Speeding up low-mass planetary microlensing simulations and modelling: the Caustic Region Of INfluence

Extensive simulations of planetary microlensing are necessary both before and after a survey is conducted: before to design and optimize the survey and after to understand its detection efficiency. The major bottleneck in such computations is the computation of lightcurves. However, for low-mass planets most of these computations are wasteful, as most lightcurves do not contain detectable planetary signatures. In this paper I develop a parameterization of the binary microlens that is conducive to avoiding lightcurve computations. I empirically find analytic expressions describing the limits of the parameter space that contain the vast majority of low-mass planet detections. Through a large scale simulation I measure the (in)completeness of the parameterization and the speed-up it is possible to achieve. For Earth-mass planets in a wide range of orbits it is possible to speed up simulations by a factor of ${\sim} 30$-$125$ (depending on the survey's annual duty-cycle) at the cost of missing ${\sim} 1$ percent of detections (which is actually a smaller loss than for the arbitrary parameter limits typically applied in microlensing simulations). The benefits of the parameterization probably outweigh the costs for planets below $100M_{\oplus}$. For planets at the sensitivity limit of AFTA-WFIRST, simulation speed-ups of a factor ${\sim} 1000$ or more are possible.Comment: 9 pages, 8 figures, 3 table

Possible Solution of the long-standing discrepancy in the Microlensing Optical Depth Toward the Galactic Bulge by correcting the stellar number count

We find that significant incompleteness in stellar number counts results in a significant overestimate of the microlensing optical depth $\tau$ and event rate per star per year $\Gamma$ toward the Galactic bulge from the first two years of the MOA-II survey. We find that the completeness in Red Clump Giant (RCG) counts $f_{\rm RC}$ decreases proportional to the galactic latitude $b$, as $f_{\rm RC}=(0.63\pm0.11)-(0.052\pm0.028)\times b$, ranging between 1 and 0.7 at $b=-6^\circ\sim-1.5^\circ$. The previous measurements using all sources by Difference Image Analysis (DIA) by MACHO and MOA-I suffer the same bias. On the other hand, the measurements using a RCG sample by OGLE-II, MACHO and EROS were free from this bias because they selected only the events associated with the resolved stars. Thus, the incompleteness both in the number of events and stellar number count cancel out. We estimate $\tau$ and $\Gamma$ by correcting this incompleteness. In the central fields with $|l|<5^\circ$, we find $\Gamma=[18.74\pm0.91]\times10^{-6}\exp[(0.53\pm0.05)(3-|b|)]$ star$^{-1}$ yr$^{-1}$ and $\tau_{200}=[1.84\pm0.14]\times10^{-6}\exp[(0.44\pm0.07)(3-|b|)]$ for the 427 events with $t_{\rm E}\leq200\,$days using all sources brighter than $I_s\leq20$ mag. Our revised all-source $\tau$ measurements are about 2-$\sigma$ smaller than the other all-source measurements and are consistent with the RCG measurements within 1-$\sigma$. We conclude that the long-standing problem on discrepancy between the high $\tau$ with all-source samples by DIA and low $\tau$ with RCG samples can probably be explained by the incompleteness of the stellar number count. A model fit to these measurements predicts $\Gamma=4.60\pm0.25\times10^{-5}$ star$^{-1}$ yr$^{-1}$ at $|b|\sim-1^\circ.4$ and $-2^\circ.25<l<3^\circ.75$ for sources with $I<20$, where the future space mission WFIRST will observe.Comment: 39 pages, 15 figures, 5 tables, accepted for publication in ApJ. arXiv admin note: substantial text overlap with arXiv:1305.018

How fast do Jupiters grow? Signatures of the snowline and growth rate in the distribution of gas giant planets

We present here observational evidence that the snowline plays a significant role in the formation and evolution of gas giant planets. When considering the population of observed exoplanets, we find a boundary in mass-semimajor axis space that suggests planets are preferentially found beyond the snowline prior to undergoing gap-opening inward migration and associated gas accretion. This is consistent with theoretical models suggesting that sudden changes in opacity -- as would occur at the snowline -- can influence core migration. Furthermore, population synthesis modelling suggests that this boundary implies that gas giant planets accrete ~ 70 % of the inward flowing gas, allowing ~ 30$ % through to the inner disc. This is qualitatively consistent with observations of transition discs suggesting the presence of inner holes, despite there being ongoing gas accretion.Comment: 7 pages, 6 figures, accepted for publication in Monthly Notices of the Royal Astronomical Societ

Is the Galactic bulge devoid of planets?

Considering a sample of 31 exoplanetary systems detected by gravitational microlensing, we investigate whether or not the estimated distances to these systems conform to the Galactic distribution of planets expected from models. We derive the expected distribution of distances and relative proper motions from a simulated microlensing survey, correcting for the dominant selection effects that affect the planet detection sensitivity as a function of distance, and compare it to the observed distribution using Anderson-Darling (AD) hypothesis testing. Taking the relative abundance of planets in the bulge to that in the disk, $f_{\rm bulge}$, as a model parameter, we find that our model is only consistent with the observed distribution for $f_{\rm bulge}<0.54$ (for a $p$-value threshold of 0.01) implying that the bulge may be devoid of planets relative to the disk. Allowing for a dependence of planet abundance on metallicity and host mass, or an additional dependence of planet sensitivity on event timescale does not restore consistency for $f_{\rm bulge}=1$. We examine the distance estimates of some events in detail, and conclude that some parallax-based distance estimates could be significantly in error. Only by combining the removal of one problematic event from our sample and the inclusion of strong dependences of planet abundance or detection sensitivity on host mass, metallicity and event timescale are we able to find consistency with the hypothesis that the bulge and disk have equal planet abundance.Comment: Revised following referee's report. 12 pages, 7 figures, 1 tabl

Caustic Structures and Detectability of Circumbinary Planets in Microlensing

Recent discoveries of circumbinary planets in Kepler data show that there is a viable channel of planet formation around binary main sequence stars. Motivated by these discoveries, we have investigated the caustic structures and detectability of circumbinary planets in microlensing events. We have produced a suite of animations of caustics as a function of the projected separation and angle of the binary host to efficiently explore caustic structures over the entire circumbinary parameter space. Aided by these animations, we have derived a semi-empirical analytic expression for the location of planetary caustics, which are displaced in circumbinary lenses relative to those of planets with a single host. We have used this expression to show that the dominant source of caustic motion will be due to the planet's orbital motion and not that of the binary star. Finally, we estimate the fraction of circumbinary microlensing events that are recognizable as such to be significant (5-50 percent) for binary projected separations in the range 0.1-0.5 in units of Einstein radii.Comment: 15 pages, 1 table, 18 figures. Accepted for publication in Ap

The Ramsey Number for 3-Uniform Tight Hypergraph Cycles

Let C(3)n denote the 3-uniform tight cycle, that is, the hypergraph with vertices v1, .–.–., vn and edges v1v2v3, v2v3v4, .–.–., vn−1vnv1, vnv1v2. We prove that the smallest integer N = N(n) for which every red–blue colouring of the edges of the complete 3-uniform hypergraph with N vertices contains a monochromatic copy of C(3)n is asymptotically equal to 4n/3 if n is divisible by 3, and 2n otherwise. The proof uses the regularity lemma for hypergraphs of Frankl and Rödl

Influence of design parameters on occurence of oil whirl

Oil whirl instability is a serious problem in oil lubricated journal bearings. The phenomenon is characterized by a subsynchronous vibration of the journal within the bush and is particularly apparent in turbogenerators, aeroengines and electric motors. A review is presented of previous papers on the subject of oil whirl, and a simple theory is described which was used to aid the design of an oil whirl test rig. Predictions of the onset of oil whirl made by the theory presented were found to agree with those of previous researchers. They showed that increasing the shaft flexibility, or the lubricant viscosity, and decreasing the bearing radial clearance tended to reduce the oil whirl onset speed thus making the system more unstable