Context. Modified Newtonian dynamics (MOND) is a promising alternative to dark matter. To further test the theory, there is a need for fluid- and particle-dynamics simulations. The force in MOND is not a direct particle-particle interaction, but derives from a potential for which a nonlinear partial differential equation (PDE) needs to be solved. Normally, this makes the problem of simulating dynamical evolution computationally expensive. Aims. We intend to develop a fast particle-mesh (PM) code for MOND (the AQUAL formalism). Methods. We transformed the nonlinear equation for MOND into a system of linear PDEs plus one algebraic equation. An iterative scheme with the fast Fourier transform (FFT) produces successively better numerical approximations. Results. The algorithm was tested for dynamical systems in MOND where analytical solutions are known: the two-body problem, a body with a circular ring, and a spherical distribution of particles in thermal equilibrium in the self-consistent potential. Conclusions. The PM code can accurately calculate the forces at subpixel scale and reproduces the analytical solutions. Four iterations are required for the potential, but when the spatial steps are small compared to the kernel width, one iteration is suffices. The use of a smoothing kernel for the accelerations is inevitable in order to eliminate the self-gravity of the point particles. Our PDE solver is 15 to 42 times as slow as a standard Poisson solver. However, the smoothing and particle propagation takes up most of the time above one particle per 10 ³ pixels. The FFTs, the smoothing, and the propagation part in the code can all be parallelized.

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Context. In a Keplerian system, a large number of bodies orbit a central mass. Accretion disks, protoplanetary disks, asteroid belts, and planetary rings are examples. Simulations of these systems require algorithms that are computationally efficient. The inclusion of collisions in the simulations is challenging but important. Aims. We intend to calculate the time of collision of two astronomical bodies in intersecting Kepler orbits as a function of the orbital elements. The aim is to use the solution in an analytic propagator (N-body simulation) that jumps from one collision event to the next. Methods. We outline an algorithm that maintains a list of possible collision pairs ordered chronologically. At each step (the soonest event on the list), only the particles created in the collision can cause new collision possibilities. We estimate the collision rate, the length of the list, and the average change in this length at an event, and study the efficiency of the method used. Results. We find that the collision-time problem is equivalent to finding the grid point between two parallel lines that is closest to the origin. The solution is based on the continued fraction of the ratio of orbital periods. Conclusions. Due to the large jumps in time, the algorithm can beat tree codes (octree and k-d tree codes can efficiently detect collisions) for specific systems such as the Solar System with N < 108. However, the gravitational interactions between particles can only be treated as gravitational scattering or as a secular perturbation, at the cost of reducing the time-step or at the cost of accuracy. While simulations of this size with high-fidelity propagators can already span vast timescales, the high efficiency of the collision detection allows many runs from one initial state or a large sample set, so that one can study statistics.

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Context. In the near future, dedicated telescopes will observe Earth-like exoplanets in reflected parent starlight, allowing their physical characterization. Because of the huge distances, every exoplanet will remain an unresolved, single pixel, but temporal variations in the pixel’s spectral flux contain information about the planet’s surface and atmosphere.

Aims. We tested convolutional neural networks for retrieving a planet’s rotation axis, surface, and cloud map from simulated single-pixel observations of flux and polarization light curves. We investigated the influence of assuming that the reflection by the planets is Lambertian in the retrieval while in reality their reflection is bidirectional, and the influence of including polarization.

Methods. We simulated observations along a planet’s orbit using a radiative transfer algorithm that includes polarization and bidirectional reflection by vegetation, deserts, oceans, water clouds, and Rayleigh scattering in six spectral bands from 400 to 800 nm, at various levels of photon noise. The surface types and cloud patterns of the facets covering a model planet are based on probability distributions. Our networks were trained with simulated observations of millions of planets before retrieving maps of test planets.

Results. The neural networks can constrain rotation axes with a mean squared error (MSE) as small as 0.0097, depending on the orbital inclination. On a bidirectionally reflecting planet, 92% of ocean facets and 85% of vegetation, deserts, and cloud facets are correctly retrieved, in the absence of noise. With realistic amounts of noise, it should still be possible to retrieve the main map features with a dedicated telescope. Except for face-on orbits, a network trained with Lambertian reflecting planets yields significant retrieval errors when given observations of bidirectionally reflecting planets, in particular, brightness artifacts around a planet’s pole. Including polarization improves the retrieval of the rotation axis and the accuracy of the retrieval of ocean and cloudy map facets.@en

Context. A double planet system or planet binary undergoes eclipses that modify the reflective light curve. In the time domain, the eclipse events are fast and weak. This would make their signal difficult to find and recognize in the phase light curve, even for small inclinations when eclipses happen frequently. However, due to the quasiperiodic nature of the phenomenon, the Fourier transform of the direct reflection signal consists of a double sum of sharp peaks. These peaks can be resolved for large close binaries and sufficiently long observation times with a star coronagraph. Aims. Eclipses modulate the phase curve, having an orbital period 2π/ω, with a contribution from the relative motion in the binary plane of a period 2π/Ω. This leads to a spectral structure with basis frequencies ω and Ω. We aim to characterize these spectra. Methods. We studied the regime of short eclipses that occur when the planet radii are small compared to the planet separation. We derived formulas for the peak amplitudes applicable to homogeneous (Lambertian) planet binaries in circular orbit with small inclination. Results. The effects of an eclipse and of double reflection appear as first- and second-order contributions (in planet radius over separation) in the reflection signal, respectively. Small peaks appear as observable side bands in the spectrum. Identical structures around mΩ are characteristic of short-duration eclipses. Deceasing side bands could indicate double reflection between companions. Conclusions. Fourier analysis of the light curve of non-transiting planets can be used to find planets and their moons. Difficulties in interpreting the structures arise for small planet separation and when there are several moons in mean-motion resonance.@en

Context. The weak orbital-phase dependent reflection signal of an exoplanet contains information on the planet surface, such as the distribution of continents and oceans on terrestrial planets. This light curve is usually studied in the time domain, but because the
signal from a stationary surface is (quasi)periodic, analysis of the Fourier series may provide an alternative, complementary approach.

Aims. We study Fourier spectra from reflected light curves for geometrically simple configurations. Depending on its atmospheric properties, a rotating planet in the habitable zone could have circular polar ice caps. Tidally locked planets, on the other hand, may have symmetric circular oceans facing the star. These cases are interesting because the high-albedo contrast at the sharp edges of the ice-sheets and the glint from the host star in the ocean may produce recognizable light curves with orbital periodicity, which could also be interpreted in the Fourier domain.

Methods. We derive a simple general expression for the Fourier coefficients of a quasiperiodic light curve in terms of the albedo map of a Lambertian planet surface. Analytic expressions for light curves and their spectra are calculated for idealized situations, and dependence of the spectral peaks on the key parameters inclination, obliquity, and cap size is studied.

Results. The ice-scattering and ocean glint contributions can be separated out, because the coefficients for glint are all positive, whereas ice sheets lead to even-numbered, higher harmonics. An in-view polar cap on a planet without axial tilt only produces a single peak. The special situation of edge-on observation, which is important for planets in transit, leads to the most pronounced spectral behavior. Then the respective spectra from planets with a circumventing ocean, a circular ocean (eyeball world), polar caps,
and rings, have characteristic power-law tails  n^-2, n^-7/2, n^-4, and (−1)ⁿ⁺¹ n^-2.

Conclusions. Promising recently discovered planetary systems may be selected as candidates for long-term (multiyear) observation: their Fourier spectra could separate the different planets and reveal or identify a water-covered planet with polar caps.@en