Galactic Dynamics

In this paper we study scalar perturbations of the metric for nonlinear f(R) models. We consider the Universe at the late stage of its evolution and deep inside the cell of uniformity. We investigate the astrophysical approach in the case of Minkowski spacetime background and two cases in the cosmological approach, the large scalaron mass approximation and the quasi-static approximation, getting explicit expressions for scalar perturbations for both these cases. In the most interesting quasi-static approximation, the scalar perturbation functions depend on both the nonlinearity function f(R) and the scale factor a. Hence, we can study the dynamical behavior of the inhomogeneities (e.g., galaxies and dwarf galaxies) including into consideration their gravitational attraction and the cosmological expansion, and also taking into account the effects of nonlinearity. Our investigation is valid for functions f(R) which have stable de Sitter points in future with respect to the present tim...

This paper is nothing short of a positive and complete verification of the unification of current physics paradigms of single field theory (SOFT) in light of recently published observational data in astronomy. An article by Sabine Hossenfelder and Stacey S. McGaugh, titled " Is Dark Matter Real?', has just appeared in the August 2018 issue of Scientific American. Every physicist, astronomer, astrophysicist and cosmologist as well as everyone interested in science needs to read this article because it portends one of the greatest scientific advances in several hundred years, the complete unification of physics and science in general. Due to new observational evidence in astronomy, they have concluded that " Astrophysicists have piled up observations that are difficult to explain with Dark Matter. It is time to consider that there may be more to gravity than Einstein taught us. " But they have understated the case. In effect, the new observational data found by McGaugh and an international group of astronomers clearly shows that specifically designated dark matter particles do not exist, so the supposed Dark Matter halos around spiral galaxies do not exist and the rotational speed discrepancies in galaxy orbiting stars and star systems that go unaccounted for by normal gravity are, in fact, due to a secondary effect of normal matter within the galaxy. This finding not only challenges the present theories of gravity (Newtonian and Einsteinian), which it demonstrates are incomplete, it also blows some big logical holes in the fundamentality of the quantum theory, the Standard Model of point-particles and quantum field theories in general that seem to rule the world of modern theoretical physics. Given other recent confirmations of Einstein's general relativity, found mostly in the detection of gravitational waves that Einstein predicted a century ago, relativity theory seems on the ascendancy and quantum theory is falling behind. The time for a unified field theory that combines the best of both paradigms is at hand.

Recent astrophysical observations hint toward the need for an extended theory of gravity to explain puzzles  presented by the standard cosmological model such as the need for dark matter and dark energy to understand the dynamics of the cosmos. This paper investigates the effect of a repulsive central universal force field on the behavior of celestial objects. Negative tidal effect on the solar and galactic orbits, like that experienced by Pioneer spacecrafts, was derived from the central force and was shown to manifest itself as dark matter and dark energy. Vertical oscillation of the sun about the galactic plane was modeled as simple harmonic motion driven by the repulsive force. The proposed universal field was used to infer the shape of dark matter halos as generated from a planar component of the universal force and to explain galactic warp, galactic halo density, and galactic rotation curves. It was found that the repulsive field addition to Newton’s gravity mimics the Yukawa potential correction employed by many current gravitational theories that modify gravity.

In this paper we study scalar perturbations of the metric for nonlinear $f(R)$ models. We consider the Universe at the late stage of its evolution and deep inside the cell of uniformity. We investigate the astrophysical approach in the case of Minkowski spacetime background and two cases in the cosmological approach, the large scalaron mass approximation and the quasi-static approximation, getting explicit expressions for scalar perturbations for both these cases. In the most interesting quasi-static approximation, the scalar perturbation functions depend on both the nonlinearity function $f(R)$ and the scale factor $a$. Hence, we can study the dynamical behavior of the inhomogeneities (e.g., galaxies and dwarf galaxies) including into consideration their gravitational attraction and the cosmological expansion, and also taking into account the effects of nonlinearity. Our investigation is valid for functions $f(R)$ which have stable de Sitter points in future with respect to the present time, that is typical for the most popular $f(R)$ models.

1I/’Oumuamua (or 1I) and 2I/Borisov (or 2I), the first InterStellar Objects (ISOs) discovered passing through the solar system, have opened up entirely new areas of exobody research. Finding additional ISOs and planning missions to intercept or rendezvous with these bodies will greatly benefit from knowledge of their likely orbits and arrival rates. Here, we use the local velocity distribution of stars from the Gaia Early Data Release 3 Catalogue of Nearby Stars and a standard gravitational focusing model to predict the velocity dependent flux of ISOs entering the solar system. With an 1I-type ISO number density of ∼0.1 AU, we predict that a total of ∼6.9 such objects per year should pass within 1 AU of the Sun. There will be a fairly large high-velocity tail to this flux, with half of the incoming ISOs predicted to have a velocity at infinity, v∞, > 40 km s . Our model predicts that ∼92% of incoming ISOs will be residents of the galactic thin disk, ∼6% (∼4 per decade) will be fr...

We investigate the phase-space structure of perturbed resonant harmonic oscillators with the method of Lie Transform Normalization. The perturbation is a quartic polynomial which can be used as a model for the central part of an elliptical galaxy and is analyzed computing approximate integrals of motion in the form of truncated series expansions. We compute surfaces of sections and compare the results with numerical integrations verifying that a global agreement is achieved at the first order which incorporates the resonance.

We present a new mechanism to explain the frequently observed and thus certainly permanent warping of spiral galaxies. We consider the possibility of non-linear coupling between the spiral wave of the galaxy and two warp waves, such that the former, which is linearly unstable and extracts energy and angular momentum from the inner regions of the galactic disk, can continuously feed the latter. We derive an expression for the coupling coefficient in the WKB approximation. We show that the coupling is too weak in the stellar disk, except at the Outer Lindblad Resonance where the spiral slows down and is efficiently coupled to warp waves. There, the spiral can be almost completely converted into " transmitted " warps, which we can observe in HI, and a " reflected " one, which we can observe as a corrugation. Our mechanism reproduces the observed amplitudes of the warp and of the corrugation, and might explain related phenomena such as the behavior of the line of nodes of the warp. Furthermore we show that the energy and momentum fluxes of observed spirals and warps are of the same order of magnitude, adding a strong point in favor of this model.

1I/'Oumuamua (or 1I) and 2I/Borisov (or 2I), the first InterStellar Objects (ISOs) discovered passing through the solar system, have opened up entirely new areas of exobody research. Finding additional ISOs and planning missions to intercept or rendezvous with these bodies will greatly benefit from knowledge of their likely orbits and arrival rates. Here, we use the local velocity distribution of stars from the Gaia Early Data Release 3 Catalogue of Nearby Stars and a standard gravitational focusing model to predict the velocity dependent flux of ISOs entering the solar system. With an 1I-type ISO number density of ∼0.1 AU −3 , we predict that a total of ∼6.9 such objects per year should pass within 1 AU of the Sun. There will be a fairly large high-velocity tail to this flux, with half of the incoming ISOs predicted to have a velocity at infinity, v ∞ , > 40 km s −1. Our model predicts that ∼92% of incoming ISOs will be residents of the galactic thin disk, ∼6% (∼4 per decade) will be from the thick disk, ∼1 per decade will be from the halo and at most ∼3 per century will be unbound objects, ejected from our galaxy or entering the Milky Way from another galaxy. The rate of ISOs with very low v ∞ 1.5 km s −1 is so low in our model that any incoming very low velocity ISOs are likely to be previously lost solar system objects. Finally, we estimate a cometary ISO number density of ∼7 × 10 −5 AU −3 for 2I type ISOs, leading to discovery rates for these objects possibly approaching once per decade with future telescopic surveys.

Scalar cosmological perturbations are investigated in the framework of a model with interacting dark energy and dark matter. In addition to these constituents, the inhomogeneous Universe is supposed to be filled with the standard noninteracting constituents corresponding to the conventional $\Lambda$CDM model. The interaction term is chosen in the form of a linear combination of dark sector energy densities with evolving coefficients. The methods of discrete cosmology are applied, and strong theoretical constraints on the parameters of the model are derived. A brief comparison with observational data is performed.

I have previously presented a geometrical explanation of Dark Matter and Dark Energy that is falsifiable, but the model was criticized for being non-mathematical. In this structure, the four-dimensional space-time of relativity is extrinsically curved in a higher spatial dimension. Dark Matter is curvature in the higher dimension that is not directly associated with local matter, but instead results from an interaction between the curvature of local matter and the global curvature due to all matter in the universe. The earlier criticism has now been overcome by looking at the possible particulate source of galactic halos and a simple yet revolutionary algebraic formula has been derived. The algebraic formula implies a five-dimensional unified field structure such as that developed by Kaluza in 1921 and extended by Einstein and his colleagues in the late 1930s. Others, such a H.T. Flint, have developed
more complete mathematical models of similar five-dimensional structures. The new equation also shows how gravity can be quantized on the basis of relativity without hypothesizing the discrete nature of matter, i.e., the existence of specific ‘particles’ of gravity, inherent in quantum mechanics, the Standard Model and other quantum models.

In accordance with long-established physical principles, and leaving aside considerations such as dark energy and the expansion of space, the galaxies could revolve in orbit around their common center of mass. This paper explores that hypothesis. We provide an estimate of the rate of rotation that would be consistent with a closed circular orbit, while recognizing that a lesser orbital velocity could lead to inward spiraling, and a greater velocity to outward spiraling. This paper also provides estimates of the maximum expanse of the population of galaxies, and of the potentially observable rate of their transverse movement. The precision of data as obtained by the Gaia spacecraft may be sufficient to provide evidence of galactic rotation.

In this paper we consider the Universe at the late stage of its evolution and deep inside the cell of uniformity. At these scales, the Universe is filled with inhomogeneously distributed discrete structures (galaxies, groups and clusters of galaxies). Supposing that a small fraction of colored objects escaped hadronization and survived up to now in the form of quark-gluon nuggets (QNs), and also taking into account radiation, we investigate scalar perturbations of the FRW metrics due to inhomogeneities of dustlike matter as well as fluctuations of QNs and radiation. In particular, we demonstrate that the nonrelativistic gravitational potential is defined by the distribution of inhomogeneities/fluctuations of both dustlike matter and QNs. Consequently, QNs can be distributed around the baryonic inhomogeneities (e.g., galaxies) in such a way that it can solve the problem of the flatness of the rotation curves. We also show that the fluctuations of radiation are caused by both the inhomogeneities in the form of galaxies and the fluctuations of quark-gluon nuggets. Therefore, if QNs exist, the CMB anisotropy should contain also the contributions from QNs. Additionally, the spatial distribution of the radiation fluctuations is defined by the gravitational potential. All these results look physically reasonable.

third video together. Yosan talks about us reaching what he calls level one, which is a level of consciousness that recognizes our interconnection with all of life. Of course we have a way to go since there is so much programming that seems to hinder us from reaching higher levels of awareness. One of the problems he points out is our limited concept of time. Are we really immortal? Can we take our bodies into other dimensions? Hopefully this will help us gain a new perspective and increased self-awareness. " Yosan: after many years of serving for the awakening mostly in person to person " as well using art works and publishing stuff, now i can not do the educative style, however a free talk is possible. in here i am trying to empower the notion about human ability to handle our impressions of the reality rather than waiting on it to change. saying that the most affective way we have on our reality is when we can alter our perceptions " including about our feeling and health. still capable and wanting to add to this discussion. thanks for watching. said .

We analyze the effects of flattening on the annihilation (J) and decay (D) factors of dwarf spheroidal galaxies with both analytic and numerical methods. Flattening has two consequences: first, there is a geometric effect as the squeezing (or stretching) of the dark matter distribution enhances (or diminishes) the J-factor; second, the line of sight velocity dispersion of stars must hold up the flattened baryonic component in the flattened dark matter halo. We provide analytic for-mulae and a simple numerical approach to estimate the correction to the J-and D-factors required over simple spherical modeling. The formulae are validated with a series of equilibrium models of flattened stellar distributions embedded in flattened dark-matter distributions. We compute corrections to the J-and D-factors for the Milky Way dwarf spheroidal galaxies under the assumption that they are prolate or oblate and find that the hierarchy of J-factors for the dwarf spheroidals is slightly altered. We demonstrate that spherical estimates of the D-factors are very insensitive to the flattening and introduce uncertainties significantly less than the uncertainties in the D-factors from the other observables for all the dwarf spheroidals. We conclude by investigating the spread in correction factors produced by triaxial figures and provide uncertainties in the J-factors for the dwarf spheroidals using different physically-motivated assumptions for their intrinsic shape and axis alignments. We find that the uncertainty in the J-factors due to triaxiality increases with the observed ellipticity and, in general, introduces uncertainties of order 25 per cent in the J-factors. We discuss our results in light of the reported gamma-ray signal from the highly-flattened ultrafaint Reticulum II.

We explore the possibility of detecting and characterizing the warp of the stellar disc of our Galaxy using synthetic Gaia data. The availability of proper motions and, for the brightest stars radial velocities, adds a new dimension to this study. A family of Great Circle Cell Counts methods is used. They are ideally suited to find the tilt and twist of a collection of rings, which allow us to detect and measure the warp parameters. To test them, we use random realizations of test particles which evolve in a realistic Galactic potential warped adiabatically to various final configurations. In some cases a twist is introduced additionally. The Gaia selection function, its errors model and a realistic 3D extinction map are applied to mimic three tracer populations: OB, A and red clump stars. We show how the use of kinematics improves the accuracy in the recovery of the warp parameters. The OB stars are demonstrated to be the best tracers determining the tilt angle with accuracy better than ∼0.5 up to Galactocentric distance of ∼16 kpc. Using data with good astrometric quality, the same accuracy is obtained for A-type stars up to ∼13 kpc and for red clump up to the expected stellar cutoff. Using OB stars the twist angle is recovered to within <3 • for all distances.

The density wave theory for the grand-design two-armed spiral pattern in galaxies is successful in explaining several observed features. However, the long-term persistence
of this spiral structure is a serious problem since the group transport would destroy it within about a billion years as shown in a classic paper by Toomre. In this paper we include the low velocity dispersion component, namely gas, on an equal footing with stars in the formulation of the density wave theory, and obtain the dispersion relation for this coupled system. We show that the inclusion of gas makes the group transport slower by a factor of few, thus allowing the pattern to persist longer - for several billion years. Though still less than the Hubble time, this helps in making the
spiral structure more long-lived. Further we show that addition of gas is essential to get a stable wave for the observed pattern speed for the Galaxy, which otherwise is
not possible for a one-component stellar disc.

We investigate the dynamics in the logarithmic galactic potential with an analytical approach. The phase-space structure of the real system is approximated with resonant detuned normal forms constructed with the method based on the Lie transform. Attention is focused on the properties of the axial periodic orbits and of low order `boxlets' that play an important role in galactic models. Using energy and ellipticity as parameters, we find analytical expressions of several useful indicators, such as stability-instability thresholds, bifurcations and phase-space fractions of some orbit families and compare them with numerical results available in the literature.

Dark Matter and Dark Energy are both regarded as anomalies that must be solved by science. Each anomaly is being independently studied while numerous solutions are being suggested for each independent of the other. However, everyone either accepts or assumes that both anomalies must eventually conform to a single common solution. With a little imagination that single solution is not that hard to find. By assuming a real macroscopically extended fifth dimension, characterized by the extrinsic curvature of the four-
dimensional portion of that space-time continuum, both DM and DE can be explained in a single model. CDM halos are a simple product of galactic formation and the DE that acts as a negative pressure to increase the expansion rate of the universe is a product of the maturity and old age of spiral galaxies. The extrinsic curvature of four-dimensional space-time, which is completely compatible with both general relativity and Kaluza’s unification of general relativity and electromagnetism, can be identified directly with all forms of DM and DE. This model thus offers a starting point for the unification of relativity and the quantum.

The use of open cluster ages is introduced as a tool to investigate the Star Formation Rate (SFR) of the galactic disk in the solar neighborhood. The measurement of the SFR so obtained is shown to be in good agreement with that derived from chromospheric ages: a burst of star formation is clearly seen within the last 4X10^8 years and a second peak at 6X10^9 years. The use of open clusters gives a much firmer statistical basis from which to explore any fine structure in the SFR, and an examination of the age histogram obtained using data for 355 open clusters from the Lund Catalog of Open Cluster Data reveals a striking periodicity in the SFR of the Galactic disk on the order of 101 years. Analysis of the measured frequencies reveals the presence of two distinct modes of star formation in the disk, with a transition between the two modes clearly seen when the Magellanic cloud made its nearest approach ~4X10^8 years ago. The most recent five measured frequencies of star formation are fit with the curve nu =(2.11X10^-8) X exp(-1.67X10^-9 X Age(y)) to within the measured errors.

An examination of the projection on the galactic plane of the location of the open clusters formed during each of the recent peaks of star formation finds clear evidence of physical correlation between the clusters formed during each starburst; further investigation suggests that the clusters were formed as the result of a spiral density wave traveling with the measured pattern speed Omega_sub_p = 32.8-48.4 km/s, in good agreement with theoretical expectations of a density wave in the Galactic disk. A detailed atlas of open cluster age histograms by galactic sector is included, and a cursory analysis indicates that observed density wavefront development in the Galactic disk is primarily radial in nature.

As an appendix, new CCD photometries in the BVRI system are presented of the largely unstudied open clusters NGC 1857 and NGC 2355, located in the Galactic anti-center.

As an appendix, new CCD photometries in the BVRI system are presented of the largely unstudied open clusters NGC 1857 and NGC 2355, located in the Galactic anti-center.

We consider lattice Universes with spatial topologies $T\times T\times T$, $\; T\times T\times R\; $ and $\; T\times R\times R$. In the Newtonian limit of General Relativity, we solve the Poisson equation for the gravitational potential in the enumerated models. In the case of point-like massive sources in the $T\times T\times T$ model, we demonstrate that the gravitational potential has no definite values on the straight lines joining identical masses in neighboring cells, i.e. at points where masses are absent. Clearly, this is a nonphysical result since the dynamics of cosmic bodies is not determined in such a case. The only way to avoid this problem and get a regular solution at any point of the cell is the smearing of these masses over some region. Therefore, the smearing of gravitating bodies in $N$-body simulations is not only a technical method but also a physically substantiated procedure. In the cases of $\; T\times T\times R\; $ and $\; T\times R\times R$ topologies, there is no way to get any physically reasonable and nontrivial solution. The only solutions we can get here are the ones which reduce these topologies to the $T\times T\times T$ one.

J-factors (or D-factors) describe the distribution of dark matter in an astrophysical system and determine the strength of the signal provided by annihilating (or decaying) dark matter respectively. We provide simple analytic formulae to calculate the J-factors for spherical cusps obeying the em- pirical relationship between enclosed mass, velocity dispersion and half-light radius. We extend the calculation to the spherical Navarro-Frenk-White (NFW) model, and demonstrate that our new formulae give accurate results in comparison to more elaborate Jeans models driven by Markov Chain Monte Carlo methods. Of the known ultrafaint dwarf spheroidals, we show that Ursa Major II, Reticulum II, Tucana II and Horologium I have the largest J-factors and so provide the most promising candidates for indirect dark matter detection experiments. Amongst the classical dwarfs, Draco, Sculptor and Ursa Minor have the highest J-factors. We show that the behaviour of the J- factor as a function of integration angle can be inferred for general dark halo models with inner slope γ and outer slope β. The central and asymptotic behaviour of the J-factor curves are derived as a function of the dark halo properties. Finally, we show that models obeying the empirical relation on enclosed mass and velocity dispersion have J-factors that are most robust at the integration angle equal to the projected half-light radius of the dSph divided by heliocentric distance. For most of our results, we give the extension to the D-factor which is appropriate for the decaying dark matter picture.

The phase-space structure of two families of galactic potentials is approximated with a resonant detuned normal form. The normal form series is obtained by a Lie transform of the series expansion around the minimum of the original Hamiltonian. Attention is focused on the quantitative predictive ability of the normal form. We find analytical expressions for bifurcations of periodic orbits and compare them with other analytical approaches and with numerical results. The predictions are quite reliable even outside the convergence radius of the perturbation and we analyze this result using resummation techniques of asymptotic series.

Low surface brightness (LSB) galaxies form a major class of galaxies, and are characterized by low disc surface density and low star formation rate. These are known to be dominated by dark matter halo from the innermost regions. Here we study the role of dark matter halo on the grand-design, m=2, spiral modes in a galactic disc by carrying out a global mode analysis in the WKB approximation. The Bohr-Sommerfeld quantization rule is used to determine how many discrete global spiral modes are permitted. First a typical superthin LSB galaxy, UGC 7321 is studied by taking only the galactic disc, modelled as fluid; and then the disc embedded in a dark matter halo. We find that both cases permit the existence of global spiral modes. This is in contrast to earlier results where the inclusion of dark matter halo was shown to nearly fully suppress local, swing-amplified spiral features. Although technically global modes are permitted in the fluid model as shown here, we argue that due to lac...

I present an alternative explanation of flat rotational curves of galaxies that does not require dark matter but rather relies on classical Newtonian dynamics and an overlooked effect of quantum tunneling. I introduce a rotational drag force, which arises from subatomic particle tunneling through potential barriers caused by energy fluctuations in rotating gravitational field, and present a model for a galactic rotational curve based on Newtonian dynamics, rotational drag force and galactic mass-density distribution. To support the rotational drag force hypothesis I supply the results of analytical modeling and numerical simulation of effects of the rotational drag force on stellar orbits and galactic morphology. The obtained results provide a clear indication that the rotational drag force can be successfully applied to explaining a wide variety of observational phenomena ranging from flat rotational curves of galaxies and Tully- Fisher relation to origination of spiral galactic ar...

We find the generalized version of the Toomre's criterion for the stability of a rotating thin disk in the context of Eddington inspired Born-Infeld (EiBI) gravity which possesses one free parameter χ. To do so we use the weak field limit of the theory and find the dispersion relation for the propagation of matter density waves on the surface of a self-gravitating and differentially rotating disk. Finally we find a new version of Toomre's stability criterion for thin disks. We show that EiBI gravity with negative χ destabilizes all the rotating thin disks. On the other hand EiBI with positive χ substantially can suppress the local fragmentation, and has stabilizing effects against axi-symmetric perturbations. More specifically, we show that only an annulus remains unstable on the surface of the disk. The width of the annulus directly depends on the magnitude of χ.

We use density and temperature profiles obtained from XMM-Newton observations to derive a potential of NGC 5846 out to 11Re, thus probing the mass distribution deep into the halo. The inferred circular velocity is significantly higher than the extrapolation of dynamical models implying a halo, more massive than previously thought. Using an I -band surfacebrightness profile and a projected velocity dispersion profile consisting of long-slit kinematic measurements and planetary nebulae (PNe) velocity dispersions, we solve the Jeans equations, assuming a non-rotating spherical system. The solutions suggest a highly radially anisotropic galaxy outside 0.7Re with β ∼ 0.75. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

We show that the propagation of warps in gaseous disks can be strongly affected by compressional effects, when the thickness of the disk is taken into account. The physical reason is that, in realistic self-gravitating disks, the sound time through the disk is comparable with the rotation time; thus the vertical hydrostatic equilibrium cannot be maintained adiabatically as the wave propagates (an implicit hypothesis in the thin-disk approximation) and the disk cannot move up and down solidly to follow the warp perturbation. There results, together with the main vertical motion, a strong horizontal one which significantly modifies the dispersion relation.
We then turn to the case of a disk composed of two fluids with different temperatures: this can correspond either to the combined motions of the gas and the stars in a galactic disk, or to their coupling with a flattened massive halo (assuming that, as for spiral waves, stars are conveniently represented by a fluid if one stays away from Lindblad resonances). We find, in addition to the usual warps, a short-wavelength wave which might explain the "corrugation" observed in many galactic disks.
Finally, as a side result of this analysis, we discuss a possible weak amplification of m>1 warps.

We present the results of two-dimensional numerical simulations of stellar galactic disks, aimed at studying the non-linear coupling between bar and spiral waves and modes, in disks with realistically peaked rotation profiles. The power spectrum analysis of the perturbed density in the disk, for az-imuthal numbers ranging from m =0t om= 4, shows an unambiguous signature of non-linear coupling between the bar and spiral waves, or between spiral waves only, with a very sharp selection of the frequencies which optimize the coupling efficiency. It turns out that non-linear coupling can be quite efficient , and even more relevant than the Swing mechanism to account for the dynamics of the galaxy beyond the corotation of the bar. Non-linear coupling is also responsible for a number of other behaviors observed in our runs, such as harmonic or sub-harmonic excitation, and the excitation of m = 1 spiral waves.

Analytic methods to investigate periodic orbits in galactic potentials. To evaluate the quality of the approximation of periodic orbits in the logarithmic potential constructed using perturbation theory based on Hamiltonian normal forms. The solutions of the equations of motion corresponding to periodic orbits are obtained as series expansions computed by inverting the normalizing canonical transformation. To improve the convergence of the series a resummation based on a continued fraction may be performed. This method is analogous to that looking for approximate rational solutions (Prendergast method). It is shown that with a normal form truncated at the lowest order incorporating the relevant resonance it is possible to construct quite accurate solutions both for normal modes and periodic orbits in general position.

In the past few decades two new ‘crises’ for fundamental physics have emerged by the observation of phenomena that indicate the existence of Dark Matter and Dark Energy. These are not problems which can be solved by quantum theory, but rather problems that are related to gravity theory as expressed by the general theory of relativity. Numerous ideas and hypotheses have been suggested to explain these problems, but no particular hypothesis or resulting model has yet proven satisfactory. No model proposed within the framework of the present paradigms seems to be able to explain either DM or DE even though most physicists agree that the two should have a single common explanation. However, a new model has been developed to explain both DM and DE by reinterpreting and unifying the present paradigms. This model includes a fundamental change in Newtonian gravity theory that expands three-dimensional space to four dimensions and thus forces the acceptance of an extrinsically curved four-dimensional space-time in relativity. The extra gravity term can then be equated to the Lambda-CDM that has already been added to Einstein’s equation describing the surface curvature of the universe and leads to a unification with the quantum. Yet this new model is not without consequences for the rest of physics and science. Accepting this new model would mean accepting the reality of a macroscopically extended fourth space-like dimension.