Telescope (refracting, reflecting, Cassegrain)

A telescope gathers light from a distant object with a large objective, a lens or a concave mirror, and forms an image at its focal plane, which is either examined through an eyepiece or recorded directly by a detector. Its two jobs are collecting more light than the eye and resolving finer angular detail. A 100 mm aperture collects about 200 times as much light as a 7 mm dark-adapted pupil and resolves about 1.4 arcseconds at 550 nm; with a 1000 mm focal length objective and a 25 mm eyepiece it magnifies 40×. Refracting telescopes use a lens objective; reflecting telescopes use a curved primary mirror, alone or with a secondary mirror that folds and extends the focal length, as in the Cassegrain family.

Objective and eyepiece

For visual use the objective and eyepiece are spaced by the sum of their focal lengths, so the instrument is afocal: parallel light from a star enters and leaves as parallel light. The angular magnification is

$$M = \frac{f_\text{obj}}{f_\text{eye}},$$

and the beam leaving the eyepiece, the exit pupil, has diameter $D/M$: 2.5 mm for $D$ = 100 mm at 40×. The exit pupil should not exceed the eye's pupil, or light is wasted. The Keplerian form (positive eyepiece) gives an inverted image with a real intermediate focus where a reticle can sit; the Galilean form (negative eyepiece) gives an upright image in a shorter tube. Used in reverse, the same two-lens layouts are the laser beam expander. For imaging, the eyepiece is removed and the detector sits at the focal plane, where the plate scale is $206\,265/f$ arcseconds per unit length: 103 arcseconds per millimeter for a 2000 mm focal length.

Refractors and reflectors

A single-lens objective focuses different colors at different distances, the chromatic aberration that limited early refractors. An achromatic doublet cancels most of it, and apochromats with low-dispersion glasses reduce the residual secondary spectrum further. Lens objectives absorb ultraviolet and infrared and can be supported only at their edges, which caps practical refractors at around a meter in diameter.

Mirrors have no chromatic aberration, work from the ultraviolet to the far infrared with suitable coatings, and can be supported from behind, so all large telescopes are reflectors. The common layouts are:

• Newtonian. A paraboloidal primary and a flat diagonal mirror that sends the focus out the side of the tube. A paraboloid is free of spherical aberration on axis but shows coma off axis.
• Cassegrain. A concave primary and a convex secondary that reflects light back through a hole in the primary. The secondary multiplies the primary's focal length: a 200 mm f/3 primary (600 mm) with a secondary magnification of 3.33 gives a 2000 mm, f/10 system in a tube much shorter than 2000 mm. The classical Cassegrain uses a paraboloidal primary and hyperboloidal secondary.
• Ritchey-Chrétien. A Cassegrain with two hyperboloidal mirrors, chosen to cancel both spherical aberration and coma, which gives a wider usable field; most large research telescopes use it.
• Gregorian. A concave ellipsoidal secondary placed beyond the primary focus, giving an upright image and an accessible intermediate focus.

The secondary mirror and its supports block part of the aperture. A secondary 33% of the primary diameter removes 11% of the collecting area and moves some energy from the central disk into the rings, lowering contrast at intermediate spatial frequencies. The straight vanes holding the secondary produce diffraction spikes on bright stars.

Resolving power

An aberration-free telescope images a star as an Airy disk, and two stars are resolved by the Rayleigh criterion when separated by

$$\theta = 1.22\,\frac{\lambda}{D}.$$

At 550 nm this is $138''/D$ with $D$ in millimeters: 1.4 arcseconds for 100 mm and 0.69 arcseconds for 200 mm. At the focal plane of an f/10 system the disk is $2.44\,\lambda N$ = 13.4 µm across, which sets the useful pixel size. Light grasp scales with $D^2$: a 200 mm aperture collects about 800 times as much light as a 7 mm pupil.

Ground-based telescopes larger than about 10–20 cm rarely reach their diffraction limit for long exposures, because atmospheric turbulence blurs images to roughly 0.5–2 arcseconds at good to average sites. Adaptive optics measures the wavefront distortion and corrects it with a deformable mirror, recovering images close to the diffraction limit in the infrared.

Common questions

What is the maximum useful magnification of a telescope?

A common rule of thumb is about twice the aperture in millimeters, 200× for 100 mm. Above that, the Airy disk is enlarged beyond what the eye needs to see it, and the image becomes dimmer without showing more detail.

Why is a Cassegrain shorter than its focal length?

The convex secondary intercepts the converging beam before the primary focus and reduces its convergence, which lengthens the effective focal length while folding the path back along the tube. The tube is roughly the primary-to-secondary spacing plus the back focus behind the primary.

Does a larger telescope magnify more?

Magnification depends only on the ratio of objective and eyepiece focal lengths. A larger aperture collects more light and resolves finer detail, which makes higher magnifications useful.

References: E. Hecht, Optics, 5th ed. (Pearson, 2017); D. J. Schroeder, Astronomical Optics, 2nd ed. (Academic Press, 2000); W. J. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2008).