CHAPTER VITHE HIGH-TEMPERATURE ABSORPTION SPECTRUM OF A GAS

IT is certain that the conditions of which we see the integrated result in the stellar spectrum are exceedingly complicated. Unfortunately, the superficial portion of the star about which direct observational evidence can be obtained is far less tractable to theory than is the interior. Progress is only made possible by treating at the outset a simplified case, by aiming merely at approximate results, and in particular by limiting the preliminary discussion to the factors which are numerically the most effective. As an introduction to the theory of thermal ionization, the present chapter aims at the reconstruction and interpretation of a stellar spectrum by applying known physical laws under very simple conditions.

The stellar reversing layer may be represented by an optically thin layer of gas, at a pressure of the order of one ten thousandth of an atmosphere; it lies between the observer and a photosphere which radiates as a black body. The observer receives the radiation from both reversing layer and photosphere, which are regarded, in the present descriptive section, as independent. A more complete treatment would take account of the temperature and pressure gradients in the atmosphere of the star, the flux of energy, and the consequent intimate connection between reversing layer and photosphere. Actually they grade imperceptibly one into the other. The photosphere is that level in the atmosphere at which the general opacity cuts off the direct light from the interior;[355]in the case discussed the reversing layer is considered to be optically so thin that the general opacity is negligible. Theselectiveopacity, depending on the natural absorption frequencies of the atoms present in the gas, gives riseto the line absorption spectrum which we are about to consider; the region of sensiblegeneralopacity, represented by the photosphere, gives rise to a continuous spectrum corresponding to the continuous background in the star.

THE ABSORPTION OF RADIATION

The light passing through the layer of gas is absorbed, in terms of atomic theory, in the shifting of an electron from one energy level in an atom to some higher level, losing in the process energy of the definite frequency which is associated with that particular atom and energy transfer. The energy levels and possible electron transfers for the hydrogen atom are reproduced inFigures 2and3. InFigure 3the horizontal lines represent the stationary states which can be assumed by the electron, and the arrows denote possible jumps from one stationary state to another. InFigure 2the electron orbits corresponding to some of the simpler corresponding transitions for the hydrogen atom are represented. Arrows denote transfers from one orbit to another. The designation of the line corresponding to each transfer is appended to the appropriate arrow. It is evident that the occurrence of a given jump requires that there shall be an electron in the stationary state from which the jump originates.

Theultimate lines[356][357]are those which arise from the lowest energy level, and are therefore those most readily absorbed by the normal (undisturbed) atom. In the hydrogen spectrum these comprise the Lyman series,[358]with the first member at 1215.68. The Balmer and Paschen series are bothsubordinateseries, requiring an initial lifting of the electron from the lowest energy level into a two and three (total) quantum orbit, respectively. The absorption of the Lyman line Lyis necessary to a hydrogen atom before it is in a fit condition to absorb any Balmer line, and for the absorption of a Paschen line, an initial absorption of Lyor His required.

It appears plausible to assume, at least for low partial pressures, that the amount of energy of any frequency that is lost by black-body radiation in passing through the absorbing layer will vary jointly with the supply of energy and the number of atoms which are in a suitable state to absorb that particular frequency. One of the problems that arise is therefore that of determining what fraction of the whole number of atoms of a given kind will be able to absorb. It is to this problem that ionization theory is able to offer a solution.

By choosing the much simplified case of very low pressure and small concentration, the effects of ionization by collision[359]and of nuclear fields are probably eliminated. The remaining factor which may influence the number of absorbing atoms is thermal ionization, and this is actually the numerically important factor, as was first pointed out by Saha.[360]It is of interest to note that Saha’s original treatment contemplated pressures of the order of one atmosphere. Under such conditions the effects of collisions and of nuclear fields are not negligible, and might well have invalidated the theory. Later work has shown conclusively, however, that the pressures in the reversing layer are probably not greater than,[361][362]and that thermal ionization is the predominant factor under these conditions.

The absorbing layer is to be regarded as consisting of a mixture of all chemical elements, without any assumption as to quantity, so long as the partial pressure of each individual element is low. In other words, no account is taken, at the present stage, of the relative abundances of different kinds of atoms—thetotaleffectiveness of the corresponding elements as absorbers. Thechangesin the absorption of the black body radiation by a given element with changing temperature will be the same whatever the partial pressure, provided it is low, and it is with these changes that the preliminary schematic discussion is concerned.

LOW TEMPERATURE CONDITIONS

At low temperatures all the elements will tend to be in their normal atomic state, unless they are aggregated into molecules or compounds. At temperatures of 2500°, which is about the lower limit encountered in dealing with stellar spectra, there is evidence of the existence of various oxides (CO, TiO₂, ZrO₂), of “cyanogen,” and of hydrocarbons, but most of the other possible compounds appear either to be dissociated or to be in very low concentration. Probably the normally polyatomic gases such as oxygen, nitrogen, and sulphur, are to some extent present in the molecular state. Even at atmospheric pressure all the metals are vaporized at 2500° excepting tantalum and the platinum metals, which boil at about 2800° under a pressure of 760 mm; at lower pressures the temperature of vaporization is, of course, lower. The metallic molecule appears normally to be monatomic, so that it will give its line spectrum unless it is in combination. The fact that silicon, the most refractory substance, excepting carbon, with which we have to deal, gives its line spectrum in the coolest stars known, indicates that all the elements may be considered to be gaseous in stellar atmospheres.

ULTIMATE LINES

The absorption spectrum given by the reversing layer when it is at a low temperature will consist of the lines given most readily by the atom in its normal state. The energy transfers which move an electron from its normal orbit to another correspond to the “ultimate lines,” and these lines will therefore be especially outstanding in the spectra of the coolest atmospheres. They are of such importance, from theoretical and from practical standpoints, that a list of them is reproduced here. Successive columns of the table give the atomic number and atom, the ionization potential, the wave-lengths of the ultimate lines, and an indication of their observed occurrence in stellar spectra. An asterisk denotes that the line has been observed, and a dash indicates that it has not been recorded.

It may be remarked that the ultimate lines of sodium, potassium, lithium, rubidium, and caesium are in the visible region—a fact which is utilized in the laboratory flame tests used in qualitative analysis.[363]The brilliancy of the flame colors obtained in the Bunsen burner, at the temperature of about 1500°C., is a striking elementary illustration of the readiness with which the atom in its normal state will take up and re-emit the frequency corresponding to the ultimate lines (second pair for K, Rb, Cs).

It is possible to predict from the table which lines are likely to appear in the spectra of the coolest stars. The ultimate lines of Al, K, Ca, Sc, Ti, V, Cr, Mn, Co, Ni, Cu, Ga, Rb, Sr, Zr, Mo, Ag, In, Ba, La, and Pb fall in the region ordinarily photographed, and Na can be reached in the yellow. All these elements in the neutral state would therefore be anticipated in the spectra of cooler stars, and they are indeed found without exception. The ultimate lines of several elements contained in the list lie in the far ultra-violet, and cannot be detected in stellar spectra. The corresponding neutral elements will therefore not be recorded unless they also give a strong subordinate series in the photographic region. The elements C, O, S, and probably N, all have ultimate lines in the ultra-violet, and possess no subordinate series in the appropriate range of wave-length. Their apparent absence in the neutral state from stellar spectra is therefore fully explained. All of these elements appear in the hotter stars in the once or twice ionized condition. The elements H, Mg, and Si have strong subordinate series in the photographic region—the Balmerseries, the “b” group, and the line[364]at 3905, respectively. They are accordingly represented in the cooler stars.

The elements contained in the table and not yet discussed are Be, B, Ne, A, F, and Cl. These elements have not been detected in stellar spectra. In seeking an explanation of their apparent absence, it has been suggested that a low relative abundance of the corresponding atoms may be responsible. Arguments from terrestrial analogy must be applied with caution, but there is reason to suspect that they may here have a legitimate application.[365]It may be suggested that boron, beryllium, neon, and argon are present in the stars in quantities too small to be detected. The halogens are unrepresented, but it is not possible to draw useful inferences until their laboratory spectra are more fully analyzed.

IONIZATION

At the lowest temperatures, then, the ultimate lines will predominate. As the temperature of the absorbing layer is raised, ionization—the complete ejection of the electron from the atom, instead of a displacement from one stationary state to another—will set in, and the tracing of the resulting spectral changes is the salient feature of the Saha theory. “Ionization can be effected in many ways. To expel an electron against the attractive force of the remainder of the molecule, work is required, and the necessary energy may be furnished by X rays orrays, or by collision with other electrons.... At high temperatures, when the conditions of maximum entropy demands an appreciable amount of ionic dissociation, the requisite energy is drawn from the environment.... The work required to ionize a single molecule, when expressed as the number of volts through which an electron must fall to acquire this energy, is theionization potential; it may be regarded as the latent heat of evaporation of the electron from the molecule” (Milne).[366]

The analogy between ionization and evaporation illustrates very well the scope of the Saha theory, in which the process is treated as a type of chemical dissociation. Corresponding to each temperature there is a definite state of equilibrium, where the forward and backward velocities of the ionization process are equal—in other words where ionization and recombination are proceeding at the same rate. The method of statistical mechanics has been applied to this problem by Fowler and Milne.[367]Here the analysis will not be reproduced, but the formulae are required in order to illustrate the process of ionization.

The number of atoms which are unionized at any given temperature is given by the expressionwhere= number of atoms ionized.= the “partition function.”,whereis the partial pressure of electrons.= absolute temperature.= Boltzmann’s constant, =.= the ionization potential.

This is the number of atoms which is effective in absorbing the ultimate lines at that temperature. For low values of,the number of unionized atoms falls off at first very slowly with rising temperature, up to a point depending only on the ionization potential. Beyond this temperature the number of neutral atoms falls off with great rapidity. The diagram (Figure 5) illustrates the fall in the number of neutral atoms, and the consequent decay in strength of the ultimate lines. So steep is the gradient ofat the higher temperatures that the quantity is best plotted logarithmically. The ultimate lines will persist, with almost undiminished intensity, up to the temperature at which the gradient ofbegins to increase.This critical temperature increases with ionization potential, and neutral atoms of high ionization potential should display very persistent ultimate lines as the temperature rises.

i005Figure 5Ultimate lines of neutral atoms. Ordinates are logarithms of computed fractional concentrations; abscissae are temperatures in thousands of degrees. The curves show the decrease in the number of neutral atoms, with rising temperature, and the consequent decay in strength of the ultimate lines, for the atoms indicated on the right margin.

Figure 5Ultimate lines of neutral atoms. Ordinates are logarithms of computed fractional concentrations; abscissae are temperatures in thousands of degrees. The curves show the decrease in the number of neutral atoms, with rising temperature, and the consequent decay in strength of the ultimate lines, for the atoms indicated on the right margin.

Figure 5

Ultimate lines of neutral atoms. Ordinates are logarithms of computed fractional concentrations; abscissae are temperatures in thousands of degrees. The curves show the decrease in the number of neutral atoms, with rising temperature, and the consequent decay in strength of the ultimate lines, for the atoms indicated on the right margin.

As ionization becomes more and more complete, the intensity of the ultimate lines falls off until so small a number of neutral atoms remains that their lines cease to appear in the absorption spectrum.

SUBORDINATE LINES

The neutral atom gives rise to other lines besides the ultimate lines, but these require the transfer of an electron from some stationary state, not the normal one, to another stationary state. The atom must receive a definite quantity of energy, equal to the excitation potential of the initial stationary state, in order to be in a condition to absorb a line of a subordinateseries which originates from that state. If there is an appreciable energy supply, a certain fraction of the neutral atoms present will have received this excitation energy, which is of course smaller than the ionization potential, and these atoms will be in a position to absorb the subordinate series.

i006Figure 6Production of the maximum of an absorption line. Ordinates are logarithms of computed fractional concentrations; abscissae are temperatures in thousands of degrees. The curves reproduced are those for the Mg + line at 4481. The upper broken curve represents the fraction of magnesium atoms that is singly ionized at the corresponding temperature; the lower broken curve represents the fraction of the Mg + atoms present that is in a suitable state for the absorption of 4481. The full line represents the sum of the ordinates of the dotted curves, and gives the fraction of the total number of magnesium atoms that is able to absorb 4481 at the various temperatures indicated by the abscissae.

Figure 6Production of the maximum of an absorption line. Ordinates are logarithms of computed fractional concentrations; abscissae are temperatures in thousands of degrees. The curves reproduced are those for the Mg + line at 4481. The upper broken curve represents the fraction of magnesium atoms that is singly ionized at the corresponding temperature; the lower broken curve represents the fraction of the Mg + atoms present that is in a suitable state for the absorption of 4481. The full line represents the sum of the ordinates of the dotted curves, and gives the fraction of the total number of magnesium atoms that is able to absorb 4481 at the various temperatures indicated by the abscissae.

Figure 6

Production of the maximum of an absorption line. Ordinates are logarithms of computed fractional concentrations; abscissae are temperatures in thousands of degrees. The curves reproduced are those for the Mg + line at 4481. The upper broken curve represents the fraction of magnesium atoms that is singly ionized at the corresponding temperature; the lower broken curve represents the fraction of the Mg + atoms present that is in a suitable state for the absorption of 4481. The full line represents the sum of the ordinates of the dotted curves, and gives the fraction of the total number of magnesium atoms that is able to absorb 4481 at the various temperatures indicated by the abscissae.

The fraction,,of the total number of neutral atoms which havebecome able to absorb the lines associated with a definite excitation potential is given by Fowler and Milne aswhere ()= excitation potential. The quantityincreases with the temperature, approaching the value unity asymptotically.

The total number of atoms active in absorbing a subordinate series at any temperature is evidently the product of the number ofneutralatoms and the quantity.The curves for these two quantities are plotted logarithmically inFigure 6, the magnesium line 4481 being used as an illustration. The total number of absorbing atoms may be obtained by adding the ordinates. It will be seen that the number of such atoms increases, passes through a maximum and decreases again, as the temperature is raised. The maximum for a subordinate line of the neutral atom may occur, as in the case of helium, when ionization is far advanced.

In the special case where,the second curve, which represents the growth of the fraction,becomes a straight line parallel to the temperature axis, and the first, or ionization, curve, approaches the zero ordinate asymptotically at low temperatures. The ordinate of the curve representingis zero, and the resultant sum gives a curve identical with the curve for the ultimate lines. Ultimate lines thus appear as the special case of subordinate lines for which the excitation potential is zero. This fits exactly with the definition of ultimate lines as the lines naturally absorbed by the cold vapor—no initial excitation is required to bring the atoms into a state in which they can absorb.

LINES OF IONIZED ATOMS

As soon as ionization sets in, the absorbing layer begins to contain a new kind of atom, derived from the neutral atoms by the complete ejection of one electron. These ionized atoms will absorb their own spectrum, which differs completely from that of the corresponding neutral atom; and the degree of absorption will again depend on the number of such ionized atoms present in the reversing layer.

The ionized atom has in general a spectrum corresponding exactly to that of the neutral atom preceding it in the periodic table, but with a different Rydberg constant.[368][369]Two types of lines arise, as before—ultimate and subordinate lines. For the number of atoms which can absorb the ultimate lines of the enhanced spectrum, the formula reduces toAccount is here taken of the residual neutral atoms by the middle term of the denominator, which is very small, and is only of sensible magnitude for the ultimate lines, when the numerator is equal to unity.

i007Figure 7Maximum of the ultimate line of an ionized atom. Ordinates are logarithms of computed fractional concentrations; abscissae are temperatures in thousands of degrees. The curve is drawn for the line 4554 of Ba+, on the assumption thatis.

Figure 7Maximum of the ultimate line of an ionized atom. Ordinates are logarithms of computed fractional concentrations; abscissae are temperatures in thousands of degrees. The curve is drawn for the line 4554 of Ba+, on the assumption thatis.

Figure 7

Maximum of the ultimate line of an ionized atom. Ordinates are logarithms of computed fractional concentrations; abscissae are temperatures in thousands of degrees. The curve is drawn for the line 4554 of Ba+, on the assumption thatis.

The following curve shows the number of absorbing atoms. The flatness of the maximum is especially to be noted, suggesting that the ultimate lines of the ionized atom, like the ultimate lines of the neutralatom, will be very persistent. Theandlines of Ca+, and the corresponding lines 4077 and 4215 of Sr+, and 4555 of Ba+, would thus be expected to show over a considerable range in temperature and spectrum, and this is actually found to be the case.

The subordinate lines behave substantially as do the subordinate lines of a neutral atom, rising to a maximum at a temperature which depends chiefly on the ionization potential. It is assumed in deriving the corresponding equations that in practice the number of surviving neutral atoms will be too small to affect the concentration of ionized atoms giving the subordinate lines. This assumption may be shown to be justified at maximum intensity of the absorption line, though possibly the neutral atoms are not always negligible at the first appearance of the ionized lines of a very abundant atom.

SUMMARY

The general results of raising the temperature of the absorbing layer have now been traced. Although a greatly simplified case has been considered, the observed changes in the stellar spectral sequence have been very satisfactorily predicted.

At low temperatures the lines of neutral atoms are strong, in particular the ultimate lines, such as 3930 of Fe, 3999 of Ti, 4254 of Cr, and 4033 of Mn, which are at maximum strength, and decrease at first slowly, then rapidly in the hotter stars. The subordinate lines of neutral atoms, 4455 of Ca and 4352 of Fe, for example, attain a maximum, and then fall off with rising temperature. For many of the metallic lines for which no maximum is recorded, like those of the subordinate series of Na, the theoretical maximum is at a temperature equal to that of the coolest stars examined. Atoms with ionization potential less than 5 volts will in general give maxima below 3000°.

As the temperature increases, the lines of ionized atoms begin to appear, the ultimate lines rising very quickly in intensity, and persisting almost at maximum over several spectral classes. Later in the sequence the subordinate series for ionized atomsappear, rise to a sharper maximum, and fade more rapidly. The 4481 line of Mg+, the 4267 line of C+, and the 4128 line of Si+, show this effect well.

As the fall of intensity of the lines of neutral atoms after maximum is the result of the progress of ionization, it would be expected that the lines of the ionized atom would appear while those of the neutral atom were still quite strong, and that the one series would rise in strength as the other decreased. The lines of the neutral atom may persist over a large part of the range of the ionized lines. This is the case with the 4227 line of Ca, which persists until Class,while theandlines of Ca+ have been visible throughout the whole spectral sequence, and have been decreasing in intensity fromonwards, owing chiefly to the rise of second ionization and the consequent formation of Ca++, which gives a spectrum in the ultra-violet and is therefore not detected in the stars.

As the temperature is further raised, the second and third ionizations set in, and presumably follow the same procedure as has been outlined for less ionized atoms. The lines of N++, C++, Si++, and Si+++ will serve as examples. The lines of the doubly ionized atoms of the metals are in general in the ultra-violet portion of the spectrum, and the corresponding elements do not therefore appear in the hotter stars, where they would otherwise be anticipated.

Qualitatively the prediction of the theory of ionization is fully satisfied. The quantitative discussion involves more rigorous treatment, and is reserved for a later chapter.

FOOTNOTES:[355]Stewart, Phys. Rev., 22, 324, 1923.[356]De Gramont, C. R., 171, 1106, 1920.[357]Russell, Pop. Ast., 32, 620, 1924.[358]A. Fowler, Report on Series in Line Spectra, 1922.[359]R. H. Fowler, Phil. Mag., 47, 257, 1924.[360]Proc. Roy. Soc., 99A, 135, 1921.[361]M. N. R. A. S., 83, 403, 1923;ibid., 84, 499, 1924.[362]Russell and Stewart, Ap. J., 59, 197, 1924.[363]Eder and Valenta, Atlas Typischer Spektren, 10, 1911.[364]See Chapter V,p. 69. A. Fowler, Bakerian Lecture, 1924, designates this an ultimate line.[365]See Chapter XIII,p. 185.[366]Milne, Observatory, 44, 264, 1921.[367]M. N. R. A. S., 83, 403, 1923; 84, 499, 1924.[368]Sommerfeld, Atombau und Spektrallinien, 457, 1922.[369]Meggers, Kiess, and Walters, Journ. Op. Soc. Am., 9, 355, 1924.

[355]Stewart, Phys. Rev., 22, 324, 1923.

[356]De Gramont, C. R., 171, 1106, 1920.

[357]Russell, Pop. Ast., 32, 620, 1924.

[358]A. Fowler, Report on Series in Line Spectra, 1922.

[359]R. H. Fowler, Phil. Mag., 47, 257, 1924.

[360]Proc. Roy. Soc., 99A, 135, 1921.

[361]M. N. R. A. S., 83, 403, 1923;ibid., 84, 499, 1924.

[362]Russell and Stewart, Ap. J., 59, 197, 1924.

[363]Eder and Valenta, Atlas Typischer Spektren, 10, 1911.

[364]See Chapter V,p. 69. A. Fowler, Bakerian Lecture, 1924, designates this an ultimate line.

[365]See Chapter XIII,p. 185.

[366]Milne, Observatory, 44, 264, 1921.

[367]M. N. R. A. S., 83, 403, 1923; 84, 499, 1924.

[368]Sommerfeld, Atombau und Spektrallinien, 457, 1922.

[369]Meggers, Kiess, and Walters, Journ. Op. Soc. Am., 9, 355, 1924.

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