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345B 2011.Pdf What a student can learn from the Saha Equation Jayant V. Narlikar IUCAA, Purie 1. Introduction sulting frequent collisions, it would be impossible The seminal contribution of Meghnad Saha was for the typical atoms to retain all of their orbital his ionization equation, now well known as Saha's electrons and so they would be ionized and the lonization Equation. To say that it was an impor- matter would be in a state of plasma. How would tant work in astrophysics would be understating it the state of equilibrium be in such circumstances? Naturally we expect some of the atoms of the mat- " : for, the subject of astrophysics really got going only as a result of the Saha Equation. Let us be- ter to remain unchanged, while some would exist gin with an examination of this assertion. For, to as ions and free electrons. But what would be 5 a student of physics the equation provides a menu the proportions of these three ingredients? The / Saha equation answered this important question "' of delicious results to be enjoyed and appreciated. by giving the following relation: Till the second decade of the last century the main observational handle on studies of stars had B been their luminosities and spectra. While the lu- N (1) minosity could give a crude estimate of the star's distance using the inverse square law of illumina- Here the number Ne, Ni, N denote the num- tion, the spectrum contained a lot more informa- ber densities of free electrons, ions and neutral tion. atoms at temperature T, B being the binding en- For example, the continuum spectrum did, in ergy of the atom. The ratio of the binding energy the first approximation resemble the black body to temperature appears in the exponential form spectrum which was well known in those days. in this equation, thus underscoring its critical ef- If the star was generating energy inside it and fect on the equilibrium abundances of these three radiating it away, then it was in a state of equi- component species. Let us try to understand it librium and provided the amount radiated was qualitatively. negligible compared to the total store of radia- What does the binding energy indicate? Re- tion being scattered within the star one expected call that an atom contains a positively charged the equilibrium state to resemble the black body nucleus surrounded by negatively charged elec- state. This enabled the astronomer to estimate trons. The latter are held close to the nucleus the star's surface temperature. by the force of electrostatic attraction. It is this With surface temperatures of the order of 3000 force that provides the binding and its energy de- K and above, it became clear that the matter at notes what work must be done to tear the elec- the surface was not likely to be in a state of neu- trons apart from the nucleus. Thus the larger the tral gas. With large thermal motions and the re- binding energy B the more likely that the atom 107 108 Physics Teacher, Vol. 53, Nos. 1 &: 2, January - June, 2011 would stay intact despite any attempt at disrup- a spherical object, made of plasma held together tion. under its own gravitation. In fact, the inward force of gravitation can be so large that unless The disruption comes from collisions of the the star has significant pressures within, it can- atom by other particles. The larger the velocity not resist gravity. < v > of a colliding particle the greater the chance of a break-up of the atom. As statistical mechan- Exercise: Assume a ball of pressure-free gas ics tells us, the measure of speed is through the of the mass the same as that of the Sun, M — temperature T of the syatem. The larger the tem- 2 x 1033g and follow its gravitational contraction perature the greater the average velocity per par- from its starting radius equal to that of the Sun, ticle. In fact, we know from this subject that the i.e., R = 7 x 1010cm. How long will it take to average kinetic energy per particle is proportional shrink to zero radius? You will be surprised to toT. see what small answer you get! In the above equation we thus see that the larger the value of B the smaller the value of the The classic equations of Eddingtonfl] are dif- ratio on the left hand side. That is, we will ex- ferential equations which give the march of physi- pect a smaller proportion of ions and free elec- cal quantities like density, temperature, pressure, trons. However, as T the temperature is raised, luminous flux etc. from the centre outwards. To the right hand side increases and we get higher solve them completely the boundary conditions at proportions of free ions and electrons. In short, the surface are required. This explains why the with rising temperature the matter moves towards Saha Equation was such an important stimulus the plasma state. for the early astrophysics. Saha's paper[2] ap- peared in around 1920 and in the next 4-5 years In Saha's equation we therefore see the broad Eddington's stellar models could be set up. That link between atomic physics, thermodynamics was the beginning of astronomers using the meth- and observational astronomy. The appearance of ods of laboratory physics and applying basic the- the Boltzmann's constant K in (1), the atomic ories of physics to understand the large scale be- binding energy, and the temperature indicate this haviour of stars, galaxies and the whole universe. tripartite relationship. This was the beginning for astrophysics : it was here that a clue was made The purpose of this article is to emphasize the available to interpret the spectrum of a star in- wide applicability of the Saha Equation to as- cluding the strengths of the emission and absorp- trophysics : for the general impression is that tion lines in it in terms of the ambient state of the equation has relevance to stellar scenarios ionization of the stellar envelope and its temper- only. I will select two scenarios to illustrate my ature. point, both of them far removed from stellar as- trophysics. The first relates to the popular theory The surface conditions of the star serve as valu- of the origin of the microwave background radia- able boundary conditions for stellar models which tion in the universe and the second to the theory seek to give details of the unseen stellar interior. of the origin of light nuclei in the early universe. In the model proposed by Eddington, the star is What a student can learn from the Saha Equation 2. The Microwave Background radiation-dominated. But, these particles espe- The presently popular big bang framework of cos- cially the lightest of them, the electrons, acted mology envisages the following sequence of events as scattering centres of the ambient radiation and since the origin of the universe in a big explo- kept it thermalized. The universe was therefore quite opaque to start with. For, with its frequent sion. In the early stages the universe was very hot, with typical particles of matter moving rel- scttering light could not travel in a straight line very far. ativistically, i.e., as photons. Such a phase was said to be radiation dominated. Even electrons However, as the universe cooled, the Coulomb and protons moved with the speeds close to that electrical attraction between the electron and the of light provided the universe had a temperature proton began to assert itself. In detailed calcu- of about ten thousand billion. As the universe lations performed by P.J.E. Peebles, the mixture expanded, it cooled and the speeds of particles of electrons and protons and of hydrogen atoms began to drop. A crude but very useful estimate was studied at varying temperatures. Because of tells us that the average energy per particle is Coulomb attraction between the electron and the comparable to kT. Thus the speeds of the more proton, the hydrogen atom has a certain binding massive particles will be lower. As the speed falls energy B. The problem of determining the rela- significantly below the light speed c, the particle tive number densities of free electrons, free pro- becomes 'non-relativistic'. So in our case, first tons (that is, ions), and neutral H-atoms in ther- the protons become non-relativistic and then the mal equilibrium is therefore analogous to that we electrons. considered earlier for stars. The only difference is Standard texts in cosmology, e.g. Ref. 3, give that the setting is cosmological rather than stel- the relevant relations describing when this hap- lar. Following (1) we arrive at the formula relat- pens. The later cooler epochs have the universe ing the number densities of electrons (Ne), pro- 'dust dominated'. That is, the universe is mainly tons (Np = Ne), and H-atoms (Npj) at a given made of matter that has negligible random mo- temperature T : tions with respect to the cosmological rest frame. Ne _ fmeKT\ _ ]^\• (9\ _ I _ i-i I C/-A. L/ I I • I ^ I Denoting the scale factor of the expanding uni- NH \27Th2 J FV «T/ v ' verse by S1, the simple rule is that the tempera- where me = electron mass. This equation is a ture of the radiation drops in inverse proportion particular case of Saha's ionization equation. to 5. Writing NB for the total baryon number den- As we shall see in the next section, during the sity, we may express the fraction of ionization by period 1-200 seconds after the big bang, the tem- the ratio perature of the universe dropped from about 10 Ne x — billion degrees to a few hundred million degrees.
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