Light as Information

Section 8 of 9

Telescopes as Light Buckets

Part 8: Telescopes as Light Buckets

Telescopes extend our reach two ways: they collect more photons and resolve finer detail.

Collecting Area: Why Bigger Is Better

For a circular aperture, the collecting area is:

Collecting area

The light-gathering area of a telescope’s aperture, $A = \pi(D/2)^2$ . The photon rate from a source scales with $A \propto D^2$ , so aperture — not magnification — sets how faint an object a telescope can detect.

The photon rate scales with area, so the faintest detectable source scales as $D^2$ : a mirror twice as wide gathers four times the light.

Two telescopes: small diameter D collects few photons (shown as dots), large diameter 2D collects four times as many photons. Caption shows Area proportional to D^2. — Figure 17Collecting area scales as diameter squared. A telescope twice as wide collects four times as much light.Illustration: A. Rosen (SVG)

Problem

Using $A \propto D^2$ : (1) if diameter doubles, area does what? (2) Triples? (3) JWST ( $D = 6.5$ m) versus Hubble ( $D = 2.4$ m)?

Angular Resolution: Why Bigger Is Sharper

Even with lots of light, you can’t see detail finer than your angular resolution. Diffraction sets the limit:

Angular resolution

The smallest angular separation a telescope can distinguish, set by diffraction: $\theta_{\min} \approx \lambda/D$ (the circular-aperture form is $1.22\,\lambda/D$ ). Larger apertures and shorter wavelengths give finer resolution.

For a circular aperture the precise form is $\theta \approx 1.22\,\lambda/D$ ; for scaling arguments we use $\lambda/D$ . Bigger $D$ and shorter $\lambda$ both sharpen the image.

Figure 18Angular resolution improves with larger diameter. The diffraction limit theta = lambda/D sets the finest detail a telescope can distinguish.Illustration: A. Rosen (SVG)

Problem

Using $\theta \approx \lambda D^{-1}$ : (1) if diameter doubles (fixed $\lambda$ ), resolution does what? (2) If wavelength doubles (fixed $D$ )? (3) Observing at $\lambda = 2\ \mu$ m instead of $0.5\ \mu$ m with the same telescope?

Ground-based optical telescopes often miss the diffraction limit because atmospheric turbulence blurs images — seeing. Adaptive optics corrects some blur in real time; space telescopes avoid the atmosphere entirely. Large modern telescopes are reflectors: mirrors can be supported from behind (even segmented), making huge apertures practical, while large lenses sag under their own weight and suffer chromatic aberration.

Seeing

The blurring of astronomical images by turbulence in Earth’s atmosphere, which makes ground-based resolution (typically $\sim 1$ arcsec) far worse than the diffraction limit. Adaptive optics or space telescopes overcome it.