Golden Elliptical Orbits in Newtonian Gravitation

In spherical symmetry with radial coordinate $r$, classical Newtonian gravitation supports circular orbits and, for $-1/r$ and $r^2$ potentials only, closed elliptical orbits [1]. Various families of elliptical orbits can be thought of as arising from the action of perturbations on corresponding circular orbits. We show that one elliptical orbit in each family is singled out because its focal length is equal to the radius of the corresponding unperturbed circular orbit. The eccentricity of this special orbit is related to the famous irrational number known as the golden ratio. So inanimate Newtonian gravitation appears to exhibit (but not prefer) the golden ratio which has been previously identified mostly in settings within the animate world.Comment: Submitted to Forum Geometricorum, 2 figures for the two golden ellipses of Newtonian dynamics. Based on the results of arXiv:1705.0935

The Two Incenters of the Arbitrary Convex Quadrilateral

For an arbitrary convex quadrilateral $ABCD$ with area ${\cal A}$ and perimeter $p$, we define two points $I_1, I_2$ on its Newton line that serve as incenters. These points are the centers of two circles with radii $r_1, r_2$ that are tangent to opposite sides of $ABCD$. We then prove that ${\cal A}=pr/2$, where $r$ is the harmonic mean of $r_1$ and $r_2$. We also investigate the special cases with $I_1\equiv I_2$ and/or $r_1=r_2$.Comment: Published on Forum Geometricoru

Gravitational Potential and Nonrelativistic Lagrangian in Modified Gravity with Varying G

We have recently shown that the baryonic Tully-Fisher (BTF) and Faber-Jackson (BFJ) relations imply that the gravitational "constant" $G$ in the force law vary with acceleration $a$ as $1/a$. Here we derive the converse from first principles. First we obtain the gravitational potential for all accelerations and we formulate the Lagrangian for the central-force problem. Then action minimization implies the BTF/BFJ relations in the deep MOND limit as well as weak-field Weyl gravity in the Newtonian limit. The results show how we can properly formulate a nonrelativistic conformal theory of modified dynamics that reduces to MOND in its low-acceleration limit and to Weyl gravity in the opposite limit. An unavoidable conclusion is that $a_0$, the transitional acceleration in modified dynamics, does not have a cosmological origin and it may not even be constant among galaxies and galaxy clusters.Comment: To appear in MNRAS

Theoretical model of HD 163296 presently forming in-situ planets and comparison with the models of AS 209, HL Tau, and TW Hya

We fit an isothermal oscillatory density model to the disk of HD 163296 in which planets have presumably already formed and they are orbiting at least within the four observed dark gaps. This 156 AU large axisymmetric disk shows various physical properties comparable to those of AS 209, HL Tau, and TW Hya that we have modeled previously; but it compares best to AS 209. The disks of HD 163296 and AS 209 are comparable in size and they share similar values of the power-law index $k\approx 0$ (a radial density profile of the form $\rho(R)\propto R^{-1}$), the rotational parameter $\beta_0$ (to within a factor of 3); a relatively small inner core radius (although this parameter for HD 163296 is exceptionally small, $R_1\simeq 0.15$ AU, presumably due to unresolved planets in the inner 50 AU); the scale length $R_0$ and the Jeans gravitational frequency $\Omega_J$ (to within factors of 1.4); the equation of state ($c_0^2/\rho_0$) and the central density $\rho_0$ (to within factors of 2); and the core angular velocity $\Omega_0$ (to within a factor of 4.5). In the end, we compare all six nebular disks that we have modeled so far.Comment: Part 10 and the final part (HD 163296

Conundrums and constraints concerning the formation of our solar system -- An alternative view

We have proposed an alternative model for the formation of our solar system that does not predict any mean-motion resonant interactions, planetary migrations, or self-gravitating instabilities in the very early isothermal solar nebula and before the protosun has formed. Within this context of nonviolent protoplanetary evolution over more than 10 million years, we examine some conundrums and constraints that have been discovered from studies of small bodies in the present-day solar system (Jupiter and Neptune's Trojans and their differences from Kuiper belt objects, the irregular satellites of gaseous giants, the stability of the main asteroid belt, and the Late Heavy Bombardment). These issues that have caused substantial difficulties to models of violent formation do not appear to be problematic for the alternative model, and the reason is the complete lack of violent events during the evolution of protoplanets.Comment: Updated version, Part 2 (Small Bodies in the Solar System

On the formation of our solar system and many other protoplanetary systems observed by ALMA and SPHERE

In view of the many recent observations conducted by ALMA and SPHERE, it is becoming clear that protoplanetary disks form planets in narrow annular gaps at various distances from the central protostars before these protostars are actually fully formed and the gaseous disks have concluded their accretion/dispersal processes. This is in marked contrast to the many multi-planet exoplanetary systems that do not conform to this pristine picture. This major discrepancy calls for an explanation. We provide such an explanation in this work, based on analytical solutions of the cylindrical isothermal Lane-Emden equation with rotation which do not depend on boundary conditions. These ``intrinsic'' solutions of the differential equation attract the solutions of the Cauchy problem and force them to oscillate permanently. The oscillations create density maxima in which dust and planetesimals are trapped and they can form protoplanetary cores during the very early isothermal evolution of such protoplanetary nebulae. We apply this model to our solar nebula that formed in-situ a minimum of eleven protoplanetary cores that have grown to planets which have survived undisturbed to the present day. We are also in the process of applying the same model to the ALMA/DSHARP disks.Comment: Updated version, Part 1 (Solar Nebula). arXiv admin note: text overlap with arXiv:0706.320

Models of Saturn's protoplanetary disk forming in-situ its regular satellites and innermost rings before the planet is formed

We fit an isothermal oscillatory density model of Saturn's protoplanetary disk to the present-day major satellites and innermost rings D/C and we determine the radial scale length of the disk, the equation of state and the central density of the primordial gas, and the rotational state of the Saturnian nebula. This disk does not look like the Jovian and Uranian disks that we modeled previously. Its power-law index is extremely steep ($k=-4.5$) and its radial extent is very narrow ($\Delta R\lesssim 0.9$ Gm), its rotation parameter that measures centrifugal support against self-gravity is somewhat larger ($\beta_0=0.0431$), as is its radial scale length (395 km); but, as was expected, the size of the Saturnian disk, $R_{\rm max}=3.6$ Gm, takes just an intermediate value. On the other hand, the central density of the compact Saturnian core and its angular velocity are both comparable to that of Jupiter's core (density of $\approx 0.3$~g~cm$^{-3}$ in both cases, and rotation period of 5.0 d versus 6.8 d); and significantly less than the corresponding parameters of Uranus' core. As with the other primordial nebulae, this rotation is sufficiently slow to guarantee the disk's long-term stability against self-gravity induced instabilities for millions of years of evolution.Comment: Updated version, Part 5 (Saturn). arXiv admin note: substantial text overlap with arXiv:1901.06448; text overlap with arXiv:1901.0513

Theoretical model of the outer disk of TW Hya presently forming in-situ planets and comparison with models of AS 209 and HL Tau

We fit an isothermal oscillatory density model to the outer disk of TW Hya in which planets have presumably already formed and they are orbiting within four observed dark gaps. At first sight, this 52 AU small disk does not appear to be similar to our solar nebula; it shows several physical properties comparable to those in HL Tau (size $R_{\rm max}=102$ AU) and very few similarities to AS 209 ($R_{\rm max}=144$ AU). We find a power-law density profile with index $k=-0.2$ (radial densities $\rho(R) \propto R^{-1.2}$) and centrifugal support against self-gravity so small that it virtually guarantees dynamical stability for millions of years of evolution to come. Compared to HL Tau, the scale length $R_0$ and the core size $R_1$ of TW Hya are smaller only by factors of $\sim$2, reflecting the disk's half size. On the opposite end, the Jeans frequency $\Omega_J$ and the angular velocity $\Omega_0$ of the smaller core of TW Hya are larger only by factors of $\sim$2. The only striking difference is that the central density ($\rho_0$) of TW Hya is 5.7 times larger than that of HL Tau, which is understood because the core of TW Hya is only half the size ($R_1$) of HL Tau and about twice as heavy ($\Omega_J$). In the end, we compare the protostellar disks that we have modeled so far.Comment: Updated version, Part 9 (TW Hya). arXiv admin note: text overlap with arXiv:1901.1064

Exact Solutions of the Isothermal Lane--Emden Equation with Rotation and Implications for the Formation of Planets and Satellites

We have derived exact solutions of the isothermal Lane--Emden equation with and without rotation in a cylindrical geometry. The corresponding hydrostatic equilibria are most relevant to the dynamics of the protosolar nebula before and during the stages of planet and satellite formation. The nonrotating solution for the mass density is analytic, nonsingular, monotonically decreasing with radius, and it satisfies easily the usual physical boundary conditions at the center. When differential rotation is added to the Lane--Emden equation, an entire class of exact solutions for the mass density appears. We have determined all of these solutions analytically as well. Within this class, solutions that are power laws or combinations of power laws are not capable of satisfying the associated boundary--value problem, but they are nonetheless of profound importance because they constitute "baselines" to which the actual solutions approach when the central boundary conditions are imposed. Numerical integrations that enforce such physical boundary conditions show that the actual radial equilibrium density profiles emerge from the center close to the nonrotating solution, but once they cross below the corresponding baselines, they cease to be monotonic. The actual solutions are forced to oscillate permanently about the baseline solutions without ever settling onto them because the central boundary conditions strictly prohibit such settling, even in the asymptotic regime of large radii. Based on our results, we expect that quasistatically--evolving protoplanetary disks should develop oscillatory density profiles in their midplanes during their isothermal phase. The peaks in these profiles correspond to local potential minima and their locations are ideal sites for the formation of protoplanets ...Comment: Modified version with longer historical introduction, discussion of model stability, and updated discussion of multi-planet extrasolar system

55 Cancri: A Laboratory for Testing Numerous Conjectures about Planet Formation

Five planets are presently believed to orbit the primary star of 55 Cnc, but there exists a large 5 AU gap in their distribution between the two outermost planets. This gap has attracted considerable interest because it may contain one or more lower--mass planets whose existence is not contradicted by long-term orbit stability analyses, in fact it is expected according to the "packed planetary systems" hypothesis and an empirical Titius--Bode relation recently proposed for 55 Cnc. Furthermore, the second largest planet is just the second farthest and it orbits very close to the star. Its orbit, the most circular of all, appears to be nearly but not quite commensurable with the orbit of the third planet, casting doubt that any migration or resonant capture of the inner planets has ever occurred and lending support to the idea of "in--situ" giant planet formation by the process of core accretion. All of the above ideas will be tested in the coming years in this natural laboratory as more observations will become available. This opportunity presents itself in conjunction with a physical model that relates the orbits of the observed planets to the structure of the original protoplanetary disk that harbored their formation at the early stages of protostellar collapse. Using only the 5 observed planets of 55 Cnc, this model predicts that the surface density profile of its protoplanetary disk varied with distance $R$ precisely as $\Sigma (R)\propto R^{-3/2}$, as was also found for the minimum--mass solar nebula. Despite this similarity, the disk of 55 Cnc was smaller, heavier, and less rotationally supported than the solar nebula, so this system represents a different mode of multi--planet formation compared to our own solar system.Comment: The physical model applied to the solar system in 0706.3205 (updated version 2) is now applied to the 5 planets of 55 Cancri. 5 tables, 2 figure