A distant galaxy has a redshift \(z=7.6\) a. What would be the observed wavelength of the Ha line \(\left(\lambda_{\text {rest }}=656.28 \mathrm{nm}\right) ?\) b. In what region of the spectrum would this line be located?

The observed wavelength \( \lambda_{\text{obs}} \) is the wavelength of light or other electromagnetic radiation observed after it has been affected by a phenomenon such as redshift. This is different from the rest wavelength \( \lambda_{\text{rest}} \) which is the wavelength of light emitted in the source's own rest frame. Redshift makes an emitted wavelength appear longer than it originally was. We can calculate the observed wavelength using the formula \ \( \lambda_{\text{obs}} = \lambda_{\text{rest}} (1 + z) \) \ where \( z \) represents the redshift factor. In simpler terms, if a light-wave source moves away from us, the waves stretch, resulting in a longer wavelength. For instance, in the given problem, the rest wavelength \( \lambda_{\text{rest}} \) is 656.28 nm, and redshift \( z \) is 7.6. Plugging these into the formula, we get the observed wavelength to be \(\approx 5633.208 \, nm\).
This calculation shows how we can infer the observed wavelength just by knowing the rest wavelength and the redshift.

The electromagnetic spectrum is the range of all types of electromagnetic radiation, from gamma rays with the shortest wavelengths to radio waves with the longest. Light that we can see is only a small part of this spectrum.
The spectrum is often divided into major regions, each with different properties:

• Gamma Rays: Shortest wavelength, highest energy
• X-Rays: Slightly longer wavelengths than gamma rays
• Ultraviolet (UV): Wavelengths shorter than visible light but longer than X-rays
• Visible Light: Wavelengths that the human eye can see (400-700 nm)
• Infrared (IR): Wavelengths longer than visible light but shorter than microwaves
• Microwaves: Wavelengths longer than infrared
• Radio Waves: Longest wavelength, lowest energy

The light from the distant galaxy in our problem falls into the infrared region because the observed wavelength, calculated to be about 5633.21 nm, is well within the infrared range (700 nm to 1 mm). This shows how understanding the electromagnetic spectrum helps us categorize and understand different types of light.

The infrared region of the electromagnetic spectrum lies between visible light and microwaves. It covers wavelengths approximately from 700 nm to 1 mm. Infrared light is not visible to the human eye but can be detected as heat. Some key points about the infrared spectrum include:

• Near-Infrared: Closest to visible light, covers wavelengths from about 700 nm to 2.5 microns
• Mid-Infrared: Wavelengths from 2.5 to 25 microns
• Far-Infrared: Further from visible light, covers from 25 microns to 1 mm

In our problem, the galaxy's redshifted Hα line, with an observed wavelength of approximately 5633.21 nm, falls within the mid-infrared region.
Studying the infrared region helps astronomers observe objects that are too cool to emit visible light, such as planets, stars in formation, and distant galaxies like the one in the problem. Understanding the infrared spectrum is crucial for various scientific applications, including astronomy, remote sensing, and even everyday uses like thermal cameras.