This paper is concerned with deblurring and spectral analysis of ground-based astronomical images of space objects. A numerical approach is provided for deblurring and sparse unmixing of ground-based hyperspectral images (HSIs) of objects taken through atmospheric turbulence. Hyperspectral imaging systems capture a three-dimensional (3D) datacube (tensor) containing two-dimensional (2D) spatial information and one-dimensional (1D) spectral information at each spatial location. Pixel intensities vary with wavelength bands, providing a spectral trace of intensity values and generating a spatial map of spectral variation (spectral signatures of materials). The deblurring and spectral unmixing problem is quite challenging since the point spread function (PSF) depends on the imaging system as well as the seeing conditions and is wavelength varying. We show how to efficiently construct an optimal Kronecker product--based preconditioner, and provide numerical methods for estimating the multiple PSFs using spectral data from an isolated (guide) star for joint deblurring and sparse unmixing of HSI datasets in order to spectrally analyze the image objects. The methods are illustrated with numerical experiments on a commonly used test example, a simulated HSI of the Hubble Space Telescope satellite.