The Stellar Nucleosynthesis Detective · AI Generator #GEN-1B6E1D16

Physics · modern-physics

Problem

Astronomers analyzing the spectrum of a distant star discover two characteristic X-ray emission lines that provide clues about the nuclear processes occurring in its core. The first line has wavelength

\lambda_1 = 0.154\;\text{nm}

and corresponds to the

K_\alpha

transition in an unknown element. The second line has wavelength

\lambda_2 = 0.071\;\text{nm}

from the same transition type in a different element. Moseley's law states that the frequency of characteristic X-ray lines follows

\nu = R(Z - \sigma)^2\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)

, where

R = 3.29 \times 10^{15}\;\text{Hz}

is the Rydberg frequency,

Z

is the atomic number, and

\sigma

is the screening constant. For

K_\alpha

transitions (

2p \to 1s

), the screening constant is

\sigma = 1.0

for both elements. During the stellar fusion process, the lighter element is being converted into the heavier one through a series of nuclear reactions. (a) Identify the two elements by determining their atomic numbers. (b) If the star converts

1.5 \times 10^{26}\;\text{kg}

of the lighter element into the heavier element per second, and each fusion sequence releases

26.7\;\text{MeV}

of energy, calculate the luminosity (power output) of this fusion process in watts.

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