1. A radiation image capturing system comprising: a plurality of image capturing apparatus for capturing a radiation image of a subject by a radiation emitted from a radiation source and applied through the subject; a detecting unit for detecting which of the image capturing apparatus the radiation source is oriented to; and a plurality of processors for setting an image capturing mode for the radiation image based on a detection result from the detecting unit and controlling a process of capturing the radiation image with the image capturing apparatus which is detected by the detecting unit in the set image capturing mode, wherein each of the processors has a plurality of image capturing modes respectively for each of the image capturing apparatus, each of the image capturing modes comprises a mode for capturing the radiation image by applying the radiation from the radiation source through the subject to one of the image capturing apparatus, the processors comprise a host console, a first console, and a second console, the host console selects one of the image capturing modes depending on one of the image capturing apparatus to which the radiation source is oriented, based on the detection result, and controls the one of the image capturing apparatus to which the radiation source is oriented to capture the radiation image, based on the selected one of the image capturing modes, the first console controls a first image capturing apparatus of the image capturing apparatus, and if the detection result indicates that the radiation source is oriented to the first image capturing apparatus, the first console selects one of the image capturing modes depending on the first image capturing apparatus and controls the first image capturing apparatus to capture the radiation image, based on the selected one of the image capturing modes, and the second console controls a second image capturing apparatus of the image capturing apparatus, and if the detection result indicates that the radiation source is oriented to the second image capturing apparatus, the second console selects one the image capturing modes depending on the second image capturing apparatus and controls the second image capturing apparatus to capture the radiation image, based on the selected one of the image capturing modes.