Astrobiology. Charles S. Cockell

      This can be converted to a wavelength using Eq. (3.1). Note that the energy in any given energy level is given as a negative number (Figure 3.25a). Hence the energy of the lower integer energy level (here n₁) is subtracted from the larger integer energy level (here n₂) to get the energy difference between them given in positive units (i.e. in this example, E(n₂)−E(n₁) = −3.4 − (−13.6) eV = 10.2 eV required to jump between these two levels.) This can cause confusion, so caution is needed. To avoid confusion, it is best to think about whether energy is being taken in (the electron is moving up energy levels) or being lost as light (the electron is dropping down energy levels and releasing energy in the process).

This equation also tells us what frequency of photon will be emitted when an electron drops an energy level (from n₂ to n₁) and emits its energy as a photon. This is also given by ν = [E(n₂)−E(n₁)]/h. This is conceptually shown in Figure 3.25a as the emission caused when an electron drops an energy level. For example, in the Lyman series for hydrogen, electrons dropping to the ground state give off energy in the ultraviolet (UV) region (and require UV radiation to make them jump from the ground state).

      So far, we have simply read off the energy associated with given energy levels from Figure 3.25 and shown how we can use those values to calculate the frequency and wavelength of light. It is possible to calculate the value of the energy associated with a given energy level from first principles by considering the structure of the atom itself. The value can be calculated by Eq. (3.4), which we won't derive here but will state. It is obtained by thinking about an electron as orbiting the atom much like a planet orbits a star. Although in reality this is not how electrons behave, it is a good enough approximation for deriving some of the basic properties of atoms. This approach to considering electrons is known as the Rutherford–Bohr model of the atom, after the physicists Ernest Rutherford (1871–1937) and Niels Bohr (1885–1962), who elaborated on this model.

      The energy of an electron at a given energy level in an atom is given by:

(3.4)

      where Z is the atomic number (or number of electrons or protons in the atom), m_e is the mass of an electron, e is the charge of the electron, k_e is Coulomb's constant, n is an integer that defines the energy level, and h is Planck's constant. Note that the energy derived in this way is expressed in electronvolts (eV). The value can be multiplied by 1.602 × 10¹⁹ to give the value in joules.

      This equation approximates to:

(3.5)

      The value, 13.6, in this equation is also known as the Rydberg unit of energy(Ry).

Taking the formula we have just seen (Eq. (3.5)), we can observe that the energy needed to make an electron jump from one energy level to another (or the energy of a photon given off when an electron drops from one energy level to another) is therefore given by:

      (3.6)

      where n₁ and n₂ are the energy levels and n₁ < n₂.

      3.14.1 The Special Case of the Hydrogen Atom

      The hydrogen atom, which is of course a common constituent of the Universe, is a special case, since its mass number, the value of Z, is 1. This leads to the general formula for the energy required for an electron to change energy levels in this atom as:

      (3.7)

      The wavelength of light that would be emitted or taken up by a transition between these two energy levels is therefore also given by:

      (3.8)

      This equation can also be expressed in a different way, known as the Rydberg formula (for hydrogen):

      (3.9)

      where the Rydberg constant (R_∞) is equal to Ry/hc and has a value of 1.097 × 10⁷ m⁻¹.

      A special case of this equation is when hydrogen is ionized. In this case, the electron is completely removed from the atom, so that n₂ = ∞. If the electron is in the ground state (i.e. n₁ = 1), then the energy required for ionization is equal to Ry eV or 13.6 eV.

      3.14.2 Uses for Astrobiology

      The implications of the above equations and ideas in relation to astrobiology can now be explained.

      When a collection of hot gases absorbs light, the electrons in its constituent gases jump energy levels and drop back down again, emitting light at discrete wavelengths. These give rise to emission spectra (Figure 3.25b). Each individual gas has a very characteristic emission spectrum that is a fingerprint, if you will, of its atomic structure, with lines arranged across the spectrum corresponding to the different energy levels of its electronic orbitals (Figure 3.25b). This emission spectrum can be used to identify the different gases in astronomical objects. These effects are pressure dependent, and at very high pressures, hot gases tend to produce continuous spectra without characteristic emission lines.

      The detection of gases in the atmospheres of distant stars, beginning in the late nineteenth century, finally confirmed that distant stars are other suns. By characterizing these gases in stars of different colors and luminosities, it became possible to systematically categorize stars into different spectral types.

      By contrast, if light travels through a collection of cold gases, the gases tend to absorb the light at the particular wavelengths corresponding to the energies needed to make electrons jump energy levels, dependent on the atomic structure of the individual gases. These create a characteristic absorption spectrum, essentially places in the spectrum where the light is “missing.” Absorption spectra from stars are characterized by very distinct lines in the spectrum called Fraunhofer lines, named after German optician Joseph von Fraunhofer (1787–1826). The temperature and density within a star affect the intensity of the lines, and so they can reveal information about the characteristics of a given star.

      As we shall see later, absorption spectroscopy allows us to investigate the gaseous composition not just of stars, but also of planetary atmospheres, such as those of extrasolar planets. By collecting light from a star that has traveled through a planetary atmosphere, we can determine the gaseous composition of the atmosphere and seek gases that are signatures of life (Chapter 20).

      We leave the summary of this last section to German physicist Gustav Kirchhoff (1824–1887) who listed, before the structure of the atom was understood, what are sometimes called the three laws of spectroscopy:

      1 A hot solid object gives off a continuous spectrum (called a blackbody – we return to this in Chapter 9).

      2 A