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Register a new userA joint analysis of the Drake equation and the Fermi paradox
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Nikos Prantzos
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I propose a unified framework for a joint analysis of the Drake equation and the Fermi paradox, which enables a simultaneous, quantitative study of both of them. The analysis is based on a simplified form of the Drake equation and on a fairly simple scheme for the colonization of the Milky Way. It appears that for sufficiently long-lived civilizations, colonization of the Galaxy is the only reasonable option to gain knowledge about other life forms. This argument allows one to define a region in the parameter space of the Drake equation where the Fermi paradox definitely holds (Strong Fermi paradox).
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In evaluating the number of technological civilizations N in the Galaxy through the Drake formula, emphasis is mostly put on the astrophysical and biotechnological factors describing the emergence of a civilization and much less on its the lifetime, which is intimately related to its demise. It is argued here that this factor is in fact the most important regarding the practical implications of the Drake formula, because it determines the maximal extent of the sphere of influence of any technological civilization. The Fermi paradox is studied in the terms of a simplified version of the Drake formula, through Monte Carlo simulations of N civilizations expanding in the Galaxy during their space faring lifetime L. In the framework of that scheme, the probability of direct contact is determined as the fraction of the Galactic volume occupied collectively by the spheres of influence of N civilizations. The results of the analysis are used to determine regions in the parameter space where the Fermi paradox holds. It is argued that in a large region of the diagram the corresponding parameters suggest rather a weak Fermi paradox. Future research may reveal whether a strong paradox holds in some part of the parameter space. Finally, it is argued that the value of N is not bound by N=1 from below, contrary to what is usually assumed, but it may have a statistical interpretation.
We discuss the hot hand paradox within the framework of the backward Kolmogorov equation. We use this approach to understand the apparently paradoxical features of the statistics of fixed-length sequences of heads and tails upon repeated fair coin flips. In particular, we compute the average waiting time for the appearance of specific sequences. For sequences of length 2, the average time until the appearance of the sequence HH (heads-heads) equals 6, while the waiting time for the sequence HT (heads-tails) equals 4. These results require a few simple calculational steps by the Kolmogorov approach. We also give complete results for sequences of lengths 3, 4, and 5; the extension to longer sequences is straightforward (albeit more tedious). Finally, we compute the waiting times $T_{nrm H}$ for an arbitrary length sequences of all heads and $T_{nrm(HT)}$ for the sequence of alternating heads and tails. For large $n$, $T_{2nrm H}sim 3 T_{nrm(HT)}$.
The Drake Equation has proven fertile ground for speculation about the abundance, or lack thereof, of communicating extraterrestrial intelligences (CETIs) for decades. It has been augmented by subsequent authors to include random variables in order to understand its probabilistic behavior. In this paper, the first model for the number of CETIs with stochastic processes governing both their emergence and quiescence is developed using the Skellam Distribution. Results from this include the possibility that there can still be substantial times multiple CETIs exist even if the Drake Equation terms are approximately zero. In addition, it can give us a basic estimate of the average CETI age gap based on their broadcast time. Finally, we will introduce a definition of how the interaction between CETIs, where possible, can be measured by statistical dependence between the terms N and L in the Drake Equation by indicating how the number of co-existing CETIs affect their relative individual lifetimes.
Within the MOJAVE VLBA program (Monitoring of Jets in AGN with VLBA Experiments), we have accumulated observational data at 15 GHz for hundreds of jets in $gamma$-ray bright active galactic nuclei since the beginning of the Fermi scientific observations in August 2008. We investigated a time delay between the flux density of AGN parsec-scale radio emission at 15 GHz and 0.1$-$300 GeV Fermi LAT photon flux, taken from constructed light curves using weekly and adaptive binning. The correlation analysis shows that radio is lagging $gamma$-ray radiation by up to 8 months in the observers frame, while in the source frame, the typical delay is about 2-3 months. If the jet radio emission, excluding the opaque core, is considered, significant correlation is found at greater time lags. We supplement these results with VLBI kinematics and core shift data to conclude that the dominant high-energy production zone is typically located within the 15 GHz VLBA core at a distance of a few parsecs from the central nucleus.
The primary emphasis of this study has been to explain how modifying a cake recipe by changing either the dimensions of the cake or the amount of cake batter alters the baking time. Restricting our consideration to the genoise, one of the basic cakes of classic French cuisine, we have obtained a semi-empirical formula for its baking time as a function of oven temperature, initial temperature of the cake batter, and dimensions of the unbaked cake. The formula, which is based on the Diffusion equation, has three adjustable parameters whose values are estimated from data obtained by baking genoises in cylindrical pans of various diameters. The resulting formula for the baking time exhibits the scaling behavior typical of diffusion processes, i.e. the baking time is proportional to the (characteristic length scale)^2 of the cake. It also takes account of evaporation of moisture at the top surface of the cake, which appears to be a dominant factor affecting the baking time of a cake. In solving this problem we have obtained solutions of the Diffusion equation which are interpreted naturally and straightforwardly in the context of heat transfer; however, when interpreted in the context of the Schrodinger equation, they are somewhat peculiar. The solutions describe a system whose mass assumes different values in two different regions of space. Furthermore, the solutions exhibit characteristics similar to the evanescent modes associated with light waves propagating in a wave guide. When we consider the Schrodinger equation as a non-relativistic limit of the Klein-Gordon equation so that it includes a mass term, these are no longer solutions.
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