The Hubble Space Telescope has excellent resolving power...

The Hubble Space Telescope has excellent resolving power because there is no atmospheric distortion of the light. The Hubble deep field camera uses the $2.4-\mathrm{m}$ diameter mirror to collect light from distant galaxies that formed very early in the history of the universe. How far apart can two galaxies be from each other if they are 10 billion light-years away from Earth and are barely resolved by the Hubble Telescope using visible light with a wavelength of $400 \mathrm{~nm}$ ?

Key Concepts

Rayleigh Criterion

The Rayleigh criterion is a widely accepted standard for determining the angular resolution limit of an optical system. It states that two point light sources are considered resolved when the principal diffraction maximum of one image coincides with the first diffraction minimum of the other. This criterion quantitatively relates the minimum resolvable angle to the wavelength of light and the diameter of the telescope's aperture.

Aperture and Distance Scaling

The aperture of a telescope is crucial in defining its light-collecting power and angular resolution. A larger aperture reduces the diffraction limit, allowing finer details to be resolved in the observed image. Additionally, converting the angular resolution into an actual physical separation when observing distant objects requires considering the distance to the objects. The further away the objects, the larger the physical separation corresponding to a given angular separation.

Diffraction Limit

The diffraction limit is the physical limit on the resolution of any optical system due to the wave nature of light. It determines the smallest angular separation at which two point sources, such as distant galaxies, can be distinctly resolved. This limit is influenced by the wavelength of light used and the aperture diameter of the telescope, setting the fundamental bound on how fine an image detail can be achieved.

Angular Resolution

Angular resolution refers to the smallest angle between two objects that an imaging system can distinguish as separate. In astronomy, this is a critical measure because it defines the ability of telescopes to resolve closely spaced objects in the sky. The angular resolution is directly affected by the physical parameters of the telescope, including its aperture and the wavelength of light employed.

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