You know from Refraction and Lenses that a lens bends light to a focus. A telescope uses a big lens (a refractor) or a curved mirror (a reflector) to bring light from a distant object to a focus, where an eye, a camera or a spectrograph collects it. Every large research telescope today is a reflector, because a mirror can be supported from behind and has no color fringes.
Why it works
Most things in the sky are faint, so a telescope's first job is to gather light. It catches every photon that lands on its mirror, so the light gathered is proportional to the area, $\pi D^2/4$: double the diameter, four times the light. Its second job is to resolve detail. Light spreading as waves (diffraction) blurs every point into a tiny disk, of angular size about $1.22\lambda/D$ radians; in arcseconds, $\theta \approx 2.5 \times 10^{5}\,\dfrac{\lambda}{D}$. A wider aperture makes a smaller blur. Magnification, $f_{\text{objective}}/f_{\text{eyepiece}}$, comes third: it only spreads out whatever light and detail the aperture captured.
Key idea
Light-gathering power grows as $D^2$; the finest resolvable angle is about $2.5 \times 10^{5}\,\lambda/D$ arcseconds. Both depend on the aperture $D$, which is why astronomers build ever larger mirrors.
Worked examples
Worked example: A big mirror against your eye
How much more light does the $10$ m Keck telescope collect than a dark-adapted eye with a $7$ mm pupil?
- Same units: $10$ m $= 10{,}000$ mm.
- Ratio of areas $= \left(\frac{10{,}000}{7}\right)^2 = 1429^2 = 2.0 \times 10^{6}$.
- About two million times the light: it sees objects millions of times fainter.
Answer: About $2 \times 10^{6}$ times.
Worked example: Hubble's sharpness
What is the finest detail the $2.4$ m Hubble Space Telescope can resolve at $500$ nm?
- $\lambda = 500$ nm $= 5.00 \times 10^{-7}$ m.
- $\theta = 2.5 \times 10^{5} \times \frac{5.00 \times 10^{-7}}{2.4} = 0.052$ arcseconds.
- Ground telescopes this size are blurred by the turbulent air to about $0.5$ to $1$ arcsecond, unless they use adaptive optics; Hubble, above the air, reaches its full sharpness.
Answer: About $0.05$ arcseconds.
Common mistake
Many people think a telescope's main job is to magnify. Stars are so far away that even the biggest telescope shows them as points; what it shows is more of them, fainter ones, and finer detail in extended objects. Extra magnification without more aperture only makes a bigger, dimmer blur. Check: telescopes are named by their diameter (a "10 m telescope"), never by their magnification.
Try lenses and focal lengths on the Optics Bench.