Main characteristics of telescopes
A telescope’s main quantity is the objective diameter D. F is the focal length of the objective and f of the eyepiece; the angular magnification is Γ = F / f. Light-gathering power is proportional to D²: for an eye pupil d ≈ 7 mm a telescope collects (D / d)² times more light than the eye, so faint stars become visible; doubling D lets us see stars ≈ 1.5 magnitudes fainter (5 lg 2 ≈ 1.5). Resolving power (seeing two close points separately) is limited by diffraction: θ ≈ 1.22 λ / D (in radians); for visible light θ ≈ 140″ / D(mm). Merely raising magnification does not improve resolution – D decides it; the useful magnification is also limited to about 1–2 · D(mm). The Earth’s atmospheric turbulence limits large telescopes to ≈ 1″ resolution, while adaptive optics and space telescopes approach the diffraction limit.
Read the marking on a pair of binoculars (e.g. 8 × 40): 8 is the magnification, 40 the objective diameter (mm). For a 7 mm pupil calculate the light-gathering power. Never look at the Sun through binoculars.