795 lines
30 KiB
Plaintext
795 lines
30 KiB
Plaintext
194 6ApJ. . .104. .446C
|
||
|
||
THE CONTINUOUS SPECTRUM OF THE SUN AND THE STARS
|
||
|
||
S. Chandrasekhar and Guido Münch1 Yerkes Observatory
|
||
Received September 5, 1946
|
||
|
||
Using the recent determination of the continuous absorption coefficient of H~ by Chandrasekhar and'Breen, we have shown that the dependence of the continuous absorption coefficient with wave length in the range 4000-24,000 A, which can be inferred from the intensity distribution in the con-
|
||
tinuous spectrum of the sun, can be quantitatively accounted for as due to E~\ and, further, that the color temperatures measured in the wave-length intervals 4100-6500 A (Greenwich) and 4000-4600 A (Barbier and Chalonge) for stars of the main sequence and of spectral types AO-GO can also be interpreted
|
||
in terms of the continuous absorption of H~ and neutral hydrogen atoms. The problem of the discontinuities at the head of the Balmer and the Paschen series is also briefly
|
||
considered on the revised physical theory of the continuous absorption coefficient.
|
||
|
||
1. Introduction.—The two principal problems in the theory of the continuous spec-
|
||
|
||
trum of the stars are, first, to identify the source of the continuous absorption in the
|
||
|
||
solar atmosphere which will account for the intensity distribution in the continuous
|
||
|
||
spectrum of the sun and the law of darkening in the different wave lengths and, second,
|
||
|
||
to account for the observed relations between the color and the effective temperatures of
|
||
|
||
the stars. In this paper we shall show that the major aspects of these two problems find
|
||
|
||
their natural solution in terms of the continuous absorption coefficient of the negative hydrogen ion as recently determined by Chandrasekhar and Breen.2 More particularly,
|
||
|
||
we shall show that the dependence of the continuous absorption coefficient with wave
|
||
|
||
length in the range 4000-24,000 A, which can be deduced from the solar data, can be
|
||
|
||
quantitatively accounted for as due to H~) and, further, that the color temperatures measured in the wave-length intervals 4100-6500 A (Greenwich3) and 4000-4600 A (Barbier and Chalonge4) for stars of the main sequence and of spectral types A0-G0
|
||
|
||
can also be interpreted in terms of the continuous absorption of H~ and neutral hydro-
|
||
|
||
genIn addition to the two problems we have mentioned, we shall also consider some
|
||
|
||
related questions concerning the discontinuities at the head of the Balmer and the
|
||
|
||
Paschen series.
|
||
|
||
2. The mean absorption coefficients of H“ and H.—As is well known, the character of
|
||
|
||
the emergent continuous radiation from a stellar atmosphere is determined in terms of the temperature distribution in the atmosphere ; and, as has recently been shown,5 the
|
||
|
||
temperature distribution in a nongray atmosphere will be given approximately by a
|
||
|
||
formula of the standard type
|
||
|
||
^ = f^(r+g[r]),
|
||
|
||
(1)
|
||
|
||
where Te denotes the effective temperature and q(r) a certain monotonie increasing function of the optical depth r, provided that the mean absorption coefficient k, in terms of which r is measured, is defined as a straight average of the monochromatic absorptior coefficient weighted according to the net flux of radiation of frequency v in a gra)
|
||
1 Fellow of the John Simon Guggenheim Memorial Foundation at the Yerkes Observatory. 2 4^.104, 430, 1946. 3 Sir Frank Dyson, Observations of Color-Temperatures of Stars, 1926-1932, London, 1932; also M.N 100, 189, 1940.
|
||
* Ann. d’ap., 4, 30, Table 4, 1941. 6 S. Chandrasekhar, Ap. J., 101, 328, 1945. This paper will be referred to as “Radiative Equilibriu VII.,,
|
||
446
|
||
|
||
© American Astronomical Society • Provided by the NASA Astrophysics Data System
|
||
|
||
194 6ApJ. . .104. .446C
|
||
|
||
CONTINUOUS SPECTRUM
|
||
|
||
447
|
||
|
||
atmosphere and if, further, k„/k is independent of depth. On this approximation, then,
|
||
the emergent intensity in a given frequency and in a given direction will depend not
|
||
only on the continuous absorption coefficient at the frequency under consideration but also on the mean absorption coefficient k over all frequencies.
|
||
As the discussion in this paper will establish for the stellar atmospheres considered, the contributions to k in the visible and the infrared regions of the spectrum are essential-
|
||
ly from only two sources: H~ and the neutral hydrogen atoms. The cross-sections for the absorption by H~ for various temperatures and wave lengths have been tabulated by
|
||
Chandrasekhar and Breen in Table 7 of their paper, while those for hydrogen can be found from the formulae of Kramers and Gaunt, standardized, for example, by B. Strömgren.6 However, the evaluation of the mean absorption coefficient for wave lengths shorter than 4000 A is made uncertain on two accounts: First, there is the absorption by the metals and the excessive crowding of the absorption lines toward the violet, which is
|
||
particularly serious for spectral types later than F0; and, second, theje is the absorption
|
||
in the Lyman continuum. On both these accounts the true values of k will be larger than those determined by ignoring them. But the exact amount by which they will be larger will be difficult to predict without a detailed theory of “blanketing,”7 on the one hand, and without going into a more exact theory8 of radiative transfer than represented by the approximations leading to equation (1), on the other. However, since in this paper our primary object is to establish only the adequacy of H~ as the source of absorption in the solar atmosphere over the entire visible and infrared regions of the spectrum and the corresponding role of H~ and H for stellar atmospheres with spectral types A2-G0, it
|
||
appeared best to ignore the refinements indicated and simply determine ic by weighting kv due to H~ and H (without the Lyman absorption) at the conditions prevailing at r = 0.6 by the flux at this level.9 For only in this way can we use the solution to the
|
||
transfer problem in the form of equation (1) in a consistent manner. It should, however, be remembered that the effects we have ignored may easily increase k, determined in
|
||
terms of H~ and H (without the absorption in the Lyman continuum) by factors of the order of 1.5 and probably not exceeding 2.10
|
||
Turning our attention, next, to the evaluation of k, we may first observe that, since our present method of averaging is a straight one, the contributions to kv from different sources are simply additive. We may, accordingly, consider the mean absorption co-
|
||
efficient of H~ and H separately. Now the absorption coefficient of H~, including both the bound-free and the free-free
|
||
transitions, is most conveniently expressed as per neutral hydrogen atom and per unit electron pressure in the unit cm4/dyne. The monochromatic coefficients /c', after allowing for the stimulated emission factor (1 — e~hv/kT), are tabulated in Chandrasekhar and
|
||
Breen’s paper for various values of 0(=5040/r). If we now denote by a{H~) the average
|
||
6 “Tables of Model Stellar Atmospheres,” Publ. mind. Meddel. Kobenhavns Obs., No. 138, 1944.
|
||
7 Cf. G. Münch, Ap. J., 104, 87, 1946.
|
||
8 Such as, e.g., the (2, 2) approximation given in “Radiative Equilibrium VII,” § 6.
|
||
9 The choice of r = 0.6 for the “representative point” was made after some preliminary trials (cf. G. Münch, Ap. /., 102, 385, 1945, esp. Table 3), though it is evident on general grounds that a level such as r = 0.6, where the local temperature is approximately the same as the effective temperature, would be the correct one in the scheme of approximations leading to eq. (1).
|
||
10 In all earlier evaluations of k the absorption in the Lyman continuum did not, indeed, play any role. This was due to the manner in which k was defined in those investigations as the Rosseland mean. But in “Radiative Equilibrium VII” it has been shown that there is no justification for taking the Rosseland means as they have been hitherto. Since the method of averaging, by which we have now replaced the Roskeland mean, is a straight one, it is no longer permissible simply to ignore the absorption in the Lyman continuum. At the same time, it is not possible to take it into account satisfactorily in the (2, 1) approximation leading to eq. (1). We should have to go at least to the (2, 2) approximation of “Radiative Equilibrium VII.”
|
||
|
||
© American Astronomical Society • Provided by the NASA Astrophysics Data System
|
||
|
||
194 6ApJ. . .104. .446C
|
||
|
||
448
|
||
|
||
S. CHANDRASEKHAR AND GUIDO MÜNCH
|
||
|
||
value of the coefficients k'v, weighted according to the flux in a gray atmosphere at T = 0.6, where the temperature is approximately the effective temperature Te, then the contribution k(H~) to the mass absorption coefficient k by H~ is given by
|
||
|
||
« (ff-) = (1 ~ *g) ^ a (S-) ,
|
||
|
||
(2)
|
||
|
||
where me is the mass of the hydrogen atom, pe the electron pressure, and xh the degree of ionization of hydrogen under the physical conditions represented by Te and pe.n The values of a(H~) found by graphical integration in accordance with the formula
|
||
|
||
a{H~) =4 F¿FW ifi.6) dv
|
||
|
||
(3)
|
||
|
||
for various values of 0 = are given in Table 1.
|
||
TABLE 1 The Mean Absorption Coefficients a(£r_) and a{E)
|
||
|
||
a(R-)
|
||
|
||
a{R-)
|
||
|
||
am
|
||
|
||
0.5
|
||
|
||
0.563 X 10~26 1.65X10“22 0.9.
|
||
|
||
6.08X10-26 5.52X10"27
|
||
|
||
0.6
|
||
0.7 0.8
|
||
|
||
1.145 X 10~26 1.22X10"23 1.0.
|
||
|
||
2.25 X10"26 3.88 XIO"26
|
||
|
||
1.13 X10-24 8.00X10-26
|
||
|
||
1.2. 1.4.
|
||
|
||
9.32X10“26 2.00X10-26 3.89X10-25
|
||
|
||
3.65 X 10~28
|
||
|
||
Similarly, the contribution to k by hydrogen can also be expressed in the form
|
||
|
||
«(H) =1—, Mh
|
||
|
||
(4)
|
||
|
||
where
|
||
|
||
a{E)
|
||
|
||
=
|
||
|
||
fœ Jq
|
||
|
||
1aR3
|
||
|
||
(1-e-°)
|
||
|
||
,(i) (0.6)
|
||
|
||
da
|
||
|
||
(5)
|
||
|
||
where a = hv/kTe and/and D are certain functions of temperature and frequency, respectively, which have been tabulated by Strömgren.12 For reasons which we have already explained, we do not include the Lyman absorption in evaluating a(H). The values of a(H) for various temperatures are also listed in Table 1.
|
||
In terms of a(H~) and a{H) given in Table 1, we can determine the combined mass absorption coefficient k according to
|
||
|
||
= Mr
|
||
|
||
pe + a(H)] .
|
||
|
||
(6)
|
||
|
||
Values of
|
||
|
||
K(H)y and k determined in accordance with the foregoing equations
|
||
|
||
for various temperatures and electron pressures are given in Table 2.
|
||
|
||
3. The continuous absorption iñ the solar atmosphere—As we have already stated in
|
||
|
||
the introduction, one of the principal problems in the interpretation of the solar spectrum
|
||
|
||
is the identification of the source of absorption which will predict the same dependence
|
||
|
||
11 It will be noted that, in writing the mass absorption coefficient in the form (2), we have assumed the preponderant abundance of hydrogen in the stellar atmosphere.
|
||
J2 See the reference quoted in n. 6.
|
||
|
||
© American Astronomical Society • Provided by the NASA Astrophysics Data System
|
||
|
||
194 6ApJ. . .104. .446C
|
||
|
||
CONTINUOUS SPECTRUM
|
||
|
||
449
|
||
|
||
of the absorption coefficient with wave length in the range 4000-24,000 A, which can be inferred from the intensity distribution in the continuous spectrum and the law of darkening in the different wave lengths. While the amount and variation of the continuous absorption in the spectral region XX 4000-24,000 A can be deduced in a variety of ways,13 it appears that, for the purposes of the identification of the physical source of absorption, it is most direct to adopt the following procedure:
|
||
TABLE 2
|
||
The Mean Mass Absorption Coefficients *(#“), k(H), and k for Various Temperatures and Electron Pressures
|
||
|
||
0« = O.5. ee=o.6. 6e=0.7. 0e=O.8. 0C=O.9. 0e=l.O. 0e=1.2. 0e=1.4.
|
||
|
||
k(H) K
|
||
k(E~) k{H) K
|
||
HE-) HE) K
|
||
HE-) HE) K
|
||
HE-) HE) K
|
||
HE-) HB) K
|
||
k{H~)
|
||
k{H~)
|
||
|
||
Pe = l
|
||
5.85X10-6 1.71X10-1 1.71X10-1
|
||
3.93X10"4 4.18X10-1 4.18X10-1
|
||
9.O8XIO-3 4.54X10-1 4.63X10"1
|
||
2.26X10"2 4.66X10-2 6.92X10-2
|
||
3.63X10-2 0.33X10-2 3.96X10-2
|
||
5.57X10-2 0.02X10-2 5.59X10-2
|
||
1.20X10"1
|
||
2.33X10-1
|
||
|
||
^ = 10
|
||
3.70X10"4 1.08 1.08
|
||
2.60X10-2 2.76 2.79
|
||
1.28X10"1 6.41X10“1 7.69X10-1
|
||
2.32X10-1 0.48X10-1 2.80X10-1
|
||
3.63X10-1 0.03X10“1 3.66X10-1
|
||
5.57X10"1 ■S.’^XIO-1
|
||
1.20
|
||
2.33
|
||
|
||
Pe = 102
|
||
4.98X10-2 14.5 14.6
|
||
5.82X10-1 6.19 6.77
|
||
1.34 0.67 2.01
|
||
2.32 0.05 2.37
|
||
3.63
|
||
3.63
|
||
5.57
|
||
5.57
|
||
12.0
|
||
23.3
|
||
|
||
Pe = 103
|
||
2.12 61.8 63.9
|
||
6.68 7.10 13.8
|
||
13.4 0.7 14.1
|
||
23.2
|
||
23.2
|
||
36.3
|
||
36.3
|
||
55.7
|
||
55.7
|
||
120
|
||
233
|
||
|
||
Pe = 104
|
||
32.1 93.5 126
|
||
68.4 7.27 75.7
|
||
134 1
|
||
135
|
||
232
|
||
232
|
||
363
|
||
363
|
||
. 557
|
||
557
|
||
1200
|
||
2330
|
||
|
||
We compare the observed, intensity distribution in the emergent solar flux F\ (obs.)
|
||
|
||
with the flux
|
||
|
||
(0) to be expected in a gray atmosphere.14 It is evident that the de-
|
||
|
||
partures,
|
||
|
||
A log Fx = log Fx (obs.) - log F\] (0) ,
|
||
|
||
(7)
|
||
|
||
must be related more or less directly with the dependence of the continuous absorption coefficient kv with wave length. Indeed, in the approximations leading to the temperature distribution (1) this relation must be one-one, since, with the adopted definition of /c, the temperature distributions in the gray and the nongray atmospheres agree.16 This suggests that, with the known value of kv due to H~, we compare the predicted de-
|
||
13 G. Miilders, Zs.f. Ap., 11,132, 1935; G. Münch, Ap. 102, 385, 1945; D. Chalonge and V. Kourganoff, Ann. d’ap. (in press).
|
||
14 The values of (0) can be readily derived from the entries along the line r = 0 in Table 2 of “Radiative Equilibrium VII.,,
|
||
16 Cf. the remarks in italics on p. 343 in “Radiative Equilibrium VII.,,
|
||
|
||
© American Astronomical Society • Provided by the NASA Astrophysics Data System
|
||
|
||
194 6ApJ. . .104. .446C
|
||
|
||
450
|
||
|
||
S. CHANDRASEKHAR AND GUIDO MÜNCH
|
||
|
||
partures from Fi1^ (0) with those observed. The only uncertaintydn these predictions will be of the nature of a “zero-point” correction, since a value of k different from the one adopted will lead to an approximately constant additive correction to A log F\.lñ
|
||
In order, then, to make the comparison suggested in the preceding paragraph, we need to determine A log F\ in terms ofjcx due to H~ and an adopted k. Assuming in the first instance that the contribution to k is only we find that
|
||
|
||
a (H~) = 5.62 X 10“26 cm4/dyne
|
||
|
||
(8)
|
||
|
||
for an adopted value of
|
||
|
||
0e = 0.8822 .
|
||
|
||
(9)
|
||
|
||
The value of for d = 0.8822 can be found by simple interpolation in Table 7 of Chandrasekhar and Breen’s paper. The ratios K\/a{E~) derived in this manner are
|
||
|
||
TABLE 3
|
||
The Predicted Departures [log F\ (obs.)-LOG /^(O)] from Grayness of the Solar Atmosphere Due to the Absorption by H~
|
||
|
||
xa
|
||
4000. 4500. 5000. 6000. 7000. 8000. 9000. 10,000. 11,000. 12,000. 13.000. 14.000. 15.000. 16.000. 17.000. 18.000. 19.000. 20.000. 21,000. 22,000. 23,000.
|
||
|
||
k{H~)
|
||
0.686 0.783 0.881 1.029 1.132 1.188 1.183 1.125 1.028 0.911 0.788 0.651 0.523 0.481 0.486 0.516 0.562 0.618 0.679 0.749 0.820
|
||
|
||
\A2k(H~)
|
||
0.483 .551 .620 .725 .797 .837 .833 .792 .724 .642 .555 .458 .368 .339 .342 .363 .396 .435 .478 .527
|
||
0.577
|
||
|
||
LOG F\ (THEO.)
|
||
|
||
LOG F^(0) =«(#-) K = \A2k{H~)
|
||
|
||
14.489 14.468 14.431 14.341 14.259 14.164 14.070 13.982
|
||
13.897 13.817 13.742 13.674 13.608 13.532
|
||
13.448 13.362 13.279 13.197 13.118 13.042
|
||
12.971
|
||
|
||
14.532
|
||
14.534 14.423 14.329 14.224 14.128 14.034 13.946 13.862 13.784 13.713 13.643 13.566 13.480 13.395 13.313 13.230 13.149 13.073 13.003
|
||
|
||
14.358
|
||
14.390 14.399
|
||
14.350 14.285
|
||
14.194 14.098
|
||
13.999 13.901
|
||
13.805 13.712
|
||
13.624 13.537
|
||
13.454 13.374
|
||
13.297 13.223 13.152
|
||
13.083 13.016 13.952
|
||
|
||
A LOG F\
|
||
|
||
k=k(H~)
|
||
+0.131 + .078 + .032 - .009 - .026 - .030 - .028 - .017 - .004 + .012 + .030 + .050 + .071 + .078 + .074 + .065 + .056 + .045 + .035 + .026 +0.019
|
||
|
||
1A2k(H-)
|
||
+0.174
|
||
+ .135 + .073 + .044 + .030 + .030 + .035 + .045 + .057 + .072 + .089 + .106 + .112 + .106 + .098 + .090 + .078 + .066 + .057 +0.051
|
||
|
||
given in Table 3 for various values of X. With these values of
|
||
|
||
the theoretical
|
||
|
||
determination of A log F\ is straightforward with the help of the nomogram of Burkhardt’s table,17 which one of us has recently published.18 The results of the determina-
|
||
|
||
tion are given in Table 3. In Figure 1 we have further compared the computed departures A log F\ with those observed.19 It is seen that the predicted variation of the
|
||
|
||
16 This is seen most directly in an approximation in which we expand the source function B\(T) as a
|
||
|
||
Taylor series about a suitable point and determine the emergent flux in terms of it (see, e.g., A. Unsold, Physik der Sternatmosphären, p. 109. eq. [31.18], Berlin, 1938).
|
||
|
||
17 Zs.f.Ap., 13, 56, 1936.
|
||
|
||
18 G. Münch, Ap. /., 102, 385, Fig. 2, 1945.
|
||
|
||
19 For X > 9000 A the observed departures were obtained from a reduction of the solar data by
|
||
|
||
M. Minnaert, B.A.N., 2, No. 51, 75, 1924; see also Unsold, op. cit., p. 32. For X < 9000 A the reduction of G. Mülders (dissertation, Utrecht, 1934) was used.
|
||
|
||
© American Astronomical Society • Provided by the NASA Astrophysics Data System
|
||
|
||
194 6ApJ. . .104. .446C
|
||
|
||
CONTINUOUS SPECTRUM
|
||
|
||
451
|
||
|
||
departures runs remarkably parallel with the observed departures over the wave-
|
||
|
||
length range 4000-20,000 A. (The observational data do not seem specially reliable for
|
||
|
||
X > 20,000 A.) However, the absolute values of the predicted departures are system-
|
||
|
||
atically less than the predicted departures by approximately a constant amount, indicat-
|
||
|
||
ing a zero-point correction in the sense that the adopted value of k as due to alone
|
||
|
||
is somewhat too_small. The calculations were accordingly repeated for other slightly
|
||
|
||
larger values of k, and it was found that with k = 1.42
|
||
|
||
the predicted and the
|
||
|
||
observed departures agree entirely within the limits of the observational uncertainties
|
||
|
||
over the whole region of the spectrum in which H~ contributes to the absorption. The
|
||
|
||
Fig. 1.—Comparison of the observed and the theoretically predicted departures [logF\ — log Fj (0)]
|
||
from a gray atmosphere due to the absorption by H~. The circles represent the observed departures of the solar emergent flux from that of a gray atmosphere, while Curves I and II are the theoretically derived departures on the two assumptions k = k(H~) and H = 1.42
|
||
|
||
agreement is, in fact, so striking that we may say that H~ reveals its presence in the solar
|
||
|
||
atmosphere by its absorption spectrum.
|
||
|
||
We may finally remark on the value of k = 1.42
|
||
|
||
indicated by the comparisons
|
||
|
||
we have just made: it can, in fact, be deduced empirically from the solar data on the
|
||
|
||
continuous spectrum that the absorption in the violet (X < 4000 A) must increase the
|
||
|
||
value of k derived from the visible and the infrared regions of the spectrum by a factor of the order of 1.5.20
|
||
|
||
4. The predicted color-effective temperature relations: comparison with observations.— All earlier attempts21 to predict the color temperatures in the region XX 4000-6500 A
|
||
|
||
for stars of spectral types A0-G0 in agreement with the observations have failed. This
|
||
|
||
failure in the past has been due to the following circumstance: The observed relation
|
||
|
||
between the color and the effective temperatures and, in particular, the fact that Tc > Te
|
||
|
||
20 Cf. G. Münch, Ap. /., 102,385, 1945, esp. the remarks preceding eq. (19) on p. 394.
|
||
|
||
21 R. Wildt, Ap. /., 93,47,1941, and Observatory, 64,195, 1942; R. E. Williamson, Ap. /., 97, 51,1943.
|
||
|
||
© American Astronomical Society • Provided by the NASA Astrophysics Data System
|
||
|
||
452
|
||
|
||
S. CHANDRASEKHAR AND GUIDO MÜNCH
|
||
|
||
implies that the continuous absorption coefficient is an increasing function of X in the
|
||
spectral region observed. But the physical theory on which the calculations were made
|
||
placed the maximum of the absorption-curve in the region of X 4500 A; this was incompatible with the observations and, moreover, predicted color temperatures less than the effective temperatures, contrary to all evidence. Indeed, on the strength of this discrepancy, it was concluded that H~ as a source of absorption was inadequate even in the region XX 4500-6500 A, and the existence of an unknown source operative in this region was further inferred. However, later evaluations22 of the bound-free transitions of H~ showed the unreliability of earher determinations and placed the maximum of the absorption-curve in the neighborhood of X 8500 A. The addition of the free-free transitions pushes this maximum only still further to the red. It is therefore evident that on the revised physical theory we should be able to remove the major discrepancies of the subject. We shall now show how complete the resolution of these past difficulties is.
|
||
From the point of view of establishing the adequacy of the physical theory in the
|
||
region XX 4000-6500 A, it is most instructive to consider the theoretical predictions for color temperatures which can be directly compared with the color determinations at Greenwich,3 for the Greenwich measures are based on the mean gradients in the wavelength interval 4100-6500 A, and it is in the prediction of these colors that the earlier calculations were most discordant.23
|
||
Now, from the Planck formula in the form
|
||
|
||
it readily follows that 1 di\
|
||
|
||
2hc2
|
||
|
||
1
|
||
|
||
X6 ec2A2:—1’
|
||
|
||
5X--p (1 - e-^T) -l.
|
||
|
||
(10) (ID
|
||
|
||
Defining the gradient
|
||
|
||
« = ^(1-
|
||
|
||
-i
|
||
|
||
(12)
|
||
|
||
in the usual manner, we can write
|
||
|
||
1 d logic i\
|
||
|
||
5X — $
|
||
|
||
(13)
|
||
|
||
M d{\/X)
|
||
|
||
If Fx, and Fx2 are the emergent fluxes at two wave lengths Xi and X2 and if <t> is the mean gradient in this wave-length interval, then we can write, in accordance with equation (13)
|
||
|
||
<£= 5X m
|
||
|
||
1 logic (F\JF\2) M X-i-X-1 ’
|
||
|
||
(14)
|
||
|
||
where \m denotes an appropriate mean wave length for the interval to which the gradient (^.refers. Equation (14) can be re-written in the following form:
|
||
|
||
1 A logio F
|
||
|
||
<j} = 5\ m ïfAG/X) *
|
||
|
||
(15)
|
||
|
||
According to equation (15), the theoretical determination of color temperatures will proceed by determining, first, the gradient 0 from the values of F\ at the end-points of the wave-length interval and then determining the temperature which will give this gradient.
|
||
|
||
22 S. Chandrasekhar, Ap. 102,223, 395,1945 23jCf. Fig. 6 in Williamson^ paper (op. cit.).
|
||
|
||
© American Astronomical Society • Provided by the NASA Astrophysics Data System
|
||
|
||
194 6ApJ. . .104. .446C
|
||
|
||
CONTINUOUS SPECTRUM
|
||
|
||
453
|
||
|
||
For the Greenwich measures \m = 0.55 ju, and equation (15) becomes
|
||
|
||
</> (Greenwich) = 2.75 — 2.56 logic
|
||
|
||
,
|
||
|
||
(16)
|
||
|
||
provided that, in determining the gradients, wave lengths are measured in microns.24 In Table 4 we have listed the values of kÍ/ k for the wave lengths 4100 A and 6500 A
|
||
for various values of de and pe- In terms of these values the determination of the fluxes at the two wave lengths 4100 A and 6500 A is straightforward with the help of BurkhardUs table. The gradient $ then follows according to equation (16) and, from that, the color temperature. The reciprocal color temperatures dc = 5040/rc derived in this manner are given in Table 5. The resulting color-effective temperature relations are illustrated
|
||
|
||
TABLE 4 kx/k in Model Stellar Atmospheres
|
||
|
||
5040/:r, 6 = 0.5. =0.6. '=0.7, 0 = 0.8. 0=0.9.
|
||
|
||
XA
|
||
X 3647 X 4000 X 4600 X 6500 X 8203
|
||
X 3647 X 4000 X 4600 X 6500 X 8203
|
||
X 3647 X 4000 X 4600 X 6500 X 8203
|
||
X 3647 X 4000 X 4600 X 6500 X 8203
|
||
X 3647 X 4000 X 4600 X 6500 X 8203
|
||
|
||
0.210 0.316 0.826
|
||
0.148 0.221 0.579
|
||
Í6.06 \0.176 0.208 0.276 0.546 /0.889 10.356
|
||
IT. 54 10.518 0.582 0.681 0.961 fl.ll 11.01
|
||
0.708 0.631 0.690 0.800 1.09 ¡1.16 \
|
||
|
||
Pe = 102
|
||
3.45 0.161 0.212 0.318 0.834 1.51 0.490
|
||
4.04 0.178 0.205 0.278 0.637 1.27 0.424
|
||
2.82 0.461 0.521 0.617 0.901 1.105 0.893
|
||
0.720 0.601 0.671 0.781 1.06 1.178\ 1.164/
|
||
0.642 0.636 0.690 0.800 1.09 1.16
|
||
|
||
P. = 10Z
|
||
3.22 0.181 0.238 0.346 0.848 1.51 0.526
|
||
2.60 0.427 0.481 0.585 0.935 1.36 0.885
|
||
0.961 0.625 0.695 0.810 1.11 1.22 1.19
|
||
0.625 0.614 0.685 0.800 1.09
|
||
1.19
|
||
0.636
|
||
0.690 0.800 1.09 1.16
|
||
|
||
¿« = 104
|
||
2.68 0.332 0.397 0.513 0.991 1.55 0.788
|
||
1.09 0.685 0.770 0.901 1.25 1.45 1.36
|
||
0.684 0.645 0.725 0.848 1.14
|
||
1.24
|
||
0.611 0.685 0.800 1.09
|
||
1.19
|
||
0.636
|
||
0.690 0.800 1.09 1.16
|
||
|
||
24 The constant c-¿ in eq. (12) then has the value 14,320.
|
||
|
||
© American Astronomical Society • Provided by the NASA Astrophysics Data System
|
||
|
||
194 6ApJ. . .104. .446C
|
||
|
||
454
|
||
|
||
S. CHANDRASEKHAR AND GUIDO MÜNCH
|
||
|
||
in Figure 2. For comparison we have also plotted in this figure the Greenwich determinations for stars on the main sequence and of spectral types A0-G0 (reduced, however, to the Morgan, Keenan, and Kellman system of spectral classification). The color temperature of the sun for this wave-length interval is also plotted in Figure 1. It is seen from Figure 1 that the agreement between the observed and the theoretical color temperatures is entirely satisfactory, particularly when it is remembered that the earlier calculations failed even to predict the correct sign for 0C — de. It will, however, be noted that the observed values of dc for spectral types later than F0 are somewhat larger than the pre-
|
||
|
||
TABLE 5
|
||
Theoretical Reciprocal Color Temperatures and the Predicted Discontinuities at the Head of the Balmer and the Paschen Series*
|
||
|
||
Pe
|
||
|
||
0.5
|
||
|
||
0.6
|
||
|
||
0.7
|
||
|
||
0.8
|
||
|
||
0.9
|
||
|
||
1.0
|
||
|
||
(ec(G) 10 í 6c(B and C)
|
||
\Db
|
||
|
||
0.68
|
||
|
||
0.80
|
||
|
||
0.88
|
||
|
||
.61
|
||
|
||
69
|
||
|
||
75
|
||
|
||
.31
|
||
|
||
015
|
||
|
||
%(G) ^MCBandC)
|
||
i Db Di
|
||
|
||
45
|
||
|
||
0.60
|
||
|
||
.73
|
||
|
||
80
|
||
|
||
.47
|
||
|
||
.63
|
||
|
||
69
|
||
|
||
[ -50]
|
||
|
||
.07
|
||
|
||
.030
|
||
|
||
.001
|
||
|
||
rec(G)
|
||
|
||
[0 31]
|
||
|
||
52
|
||
|
||
.65
|
||
|
||
.74
|
||
|
||
80
|
||
|
||
103J ÜB and C)
|
||
|
||
58]
|
||
|
||
42
|
||
|
||
.56
|
||
|
||
34]
|
||
|
||
.13
|
||
|
||
.64
|
||
|
||
69
|
||
|
||
114
|
||
|
||
051
|
||
|
||
.003
|
||
|
||
%(G)
|
||
|
||
39]
|
||
|
||
59
|
||
|
||
.66
|
||
|
||
.74
|
||
|
||
80
|
||
|
||
lov dAB and ^ • •
|
||
|
||
34]
|
||
|
||
40]
|
||
|
||
50
|
||
|
||
.58
|
||
|
||
.64
|
||
|
||
12
|
||
|
||
.018
|
||
|
||
69
|
||
|
||
071]
|
||
|
||
006
|
||
|
||
Purei0c(G)
|
||
|
||
52
|
||
|
||
H-\ec(B and C)
|
||
|
||
41
|
||
|
||
59
|
||
|
||
.66
|
||
|
||
.74
|
||
|
||
50
|
||
|
||
0.58
|
||
|
||
0.64
|
||
|
||
alinmdit*4s,0d00a(0rGe-)4ta6hn0ed0edAxcp(, BerceatsenpddecBCti)avlamerleyer;thaDenBdreacPniapdsrcoDhcepanl,
|
||
|
||
color temperatures, 5040/Tc, appropriate for the wave-length rdeipscreosnetnintiunigtietsh,erelsopgeacrtiitvhemlyo. f the ratio of the fluxes at the
|
||
|
||
intervals two sides
|
||
|
||
4100-6500 A of the series
|
||
|
||
dieted values, though the agreement is as good as can be expected in the case of the sun. The reason for this must undoubtedly be the crowding of the absorption lines toward the violet in the later spectral types and the consequent depression of the continuous spectrum in this región.‘The correctness of this explanation is apparent when it is noted that in the case of the sun, in which allowance has been made for this effect of the lines
|
||
on the continuum, the discordance is not present. Comparisons similar to those we have just made also can be made with the measure-
|
||
ments of Barbier and Chalonge4 on the color temperatures based on the observed gradients in the wave-length interval 4000-4600 A. The formula giving the theoretical gradient for this wave-length interval takes the form
|
||
|
||
= 2.175- 7.06 log10(^).
|
||
|
||
(17)
|
||
|
||
© American Astronomical Society • Provided by the NASA Astrophysics Data System
|
||
|
||
194 6ApJ. . .104. .446C
|
||
|
||
CONTINUOUS SPECTRUM
|
||
|
||
455
|
||
|
||
The values of k\/k at X 4000 A and X 4600 A are given in Table 4, and the reciprocal color temperatures derived from these in Table 5. The results are further illustrated in Figure 3, where the theoretical relations for various electron pressures are compared with the measures of Barbier and Chalonge (reduced also to the Morgan, Keenan, and Kellman system of spectral classification). It is seen that the general agreement is again good, though there are now somewhat larger differences between the computed and the observed color temperatures for spectral types later than F0 than were encountered in the comparison with the Greenwich colors. This must again be due to the crowding of the absorption lines toward the violet in the later spectral types and the further fact that
|
||
|
||
Fig. 2.—Comparison of the observed and the theoretical color effective-temperature relations for the wave-length interval 4100-6500 A. The ordinates denote the reciprocal color temperatures and the abscissae denote the reciprocal effective temperatures (6 = 5040/T). The circles represent the Greenwich color determinations reduced to the Morgan, Keenan, and Kellman system of spectral classification. The double circle represents the sun.
|
||
the base line for the Barbier and Chalonge colors is much shorter than that for the Greenwich colors.
|
||
5. The discontinuities at the head of the Balmer and the Paschen series.—With the physical theory of the continuous absorption coefficient now available, we can also predict the extent of the discontinuities which we may expect at the head of the Balmer and the Paschen series of hydrogen. For this purpose the values of k^/k on the two sides of the series limits are also given in Table 4. From these values it is a simple matter to estimate the discontinuities which will exist at the head of the Balmer and the Paschen series, and they are given in Table 5. The results for the Balmer discontinuities are further illustrated in Figure 4, in which the discontinuities measured by Barbier ànd Chalonge4 for various stars are also plotted. The progressive increase of the electron pressure as we go from the later to the earlier spectral types is particularly apparent
|
||
© American Astronomical Society • Provided by the NASA Astrophysics Data System
|
||
|
||
194 6ApJ. . .104. .446C
|
||
|
||
Fig. 3.—Comparison of the observed and the theoretical color effective-temperature relations for the wave-length interval 4000-4600 A. The ordinates denote the reciprocal color temperatures, and the abscissae denote the reciprocal effective temperatures 6 = 5040/T). The circles represent the color determinations of Barbier and Chalonge for the wave-length interval 4000-4600 A, reduced to the
|
||
Morgan, Keenan, and Kellman system of spectral classification.
|
||
|
||
Sp
|
||
|
||
A0 AI A2 A3 AS A7 FO F2
|
||
|
||
F5
|
||
|
||
GO
|
||
|
||
Fig. 4.—The predicted discontinuities (D) at the head of the Bahner series for various effective temperatures and electron pressures. The circles represent the discontinuities as measured by Barbier and Chalonge. (The double circles represent the observations for supergiants.)
|
||
© American Astronomical Society • Provided by the NASA Astrophysics Data System
|
||
|
||
194 6ApJ. . .104. .446C
|
||
|
||
CONTINUOUS SPECTRUM
|
||
|
||
457
|
||
|
||
from Figure 4; this progression is, moreover, in agreement with what is indicated by the color determinations (cf. Figs. 2 and 3). In Table 6 we give the electron pressures for the main-sequence stars of various spectral types estimated in this manner.
|
||
6. Concltiding remarks.—While our discussion in the preceding sections has established the unique role which H~ plays in determining the character of the continuous spectrum of the sun and the stars, it should not be concluded that the various other
|
||
TABLE 6
|
||
Electron Pressures for Stars on the Main Sequence
|
||
|
||
Type
|
||
A1 A2 A3 A5 A7
|
||
|
||
log Pe
|
||
3.7 3.3 3.0 2.8 2.6
|
||
|
||
Type
|
||
F0. F2. F4. F5. F6.
|
||
|
||
log pe
|
||
2.4 2.2 2.0 1.8 1.6
|
||
|
||
Type
|
||
F8. GO. G2.
|
||
|
||
log Pe
|
||
1.4 1.2 1.0
|
||
|
||
astrophysical elements of the theory are equally well established. Indeed, the theory of model stellar atmospheres as developed by Strömgren in recent years must not only be revised on the basis of the new absorption coefficients of H~ but also be advanced still further before we can be said to have a completely satisfactory account of all the classical problems of the theory of stellar atmospheres. But our discussion in this paper does give us the confidence that the continuous absorption by H~ discovered by Wildt must provide the key to the solution of many of these problems.
|
||
|
||
© American Astronomical Society • Provided by the NASA Astrophysics Data System
|
||
|