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Register a new userSolving the Alhazen-Ptolemy Problem: Determining Specular Points on Spherical Surfaces for Radiative Transfer of Titans Seas
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Given a light source, a spherical reflector, and an observer, where on the surface of the sphere will the light be directly reflected to the observer, i.e. where is the the specular point? This is known as the Alhazen-Ptolemy problem, and finding this specular point for spherical reflectors is useful in applications ranging from computer rendering to atmospheric modeling to GPS communications. Existing solutions rely upon finding the roots of a quartic equation and evaluating numerically which root provides the real specular point. We offer a formulation, and two solutions thereof, for which the correct root is predeterminable, thereby allowing the construction of the fully analytical solutions we present. Being faster to compute, our solutions should prove useful in cases which require repeated calculation of the specular point, such as Monte-Carlo radiative transfer, including reflections off of Titans hydrocarbon seas.
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Seismology is the main tool for inferring the deep interior structures of Earth and potentially also of other planetary bodies in the solar system. Terrestrial seismology is influenced by the presence of the ocean-generated microseismic signal, which sets a lower limit on the earthquake detection capabilities but also provides a strong energy source to infer the interior structure on scales from local to continental. Titan is the only other place in the solar system with permanent surface liquids and future lander missions there might carry a seismic package. Therefore, the presence of microseisms would be of great benefit for interior studies, but also for detecting storm-generated waves on the lakes remotely. We estimated the strength of microseismic signals on Titan, based on wind speeds predicted from modeled global circulation models interior structure. We find that storms of more than 2 m/s wind speed, would create a signal that is globally observable with a high-quality broadband sensor and observable to a thousand kilometer distance with a space-ready seismometer, such as the InSight instruments currently operating on the surface of Mars.
The discontinuous Galerkin finite element method (DG-FEM) is successfully applied to treat a broad variety of transport problems numerically. In this work, we use the full capacity of the DG-FEM to solve the radiative transfer equation in spherical symmetry. We present a discontinuous Galerkin method to directly solve the spherically-symmetric radiative transfer equation as a two-dimensional problem. The transport equation in spherical atmospheres is more complicated than in the plane-parallel case due to the appearance of an additional derivative with respect to the polar angle. The DG-FEM formalism allows for the exact integration of arbitrarily complex scattering phase functions, independent of the angular mesh resolution. We show that the discontinuous Galerkin method is able to describe accurately the radiative transfer in extended atmospheres and to capture discontinuities or complex scattering behaviour which might be present in the solution of certain radiative transfer tasks and can, therefore, cause severe numerical problems for other radiative transfer solution methods.
Occurring in protoplanetary discs composed of dust and gas, streaming instabilities are a favoured mechanism to drive the formation of planetesimals. The Polydispserse Streaming Instability is a generalisation of the Streaming Instability to a continuum of dust sizes. This second paper in the series provides a more in-depth derivation of the governing equations and presents novel numerical methods for solving the associated linear stability problem. In addition to the direct discretisation of the eigenproblem at second order introduced in the previous paper, a new technique based on numerically reducing the system of integral equations to a complex polynomial combined with root finding is found to yield accurate results at much lower computational cost. A related method for counting roots of the dispersion relation inside a contour without locating those roots is also demonstrated. Applications of these methods show they can reproduce and exceed the accuracy of previous results in the literature, and new benchmark results are provided. Implementations of the methods described are made available in an accompanying Python package psitools.
This and companion papers by Harrington et al. and Blecic et al. present the Bayesian Atmospheric Radiative Transfer ({BART}) code, an open-source, open-development package to characterize extrasolar-planet atmospheres. {BART} combines a thermochemical equilibrium abundances ({TEA}), a radiative-transfer ({transit}), and a Bayesian statistical (MC3) module to constrain atmospheric temperatures and molecular abundances for given spectroscopic observations. Here, we describe the {transit} radiative-transfer package, an efficient line-by-line radiative-transfer C code for one-dimensional atmospheres, developed by P. Rojo and further modified by the UCF exoplanet group. This code produces transmission and hemisphere-integrated emission spectra. {transit} handles line-by-line opacities from HITRAN, Partridge & Schwenke ({water}), Schwenke (TiO), and Plez (VO); and collision-induced absorption from Borysow, HITRAN, and ExoMol. {transit} emission-spectra models agree with models from C. Morley (priv. comm.) within a few percent. We applied {BART} to the {Spitzer} and {Hubble} transit observations of the Neptune-sized planet HAT-P-11b. Our results generally agree with those from previous studies, constraining the {water} abundance and finding an atmosphere enhanced in heavy elements. Different conclusions start to emerge when we make different assumptions from other studies. The {BART} source code and documentation are available at https://github.com/exosports/BART.
An accurate and efficient method dealing with the few-body dynamics is important for simulating collisional N-body systems like star clusters and to follow the formation and evolution of compact binaries. We describe such a method which combines the time-transformed explicit symplectic integrator (Preto & Tremaine 1999; Mikkola & Tanikawa 1999) and the slow-down method (Mikkola & Aarseth 1996). The former conserves the Hamiltonian and the angular momentum for a long-term evolution, while the latter significantly reduces the computational cost for a weakly perturbed binary. In this work, the Hamilton equations of this algorithm are analyzed in detail. We mathematically and numerically show that it can correctly reproduce the secular evolution like the orbit averaged method and also well conserve the angular momentum. For a weakly perturbed binary, the method is possible to provide a few order of magnitude faster performance than the classical algorithm. A publicly available code written in the c++ language, SDAR, is available on GitHub (https://github.com/lwang-astro/SDAR). It can be used either as a stand alone tool or a library to be plugged in other $N$-body codes. The high precision of the floating point to 62 digits is also supported.
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