The Principle of Similitude in Biology: From Allometry to the Formulation of Dimensionally Homogenous `Laws'

Meaningful laws of nature must be independent of the units employed to measure the variables. The principle of similitude (Rayleigh 1915) or dimensional homogeneity, states that only commensurable quantities (ones having the same dimension) may be compared, therefore, meaningful laws of nature must be homogeneous equations in their various units of measurement, a result which was formalized in the $\rm \Pi$ theorem (Vaschy 1892; Buckingham 1914). However, most relations in allometry do not satisfy this basic requirement, including the `3/4 Law' (Kleiber 1932) that relates the basal metabolic rate and body mass, which it is sometimes claimed to be the most fundamental biological rate (Brown et al. 2004) and the closest to a law in life sciences (West \& Brown 2004). Using the $\rm \Pi$ theorem, here we show that it is possible to construct a unique homogeneous equation for the metabolic rates, in agreement with data in the literature. We find that the variations in the dependence of the metabolic rates on body mass are secondary, coming from variations in the allometric dependence of the heart frequencies. This includes not only different classes of animals (mammals, birds, invertebrates) but also different exercise conditions (basal and maximal). Our results demonstrate that most of the differences found in the allometric exponents (White et al. 2007) are due to compare incommensurable quantities and that our dimensionally homogenous formula, unify these differences into a single formulation. We discuss the ecological implications of this new formulation in the context of the Malthusian's, Fenchel's and the total energy consumed in a lifespan relations.Comment: A accepted for publication in Theoretical Ecology. Comments are welcome ([email protected]

Multiscale mass transport in z~6 galactic discs: fueling black holes

By using AMR cosmological hydrodynamic N-body zoom-in simulations, with the RAMSES code, we studied the mass transport processes onto galactic nuclei from high redshift up to $z\sim6$. Due to the large dynamical range of the simulations we were able to study the mass accretion process on scales from $\sim50[kpc]$ to $\sim$ few $1[pc]$. We studied the BH growth on to the galactic center in relation with the mass transport processes associated to both the Reynolds stress and the gravitational stress on the disc. Such methodology allowed us to identify the main mass transport process as a function of the scales of the problem. We found that in simulations that include radiative cooling and SNe feedback, the SMBH grows at the Eddington limit for some periods of time presenting $\langle f_{EDD}\rangle\approx 0.5$ throughout its evolution. The $\alpha$ parameter is dominated by the Reynolds term, $\alpha_R$, with $\alpha_R\gg 1$. The gravitational part of the $\alpha$ parameter, $\alpha_G$, has an increasing trend toward the galactic center at higher redshifts, with values $\alpha_G\sim 1$ at radii <$\sim$ few $10^1[pc]$ contributing to the BH fueling. In terms of torques, we also found that gravity has an increasing contribution toward the galactic center at earlier epochs with a mixed contribution above $\sim 100 [pc]$. This complementary work between pressure gradients and gravitational potential gradients allows an efficient mass transport on the disc with average mass accretion rates of the order $\sim$ few $1 [M_{\odot}/yr]$. These level of SMBH accretion rates found in our cosmological simulations are needed in all models of SMBH growth that attempt to explain the formation of redshift $6-7$ quasars

Towards a Comprehensive Fueling-Controlled Theory on the Growth of Massive Black Holes and Host Spheroids

We study the relation between nuclear massive black holes and their host spheroid gravitational potential. Using AMR numerical simulations, we analyze how gas is transported in the nuclear (central kpc) regions of galaxies. We study the gas fueling onto the inner accretion disk (sub-pc scale) and the star formation in a massive nuclear disk like those generally found in proto-spheroids (ULIRGs, SCUBA Galaxies). These sub-pc resolution simulation of gas fueling that is mainly depleted by star formation naturally satisfy the `M_BH - $M_virial' relation, with a scatter considerably less than the observed one. We found a generalized version of Kennicutt-Schmidt Law for starbursts is satisfied, in which the total gas depletion rate (dot{M}_gas = dot{M}_BH + dot{M}_SF) is the one that scales as M_gas/t_orbital. We also found that the `M_BH - sigma' relation is a byproduct of the `M_BH - M_virial' relation in the fueling controlled scenario.Comment: 12 pages, figures, submited to ApJ, email: [email protected]

Near-infrared adaptive optics imaging of infrared luminous galaxies: the brightest cluster magnitude - star formation rate relation

We have established a relation between the brightest super star cluster magnitude in a galaxy and the host star formation rate (SFR) for the first time in the near infrared (NIR). The data come from a statistical sample of ~ 40 luminous IR galaxies (LIRGs) and starbursts utilizing K-band adaptive optics imaging. While expanding the observed relation to longer wavelengths, less affected by extinction effects, it also pushes to higher SFRs. The relation we find, M_K ~ - 2.6 log SFR, is similar to that derived previously in the optical and at lower SFRs. It does not, however, fit the optical relation with a single optical to NIR color conversion, suggesting systematic extinction and/or age effects. While the relation is broadly consistent with a size-of-sample explanation, we argue physical reasons for the relation are likely as well. In particular, the scatter in the relation is smaller than expected from pure random sampling strongly suggesting physical constraints. We also derive a quantifiable relation tying together cluster-internal effects and host SFR properties to possibly explain the observed brightest SSC magnitude vs. SFR dependency.Comment: 6 pages, 4 figures, accepted for publication in ApJ Letter