THE ECONOMY OF PROPERTY FORMS

This essay explores a puzzle from the world of property theory, that is from the world of mine and yours, the basic social organizational molecules with which we build our sense of justice. The puzzle is this: why is there so little variety in the forms of property people use across the world? We lack a convincing theory for the "economy of property forms," where economy is understood in the sense of parsimony. Three partial answers have been suggested. First, the limited number of forms may keep people from wasting property through over-fragmentation. Second, the limit may economize on communication costs for third parties who want to buy or sell property. Third, the limit may be an inexpensive way to help verify ownership. But none of these theories accounts for why obsolete forms persist in many economies, and why value-increasing forms fail to be created. Perhaps a more satisfying answer will require looking to political economy and to cognitive psychology. For now, the economy of property forms remains a provocative question.

State Vector Reduction as a Shadow of a Noncommutative Dynamics

A model, based on a noncommutative geometry, unifying general relativity with quantum mechanics, is further develped. It is shown that the dynamics in this model can be described in terms of one-parameter groups of random operators. It is striking that the noncommutative counterparts of the concept of state and that of probability measure coincide. We also demonstrate that the equation describing noncommutative dynamics in the quantum gravitational approximation gives the standard unitary evolution of observables, and in the "space-time limit" it leads to the state vector reduction. The cases of the spin and position operators are discussed in details.Comment: 20 pages, LaTex, no figure

Transit Least Squares: Optimized transit detection algorithm to search for periodic transits of small planets

We present a new method to detect planetary transits from time-series photometry, the Transit Least Squares (TLS) algorithm. TLS searches for transit-like features while taking the stellar limb darkening and planetary ingress and egress into account. We have optimized TLS for both signal detection efficiency (SDE) of small planets and computational speed. TLS analyses the entire, unbinned phase-folded light curve. We compensate for the higher computational load by (i.) using algorithms like "Mergesort" (for the trial orbital phases) and by (ii.) restricting the trial transit durations to a smaller range that encompasses all known planets, and using stellar density priors where available. A typical K2 light curve, including 80d of observations at a cadence of 30min, can be searched with TLS in ~10s real time on a standard laptop computer, as fast as the widely used Box Least Squares (BLS) algorithm. We perform a transit injection-retrieval experiment of Earth-sized planets around sun-like stars using synthetic light curves with 110ppm white noise per 30min cadence, corresponding to a photometrically quiet KP=12 star observed with Kepler. We determine the SDE thresholds for both BLS and TLS to reach a false positive rate of 1% to be SDE~7 in both cases. The resulting true positive (or recovery) rates are ~93% for TLS and ~76% for BLS, implying more reliable detections with TLS. We also test TLS with the K2 light curve of the TRAPPIST-1 system and find six of seven Earth-sized planets using an iterative search for increasingly lower signal detection efficiency, the phase-folded transit of the seventh planet being affected by a stellar flare. TLS is more reliable than BLS in finding any kind of transiting planet but it is particularly suited for the detection of small planets in long time series from Kepler, TESS, and PLATO. We make our Python implementation of TLS publicly available.Comment: A&A accepted. Code, documentation and tutorials at https://github.com/hippke/tl

Noncommutative Dynamics of Random Operators

We continue our program of unifying general relativity and quantum mechanics in terms of a noncommutative algebra ${\cal A}$ on a transformation groupoid $\Gamma = E \times G$ where $E$ is the total space of a principal fibre bundle over spacetime, and $G$ a suitable group acting on $\Gamma$. We show that every $a \in {\cal A}$ defines a random operator, and we study the dynamics of such operators. In the noncommutative regime, there is no usual time but, on the strength of the Tomita-Takesaki theorem, there exists a one-parameter group of automorphisms of the algebra ${\cal A}$ which can be used to define a state dependent dynamics; i.e., the pair $({\cal A}, \phi)$, where $\phi$ is a state on ${\cal A}$, is a ``dynamic object''. Only if certain additional conditions are satisfied, the Connes-Nikodym-Radon theorem can be applied and the dependence on $\phi$ disappears. In these cases, the usual unitary quantum mechanical evolution is recovered. We also notice that the same pair $({\cal A}, \phi)$ defines the so-called free probability calculus, as developed by Voiculescu and others, with the state $\phi$ playing the role of the noncommutative probability measure. This shows that in the noncommutative regime dynamics and probability are unified. This also explains probabilistic properties of the usual quantum mechanics.Comment: 13 pages, LaTe

Optimized trajectories to the nearest stars using lightweight high-velocity photon sails

New means of interstellar travel are now being considered by various research teams, assuming lightweight spaceships to be accelerated via either laser or solar radiation to a significant fraction of the speed of light (c). We recently showed that gravitational assists can be combined with the stellar photon pressure to decelerate an incoming lightsail from Earth and fling it around a star or bring it to rest. Here, we demonstrate that photogravitational assists are more effective when the star is used as a bumper (i.e. the sail passes "in front of" the star) rather than as a catapult (i.e. the sail passes "behind" or "around" the star). This increases the maximum deceleration at $\alpha$ Cen A and B and reduces the travel time of a nominal graphene-class sail (mass-to-surface ratio 8.6e-4 gram m$^{-2}$) from 95 to 75 yr. The maximum possible velocity reduction upon arrival depends on the required deflection angle from $\alpha$ Cen A to B and therefore on the binary's orbital phase. Here, we calculate the variation of the minimum travel times from Earth into a bound orbit around Proxima for the next 300 yr and then extend our calculations to roughly 22,000 stars within about 300 ly. Although $\alpha$ Cen is the most nearby star system, we find that Sirius A offers the shortest possible travel times into a bound orbit: 69 yr assuming 12.5% c can be obtained at departure from the solar system. Sirius A thus offers the opportunity of flyby exploration plus deceleration into a bound orbit of the companion white dwarf after relatively short times of interstellar travel.Comment: 14 pages, 7 figures (5 col, 2 b/w), 2 table

Transit least-squares survey -- II. Discovery and validation of 17 new sub- to super-Earth-sized planets in multi-planet systems from K2

The extended Kepler mission (K2) has revealed more than 500 transiting planets in roughly 500,000 stellar light curves. All of these were found either with the box least-squares algorithm or by visual inspection. Here we use our new transit least-squares (TLS) algorithm to search for additional planets around all K2 stars that are currently known to host at least one planet. We discover and statistically validate 17 new planets with radii ranging from about 0.7 Earth radii to roughly 2.2 Earth radii and a median radius of 1.18 Earth radii. EPIC201497682.03, with a radius of 0.692 (-0.048, +0.059) Earth radii, is the second smallest planet ever discovered with K2. The transit signatures of these 17 planets are typically 200 ppm deep (ranging from 100 ppm to 2000 ppm), and their orbital periods extend from about 0.7 d to 34 d with a median value of about 4 d. Fourteen of these 17 systems only had one known planet before, and they now join the growing number of multi-planet systems. Most stars in our sample have subsolar masses and radii. The small planetary radii in our sample are a direct result of the higher signal detection efficiency that TLS has compared to box-fitting algorithms in the shallow-transit regime. Our findings help in populating the period-radius diagram with small planets. Our discovery rate of about 3.7 % within the group of previously known K2 systems suggests that TLS can find over 100 additional Earth-sized planets in the data of the Kepler primary mission.Comment: published in A&A, 12 pages, 6 colored Figures, 1 Table; minor textual corrections; Fig. 5 corrected for the distance scalin

Transit least-squares survey - I. Discovery and validation of an Earth-sized planet in the four-planet system K2-32 near the 1:2:5:7 resonance

We apply, for the first time, the Transit Least Squares (TLS) algorithm to search for new transiting exoplanets. TLS is a successor to the Box Least Squares (BLS) algorithm, which has served as a standard tool for the detection of periodic transits. In this proof-of-concept paper, we demonstrate how TLS finds small planets that have previously been missed. We showcase TLS' capabilities using the K2 EVEREST-detrended light curve of the star K2-32 (EPIC205071984) that was known to have three transiting planets. TLS detects these known Neptune-sized planets K2-32b, d, and c in an iterative search and finds an additional transit signal with a high signal detection efficiency (SDE_TLS) of 26.1 at a period of 4.34882 (-0.00075, +0.00069) d. We show that this signal remains detectable (SDE_TLS = 13.2) with TLS in the K2SFF light curve of K2-32, which includes a less optimal detrending of the systematic trends. The signal is below common detection thresholds, however, if searched with BLS in the K2SFF light curve (SDE_BLS = 8.9) as in previous searches. Markov Chain Monte Carlo sampling shows that the radius of this candidate is 1.01 (-0.09, +0.10) Earth radii. We analyze its phase-folded transit light curve using the vespa software and calculate a false positive probability FPP = 3.1e-3, formally validating K2-32e as a planet. Taking into account the multiplicity boost of the system, FPP < 3.1e-4. K2-32 now hosts at least four planets that are very close to a 1:2:5:7 mean motion resonance chain. The offset of the orbital periods of K2-32e and b from a 1:2 mean motion resonance is in very good agreement with the sample of transiting multi-planet systems from Kepler, lending further credence to the planetary nature of K2-32e. We expect that TLS can find many more transits of Earth-sized and smaller planets in the Kepler data that have hitherto remained undetected with BLS and similar algorithms.Comment: published in A&A, Vol. 625, id. A31 , 8 pages, 6 colored figure