When observing the spectral signal of an artificial space target, because of the long observation distance and the low spatial resolution of observation equipment, the spectral signatures of multiple pure materials in a certain instantaneous scene is combined in one pixel to form a “mixed spectrum”. Therefore, unmixing these mixed spectra into the collection of pure material spectra and estimating the corresponding fractional abundances have been increasingly significant in the field of spectral analysis for artificial space targets. Most existing spectral unmixing methods assume that the number of pure materials (that is, “the number of endmembers”) contained in mixed spectra of an artificial space target is known as a priori, which is unrealistic for unknown artificial space targets. Therefore, the exact estimation of the number of endmembers plays a significant role in the accuracy of subsequent spectral analysis and processing. At present, the existing methods of endmember number estimation are mostly proposed under the assumption of Gaussian white noise interference. However, when the distribution of the noise signal is a spectral correlation, poor estimation results will be provided. In this paper, based on the intrinsic dimensions of data and the theory of maximum likelihood, a Robust Eigenvalue Maximum Likelihood (REML) method is proposed. By analyzing the statistical distribution characteristics of differences between the eigenvalues of the signal covariance matrix and those of signal correlation matrix, a maximum likelihood function can be established to estimate the number of endmembers contained in mixed spectra. This method consists of two steps: first, the original spectral data is pre-processed using multiple regression and a modified minimum noise fraction method to complete the noise estimation and whitening process, thereby effectively suppressing the interference of spectrally correlated noise. Then, the number of endmembers is estimated by solving a discrete logarithmic maximum likelihood function. This method does not require any input parameters and runs fast. Simulation experiments are based on synthetic mixed spectral data generated by the visible/near-infrared spectral signatures of five varieties of artificial space target materials measured in the laboratory and U. S. Geological Survey spectral dataset. And, experimental results demonstrate that this method can effectively suppress the interference of spectrally correlated noise and Gaussian white noise, and the estimation results have good accuracy and stability.