Resolving Power and Diffraction
Even a perfectly made telescope, with every aberration corrected, still cannot form an ideally sharp point image of a star. Diffraction — the same bending of waves around obstacles and through gaps seen with sound and water waves — sets a hard limit on how much fine detail any telescope can ever resolve.
Diffraction at the aperture
Light entering a telescope has to pass through a circular opening — the objective lens or the primary mirror, whichever collects the incoming radiation. Any wave passing through a finite-sized opening diffracts, spreading out slightly rather than continuing in perfectly straight lines. Because of this, even light from an ideal, infinitely distant point source is never brought to a perfect point image. Instead it forms a small diffraction pattern: a central bright spot surrounded by faint concentric rings, spreading over a small but non-zero angular size.
This matters enormously for observing two objects that are close together in the sky, such as a close double star or two adjacent surface features on a planet. Each source produces its own diffraction pattern, and if the two patterns overlap too much, they merge into a single blur that can no longer be told apart as two distinct sources — no matter how much the resulting image is magnified afterwards.
The Rayleigh criterion
The generally accepted criterion for when two point sources can still just be told apart is the Rayleigh criterion: two sources are said to be "just resolved" when the central maximum of the diffraction pattern from one source falls exactly on the first minimum of the diffraction pattern from the other. Any closer together than this, and the two patterns overlap so much that they can no longer be distinguished as separate.
For a circular aperture, this condition gives a minimum angular separation, $\theta$, below which two sources cannot be resolved:
where $\theta$ is the minimum resolvable angular separation in radians, $\lambda$ is the wavelength of the radiation being observed, and $D$ is the diameter of the telescope's aperture (the objective lens or primary mirror). The radian is the natural unit of angle here, and you should be comfortable converting between radians and degrees when applying this relationship, since $\theta$ is almost always required in radians for the formula to hold.
Since $\theta \approx \lambda/D$, achieving a smaller (better) minimum resolvable angle requires either observing at a shorter wavelength or using a larger aperture diameter $D$. The observing wavelength is usually fixed by what's being studied, so in practice the main lever available to telescope designers is aperture size — which is the central reason astronomical telescopes, across every part of the spectrum, are built as large as engineering and cost allow.
Collecting power: the second reason size matters
Aperture size affects a telescope in a second, independent way besides resolving power. Collecting power describes how much radiation a telescope can gather from a source — essentially how bright or faint an object needs to be for the telescope to detect it at all. Collecting power is proportional to the area of the aperture, and since the area of a circular aperture is proportional to $D^2$:
Doubling a telescope's aperture diameter doesn't just improve its resolving power (via $\theta \propto 1/D$) — it also quadruples the amount of radiation it collects, since area scales with the square of diameter. These two effects are independent but reinforce each other: a larger aperture produces both sharper images and the ability to detect fainter, more distant objects, which is why the drive toward ever-larger telescope apertures runs through the entire history of observational astronomy.
Aperture diameter $D$ governs two independent aspects of telescope performance: resolving power, through $\theta \approx \lambda/D$ (smaller $\theta$ is better, so larger $D$ helps), and collecting power, through area $\propto D^2$ (larger $D$ gathers more radiation). Both push telescope design in the same direction — bigger apertures are better on both counts — which is the fundamental reason observatories and space telescopes keep getting larger.
Key takeaways
- Diffraction at a telescope's circular aperture means even a point source forms a small diffraction pattern, not a perfect point image.
- The Rayleigh criterion: two sources are just resolved when the central maximum of one diffraction pattern coincides with the first minimum of the other.
- The minimum resolvable angular separation is $\theta \approx \lambda/D$, with $\theta$ in radians, $\lambda$ the observing wavelength, and $D$ the aperture diameter.
- Better resolution (smaller $\theta$) requires a shorter wavelength or, more practically, a larger aperture $D$.
- Collecting power — how much radiation is gathered, setting how faint an object can be detected — is proportional to $D^2$, the aperture's area.
- Resolving power and collecting power are independent benefits of a larger aperture, both favouring bigger telescopes.
- The radian is the standard unit of angle used throughout resolving-power calculations.