Animated Solution for Physics - Optics: Diameter of the objective lens of a telescope is $250\text{ cm}$. For light of wavelength $600\text{ nm}$ coming from a distant object, the limit of resolution of the telescope is close to

Understanding the Limit of Resolution of a Telescope

Imagine you are an astronomer peering through a telescope at a distant binary star system. To the naked eye, they might look like a single point of light. But through the telescope, you hope to see two distinct stars. However, there is a fundamental physical limit to how well any telescope can separate, or "resolve," these two stars. This limit is not due to imperfect glass or bad engineering, but rather the wave nature of light itself.

When light from a distant star passes through the circular aperture (the objective lens) of a telescope, it doesn't focus into a perfect geometric point. Instead, it diffracts, creating a central bright spot surrounded by fainter concentric rings. This pattern is known as an Airy disk. If two stars are very close together, their Airy disks will overlap on the telescope's focal plane.

The Master Equation

To determine if we can distinguish the two stars, we use Rayleigh's Criterion. It states that two point sources are just resolved when the central maximum of one diffraction pattern coincides with the first minimum of the other. Mathematically, the minimum angular separation $\Delta \theta$ required to resolve the two objects is given by the formula:

Here, $\lambda$ is the wavelength of the light being observed, and $D$ is the diameter of the telescope's objective lens. The factor of $1.22$ arises from the complex mathematics of diffraction through a circular aperture (specifically, the first root of a Bessel function).

In our specific problem, we are given: - The diameter of the objective lens, $D=250\text{ cm}$. - The wavelength of the light, $\lambda =600\text{ nm}$.

Before we plug these numbers into our master equation, we must ensure our units are consistent. The standard SI unit for length is the meter ($\text{m}$). Let's convert both values:

Now, we substitute these pristine values into Rayleigh's formula:

Let's break down the arithmetic step-by-step to avoid any silly mistakes. First, we multiply the constants in the numerator:

So, our expression becomes:

Next, we divide $732$ by $2.5$. A quick mental math trick is to multiply both the numerator and denominator by $2$ to get a denominator of $5$, or multiply by $4$ to get a denominator of $10$:

Bringing back our power of ten, we have:

Finally, we convert this into standard scientific notation by shifting the decimal point two places to the left, which increases the exponent by $2$:

Rounding to two significant figures, we get our final answer:

Look closely at the formula $\Delta \theta =\frac{1.22\lambda }{D}$. The limit of resolution $\Delta \theta$ is inversely proportional to the diameter $D$. A smaller $\Delta \theta$ means the telescope can resolve finer details. This is the exact reason why modern observatories build massive telescopes with mirrors spanning several meters across. The larger the aperture, the sharper the view of the universe!