Saha's Equation And Its Importance

In the series of Basics of Astrophysics, today we are going to discuss a very important mathematical tool. This tool is known as Saha's Equation. This equation laid the foundation of a very prominent branch of Astrophysics and also acted as a milestone in the study of plasmas. Being a plasma physicist, I am glad to discuss and explain this equation with you and some history about it. So, let us try to understand how Saha's equation helped in the spectral studies of stars.

A Brief History

The discovery of Fraunhofer lines in 1814, gave birth to the spectral studies of stars. The spectrum of stars manifests the general characteristics of the Fraunhofer spectrum. All stellar spectra indicate lines of certain elements to be much stronger as compared to the lines of other elements. Interestingly, the strength of the same element is also found to vary continuously in the spectra of different stars.

Also Read: What Is The Most Important Tool To Study The Universe?

The Fraunhofer Lines

When the atomic and radiation theories were still unknown, the astrophysicists were tempted to interpret the spectral sequence as a result of a difference in the initial compositions of the stellar material. But today, we know that this discrepancy is due to temperature differences. Let us today take a deep dive in the history to know how the puzzle of stellar spectral variety was solved by an Indian Astrophysicist Meghnad Saha.

In 1920, Saha's Ionization Theory depicted an important application of the atomic theory of Bohr. Ionization is basically the phenomenon in which the electron orbiting the nucleus in an atom gains enough energy and gets stripped off from the atom (gets loosely bound to the nucleus). Saha gave a mathematical formula that described how the excitation and ionization in the stellar atmospheres are actually dependent on the conditions of temperature and pressure prevailing in those stars and not just the composition. Saha's equation actually laid the foundation of a very important branch of Astrophysics known as - Stellar Spectroscopy. Let us now try to understand his famous work called Saha's Ionization Equation.

Meghnad Saha

Meaning of Saha's equation:

Saha's equation is a useful result of combining quantum mechanics and statistical mechanics to explain the spectral classification of stars.
It tells the degree of ionization of a gas in thermal equilibrium by relating it to the pressure and temperature of the gas.

Saha's Equation
The Saha's Equation

This equation depicts the dependence of ionization of gas on various physical parameters such that :

  1. Ionization energy: As the temperature of a gas is raised, the degree of ionization of gas remains low until the ionization energy is greater than the gas temperature (evident from the exponential factor).
  1. Temperature: Afterwards, the degree of ionization i.e., the ratio of the number density of ions to the number density of neutral atoms of a gas in thermal equilibrium, increases abruptly with an increase in temperature. The gas then becomes a plasma (composed of ions, electrons and few neutral atoms).
  1. Number density of ions: When an atom becomes charged, it may recombine with an electron and become a neutral again. So, as the number of electrons increases, the ionization ratio decreases. In the simplest hydrogen plasma, the number of electrons is considered to be equal to the number of ions. Hence, as the number density of ions increases in a plasma, the rate of neutralization of ions is enhanced too. This leads to a decrease in the ionization ratio.

Let us now try to understand the physical significance of this equation:

Saha pointed out that pressure has a great influence on the degree of ionization of a gas. This fact had not been anticipated so far. He published his paper in 1921 in the Proceedings of the Royal Society. In this paper, Saha employed his theory to explain the stellar spectral sequence. In his words, he argued,

We are not justified in speaking of a star as a hydrogen, helium or carbon star, thereby suggesting that these elements for the chief ingredients in the chemical composition of the star. The proper conclusion would be that under the stimulus prevailing in the star, the particular element or elements are excited by radiation of their characteristic lines, while other elements are either ionized or the stimulus is too weak to excite the lines by which we can detect the element.

Saha's equation also depicts that a gas attains a plasma state at extremely high temperatures and low number densities of charged particles. It is because of this reason that plasmas exist naturally in astronomical objects with a temperature of millions of degrees and very low number densities of atoms around 1 per cm cube. Due to their natural occurrence, plasma is considered to be the fourth state of matter.

Author's Message

Being a plasma physicist, I can say that the stellar atmospheres are one of the active research topics in Astrophysics. The spectrum of a star really gives tonnes of information which an Astrophysicist requires to decode the Universe. We have covered one-third of the series. In the next article, we will be learning more about the atmosphere of stars and the importance of ionization theories in Astrophysics. After that, we will study the basic structure of the nearest and the most studied star, the Sun.

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