Life cycle of stars：練習問題

Two stars, Alpha and Beta, are observed to have the same total luminosity. Star Alpha has a surface temperature of \(20000 \text{ K}\) and Star Beta has a surface temperature of \(5000 \text{ K}\). Assuming both stars radiate as blackbodies, what is the ratio of their radii, \(R_{\text{Beta}} / R_{\text{Alpha}}\)?

Stars with initial masses significantly greater than the Sun, for example, eight times the Sun's mass (\(8 M_\odot\)), have different life cycles. Which of the following is the most probable ultimate fate for such a high-mass star after exhausting its nuclear fuel?

An O-type star has a surface temperature of \(40000 \text{ K}\) and a radius 15 times that of the Sun. Given that the Sun's surface temperature is \(5800 \text{ K}\), calculate the approximate ratio of the luminosity of the O-type star to the Sun's luminosity, \(L_{\text{O-star}} / L_{\odot}\).

What is the primary factor that determines the main sequence lifetime of a star?

A supergiant star is 100 times more luminous than a main sequence star of the same spectral class G (meaning they have the same surface temperature \(T\)). If the main sequence star has a radius \(R_M\), express the radius of the supergiant star \(R_S\) in terms of \(R_M\).

Consider two main sequence stars, Star X and Star Y. Star X has a mass $$M_X$$ and a main sequence lifetime $$T_X$$. Star Y has a mass $$M_Y = 3M_X$$. Given that a star's luminosity is approximately proportional to the cube of its mass ($$L \propto M^3$$) and its main sequence lifetime is inversely proportional to its luminosity and directly proportional to its mass ($$T \propto M/L$$), calculate the main sequence lifetime of Star Y in terms of $$T_X$$.

(a) The Hertzsprung-Russell (H-R) diagram plots stars according to their luminosity and surface temperature. Describe the general location of main-sequence stars and red giants on this diagram, and explain the key physical difference in how they generate energy. (3 points)

(b) After a star like our Sun exhausts the hydrogen fuel in its core, it evolves into a red giant. Describe the internal changes (e.g., core contraction, shell burning, outer layers expansion) that lead to this transformation. (3 points)

(c) A star X has a luminosity $$L_X$$ and a surface temperature $$T_X$$. Another star Y has a luminosity $$L_Y = 16 L_X$$ and a surface temperature $$T_Y = 2T_X$$. Using Stefan's Law ($$L = 4\pi R^2 \sigma T^4$$), determine the ratio of their radii, $$R_Y / R_X$$. (2 points)