X. Bakerian lecture.—The structure and opacity of a stellar atmosphere

tu r e

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[

421

]

The Structure and Opacity o f a Stellar Atmosphere.

.—

By Prof. E. A. M il n e , (R eceived J u n e 6,— R ead Ju n e 6, 1929.)

CONTENTS. P age 1. The general problem

.........................................................................................................................................421 I. Analysis of thermal structure of stellar atmospheres.

2. Schwarzschild ’s p r o b l e m ............................................................................................................................... 424 3. Schuster ’s p r o b l e m .................................................................................................................................................. 426 4. R elatio n betw een Schwarzschild ’s problem and Schuster ’s p r o b l e m ................................................. 427 5. Solution for sim ultaneous sc a tte rin g an d a b s o r p t i o n .................................................................................... 427 6. Selective ra d ia tio n pressure

................................................................................................................................431

7. D eterm in atio n of eq u iv alen t photospheric s u r f a c e ........................................................................................ 432 II.

Determination of photospheric opacity.

8. B ehaviour of line-w idth from s ta r to s ta r 9. In te g ra te d io n izatio n formulae

..................................................................................................... 434

........................................................................................................................... 435

10. E lectron-pressure an d optical d e p t h ...................................................................................................................436

10a . “ Mass above th e photosphere ” a n d photospheric opacity

............................................................. 437

11. The “ m ethod of m ax im a ” for d e te rm in a tio n of photospheric o p a c i t y .................................................438 12. N um erical ev alu atio n of th e ab so rp tio n coefficient

....................................................................................440

13. D e term in atio n from th e m axim um of th e B alrner l i n e s ............................................................................... 441 III.

Applications of the absolute value of the photospheric absorption coefficient.

14. A pplication to ionization of calcium a t low tem p eratu res

...................................................................... 443

15. A pplication to th e B alm er lines ...................................................................................................................... 448 16. A pplication to determ ine th e physical s ta te of th e photospheric l a y e r s ................................................ 451 17. Im perfections of th e m e t h o d ............................................................................................................................... 459 18. S um m ary

..................................................................................................................................................................460

1. The general problem.—“ Stellar atmosphere ' is the name given loosely to the outer portions of a star. The stellar atmosphere is divided observationally into three superincumbent layers, named the photospheric layers, the reversing layer and the chromosphere, in order of increasing level. The boundaries between these are only roughly defined, but broadly speaking the photospheric layers give rise to the con­ tinuous spectrum of the star, the reversing layer to the absorption-line spectrum and the chromosphere (when seen edgeways) to the flash spectrum.* Mathematical analysis of the way in which gaseous material comprising the outer portions of a star may be expected to thin out into space confirms this threefold division. It also brings to light * A ctually only so studied for the Sun, b u t the term is useful generally. VOL. C C X X V III.— A 6 6 8

3 L

[Published, November 12, 1929.

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E. A. MILNE ON STRUCTURE AND OPACITY OF A STELLAR ATMOSPHERE.

certain dynamical and thermal characteristics of the three layers. For example a definite temperature gradient in the photospheric layers shades off into an approximately isothermal state in the chromosphere ; “ local thermodynamic equilibrium ” in the photospheric layers shades off into “ monochromatic radiative equilibrium ” in the upper chromosphere ; and a somewhat unimportant general radiation pressure in the photospheric layers augments to a strong selective radiation-pressure in the reversing layer and chromosphere. The reversing layer is in most cases the transition layer. Assumptions valid for either photospheric layers or chromosphere separately cease to be so near their upper and lower boundaries respectively and so far it has not been possible to give a treatm ent which accurately deals with the regions of transition. In the present lecture it is proposed to consider chiefly the photospheric layers and the reversing layer. For these regions the dominant need is the determination of the general opacity—the fogginess—for this determines the depth we see into the star and so the pressures, densities, etc., at which the observed spectral phenomena originate. The abstract problem of the stellar atmosphere may be stated as follows. For many purposes the curvature of the outer regions of a star may be neglected and we consider only material stratified in parallel planes. The material is subject to (a) a gravitational field of acceleration g,( b)a net flux of energy of amou it from below and emergent into space above. This is determined by the evolution of energy in the interior of the star. The amount of energy actually incident on the atmospheric layers from below exceeds wF, but a portion is re-radiated downwards by the atmospheric layers, ttF being the net amount passing through. If the atmospheric layers are in a steady state* there is no accumulation of energy, and the net amount of energy crossing any surface of stratification is equal to th at crossing any parallel surface, namely ttF. The quantity F itself is the mean value of the emergent intensity of radiation at any point, or, what is the same thing, the mean intensity of radiation over the stellar disc. The abstract problem is :—Given the two parameters g and F, and given also the ultimate chemical composition of the material, to determine the distribution of temperature, pressure, density, ionization and chemical composition in the layers, and to determine also the complete intensity-distribution both in angle and in frequency, of the emergent radiation. The practical problem is to some extent the converse one of inferring the temperature and other physical quantities from the observed emergent radiation, i.e., from the observed spectra, measured if possible spectro-photometrically. In many cases we do not know either g or F, and these also may have to be determined from the observed spectra. The dependence on two parameters g and F (assuming similar chemical composition) corresponds to the observed two-fold classification of stellar spectra. On the one hand we have the grand sequence of types of spectra, principally com­ prised in the Draper classes M, K, G, F, A, B, 0 , and correlated with systematic changes of colour. This is taken to correspond to changes in the variable F, or, as is more * In this lecture only steady states are considered.

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F. A. MILNE ON STRUCTURE AND OPACITY OF A STELLAR ATMOSPHERE.

423

usually stated, to change in elective tem p eratu re/’ A full radiator emitting energy ttF per cm.2 would have a tem perature Tx given by ttF = aTF, where a is Stefan ’s c o n sta n t; T1} defined in this way, is called the “ effective tem perature ” of the star. Strictly speaking, only for a very few stars are the effective temperatures known. If L is the absolute bolometric brightness of a star, l its apparent bolometric brightness, rx its radius, E its distance, a its angular diameter, then Loc r ^ T ^ , l x L /E 2, a x rx/E, whence Txx Thus a determination of effective tem perature requires a brightness measure combined with a measure of angular diameter. For the few stars for which a has been determined experimentally the effective tem perature Tx agrees with the " colour tem perature ” or tem perature determined by the shape of the energy-curve for the continuous spectrum. The exact relation between colour-temperature and effective tem perature remains for future investigation,* but provisionally it is assumed th at the observed correlation of type of spectrum with colour-temperature indicates a correlation of type with effective tem perature. On the other hand for stars of given type there are smaller but well-marked variations of spectrum—variations in relative intensity of pairs of lines—which have been found to be correlated with absoute lum inosity.! It is generally agreed! th at this correlation must in fact be a correlation of spectrum with the second parameter, surface gravity The explanation of the observed correlations in its broad outline is due to Saha,§ who proposed the theory of the high-temperature ionization of the elements and traced its consequences. Variations of spectral type are due to variations of ionization and excitation consequent on variations of tem perature—actual temperature, not necessarily effective temperature. Variations of spectrum inside a given spectral type are due to variations of ionization consequent on variations of pressure, which in turn are con­ sequences of variations of surface gravity. The researches summarized in this lecture are attem pts to obtain definite quantitative laws for the calculation of intensities of spectral lines as functions of the two fundamental parameters F (or T\) and g. The quantitative applications have resulted first in a knowledge of the pressures in stellar atmospheres, secondly, in a knowledge of the coefficient of general opacity of the material forming the photospheric layers, and they bid fair to give ultimately a * A determ in atio n of effective tem p eratu re m ight be possible for an eclipsing binary of ascertained surface brightness and p arallax. The problem of the low colour-tem peratures of m any early-type stars m ight be investigated in th is w ay, by com paring th e observed colour-tem perature (deduced from shape of energy-curve) w ith th e observed effective tem p eratu re (deduced from the absolute values of the ordinate of th e energy-curve). f A dams an d K ohlschutter , ‘ A strophys. J . , ’ vol. 40, p. 385 (1914); A dams and J oy , ibid., vol. 46, p. 3 1 3 (1 9 1 7 ); etc. J e.g., P annekoek , ‘ Bull. A st. N e th .,’ vol. 1 (No. 19), p. 107 (1922); ‘ O bservatory, vol. 46, p. 304 (1923). § ‘ Roy. Soc. P ro c.,’ A, vol. 99, p. 135 (1921). ‘ Phil. Mag., vol. 40, pp. 472, 809 (1920). A p articu lar case of h igh-tem perature ionization had been considered earlier by L indemann , Phil. Mag., vol. 38, p. 676 (1919).

3

L

2

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E. A. MILNE ON STRUCTURE AND OPACITY OF A STELLAR ATMOSPHERE.

complete scheme of the chemical composition (in proportion by mass) of stellar atm o­ spheres. Recent progress depends on a combination of a fundamental idea due to Stewart and U nsold* with the ideas of Saha, along lines of investigation originated by Sir Arthur Schuster and Schwarzschild*)* on the thermal structure of stellar atm o­ spheres. Stewart and Unsold have developed the idea of the breadth and intensity of a spectral line as arising from the “ number of atoms above the photosphere, per cm.2.” Development of the ideas of Schuster and Schwarzschild permits us to give a precise meaning to the phrase ” above the photosphere. Refinements of the methods of calculation originated by Saha permit the calculation of the “ number of atoms in the column ” as a function of temperature and pressure at the base of the column. It appears th at the fundamental desideratum is the intrinsic opacity of the photospheric material, as this determines the pressure at the base of the column. Observations of maxima of lines in the stellar sequence of spectra permit the determination of the opacity without introducing U nsold’s formula for the atomic (selective) scattering coefficient. Determination of the absolute value of the coefficient of opacity of photospheric material sets the way open for calculations of “ numbers of atoms above the photosphere ” in terms of the absolute proportions by mass of the constituents. Com­ parison of the calculated values with those derived by observation of line-contours and reduction by U nsold’s method gives either a check on U nsold ’s formula for the atomic scattering coefficient or a determination of an absolute abundance (proportion by mass). The observations of Miss P ayneJ and her co-workers at the Harvard College Observatory give precisely the information required to test the whole theory, which is in general confirmed. They indicate the order of magnitude of the proportion of the element observed (calcium) in stellar atmospheres, and will probably eventually determine “ effective gravity ” as compared with true gravity, i.e., will determine the importance of selective radiation pressure. I .— Analysis of thermal structure of stellar atmospheres.

2. Schwarzschild’s problem— A ssociated with the names of S chw Schuster are two idealised problems which have had great influence on the develop­ ment of the theory. Mathematically they are completely equivalent. It is necessary to state the solutions of these two problems. Schwarzschild’s problem is th at of finding the temperature distribution for material in radiative equilibrium possessing an absorption coefficient independent of wave-length, it being given th at the material is traversed by a net-flux 7rF. The material is supposed stratified in parallel planes, and extends to infinity in the direction * Stewart , ‘ A strophys. J . , ’ vol. 59, p. 30 (1924); U nsold , ‘ Z. Physik, vol. 44. p. 793 (1927), and vol. 46, p. 765 (1928). t Schuster , ‘ A strophys. J . , ’ vol. 21, p. 1 (1905); Schwarzschild , ‘ G ott. N ach .,” p. 41 (1906).

t P ayne and H ogg, ‘ H arv ard C irc.,’ p. 334 (1928); P ayne and W illiams, ‘ M .N .R .A .S .,’ vol. 89, p. 526 (1929).

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E. A. MILNE ON STRUCTURE AND OPACITY OF A STELLAR ATMOSPHERE.

from which the net flux is coming. equations

425

If B — (a/n) T4, the problem is embodied by the

cos 0 i l ( j ’ (It

= I (T, 0) - B .........................................(1)

11 (t , 0)

d(* =

dx — Kp

,

4 7 tB ,

...................

where h is the depth measured inwards, p is the density, k the absorption coefficient, t is the optical depth and I ( t , 0 ) is the intensity of radiation in a direction making 0 with the outward normal to the planes of stratification. Equation (1) is the “ equation of transfer,” * equation (2) the equation of radiative equilibrium. The problem is to determine B as a function of x.The function B can be shownf to be in the solutions of the integral equation B ( t) = J [ B ($) Ei ( t — Jo

t)-f-

Jr

where rco p - n x

mix)=

J 1 {X

—

dp.

This, however, does not put in evidence the net flux uF. To exhibit the dependence of B on F, and to pick out the solution of (4) which we require, we replace (2) by the equivalent equationj expressing the constancy of net flux, COS 0

dlO = 7 c F .

(5)

It is then found th a t B satisfies§ F = 2 p B (<) Ei2 (t — t ) dt — 2 £ b (t) Ei2 (x - t)dt,

................(6)

where fC O

E i2 ( » ) =

p -/x X

— Ji f

An approximate solution of (6), satisfying it exactly for

t

= 0 and for x large, is|

B = J F ( l + f r ) ........................................................(7) * The te rm is due to Sir J ames J eans , ‘ M .N .R .A .S., vol. 78, p. 28 (1917). t e M .N .R .A .S .,’ vol. 81, p. 361 (1921). X E q u a tio n (5) follows from (1) by m ultiplying by dw, in teg ratin g over th e solid angle 4tt, using (2), and in teg ratin g w ith respect to t . I t can also be w ritten down from first principle, by the conditions of th e problem . § E q u atio n (6) does n o t ap p ear to have been given before.

I t is readily derived by solving (1) for

I ( t , 6) and inserting in (5), an d th e n inverting the order of integration. || ‘ M .N .R .A .S.,’ vol. 81, p. 361 (1921) ; ‘ Phil. T ran s.,’ A, vol. 223, p. 202 (1922).

( B

E

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E. A. MILNE ON STRUCTURE AND OPACITY OF A STELLAR ATMOSPHERE.

An approximate solution of Schwarzschild’s problem is accordingly T4 =

iTj4 (1 + f x)....(8)

Associated with this temperature distribution is the law of distribution of the emergent radiation 1(0, 6) = | F ( l + f c o s 0 ) .............................................(9) Relations (7), (8) and (9) are solutions of S chwarzschild’s problem sufficiently accurate for ordinary purposes. (8) implies a boundary tem perature T0 = 2~^T1 = 0*84 1\. More accurate investigations show th at there is a sudden drop near the boundary, dB (r)/d t becoming infinite like log (1/x), and the boundary tem perature is very close to 0-813 x. The law of darkening (9), corresponding to a coefficient of darkening §, is fairly well confirmed by observations of the integrated radiation at various points of the sun’s disc ; observed departures from (9) can be shown to be accounted for by departures from the ideal conditions postulated in S chwarzschild’s problem, namely the absence of line absorption.* The verification of (9) is observa­ tional proof of the existence of radiative equilibrium in the sun’s photospheric layers. The utility of the solution to Schwarzschild’s problem is th a t it relates actual temperatures to the effective temperature. 3. Schuster’s Problem.—Schuster’s problem can be stated as follows. A layer of gas, capable of scattering monochromatic radiation of frequency v with scattering coefficient s„, is placed in front of a bright background radiating with a given flux tcG„ ; given the optical thickness av of the scattering material, to determine the emergent flux. The solution is readily derived from the solution to Schwarzschild’s problem. The equation of transfer of v-radiation is cos 0

dJ.„— [g„, 0 ) _tXj, d
I It± yd~~co ? J 471

( 11)

where dav =

st,

p

Thus the law (11) governing the monochromatic radiation in Schuster’s problem is the same as the combined laws (1) and (2) of S chwarzschild’s problem. The mean intensity j I,, ( 0)

P ’,(>

..................(9)

but in Schuster’s problem F„ is so far unknown. Now in Schwarzschild’s problem the outward and inward fluxes at x are readily found to be approximately 7tF + = 7rF(l + fx ) 7xF_ = 7rF ( | t), * ‘ O bservatory,’ vol. 51, p. 88 (1928).

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E. A. MILNE ON STRUCTURE AND OPACITY OF A STELLAR ATMOSPHERE.

427

the difference being uF. Replacing 7tF+ by 7cG„ the given incident flux in Schuster’s problem, and replacing t by
<7V = N a „ ..................................

Hence if a„ is known from physical theory, N can be determined from the observed line-contour. This is the method of U nsold. But it does not follow from this that N will be independent of v, for the observed shape may not agree exactly with the theoretical a„. In fact until we know how deep to place the photospheric surface there is con­ siderable uncertainty as to what we mean by N . It may be different for different parts of the line-contour. We require a fusion of the Schuster and Schwarzschild models. 5. Solution for Simultaneous Scattering and us follow the Schwarz­ schild model in having no definite background surface ; the material is to extend inwards

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E. A. MILNE ON STRUCTURE AND OPACITY OF A STELLAR ATMOSPHERE.

indefinitely. Let us follow the Schuster model in retaining monochromatic scattering. The equation of transfer, for a scattering coefficient s„ and an absorption coefficient k v now takes the form cos 9 prk =

+

I" ~~ K

I^

( 12 )

where o _ 2Av3

'

1

_ l ..................

(13)

Equation (12) follows from purely macroscopic considerations, assuming the material is in local thermodynamic equilibrium. It can however be derived—or rather a similar equation, from entirely different considerations—from the quantum theory of emission and absorption as modified by collisions.* It is not necessary to assume local therm o­ dynamic equilibrium, and the function B„ makes its appearance not from an appeal to K irchhoff’s or P lanck’s law but from an application of the principle of detailed balancing to enclosure conditions. All th at is necessary to assume is a Maxwellian velocity distribution ; the effect of collisions in exciting and de-exciting atoms is then the same as in an enclosure, though enclosure conditions are not assumed in calculating the relative numbers of normal and excited atoms. The ratio kv/ sv is replaced by a ratio in which the denominator is the E instein B coefficient and the numerator measures the relative importance of collisions. It follows th at near the surface, where the density is low and collisions are negligible, the terms in kv in (12) may be neglected in com­ parison with those in sv; in the deep interior, where collisions are all important, the terms in svin (12) may be neglected in comparison with those in #c„. Near the surface the conditions are those of pure monochromatic scattering—Schuster’s problem ; hence the solution of Schuster’s problem is strictly apposite for the chromosphere. In the deep interior (photospheric layers) the conditions are those of local thermodynamic equilibrium—Schwarzschild’s problem. It is evident th at in writing down (12) we have successfully fused the two problems. It would be desirable to solve (12) with kJ sv an increasing function of density (approximately, kJ s„o c)p. This has not yet been accomplished. Failing this w a mean value for kv/ sv,a nd we expect the results to be satisfactory for the t region—the reversing layer. Actually the significant ratio is kvJ{kv -f- $„) which varies from 0 to 1 between boundary and interior, so the assumption of a mean value for *„/(*„ + sv) is not too violent an approximation. Formally the result is equivalent to taking kvand sv independent of hin (12). Following a procedure due to E ddington ,j- we now write J„ = J I„

, rcF„ =

J

I„ cos e

* ‘ M .N .R .A .S.,’ vol. 88, p. 493 (1928). t ‘ In tern al C onstitution of th e S tars,’ p. 322 (1926).

da, = j*I„ cos2 6

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E. A. MILNE ON STRUCTURE AND OPACITY OF A STELLAR ATMOSPHERE.

429

and put p

dk — dm...... ( 15)

J„, F„, L„ are functions of the depth h, or of m. Then (12) on multiplication in turn by 1 and cos 0 and integration over all directions gives L

(*F.) =

(J, - B„),

+

.................................... (16)

............................................... (i7)

Introduce as an approximation B„ ==

.................................................................(18)

which holds good exactly if 1^.(0) is expansible in odd powers of cos 0 in 0 < 0 < rc. This condition is not fulfilled at the boundary, but approximation (18), which avoids the occurrence of integral equations, is found to be of an accuracy comparable with the solutions of Schwarzschild’s and S chuster’s problems already quoted. Eliminating F„ we find f]2 i p - zJ , K + 0 ( J , - B„)...(19) Let k and s denote certain mean values of will be defined later. Put K ~i~ = v) k-f- s +

kv

and s„ through the spectrum which ( 20 ) .

( 21 )

( k + s) dm = Then (19) becomes f

d2J, p x f—

(J„

( 22 )

B„).

Equation (22) has to be solved with the two boundary conditions (1) for r large, J„ is at most of the order of magnitude of B,„ (2) at the boundary 0 the inward radiation is zero; this latter condition is equivalent, to a degree of approximation consistent with (18), to the condition J F ,( 0 ) = U (o)...................................................(23> After some algebra we arrive at the results

J,

(t )

i l l ------ le —SsS* r B„ ( i . Ll J° V3 ' 2

t)e~n^

_p i e+'vW 3 [*Br (t) e-W-s* n £ v \/S VOL. CCXXVIII.— A

3 M

+

z

J^B,

dt

V 3 dt> (23)

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E. A. MILNE ON STRUCTURE AND OPACITY OF A STELLAR ATMOSPHERE.

25,

V3

j - A . ------ Y 2 e—AT'fl’ f“ B, (t) Jo 1 4.2 +

n ,l,

3

V3

-f- ewp| pT'/3 I B„ (£) e“ “A*^3w„£„ \/3

dt —B„ (£) eV.-^3 y'S (24)

We have not yet determined the temperature distribution ; we have not yet in fact imposed the condition of constant net flux for the integrated radiation of all frequencies. To make progress at this point we will assume that B„ ( can be replaced by its two-term Taylor expansion B„ (t) =

a+

where _ 2Av3


1

^2 • gAp/tT„

__ 2Av3


j > ........................................

Av 1 ^ dT ^ /O

It will be found th at this approximation preserves all the essential features of the investigation. Equation (23) and (24) now give a

3

J„ (T) = a* +

a'

-f- AJ? - 3 :----- e ^

(28)

E a_ V j? 1

2

a„ —

(29)

F „ ( t ) = f — + 2?,

^£p -+t —2

We now impose the condition of constant net flux, in the form P

Jo

F„

(30)

F .....................................

All we can hope for in accordance with the general tenour of our approximations is th at (30) shall be satisfied exactly for t = 0 and for t large. For r large the exponential term in (29) vanishes and we require therefore dv

J0

If we now put

B ( t ) = I B„dv = f Jo

Jo

.

(a+

t)

(31)

+ bz,

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E. A. MILNE ON STRUCTURE AND OPACITY OF A STELLAR ATMOSPHERE.

and define

kj--

431

sby the relation* * i — kdv,

(32)

6 = fF . . . . (dT/dx)0. 0 requires

(33)

+

=

Jo

-J- S„

relation (31) gives This determines (1/T0) The satisfaction of (30) at

(34)

This determines the boundary tem perature T0, which clearly depends on the behaviour of as a function of v.*j* It is equally clear, however, th a t the order of magnitude of T0 is independent of the absolute values of and if this is constant or varies only slowly through the spectrum or has its rapid variation confined to small ranges of v, (34) reduces approximately to or by (31) to a = J F ................................................................ (35)

Hence B ( t ) = i F (1 + | t )

................................................(36)

and T* = J V (1 +

h),

as in S chwarzschild’s problem. This result is well known for the case when and are independent of v. We now see th at it is true in general (apart from possible modifications in the boundary temperature) provided x is calculated from the R osseland mean sum of the scattering and absorption coefficients. We can make two im portant applications of (29). 6. Selective Radiation Pressure.—The first is to selective radiation pressure, which inside a spectral interval Av on a layer p dh gives rise to a force per unit area t:F„ ( t)

At the boundary

Av (

k„j-- S„) p dll.

r = 0 this will be found to reduce to B„ (0)

n Av p ^5 ~

1 / dBy\ k } (K + + -\/3 \ p (

2

* This is an exam ple of R osseland ’s theorem , ‘ M .N .R .A .S.,’ vol. 84, p. 525 (1924). t I t would be interesting to stu d y T0 as given by (34) for various forms of

3 M2

(37)

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432

As

E. A. MILNE ON STRUCTURE AND OPACITY OF A STELLAR ATMOSPHERE. t

increases it rapidly approaches a value which will be found to be

" * ' ' < ’ * * ( S O .................................................(38)

In expression (3 8 ) the selective effect of radiation pressure has entirely disappeared ; at great depths the effect of a large opacity ( -f- s„) is neutralised by a small F„. But in (37) the term is k} (kv -f- svf will become large if kv and sv become large, whilst B„ (0) is unaffected ; at the surface selective radiation pressure exists and may become very large. It is this phenomenon which gives rise to the existence of the chromosphere.* 7. Determination of Equivalent PhotosphericSurface.—T of our main objects, namely to determine the optical depths at which absorption lines may be said to originate. We will first obtain the intensity-ratio (line/continuous spectrum). The absorption coefficient kvrepresents the degree to which energy is truly absorbed, i.e., available for conversion into other frequencies and not merely held pending re-radiation in the same frequency. It includes the part played by the photo-electric effect in giving rise to the continuous spectrum—-it includes the effect of every kind of collision. We wish our formulae to hold both for line absorption and for the continuous spectrum between the lines. We will therefore assume to be constant throughout the spectrum, and denote it simply by k. We will take s„ to be zero save in the vicinity of a line, and hence we may take s—- 0. In the vicinity of a l K+

*+

whence K= Then from (29) F„ (0)

(39)

^ v l ( a- + V s 1" 'iv +

In the adjacent continuous spectrum sv = 0, and flux here by tcG„ (0) we have from (29)

6r (0) -

= 1=

Denoting the net

( a~+ v s ) ...................................... (40)

Accordingly the intensity-ratio rv is given by

Vy —

F, (0) a (0)

’ * M .N .R .A .S.,’ vol. 87, p. 697 (1927).

14 +

A/3 2

1 ~r c. ^ / a/3 A/3 1 -|- c,j a/3 ’

The proof given in the te x t is, how ever, m ore general.

(41)

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E. A. MILNE ON STRUCTURE AND OPACITY OF A STELLAR ATMOSPHERE.

433

where c - h - h . 1 f dT \ " ^T oU /o,

3 8 &T0’

(42)

approximately, by (36a). We now return temporarily to S c h u s t e r ’s problem and ask what is the amount of m atter per cm.2 which, scattering monochromatically with coefficient sv and otherwise transparent, would require to be placed in front of a surface radiating with intensity Gr„ (0) to give the same intensity-ratio r„ as is given by (41). The answer is th at an amount f p

dhmust be so placed, where 1

r„

(43)

1 4-r v t( sv 9

by (10).

B ut £,

I

K

E2

— 1

and k |p

is the optical thickness r of this column measured in general absorption (not selective scattering). Hence from (43) (44) rv — 1} + i T ( l - U ) In this formula t is the optical thickness of the column in the real atmosphere which if placed in front of a photospheric radiating surface and deprived of all opacity except the power of scattering monochromatically, would produce an intensity-ratio at frequency v equal to th at in the actual stellar atmosphere. Thus t is the optical thick­ ness, or opacity, or fogginess, which the atmosphere would have to lose in order that the atoms, down to this level, when now fully viewed against a background, would produce the proper line-contour. Actually the atoms below t are having some effect in causing conditions leading to the formation of the actual absorption line, but the atoms down to t, “ fully viewed,” are equivalent to the atoms down to r <*>, obscured at all depths as they are by general opacity. So far as I can see, this is what is meant in the literature of the subject by speaking of a line being formed by the atoms down to a particular optical depth. For equations (41) and (44) to give the same line contour, t must be a function oi It is more convenient, however, to have t as a function of r„, the observed intensityratio at v. Eliminating £„ between (41) and (44) we find a quadratic for t . A simple approximation to the solution is obtained by writing i s hv l - R _ 1 T e*T„ 1 -j- c„/a/3 . . -\/3 h ' j ^ 8 ifcTo

(4 5 )

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E. A. MILNE ON STRUCTURE AND OPACITY OF A STELLAR ATMOSPHERE.

and noting th at mately

dvis in practice small compared with unity.*

We then find

T = * n + r l } ~ (r, + 1) (r ’+ i V 8 )J .............................. (46) In applications it seems sufficient to takef T =

|

rv rv + 1 ’

(47)

The inner consistency of this with our solution to Schwarzschild’s problem may be noted. In between the absorption lines we are viewing the equivalent photospheric surface. Here rv= 1, and accordingly by (47)= §-. The tem perature at this deduced from T4 = JT \4 (1 + f t ) is T = Tx. Thus the equivalent photospheric radiating surface between the lines has to be placed at a depth where the true tem perature is equal to the effective tem perature—a most satisfactory result. More generally, the photosphere for intensity-ratio r„ has to be placed at a depth where the tem perature is T, given by T4 = T,4 .................................................... (48) ‘ 1 2r„ + 2 V ' Our formula for the depth at which to place the “ photosphere 55 has been obtained on a special assumption, namely the constancy of k J s v . But it may evidently be taken to give the right order of magnitude in general— varies with in just the way to be expected. We shall therefore use our formula in cases where our original assumptions do not necessarily hold. We shall make the assumption—and we argue th at it is a fair assumption—th at if we observe a point on an absorption-line contour where the intensityratio is r„, then study of the atoms giving rise to this absorption line may be confined to those contained in the uppermost -fr„/(r„ + 1) of optical thickness. This assumption is only to be retained until more thorough mathematical analysis determines absorption” line contours under more general conditions. The advantage of the assumption is th at it is easier to deal mathematically with a column (of finite length) of fully-viewed atoms than an infinite column of partially-viewed atoms lost at great depths in the general fog. II.—Determination of Photospheric Opacity. 8. Behaviour of Line-width from Star to Star.—Let us now consider the same absorption line in a number of stellar spectra. Select in each case the point of the line-contour where the intensity-ratio r has some previously assigned value, say J. Determine the corresponding frequency v. Then |v — v0 |, where v 0 is the line-centre, measures the half-width of the line associated with this value of r. In general, |v — v 0 | will be * Its m ean value th ro u g h th e spectrum is ab o u t zero. t This is a revision of a form ula r = r/(r vol. 89, p. 3 (1928).

1), of inferior accuracy, given previously, * M .N .R .A .S.,’

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435

different for the different spectra. B ut by S chuster’s formula the selective optical thickness a is the same for all, since r is the same. Hence by (11), if N increases, a„ must decrease (since Na„ remains constant) and hence |v — v0| must increase. Thus increasing N down to given t is associated with increasing line-width for given r. Conse­ quently study of line-widths for given r provides information about the behaviour of N down to given t . In particular the line will be widest, for given r, when N is a maximum, for given t . Accordingly we m ust now show how to calculate the number of atoms in a given state of excitation and ionization down to given t . Saha showed* th at the degree of ioniza­ tion was calculable in terms of the tem perature T and partial pressure of free electrons P. We know the range of tem perature down to given t by the solution to Schwarzschild ’s problem. We do not yet know how to calculate P. 9. Integrated Ionization Formulce.—I t is a comparatively simple m atter to calculate the number of ionized or excited atoms in a column given the value of P at its base. The chief point requiring attention is th a t since P varies through the column from 0 at its upper end to say P 0 at its base, therefore the degree of ionization varies from point to point in the column. The variation of P along the column must be found from the mechanical equilibrium of the column. This depends on the value of , which thus makes its first appearance in our analysis. The method of calculation! is best shown by a simple example. Let x denote the degree of ionization of an element present to the extent of a pro­ portion s by mass and surrounded by atoms of the same mean ionization potential. Let xi be the ionisation potential, K x the corresponding equilibrium constant calculated from the formula K — -(27rme)3/2 (^T)5/2 e~Xl/fcx q2 in a well-known notation.

Then the degree of ionization x is given by —— P = K , .................................................... (50) 1

—

X

and the number of neutral and ionized atoms, N0 and Nl3 down to level P = P 0 are given by N„ = - f"(l - *) p d h ....................................... (51)

mJ

N , = — P * p d h , ........................................................(52)

mJ

m being the atomic mass. Now by the equation of mechanical equilibrium the total pressure p is determined by .................................................................... (53) * ‘ Phil. M ag.,’ vol. 40, p. 472 (1920). f ‘ M .N .R .A .S.,’ vol. 89, p. 17 (1928).

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where gis the value of surface gravity, modified if necessary so as to take account of radiation pressure. Since the ionization of the surrounding atoms is the same as th a t of the atoms in question, the pressure p is connected wi P by the relation

- = —

p

.................................................. (54)

1 +

Hence by (50), (53) and (54), (51) and (52) become

_ 0

£ rl0 P ^ rp (p + 2K1n J m ^JoP + K,

K > d i p (p + 2 K ,r = — 2P„ mg

(55) (56)

1

T— i

,)gmP+ K . /

w

1

_1

N - J - P

mg K j’

on taking a mean value for T so th a t Kx may be treated as a constant. More complicated cases may be dealt with similarly. For example when the element is surrounded by an excess of atoms just once-ionized, as may be taken to occur for- stars of type G0 and earlier, (55) and (56) become replaced by

which approximate to (55) and (56) when Kx is large. The main point to note about these formulae is that variations of electron-pressure in the column have been accurately allowed for and th at the resulting formulae are of the type

N °’ N i

=

T ) .......................................................(57)

10. Electron-pressure and Optical Depth.—It remains to calculate the electron-pressure P 0 at the optical depth t . This can only be done when the absorption coefficient of the photospheric material is known. We require the physical law determining the opacity of unit mass of the photospheric material in terms of its tem perature and pressure. This is at present unknown from pure physics, and we seek to use our analysis of stellar atmospheres to determine it from observation. Clearly the greater is the absorption coefficient k, the smaller the amount of mass required to establish a given total opacity t , and the smaller is the pressure at the base of the column. The order of magnitude of this pressure, and the complete structure of the photospheric layers as a function of depth measured in kilometres, depend on the absolute value of A knowledge of the value of k is the key which would unlock the whole position.

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A simple assumption would be to take k constant, say k, independent of the physical state of the material. B ut in the interior of a star, E ddington * has shown th at opacity is due to the photo-electric effect. If we make the same assumption for the photospheric layers, we are led to the formula ocP (58) ip9/2’ where a is roughly independent of P and T but depends on the chemical composition. I t is interesting to work out the results of both assumptions. If K = k, we havef j ? P
9

(59)

on using (53), (50) and (54) as before. If (58) holds, we have * = f | £ i > < » = 5!F j ! P . , ( i + * s j ) .......................... (59')

on taking a mean value for T. We have now completed the determination of the members of relevant atoms concerned in producing an absorption line of given r. Provided or a were known, (59) or (59r) determines the base pressure P 0 given t and T, and then (55) and (56) or similar formula) determine N0 and Nx. 10a . “ Mass above the Photosphere ” and Opacity.— Before we show how to determine the absolute value of the photospheric absorption coefficient we will enquire whether it is possible to distinguish observationally between k constant and k — aP/T 9/2. Let us calculate the total mass (x down to level t on the two assumptions. In each case [x = p 0/g. Hence by (59), when k = constant = (x = When

k— aP/T 9/2,

t

/ k ,................

by (59'), (50) and (54)

T9/2 2 + Bq/Iv!.... ................................(60') * “ a Po (1 + fPo/Ki) We see th at when k = constant, (x is independent of g. But when k — aP/T 9/2, by (60') [x increases as P 0 decreases ; by (59'), P 0 decreases as g decreases ; and thus [x increases as g decreases. The fundamental difference between k = constant and k = aP/T 9/2 is th a t in the former case the total “ mass above the photosphere " is independent of g and therefore the same for giants and dwarfs, but in the latter case the total “ mass above the photosphere ’ increases, at constant I, as we pass from dwarfs to giants. The observed fact is th at all lines are strengthened from dwarfs to giants. We cannot immediately hail this as evidence in favour of the law k ^ P, for the decrease of 1 0 * ‘ M .N .R .A .S.,’ vol. 83, p. 32 (1922) and vol. 84, p. 104 (1924). f Formulae (59) and (59') are calculated for an elem ent surrounded by atom s of the same m ean 1.1'. VOL. CCXXVIII.— A

3 N

(60)

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E. A. MILNE ON STRUCTURE AND OPACITY OF A STELLAR ATMOSPHERE.

from dwarfs to giants causes increased ionization, which affects neutral and ionized atoms differently. The effect is however calculable, and it is found* th a t the law k = aP/T 9/2 in every case predicts the right change of intensity between dwarfs and giants, whilst in many cases /< = constant predicts a change in the opposite direction. Full details have been given elsewhere. The main point of physical interest is th at the reduced pressure gradients in giants as compared with dwarfs do not of themselves lead to an increased “ mass above the photosphere “ ; to secure the latter, the material must be j* more trans­ parent under giant conditions. This suggests th at the coefficient of opacity must decrease with decreasing pressure. We do not establish the detailed law k aP /T 9/2, but we obtain confirmation of its main feature, the proportionality with P. 11. The “ Method of Maxima ” for Determination of P settled this point, we will now show how the absolute value of the constant a may be derived from stellar data. The principle of the m ethodj—conveniently described as the ‘ method of maxima ”—is as follows. Maxima of certain absorption lines in the stellar sequence of spectra are well known. It is found th at such lines originate from either normal states of ionized atoms, or excited states of atoms either neutral or ionized. For simplicity we will confine the argument to the excited state of a neutral atom. The maximum results from the interplay of ionization and excitation with change of temperature. Increasing temperature decreases the proportion of neutral atoms, but increases the fraction of these which are excited, and the result is a rise to a maximum followed by a steady decline. At low temperatures the fraction of atoms neutral is large, but the fraction of these excited is sm all; at high temperatures the fraction excited is larger but the proportion of neutral is small. In between there is a maximum. It was shown by F owler and Milne § th at the maximum concentration of excited neutral atoms reaches a maximum at a temperature determined by P, the electronpressure. Conversely, from an observed temperature of maximum, (Tmax), P could be deduced. It was supposed th at the value P so obtained was some kind of average through the stellar atmosphere, though it was quite uncertain to what level it referred. The formulae now developed permit this uncertainty to be removed. In accordance with the ideas of Stewart and U nsold, as here extended, what determines the maximum width of a line for given intensity-ratio r is not the maximum concentration of excited atoms but the maximum number of excited atoms down to the associated t . If A0(s) is the excitation factor for neutral atoms (a function of the tem perature T and the excitation potential), then the number of excited neutral atoms in the column is N0(,), given by N0W= A0(s)N0, * ' M .N .R .A .S.,’ vol. 89, p. 157 (1928).

i.e .,more tran sp a re n t per u n it m ass per cm .2, t ‘ M .N .R .A .S.,’ vol. 89, p. 17 (1928).

t

§ ‘ M .N .R .A .S.,’ vol. 83, p. 403 (1923), and vol. 84, p. 499 (1924).

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439

where N 0 is to be calculated from a formula of the type (55) or (57), P 0 being determined by a formula of the type (59') and r being determined from obtained from the linecontour. We have thus an expression

N0(s)

— — (f)T, P 0) , ... (61)

mg

and we have to make this a maximum for variation of T, P 0 being a function of T varying according to a relation of the type t

= “
The ordinary process of differentiating for a maximum leads to an equation

H

_ _

8T a p 0

H __ n ^

ap„ aT

5

........................................v *'

valid at T = Tmax This equation, which is explicitly independent of e, a and g, can be solved for P 0 if Tmax is known from observation, and thus the partial electron pressure at the base of the column is determined. Introduction of this value of P 0 into (62), t being already known from r,evaluates a. Thus the cons for the coefficient of opacity is derived from observation. Once a is determined, the coefficient of opacity may be calculated at any desired level in the atmosphere. The explicit form of (63) has been given elsewhere* for a variety of cases. The solution is found to be expressible in all cases in the form P 0u = [e K JT._T L x max.

............................................(64)

where 6 depends on the excitation and ionization potentials concerned but varies only very slowly with Tmax The applicability of the method is closely connected with the fact th at (64) is explicitly independent of s, the proportion by mass of the element con­ sidered. It has merely been necessary to assume s constant in differentiating. The physical reason why observation of a maximum determines the general opacity coefficient is worth considering. The column of atoms has to contain a given total opacity, t , and at the same time to contain a maximum number of excited atoms, in order th at a line shall be widest for given intensity-ratio. The temperature of maximum depends on the partial electron-pressure, and this can be thus found from an observed Tmax. We thus know two things about the column—its total opacity and the electron pressure at its base. From the latter we can infer from consideration of mechanical equilibrium what is the total mass of the column. Dividing the total mass into the total opacity we get the opacity per gram, which is precisely the coefficient of opacity. Using once again the value of P 0, we find the constant a. Free electrons play two distinct parts—they determine the ionization and they determine the opacity. But the method does not depend on this two-fold appearance of free electrons. Whatever the so u k e of * ‘ M .N .R .A .S.,’ vol. 89, p. 17 (1928).

3

N

2

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E. A. MILNE ON STRUCTURE AND OPACITY OF A STELLAR ATMOSPHERE.

opacity, the method gives the opacity coefficient. But the law k = aP /T 9/2 makes the details of the determination both simple and interesting. The necessity for an appeal to the condition of mechanical equilibrium shows th at must be known. Consequently it is necessary to determine a maximum for stars of constant g, or else to introduce the actual variation of g in performing differentiations. 12. Numerical Evaluation of the Absorption Coefficient.—We now apply the whole of the preceding theory to actual observations. I t is recorded by Menzel and by Miss P ayne* th at the Zn triplet 1 3P—1 3S XX 4810, 4722, (4680) has a maximum in stars of type G0. Amongst many recorded maxima we choose this as a fundamental one, because we have two stars of type G0, namely the Sun and Capella, for which the values of g and T are fairly accurately known. We shall take T to be equal to Tl5 the effective temperature. It would be easy to determine the true mean tem perature T of the layers down to t , but it is not certain th at the ionization would be accurately calculable from T, since the material is traversed by radiation of temperature Tx. Whilst recognising th a t the best temperature to take is one lying between T and T1} we shall adopt T = Tx for simplicity.! (The point is discussed further below in relation to the B almer maximum.) We adopt t = J, corresponding to r — J. Actually line-contour determinations of these lines are not available but r = J cannot be far wrong. The formulae employed are those of “ Problem III ” of a previous paper,{ namely those appropriate to “ atoms becoming ionized in the presence of an excess of atoms already once ionized ” ; for this problem 6 = P q/K x is the root of the equation log (1 + 6 ) - 6/(1 + 6) _ f&T 6V(1 + 6) = Xl ~ XiU) 6 - log (1 + 6) xi + *AjT 6 - log (1 + 6) Xi + # T ’ where T is to be put equal to Tmax.

The results of the calculations are as follows.§

T able I.—Maximum of Zn lines 1 3P—1 3S, XX 4810, 4722 (4680), Type G0. Xi = 9-34 volts, Xi(s) = 5*33 volts, 1, — 2. S tar.

Sun.

Capella (6).

T em perature T x = Tmax. ................ ... ... ... ............................... ... ................. Surface g ra v ity — g E quilibrium co n stan t = K x ... 6 Degree of ionization x0 = 1/(1 + 6) ... ................ E lectron-pressure a t base P 0 = 0 K j A bsorption coefficient k0 = t /P 0 j a X 10-18 = k0 (T/104)°/2/P 0 '.......................................................

5,740° 2-74 X 104 20-8 1-48 0-403 30-8 297 0-794

5,200° 6-08 X 10a cm. sec.-2 2-20 1-56 0-390 3 • 43 dyne cm .-2 59 -0 g ram .-1 cm .-2 0-911

[The suffix 0 refers th ro u g h o u t to th e level t = |.]

* Menzel , ‘ H a rv a rd C irc.,’ p. 258 (1924); P ayne , f Stellar A tm ospheres,’ p. 130 (1925). t Cf. Gerasimovic, ‘ Proc. N at. Acad. Sci.,’ vol. 13, p. 180 (1927). %‘ M .N .R .A .S.,’ vol. 89, p. 17 (1928). § In ‘ M .N .R .A .S.,’ vol. 89, p. 35 (1928), I adopted Xi = 7-5 volts, w hilst pointing o u t th a t th is was probably too low .

The opacities now calculated are considerably higher, b u t seem m uch more tru stw o rth y .

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441

If the data and theory were quite accurate the two values of a should agree. The two values are a = 0*79 X 1018 and a = 0*91 X 1018. Considering the differences in T and gbetween the two stars, the agreement seems satisfactory. It should be borne in mind th a t the method can only be expected to give an accurate value of a if either yA for given Tmax or Tmax for given yA are known very accurately. For (K j )t max is chiefly determined by the factor e~Xi!kTmax-, which is sensitive to small errors in xi or Tmax ; P 0 is determined by K x ; and lastly the formula a gT/*» P 02 shows th a t a is determined by the square of P 0. In what follows, we shall adopt the value (the mean of the two determinations) a = 0-85

X

1018,

so th a t the photospheric absorption coefficient is given by P K " °* 85 (T / 104)972'

13. Determination from the Maximum of the B almer Lines. Miss P ayne and Miss W illiams have recently* made an accurate determination of the maximum of the B almer lines in A-type stars. They find th a t for 0*48 it occurs at type A0, for r —0*96 at types A3-A 5 . (We will later discuss the variation of maximum with There is some uncertainty as to the proper temperature-scale for the A-type stars. We shall average the observations by assuming th at at 0-7, the maximum occurs at A2 , Tx = 10,000 °. For r = 0 - 7 , t = 0 - 55 . For g we adopt the value for Sinus deduced from the mass and luminosity recorded by E ddington . (It is not suggested th at Sirius is of type A2 ; we merely use Sirius to provide a 0 -value.) The formulae employed are those for “ Problem III ” as before (material on the whole once ionized). The results of the calculation are given in Table II. The accuracy of this calculation is capable of considerable impro\ement when bettei values of Tmax and g are available. We note however that a comes out of the same order of magnitude as for the photospheres of the Sun and Capella about twice as lar&e. The agreement is on the whole as close as could be expected. We now consider the variation in the position of maximum as we move fiom the centre of the line to the wings. The observations of Miss P ayne and Miss Williams show th at as we move out to the wings (r and t increasing) l max. decreases. 1 origina j predictedf th at Tm»x. should increase with increasing Increasing r or t means seeing to increased depth, i.e,increased pressure ; increased pressure retards ionization * ‘ M .N .R .A .S.,’ vol. 89, p. 526 (1929). f ‘ M .N .R .A .S.,’ vol. 89, p. 17 (1928).

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therefore would appear to require increased tem perature to compensate. This was the crude argument. But I omitted to notice th a t increased depth t means not only increased pressure but also increased temperature. Miss P a y n e has suggested to me in correspondence th at this might cause an effect of the opposite sign. The argument T a b le II.—Maximum of B a lm er Lines for

Xj. = 13*54 volts, T em p eratu re = Tx = (T)max. Surface g ra v ity = g

— 0-7, taken as Type A2. Xi{,)~ 3*385 volts, = 1, 10, 000°

•••

2-64 X 104

E q u ilib riu m c o n sta n t = K x

4-82 X 10a

0*

0-203

........................................................................................................

Degree of ion izatio n =

0-830

— 1/(1 + 0 )

0978 X 102 d y n e cm .-2

E lectro n -p ressu re a t base = P 0 = 0 K ,

148 g ra m -1 c m .-2

A bsorption coefficient = k0 = t# /P 0 ... a

x

10-18 =

#c0 (T/104)9/2/P 0 ................

* Form erly I estim ated 0 = 1. un ity .

2.

1*51

A ctual solution of the equation for 0 shows it to be m uch less th a n

In m y previous analysis of th e B almer m axim um I found /
In th e presen t analysis I

tak e
H ence

k0

—148.

is th at the decreased ionization consequent on increased pressure may be more than compensated by the increased ionization consequent on increased temperature, so th at on balance the temperature of maximum may be decreased and not increased, as observed. Mathematical analysis confirms Miss P a y n e ’s conjecture. We have in fact an interesting application of the solution to S c h w a r z sc h ild ’s problem. The temperature of maximum is determined by the equation P 0 = [0 K i ]t where 0 varies only slowly with Tmax and K xvaries principally thro containing Tmax. W hat we actually observe is not, however, Tmax but ( T j ) ^ , where (T m a x .)4 =

i (T i)m a x . ( 1 +

f T )>

and t is related to P 0 by an equation already given. Ignoring slowly varying factors, we have in principle P 0 a e-WThmax. (W)*, T QC P 1 025

or

T OC e-2Xi/*(Tl)max. O+’O4The crude argument originally put forward by myself ignores the occurrence of t on the right-hand side of this equation, for it ignores the variation of true temperature with depth. In th at case (Ti)max would increase with increasing t . But differentiating the proportionality as it stands, we have d 1— '

2xi (T i)max. (TxXnax. (1 + | T)* L (Tj)max. ^

f 1 + f tJ '

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E. A. MILNE ON STRUCTURE AND OPACITY OF A STELLAR ATMOSPHERE.

This shows th a t d (T1)m

a

x

443

f dx will be negative provided

T or xi > 1 ( 1 1 1 1 ) kT

(65)

For the B almer maximum at 10,000°, i/kTertainly X c exceeds 15 ri \ . For r = 0*30 (the observed central residual intensity) t is | (0 *3)/(l •3) = 0*308 and | ( I -f | x)[x = 6* 3. Thus the inequality is satisfied, and d (T1)max/c?T is ne as observed. Increased tem perature with increased depth causes more ionization than is necessary to compensate for the increased pressure, and the maximum occurs at a lower effective temperature. The foregoing amounts to a second approximation to the theory. In the first approxi­ mation we took a mean value for T through the layer contributing the absorption line, and put it equal to T\. In the second approximation we take into account the probable variation of T through the layer as given by the solution to Schwarzschild’s problem. For simplicity we have taken the actual temperature at the base of the layer as deter­ mining the ionisation, in this second approximation. III.—Applications of the Absolute Value of the

Absorption Coefficient.

Application to Ionization of Calcium at Low Temperatures.—Once a is determined, the number of atoms per cm.2 down to given may be calculated for any element in terms of an assumed composition by mass. We illustrate by calculating the numbers of Ca and Ca+ atoms (N0 and Nx) down to t = in atmospheres of various Tx and assuming calcium forms 5 per cent, of the mass of the atmosphere. The formulae used are (55), (56) and (59') with a = 0*85 X 1018, x = e = 0-05, 40 X 1 *66 X 1024, Xi — 6*08. The calculations are confined to low temperatures (3500°, 4000°, 4500°) so th at second-stage ionization of calcium may be ignored. The following tables (Tables III, IV and V) show the calculated values of P 0, log10N0, log10 Nx, log10 (Ni/N0) and log10 (N0 -j- Nx) for various values of g, for three different temperatures. The calculations have been performed by choosing the values of P 0 in the first instance, so as to avoid the solution of a cubic equation, but in each table g should be regarded as the actual independent variable. The results are plotted as curves in figs. 1, 2. Fig. 1 shows log P 0 as a function of g at temperatures 3500°, 4000°, 4500°. It will be seen that the variation of P 0 with g is much more pronounced than the variation with T. Fig. 2 shows logN 0, log Nx, log (Nx/Nj) and log (N0 + Nx) as functions of g for the three temperatures. These may be regarded as the predicted “ absolute magnitude ” effects at various temperatures for the relative intensities of X 4227 (Ca) and XX 3933, 3968 (Ca+). It will be seen that N0 14.

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E. A. MILNE ON STRUCTURE AND OPACITY OF A STELLAR ATMOSPHERE. T able

III. — Ionization of Calcium in Stellar Atmospheres.

N0 = Number of Ca atoms down to Nj = Number of Ca+ atoms down to P0 = electron pressure at

Log10 g (gin cm. sec. 2)

= -J \ Assumed composition, 5 per cent, of x= J J calcium by x= J. T = 3500°.

t

P0 dynes cm . 2

L°gio N0

Log10 N x

Log10 (N x/N 0)

lO-2 1 0 -1 5 X 10"1 10° 5 X 10° 10 5 X 10

18-21 18-19 18-13 18-06 17-72 17-50 16-87

20-72 19-70 18-94 18-57 17-54 17-01 15-69

2-51 1-51 0-81 0-51 1-82 1 *51 2-82

2-46 0 -4 8 1-94 2-61 4-34 517 7-19

Log10 (N0 + Nj)

20-72 19-71 19-00 18-69 17-94 17-62 16-90

IV.— Ionization of Calcium in Stellar Atmospheres.

T able

T = 4000°. Log10 g

P0 dynes cm. 2

Logio N 0

L °gi0 N j

2-20 0-20 2-21 3-64 4-29 5-93

10“ 2 lO "1 10° 5 X 10° 10 5 X 10

17 -23 17-23 17-22 17-18 17-13 16-89

20-98 19-98 18-97 18-23 17-89 16-94

T able

Log10 (Nj/No)

3-75 2-75 1 75 1-05 0-76 0-05

Logxo (N0 + N j)

20-98 19-98 18-98 18-27 17-96 17-22

V.—Ionization of Calcium in Stellar Atmospheres. T = 4500°.

Log10 g

3-97 1-97 1 37 1-97 3-37 3-98 5-42 6 06 8-51

Po dynes cm .-2

Log10 N0

10~2 1 0 "1 5 X 10"1 10° 5 X 10° 10 5 x 10 102 103

16-48 16-48 16-48 16-47 16-47 16-47 16-43 16-38 15-94

Logio N i

21-21 20-21 19-51 19-21 18-51 18-20 17-46 17-11 15-67

P°gio (N i/N 0)

4-73 3-73 3-03 2-74 2-04 1-73 1-03 0-73 1-73

L°gio (No + Nj)

21-21 20-21 19-51 19-21 18-51 18-21 17-50 17-19 16-13

:

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E. A. MILNE ON STRUCTURE AND OPACITY OF A STELLAR ATMOSPHERE.

445

decreases slowly, N t quickly, with increasing The curve for log (Nx/Nx) is independent of the assumed value of z. The other curves may be displaced bodily up correspond to different values of e. The curves for log (N0 + Ny) represent the total amount of calcium. “ above the photosphere.” By adjustm ent of e they could be taken to represent the total amount of any other element “ above the photosphere.” In particular, for an element only slightly ionized at low temperatures they represent the strength of the neutral lines as a function of g. Thus, for example, they give the pre­ dicted behaviour of hydrogen intensities at low temperatures as a function of g. PARTIAL

ELECTRONPRESSURE IN STELLAR. ATMOSPHERE A T

'©420Cf

V c f /u o A

u c u u u x i

u y r a yn ts cuiu i i v y y ,

//

o b serv a tio n s o f Line-contours an d reduction b y UnsoLds m eth o d . (F u rth e r p o in ts n o t h ere p lo tte d , g iv in g considerably Low er p r e s s u r e s , were o b ta in ed b y Payne a n d H ogg)

F ig . 1.— Partial electron pressures P 0 in stellar atmospheres— at low temperatures at depth t — as a function of surface gravity g.

calculated

Note .— The observed points correspond to pressures P deduced by P ayne and H ogg from the values of N 0 and N x for Ca, as o btained b y U nsold ’s m ethod. They show a m uch more rapid decrease of pressure w ith g rav ity th a n shown b y th e calculated curves.

It should be remembered th at the only astrophysical datum employed in calculating these curves is the value a = 0*85 X 1018, derived from the observed maximum of the Zn lines in Gr0 type stars. No observed intensities of calcium lines have been introduced. It is therefore of great interest to compare these predicted effects with actual observa­ tions. P a y n e and H o g g ( loc.cit.) have applied U n so l d ’s method to the contours of and A 4227 in the cooler stars. Using U n so l d ’s formula for the atomic scattering coefficient—which has not been required at any point in our work—they have derived values of logN 0, logN l5 log (N0 + NT) and log (Ni/No). From the latter they have VOL. CCXXVIII.— A

3 O

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446

E. A. MILNE ON STRUCTURE AND OPACITY OF A STELLAR ATMOSPHERE. IONIZATION OF CALCIUM IN STELLAR ATMOSPHERES

>Observed points, Payne and Hogq N0 -rNl Departures o f observed points mom curves may be a ttribu ted either bo an e rro r in the assumed proportion e = (verb'cal sh ift) or to the efFect o f selective radiation pressure in decreasing qravitu (Horizontal s h ift)

0-QS

>Observed points , Payne an d

Hogg Ni/No. Curvesxhere independent o f the assumed value e mO'OS

300,0° a9Js(

Observed points (Payne and Hogg) Ca^(Ni) The departures o f the observed points from the curves can be attributed ej'ther to an error in the assumed proportion o f calcium (verbcal shift) or to the 'effect o f selective radiation pressure on g (horizontal sh ift)

F ig . 2. Ionization of calcium in stellar atmospheres at low temperatures, as a f unction of surface gravity g.

Aote (1). N0 + is the total number of atoms above the photosphere 55 (actually above t = A). ^ _rePresen^s ^be total number of neutral atoms of any constituents ionizable only with difficulty, and thus gives, for example, the behaviour of Balmer lines with gravity at low temperatures. ^ ote (2). The single datum from which the curves are calculated is the observed Zn maximum in Gr0 type stars. Aote (3)^—The observed points were all obtained by an entirely independent method, the line-contours method of Unsold, assuming a formula for the atomic scattering coefficient. The fact that the calcu­ lated curves pass through the neighbourhoods of the observed points checks the general correctness of the aggregate of assumptions made and formulae used.

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E. A. MILNE ON STRUCTURE AND OPACITY OF A STELLAR ATMOSPHERE.

determined log B. are as follows.

447

Their results, transcribed from H arvard Circular, p. 334 (1928)

T able VI.—Observed Ionization of Calcium in Stellar Atmospheres. (P ayne and H ogg.) j

Spectral Class,

G„ 5 K„ g

k2 k6 M0

Mx M,

1

m3 m5 m7

j

Tx

5600 4800 4200 3850 3400 3100 3050 3000 2950 2850 2800

log£ 2-7 2-5 2-3 2-1 1-9 1-7 1-7 11 1-6 1-5 1-5

log N0

log Ni

log (N0+ Nj).

16-44 17-56 18-18 18-39 18-66 19-00 19-05 19-11 19-16 19-26 19-35

19-55 19-61 19-61 19-63 19-72 19-90 19-87 19-82 19-78 19-67 19-55

19-56 19-61 19-63 19-66 19-76 19-95 19-93 19-90 19-90 19-81 19-76

log P + 0-6 + 0-8 -0 -2 -0 -8 -1 -7 - 2-2 -2 -4 -2 -5 - 2-7 - 2-6 - 2-8

The whole of these observations (with the exception of some of the smaller P ’s) are plotted on the appropriate diagrams in figs. 1, 2. Actually the values of P should have been doubled ( i.e,. 0-30 added to their logarithms) before being plotted, as P ayne an H ogg calculate P from the formula whilst (55) and (56) give

so th at taking x t \xx —N2/Nl5 P () — 2P. Our curves give P„, P ayne an points refer to P. The observed values of P decrease with decreasing g much more rapidly than pre­ dicted. This is an outstanding discrepancy. Apart from this the observed points cluster well round the calculated curves. This shows th at s = 0*05 is roughly correct for the calcium content. But it is undesirable to press this. The main conclusion is th at the theory of opacity put forward in this lecture, and the determination of its absolute amount, is consistent with the determinations of content by U nsold’s method, based on entirely different data, both theoretical and observational. Developments of the method at once suggest themselves. Once the value of s is determined with some approach to accuracy for a few typical stars, it may presumably be used for other stars. A determination of Nt or N0 by U nsold’s method for a star of known T but unknown g can then be plotted on the appropriate curve and the corre­ sponding value of g read off. We have thus a method of determining (/-values for separate stars, independent of empirical calibration curves such as are used in the method of spectroscopic parallaxes. The method is applicable to any line for which line-contours can be determined. The ratio Nx/N0 can be used, alternatively, instead of the separate values of Nx and N0, and this has certain advantages. 15. Application to B the almer Ames.—Tables YII and V III show the num 3 o 2.

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448

E. A. MILNE ON STRUCTURE AND OPACITY OF A STELLAR ATMOSPHERE.

neutral atoms of hydrogen in a stellar atmosphere down to t = J, and the numbers of excited atoms capable of absorbing B almer lines, for various and T, on the two laws (1) * = 0-85P/(T/104)9/2, (2) k= constant = 100. The first s actual behaviour of B almer lines in stellar atmospheres ; the second is merely inserted for contrast. The values of log N0W (numbers of excited atoms) are plotted in figs. 3 and 4. T able V II.—Hydrogen. Calculated numbers of atoms per cm.2 down to t = J capable of absorbing B almer lines (N0(s)) for various g and T, assuming 5 per cent, by mass of hydrogen, for k— 0*85 P/(T/104)9/2. (N0 = number of neutral a I

9— 10-

9 = : 102.

9 = 103.

9

: 104.

T.

4,500 5,000 5,500 6,000 6,500 7,000 8,000 9,000 10,000 11,000 12,000 13,000 14,000

LogN 0

Log N„<*>

Log N0

Log N0(®)

LogN 0

Log N 0<*>

Log N 0

Log N,<*>

21 -30 21-40 21-49 21-56 21-59 21-50 20-84 20-05 19-39 18-85 18-41 18-04 17-73

10-51 11-76 12-77 13-63 14-31 14-78 15-04 14-96 14-87 14-79 14-74 14-70 14-67

20-80 20-90 20-99 21-07 21-13 21-13 20-74 20-03 19-38 18-85 18-41 18-04 17-73

10-01 11-26 12-27 13-13 13-85 14-41 14-94 14-95 14-87 14-79 14-74 14-70 14-67

20-30 20-40 20-49 20-58 20-64 20-68 20-53 19-99 19-37 18-85 18-41 18-04 17-73

9-51 10-76 11-78 12-64 13-37 13-97 14-74 14-90 14-85 14-79 14-74 14-70 14-67

19-80 19-90 19-99 20-08 20-15 20-21 20-21 19-89 19-36 18-84 18-41 18-04 17-73

9-01 10-26 11-27 12-14 12-87 13-49 14-41 14-80 14-84 14-78 14-74 14-70 14-67

|

|

;

T able V III.—Hydrogen. Calculated numbers of atoms per cm.2 down to t = capable of absorbing B almer lines (N0W) for various g and T, assuming 5 per cent, by mass of hydrogen for k = constant = 100. (N0 = number of neutral atoms.) ]

T

4,500 5,000 5,500 6,000 6,500 1 7,000 8,000 9,000 10,000 11,000 12,000 13,000

© r-H

10

9

103

104

Log N 0

Log N 0«

Log N 0

Log N0<*>

Log N 0

Log N0<»

Log N 0

Log N 0(s>

20-00 20-00 19-97 19-79 19-27 18-53 17-19 16-11 15-24 14-51 13-90 13-37

9-21 10-36 11-25 11-86 11-99 11-82 11-39 11-02 10-72 10-45 10-23 10-04

20-00 20-00 20-00 19-96 19-81 19-39 18-12 17-11 16-24 15-51 14-90 14-37

9-21 10-36 11-28 12-02 12-53 12-67 12-32 12-02 11-72 11-45 11-23 11-04

20-00 20-00 20-00 20-00 19-97 19-85 19-12 18-10 17-24 16-51 15-90 15-37

9-21 10-36 11-28 12 -06 12-69 13-13 13-32 13-02 12-72 12-45 12-23 12-04

20 -00 20-00 20-00 20-00 20-00 19-98 19-74 19-04 18-22 17-51 16-90 16-37

9-21 10-36 11-28 12-06 12-72 13-26 13-94 13-96 13-71 13-45 13-23 13-04

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E. A. MILNE ON STRUCTURE AND OPACITY OF A STELLAR ATMOSPHERE.

449

These curves demonstrate in the clearest possible way the difference between >
t

3

—)

13000°

10000°

F ig . 3.— N um bers of excited hydrogen atom s (capable of absorbing B almer lines) above

t

k

= 0 - 8 5 P /(T /1 0 4)9/2, where P = electron pressure.

for various

g, calculated for

values of tem p eratu re T an d surface g ra v ity photospheric opacity

=

14000°

F or this law of photo-

spheric opacity th e curve show no “ absolute m agnitude effect ” above T = 10,000 .

The horizontal

distance betw een an y tw o curves a t lower tem p eratu res gives th e com puted difference of tem perature betw een g ian t (low g) an d dw arf (high g) showing B almer lines of same strength.

lines. But if kis taken as constant, there should be no “ absolute magnitude at low temperatures ; instead a strong one should set in at high temperatures , the curves of fig. 4 coincide at low temperatures but separate at high temperatures.* * The law /c oc P p red icts a null absolute m agnitude effect a t high te m p eratu res as arising from changes

of atmospheric thickness and of ionization; th e tw o effects neutralise one an o ther. B ut th is does not preclude o th e r effects of (/-changes. F o r exam ple reduced g, which causes ieduced I , will cause a decreased S ta rk effect, w hich m ig h t be used as an absolute m agnitude criterion. I owe this lem ark to discussions w ith Miss W illiams , of H a rv a rd College O bservatory. See “ H a rv a rd College O bservatory B u lle tin ,” No. 870 (in press.)

eff

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450

E. A. MILNE ON STRUCTURE AND OPACITY OF A STELLAR ATMOSPHERE.

The observed facts are th a t hydrogen shows a strong absolute magnitude effect at temperatures below A0* (increasing intensity accompanying increasing luminosity) and the c-effect in classes A and B (increasing sharpness accompanying increasing luminosity). The second effect is well shown in P a y n e and W il l ia m ’s line-contours (,loc. cit.)for stars of type B5. If following U n so ld we take increasing sharpness to indicate a decreasing number of fully-viewed atoms, then the observed facts correspond to a kind of mixture of the effects shown by our two diagrams. At low temperatures, the hypothesis k x P is confirmed ; the curves for the lower gr-values lie above th e NUMBERS OF EXCITED HYDROGEN ATOMS K = I0 0

4000°

5000°

6000°

7000°

8000°

9000°

10000°

11000°

12000°

13000°

F ig . 4.— N um bers of excited hydrogen atom s (capable of absorbing B almer lines) above t = for various values of tem p eratu re T and surface g rav ity g, calculated for e == 0 -05 (proportion by mass) a n d photospheric opacity k = co n stan t = 100. These curves are inserted sim ply for c o of Fig. 3. I t will be seen th a t for k — co n stan t th ere should be a strong “ absolute m ag n itu d e ” effect a t high tem p eratu re b u t none a t low tem p eratu res.

curves for the higher values, as observed. At high temperatures the reverse effect is show n; observation seems to favour k = constant. This is a point for future investiga­ tion. In the past hydrogen lines have not been used as a quantitative criterion for absolute magnitude at low temperatures ; and at high temperatures, they have been only partially so used, as there appears to be a correlation of type with absolute magnitude which is hard to disentangle.*j* But it is quite clear th at ultimately the behaviour of the hydrogen lines with g and T will afford delicate tests of any theory of the photospheric absorption coefficient. * ‘ T rans. In te rn a t. A stron. U nion,’ vol. 1, p. 98 (1922). t Adams and J oy , ‘ A strophys. J .,’ vol. 56, p. 242 (1922), an d vol. 57, p. 294 (1923).

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E. A. MILNE ON STRUCTURE AND OPACITY OF A STELLAR ATMOSPHERE.

451

The calculated values of the numbers of fully-vie wed excited hydrogen atoms at low tem peratures scarcely seem adequate to account for the great strength of the B a lm er lines in late-type giants ; Adams and B u s s e l l * have found th a t the number of excited hydrogen atoms is much larger at low tem peratures than can be accounted for by therm o­ dynamic theory. Perhaps the excited fraction is much larger than is given by the B oltzmann factor. Once again it may be mentioned th at the calculated numbers of hydrogen atoms given in our tables and curves do not involve any datum of observation referring to hydrogen in stellar atmospheres ; they are derived from the observed Zn maximum in the case k oc P, and from an assumed order of magnitude of k in the case k = constant. Our proportion by mass e = 0*05 may be of course too small, in view of K osseland’s suggestion of an excess of hydrogen in stellar atmospheres.•)* These two examples, Ca and IT, will serve to show the fundamental importance of an accurate knowledge of the behaviour and order of magnitude of the photospheric absorp­ tion coefficient. A few further examples are given, relating to Mg, Mg+, He, He+, Si+, Si++, Si+++. The numbers of silicon atoms responsible for various lines are shown, calculated for two values of g(g = 103 and g — 104), s = 0*05, and The calculated electron pressures are also shown at certain places on each curve, at the depth t = J. For Mg, He and H e+ e is taken as 0 •05 for definiteness, but of course the curves require simple vertical displacement to correspond to other values of e ; for these latter curves g has been taken to be 6-08 X 102, the value for Capella. See figs. 5 and 6. 16. Application to Determine the Physical State of the Photospheric .—Our last application of the determination of photospheric opacity is to calculate the distribution of pressure, tem perature and opacity through the photospheric layers of two typical stars. The theory is a slight modification of one previously given,J but the approxi­ mations now made are of a much more accurate nature. We have previously calculated the mechanical equilibrium of the layer in which general opacity sets in, taking account of the variation of ionization with depth but ignoring radiation pressure. We now propose to allow for radiation pressure, but we have to pay the penalty in th at we are compelled to take a mean value for the ionization. Selective radiation pressure is not allowed for, so that the investigation which follows may be taken to apply to the lower reversing layer and the photospheric layers. Let p ' denote the pressure of radiation at depth h in the photospheric layers, measured from some convenient reference level. Then since a net flux tcF traversing a layer p of absorption coefficient k communicates momentum tcF «p dhjc we have dp = dh

k 9 ttF

............................. ... ................... (66)

c

* ‘ A strophys. J .,’ vol. 68, p. 9 (1928). t ‘ M .N .R .A .S.,’ vol. 85, p. 541 (1925). % ‘ M .N .R .A .& ,’ vol. 85, p. 768 (1925).

8000°

10000°

12000°

14000°

16000°

20000°

T —^

18000°

22000°

24000°

26000°

28000°

200 9

30000°

32000°

34000°

g,

g= 103 and g — 104 cms. sec.-2 .

t

ionization.

The descending b ranch es should really be steeper (see R . H . Fow ler, ‘ S tatistical M echanics,’ p. 385).

Note .—The descending branches of th e curves for Si+ and Si++ are n o t accurate, as no account has been ta k e n of th e on set of th e succeeding

peaks of th e curves are th e ty p es of observed m axim a.

The sp ectral types m arked near the

E. A. M ILN E ON ST R U C T U R E AND O PA CITY O F A S T E L L A R A T M O S PH E R E .

The calculated cu

The sm all figures near th e

—| a t th e corresponding tem p eratu res, in dynes cm .-2 .

Assum ed p ro p ortion of silicon, 5 per cent, by m ass (e = 0 - 0 5 ) .

w ith Miss P a y n e ’s estim ates of line-in ten sity p lo tte d in H a rv a rd C ircular No. 252 (1924), p. 8.

curves are th e electron pressures a t

of

atom s, and N 3(s) of excited Si+ + + atom s p er cm .2 above t = J, a t various tem p eratu res and for tw o different values of surface g ra v ity

F ig . 5 .— Ionization of silicon in the hotter stars .— The curves show th e calculated num bers N x(s) of excited S i+ atom s, N 2
6000°

SILICON IN THE HOTTER STARS

452

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E. A. MILNE ON STRUCTURE AND OPACITY OF A STELLAR ATMOSPHERE. NUMBERS

OF

E XC ITE D

ATOMS

6000°

2000°

ABOVE

8000°

r- i

FOR

G = 6 0 8 x l0 S

14000°

10000 °

16000°

18000°

20000°

F ig . 6.— N um bers of excited atom s (capable of absorbing certain lines) above x — for

453

y — 6 -08 X 10-2 cm. sec.-2.

22000°

24000°

26000“

per cm .2, calculated

In each case th e assum ed proportion is 5 per cent, by mass (s = 0 -05).

D ifferent proportions are ob tain ed by displacing th e curves rigidly up or down.

(N .B.— Owing to an

error, inaccu rate sta tistic a l w eights were em ployed in the calculation of th e curves of this diagram , b u t th is sim ply corresponds to sm all vertical displacem ents, n o t necessarily the same for each curve). (Second-stage ionization has n o t been allowed for in th e descending branch of the Mg curve).

This is really the definition of may he written

p'. To connect it with the temperature d £

dx

By the solution to differentiation

G

S c h w a r z s c h i l d ’s

p r o b le m , g e n e r a lis e d

(TP_ dx

=

tcF

o iy

C a s in §5, w e h a v e

by

f'JV

Hence dp' dx

T4a dT *

6 c (I t

or integrating P - JaT 4, provided a — 4 [c. VOL. CCXXVIII.— A

3 P

(67)

i

O

r—H

II

8-9 12-0 15-6 19-8 24-5 29-8 35-7 42-2 57-0 74-3 94-2 116-7 142-0 170-0 200-9 234-6 271 -2 310-9

28-1 37-9 49-4 62-6 77-6 94-4 113-0 133-5 180-3 235-0 297-9 369-1 449 -0 537-6 635-1 741 -8 857-6 983-1

—

—

—

—

—

—

—

—

—

—

—

—

—

-—

—

—

-

11*94 12*95 13*73 14*28 14*55 14*59 14*52 14*41

■ —

—

—

•

19-95 20 -05 19-93 19-44 18-82 18-27 17-80

—

—

—

—

13-61 14-55 15-10 15-14 14-95 14-77 14-61

—

12-33

—

—

—

19*45 19*57 19*57 19*30 18*79 18*27 17*80

—

19*30

—

—

19-80

— —

—

-—

— -—

—

—

—

-—

-

13-11 14-08 14-75 15-00 14-92 14-76 14-60

—

11-83

—

—

—

— —

—

—

—

|

: 104

9 = : 103

9

20-08 20-20 20-30 20-37 20-39 20-36 19-96 19-58 19-18 18-83

—

__

—

__ __

_

_ —

12-53 13-49 14-25 14-87 15-36 15-71 15-64 15-55 15-40 15-28

__ _ __ __ __ __ __

■

19-58 19-70 19-80 19-89 19-94 20-05 19-77 19-49 19-16 18-82

__ __ __ __ __ __ __

!

12-03 12-99 13-76 14-39 14-91 15-40 i 15-45 15-46 ! 15-38 1 15-27 ;

Z

Log N , Log N 2(s> Log N 3 Log N #W Log N 3 Log N ,«

9

S i+ + + (XX 4089, 4116)

w tel W tel

s

H3 g

E . A. M ILN E ON ST R U C T U R E AND O PA CITY OF A S T E L L A R

—

—

—

—

—

■--------

—

—

—

—

—

18*78 18*90 19*00 19*00 18*83 18*49 18*10 17*73

9 = 103 *

Log N xO) Log N 2 Log N s« *

9 = 10* Log N x

—

—

—

—

^

—

—

—

-—

—

—

12-44 13-44 14-20 14-63 14-72 14-65 14-54

Log N x«

18*28 18*40 18*46 18*35 18*00 17*54 17*11

'

Log N x

= 103

Si++ (XX 4552, 4567, 4574)

r*H

7,000 8,000 9,000 10,000 11,000 12,000 13,000 14,000 16,000 1 18,000 20,000 22,000 24,000 26,000 28,000 30,000 32,000 34,000

T

Po g — io*

Si+ (XX 4131, 4128)

VIII a .—Behaviour of Ionized Silicon in Stellar Atmospheres. Calculated for a = 0*85 P / ( T / 1 0 4)9/2 and £ = 0-05. (For spectroscopic data here used, see R. H. F o w l e r , £Statistical Mechanics,’ pp. 377, 378). P 0 in dynes cm.-2.

T able

454

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E. A. MILNE ON STRUCTURE AND OPACITY OF A STELLAR ATMOSPHERE.

455

Behaviour of Normal Excited O Atoms in Stellar Atmospheres calculated for k= 0-85 P /(T / 104)9/2. The values of Nx(s) should strength of the line X 4267, 1 2D— 1 2F. Observed maximum is at type B3. g = 104 cans. sec.-2.

T able

1

V I I I b .—

T

12,000 14,000 16,000 18,000 20,000 22,000 24,000

Po d ynes cm .-2

Log N x

94-38 133-51 180-30 235-01 297-89 369-13 448-97

19-67 19-77 19-64 19-19 18-66 18-20 17-82

Log NG>

12-34 13-53 14-21 14-38 14-36 14-31 14-27

Behaviour of Normal and Excited N + Atoms in Stellar Atmospheres calculated for L = 0-85 P /(T / 104)9/2. The values of Nxw should give the strength of the line X 3995 , 1 4P '— 1 4P. The observed maximum is at type B;!_5. g = 104 cans. sec.-2.

T a b l e Y I I I c .—

Po dynes cm. 2

Log N t

1

Log N ,w ------ ----------------------------- -

14,000 16,000 18,000 20,000 22,000 24,000 26,000

19-76 19-85 19-80 19-47 18-99 18-53 18-13

133-51 180-30 235-01 297-89 369-13 448-97 537-56

12-65 13-57 14-16 14-35 14-29 14-18 14-08

This is the usual definition of the constant a, and (67) agrees with the formula for p' for thermo-dynamic equilibrium. Thus (67) holds to the extent to which T 4 = JT j4 (1 + h ) is an accurate solution of S ch w a r zsc h ild ’s problem. The equation of mechanical equilibrium is %

+ %

- » ............................................................... < • >

Dividing (68) by (66),

J.l....................... ^

Now if M is the mass of the star, and ttE = L / 4tcrp. Hence

g/nY

t dp'zk T

rxits radius, L its bolometric lu = 4ttGM/L. Further

.....................................................™

3

p

2

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456

E. A. MILNE ON STRUCTURE AND OPACITY OF A STELLAR ATMOSPHERE.

We shall now suppose there are on the average free electrons per atom in the photospheric layers.* Then P I + £C P Then (69) becomes dp ___ 4ttcGtM (1 + 7 / 9/8 ................ (71) dp' L a (J a )9/8 p Make the substitution! ,v /1 7 /1 6

................................ ................ (72) where Pi - io T ,1...................................

................ (73)

Then (71) becomes ( V x \ xlx\ * ( / )

, 17

47teGM (1 +

/

x )[x

1/8

L

+

4itcGM (1 ~t~ x) L a (i

ri'

(74)

1

Denote the coefficient of u~x on the right-hand side by A. The term (P iIp'V1C/ varies as T_i and is equal to unity for T — rJ \. This variation is so slight for the range of values of T we shall have to consider th at we shall ignore it, replacing the term by unity. Equation (74) is then soluble. It may be written p'd_

u du p' A — u —

(75)

Writing A—

u—

id EE —

(u — tq)

-j-

(76)

+ — ) + U21log u2

(77)

we have on integration! loo- JL = ___ % log ( l — — U-y / & -f w2

since

u = 0 when p ' = p 0' We have also by the solution to Schwarzschild’s problem

= aT04.

p'lpo'

=1 + f

T

We see th at as t increases, and accordingly increases, u rapidly increases§ and approaches the limit ux.\Hence by (72) the ratio of gas-pressure to radiation-pr pip', slowly increases in the deeper layers, being equal to Thus ultimately increases as Th * The equation is ra th e r in tractab le if we try to calculate th e v ariatio n of x th ro u g h th e layer. t The physical m eaning of u is m o st read ily u n derstood by n o tin g th a t a t th e level T = T l5 where

V' — V 'i> u is equal to th e ra tio of gas-pressure to ra d ia tio n pressure. x W oltjer , B .A .N ., vol. 2, p. 171 (No. 66) (1924). § J eans , ‘ M .N .R .A .S.,’ vol. 85, p. 199 (1925).

|| This argument is due in principle to J eans, whose analysis we have slightly modified.

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E. A. MILNE ON STRUCTURE AND OPACITY OF A STELLAR ATMOSPHERE.

Equations (77), (78) and (72) determine tem perature T. I t remains to determine the heights to which these values of From (66) we have, on putting p = yi/(R /g)T and inserting the formula for dp'

x Aa \ 9/8 (R/g) T 47tcGM ’

a

-r+ lV /J

457

pr and p for refer.

ffL

Hence using (72) 1 dT _ T l "

1 dp' _

Jax (Ja )9/8

p2

gh dh (R/g)T 4tu'cGM

z p' =

A (R /g)T * The value of this formula is th a t T cancels from each side and we obtain dT dh

______ iff w2, A (R/g)

or by (76) dT d/i

id

(79)

R /g ' (17/16)

This is an exact formula for the temperature gradient involving no approximations except the use of x instead of variable x. We observe th a t as u -> uu dT dh

iff R /g * (17/16)

’

The temperature gradient approaches a definite limit.

Since u2

16/17, the

value of this limit is

df \ T \dh

_i f f _1 )oo

R /g *(17/16) + f i r 1'

Thus provided tq is not too small, the temperature gradient at great depths is roughly independent of A, and so of a and L/M. It can never exceed (4/17) g[(R/ g). To obtain the height for given u, we write (79) in the form 4 dT _ 1 _ _ T dh p' and substitute for dp'/p' from (75). g d/i (R /g ) T

______ ,_______ 1112 dh (R/g) T (17/16) w,?/

d p '_

u~

The result is UiU2 u (u — ux)

(80) +

)

To avoid complicated integrals it_is convenient to take a mean value for T on the left hand side here. l l ---lli

Denoting it by T we have

( P M I f l o g t , --------^ - l o g 11■ 1 u* 9

where /i0 is arbitrary.

u Ui

— III

+

— log ( i + ^2 \ ^2

( l (81)

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458

E. A. MILNE ON STRUCTURE AND OPACITY OF A STELLAR ATMOSPHERE.

Tables IX and X show the physical conditions of the photospheric layers of the Sun and Capella (bright component) calculated from the above formulae. The value T required in (81) has been taken to be \ (T 0 + T). Actually a sort of harmonic mean is T able

u

0 5 10 15 20 25 27 28 28-3 28-6 28-8 ■ 28-9 28-95

IX . — The Sun ( # = 2 * 7 4 X 104 cm. sec.-2, photospheric layers. tq = 29*0, 29*9,

R ad iatio n O ptical pressure thickness p' dynes T cm .-2

0 0 -009 0-040 0-103 0-231 0-582 1 -027 1-650 2 -054 2-854 4-185 6-03 8-57

1-386 1 -405 1 -468 1-600 1-862 2-595 3-515 4-81 5-65 7-30 10 -09 13-91 19-17

Gaspressure p dynes cm .-2

0 6-72 14-11 23-2 36-3 64-5 96-3 139-6 168-0 222 315 444 625

T em pera­ tu re T degrees

:

4,830 4,840 4,900 5,000 5,200 5,650 6,090 6,590 6,860 7,310 7,920 8,540 9,300

T em pera­ tu re grad ien t

dT/dh degrees k m -1

0 4-5 18-0 40-5 72-0 112-7 131-3 141-1 144-2 147-3 149-2 150-3 151 -0

X. -Capella (bright component) Physical state of photospheric layers. tq = 4-80,

= 5740°). Physical state of 1/(1 -j~ tq) = 0*0333.

D ep th

D e n sity

h

P gram c m .-3

km s.

— oo 11-9 17-5 21-4 25-3 31-1 35-8 40-7 43-3 47-7 53-6 59-8 67-0

0 3-36 6-95 u -2 16-85 27-7 38-3 51 -0 59-1 73-3 1 96-4 !126 613

X 10“ 10 X lO-™ x :io-'<> x lO " 10 x 1 0 - 10 X 10“ 10 x 10“ 10 x 1 0 - 10 x 10“ 10 x 10“ 10 x 10“ 10 X 10“ 10

A bsorp­ tio n coefficient

K

0 74-6 149 222 293 359 381 388 390 385 380 375 367

( 0-08

T able

U

0 1 2 3 3-5 4-0 4-2 4-4 4-6 4-7 4-75 4-78 4 79

R ad iation O ptical pressure thickness p ' dynes T cm ,-2

0 0-013 0 -055 0-153 0-249 0-443 0-559 0-780 1 -261 1-909 2-799 4-462 6-247

0-932 0-950 1 -010 1 -146 1 -281 1-553 1-715 2 -025 2-695 3-61 4-85 7-17 9-68

Gaspressure p dynes cm .-2

0 0-910 1 -94 3-33 4-37 6-14 7-16 8-95 12-70 17-70 24-4 37-2 51 -5

T em pera­ tu re T degrees

4,370 4,390 4,460 4,600 4,730 4,960 5,090 5,300 5,700 6,120 6,590 7,270 7,830

T em pera­ tu re g radient

^ dT/dh degrees k m .-1

0 0-13 0-50 1 -13 1-54 2 -01 2-22 2-43 2-66 2-78 2-83 2-87 2-88

w2 = 5*74,

1/(1 -f-

D ep th

D en sity

h

P gram cm.- 3

km s.

0-172.

A bsorp­ tio n coefficient K

----CO

16 255 440 540 672 745 844 1,023 1,205 1,403 1,690 1,928

0 0-501 1-05 1-75 2-24 3-00 3-40 4-08 5-38 6-99 8-95 12-40 15-9

X 10“ 10 x 1 0 - :10 x 10“ ]° x l o -10 x i o _1° X 10“ 10 x lO-10 X 10“ 10 X 10“ 10 X 10“ 10 x t o - 10 X 10“ 10

0 15-7 31-2 46-5 53-8 60-9 63-7 65-8 67-7 67-9 67-1 66-0 65-3

XI

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E. A. MILNE ON STRUCTURE AND OPACITY OF A STELLAR ATMOSPHERE.

450

required. In fact once the temperatures have been obtained from (77) and the gradients from (79), the heights could be found accurately by quadrature. This did not seem worth while, so th at the heights given in the tables, though approximately correct, do not exactly tally with the tabulated temperature-gradients, which are accurate. The tables are calculated for an assumed molecular weight [x 20 , corresponding to once-ionized calcium, which would seem to be typical for the Sun and Capella ; x is taken to be unity. The values of the absorption coefficient k at each level are also calculated. The whole calculation is based on the value a = 0-85 X 1018. It will be seen th a t the tables reproduce, as is only natural, values for k of the same order as found by the method of maxima. The temperature gradient in the photosphere of Capella never exceeds 3° per kilometre ; th at in the Sun rises to 150° per kilometre. The heights tabulated contain an arbitrary additive constant. We see that for the Sun, the layer x = 0*01 to t = 1 *03 is contained in 24 km. ; to x = 4, which is about the furthest we can see at the centre of the disc (e'~4 = 0-018), in 42 km. ; for Capella, from x = 0-01 to t = 0*78 is contained in 830 km. ; to t = 4, in about 1600 km. The distances are roughly in the ratios of the (/-values. These tables do not take into account the selective radiation pressure which rises into importance in the upper reversing layer. Consequently the actual thickness of the reversing layer will be greater than the values found. The ratio of radiation-pressure to total pressure at the depth where T — Tx is equal to 1/(1 + uf). This is tabulated at the head of each table. 17. Imperfections of the Method.—Both the theory of the present method and its numerical applications must for various reasons be regarded as provisional. The theory itself is open to criticism on a number of grounds, which may be divided into mathe­ matical, (b) physical. (а) The formula for the depth of the equivalent photosphere, x = | + 1) is derived only for a special model, and then applied to cases in which the assumption of the special model no longer hold. It assumes that lines are formed by strict mono­ chromatic scattering by atoms embedded in a gaseous medium contributing a con­ tinuous background. In deriving the formula, k is assumed to bear a constant ratio to sv. We then apply it to the case in which *aP whilst 5, varies with P according to a different law determined by the variation of ionization with P. We assume that in the actual atmosphere study may be confined to the atoms contained in the uppermost x of optical thickness. However, all these assumptions are equivalent to the assumption th at failing mathematically rigorous solutions of the general equations we are entitled to introduce averaging processes. The averaging processes here introduced are pro­ visional, to be replaced in due course by progress in the mathematical analysis. (б) The method of deriving the absorption coefficient of the photospheric layers seems at first sight to be independent of any assumption about their chemical composition ; factors we have called e disappear. We have, however, implicitly assumed that something is known about the mean ionization potential of the material. F or example in analysing

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460

E. A. MILNE ON STRUCTURE AND OPACITY OF A STELLAR ATMOSPHERE.

the Zn maximum we have assumed th at in solar type stars the material may be assumed to be on the whole just once ionized ; alternatively, we have assumed th a t the mean ionization potential of the material is of the order of the comparatively low first-stage ionization potential of the metals. The results would require re-calculation should it be found that there is, in solar stars, a considerable proportion of neutral atoms, atoms of hydrogen.* The average ionization-potential would then exceed th at of the metals. Neither of these classes of imperfections appear, however, to destroy the general principles of the method. Accumulating measures of line-contours will ultimately permit our provisional assumptions to be replaced by more accurate ones. The details of the application of the method are flexible ; the method itself appears to the author to remain valid. The advantage of the method of maxima is that it avoids any appeal to a knowledge of the exact formula for the atomic scattering coefficient. In fact if the chemical composition factors e could be assumed known, the atomic scattering coefficients could themselves be determined from observed line-contours. Probably a complete knowledge of all line-contours in a sufficient number of stars of known T and g would provide more equations than unknowns—the unknowns being the c’s and the atomic sjs. After that the method could be applied to determine for stars of known T but unknown g. An important need at the moment is a mathematical treatm ent of the effect of selective radiation-pressure in the transition region the reversing layer. This would in effect enable us to calculate effective gravity in terms of actual gravity and the atomic scattering coefficients. The various unknowns will be entangled with one another in increased complication. But this simply lends an increased attractiveness to the whole subject. 18. Summary.—(1) Formulae are presented for calculating the net outward flux, at any depth, in material of absorption coefficient kvand scattering coefficient s,, arbitrary functions of frequency v in radiative equilibrium. (2 ) The temperature distribution is of the type T4 = JT\ 4 (1 + f t ), where Tx is the effective temperature, provided rthe optical thickness is calculated mean ” of the sum of the scattering and absorption coefficients. The boundary tempera­ ture may be affected by the behaviour of K vj s vas a function (3) The results apply generally to the formation of absorption lines by monochromatic scattering as modified by collisions, in the presence of a continuous spectrum formed in any way by true absorption. It is shown th at an intensity-ratio at a point of an absorption line contour corresponds to simple S c h u s t e r scattering by an otherwise transparent medium if the photospheric radiating surface ” is placed at an optical depth t = fr/(r -j- 1). Study of the atoms producing an absorption line of given r can be confined to the uppermost t of optical thickness. (4) If a column of equivalent “ fully-viewed ” N atoms produce an intensity-ratio r, the line is widest for given r when N is a maximum for given t . * I owe this rem ark in principle to discussions with Mr. W. H. McCrea, of Trinity College, Cambridge.

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E. A. MILNE ON STRUCTURE AND OPACITY OF A STELLAR ATMOSPHERE.

461

(5) I lie number of atoms Js may be calculated as a function of g, T and P 0, where g is the surface giavity, I the tem perature and P 0 the pressure of free electrons at depth t . P 0 can only be calculated when the absolute value of the absorption coefficient of photospheric material is known. (6) An observed tem perature I max. of maximum width of an absorption line for given ) determines the electron-pressure P 0 at the base of the column. Consideration of mechanical equilibrium determines the amount of m atter in the column of given t , and hence the absolute value of the absorption coefficient is found. (7) I he absorption coefficient for photospheric material, determined from the observed maximum of Zn lines in G-type stars, is found to be given by the formula 0-85 P (T / 104)9/2 where P is the electron pressure in dynes cm.-2. This is a revised value, and the two determinations from the Sun and Capella, corresponding to giant and dwarf conditions, are in good agreement. (8) Kevised calculations from the observed maximum of B alm er lines in A-type stars according to the latest determinations of Miss P a y n e and Miss W illia m s are in fair agreement with the above value of the opacity. Observed variations of Tniax. with rare accounted for. (9) At depth t = A, the photospheric absorption coefficients in the Sun and Capella (bright component) are about 300 and 60 respectively. (10) The photospheric absorption coefficient derived from the Zn maximum is applied to calculate the behaviour of the lines of Ca and Ca+ in stars of low temperature. Com­ parison is made with the observations of P a y n e and H ogg reduced by U n s o l d ’s method. (11) The behaviour of the B alm er lines is calculated. It is shown that with the law kqcP for photospheric opacity there should be a strong £