Animated Solution for Physics - Optics: Calculate the limit of resolution of a telescope objective having a diameter of $200\text{ cm}$, if it has to detect light of wavelength $500\text{ nm}$ coming from a star.

Imagine you are an astronomer, peering through a massive telescope into the deep, dark abyss of the night sky. You spot what looks like a single, bright point of light. But is it really just one star? Often, what appears as a single star to the naked eye is actually a binary star system—two stars orbiting each other, separated by vast cosmic distances, yet appearing incredibly close from our vantage point on Earth.

The ability of your telescope to reveal these two distinct points of light instead of a single blurred blob is governed by a fundamental principle of wave optics. This principle is known as the limit of resolution.

When light from a distant star passes through the circular aperture of a telescope's objective lens, it doesn't just form a perfect point. Instead, due to the phenomenon of diffraction, it forms a central bright spot surrounded by faint, concentric rings. This pattern is called an Airy disk. When two stars are very close, their Airy disks can overlap. If they overlap too much, they blend into one, and we cannot resolve them.

Our Mathematical Tool

To determine exactly when we can just barely distinguish two stars, we rely on the Rayleigh Criterion. Lord Rayleigh proposed that two point sources are just resolved when the principal diffraction maximum of one image coincides with the first minimum of the other.

Mathematically, this gives us a beautiful and elegant formula for the limit of resolution, denoted by the angular separation $\Delta \theta$:

Here, $\lambda$ represents the wavelength of the light we are observing, and $D$ is the diameter of the telescope's objective lens. The factor of $1.22$ arises from the complex mathematics of Bessel functions used to describe diffraction through a circular aperture.

This equation tells us a profound story: to get a smaller, better limit of resolution (meaning we can distinguish closer objects), we need either a smaller wavelength or a larger telescope diameter.

The Importance of Units

Now, let's bring this cosmic concept down to Earth and apply it to our specific problem. We are given the parameters of our telescope and the light it's detecting.

The wavelength of the light from the star is given as:

The diameter of the telescope's objective lens is:

Before we rush into plugging these numbers into our formula, we must heed a critical warning: always ensure your units are consistent! Mixing nanometers and centimeters is a guaranteed recipe for a silly mistake. Let's convert everything into the standard SI unit of meters.

For the wavelength:

For the diameter:

Now, our stage is set. We can substitute these pristine values into the Rayleigh criterion equation:

Let's break down the computation step-by-step to avoid any algebraic traps. First, let's handle the numerical coefficients separate from the powers of ten.

We have $500$ in the numerator and $200$ in the denominator. We can easily simplify this fraction:

Now, let's substitute this back into our expression:

Next, we multiply the coefficients:

And we resolve the powers of ten using the laws of exponents. When dividing, we subtract the exponents:

Combining these results, we arrive at our raw angular resolution:

We have our answer, but we must present it in the format requested by the options. The options are all expressed in terms of $10^{-9}\text{ rad}$.

To convert our answer, we need to shift the decimal point two places to the right, which means we must decrease the exponent by two:

Looking at our choices, this perfectly matches option (b).

This problem is more than just a calculation; it's a window into the engineering of modern astronomy. Why do space agencies spend billions of dollars to build telescopes with massive mirrors, like the James Webb Space Telescope? Because a larger diameter $D$ directly shrinks the limit of resolution $\Delta \theta$, allowing us to peer deeper into the universe and resolve the intricate details of distant galaxies and nebulae. The math we just performed is the exact same math that guides the exploration of the cosmos!