Spectral Lines

Stars are classified based on their spectra, that is: the frequency and intensity of the light that comes from them. First, each frequency corresponds to different photon absorption and emission phenomena. Roughly speaking, each frequency corresponds to a particular element in a particular electronic state and ionization state. Second, each intensity is proportional the quantity, or concentration, of such elements in the star, which at the same time is related to the temperature of the star.

This means that we can get a lot of info by just looking at (and interpreting) the spectra!

Stars are classified on spectral types.

Oxford classification: “O B A F G K M” (mnemotechnic rule: “Oh Be A Fine Guy/Girl, Kiss Me”).

  • It is a Temperature sequence based on H absorption lines.
  • O is hottest. O, B, A… stars are called “early type” stars. M is coolest. …, G, K, M, are called “late type” stars.
  • Each Spectral Type is subdivided into 10 ranges numerically, from hottest (0) to coolest (9). Example: F0, F1, F2, …, F9. Same nomenclature applies. F0 is a “early F star“, F9 is a “late F star“.

However, spectra are more complex than just H absorption lines, and can contains electronic transitions between many different orbitals (i.e. electronic states) and involve many different ionic states.

Strength of different spectral lines depending on T. The spectral types are indicated in the lower axis. Source.

Strength of different spectral lines depending on T. The spectral types are indicated in the lower axis. Source.

Strength of spectral lines: I don’t want to repeat myself, but the strength of each line depends on the proportion of atoms in the specific ionization state and electronic state which can generate that line. To a good approximation, this depends on T, since only minor changes in star composition occur.

Examples:

  • The strength of the Balmer lines depends on the fraction of all H atoms (i.e. not ionized) which are in the first excited state.
  • The strengths of the Ca II H and K lines depend on the fraction of all singly ionized Ca atoms in the electronic ground state.

Note: An atom’s stage of ionization is denoted by the symbol of the atom followed by a Roman numeral. Examples:

  • H I = Neutral H
  • H II = Ionized H (H+)
  • Si III = Si2+

Now, how do we know the proportion of atoms on a given electronic state and on a given ionization state, depending on T? For this, we use Statistical mechanics. Statistical mechanics is about the properties of large ensembles of particles as a whole, without caring about the behavior of individual particles. The properties of ensembles of particles are average quantities, examples are temperature and pressure.

Boltzmann Equation: gives the ratio of the number of atoms in different states of electronic excitation.

boltzmann-eq

The degeneracy gi is the number of states that have energy Ei. For example, the ground state of the H atom has energy E0 and is doubly degenerate, go = 2. These are the electron with spin 1/2, and the electron with spin -1/2 configurations. The degeneracy the states of a H atom depend on the principal quantum number n as gn = 2n2. bla

Saha Equation: Gives the ratio between different stages of ionization.

saha-eq

The electron density number can also be expressed as a function of the pressure of the free electron, according to the ideal gas equation:

electron-density

Z is the partition function. It is the weighted sum of the number of ways the atom can arrange is electrons with the same energy. The more energetic terms have less weight in an exponential factor, thus as the energy increases, the corresponding term becomes smaller very fast (tip: and can be neglected).

partition-function

The results of these equations do not reproduce reality exactly, due to approximations used:

  • Stars are formed by a multitude of species that may shift the ratios between states. E.g. helium can provide electrons to H ions, thus for a given ratio between the populations of H II and H I the actual T required may be higher than the calculated one.
  • We are assuming thermal equilibrium.
  • If the density is too high (> 10-3 g/cm3), orbital distortion by neighboring atoms may alter populations.
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