(For all of these problems, where necessary assume the normal ratio of total- to-selective extinction.) For an excitation temperature of \(100 \mathrm{K},\) what is the ratio of populations for the two levels in the \(21 \mathrm{cm}\) transition? (Take the statistical weight of the lower level to be 1 and the upper level to be \(3 .\))

The Boltzmann equation is a key concept in statistical mechanics and used to determine the distribution of particles across various energy levels in a system. It shows how the population of particles in different energy states is related to temperature and the energy difference between the states. The equation is usually expressed as:

\[ \frac{n_i}{n_0} = \frac{g_i}{g_0} \times e^{-\frac{E_{ij}}{kT}} \]

Here:

• \(n_i\): Population of the higher energy state


• \(n_0\): Population of the ground state


• \(g_i\): Statistical weight of the higher energy state


• \(g_0\): Statistical weight of the ground state


• \(E_{ij}\): Energy difference between the two states


• \(k\): Boltzmann constant (\(1.381 \times 10^{-23} J/K\))


• \(T\): Absolute temperature (in Kelvin)

This equation informs us that the higher the energy difference \(E_{ij}\) between two states, the fewer particles will populate the higher energy state at a given temperature \(T\).

The 21 cm transition, also known as the hydrogen line, is a specific change in the energy level of neutral hydrogen atoms, which emit or absorb electromagnetic radiation at a wavelength of 21 centimeters (around 1420 MHz). This transition occurs when the spins of the proton and electron in a hydrogen atom change from being aligned (parallel) in the same direction to anti-aligned (antiparallel).

The 21 cm line is important in astrophysics as it is used to study a variety of cosmic features and phenomena:

• Mapping the distribution and density of neutral hydrogen in our galaxy and others


• Understanding the structure and dynamics of galaxies


• Tracing the evolution of matter throughout cosmic history

This spectral line's significance lies in its relatively long wavelength, which can penetrate dust clouds and allow astronomers to observe star-forming regions and other obscured areas.

In the context of astrophysics, energy levels refer to the specific quantized states that electrons can occupy within an atom. Each energy level corresponds to a different electron configuration and is associated with a certain amount of energy. When an electron transitions between two energy levels, it either absorbs or emits energy in the form of electromagnetic radiation.

Key aspects to understand about energy levels include:

• Electrons in higher energy levels are less stable and are more likely to drop to lower energy levels, releasing energy.


• Energy levels are often described by quantum numbers which ensure the unique configuration of electrons.


• The difference in energy between these levels determines the wavelength and frequency of the resultant radiation.

In the example of the 21 cm transition, the energy difference between the parallel and antiparallel spin states of hydrogen leads to the emission or absorption of microwave radiation.

Calculating the population ratio of two energy levels involves using the Boltzmann equation, which allows us to quantify the number of particles populating one energy state compared to another. Consider the exercise where we calculate this ratio for the 21 cm transition at an excitation temperature of 100K, with statistical weights given as: \(g_0 = 1\) and \(g_i = 3\).

The steps are:

• Convert the 21 cm wavelength to energy: \(E_{ij} = \frac{hc}{\text{wavelength}} = \frac{6.626 \times 10^{-34} × 3 \times 10^8}{21 \times 10^{-2}} = 5.7 \times 10^{-24} J\)


• Calculate the exponent: \(\frac{E_{ij}}{kT} = \frac{5.7 \times 10^{-24}}{1.381 \times 10^{-23} \times 100} = 0.04125\)


• Substitute these values into the Boltzmann equation: \(\frac{n_i}{n_0} = \frac{g_i}{g_0} \times e^{-\frac{E_{ij}}{kT}} = 3 \times e^{-0.04125} ≈ 3 \times 0.9596 ≈ 2.8788\)

This value denotes the ratio of particles in the higher energy state to the ground state at the given temperature.