A star too small and cold to shine has been found by S. Kulkarni, K. Matthews, B.R. Oppenheimer, and T. Nakajima (Science 270, 1478 (1995)). The spectrum of the object shows the presence of methane, which, according to the authors, would not exist at temperatures much above 1000 K. The mass of the star, as determined from its gravitational effect on a companion star, is roughly 20 times the mass of Jupiter. The star is considered to be a brown dwarf, the coolest ever found.
(a) From available thermodynamic data, test the stability of methane at temperatures above 1000 K.
(b) What is Amax for this star?
(c) What is the energy density of the star relative to that of the Sun (6000 K)?
(d) To determine whether the star will shine, estimate the fraction of the energy density of the star in the visible region of the spectrum.