Kepler's Law Research Articles

Slowly rotating black holes in Einstein-Dilaton-Gauss-Bonnet gravity: Quadratic order in spin solutions

We derive a stationary and axisymmetric black hole solution in Einstein-Dilaton-Gauss-Bonnet gravity to quadratic order in the ratio of the spin angular momentum to the black hole mass squared. This solution introduces new corrections to previously found nonspinning and linear-in-spin solutions. The location of the event horizon and the ergosphere are modified, as well as the quadrupole moment. The new solution is of Petrov type I, although lower order in spin solutions are of Petrov type D. There are no closed timelike curves or spacetime regions that violate causality outside of the event horizon in the new solution. We calculate the modifications to the binding energy, Kepler's third law, and properties of the innermost stable circular orbit. These modifications are important for determining how the electromagnetic properties of accretion disks around supermassive black holes are changed from those expected in general relativity.

The Effect of Meaning Making Instruction about Kepler's Laws at High School students’ Conceptual Understanding

It is determined that ideas that were encountered in students are not generally compatible with scientific knowledge and these ideas were named as alternative conceptions. A large number of conceptual change studies were done intended for the change of alternative concepts that were encountered in students towards scientific knowledge. But no study was encountered intended for revealing students’ alternative ideas related to Kepler's laws and for the conceptual change of these ideas in literature. The purpose of this study is to search the effects of education intended for creating 11th grade students’ ideas related to Kepler's laws. This study was conducted with twenty-five 11th grade students. With the aim of provide students creating meaning of the concept forty-five minutes teaching were made. As a means of collecting data on the Kepler's laws a test consisting of two open-ended questions were used. This concept test was administered before instruction, after instruction and 15 weeks later after instruction. In the analysis of data analysis six categories were used. It is observed that students generally have no scientific ideas related to these concepts and have alternative ideas that are not related to scientific ideas. More than half of the students gave the scientific response after taking instruction. Rate of encountering alternative answers decreased after the instruction. At the delayed post-test responses in the category of scientific response rate increased. However, at students a higher proportion of alternative conceptions encountered according to before instruction.

Bernoulli's Light Ray Solution of the Brachistochrone Problem Through Hamilton's Eyes

This article contains a review of the brachistochrone problem as initiated by Johann Bernoulli in 1696–1697. As is generally known, the cycloid forms the solutions to this problem. We follow Bernoulli's optical solution based on the Fermat principle of least time and later rephrase this in terms of Hamilton's 1828 paper. Deliberately an anachronistic style is maintained throughout. Hamilton's solution recovers the cycloid in a way that is reminiscent of how Newton's mathematical principles imply Kepler's laws.

General Keplerian Dynamics (GKD): A Testable Grand Unified Theory

Newton Generalized Kepler's Laws of Planetary Motion when he Developed his Laws of Universal Gravitation. Additional Generalizations are Submitted and an Auspicious Unified Model that can Be Tested Experimentally is Disclosed.

Towards a new generalized space expansion dynamics applied to the rotation of galaxies and Tully Fisher law

Up to now, the rotational velocities of galaxies are not clearly understood and the experimental Tully Fisher rule, linking the total galactic mass to the fourth power of the velocity, through an acceleration coefficient of about 10−10 m/s2 has not found a deep theoretical explanation. Tentative proposals (MOND theory of a modified Newton’s law and extraneous dark matter) do not bring a definite clarification. We propose here a new approach to this problem, without exotic matter and using the classical Newton force. But we introduce a new additional universal acceleration, which could represent a universal expansion law valid at the scale level of a galaxy. We show that this hypothesis leads to a good description of the observed variations of the galactic transverse velocity. It can be considered as a consequence of the Scale Expansion Cosmos theory (SEC) introduced by J. Masreliez, but we postulate that the space expansion acceleration universally applies at any scale. We obtain a formal derivation of the Tully Fisher law, linking the constant galactic transverse velocity to its total mass, via the universal minimum acceleration. We derive a good estimate of the TF acceleration coefficient and show that expansion should be proportional to the square root of the local volumic mass density. Our conjecture is in fact a new dynamics principle which could be applied to many other physical problems at different scales. Applying it to the range of the solar planet system confirms the well known Kepler laws, at least as a valid approximation for the order of magnitude of the solar system.

On trajectories of rolling marbles in cones and other funnels

We report on theoretical and experimental results for a ball that rolls without slipping on a surface of revolution, whose symmetry axis is aligned with a uniform gravitational field, particularly investigating both near-circular orbits and scattering-type orbits in cones. The experimental data give support for the theoretical treatment, a non-trivial application of Newton's second law that expands on our previous work and related work of others. Our findings refine those from a recent article in this journal, and largely replicate those obtained from an earlier Lagrangian approach, adding some new details and commentary. While the orbits of marbles rolling in cones do not match inverse-square-law orbits quantitatively (e.g., instead of Kepler's 3rd law, we have T2∝R), we argue that students should experience these qualitative phenomena—precession of orbits, escape velocity behavior, spin-orbit coupling, conservation laws for angular momentum, energy, and spin projection—as much for the fun and kinesthetic impressions as for the raw learning. We also report on a heretofore largely ignored variable in the exploration of rolling orbits in a gravity well: the marble's spin about its own axis as it rolls. Experimenters can, intentionally or not, vary this initial condition and produce different orbital periods for a given orbital radius—a distinctly non-celestial behavior. Careful selection of the initial spin direction and speed for a particular cone can result in marble orbits that mimic the planetary ellipses.

Correction to “Galilean moons, Kepler's third law, and the mass of Jupiter,” Phys. Teach. 51, 428 (Oct. 2013).

Correction to “Galilean moons, Kepler's third law, and the mass of Jupiter,” <i>Phys. Teach.</i> 51, 428 (Oct. 2013).

MOND laws of galactic dynamics

MOND predicts a number of laws that galactic systems should obey irrespective of their complicated, haphazard, and mostly unknowable histories -- as Kepler's laws are obeyed by planetary systems. The main purpose of this work is to show how, and to what extent, these MOND laws follow from only the paradigm's basic tenets: departure from standard dynamics at accelerations a<~a0, and space-time scale invariance in the limit a<<a0. Such predictions will be shared by all MOND theories that embody these premises. This is important because we do not know which of the existing MOND theories, if any, is a step in the right direction. In the Newtonian-dynamics-plus-dark-matter paradigm, the validity of such clear-cut laws -- which tightly constrain baryons, `dark matter', and their mutual relations -- is contrary to expectations.

Gravitomagnetic effects in conformal gravity

Gravitomagnetic effects are characterized by two phenomena: first, the geodetic effect which describes the precession of the spin of a gyroscope in a free orbit around a massive object, second, the Lense-Thirring effect which describes the precession of the orbital plane about a rotating source mass. We calculate both these effects in the fourth-order theory of conformal Weyl gravity for the test case of circular orbits. We show that for the geodetic effect a linear term arises which may be interesting for high radial orbits, whereas for the Lense-Thirring effect the additional term has a diminishing effect for most orbits. Circular orbits are also considered in general leading up to a generalization of Kepler's third law.

Galilean Moons, Kepler's Third Law, and the Mass of Jupiter

Simulations of physical systems are widely available online, with no cost, and are ready to be used in our classrooms.1,2 Such simulations offer an accessible tool that can be used for a range of interactive learning activities. The Jovian Moons Applet2 allows the user to track the position of Jupiter's four Galilean moons with a variety of viewing options. For this activity, data are obtained from the orbital period and orbital radii charts. Earlier experiments have used telescopes to capture the orbital motion of the Galilean moons,3 although observation of astronomical events and the measurement of quantities may be difficult to achieve due to a combination of cost, training, and observing conditions. The applet allows a suitable set of data to be generated and data analysis that verifies Kepler's third law of planetary motion, which leads to a calculated value for the mass of Jupiter.

Gravitational field of a slowly rotating black hole with a phantom global monopole

We present a slowly rotating black hole solution with a phantom global monopole by solving Einstein's field equation and find that the presence of the global monopole changes the black hole structure. The metric coefficient gtϕ contains the hypergeometric function of the polar coordinate r, which is more complex than that in the usual slowly rotating black hole. The energy scale of symmetry breaking η affects the black hole horizon and the deficit solid angle. In particular, the solid angle is surplus rather than deficit for the black hole with the phantom global monopole. We also study the effects of the global monopole on the angular velocity of the horizon ΩH, Kepler's third law, the innermost stable circular orbit and the radiative efficiency ϵ in the thin accretion disc model. Our results also show that for a phantom black hole, the radiative efficiency ϵ is positive only in the case η ⩽ ηc. The threshold value ηc increases with the rotation parameter a.

Calculating Earth–Moon system parameters from sub-yearly tidal deposit records: An example from the carboniferous tradewater formation

The secular evolution of the Earth–Moon system remains poorly constrained, largely because there are few continuous tidal rhythmite or foreset bundle sequences that preserve deposition over more than several neap–spring tidal cycles. Deposits recording less than one year of deposition do not facilitate direct calculation of past lunar distance directly from Kepler's Laws, but may prove necessary to reconstruct lunar orbital evolution because they are far more common than longer records. A method is demonstrated to make use of shorter tidal deposit sequences by utilizing conservation of angular momentum between the Earth and Moon and estimating the solar component of tidal deposition, while assuming a constant moment of inertia for Earth since the Proterozoic. The precision and accuracy of spectral estimates obtained from short records are considered, as are the limitations of subsequent calculations of Earth–Moon parameters.The Late Carboniferous Abbott sandstone of the Tradewater Formation in the Illinois Basin preserves just over 6 apparently continuous neap–spring cycles in its semidiurnal deposits. The quality of these data, as assessed via sedimentological evidence and statistical time series properties, produces spectral estimates that are likely within at least ±5% of the actual underlying periodicity (90% accuracy). To test the usefulness of such records, we assessed the possible scenarios of 90% and 95% accuracy. At 90% accuracy, the error bounds on Earth–Moon parameter estimates become rather large and render individual data sets to be of limited use. At 95% accuracy, very general inferences about the evolution of the Earth–Moon system may be made. Calculated mean lunar orbital distance at 315Ma is 3.798×108m with error bounds of −0.086×108m and +0.046×108m. We conclude that short sequences of cyclic tidal deposits offer rather limited resolution of lunar distance estimates. Utilizing multiple sub-yearly data of similar age in ensemble may prove necessary if further investigations of secular changes in Earth–Moon parameters are to proceed.

큐잉 데이터 기반 하층방어 요격체계의 초고속 표적 탐지 방향 지정을 위한 정밀 궤적예측 기법

A recent air defense missile system is required to have a capability to intercept short-range super-high speed targets such as tactical ballistic missile(TBMs) by performing engagement control efficiently. Since flight time and distance of TBM are very short, the missile defense system should be ready to engage a TBM as soon as it takes an indication of the TBM launch. As a result, it has to predict TBM trajectory accurately with cueing information received from an early warning system, and designate search direction and volume for own radar to detect/track TBM as fast as it can, and also generate necessary engagement information. In addition, it is needed to engage TBM accurately via transmitting tracked TBM position and velocity data to the corresponding intercept missiles. In this paper, we proposed a method to estimate TBM trajectory based on the Kepler's law for the missile system to detect and track TBM using the cueing information received before the TBM arrives the apogee of the ballistic trajectory, and analyzed the bias of prediction error in terms of the transmission period of cueing data between the missile system and the early warning system.

Did the Gravity Probe B Actually Prove the Einstein's General Relativity Theory?

It has been shown in an earlier publication (Hynecek, 2013) that the frame dragging effect due to the Earth's rotation, which is claimed that the Gravity Probe B (GP-B) has detected, cannot exist. In this paper it is shown that the Geodetic Precession, which the GP-B has also measured, and is typically calculated from the Einstein's General Relativity Theory (GRT), can also be calculated from a different metric. When the results of calculations are compared side by side the differences are very small, but it becomes clear that the GRT actually may not be providing the correct derivation of formula for this effect. The one reason among several others is due to the fact that the GRT derivation, as is shown in this paper, also leads to the derivation of the classical Kepler's third law for circular orbits, which is easily derived from the Newtonian physics, and should therefore hold true only in the flat space-time geometry.

Some learning objects to explain Kepler's laws

AbstractIn this paper, we present some learning objects for the study of Kepler's laws that graphically show the orbits and the movements of various planets. One of them shows the orbit of a planet from the point of view of a fixed planet, showing that the orbit is quite involved. No differentials equations are required, but only elementary vector calculus. The learning objects have been implemented in Matlab. © 2010 Wiley Periodicals, Inc. Comput Appl Eng Educ 21: 1–7, 2013

Visual Angular Momentum: Mamikon Meets Kepler

A new areal proof of Kepler's second law of planetary motion is presented, based on Mamikon's sweeping-tangent theorem.

Wide binaries as a critical test for Gravity theories

Assuming Newton's gravity and GR to be valid at all scales leads to the dark matter hypothesis as a requirement demanded by the observed dynamics and measured baryonic content at galactic and extragalactic scales. Alternatively, modified gravity scenarios where a change of regime appears at acceleration scales a < a0 have been proposed. This modified regime at a < a0 will generically be characterised by equilibrium velocities which become independent of distance. Here we identify a critical test in this debate and we propose its application to samples of wide binary stars. Since for 1M⊙ systems the acceleration drops below a0 at scales of around 7000 AU, a statistical survey of wide binaries with relative velocities and separations reaching 104 AU and beyond should prove useful to the above debate. We apply the proposed test to the best currently available data. Results show a constant upper limit to the relative velocities in wide binaries which is independent of separation for over three orders of magnitude, in analogy with galactic flat rotation curves in the same a < a0 acceleration regime. Our results are suggestive of a breakdown of Kepler's third law beyond a ≈ a0 scales, in accordance with generic predictions of modified gravity theories designed not to require any dark matter at galactic scales and beyond.

Spacetime symmetries and Kepler's third law

The curved spacetime geometry of a system of two point masses moving on a circular orbit has a helical symmetry. We show how Kepler’s third law for circular motion, and its generalization in post-Newtonian theory, can be recovered from a simple, covariant condition on the norm of the associated helical Killing vector field. This unusual derivation can be used to illustrate some concepts of prime importance in a general relativity course, including those of Killing field, covariance, coordinate dependence and gravitational redshift.

Slowly rotating black holes in dynamical Chern-Simons gravity: Deformation quadratic in the spin

We derive a stationary and axisymmetric black hole solution to quadratic order in the spin angular momentum. The previously found, linear-in-spin terms modify the odd-parity sector of the metric, while the new corrections appear in the even-parity sector. These corrections modify the quadrupole moment, as well as the (coordinate-dependent) location of the event horizon and the ergoregion. Although the linear-in-spin metric is of Petrov type D, the quadratic-order terms render it of type I. The metric does not possess a second-order Killing tensor or a Carter-like constant. The new metric does not possess closed timelike curves or spacetime regions that violate causality outside of the event horizon. The new, even-parity modifications to the Kerr metric decay less rapidly at spatial infinity than the leading order in spin, odd-parity ones, and thus, the former are more important when considering black holes that are rotating moderately fast. We calculate the modifications to the Hamiltonian, binding energy and Kepler's third law. These modifications are crucial for the construction of gravitational wave templates for black hole binaries, which will enter at second post-Newtonian order, just like dissipative modifications found previously.

Simple derivations of Kepler's first law: use of complex variables

By making use of complex variables, simple derivations of Kepler's first law are introduced. Such derivations might be of interest for undergraduate students in an introductory physics course.