The Effect of Multiple Heat Sources on Exomoon Habitable Zones

With dozens of Jovian and super-Jovian exoplanets known to orbit their host stars in or near the stellar habitable zones, it has recently been suggested that moons the size of Mars could offer abundant surface habitats beyond the solar system. Several searches for such exomoons are now underway, and the exquisite astronomical data quality of upcoming space missions and ground-based extremely large telescopes could make the detection and characterization of exomoons possible in the near future. Here we explore the effects of tidal heating on the potential of Mars- to Earth-sized satellites to host liquid surface water, and we compare the tidal heating rates predicted by tidal equilibrium model and a viscoelastic model. In addition to tidal heating, we consider stellar radiation, planetary illumination and thermal heat from the planet. However, the effects of a possible moon atmosphere are neglected. We map the circumplanetary habitable zone for different stellar distances in specific star-planet-satellite configurations, and determine those regions where tidal heating dominates over stellar radiation. We find that the `thermostat effect' of the viscoelastic model is significant not just at large distances from the star, but also in the stellar habitable zone, where stellar radiation is prevalent. We also find that tidal heating of Mars-sized moons with eccentricities between 0.001 and 0.01 is the dominant energy source beyond 3--5 AU from a Sun-like star and beyond 0.4--0.6 AU from an M3 dwarf star. The latter would be easier to detect (if they exist), but their orbital stability might be under jeopardy due to the gravitational perturbations from the star.Comment: accepted for publication in A&A, 8 pages, 4 figure

Conceptual Unification of Gravity and Quanta

We present a model unifying general relativity and quantum mechanics. The model is based on the (noncommutative) algebra \mbox{{\cal A}} on the groupoid \Gamma = E \times G where E is the total space of the frame bundle over spacetime, and G the Lorentz group. The differential geometry, based on derivations of \mbox{{\cal A}}, is constructed. The eigenvalue equation for the Einstein operator plays the role of the generalized Einstein's equation. The algebra \mbox{{\cal A}}, when suitably represented in a bundle of Hilbert spaces, is a von Neumann algebra \mathcal{M} of random operators representing the quantum sector of the model. The Tomita-Takesaki theorem allows us to define the dynamics of random operators which depends on the state \phi . The same state defines the noncommutative probability measure (in the sense of Voiculescu's free probability theory). Moreover, the state \phi satisfies the Kubo-Martin-Schwinger (KMS) condition, and can be interpreted as describing a generalized equilibrium state. By suitably averaging elements of the algebra \mbox{{\cal A}}, one recovers the standard geometry of spacetime. We show that any act of measurement, performed at a given spacetime point, makes the model to collapse to the standard quantum mechanics (on the group G). As an example we compute the noncommutative version of the closed Friedman world model. Generalized eigenvalues of the Einstein operator produce the correct components of the energy-momentum tensor. Dynamics of random operators does not ``feel'' singularities.Comment: 28 LaTex pages. Substantially enlarged version. Improved definition of generalized Einstein's field equation

Noncommutative Unification of General Relativity and Quantum Mechanics. A Finite Model

We construct a model unifying general relativity and quantum mechanics in a broader structure of noncommutative geometry. The geometry in question is that of a transformation groupoid given by the action of a finite group G on a space E. We define the algebra of smooth complex valued functions on the groupoid, with convolution as multiplication, in terms of which the groupoid geometry is developed. Owing to the fact that the group G is finite the model can be computed in full details. We show that by suitable averaging of noncommutative geometric quantities one recovers the standard space-time geometry. The quantum sector of the model is explored in terms of the regular representation of the groupoid algebra, and its correspondence with the standard quantum mechanics is established.Comment: 20 LaTex pages, General Relativity and Gravitation, in pres

Geometry and General Relativity in the Groupoid Model with a Finite Structure Group

In a series of papers we proposed a model unifying general relativity and quantum mechanics. The idea was to deduce both general relativity and quantum mechanics from a noncommutative algebra ${\cal A}_{\Gamma}$ defined on a transformation groupoid $\Gamma$ determined by the action of the Lorentz group on the frame bundle $(E, \pi_M, M)$ over space-time $M$. In the present work, we construct a simplified version of the gravitational sector of this model in which the Lorentz group is replaced by a finite group $G$ and the frame bundle is trivial $E=M\times G$. The model is fully computable. We define the Einstein-Hilbert action, with the help of which we derive the generalized vacuum Einstein equations. When the equations are projected to space-time (giving the "general relativistic limit"), the extra terms that appear due to our generalization can be interpreted as "matter terms", as in Kaluza-Klein-type models. To illustrate this effect we further simplify the metric matrix to a block diagonal form, compute for it the generalized Einstein equations and find two of their "Friedmann-like" solutions for the special case when $G =\mathbb{Z}_2$. One of them gives the flat Minkowski space-time (which, however, is not static), another, a hyperbolic, linearly expanding universe.Comment: 32 page