Relation of internal attenuation, dust emission, and the size of spiral
  galaxies. Calibration at low-z and how to use it as a cosmological test at
  high-z
  Dust in spiral galaxies produces emission in the far-infrared (FIR) and
internal absorption in visible wavelengths. However, the relation of the two
amounts is not trivial because optical absorption may saturate, but the FIR
emission does not. Moreover, the volume concentration of dust plays a role in
the relation of absorption and emission, which depends on the size of the
galaxy. We explore the relation of these three quantities. In order to
understand the geometrical problem, we developed a model of dust distribution.
We also investigated the relation of the three variables with real data of
spiral galaxies at z<0.2 using the spectroscopic SDSS and FIR AKARI surveys.
Internal absorptions were derived with two different methods: the ratio of
emission lines H$_\alpha $ and H$_\beta $, and a previously calibrated relation
based on the color variations as a function of absolute magnitude and
concentration index.
  We find that in our low-z sample, the dependence of the average internal
attenuation on galaxy size is negligible on average. It allows us to derive the
internal attenuation of the galaxy, $A_V$, even when we only know its FIR flux.
This attenuation approximately depends on the inclination of the galaxy $i$ as
$\overline {A_V}=\gamma_V \log _{10}\left(\frac{1}{\cos i}\right)$, where
$\gamma_V$ is a constant. We found that $\gamma_V$ has a maximum value of
$1.45\pm 0.27$ magnitudes.
  When similar properties of dust are assumed, a general expression can be used
at any $z$. For cases of nonsaturation, this might be used as a cosmological
test. Although the present-day sensitivity of FIR or mm surveys does not allow
us to carry out this cosmological test at z>2 within the standard model, it may
be used in the future. For much lower z or different cosmological models, a
test might be feasible at present.
