Remarkable analytic relations among greybody parameters
  In this paper we derive and discuss several implications of the analytic form
of a modified blackbody, also called greybody, which is widely used in
Astrophysics, and in particular in the study of star formation in the
far-infrared/sub-millimeter domain. The research in this area has been greatly
improved thanks to recent observations taken with the Herschel satellite, so
that it became important to clarify the sense of the greybody approximation, to
suggest possible further uses, and to delimi its intervals of validity. First,
we discuss the position of the greybody peak, making difference between the
optically thin and thick regimes. Second, we analyze the behavior of bolometric
quantities as a function of the different greybody parameters. The ratio
between the bolometric luminosity and the mass of a source, the ratio between
the so-called "sub-millimeter luminosity" and the bolometric one, and the
bolometric temperature are observables used to characterize the evolutionary
stage of a source, and it is of primary importance to have analytic equations
describing the dependence of such quantities on the greybody parameters. Here
we discuss all these aspects, providing analytic relations, illustrating
particular cases and providing graphical examples. Some equations reported here
are well-known in Astrophysics, but are often spread over different
publications. Some of them, instead, are brand new and represent a novelty in
Astrophysics literature. Finally we indicate an alternative way to obtain,
under some conditions, the greybody temperature and dust emissivity directly
from an observing spectral energy distribution, avoiding a best-fit procedure.
