- Email: [email protected]
Some aspects of the non-LTE physics of the helium atom in hot stars
J. Quant. Spectrosc. Radiat. Transfer. Vol, 1I, pp. 617~622. Pergamon Press 1971. Printed in Great Britain
SOME
ASPECTS
OF
THE ATOM
NON-LTE IN
HOT
PHYSICS
OF
THE
HELIUM
STARS
A. B. UNDERHILL* and A. G. HEARN Sonneborgh Observatory, Utrecht, and Observatoire de Nice Abstract--In the atmospheres o f the hot stars (types O and B), the populations o f the states of the helium atom are significantly different from the LTE populations. This not only gives a change in the line source function with a consequent change in the predicted line depths, but it also changes the scale of the optical depths in the lines, with a consequent change in the deduced abundance o f helium. Some exploratory numerical results are given to illustrate these effects, which could explain the observed variations of the strengths of the He i and He I1 lines in O and B type stars. There is an important coupling between the intensities of the He I lines in the visible region of the spectrum and radiation densities at wavelengths less than 584 A.
INTRODUCTION
THE EFFECTof departures from local thermodynamic equilibrium on absorption lines are frequently expressed in terms of the departures of the line source function from the Planck function, that is as a function of the difference of the ratio of the population of the upper and lower levels of the line from the ratio given by the Boltzmann relation. Such differences for an absorption line in the visible region of the spectrum generate a comparatively small (though not negligible) difference between the line source function and the Planck function. For He I lines, normally observed in stellar spectra, the ratio of the line source function to the Planck function lies in the range 0.5 ~-
SL/B~ < 1.
(1)
The change is relatively small because the energy separation of the two levels giving an absorption line in the observable region of the spectrum is between 1-5 and 3 V and the collisional coupling between the levels is quite strong at the electron densities and energies found in the atmospheres of hot stars. A change in the source function (1) will result in central absorptions for strong lines which are up to 10 or 15 per cent greater than those calculated for LTE. In the case of weak lines the difference in depth is only a few per cent. No difference in line shape due to the fact that St. ¢: B~ is detectable in the wings of the lines. The equivalent widths of strong lines are increased by a small amount over their LTE values, t6) These facts have been interpreted to mean that it is not necessary to use non-LTE physics of the helium atom when estimating the helium abundance from the equivalent * Now at NASA, G o d d a r d Space Flight Center, Greenbelt, Maryland. 617
618
A.B. UNDERHILLand A. G. HEARN
widths of the He I lines or from matching line wings, t4'7'8~ This conclusion is false because abundances are deduced from the relative optical-depth scales in the lines of different elements and the optical-depth scales depend on the actual populations of the lower levels of the observed lines. In the case of He I this means on the actual populations of the 2UP and 21P levels. The non-LTE calculations of JOHNSONand POLAND (6) and of HEARN (see SNIJDERSand UNDERHILL(9)) indicate that in the deeper layers of B type atmospheres, where the electron density is of the order of l0 TM but not much greater, the population of 23p can be about twice its LTE population. How this comes about has significant implications for the theory and interpretation of lines of O and B stars. We shall outline the principal factors in the problem because they illustrate important concepts which have received too little attention in the theory of stellar spectra. GROUND
STATE POPULATIONS
The departure of the population of the ground level of neutral helium from the population given by the Saha-Boltzmann relation at electron and ion densities typical of the atmospheres of O and B stars is expected to be larger than that of levels with n > 2 because the collisional coupling between the ground level and the once-ionized state of helium is much weaker than is the effective coupling between states with n > 2 and the once-ionized state. What the departure is depends sensitively upon the radiation field present at wavelengths energetic enough to cause radiative transitions from the ground state, that is at 2 < 584/~. The importance of departures in the case of the ground level of hydrogen and their effect on the temperature structure of the atmosphere of a hot star have been shown by FEAUTRIER,(3~ and more recently by AUER and MIHALAS(1) who have also investigated the effects on the profiles and strengths of some Balmer and Paschen lines of hydrogen and have shown them to be significant. In the case of the helium atom one must consider not only the population of the ground state of neutral helium but also the population of the ground level of He +. The population of the ground level of He + relative to that of the He + + ions will affect the calculated neutral helium absorption line profiles, even if the populations of the upper and lower levels of the absorption line are related by the Boltzmann equation, that is even if the line source function is equal to the Planck function, because the populations of the excited levels of neutral
helium are determined mainly by the population of He* ions and not by the population of the ground state of the helium atom. An increase in the He + ground-level population will increase nearly proportionately the populations of the lower levels of the observed lines. The result is a change in the relation between the optical-depth scale in the observed He I absorption lines and the other optical-depth scales in the atmosphere. An estimate of the change in the ground population of the He + ion can be obtained by the following reasoning. Feautrier's calculations show that in a star with an effective temperature of 25,000°K and a gravity of 10* cm sec-2, the population of the ground-level of hydrogen at the edge of the atmosphere is 10 times greater than the Saha-Boltzmann population and that departures from the Saha-Boltzmann population extend to a Rosseland mean optical depth of 10-2. The departures from the Saha-Boltzmann relation will be very much greater for the ground level of He + relative to the He + + ions because the radiative rates are proportional to Z 4 where Z is the nuclear charge for a hydrogen-like
N o n - L T E physics of the helium atom in hot stars
619
ion and the collisional rates (excitation or de-excitation) are proportional to Z -2. The simple theory of the transfer of a continuum in a two-level model atom in an isothermal atmosphere by DIETZ and House t2) shows that the departures from the Saha-Boltzmann population and the optical thickness of the region where departures from the Saha-Boltzmann population are significant both increase as e- 1/2, where e is the ratio of the collisional de-excitation and radiative de-excitation rates. Accordingly, for a hydrogen-like ion the factor e- 1/2 is proportional to Z 3, so that from the simple theory of Dietz and House one may expect the departures of the population of the ground level of He ÷ from the SahaBoltzmann population to be 8 times greater and the region where significant departures exist to be 8 times greater than for hydrogen. This theory does not include the coupling between the hydrogen and He + continua which is difficult to consider in any simple way.
NUMERICAL
ESTIMATES
O F N(He+)/N(He) A N D O F T H E He I L E V E L P O P U L A T I O N S IN B TYPE ATMOSPHERES
In a stellar atmosphere the level populations should be found by simultaneous solution of the equations of statistical equilibrium and of the equations of transfer for radiation in frequencies relevant to the radiative processes which occur in the atom or ion under consideration. A model atmosphere is adopted giving the run of electron temperature and density with geometrical depth. JOHNSON and P O L A N D (6) have written down the appropriate set of equations for He I and have given some results for three model atmospheres in the case of some restricted model helium atoms. The equations of statistical equilibrium state that the number of atoms entering any state per unit time by all possible processes is equal to the number leaving. Atoms can enter a state from states of lower energy by absorption of radiation, which requires knowing the radiation field in the appropriate frequency interval, and as a result of collisions with electrons. Atoms can enter a state from states of higher energy as a result of spontaneous emission and of stimulated emission, which requires knowing the radiation intensity in the appropriate energy range, and as a result of collisions with electrons. Atoms leave energy states by the same processes. Thus, in order to solve the equations of statistical equilibrium for a model atom consisting of a specified number of levels, one must know the electron density and the collision cross-sections for excitation and de-excitation as well as the radiative cross-sections for spontaneous and stimulated emission and for absorption. In addition one must know the radiation field in the frequency ranges concerned in the radiative transitions. In the case of the helium atom the required radiation field can be divided into two parts: (i) wavelengths shorter than 584A and (ii) wavelengths longer than about 2500A. The radiation density in the first part of the spectrum is needed only for transitions in and out of the ground level 11S of He I; the second part of the spectrum is required for transitions between excited levels, including transitions to the ionized state from excited levels. A first attempt at solving the full set of equations and finding a set of level populations for He I consistent with the non-LTE physics of the helium atom and with a model atmosphere for a B star was done by JOHNSON and POLAND.~6~ The numerical problem is difficult and their results are not entirely satisfactory nor do they cover as wide a range of conditions, as one would like, in order to explore the meaning of the He I spectrum in O and B stars.
620
A.B.
UNDERHILL a n d A. G . HEARN
A survey of the situation can be obtained by using a computing program developed by HEARN.~5~Here the equations of statistical equilibrium are solved for a model helium atom consisting of 41 levels; the atmosphere is postulated to be at one electron temperature and electron density and the radiation density is everywhere placed equal to WB,,(Trad), where By is the Planck function and W is a dilution factor. Since in a real stellar atmosphere the electron temperature and density are determined chiefly by what is happening with the hydrogen atom, it is adequate to consider Te and Ne to be assigned parameters and to use Hearn's program to explore the effect of changing the radiation field by varying W and/or Trad . The new non-LTE model atmospheres of AUER and MIHALAS t 1~ indicate that adopting a layer at one temperature and electron density is not a poor approximation for the outer layers of an O or B star. The departures from LTE values are not strongly dependent on electron density when 1012 _< N e <_ 101., which is a range of values typical for the atmospheres of O and B stars. In Table 1 we present results for the case N~ = 1014 and W = 0-5 in order to demonstrate what will occur in the deeper layers of the atmosphere of a main-sequence star. It is here that the line wings, the weak lines and the continuous spectrum are formed. The quantities b(11S), b(23p) and b(2~P) represent the ratio of the computed population of the indicated level to that which will be found if local thermodynamic equilibrium existed at the assigned electron temperature and electron density. These results confirm the statements made above. They can be summarized as follows: (1) In a normal stellar atmosphere the fraction of once-ionized helium differs greatly from what is expected in LTE at the assigned electron temperature, the fraction increasing with the radiation temperature, that is as the radiation density at wavelengths energetic enough to ionize helium increases with respect to the electron temperature.
TABLE 1. SOME RESULTS OF THE NON-LTE PHYSICS OF THE HELIUM ATOM, CASE W = 0"5,
Te
T,. d
N ( H e +)
L T E at Te
C K)
~ K)
N(He)
N ( H e + )/N(He)
10,000
10,000 15,000 20,000 30,000 I0,000 15,000 20,000 30,000 10,000 15,000 20,000 30,000 10,000 15,000 20,000 30,000 10,000 15,000 20,000 30,000
1.350-5 0-266+0 0.980 + 0 1.000 + 0 4-830 - 4 0-267+0 0.983 + 0 1.000+ 0 3-710 - 2 0-344 + 0 0.986 + 0 1.000 + 0 0-818 + 0 0.887 + 0 0.990+0 1.000 + 0 0-983 + 0 0.988 + 0 0-995 + 0 1.000 + 0
15,000
20,000
30,000
40,000
3.94 - 5
0-494 + 0
0.994 + 0
0.917 + 0
3.57 - 2
N e
= 10 ~4
b(l ~S)
b(2 3p)
b(2 ~P)
0-292+ 1 0.135-3 0.806 - 6 0.433 - 8 0,202 + 4 0.267+ 1 0-164 - 1 0.878 - 4 0.452 + 4 0-333 + 3 0.249 + 1 0.135 - 1 0.828 + 4 0.474+4 0.373+3 0-230 + 1 0-108 + 5 0.768 + 4 0.296 + 4 0-325 + 2
0.147+ 1 0.328+0 0.147 + 0 0.573 - 1 0.208 + 2 0-168+ 1 0-779 + 0 0.314+0 0-339 + 2 0.532 + 1 0.188 + 1 0.777 + 0 0.486 + 2 0.233 + 2 0-561 + 1 0.206 + 1 0.594 + 2 0.339 + 2 0-137 + 2 0.356+ 1
0-144+ 1 0.247+0 0-910 - 1 0-303 - 1 0.732 + 0 0.134+ 1 0-503 + 0 0-167+0 0.765 + 0 0-278 + 1 0.126 + 1 0.424 + 0 0-106 + 1 0.116+ 1 0.313+ 1 0.118 + 1 0.126 + 1 0.915 + 0 0-339 + 0 0.214+0
N o n - L T E physics of the helium a t o m in hot stars
621
(2) Within the temperature range investigated, the fraction of once-ionized helium does not decrease from unity. However in LTE an appreciable decrease due to the second ionization of helium has set in by the time Te is 40,000°K. (3) The population of the ground state of He I may greatly exceed or be greatly less than the LTE population depending upon the relationship of the density of ionizing radiation to the electron temperature. The population of 1IS seems to be very sensitive to the amount of ionizing radiation available. This implies that the strengths of the resonance lines of He I will show a similar variation with the intensity of far u.v. radiation. (4) The populations of 23p and of 21P are not very sensitive to the relation of the density of ionizing radiation to the electron temperature but the 23P level is consistently overpopulated with respect to the 2~P level. On the average for a modest difference between Te and Tra d the 23p level is overpopulated with respect to what would be expected in LTE at the electron temperature by a factor near 2. This means that if the optical-depth scale in lines from 23P is interpreted by LTE physics the helium abundance will be overestimated by a factor close to 2. (5) When the electron temperature is high and the radiation temperature is also high, the 23p level is overpopulated with respect to the 21P level by a large factor in comparison to what is expected in LTE. (6) When the electron temperature is low and the radiation temperature is moderately high both the 2aP and the 2~P level are underpopulated in comparison to what would be expected in LTE at the electron temperature.
ASTROPHYSICAL
SIGNIFICANCE OF THE NON-LTE HELIUM ATOM
PHYSICS
OF THE
The calculations discussed above demonstrate that the hypothesis of LTE is not valid with the densities and temperatures deduced to occur in the atmospheres of O and B stars. Were this hypothesis valid the b-factors given in Table 1 should be all close to unity. Although the computations are schematic particularly in the manner in which they handle the radiation field, the results obtained indicate the true meaning to be attached to the following observed phenomena in O and B type spectra: (1) the persistence of the lines of He II at great strength in O stars, (2) the relative great strength of lines of the triplet series of He I and the weakness of lines of the singlet series in atmospheres where the electron temperature is believed to be greater than 20,000°K, and (3) the weakness of lines of He I in some middle and late B type stars which seem to have the same continuous intensity distribution in the blue-violet spectral region, that is the existence of 'helium-weak' and 'helium-strong' B stars. The present calculations, particular for Te = 10,000°K, demonstrate how sensitive the populations of the 23p and 21p levels are to the density of radiation at wavelengths shorter than 584 A. Small differences in the stellar model structure might easily cause changes in the far u.v. radiation field which would affect the strength of the He I lines from 23p and 21P but not the distribution of the continuous spectrum in the blue-violet spectral region. The present work, however, is too schematic to permit detailed investigation of this point now.
622
A . B . UNDERHILL and A. G. HEARN
The importance for determining abundances of knowing correct absolute values of the optical-depth scales in lines has been commented upon. The relative optical-depth scales affect the apparent relative strengths of the triplet and singlet lines, in short the singlettriplet ratio. Any interpretation of the singlet-triplet ratio must take into account the relative variation of b(23P) to b(21P) as the electron temperature and radiation density in the atmosphere are varied. The present results indicate that when the electron temperature is decreased say from 20,000° to 10,000 °, but the radiation density does not change much, say from T~ad = 30,000° to 20,000°, the triplets will weaken with respect to the singlets. When the electron temperature becomes high in comparison to the radiation temperature, the triplets increase in strength greatly. Clearly interpretations of the singlet-triplet ratio based solely on the position of the lines on the curve of growth and on the hypothesis of LTE are insufficient. There is a real non-LTE effect appearing in the observed singlettriplet ratio. The intensities of the He I lines in the observable spectral region and the radiation intensities at wavelengths shorter than about 584 A are coupled. In very hot stars a coupling with radiation intensities at wavelengths shorter than 303 A, which are needed to cause radiative transitions out of the ground level of He ÷ should occur. The present calculations do not go to high enough temperatures to show this coupling. At temperatures near 50,000°K, which are temperatures near the values usually suggested for 0 5 stars, the He + ion should be almost completely doubly ionized according to LTE calculations. However, in non-LTE the helium remains dominantly in the singlyionized state. This fact gives a satisfying explanation for the persistence at great strength of the lines of He II to type 05. It was noted by UNDERHILL(1°) that LTE calculations using model atmospheres predicted He II lines in early O type stars which were much weaker than are ever observed. In fact the calculations indicated that He II should be past maximum strength at type 05, whereas observation shows that the He II lines increase in strength to type 05. The use of the non-LTE physics of the helium atom gives a straightforward interpretation for the observed facts and there is no need to postulate a cool outer atmosphere for hot O stars in order to account for the observed strengths of lines of He I and He II. Acknowledgements--One o f us (A. G. HEARN) is very grateful to the European Space Research Organization for a fellowship held at the Observatoire de Nice.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
L. H. AUER and D. MIHALAS,Astrophys. J. 160, 233 (1970). R. D. DmTZ and L. L. HousE, Astrophys. J. 141, 1393 (1965). P. FEAUTRmR, Annals. Astrophys. 31,257 (1968). J. HARDORP and M. SCHOLZ, Astrophys. J. Supp. 19, 193 (1970). A. G. HEARN, Mon. Not. R. astr. Soc. 142, 53 (1969). H. R. JOHNSON and A. I. POLAND, J Q S R T 9 , 1151 (1969). J. NORRIS and B. BASCrlEK,Astrophys. J. Supp. 19, 305 (1970). H. L. SHWMANand S. E. STROM, Astrophys. J. 158, 183 (1970). M. A. J. SNUDERSand A. B. UNDERHILL, Mon. Not. R. astr. Soc. in press. A. B. UNDERHILL, Can. J. Phys. 29, 447; Contrib. Dora. Astrophys. Obs. No. 23.
SOME
ASPECTS
OF
THE ATOM
NON-LTE IN
HOT
PHYSICS
OF
THE
HELIUM
STARS
A. B. UNDERHILL* and A. G. HEARN Sonneborgh Observatory, Utrecht, and Observatoire de Nice Abstract--In the atmospheres o f the hot stars (types O and B), the populations o f the states of the helium atom are significantly different from the LTE populations. This not only gives a change in the line source function with a consequent change in the predicted line depths, but it also changes the scale of the optical depths in the lines, with a consequent change in the deduced abundance o f helium. Some exploratory numerical results are given to illustrate these effects, which could explain the observed variations of the strengths of the He i and He I1 lines in O and B type stars. There is an important coupling between the intensities of the He I lines in the visible region of the spectrum and radiation densities at wavelengths less than 584 A.
INTRODUCTION
THE EFFECTof departures from local thermodynamic equilibrium on absorption lines are frequently expressed in terms of the departures of the line source function from the Planck function, that is as a function of the difference of the ratio of the population of the upper and lower levels of the line from the ratio given by the Boltzmann relation. Such differences for an absorption line in the visible region of the spectrum generate a comparatively small (though not negligible) difference between the line source function and the Planck function. For He I lines, normally observed in stellar spectra, the ratio of the line source function to the Planck function lies in the range 0.5 ~-
SL/B~ < 1.
(1)
The change is relatively small because the energy separation of the two levels giving an absorption line in the observable region of the spectrum is between 1-5 and 3 V and the collisional coupling between the levels is quite strong at the electron densities and energies found in the atmospheres of hot stars. A change in the source function (1) will result in central absorptions for strong lines which are up to 10 or 15 per cent greater than those calculated for LTE. In the case of weak lines the difference in depth is only a few per cent. No difference in line shape due to the fact that St. ¢: B~ is detectable in the wings of the lines. The equivalent widths of strong lines are increased by a small amount over their LTE values, t6) These facts have been interpreted to mean that it is not necessary to use non-LTE physics of the helium atom when estimating the helium abundance from the equivalent * Now at NASA, G o d d a r d Space Flight Center, Greenbelt, Maryland. 617
618
A.B. UNDERHILLand A. G. HEARN
widths of the He I lines or from matching line wings, t4'7'8~ This conclusion is false because abundances are deduced from the relative optical-depth scales in the lines of different elements and the optical-depth scales depend on the actual populations of the lower levels of the observed lines. In the case of He I this means on the actual populations of the 2UP and 21P levels. The non-LTE calculations of JOHNSONand POLAND (6) and of HEARN (see SNIJDERSand UNDERHILL(9)) indicate that in the deeper layers of B type atmospheres, where the electron density is of the order of l0 TM but not much greater, the population of 23p can be about twice its LTE population. How this comes about has significant implications for the theory and interpretation of lines of O and B stars. We shall outline the principal factors in the problem because they illustrate important concepts which have received too little attention in the theory of stellar spectra. GROUND
STATE POPULATIONS
The departure of the population of the ground level of neutral helium from the population given by the Saha-Boltzmann relation at electron and ion densities typical of the atmospheres of O and B stars is expected to be larger than that of levels with n > 2 because the collisional coupling between the ground level and the once-ionized state of helium is much weaker than is the effective coupling between states with n > 2 and the once-ionized state. What the departure is depends sensitively upon the radiation field present at wavelengths energetic enough to cause radiative transitions from the ground state, that is at 2 < 584/~. The importance of departures in the case of the ground level of hydrogen and their effect on the temperature structure of the atmosphere of a hot star have been shown by FEAUTRIER,(3~ and more recently by AUER and MIHALAS(1) who have also investigated the effects on the profiles and strengths of some Balmer and Paschen lines of hydrogen and have shown them to be significant. In the case of the helium atom one must consider not only the population of the ground state of neutral helium but also the population of the ground level of He +. The population of the ground level of He + relative to that of the He + + ions will affect the calculated neutral helium absorption line profiles, even if the populations of the upper and lower levels of the absorption line are related by the Boltzmann equation, that is even if the line source function is equal to the Planck function, because the populations of the excited levels of neutral
helium are determined mainly by the population of He* ions and not by the population of the ground state of the helium atom. An increase in the He + ground-level population will increase nearly proportionately the populations of the lower levels of the observed lines. The result is a change in the relation between the optical-depth scale in the observed He I absorption lines and the other optical-depth scales in the atmosphere. An estimate of the change in the ground population of the He + ion can be obtained by the following reasoning. Feautrier's calculations show that in a star with an effective temperature of 25,000°K and a gravity of 10* cm sec-2, the population of the ground-level of hydrogen at the edge of the atmosphere is 10 times greater than the Saha-Boltzmann population and that departures from the Saha-Boltzmann population extend to a Rosseland mean optical depth of 10-2. The departures from the Saha-Boltzmann relation will be very much greater for the ground level of He + relative to the He + + ions because the radiative rates are proportional to Z 4 where Z is the nuclear charge for a hydrogen-like
N o n - L T E physics of the helium atom in hot stars
619
ion and the collisional rates (excitation or de-excitation) are proportional to Z -2. The simple theory of the transfer of a continuum in a two-level model atom in an isothermal atmosphere by DIETZ and House t2) shows that the departures from the Saha-Boltzmann population and the optical thickness of the region where departures from the Saha-Boltzmann population are significant both increase as e- 1/2, where e is the ratio of the collisional de-excitation and radiative de-excitation rates. Accordingly, for a hydrogen-like ion the factor e- 1/2 is proportional to Z 3, so that from the simple theory of Dietz and House one may expect the departures of the population of the ground level of He ÷ from the SahaBoltzmann population to be 8 times greater and the region where significant departures exist to be 8 times greater than for hydrogen. This theory does not include the coupling between the hydrogen and He + continua which is difficult to consider in any simple way.
NUMERICAL
ESTIMATES
O F N(He+)/N(He) A N D O F T H E He I L E V E L P O P U L A T I O N S IN B TYPE ATMOSPHERES
In a stellar atmosphere the level populations should be found by simultaneous solution of the equations of statistical equilibrium and of the equations of transfer for radiation in frequencies relevant to the radiative processes which occur in the atom or ion under consideration. A model atmosphere is adopted giving the run of electron temperature and density with geometrical depth. JOHNSON and P O L A N D (6) have written down the appropriate set of equations for He I and have given some results for three model atmospheres in the case of some restricted model helium atoms. The equations of statistical equilibrium state that the number of atoms entering any state per unit time by all possible processes is equal to the number leaving. Atoms can enter a state from states of lower energy by absorption of radiation, which requires knowing the radiation field in the appropriate frequency interval, and as a result of collisions with electrons. Atoms can enter a state from states of higher energy as a result of spontaneous emission and of stimulated emission, which requires knowing the radiation intensity in the appropriate energy range, and as a result of collisions with electrons. Atoms leave energy states by the same processes. Thus, in order to solve the equations of statistical equilibrium for a model atom consisting of a specified number of levels, one must know the electron density and the collision cross-sections for excitation and de-excitation as well as the radiative cross-sections for spontaneous and stimulated emission and for absorption. In addition one must know the radiation field in the frequency ranges concerned in the radiative transitions. In the case of the helium atom the required radiation field can be divided into two parts: (i) wavelengths shorter than 584A and (ii) wavelengths longer than about 2500A. The radiation density in the first part of the spectrum is needed only for transitions in and out of the ground level 11S of He I; the second part of the spectrum is required for transitions between excited levels, including transitions to the ionized state from excited levels. A first attempt at solving the full set of equations and finding a set of level populations for He I consistent with the non-LTE physics of the helium atom and with a model atmosphere for a B star was done by JOHNSON and POLAND.~6~ The numerical problem is difficult and their results are not entirely satisfactory nor do they cover as wide a range of conditions, as one would like, in order to explore the meaning of the He I spectrum in O and B stars.
620
A.B.
UNDERHILL a n d A. G . HEARN
A survey of the situation can be obtained by using a computing program developed by HEARN.~5~Here the equations of statistical equilibrium are solved for a model helium atom consisting of 41 levels; the atmosphere is postulated to be at one electron temperature and electron density and the radiation density is everywhere placed equal to WB,,(Trad), where By is the Planck function and W is a dilution factor. Since in a real stellar atmosphere the electron temperature and density are determined chiefly by what is happening with the hydrogen atom, it is adequate to consider Te and Ne to be assigned parameters and to use Hearn's program to explore the effect of changing the radiation field by varying W and/or Trad . The new non-LTE model atmospheres of AUER and MIHALAS t 1~ indicate that adopting a layer at one temperature and electron density is not a poor approximation for the outer layers of an O or B star. The departures from LTE values are not strongly dependent on electron density when 1012 _< N e <_ 101., which is a range of values typical for the atmospheres of O and B stars. In Table 1 we present results for the case N~ = 1014 and W = 0-5 in order to demonstrate what will occur in the deeper layers of the atmosphere of a main-sequence star. It is here that the line wings, the weak lines and the continuous spectrum are formed. The quantities b(11S), b(23p) and b(2~P) represent the ratio of the computed population of the indicated level to that which will be found if local thermodynamic equilibrium existed at the assigned electron temperature and electron density. These results confirm the statements made above. They can be summarized as follows: (1) In a normal stellar atmosphere the fraction of once-ionized helium differs greatly from what is expected in LTE at the assigned electron temperature, the fraction increasing with the radiation temperature, that is as the radiation density at wavelengths energetic enough to ionize helium increases with respect to the electron temperature.
TABLE 1. SOME RESULTS OF THE NON-LTE PHYSICS OF THE HELIUM ATOM, CASE W = 0"5,
Te
T,. d
N ( H e +)
L T E at Te
C K)
~ K)
N(He)
N ( H e + )/N(He)
10,000
10,000 15,000 20,000 30,000 I0,000 15,000 20,000 30,000 10,000 15,000 20,000 30,000 10,000 15,000 20,000 30,000 10,000 15,000 20,000 30,000
1.350-5 0-266+0 0.980 + 0 1.000 + 0 4-830 - 4 0-267+0 0.983 + 0 1.000+ 0 3-710 - 2 0-344 + 0 0.986 + 0 1.000 + 0 0-818 + 0 0.887 + 0 0.990+0 1.000 + 0 0-983 + 0 0.988 + 0 0-995 + 0 1.000 + 0
15,000
20,000
30,000
40,000
3.94 - 5
0-494 + 0
0.994 + 0
0.917 + 0
3.57 - 2
N e
= 10 ~4
b(l ~S)
b(2 3p)
b(2 ~P)
0-292+ 1 0.135-3 0.806 - 6 0.433 - 8 0,202 + 4 0.267+ 1 0-164 - 1 0.878 - 4 0.452 + 4 0-333 + 3 0.249 + 1 0.135 - 1 0.828 + 4 0.474+4 0.373+3 0-230 + 1 0-108 + 5 0.768 + 4 0.296 + 4 0-325 + 2
0.147+ 1 0.328+0 0.147 + 0 0.573 - 1 0.208 + 2 0-168+ 1 0-779 + 0 0.314+0 0-339 + 2 0.532 + 1 0.188 + 1 0.777 + 0 0.486 + 2 0.233 + 2 0-561 + 1 0.206 + 1 0.594 + 2 0.339 + 2 0-137 + 2 0.356+ 1
0-144+ 1 0.247+0 0-910 - 1 0-303 - 1 0.732 + 0 0.134+ 1 0-503 + 0 0-167+0 0.765 + 0 0-278 + 1 0.126 + 1 0.424 + 0 0-106 + 1 0.116+ 1 0.313+ 1 0.118 + 1 0.126 + 1 0.915 + 0 0-339 + 0 0.214+0
N o n - L T E physics of the helium a t o m in hot stars
621
(2) Within the temperature range investigated, the fraction of once-ionized helium does not decrease from unity. However in LTE an appreciable decrease due to the second ionization of helium has set in by the time Te is 40,000°K. (3) The population of the ground state of He I may greatly exceed or be greatly less than the LTE population depending upon the relationship of the density of ionizing radiation to the electron temperature. The population of 1IS seems to be very sensitive to the amount of ionizing radiation available. This implies that the strengths of the resonance lines of He I will show a similar variation with the intensity of far u.v. radiation. (4) The populations of 23p and of 21P are not very sensitive to the relation of the density of ionizing radiation to the electron temperature but the 23P level is consistently overpopulated with respect to the 2~P level. On the average for a modest difference between Te and Tra d the 23p level is overpopulated with respect to what would be expected in LTE at the electron temperature by a factor near 2. This means that if the optical-depth scale in lines from 23P is interpreted by LTE physics the helium abundance will be overestimated by a factor close to 2. (5) When the electron temperature is high and the radiation temperature is also high, the 23p level is overpopulated with respect to the 21P level by a large factor in comparison to what is expected in LTE. (6) When the electron temperature is low and the radiation temperature is moderately high both the 2aP and the 2~P level are underpopulated in comparison to what would be expected in LTE at the electron temperature.
ASTROPHYSICAL
SIGNIFICANCE OF THE NON-LTE HELIUM ATOM
PHYSICS
OF THE
The calculations discussed above demonstrate that the hypothesis of LTE is not valid with the densities and temperatures deduced to occur in the atmospheres of O and B stars. Were this hypothesis valid the b-factors given in Table 1 should be all close to unity. Although the computations are schematic particularly in the manner in which they handle the radiation field, the results obtained indicate the true meaning to be attached to the following observed phenomena in O and B type spectra: (1) the persistence of the lines of He II at great strength in O stars, (2) the relative great strength of lines of the triplet series of He I and the weakness of lines of the singlet series in atmospheres where the electron temperature is believed to be greater than 20,000°K, and (3) the weakness of lines of He I in some middle and late B type stars which seem to have the same continuous intensity distribution in the blue-violet spectral region, that is the existence of 'helium-weak' and 'helium-strong' B stars. The present calculations, particular for Te = 10,000°K, demonstrate how sensitive the populations of the 23p and 21p levels are to the density of radiation at wavelengths shorter than 584 A. Small differences in the stellar model structure might easily cause changes in the far u.v. radiation field which would affect the strength of the He I lines from 23p and 21P but not the distribution of the continuous spectrum in the blue-violet spectral region. The present work, however, is too schematic to permit detailed investigation of this point now.
622
A . B . UNDERHILL and A. G. HEARN
The importance for determining abundances of knowing correct absolute values of the optical-depth scales in lines has been commented upon. The relative optical-depth scales affect the apparent relative strengths of the triplet and singlet lines, in short the singlettriplet ratio. Any interpretation of the singlet-triplet ratio must take into account the relative variation of b(23P) to b(21P) as the electron temperature and radiation density in the atmosphere are varied. The present results indicate that when the electron temperature is decreased say from 20,000° to 10,000 °, but the radiation density does not change much, say from T~ad = 30,000° to 20,000°, the triplets will weaken with respect to the singlets. When the electron temperature becomes high in comparison to the radiation temperature, the triplets increase in strength greatly. Clearly interpretations of the singlet-triplet ratio based solely on the position of the lines on the curve of growth and on the hypothesis of LTE are insufficient. There is a real non-LTE effect appearing in the observed singlettriplet ratio. The intensities of the He I lines in the observable spectral region and the radiation intensities at wavelengths shorter than about 584 A are coupled. In very hot stars a coupling with radiation intensities at wavelengths shorter than 303 A, which are needed to cause radiative transitions out of the ground level of He ÷ should occur. The present calculations do not go to high enough temperatures to show this coupling. At temperatures near 50,000°K, which are temperatures near the values usually suggested for 0 5 stars, the He + ion should be almost completely doubly ionized according to LTE calculations. However, in non-LTE the helium remains dominantly in the singlyionized state. This fact gives a satisfying explanation for the persistence at great strength of the lines of He II to type 05. It was noted by UNDERHILL(1°) that LTE calculations using model atmospheres predicted He II lines in early O type stars which were much weaker than are ever observed. In fact the calculations indicated that He II should be past maximum strength at type 05, whereas observation shows that the He II lines increase in strength to type 05. The use of the non-LTE physics of the helium atom gives a straightforward interpretation for the observed facts and there is no need to postulate a cool outer atmosphere for hot O stars in order to account for the observed strengths of lines of He I and He II. Acknowledgements--One o f us (A. G. HEARN) is very grateful to the European Space Research Organization for a fellowship held at the Observatoire de Nice.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
L. H. AUER and D. MIHALAS,Astrophys. J. 160, 233 (1970). R. D. DmTZ and L. L. HousE, Astrophys. J. 141, 1393 (1965). P. FEAUTRmR, Annals. Astrophys. 31,257 (1968). J. HARDORP and M. SCHOLZ, Astrophys. J. Supp. 19, 193 (1970). A. G. HEARN, Mon. Not. R. astr. Soc. 142, 53 (1969). H. R. JOHNSON and A. I. POLAND, J Q S R T 9 , 1151 (1969). J. NORRIS and B. BASCrlEK,Astrophys. J. Supp. 19, 305 (1970). H. L. SHWMANand S. E. STROM, Astrophys. J. 158, 183 (1970). M. A. J. SNUDERSand A. B. UNDERHILL, Mon. Not. R. astr. Soc. in press. A. B. UNDERHILL, Can. J. Phys. 29, 447; Contrib. Dora. Astrophys. Obs. No. 23.