Absorption in the surface of a star involves both discrete...

Absorption in the surface of a star involves both discrete lines, such as the Balmer absorption series for hydrogen, and continuum absorption. An important source of continuum absorption is boundfree transitions where an electron is excited from a bound state to an unbound state (ionization). Since the final state contains an essentially free electron, bound-free transitions produce continuous absorption for wavelengths shorter than the threshold wavelength corresponding to the energy difference between the bound state and the ionization threshold. Show that Balmer bound-free continuum absorption (corresponding to ionization from the $n=2$ hydrogen level at $10.2 \mathrm{eV}$ ) and Lyman bound-free continuum absorption (corresponding to ionization from the $n=1$ ground state of hydrogen) both occur at UV wavelengths where the Sun outputs little light. Show that Paschen bound- free absorption (corresponding to ionization from the $n=3$ level of hydrogen) occurs primarily in the visible spectrum where the solar light output peaks and therefore is expected to dominate the contribution of neutral hydrogen to continuum absorption at visible wavelengths. (However, note from Problem 2.11 that this is not the dominant overall contribution to continuum absorption for the Sun.)

Key Concepts

Bound-Free Transitions

Bound-free transitions occur when an electron bound to an atom absorbs enough energy to escape entirely from the atom, resulting in ionization. This process does not lead to the production of a single discrete spectral line; instead, it produces a continuous absorption spectrum for photon energies greater than the ionization threshold of the particular bound state.

Atomic Energy Levels

Atomic energy levels, such as the n=1, n=2, and n=3 states in hydrogen, define the quantized energies that an electron can have while bound to the nucleus. Transitions from these states, including those that result in ionization (bound-free transitions), depend on the initial energy level, determining the threshold energy – and thus the corresponding threshold wavelength – required for the electron to become unbound.

Ionization Threshold and Continuum Absorption

The ionization threshold is the minimum energy needed to free an electron from a particular bound state. For photons with energies exceeding this threshold, the absorption occurs continuously rather than discretely, leading to what is known as continuum absorption. The threshold wavelength (the inverse of the energy required) determines the spectral region in which the bound-free absorption can occur.

Spectral Series and Wavelength Dependence

The different spectral series in hydrogen (Lyman, Balmer, Paschen, etc.) correspond to electron transitions from different initial energy states. Each series has a distinct threshold wavelength for ionization. For example, ionization from the hydrogen ground state (Lyman series) and from the n=2 level (Balmer series) leads to absorption in the ultraviolet, while ionization from the n=3 level (Paschen series) occurs within the visible spectrum, aligning with the peak emission of the Sun.

Stellar Spectral Energy Distribution

The spectral energy distribution of a star, such as the Sun, shows the intensity of light output as a function of wavelength. A star's thermal (blackbody) spectrum typically peaks at a specific wavelength range, which for the Sun is in the visible region. Therefore, absorption processes that occur at wavelengths near this peak are more influential in shaping the overall opacity and observational appearance of the star's spectrum.

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