Problem — Stellar spectrum and the 397 nm line

Problem

A star’s spectrum shows a dark line at $\lambda = 397$ nm. Which hydrogen transition produces it (knowing it ends on $n_f = 2$), and what physical information does it carry about the star?

Solution

Step 1 — Energy of the absorbed photon.

$\Delta E = \frac{1240}{\lambda(\text{nm})} = \frac{1240}{397} \approx 3{,}12\;\text{eV}$

Step 2 — Initial level. The dark line is an absorption from level $n_f = 2$ (Balmer series) to a higher level $n_i$. The energy of the arrival level is

$E_i = E_2 + \Delta E = -3{,}40 + 3{,}12 = -0{,}28\;\text{eV}$

Step 3 — Quantum number. From $E_i = -13{,}6/n_i^2$:

$n_i^2 = \frac{13{,}6}{0{,}28} \approx 49 \quad\Longrightarrow\quad n_i = 7$

$\ev{n_i = 7 \quad (2\to 7,\ \text{Balmer}\ H_\epsilon\ \text{line})}$

Step 4 — Physical information. The presence of a Balmer line in absorption indicates hydrogen in the stellar atmosphere with many atoms already in level $n=2$ (first excited state). For level $n=2$ to be populated a high temperature is needed: pronounced Balmer lines are typical of hot stars (type A, $\sim 10^4$ K). Analysis of the lines thus reveals the star’s composition and temperature, without ever visiting it.

Collegamenti

Argomenti: Quantum physics Concetti: Bohr model · Photon Competenze: Unit conversion · Symbolic setup