See how a telescope works, ray by ray

1. A converging lens has a focal length

Rays that arrive parallel to the axis leave a convex lens heading for one point, the focus, one focal length f behind it. Rays that arrive parallel but tilted by a small angle θ meet in the same plane, a distance f·tan θ off the axis. That plane is where the objective paints a tiny real picture of the sky.

2. Why you need an eyepiece

Your eye is built to focus parallel light. The eyepiece is a small lens placed one eyepiece focal length from the objective's focal plane: in a Keplerian telescope or a reflector it receives the rays after they have crossed at the real image, so they reach it diverging; in a Galilean telescope it intercepts them while they are still converging. Either way it makes them parallel again. Two focal points coincide, which is why the tube length is f_obj + f_eye, with the sign of f_eye deciding whether that means longer or shorter.

3. Angular magnification

A star one degree from another enters the objective one degree apart. After the eyepiece the two directions are M times further apart, with M = -f_obj / f_eye. The sign is the orientation: negative means the picture is rotated by 180°. A narrower exit beam is not a smaller image; it is the same image seen by a smaller pupil. The lab computes the exit angle exactly within the model and shows the small-angle value M·θ next to it, because above about 20° the two part ways and the paraxial model used here stops being valid.

4. Real image, virtual crossing

In a Keplerian telescope or a reflector the rays really cross inside the instrument: put a screen there and you see the picture. That is a real image, and because the rays crossed, it is inverted. In a Galilean telescope the diverging eyepiece sits before that crossing; the dashed rays in the lab show where the crossing would have been. No real image, no inversion, and no place to put a measuring reticle.

5. Mirrors reflect, they do not refract

A curved mirror focuses by reflection: at every point the ray leaves at the same angle to the surface normal at which it arrived, and for a parabola all the parallel rays meet exactly at one focus. No glass is crossed, so every colour focuses at the same place, and the mirror can be supported from behind and made huge. The price is that something has to sit in the beam to bring the focus out: a flat diagonal in the Newtonian, a convex secondary and a hole in the primary in the Cassegrain. That secondary shadows the centre of the beam, which costs a little light and some contrast (it changes the diffraction pattern), but it does not blur the image.

6. Refraction happens at both surfaces

The single bend in the main drawing is the thin-lens model. A real lens bends the light twice: entering the glass (air to glass, the ray turns towards the normal) and leaving it (glass to air, it turns away). Both follow Snell's law, n₁·sin θ₁ = n₂·sin θ₂, where the normal at each point is the line to the centre of curvature of that surface. Open a lens detail in the lab to see both bends, and notice that rays near the edge of a spherical lens focus a little short: that is spherical aberration, and it is why real objectives are more than one piece of glass.

7. What this model leaves out

Paraxial means small angles: slopes are treated as tangents and every element is a plane. Colour, aberrations, diffraction and the eye's own pupil are not modelled. Lengths are real millimetres and every surface is finite, so a ray that misses a mirror or the eyepiece rim stops where it hits; but the picture stretches the vertical scale so that a 0.5° star is visible at all. Treat the numbers as the textbook formulas made visible, not as a lens designer's ray trace.