Non-zero Eccentricity Research Articles

Horizon fluxes of binary black holes in eccentric orbits

We compute the rate of change of mass and angular momentum of a black hole, namely tidal heating, in an eccentric orbit. The change is caused due to the tidal field of the orbiting companion. We compute the result for both the spinning and non-spinning black holes in the leading order of the mean motion, namely ξ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi $$\\end{document}. We demonstrate that the rates get enhanced significantly for nonzero eccentricity. Since eccentricity in a binary evolves with time we also express the results in terms of an initial eccentricity and azimuthal frequency ξϕ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi _{\\phi }$$\\end{document}. In the process, we developed a prescription that can be used to compute all physical quantities in a series expansion of initial eccentricity, e0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$e_0$$\\end{document}. These results are computed taking account of the spin of the binary components. The prescription can be used to compute very high-order corrections of initial eccentricity. We use it to find the contribution to eccentricity up to O(e05)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {O}}(e_0^5)$$\\end{document} in the spinning binary. Using the computed expression of eccentricity, we derived the rate of change of mass and angular momentum of a black hole, both rotating and non-rotating, in terms of initial eccentricity and azimuthal frequency up to O(e06)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {O}}(e_0^6)$$\\end{document}. With the computed fluxes we also compute for the first time the leading order dephasing in both cases analytically up to O(e06)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {O}}(e_0^6)$$\\end{document} and study its impact. We argue that for high signal-to-noise ratio sources, these contributions require inclusion in the waveform modeling.

The K2-24 planetary system revisited by CHEOPS

The planetary system K2-24 is composed of two transiting low-density Neptunians locked in an almost perfect 2:1 resonance and showing large transit time variations (TTVs), and it is an excellent laboratory to search for signatures of planetary migration. Previous studies performed with K2, Spitzer, and RV data tentatively claimed a significant non-zero eccentricity for one or both planets, possibly high enough to challenge the scenario of pure disk migration through resonant capture. With 13 new CHEOPS light curves (seven of planet b, six of planet c), we carried out a global photometric and dynamical re-analysis by including all the available literature data as well. We obtained the most accurate set of planetary parameters to date for the K2-24 system, including radii and masses at 1% and 5% precision (now essentially limited by the uncertainty on stellar parameters) and non-zero eccentricities eb = 0.0498−0.0018+0.0011, ec = 0.0282−0.0007+0.0003 detected at very high significance for both planets. Such relatively large values imply the need for an additional physical mechanism of eccentricity excitation during or after the migration stage. Also, while the accuracy of the previous TTV model had drifted by up to 0.5 days at the current time, we constrained the orbital solution firmly enough to predict the forthcoming transits for the next ~15 years, thus enabling efficient follow-up with top-level facilities such as JWST or ESPRESSO.

Fate of initially bound timelike geodesics in spherical boson stars

Boson stars are horizonless compact objects and they could possess novel geodesic orbits under the equilibrium assumption, which differ from those in black hole backgrounds. However, unstable boson stars may collapse into black holes or migrate to stable states, resulting in an inability to maintain the initially bound geodesic orbits within the backgrounds of unstable boson stars. To elucidate the fate of initially bound geodesic orbits in boson stars, we present a study of geodesics within the spherical space times of stable, collapsing, and migrating boson stars. We focus on timelike geodesics that are initially circular or reciprocating. We verify that orbits initially bound within a stable boson star persist in their bound states. For a collapsing boson star, we show that orbits initially bound and reciprocating finally either become unbound or plunge into the newly formed black hole, depending on their initial maximal radii. For initially circular geodesics, we have discovered the existence of a critical radius. Orbits with radii below this critical value are found to plunge into the newly formed black hole, whereas those with radii larger than the critical radius continue to orbit around the vicinity of the newly formed black hole, exhibiting nonzero eccentricities. For the migrating case, a black hole does not form. In this case, the reciprocating orbits span a wider radial range. For initially circular geodesics, orbits with small radii become unbound, and orbits with large radii remain bound with nonvanishing eccentricities. This geodesic study provides an approach to investigating the gravitational collapse and migration of boson stars. Published by the American Physical Society 2024

Evidence of Water Vapor in the Atmosphere of a Metal-rich Hot Saturn with High-resolution Transmission Spectroscopy

Transmission spectroscopy presents one of the most successful approaches for investigating the atmospheres of exoplanets. We analyzed the near-infrared high-resolution transmission spectrum of a hot Saturn, HD 149026 b, taken using CARMENES spectrograph ( R∼80,400 ). We found evidence of H2O at a signal-to-noise ratio (S/N) of ∼4.8. We also performed grid search using a Bayesian framework and constrained the orbital velocity K p and rest velocity V rest to 158.17−7.90+8.31 km s−1 and 2.57−0.57+0.54 km s−1, respectively. While the retrieved K p value is consistent with theoretical prediction, the retrieved V rest value is highly redshifted (>3σ). This might be an indication of either anomalous atmospheric dynamics at play or an orbit with nonzero eccentricity. Additionally, we searched for HCN but no successful detection has been made, possibly due to the relatively low-S/N data set. The detection of H2O and subsequent retrieval of its abundance, coupled with analysis of other species such as CO in the K band, for example, might help us to get some information about the atmospheric C/O ratio and metallicity, which in turn could give us some insight into the planet’s formation scenario.

Extreme Resonant Eccentricity Excitation of Stars around Merging Black-Hole Binary.

We study the dynamics of a star orbiting a merging black-hole binary (BHB) in a coplanar triple configuration. During the BHB's orbital decay, the system can be driven across the apsidal precession resonance, where the apsidal precession rate of the stellar orbit matches that of the inner BHB. As a result, the system gets captured into a state of resonance advection until the merger of the BHB, leading to extreme eccentricity growth of the stellar orbit. This resonance advection occurs when the inner binary has a nonzero eccentricity and unequal masses. The resonant driving of the stellar eccentricity can significantly alter the hardening rate of the inner BHB and produce observational signatures to uncover the presence of nearby merging or merged BHBs.

Constraints on Bianchi-I type universe with SH0ES anchored Pantheon+ SNIa data

We study the Bianchi-I cosmological model motivated by signals of statistical isotropy violation seen in cosmic microwave background (CMB) observations and others. To that end, we consider various kinds of anisotropic matter that source anisotropy in our model, specifically Cosmic strings, Magnetic fields, Domain walls and Lorentz violation generated magnetic fields. These anisotropic matter sources, taking one at a time, are studied for their co-evolution with standard model (isotropic) sources viz., dust-like (dark/normal) matter, and dark energy modelled as cosmological constant. We constrain the Hubble parameter, density fractions of anisotropic matter, cold dark matter (CDM), and dark energy (Λ) in a Bianchi-I universe with planar symmetry i.e., which has a global ellipsoidal geometry, and try to find signatures of a cosmic preferred axis if any.The latest compilation of Type Ia Supernova (SNIa) data from Pantheon+SH0ES collaboration is used in our analysis to obtain constraints on cosmological parameters and any preferred axis for our universe.In our analysis, we found mild evidence for a cosmic preferred axis. It is interesting to note that this preferred axis lies broadly in the vicinity of other prominent cosmic anisotropy axes reported in the literature from diverse data sets.Also we find some evidence for non-zero (negative) cosmic shear and eccentricity that characterize different expansion rates in different directions and deviation from an isotropic scale factor respectively.The energy density fractions of two of the sources considered are found to be non-zero at a2σ confidence level.To be more conclusive, we require more SNIa host galaxy data for tighter constraints on distance and absolute magnitude calibration which are expected to be available from the future JWST observations and others.

How many random observations are needed for good phase coverage of a periodic source?

The specific problem considered is the number of radial velocity measurements required to obtain good estimates of physical parameters of binary star. It is assumed that observations are made at random binary phases. The loss of information due to poor phase coverage is explored, and a suggested limit on the largest acceptable gap introduced. The statistical distribution of maximum gap lengths can then be used to specify the minimum number of velocity measurements to obtain good phase coverage with a specified confidence limit. The effects of non-zero orbital eccentricity are discussed, as are the ramifications of having multiple binary targets. The theory is also applicable to the characterisation of the radial velocity curves induced by exoplanets on their host stars, provided that the periods and eccentricities are known (from e.g. transit observations).

Spectroastrometric Survey of Protoplanetary Disks with Inner Dust Cavities

We present high-resolution spectra and spectroastrometric (SA) measurements of fundamental rovibrational CO emission from nine nearby (≲300 pc) protoplanetary disks where large inner dust cavities have been observed. The emission-line profiles and SA signals are fit with a slab disk model that allows the eccentricity of the disk and intensity of the emission to vary as power laws. Six of the sources are well fit with our model, and three of these sources show asymmetric line profiles that can be fit by adopting a nonzero eccentricity. The three other sources have components in either their line profile or SA signal that are not captured by our disk model. Two of these sources (V892 Tau and CQ Tau) have multi-epoch observations that reveal significant variability. CQ Tau and AB Aur have CO line profiles with centrally peaked components that are similar to line profiles which have been interpreted as evidence of molecular gas arising from a wide-angle disk wind. Alternatively, emission from a circumplanetary disk could also account for this component. The interpretations of these results can be clarified in the future with additional epochs that will test the variability timescale of these SA signals. We discuss the utility of using high-resolution spectroscopy for probing the dynamics of gas in the disk and the scenarios that can give rise to profiles that are not fit with a simple disk model.

Resolving the eccentricity of stellar mass binary black holes with next generation ground-based gravitational wave detectors

ABSTRACT Next generation ground-based gravitational wave (GW) detectors are expected to detect ∼104–105 binary black holes (BBHs) per year. Understanding the formation pathways of these binaries is an open question. Orbital eccentricity can be used to distinguish between the formation channels of compact binaries, as different formation channels are expected to yield distinct eccentricity distributions. Due to the rapid decay of eccentricity caused by the emission of GWs, measuring smaller values of eccentricity poses a challenge for current GW detectors due to their limited sensitivity. In this study, we explore the potential of next generation GW detectors such as Voyager, Cosmic Explorer (CE), and Einstein Telescope (ET) to resolve the eccentricity of BBH systems. Considering a GWTC-3 like population of BBHs and assuming some fiducial eccentricity distributions as well as an astrophysically motivated eccentricity distribution (Zevin et al. 2021), we calculate the fraction of detected binaries that can be confidently distinguished as eccentric. We find that for Zevin eccentricity distribution, Voyager, CE, and ET can confidently measure the non-zero eccentricity for ${\sim} 3\%$, 9%, and 13% of the detected BBHs, respectively. In addition to the fraction of resolvable eccentric binaries, our findings indicate that Voyager, CE, and ET require typical minimum eccentricities ≳0.02, 5 × 10−3, and 10−3 at 10 Hz GW frequency, respectively, to identify a BBH system as eccentric. The better low-frequency sensitivity of ET significantly enhances its capacity to accurately measure eccentricity.

Potential Melting of Extrasolar Planets by Tidal Dissipation

Tidal heating on Io due to its finite eccentricity was predicted to drive surface volcanic activity, which was subsequently confirmed by the Voyager spacecraft. Although the volcanic activity in Io is more complex, in theory volcanism can be driven by runaway melting in which the tidal heating increases as the mantle thickness decreases. We show that this runaway melting mechanism is generic for a composite planetary body with liquid core and solid mantle, provided that (i) the mantle rigidity, μ, is comparable to the central pressure, i.e., μ/(ρ gR P) ≳ 0.1 for a body with density ρ, surface gravitational acceleration g, and radius R P; (ii) the surface is not molten; (iii) tides deposit sufficient energy; and (iv) the planet has nonzero eccentricity. We calculate the approximate liquid core radius as a function of μ/(ρ gR P), and find that more than 90% of the core will melt due to this runaway for μ/(ρ gR P) ≳ 1. From all currently confirmed exoplanets, we find that the terrestrial planets in the L 98-59 system are the most promising candidates for sustaining active volcanism. However, uncertainties regarding the quality factors and the details of tidal heating and cooling mechanisms prohibit definitive claims of volcanism on any of these planets. We generate synthetic transmission spectra of these planets assuming Venus-like atmospheric compositions with an additional 5%, 50%, and 98% SO2 component, which is a tracer of volcanic activity. We find a ≳3σ preference for a model with SO2 with 5–10 transits with JWST for L 98-59bcd.

Long-term Evolution of Warps in Debris Disks—Application to the Gyr-old System HD 202628

We present the results of N-body simulations meant to reproduce the long-term effects of mutually inclined exoplanets on debris disks, using the HD 202628 system as a proxy. HD 202628 is a Gyr-old solar-type star that possesses a directly observable, narrow debris ring with a clearly defined inner edge and nonzero eccentricity, hinting at the existence of a sculpting exoplanet. The eccentric nature of the disk leads us to examine the effect on it over Gyr timescales from an eccentric and inclined planet, placed on its orbit through scattering processes. We find that, in systems with dynamical timescales akin to that of HD 202628, a planetary companion is capable of completely tilting the debris disk. This tilt is preserved over the Gyr age of the system. Simulated observations of our models show that an exoplanet around HD 202628 with an inclination misalignment ≳10° would cause the disk to be observably diffuse and broad, which is inconsistent with Atacama Large Millimeter Array (ALMA) observations. With these observations, we conclude that, if there is an exoplanet shaping this disk, it likely had a mutual inclination of less than 5° with the primordial disk. The conclusions of this work can be applied either to debris disks appearing as narrow rings (e.g., Fomalhaut and HR 4796) or to disks that are vertically thick at ALMA wavelengths (e.g., HD 110058).

Spin evolution of Venus-like planets subjected to gravitational and thermal tides

Context. The arrival of powerful instruments will provide valuable data for the characterization of rocky exoplanets. Rocky planets are mostly found in close-in orbits. They are therefore usually close to the circular-coplanar orbital state and are thus considered to be in a tidally locked synchronous spin state. For planets with larger orbits, however, exoplanets should still have nonzero eccentricities and/or obliquities, and realistic models of tides for rocky planets can allow for higher spin states than the synchronization state in the presence of eccentricities or obliquities. Aims. This work explores the secular evolution of a star–planet system under tidal interactions, both gravitational and thermal, induced by the quadrupolar component of the gravitational potential and the irradiation of the planetary surface, respectively. We show the possible spin–orbit evolution and resonances for eccentric orbits and explore the possibility of spin-orbit resonances raised by the obliquity of the planet. Then, we focus on the additional effect of a thick atmosphere on the possible resulting spin equilibrium states and explore the effect of the evolution of the stellar luminosity. Methods. We implemented the general secular evolution equations of tidal interactions in the secular code called ESPEM. In particular, we focus here on the tides raised by a star on a rocky planet and consider the effect of the presence of an atmosphere, neglecting the contribution of the stellar tide. The solid part of the tides was modeled with an anelastic rheology (Andrade model), while the atmospheric tides were modeled with an analytical formulation that was fit using a global climate model simulation. We focused on a Sun-Venus-like system in terms of stellar parameters, orbital configuration and planet size and mass. The Sun-Venus system is a good laboratory for studying and comparing the possible effect of atmospheric tides, and thus to explore the possible spin state of potential Venus-like exoplanets. Results. The formalism of Kaula associated with an Andrade rheology allows spin orbit resonances on pure rocky worlds. Similarly to the high-order spin–orbit resonances induced by eccentricity, the spin obliquity allows the excitation of high-frequency Fourier modes that allow some spin-orbit resonances to be stable. If the planet has a dense atmosphere, like that of Venus, another mechanism, the thermal tides, can counterbalance the effect of the gravitational tides. We found that thermal tides change the evolution of the spin of the planet, including the capture in spin–orbit resonances. If the spin inclination is high enough, thermal tides can drive the spin toward an anti-synchronization state, that is, a the 1:1 spin–orbit resonance with an obliquity of 180 degrees. Conclusions. Through our improvement of the gravitational and thermal tidal models, we can determine the dynamical state of exo-planets better, especially if they hold a thick atmosphere. In particular, the contribution of the atmospheric tides allows us to reproduce the spin state of Venus at a constant stellar luminosity. Our simulations have shown that the secular evolution of the spin and obliquity can lead to a retrograde spin of the Venus-like planet if the system starts from a high spin obliquity, in agreement with previous studies. The perturbing effect of a third body is still needed to determine the current state of Venus starting from a low initial obliquity. When the luminosity evolution of the Sun is taken into account, the picture changes. We find that the planet never reaches equilibrium: the timescale of the rotation evolution is longer than the luminosity variation timescale, which suggests that Venus may never reach a spin equilibrium state, but may still evolve.

Non-integrability of the planar elliptic restricted three-body problem

We present the non-integrability proof for the planar elliptic restricted three-body problem. Two versions of this problem are considered: the classical one when only gravitational interactions are taken into account, and the photo-gravitational version where radiation pressure from the primaries is also included. Our result is valid for nonzero eccentricity and arbitrary mass ratio of the primaries. In the proof, we apply the differential Galois approach to study the integrability.

Long-Term Evolution of the Space Environment Considering Constellation Launches and Debris Disposal

Increasingly frequent launch activities, as well as the development of mega constellations, would cause a drastic increase in the number of space objects, which will then alter the evolution of outer space. To reveal this long-term change, an accurate space-environment model is required. There are two main approaches to building this model, one of which is to track the state of space objects individually, which will use significant computing resources; the other is to take macroscopic variables, such as spatial density, as the state variable to depict a group of space debris, which requires less computational effort. In this study, a space debris environment evolution model with spatial density as the state variable is established, which considers the nonzero eccentricity of the debris orbit and utilizes the NASA breakup model to ensure accuracy. In addition, the Gaussian mixture model (GMM) is applied to take the uncertainty of launch activities into account. The long-term impacts of mega constellations and their post-mission disposal (PMD) on the debris environment are discussed based on the evolution model. It was found that constellations with high orbit altitude, such as OneWeb, will lead to an exponential increase in space objects in low Earth orbit (LEO). In addition, deorbit time is the main factor affecting the PMD efficiency, followed by deorbit strategies.

Inner Habitable Zone Boundary for Eccentric Exoplanets

The climate of a planet can be strongly affected by its eccentricity due to variations in the stellar flux. There are two limits for the dependence of the inner habitable zone boundary (IHZ) on eccentricity: (1) the mean stellar flux approximation (), in which the temperature is approximately constant throughout the orbit, and (2) the maximum stellar flux approximation (S IHZ ∝ (1 − e)2), in which the temperature adjusts instantaneously to the stellar flux. Which limit is appropriate is determined by the dimensionless parameter , where C is the heat capacity of the planet, P is the orbital period, and , where Ω is the outgoing long-wave radiation and T s is the surface temperature. We use the Buckingham Π theorem to derive an analytical function for the IHZ in terms of eccentricity and Π. We then build a time-dependent energy balance model to resolve the surface temperature evolution and constrain our analytical result. We find that Π must be greater than about ∼1 for the mean stellar flux approximation to be nearly exact and less than about ∼0.01 for the maximum stellar flux approximation to be nearly exact. In addition to assuming a constant heat capacity, we also consider the effective heat capacity including latent heat (evaporation and precipitation). We find that for planets with an Earthlike ocean, the IHZ should follow the mean stellar flux limit for all eccentricities. This work will aid in the prioritization of potentially habitable exoplanets with nonzero eccentricity for follow-up characterization.

Uncovering a hidden black hole binary from secular eccentricity variations of a tertiary star

We study the dynamics of a solar-type star orbiting around a black hole binary (BHB) in a nearly coplanar system. We present a novel effect that can prompt a growth and significant oscillations of the eccentricity of the stellar orbit when the system encounters an ``apsidal precession resonance,'' where the apsidal precession rate of the outer stellar orbit matches that of the inner BHB. The eccentricity excitation requires the inner binary to have a nonzero eccentricity and unequal masses, and can be created even in noncoplanar triples. We show that the secular variability of the stellar orbit's apocenter, induced by the changing eccentricity, could be potentially detectable by Gaia. Detection is favorable for BHBs emitting gravitational waves in the frequency band of the Laser Interferometer Space Antenna, hence providing a distinctive, multimessenger probe of the existence of stellar-mass BHBs in the Milky Way.

Parameter distributions of binary black hole mergers near supermassive black holes as seen by advanced gravitational wave detectors

ABSTRACT The environment surrounding supermassive black holes (SMBHs) in galactic nuclei (GNs) is expected to harbour stellar-mass binary black hole (BBH) populations. These binaries were suggested to form a hierarchical triple system with the SMBH, and gravitational perturbations from the SMBH can enhance the mergers of BBHs through Lidov–Kozai (LK) oscillations. Previous studies determined the expected binary parameter distribution for this merger channel in single GNs. Here, we account for the different spatial distribution and mass distribution models of BBHs around SMBHs and perform direct high-precision regularized N-body simulations, including Post-Newtonian (PN) terms up to order PN2.5, to model merging BBH populations in single GNs. We use a full inspiral-merger-ringdown waveform model of BBHs with non-zero eccentricities and take into account the observational selection effect to determine the parameter distributions of LK-induced BBHs detected with a single advanced gravitational-wave (GW) detector from all GNs in the Universe. We find that the detected mergers’ total binary mass distribution is tilted towards lower masses, and the mass ratio distribution is roughly uniform. The redshift distribution peaks between ∼0.15 and 0.55, and the vast majority of binaries merge within redshift ∼1.1. The fraction of binaries entering the LIGO/Virgo/KAGRA band with residual eccentricities >0.1 is below $\sim 10 {{\ \rm per\ cent}}$. We identify a negative correlation between residual eccentricity and mass parameters and a negative correlation between residual eccentricity and source distance. Our results for the parameter distributions and correlations among binary parameters may make it possible to disentangle this merger channel from other BBH merger channels statistically.

Study of five eccentric eclipsing binary systems

In this paper we present the comprehensive analysis of five binary systems: GX Lac, TT Lyr, OGLE LMC-ECL-7641, OGLE LMC-ECL-17660, and OGLE LMC-ECL-17411. With new photometric observations combined with other available data mostly from TESS, MACHO, and OGLE, we calculated new minima timings and study the O−C diagrams. We fitted the light curves and radial velocity curves using PHOEBE software and obtain absolute parameters of these objects. All five binary stars are eccentric. We present the first analysis of TT Lyr, which reveals its nonzero eccentricity. The calculated distance of OGLE LMC-ECL-7641 indicates that this possibly triple system could be located in the Magellanic Stream.

Tidal excitation of the obliquity of Earth-like planets in the habitable zone of M-dwarf stars

Close-in planets undergo strong tidal interactions with the parent star that modify their spins and orbits. In the two-body problem, the final stage for tidal evolution is the synchronisation of the rotation and orbital periods, and the alignment of the planet spin axis with the normal to the orbit (zero planet obliquity). The orbital eccentricity is also damped to zero, but over a much longer timescale, that may exceed the lifetime of the system. For non-zero eccentricities, the rotation rate can be trapped in spin–orbit resonances that delay the evolution towards the synchronous state. Here we show that capture in some spin–orbit resonances may also excite the obliquity to high values rather than damp it to zero. Depending on the system parameters, obliquities of 60º–80º can be maintained throughout the entire lifetime of the planet. This unexpected behaviour is particularly important for Earth-like planets in the habitable zone of M-dwarf stars, as it may help to sustain temperate environments and thus more favourable conditions for life.

Challenges in Forming Phobos and Deimos Directly from a Splitting of an Ancestral Single Moon

The origin and evolution of Martian moons have been intensively debated in recent years. It is proposed that Phobos and Deimos may originate directly from the splitting of an ancestral moon orbiting at around the Martian synchronous orbit. At this hypothetical splitting, the apocenter of the inner moon (presumed as Phobos) and the pericenter of the outer moon (presumed as Deimos) would coincide, in that, their semimajor axes would reside inside and outside the Martian synchronous orbit with nonzero eccentricities, respectively. However, the successive orbital evolution of the two moons is not studied. Here, we perform direct N-body orbital integrations of the moons, including the Martian oblateness of the J 2 and J 4 terms. We show that the two moons, while they precess, likely collide within ∼104 yr with an impact velocity of v imp ∼ 100–300 m s−1 (∼10–30 times moons’ escape velocity) and with an isotropic impact direction. The impact occurs around the apocenter and the pericenter of the inner and outer moons, respectively, where the timescale of this periodic orbital alignment is regulated by the precession. By performing additional impact simulations, we show that such a high-velocity impact likely results in a disruptive outcome, forming a debris ring at around the Martian synchronous orbit, from which several small moons would accrete. Such an evolutionary path would eventually form a different Martian moon system from the one we see today. Therefore, it seems unlikely that Phobos and Deimos are split directly from a single ancestral moon.