Construction of Non-Perturbative, Unitary Particle-Antiparticle Amplitudes for Finite Particle Number Scattering Formalisms

Starting from a unitary, Lorentz invariant two-particle scattering amplitude , we show how to use an identification and replacement process to construct a unique, unitary particle-antiparticle amplitude. This process differs from conventional on-shell Mandelstam s,t,u crossing in that the input and constructed amplitudes can be off-diagonal and off-energy shell. Further, amplitudes are constructed using the invariant parameters which are appropriate to use as driving terms in the multi-particle, multichannel non-perturbative, cluster decomposable, relativistic scattering equations of the Faddeev-type integral equations recently presented by Alfred, Kwizera, Lindesay and Noyes. It is therefore anticipated that when so employed, the resulting multi-channel solutions will also be unitary. The process preserves the usual particle-antiparticle symmetries. To illustrate this process, we construct a J=0 scattering length model chosen for simplicity. We also exhibit a class of physical models which contain a finite quantum mass parameter and are Lorentz invariant. These are constructed to reduce in the appropriate limits, and with the proper choice of value and sign of the interaction parameter, to the asymptotic solution of the non-relativistic Coulomb problem, including the forward scattering singularity, the essential singularity in the phase, and the Bohr bound-state spectrum

On the relevance of the mathematics curriculum to young people

In this paper we draw upon focus group data from a large study of learner trajectories through 14-19 mathematics education to think about the notion of relevance in the mathematics curriculum. Drawing on data from three socially distanced sites we explore how different emphases on what might be termed practical, process and/or professional forms of relevance affect the experiences and aspirations of learners of mathematics. We consider whether an emphasis on practical relevance in schools serving relatively disadvantaged communities might aid the reproduction of students’ social position. This leads us to suggest that a fourth category of curriculum relevance – political relevance – is largely missing from classrooms

A High-Eccentricity Low-Mass Companion to HD 89744

HD 89744 is an F7 V star with mass 1.4 M, effective temperature 6166 K, age 2.0 Gy and metallicity [Fe/H]= 0.18. The radial velocity of the star has been monitored with the AFOE spectrograph at the Whipple Observatory since 1996, and evidence has been found for a low mass companion. The data were complemented by additional data from the Hamilton spectrograph at Lick Observatory during the companion's periastron passage in fall 1999. As a result, we have determined the star's orbital wobble to have period P = 256 d, orbital amplitude K = 257 m/s, and eccentricity e = 0.7. From the stellar mass we infer that the companion has minimum mass m2 sin i = 7.2 MJup in an orbit with semi-major axis a2 = 0.88 AU. The eccentricity of the orbit, among the highest known for extra-solar planets, continues the trend that extra-solar planets with semi-major axes greater than about 0.15 AU tend to have much higher eccentricities than are found in our solar system. The high metallicity of the parent star reinforces the trend that parent stars of extra-solar planets tend to have high metallicityComment: AASTEX-LateX v5.0, 7 pages w/ 3 figures, to be published in ApJ

HATS-3b: An inflated hot Jupiter transiting an F-type star

We report the discovery by the HATSouth survey of HATS-3b, a transiting extrasolar planet orbiting a V=12.4 F-dwarf star. HATS-3b has a period of P = 3.5479d, mass of Mp = 1.07MJ, and radius of Rp = 1.38RJ. Given the radius of the planet, the brightness of the host star, and the stellar rotational velocity (vsini = 9.0km/s), this system will make an interesting target for future observations to measure the Rossiter-McLaughlin effect and determine its spin-orbit alignment. We detail the low/medium-resolution reconnaissance spectroscopy that we are now using to deal with large numbers of transiting planet candidates produced by the HATSouth survey. We show that this important step in discovering planets produces logg and Teff parameters at a precision suitable for efficient candidate vetting, as well as efficiently identifying stellar mass eclipsing binaries with radial velocity semi-amplitudes as low as 1 km/s.Comment: 11 pages, 10 figures, submitted to A

HATS-5b: A Transiting hot-Saturn from the HATSouth Survey

We report the discovery of HATS-5b, a transiting hot-Saturn orbiting a G type star, by the HAT-South survey. HATS-5b has a mass of Mp=0.24 Mj, radius of Rp=0.91 Rj, and transits its host star with a period of P=4.7634d. The radius of HATS-5b is consistent with both theoretical and empirical models. The host star has a V band magnitude of 12.6, mass of 0.94 Msun, and radius of 0.87 Rsun. The relatively high scale height of HATS-5b, and the bright, photometrically quiet host star, make this planet a favourable target for future transmission spectroscopy follow-up observations. We reexamine the correlations in radius, equilibrium temperature, and metallicity of the close-in gas-giants, and find hot Jupiter-mass planets to exhibit the strongest dependence between radius and equilibrium temperature. We find no significant dependence in radius and metallicity for the close-in gas-giant population.Comment: 10 pages, submitted to A

Stellar Pollution in the Solar Neighborhood

We study spectroscopically determined iron abundances of 642 solar-type stars to search for the signature of accreted iron-rich material. We find that the metallicity [Fe/H] of a subset of 466 main sequence stars, when plotted as a function of stellar mass, mimics the pattern seen in lithium abundances in open clusters. Using Monte Carlo models we find that, on average, these stars have accreted about 0.4 Earth masses of iron while on the main sequence. A much smaller sample of 19 stars in the Hertzsprung gap, which are slightly evolved and whose convection zones are significantly more massive, have lower average [Fe/H], and their metallicity shows no clear variation with stellar mass. These findings suggest that terrestrial-type material is common around solar type stars.Comment: 33 pages, 11 figures. Submitted to Ap

HATS-11b and HATS-12b: Two transiting Hot Jupiters orbiting sub-solar metallicity stars selected for the K2 Campaign 7

We report the discovery of two transiting extrasolar planets from the HATSouth survey. HATS-11, a V=14.1 G0-star shows a periodic 12.9 mmag dip in its light curve every 3.6192 days and a radial velocity variation consistent with a Keplerian orbit. HATS-11 has a mass of 1.000 $\pm$ 0.060 M$_{\odot}$, a radius of 1.444 $\pm$ 0.057 M$_{\odot}$ and an effective temperature of 6060 $\pm$ 150 K, while its companion is a 0.85 $\pm$ 0.12 M$_J$, 1.510 $\pm$ 0.078 R$_J$ planet in a circular orbit. HATS-12 shows a periodic 5.1 mmag flux decrease every 3.1428 days and Keplerian RV variations around a V=12.8 F-star. HATS-12 has a mass of 1.489 $\pm$ 0.071 M$_{\odot}$, a radius of 2.21 $\pm$ 0.21 R$_{\odot}$, and an effective temperature of 6408 $\pm$ 75 K. For HATS-12, our measurements indicate that this is a 2.38 $\pm$ 0.11 M$_J$, 1.35 $\pm$ 0.17 R$_J$ planet in a circular orbit. Both host stars show sub-solar metallicity of -0.390 $\pm$ 0.060 dex and -0.100 $\pm$ 0.040 dex, respectively and are (slightly) evolved stars. In fact, HATS-11 is amongst the most metal-poor and, HATS-12 is amongst the most evolved stars hosting a hot Jupiter planet. Importantly, HATS-11 and HATS-12 have been observed in long cadence by Kepler as part of K2 campaign 7 (EPIC216414930 and EPIC218131080 respectively).Comment: 14 pages, 7 figures, 6 tables, submitted to A

HAT-P-6b: A Hot Jupiter transiting a bright F star

In the ongoing HATNet survey we have detected a giant planet, with radius 1.33 +/- 0.06 RJup and mass 1.06 +/- 0.12 MJup, transiting the bright (V = 10.5) star GSC 03239-00992. The planet is in a circular orbit with period 3.852985 +/- 0.000005 days and mid-transit epoch 2,454,035.67575 +/- 0.00028 (HJD). The parent star is a late F star with mass 1.29 +/- 0.06 Msun, radius 1.46 +/- 0.06 Rsun, Teff ~ 6570 +/- 80 K, [Fe=H] = -0.13 +/- 0.08 and age ~ 2.3+/-^{0.5}_{0.7}Gy. With this radius and mass, HAT-P-6b has somewhat larger radius than theoretically expected. We describe the observations and their analysis to determine physical properties of the HAT-P-6 system, and briefly discuss some implications of this finding.Comment: Accepted for publication in ApJL, 5 pages, minor changes compared to V

HATS-13b and HATS-14b: two transiting hot Jupiters from the HATSouth survey

We report the discovery of HATS-13b and HATS-14b, two hot-Jupiter transiting planets discovered by the HATSouth survey. The host stars are quite similar to each other (HATS-13: V = 13.9 mag, M* = 0.96 Msun, R* = 0.89 Rsun, Teff = 5500 K, [Fe/H] = 0.05; HATS-14: V = 13.8 mag, M* = 0.97 Msun, R* = 0.93 Rsun, Teff = 5350 K, [Fe/H] = 0.33) and both the planets orbit around them with a period of roughly 3 days and a separation of roughly 0.04 au. However, even though they are irradiated in a similar way, the physical characteristics of the two planets are very different. HATS-13b, with a mass of Mp = 0.543 MJ and a radius of Rp = 1.212 RJ, appears as an inflated planet, while HATS-14b, having a mass of Mp = 1.071 MJ and a radius of Rp = 1.039 RJ, is only slightly larger in radius than Jupiter.Comment: 13 pages, 7 figures, Submitted to Astronomy & Astrophysics. arXiv admin note: text overlap with arXiv:1503.0006