Perihelion Precession Research Articles

SOFT X-RAY TEMPERATURE TIDAL DISRUPTION EVENTS FROM STARS ON DEEP PLUNGING ORBITS

One of the puzzles associated with tidal disruption event candidates (TDEs) is that there is a dichotomy between the color temperatures of ${\rm few}\times 10^4$~K for TDEs discovered with optical and UV telescopes, and the color temperatures of ${\rm few}\times 10^5 - 10^6$~K for TDEs discovered with X-ray satellites. Here we propose that high-temperature TDEs are produced when the tidal debris of a disrupted star self-intersects relatively close to the supermassive black hole, in contrast to the more distant self-intersection that leads to lower color temperatures. In particular, we note from simple ballistic considerations that greater apsidal precession in an orbit is the key to closer self-intersection. Thus larger values of $\beta$, the ratio of the tidal radius to the pericenter distance of the initial orbit, are more likely to lead to higher temperatures of more compact disks which are super-Eddington and geometrically and optically thick. For a given star and $\beta$, apsidal precession also increases for larger black hole masses, but larger black hole masses imply a lower temperature at the Eddington luminosity. Thus the expected dependence of the temperature on the mass of the black hole is non-monotonic. We find that in order to produce a soft X-ray temperature TDE, a deep plunging stellar orbit with $\beta> 3$ is needed and a black hole mass of $\lesssim 5\times 10^6 M_\odot$ is favored. Although observations of TDEs are comparatively scarce and are likely dominated by selection effects, it is encouraging that both expectations are consistent with current data.

Minimal length in quantum gravity and gravitational measurements

The existence of a minimal length is a common prediction of various theories of quantum gravity. This minimal length leads to a modification of the Heisenberg uncertainty principle to a Generalized Uncertainty Principle (GUP). Various studies showed that a GUP modifies the Hawking radiation of black holes. In this paper, we propose a modification of the Schwarzschild metric based on the modified Hawking temperature derived from the GUP. Based on this modified metric, we calculate corrections to the deflection of light, time delay of light, perihelion precession, and gravitational redshift. We compare our results with gravitational measurements to set an upper bound on the GUP parameter.

Motion of particles on a four-dimensional asymptotically AdS black hole with scalar hair

Motivated by black hole solutions with matter fields outside their horizon, we study the effect of these matter fields on the motion of massless and massive particles. We consider as background a four-dimensional asymptotically AdS black hole with scalar hair. The geodesics are studied numerically and we discuss the differences in the motion of particles between the four-dimensional asymptotically AdS black holes with scalar hair and their no-hair limit, that is, Schwarzschild AdS black holes. Mainly, we found that there are bounded orbits like planetary orbits in this background. However, the periods associated to circular orbits are modified by the presence of the scalar hair. Besides, we found that some classical tests such as perihelion precession, deflection of light, and gravitational time delay have the standard value of general relativity plus a correction term coming from the cosmological constant and the scalar hair. Finally, we found a specific value of the parameter associated to the scalar hair, in order to explain the discrepancy between the theory and the observations, for the perihelion precession of Mercury and light deflection.

Classroom reconstruction of the Schwarzschild metric

A promising way to introduce general relativity (GR) in the classroom is to study the physical implications of certain given metrics, such as the Schwarzschild one. This involves lower mathematical expenditure than an approach focusing on differential geometry in its full glory and permits to emphasize physical aspects before attacking the field equations. Even so, in terms of motivation, lacking justification of the metric employed may pose an obstacle. The paper discusses how to establish the weak-field limit of the Schwarzschild metric with a minimum of relatively simple physical assumptions, avoiding the field equations but admitting the determination of a single parameter from experiment. An attractive experimental candidate is the measurement of the perihelion precession of Mercury, because the result was already known before the completion of GR. It is shown how to determine the temporal and radial coefficients of the Schwarzschild metric to sufficiently high accuracy to obtain quantitative predictions for all the remaining classical tests of GR.

Gravitational tests of the generalized uncertainty principle

We compute the corrections to the Schwarzschild metric necessary to reproduce the Hawking temperature derived from a Generalized Uncertainty Principle (GUP), so that the GUP deformation parameter is directly linked to the deformation of the metric. Using this modified Schwarzschild metric, we compute corrections to the standard General Relativistic predictions for the light deflection and perihelion precession, both for planets in the solar system and for binary pulsars. This analysis allows us to set bounds for the GUP deformation parameter from well-known astronomical measurements.

Gravitational mass and Newton's universal gravitational law under relativistic conditions

We discuss the predictions of Newton's universal gravitational law when using the gravitational, mg, rather than the rest masses, mo, of the attracting particles. According to the equivalence principle, the gravitational mass equals the inertial mass, mi, and the latter which can be directly computed from special relativity, is an increasing function of the Lorentz factor, γ, and thus of the particle velocity. We consider gravitationally bound rotating composite states, and we show that the ratio of the gravitational force for gravitationally bound rotational states to the force corresponding to low (γ ≈ 1) particle velocities is of the order of (mPl/mo)2 where mpi is the Planck mass (ħc/G)1/2. We also obtain a similar result, within a factor of two, by employing the derivative of the effective potential of the Schwarzschild geodesics of GR. Finally, we show that for certain macroscopic systems, such as the perihelion precession of planets, the predictions of this relativistic Newtonian gravitational law differ again by only a factor of two from the predictions of GR.

Azimuthally symmetric theory of gravitation – II. On the perihelion precession of solar planetary orbits

Azimuthally symmetric theory of gravitation – II. On the perihelion precession of solar planetary orbits

Implications of the eccentric Kozai–Lidov mechanism for stars surrounding supermassive black hole binaries

An enhanced rate of stellar tidal disruption events (TDEs) may be an important characteristic of supermassive black hole (SMBH) binaries at close separations. Here we study the evolution of the distribution of stars around a SMBH binary due to the eccentric Kozai-Lidov (EKL) mechanism, including octupole effects and apsidal precession caused by the stellar mass distribution and general relativity. We identify a region around one of the SMBHs in the binary where the EKL mechanism drives stars to high eccentricities, which ultimately causes the stars to either scatter off the second SMBH or get disrupted. For SMBH masses 10^7 Msun and 10^8 Msun, the TDE rate can reach 10^{-2} yr and deplete a region of the stellar cusp around the secondary SMBH in ~0.5 Myr. As a result, the final geometry of the stellar distribution between 0.01 and 0.1 pc around the secondary SMBH is a torus. These effects may be even more prominent in nuclear stellar clusters hosting a supermassive and an intermediate mass black hole.

Lindblad Zones: resonant eccentric orbits to aid bar and spiral formation in galaxy discs

The apsidal precession frequency in a fixed gravitational potential increases with the radial range of the orbit (eccentricity). Although the frequency increase is modest, it can have important implications for wave dynamics in galaxy discs, which have not been previously explored in detail. One of the most interesting consequences is that for a given pattern frequency, each Lindblad resonance does not exist in isolation, but rather is the parent of a continuous sequence of resonant radii, a Lindblad Zone, with each radius in this zone characterized by a specific eccentricity. In the epicyclic approximation, the precession or epicyclic frequency does not depend on epicycle size, and this phenomenon is not captured. A better approximation for eccentric orbits is provided by p-ellipse curves, which do exhibit this effect. Here, the p-ellipse approximation and precession–eccentricity relation are used as tools for finding the resonant radii generated from various Lindblad parent resonances. Simple, idealized examples, in flat rotation curve and near solid-body discs, are used to show that ensembles of eccentric resonant orbits excited in Lindblad Zones can provide a backbone for generating a variety of (kinematic) bars and spiral waves. In cases balancing radius-dependent circular frequencies and eccentricity-dependent precession, a range of resonant orbits can maintain their form in the pattern frame, and do not wind up. Eccentric resonance orbits require a strong perturbation to excite them, and may be produced mostly in galaxy interactions or by strong internal disturbances.

Gravitational anomalies in the solar system?

Mindful of the anomalous perihelion precession of Mercury discovered by Le Verrier in the second half of the nineteenth century and its successful explanation by Einstein with his General Theory of Relativity in the early years of the twentieth century, discrepancies among observed effects in our Solar system and their theoretical predictions on the basis of the currently accepted laws of gravitation applied to known matter-energy distributions have the potential of paving the way for remarkable advances in fundamental physics. This is particularly important now more than ever, given that most of the universe seems to be made of unknown substances dubbed Dark Matter and Dark Energy. Should this not be directly the case, Solar system's anomalies could anyhow lead to advancements in either cumulative science, as shown to us by the discovery of Neptune in the first half of the nineteenth century, and technology itself. Moreover, investigations in one of such directions can serendipitously enrich the other one as well. The current status of some alleged gravitational anomalies in the Solar system is critically reviewed. They are: (a) Possible anomalous advances of planetary perihelia. (b) Unexplained orbital residuals of a recently discovered moon of Uranus (Mab). (c) The lingering unexplained secular increase of the eccentricity of the orbit of the Moon. (d) The so-called Faint Young Sun Paradox. (e) The secular decrease of the mass parameter of the Sun. (f) The Flyby Anomaly. (g) The Pioneer Anomaly. (h) The anomalous secular increase of the astronomical unit.

Chaos in navigation satellite orbits caused by the perturbed motion of the Moon

Numerical simulations carried out over the past decade suggest that the orbits of the Global Navigation Satellite Systems are unstable, resulting in an apparent chaotic growth of the eccentricity. Here we show that the irregular and haphazard character of these orbits reflects a similar irregularity in the orbits of many celestial bodies in our Solar System. We find that secular resonances, involving linear combinations of the frequencies of nodal and apsidal precession and the rate of regression of lunar nodes, occur in profusion so that the phase space is threaded by a devious stochastic web. As in all cases in the Solar System, chaos ensues where resonances overlap. These results may be significant for the analysis of disposal strategies for the four constellations in this precarious region of space.

The monotonicity of the apsidal angle in power-law potential systems

In a central force system the apsidal angle is the angle at the centre of force between two consecutive apsides and measures the precession rate of the line of apsis. The apsidal angle has applications in different fields and Newton's apsidal precession theorem has been extensively studied by astronomers, physicist and mathematicians. The perihelion precession of Mercury, the dynamics of galaxies, the vortex dynamics, the JWKB quantisation condition are some examples where the apsidal angle is of interest. In case of eccentric orbits and forces far from inverse square, numerical investigations provide evidence of the monotonicity of the apsidal angle with respect to the orbit parameters, such as the orbit eccentricity. However, no proof of this statement is available. In this paper central force systems with f(r)∼μr−(α+1) are considered. We prove that for any −2<α<1 the apsidal angle is a monotonic function of the orbital eccentricity, or equivalently of the angular momentum. As a corollary, the conjecture stating the absence of isolated cases of zero precession is proved.

A numerical study of vector resonant relaxation

Stars bound to a supermassive black hole interact gravitationally. Persistent torques acting between stellar orbits lead to the rapid resonant relaxation of the orbital orientation vectors ("vector" resonant relaxation) and slower relaxation of the eccentricities ("scalar" resonant relaxation), both at rates much faster than two-body or non-resonant relaxation. We describe a new parallel symplectic integrator, N-ring, which follows the dynamical evolution of a cluster of N stars through vector resonant relaxation, by averaging the pairwise interactions over the orbital period and periapsis-precession timescale. We use N-ring to follow the evolution of clusters containing over 10^4 stars for tens of relaxation times. Among other results, we find that the evolution is dominated by torques among stars with radially overlapping orbits, and that resonant relaxation can be modelled as a random walk of the orbit normals on the sphere, with angular step size ranging from 0.5-1 radian. The relaxation rate in a cluster with a fixed number of stars is proportional to the RMS mass of the stars. The RMS torque generated by the cluster stars is reduced below the torque between Kepler orbits due to apsidal precession and declines weakly with the eccentricity of the perturbed orbit. However since the angular momentum of an orbit also decreases with eccentricity, the relaxation rate is approximately eccentricity-independent for e<0.7 and grows rapidly with eccentricity for e>0.8. We quantify the relaxation using the autocorrelation function of the spherical multipole moments; this decays exponentially and the e-folding time may be identified with the vector resonant relaxation timescale.

The Mechanism and Timescale of Nodal Precession: Two Nuclear Stellar Disks in the Galactic Center

Abstract A dynamical model of interacting nuclear stellar rings in the central parsec of our Galaxy is constructed. We discuss the physical sources of nodal precession and of the associated time scales. For approximate study of the mutual orbital precession, we replace broad nuclear rings by weighted average narrow circular rings. The model with narrow circular rings is shown to adequately describe the nodal precession. The period of relativistic apsidal precession in the center of the Galaxy, T ap • ≈ 5 ⋅ 10 8 yr $T_{{\rm{ap}}}^ \bullet \approx 5 \cdot 10^8 \, {\rm{yr}}$ , is almost an order of magnitude longer that the period of nodal precession, T nod ≈ 7 · 107 yr, due to gravitational perturbations of nuclear disks (or rings). An important property of the nodal precession of nuclear rings is established: the lines of nodes of the two rings rotate uniformly with the same angular velocity, but in different directions. This explains the important observational fact that the lines of nodes of nuclear disks are not collinear, but are directed at large angles to each other.

The Apsidal Precession for Low Earth Sun Synchronized Orbits

By nodal regression and apsidal precession, the Earth flattering at satellite low Earth orbits (LEO) is manifested. Nodal regression refers to the shift of the orbit’s line of nodes over time as Earth revolves around the Sun. Nodal regression is orbit feature utilized for circular orbits to be Sun synchronized. A sun¬-synchronized orbit lies in a plane that maintains a fixed angle with respect to the Earth-Sun direction. In the low Earth Sun synchronized circular orbits are suited the satellites that accomplish their photo imagery missions. Nodal regression depends on orbital altitude and orbital inclination angle. For the respective orbital altitudes the inclination window for the Sun synchronization to be attained is determined. The apsidal precession represents major axis shift, respectively the argument of perigee deviation. The apsidal precession simulation, for inclination window of sun synchronized orbital altitudes, is provided through this paper.

On the Intrinsic Precession of the Perihelion of Mercury

The longitude of the perihelion advance of Mercury was calculated for the two and ten-body problem by using a correction to the balance between the force given by the Newton 2nd law of motion and the Newton gravitational force. The corresponding system of differential equations was solved numerically. The correction, that expresses the apparent mass variation with the body speed, has a trend that is different from those that usually appear in the electron theory and in the special theory of relativity. The calculated intrinsic precession was ~42.95 arc-sec/cy for the Sun-Mercury system and ~42.98 arc-sec/cy when the difference between the corrected model and the Newtonian model, for the 10-body problem, is taken.

Newtonian Computation of Perihelion Precession of Mercury—An Update

A updated numerical evaluation of the Newtonian component of Mercury’s perihelion advance over more than two centuries starting from about the year 2000 is made using the Euler’s algorithm as well as a modified Euler algorithm. Results are given for about the last 30 years of this interval which show that the precession rate may be substantially higher than what it is believed to be.

Natural orbit approximations in single power-law potentials

In a previous paper, I demonstrated the accuracy of simple, precessing, power ellipse (p-ellipse) approximations to orbits of low-to-moderate eccentricity in power-law potentials. Here I explore several extensions of these approximations to improve accuracy, especially for nearly radial orbits. 1) It is found that moderately improved orbital fits can be achieved with higher order perturbation expansions (in eccentricity), with the addition of `harmonic' terms to the solution. 2) Alternately, a matching of the extreme radial excursions of an orbit can be imposed, and a more accurate estimate of the eccentricity parameter is obtained. However, the error in the precession frequency is usually increased. 3) A correction function of small magnitude corrects the frequency problem. With this correction, even first order approximations yield excellent fits at quite high eccentricity over a range of potential indices that includes flat and falling rotation curve cases. 4) Adding a first harmonic term to fit the breadth of the orbital loops, and determining the fundamental and harmonic coefficients by matching to three orbital positions further improves the fit. With a couple of additional small corrections one obtains excellent fits to orbits with radial ranges of more than a thousand for some potentials. These simple corrections to the basic p-ellipse are basically in the form of several successive approximations, and can provide high accuracy. They suggest new results including that the apsidal precession rate scales approximately as $log(1-e)$ at very high eccentricities $e$. New insights are also provided on the occurrence of periodic orbits in various potentials, especially at high eccentricity.

Orbital Motions and the Conservation-Law/Preferred-Frame α3 Parameter

We analytically calculate some orbital effects induced by the Lorentz-invariance/momentum-conservation PPN parameter $\alpha_3$ in a gravitationally bound binary system made of a compact primary orbited by a test particle. We neither restrict ourselves to any particular orbital configuration nor to specific orientations of the primary's spin axis $\boldsymbol{\hat{\psi}}$. We use our results to put preliminary upper bounds on $\alpha_3$ in the weak-field regime by using the latest data from Solar System's planetary dynamics. By linearly combining the supplementary perihelion precessions $\Delta\dot\varpi$ of the Earth, Mars and Saturn, determined with the EPM2011 ephemerides, we infer $|\alpha_3|\lesssim 6\times 10^{-10}$. Our result is about 3 orders of magnitude better than the previous weak-field constraints existing in the literature, and of the same order of magnitude of the bound expected from the future BepiColombo mission to Mercury. It is, by construction, independent of the other preferred-frame PPN parameters $\alpha_1,\alpha_2$, both preliminarily constrained down to a $\approx 10^{-6}$ level. The wide pulsar-white dwarf binary PSR J0407+1607 yields a preliminary upper bound on the strong-field version $\hat{\alpha}_3$ of the Lorentz-invariance/momentum-conservation PPN parameter of the order of $3\times 10^{-17}$. It relies upon certain assumptions on the unknown values of the pulsar's spin axis orientation $\boldsymbol{\hat{\psi}}$, the orbital node $\Omega$ and the inclination $I$. Neither the pulsar's proper motion, still undetected, nor a possible value of the pulsar's mass $m_{\rm p}$ up to two solar masses substantially affect our result.

Constraints on Non-Standard Gravitomagnetism by the Anomalous Perihelion Precession of the Planets

In 2008, a team of astronomers reported an anomalous retrograde precession of the perihelion of Saturn amounting to \(\Delta \dot{\omega}_{\mathrm{SATURN}}=-0.006(2)\) arcsec per century (arcsec cy\(^{-1}\)). This unexplained precession was obtained after taking into account all classical and relativistic effects in the context of the highly refined EPM2008 ephemerides. More recent analyzes have not confirmed this effect, but they have found similar discrepancies in other planets. Our objective in this paper is to discuss a non-standard model involving transversal gravitomagnetism generated by the Sun as a possible source of these potential anomalies, to be confirmed by further data analyses. In order to compute the Lense–Thirring perturbations induced by the suggested interaction, we should consider the orientation of the Sun's rotational axis in Carrington elements and the inclination of the planetary orbits with respect to the ecliptic plane. We find that an extra component of the gravitomagnetic field not predicted by General Relativity could explain the reported anomalies without conflicting with the Gravity Probe B experiment and the orbits of the geodynamics satellites.