Question

Which close pair of stars will be more easily resolvable with a telescope: two red stars or two blue ones? Assume the binary star systems are the same distance from Earth and are separated by the same angle.

Short answer

Explain your answer. Answer: A close pair of blue stars is more easily resolvable with a telescope than a close pair of red stars. This is because blue light has a shorter wavelength compared to red light, leading to a better angular resolution according to the Rayleigh Criterion.

Step-by-step solution

Understanding the Rayleigh Criterion

The Rayleigh Criterion states that the minimum angular separation (θ) between two objects that can be resolved by a telescope is given by θ = 1.22 * (λ / D), where λ is the wavelength of light being observed and D is the aperture size of the telescope.

Comparing wavelengths of red and blue light

Red light has a longer wavelength (approximately 700 nm) compared to blue light (approximately 450 nm). Since the Rayleigh Criterion states that the minimum angular separation is directly proportional to the wavelength, the larger the wavelength, the larger the minimum angular separation will be between objects.

Applying the Rayleigh Criterion for red and blue stars

Now, we will apply the Rayleigh Criterion for both red and blue stars. Since the binary star systems are separated by the same angle and are at the same distance from Earth, the aperture size of the telescope (D) remains the same for both cases. For red stars: θ_red = 1.22 * (λ_red / D) For blue stars: θ_blue = 1.22 * (λ_blue / D)

Comparing angular resolutions of red and blue stars

We can now compare the angular resolutions for red and blue stars. Since λ_red > λ_blue, θ_red > θ_blue. This shows that the angular resolution for red stars is worse than the resolution for blue stars.

Concluding which pair of stars is more easily resolvable

Since the angular resolution for blue stars is better than the resolution for red stars (θ_blue < θ_red), it is easier to resolve a close pair of blue stars with a telescope than a close pair of red stars.

Key concepts

Angular Separation

When we observe the night sky with a telescope, we often want to distinguish between two closely spaced objects, such as stars in a binary system. The ability of a telescope to tell two such objects apart is quantified by the concept of angular separation. The smaller the angle between the two objects as seen from the observer's viewpoint, the greater the challenge it is to resolve them individually.

According to the Rayleigh Criterion, the minimum angular separation \( \theta \) that can be distinguished is related to both the wavelength of light \( \lambda \) and the telescope aperture size \( D \). The criterion is expressed as \[ \theta = 1.22 \times \frac{\lambda}{D} \. \
This principle is essential for astronomers who need to differentiate between celestial objects in the vast expanse of the universe. Understanding the relationship between the wavelength of light and the aperture size helps in choosing the right telescope for specific observations and also explains why certain objects may be easier to resolve than others.

Telescope Aperture Size

When discussing telescopes and their capabilities, the term aperture size frequently comes up. The aperture of a telescope is the diameter of its primary light-collecting element, which can be a lens or a mirror. This size is crucial because, in general, a larger aperture allows a telescope to gather more light, leading to brighter and sharper images.

However, the telescope's aperture also plays a pivotal role in determining its resolving power—that is, the ability to see fine details and distinguish between objects that are close together in the sky. According to the Rayleigh Criterion, mentioned earlier, there is an inverse relationship between the resolving power and the aperture size \( D \): a larger \( D \) results in a smaller minimum angular separation \( \theta \).

This is why larger telescopes are highly sought after in astronomy. They not only deliver more detailed images, but their increased resolving power enables astronomers to observe and separate objects that would otherwise appear as a single point of light in smaller telescopes.

Wavelength of Light

Light, the primary carrier of information from the cosmos to our eyes and instruments, has different wavelengths that correspond to different colors. The wavelength of light \( \lambda \) is a fundamental concept in understanding optical phenomena, including the operation of telescopes.

Red light has a longer wavelength than blue light—approximately 700 nanometers (nm) for red, compared to about 450 nm for blue. The wavelength affects not only the color we perceive but also the angular resolution according to the Rayleigh Criterion. Since the criterion stipulates that minimum angular separation is proportional to \( \lambda \), light of a longer wavelength will have a larger \( \theta \) and thus worse angular resolution.

This explains why a telescope would more easily resolve a pair of blue stars compared to a pair of red stars, as demonstrated in the exercise provided. The shorter the wavelength of light we are observing, the finer the details we can discern. This principle guides astronomers in selecting different types of telescopes and observational techniques depending on the nature of the objects they are studying and the level of detail required.