About Angular Resolution
Angular resolution is the smallest angle between two point sources of light that an optical system — a telescope, camera lens, microscope, or the human eye — can still record as two separate points instead of one blurred blob. It is a diffraction limit: even a flawless, perfectly focused lens cannot beat it, because light bends slightly as it passes through any finite opening (aperture) and spreads each point source into a small disk-shaped pattern called an Airy disk. When two Airy disks overlap too much, the two sources merge into one.
The Rayleigh criterion
The standard formula for the diffraction limit of a circular aperture is the Rayleigh criterion:
θ = 1.22 × λ / D
Here θ is the angular resolution in radians, λ (lambda) is the wavelength of the light being observed, and D is the diameter of the aperture — the objective lens or mirror, or the pupil of an eye. λ and D must be expressed in the same length unit before dividing; the constant 1.22 comes from the position of the first dark ring in the diffraction pattern of a circular opening. Because θ is normally a tiny fraction of a radian, it is usually converted to arcseconds by multiplying by 206,265 (the number of arcseconds in a radian), or to arcminutes by dividing that result by 60.
Reading the result
A smaller θ means finer resolution — the system can separate closer-together details. Two things shrink θ: a larger aperture D (more light-gathering opening narrows the diffraction pattern) or a shorter wavelength λ (blue light resolves better than red light through the same aperture). This calculator also converts θ into a linear distance at a chosen target range, using linear resolution = θ (radians) × distance, which answers a practical question: how far apart do two features need to be, at that distance, to appear as separate rather than blurred together.
Working with units
- Wavelength is entered in nanometers (nm); visible light spans roughly 380–700 nm, with 550 nm representing the middle of the visible spectrum (green-yellow light, near where the eye is most sensitive).
- Aperture diameter can be entered in millimeters, centimeters, or inches — the calculator converts internally to meters before applying the formula.
- Distance to the target is entered in kilometers and is only used to compute the optional resolvable-separation result; it does not affect the angular resolution itself.
Knowing the limits of this formula
The Rayleigh criterion assumes a circular, unobstructed, diffraction-limited aperture and light of a single wavelength — it is a theoretical best case. Real optical systems can fall short of it due to lens aberrations, atmospheric turbulence (the "seeing" that blurs ground-based astronomical images), sensor pixel size, or focus error, so the calculated θ is a floor on blur, not a guarantee of achieving it. Obstructed apertures (such as reflecting telescopes with a secondary mirror) and non-circular openings shift the diffraction pattern slightly, and the Rayleigh criterion itself is a convention — a closely related quantity, the Abbe diffraction limit used in microscopy, defines resolution slightly differently.
Frequently Asked Questions
What formula does this calculator use?
Why does a bigger aperture improve resolution?
How does wavelength affect angular resolution?
What does the resolvable separation at distance mean?
Common Tip Percentages
| Service | Typical Tip Range | Notes |
|---|---|---|
| Restaurant (sit-down) | 15–20% | 20%+ for exceptional service |
| Food delivery | 10–20% | Consider extra for long distances |
| Taxi / rideshare | 10–20% | More for assistance with luggage |
| Hotel housekeeping | $2–5/night | Leave daily, not just at checkout |
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General Household Budget Ratios (50/30/20 Rule)
| Category | % of Net Income | Examples |
|---|---|---|
| Needs | 50% | Rent, groceries, utilities, insurance |
| Wants | 30% | Dining out, entertainment, hobbies |
| Savings / Debt | 20% | Emergency fund, retirement, extra payments |
Common Measurement Conversions for Everyday Use
| Quantity | US / Imperial | Metric Equivalent |
|---|---|---|
| 1 cup | 8 fl oz | 237 mL |
| 1 pint | 16 fl oz | 473 mL |
| 1 quart | 32 fl oz | 946 mL |
| 1 gallon | 128 fl oz | 3.785 L |
| 1 lb | 16 oz | 453.6 g |
| 1 ft | 12 in | 30.48 cm |