A spectral analysis is presented to investigate the onset of the convective instability in a fluid subject to a linear temperature gradient. The hydrodynamic theory is developed for the case of a binary system, where the concentration of one of the components is small. Therefore the present results will be applicable to the case of a Brownian system. We consider exclusively those modes which correspond to the central components of the spectral distribution, i.e., the diffusion mode and the thermal diffusivity mode. One finds that those modes are effected by the presence of the external temperature gradient in such a way that the spectrum of the scattered light should exhibit an important narrowing of the thermal diffusivity peak and a slight narrowing of the diffusion peak when approaching the convective instability critical point. Only the thermal diffusivity mode is affected in the limit of a pure fluid