Question

When using a telescope with an objective of diameter \(12.0 \mathrm{~cm},\) how close can two features on the Moon be and still be resolved? Take the wavelength of light to be \(550 \mathrm{nm}\), near the center of the visible spectrum.

Short answer

Answer: The closest distance between two features on the Moon that can be resolved using this 0.12 m diameter telescope observing at a wavelength of 550 nm is approximately 2.16 kilometers.

Step-by-step solution

Calculate the angular resolution using the Rayleigh Criterion formula

The Rayleigh Criterion formula for calculating the angular resolution is: $$θ = \dfrac{1.22 * λ}{D}$$ where \(θ\) represents the angular resolution, \(λ\) represents the wavelength of light, and \(D\) represents the diameter of the telescope's objective. We are given the wavelength of light, \(λ = 550 \times 10^{-9} \mathrm{m}\), and the diameter of the objective, \(D = 0.12 \mathrm{m}\). Plugging these values into the Rayleigh Criterion formula, we get the angular resolution: $$θ = \dfrac{1.22 * (550 \times 10^{-9}\mathrm{m})}{0.12\mathrm{m}}$$

Convert the angular resolution from radians to arcseconds

Before calculating the distance between two features on the Moon using the angular resolution, we need to convert the angular resolution from radians to arcseconds, because arcseconds are a more commonly used unit for measuring angular distances in astronomy. To convert from radians to arcseconds, use the following conversion factor: 1 radian = 206265 arcseconds First, we need to evaluate the angular resolution \(θ\) we calculated: $$θ = \dfrac{1.22 * (550 \times 10^{-9}\mathrm{m})}{0.12\mathrm{m}} = 5.61 \times 10^{-6} \mathrm{radians}$$ Now, we convert this value to arcseconds: $$θ_{arcseconds} = 5.61 \times 10^{-6} \mathrm{radians} * 206265 \mathrm{\dfrac{arcseconds}{radian}} = 1.16 \mathrm{arcseconds}$$

Calculate the distance between two features on the Moon using the angular resolution and the distance to the Moon

Now we can use the angular resolution in arcseconds to calculate the distance between two features on the Moon's surface that can be resolved by this telescope. We will assume an average distance from Earth to the Moon of 384000 km. We can use the small angle approximation formula to relate the angular size, actual size, and distance: $$θ = \dfrac{d}{D}$$ where \(θ\) is the angular size in radians, \(d\) is the actual size of the features, and \(D\) is the distance between the observer and the features. We already converted the angular resolution into arcseconds, so we'll first convert it back to radians: $$θ = \dfrac{1.16 \mathrm{arcseconds}}{206265 \mathrm{\dfrac{arcseconds}{radian}}} = 5.62 \times 10^{-6} \mathrm{radians}$$ Now we can plug the values of \(θ\) and \(D\) (the Earth-Moon distance) into the small angle approximation formula and solve for \(d\): $$5.62 \times 10^{-6} \mathrm{radians} = \dfrac{d}{384000 \mathrm{km}}$$ $$d = (5.62 \times 10^{-6} \mathrm{radians}) * (384000 \mathrm{km}) = 2.16 \mathrm{km}$$ The closest distance between two features on the Moon that can still be resolved using this telescope is approximately 2.16 kilometers.

Key concepts

Rayleigh Criterion

The Rayleigh Criterion is a pivotal concept in optics, dealing with the ability of an imaging system, such as a telescope, to resolve fine details. Specifically, it defines the minimum angular separation at which two point light sources can be distinguished as separate when observed through an optical instrument.

The formula to calculate this critical separation is given by:
\r\[\theta = \frac{1.22 \times \lambda}{D}\]
\rwhere \( \theta \) is the angular resolution in radians, \( \lambda \) is the wavelength of the light being observed, and \( D \) is the diameter of the lens or mirror—known as the aperture—of the telescope rendering the image.

This criterion is named after the British physicist Lord Rayleigh, who provided a theoretical explanation for the expression. The factor 1.22 in the formula originates from a calculation involving the first zero of the Bessel function, which describes the diffraction pattern produced by a circular aperture. The Rayleigh Criterion is a fundamental limit to the detail achievable by optical systems, dictated by the physics of light diffraction.

Telescope Objective Diameter

The telescope's objective diameter, often referred to as aperture size, represents one of the main factors influencing the power of a telescope. Essentially, the objective diameter is the size of the main light-collecting lens or mirror of the telescope. A larger diameter means that the telescope can gather more light, improving its light-gathering capability and its resolving power—the ability to see fine details.

A larger aperture accomplishes two main objectives:

• It allows more light to enter the telescope, which enhances image brightness and allows for the observation of fainter objects.
• It increases resolving power, which enables the telescope to separate celestial objects that are close together, as determined by the Rayleigh Criterion.

When applying the Rayleigh Criterion to telescope observations, as seen in the angular resolution exercise, the diameter of the telescope's objective is a critical factor in the calculation and ultimately influences the level of detail that can be discerned in astronomical images. This makes the objective diameter of a telescope an important specification for amateur and professional astronomers alike.

Wavelength of Light

The wavelength of light is a fundamental concept in the study of optics and physics, relating to the nature of electromagnetic radiation. Light behaves both as a wave and as a particle, and its wavelength is the distance between two consecutive peaks or troughs in the light wave.

Visible light is simply a small portion of the electromagnetic spectrum, with wavelengths ranging from about 380 nm (violet) to 740 nm (red). The wavelength of light affects its color, with shorter wavelengths corresponding to blue and violet colors and longer wavelengths to red colors.

In the context of telescopes and resolving power, the wavelength of light being observed is crucial because it determines the diffraction limit of the system. Diffraction is the bending of light around the edges of an object or opening, such as a telescope's aperture. As demonstrated in the exercise, observing light with a wavelength of \(550 \mathrm{nm}\), near the center of the visible spectrum, allows for the calculation of angular resolution using the Rayleigh Criterion. The choice of wavelength directly influences the result, as a shorter wavelength yields better angular resolution and thus, a greater ability to discern finer details.