ASTR 3130, Majewski [Spring 2015]. Lecture Notes

## ASTR 3130 (Majewski) Lecture Notes

### A. OPTICS - GEOMETRY

Two principles are the basis of all standard optical systems:
1. The Law of Reflection states that when light hits a reflecting surface, the light ray will be reflected from the surface at an angle that is equal to the angle of the incident ray.

• We normally measure the angles from the normal to the surface, which is the line perpendicular to the surface at the point of reflection.
• Thus, mirrors can be used to reflect light in a way to divert it to different angles.
• In astronomical optics, we often use curved mirrors to redirect light more cleverly, e.g., bringing collimated (i.e. parallel) light to a focus.

This is done by an appropriately shaped reflective surface.

2. Refraction: In general, light moves slower through media of higher density. The index of refraction, n, measures the amount by which speed is diminished in a medium.
3. The Law of Refraction states that when light passes from one medium to another, the angle that the light beam makes with the normal to the interface between the two media is always less in the medium of higher index of refraction (i.e. density).

• Thus, for example, light becomes more "vertical" in glass than in air.
• More accurately, Snell's Law:

• By clever use of the law of refraction, we can bend, or redirect light by a range of angles:

• Prisms have a net effect on the bending of light that varies with the size of the apex angle.

• We can also use refraction to focus light.
•  In this simple example, only two rays are shown. The real situation is more complex. Rays at different angles focus at different points. To remedy this problem (which makes a blurry image at any distance from the glass), the angle of the glass must slowly change. We form a lens which can be thought of as the combination of different prism segments, where each prism has a different apex angle.

• The perfect lens (or mirror) shape, one that brings parallel light to a common focus, has a parabolic shape:

• Unfortunately, while it is much easier to make a lens with a spherical surface, this type of lens does not bring parallel light rays to a common focus.

Because the rays from different parts of the lens have slightly different focal points, we say that a spherical optical surface has spherical aberration. In this case, trying to capture an image of the source in one single plane will result in some fuzziness of the image.

### B. IMAGE FORMATION BY A LENS (IN MORE DETAIL)

The figure below shows the situation for image formation of an object relatively near the lens (top) and for an object at infinity (bottom).

• The Focal Length of a lens is the distance between the lens and the point where parallel incoming rays are brought to a focus.

• For images of astronomical sources: distant objects are so far away, that their received rays are essentially parallel.

Thus the focal plane for astronomical viewing is at a distance = focal length of "objective lens", which is a lens that forms the image of an object extremely far away.

• We use the expression Focal Ratio to describe the rate of beam convergence, which is given by the ratio of the focal length, Fo, to the diameter, Do of a lens or parabolic mirror :

FOCAL RATIO = "f/#" = Fo / Do

• This ratio is also called the "speed" of the lens.

• A "fast" optic brings light to a focus near that optic (i.e., "more quickly") and a "slow" optic brings light to a focus farther away (i.e., "more slowly").

Some examples:
• The telescopes in the non-majors Student Observatory are 8-inch Celestron telescopes with 80-inch focal lengths:

(80" Focal length) / (8" Objective) = f/10

• For the 6" Clark refractor at McCormick Observatory:

Focal Length = 1830 mm

Objective = 152 mm

1830 mm / 152 mm = f/12

This telescope would be "slower" than the 8-inch telescope example above.

• For the 26" Clark refractor at McCormick Observatory:

Focal Length = 391 inches

Objective = 26.25 inches

391 inches / 26.25 inches = f/14.9 (rather slow; larger images, good for astrometry)

This telescope is the "slowest" of the three examples here (and, frankly, all three would be considered "slow" in the grand scheme of things).

Law of reflection image from http://bhs.broo.k12.wv.us/homepage/staff/gbirkhim/waves_4.htm. Other figures copyright © 2002 Prentice Hall, copyright © 2002 W. H. Freeman and Company. Other material copyright © 2002 Steven R. Majewski. All rights reserved. These notes are intended for the private, noncommercial use of students enrolled in Astronomy 3130 at the University of Virginia.