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The possibility of setting up a space experiment aimed at detecting ultra-small deviations of the law of gravity of the central star of the planetary system from the Newton and Schwarzschild formats is demonstrated. The experiment comprises a ballistic flight of a space-probe along a trajectory composed of three treks divided by two gravity assist (GA) maneuvers at different planets, each GA acting as a powerful amplifier of the probe deflections caused by small deviations of the gravitational law. It is shown that the proposed (different) analytical and numerical computations of the probe's flight in a model of a planetary system simulating the Sun, Earth, and Venus, give similar results confirming a certain possibility to distinguish the forces of gravity described by the Schwarzschild and Eddington-Robertson metrics. • A space experiment to detect deviations from a 'standard' gravity law is discussed. • The Sun, Earth, Venus in Eddington or Schwarzschild gravity make up the space lab. • The experiment scheme of a probe flight with two gravity assists (GA) is suggested. • Analytical & numerical computations of the probe flight are made with close results. • Two GAs at Venus and Earth definitely distinguish between the regarded gravity laws. [ABSTRACT FROM AUTHOR]

This paper defines Mathematical Modelling as the art of translating or converting a real world system or situation (problem) into a mathematical form or statement. That is, Mathematical Modelling is the process of how to use mathematics in solving problems of real life situations. Whereas, Mathematical Model is a simplified mathematical representation (equations, graphs, functions, relations) of the basic characteristics of the real situation. This is the product of the process. That is, the real life problem that we come across is transformed into mathematical statements and then solved using mathematical techniques. Few examples (in particular, the model of Planetary system) are used to concretize this concept of Mathematical Modelling. It is also demonstrated that to some extent, Mathematical Modelling is not only related to problem solving but a potent problem solving tool. [ABSTRACT FROM AUTHOR]

Despite its remarkable growth in recent years, the study of populism has long suffered from the lack of a comprehensive theory about its causes, development, and outcomes. Yet, no such theory can be arrived at before previously having resolved what "populism" really is (and what it is not). Good definitions, therefore, require a prior conceptualization of the phenomena we classify as populist, and want explained. Like planets, which are hard to define but easier to conceptualize as the component parts of our solar system, modern populism is better understood as a novel political system that maintains electoral democracy while also working against the principles of political liberalism. Based on such a conceptualization, we may define populism simply as "democratic illiberalism," which opens the door for new empirical research and worthwhile theory-building.

Low‐frequency waves are closely related to magnetospheric energy dissipation processes. The Cassini spacecraft explored Saturn's magnetosphere for over 13 years, until September 2017, covering a period of more than a complete solar cycle. Using this rich heritage data set, we systematically investigated key physical parameters of low‐frequency waves in Saturn's magnetosphere, including their local time distribution and the dependence on solar activity. We found that the wave activity peaked in the near noon sector. For the nightside, the wave intensity also appeared to peak pre and postmidnight. Due to the limited local time coverage for each solar phase, we were not able to draw a firm conclusion on the wave's dependence on solar activity. In general, the wave power showed a monotonically decreasing trend toward larger distances in nightside sectors especially during the declining phase, which implied that low‐frequency waves mainly originate from the relatively inner regions of the magnetosphere. On the dayside, stronger waves were mostly located at/within ∼25 Rs, near the magnetopause. The study shows a global picture of low‐frequency waves in Saturn's magnetosphere, providing important implications for how magnetospheric energy dissipates into Saturn's polar ionosphere and atmosphere.Key Points: Using the complete Cassini magnetometer data set, we present the global picture of ultralow frequency waves in Saturn's magnetosphereThe wave power shows a rapid decrease beyond 25 RS in both the morning and afternoon sectorsThe wave activity peaks in noon sector, implying that these waves could be driven by the solar wind interaction with Saturn's magnetopause [ABSTRACT FROM AUTHOR]

A mathematical model of the early stage of the formation of interstellar clouds resulting from the interaction of oppositely directed cosmic particle flows and evolution of the clouds into planetary systems is proposed. An important role in the evolution is played by mechanical energy dissipation caused mainly by particle collisions. The simulation is based on the classical n-body problem and the Newtonian theory of instantaneous collisions of rigid bodies with a relative velocity recovery factor smaller than unity. The performed computations suggest that this model is applicable in cosmogonic theories of planetary system formation. [ABSTRACT FROM AUTHOR]

A type of compactly spaced and comparably sized multi-exoplanet system similar to TRAPPIST-1 has been discovered recently. The stability of these systems is an important issue, requiring further study. We examined how the initial inclinations influence the stability of multi-planet systems and derived an empirical formula describing the dependence of the instability time-scale on planetary mass, space separation and initial inclination. We find the following. (i) If space separations between planets are greater than 12 R H (mutual Hill radius), coplanar multi-planet systems with 10−6 ≤ μ ≤ 10−3 (reduced planetary mass μ  = m/M *) will remain stable within 1010 T in (the innermost orbital period). (ii) If initial inclinations of planets are smaller than 10° and space separations are greater than 10 R H, multi-planet systems consisting of ≥5 planets with μ ≥ 10−5 will remain stable within 1010 T in. (iii) Initial inclinations in [0°, 10°] have inconsequential effects on the instability time-scales of massive multi-planet systems (μ ≥ 10−5), because eccentricities (excited during evolution) dominate the stability of these systems. (iv) If the initial inclinations are large enough (≥10°), sharp increases of instability time-scales in groups with 10−3 ≥ μ ≥ 10−5 will be moderated. This article presents a comprehensive study of the influence of inclination on the stability of multi-planet systems and discusses critical space separations for a multi-planet system becoming unstable. [ABSTRACT FROM AUTHOR]

The high-resolution Jet Propulsion Laboratory DE431 and DE432 planetary ephemeris are used to evaluate the instantaneous eccentricity functions of the orbits of the planets of the solar system from 13 000 BC to 17 000 AD at different time scales. Spectral analyses are performed to determine the frequencies and the amplitudes of the detected perturbations from 0.1-year to 10 000-year periods. Taken as contiguous pairs (Mercury-Venus, Earth-Mars, Jupiter-Saturn, and Uranus-Neptune), we found anti-phase patterns between contiguous planets at specific time scales (30 000 years): namely, the eccentricity of one planet increases while the other decreases. Venus and Earth instead appear in phase. However, on shorter time scales the phase coupling becomes more complex and irregular. Yet, Jupiter and Saturn present a π/2 phase coupling at the 1000-year scale. Periodogram analysis of the planetary eccentricity functions shows several fast fluctuations whose magnitudes indicate the strength of their mutual interactions where the Jovian planets significantly perturb the orbits of the inner planets. Besides, the wavelet power spectrum and wavelet squared coherence spectrum analyses are adopted to examine the relationships in time-frequency space between the eccentricity functions of each couple of terrestrial and Jovian planets. The analysis reveals the complexity and the evolution of the gravitational couplings perturbing each other planetary orbits. In some cases and for specific frequencies, this analysis technique also led to the discovery that the coherence phase can rotate clockwise or anticlockwise. The eccentricity function of the orbit of Jupiter presents large oscillations with periods of about 60 and 900- 960 years, mostly due to the interaction with Saturn. These oscillations also correspond to oscillations found in several geophysical records. The eccentricity functions of Uranus and Neptune are characterized by a large 4300-year oscillation. The eccentricity function of Pluto is characterized by a large nearly 20 000-year modulation. [ABSTRACT FROM AUTHOR]