Establishing a department of community affairs in Illinois

"This paper was filed with the Illinois Cities and Villages Municipal Problems Commission of the Illinois General Assembly on February 10, 1969."Cover title

On the complete integrability of the discrete Nahm equations

The discrete Nahm equations, a system of matrix valued difference equations, arose in the work of Braam and Austin on half-integral mass hyperbolic monopoles. We show that the discrete Nahm equations are completely integrable in a natural sense: to any solution we can associate a spectral curve and a holomorphic line-bundle over the spectral curve, such that the discrete-time DN evolution corresponds to walking in the Jacobian of the spectral curve in a straight line through the line-bundle with steps of a fixed size. Some of the implications for hyperbolic monopoles are also discussed

Circle actions, central extensions and string structures

The caloron correspondence can be understood as an equivalence of categories between $G$-bundles over circle bundles and $LG \rtimes_\rho S^1$-bundles where $LG$ is the group of smooth loops in $G$. We use it, and lifting bundle gerbes, to derive an explicit differential form based formula for the (real) string class of an $LG \rtimes_\rho S^1$-bundle.Comment: 25 page

Hot Jupiters in binary star systems

Radial velocity surveys find Jupiter mass planets with semi-major axes a less than 0.1 AU around ~1% of solar-type stars; counting planets with $a$ as large as 5 AU, the fraction of stars having planets reaches ~ 10% {Marcy,Butler}. An examination of the distribution of semi-major axes shows that there is a clear excess of planets with orbital periods around 3 or 4 days, corresponding to a~0.03$ AU, with a sharp cutoff at shorter periods (see Figure 1). It is believed that Jupiter mass planets form at large distances from their parent stars; some fraction then migrate in to produce the short period objects. We argue that a significant fraction of the `hot Jupiters' (a<0.1 AU) may arise in binary star systems in which the orbit of the binary is highly inclined to the orbit of the planet. Mutual torques between the two orbits drive down the minimum separation or periapse r_p between the planet and its host star (the Kozai mechanism). This periapse collapse is halted when tidal friction on the planet circularizes the orbit faster than Kozai torque can excite it. The same friction then circularizes the planet orbit, producing hot Jupiters with the peak of the semimajor axis distribution lying around 3 days. For the observed distributions of binary separation, eccentricity and mass ratio, roughly 2.5% of planets with initial semimajor axis a_p ~ 5au will migrate to within 0.1au of their parent star. Kozai migration could account for 10% or more of the observed hot Jupiters.Comment: accepted to ApJ main journal, added one figure and expanded discussion