Kepler's Law Research Articles

A relativistic description of MOND using the Palatini formalism in an extended metric theory of gravity

We construct a relativistic metric description of MOND using the Palatini formalism following the \( f(\chi)=\chi^b \) description of \citet{mendozatula}. We show that in order to recover the non-relativistic MOND regime where, for circular orbits the Tully-Fisher law replaces Kepler's third law, the value of the parameter $ b = 3/2 $, which is coincident with the value found using a pure metric formalism Capozziello et al. (2011). Unlike the pure metric formalism, which yields 4th order field equations, the Palatini approach yields second order field equations, which is a desirable requirement from a theoretical perspective. Thus, the phenomenology associated to astrophysical phenomena with Tully-Fisher scalings can be accounted for using this proposal, without the need to introduce any non-baryonic dark matter particles.

Keplerian planetary orbits in multidimensional Euclidian spaces

Newton's laws of motion are three physical laws that together, laid the foundation for classical three dimensional mechanics. They describe the relationship between a body and the forces acting upon it, and its motion in response to those forces. Kepler's laws of planetary motion are also three scientific laws describing the motion of planets around the Sun. Although the differential calculus was invented by Newton, Kepler established his famous laws 70 years earlier by using the same idea, namely to find a path in a non uniform field of force by small steps. Even though neither the force was known nor its relation to motion, he could determine the differential equations of motion from observation. This is one of the most important achievements in the history of physics. In this paper, we will see that these laws are a consequence of Newton’s second law even in multidimensional Euclidian spaces.

N-body dynamics on closed surfaces: the axioms of mechanics.

A major challenge for our understanding of the mathematical basis of particle dynamics is the formulation of N-body and N-vortex dynamics on Riemann surfaces. In this paper, we show how the two problems are, in fact, closely related when considering the role played by the intrinsic geometry of the surface. This enables a straightforward deduction of the dynamics of point masses, using recently derived results for point vortices on general closed differentiable surfaces M endowed with a metric g. We find, generally, that Kepler's Laws do not hold. What is more, even Newton's First Law (the law of inertia) fails on closed surfaces with variable curvature (e.g. the ellipsoid).

Electronic solar compass for high precision orientation on any planet

A compact, fully automatic electronic solar compass has been developed at the ENEA Frascati Laboratories. The compass is inspired to ``camera obscura'' sundials like those inside churches. Sun ephemerides are calculated using an approximate but effective analytical solution of Kepler's laws, where the Earth (or other planets) orbit main parameters are introduced. The instrument is light, cheap and it has an accuracy better than 1 arcmin. Some examples of application of the device as well as the possibility to use it on Mars are presented.

Z Draconis with two companions in a 2:1 mean-motion resonance

All available mid-eclipse times of the eclipsing binary Z Draconis are analyzed, and three sets of cyclic variations with periods of 20.1, 29.96 and 59.88 yr are found. The low-amplitude variations with a period of 20.1 yr may be attributed to the unavoidable slight imperfection in the double-Keplerian model, which gives periods of 29.96 and 59.88 yr. Interestingly, the Z Draconis system is close to a 2:1 mean-motion resonance, or a 6:3:2 mean-motion resonance if the 20.1 yr period really exists. We also find that the best solutions tend to give the minimum eccentricities. Based on Kepler's third law, the outermost companion has a minimum mass of ∼ 0.77 M⊙, whereas the middle companion is an M dwarf star with a mass of ∼ 0.40 M⊙, suggesting that Z Draconis is a general N-body system.

Improved orbits and parallaxes for eight visual binaries with unrealistic previous masses using theHipparcosparallax

Improved orbits are presented for the visual binaries WDS 02366+1227, WDS 02434-6643, WDS 03244-1539, WDS 08507+1800, WDS 09128-6055, WDS 11532-1540, WDS 17375+2419, and WDS 22408-0333. The latest orbits for these binaries were demonstrating a great inconsistency between the systemic mass obtained through Kepler's Third Law and that calculated as a sum of their components' mass through standard mass-luminosity and mass-spectrum relationships. Our improvement allowed us to obtain consistent systemic masses for WDS 02434-6643 and WDS 09128-6055 without a need for changing the Hipparcos parallax. For the remaining 6 pairs, we suggest the use of their dynamical parallax as a reliable distance estimate unless more precise parallax is reported. Astrophysical and dynamical properties of individual objects are discussed.

New distances for a selected set of visual binaries with inconsistent dynamical masses

We have selected a set of 17 visual binaries that demonstrate great inconsistency between the systemic mass obtained through Kepler's Third Law as compared to that calculated through standard mass-luminosity and mass-spectrum relationships. A careful inspection of orbital data and parallaxes showed that the current orbits of nine binaries (WDS 00155-1608, WDS 00174+0853, WDS 05017+2050, WDS 06410+0954, WDS 16212-2536, WDS 17336-3706, WDS 19217-1557, WDS 20312+1116, and WDS 21118+5959) do not need to be improved, instead we recommend different parallax (distance) value for them. On the other hand, we considered that eight orbits (WDS 02366+1227, WDS 02434-6643, WDS 03244-1539, WDS 08507+1800, WDS 09128-6055, WDS 11532-1540, WDS 17375+2419, and WDS 22408-0333) had to be improved. Due to various reasons mentioned in this paper, their distances should most likely be corrected unless better orbital solutions and/or more precise parallaxes are reported. To obtain consistent mass values, the use of the dynamical parallax is still recommended for 5 out of the 8 improved orbits. For WDS 02434-6643, WDS 09128-6055, and WDS 11532-1540, the improvement itself yields reasonable mass sums while maintaining {\pi}_{Hip} within a 1-2{\sigma} margin. New distance estimates for 16 stars (mainly based on the obtained dynamical parallaxes) and individual comments for all objects are presented and discussed.

Introducing Earth's Orbital Eccentricity

Most students know that planetary orbits, including Earth's, are elliptical; that is Kepler's first law, and it is found in many science textbooks. But quite a few are mistaken about the details, thinking that the orbit is very eccentric, or that this effect is somehow responsible for the seasons. In fact, the Earth's orbital eccentricity is small, and its only effect on the seasons is their unequal durations. Here I show a pleasant way to guide students to the actual value of Earth's orbital eccentricity, starting from the durations of the four seasons. The date of perihelion is also found.

Examples of Complete Solvability of 2D Classical Superintegrable Systems

Classical (maximal) superintegrable systems in $n$ dimensions are Hamiltonian systems with $2n-1$ independent constants of the motion, globally defined, the maximum number possible. They are very special because they can be solved algebraically. In this paper we show explicitly, mostly through examples of 2nd order superintegrable systems in 2 dimensions, how the trajectories can be determined in detail using rather elementary algebraic, geometric and analytic methods applied to the closed quadratic algebra of symmetries of the system. We treat a family of 2nd order degenerate systems: oscillator analogies on Darboux, nonzero constant curvature, and flat spaces, related to one another via contractions, and obeying Kepler's laws. Then we treat two 2nd order nondegenerate systems, an analogy of a caged Coulomb problem on the 2-sphere and its contraction to a Euclidean space caged Coulomb problem. In all cases the symmetry algebra structure provides detailed information about the trajectories. An interesting example is the occurrence of ''metronome orbits'', trajectories confined to an arc rather than a loop, which are indicated clearly from the structure equations but might be overlooked using more traditional methods. We also treat the Post-Winternitz system, an example of a classical 4th order superintegrable system that cannot be solved using separation of variables. Finally we treat a superintegrable system, related to the addition theorem for elliptic functions, whose constants of the motion are only rational in the momenta, a system of special interest because its constants of the motion generate a closed polynomial algebra. This paper contains many new results but we have tried to present most of the materials in a fashion that is easily accessible to nonexperts, in order to provide entr\'ee to superintegrablity theory.

PROBING THE MASSES AND RADII OF DONOR STARS IN ECLIPSING X-RAY BINARIES WITH THESWIFTBURST ALERT TELESCOPE

Physical parameters of both the mass donor and compact object can be constrained in X-ray binaries with well-defined eclipses, as our survey of wind-fed supergiant X-ray binaries (SGXBs) IGR J16393-4643, IGR J16418-4532, IGR J16479-4514, IGR J18027-2016 and XTE J1855-026 reveals. Using the orbital period and Kepler's third law, we express the eclipse half-angle in terms of radius, inclination angle and the sum of the masses. Pulse-timing and radial velocity curves can give masses of both the donor and compact object as in the case of the "double-lined" binaries IGR J18027-2016 and XTE J1855-026. The eclipse half angles are 15 (+3,-2), 31.7 (+0.7,-0.8), 32 +/- 2, 34 +/- 2 and 33.6 +/- 0.7 degrees for IGR J16393-4643, IGR J16418-4532, IGR J16479-4514, IGR J18027-2016 and XTE 1855-026, respectively. In wind-fed systems, the primary not exceeding the Roche-lobe size provides an upper limit on system parameters. In IGR J16393-4643, spectral types of B0 V or B0-5 III are found to be consistent with the eclipse duration and Roche-lobe, but the previously proposed donor stars in IGR J16418-4532 and IGR J16479-4514 were found to be inconsistent with the Roche-lobe size. Stars with spectral types O7.5 I and earlier are possible. For IGR J18027-2016, the mass and radius of the donor star lie between 18.6--19.4 $M_\odot$ and 17.4--19.5 $R_\odot$. We constrain the neutron star mass between 1.37--1.43 $M_\odot$. We find the mass and radius of the donor star in XTE J1855-026 to lie between 19.6--20.2 $M_\odot$ and 21.5--23.0 $R_\odot$. The neutron star mass was constrained to 1.77--1.82 $M_\odot$. Eclipse profiles are asymmetric in IGR J18027-2016 and XTE J1855-026, which we attribute to accretion wakes.

ECCENTRICITY FROM TRANSIT PHOTOMETRY: SMALL PLANETS IN KEPLER MULTI-PLANET SYSTEMS HAVE LOW ECCENTRICITIES

Solar system planets move on almost circular orbits. In strong contrast, many massive gas giant exoplanets travel on highly elliptical orbits, whereas the shape of the orbits of smaller, more terrestrial, exoplanets remained largely elusive. Knowing the eccentricity distribution in systems of small planets would be important as it holds information about the planet's formation and evolution, and influences its habitability. We make these measurements using photometry from the Kepler satellite and utilizing a method relying on Kepler's second law, which relates the duration of a planetary transit to its orbital eccentricity, if the stellar density is known. Our sample consists of 28 bright stars with precise asteroseismic density measurements. These stars host 74 planets with an average radius of 2.6 $R_\oplus$. We find that the eccentricity of planets in Kepler multi-planet systems is low and can be described by a Rayleigh distribution with $\sigma$ = 0.049 $\pm$ 0.013. This is in full agreement with solar system eccentricities, but in contrast to the eccentricity distributions previously derived for exoplanets from radial velocity studies. Our findings are helpful in identifying which planets are habitable because the location of the habitable zone depends on eccentricity, and to determine occurrence rates inferred for these planets because planets on circular orbits are less likely to transit. For measuring eccentricity it is crucial to detect and remove Transit Timing Variations (TTVs), and we present some previously unreported TTVs. Finally transit durations help distinguish between false positives and true planets and we use our measurements to confirm six new exoplanets.

A novel hybrid algorithm of GSA with Kepler algorithm for numerical optimization

It is now well recognized that pure algorithms can be promisingly improved by hybridization with other techniques. One of the relatively new metaheuristic algorithms is Gravitational Search Algorithm (GSA) which is based on the Newton laws. In this paper, to enhance the performance of GSA, a novel algorithm called “Kepler”, inspired by the astrophysics, is introduced. The Kepler algorithm is based on the principle of the first Kepler law. The hybridization of GSA and Kepler algorithm is an efficient approach to provide much stronger specialization in intensification and/or diversification. The performance of GSA–Kepler is evaluated by applying it to 14 benchmark functions with 20–1000 dimensions and the optimal approximation of linear system as a practical optimization problem. The results obtained reveal that the proposed hybrid algorithm is robust enough to optimize the benchmark functions and practical optimization problems.

Topological dependence of Kepler's third law for collisionless periodic three-body orbits with vanishing angular momentum and equal masses

We present results of numerical calculations showing a three-body orbit's period's T dependence on its topology. This dependence is a simple linear one, when expressed in terms of appropriate variables, suggesting an exact mathematical law. This is the first known relation between topological and kinematical properties of three-body systems. We have used these results to predict the periods of several sets of as yet undiscovered orbits, but the relation also indicates that the number of periodic three-body orbits with periods shorter than any finite number is countable.

Towards an alternative gravitational theory

In 1680, Cassini proposed oval curves as alternative trajectories for the visible planets around the sun. The Cassini ovals were of course overshadowed by the Kepler's first law (1609), namely the planets move around the sun describing conic orbits. Here, we describe the possibility that the Cassini's idea works at larger or smaller scales. Indeed, we consider the Spiric curves (which are a generalization of the Cassini oval) and present the first steps towards a Spiric gravitational theory. We show that from our formalism an ellipse associated with a planet can be obtained as a particular case.

Multiple applications of a derived formula for Kepler's law of areas

Multiple applications of a derived formula for Kepler's law of areas

DETERMINANDO A FORMA DA ÓRBITA DE MARTE NO ENSINO MÉDIO

No presente trabalho, visando suprir a deficiência de atividades práticas relacionadas ao conteúdo de Leis de Kepler nos livros-textos de Física do 1º ano do Ensino Médio, apresentamos uma atividade prática de determinação da órbita de Marte. O aluno, combinando conceitos físicos com a geometria poderá vivenciar a experiência da descoberta da forma da órbita de Marte de modo similar ao realizado por Johannes Kepler. Aplicamos a metodologia proposta junto a dezoito professores do Curso de Especialização em Educação em Ciências e obtivemos em um trabalho de duas horas de duração o traçado da órbita de Marte com resultados numéricos médios que reproduzem com erro inferior a 14% os parâmetros orbitais de referência da órbita de Marte.

Large-Scale Virtual Simulation of the Solar System Based on Java3D

As the modeling of the solar system is difficult and inaccurate, the 3D modeling of the planets in the solar system is studied by using Java3D technology. To make the revolution of simulated planets meet Kepler's laws while traveling around the sun, an improved approximate algorithm to calculate the optimum angle and the vertex coordinates of each planet is proposed. Using java3D perspective projection technique, the observation coordinate system is determined, and real-time viewpoint setting as well as roaming processing is realized. Through multiple view technique, we can observe the movement of celestial bodies from multiple aspects to obtain vivid simulation effects of planets' behavior in the solar system. Simulation results show that the virtual simulation platform developed based on java3D can improve the accuracy and naturalness of solar system simulation.

الاستثمار الإبداعی والتعلیم عن بعُد

Witness the current era of many successive scientific and technological level developments, which makes it imperative for educators to search for new forms of education commensurate with the revolution in communications and information technology, particularly in the service sector, which represents one of education of the most important services provided by communities to individual sand return them benefit in the short and long term,and what was the creative one of the most important variables that developed and developing countries seeking to develop both; due for submission unconventional solutions to the problems faced by these countries in various aspects of political and economic life, social, and environmental. The problems of poverty, hunger, disease, ignorance and street children are a product of the accumulation of conventional solutions that have failed to deal with. Sternberg & Lubart( 1996) Defined creativity as the ability to produce a work of unexpected novelty and originality, and high quality, and useful, and brings important limitations. Creativity is an important subject and widespread at the individual and community levels in plenty of areas of different tasks. At the individual level, creativity is associated with, for example, to dissolve the work and daily life problems. At the community level, it could lead to the creation of new scientific results, and new movements in art and new inventions, new social programs. The economic significance of creativity and clear because of the new products and services, creating jobs, moreover, individuals, organizations, and communities must adapt existing resources to change the requirements of tasks in order to remain competitive. As the e-learning one of the most remote learning formats and its relationship to creativity entrances study based on computer simulation programs have made (Boden, 1992,1999) reviewed, and it was aimed at the production of creative thinking CNC manner that simulates what people do. For example, he developed Zytkow, 1987) & Langley, Simon, Bradshaw) a set of programs that will restore basic scientific laws, and adopt these computer models to infer by the guiding principles to solve the problem of the search for a set of data or space concepts to get the subtle relationships between the input variables . The initial program is called Bacon BACON used in reasoning associated with space-like if the value of two Addetyn getting together, look for the ratio between them to search for data in order forms. Among the most important achievements of the program BACON observation test data on the orbits of the planets, which were available to Kepler, and put through the laws of planetary motion. The program also contributed to the re-discovery of Kepler's third law. And maligned on this program is that it differs from the creative function, where gives him problems in the form builders, while the creative function depends on the detection of problems.

PLANET HUNTERS. VII. DISCOVERY OF A NEW LOW-MASS, LOW-DENSITY PLANET (PH3 C) ORBITING KEPLER-289 WITH MASS MEASUREMENTS OF TWO ADDITIONAL PLANETS (PH3 B AND D)

We report the discovery of one newly confirmed planet ($P=66.06$ days, $R_{\rm{P}}=2.68\pm0.17R_\oplus$) and mass determinations of two previously validated Kepler planets, Kepler-289 b ($P=34.55$ days, $R_{\rm{P}}=2.15\pm0.10R_\oplus$) and Kepler-289-c ($P=125.85$ days, $R_{\rm{P}}=11.59\pm0.10R_\oplus$), through their transit timing variations (TTVs). We also exclude the possibility that these three planets reside in a $1:2:4$ Laplace resonance. The outer planet has very deep ($\sim1.3%$), high signal-to-noise transits, which puts extremely tight constraints on its host star's stellar properties via Kepler's Third Law. The star PH3 is a young ($\sim1$ Gyr as determined by isochrones and gyrochronology), Sun-like star with $M_*=1.08\pm0.02M_\odot$, $R_*=1.00\pm0.02R_\odot$, and $T_{\rm{eff}}=5990\pm38$ K. The middle planet's large TTV amplitude ($\sim5$ hours) resulted either in non-detections or inaccurate detections in previous searches. A strong chopping signal, a shorter period sinusoid in the TTVs, allows us to break the mass-eccentricity degeneracy and uniquely determine the masses of the inner, middle, and outer planets to be $M=7.3\pm6.8M_\oplus$, $4.0\pm0.9M_\oplus$, and $M=132\pm17M_\oplus$, which we designate PH3 b, c, and d, respectively. Furthermore, the middle planet, PH3 c, has a relatively low density, $\rho=1.2\pm0.3$ g/cm$^3$ for a planet of its mass, requiring a substantial H/He atmosphere of $2.1^{+0.8}_{-0.3}%$ by mass, and joins a growing population of low-mass, low-density planets.

KIC 2856960: the impossible triple star

KIC 2856960 is a star in the Kepler field which was observed by Kepler for four years. It shows the primary and secondary eclipses of a close binary of period 0.258 d as well as complex dipping events that last for about 1.5 d at a time and recur on a 204 d period. The dips are thought to result when the close binary passes across the face of a third star. In this paper, we present an attempt to model the dips. Despite the apparent simplicity of the system and strenuous efforts to find a solution, we find that we cannot match the dips with a triple star while satisfying Kepler's laws. The problem is that to match the dips, the separation of the close binary has to be larger than possible relative to the outer orbit given the orbital periods. Quadruple star models can get round this problem but require the addition of a so-far undetected intermediate period of the order of 5–20 d that has been a near-perfect integer divisor of the outer 204 d period. Although we have no good explanation for KIC 2856960, using the full set of Kepler data we are able to update several of its parameters. We also present a spectrum showing that KIC 2856960 is dominated by light from a K3- or K4-type star.