The Glowing Galaxy's Missing Mass · AI Generator #GEN-0556EA98

Physics · modern-physics

Problem

Astronomers observe a distant galaxy emitting characteristic X-rays with wavelength λ=0.154  nm\lambda = 0.154\;\text{nm}, which they identify as coming from highly ionized iron atoms that have lost 25 electrons (Fe25+^{25+}). Using Moseley's law and the Rydberg formula, they want to determine the rest mass of this galaxy. The observed X-ray photons are redshifted due to the galaxy's recession velocity v=0.12cv = 0.12c relative to Earth. The power output of the galaxy in this X-ray line is measured to be P=3.6×1031  WP = 3.6 \times 10^{31}\;\text{W}. If the galaxy converts matter to energy with an efficiency of η=0.7%\eta = 0.7\% (similar to nuclear fusion processes), what is the rest mass M0M_0 of the galaxy? Given constants: Rydberg constant R=1.097×107  m−1R = 1.097 \times 10^7\;\text{m}^{-1}, Planck's constant h=6.626×10−34  J⋅sh = 6.626 \times 10^{-34}\;\text{J⋅s}, speed of light c=3.0×108  m/sc = 3.0 \times 10^8\;\text{m/s}. For the X-ray transition, assume it corresponds to n=2→n=1n = 2 \to n = 1 in the hydrogen-like ion.

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