Today, in the series of Basics of Astrophysics, we will discuss an essential mathematical equation: The 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. As a plasma physicist, I am glad to discuss and explain this equation with you and share some history. So, let us try to understand how Saha’s equation helped in the spectral studies of stars.

In the second article of the series, I wrote about the importance of the EM spectrum in Astrophysics. Then we later learned how atomic physics and thermodynamics play a crucial role in this subject. Finally, in today’s article, we will learn the role of plasma physics in astrophysics. So let’s start!

Read all the articles on the Basics of Astrophysics here.

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 than the lines of other elements. Interestingly, the strength of the same element is also found to vary continuously in the spectra of different stars.

Solar spectrum with Fraunhofer lines as it appears visually.
Image Courtesy :Wikipedia
Solar spectrum with Fraunhofer lines as it appears visually

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

Meghnad Saha
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 significant branch of Astrophysics known as – Stellar Spectroscopy. Let us now try to understand his famous work called Saha’s Ionization Equation.

Meaning of the 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.

The Saha's equation
Saha’s Equation

Saha’s equation depicts the dependence of ionization of gas on various physical parameters such as:

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).

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).

The number density of ions: When an atom becomes charged, it may recombine with an electron and become neutral again. So, as the number of electrons increases, the ionization ratio decreases. In the simplest hydrogen plasma, the number of electrons is 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 Saha’s 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. Because of this, 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 stellar atmospheres are one of the active research topics in Astrophysics. The spectrum of a star really gives tonnes of information that an Astrophysicist requires to decode the Universe. We have covered more than one-third of the series. In the next couple of articles, I will be writing about the structure of the Sun. In the previous few articles, Rishabh had written about the basic concepts of stellar astrophysics. Now to understand stellar evolution, it is important to learn about our nearest star. After we are done with some basics of solar physics, we’ll be fully equipped to study the life and death of stars.

I hope you are enjoying the series. Stay tuned!

This article is written by Dr. Yashika Ghai, Theory and Computation Plasma Physicist at the Oak Ridge National Laboratory, US.

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[…] Stars”. What she did was relating the spectral data from stars to their temperatures, by applying Saha’s ionization theory. She found that helium and hydrogen were greatly abundant, concluding her thesis with the […]


[…] Also read in series:Chapter 1 – What is Astrophysics?Chapter 10 – The Hertzsprung Russell DiagramChapter 11: The Saha’s Equation […]


Nice article.

To know more about the topic, one shall refer the popular science book,
*Saha and his formula* by Prof G. Venkatraman from Universities Press, India.

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