Understanding Diffraction in Telescopes

When light passes through a circular aperture it spreads into a diffraction pattern rather than forming a point. The central bright region, known as the Airy disk, has an angular radius $\theta$_R=1.22λD. Two stars separated by less than this angle blur together. This page upgrades the classic Rayleigh criterion calculator by coupling the formula to a dynamic canvas that renders two moving Airy disks. Users can watch as the star separation oscillates and determine whether the peaks are distinguishable. Such visual feedback makes the abstract resolution limit intuitive.

Variables and Assumptions

The simulation assumes a circular unobstructed aperture of diameter $D$ and monochromatic light of wavelength $\lambda$. The stars are equally bright point sources. Their angular separation $\theta$ follows a simple harmonic motion defined by and where $\omega$ is set to 1 rad/s for clarity. Numerical integration uses a fourth-order Runge–Kutta (RK4) scheme for stability.

Core Equations

Each Airy intensity pattern is computed with the Fraunhofer diffraction formula $I=(\frac{2J}{}$₁(x)x)² where and $J$₁ is the Bessel function approximated as $J$₁(x)=-. The combined intensity is the sum of the two star patterns. We also track the oscillator energy $E=\frac{1}{2}$ to expose numerical drift.

Numerical Scheme

The star separation dynamics are advanced with RK4. At each frame the state vector $(\theta ,v)$ is updated using the time step $\mathrm{\Delta t}$. RK4 combines four slope evaluations: $k$₁=f(θ,v), $k$₂=f(θ+{Δt}{2}k_1), and so on, leading to excellent energy conservation for modest $\mathrm{\Delta t}$. Users can experiment with the time step to see how coarse integration increases the energy error $\Delta E\backslash /E\_0$.

Worked Example

Suppose $D=1\backslash mathrm\{m\}$, $\lambda =550\backslash mathrm\{nm\}$, and separation amplitude equals one Rayleigh angle. The simulation begins with the stars at maximum separation and zero velocity. As it runs, the stars approach, momentarily overlap, and recede. The canvas shows the twin Airy disks merging into one when . The energy overlay confirms that RK4 conserves $E$ to better than 0.1% for $\mathrm{\Delta t}=0.016$ s. A CSV download captures time, separation, and energy for further analysis.

Comparison Table

How to Read the Animation

The canvas renders intensity with brightness. Two bright spots signify resolvable stars, while a single blob indicates blending. A vertical dashed line marks one Rayleigh radius. Energy is displayed as a horizontal bar: green shows kinetic portion, blue potential, and a thin red band marks drift from the initial total.

Limitations

This model neglects atmospheric seeing, optical aberrations, and finite star size. The Bessel approximation breaks down for very large angles. The oscillator is an idealization; real binaries may have eccentric orbits or unequal brightness. Numerical errors grow if $\mathrm{\Delta t}$ exceeds about 0.05 s.

Extensions

Future versions could include adjustable damping, different aperture shapes, or a phase-space plot of $(\theta ,v)$. Adding photon noise would illustrate signal-to-noise limits.

References

• Born & Wolf, Principles of Optics
• Goodman, Introduction to Fourier Optics

Related calculators: double-slit interference, wavelength-frequency converter, Rayleigh–Taylor growth rate.