Telescope Magnification
The relationship $M = f_o/f_e$ applies to both refracting and reflecting telescopes in normal adjustment, and exam questions routinely test it with direct numerical calculations. But chasing the biggest possible number for M turns out to be a trap if the rest of the telescope can't back it up.
Applying M = f_o / f_e
In normal adjustment, the angular magnification of an astronomical telescope — refracting or reflecting — is given by the ratio of the objective's focal length to the eyepiece's focal length:
For a reflecting telescope, $f_o$ is the effective focal length of the primary mirror system rather than a single lens, but the same ratio applies once light reaches the eyepiece from the equivalent intermediate image. This makes the calculation identical in form for both telescope types, which is why exam questions often test the two designs with the same style of numerical problem.
The same approach works in reverse: given a required magnification and one focal length, the other focal length follows immediately by rearranging $M = f_o/f_e$. Questions may also give the tube length and one focal length and ask for the other, using tube length $= f_o + f_e$ alongside the magnification equation.
Why magnification alone isn't the whole story
It's tempting to treat magnification as the single number that describes how "powerful" a telescope is, but a telescope with a very high magnification and a small aperture can actually produce a worse image than one with lower magnification and a larger aperture. Two separate problems appear if the aperture is too small for the magnification being used:
- Dimness. A small aperture collects relatively little light. Spreading that limited light over a highly magnified image spreads its brightness thinner still, so high-magnification views through a small telescope often look dim and washed out, especially for faint objects.
- Poor resolution. Magnifying an image doesn't add detail that isn't there to begin with. If the telescope's aperture is too small to resolve fine detail in the first place, increasing the eyepiece magnification just enlarges the existing blur rather than revealing anything new — sometimes called "empty magnification."
A genuinely good telescope image needs the magnification to be matched to what the aperture can actually resolve and how much light it can gather. That second factor — exactly how an aperture's size limits the fineness of detail a telescope can show — is the subject of the next lesson.
Magnification describes how much an image is enlarged, but it says nothing on its own about image quality. A telescope's usable magnification is limited by its aperture diameter, because aperture governs both how much light is collected and how finely detail can be resolved — pushing magnification beyond what the aperture supports just produces a bigger, dimmer, blurrier image.
Key takeaways
- $M = f_o/f_e$ applies to both refracting and reflecting telescopes in normal adjustment.
- Given any two of $M$, $f_o$, and $f_e$, the third can be found by rearranging the magnification equation.
- Tube length in normal adjustment is $f_o + f_e$, letting problems link magnification to physical telescope length.
- A high magnification with a small aperture produces a dim image, since limited collected light is spread over a larger, more magnified image.
- A high magnification with a small aperture also cannot reveal detail the aperture isn't able to resolve — "empty magnification."
- Image quality depends on matching magnification to the telescope's aperture, not on maximising magnification alone.
- Aperture size is the deciding factor for both light-gathering and resolving power, covered next.