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1+1+2 Electromagnetic perturbations on general LRS space-times: Regge-Wheeler and Bardeen-Press equations

We use the, covariant and gauge-invariant, 1+1+2 formalism developed by Clarkson and Barrett, and develop new techniques, to decouple electromagnetic (EM) perturbations on arbitrary locally rotationally symmetric (LRS) space-times. Ultimately, we derive 3 decoupled complex equations governing 3 complex scalars. One of these is a new Regge-Wheeler (RW) equation generalized for LRS space-times, whereas the remaining two are new generalizations of the Bardeen-Press (BP) equations. This is achieved by first using linear algebra techniques to rewrite the first-order Maxwell equations in a new complex 1+1+2 form which is conducive to decoupling. This new complex system immediately yields the generalized RW equation, and furthermore, we also derive a decoupled equation governing a newly defined complex EM 2-vector. Subsequently, a further decomposition of the 1+1+2 formalism into a 1+1+1+1 formalism is developed, allowing us to decompose the complex EM 2-vector, and its governing equations, into spin-weighted scalars, giving rise to the generalized BP equations

The Arches Cluster: Extended Structure and Tidal Radius

At a projected distance of ~26 pc from Sgr A*, the Arches cluster provides insight to star formation in the extreme Galactic Center (GC) environment. Despite its importance, many key properties such as the cluster's internal structure and orbital history are not well known. We present an astrometric and photometric study of the outer region of the Arches cluster (R > 6.25") using HST WFC3IR. Using proper motions we calculate membership probabilities for stars down to F153M = 20 mag (~2.5 M_sun) over a 120" x 120" field of view, an area 144 times larger than previous astrometric studies of the cluster. We construct the radial profile of the Arches to a radius of 75" (~3 pc at 8 kpc), which can be well described by a single power law. From this profile we place a 3-sigma lower limit of 2.8 pc on the observed tidal radius, which is larger than the predicted tidal radius (1 - 2.5 pc). Evidence of mass segregation is observed throughout the cluster and no tidal tail structures are apparent along the orbital path. The absence of breaks in the profile suggests that the Arches has not likely experienced its closest approach to the GC between ~0.2 - 1 Myr ago. If accurate, this constraint indicates that the cluster is on a prograde orbit and is located front of the sky plane that intersects Sgr A*. However, further simulations of clusters in the GC potential are required to interpret the observed profile with more confidence.Comment: 24 pages (17-page main text, 7-page appendix), 24 figures, accepted to Ap

1+1+2 Electromagnetic perturbations on non-vacuum LRS class II space-times: Decoupling scalar and 2-vector harmonic amplitudes

We use the covariant and gauge-invariant 1+1+2 formalism of Clarkson and Barrett \cite{Clarkson2003} to analyze electromagnetic (EM) perturbations on non-vacuum {\it locally rotationally symmetric} (LRS) class II space-times. Ultimately, we show how to derive six real decoupled equations governing the total of six EM scalar and 2-vector harmonic amplitudes. Four of these are new, and result from expanding the complex EM 2-vector which we defined in \cite{Burston2007} in terms of EM 2-vector harmonic amplitudes. We are then able to show that there are four precise combinations of the amplitudes that decouple, two of these are polar perturbations whereas the remaining two are axial. The remaining two decoupled equations are the generalized Regge-Wheeler equations which were developed previously in \cite{Betschart2004}, and these govern the two EM scalar harmonic amplitudes. However, our analysis generalizes this by including a full description and classification of energy-momentum sources, such as charges and currents.Comment: 9 page

The Quintuplet Cluster: Extended Structure and Tidal Radius

The Quintuplet star cluster is one of only three known young ($<10$ Myr) massive (M $>10^4$ M$_\odot$) clusters within $\sim100$ pc of the Galactic Center. In order to explore star cluster formation and evolution in this extreme environment, we analyze the Quintuplet's dynamical structure. Using the HST WFC3-IR instrument, we take astrometric and photometric observations of the Quintuplet covering a $120''\times120''$ field-of-view, which is $19$ times larger than those of previous proper motion studies of the Quintuplet. We generate a catalog of the Quintuplet region with multi-band, near-infrared photometry, proper motions, and cluster membership probabilities for $10,543$ stars. We present the radial density profile of $715$ candidate Quintuplet cluster members with $M\gtrsim4.7$ M$_\odot$ out to $3.2$ pc from the cluster center. A $3\sigma$ lower limit of $3$ pc is placed on the tidal radius, indicating the lack of a tidal truncation within this radius range. Only weak evidence for mass segregation is found, in contrast to the strong mass segregation found in the Arches cluster, a second and slightly younger massive cluster near the Galactic Center. It is possible that tidal stripping hampers a mass segregation signature, though we find no evidence of spatial asymmetry. Assuming that the Arches and Quintuplet formed with comparable extent, our measurement of the Quintuplet's comparatively large core radius of $0.62^{+0.10}_{-0.10}$ pc provides strong empirical evidence that young massive clusters in the Galactic Center dissolve on a several Myr timescale.Comment: 25 pages (21-page main text, 4-page appendix), 18 figures, submitted to Ap

On detecting terrestrial planets with timing of giant planet transits

The transits of a planet on a Keplerian orbit occur at time intervals exactly equal to the period of the orbit. If a second planet is introduced the orbit is not Keplerian and the transits are no longer exactly periodic. We compute the magnitude of these variations in the timing of the transits, dt. We investigate analytically several limiting cases: (i) interior perturbing planets with much smaller periods; (ii) exterior perturbing planets on eccentric orbits with much larger periods; (iii) both planets on circular orbits with arbitrary period ratio but not in resonance; and (iv) planets on initially circular orbits locked in resonance. Case (iv) is perhaps the most interesting case since some systems are known to be in resonances and the perturbations are the largest. As long as the perturber is more massive than the transiting planet, the timing variations would be of order of the period regardless of the perturber mass! For lighter perturbers, we show that the timing variations are smaller than the period by the perturber to transiting planet mass ratio. An earth mass planet in 2:1 resonance with a 3-day period transiting planet (e.g. HD 209458b) would cause timing variations of order 3 minutes, which would be accumulated over a year. These are easily detectable with current ground-based measurements. For the case of both planets on eccentric orbits, we compute numerically the transit timing variations for several cases of known multiplanet systems assuming they were edge-on. Transit timing measurements may be used to constrain the masses and radii of the planetary system and, when combined with radial velocity measurements, to break the degeneracy between mass and radius of the host star. (abstract truncated)Comment: 21 pages, 9 figures, submitted to MNRA

A generalized linear Hubble law for an inhomogeneous barotropic Universe

In this work, I present a generalized linear Hubble law for a barotropic spherically symmetric inhomogeneous spacetime, which is in principle compatible with the acceleration of the cosmic expansion obtained as a result of high redshift Supernovae data. The new Hubble function, defined by this law, has two additional terms besides an expansion one, similar to the usual volume expansion one of the FLRW models, but now due to an angular expansion. The first additional term is dipolar and is a consequence of the existence of a kinematic acceleration of the observer, generated by a negative gradient of pressure or of mass-energy density. The second one is quadrupolar and due to the shear. Both additional terms are anisotropic for off-centre observers, because of to their dependence on a telescopic angle of observation. This generalized linear Hubble law could explain, in a cosmological setting, the observed large scale flow of matter, without to have recourse to peculiar velocity-type newtonian models. It is pointed out also, that the matter dipole direction should coincide with the CBR dipole one.Comment: 9 pages, LaTeX, to be published in Class. Quantum Gra

Existence and Uniqueness of Tri-tronqu\'ee Solutions of the second Painlev\'e hierarchy

The first five classical Painlev\'e equations are known to have solutions described by divergent asymptotic power series near infinity. Here we prove that such solutions also exist for the infinite hierarchy of equations associated with the second Painlev\'e equation. Moreover we prove that these are unique in certain sectors near infinity.Comment: 13 pages, Late