New intrinsic-colour calibration for uvby-β photometry

A new intrinsic-colour calibration $((b-y{)}_o-\beta )$ is presented for the uvby-β photometric system, making use of re-calibrated Hipparcos parallaxes and published reddening maps. This new calibration for $(b-y{)}_o-\beta$, our Eq. (1), has been based upon stars with $d_{\mathit{Hip}}<70\text{pc}$ in the photometric catalogues of Schuster and Nissen (1988), Schuster et al., 2004, Schuster et al., 2006, provides a small dispersion, $\pm 0.009$, and has a positive “standard” $+2.239\mathrm{\Delta }\beta$ coefficient, which is not too different from the coefficients of Crawford (1975a, +1.11) and of Olsen (1988, +1.34). For 61 stars with spectra from CASPEC, UVES/VLT, and FIES/NOT databases, without detectable Na I lines, the average reddening value $〈E(b-y)〉=-0.001\pm 0.002$ shows that any zero-point correction to our intrinsic-colour equation must be minuscule.     

An application of the tensor virial theorem to hole + vortex + bulge systems

The tensor virial theorem for subsystems is formulated for three-component systems and further effort is devoted to a special case where the inner subsystems and the central region of the outer one are homogeneous, the last surrounded by an isothermal homeoid. The virial equations are explicitly written under the additional restrictions: (i) similar and similarly placed inner subsystems, and (ii) spherical outer subsystem. An application is made to hole + vortex + bulge systems, in the limit of flattened inner subsystems, which implies three virial equations in three unknowns. Using the Faber-Jackson relation, $R_{\text{e}}\propto {\sigma }_0^2$, the standard $M_{\text{H}}$-${\sigma }_0$ form $(M_{\text{H}}\propto {\sigma }_0^4)$ is deduced from qualitative considerations. The projected bulge velocity dispersion to projected vortex velocity ratio, $\eta =({\sigma }_{\text{B}}{)}_{33}/\{[(v_{\text{V}}{)}_{\mathit{qq}}{]}^2+[({\sigma }_{\text{V}}{)}_{\mathit{qq}}{]}^2{\}}^{1/2}$, as a function of the fractional radius, $y_{\text{BV}}=R_{\text{B}}/R_{\text{V}}$, and the fractional masses, $m_{\text{BH}}=M_{\text{B}}/M_{\text{H}}$ and $m_{\text{BV}}=M_{\text{B}}/M_{\text{V}}$, is studied in the range of interest, $0?m_{\text{VH}}=M_{\text{V}}/M_{\text{H}}?5$ [Escala, A., 2006. ApJ, 648, L13] and $229?m_{\text{BH}}?795$ [Marconi, A., Hunt, L.H., 2003. ApJ 589, L21], consistent with observations. The related curves appear to be similar to Maxwell velocity distributions, which implies a fixed value of $\eta$ below the maximum corresponds to two different configurations: a compact bulge on the left of the maximum, and an extended bulge on the right. All curves lie very close one to the other on the left of the maximum, and parallel one to the other on the right. On the other hand, fixed $m_{\text{BH}}$ or $m_{\text{BV}}$, and $y_{\text{BV}}$, are found to imply more massive bulges passing from bottom to top along a vertical line on the $(\mathsf{O}{y}_{\text{BV}}\eta )$ plane, and vice versa. The model is applied to NGC 4374 and NGC 4486, taking the fractional mass,$m_{\text{BH}}$, and the fractional radius, $y_{\text{BV}}$, as unknowns, and the bulge mass is inferred from the knowledge of the hole mass, and compared with results from different methods. In presence of a massive vortex $(m_{\text{VH}}=5)$, the hole mass has to be reduced by a factor 2–3 with respect to the case of a massless vortex, to get the fit. Finally, the assumptions of homogeneous inner bulge and isotropic stress tensor are discussed.     

GRBs as standard candles: There is no “circularity problem” (and there never was)

Beginning with the 2002 discovery of the “Amati Relation” of GRB spectra, there has been much interest in the possibility that this and other correlations of GRB phenomenology might be used to make GRBs into standard candles. One recurring apparent difficulty with this program has been that some of the primary observational quantities to be fit as “data” – to wit, the isotropic-equivalent prompt energy E_iso and the collimation-corrected “total” prompt energy E_γ – depend for their construction on the very cosmological models that they are supposed to help constrain. This is the so-called “circularity problem” of standard candle GRBs. This paper is intended to point out that the circularity problem is not in fact a problem at all, except to the extent that it amounts to a self-inflicted wound. It arises essentially because of an unfortunate choice of data variables – “source-frame” variables such as E_iso, which are unnecessarily encumbered by cosmological considerations. If, instead, the empirical correlations of GRB phenomenology which are formulated in source-variables are mapped to the primitive observational variables (such as fluence) and compared to the observations in that space, then all taint of circularity disappears. I also indicate here a set of procedures for encoding high-dimensional empirical correlations (such as between E_iso, $E_{\mathit{pk}}^{(\mathit{src})}\text{,}{t}_{\mathit{jet}}^{(\mathit{src})}$, and $T_{45}^{(\mathit{src})}$) in a “Gaussian Tube” smeared model that includes both the correlation and its intrinsic scatter, and how that source-variable model may easily be mapped to the space of primitive observables, to be convolved with the measurement errors and fashioned into a likelihood. I discuss the projections of such Gaussian tubes into sub-spaces, which may be used to incorporate data from GRB events that may lack some element of the data (for example, GRBs without ascertained jet-break times). In this way, a large set of inhomogeneously observed GRBs may be assimilated into a single analysis, so long as each possesses at least two correlated data attributes.