Structure and scale of the universe：練習問題

Which of the following celestial structures represents the largest scale in the hierarchy of the universe?

A star is observed to have a very high surface temperature, typically above \(30000 \text{ K}\), and emits a significant amount of its light in the blue-violet part of the spectrum. To which of the following spectral classes would this star most likely belong?

On a Hertzsprung-Russell (H-R) diagram, a star is found to be located in the top-right region. This region is characterized by high luminosity and relatively low surface temperature. Which of the following types of stars best fits this description?

An astronomer measures the parallax angle of a distant star to be \(0.1 \text{ arcsecond}\). What is the distance to this star from Earth, expressed in parsecs (pc)?

Three stars, P, Q, and R, are observed to have the following properties:

• Star P has a surface temperature of \(T_0\) and a luminosity of \(L_0\).
• Star Q has a surface temperature of \(2T_0\) and a luminosity of \(L_0\).
• Star R has a surface temperature of \(T_0\) and a luminosity of \(4L_0\).

Given that a star's luminosity \(L\), radius \(R\), and surface temperature \(T\) are related by the Stefan-Boltzmann law, \(L = 4\pi R^2 \sigma T^4\), where \(\sigma\) is the Stefan-Boltzmann constant, which of the following statements correctly ranks the stars by their radius from smallest to largest?

Explain the difference between a galaxy and a cluster of galaxies in terms of their primary components, and identify which structure occupies a higher level in the hierarchy of the universe.

Define the term light year and calculate its value in astronomical units (AU), assuming the distance of \(1 \text{ AU} = 1.50 \times 10^8 \text{ km}\) and the speed of light is \(3.00 \times 10^8 \text{ m s}^{-1}\). Briefly explain why the AU is unsuitable for measuring the spatial scale of galaxies.

If a star is located at a distance of 4.0 parsecs ( ext{pc}) from Earth, approximately how far is this star measured in light-years ( ext{ly})? (Given that 1 \(\text{ pc} \approx 3.26 \text{ ly}\)\).

The hierarchical structure of the universe is organized such that celestial bodies of smaller scales are constituents of larger-scale structures.

(a) Calculate the value of 1 astronomical unit (AU) in light-years (ly). Take \(1 \text{ AU} = 1.50 \times 10^{11} \text{ m}\), the speed of light \(c = 3.00 \times 10^8 \text{ m s}^{-1}\), and \(1 \text{ year} = 3.15 \times 10^7 \text{ s}\). (2 points)
(b) Describe the hierarchy of the following four entities by arranging them from the smallest to the largest spatial scale: star cluster, planet, galaxy, and cluster of galaxies. (2 points)
(c) Using the "Powers of Ten" concept, determine how many orders of magnitude larger a typical galaxy (diameter \(\approx 10^5 \text{ ly}\)) is compared to the distance between the Sun and Proxima Centauri (distance \(\approx 4.2 \text{ ly}\)). Explain why a nebula is classified as a structure within a galaxy rather than a structure that contains galaxies. (3 points)