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17 pages, 2430 KiB  
Article
PyAMARES, an Open-Source Python Library for Fitting Magnetic Resonance Spectroscopy Data
by Jia Xu, Michael Vaeggemose, Rolf F. Schulte, Baolian Yang, Chu-Yu Lee, Christoffer Laustsen and Vincent A. Magnotta
Diagnostics 2024, 14(23), 2668; https://doi.org/10.3390/diagnostics14232668 - 27 Nov 2024
Viewed by 874
Abstract
Background/Objectives: Magnetic resonance spectroscopy (MRS) is a valuable tool for studying metabolic processes in vivo. While numerous quantification methods exist, the advanced method for accurate, robust, and efficient spectral fitting (AMARES) is among the most used. This study introduces pyAMARES, an open-source [...] Read more.
Background/Objectives: Magnetic resonance spectroscopy (MRS) is a valuable tool for studying metabolic processes in vivo. While numerous quantification methods exist, the advanced method for accurate, robust, and efficient spectral fitting (AMARES) is among the most used. This study introduces pyAMARES, an open-source Python implementation of AMARES, addressing the need for a flexible, user-friendly, and versatile MRS quantification tool within the Python ecosystem. Methods: PyAMARES was developed as a Python library, implementing the AMARES algorithm with additional features such as multiprocessing capabilities and customizable objective functions. The software was validated against established AMARES implementations (OXSA and jMRUI) using both simulated and in vivo MRS data. Monte Carlo simulations were conducted to assess robustness and accuracy across various signal-to-noise ratios and parameter perturbations. Results: PyAMARES utilizes spreadsheet-based prior knowledge and fitting parameter settings, enhancing flexibility and ease of use. It demonstrated comparable performance to existing software in terms of accuracy, precision, and computational efficiency. In addition to conventional AMARES fitting, pyAMARES supports fitting without prior knowledge, frequency-selective AMARES, and metabolite residual removal from mobile macromolecule (MM) spectra. Utilizing multiple CPU cores significantly enhances the performance of pyAMARES. Conclusions: PyAMARES offers a robust, flexible, and user-friendly solution for MRS quantification within the Python ecosystem. Its open-source nature, comprehensive documentation, and integration with popular data science tools enhance reproducibility and collaboration in MRS research. PyAMARES bridges the gap between traditional MRS fitting methods and modern machine learning frameworks, potentially accelerating advancements in metabolic studies and clinical applications. Full article
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Figure 1
<p>Flowchart of pyAMARES. The workflow starts with importing prior knowledge from spreadsheets (1a) and loading the FID signal (1b) to establish initial values and constraints for fitting (3). If the initial parameters are far from the actual values, users can optionally employ Hankel singular value decomposition (HSVD) or Levenberg–Marquardt (LM) initializers to optimize these starting values (2a). The FID signal can be processed directly or optionally filtered using MPFIR to focus on specific spectral regions (2b). The non-linear least-squares minimization (4) using either trust region reflective (TRR) or LM, with either default or user-defined objective functions (1c). The fitting process can be iterative—the output can be fine-tuned and used as initial parameters for subsequent iterations (7). The Cramér–Rao lower bound (CRLB) estimation (5) integrates information from both the fitting results and the linear relationships between parameters (2b). These relationships include constraints like fixed amplitude ratios or chemical shift differences between multiplet peaks. The final output (6) includes fitted parameters, their uncertainties (CRLB), and signal-to-noise ratios. Solid arrows indicate the main workflow, while dashed arrows and boxes represent optional processing steps.</p>
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<p>PyAMARES plotting outputs. The default output figure from the <span class="html-italic">plotAMARES</span> function shows the fit of (<b>A</b>) an in vivo brain <sup>31</sup>P MRS spectrum acquired at 7T [<a href="#B34-diagnostics-14-02668" class="html-bibr">34</a>], (<b>B</b>) a voxel of hyperpolarized <sup>129</sup>Xe MRSI acquired from healthy porcine lungs at 3T, and (<b>D</b>) a voxel of in vivo brain <sup>2</sup>H 3D MRSI spectra acquired at 3T. In (<b>A</b>,<b>B</b>,<b>D</b>), the top panels display the original spectrum (gray), the fitted spectrum (red), and the residual (green dash), with individual fitted components shown in the bottom panels. Panel (<b>A</b>) is shown with phase correction applied (<span class="html-italic">ifphase = True</span> for the <span class="html-italic">plotAMARES</span> function) for display purposes, while (<b>B</b>,<b>D</b>) are not phased. The prior knowledge for the fitting (<b>A</b>) is in <a href="#diagnostics-14-02668-t001" class="html-table">Table 1</a>. The fitting results for <sup>31</sup>P MRS (<b>A</b>), including metabolite concentrations and their respective Cramér–Rao lower bounds (CRLBs), are presented in (<b>C</b>), where green grows indicate reliable fits with CRLB &lt; 20% and red rows indicate less reliable fits. The fitting results of (<b>B</b>,<b>D</b>) are shown in <a href="#app1-diagnostics-14-02668" class="html-app">Figure S2</a>. Abbreviations: RBC, red blood cells; DHO, deuterated water; Glx, combined signals of glutamate and glutamine; PCr: phosphocreatine; PE: phosphoethenolamine; GPE: glycerophosphoethanolamine; GPC: glycerophosphocholine; Pi: inorganic phosphate; NAD, nicotinamide adenine dinucleotide; UDPG, uridine diphosphoglucose.</p>
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<p>Comparison of Monte Carlo simulated single-peak spectra fitting using OXSA and pyAMARES. (<b>A</b>) Ground truth for spectra simulation with fixed (red) and 3000 perturbed (various colors) parameters. Gaussian noise is omitted for clarity. (<b>B</b>) Relative bias of fitted amplitude compared to ground truth at different SNR levels. (<b>C</b>) Bias of fitted chemical shift compared to ground truth at different SNRs. (<b>D</b>) CRLB of fitted amplitude at each SNR, with the 20% threshold indicated by a green dashed line. In (<b>B</b>–<b>D</b>), blue and red represent pyAMARES and OXSA fitted results, respectively; solid patterns indicate results from spectra simulated with perturbed parameters, while hatched patterns show results from spectra simulated with fixed parameters.</p>
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<p>Comparison of Monte Carlo simulated in vivo human brain <sup>31</sup>P MRS spectra fitting at 7T using OXSA and different algorithms implemented in pyAMARES. (<b>A</b>) Ground truth for spectra simulation with slightly perturbed parameters. Gaussian noise is omitted for clarity. (<b>B</b>) Relative bias of peak amplitude quantification compared to ground truth. (<b>C</b>) CRLB of fitted amplitude for each peak, with the 20% threshold indicated by a green dashed line. (<b>D</b>) Pearson’s correlation coefficient (R) between OXSA and pyAMARES quantified amplitudes. Abbreviations: LM: Levenberg–Marquardt algorithm; TRR: trust region reflective algorithm; Init: Initializer using LM; PCr: phosphocreatine; PE: phosphoethenolamine; GPE: glycerophosphoethanolamine; GPC: glycerophosphocholine; Pi: inorganic phosphate; NAD, nicotinamide adenine dinucleotide; UDPG, uridine diphosphoglucose.</p>
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<p>Multiprocessing fitting of dynamic unlocalized <sup>31</sup>P MRS spectra of the tibialis anterior muscle at 3T using pyAMARES and comparison to OXSA. (<b>A</b>). Representative fitting results from pyAMARES (blue solid line) and OXSA (red dash line), with the differences between them shown as green dashed line. The metabolites of interest (PCr and Pi) are labeled. (<b>B</b>). Linear correlations between fitted amplitudes (a.u.), linewidths (Hz), and CRLBs obtained by pyAMARES and OXSA. Pearson’s R and the <span class="html-italic">p</span>-value for each dataset are shown in the plots. (<b>C</b>,<b>D</b>) Time courses of PCr (blue) and Pi (orange) amplitudes fitted by pyAMARES (<b>C</b>) and OXSA (<b>D</b>). The time points at which exercise and recovery start are indicated by dotted and dashed vertical lines, respectively. (<b>E</b>,<b>F</b>) Mono-exponential fitting of the PCr recovery kinetics using pyAMARES (<b>E</b>) and OXSA (<b>F</b>). The fitted equations are PC<sub>recover</sub> = 0.435 − 0.173 × e<sup>−time/44.171</sup>, R<sup>2</sup> = 0.914 for pyAMARES, and PC<sub>recover</sub> = 0.435 − 0.165 × e<sup>−time/42.523</sup>, R<sup>2</sup> = 0.928 for OXSA.</p>
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<p>Using AMARES for post-processing: Removal of metabolite residuals from a short echo time (TE) <sup>1</sup>H MR spectrum at 9.4T. (<b>A</b>) Upper panel: Fitting of residual metabolites (red) and the resulting macromolecule (MM) spectrum after subtraction of residual metabolite signals from the original spectrum (green). Lower panel: AMARES modeling of residual metabolite signals. (<b>B</b>) Comparison of metabolite-free MM spectra obtained by jMRUI (red) and pyAMARES (blue), showing identical results as confirmed by the flat difference spectrum (black).</p>
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17 pages, 3835 KiB  
Article
Clinical Improvement and P63-Deficiency Correction in OLP Patients After Photobiomodulation
by Maria Zaharieva Mutafchieva, Milena Nenkova Draganova, Blagovesta Konstantinova Yaneva, Plamen Ivanov Zagorchev and Georgi Tomchev Tomov
Dent. J. 2024, 12(11), 338; https://doi.org/10.3390/dj12110338 - 22 Oct 2024
Viewed by 884
Abstract
Background: Oral lichen planus (OLP) is a chronic inflammatory disease associated with the formation of symptomatic lesions in the mouth. P63 is essential for epidermal development and regeneration. Weak expression of this protein has been shown in OLP lesions. Photobiomodulation (PBM) therapy has [...] Read more.
Background: Oral lichen planus (OLP) is a chronic inflammatory disease associated with the formation of symptomatic lesions in the mouth. P63 is essential for epidermal development and regeneration. Weak expression of this protein has been shown in OLP lesions. Photobiomodulation (PBM) therapy has been reported to reduce OLP symptoms, but its ability to correct the molecular perturbations of the disease has not been studied. This study aimed to evaluate the efficacy of PBM in OLP treatment by evaluating changes in p63 expression and their association with clinical response. Methods: Twenty OLP patients underwent PBM with a diode laser (810 nm), (0.50 W, 30 s, 1.2 J/cm2), 3 times weekly for a month. The treatment efficacy index (EI) was calculated based on pain-level values and clinical scores of lesions before and after therapy. Biopsies were taken before and after therapy, analyzed immunohistochemically for p63 expression, and compared with 10 healthy controls. Results: P63 levels in OLP lesions were significantly lower than those in normal oral mucosa. After treatment, the pain level and clinical scores of the lesions decreased significantly. The calculated EI showed PBM effectiveness in 90% of cases. Increased p63 positivity and staining intensity were observed after therapy. Conclusions: The established p63 deficiency in OLP lesions is likely an important molecular mechanism in the pathogenesis of the disease. Laser irradiation at 810 nm increased p63 expression to a level close to that found in the healthy epithelium and significantly improved the symptoms and clinical signs of OLP. All of this determines the effectiveness of PBM therapy in the management of OLP. Full article
(This article belongs to the Section Lasers in Dentistry)
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<p>Efficacy index of PBM therapy in patients with OLP.</p>
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<p>(<b>A</b>) Reticular form of OLP on the gingiva with scattered areas of atrophy; (<b>B</b>) mild improvement after PBM therapy (<b>C</b>) erosive lesion on the hard palate, surrounded by atrophic and keratotic fields; (<b>D</b>) healing of the wound at the end of the third week of PBM therapy; (<b>E</b>) partial deletion of the keratotic striae at the end of the fourth week of PBM therapy.</p>
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<p>P63 expression in OLP patients (<span class="html-italic">n</span> = 20) and healthy controls (<span class="html-italic">n</span> = 10).</p>
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<p>(<b>A</b>) Negative control of p63 staining with a slide incubated with PBS instead of primary antibody; (<b>B</b>) section from the prostate used as a positive control for the expression of p63; (<b>C</b>) strong expression (+++) of p63 in healthy oral mucosa; (<b>D</b>) lack of p63 positivity in the epithelium of OLP lesion before therapy (−); (<b>E</b>) increase in p63 immunoreactivity after PBM therapy with staining intensity comparable to that in healthy controls (+++); (<b>F</b>) corresponding Ki-67 expression for the same patient before (+) and (<b>G</b>) after (+++) treatment.</p>
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<p>Association between p63- and Ki-67 levels (Spearman correlation test) in 20 patients with OLP. Light-blue dots represent a single case (patient) (<span class="html-italic">n</span> = 1), with corresponding p63 and Ki-67 levels shown; medium-blue dots represent two cases (patients) (<span class="html-italic">n</span> = 2); and dark-blue dots represent more than two cases (patients, <span class="html-italic">n</span> = 3 and <span class="html-italic">n</span> = 7), with p63 and Ki-67 levels shown.</p>
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<p>Change in p63 expression for all 20 OLP patients after PBM therapy.</p>
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18 pages, 2442 KiB  
Article
Cytotoxic Potencies of Zinc Oxide Nanoforms in A549 and J774 Cells
by Nazila Nazemof, Dalibor Breznan, Yasmine Dirieh, Erica Blais, Linda J. Johnston, Azam F. Tayabali, James Gomes and Premkumari Kumarathasan
Nanomaterials 2024, 14(19), 1601; https://doi.org/10.3390/nano14191601 - 3 Oct 2024
Viewed by 1428
Abstract
Zinc oxide nanoparticles (NPs) are used in a wide range of consumer products and in biomedical applications, resulting in an increased production of these materials with potential for exposure, thus causing human health concerns. Although there are many reports on the size-related toxicity [...] Read more.
Zinc oxide nanoparticles (NPs) are used in a wide range of consumer products and in biomedical applications, resulting in an increased production of these materials with potential for exposure, thus causing human health concerns. Although there are many reports on the size-related toxicity of ZnO NPs, the toxicity of different nanoforms of this chemical, toxicity mechanisms, and potency determinants need clarification to support health risk characterization. A set of well-characterized ZnO nanoforms (e.g., uncoated ca. 30, 45, and 53 nm; coated with silicon oil, stearic acid, and (3-aminopropyl) triethoxysilane) were screened for in vitro cytotoxicity in two cell types, human lung epithelial cells (A549), and mouse monocyte/macrophage (J774) cells. ZnO (bulk) and ZnCl2 served as reference particles. Cytotoxicity was examined 24 h post-exposure by measuring CTB (viability), ATP (energy metabolism), and %LDH released (membrane integrity). Cellular oxidative stress (GSH-GSSG) and secreted proteins (targeted multiplex assay) were analyzed. Zinc oxide nanoform type-, dose-, and cell type-specific cytotoxic responses were seen, along with cellular oxidative stress. Cell-secreted protein profiles suggested ZnO NP exposure-related perturbations in signaling pathways relevant to inflammation/cell injury and corresponding biological processes, namely reactive oxygen species generation and apoptosis/necrosis, for some nanoforms, consistent with cellular oxidative stress and ATP status. The size, surface area, agglomeration state and metal contents of these ZnO nanoforms appeared to be physicochemical determinants of particle potencies. These findings warrant further research on high-content “OMICs” to validate and resolve toxicity pathways related to exposure to nanoforms to advance health risk-assessment efforts and to inform on safer materials. Full article
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<p>Cell morphology observed after exposure of A549 and J774 to the different doses of ZnO nanoparticles (e.g., (<b>A</b>) UC-2 and (<b>B</b>) AM): light microscopy images (40× magnification).</p>
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<p>Cytotoxicity in A549 cells (mean ± SEM) after exposure (24 h) to ZnO nanoforms and the reference particles. Exposure experiments were conducted three times (n = 3), with duplicate samples per treatment group in each exposure experiment. (<b>A</b>) LDH Release, (<b>B</b>) CTB (Resazurin) Reduction, (<b>C</b>) Cellular ATP Levels.</p>
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<p>Cytotoxicity in J774 cells (mean ± SEM) after exposure (24 h) to ZnO nanoforms and reference particles. Exposure experiments were conducted three times (n = 3), with duplicate samples per treatment group in each exposure experiment. (<b>A</b>) LDH Release, (<b>B</b>) CTB (Resazurin) Reduction, (<b>C</b>) Cellular ATP Levels.</p>
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<p>Cellular oxidative stress status in (<b>A</b>) A549 and (<b>B</b>) J774 cells after exposure to ZnO NPs, as well as to the reference particles (30 µg/cm<sup>2</sup>).</p>
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<p>Heatmap and hierarchical clustering of secreted protein fold changes normalized to control (24 h post exposure of cells to ZnO nanoforms and reference particles: (<b>A</b>) A549 and (<b>B</b>) J774). Red—increased; green—decreased.</p>
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<p>Pathway analysis results for in vitro cellular exposure (24 h) to ZnO nanoforms and the reference particles ((<b>A</b>) A549 and (<b>B</b>) J774). Orange—increased; blue—decreased.</p>
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26 pages, 4402 KiB  
Article
Fuel-Efficient and Fault-Tolerant CubeSat Orbit Correction via Machine Learning-Based Adaptive Control
by Mahya Ramezani, Mohammadamin Alandihallaj and Andreas M. Hein
Aerospace 2024, 11(10), 807; https://doi.org/10.3390/aerospace11100807 - 30 Sep 2024
Cited by 1 | Viewed by 1158
Abstract
The increasing deployment of CubeSats in space missions necessitates the development of efficient and reliable orbital maneuvering techniques, particularly given the constraints on fuel capacity and computational resources. This paper presents a novel two-level control architecture designed to enhance the accuracy and robustness [...] Read more.
The increasing deployment of CubeSats in space missions necessitates the development of efficient and reliable orbital maneuvering techniques, particularly given the constraints on fuel capacity and computational resources. This paper presents a novel two-level control architecture designed to enhance the accuracy and robustness of CubeSat orbital maneuvers. The proposed method integrates a J2-optimized sequence at the high level to leverage natural perturbative effects for fuel-efficient orbit corrections, with a gated recurrent unit (GRU)-based low-level controller that dynamically adjusts the maneuver sequence in real-time to account for unmodeled dynamics and external disturbances. A Kalman filter is employed to estimate the pointing accuracy, which represents the uncertainties in the thrust direction, enabling the GRU to compensate for these uncertainties and ensure precise maneuver execution. This integrated approach significantly enhances both the positional accuracy and fuel efficiency of CubeSat maneuvers. Unlike traditional methods, which either rely on extensive pre-mission planning or computationally expensive control algorithms, our architecture efficiently balances fuel consumption with real-time adaptability, making it well-suited for the resource constraints of CubeSat platforms. The effectiveness of the proposed approach is evaluated through a series of simulations, including an orbit correction scenario and a Monte Carlo analysis. The results demonstrate that the integrated J2-GRU system significantly improves positional accuracy and reduces fuel consumption compared to traditional methods. Even under conditions of high uncertainty, the GRU-based control layer effectively compensates for errors in thrust direction, maintaining a low miss distance throughout the maneuvering period. Additionally, the GRU’s simpler architecture provides computational advantages over more complex models such as long short-term memory (LSTM) networks, making it more suitable for onboard CubeSat implementations. Full article
(This article belongs to the Special Issue Small Satellite Missions)
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<p>Orbital elements’ definition.</p>
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<p>The LVLH coordinate reference frame (<math display="inline"><semantics> <mrow> <mi mathvariant="bold-italic">x</mi> <mi mathvariant="bold-italic">y</mi> <msup> <mrow> <mi mathvariant="bold-italic">z</mi> </mrow> <mrow> <mi mathvariant="bold-italic">L</mi> </mrow> </msup> </mrow> </semantics></math>) definition with respect to the Earth-centered inertial reference frame (<math display="inline"><semantics> <mrow> <mi mathvariant="bold-italic">x</mi> <mi mathvariant="bold-italic">y</mi> <msup> <mrow> <mi mathvariant="bold-italic">z</mi> </mrow> <mrow> <mi mathvariant="bold-italic">I</mi> </mrow> </msup> </mrow> </semantics></math>).</p>
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<p>The schematic block diagram of the system.</p>
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<p>The relative plane change error with respect to <math display="inline"><semantics> <mrow> <msub> <mrow> <mi mathvariant="bold-italic">B</mi> </mrow> <mrow> <mi mathvariant="bold-italic">t</mi> </mrow> </msub> </mrow> </semantics></math>.</p>
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<p>Schematic diagram of the <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>J</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> </semantics></math><span class="html-italic">-optimized sequence</span>.</p>
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<p>The time histories of orbital elements’ errors using the <span class="html-italic">classic</span> and <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>J</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> </semantics></math><span class="html-italic">-optimized sequence</span>.</p>
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<p>The time histories of the miss distance using the <span class="html-italic">classic</span> and <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>J</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> </semantics></math><span class="html-italic">-optimized sequence</span>.</p>
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<p>Comparison of required <math display="inline"><semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> <mi>V</mi> </mrow> </semantics></math> using the <span class="html-italic">classic</span> and <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>J</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> </semantics></math><span class="html-italic">-optimized sequence</span>.</p>
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<p>Normalized loss vs. iterations for training and validation.</p>
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<p>Miss distance as a function of pointing accuracy <math display="inline"><semantics> <mrow> <mi>α</mi> </mrow> </semantics></math> for the system using only the <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>J</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> </semantics></math><span class="html-italic">-optimized sequence</span>.</p>
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<p>Miss distance as a function of pointing accuracy <math display="inline"><semantics> <mrow> <mi>α</mi> </mrow> </semantics></math> for the combined system using the <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>J</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> </semantics></math><span class="html-italic">-optimized sequence</span>, GRU, and Kalman filter.</p>
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20 pages, 7721 KiB  
Article
Role of Multiple Intermolecular H-Bonding Interactions in Molecular Cluster of Hydroxyl-Functionalized Imidazolium Ionic Liquid: An Experimental, Topological, and Molecular Dynamics Study
by Sumit Kumar Panja, Sumit Kumar, Boumediene Haddad, Abhishek R. Patel, Didier Villemin, Hakkoum-Mohamed Amine, Sayantan Bera and Mansour Debdab
Physchem 2024, 4(4), 369-388; https://doi.org/10.3390/physchem4040026 - 24 Sep 2024
Cited by 1 | Viewed by 1303
Abstract
Multiple intermolecular H-bonding interactions play a pivotal role in determining the macroscopic state of ionic liquids (ILs). Hence, the relationship between the microscopic and the macroscopic properties is key for a rational design of new imidazolium ILs. In the present work, we investigated [...] Read more.
Multiple intermolecular H-bonding interactions play a pivotal role in determining the macroscopic state of ionic liquids (ILs). Hence, the relationship between the microscopic and the macroscopic properties is key for a rational design of new imidazolium ILs. In the present work, we investigated how the physicochemical property of hydroxyl-functionalized imidazolium chloride is connected to the molecular structure and intermolecular interactions. In the isolated ion pair, strong N-H···Cl H-bonding interactions are observed rather than H-bonding interactions at the acidic C2-H site and alkyl-OH···Cl of the hydroxyl-functionalized imidazolium chloride. However, the N-H···Cl H-bonding interaction of the cation plays a significant role in ion-pair formations and polymeric clusters. For 3-(2-Hydroxy)-1H-imidazolium chloride (EtOHImCl), the oxygen atom (O) engages in two significant interactions within its homodimeric ion-pair cluster: N-H···O and alkyl OH···Cl. Vibrational spectroscopy and DFT calculations reveal that the chloride ion (Cl) forms a hydrogen bond with the C2-H group via a C2-H···Cl interaction site. Natural Bond Orbital (NBO) analysis indicates that the O-H···Cl hydrogen-bonding interaction is crucial for the stability of the IL, with a second-order perturbation energy of approximately 133.8 kJ/mol. Additional computational studies using Atoms in Molecules (AIMs), non-covalent interaction (NCI) analysis, Electron Localization Function (ELF), and Localized Orbital Locator (LOL) provide significant insights into the properties and nature of non-covalent interactions in ILs. Ab initio molecular dynamics (AIMD) simulations of the IL demonstrate its stable states with relatively low energy values around −1680.6510 atomic units (a.u.) at both 100 fs and 400 fs due to O-H···Cl and C-H···Cl interactions. Full article
(This article belongs to the Section Experimental and Computational Spectroscopy)
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<p>(<b>a</b>) C2-H···Cl, (<b>b</b>) Alkyl-OH···Cl, and (<b>c</b>) N-H···Cl H-bonding interactions of ion pairs in EtOHImCl. (<b>d</b>) Intermolecular interactions of dimeric ion pairs from DFT calculation.</p>
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<p>Experimental FTIR spectra of IM and ClEtOH (region: 2000–4000 cm<sup>−1</sup>).</p>
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<p>Experimental FTIR spectra of EtOHImCl (region: 400–4000 cm<sup>−1</sup>).</p>
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<p>Temperature-dependent FTIR spectra of EtOHImCl (region: 400–4000 cm<sup>−1</sup>).</p>
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<p>Experimental Raman spectra of EtOHImCl (region: 2500–3600 cm<sup>−1</sup>). Colored lines: elementary adsorption decomposition.</p>
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<p>Low-frequency Raman spectra of EtOHImCl (region: 350–800 cm<sup>−1</sup>).</p>
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<p>(<b>a</b>) Experimental and (<b>b</b>) theoretical Raman spectra of EtOHImCl (region: 500–3600 cm<sup>−1</sup>).</p>
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<p>NBO interaction of (<b>a</b>) n(1)O1→σ*(1)N21-H34, (<b>b</b>) n(2)O1→σ*(1)N21-H34, (<b>c</b>) n(1)O17→σ*(1)N4-H19, (<b>d</b>) n(1)O17→σ*(1)N4-H19, (<b>e</b>) n(3)Cl35→σ*(1)C11-H12, (<b>f</b>) n(4)Cl35→σ*(1)C11-H12, (<b>g</b>) n(1)Cl35→σ*(1)O17-H18, (<b>h</b>) n(4)Cl35→σ*(1)O17-H18, (<b>i</b>) n(1)Cl36→σ*(1)C1-H2, (<b>j</b>) n(4)Cl36→σ*(1)C1-H2, (<b>k</b>) n(3)Cl36→σ*(1)C28-H29, and (<b>l</b>) n(4)Cl36→σ*(1)C28-H29 for EtOHImCl IL dimer. Atom numbering is shown in <a href="#app1-physchem-04-00026" class="html-app">Figure S2 in Supporting Information</a>.</p>
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<p>AIM representation of dimer of EtOHImCl PIL. Small red spheres show bond critical points (BCPs) whereas the paths joining these spheres are bond paths. Small yellow and green spheres show ring critical points (RCPs) and cage critical points (CCPs), respectively.</p>
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<p>Plot of (blue) electron density <span class="html-italic">ρ</span><sub>bcp</sub>(r) and (red) Laplacian of electron density ∇<sup>2</sup><span class="html-italic">ρ</span><sub>bcp</sub> (r) of the dimer of EtOHImCl PIL calculated using the structure optimized at the B3LYP/6-311++G(d,p) level of theory. Atom numbering is shown in <a href="#app1-physchem-04-00026" class="html-app">Figure S2 in Supporting Information</a>.</p>
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<p>(<b>a</b>) ELF and (<b>b</b>) LOL analysis of the dimer of EtOHImCl PIL calculated using the structure optimized at the B3LYP/6-311++G(d,p) level of theory. Atom numbering is shown in <a href="#app1-physchem-04-00026" class="html-app">Figure S2 in Supporting Information</a>.</p>
Full article ">Figure 12
<p>(<b>red</b>) ELF and (<b>blue</b>) LOL data of the dimer of EtOHImCl PIL calculated using the structure optimized at the B3LYP/6-311++G(d,p) level of theory. Atom numbering is shown in <a href="#app1-physchem-04-00026" class="html-app">Figure S2 in Supporting Information</a>.</p>
Full article ">Figure 13
<p>(<b>a</b>) RDG plot and (<b>b</b>) isosurface extractions of RDG plots of the NCI analysis of the EtOHImCl PIL dimer calculated using the structure optimized at the B3LYP/6-311++G(d,p) level of theory.</p>
Full article ">Figure 14
<p>(<b>a</b>–<b>c</b>): From top to the bottom: Energy drift and important hydrogen bond distances during AIMD simulations of the dimer of EtOHImCl PIL. Atom numbering is shown in <a href="#app1-physchem-04-00026" class="html-app">Figure S2 in Supporting Information</a>.</p>
Full article ">Scheme 1
<p>Structural formula and abbreviation for investigated system.</p>
Full article ">Scheme 2
<p>Synthetic route of EtOHImCl.</p>
Full article ">
15 pages, 6223 KiB  
Article
Revisiting the Most Stable Structures of the Benzene Dimer
by Jiří Czernek and Jiří Brus
Int. J. Mol. Sci. 2024, 25(15), 8272; https://doi.org/10.3390/ijms25158272 - 29 Jul 2024
Cited by 1 | Viewed by 1223
Abstract
The benzene dimer (BD) is an archetypal model of π∙∙∙π and C–H∙∙∙π noncovalent interactions as they occur in its cofacial and perpendicular arrangements, respectively. The enthalpic stabilization of the related BD structures has been debated for a long time and is revisited here. [...] Read more.
The benzene dimer (BD) is an archetypal model of π∙∙∙π and C–H∙∙∙π noncovalent interactions as they occur in its cofacial and perpendicular arrangements, respectively. The enthalpic stabilization of the related BD structures has been debated for a long time and is revisited here. The revisit is based on results of computations that apply the coupled-cluster theory with singles, doubles and perturbative triples [CCSD(T)] together with large basis sets and extrapolate results to the complete basis set (CBS) limit in order to accurately characterize the three most important stationary points of the intermolecular interaction energy (ΔE) surface of the BD, which correspond to the tilted T-shaped (TT), fully symmetric T-shaped (FT) and slipped-parallel (SP) structures. In the optimal geometries obtained by searching extensive sets of the CCSD(T)/CBS ΔE data of the TT, FT and SP arrangements, the resulting ΔE values were −11.84, −11.34 and −11.21 kJ/mol, respectively. The intrinsic strength of the intermolecular bonding in these configurations was evaluated by analyzing the distance dependence of the CCSD(T)/CBS ΔE data over wide ranges of intermonomer separations. In this way, regions of the relative distances that favor BD structures with either π∙∙∙π or C–H∙∙∙π interactions were found and discussed in a broader context. Full article
(This article belongs to the Special Issue Feature Papers in 'Physical Chemistry and Chemical Physics' 2024)
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Graphical abstract

Graphical abstract
Full article ">Figure 1
<p>The tilted T-shaped (in magenta, to the left) and slipped-parallel (in cyan, to the right) structures of the benzene dimer, shown at the same scale. Also depicted is a schematic representation of the geometric parameters defining the respective angles <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>ϑ</mi> </mrow> <mrow> <mi>A</mi> </mrow> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>ϑ</mi> </mrow> <mrow> <mi>B</mi> </mrow> </msub> </mrow> </semantics></math>.</p>
Full article ">Figure 2
<p>Surface plot of the intermolecular interaction energy in the region around the global minimum of the benzene dimer. The <math display="inline"><semantics> <mrow> <mfenced separators="|"> <mrow> <msub> <mrow> <mi>β</mi> </mrow> <mrow> <mi>A</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mi>γ</mi> </mrow> <mrow> <mi>B</mi> </mrow> </msub> </mrow> </mfenced> </mrow> </semantics></math> angular cuts generated by an interpolation of the <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>E</mi> <mfenced separators="|"> <mrow> <msub> <mrow> <mi>β</mi> </mrow> <mrow> <mi>A</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mi>γ</mi> </mrow> <mrow> <mi>B</mi> </mrow> </msub> <mo>,</mo> <mi>R</mi> </mrow> </mfenced> </mrow> </semantics></math> data points for <span class="html-italic">R</span> values of 490.00 and 498.75 pm are shown with black and white mesh, respectively.</p>
Full article ">Figure 3
<p>Plot of the distance dependence of the intermolecular interaction energy of the tilted T-shaped dimer of benzene.</p>
Full article ">Figure 4
<p>Plot of the distance dependence of the intermolecular interaction energy of the canonical T-shaped dimer of benzene.</p>
Full article ">Figure 5
<p>Surface plot of the intermolecular interaction energy in the region around a minimum of the slipped-parallel configuration of the benzene dimer.</p>
Full article ">Figure 6
<p>Plot of the distance dependence of the intermolecular interaction energy of the slipped-parallel dimer of benzene.</p>
Full article ">Figure 7
<p>Plot of the distance dependence of the differences between predicted values of various components of the interaction energy of the tilted and canonical T-shaped configurations of the benzene dimer.</p>
Full article ">Figure 8
<p>Plot of the intrinsic strength of intermolecular interactions in three configurations of the benzene dimer.</p>
Full article ">
32 pages, 992 KiB  
Article
When the Anomalistic, Draconitic and Sidereal Orbital Periods Do Not Coincide: The Impact of Post-Keplerian Perturbing Accelerations
by Lorenzo Iorio
Time Space 2025, 1(1), 2; https://doi.org/10.3390/timespace1010002 - 5 Jul 2024
Cited by 2 | Viewed by 1082
Abstract
In a purely Keplerian picture, the anomalistic, draconitic and sidereal orbital periods of a test particle orbiting a massive body coincide with each other. Such degeneracy is removed when post-Keplerian perturbing acceleration enters the equations of motion, yielding generally different corrections to the [...] Read more.
In a purely Keplerian picture, the anomalistic, draconitic and sidereal orbital periods of a test particle orbiting a massive body coincide with each other. Such degeneracy is removed when post-Keplerian perturbing acceleration enters the equations of motion, yielding generally different corrections to the Keplerian period for the three aforementioned characteristic orbital timescales. They are analytically worked out in the case of the accelerations induced by the general relativistic post-Newtonian gravitoelectromagnetic fields and, to the Newtonian level, by the oblateness of the central body. The resulting expressions hold for completely general orbital configurations and spatial orientations of the spin axis of the primary. Astronomical systems characterized by extremely accurate measurements of orbital periods like transiting exoplanets and binary pulsars may offer potentially viable scenarios for measuring such post-Keplerian features of motion, at least in principle. As an example, the sidereal period of the brown dwarf WD1032 + 011 b is currently known with an uncertainty as small as ≃105s, while its predicted post-Newtonian gravitoelectric correction amounts to 0.07s; however, the accuracy with which the Keplerian period can be calculated is just 572 s. For double pulsar PSR J0737–3039, the largest relativistic correction to the anomalistic period amounts to a few tenths of a second, given a measurement error of such a characteristic orbital timescale as small as 106s. On the other hand, the Keplerian term can be currently calculated just to a 9 s accuracy. In principle, measuring at least two of the three characteristic orbital periods for the same system independently would cancel out their common Keplerian component, provided that their difference is taken into account. Full article
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Figure 1

Figure 1
<p>Perturbed 1 pN trajectory (continuous blue curve) and its osculating Keplerian ellipse (dashed red curve) at the initial instant of time (<math display="inline"><semantics> <msub> <mi>t</mi> <mn>0</mn> </msub> </semantics></math>) of a restricted two-body system characterized by <math display="inline"><semantics> <mrow> <mi>e</mi> <mo>=</mo> <mn>0.95</mn> <mo>,</mo> <mspace width="0.166667em"/> <mi>I</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="0.166667em"/> <mi mathvariant="sans-serif">Ω</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>ω</mi> <mo>=</mo> <msup> <mn>90</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <msub> <mi>f</mi> <mn>0</mn> </msub> <mo>=</mo> <msup> <mn>180</mn> <mo>∘</mo> </msup> </mrow> </semantics></math> as seen from above the fixed orbital plane. Here, it is assumed that both <math display="inline"><semantics> <mi>ω</mi> </semantics></math> and <math display="inline"><semantics> <mi>η</mi> </semantics></math> undergo their known 1 pN gravitoelectric secular precessions due to the mass (<span class="html-italic">M</span>) of the primary [<a href="#B56-timespace-01-00002" class="html-bibr">56</a>]. For a better visualization of their effects, their sizes are suitably rescaled. The positions on the perturbed trajectory after one, two and three Keplerian periods (<math display="inline"><semantics> <msub> <mi>T</mi> <mi mathvariant="normal">K</mi> </msub> </semantics></math>) are marked in gray. On each orbit, the passages at the precessing dashed green line of apsides always occur later than in the Keplerian case with amount given by Equation (<a href="#FD61-timespace-01-00002" class="html-disp-formula">61</a>), which is always positive.</p>
Full article ">Figure 2
<p>Numerically produced time series of the cosine (<math display="inline"><semantics> <mrow> <mover accent="true"> <mi mathvariant="bold-italic">r</mi> <mo stretchy="false">^</mo> </mover> <mo>·</mo> <mover accent="true"> <mi mathvariant="bold-italic">C</mi> <mo stretchy="false">^</mo> </mover> </mrow> </semantics></math>) of the angle between the position vector (<math display="inline"><semantics> <mi mathvariant="bold-italic">r</mi> </semantics></math>) and the Laplace–Runge–Lenz vector (<math display="inline"><semantics> <mi mathvariant="bold-italic">C</mi> </semantics></math>) versus time (<span class="html-italic">t</span>) in units of <math display="inline"><semantics> <msub> <mi>T</mi> <mi mathvariant="normal">K</mi> </msub> </semantics></math> obtained by integrating the equations of motion of a fictitious test particle with (continuous ocher-yellow curve) and without (dashed azure curve) the 1 pN gravitoelectric acceleration of Equation (<a href="#FD14-timespace-01-00002" class="html-disp-formula">14</a>) for an elliptical (<math display="inline"><semantics> <mrow> <mi>e</mi> <mo>=</mo> <mn>0.665</mn> </mrow> </semantics></math>) orbit arbitrarily oriented in space (<math display="inline"><semantics> <mrow> <mi>I</mi> <mo>=</mo> <msup> <mn>40</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <mi mathvariant="sans-serif">Ω</mi> <mo>=</mo> <msup> <mn>45</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <mi>ω</mi> <mo>=</mo> <msup> <mn>50</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>) starting from the periapsis (<math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>), i.e., <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi mathvariant="bold-italic">r</mi> <mo stretchy="false">^</mo> </mover> <mn>0</mn> </msub> <mo>·</mo> <msub> <mover accent="true"> <mi mathvariant="bold-italic">C</mi> <mo stretchy="false">^</mo> </mover> <mn>0</mn> </msub> <mo>=</mo> <mo>+</mo> <mn>1</mn> </mrow> </semantics></math>; the semimajor axis is <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>6</mn> <msub> <mi>R</mi> <mi mathvariant="normal">e</mi> </msub> </mrow> </semantics></math>. The physical parameters of the Earth are adopted. The 1 pN acceleration is suitably rescaled in such a way that <math display="inline"><semantics> <mrow> <mo>Δ</mo> <msubsup> <mi>T</mi> <mi>ano</mi> <mrow> <mn>1</mn> <mi>pN</mi> </mrow> </msubsup> <mo>/</mo> <msub> <mi>T</mi> <mi mathvariant="normal">K</mi> </msub> <mo>=</mo> <mn>0.001</mn> </mrow> </semantics></math>. The time needed to come back to the initial position on the (moving) line of apsides so that <math display="inline"><semantics> <mrow> <mover accent="true"> <mi mathvariant="bold-italic">r</mi> <mo stretchy="false">^</mo> </mover> <mo>·</mo> <mover accent="true"> <mi mathvariant="bold-italic">C</mi> <mo stretchy="false">^</mo> </mover> <mo>=</mo> <mo>+</mo> <mn>1</mn> </mrow> </semantics></math> again is longer than in the Keplerian case by an amount equal to <math display="inline"><semantics> <mrow> <mo>Δ</mo> <msubsup> <mi>T</mi> <mi>ano</mi> <mrow> <mn>1</mn> <mi>pN</mi> </mrow> </msubsup> <mo>=</mo> <mo>+</mo> <mn>0.001</mn> <msub> <mi>T</mi> <mi mathvariant="normal">K</mi> </msub> </mrow> </semantics></math>, as shown by the shaded area, in agreement with Equation (<a href="#FD61-timespace-01-00002" class="html-disp-formula">61</a>).</p>
Full article ">Figure 3
<p>Perturbed LT trajectory (continuous blue curve) and its osculating Keplerian ellipse (dashed red curve) at the initial instant of time (<math display="inline"><semantics> <msub> <mi>t</mi> <mn>0</mn> </msub> </semantics></math>), characterized by <math display="inline"><semantics> <mrow> <mi>e</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> <mspace width="0.166667em"/> <mi>I</mi> <mo>=</mo> <msup> <mn>30</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <mi mathvariant="sans-serif">Ω</mi> <mo>=</mo> <msup> <mn>72</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <mi>ω</mi> <mo>=</mo> <msup> <mn>50</mn> <mo>∘</mo> </msup> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mn>0</mn> </msub> <mo>=</mo> <msup> <mn>180</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>. The orientation of the spin axis (<math display="inline"><semantics> <mover accent="true"> <mi mathvariant="bold-italic">J</mi> <mo stretchy="false">^</mo> </mover> </semantics></math>) of the central body is set by <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mi>J</mi> </msub> <mo>=</mo> <msup> <mn>45</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <msub> <mi>δ</mi> <mi>J</mi> </msub> <mo>=</mo> <msup> <mn>60</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>. In this example, <span class="html-italic">I</span>, <math display="inline"><semantics> <mi mathvariant="sans-serif">Ω</mi> </semantics></math> and <math display="inline"><semantics> <mi>ω</mi> </semantics></math> undergo their known LT precessions due to the spin angular momentum (<math display="inline"><semantics> <mi mathvariant="bold-italic">J</mi> </semantics></math>) of the primary [<a href="#B56-timespace-01-00002" class="html-bibr">56</a>]; their magnitudes are suitably rescaled by enhancing them for a better visualization. The initial position is chosen at the apocenter instead of the pericenter solely for the sake of better visualization. The positions on the perturbed trajectory after one, two and three Keplerian periods are marked. In each orbit, the passage at the drifting dashed green line of apsides always occurs as in the Keplerian case because, according to Equation (<a href="#FD63-timespace-01-00002" class="html-disp-formula">63</a>), <math display="inline"><semantics> <mrow> <mo>Δ</mo> <msubsup> <mi>T</mi> <mi>ano</mi> <mi>LT</mi> </msubsup> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 4
<p>Numerically produced time series of the cosine (<math display="inline"><semantics> <mrow> <mover accent="true"> <mi mathvariant="bold-italic">r</mi> <mo stretchy="false">^</mo> </mover> <mo>·</mo> <mover accent="true"> <mi mathvariant="bold-italic">C</mi> <mo stretchy="false">^</mo> </mover> </mrow> </semantics></math>) of the angle between the position vector (<math display="inline"><semantics> <mi mathvariant="bold-italic">r</mi> </semantics></math>) and the Laplace–Runge–Lenz vector (<math display="inline"><semantics> <mi mathvariant="bold-italic">C</mi> </semantics></math>) versus time (<span class="html-italic">t</span>) in units of <math display="inline"><semantics> <msub> <mi>T</mi> <mi mathvariant="normal">K</mi> </msub> </semantics></math> obtained by integrating the equations of motion of a fictitious test particle with (continuous ocher-yellow curve) and without (dashed azure curve) the LT acceleration of Equation (<a href="#FD22-timespace-01-00002" class="html-disp-formula">22</a>) for an elliptical (<math display="inline"><semantics> <mrow> <mi>e</mi> <mo>=</mo> <mn>0.665</mn> </mrow> </semantics></math>) orbit arbitrarily oriented in space (<math display="inline"><semantics> <mrow> <mi>I</mi> <mo>=</mo> <msup> <mn>40</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <mi mathvariant="sans-serif">Ω</mi> <mo>=</mo> <msup> <mn>45</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <mi>ω</mi> <mo>=</mo> <msup> <mn>50</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>) starting from the periapsis (<math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>), i.e., <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi mathvariant="bold-italic">r</mi> <mo stretchy="false">^</mo> </mover> <mn>0</mn> </msub> <mo>·</mo> <msub> <mover accent="true"> <mi mathvariant="bold-italic">C</mi> <mo stretchy="false">^</mo> </mover> <mn>0</mn> </msub> <mo>=</mo> <mo>+</mo> <mn>1</mn> </mrow> </semantics></math>; the semimajor axis is <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>6</mn> <msub> <mi>R</mi> <mi mathvariant="normal">e</mi> </msub> </mrow> </semantics></math>. The physical parameters of the Earth are adopted, apart from the spin axis position set by <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mi>J</mi> </msub> <mo>=</mo> <msup> <mn>45</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <msub> <mi>δ</mi> <mi>J</mi> </msub> <mo>=</mo> <msup> <mn>60</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>. The time needed to come back to the initial position on the (moving) line of apsides so that <math display="inline"><semantics> <mrow> <mover accent="true"> <mi mathvariant="bold-italic">r</mi> <mo stretchy="false">^</mo> </mover> <mo>·</mo> <mover accent="true"> <mi mathvariant="bold-italic">C</mi> <mo stretchy="false">^</mo> </mover> <mo>=</mo> <mo>+</mo> <mn>1</mn> </mrow> </semantics></math> again is the same as in the Keplerian case.</p>
Full article ">Figure 5
<p>Perturbed <math display="inline"><semantics> <msub> <mi>J</mi> <mn>2</mn> </msub> </semantics></math> trajectory (continuous blue curve) and its osculating Keplerian ellipse (dashed red curve at the initial instant of time (<math display="inline"><semantics> <msub> <mi>t</mi> <mn>0</mn> </msub> </semantics></math>), characterized by <math display="inline"><semantics> <mrow> <mi>e</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> <mspace width="0.166667em"/> <mi>I</mi> <mo>=</mo> <msup> <mn>30</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <mi mathvariant="sans-serif">Ω</mi> <mo>=</mo> <msup> <mn>45</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <mi>ω</mi> <mo>=</mo> <msup> <mn>50</mn> <mo>∘</mo> </msup> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mn>0</mn> </msub> <mo>=</mo> <msup> <mn>180</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, as seen from the <span class="html-italic">z</span>-axis. The orientation of the spin axis (<math display="inline"><semantics> <mover accent="true"> <mi mathvariant="bold-italic">J</mi> <mo stretchy="false">^</mo> </mover> </semantics></math>) of the central body is set by <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mi>J</mi> </msub> <mo>=</mo> <msup> <mn>45</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <msub> <mi>δ</mi> <mi>J</mi> </msub> <mo>=</mo> <msup> <mn>60</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>. In this example, <span class="html-italic">I</span>, <math display="inline"><semantics> <mi mathvariant="sans-serif">Ω</mi> </semantics></math>, <math display="inline"><semantics> <mi>ω</mi> </semantics></math> and <math display="inline"><semantics> <mi>η</mi> </semantics></math> undergo their own Newtonian precessions due to the quadrupole mass moment (<math display="inline"><semantics> <msub> <mi>J</mi> <mn>2</mn> </msub> </semantics></math>) of the primary [<a href="#B56-timespace-01-00002" class="html-bibr">56</a>]; their magnitudes are suitably rescaled by enhancing them for a better visualization. The positions on the perturbed trajectory after one, two and three Keplerian periods (<math display="inline"><semantics> <msub> <mi>T</mi> <mi mathvariant="normal">K</mi> </msub> </semantics></math>) are marked in gray. In each orbit, the passages at the drifting dashed green line of apsides always occur later than in the Keplerian case with the amount given by Equation (<a href="#FD66-timespace-01-00002" class="html-disp-formula">66</a>), which is positive for the given values of the spin and orbital parameters.</p>
Full article ">Figure 6
<p>Numerically produced time series of the cosine (<math display="inline"><semantics> <mrow> <mover accent="true"> <mi mathvariant="bold-italic">r</mi> <mo stretchy="false">^</mo> </mover> <mo>·</mo> <mover accent="true"> <mi mathvariant="bold-italic">C</mi> <mo stretchy="false">^</mo> </mover> </mrow> </semantics></math>) of the angle between the position vector (<math display="inline"><semantics> <mi mathvariant="bold-italic">r</mi> </semantics></math>) and the Laplace–Runge–Lenz vector (<math display="inline"><semantics> <mi mathvariant="bold-italic">C</mi> </semantics></math>) versus time (<span class="html-italic">t</span>) in units of <math display="inline"><semantics> <msub> <mi>T</mi> <mi mathvariant="normal">K</mi> </msub> </semantics></math> obtained by integrating the equations of motion of a fictitious test particle with (continuous ocher-yellow curve) and without (dashed azure curve) the <math display="inline"><semantics> <msub> <mi>J</mi> <mn>2</mn> </msub> </semantics></math> acceleration of Equation (<a href="#FD34-timespace-01-00002" class="html-disp-formula">34</a>) for an elliptical (<math display="inline"><semantics> <mrow> <mi>e</mi> <mo>=</mo> <mn>0.665</mn> </mrow> </semantics></math>) orbit arbitrarily oriented in space (<math display="inline"><semantics> <mrow> <mi>I</mi> <mo>=</mo> <msup> <mn>40</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <mi mathvariant="sans-serif">Ω</mi> <mo>=</mo> <msup> <mn>45</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <mi>ω</mi> <mo>=</mo> <msup> <mn>50</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>) starting from the periapsis (<math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>), i.e., <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi mathvariant="bold-italic">r</mi> <mo stretchy="false">^</mo> </mover> <mn>0</mn> </msub> <mo>·</mo> <msub> <mover accent="true"> <mi mathvariant="bold-italic">C</mi> <mo stretchy="false">^</mo> </mover> <mn>0</mn> </msub> <mo>=</mo> <mo>+</mo> <mn>1</mn> </mrow> </semantics></math>; the semimajor axis is <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>6</mn> <msub> <mi>R</mi> <mi mathvariant="normal">e</mi> </msub> </mrow> </semantics></math>. The physical parameters of the Earth are adopted, apart from the spin axis position set by <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mi>J</mi> </msub> <mo>=</mo> <msup> <mn>45</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <msub> <mi>δ</mi> <mi>J</mi> </msub> <mo>=</mo> <msup> <mn>60</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>. The <math display="inline"><semantics> <msub> <mi>J</mi> <mn>2</mn> </msub> </semantics></math> acceleration is suitably rescaled in such a way that <math display="inline"><semantics> <mrow> <mfenced separators="" open="|" close="|"> <mo>Δ</mo> <msubsup> <mi>T</mi> <mi>ano</mi> <msub> <mi>J</mi> <mn>2</mn> </msub> </msubsup> </mfenced> <mo>/</mo> <msub> <mi>T</mi> <mi mathvariant="normal">K</mi> </msub> <mo>=</mo> <mn>0.001</mn> </mrow> </semantics></math>. The time needed to come back to the initial position on the (moving) line of apsides so that <math display="inline"><semantics> <mrow> <mover accent="true"> <mi mathvariant="bold-italic">r</mi> <mo stretchy="false">^</mo> </mover> <mo>·</mo> <mover accent="true"> <mi mathvariant="bold-italic">C</mi> <mo stretchy="false">^</mo> </mover> <mo>=</mo> <mo>+</mo> <mn>1</mn> </mrow> </semantics></math> again is longer than in the Keplerian case by an amount equal to <math display="inline"><semantics> <mrow> <mo>Δ</mo> <msubsup> <mi>T</mi> <mi>ano</mi> <msub> <mi>J</mi> <mn>2</mn> </msub> </msubsup> <mo>=</mo> <mo>+</mo> <mn>0.001</mn> <msub> <mi>T</mi> <mi mathvariant="normal">K</mi> </msub> </mrow> </semantics></math>, as shown by the shaded area, in agreement with Equation (<a href="#FD66-timespace-01-00002" class="html-disp-formula">66</a>).</p>
Full article ">Figure 7
<p>Perturbed 1 pN trajectory (continuous blue curve) and its osculating Keplerian ellipse (dashed red curve) at the initial instant of time (<math display="inline"><semantics> <msub> <mi>t</mi> <mn>0</mn> </msub> </semantics></math>), characterized by <math display="inline"><semantics> <mrow> <mi>e</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> <mspace width="0.166667em"/> <mi>I</mi> <mo>=</mo> <msup> <mn>30</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <mi mathvariant="sans-serif">Ω</mi> <mo>=</mo> <msup> <mn>45</mn> <mo>∘</mo> </msup> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>=</mo> <msup> <mn>50</mn> <mo>∘</mo> </msup> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mn>0</mn> </msub> <mo>=</mo> <msup> <mn>180</mn> <mo>∘</mo> </msup> <mo>−</mo> <mi>ω</mi> </mrow> </semantics></math>. In this example, it is assumed that both <math display="inline"><semantics> <mi>ω</mi> </semantics></math> and <math display="inline"><semantics> <mi>η</mi> </semantics></math> undergo their own 1 pN gravitoelectric secular precessions due to the mass (<span class="html-italic">M</span>) of the primary [<a href="#B56-timespace-01-00002" class="html-bibr">56</a>]. For a better visualization of their effects, their sizes are suitably rescaled. The positions on the perturbed trajectory after one, two and three Keplerian periods (<math display="inline"><semantics> <msub> <mi>T</mi> <mi mathvariant="normal">K</mi> </msub> </semantics></math>) are marked in gray. At each orbit, the passages at the fixed dashed cyan line of nodes always occurs later than in the Keplerian case by the amount given by Equation (<a href="#FD79-timespace-01-00002" class="html-disp-formula">79</a>), which is always positive.</p>
Full article ">Figure 8
<p>Numerically produced time series of the cosine (<math display="inline"><semantics> <mrow> <mover accent="true"> <mi mathvariant="bold-italic">r</mi> <mo stretchy="false">^</mo> </mover> <mo>·</mo> <mover accent="true"> <mi mathvariant="bold-italic">l</mi> <mo stretchy="false">^</mo> </mover> </mrow> </semantics></math>) of the angle between the position vector (<math display="inline"><semantics> <mi mathvariant="bold-italic">r</mi> </semantics></math>) and the node unit vector (<math display="inline"><semantics> <mover accent="true"> <mi mathvariant="bold-italic">l</mi> <mo stretchy="false">^</mo> </mover> </semantics></math>) versus time (<span class="html-italic">t</span>) in units of <math display="inline"><semantics> <msub> <mi>T</mi> <mi mathvariant="normal">K</mi> </msub> </semantics></math> obtained by integrating the equations of motion of a fictitious test particle with (continuous ocher-yellow curve) and without (dashed azure curve) the 1 pN gravitoelectric acceleration of Equation (<a href="#FD14-timespace-01-00002" class="html-disp-formula">14</a>) for an elliptical (<math display="inline"><semantics> <mrow> <mi>e</mi> <mo>=</mo> <mn>0.665</mn> </mrow> </semantics></math>) orbit arbitrarily oriented in space (<math display="inline"><semantics> <mrow> <mi>I</mi> <mo>=</mo> <msup> <mn>40</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <mi mathvariant="sans-serif">Ω</mi> <mo>=</mo> <msup> <mn>45</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <mi>ω</mi> <mo>=</mo> <msup> <mn>50</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>) starting from the ascending node (<span class="html-fig-inline" id="timespace-01-00002-i001"><img alt="Timespace 01 00002 i001" src="/timespace/timespace-01-00002/article_deploy/html/images/timespace-01-00002-i001.png"/></span> (<math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mn>0</mn> </msub> <mo>=</mo> <mo>−</mo> <mi>ω</mi> <mo>+</mo> <msup> <mn>360</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>)), i.e., <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi mathvariant="bold-italic">r</mi> <mo stretchy="false">^</mo> </mover> <mn>0</mn> </msub> <mo>·</mo> <msub> <mover accent="true"> <mi mathvariant="bold-italic">l</mi> <mo stretchy="false">^</mo> </mover> <mn>0</mn> </msub> <mo>=</mo> <mo>+</mo> <mn>1</mn> </mrow> </semantics></math>; the semimajor axis is <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>6</mn> <msub> <mi>R</mi> <mi mathvariant="normal">e</mi> </msub> </mrow> </semantics></math>. The physical parameters of the Earth are adopted. The 1 pN acceleration is suitably rescaled in such a way that <math display="inline"><semantics> <mrow> <mo>Δ</mo> <msubsup> <mi>T</mi> <mi>dra</mi> <mrow> <mn>1</mn> <mi>pN</mi> </mrow> </msubsup> <mo>/</mo> <msub> <mi>T</mi> <mi mathvariant="normal">K</mi> </msub> <mo>=</mo> <mn>0.001</mn> </mrow> </semantics></math>. The time needed to come back to the initial position on the (fixed) line of nodes so that <math display="inline"><semantics> <mrow> <mover accent="true"> <mi mathvariant="bold-italic">r</mi> <mo stretchy="false">^</mo> </mover> <mo>·</mo> <mover accent="true"> <mi mathvariant="bold-italic">l</mi> <mo stretchy="false">^</mo> </mover> <mo>=</mo> <mo>+</mo> <mn>1</mn> </mrow> </semantics></math> again is longer than in the Keplerian case by an amount equal to <math display="inline"><semantics> <mrow> <mo>Δ</mo> <msubsup> <mi>T</mi> <mi>dra</mi> <mrow> <mn>1</mn> <mi>pN</mi> </mrow> </msubsup> <mo>=</mo> <mo>+</mo> <mn>0.001</mn> <msub> <mi>T</mi> <mi mathvariant="normal">K</mi> </msub> </mrow> </semantics></math>, as shown by the shaded area, in agreement with Equation (<a href="#FD79-timespace-01-00002" class="html-disp-formula">79</a>).</p>
Full article ">Figure 9
<p>Perturbed LT trajectory (continuous blue curve) and its osculating Keplerian ellipse (dashed red curve) at the initial instant of time (<math display="inline"><semantics> <msub> <mi>t</mi> <mn>0</mn> </msub> </semantics></math>), characterized by <math display="inline"><semantics> <mrow> <mi>e</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> <mspace width="0.166667em"/> <mi>I</mi> <mo>=</mo> <msup> <mn>30</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <mi mathvariant="sans-serif">Ω</mi> <mo>=</mo> <msup> <mn>72</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <mi>ω</mi> <mo>=</mo> <msup> <mn>50</mn> <mo>∘</mo> </msup> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mn>0</mn> </msub> <mo>=</mo> <msup> <mn>180</mn> <mo>∘</mo> </msup> <mo>−</mo> <mi>ω</mi> </mrow> </semantics></math>. The orientation of the spin axis (<math display="inline"><semantics> <mover accent="true"> <mi mathvariant="bold-italic">J</mi> <mo stretchy="false">^</mo> </mover> </semantics></math>) of the central body is set by <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mi>J</mi> </msub> <mo>=</mo> <msup> <mn>45</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <msub> <mi>δ</mi> <mi>J</mi> </msub> <mo>=</mo> <msup> <mn>60</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>. In this example, <span class="html-italic">I</span>, <math display="inline"><semantics> <mi mathvariant="sans-serif">Ω</mi> </semantics></math> and <math display="inline"><semantics> <mi>ω</mi> </semantics></math> undergo their own LT precessions due to the spin angular momentum (<math display="inline"><semantics> <mi mathvariant="bold-italic">J</mi> </semantics></math>) of the primary [<a href="#B56-timespace-01-00002" class="html-bibr">56</a>]; their magnitudes are suitably rescaled by enhancing them for a better visualization. The positions on the perturbed trajectory after one, two and three Keplerian periods (<math display="inline"><semantics> <msub> <mi>T</mi> <mi mathvariant="normal">K</mi> </msub> </semantics></math>) are marked as well. In each orbit, the passages at the precessing dashed cyan line of nodes always occur later than in the Keplerian case by the amount given by Equation (<a href="#FD81-timespace-01-00002" class="html-disp-formula">81</a>), which is positive for the given values of the spin and orbital parameters.</p>
Full article ">Figure 10
<p>Numerically produced time series of the cosine (<math display="inline"><semantics> <mrow> <mover accent="true"> <mi mathvariant="bold-italic">r</mi> <mo stretchy="false">^</mo> </mover> <mo>·</mo> <mover accent="true"> <mi mathvariant="bold-italic">l</mi> <mo stretchy="false">^</mo> </mover> </mrow> </semantics></math>) of the angle between the position vector (<math display="inline"><semantics> <mi mathvariant="bold-italic">r</mi> </semantics></math>) and the node unit vector (<math display="inline"><semantics> <mover accent="true"> <mi mathvariant="bold-italic">l</mi> <mo stretchy="false">^</mo> </mover> </semantics></math>) versus time (<span class="html-italic">t</span>) in units of <math display="inline"><semantics> <msub> <mi>T</mi> <mi mathvariant="normal">K</mi> </msub> </semantics></math> obtained by integrating the equations of motion of a fictitious test particle with (continuous ocher-yellow curve) and without (dashed azure curve) the LT acceleration of Equation (<a href="#FD22-timespace-01-00002" class="html-disp-formula">22</a>) for an elliptical (<math display="inline"><semantics> <mrow> <mi>e</mi> <mo>=</mo> <mn>0.665</mn> </mrow> </semantics></math>) orbit arbitrarily oriented in space (<math display="inline"><semantics> <mrow> <mi>I</mi> <mo>=</mo> <msup> <mn>40</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <mi mathvariant="sans-serif">Ω</mi> <mo>=</mo> <msup> <mn>45</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <mi>ω</mi> <mo>=</mo> <msup> <mn>50</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>) starting from the ascending node <span class="html-fig-inline" id="timespace-01-00002-i001"><img alt="Timespace 01 00002 i001" src="/timespace/timespace-01-00002/article_deploy/html/images/timespace-01-00002-i001.png"/></span> (<math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mn>0</mn> </msub> <mo>=</mo> <mo>−</mo> <mi>ω</mi> <mo>+</mo> <msup> <mn>360</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>), i.e., <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi mathvariant="bold-italic">r</mi> <mo stretchy="false">^</mo> </mover> <mn>0</mn> </msub> <mo>·</mo> <msub> <mover accent="true"> <mi mathvariant="bold-italic">l</mi> <mo stretchy="false">^</mo> </mover> <mn>0</mn> </msub> <mo>=</mo> <mo>+</mo> <mn>1</mn> </mrow> </semantics></math>; the semimajor axis is <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>6</mn> <msub> <mi>R</mi> <mi mathvariant="normal">e</mi> </msub> </mrow> </semantics></math>. The physical parameters of the Earth are adopted, apart from the spin axis position set by <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mi>J</mi> </msub> <mo>=</mo> <msup> <mn>45</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <msub> <mi>δ</mi> <mi>J</mi> </msub> <mo>=</mo> <msup> <mn>60</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>. The LT acceleration is suitably rescaled in such a way that <math display="inline"><semantics> <mrow> <mfenced separators="" open="|" close="|"> <mo>Δ</mo> <msubsup> <mi>T</mi> <mi>dra</mi> <mi>LT</mi> </msubsup> </mfenced> <mo>/</mo> <msub> <mi>T</mi> <mi mathvariant="normal">K</mi> </msub> <mo>=</mo> <mn>0.001</mn> </mrow> </semantics></math>. The time needed to come back to the initial position on the (moving) line of nodes so that <math display="inline"><semantics> <mrow> <mover accent="true"> <mi mathvariant="bold-italic">r</mi> <mo stretchy="false">^</mo> </mover> <mo>·</mo> <mover accent="true"> <mi mathvariant="bold-italic">l</mi> <mo stretchy="false">^</mo> </mover> <mo>=</mo> <mo>+</mo> <mn>1</mn> </mrow> </semantics></math> again is longer than in the Keplerian case by an amount equal to <math display="inline"><semantics> <mrow> <mo>Δ</mo> <msubsup> <mi>T</mi> <mi>dra</mi> <mi>LT</mi> </msubsup> <mo>=</mo> <mo>+</mo> <mn>0.001</mn> <msub> <mi>T</mi> <mi mathvariant="normal">K</mi> </msub> </mrow> </semantics></math>, as shown by the shaded area, in agreement with Equation (<a href="#FD81-timespace-01-00002" class="html-disp-formula">81</a>).</p>
Full article ">Figure 11
<p>Perturbed <math display="inline"><semantics> <msub> <mi>J</mi> <mn>2</mn> </msub> </semantics></math> trajectory (continuous blue curve) and its osculating Keplerian ellipse (dashed red curve) at the initial instant of time (<math display="inline"><semantics> <msub> <mi>t</mi> <mn>0</mn> </msub> </semantics></math>), characterized by <math display="inline"><semantics> <mrow> <mi>e</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> <mspace width="0.166667em"/> <mi>I</mi> <mo>=</mo> <msup> <mn>30</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <mi mathvariant="sans-serif">Ω</mi> <mo>=</mo> <msup> <mn>45</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <mi>ω</mi> <mo>=</mo> <msup> <mn>50</mn> <mo>∘</mo> </msup> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mn>0</mn> </msub> <mo>=</mo> <msup> <mn>180</mn> <mo>∘</mo> </msup> <mo>−</mo> <mi>ω</mi> </mrow> </semantics></math>, as seen from the <span class="html-italic">z</span>-axis. The orientation of the spin axis (<math display="inline"><semantics> <mover accent="true"> <mi mathvariant="bold-italic">J</mi> <mo stretchy="false">^</mo> </mover> </semantics></math>) of the central body is set by <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mi>J</mi> </msub> <mo>=</mo> <msup> <mn>45</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <msub> <mi>δ</mi> <mi>J</mi> </msub> <mo>=</mo> <msup> <mn>60</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>. In this example, <span class="html-italic">I</span>, <math display="inline"><semantics> <mi mathvariant="sans-serif">Ω</mi> </semantics></math>, <math display="inline"><semantics> <mi>ω</mi> </semantics></math> and <math display="inline"><semantics> <mi>η</mi> </semantics></math> undergo their own Newtonian shifts due to the quadrupole mass moment (<math display="inline"><semantics> <msub> <mi>J</mi> <mn>2</mn> </msub> </semantics></math>) of the primary [<a href="#B56-timespace-01-00002" class="html-bibr">56</a>]; their magnitudes are suitably rescaled for better visualization of their effects. The positions on the perturbed trajectory after one, two and three Keplerian periods (<math display="inline"><semantics> <msub> <mi>T</mi> <mi mathvariant="normal">K</mi> </msub> </semantics></math>) are marked in gray. In each orbit, the passages at the precessing dashed cyan line of nodes always occur earlier than in the Keplerian case by the amount given by Equation (<a href="#FD86-timespace-01-00002" class="html-disp-formula">86</a>), which is negative for the given values of the spin and orbital parameters.</p>
Full article ">Figure 12
<p>Numerically produced time series of the cosine (<math display="inline"><semantics> <mrow> <mover accent="true"> <mi mathvariant="bold-italic">r</mi> <mo stretchy="false">^</mo> </mover> <mo>·</mo> <mover accent="true"> <mi mathvariant="bold-italic">l</mi> <mo stretchy="false">^</mo> </mover> </mrow> </semantics></math>) of the angle between the position vector (<math display="inline"><semantics> <mi mathvariant="bold-italic">r</mi> </semantics></math>) and the node unit vector (<math display="inline"><semantics> <mover accent="true"> <mi mathvariant="bold-italic">l</mi> <mo stretchy="false">^</mo> </mover> </semantics></math>) versus time (<span class="html-italic">t</span>) in units of <math display="inline"><semantics> <msub> <mi>T</mi> <mi mathvariant="normal">K</mi> </msub> </semantics></math> obtained by integrating the equations of motion of a fictitious test particle with (continuous ocher-yellow curve) and without (dashed azure curve) the <math display="inline"><semantics> <msub> <mi>J</mi> <mn>2</mn> </msub> </semantics></math> acceleration of Equation (<a href="#FD34-timespace-01-00002" class="html-disp-formula">34</a>) for an elliptical (<math display="inline"><semantics> <mrow> <mi>e</mi> <mo>=</mo> <mn>0.665</mn> </mrow> </semantics></math>) orbit arbitrarily oriented in space (<math display="inline"><semantics> <mrow> <mi>I</mi> <mo>=</mo> <msup> <mn>40</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <mi mathvariant="sans-serif">Ω</mi> <mo>=</mo> <msup> <mn>45</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <mi>ω</mi> <mo>=</mo> <msup> <mn>50</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>) starting from the ascending node <span class="html-fig-inline" id="timespace-01-00002-i001"><img alt="Timespace 01 00002 i001" src="/timespace/timespace-01-00002/article_deploy/html/images/timespace-01-00002-i001.png"/></span> (<math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mn>0</mn> </msub> <mo>=</mo> <mo>−</mo> <mi>ω</mi> <mo>+</mo> <msup> <mn>360</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>), i.e., <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi mathvariant="bold-italic">r</mi> <mo stretchy="false">^</mo> </mover> <mn>0</mn> </msub> <mo>·</mo> <msub> <mover accent="true"> <mi mathvariant="bold-italic">l</mi> <mo stretchy="false">^</mo> </mover> <mn>0</mn> </msub> <mo>=</mo> <mo>+</mo> <mn>1</mn> </mrow> </semantics></math>; the semimajor axis is <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>6</mn> <msub> <mi>R</mi> <mi mathvariant="normal">e</mi> </msub> </mrow> </semantics></math>. The physical parameters of the Earth are adopted, apart from the spin axis position set by <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mi>J</mi> </msub> <mo>=</mo> <msup> <mn>45</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <msub> <mi>δ</mi> <mi>J</mi> </msub> <mo>=</mo> <msup> <mn>60</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>. The <math display="inline"><semantics> <msub> <mi>J</mi> <mn>2</mn> </msub> </semantics></math> acceleration is suitably rescaled in such a way that <math display="inline"><semantics> <mrow> <mfenced separators="" open="|" close="|"> <mo>Δ</mo> <msubsup> <mi>T</mi> <mi>dra</mi> <msub> <mi>J</mi> <mn>2</mn> </msub> </msubsup> </mfenced> <mo>/</mo> <msub> <mi>T</mi> <mi mathvariant="normal">K</mi> </msub> <mo>=</mo> <mn>0.001</mn> </mrow> </semantics></math>. The time needed to come back to the initial position on the (moving) line of nodes so that <math display="inline"><semantics> <mrow> <mover accent="true"> <mi mathvariant="bold-italic">r</mi> <mo stretchy="false">^</mo> </mover> <mo>·</mo> <mover accent="true"> <mi mathvariant="bold-italic">l</mi> <mo stretchy="false">^</mo> </mover> <mo>=</mo> <mo>+</mo> <mn>1</mn> </mrow> </semantics></math> again is longer than in the Keplerian case by an amount equal to <math display="inline"><semantics> <mrow> <mo>Δ</mo> <msubsup> <mi>T</mi> <mi>dra</mi> <msub> <mi>J</mi> <mn>2</mn> </msub> </msubsup> <mo>=</mo> <mo>+</mo> <mn>0.001</mn> <msub> <mi>T</mi> <mi mathvariant="normal">K</mi> </msub> </mrow> </semantics></math>, as shown by the shaded area, in agreement with Equation (<a href="#FD86-timespace-01-00002" class="html-disp-formula">86</a>).</p>
Full article ">Figure 13
<p>Numerically produced time series of the cosine (<math display="inline"><semantics> <mrow> <mo form="prefix">cos</mo> <mi>ϕ</mi> <mfenced open="(" close=")"> <mi>t</mi> </mfenced> </mrow> </semantics></math>) of the azimuthal angle (<math display="inline"><semantics> <mrow> <mi>ϕ</mi> <mfenced open="(" close=")"> <mi>t</mi> </mfenced> </mrow> </semantics></math>) normalized to its initial value (<math display="inline"><semantics> <mrow> <mo form="prefix">cos</mo> <msub> <mi>ϕ</mi> <mn>0</mn> </msub> </mrow> </semantics></math>) versus time (<span class="html-italic">t</span>) in units of <math display="inline"><semantics> <msub> <mi>T</mi> <mi mathvariant="normal">K</mi> </msub> </semantics></math> obtained by integrating the equations of motion of a fictitious test particle with (continuous ocher-yellow curve) and without (dashed azure curve) the 1 pN gravitoelectric acceleration of Equation (<a href="#FD14-timespace-01-00002" class="html-disp-formula">14</a>) for an elliptical (<math display="inline"><semantics> <mrow> <mi>e</mi> <mo>=</mo> <mn>0.665</mn> </mrow> </semantics></math>) orbit arbitrarily oriented in space (<math display="inline"><semantics> <mrow> <mi>I</mi> <mo>=</mo> <msup> <mn>40</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <mi mathvariant="sans-serif">Ω</mi> <mo>=</mo> <msup> <mn>45</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <mi>ω</mi> <mo>=</mo> <msup> <mn>50</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>) starting from the ascending node <span class="html-fig-inline" id="timespace-01-00002-i001"><img alt="Timespace 01 00002 i001" src="/timespace/timespace-01-00002/article_deploy/html/images/timespace-01-00002-i001.png"/></span> (<math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mn>0</mn> </msub> <mo>=</mo> <mo>−</mo> <mi>ω</mi> <mo>+</mo> <msup> <mn>360</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>); the semimajor axis is <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>6</mn> <msub> <mi>R</mi> <mi mathvariant="normal">e</mi> </msub> </mrow> </semantics></math>. The physical parameters of the Earth are adopted. The 1 pN acceleration is suitably rescaled in such a way that <math display="inline"><semantics> <mrow> <mo>Δ</mo> <msubsup> <mi>T</mi> <mi>sid</mi> <mrow> <mn>1</mn> <mi>pN</mi> </mrow> </msubsup> <mo>/</mo> <msub> <mi>T</mi> <mi mathvariant="normal">K</mi> </msub> <mo>=</mo> <mn>0.001</mn> </mrow> </semantics></math>. The time needed for <math display="inline"><semantics> <mrow> <mo form="prefix">cos</mo> <mi>ϕ</mi> <mfenced open="(" close=")"> <mi>t</mi> </mfenced> </mrow> </semantics></math> to assume its initial value of <math display="inline"><semantics> <mrow> <mo form="prefix">cos</mo> <msub> <mi>ϕ</mi> <mn>0</mn> </msub> </mrow> </semantics></math> again is longer than in the Keplerian case by an amount equal to <math display="inline"><semantics> <mrow> <mo>Δ</mo> <msubsup> <mi>T</mi> <mi>sid</mi> <mrow> <mn>1</mn> <mi>pN</mi> </mrow> </msubsup> <mo>=</mo> <mo>+</mo> <mn>0.001</mn> <msub> <mi>T</mi> <mi mathvariant="normal">K</mi> </msub> </mrow> </semantics></math>, as shown by the shaded area, in agreement with the sum of Equation (<a href="#FD79-timespace-01-00002" class="html-disp-formula">79</a>).</p>
Full article ">Figure 14
<p>Projections of the perturbed LT trajectory (continuous blue curve) and its osculating Keplerian ellipse (dashed red curve) in the reference plane (<math display="inline"><semantics> <mfenced separators="" open="{" close="}"> <mi>x</mi> <mo>,</mo> <mspace width="0.166667em"/> <mi>y</mi> </mfenced> </semantics></math>) at the initial instant of time (<math display="inline"><semantics> <msub> <mi>t</mi> <mn>0</mn> </msub> </semantics></math>), characterized by generic initial conditions of <math display="inline"><semantics> <mrow> <mi>e</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> <mspace width="0.166667em"/> <mi>I</mi> <mo>=</mo> <msup> <mn>30</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <mi mathvariant="sans-serif">Ω</mi> <mo>=</mo> <msup> <mn>45</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <mi>ω</mi> <mo>=</mo> <msup> <mn>50</mn> <mo>∘</mo> </msup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mspace width="0.166667em"/> <msub> <mi>f</mi> <mn>0</mn> </msub> <mo>=</mo> <msup> <mn>285</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>. The orientation of the spin axis (<math display="inline"><semantics> <mover accent="true"> <mi mathvariant="bold-italic">J</mi> <mo stretchy="false">^</mo> </mover> </semantics></math>) of the central body, whose projection in the fundamental plane is depicted as well, is set by <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mi>J</mi> </msub> <mo>=</mo> <msup> <mn>45</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <msub> <mi>δ</mi> <mi>J</mi> </msub> <mo>=</mo> <msup> <mn>60</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>. In this example, <span class="html-italic">I</span>, <math display="inline"><semantics> <mi mathvariant="sans-serif">Ω</mi> </semantics></math> and <math display="inline"><semantics> <mi>ω</mi> </semantics></math> undergo their own LT shifts due to the spin angular momentum (<math display="inline"><semantics> <mi mathvariant="bold-italic">J</mi> </semantics></math>) of the primary [<a href="#B56-timespace-01-00002" class="html-bibr">56</a>]; their sizes are suitably rescaled for better visualization of their effects. The positions on the perturbed trajectory after one, two and three Keplerian periods (<math display="inline"><semantics> <msub> <mi>T</mi> <mi mathvariant="normal">K</mi> </msub> </semantics></math>) are marked as well. In each orbit, the passages at the generic fixed dashed brown line characterized by <math display="inline"><semantics> <msub> <mi>ϕ</mi> <mn>0</mn> </msub> </semantics></math> always occur earlier than in the Keplerian case by the amount given by the sum of Equations (<a href="#FD81-timespace-01-00002" class="html-disp-formula">81</a>) and (<a href="#FD108-timespace-01-00002" class="html-disp-formula">108</a>). It is so because for the given values of the spin and orbital parameters, <math display="inline"><semantics> <mrow> <mo>Δ</mo> <msubsup> <mi>T</mi> <mi>dra</mi> <mi>LT</mi> </msubsup> <mo>+</mo> <mo>Δ</mo> <msubsup> <mi>T</mi> <mrow> <mi>sid</mi> <mi>II</mi> </mrow> <mi>LT</mi> </msubsup> <mo>&lt;</mo> <mn>0</mn> </mrow> </semantics></math>, as per Equations (<a href="#FD81-timespace-01-00002" class="html-disp-formula">81</a>) and (<a href="#FD108-timespace-01-00002" class="html-disp-formula">108</a>).</p>
Full article ">Figure 15
<p>Numerically produced time series of the cosine (<math display="inline"><semantics> <mrow> <mo form="prefix">cos</mo> <mi>ϕ</mi> <mfenced open="(" close=")"> <mi>t</mi> </mfenced> </mrow> </semantics></math>) of the azimuthal angle (<math display="inline"><semantics> <mrow> <mi>ϕ</mi> <mfenced open="(" close=")"> <mi>t</mi> </mfenced> </mrow> </semantics></math>) normalized to its initial value (<math display="inline"><semantics> <mrow> <mo form="prefix">cos</mo> <msub> <mi>ϕ</mi> <mn>0</mn> </msub> </mrow> </semantics></math>) versus time (<span class="html-italic">t</span>) in units of <math display="inline"><semantics> <msub> <mi>T</mi> <mi mathvariant="normal">K</mi> </msub> </semantics></math> obtained by integrating the equations of motion of a fictitious test particle with (continuous ocher-yellow curve) and without (dashed azure curve) the LT acceleration of Equation (<a href="#FD22-timespace-01-00002" class="html-disp-formula">22</a>) for an elliptical (<math display="inline"><semantics> <mrow> <mi>e</mi> <mo>=</mo> <mn>0.665</mn> </mrow> </semantics></math>) orbit arbitrarily oriented in space (<math display="inline"><semantics> <mrow> <mi>I</mi> <mo>=</mo> <msup> <mn>40</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <mi mathvariant="sans-serif">Ω</mi> <mo>=</mo> <msup> <mn>45</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <mi>ω</mi> <mo>=</mo> <msup> <mn>310</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>) starting from <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mn>0</mn> </msub> <mo>=</mo> <msup> <mn>50</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>; the semimajor axis is <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>6</mn> <msub> <mi>R</mi> <mi mathvariant="normal">e</mi> </msub> </mrow> </semantics></math>. The physical parameters of the Earth are adopted, apart from the spin axis position set by <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mi>J</mi> </msub> <mo>=</mo> <msup> <mn>45</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <msub> <mi>δ</mi> <mi>J</mi> </msub> <mo>=</mo> <msup> <mn>60</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>. The LT acceleration is suitably rescaled in such a way that <math display="inline"><semantics> <mrow> <mfenced separators="" open="|" close="|"> <mo>Δ</mo> <msubsup> <mi>T</mi> <mi>sid</mi> <mi>LT</mi> </msubsup> </mfenced> <mo>/</mo> <msub> <mi>T</mi> <mi mathvariant="normal">K</mi> </msub> <mo>=</mo> <mn>0.001</mn> </mrow> </semantics></math>. The time needed for to <math display="inline"><semantics> <mrow> <mo form="prefix">cos</mo> <mi>ϕ</mi> <mfenced open="(" close=")"> <mi>t</mi> </mfenced> </mrow> </semantics></math> to assume its initial value (<math display="inline"><semantics> <mrow> <mo form="prefix">cos</mo> <msub> <mi>ϕ</mi> <mn>0</mn> </msub> </mrow> </semantics></math>) again is longer than in the Keplerian case by an amount equal to <math display="inline"><semantics> <mrow> <mo>Δ</mo> <msubsup> <mi>T</mi> <mi>sid</mi> <mi>LT</mi> </msubsup> <mo>=</mo> <mo>+</mo> <mn>0.001</mn> <msub> <mi>T</mi> <mi mathvariant="normal">K</mi> </msub> </mrow> </semantics></math>, as shown by the shaded area, in agreement with the sum of Equations (<a href="#FD81-timespace-01-00002" class="html-disp-formula">81</a>) and (<a href="#FD108-timespace-01-00002" class="html-disp-formula">108</a>).</p>
Full article ">Figure 16
<p>Projections of the perturbed <math display="inline"><semantics> <msub> <mi>J</mi> <mn>2</mn> </msub> </semantics></math> trajectory (continuous blue curve) and its osculating Keplerian ellipse (dashed red curve) in the reference plane (<math display="inline"><semantics> <mfenced separators="" open="{" close="}"> <mi>x</mi> <mo>,</mo> <mspace width="0.166667em"/> <mi>y</mi> </mfenced> </semantics></math>) at the initial instant of time (<math display="inline"><semantics> <msub> <mi>t</mi> <mn>0</mn> </msub> </semantics></math>), characterized by the generic initial conditions of <math display="inline"><semantics> <mrow> <mi>e</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> <mspace width="0.166667em"/> <mi>I</mi> <mo>=</mo> <msup> <mn>30</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <mi mathvariant="sans-serif">Ω</mi> <mo>=</mo> <msup> <mn>45</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <mi>ω</mi> <mo>=</mo> <msup> <mn>50</mn> <mo>∘</mo> </msup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mspace width="0.166667em"/> <msub> <mi>f</mi> <mn>0</mn> </msub> <mo>=</mo> <msup> <mn>285</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>. The orientation of the spin axis (<math display="inline"><semantics> <mover accent="true"> <mi mathvariant="bold-italic">J</mi> <mo stretchy="false">^</mo> </mover> </semantics></math>) of the central body, whose projection in the fundamental plane is depicted as well, is set by <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mi>J</mi> </msub> <mo>=</mo> <msup> <mn>45</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <msub> <mi>δ</mi> <mi>J</mi> </msub> <mo>=</mo> <msup> <mn>60</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>. In this example, <span class="html-italic">I</span>, <math display="inline"><semantics> <mi mathvariant="sans-serif">Ω</mi> </semantics></math>, <math display="inline"><semantics> <mi>ω</mi> </semantics></math> and <math display="inline"><semantics> <mi>η</mi> </semantics></math> undergo their own Newtonian shifts due to the quadrupole mass moment (<math display="inline"><semantics> <msub> <mi>J</mi> <mn>2</mn> </msub> </semantics></math>) of the primary [<a href="#B56-timespace-01-00002" class="html-bibr">56</a>]; their magnitudes are suitably rescaled for better visualization of their effects. The positions on the perturbed trajectory after one, two and three Keplerian periods (<math display="inline"><semantics> <msub> <mi>T</mi> <mi mathvariant="normal">K</mi> </msub> </semantics></math>) are marked as well. In each orbit, the passages at the generic fixed dashed brown line characterized by <math display="inline"><semantics> <msub> <mi>ϕ</mi> <mn>0</mn> </msub> </semantics></math> always occur earlier than in the Keplerian case by the amount given by the sum of Equations (<a href="#FD86-timespace-01-00002" class="html-disp-formula">86</a>) and (<a href="#FD113-timespace-01-00002" class="html-disp-formula">113</a>). It is so because for the given values of the spin and orbital parameters, <math display="inline"><semantics> <mrow> <mo>Δ</mo> <msubsup> <mi>T</mi> <mi>dra</mi> <msub> <mi>J</mi> <mn>2</mn> </msub> </msubsup> <mo>+</mo> <mo>Δ</mo> <msubsup> <mi>T</mi> <mrow> <mi>sid</mi> <mi>II</mi> </mrow> <msub> <mi>J</mi> <mn>2</mn> </msub> </msubsup> <mo>&lt;</mo> <mn>0</mn> </mrow> </semantics></math>, as per Equations (<a href="#FD86-timespace-01-00002" class="html-disp-formula">86</a>) and (<a href="#FD113-timespace-01-00002" class="html-disp-formula">113</a>).</p>
Full article ">Figure 17
<p>Plot of the numerically produced time series of the cosine (<math display="inline"><semantics> <mrow> <mo form="prefix">cos</mo> <mi>ϕ</mi> <mfenced open="(" close=")"> <mi>t</mi> </mfenced> </mrow> </semantics></math>) of the azimuthal angle (<math display="inline"><semantics> <mrow> <mi>ϕ</mi> <mfenced open="(" close=")"> <mi>t</mi> </mfenced> </mrow> </semantics></math>) normalized to its initial value (<math display="inline"><semantics> <mrow> <mo form="prefix">cos</mo> <msub> <mi>ϕ</mi> <mn>0</mn> </msub> </mrow> </semantics></math>) versus time (<span class="html-italic">t</span>) in units of <math display="inline"><semantics> <msub> <mi>T</mi> <mi mathvariant="normal">K</mi> </msub> </semantics></math> obtained by integrating the equations of motion of a fictitious test particle with (continuous ocher-yellow curve) and without (dashed azure curve) the <math display="inline"><semantics> <msub> <mi>J</mi> <mn>2</mn> </msub> </semantics></math> acceleration of Equation (<a href="#FD34-timespace-01-00002" class="html-disp-formula">34</a>) for an elliptical (<math display="inline"><semantics> <mrow> <mi>e</mi> <mo>=</mo> <mn>0.665</mn> </mrow> </semantics></math>) orbit arbitrarily oriented in space (<math display="inline"><semantics> <mrow> <mi>I</mi> <mo>=</mo> <msup> <mn>40</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <mi mathvariant="sans-serif">Ω</mi> <mo>=</mo> <msup> <mn>45</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <mi>ω</mi> <mo>=</mo> <msup> <mn>50</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>) starting from the ascending node (<span class="html-fig-inline" id="timespace-01-00002-i001"><img alt="Timespace 01 00002 i001" src="/timespace/timespace-01-00002/article_deploy/html/images/timespace-01-00002-i001.png"/></span>) (<math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mn>0</mn> </msub> <mo>=</mo> <mo>−</mo> <mi>ω</mi> <mo>+</mo> <msup> <mn>360</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>); the semimajor axis is <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>6</mn> <msub> <mi>R</mi> <mi mathvariant="normal">e</mi> </msub> </mrow> </semantics></math>. The physical parameters of the Earth are adopted, apart from the spin axis position set by <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mi>J</mi> </msub> <mo>=</mo> <msup> <mn>45</mn> <mo>∘</mo> </msup> <mo>,</mo> <mspace width="0.166667em"/> <msub> <mi>δ</mi> <mi>J</mi> </msub> <mo>=</mo> <msup> <mn>60</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>. The <math display="inline"><semantics> <msub> <mi>J</mi> <mn>2</mn> </msub> </semantics></math> acceleration is suitably rescaled in such a way that <math display="inline"><semantics> <mrow> <mfenced separators="" open="|" close="|"> <mo>Δ</mo> <msubsup> <mi>T</mi> <mi>sid</mi> <msub> <mi>J</mi> <mn>2</mn> </msub> </msubsup> </mfenced> <mo>/</mo> <msub> <mi>T</mi> <mi mathvariant="normal">K</mi> </msub> <mo>=</mo> <mn>0.001</mn> </mrow> </semantics></math>. The time needed for <math display="inline"><semantics> <mrow> <mo form="prefix">cos</mo> <mi>ϕ</mi> <mfenced open="(" close=")"> <mi>t</mi> </mfenced> </mrow> </semantics></math> to assume its initial value (<math display="inline"><semantics> <mrow> <mo form="prefix">cos</mo> <msub> <mi>ϕ</mi> <mn>0</mn> </msub> </mrow> </semantics></math>) again is longer than in the Keplerian case by an amount equal to <math display="inline"><semantics> <mrow> <mo>Δ</mo> <msubsup> <mi>T</mi> <mi>sid</mi> <msub> <mi>J</mi> <mn>2</mn> </msub> </msubsup> <mo>=</mo> <mo>+</mo> <mn>0.001</mn> <msub> <mi>T</mi> <mi mathvariant="normal">K</mi> </msub> </mrow> </semantics></math>, as shown by the shaded area, in agreement with the sum of Equations (<a href="#FD86-timespace-01-00002" class="html-disp-formula">86</a>) and (<a href="#FD113-timespace-01-00002" class="html-disp-formula">113</a>).</p>
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22 pages, 4082 KiB  
Article
A Multistep Method for Integration of Perturbed and Damped Second-Order ODE Systems
by Fernando García-Alonso, José Antonio Reyes and Mónica Cortés-Molina
Mathematics 2024, 12(13), 2018; https://doi.org/10.3390/math12132018 - 28 Jun 2024
Viewed by 861
Abstract
Based on the Ψ-functions series method, a new numerical integration method for perturbed and damped second-order systems of differential equations is presented. This multistep method is defined for variable step and variable order (VSVO) and maintains the good properties of the Ψ-functions series [...] Read more.
Based on the Ψ-functions series method, a new numerical integration method for perturbed and damped second-order systems of differential equations is presented. This multistep method is defined for variable step and variable order (VSVO) and maintains the good properties of the Ψ-functions series method. In addition, it incorporates a recurring algebraic procedure to calculate the algorithm’s coefficients, which facilitates its implementation on the computer. The construction of Ψ-functions and the Ψ-functions series method are presented to address the construction of both explicit and implicit multistep methods and a predictor–corrector method. Three problems analogous to those solved by the Ψ-functions series method are analyzed, contrasting the results obtained with the exact solution of the problem or with its first integral. The first example is the integration of a quasi-periodic orbit. The second example is a Structural Dynamics problem associated with an earthquake, and the third example studies an equatorial satellite with perturbation J2. This allows us to compare the good behavior of the new code with other prestige codes. Full article
(This article belongs to the Section E2: Control Theory and Mechanics)
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<p>The decimal logarithm of the module of the relative error of the position <math display="inline"><semantics> <mrow> <mi mathvariant="bold-italic">x</mi> <mfenced separators="|"> <mrow> <mi>t</mi> </mrow> </mfenced> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>18</mn> </mrow> </semantics></math>.</p>
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<p>Problem 1, efficiency plot for the integration of <math display="inline"><semantics> <mrow> <mi mathvariant="bold-italic">x</mi> <mfenced separators="|"> <mrow> <mi>t</mi> </mrow> </mfenced> </mrow> </semantics></math> at the last point.</p>
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<p>Two-story frame.</p>
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<p>The decimal logarithm of the module of the relative error of the position <math display="inline"><semantics> <mrow> <mi mathvariant="bold-italic">x</mi> <mfenced separators="|"> <mrow> <mi>t</mi> </mrow> </mfenced> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>16</mn> </mrow> </semantics></math>.</p>
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<p>The decimal logarithm of the module of the relative error of the position <math display="inline"><semantics> <mrow> <mi mathvariant="bold-italic">x</mi> <mfenced separators="|"> <mrow> <mi>t</mi> </mrow> </mfenced> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>=</mo> <mn>0.001</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>16</mn> </mrow> </semantics></math>.</p>
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<p>Trajectory error of a circular equatorial <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>J</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> </semantics></math>-perturbed satellite orbit, integrated with <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>15</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>e</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p>
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<p>Trajectory error of a circular equatorial <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>J</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> </semantics></math>-perturbed satellite orbit, integrated with <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>15</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>e</mi> <mo>=</mo> <mn>0.99</mn> </mrow> </semantics></math>.</p>
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<p>Trajectory error of a circular equatorial <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>J</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> </semantics></math>-perturbed satellite orbit, integrated with <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>e</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>=</mo> <mn>0.001</mn> </mrow> </semantics></math>.</p>
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<p>Trajectory error of a circular equatorial <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>J</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> </semantics></math>-perturbed satellite orbit, integrated with <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>e</mi> <mo>=</mo> <mn>0.99</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>=</mo> <mn>0.001</mn> </mrow> </semantics></math>.</p>
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15 pages, 3075 KiB  
Article
Decadal Changes in Benthic Community Structure and Function in a Coral Community in the Northeastern Tropical Pacific
by Cassandra de Alba-Guzmán, Rafael Andrés Cabral-Tena, Fabián Alejandro Rodríguez-Zaragoza, José de Jesús Adolfo Tortolero-Langarica, Amílcar Leví Cupul-Magaña and Alma Paola Rodríguez-Troncoso
Diversity 2024, 16(7), 372; https://doi.org/10.3390/d16070372 - 27 Jun 2024
Viewed by 2034
Abstract
The high diversity and biomass of organisms associated with coral communities depend directly on the maintenance or changes in the benthic composition. Over a decade, we evaluated the spatiotemporal variation in the benthic structure and composition of an insular coral community in the [...] Read more.
The high diversity and biomass of organisms associated with coral communities depend directly on the maintenance or changes in the benthic composition. Over a decade, we evaluated the spatiotemporal variation in the benthic structure and composition of an insular coral community in the Northeastern Tropical Pacific. Our results show that local conditions drive spatiotemporal differences, and benthic organisms such as sponges, crustose coralline algae, octocorals, and hydrocorals all increased in abundance (cover) in response to negative thermal anomalies caused by the 2010–2011 La Niña event. In contrast, abnormally high temperatures, such as those recorded during the 2015–2016 El Niño Southern Oscillation (ENSO) event, explain the loss of scleractinian corals and crustose coralline algae coverage, which reduced the benthic groups’ richness (BGR), diversity (H’BG), and evenness (J’BG), with evidence of a consequent decrease in ecosystem function recorded the following year. Our analysis also showed that sites with high habitat heterogeneity harbored higher average BRG and H’BG values and were less affected by environmental fluctuations than sites with high live scleractinian coral cover and lower BRG and H’BG values. Therefore, the benthic structure was impacted differently by the same perturbation, and changes in the benthic community composition affected the groups associated with the community and ecological functions. More importantly, regional stressors such as the ENSO event caused only temporary changes in the benthic community structure, demonstrating the high resilience of the community to annual and interannual stressors. Full article
(This article belongs to the Special Issue Biodiversity and Conservation of Coral Reefs)
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<p>Study area and sampling sites at Islas Marietas National Park in the Mexican Pacific. Codes: IL = Isla Larga, CM = Cueva del Muerto, ZR = Zona de Restauración, ZRS = Zona de Restauración Sur, IR = Isla Redonda, TA = Túnel-Amarradero, PP = Plataforma Pavonas, and PA = Playa del Amor.</p>
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<p>Spatiotemporal variation in the benthic communities at Isla Larga (<b>A</b>,<b>C</b>) and Isla Redonda (<b>B</b>,<b>D</b>).</p>
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<p>Spatiotemporal comparison of benthic structural and functional metrics (<b>A</b>–<b>D</b>; mean ± std. error). Abbreviations: BGR = benthic group richness, H’<sub>BG</sub> = Shannon entropy, J’<sub>BG</sub> = Pielou evenness, D<sub>BG</sub> = Simpson dominance, FRic<sub>BG</sub> = functional richness, FDis<sub>BG</sub> = functional dispersion, FDiv<sub>BG</sub> = functional divergence, IL = Isla Larga, and IR = Isla Redonda.</p>
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<p>Temporal variability (±std. error) in diffuse attenuation coefficient (Kd 490), sea surface temperature (SST), and photosynthetically active radiation (PAR) at Islas Marietas National Park.</p>
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<p>Linear regression between photosynthetically active radiation (PAR) and Shannon entropy (H’<sub>BG</sub>).</p>
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13 pages, 5285 KiB  
Communication
Microbial Community and Metabolome Analysis of the Porcine Intestinal Damage Model Induced by the IPEC-J2 Cell Culture-Adapted Porcine Deltacoronavirus (PDCoV) Infection
by Ying Shi, Benqiang Li, Jinghua Cheng, Jie Tao, Pan Tang, Jiajie Jiao and Huili Liu
Microorganisms 2024, 12(5), 874; https://doi.org/10.3390/microorganisms12050874 - 27 Apr 2024
Cited by 1 | Viewed by 1229
Abstract
This study was conducted to elucidate the intestinal damage induced by the IPEC-J2 cell culture-passaged PDCoV. The results showed that PDCoV disrupted the intestinal structure and increased intestinal permeability, causing abnormalities in mucosal pathology. Additionally, PDCoV induced an imbalance in the intestinal flora [...] Read more.
This study was conducted to elucidate the intestinal damage induced by the IPEC-J2 cell culture-passaged PDCoV. The results showed that PDCoV disrupted the intestinal structure and increased intestinal permeability, causing abnormalities in mucosal pathology. Additionally, PDCoV induced an imbalance in the intestinal flora and disturbed its stability. Microbial community profiling revealed bacterial enrichment (e.g., Proteobacteria) and reduction (e.g., Firmicutes and Bacteroidetes) in the PDCoV-inoculated piglet model. In addition, metabolomics analysis indicated that 82 named differential metabolites were successfully quantified, including 37 up-regulated and 45 down-regulated metabolites. Chenodeoxycholic acid, sphingosine, and oleanolic aldehyde levels were reduced in PDCoV-inoculated piglets, while phenylacetylglycine and geranylgeranyl-PP levels were elevated. Correlation analysis indicated a negative correlation between Escherichia-Shigella and choline, succinic acid, creatine, phenyllactate, and hippuric acid. Meanwhile, Escherichia-Shigella was positively correlated with acetylcholine, L-Glutamicacid, and N-Acetylmuramate. Roseburia, Lachnospiraceae_UCG-010, Blautia, and Limosilactobacillus were negatively and positively correlated with sphingosine, respectively. These data suggested PDCoV-inoculated piglets exhibited significant taxonomic perturbations in the gut microbiome, which may result in a significantly altered metabolomic profile. Full article
(This article belongs to the Section Veterinary Microbiology)
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<p>Gross and histological lesions in the small intestine of 5-day-old piglets inoculated orally with the IPEC-J2 cell culture-passaged PDCoV (PD-F6) or mock. (<b>A</b>) The gross intestinal changes in PDCoV-inoculated piglet at post-inoculation day (PID) 4. (<b>B</b>) Intestine appearance of control piglet at post-inoculation day (PID) 4. (<b>C</b>) The histological lesions in the small intestine of PDCoV-inoculated piglet by hematoxylin and eosin-staining. (<b>D</b>) The histological observation in small intestine of control piglet by hematoxylin and eosin staining.</p>
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<p>Microbial community profiling. (<b>A</b>,<b>B</b>) The community structure of intestinal bacteria in group A1 and A2 at the phylum (<b>A</b>) and genus (<b>B</b>) levels. The top species with the highest abundance for each group at each taxonomic level (phylum and genus) were selected. Different colors represent different microbes, and the ordinate indicates the relative abundance, <span class="html-italic">n</span> = 5. A1 (PDCoV–inoculated group) and A2 (control group). (<b>C</b>,<b>D</b>) Cladogram (<b>C</b>) and Histogram (<b>D</b>) generated from the LEfSe analysis of intestinal microbiota. <span class="html-italic">n</span> = 5. Only taxa of an LDA significant threshold of 2.0 were shown. A1 group (in red), taxa enrichment with a negative LDA score; A2 group (in blue), taxa enrichment with a positive LDA score.</p>
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<p>The metabolomic signatures of gut digesta from piglets. (<b>A</b>) PCA map. The distance of each coordinate point represents the degree of aggregation and dispersion between samples. (<b>B</b>) OPLS–DA map. (<b>C</b>) Model verification map of OPLS–DA. The abscissa represents the replacement retention of the replacement test; the ordinate represents the R2 (green dot) and Q2 (blue square) replacement test values; and the two dashes represent the regression lines of R2 and Q2, respectively, and positive ion mode. (<b>D</b>) Volcanic map of differential metabolites. Each point in the figure represents a specific metabolite. The abscissa represents the multiple change value; and the ordinate represents the statistical test value: that is, <span class="html-italic">p</span>-value. All the values are processed logarithmically. (<b>E</b>) Volcanic map of differential metabolites. (<b>F</b>) Heatmap of differential metabolites between groups. The color represents the relative abundance of the metabolite in samples.</p>
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<p>Metabolic pathways and altered intestinal metabolites in PDCoV–inoculated piglets. (<b>A</b>) Pathway analysis using MetaboAnalyst. (<b>B</b>) Metabolic pathways and altered metabolites. Blue dots indicate pathways, and other dots indicate metabolites. Red–labeled metabolites were up-regulated, and blue–labeled metabolites were down-regulated, and the darker the color, the greater the difference. Green blocks represent disturbed metabolic pathways.</p>
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<p>Spearman’s correlation analysis between microbiota and the top 20 differential metabolites. Positive and negative correlations are shown as red and blue in the heat map, respectively. Significant microbiota–metabolite correlations were determined based on an |r| ≥ 0.7 and <span class="html-italic">p</span> &lt; 0.05 (* <span class="html-italic">p</span> &lt; 0.0, ** <span class="html-italic">p</span> &lt; 0.01, *** <span class="html-italic">p</span> &lt; 0.001).</p>
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13 pages, 277 KiB  
Article
The Strong Ekeland Variational Principle in Quasi-Pseudometric Spaces
by Ştefan Cobzaş
Mathematics 2024, 12(3), 471; https://doi.org/10.3390/math12030471 - 1 Feb 2024
Viewed by 971
Abstract
Roughly speaking, Ekeland’s Variational Principle (EkVP) (J. Math. Anal. Appl. 47 (1974), 324–353) asserts the existence of strict minima of some perturbed versions of lower semicontinuous functions defined on a complete metric space. Later, Pando Georgiev (J. Math. Anal. Appl. 131 (1988), no. [...] Read more.
Roughly speaking, Ekeland’s Variational Principle (EkVP) (J. Math. Anal. Appl. 47 (1974), 324–353) asserts the existence of strict minima of some perturbed versions of lower semicontinuous functions defined on a complete metric space. Later, Pando Georgiev (J. Math. Anal. Appl. 131 (1988), no. 1, 1–21) and Tomonari Suzuki (J. Math. Anal. Appl. 320 (2006), no. 2, 787–794 and Nonlinear Anal. 72 (2010), no. 5, 2204–2209)) proved a Strong Ekeland Variational Principle, meaning the existence of strong minima for such perturbations. Please note that Suzuki also considered the case of functions defined on Banach spaces, emphasizing the key role played by reflexivity. In recent years, an increasing interest was manifested by many researchers to extend EkVP to the asymmetric case, i.e., to quasi-metric spaces (see references). Applications to optimization, behavioral sciences, and others were obtained. The aim of the present paper is to extend the strong Ekeland principle, both Georgiev’s and Suzuki’s versions, to the quasi-pseudometric case. At the end, we ask for the possibility of extending it to asymmetric normed spaces (i.e., the extension of Suzuki’s results). Full article
15 pages, 1409 KiB  
Article
Reliable Dimerization Energies for Modeling of Supramolecular Junctions
by Jiří Czernek and Jiří Brus
Int. J. Mol. Sci. 2024, 25(1), 602; https://doi.org/10.3390/ijms25010602 - 2 Jan 2024
Cited by 3 | Viewed by 1562
Abstract
Accurate estimates of intermolecular interaction energy, ΔE, are crucial for modeling the properties of organic electronic materials and many other systems. For a diverse set of 50 dimers comprising up to 50 atoms (Set50-50, with 7 of its members being models [...] Read more.
Accurate estimates of intermolecular interaction energy, ΔE, are crucial for modeling the properties of organic electronic materials and many other systems. For a diverse set of 50 dimers comprising up to 50 atoms (Set50-50, with 7 of its members being models of single-stacking junctions), benchmark ΔE data were compiled. They were obtained by the focal-point strategy, which involves computations using the canonical variant of the coupled cluster theory with singles, doubles, and perturbative triples [CCSD(T)] performed while applying a large basis set, along with extrapolations of the respective energy components to the complete basis set (CBS) limit. The resulting ΔE data were used to gauge the performance for the Set50-50 of several density-functional theory (DFT)-based approaches, and of one of the localized variants of the CCSD(T) method. This evaluation revealed that (1) the proposed “silver standard” approach, which employs the localized CCSD(T) method and CBS extrapolations, can be expected to provide accuracy better than two kJ/mol for absolute values of ΔE, and (2) from among the DFT techniques, computationally by far the cheapest approach (termed “ωB97X-3c/vDZP” by its authors) performed remarkably well. These findings are directly applicable in cost-effective yet reliable searches of the potential energy surfaces of noncovalent complexes. Full article
(This article belongs to the Collection Feature Papers in 'Macromolecules')
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<p>Comparison of computational results for dimers from the Set50-50. The regression line plotted in blue color is <math display="inline"><semantics> <mrow> <mfenced open="{" close="}" separators="|"> <mrow> <mi>y</mi> </mrow> </mfenced> <mo>=</mo> <mn>0.991</mn> <mo>×</mo> <mfenced open="{" close="}" separators="|"> <mrow> <mi>x</mi> </mrow> </mfenced> <mo>+</mo> <mn>0.080</mn> </mrow> </semantics></math> kJ/mol, while the regression line plotted in red color is <math display="inline"><semantics> <mrow> <mfenced open="{" close="}" separators="|"> <mrow> <mi>y</mi> </mrow> </mfenced> <mo>=</mo> <mn>1.097</mn> <mo>×</mo> <mfenced open="{" close="}" separators="|"> <mrow> <mi>x</mi> </mrow> </mfenced> <mo>+</mo> <mn>1.714</mn> </mrow> </semantics></math> kJ/mol.</p>
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<p>Computational results for azulene- and naphthalene-based dimers that are discussed in the text. The linear regression visualized in red and blue color has the adjusted <span class="html-italic">R</span><sup>2</sup> value of 0.980 and 0.964, respectively.</p>
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16 pages, 3986 KiB  
Article
Accelerating Convergence of Langevin Dynamics via Adaptive Irreversible Perturbations
by Zhenqing Wu, Zhejun Huang, Sijin Wu, Ziying Yu, Liuxin Zhu and Lili Yang
Mathematics 2024, 12(1), 118; https://doi.org/10.3390/math12010118 - 29 Dec 2023
Viewed by 1208
Abstract
Irreversible perturbations in Langevin dynamics have been widely recognized for their role in accelerating convergence in simulations of multi-modal distributions π(θ). A commonly used and easily computed standard irreversible perturbation is Jlogπ(θ), [...] Read more.
Irreversible perturbations in Langevin dynamics have been widely recognized for their role in accelerating convergence in simulations of multi-modal distributions π(θ). A commonly used and easily computed standard irreversible perturbation is Jlogπ(θ), where J is a skew-symmetric matrix. However, Langevin dynamics employing a fixed-scale standard irreversible perturbation encounter a trade-off between local exploitation and global exploration, associated with small and large scales of standard irreversible perturbation, respectively. To address this trade-off, we introduce the adaptive irreversible perturbations Langevin dynamics, where the scale of the standard irreversible perturbation changes adaptively. Through numerical examples, we demonstrate that adaptive irreversible perturbations in Langevin dynamics can enhance performance compared to fixed-scale irreversible perturbations. Full article
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Figure 1

Figure 1
<p>The paths of a pair of Langevin dynamics, driven by two different temperatures, are depicted. The orange and red traces represent the lower and higher temperatures, respectively. The background corresponds to the contours of the objective function. It is worth noting that lines of the same color are separate from each other due to the swap operation, indicated by the dashed line.</p>
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<p>MSE, squared bias, and variance of the resulting estimator for <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>μ</mi> <mo>,</mo> <mi>σ</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>μ</mi> <mo>+</mo> <mi>σ</mi> </mrow> </semantics></math>. Real gradients are computed.</p>
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<p>MSE, squared bias, and variance of the resulting estimator for <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>μ</mi> <mo>,</mo> <mi>σ</mi> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>μ</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>σ</mi> <mn>2</mn> </msup> </mrow> </semantics></math>. Real gradients are computed.</p>
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<p>MSE, squared bias, and variance of the resulting estimator for <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>μ</mi> <mo>,</mo> <mi>σ</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>μ</mi> <mo>+</mo> <mi>σ</mi> </mrow> </semantics></math>. Stochastic gradients are computed.</p>
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<p>MSE, squared bias, and variance of the resulting estimator for <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>μ</mi> <mo>,</mo> <mi>σ</mi> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>μ</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>σ</mi> <mn>2</mn> </msup> </mrow> </semantics></math>. Stochastic gradients are computed.</p>
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<p>Empirical behavior on 25 two-dimensional Gaussian distributions. Stochastic gradients are computed.</p>
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<p>KL divergence of different sampling algorithms. Stochastic gradients are computed.</p>
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<p>Trajectory of various sampling algorithms. Stochastic gradients are computed.</p>
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<p>Empirical behavior on multi-modal distribution. Stochastic gradients are computed.</p>
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<p>KL divergence of different sampling algorithms. Stochastic gradients are computed.</p>
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<p>Log-loss calculated on test dataset for different algorithms. Stochastic gradients are computed.</p>
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14 pages, 3486 KiB  
Article
A Triplet/Singlet Ground-State Switch via the Steric Inhibition of Conjugation in 4,6-Bis(trifluoromethyl)-1,3-phenylene Bisnitroxide
by Nagito Haga and Takayuki Ishida
Molecules 2024, 29(1), 70; https://doi.org/10.3390/molecules29010070 - 21 Dec 2023
Cited by 1 | Viewed by 1371
Abstract
Ground triplet 4,6-bis(trifluoromethyl)-1,3-phenylene bis(tert-butyl nitroxide) (TF2PBN) reacted with [Y(hfac)3(H2O)2] (hfac = 1,1,1,5,5,5-hexafluoropentane-2,4-dionate), affording a doubly hydrogen-bonded adduct [Y(hfac)3(H2O)2(TF2PBN)]. The biradical was recovered from the adduct through recrystallization. Crystallographic analysis [...] Read more.
Ground triplet 4,6-bis(trifluoromethyl)-1,3-phenylene bis(tert-butyl nitroxide) (TF2PBN) reacted with [Y(hfac)3(H2O)2] (hfac = 1,1,1,5,5,5-hexafluoropentane-2,4-dionate), affording a doubly hydrogen-bonded adduct [Y(hfac)3(H2O)2(TF2PBN)]. The biradical was recovered from the adduct through recrystallization. Crystallographic analysis indicates that the torsion angles (|θ| ≤ 90°) between the benzene ring and nitroxide groups were 74.9 and 84.8° in the adduct, which are larger than those of the starting material TF2PBN. Steric congestion due to o-trifluoromethyl groups gives rise to the reduction of π-conjugation. Two hydrogen bonds enhance this deformation. Susceptometry of the adduct indicates a ground singlet with 2J/kB = −128(2) K, where 2J corresponds to the singlet–triplet gap. The observed magneto-structure relation is qualitatively consistent with Rajca’s pioneering work. A density functional theory calculation at the UB3LYP/6-311+G(2d,p) level using the atomic coordinates determined provided a result of 2J/kB = −162.3 K for the adduct, whilst the corresponding calculation on intact TF2PBN provided +87.2 K. After a comparison among a few known compounds, the 2J vs. |θ| plot shows a negative slope with a critical torsion of 65(3)°. The ferro- and antiferromagnetic coupling contributions are balanced in TF2PBN, being responsible for ground-state interconversion by means of small structural perturbation like hydrogen bonds. Full article
(This article belongs to the Special Issue Computational Studies of Novel Function Materials)
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Graphical abstract

Graphical abstract
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<p>(<b>a</b>) Crystal structure of <b>1</b> with thermal ellipsoids at the 50% probability level. Major conformations are drawn for disordered CF<sub>3</sub> groups at C17 (occupancy 0.640(4)), C26 (0.770(5)), and C31 (0.68(2)). Dotted lines stand for H bonds, denoted between the oxygen atoms. Selected atomic numbering is also shown. Atomic color codes: C, gray; H, turquoise; N, blue; O, red; F, yellow; Y, green. (<b>b</b>) An important portion after the C, H, and F atoms in hfac is omitted for the sake of clarity. (<b>c</b>) Simulated PXRD profiles for <b>1</b> and [Y(hfac)<sub>3</sub>(H<sub>2</sub>O)<sub>2</sub>] and experimental PXRD data for the product, <b>1</b>·[Y(hfac)<sub>3</sub>(H<sub>2</sub>O)<sub>2</sub>]<sub>1.1</sub>.</p>
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<p>Crystal structure of (<b>a</b>) the [Y(hfac)<sub>3</sub>(H<sub>2</sub>O)<sub>2</sub>] moiety in <b>1</b> and (<b>b</b>) [Y(hfac)<sub>3</sub>(H<sub>2</sub>O)<sub>2</sub>] with thermal ellipsoids at the 50% probability level. Major conformations are drawn for disordered CF<sub>3</sub> groups. Selected atomic numbering is also shown. For the atomic color codes, see <a href="#molecules-29-00070-f001" class="html-fig">Figure 1</a>.</p>
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<p>The <span class="html-italic">χ</span><sub>m</sub><span class="html-italic">T</span> vs. <span class="html-italic">T</span> plot for TF2PBN and <b>1</b>. The solid line represents the theoretical fit to the data on <b>1</b>. For the equation and parameters, see the text.</p>
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<p>Relative energy levels of triplet and broken-symmetry singlet states, calculated at the UB3LYP/6-311+G(2d,p) level. Spin density surfaces are drawn at the 0.002 e<sup>−</sup> Å<sup>−3</sup> level with blue and white lobes for the positive and negative spin densities, respectively. (<b>a</b>) The self-consistent field (SCF) energies of TF2PBN are −1480.8374662573 au with &lt;<span class="html-italic">S</span><sup>2</sup>&gt; = 2.0114 and −1480.8373276154 au with &lt;<span class="html-italic">S</span><sup>2</sup>&gt; = 1.0080 for the triplet and singlet states, respectively. (<b>b</b>) (<b>Left</b>) The SCF energies of <b>1</b> are −4494.9085419557 au with &lt;<span class="html-italic">S</span><sup>2</sup>&gt; = 2.0094 and −4494.9088003966 au with &lt;<span class="html-italic">S</span><sup>2</sup>&gt; = 1.0041 for the triplet and singlet states, respectively. (<b>Right</b>) The SCF energies of the TF2PBN portion in <b>1</b> are −1480.9064795589 au with &lt;<span class="html-italic">S</span><sup>2</sup>&gt; = 2.0094 and −1480.9067334369 au with &lt;<span class="html-italic">S</span><sup>2</sup>&gt; = 1.0045 for the triplet and singlet states, respectively. Ferro- and antiferromagnetic <span class="html-italic">J</span> values are marked in blue and red, respectively. For the atomic color codes, see <a href="#molecules-29-00070-f001" class="html-fig">Figure 1</a>.</p>
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<p>Relative energy levels of triplet and broken-symmetry singlet states, calculated at the UB3LYP/6-311+G(2d,p)//UB3LYP/6-31G(d) level. Optimized geometries were also shown. (<b>a</b>) The SCF energies of <span class="html-italic">syn</span>-MesBN are −924.7811231198 au with &lt;<span class="html-italic">S</span><sup>2</sup>&gt; = 2.0095 and −924.7812718628 au with &lt;<span class="html-italic">S</span><sup>2</sup>&gt; = 1.0060 for the triplet and singlet states, respectively. (<b>b</b>) The SCF energies of <span class="html-italic">anti</span>-MesBN are −924.7809370560 au with &lt;<span class="html-italic">S</span><sup>2</sup>&gt; = 2.0096 and −924.7811143077 au with &lt;<span class="html-italic">S</span><sup>2</sup>&gt; = 1.0056 for the triplet and singlet states, respectively. Spin density surfaces are drawn at the 0.002 e<sup>−</sup> Å<sup>−3</sup> level with blue and white lobes for the positive and negative spin densities, respectively. Antiferromagnetic <span class="html-italic">J</span> values are marked in red. For the atomic color codes, see <a href="#molecules-29-00070-f001" class="html-fig">Figure 1</a>.</p>
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<p>Plot of the exchange coupling constant, 2<span class="html-italic">J</span>, vs. the averaged out-of-conjugation torsion angle, <span class="html-italic">θ</span> = (|<span class="html-italic">θ</span><sub>1</sub>| + |<span class="html-italic">θ</span><sub>2</sub>|)/2. For the definition of <span class="html-italic">θ</span><sub>1</sub> and <span class="html-italic">θ</span><sub>2</sub>, see <a href="#molecules-29-00070-sch002" class="html-scheme">Scheme 2</a>, and the angular error bar implies |<span class="html-italic">θ</span><sub>1</sub>| and |<span class="html-italic">θ</span><sub>2</sub>|. The experimental and calculated 2<span class="html-italic">J</span> data are marked in filled and open circles, respectively. The lines represent an empirical linear relationship.</p>
Full article ">Scheme 1
<p>(<b>a</b>) Structural formulas of BPBN, tBuPBN, and TF2PBN. (<b>b</b>) Reaction Scheme for [Y(hfac)<sub>3</sub>(H<sub>2</sub>O)<sub>2</sub>(TF2PBN)] (<b>1</b>).</p>
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<p>The torsion angles (<b>left</b>) before and (<b>right</b>) after adduct formation. <span class="html-italic">θ</span><sub>1</sub> and <span class="html-italic">θ</span><sub>2</sub> are shown.</p>
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<p>Structural formulas of MO2PBN, BrMO2PBN, <span class="html-italic">syn-C</span><sub>S</sub>-MesBN, and <span class="html-italic">anti-C</span><sub>2</sub>-MesBN.</p>
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15 pages, 2467 KiB  
Article
Insight into the Binding of Argon to Cyclic Water Clusters from Symmetry-Adapted Perturbation Theory
by Carly A. Rock and Gregory S. Tschumper
Int. J. Mol. Sci. 2023, 24(24), 17480; https://doi.org/10.3390/ijms242417480 - 14 Dec 2023
Cited by 1 | Viewed by 1105
Abstract
This work systematically examines the interactions between a single argon atom and the edges and faces of cyclic H2O clusters containing three–five water molecules (Ar(H2O)n=35). Full geometry optimizations and subsequent harmonic vibrational frequency [...] Read more.
This work systematically examines the interactions between a single argon atom and the edges and faces of cyclic H2O clusters containing three–five water molecules (Ar(H2O)n=35). Full geometry optimizations and subsequent harmonic vibrational frequency computations were performed using MP2 with a triple-ζ correlation consistent basis set augmented with diffuse functions on the heavy atoms (cc-pVTZ for H and aug-cc-pVTZ for O and Ar; denoted as haTZ). Optimized structures and harmonic vibrational frequencies were also obtained with the two-body–many-body (2b:Mb) and three-body–many-body (3b:Mb) techniques; here, high-level CCSD(T) computations capture up through the two-body or three-body contributions from the many-body expansion, respectively, while less demanding MP2 computations recover all higher-order contributions. Five unique stationary points have been identified in which Ar binds to the cyclic water trimer, along with four for (H2O)4 and three for (H2O)5. To the best of our knowledge, eleven of these twelve structures have been characterized here for the first time. Ar consistently binds more strongly to the faces than the edges of the cyclic (H2O)n clusters, by as much as a factor of two. The 3b:Mb electronic energies computed with the haTZ basis set indicate that Ar binds to the faces of the water clusters by at least 3 kJ mol1 and by nearly 6 kJ mol1 for one Ar(H2O)5 complex. An analysis of the interaction energies for the different binding motifs based on symmetry-adapted perturbation theory (SAPT) indicates that dispersion interactions are primarily responsible for the observed trends. The binding of a single Ar atom to a face of these cyclic water clusters can induce perturbations to the harmonic vibrational frequencies on the order of 5 cm1 for some hydrogen-bonded OH stretching frequencies. Full article
(This article belongs to the Special Issue Noncovalent Interactions: New Developments in Experiment and Theory)
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Figure 1
<p>Minima identified for the Ar(H<math display="inline"><semantics> <msub> <mrow/> <mn>2</mn> </msub> </semantics></math>O)<math display="inline"><semantics> <msub> <mrow/> <mrow> <mi>n</mi> <mo>=</mo> <mn>3</mn> <mo>–</mo> <mn>5</mn> </mrow> </msub> </semantics></math> complexes (H white; O red; Ar cyan).</p>
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