## Quality of seeing

Resolution: There is a problem of interference given the wave nature of light. It is analogous to the interference pattern caused by a single slit.

f = focal length
D = width of the slit
m = positive integer. 1, 2, 3, …

Rayleigh criterion for two sources: Arbitrary criterion to decide if two light sources can be resolved. If the central maximum of one pattern falls into the location of the first minimum of the other, the two images are said to be unresolved

Thus, larger D and smaller λ are good for resolution. Despite of this, the resolution of Earth-based telescopes is limited by the turbulence of the atmosphere. Local changes in atmospheric temperature and density over small distances create regions where the light is refracted in nearly random directions, causing the image of a point source to become blurred.

• Stars appear blurred since most of them act like point sources. This produces the twinkling.
• Planets do not twink. The angular size of most planets is larger than the scale of atmospheric turbulence. Distortions are averaged out over the size of the image (so it is not because stars really twink, that they twink!)

Aberrations: Inherent average distortions of lens (refracting telescopes) and mirror (reflecting telescopes).

• Chromatic (only in refracting telescopes): the focal length of a lens is λ-dependent. The index of refraction of a material varies with λ, thus the angle of refraction at the interface lens/air (or lens/vacuum) depends on λ. The focal length fλ is λ-dependent.

Chromatic aberration. Source: wikipedia. Also see in link how this can be fixed, if you want.

• Spherical: Results from the shape of the reflecting/refracting surface. Only paraboloids focus a parallel set of light rays to a single point (focus) in all areas of their surface. Making spherical mirrors and lenses is however cheaper and easier. The crafting must be made carefully, as small deviations can cause large distortions in the position of the focal point of two different areas of the reflecting/refracting surface.
• Coma: Elongated images of point sources that lie off the optical axis, caused by the focal length of paraboloids depending on the angle of incidence of light relative to the optical axis (y = f tanθ)
• Astigmatism: Different parts of a lens or a mirror converge an image at slightly different locations of the focal plane.
• Curvature of field: Due to the focusing of images on a curve rather than on a plane.
• Distortion of field: The plate scale

depends on the distance from the optical axis.

Brightness or intensity does not depend on the size (surface area) of the lens/mirror. The image intensity is identical to the object intensity, independent on the size of the aperture.

What does describe then the light-gathering power of the telescope?

Illumination (J): the amount of light energy per second focused onto a unit area of the resolved image. It is proportional to the diameter of the aperture D squared, and inversely proportional to the focal ratio F squared.

In sum:

• The larger is D, the better the resolution and the illumination.
• The larger is f, the larger the size of the image and the smaller the illumination.
• For a fixed F, increasing D produces an increase in resolution, but the illumination is constant.