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Register a new userModified Hermite Integrators of Arbitrary Order
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Alexander J. Dittmann
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We present a family of modified Hermite integrators of arbitrary order possessing superior behaviour for the integration of Keplerian and near-Keplerian orbits. After recounting the derivation of Hermite N-body integrators of arbitrary order, we derive a corrector expression that minimises integrated errors in the argument of periapsis for any such integrator. In addition to providing an alternate derivation of the modified corrector for the 4th-order Hermite integrator, we focus on improved correctors for the 6th- and 8th-order integrators previously featured in the literature. We present a set of numerical examples and find that the higher-order schemes improve performance, even when considering their slightly higher cost in floating point operations. The algorithms presented herein hold promise for systems dominated by central potentials, such as planetary systems and the centres of galaxies. Existing Hermite integrators of any order can be modified to use the expressions presented here with minimal effort. Accordingly the schemes presented herein can be easily implemented on massively parallel architectures.
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In most of mesh-free methods, the calculation of interactions between sample points or particles is the most time consuming. When we use mesh-free methods with high spatial orders, the order of the time integration should also be high. If we use usual Runge-Kutta schemes, we need to perform the interaction calculation multiple times per one time step. One way to reduce the number of interaction calculations is to use Hermite schemes, which use the time derivatives of the right hand side of differential equations, since Hermite schemes require smaller number of interaction calculations than RK schemes do to achieve the same order. In this paper, we construct a Hermite scheme for a mesh-free method with high spatial orders. We performed several numerical tests with fourth-order Hermite schemes and Runge-Kutta schemes. We found that, for both of Hermite and Runge-Kutta schemes, the overall error is determined by the error of spatial derivatives, for timesteps smaller than the stability limit. The calculation cost at the timestep size of the stability limit is smaller for Hermite schemes. Therefore, we conclude that Hermite schemes are more efficient than Runge-Kutta schemes and thus useful for high-order mesh-free methods for Lagrangian Hydrodynamics.
In this paper, we present a generalization of the super-twisting algorithm for perturbed chains of integrators of arbitrary order. This Higher Order Super-Twisting (HOST) controller, which extends the approach of Moreno and als., is homegeneous with respect to a family of dilations and can be continuous. Its design is derived from a first result obtained for pure chains of integrators, the latter relying on a geometric condition introduced by the authors. The complete result is established using a homogeneous strict Lyapunov function which is explicitely constructed. The effectiveness of the controller is finally illustrated with simulations for a chain of integrator of order four, first pure then perturbed, where we compare the performances of two HOST controllers.
We investigate the microlensing event KMT-2021-BLG-0322, for which the light curve exhibits three distinctive sets of caustic-crossing features. It is found that the overall features of the light curve are approximately described by a binary-lens (2L1S) model, but the model leaves substantial residuals. We test various interpretations with the aim of explaining the residuals. We find that the residuals can be explained either by considering a nonrectilinear lens-source motion caused by the microlens-parallax and lens-orbital effects or by adding a low-mass companion to the binary lens (3L1S model). The degeneracy between the higher-order 2L1S model and the 3L1S model is very severe, making it difficult to single out a correct solution based on the photometric data. This degeneracy was known before for two previous events (MACHO-97-BLG-41 and OGLE-2013-BLG-0723), which led to the false detections of planets in binary systems, and thus the identification of the degeneracy for KMT-2021-BLG-0322 illustrates that the degeneracy can be not only common but also very severe, emphasizing the need to check both interpretations of deviations from 2L1S models. From the Bayesian analysis conducted with the measured lensing observables of the event timescale, angular Einstein radius, and microlens parallax, it was estimated that the binary lens components have masses $(M_1, M_2) =(0.62^{+0.25}_{-0.26}~M_odot,0.07^{+0.03}_{-0.03}~M_odot)$, for both 2L1S and 3L1S solutions, and the mass of the tertiary lens component according to the 3L1S solution is $M_3=6.40^{+2.64}_{-2.78}~M_{rm J}$.
We use a large data-set of realistic synthetic observations (PaperI) to assess how observational techniques affect the measurement of physical properties of star-forming regions. In this paper (PaperII), we explore the reliability of the measured total gas mass, dust surface density and dust temperature maps derived from modified blackbody fitting of synthetic Herschel observations. We found from our pixel-by-pixel analysis of the measured dust surface density and dust temperature a worrisome error spread especially close to star-formation sites and low-density regions, where for those contaminated pixels the surface densities can be under/overestimated by up to three orders of magnitude. In light of this, we recommend to treat the pixel-based results from this technique with caution in regions with active star formation. In regions of high background typical in the inner Galactic plane, we are not able to recover reliable surface density maps of individual synthetic regions, since low-mass regions are lost in the FIR background. When measuring the total gas mass of regions in moderate background, we find that modified blackbody fitting works well (absolute error:+9%;-13%) up to 10kpc distance (errors increase with distance). Commonly, the initial images are convolved to the largest common beam-size, which smears contaminated pixels over large areas. The resulting information loss makes this commonly-used technique less verifiable as now chi^2-values cannot be used as a quality indicator of a fitted pixel. Our control measurements of the total gas mass (without the step of convolution to the largest common beam size) produce similar results (absolute error:+20%;-7%) while having much lower median errors especially for the high-mass stellar feedback phase. In upcoming papers (III&IV) we test the reliability of measured star-formation rate with direct and indirect techniques.
We present results from a near-infrared (NIR) adaptive optics (AO) survey of pre-main-sequence stars in the Lupus Molecular Cloud with VLT/NACO to identify (sub)stellar companions down to $sim$20 au separation and investigate the effects of multiplicity on circumstellar disc properties. We observe for the first time in the NIR with AO a total of 47 targets and complement our observations with archival data for another 58 objects previously observed with the same instrument. All 105 targets have millimetre ALMA data available, which provide constraints on disc masses and sizes. We identify a total of 13 multiple systems, including 11 doubles and 2 triples. In agreement with previous studies, we find that the most massive (M$_{rm dust}$ $>$ 50 M$_{oplus}$) and largest ($R_{rm dust}>$ 70 au) discs are only seen around stars lacking visual companions (with separations of 20-4800 au) and that primaries tend to host more massive discs than secondaries. However, as recently shown in a very similar study of $>$200 PMS stars in the Ophiuchus Molecular Cloud, the distribution of disc masses and sizes are similar for single and multiple systems for M$_{rm dust} < 50$ M$_{oplus}$ and radii $R_{rm dust}<$ 70 au. Such discs correspond to $sim $80-90% of the sample. This result can be seen in the combined sample of Lupus and Ophiuchus objects, which now includes more than 300 targets with ALMA imaging and NIR AO data, and implies that stellar companions with separations $>$20 au mostly affect discs in the upper 10$%$ of the disc mass and size distributions.
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