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1961 Profile of the H-Gamma Line in the 40 Eridani B. Jerry Don Fuller Louisiana State University and Agricultural & Mechanical College

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FULLER, Jerry Don. PROFILE OF THE H-GAMMA LINE IN THE WHITE DWARF 40 ERIDANI B.

Louisiana State University, Ph.D., 1961 A stronom y

University Microfilms, Inc., Ann Arbor, Michigan PROFILE OF THE H-GAMMA. LINE IN THE

WHITE DWARF 40 ERIDANI B

A Dissertation

Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy

in

The Department of Physics and Astronomy

by Jerry Don Fuller B.S., Baylor University, 1954 M. S. , Baylor University, 1955 January, 1961 ACKNOWLEDGMENT

The author is greatly indebted to Dr. Raymond Grenchik, under whose patient and very helpful guidance this work was done.

Thanks are. also in order to Dr. Pierre Demarque for many helpful suggestions. The author also wishes to express appreciation to

Dr. Bill Townsend and the staff of the Louisiana State University

Computer Center, where a great part of the research was performed.

ii TABLE OF CONTENTS

CHAPTER PAGE

I. INTRODUCTION...... 1

II. STARK BROADENING AND THE LINE ABSORPTION COEFFICIENT ...... 8

III. NORMALIZATION OF THE LINE ABSORPTION COEFFICIENT 17

IV. CALCULATION OF THE LINE PROFIIE...... 22

V. RESULTS AND CONCLUDING REMARKS ...... 27

APPENDIX

I. SOAP STATEMENTS AND EXPLANATION FOR LINE COEFFICIENT CALCULATION ...... 33

II. SOAP STATEMENTS AND EXPLANATION FOR RESIDUAL INTENSITY CALCULATION ...... 43

III. CORRECTED, NORMALIZED ABSORPTION COEFFICIENTS FOR HE/H - 0 54

SEIECTED BIBLIOGRAPHY ...... 56

VITA ...... 57

ill LIST OF TABUS S

TABLE PAGE

I. Optical Depth, Electron Pressure and Continuous Absorption Coefficient at 4340.5A 5

II. Corrected, Normalized Absorption Coefficients for He/H - 0 54

iv LIST OF FIGURES

FIGURE PAGE

1. Optical Depth at 4340.5A vs. Optical Depth at 5000A 7

2. Flow Diagram for Line Absorption Coefficient Calculation 16

3. Optical Depth in H-gamma Line vs. Optical Depth in Continuum for Eight Points of the Line 24

4. Flow Diagram for Residual Intensity Calculation 26

5. Line Profiles of H-gamma Line 28

v ABSTRACT

The ultimate test of a model stellar atmosphere is the compari­ son of resulting theoretical line profiles with the observed spectrum.

Line profiles of the Balmer H-gamma line have been calculated for two model atmospheres, one of pure hydrogen, the other with equal numbers of hydrogen and helium atoms. In both cases the broadening of the absorption line was assumed to result solely from the first order

Stark effect in hydrogen, with both hydrogen ions and plasma electrons being included as perturbers.

The extremely involved computations, especially of the line absorption coefficients, were programmed and performed on an IBM 650 digital computer. Even so, the calculation of a single line profile requires approximately twenty-four hours of machine time.

The theoretical profiles obtained are found to be broader and deeper than the observed one, though in the far wings and near the center the agreement is fairly good. The profile obtained from the atmosphere containing helium is slightly broader and deeper than the pure hydrogen profile, in agreement with earlier predictions. CHAPTER I

INTRODUCTION

The study of stellar atmospheres has as its ultimate goal the determination of some of the fundamental physical properties of the , viz., the chemical composition of the atmosphere and the surface gravity of the star. After mathematically constructing a model stel­ lar atmosphere, one can calculate theoretical absorption line profiles to be compared with the observed spectrum. In principle, then, one can, by changing the mechanism of line broadening and varying the properties of the model atmosphere, account for all the features of the observed spectral lines.

Current Interest in the study of white dwarf atmospheres has 1 prompted the calculation of model atmospheres for the white dwarf 40

Eridani B. In this particular calculation, three atmospheres with different values of helium content were compared, as to emergent flux, with a pure hydrogen atmosphere. Furthermore, the pure hydrogen at- mosphere has been checked for constancy of flux, which has been found to be constant within 3% through optical depth 1.0

As a final check on the model atmosphere of Nicola and Grenchik, the profile of the Balmer H-gamma line has been calculated for helium

*L. Nicola, The Effects of Helium on a White Dwarf Atmosphere, Master of Science thesis completed at Louisiana State University (1959). 2Randolph Lesseps, Computer Program for the Constancy of Flux in a Stellar Atmosphere, Master of Science thesis completed at Louisiana State University (1960). 1 2

to hydrogen ratios of 0 and 1.0. The remainder of the present chapter describes preliminary calculations preparatory to the determination of the line absorption coefficients, which are then discussed in Chapter

II. The third chapter is a discussion of the normalization procedure for the line absorption coefficients. Following a description of the residual intensity calculation in Chapter IV, the final section of the paper gives the resulting line profiles and some concluding comments.

The model atmosphere used in the present line profile calcula­ tion was computed for a standard wavelength of 5000A. In it are tabu- 3 lated total gas pressure, temperature, fraction of unionized hydrogen and opacity, all as functions of optical depth.

First it was necessary to transform the optical depths at

5000A to the corresponding optical depths at 4340.5A, the wavelength of the H-gamma line. These new optical depths were found by numerically integrating the differential equation

^4-3 4-0 _ K 454 o (l-i)

^ ^Sooo ^ 5 0 0 0

The opacity £ 5000* ta^ulated in the atmosphere is the effective hydrogen absorption coefficient per gram, which includes bound-free and free-free transitions of neutral hydrogen.

%icola, o£. cit. , p. 12. 3

Its functional fomA is 3z TTC4 ftZ4 a~~^ f** - X 7/. “PF \ n ^ K(») = 3 V T cKr M ^ ^ v l«r 9 " + | ^ 9 j 0 “ c / ’ where the term (1 - e“ i*• r ) has been included to account for induced emission. For both X 4340 and X 5000, only photoionizations from levels with n - 3 contribute to the absorption. Accordingly, the first summation within the braces of equation (1-2) has precisely the same value for both wavelengths. Therefore, substitution of (1-2) into

(1-1) yields r - h e I

- (i*S±o\S 1 -"UCAJMoJhTj

dC,+5-W = (Asooo ) “ |- -he ~1 dCsoo° ' ( 1 ’ 3 ) 1 - « f [ ' (\5ooo)kT J The numerical integration was performed in the following manner: each increment of optical depth d C "5000 was multiplied by the arithmetic mean of the slopes at the corresponding endpoints, to yield an incre­ ment of optical depth ^€"4340* The relation between the optical depth at 4340.5A and the optical depth at 5000A is approximately linear, and is shown in Fig. 1.

Secondly, a table of continuous absorption coefficients, cor­ responding to a wavelength of 4340.5A, was calculated using equation

(1-2). The Gaunt factors in the expression were set equal to 1.0.

^Lawrence H. Aller, The Atmospheres of the and (New York: Ronald Press Co., 1953), p. 181. 4

One final preliminary calculation was required. The line absorption coefficients, to be discussed later, are functions of tem­ perature and electron pressure. The latter of the two quantities is not tabulated in the original model atmosphere, but is easily available

from the values of (1 - x), which is tabulated - the quantity (1 - x) represents the fraction of unionized hydrogen. The electron pressure

is obtained from

(1-4) k5 a useful variation** of the Saha ionization equation.

Table I is a tabulation of optical depth at 4340.5A, opacity at 4340.5A, and electron pressure, as functions of optical depths at

5000A.

^Nicola, op. cit., p. 7. 5

TABLE I

OPTICAL DEPTH, ELECTRON PRESSURE AND CONTINUOUS ABSORPTION COEFFICIENT AT 4340.5A, AS FUNCTIONS OF OPTICAL DEPTH AT 5000A (He/H - 0)

TAU (5000A) TAU (4340. 5A) Pe (4340.5A) K (4340. 5A)

0.0000 ( 0) 0.0000 ( 0) 4.9895 (0) 1.5152 (-1 2.0000 (-5) 1.4174 (“5) 1.2513 (3) 2.7660 (1) 4.0000 (-5) 2.8348 (-5) 1.2769 (3) 2.8069 (1) 6.0000 (-5) 4.2522 (-5) 1.3020 (3) 2.8465 (1) 8.0000. C-5) 5.6696 (-5) 1.3286 (3) 2.8883 (1) 1.0000 (-4) 7.0870 (-5) 1.3527 (3) 2.9256 (1) 2.0000 (-4) 1.4174 (-4) 1.4702 (3) 3.1023 (1) 3.0000 (-4) 2.1261 (-4) 1.5773 (3) 3.2558 (1) 4.0000 (-4) 2.8348 (-4) 1.6787 (3) 3.3956 (1) 5.0000 (-4) 3.5435 (-4) 1.7754 (3) 3.5240 (1) 6.0000 (-4) 4.2522 (-4) 1.8661 (3) 3.6404 (1) 7.0000 (-4) 4.9610 (-4) 1.9509 (3) 3.7450 (1) 8.0000 (-4) 5.6697 (-4) 2.0319 (3) 3.8418 (1) 9.0000 (-4) 6.3784 (-4) 2.1115 (3) 3.9351 (1) 1.0000 (-3) 7.0871 (-4) 2.1895 (3) 4.0241 (1) 2.0000 (-3) 1.4174 (-3) 2.8668 (3) 4.7076 (1) 3.0000 (-3) 2.1262 (-3) 3.4088 (3) 5.1667 (1) 4.0000 (-3) 2.8349 (-3) 3.8769 (3) 5.5175 (1) 5.0000 (-3) 3.5438 (-3) 4.2944 (3) 5.8027 (1) 6.0000 (-3) 4.2526 (-3) 4.6791 (3) 6.0470 (1) 7.0000 (-3) 4.9614 (-3) 5.0304 (3) 6.2546 (1) 8.0000 (-3) 5.6703 (-3) 5.3678 (3) ' ' 6.4472 (1) 9.0000 (-3) 6.3792 (-3) 5.6855 (3) 6.6218 (1) 1.0000 (-2) 7.0881 (-3) 5.9915 (3) 6.7843 (1) 2.0000 (-2) 1.4178 (-2) 8.6169 (3) 8.0331 (1) 3.0000 (-2) 2.1271 (-2) 1.0746 (4) 8.9834 (1) 4.0000 (“2) 2.8365 (-2) 1.2646 (4) 9.8356 (1) 5.0000 (-2) 3.5462 (-2) 1.4404 (4) 1.0643 (2) 6.0000 (-2) 4.2560 (-2) 1.6088 (4) 1.1440 (2) 7.0000 (-2) 4.9661 (“2) 1.7689 (4) 1.2221 (2) 8.0000 (-2) 5.6763 (-2) 1.9256 (4) 1.3007 (2) 9.0000 (-2) 6.3867 (-2) 2.0802 (4) 1.3801 (2) 1.0000 C-i) 7.0973 (-2) 2.2289 (4) 1.4587 (2) 2.0000 (-1) 1.4212 (-1) 3.6668 (4) 2.2845 (2) 3.0000 (-1) 2.1340 (-1) 4.9872 (4) 3.1523 (2) 4.0000 (-1) 2.8480 (-1) 6.2318 (4) 4.0388 (2) 5.0000 (“1) 3.5631 (-1) 7.3957 (4) 4.9145 (2) 6.0000 (-1) 4.2790 (-1) 8.4954 (4) 5.7670 (2) 7.0000 (-1) 4.9959 (-1) 9.5348 (4) 6.5828 (2) 8.0000 (-1) 5.7134 (-D 1.0511 (5) 7.3479 (2) 9.0000 (-D 6.4317 (-1) 1.1435 (5) 8.0601 (2) TAD (5000A) TAU (4340.5A) Pe (4340.5A) K (4340. 5A)

1.0000 (0) 7.1507 (-D 1.2303 (5) 8.7115 (2) 1.5000 (0) 1.0754 (0) 1.6027 (5) 1.1127 (3) 2.0000 (0) 1.4369 (0) 1.8977 (5) 1.2427 (3) 2.5000 (0) 1.7995 (0) 2.1491 (5) 1.3086 (3) 3.0000 (0) 2.1628 (0) 2.3762 (5) 1.3415 (3) 4.0000 (0) 2.8916 (0) 2.7998 (5) 1.3694 (3) 5.0000 (0) 3.6226 (0) 3.2009 (5) 1.3795 (3) 6.0000 (0) 4.3555 (0) 3.5896 (5) 1.3855 (3) 7.0000 (0) 5.0899 (0) 3.9408 (5) 1.3840 (3) 8.0000 (0) 5.8257 (0) 4.3552 (5) 1.3998 (3) 9.0000 (0) 6.5627 (0) 4.7274 (5) 1.4070 (3) 1.0000 (1) 7.3007 (0) 5.1007 (5) 1.4158 (3)

Note: Numbers are multiplied by ten to the power in parentheses. FIG. 1. RELATION RELATION 1. FIG.

434015A T 00 AD 5A. .5 0 4 3 4 AND 5000A AT EWE THEBETWEEN tr 5 a o o o OPTICAL DEPTHS 7 CHAPTER II

STARK BROADENING AND THE LINE ABSORPTION COEFFICIENT

The atmospheres of white dwarf stars possess relatively quite high ionic and electronic densities, as evinced by the large electron pressures tabulated in Chapter I. As a result, the broadening of the hydrogen lines in a white dwarf atmosphere is considered to arise pri­ marily from the linear Stark effect in neutral hydrogen, Doppler and collisional broadening being negligible. The early theories of line broadening, notably that of Holtsmark, considered the Stark field to be produced by ions alone. The theory upon which the present calcula­ tion is based takes into account both ions and plasma electrons.

The change in the energy levels of a hydrogen atom subject to an external electric field, as calculated by first order perturbation theory, is

A E = -f-n (It, -JiJ ea„F . (2-1)

In the above expression e is the charge on the electron, aQ is the

Bohr radius and F represents the external electric field. The princi­ pal quantum number n and the electric quantum numbers k^ and k^ are related by the condition k^, k2 * 0, 1, 2, ...., n - 1.

Since the energy shift A E i s , to a first order approximation. 9 directly proportional to the external field F, an exact treatment of the Stark broadening would require the application of a time-dependent perturbation theory involving the positions and velocities of all the ions and electrons. The resulting distribution of radiation intensity would then have to be averaged over all possible configurations of the ions and electrons, in order to calculate the absorption coefficient per radiating atom. To make the problem mathematically feasible, sim­ plifying assumptions must be made.

The subsequent cursory discussion of the basic line broadening assumptions applicable to this calculation follows the treatment by

Kolb.^ The first and most fundamental assumption made in this theory of Stark broadening is the classical path approximation, which states, briefly, that the_perturbing particles are considered to move with con­ stant velocity along trajectories which can be described classically.

Under the conditions of temperature and pressure which are being con­ sidered, the perturbers possess angular momentum quantum numbers I) mnr/i 2 3 X x — ^ of the order of 10 or 10 . Such large quantum numbers cor­ respond physically to distant collisions which involve very little exchange of energy between perturber and radiator.

As an extension of the same idea, the interactions are con­ sidered to obey the adiabatic approximation, by which collision-induced

*Alan C. Kolb, Theory of Hydrogen Line Broadening in High- Temperature Partially Ionized Gases, AFOSR-TN-57-8, ASTIA Document No. AD 115 040, Univ. of Mich. Eng. Res. Inst., (1957); Dissertation, Univ. of Mich., (1957). 10

transitions between states of different principal quantum number are neglected.

A great simplification of the basic quantum theory can be effected if the perturbation can be considered time independent. For

ions which move so slowly that their associated electric field does not change appreciably during the time of a collision,•such a restric­

tion is valid, and is called the statistical, or static, approximation.

The criterion for the above approximation is that the duration of a collision must be large compared to , where represents the variation from line center of the radiation frequency. For the much

faster plasma electrons, however, a contrasting approximation is made.

In the case of the electrons, if the collision duration is small com­

pared to the so-called phase shift approximation is made. Thus

the phase factor

e which arises in the equation for the absorption coefficient, 2 is re­

placed by oo

e 1 where in the former expression refers to the time during which the major part of the phase change takes place.

^Kolb, ££. cit. , p. 120. 11

Therefore, the statistical and phase shift approximations to

the classical path theory are designed to apply to slow ions and fast

electrons, the meanings of fast and slow being partially defined above.

The effect of the ions is to remove the normal hydrogen degeneracy by means of the linear Stark effect. The plasma electrons give rise to

two effects. First, they can cause phase changes in the hydrogenic wave functions. Secondly, the electrons induce transitions between

the degenerate substates of the hydrogen atoms. The former of the two

effects results in a further spreading of the levels already shifted

by the static field of the ions.

Since the external electric field acting upon the radiating

atom depends on the Instantaneous configuration of ions and electrons,

there must be an average taken over all such possible configurations.

This averaging process is accomplished by a distribution function, which Holtsmark originally derived considering only ions as perturbers.

The modified distribution function in the Kolb theory Includes a term

giving the relative importance of ions and electrons. Inclusion of

the new distribution function into the equation for the absorption

coefficient results in an integral which is not soluble in closed form. 3 The solution is expressed, therefore, in terms of two series expansions.

For purposes of the present discussion it is sufficient to write

the two above-mentioned series expansions, explaining the terms and

the manner in which the equations are applied.

3 Kolb, op. cit.. pp. 123-126. 12

Each expansion Involves two argumentsand R, which first must be defined.

where A oo^ * 2TTZXv(u< and ^ is the change in frequency from line center mentioned previously. The subscripts refer to the initial and final states of the atom. A ao< *-8 t^ie *-on halfwidth given by

A- = 4-52*a*^ , (2-3) where is the number of ions per cubic centimeter and X ^ ° [n(kL - k^)J - £n(kl “ t*Ie Stark displacement coefficient. Y' ^ is •a u ■*« the electron halfwidth given by

r.- = where v* is the mean relative velocity between perturber and radiator 3 2 X (obtained from kinetic theory), A ■ — e a0 — , and J represents the 2 u fi >m maximum phase change due to the electron. 1 2 rAcl , « -6 v Ni (2-5)

The ratio of ion and electron halfwidths, R, expresses the relative importance of the ion and electron fields. The mean relative velocity 13

Is directly proportional to the square root of the temperature, T, and is expressible in terms of electron pressure and temperature by means of the equation

(2-6) in which k is Boltzmann's constant and is assumed equal to Ne, the electron density. Thus, for each value of the Stark displacement co­ efficient Katl , bothp and R, and hence the following expressions for the absorption coefficient are ultimately functions only of T and Pe»

For values of & less than 1.0, i.e., for small the following expression for the absorption coefficient per atom converges:

A

(2-7)

For p greater than 5.0, the absorption coefficient per atom is expressed by

(2- 8) 14

In the preceding equations y U ^ Is an element o£ the unperturbed dipole matrix, or the unperturbed line strength, corresponding to a particular

Stark component. To avoid the difficult and tedious calculation of the matrix elements , they are replaced by the relative oscillator strengths f ^ . The fa#< , which are proportional to the , have

• * 4 been calculated by Schrodinger for the first four lines of the Balmer series. The use of these relative f-numbers necessitates the use of a special method of normalization, to be discussed in the following chapter.

In order to obtain the absorption coefficient per atom at a single point in the absorption line, it is necessary to determine Ie< for each of the Stark components - 13 of them for H-gamma - and to form the sum

(2-9) in which />(«) denotes the Boltzmann factor of the level from which absorption occurs. For the H-gamma line the non-zero values of co­ efficients X ft^ are 2, 3, 5, 7, 8, 10, 12, 13, 15, 17, 18, 20 and 22.

In the rangeyg » 1.0 toyfl « 5.0, neither of the series expan­ sions for the absorption coefficient is convergent. For values ofJ3 in this range the absorption coefficient is roughly estimated as being

^E. Schrodinger, Annalen der Physik, 80, (1926), p. 437. 15 proportional to . Fortunately, the few values of B for which convergence is not obtained occur at optical depths which do not contribute strongly to the flux integral to be mentioned in Chapter IV.

The calculation of the line absorption coefficients is performed in the following manner. For each of the 13 Stark components (values of X__), an absorption coefficient is calculated by means of (2-7),

(2-8) or the estimate. By equation (2-9) these values are summed to yield the total absorption coefficient for the particular value of

Such a total absorption coefficient is found for each of eight values of corresponding to

« 10, 20, 30 80A.

The above procedure is followed for each of the 53 values of optical depth in the stellar atmosphere, using the values of temperature and electron pressure mentioned in Chapter I.

As outlined above, the absorption coefficient calculation en­ tails the evaluation of more than 5000 infinite series, many of which require eight or more terms for convergence. Therefore, it necessarily is performed on a high speed electronic machine, the IBM 650 digital computer. The computer program was written in Symbolic Optimum

Assembly Program (SOAP) instructions. A list of the SOAP statements and a detailed explanation of the program is given in Appendix I. 16

INITIAL­ READ SET SET INPEX SET INDEX H A CARP * X » /0“ f A E G .8 - 0 J 1 r E G . A « IZ IZATION T I SET SUM I = O

COMPUTE PUNCU BETA j R POOOl STORE SETA I R

- - I f yes

COMPUTE I FOR SMALL BeTA M A K E £37. COMP JTB J FOR LARGE BAT A

SUM I = A X a SUM1 + I IO"7 SUBTRACTi FROM INDEX R E G ^

ADD L TO STORK AN$. ANS.c SUm I COMPUTE INDEX REG. 1— 1 N TIMES — 0OLTJTMANN INDEX 0 POOOIB BOLTZ. FACTOR FACTOR

FIG. 2. FLOW DIAGRAM FOR LINE ABSORPTION

COEFFICIENT CALCULATION. CHAPTER III

NORMALIZATION OF THE LINE ABSORPTION COEFFICIENT

The unperturbed line strengths yU #ot in equations (2-7) and

(2-8) are each proportional to the strength of the corresponding single Stark component. Stark effect splitting is symmetrical about the original unperturbed energy level; hence, corresponding to each

Stark component with energy shift A E » there is an oppositely situ­ ated component with energy shift -AE. Therefore, the quantity is proportional to the strength of such a symmetrically shifted pair of components. Use of theyli*^ leads, consequently, to an apparently straightforward normalization equation

c I M do* « X [/» (2/£,)] , (3.i) o in which I(u>) refers to the total absorption coefficient at some point,U), of the line, and p is the previously mentioned Boltzmann factor. The right member of (3-1) is the total absorption coefficient for a hydrogen line. The limits of the integral in (3-1) apparently take account of only half the absorption line, u) being the frequency displacement from line center. This is not the case, however, because the distribution function leading to the expressions for 1(a)) Includes

17 18 both high and low energy Stark components, thus doubling the absorp­

tion coefficient at each value of Ui to account for the symmetry Of

the line.

Substitution of the relative oscillator strengths faa for

the unperturbed strengths precludes the possible use of (3-1) for

the normalization of the line absorption coefficients. It is neces­

sary to use an alternate expression for the total absorption of the

line.

According to the classical theory of an oscillating electron,

the line absorption coefficient per oscillator, at a frequency p , is

In equation (3-2) e and m refer to the charge and , respectively,

of the electron and c to the velocity of light; Y t*ie frictional

damping constant. An integration over V yields the total absorption

coefficient per oscillator

/ cX (3-3)

The oscillator strength f for a particular transition, obtained from

quantum considerations, is a measure of the number of classical oscil­

lators per atom. This is not to be confused with the relative 19 f-numbers f a * , which give the strengths of the individual Stark components. Thus the absorption coefficient per atom for a parti­ cular transition is given by

* - f •

The numerical value of the oscillator strength for the transition n - 2 to n - 5 in hydrogen (Balmer H-gamma) is 0.0443.*

In further reference to equation (3-1), one should note a fundamental limitation to the line broadening theory being employed here. The integrand of the left member of (3-1) cannot be evaluated at the lower limit, because of the divergence of equations (2-7) and

(2-8) for " 0. This is particularly unfortunate since the major contribution to the integral is in the region of UJ m 0 (line center).

The limitation is still present, of course, when one normalizes by means of (3-4): oo

j ” I ( w ) d < D = f • (3-5) o

To breach the difficulty at CO "0, resort is made to the 2 fictitious absorption coefficient per atom at line center,

*R. v. d. R. Woolley and D. W. N. Stlbbs.The Outer Layers of a Star (Oxford: at the Clarendon Press, 1953), p. 111.

^Aller, ojj. clt., p. 252. 20

- K j l f ( M r 0 ~ m va ( z k j ) > (3_6)

7/0 denoting the central line frequency and M being the mass of a hydrogen atom, The value of o(0 , however, does not require normaliza­ tion and hence cannot correctly be Included in a numerical Integration of the left member of (3-5).

The normalization procedure deemed most reliable is actually a double normalization which will now be explained. At each point of the atmosphere the eight values of I(u)), calculated for a A “ 10, 20,

...., 80A, are extended to nine values by extrapolation to A A ■ 90 A.

This is done simply by letting I ( a A “ 90A) ■ %I( A A “ 80A), since at this point in the wings the absorption coefficient is virtually zero in comparison to values in the body of the line. Using these nine values, I(u>) is numerically integrated by Simpson's rule fromAA- 10A to a A ■ 90A. Comparison with planlmeter measurement shows this numeri­ cal integration to be accurate to within approximately 9%. The inac­ curacy in the total absorption coefficient, however, is only of the order of 3%, because less than half the integral comes from the Interval

A A - 10A to A A * 90A. Between A A ■ 0 and A A ■ 10A, the Integral is crudely approximated by setting I( A A “ 0) ■ I( A A “ 10A) and simply adding the resulting rectangle to the contribution from the wings,

’ftie' primary normalization then follows from equation (3-5). 21

The contribution to the total absorption from the region near line center is obviously underestimated by equating I ( A A “ 0) to

I( A A ■ 10A), since the absorption is greatest at line center. To better evaluate the integral, a second normalization is performed - in the same manner as the first - using the new values of I(u))> which are now at least roughly normalized. At this point, however, the con­ tribution between A A ■ 0 and A A * 10A is obtained by adding to the rectangle I ( a A ■ 10A) d A the triangle - I ( a A m 10A)J d A where d A is the 10 angstrom abcissa Interval (actually used in terms of a frequency Interval rather than a wavelength Interval). Being a function of temperature, the fictitious absorption coefficient o(0 is also calculated for each point of the atmosphere.

In addition to the normalization described above, the absorp­ tion coefficients are multiplied by a factor of 1 - X M ’ yielding absorption coefficients per gram of stellar material. As mentioned in Chapter I, (1 - x) is the fraction of unionized hydrogen.

Hydrogen being assumed the only absorbing medium in the atmosphere, the reciprocal of M, the mass of the hydrogen atom, is equivalent to the total number of absorbers per gram of material.

For the pure hydrogen atmosphere, the corrected, normalized absorption coefficients per gram are tabulated in Appendix III. CHAPTER IV

CALCULATION OF THE LINE PROFILE

The present chapter deals with the actual determination of the H-gamma profile, i.e., the calculation of the residual flux at each of the points for which the line absorption coefficient has been obtained. Residual flux is defined by the equation

f B„(t) E ^W dt n = ------| Ei, (r) E, (c)dc (4-D or, in words, r is the ratio of the radiation flux at a point in the line to that in the continuum.

3 i Bp( t) ti xl kT , (4-2) e - 1 is the Planck law for black body radiation at a frequencyU and temper­ ature T, and /°° f -crx -2 E (C) = e x d X z \ <4“3> is the second exponential integral function at an optical depth C .

The optical depths t refer to the absorption line and the optical depths

C to the continuum.

22 23

The next step is the determination of the optical depths t in the absorption line. Equation (1-1) states the direct relation between optical depth and absorption coefficient. A similar equa­ tion may be used to obtain the depths t. Thus

dt = dc43+0 , <«> 4 3 4 o where ^ represents the corrected, normalized line absorption coef­ ficient obtained by the methods of Chapter III. The continuous absorption coefficient that given in Table I. The slope

^ * 4 3 4 0 * 4 3 4 0 of equation (4-4) is indicative of the fact that in an absorption line the absorption is due to both the line and the continuum. Equa­ tion (4-4) is numerically integrated for each of the eight sets of line absorption coefficients j , resulting in eight sets of optical depth t in the H-gamma line.

After obtaining the optical depths in the line it remains to do the integrals of (4-1) and form the ratios, i.e., the residual fluxes. Because of the unequal Increments of optical depth, these numerical integrations for the flux must be done by trapezoidal rule or by some means of linear interpolation. The results from the 24

INI TIA L- SET SUM IK) SET SUH READ CARDS CON­ \ZAT10 N CONTItiUUM'O IN LINE-O TAINING tr,dclNCOKl AUOVtdv IN L/NE

C0MPUTE^^;

A n s w e r a t bach aA SET INDEX REG.A»8 -L IN E SUM DlVIDSP BY CONTINUUM SUN\ COMPUTE ExCC)lNL\UE F o r ^ X co ntrolled

5ET INDEX REG. A* 8 b tin p e x Re g . A YES COMP. VALUE OFTRAR NAS LAST S E T (NT PANEL IN LINE

o f c a r p s been I READ? LINE SUM = LINE SUM +• LINE PANEL VALUE AT EACH a \

PUNCH CONT. SUBTRACT i FROM PUNCH E>.W/N LINE INDEX RgG.A

FIG. 3. FLOW DIAGRAM FOR RESIDUAL

INTENSITY CALCULATION. 25 trapezoidal rule and a three-point linear interpolation formula differ by less than 3%. Accordingly, the former is used for simpli­ city.

The exponential integrals arising within the flux integrals are calculated by Simpson's Rule, and are compared for accuracy with the tables of Kourganoff.*

Residual intensity calculations for all of the eight points of the line are done simultaneously on an IBM 650 digital computer.

The program (in SOAP) is listed and discussed in Appendix II.

*V. Kourganoff, Basic Methods in Transfer Problems (Oxford: at the Clarendon Press, 1952), p. 266. < z z < 0 1 I

-2, i m il IO -5

FIG.+. RELATION BETWEEN OPTICAL DEPTHS IN THE CONTINUUM (4 5 4 -0 .5 A ) AND IN THE H-GAMMA LIME CHAPTER V

RESULTS AMD CONCLUDING REMARKS

Employing the methods described in the previous chapters, the residual £lux is determined at distances from the line center equal to 10, 20, 80A. In Fig. 5 are shown the two obtained profiles of the H-gamma line, as calculated from a pure hydrogen atmosphere and from an atmosphere with equal numbers of hydrogen and helium atoms. For comparison are shown also the most recent observed curve, by Greenstein,^ and a previous theoretical profile 2 by Grenchik.

As is observable from the figure, the profile derived from an atmosphere containing helium is somewhat broader and deeper than that derived from the pure hydrogen atmosphere. Such a result agrees with expectations. Rie calculations of Nicola and Grenchik reveal that, for a given optical depth, the value of electron pressure in­ creases with atmospheric helium content. The final analytic form for p ,

8 )3 = 5.9067 *10

^Jesse L. Greenstein, Handbuch der Fhysik, 50, (1958), p. 169. 2 Raymond Grenchik, Ph.D. thesis, Indiana University, (1956).

27 RESIDUAL INTENSITY 0.7 Q9 0-5 0.6 03 10 I.5 P IE FTHE H- IE N 0 4 IN LINE A M M A -G H E H T OF FILE O PR 5. FIG. 20 30 a X 0 4 INANGSTROMS 0 5 0 6 — : ------CURVE2. CURVE ------0 7 - OBSERVED ERIDANI B 1 HE/H = = HE/H 1 GRENCHIK E/H H = 1 = 90 O 2 9

-2/3 shows that it is proportional to P . For greater than 1.0 the line absorption coefficient is proportional to inverse powers of

P (cf. eqn. 2-8). Thus it is easily seen that increasing electron pressures are accompanied by greater line absorption, which depresses the profile. The physical reasoning appears quite naturally to be that the line broadening is increased due to stronger Stark fields.

Compared with the observed profile, however, the profile for a pure hydrogen atmosphere is too broad and deep, being only a few per cent higher in the wings than that of Grenchik. Also, the behavior of both the present curves in the region a X- 20-50A is quite unex­ pected. Judging from several plots made of the line absorption coef­ ficient (at different optical depths), the profile should rise more steeply in this region, as does the obsepratlonal profile. This definitely suggests the possibility of some inadequacy in the method of normalization of the line coefficients. The crossover of the two theoretical curves between 20A and 30A is probably due to the same hitherto undetermined cause.

Were the irregularities in the line shape corrected, however, the pure hydrogen profile would still be deeper than the observed one by several per cent, except neai: line center. One is apparently forced to one of two conclusions. Either the broadening theory applied over­ estimates the line absorption, or the model atmosphere in some way fails to account for all the continuous absorption. The former appears to be very doubtful. Results of the Kolb theory of line broadening 30 have been favorably compared with experimental measurements from shock tubes.

Regarding possible other sources of continuous absorption, a rough calculation is made of the contribution of negative hydrogen . ions. The probability of H*" ion formation in a star with the effec­ tive temperative of 40 Erldani B is quite small, and decreases with increasing temperature. Based upon tabulated values of the contin­ uous absorption coefficient of H ”, the contribution from this cause is less than 2.5% of that from neutral hydrogen, at a temperature of 4 10 degrees. Since even the boundary temperature of the atmosphere 4 under consideration is slightly greater than 10 degrees, the effect of H” ions may very safely be neglected.

Likewise, there is an exceedingly small amount of molecular 4 hydrogen present in the atmosphere of 40 Eridani B. Weidemann, in doing calculations of metal lines in the white Dwarf , has obtained a plot of log ejj vs. log y, where e^ is hydrogen abun­ dance and y is the ratio of molecular hydrogen to neutral hydrogen, for four different values of . An approximate extrapolation to the effective temperature of 40 Eridani B yields

- 2.0 as an upper limit for log y. The abundance of molecular hydro­ gen in the Eridani B atmosphere is therefore less than 1% of the neutral hydrogen abundance.

O JWoolley and Stibbs, op. clt., p. 65. 4 Volker Weidemann, Astrophysical Journal, 131, (1960), p. 656. 31

If one supposes convection to occur to some appreciable degree

In the atmosphere, the Chandrasekhar grey body temperature distribu­

tion becomes a less valid assumption. Although no convective white

dwarf atmospheres have been calculated as yet, the temperature gradient would clearly be less steep than that of the grey body distribution.

The movement of hot gases from-the Interior of the star toward the sur­

face would tend to equalize the temperatures In the various strata of

the atmosphere, thus producing higher overall temperatures at optical

depths corresponding to radiative equilibrium. Equation (l-4)reveals 5/2 that Pe oC T , so that, from equation (5-1),

ft °C - T . (5-2)

An increase in temperature will therefore cause an increase in line

absorption. Qualitatively speaking, the greater temperatures would,

however, seem to lower the continuous absorption by decreasing the

fraction of neutral hydrogen, which is the primary source of continuous

absorption. Any quantitative determination of the effect of a convec­

tive atmosphere is almost certain to result in an even deeper line

profile.

One further line of thought is determination of the amount the

continuum is depressed by the overlapping wings of the Balmer lines.

The H-gamma and H-delta lines cross each other at approximately 150A 32

from the center of the H-gamma line. Having been computed no farther

than 80A from line center, the absorption (1-R) in the far wing region

can only be estimated roughly. However, the contribution of H-gamma

and H-delta together might' conceivably depress the continuum by as much

as 10Z between the lines. The calculation of profiles for several higher Balmer lines would afford several crossover points, from which

an improved value of the continuous absorption coefficient could be

determined. Such a correction to the continuous absorption would pro­

duce a shift of the line profile in the direction of better agreement with observation. It is not possible, without a residual flux calcu­

lation, to quantitatively estimate this Improvement.

In view of the uncertainty attached to the method of normali­

zation of the line absorption coefficients, there is definitely justi­

fication for a recalculation of the H-gamma profiles, using dipole

matrix elements rather than the relative f-numbers in the determination

of the line coefficients. With the advantages of high speed computa­

tion, it is perhaps now desirable to vary the boundary temperature and

surface gravity of the model to determine their effect upon the profile.

An increased surface gravity will Increase the pressures within the

atmosphere and doubtless broaden the lines even more.

By certain additional refinements, the IBM 650 computer programs

used in this calculation may be substantially reduced in.required ma­

chine time. Nevertheless, the computations involved will require

several hours, and could very profitably be programmed for an even

faster machine. APPENDIX I

SOAP STATEMENTS AND EXPLANATION POR LINE COEFFICIENT CALCULATION

Statement Location Operation Data Instruction Number Address Code Address Address Comments

1 REG R1951 1960 2 REG P1927 1936 3 REG C1937 1949 4 REG K1961 1973 5 START RCD R0001 6 RAB 0000 7 LDD C14 8 STD DLAMB STPLA 9 STPLA RAA 0012 10 LDD ZERO 11 STD SUMI COMP 12 COMP RAU R0003 CALCULATION 13 FDV R0006 14 LDD NXTl FLLOG 15 NXTl FMP TU3DS 16 LDD NXT2 FLEXP 17 NXT2 FMPDLAMB 18 FDV C0001A 19 FMP C15 OF 20 STU BETA 21 RAU R0006 22 FDV R0003 23 LDD NXT3 FLLOG 24 NXT3 STU LNPOT 25 FDV TWO 26 LDD NXT4 FLEXP. BETA. 27 NXT4 FDV R0003 28 FMP C0001A 29 LDD NXT5 FLLOG 30 NXT5 FAD C16 31 RSU 8003 32 STU BRKT 33 RAU LNPOT AND 34 FDV THREE 35 LDD NXT6 FLEXP 36 NXT6 FMP C0001A 37 FMP BRKT 38 STU DEN 39 RAU R0003 40 LDD NXT7 FLSQR 41 NXT7 FMP C17 42 FDV DEN 43 STU R 44 RAU BETA

33 34 Statement Location Operation Data Instruction Number Address Code Address Address Comments

45 FSB FIVE TEST FOR 46 BMI LBETA APPROPRIATE 47 FAD FOUR SERIES EXPANSION 48 BMI EST 49 RAU F03DS BEGINNING 50 LDD NXT8 FLLGR OF SMALL BETA 51 NXT8 LDD NXT9 FLEXP EXPANSION 52 NXT9 STU GAM1 53 RAUBETA 54 FMPBETA 55 STUBETSQ 56 RAUR 57 FMPR 58 STU RSQ 59 RAU ONE 60 FDV RSQ 61 FAD BETSQ 62 LDD NXT10 FLLOG 63 NXT10 STU LNPAR 64 FDVTWO 65 LDD NXTll FLEXP 66 ““ NXTll STU PAR 67 RAUBETA 68 FMPR 69 LDD NXTl 2 FLATN 70 NXTl 2 STU ARCTG 71 LDD NXT13 FLCOS 72 NXT13 FMP PAR 73 FMP GAM1 74 STU SUM 75 LDD TWO 76 STD N 77 RAU FIXD2 78 STUFIXDN LOOPl 79 LOOPl SUP FIXDl — 80 RAC 8003 81 RAU ONE REP 82 REP FMP MIONE 83 SXC 0001 84 NZCREP ENUFF 85 ENUFF STU FI 86 RAUN 87 FSB ONE 88 STU M RET 89 RET STU FACT 90 RAUM 91 FSB ONE 92 STU M 93 NZU ALL 94 FMPFACT RET 95 ALL LDDFACT 35 Statement Location Operation Data Instruction Number Address Code Address Address Comments

96 STD DEN 97 RAU N 98 FAD ONE 99 FMP TU3DS 100 LDD NXT14 FLLGR 101 NXT14 LDD NXTl 5 FLEXP 102 NXTl 5 STU GAMMA 103 RAU LNPAR 104 FMP N 105 FDV TWO 106 LDD NXTl 6 FLEXP 107 NXT16 STU F2 108 RAU ARCTG 109 FMP N 110 LDD NXTl 7 FLCOS 111 NXTl 7 FMP GAMMA. 112 FMP F2 113 FMP FI 114 FDV DEN 115 STU TERM 116 FAD SUM 117 STU SUM 118 RAU TERM 119 BMI NEXT 120 RSU 8003 NEXT 121 NEXT FSB C18 122 BMI THRUSTPN 123 STPN RAU N 124 FAD ONE 125 STU N 126 RAU FIXDN 127 AUP FIXDl 128 STU FIXDN LOOPl 129 THRU RAU SUM 130 FMP C19 ENTER 131 ENTER FMP KOOOlA CMPLT 132 ESTRAU BETA 133 LDD FLLOG 134 FMP TUPT3 135 LDD FLEXP 136 STU LOWER 137 RAU APPRX 138 FDV LOWER ENTER 139 LBETA LDD TWO BEGINNING 140 STD N OF 141 LDD ZERO EXPANSION 142 STD SUM FOR 143 RAU FIXD2 LARGE 144 STU FIXDN LOOP2 BETA 145 L00P2 SUP FIXDl 146 RAC 8003 - 36 Statement Location Operation Data Instruction Number Address Code Address Address Comments

147 RAD ONE REPl 148 RE Pi FMP MI ONE 149 SXC 0001 150 NZC REP1 ENDF 151 ENUF STD FI 152 RAD N 153 FSB ONE 154 STD M “ RETN 155 RETN STD FACT 156 RAD M 157 FSB ONE 158 STD M 159 NZD ALL1 160 FMP FACT RETN 161 ALL1 LDD FACT 162 STD DEN 163 RAD N 164 FMP THREE 165 FAD ONE 166 FDV TWO 167 LDD NXT18 FLLGR 168 NXT18 LDD NXTl 9 FLEXP 169 NXT19 STD GAMMA 170 RAD N 171 FMP THREE 172 FSB ONE 173 STD NNNMl 174 RAD BETA 175 LDD NXT20 FLLOG 176 NXT20 FMP NNNMl 177 FDV TWO 178 LDD NXT21 FLEXP 179 NXT21 STD DENA 180 RAD ONE 181 FDV BETA 182 FDVBETA 183 FDVR 184 FDV R 185 FAD ONE 186 LDD NXT22 FLLOG 187 NXT22 FMP NNNMl 188 FDVFOUR 189 LDD NXT23 FLEXP 190 NXT23 STD DENB 191 RADBETA 192 FMP R 193 - LDD NXT24 FLATN 194 NXT24 FMP NNNMl 195 FDV TWO 196 LDD NXT25 FLCOS 197 NXT25_ . FDV DENB 198 FDVDENA 37 Statement Location Operation Data Instruction Number Address Code Address Address Comments

199 FMPGAMMA 200 FDV DEN 201 FMP 'FI 202 STU TERM 203 FAD SUM 204 STU SUM 205 RAUTERM 206 BMI SUBR 207 RSU 8003 SUBR 208 SUBR FSB C18 209 BMI THRU1 STPNl 210 STPNl RAU N 211 . FAD ONE 212 STU N 213 RAUFIXDN 214 AUP FIXDl 215 STU FIXDN LOOP2 216 THRU1 RAU ONE 217 FDVR 218 FDVR 219 STURCRSQ 220 RAUBETA 221 FMPBETA 222 FAD RCRSQ 223 FMP R 224 STU BOTM 225 RAU ONE 226 FDVBOTM 227 FAD SUM 228 FMP C20 229 FMP K0001A CMPLT 230 CMPLT FAD SUMI 231 STU s u m 232 SXA 0001 233 BMA COMP 234' RSU DLAMB CALCULATION 235 FMP C21 236 FSB ENGY2 OF 237 FDV BOLTZ BOLTZMANN 238 FDV R0003 FACTOR 239 LDD NXT27 FLEXP 240 NXT27 FMP SUMI 241 STU P0001B 242 AXB 0001 243 RAU DLAMB 244 FSB LIMIT 245 NZU PUNCH 246 RAU DLAMB 247 FAD C14 248 STU DLAMB STPLA. 249 PUNCH PCH P0001 START 38 Statement Location Operation Data Instruction Number Address Code Address Address Comments

250 ZERO 00 0000 0000 LIST OF 251 ONE 10 0000 0051 CONSTANTS 252 HI ONE 10 0000 0051 253 TWO 20 0000 0051 254 THREE 30 0000 0051 255 FOUR 40 0000 0051 256 TU3DS 66 6666 6650 257 F03DS 13 3333 3351 258 FIXDl 00 0000 0001 259 FIXD2 00 0000 0002 260 C14 10 0000 0044 261 . C15 59 0674 1059 262 C16 26 7192 0051 263 C17 90 2826 0650 264 C18 50 0000 0044 265 C19 42 4413 1850 266 C20 63 6619 7650 267 C21 10 5399 2944 268 . ENGY2 16 2606 0440 269 BOLTZ 13 8000 0035 270 LIMIT 80 0000 0044 271 C0001 22 0000 0052 272 C0002 20 0000 0052 _ . 273 C0003 18 0000 0052 274 C0004 17 0000 0052 275 C0005 15 0000 0052 276 C0006 13 0000 0052 277 C0007 12 0000 0052 278 C0008 10 0000 0052 279 C0009 80 0000 0051 280 C0010 70 0000 0051 281 C0011 50 0000 0051 282 C0012 30 0000 0051 283 C0013 20 0000 0051 284 K0001 70 0000 0051 285 K0002 80 - 0000 0052 286 K0003 13 1800 0054 287 K0004 52 0000 0052 288 K0005 11 5200 0054 289 K0006 16 6400 0054 290 K0007 16 6000 0053 291 K0008 17 6000 0054 292 K0009 15 0000 0052 293 R0010 11 6000 0053 294 K0011 19 2000 0053 295 R0012 93 2000 0053 296 K0013 15 6000 0053 297 TUPT3 23 0000 0051 298 APFRX 42 7200 0051 299 FIVE 50 0000 0051 300 FOUR 40 0000 0051 39

The preceding symbolic optimum assembly program was assembled with the Wisconsin Floating Point Subroutine Package Deck, plus a real gamma function subroutine.^ The gamma function subroutine actually computes the logarithm to base e of the real gamma function, and is used in conjunction with the ex subroutine. The following list of the sub­ routines gives the entry symbols in parentheses.

Exponential ex (FLEXP) Logarithm to base e (FLLOG) Square root (FLSQR) Arctangent (FLATN) Cosine (FLCOS) Gamma function (FLLGR)

Initialization for the absorption coefficient program (statements

1-4) consists of specifying read and punch bands and thirteen locations each for the Stark displacement coefficients and relative oscillator strengths.

The program proper begins in statement 5, the read command. The following statement zeros index register B, which specifies the appropri­ ate word of the output card. Statements 7 and 8 store the constant C14

(10"7 or 10A) in DLAMB ( ), and direct the computer to STPLA (step A ) where is located the first instruction of the outermost loop of the pro­ gram. This loop causes the calculation of an absorption coefficient for each of eight values of A A , ranging from 10A through 80A. The instruc­ tion located in STPLA places 12 in index register A, since index register

A is used to vary both the Stark displacement coefficients and the f-numbers in the computation of absorption coefficients for individual

Stark components. The summation over the thirteen components is

iRoy A. Parker, "Floating Point Gamma Function" (unpublished subroutine for the IBM 650 Computer), Louisiana State University, 1960. 4° accomplished by a running sum (SUMI) that is originally set to zero in statements 10 and 11.

COMP (statement 12) begins the second major loop of the program, which calculates absorption coefficients for the individual Stark com­ ponents. The loop begins by computing and storing the values of and

R by means of the following equations: 8 A \ p - 5.9067 xio ~Y±-

„ 0.90282606

The temperature and electron pressure are denoted by R0003 and R0006, respectively, being in word 3 and word 6, respectively, of the input card. The Stark coefficients, X , are given the regionalized name uot C0001A in order that they can be controlled by index register A. C15,

C16 and C17 are the three constants involved in the two equations above.

Statements 44-48 are the first branching Instructions of the program, and they choose the appropriate asymptotic expansion for the absorption coefficient. For yfl = 5 or greater the series expansion for large^ (LBETA) is used. This expansion begins in statement 139. The range m 1 to ^ . ■ 5 was found to be Inaccessible to either of the asymptotic forms, and so was bypassed with a simple rough estimate of the absorption coefficient in that range. Statement 132, named EST, -2.3 begins this short approximation, which is proportional to j3 . For

less than one, the series expansion for smalljB is used beginning in statement 49. 41

Hie bulk of the Instructions in the program are involved with calculating the two aforementioned series expansions for the absorption coefficient. Each term involves several subroutines to compute such

terms of a particular series are summed as they are computed, using a running sum, and the series is cut off after calculating a term less than 5 X 10“^. The convergence criterion is applied by obtaining the absolute value of the series term, subtracting 5 X 10 ^ (C18) and using a branch on minus (BMI) command. For example, the convergence test for the small ^ series is located in statements 118 through 122. After convergence in either series, the series sum (SUM) is multiplied by the appropriate relative f-number, denoted by K0001A.

From there the program proceeds to the completion (CMPLT) of the loop for a particular Stark component. The statements 230 (CMPLT) and 231 develop the sum over the different components. The instructions in statements 232 and 233 test index register A-to see if all thirteen components have been analyzed. If not, the computer is returned to

COMP in statement 12. Upon subsequently leaving the COMP loop, the program has developed the total absorption coefficient for the particu­ lar a A . It now calculates a Boltzmann factor (statements 234-239) and multiplies by this total absorption coefficient (SUMI). By statement

241 this result is stored in the proper punch location (P0001B).

Statements 243-245 compare AX(DLAMB) with 8 X 10“^ (80A). If

A A l 8 less than 80A, it is Incremented by 10A (C14) , and the program returns to STPLA (statement 9). If A X 8 80A, then all eight words 42 of the punch band are filled. At this point the computer is made to PUNCH (statement 249) and then sent to START to read another card.

Beginning in statement 250, 51 necessary constants are listed, including the Stark displacement coefficients (C0001-C0013) and the relative oscillator strengths (K0001-K0013). APPENDIX II

SOAP STATEMENTS AND EXPLANATION FOR RESIDUAL INTENSITY CALCULATION

Statement Location Operation Data Instruction Number Address Code Address Address Comments

1 REG R1951 1960 2 REG S1851 1860 3 REG T1751 1760 4 REG P1927 1936 5 REG Q1827 1836 6 REG 01727 1736 7 BEG H1882 1889 8 REG J1908 1915 9 REG W1916 1923 10 REG B1890 1897 11 REG 11900 1907 12 REG X1600 . 1695 13 START RAU 8002 ZEROS IN 14 STL N CARD COUNT 15 RAA 0008 RE 16 RE STD IOOOOA IN RUNNING 17 STD W0000A 18 SXA 0001 INTEGRATION 19 NZA RE 20 STD SCONT SUMS 21 STD I 22 STD Q0002 AND IN 23 - STD Q0003

24 STD Q0004 . UNUSED 25 STD Q0005 26 STD Q0006 PUNCH 27 STD Q0007 28 STD Q0008 READ LOCATIONS 29 READ BCD R0001 READ 30 BCD S0001 DATA CARDS 31 ROD T0001 32 RSU T0005 CALCULATION 33 FMP C4 34 LDD FLEXP 35 FMP C5 36 STU ADDl OF 37 RSU T0005 38 FMP C6 39 LDD FLEXP 40 FMP C7 CHANDRASEKHAR 41 STU ADD2 42 RSU T0005 43 FMP C8 44 LDD FLEXP Q (TAU)

43 Statement Location Operation Data Instruction Number Address Code Address Address Comments

45 FMP C9 46 FAD ADD2 47 FAD ADD1 48 FSB CIO FUNCTION 49 RSU 8003“ 50 FAD T0005 51 FMP Cl -52 LDD FLLOG 53 FDV FOUR 54 LDD FLEXP PHYSICAL 55 STU TCONT TEMPERATURE 56 RAU C2 57 FDV TCONT 58 LDD FLEXP 59 FSB ONE 60 STU DEN 61 RAU C3 62 FDV DEN PLANCK 63 STU BCONT FUNCTION 64 LDD THREE FIRST 65 STD M 66 RAA 0041 LOOPl 67 LOOPl RSU T0005 PART OF 68 FMP M • 69 LDD FLEXP 70 STU EXP SIMPSON'S 71 RAU ONE RULE 72 FDV M 73 FDVM 74 FMP EXP STO 75 STO STUXOOOOA INTEGRATION 76 SXA 0001 77 RAU M 78 FSB PT05 79 STUM TO 80 FSB ONE 81 BMI LOOPl 82 RAA 0040 83 RAU ZERO MORE OBTAIN 84 MORE FAD XOOOOA 85 SXA 0002 86 NZA MORE 87 FMP TWO E2 (Tau) 88 STU PARTI 89 RAA 0019 FOR THE 90 RAB 0039 91 RAU ZERO MOREl CONTINUUM 92 MOREl FAD XOOOOB 93 SXB 0002 45

Statement Location Operation Data Instruction Number Address Code Address Address Comments

95 NZA MOREl FIRST 96 FMP FOUR 97 FAD PARTI SECTION 98 FAD X0001 INTEGRATES 99 FAD X0041 100 FMP INCR2 FROM X - 1 101 STU INTI TO X - 3 102 LDD TEN SECOND 103 STD M 104 RAA 0015 LUPl 105 LUPl RSU T0005 106 FMP M 107 LDD FLEXP . 108 STU EXP SECTION 109 RAU ONE 110 FDV M 111 FDV M 112 FMP EXP 113 STU XOOOOA 114 SXA 0001 INTEGRATES 115 RAU M 116 FSB PT5 117 STU M 118 FSB THREE 119 BMI LUPl 120 RAA 0014 121 RAU ZERO MO 122 MO FAD XOOOOA FROM 123 SXA 0002 124 NZA MO 125 FMP TWO 126 STU PARTI 127 RAA 0006 128 RAB 0013 X “ 3 129 RAU ZERO MOl 130 MOl FAD XOOOOB 131 SXB 0002 132 SXA 0001 133 NZA MOl 134 FMP FOUR TO 135 FAD PARTI 136 FAD X0001 137 FAD X0015 138 FMP INCR3 X => 10 139 - FAD INTI 140 STU Q0001 141 FMP BCONT COMPUTATION 142 STU J OF 143 FAD I TRAPEZOIDAL 144 FDV TWO INTEGRATION 46 Statement Location Operation Data Instruction Number Address Code Address Address Comments

145 FMP T0003 PANEL FOR 146 FAD SCONT RESIDUAL 147 STU SCONT INTENSITY 148 LDD J 149 STD I 150 RAA 0008 LOOP2 151 L00P2 RAU R0000A CHOOSE 152 FSBTENAPPROPRIATE 153 BMI STAU INTEGRATION 154 FSB TEN RANGE FOR 155 BMI LTAU VLTAU E 2 (Tau) IN LINE 156 STAU LDD THREE 157 STD M SIMPSON'S 158 RAB 0041 LOOP3 RULE 159 L00P3 RSU R0000A FOR SMALL 160 FMP M TAU 161 LDD FLEXP 162 STU EXP 163 RAU ONE 164 FDV M 165 FDV MFROM 166 FMP EXP STOR 167 STOR STU X0000B 168 SXB 0001 169 RAU M 170 FSB PT05 171 STU M 172 FSB ONE 173 BMI LOOP3 X - 1 174 RAB 0040 175 RAU ZERO MORE 2 176 MORE 2 FAD X0000B 177 SXB 0002 178 NZB MORE 2 179 FMPTWO 180 STU PARTI 181 RAC 0019 TO 182 RAB 0039 183 RAU ZERO MORE 3 184 MORE 3 FAD X0000B 185 SXB 0002 186 SXC 0001 187 NZC MORE 3 188 FMPFOUR 189 FAD PARTI X - 3 190 FAD X0001 191 FAD X0041 192 FMP INCR2 193 STU INTI 194 LDD TENSIMPSON'S 47 Statement Location Operation Data Instruction Number Address Code Address Address Comments

195 STD M RULE 196 RAB 0015 LUP3 197 LUP3 RSUROOOOA 198 FMPM 199 LDD FLEXP 200 STU EXP 201 RAU ONE 202 FDV M 203 FDV M FROM 204 FMP EXP 205 STU XOOOOB 206 SXB 0001 207 RAUM 208 FSB PT5 209 STU M 210 FSBTHREE 211 BMI LUP3 X - 3 212 RAB 0014 213 RAU ZERO M02 214 M02 FAD XOOOOB 215 SXB 0002 216 NZB M02 217 FMP TWO 218 STU PARTI TO 219 RAC 0006 220 RAB 0013 221 RAU ZERO M03 222 M03 FAD XOOOOB 223 SXB 0002 224 SXC 0001 225 NZC M03 226 FMP FOUR X - 10 227 FAD PARTI 228 FAD X0001 229 FAD X0015 230 FMP INCR3 231 FAD INTI STORE 232 STORE STU OOOOOA 233 FMPBCONT TRAPEZOIDAL 234 STU JOOOOA 235 FAD I0000A INTEGRATION 236 FDV TWO PANEL 237 FMP SOOOOA 238 FAD WOOOOA FOR 239 STU WOOOOA RESIDUAL 240 LDD JOOOOA INTENSITY 241 STD IOOOOA OUT IN LINE 242 LTAU LDD ELEVN SIMPSON'S 243 STD M RULE Location Operation Data Instruction Address Code Address Address

244 RAB 0011 LOOP4 245 L00P4 RSU R0000A 246 FMPM 247 STD TEMP 248 RSU 8003 249 FSBFIFTY 250 BMI NO 251 RAU TEMP YES 252 NO RAU ZERO STOl 253 YES LDD FLEXP 254 STUEXP 255 RAU ONE 256 FDV M 257 FDV M 258 FMP EXP STOl 259 STOl STU XOOOOB 260 SXB 0001 261 RAU M 262 FSB ONE 263 STU M 264 FSB ONE 265 BMI LOOP4 266 RAB 0010 267 RAU ZERO MORE4 268 M0RE4 FAD XOOOOB 269 SXB 0002 270 NZB MORE4 271 FMP TWO 272 STU PARTI 273 RAC 0004 274 RAB 0009 275 RAUZERO MORE 5 276 MORE 5 FADXOOOOB 277 SXB 0002 278 SXC 0001 279 NZC MORE 5 280 FMPFOUR 281 FADPARTI 282 FAD X0001 283 FAD X0011 284 FMP INCR4 STORE 285 VLTAU RAUZERO STORE 286 OUT SXA 0001 287 NZA LOOP2 288 PCH Q0001 289 PCH 00001 290 RAUN 291 AUP FONE 49

Statement Location Operation Data Instruction Number Address Code Address Address Comments

292 STU N IF ALL 293 SUP FTY3 CARDS 294 NZU READ READ 295 RAA 0008 L00P5 296 L00P5 RAU WOOOOA RATIO 297 FDV SCONT OF 298 STU POOOOA FLUXES 299 SXA 0001 YIELD 300 NZA LQOP5 RESIDUAL 301 PCH P0001 INTENSITIES 302 HLT 303 C3 48 . 5887 1648 LIST OF 304 C4 11 0319 0051 305 C5 94 6000 0048 CONSTANTS 306 C6 15 9178 0051 307 C7 36 1900 0049 308 C8 44 5808 0051 309 C9 83 9200 0049 310 CIO 70 6920 0059 311 TWO 20 0000 0051 312 FOUR 40 0000 0051 313 Cl 27 1950 8767 314 C2 33 1511 3055 315 TEN 10 0000 0052 316 FIFTY 50 0000 0052 317 PTONE 10 0000 0050 318 ONE 10 0000 0051 319 ZERO 00 0000 0000 320 INCR1 33 3333 3349 321 INCR2 16 6666 6649 322 PT05 50 0000 0049 323 FTY3 00 0000 0053 j 324 FONE 00 0000 0001 325 THREE 30 0000 0051 326 PT5 50 0000 0050 327 INCR3 16 6666 6650 328 . ELEVN 11 0000 0052 329 INCR4 33 3333 3350 50

The IBM 650 Computer program for calculating residual intensi­ ties is also written in SOAP, and requires only two subroutines. They are the floating exponential (FIJJXP) and the floating Naperian loga­ rithm (FLLOG) from the Wisconsin package deck.

Input cards are fed in three at a time because o f ■the large volume of data necessary for each calculation. Referring to the first three program statements, R-cards contain previously calculated optical depths' in the H-gamma line for each of eight values of A A , S-cards contain the increments of these line optical depths, and T-cards con­ tain both optical depths and Increments in the continuum corresponding to 4340.5A. Initialization requires the first 28 statements of the program, due to the fact that nine numerical integrations - each being continuously summed - are performed simultaneously. Zeros initially must be stored in all the integration sums, the card count, N, and several unused punch locations.

The read commands are statements 29-31, followed directly by calculation of the Chandrasekhar q(tau) function in statements 32-49.

At statement 55 the computer has obtained the physical temperature

(TCONT) corresponding to 4340.5A, and in statement 63 has completed calculation of the resulting Planck function (BCONT). As the paren­ thetical code names imply, these quantities refer to the continuum. -

A numerical integration to compute the exponential integral

E 2 (tau) begins in statement 64. The integration (by Simpson's Rule) continues through statement 139, being in two separate parts. The integral can be approximated to satisfactory 51 accuracy by the sum

( e-t* i ~ z dx + [ e-cS - 2 dx . t 3 The major contribution Is from the region near x = 1, where, for tau less than 0.1, the function decreases very-rapidly. For this part of the integral the increment of x should be quite small, 0.05 In this case. From x ■ 3 outward the function is smooth and has a very small second derivative. In this region, therefore, the increment of x is increased by a factor of ten, to 0.5. The first of the two contribu­ tions is reached at statement 101 of the program. In statement 140 the complete exponential integral is stored in punch band Q.

From statement 141 to statement 149 a trapezoidal integration panel for the continuum flux is computed, as follows:

In the above equation Cj, and refer to the endpoints of an inter­ val, and B and E2 denote the Planck function and exponential integral, respectively. The quantity B C c J E t(q) is stored in J by instruc­ tion 142. The quantity I, which is then added to J (statement 143), represents the product I is equal to zero for the first calculation, having been set to zero in the initialization.

But in statements 148 and 149 the value of I is replaced by that of J.

In this manner the second ordinate of every integration panel auto­ matically becomes the first ordinate of the succeeding panel. A panel is evaluated for each set of input cards, and each value added to the growing sum SCONT by Instructions 146 and 147. By this means the flux is calculated. 52

Heretofore, only the continuum has been considered. State­ ment 150 sets index register A equal to eight, and the loop thus created performs the i'lux calculations - in the H-gamma line - for each of the eight values of The exponential integrals as func- - tions of the optical depths in the line are programmed in a manner similar to those for the continuum. This calculation occupies in­ structions 151-231 and instructions 242-285. The large number of instructions actually effects a considerable saving of computer time.

Because the optical depths in the absorption line increase quite rapidly toward line center, and attain magnitudes of several hundred, the numerical integration for the exponential integral function need not be nearly so strict. Statements 151-155 examine the optical depth in the line (R0000A) and send the program to one of three branches to calculate (tau). For optical depths less than 10 the same procedure is used as was used for the small optical depths in the continuum. The beginning of this branch of the program is in statement 156, the instruction being located in STAU (small tau). In -tX -2 the range of optical depths between 10 and 20, the function G X never exceeds 5 X 10-^. Accordingly, very fine increments for Simpson's

Rule are not necessary. The routine - starting at LTAU in statement

242 - performs the integral from x ■ 1 to x ■ 11 in steps of one. For optical depths greater than 20, the flux is neglected, since E£ (tau) is exceedingly small. This is done in statement 285 (VLTAU) by plac­ ing a zero in the upper accumulator and going to STORE in statement

232. All three routines for E£ (tau) proceed to this STORE instruction

) 53

where the value of the exponential integral Is stored in punch

band 0.

In statements 233-241 the trapezoidal integration panel is

evaluated exactly as before. All data addresses here, however, are

regionalized and tagged with index register A, because the calcula­

tion is performed eight times for each card set. Instruction 241 sends

.the computer to OUT in statement 286. From here through statement 294

the program tests for exit from the line flux loop (L00P2), punches

the exponential Integrals (0 0001 and Q0001), and determines whether

or not all 53 sets of data cards have been read and calculated. When

this latter condition obtains, the flux in the continuum is complete

in the running sum SCONT, and the line fluxes for the eight values

of ^ A a r e in the eight sums WOOOOA. Statements 295-301 produce the

ratios between the line fluxes and the flux in the continuum, i.e.,

the residual intensities, and punch them from band P.

Statement 302 is a halt command and the remainder of the state­

ments are constants of the program. APPENDIX III Table II.

CORRECTED, NORMALIZED ABSORPTION COEFFICIENT FOR 8 VALDES OF A A , AS A FUNCTION OF OPTICAL DEPTH (He/H = 0)

TAU (4340.5A) a X = 10A a \ = 20A A.X = 30A a A = 40A A X = 50A a X = 60A A A = 70A A A = 80A

0.000 ( 0) 1.994 (1) 4.244 (0) 1.654 (0) 8.569 (-D 5.208 2.738 f .1 ^ 1.993 (-1) 1.516 (-1 1.417 (-5) 3.690 (3) 7.193 (2) 2.839 (2) 1.479 ( 2) 8.953 ( 1) 5.949 ( 1) 4.214 ( 1) 3.127 ( 1 2.835 (-5) 3.746 (3) 7.293 (2) 2.878 (2) 1.499 ( 2) 9.073 ( 1) 6.028 ( 1) 4.269 ( 1) 3.168 ( 1 4.252 (-5) 3.800 (3) 7.391 (2) 2.915 (2) 1.519 ( 2) 9.188 ( 1) 6.104 ( 1) 4.323 ( 1) 3.208 ( 1 5.670 (-5) 3.856 (3) 7.492 (2) 2.954 (2) 1.539 ( 2) 9.308 ( 1) 6.183 ( 1) 4.379 ( 1) 3.249 ( 1 7.087 (-5) 3.907 (3) 7.583 (2) 2.989 (2) 1.556 ( 2) 9.415 ( 1) 6.254 ( 1) 4.429 ( 1) 3.286 ( 1 1.417 (-4) 4.148 (3) 8.012 (2) 3.153 (2) 1.641 ( 2) 9.919 ( 1) 6.586 ( 1) 4.663 ( 1) 3.459 ( 1 2.126 (-4) 4.359 (3) 8.382 (2) 3.294 (2) 1.713 ( 2) 1.035 ( 2) 6.871 ( 1) 4.864 ( 1) 3.607 ( 1 2.835 (-4) 4.550 (3) 8.715 (2) 3.420 (2) 1.777 ( 2) 1.074 ( 2) 7.126 ( 1) 5.043 ( 1) 3.739 ( 1 3.543 (-4) 4.726 (3) 9.018 (2) 3.535 (2) 1.836 ( 2) 1.109 ( 2) 7.356 ( 1) 5.204 ( 1) 3.859 ( 1 4.252 (-4) 4.886 (3) 9.291 (2) 3.638 (2) 1.888 (2) 1.140 ( 2) 7.561 ( 1) 5.349 ( 1) 3.965 ( 1 4.961 (-4) 5.031 (3) 9.536 (2) 3.730 (2) 1.935 ( 2) 1.168 ( 2) 7.744 ( 1) 5.478 ( 1) 4.060 ( 1 5.670 (-4) 5.166 (3) 9.761 (2) 3.815 (2) 1.978 ( 2) 1.193’( 2) 7.912 ( 1) 5.595 ( 1) 4.147 ( 1 6.378 (-4) 5.294 (3) 9.976 (2) 3.895 (2) 2.019 ( 2) 1.217 ( 2) 8.071 ( 1) 5.707 ( 1) 4.229 ( 1 7.087 (-4) 5.417 (3) 1.018 (3) 3.970 (2) 2.057 ( 2) 1.240 ( 2) 8.220 ( 1) 5.812 ( 1) 4.306 ( 1 1.417 (-3) 6.368 (3) 1.170 (3) 4.530 (2) 2.338 ( 2) 1.407 ( 2) 9.309 ( 1) 6.574 ( 1) 4.867 ( 1 2.126 (-3) 7.009 (3) 1.269 (3) 4.884 (2) 2.514 ( 2) 1.510 ( 2) 9.979 ( 1) 7.042 ( 1) 5.210 ( 1 2.835 (-3) 7.498 (3) 1.342 (3) 5.143 (2) 2.641 ( 2) 1.584 ( 2) 1.046 ( 2) 7.376 ( 1) 5.455 ( 1 3.544 (-3) 7.895 (3) 1.401 (3) 5.348 (2) 2.825 ( 2) 1.642 ( 2) 1.083 ( 2) 7.636 ( 1) 5.645 ( 1 4.253 (-3) 8.232 (3) 1.452 (3) 5.522 (2) 2.889 ( 2) 1.69.1 ( 2) 1.115 ( 2) 7.854 ( 1) 5.803 ( 1 4.961 (-3) 8.523 (3) 1.492 (3) 5.656 (2) 2.955 ( 2) 1.727 ( 2) 1.138 ( 2) 8.016 ( 1) 5.921 ( 1 5.670' (-3) 8.784 (3) 1.532 (3) 5.793 (2) 2.954 ( 2) 1.765 ( 2) 1.162 ( 2) 8.182 ( 1) 6.042 ( 1 6.379, (-3) 9.063 (3) 1.538 (3) 5.801 (2) 3.011 ( 2) 1.763 ( 2) 1.161 ( 2) 8.168 ( 1) 6.029 ( 1 7.088 (-3) 9.279 (3) 1.573 (3) 5.918 (2) 2.783 ( 2) 1.796 ( 2) 1.181 ( 2) 8.309 ( 1) 6.132 ( 1 1.418 (-2) 1.138 (4) 1.495 (3) 5.525 (2) 2.798 ( 2) 1.650 ( 2) 1.081 ( 2) 7.583 ( 1) 5.584 ( 1 i /< 4 TAU (4340.5A) a X « 10A ii 20A a A = 30A 40A a A = 50A a A - 60A a A= 70A a A= 80A

2.127 (-2) 1.277 (4) 1.527 (3) 5.592 (2) 2.758 (2) 1.651 (2) 1.079 (2) 7.551 (1) 5.553 (1) 2.836 (-2) 1.403 (4) 1.521 (3) 5.542 (2) 2.993 (2) 1.622 (2) 1.058 (2) 7.389 (1) 5.426 (1) 3.546 (-2) 1.498 (4) 1.665 (3) 6.039 (2) 2.993 (2) 1.755 (2) 1.142 (2) 7.967 (1) 5.844 CD 4.256 (-2) 1.590 (4) 1.798 (3) 6.525 (2) 3.224 (2) 1.886 (2) 1.225 (2) 8.532 (1) 6.252 (1) 4.966 (-2) 1.690 (4) 1.862 (3) 6.626 (2) 3.265 (2) 1.906 (2) 1.236 (2) 8.598 CD 6.295 (1) 5.676 (-2) 1.750 (4) 2.505 (3) 6.964 (2) 3.425 (2) 1.995 (2) 1.292 (2) 8.978 (1) 6.567 (1) 6.387 (-2) 1.836 (4) 2.665 (3) 7.405 (2) 3.636 (2) 2.115 (2) 1.367 (2) 9.492 (1) 6.938 (1) 7.097 (-2) 1.921 (4) 2.802 (3) 7.837 (2) 3.845 (2) 2.233 (2) 1.441 (2) 9.999 (1) 7.303 CD 1.421 (-D 2.677 (4) 5.054 (3) 1.517 (3) 5.780 (2) 3.334 (2) 2.135 (2) 1.470 (2) 1.068 (2) 2.134 (-D 3.421 (4) 6.876 (3) 2.229 (3) 7.847 (2) 4.446 (2) 2.840 (2) 1.949 (2) 1.410 (2) 2.848 (-D 4.118 (4) 8.304 (3) 3.020 (3) 1.224 (3) 5.532 (2) 3.532 (2) 2.421 (2) 1.748 (2) 3.563 (-D 4.765 CD 9.886 (3) 3.634 (3) 1.470 (3) 6.827 (2) 4.292 (2) 2.943 (2) 2.123 (2) 4.279 (-D 5.952 (4) 6.618 (3) 2.433 (3) 1.105 (3) 5.773 (2) 2.918 (2) 1.999 (2) .1.442 (2) 4.996 (-D 6.552 (4) 7.284 (3) 2.843 (3) 1.220 (3) 6.439 (2) 3.308 (2) 2.228 (2) 1.608 (2) 5.713 (-D 7.087 (4) 7.879 (3) 3.076 (3) 1.470 (3) 6.996 (2) 3.603 (2) 2.439 (2) 1.758 (2) 6.432 (-D 7.547 (4) 8.390 (3) 3.276 (3) 1.581 (3) 8.348 (2) 4.810 (2) 2.617 (2) 1.888 (2) 7.151 (-D 7.951 (4) 8.840 (3) 3.452 (3) 1.667 (3) 8.815 (2) 5.087 (2) 2.773 (2) 2.004 (2) 1.075 ( 0) 9.134 (4) 1.015 (4) 3.968 (3) 1.921 (3) 1.131 (3) 5.946 (2) 4.095 (2) 2.354 (2) 1.437 ( 0) 9.419 (4) 1.047 (4) 4.095 (3) 2.098 (3) 1.177 (3) 6.858 (2) 4.274 (2) 2.490 (2) 1.799 ( 0) 9.302 (4) 1.034 (4) 4.057 (3) 2.079 (3) 1.167 (3) 6.811 (2) 4.250 (2) 3.074 (2) 2.163 ( 0) 9.030 (4) 1.004 (4) 3.939 (3) 2.019 (3) 1.135 (3) 7.349 (2) 4.609 (2) 3.016 (2) 2.892 ( 0) 8.421 (4) 9.362 (4) 3.778 (3) 1.937 (3) 1.154 (3) 7.120 (2) 4.436 (2) 2.907 (2) 3.623 ( 0) 7.899 (4) 8.782 (4) 3.545 (3) 1.818 (3) 1.084 (3) 6.691 (2) 4.627 (2) 3.047 (2) 4.355 ( 0) 7.542 (4) 8.385 (4) 3.041 (3) 1.560 (3) 9.299 (2) 5.746 (2) 3.975 (2) 2.621 (2) 5.090 ' 0) 7.160 (4) 7.961 (4) 2.887 (3) 1.484 (3) 8.831 (2) 5.460 (2) 3.809 (2) 2.493 (2) 5.826 ( 0) 6.904 (4) 7.676 (4) 2.784 (3) 1.429 (3) 8.505 (2) 5.571 (2) 3.672 (2) 2.664 (2) 6.563 ( 0) 6.657 (4) 7.401 (4) 2.684 (3) 1.378 (3) 8.205 (2) 5.373 (2) 3.544 (2) 2.572 (2) 7.301 ( 0) 6.447 (4) 7.168 (4) 2.599 (3) 1.335 (3) 7.947 (2) 5.205 (2) 3.435 (2) 2.493 (2)

Note: Numbers are multiplied by ten to the power in parentheses.

Ln Cn SELECTED BIBLIOGRAPHY

Aller, Lawrence. The Atmospheres of the Sun and Stars. New York: Vie Ronald Press Company, 1953.

Greensteln, Jesse. "The Spectra of White Dwarfs," Handbuch Per Physlk. vol. 50 (1958), 161-186.

Grenchik, Raymond. A Model Atmosphere for the White Dwarf 40 Eridani B. Ph. D. Thesis at Indiana University, 1956.

Kolb, Alan C. Theory of Hydrogen Line Broadening in High-Temperature Partially Ionized Gases. AFOSR-TN-57-8, ASTIA Document No. AD 115 040, Univ. of Mich. Eng. Res. Inst. (1957); Dissertation Univ. of Mich. (1957).

. Kourganoff, V. Basic Methods in Transfer Problems. London: Oxford University-Press, 1952.

Lesseps, Randolph. Computer Program for Constancy of Flux in a Stellar Atmosphere. M. S. Thesis at Louisiana State University, 1960.

Nicola, Lawrence. Effects of Helium on a White Dwarf Atmosphere. M. S. Thesis at Louisiana State University, 1959.

Schatzman, E. White Dwarfs. New York: Interscience Publishers, 1958.

Schrodlnger, E. "Stark Effect," Annalen der Physik, vol. 80 (1926), 437.

Weidemann, Volker. "Spectra of the White Dwarf Van Maanen 2," Astrophysical Journal, vol. 131 (1960), 656.

Woolley, R. R. and Stlbbs, D. W. N. The Outer Layers of a Star. London: Oxford University Press, 1953.

56 VITA

Jerry Fuller was born June 2, 1934, in Palestine, Anderson

County, Texas. He completed his elementary schooling, through the eighth grade, at Swanson Hill Elementary School of Anderson County.

His high-school diploma was received from Davey Crockett High School of Parestine, Texas, in June of 1951. During the same month he entered Baylor University, located in Waco, Texas. By attending summer school, he was graduated in February, 1954, with a Bachelor of Science degree in physics. Fifteen months later, in June, 1955, he received the Master of Science degree in physics from Baylor

University. Since September, 1955, he has been a graduate student

in the Department of Physics and Astronomy at Louisiana State Univer­ sity. He is now a candidate for the degree of Doctor of Philosophy

in that department.

57 EXAMINATION AND THESIS REPORT

Candidate: Jerrv Don ’ru l! or

Major Field: J h ric :

Title of Thesis: • * ' ' r* Q ” * 1 c- f) * V i • *»■ H v 1 r.

Approved:

r Professor and Chairman

e Graduate Schoo

EXAMINING COMMITTEE:

Date of Examination:

______3 -|o-lc>