa \upsilon } $$
(2)

where \(\alpha\) is the thermal expansion coefficient, \(g\) is gravity, \(\Delta T\) the superadiabatic temperature difference across the mantle, \(D\) its thickness, \(\kappa\) the thermal diffusivity, and \(\upsilon\) the kinematic viscosity (e.g., Breuer and Moore 2015). With the parameterized models, the evolution of the average mantle and core temperatures, heat fluxes and crust production rates are calculated.

Two main types of heat transport mechanisms and tectonic regimes can be distinguished: the plate tectonics or mobile lid regime and the stagnant lid regime. Plate tectonics is known from the Earth and efficiently cools the interior through the subduction of cold surface plates. In the stagnant lid regime, convection takes place below a rigid lid and results in a less efficient cooling of the planet interior by conduction through this outer lid. The stagnant lid regime is the natural regime because of the large viscosity contrast between the surface and the mantle caused by the strong temperature dependence of the viscosity of mantle rock, and is observed for all terrestrial planets, except Earth. To explain the occurrence of plate tectonics, complex rheologies must be considered, such as those that allow for brittle deformation (e.g., Moresi and Solomatov 1998) or even the memory of failure (Bercovici and Ricard 2014). Of course, such deformation mechanisms must plausibly be inoperative or ineffective on the other terrestrial planets where plate tectonics is absent.

The difference in heat transfer rate between plate tectonics and stagnant lid convection is reflected in the scaling laws for the heat flow (Breuer and Moore 2015). The parametrization of stagnant lid convection has been improved in recent years and was tested against fully dynamic evolution models in 3D (Thiriet et al. 2019). It should be noted here that due to the complex rheology of silicates, other tectonic regimes can occur. This includes the so-called transitional regime, when the viscosity contrast between mantle and surface is not too large (for instance in case of high surface temperatures), but also the episodic regime and the so-called slushy lid regime. The latter becomes active when the intrusion rate is very large and is characterized by local resurfacing events: Magma is not entirely transported to the surface, but a large part intrudes into the crust and weakens the material there. In these regions, the crust can become locally destabilized, detaches and sinks with the convection. Especially for Venus and also the early Earth these mechanisms are discussed (Gerya et al. 2015).

The models (parameterized or full 2D or 3D convection) can be extended to include important physical processes, such as the growth of a solid inner core that is important for magnetic field generation (Stevenson et al. 1983), crust formation and associated redistribution of radioactive elements, as well as outgassing of volatile elements (Morschhauser et al. 2011). The latter may have a strong influence on the mantle viscosity, the higher the volatile content the lower the viscosity and the stronger the mantle convection and more effective cooling of the planet.

The outgassing of the mantle also has an influence on the atmospheric composition and pressure, which in turn can influence the surface temperature and thus also the internal dynamics. The interior-atmosphere coupling has proven important for Venus (Gillmann and Tackley 2014), for example, but also for the early evolution of other terrestrial planets—in the phase where strong outgassing is expected. Even more advanced models couple both accretion and core formation and the subsequent thermochemical convection (e.g., Golabek et al. 2018 for Mars) or the influence of impacts on the thermochemical evolution (e.g., Padovan et al. 2017 for Mercury). For planets and moons for which tidal effects in the interior play a major role as a heat source, coupled models of the interior with orbital dynamics are needed [e.g., Husmann and Spohn (2004) for Io and Europa, Bland et al. (2009) for Ganymede].

To model the thermochemical evolution of a planet, model parameters and initial and boundary conditions must be defined and observations are then used as constraints for successful models. An important input is the internal structure and composition. Each body has its own peculiarities here, which are partly due to the fact that they have different sizes (Moon versus Venus), a different core to mantle ratio (Moon versus Mercury) and a different composition (e.g., Mg number, water content in the mantle, abundances of light elements in the core or radioactive elements in the silicate shell).

Here, the observational and laboratory-based knowledge about the interiors of the respective planets, as described in Sects. 2.1 and 2.2, serve as input for the models. Whereas knowledge about the present and past magnetic field, as described in Sect. 2.3, can serve as a constraint for successful models. Via planetary missions we obtain further important constraints that have to be fulfilled. Indeed, several quantities, which can be inferred from remote or in-situ observations, are tightly related to the thermochemical evolution of the interior. Measurements of topography, gravity, magnetic and seismic fields, surface spectra and surface images can all be employed to infer a series of fundamental constraints for the evolution of the interior in addition to those describe in Sects. 2.1 and 2.2 (see Tosi and Padovan 2021, for a review).

The heat flow from the deep interior of a planetary body is an important quantity to constrain its thermal evolution, interior dynamics and composition (e.g., Breuer and Moore, 2015; Spohn et al., 2018). The heat flow can be used as a boundary condition for thermal evolution models that would be required to satisfy the present-day value. Because a significant part—measured by the Urey ratio—of the heat flow is due to radiogenic heat from the mantle and the crust, the concentration of radiogenic elements in the interior can be constrained if the Urey ratio and the surface heat flow are known (Plesa et al. 2015 for Mars).

The surface heat flow varies over the surface of a planet, having relative maxima where the temperature at depth is high as in volcanic provinces and relative minima where the latter is low, such as in regions of anomalously thick lithosphere. The heat flow data base for Earth (https://ihfc-iugg.org/products/global-heat-flow-database/data) contains almost 75,000 data points covering most of the oceanic and continental areas. The heat flow map (e.g., Lucazeau 2019) mirrors the elements of plate tectonics such as mid-ocean ridges with the highest heat flow values, subduction zones, volcanic arcs and continents.

To measure the heat flow one usually measures the thermal conductivity and the temperature gradient, the product of the two giving the heat flow. The difficulty with the measurement comes from two sources: First, the temperature gradient and the conductivity are mostly small values of order 1 K/m and 10–2 W/mK. But more severe is the disturbance of the near-surface temperature gradient by heat transferred from the surface to depth. The depth of penetration depends on the period of the disturbance with daily variations penetrating to centimeters depth while annual perturbations can be measured even at the depth of several meters. On Earth even climatic changes are taken into consideration and the heat flow is measured in boreholes extending to 10 s of meters depth or deeper.

Heat flow has been measured at two locations on the Moon on which the Apollo astronauts drilled to 1.4 m (Apollo 15) and 2.4 m (Apollo 17) depth. The heat flow values measured were 21 mW/m2 at the Apollo 15 and 16 mW/m2 at the Apollo 17 site (Langseth et al. 1976). The difference between the two reflects differences in subsurface radiogenic Thorium content (Wieczorek and Phillips 2000). The Apollo 15 site is closer to the center of the Procellarum KREEP terrain with its high Th content than the Apollo 15 site. The InSight mission has a Heat Flow and Physical Properties Package (Spohn et al. 2018) that was planned to measure the heat flow at Homestead Hollow in Elysium Planitia at a depth of 3–5 m, targeted to avoid the daily and annual temperature disturbances. The package deployed a small penetrator with thermal conductivity and temperature sensors. Unfortunately, the attempts to penetrate sufficiently deep had to be given up in January 2021. The penetrator reached only a depth of about 40 cm being inclined to the vertical at about 30° (it is 40 cm long) and serves as a thermal probe to study surface atmosphere interactions and the properties of the Martian soil (Spohn et al. 2021a,b).

In the absence of direct heat flux measurements, the thickness of the elastic lithosphere can be used to constrain the surface heat flow. The evolution of the elastic thickness is considered an important quantity to reconstruct the thermal history of a planet (e.g., Grott and Breuer 2008). The elastic thickness can be estimated by analyzing the response of the lithosphere to loading stresses such as volcanoes or by analyzing fault structures. The mechanical thickness of the lithosphere can then be inferred from the elastic thickness using a rheological model and the mechanical thickness can, in turn, be identified with an isotherm (McNutt 1984). Assuming a crustal thermal conductivity, an estimate of the surface heat flow can be made. Most elastic thickness estimates have been derived from gravity and topography data (e.g., McGovern et al. 2004), but some geological features allow more direct approaches such as studies of lithosphere deflection due to volcanic or polar cap loading. The heat flow thus obtained, however, is not the present-day value but is related to the time when the surface loading took place. This requires dating of the age of the structure mostly by crater counting.

Other constraints that can be gained with the help of planetary missions include crustal thickness, duration and timing of volcanism, amount of accumulated radial contraction, surface concentration of radioactive elements, or evolution of the mantle potential temperature. Knowledge of input parameters and observations for determining the themo-chemical evolution of terrestrial bodies is available in varying degrees of accuracy and depends on the particular body considered and the missions performed. Examples are given in the following section.

3 Selected Planets and Moons

3.1 Mercury

Our knowledge of Mercury’s interior has greatly improved due to recent analyses of Mercury’s rotational dynamics, gravity field, magnetic field, and surface properties, mostly from MESSENGER data. This complements recent high-pressure experiments on relevant core compositions and theoretical modeling of the planet’s thermal evolution, mantle dynamics and core dynamo. Detailed reviews of results from the MESSENGER mission and the implications for the planet’s interior can be found in Margot et al. (2018), Johnson et al. (2018), and Hauck et al. (2018), see also Genova et al. (2021).

Geodetic data for Mercury (e.g., obliquity, libration amplitude, gravity coefficients) have constrained the interior structure of the planet. Earth–based radar observations (Margot et al. 2012) and MESSENGER radio science, imaging, and laser altimeter data (Mazarico et al. 2014, Stark et al. 2015, Verma and Margot 2016, Genova et al. 2019, Konopliv et al. 2020a, b) have accurately determined Mercury’s orientation in space and indicate that Mercury closely occupies the equilibrium Cassini state. The normalized polar moment of inertia can then be determined from the orientation and is between 0.32 and 0.35. These geodetic data also show that Mercury’s librations have an amplitude between about 37 arcsec and 40 arcsec. The librations, periodic variations in Mercury’s rotation, strongly depend on the size of the core and also can contain a signature of an inner core (Rambaux et al. 2007; Van Hoolst et al. 2012). Together with the orientation of Mercury, they provide an estimate of the polar moment of inertia of the silicate shell of Mercury. Both polar moments of inertia can be used as constraints for interior structure models to determine allowable layer thicknesses and densities for Mercury’s crust, mantle, and core.

Mercury has long been known to contain a large core-mass fraction (e.g., Siegfried and Solomon 1974). By using the geodetic data as constraints on interior structure models based on likely compositions of Mercury’s core, mantle and crust, the core’s radius is shown to be around 2000 km, i.e. ~ 80% of the planet’s radius (e.g., Hauck et al. 2013; Rivoldini and Van Hoolst 2013; Margot et al. 2018; Knibbe et al. 2021). Combined with estimates for Mercury’s crustal thickness of ~ 35 km (James et al. 2015; Padovan et al. 2015; Margot et al. 2018) this implies a very thin mantle layer in Mercury of ~ 400 km (Fig. 1). Such a thin mantle suggests that mantle convection is challenging, and it is currently debated whether the planet undergoes very sluggish convection under a stagnant lid or, rather, is purely conductive (e.g., Hauck et al. 2018). If convection is occurring, then the planform is likely to be small-scale, i.e. length scales of convection are likely on the order of the mantle thickness (Tosi et al. 2013). This suggests that there may be significant heterogeneity in Mercury’s mantle due to lack of mixing over large lateral distances.

Mercury has a liquid layer in its core, as revealed by the presence of an intrinsic magnetic field (Anderson et al. 2012) and observations of its 88‐day longitudinal libration forced by the periodically reversing solar torque (Peale 1976; Margot et al. 2007). Independent evidence in support of a liquid core is provided by the measurement of the tidal Love number k2 (e.g., Genova et al. 2019) since it is significantly larger than that expected for a fully solid Mercury (Rivoldini et al. 2009; Padovan et al. 2014). There is no direct observational data on the existence or size of Mercury’s solid inner core. Inferring the presence and size of an inner core based on geodetic and geochemical data is strongly dependent on the knowledge of the core’s composition and thermal state, which are not well-constrained. However, some constraints come from implications of inner core sizes on the thermal contraction Mercury should have experienced from core solidification, from compensating densities required in other regions of Mercury if the planet has a solid iron core in order to match the polar moments of inertia, as well as from dynamo models. A range of inner core sizes is consistent with the data. Monte Carlo solutions using the lowest estimate of the polar moment of inertia suggest that the solid inner core most likely occupies between 30 and 70% of Mercury’s total core radius (Genova et al. 2019). Steinbrügge et al. (2021) studied the implications of the lower polar moment of inertia value from Genova et al. (2019) on the size of the inner core using interior structure models. Models with a larger normalized moment of inertia (0.346) favor a small inner core whereas models with the smaller value (0.333) require an inner core > 600 km. Furthermore, the smaller MoI suggests a smaller core radius by about 75 km and a smaller mantle density than previous estimates (Fig. 4).

Fig. 4
figure 4

Core‐mantle boundary (CMB) radius and temperature, sulfur content, and mantle density as a function of inner core boundary (ICB) radius, for Mercury models with a Fe‐S core composition and constrained to match C/MR2 = 0.346 (red) or C/MR2 = 0.333 (blue). The blue dashed lines bracket the range of models matching C/MR2 = 0.333 + 0.005, the 1σ error bar given in Genova et al. (2019). The red dashed line corresponds to C/MR2 = 0.346–0.014 = 0.332, the lower 1σ error bar given in Margot et al. (2012). Figure from Steinbrügge et al. (2021)

Full size image

Studies of Mercury’s surface composition demonstrate that the interior is chemically reduced (Nittler et al. 2011). This implies that silicon would have been more soluble in iron during planet differentiation and hence that there may be significant silicon in Mercury’s core. Although sulfur has long been considered a potential light element for Mercury’s core, the observed high sulfur to silicon ratios on Mercury’s surface suggest that sulfur may be much less abundant in Mercury’s core than previously thought. This suggests that the Fe-S-Si system may be an important composition to understand at Mercury’s core pressures and temperatures. Interestingly, there are hints that the Fe-S-Si system might have unusual melting properties and experience immiscibility at certain pressures and temperatures relevant for Mercury’s core, both of which may have important implications for Mercury’s dynamo (e.g., Chabot et al. 2014). Other light elements may also be present in the core and hence further high-pressure studies of iron mixed with light elements such as carbon and phosphorus are also needed (e.g., Knibbe et al. 2021). An inner core size less than about 1200 km likely requires the presence of another light element than Si, such as S or C, at the wt% level in the core (Knibbe et al. 2021).

MESSENGER data confirmed the Mariner 10 results that Mercury has an internally generated magnetic field (Anderson et al. 2012). The mission also discovered crustal magnetization in the north polar region implying that Mercury’s dynamo was also active around 3.9–3.7 Ga (Johnson et al. 2015). This provides constraints on the thermal evolution of Mercury since the planet must have been sufficiently convective to produce a dynamo at both of these times. Whether the field has been active in the interim time, or earlier than 3.9–3.7 Ga, is currently unknown.

The morphology and intensity of Mercury’s magnetic field seem to require some special circumstances in Mercury’s core. In particular, the low field strength is a challenge for dynamo scaling laws and the ability for the field to maintain a large dipole offset (equivalent to a significant axial quadrupole moment) while also appearing quite axisymmetric is a challenge for dynamo generation theory. Studies of dynamo models for Mercury have appealed to the presence of stably stratified layers at the top of the core (Christensen 2006; Christensen and Wicht 2008), as well as thermal perturbations at the core-mantle boundary (Cao et al. 2014; Tian et al. 2015) or double diffusive convection (Takahashi et al. 2019) to explain these field characteristics. However, discerning between these possibilities will likely require better understanding of the composition of Mercury’s core (i.e. to constrain the stability profile) and the role of the mantle in linking surface features to thermal signatures at the core-mantle boundary (e.g., Beuthe et al. 2020). Magnetic induction studies have also provided an estimate of the radius of Mercury’s core, independent from that determined from geodetic studies which is consistent with the value based on Mercury’s geodetic data (Wardinski et al. 2019; Katsura et al. 2021).

Resolving smaller scales in Mercury’s internally generated field from spacecraft data is made particularly challenging because of the prevalence of external fields in the data. External fields are particularly impactful at Mercury because of the small magnetosphere and relatively strong solar wind at Mercury. Future work to model and understand these external fields may therefore lead to further insights on the morphology and generation of Mercury’s field. More details about the magnetic field but also the internal structure are expected from the upcoming BepiColombo mission, a joint mission of ESA and JAXA launched in 2018 (Benkhoff et al. 2010). BepiColombo consists of two individual orbiters: the Mercury Planetary Orbiter (MPO) to map the planet, and the Mercury Magnetospheric Orbiter (MMO) to study its magnetosphere.

3.2 Venus

Venus is the closest analog to Earth in the solar system in terms of mass, radius, and density (Table 1) but differs significantly in surface tectonics and atmosphere composition. However, of all terrestrial planets we know the least about Venus, in particular about its interior. Firstly, the conditions for investigation are more difficult due to Venus’ dense atmosphere, that is opaque to visible light, and the high surface temperatures, which poses a particular challenge for lander science. Secondly, there have been comparatively few missions that have studied the deep interior and surface of Venus such as the Russian Venera landers (1970–1981) (Keldysh 1977), Pioneer Venus (1977–1992) (Colin 1980), Magellan (1990–1994) (Saunders et al 1992) and Venus Express (2006–2014) (Svedhem et al. 2009). In this subsection, we summarize what we have learned so far about the internal structure and dynamics with a focus on the knowledge we have gained through the analysis of gravity and topography data.

The diameter of Venus is 5% smaller than that of Earth and for the same mantle and core composition the Venusian core radius would be 0.51 of the planetary radius as compared with Earth, for which the core radius is 0.55 of the planetary radius. However, the composition of Venus is not well known and previous models based on cosmochemical constraints vary strongly with the amount of FeO in the mantle between almost no iron and FeO of 18.7% (Earth value corresponds to 8.4 wt%) (see Fegley 2014 for a review). To fit the observed mass and radius of Venus, models show then a 500 km uncertainty in the core size with values between 2940 and 3425 km (Dumoulin et al. 2017). As the observed gravity coefficient J2 is 25 times larger than the hydrostatic one, the estimation of the mean moment of inertia from gravity field measurements alone is prevented and measurement of the precession rate are required. Recently, Margot et al. (2021) used Earth based observations of radar speckles tied to the rotation of Venus obtained in 2006–2020 to measure its spin axis orientation and precession rate. They obtained a mean moment of inertia normalization factor of 0.337 ± 0.024, but the large uncertainty does not allow better constraint of the core size. Similarly, the tidal Love number k2 derived from Doppler tracking of the Magellan and Pioneer Venus spacecraft of k2 = 0.295 ± 066 (Konopliv and Yoder 1996) shows a large uncertainty which implies that even the state of the core, i.e. whether Venus’ present core is solid or fluid, cannot be constrained (Dumoulin et al. 2017, Fig. 5). The absence of an observation of a planetary magnetic field is consistent with both implied extreme core states but does not help to place further constraints on interior structure. Knowing the present-day state of the core is essential because it would provide constraints on the efficiency of heat extraction from the core by mantle convection through Venus’ history, in particular, on the possible cessation of a core dynamo. It should also be noted that no magnetization has yet been measured on the surface of Venus either, so it is not clear whether Venus once had an early magnetic field. A scenario that can certainly not be excluded at present (Stevenson et al. 1983).

Fig. 5
figure 5

Potential Love number, k2, as a function of the dissipation factor Q. The case of a solid core of pure iron is illustrated using hexagons, and the case of a 2500 km radius solid core and a liquid outer core with diamonds. The black triangle stands for the case with an entirely liquid core. Colors of hexagons depict the chosen viscosity of the solid core. The star on the right side shows the result obtained for the elastic solid core case. The long-dashed line represents the lowest estimate of k2 from Konopliv and Yoder (1996), the upper one being too large to be drawn on the figure due to the chosen scale. This figure illustrates that the state of the core, i.e. whether Venus present core is solid or fluid, cannot be constrained with the measured k2 value. Figure from Dumoulin et al (2017)

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The present convective regime seems to be consistent with convection underneath a stagnant lid–no clear present-day plate tectonic features have been observed. However, in some locations, this lid does not seem to be entirely “stable”, but a so-called sluggish lid is formed in some places, which may be an instability of the lower crust like a delamination or even the beginning of subduction similar to plate tectonics, possibly connected to upwelling mantle plumes (Davaille et al. 2017). In fact, the analysis of gravity and topographic data indicates that Venus has a comparable number of active large mantle plumes to Earth, as well as many hundreds of smaller plumes that may also be active (Smrekar and Phillips 1991). Active volcanism has also possibly been detected by Venus Express via a thermal signature (Shalygin et al. 2012) and recent volcanism from surface emissivity near 1 micron that distinguishes fresh from weathered basalt (Smrekar et al. 2010).

The strong correlation of the long-wavelength global gravity with the global topography spectrum is a particularity of Venus and is probably caused by mantle convection. The analysis of the long-wavelength spectra thus allows conclusions to be drawn about internal dynamic processes. It has been used, for example, to constrain the viscosity structure of the Venusian mantle (Rolf et al. 2018). General features of the viscosity structure, which are similar for various models, show that there is no pronounced asthenosphere and viscosity jump between upper and lower mantle—in contrast to Earth. Global estimates of elastic thickness from the gravity and topography data find a wide range of values, from 0 to 100 km (Anderson and Smrekar 2006; Jiménez-Díaz et al. 2015), with more than half of the planet having best-fit values of 20 km (or less, indistinguishable from isostatic compensation). Thus, these regions might have a quite high heat flux and exclude models of a cold Venus.

In general, Venus shows a relatively young surface with a mean crustal thickness of less than 50 km (e.g., James et al. 2013) and an average age younger than 1 Ga (McKinnon et al. 1997). Venus’ geologically young surface requires extensive volcanic resurfacing, but it is not clear whether this happens in a global-scale resurfacing event (e.g., Strom et al. 1994) or whether a more continuous resurfacing occurs (e.g., Bjonnes et al. 2012). Currently discussed evolution scenarios include episodic catastrophic resurfacing (e.g., Armann and Tackley 2012), gradual decay of volcanism over time in a stagnant lid regime, but also mixed forms and transitions between the different tectonic regimes, i.e. stagnant lid regime, mobile lid regime, episodic regime and transitional regime (e.g., Gillmann and Tackley 2014; Weller and Kiefer 2020). A key parameter controlling the tectonic regime may be surface temperature, which is influenced by the atmosphere evolution and thus by the coupling between interior and the atmosphere (Gillmann and Tackley 2014).

Distinguishing these various evolution scenarios is not possible with today's data and will be a task of upcoming missions such as DAVINCI + and VERITAS (NASA Discovery Missions) or EnVision (ESA M5 Mission). VERITAS and EnVision are specifically dedicated to studying the surface and deep interior of Venus with radio science, radar and infrared spectroscopy. Among other things, the combined processing of Earth Doppler tracking and radar data will allow accurate determination of the planet's rotational state and moment of inertia factor in addition to the much more accurate determination of the Love number k2 from tracking alone (Rosenblatt et al. 2021; Cascioli et al. 2021). VERITAS will provide accuracies (\(3\sigma )\) in the estimates of the tidal Love number \(k_{2}\) to \(4.6 \times 10^{ - 4}\), its tidal phase lag to \(0.05^\circ\), and the moment of inertia factor to \({9}.{8} \times 10^{ - 4}\) (0.3% of the expected value) (Cascioli et al. 2021).Those results will place much tighter constraints on the state and size of the core and the viscous response of the planet than is currently the case as shown in the analysis by Dumoulin et al. (2017).

3.3 Moon

More than fifty years after the first crewed mission to the lunar surface, the Moon remains a cornerstone in planetary science. The amount and quality of data that is available from returned samples, remote sensing observations from Earth and from space, instruments placed on the lunar surface, high-pressure experiments, and thermochemical modeling is still unrivaled. Almost 400 kg of lunar rock and soil samples were returned by American, Soviet, and Chinese robotic and manned missions over the past 5 decades, dozens of remote missions have orbited the Moon (and continue to do so), and its proximity to Earth means the Moon is a favorite target for new space-faring nations. It is likely that the coming decades will see a further resurgence of lunar exploration, with advanced plans for the return of humans to the lunar surface as well as a plethora of robotic and sample return missions, including many that intend to explore areas of the surface not covered previously. In this short overview, three first order aspects of the lunar interior and its evolution are discussed: (1) Moon formation; (2) the lunar core and lunar magnetic field generation, and (3) the hypothesis of an initial global lunar magma ocean (LMO), that can tie together many of the geochemical and geophysical constraints on the interior of the Moon.

The scientific goal of the Apollo missions was to figure out how the Moon formed. Although the giant impact model, in which the Moon formed from the debris of a giant collision between Earth and another planet, is almost universally accepted, many unresolved issues, some of them fundamental, remain (e.g., review by Canup et al. 2021). Even with the vast amount of geochemical and geophysical data obtained during and since these missions, the origin of the Moon therefore remains hotly debated. The core of the problem with the giant impact model lies in the observation that, on the one hand, the Moon’s chemical composition, for all elements except the volatile elements, very closely resembles the composition of the bulk silicate Earth, whereas on the other hand physical models of classic versions of the giant impact show that the Moon should mainly consist of material originating from the planet that hit Earth (e.g., De Meijer et al. 2013). The only ways out of this conundrum seem to require significant loss of angular momentum after the giant impact (e.g., Cuk and Stewart 2012), complete equilibration of both light and very heavy isotopic systems between silicate Earth and Moon after the giant impact (e.g., Pahlevan and Stevenson 2007) or a similarity between Earth and giant impactor of a level currently not seen anywhere in the solar system.

In view of models for the interior evolution of the Moon, this makes it difficult to accurately quantify the thermal state of the Moon just after formation. One way of estimating the initial interior conditions in the Moon is to study the conditions under which the small lunar iron-rich core formed. There are several lines of evidence to suggest the Moon has a small core. An ancient lunar magnetic field generated by dynamo action in the metallic molten core has been identified in lunar rock samples (e.g., Tikoo et al. 2017); siderophile (iron-metal loving) elements are depleted in lunar samples relative to realistic lunar bulk composition models, suggesting they have been sequestered into a core (e.g., Steenstra et al. 2016; Righter 2019); and lunar seismic data (e.g., Weber et al. 2011; Garcia et al. 2011, 2012) and lunar laser ranging observations (e.g., Williams et al. 2001, 2014; Viswanathan et al. 2019) also require the presence of a small core. Recent analyses of Apollo-era seismic data suggest a core radius of 330 ± 20 km radius, possibly with a solid inner core with a radius of 240 ± 10 km covered by a molten outer core (Weber et al. 2011) or 380 ± 40 km (Garcia et al 2011, 2012). Based on results from the GRAIL gravity field mission and lunar laser ranging results, Williams et al. (2014) suggested that the moment of inertia, k2 Love number, and density of the core are consistent with the presence of 0–280 km solid inner core and/or a 200–380 km molten core. Based on a combination of high-pressure experimental measurements and seismic travel times from Apollo seismometers, Antonangeli et al. (2015) derive core radii close to ~ 240 km for the inner core and ~ 325 km for the outer core, close to the GRAIL-based estimates. Recent analysis of Lunar laser ranging observations combined with the lunar gravity field of the Moon derived from GRAIL mission provided a lunar core radius of 380 ± 12 km (Viswanathan et al. 2019). Overall these results suggest that the core makes up less than 1.5 wt. % of the Moon (Fig. 1a). Some of the geophysical models of the deep lunar interior argued in favor of the presence of a partially molten rock layer on top of the core-mantle boundary (e.g., Weber et al. 2011; Harada et al. 2014, 2016) but its geophysical signature is still debated (Matsuyama et al. 2016). The melt percentages in this layer could exceed 10 per cent, requiring special circumstances to prevent transport of this magma towards the surface (Van Kan Parker et al. 2012).

The chemical composition of the lunar core strongly affects its physical properties and magnetic field evolution. The current fluid state of the outer core requires the presence of one or more light elements along with iron to reduce the solidus (temperature above which molten metal is presence). Sulfur and carbon are the likeliest candidates, but there is currently no agreement on their core concentrations. One way to estimate the light element content comes indirectly from observing the remanent magnetization of the lunar crust. The Moon does not have an internally generated global magnetic field today, but the remanent magnetization in the crust and in returned Apollo samples (e.g., Weiss and Tikoo 2014) suggests that a core dynamo probably existed in the Moon at least 4.25 billion to 3.56 billion years ago, with an intensity of ~ 100 \(\mu T\), similar to that at the surface of the Earth today. The field then decreased by at least an order of magnitude until 3.3 billion years ago, possibly remaining until 2.25–1 Ga b.p. (Tikoo et al. 2014; Tikoo et al. 2017) persisting at a low level of a few \(\mu T\) (Fig. 1b). The early strong field, and later rapid decay, is not well understood so far and may require two different mechanisms (Wieczorek et al. 2021). To explain the strong early magnetic field, mechanisms such as the precession driven dynamo (e.g., Dwyer et al. 2011; Cebron et al. 2019) or impact induced dynamos (LeBars et al. 2011) are proposed. The long duration, however, seems to require that the core had a chemical dynamo. For example, Zhang et al. (2013) and Laneuville et al. (2014) suggested, on the basis of thermochemical evolution models, that > 5 wt.% S should be present in the lunar core to fit the magnetic field evolution, whereas Steenstra et al. (2017a) and Righter et al. (2017) suggested on the basis of experimental work on the partitioning of sulfur between metal and silicate melt that the amount of S in the lunar core is < 0.5 wt.%.

Measurements of the abundances of siderophile element abundances in lunar returned samples and meteorites can be combined with high-pressure, high-temperature experiments (Sect. 2.1) to estimate the pressures and temperatures at which core and mantle last equilibrated. Results of this exercise are non-unique, but one consistent recent result appears to be that molten iron-rich metal and molten lunar mantle have equilibrated at a pressure of approximately 5 GPa (Rai and van Westrenen 2012, Steenstra et al. 2016, 2020). This pressure is equivalent to the pressure at the present-day core-mantle boundary of the Moon as derived from seismological data (e.g., Garcia et al. 2011). This result implies that the Moon was fully molten at the time the core formed, consistent with high-energy formation scenarios for the Moon. Some core formation models suggest that the Moon formed super-hot, at temperatures above 3000 K–temperatures that far exceed what is needed to fully form the Moon. On the other hand, high-pressure experiments have been used to suggest the Moon never melted completely, using the average thickness of the lunar anothositic crust as constraint (e.g., Charlier et al. 2018). Its thickness is between 34 and 43 km and was derived from seismic data near the Apollo landing sites (Lognonné et al. 2003, Khan and Mosegaard 2002) in combination with the inversion of gravity topography data from Gravity Recovery and Interior Laboratory (GRAIL) mission (Wieczorek et al. 2013). According to this line of thought, the lower part of the mantle of the Moon can be still primitive (i.e., essentially the same as when the Moon formed), with perhaps the top ~ 600 km formerly fully molten early in lunar history, forming a global magma ocean. This is also consistent with a discontinuity in seismic velocities at depths of about 500–550 km (e.g., Nakamura 1983; Khan et al. 2000, 2006; Lognonné 2005). However, a wide range of acceptable models have been proposed and care should be taken in interpreting this seismic discontinuity (e.g., Gagnepain-Beyneix et al. 2006; Wieczorek et al. 2006). It should also be noted that these mantle reservoirs may have been mixed by convection after their formation (e.g., Laneuville et al. 2014) and may not exist in their original structure today.

Although opinions vary as to its initial depth (e.g., Gaffney et al. 2021), a global lunar magma ocean (LMO) is thought to have played a crucial role in the interior evolution of the Moon. Several recent experimental studies have focused on the cooling history of such an ocean (e.g., Lin et al. 2017a, b, 2001) on a great circle arc from InSight through the Mars Orbiter Laser Altimeter (MOLA) map. The S-wave shadow zone is minimal and probably filled by diffracted S waves, whereas the P-wave shadow zone is large and contains specifically the Tharsis region. The existence of an inner core cannot be determined by current data, and the seismic ray paths shown assume no inner core. Topography and InSight lander are exaggerated in scale. Figure from Stähler et al. (2021)

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