RESONANCE LINE FORMATION IN EXPANDING DECELERATING ATMOSPHERES Dissertation for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY FELIX MARTI 1977 This is to certify that the thesis entitled RESONANCE LINE FORMATION IN EXPANDING DECELERATING ATMOSPHERES presented by Felix Marti has been accepted towards fulfillment of the requirements for Jim—degree in ML 77% Major professor Date El 11'] II 0-7639 L7":‘ .“" 1L: .. ‘- MiCIiigan 5 Late University ABSTRACT RESONANCE LINE FORMATION IN EXPANDING DECELERATING ATMOSPHERES BY Felix Marti We study the formation of resonance lines in a stellar atmosphere that is expanding but decelerating, or infalling and accelerating as it falls in. Spherical geometry is taken into account, and we assume the supersonic approxima- tion with complete redistribution over the line profile in the fluid frame. Various layers in the atmosphere become radiatively coupled or interconnected, because of Doppler shifts and effects of the projection of the velocity on the ray direction. The line profiles show sharper and sharper peaks as the outer cutoff radius is increased, or the velo- city law is steepened. It is possible to obtain profiles with as sharp a drop-off at the blue of the peak as is seen in the QSO PHL 5200. Some profiles with emission in the envelope are qualitatively like those found by Walker for young stars with infalling matter. The profiles disagree with recent observations by Hutchings for one sample of P Cygni stars, showing that the decelerating model does not apply to this sample. The approximate method of Kuan and Felix Marti Kuhi, which neglects the coupling between layers, turns out to be generally acceptable for finding the source function, as they proposed, but would lead to qualitatively wrong re— sults if applied to find the radiative acceleration. We find that their suggested criterion for validity of the ap- proximation is questionable, and we give better criteria. We construct "pictures" of what the star would look like in two dimensions, in anticipation of interferometry or occultation work, but find that the appearance is annoy- ingly similar in different wavelength bands. RESONANCE LINE FORMATION IN EXPANDING DECELERATING ATMOSPHERES By Felix Marti A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics 1977 ACKNOWLEDGEMENTS I am most grateful to Professor Peter D. Noerdlinger for suggesting the topic of this dissertation, and for his ideas, patient guidance, and encouragement throughout the whole work, making it an enjoyable learning process. I would also like to express my gratitude to Dr. Morton M. Gordon for his support and friendship during my graduate study years at Michigan State. I thank the College of Natural Science for financial support during the last period of this work. I am indebted to Drs. David Hummer and George B. Rybicki for communicating results of their unpublished work that en- abled me to detect a programming error. I thank The University of Chicago Press for granting us authorization to reproduce Figures 2 through 6, 9 through 19, and 21 through 24. And finally many thanks to all the people who contri- buted in many ways to make this possible, and made my stay in East Lansing a very enjoyable experience, and especially to my wife Elida and my family. ii TABLE OF CONTENTS LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . iv CHAPTER I. INTRODUCTION. . . . . . . . . . . . . . . . . 1 Previous Work Our Approach. (NH II. SOLUTION OF THE EQUATION OF TRANSFER. . . . . 9 Fluid Frame Picture . . . . . . . . . . . . . 9 The Formal Integral . . . . . . . . 14 Mean Intensity and Source Function. . . . . . 29 Radiation Force . . . . . . . . . . . . . . . 34 Line Profile. . . . . . . . . . . . . . . . . 37 III. RESULTS OF THE COMPUTATIONS AND COMPARISON WITH OBSERVATIONS . . . . . . . . . . . . . . 39 The Model Parameters. . . . . . . . . . . . . 39 Source Function . . . . . . . . . . . . . . 4O Radiative Acceleration. . . . . . . . . . . . 50 Line Profile. . . . . . . . . . . . . . . . . 60 Image . . . . . . . . . . . . . . . . . . . . 82 IV. CONCLUSIONS AND FUTURE WORK . . . . . . . . . 93 Conclusions . . . . . . . . . . . . . . . . . 93 Future Work . . . . . . . . . . . . . . . . . 95 APPENDIX A. . . . . . . . . . . . . . . . . . . . . . . . 97 B. . . . . . . . . . . . . . . . . . . . . . . . 104 C. . . . . . . . . . . . . . . . . . . . . . . . 107 REFERENCES. . . . . . . . . . . . . . . . . . . . . . . 110 iii FIGURE 1. LIST OF FIGURES Page a, b, and c, are the surfaces able to inter- act with the atoms at A, B, and C reSpective- 1y. The velocity law is V = vo(rC/r) . . . . . l3 Constant velocity surfaces for an accelera- ting atmosphere. The velocity 1aw is v(r) = vm(l-rC/r)I. The numbers indicate the ratio vZ/voo for each curve. . . . . . . . . . . . . . 18 Constant velocity surfaces for a decelerating atmosphere. The velocity law is v(r) = VO(rC/r). The numbers indicate vZ/vo. A line parallel to the observing direction intersects these surfaces once, twice, or not at all. . . . . . . . . . . . . . . . . . . 21 The straight line defined by the observer and the center of the star is the z axis, with its origin at the center of the star and the positive direction away from the observer. The intersection of the photo- spheric radius r and line with impact para- meter p has abscissa zc. For each sphere of radius r, there are two surfaces of constant velocity vZ that have tangents at the inter- section points parallel to the z axis; those intersections have abscissas z'd and z"d. Z6 is the abscissa of the intersection of the line with impact parameter rC and the sphere of radius r on the negative side of the z axis. . . . . . . . . . . . . . . . . . . 23 iv Figure 10. Page The star core, and a constant velocity sur- face and its intersection with a constant p line (horizontal dashed) are shown. The in- tersection closest to the observer has ab- scissa 21, the farthest, 22. On top there is a graph showing the optical depth as a function of z. The values of the auxiliary functions Y1 and Y2 in the different regions are given . . . . . . . . . . . . . . . . . . . 25 Comparison between the exact solution and the disconnected approximation for a star with RR = 10, r = 10 R0, To = 5, e = 0, l = 1. In general, "exact solution" refers to in- cluding interconnections; the supersonic approximation is always used. . . . . . . . . . 43 Effect of the outer radius on the source function. . . . . . . . . . . . . . . . . . . . 46 Effect of the optical depth on the source function. . . . . . . . . . . . . . . . . . . . 49 The ratio of the radiative acceleration to that of gravity for the case shown in Figure 6. We assume a star of 10 solar masses, effective temperature 30000 K, . -6 To = 5, corresponding to Xion = 10 , and other parameters as described in the text. . . . . . . . . . . . . . . . . . . . 52 Force law for the case RR = 10, rC = 10 R0, 8 = 0, A = 0.5, and three values of To as shown. In runs a, b, and c, X was ion 5 x 10'6, 10'6 , and 10-7 respectively . . . . . 55 Figure 11. 12. 13. 14. 15. 16. 17. Page Comparison between the exact solution (solid curve) and the disconnected ap- proximation of the force for a star with RR = 10, rC = 10 R0, 194 100, B = 5 Ic’ 2 = 0.5, and Xion = 10 . . . . . . . . . . . . 57 Comparison between the exact solution (solid curve) and the disconnected ap- proximation of the force for a star with RR = 10, rC = 10 R0, To = 1000, €3= 0.001, B = 5 IC, 2 = 0.5, and xion = 10 59 Comparison of the line profile in the exact solution (solid curve) and the disconnected approximation. The t given is To“ The ab- scissa is (O-OO)/Avm where Avm = (vo/c)\)O . ‘.' 62 Comparison of two line profiles for atmo- spheres with To = 50, e = 0.001, B(r) = IC(rC/r)I, and R = 0.5. Curve a corre- sponds to an atmosphere with RR = 10, (equivalent width (EW) = -0.410); curve b has RR = 3, (EW = -0.0739). Note that the case 1 - 1 is expected to be more sensitive to the value of RR (see text) . . . . . . . . . 66 Effect of the exponent 2 in the velocity law on the profile. Curve a: 2 = 0.5 (EW = 0.0086), curve 9: R = 1 (EW = 0.0053). In both cases RR = 10, To = 1, e = O . . . . . . . 68 Profiles for three atmospheres with RR = 10, e = 0, L = 0.5, and To as indicated. Curve 3: EW 0.0050, curve b: EW = 0.0086, curve 9: EW 0.0028 . . . . . . . . . . . . . . . . . . 71 The Si IV A 1400 profiles for several stars . . 73 vi Figure 18. 19. 20. 21. 22. 23. 24. Upper: Copernicus scans of individualostars from 1165 to 1255 A, smoothed over 1 A. Center: unsmoothed mean spectrum of all stars in upper section, with continuum drawn in. Lower: mean of 11 scans of C III A 1175 for P cyg and g1 Sco.. The dots represent the observed profiles of the resonance lines (normalized to voo = 2660 km/s) of Zeta Puppis. . . The C IV A 1550 resonance line profile observed for the QSO PHL 5200 p Itot and p Ir versus p/rm for a star W1th RR = 3, To = 50, e = 0.001, B = IC, and Q = 0.5 . . . . . . . . . . . . p Itot and p II. for a star w1th RR = 10, To = 100, 6 = 0.001, B = 5 Ic’ and l = 0.5. p Itot and p Ir for a star with RR = 10, To = 1000, e = 0.001, B = 5 Ic’ and 1 = 0.5 p Itot and p II. for a star with RR = 10, To = 1000, 8 0.001, B(r) = 5 IC (re/r), and l = 0.5. Note that all the parameters but B coincide with the parameters in Figure 16 vii Page 76 79 81 84 86 88 90 CHAPTER I INTRODUCTION Previous Work There has been a permanent interest among astronomers on the problem of spectral line formation in expanding or contracting media. We have numerous examples of objects that are apparently expelling material in a steady way or in bursts: Wolf-Rayet, P Cygni, Be, Of stars, Novae, Seyfert galaxies, and possibly some QSOS. The presence of a P Cygni type profile (emission peaked at the central frequency of the line and a violet displaced absorption feature) in the spectrum of a star is inter- preted as an indication of the existence of material around the star that is moving away from it. Due to the Doppler shift this material will see radiation emitted at the cen- tral frequency of the line in the observer's (laboratory) frame displaced toward the red and will absorb more of the radiation in the violet wing of the central peak, producing the absorption dip observed in the high frequency side of the line. Milne (1926) was the first one to point out the possi- bility of ejection of particles by hot stars due to unbal- anced radiation pressure. He assumed that initially the atom is in equilibrium with the radiation in the deep part of the absorption line, then an accidental motion outward makes the atom absorb in the violet wing where the intensity is larger, thus accelerating it further. After Beals (1929,1930,1934) established the model of an expanding extended atmosphere to explain the spectra of Wolf-Rayet stars, Gerasimovic (1933), Chandrasekhar (1934, 1945), and McCrea and Mitra (1936), made the first attempts to develop a theory that could explain quantitatively the observed profiles. We must notice that both Gerasimovic and Chandrasekhar considered the case of a decelerating atmo- sphere as well as the accelerating one. Gerasimovic found a velocity law of the form r-£ for the expansion of the hydro- gen shell in Nova Aquilae four days after the maximum. Both authors assumed complete transparency and that the emission per unit volume was known, neglecting then the transfer problem. The next step toward the solution came when Sobolev (1958,1960) developed a simple theory applicable in cases where the macroscopic velocity of the material in the atmo- sphere is much larger than the mean thermal speed. Rublev (1961,1963) applied this formulation to the interpretation of the spectra of Wolf-Rayet stars and also studied the case of decelerating flows (Rublev 1964). He computed the line profiles assuming that the emission and absorption compo- nents of the bright lines are formed in the same spherical layer of the envelope, and that the corresponding 3 coefficients vanished everywhere else. Kuhi (1964) applied Chandrasekhar's method to T-Tauri stars considering that after leaving the stellar surface, the atoms were subjected only to the force of gravity, de- celerating the flow after the initial thrust. The coeffi- cient of emission was assumed known. Castor (1970) gave the first solution that allowed one to determine the emission coefficient in a self-consistent way, in the case of an accelerating atmosphere. Castor and Van Blerkom (1970) applied it to the He II lines in Wolf- Rayet stars, and Castor and Nussbaumer (1972) to C III in the same type of stars. Oegerle and Van Blerkom (1976a,b) have studied the neutral Helium lines in Wolf-Rayet and P Cygni envelopes using the same formalism. Our Approach As we see, most of the recent work done using Sobolev and Castor's escape probability method is restricted to ac- celerating flows, probably because these flows are by far the most common and the most relevant to understanding stel- lar winds and mass loss from stars. Nevertheless, there re- mains considerable interest in studying other possibilities, such as outflows that decelerate, inflows that speed up un- der the influence of gravity as they near the star or cen- tral object, and more complicated velocity fields. In the flows just mentioned, and more complex ones, such as ro- tating flows or those with shocks, there will generally be fairly distant parts of the flow that are in touch with each 4 other rather closely through the radiation field, since the relevant radiation travels freely through the intervening material. The reasons for this are shown in some detail in Chapter II, but rest simply on the fact that Doppler shifts can cause distant parts of the gas to contribute opacity at the same laboratory frame frequency, while in simpler mono- tonically accelerating flows, the radiation at one labora- tory frequency can interact with the gas only in one small connected region. The coupling of distant regions has been emphasized by Hummer (1976). There are at least three reasons for our interest in such flows: the decelerating flow has been claimed by Kuan and Kuhi (1975) to explain the hydrogen profiles in P Cygni stars better than the standard accelerating flows; there is evidence for infall of matter in such objects as 61 Ori C (Conti 1972) and several stars described by Walker (1968, 1972); and finally, systems such as we study here are of in- termediate complexity between the accelerating outflow and the cases with rotation or shocks; hence we may gain expe- rience useful in the harder cases. Kuan and Kuhi (1975) coupled a multi-level populations calculation with the escape probability method of Sobolev and Castor to find the source function S(r). In common with Kuan and Kuhi, we adopt the supersonic approximation. In this approximation, assuming complete redistribution over the line profile, one finds (Castor 1970) that the source function is frequency independent, and can be related in a 5 simple way to the incident stellar continuum IC and the quantity 3 defined as the mean intensity Jx’ integrated over the fluid-frame line profile ¢(x) _ _ m _.l_ J — L)Jx ¢(x)dx, JX — 4n I dew , (I-l) where x is the frequency in the fluid frame. (In multi- level problems, 3 and S will have subscripts, usually sup- pressed here.) An analysis of populations, or a simplify- ing assumption such as that of the two-level atom (adopted here), leads to relationships among the various Jij’ S. , 11 and the continua, but the transfer equation itself must be applied to close the system of equations, because the local excitation depends on transfer to and from the region under study. For an accelerating expanding atmosphere, Castor (1970) showed that this closure is obtained locally (zone by zone, i.e. at each radius r in the atmosphere separate— ly) by the equation 3 = (1 - E)s + BCIC , (1-2) where 8 is the escape probability and BC the escape proba- bility in the cone of solid angle occupied by the stellar photosphere (see Castor (1970) and Chapter 11 below). Equation (I-Z) shows that the integrated mean intensi- ty at a point is a weighted combination of the local source function and a contribution from the photospheric continuum; i.e. it contains light emitted locally and incident light. Clearly, when Doppler shifts allow light from one part of the atmosphere to interact again in another, additional 6 terms coupling the layers must be present. In Chapter II, we show explicitly how a coupled set of linear equations, relating S(r) in the various layers of different r, re- placed (I-Z) for a decelerated atmosphere. In a later paper, Kuan (1975) alluded to this interconnection, but stated that it was a "justifiable approximation" to ignore it "if the radiation field in the envelope is weak in comparison with the photospheric intensity, which is usually the case". We point out here that this statement is valid insofar as one has radiation from one part of the envelope adding to the excitation in other parts; however, one also has parts of the extended atmosphere masking other parts from the central continuum. Our primary thrust is to study the influence of the in- terlayer radiative coupling, and so verify in what cases the approximation of Kuan and Kuhi (hereinafter called the "dis- connected approximation") applies. Furthermore, we evaluate the radiation force on the material, and comment on the ap- plicability of decelerating atmospheres to real stars. In order to test the effect of the "disconnected approximation" we have arranged our analysis to turn the interlayer radia- tive coupling on and off at will. One naturally asks how valid the assumption of the two level atom is for real stars. We note that in the cases considered by Kuan and Kuhi as strong candidates for de- celerating flow, the Balmer lines have large emission, with small absorption features. Under their assumptions such net 7 emission must come from the expanding envelope. No matter what the excitation process, then for Balmer lines the ra- diation field due to the envelope is a fortiori strong com- pared with the photospheric continuum; furthermore, if the excitation is partly Lyman flouresence, as is likely, the interlayer coupling for Lyman radiation is also strong, and could importantly affect the Balmer profiles. The specific cases we study are the velocity laws v(r) = +v (r M)" (1-3) ‘ o c ’ where rC is the radius of the opaque core (photosphere), v0 the largest velocity in the atmosphere, and the two choices R = l and t = i were considered. The case 1 = i and v nega- tive, corresponds to free fall from rest at infinity. Note that all our computations were done with expanding atmo- spheres, and if we want to apply them to contracting ones (minus sign in equation I-3) we just have to change the sign in the abscissa in our profile graphs, where the red side will now be the blue and vice versa. In all cases, mass conservation 2 _ Nir v(r) — const., (1-4) was assumed, where Ni is the number density. (This fails to allow for changes in the state of ionization, since Ni ac- tually will represent a specific ion.) According to equations (1-3) and (I-4) the material does not thin out as fast with increasing radius as in an accelerating flow. For computational reasons, a cutoff is 8 introduced at some radius rm. This cutoff, which seems ar- tificial, has considerable influence on the resonance line profiles, in contrast to Kuan and Kuhi's case of subordinate lines. Typical times to traverse an atmosphere of radius 1013 cm at a thousand km/s are of the order of a day. Therefore, the true radius is much larger than what is used, and we must regard the cutoff as due to a change in state of ionization. If the velocities accelerated again past rm this could help thin out the material and so reduce the in- fluence of the cutoff. In summary, we will be studying the formation of reso- nance lines in expanding or contracting atmospheres with velocity fields given by equation (I—3). We assume the photospheric core where the continuum (with no limb darken— ing) is formed to have radius re, and the expanding envelope extends from rC up to rm. We will use the two level atom approximation of the source function. The absorption pro- file is taken to be extremely sharp (the macroscopic velo- cities are much larger than the thermal velocities). We tried to express our results in such a way that allows us to use them with different sets of parameters that satisfy cer- tain scaling laws. The atomic data we use correspond to the C IV A1550 resonance line. CHAPTER II SOLUTION OF THE EQUATION OF TRANSFER Fluid Frame Picture In this work we will assume that the thermal width of the line is negligible and take it as practically zero. This assumption allows us to study the formidable problem of transfer of radiation in a simple formulation similar to Sobolev and Castor's. Several papers have been published on the problem of an accelerating atmosphere using a more accurate formulation than ours, without making the assumption of zero thermal width. In such a case it is convenient to solve the equa- tion of transfer in the frame co-moving with the fluid as was indicated already by McCrea and Mitra (1936). Consider for example (Hummer 1976) the integrated mean intensity J = 4? [madx LJ-dw ¢(x - u-n)IX(n), (II-l) where x is a dimensionless frequency referred to the line center (see formula A-17), u is the fluid velocity in units of the thermal velocity (formula A-20), and 6-3 = ulul. Even if ¢(y) is different from zero in a small region about y = 0, in the observer's frame we have to consider a large region in the (x,u) plane that will contribute to the inte- gral for large u. The size of the mesh of points required 9 10 to solve the equation of transfer is therefore much larger in the observer's frame than in the co-moving frame. The differential equation in the co-moving frame (see equation A-30) is significantly more complicated than the corresponding one in the observer's frame (in the stationary frame, the frequency derivative, BIX,/ax', is absent). An additional difficulty appears when we try to find the inten- sity emitted by the envelope at a fixed observer's frequency, because the solution was obtained in principle with the co- moving frequency as a variable. Noerdlinger and Rybicki (1974) gave the first stable scheme for the numerical solution in the co-moving frame for the case of a plane parallel atmosphere, that was developed later to the spherically symmetric case by Mihalas, Kunasz, and Hummer (1975,1976a,1976b). Consider an observer carried outward in a spherically symmetric velocity field v(r). He observes his neighbors located a distance ds away in the radial direction to have velocity relative to him dg = (dv/dr)ds, while neighbors at the same distance from the center are departing at a rate dn = (v/r)ds. If dv/dr > 0, then dg and dn have the same Sign, and the fluid expands in all directions. If not, the fluid is compressed in one dimension and it expands in the other; for some intermediate directions neighboring parti- cles maintain constant separation to first order. These two possibilities lead to vast differences in radiative transfer. In the former case, a light quantum repeatedly absorbed and ll emitted in a region will constantly encounter fluid moving away from it; hence it must eventually escape in the red wing of the line in the fluid frame (if it does not escape due to density gradients, or become buried back in the cen- tral source). (If dg and dn are both negative, replace "away" by "toward" and "red" by "blue" in the foregoing.) Figure 1 shows the loci of points that have zero velo- city of approach with respect to the moving points A, B, C. The velocity law is given by equation (I-3) with l = 1. Consider a set of observers that are moving with the fluid, like the ones sitting on points A, B, and C. If they can detect light only through a narrow band filter centered at the line frequency v they will see light from that transi- 0’ tion coming only from points in the atmosphere that have zero relative velocity in the direction that joins them with the emitting points. Figure 1 shows the surfaces a, b, c, of zero relative velocity in the connecting direction with respect to observers A, B, and C respectively. The velocity law is given by equation (1-3) with L = 1. In the case of an accelerated atmOSphere the picture is completely different. The observers will see light coming only from a small region around them, they will not be "connected" to the rest of the atmosphere, and the excita- tion at each point can be found locally (it will also depend on the continuum intensity Ic)' Our next step will be to find an integral equation for the source function, but before, assuming S(r) as known we 12 Figure l. a, b, and c, are the surfaces able to interact with the atoms at A, B, and C respectively. The velocity law is v = v0(rC/r). 13 Figure 1 14 find an expression for the specific intensity I, the so called "formal integral". The Formal Integral Following Castor's (1970) notation, we use a cylindri- cal coordinate system (p, e, z) where the z axis is the line joining the center of symmetry of the atmOSphere and the ob- server, with the positive direction away from the observer and the origin at the center of the star. Note that p is the impact parameter. We define r = p + z . (II-2) That is, r is the radial distance measured from the center of the star. If the absorption and emission of radiation within the line are uncorrelated both in angle and frequency (complete redistribution) in the fluid frame, the source function will be independent of frequency. The equation of transfer takes the form (Castor 1970) “(317,27 = k(v,p,Z)II(v,P,Z) - S(r)]. (II-3) where I is the specific intensity directed toward the ob- server as seen at the point with impact parameter p and ab- scissa z. This equation is a simple modification of our (A-7). The photosphere lies at r = rC and is supposed to radi- ate a continuous spectrum IC with no limb darkening. Then the formal solution of the transfer equation is 15 r(m) ’e-T(V’p’Z)I(V,p,Z) = 1(2) ice . I S[(p‘7‘+z'27‘1’1e‘T9’4”Z )dT(v,P,Z'), (II-4) 1(2) where the integral is done at constant v and p, and dr(o, p, z) = k(O, p, 2) dz. If p > rC, or z > 0 the integration path will not in- tersect the core and we get 1(Vspsz) = I S[(p2+2'2)*] eprrIv,p.Z)-r(v,p.2')1dr. (II-5) 2 But if p < rC and z < 0 1(v,p,Z) = C 2 .2 i . . I S[(p +2 ) TeXP[r(v,p,2)-r(v,p.z )IdTCv,p,z ) Z + IC eXpIT(V9paz)'T(VspszC)]9 (II-6) where zC = -(rCZ - p2)%. (II~7) The optical depth along a line of constant impact parameter p, from the location 2 to the observer is given by Z T(v.p.2) = I k(v,p,z') dz'. (II-8) -00 where the absorption coefficient is 1 k(v,p,Z) = k£(r) ¢[v-v0+v08(r) z r- ]- (II-9) Note that the zr-1 term comes from the projection of the l6 radial velocity in the observer's direction. Here, B(r) = v(r)/c, and ¢ is the line profile in the fluid frame, normalized as I ¢(x) dx = l ; x = v - v + v 8 z r , (II—10) and kg, the line absorption coefficient between levels 1 (lower) and 2 (upper), depends on r only: h\) N N h\) N B 1 2 B12 2 21) 1hr g2 B21(g1 g2)Zn-’ k = (N g (II-ll) l where the B's are the Einstein's coefficients and the g's the statistical weights of the respective levels. Intro— ducing the absorption oscillator strength f12 (Mihalas 1970) 2 2 4n e _ f12 hvomc ’ (II 12) and finally n62 N N _ __ _1 - __2_ _ k, - me (gmgl g2). (II 13) We shall assume that the line profile is extremely sharply peaked at x = 0 (frequency = v0 in the rest frame of the gas), so that r is essentially a combination of step func- tions. The number of "steps” to the step function is the number of intersections of the line of sight with the sur- face of constant 2 velocity. Chandrasekhar (1934) seems to have been the first one to construct surfaces of constant line of sight velocities as seen by an external observer looking at a monotonically accelerating atmosphere, Figure 2. Note that a line paral- lel to the line of sight intersects such a surface just at 17 Figure 2. Constant velocity surfaces for an accelerating atmosphere. The velocity law is v(r) = vm(l-rC/r)I. The numbers indicate the ratio vz/v0° for each curve. 18 $238 2 «use ‘W .k L 19 one point. If the flow is a decelerating expansion, the constant velocity surfaces appear as in Figures 3 and 4. We see that in this case there can be 0, 1, or 2 intersections. In the region close to each intersection, the contribu- tion to the integral in equation (II-8) is zi+5 ri(p.v) = I kcv.p.z')dz' = Zi'O z.+6 1 kg[(p2+ziz)II I ¢(v'v0+v08(r)zr'l)dz = 21-6 k,[(p2+ziz)i1 I(%§)p,vIz=zi I ¢(x) dx = I(%§Jp,v where 6 is selected to pick up all the contribution to the i=1,2 , (II-l4) opacity from each intersection. The derivative in the de- nominator is the Jacobian of the change of variables from 2 to X. Figure 5 shows the definition of some unit step func- tions Yi’ that we find useful in describing the variation of optical depth with 2. When some of the intersections are absent, the corresponding T should not be included, and in the following formulae this should be done by setting the corresponding r equal to zero. When there are two intersec- tions, T has the form TIVspaz) = T1(p:V) Y1(Z) + T2(p,V) Y2(Z). (11'15) Note that v(z) stands for y[x(z)] where x is the argument of O, that is the frequency in the fluid frame. The behavior 20 Figure 3. Constant velocity surfaces for a decelerating atmosphere. The velocity law is v(r) = v0(rC/r). The numbers indicate vz/vo. A line parallel to the observing direction intersects these surfaces once, twice, or not at all. Nd .. c3530 2 A $0 . md .. Nd 22 Figure 4. The straight line defined by the observer and the center of the star is the z axis, with its origin at the center of the star and the positive direction away from the observer. The intersection of the photospheric radius rC and the line with impact parameter p has abscissa 2C. For each sphere of radius r, there are two surfaces of constant line of sight velocity, equal to ivz, that have tangents at the intersection points parallel to the z axis; those inter- sections have abscissas z'd and z"d. 26 is the abscissa of the intersection of the line with impact parameter rC and the sphere of radius r on the negative side of the z axis. 23 sN _—--———_—_' Figure 4 24 Figure 5. The star core, and a constant velocity surface and its intersection with a constant p line (horizontal dashed) are shown. The intersection closest to the observer has abscissa 21, the farthest, ZZ' On top there is a graph showing the optical depth as a function of z. The values of the auxiliary functions Y1 and Y2 in the different regions are given. 25 .2me0 0., ‘ _ -A-._A :Ndxv C. Figure 5 26 of r for a case with two intersections of equal T1 is shown in Figure 5. Evaluating the derivative in equation (II-l4) along a line of constant p gives V V (3925mm = :9 fitvf.“ 2] = 29W”) + ztl 91’- - 3111151 = v ( ) + 2 d ) 29W r (g) v _ v(r )] V r (a? r = _9[(1-u2)1§£l + “2 dV]. (II-l6) C In evaluating the different ri's in equation (II-l4) for use in equation (II-15), we can consider two extreme cases in equation (II-l6). If our line of constant p passes through the center of the star, the optical depth that we will be determining is the optical depth in the radial direction. For p = 0, z = -r, U = 1 k (r) T = 1 , r = -z (II-l7) r :9 dv(r) c dr Note that it is the optical depth as measured at location r toward the external observer in the radial direction. If instead, we take v = Vo’ the intersection of the p = constant line with the surface of constant vz (= 0), will be on the z = 0 axis (p = 0), giving the optical depth in the tangen- tial direction, perpendicular to r I' k T = '(r) (II-l8) " :9 v(r) C I‘ The optical depth in the direction 8, making an angle 1 a = cos- p with the radial direction is then 27 A k2(r) T(n) = v 2 ( ) 2 d . (II-l9) 0 V r v 1? (l-u )‘j:— + U a; Note that these r's were derived using the observer's direc— tion as a preferred direction, but due to the symmetry of the problem they are completely general. For a power law velocity field like ours, (equation 1-3), (II-17), (II-18), and (II-19) can be expressed as k£(r) riIl r = , (II-20) r V0 2 v —— L r o c: c It — R: Tr , (II-21) L r T = r . (II-22) Il-u2(1+l) The probability of a single emitted photon being in the solid angle dw about n, with frequency in the range x to x+dx is 99 ¢(x) dx (II-23) 4n ' Only the fraction exp[-T(n,x)] of those emitted will escape the surrounding region. Therefore the net escape probabil- ity is (Rybicki 1970) k2(r) Ix . . ¢(X )dX ], (II-24) g = £% Ide::dx ¢(x) exp[- I (%§)p.v where we have used the value of r from equation (II-l4), and 28 then 3 a 1 Id NIIIP’VI [1 ( kw) 1 (II 25) =—— r — exp — ’ - 4Ir k ’ 8(r) I(%§)p.vI or simply l 1 1-expfrr) B(r) = 7 I d l I, (II-26) 1-1 T where T is given by equation (II-22). In the following, we shall often suppress the frequency v and impact parameter p, which are always fixed during formal integration of S to ob- tain 1, (although not when I is integrated to get the mean intensity 3). Also we may denote, for example, (p2+z'2)I 2 2 i +21 ) as r1, etc. as r', (p Substituting expression (II-15) into equation (II-5), we find for the case p > rC 1(2) = S(r1){l - exp[-T1 71 (z)T} + S(rz) epr'Tl -Y-1 (2)1{1'epr'T2 TY-2(Z)]}, (11'27) where 71 = 1 - Yi' One must remember that either term in equation (II-27) may vanish if the intersection of the ray with the surface of constant 2 velocity falls outside the assumed outer radi- us of the atmosphere, rm. For frequencies to the blue of line center, Ile may be so large that rl exceeds rm, in which case T1 is defined to be zero. For frequencies to the 29 red of line center (left half of Figure 3), 2 may fall out- 2 side the atmosphere, and the term in S(rz) will be zero. (22 is always to the left, i.e. larger than 2 by defini- 1 tion, as shown in Figure 5.) For the case p < rC, the cases 2 > 0 and z < 0 must be handled separately. In the former, there is no contribution from the photosphere, and the ray does not reach the observ- er, but is intercepted by the star. Such rays are perceived indirectly by the observer, because they affect the source function, however. For the case 2 > 0, the same equation (II-27) applies, with T2 still set equal to zero if r2 > rm; however, must be set equal to zero, causing omission of T1 the S(rl) term, 1f r < rC 1 Finally, in the case p < re, 2 < 0, we have 1(2) = S(rl) {l-eXPI-Tl 71 (2)]} + sag) epr-T, 71 (2)] {I-epr-TZ 72 (2m + IC eXp[-r1 71 (Z) - 12 72 (2)] . (II-28) with the understanding that T2 vanishes if r2 < rC or r >1‘ and r vanishes if r > r 2 1 m m’ 1 Mean Intensity and Source Function The mean intensity itself is, of course, frequency de- pendent, and this dependence can differ considerably in the laboratory and fluid frames. From the work of Castor (1970) however, one expects the integral of the mean intensity over 30 the line profile, 3(r) = 5% Ir dzIm ¢(x) I[O, (r2-22)I, z] dO , (II-29) -r 0 where x is defined in equation (II-10), to have an especial- ly simple relationship to S. The integration over 2 in equation (II-29) actually represents an integral over p; spherical symmetry has been used to rotate rays that would ordinarily pass through a common point on the shell of radi- us r so that they are instead parallel. We wish to manipu- late equation (II-29) into an integral equation relating j in the various layers and IC. It can then be combined with the excitation law for the two-level atom: S(r) = e B(T) + (l - e) 3(r) , (II-30) where B is the Planck function evaluated at the line fre- quency and depends on the temperature T at radius r; e is the ratio of collisional de-excitation rate to total de- excitation rate. The term in 3 corresponds to the scatter- ing contribution to the source function, while the term in B is produced by collisional excitation followed by radiative de-excitation (Mihalas 1970). Returning to equation (II-29), the integration over V is done first, leading to 6-functions that select the chosen layer at r and all other layers that couple to it radiative- 1y. The results are best described in terms of Figure 4. The locus of the points in the constant line of sight velo- city surfaces that have tangent parallel to the z axis can 31 be found from 3v 8 T: = 52- [v(r) g] = o (II-31) z dv zv dr v _ (;3;-;7)g;+;-0. (II-32) substituting dr/dz = z/r, and dv/dr = -£ v/r gives .127: :(2 + 1)-I. (II-33) For a given r, the intersections with the locus are then n=_v= ’2' - z d z d r(£ + 1) . (II 34) Also, clearly, -_2_ 2% _ ze - (r rC ) . (II 35) The integration in equation (II-29) is thus broken up at ze, for the core cutoff, i.e. for z < 28 p is smaller than rc, while for z > ze, p > rC or z > 0. We also separate the integration at z'd and z"d to distinguish the cases where the intersections are single or double. The results are: for z' > z d e 32 1 r lT(r) = S(r) [l - §;-I C(T) dz] -r 2e + 1c §%-Ifirdz exp(-T2) G(r) 1 Z'd + 7?.Ifr dz S(rz) S(r) [1 - exp(-T2)] znd + g? I) dz S(rz) G(T) [1 - eXp(-rz)], (II-36) and for z' < 2 d e l I‘ 3(r) = S(r) (I - 7; I 6(1) dz] '1' Z + IC [%; Ir dz eXp("T2) GIT) + f%‘I f dz G(T)] 1 ZHd—I. Z d + 7; I) dz S(rz) G(T) [1 - epr-r271 Z'd + %% I.r dz S(rz) S(r) [1 - epr-rz)] , (11‘37) where G(T) is given by G(T(u)] = 1 ‘ :XPI'T) . (II-38) In equations (II—36) and (II-37), unsubscripted r's are evaluated at radius r and at u = -z/r, while quantities with subscript 2 must be evaluated by finding the other intersection of the ray with a surface of the same line of sight velocity, as in Figure 5. It should be noted that d8 = C(T) %% , (II-39) 33 is the increment of escape probability. When equations (II-30), (II-36) and (II-37) are com- bined, there follows an integral equation for S(r) of the form 1 Z'd S(r)(1§€ + B(T)] - 7; I dz S(rZ) G(T)II-exp(-T2)1 'I' z" --l— I d dx S(r ) G( )[1-ex (- )] 2r ,0 2 T P T2 = e B/(l - e) + H(r) IC , (II-40) where H(r) has different expressions, depending on z'd and z for a given r. For z'd > z e e Z 1 e H(r) = 7; I dz G(T) exp(-r2) . (II-41) ‘1' and for z'd < z e zId Ze H(r) = 4L—[ dz C(T) exp(-T ) + dz G(r)]. (II-42) 2r r 2 z' ' d If we ignore the interconnections, letting T = O in equa- 2 tions (II-40), (II—41) and (II—42), we obtain S(r)[l‘_:E + B(r)] = e B/(1 - 6) + Bc(r) 1C , (II-43) where 1 Ze ec(r) = 7; I dz G(T) . (II-44) '1‘ 34 and can be approximated by 1 -(r2-rc2)if ec(r) = 7; Ir dz 6(T) = W(r) B(T) , (II-45) where W(r) is the dilution factor [I - (I - rCZ/rz)*], (II-46) NIT—l W(r) = that is the probability of a ray emitted in a random direc- tion striking the core. We must notice that in this approx- imation (12 = 0), we recover in our equations (II-30) and (II-43) Castor's (1970) result. The description of the method of solution of the inte- gral equation (II-40) is outlined in Appendix B, and some approximations useful in limiting cases are discussed in Appendix C. Radiation Force If the density is p(r), the absorption coefficient may be written k(V,P,Z) = k2(r) OIX) = 0(r) 0(r) ¢(X), (II-47) which defines the cross section per unit mass, 0. Then the radiation force per gram is (X) 1 a(r) = 319 uduI ¢(x) 1(v,p,Z) dd. (II-48) C -1 J0 where x = v - v + 8 z r-1 and u = -z/r, and I(v,p,z) is 0 given by equations (II-27) and (II-28). Castor (1974) has 35 derived higher order corrections to equation (II—48) em- bodying the effects of sphericity and finite thermal line width, but he recovers the intuitive result (II-48) in the extreme supersonic limit. For resonance lines we approxi- mate (see equation II-l3) 2 N k,(r) = IE; (gf) g; . (11-49) and set p = N mav where N is the number density of all ions and m their average mass in grams. If X10 then stands av n for the fraction by number of all ions in the state under consideration, we write X 2 . o = k /p = IITf'c— (gf) E431— . (II-50) i 1 av Substituting I into equation (II-48) we get for z' > 2 d e Z! d a(r) = 31% [I 2 dz S(r) [l-exp(-r2)] S(rZ) CI‘ ~r an + I 2 dz G(r) [l-exp(-r2)] S(rz) 0 + IC I{: z dz exp(-rz) G(r)], (II-51) f r z' < Z and o d e 36 d acr) = 319 [I 2 dz GIT) II-exp(-T,)] S(rz) Z" d I 2 dz GIT) II-exp(-T2)1 S(rz) 0 + z'd + IC I 2 dz G(T) exp(-T2) '1” Z + IC I f 2 dz C(T) . (II-52) Z d Fortunately, the coefficients needed for the acceleration a may be calculated in a similar manner and at the same time as those needed to formulate integral equation (II~40). When we considered the disconnected approximation we also omitted terms in S(rz) for the force, so that both the source function and the algorithm for obtaining the force from it were different. This is not intended as representa- tive of Kuan and Kuhi's approach, because they did not con- sider the force; also they took into account the possibility for double absorptions once S was found in the disconnected approximation. Here it would likewise be possible to use an approximate S but then perform the integrals in equations (II—51) and (II-52) exactly. This is not particularly more consistent than what we used for the disconnected case. Furthermore, the labor of finding the second intersection in a rigorous integration for the force is tantamount to doing the labor for the whole calculation. We therefore assume anyone considering this kind of calculation of the force would solve the whole transfer problem rigorously, and thus 37 our "disconnected approximation" for the force is appropri- ate as an example of leaving out interconnections. Line Profile The power emitted by the star plus envelope, per unit frequency is r m Fv = 4n I) 2n p dp I(u,p,-w) . (II-53) In the case rC < p < rm, we have I(v,p,-w) = S(r1)[1-eXR(-T1)I + S(rz) eXP(-rl)[1-eXP(-I2)] , (II-54) and in the case 0 < p < rC the result is; I(v.p.-m) = S(rlitl-exp(-Tl)1 + S(rZ) exp(-T1)I1-exp(-Tz)] + IC exp (-r1 - T2) 9 (11'55) where, as usual, either r is set equal to zero if that in- tersection is absent. If we combine these and normalize by the unattenuated continuum intensity I‘ C F = IC 4"I Zn p dp = 4n r I (II-S6) o C we obtain 38 2 -1 rC Fv/Fc = (IC rC ) I) (S(r1)[I-exp(-Tl)1 + S(rz) exp(-r1)[1-exp(‘T2)I + IC exp(-T1 - 12)} 2 p dp r m + (IC rczi‘l I) (S(r1)[1-exp(-T1) + S(rz) eXPI‘T1)I1’eXp('T2)I} 2 p d p . (II-57) CHAPTER III RESULTS OF THE COMPUTATIONS AND COMPARISON WITH OBSERVATIONS The Model Parameters We have carried out the calculations with some atten- tion to actual dimensional quantities relevant to stars and QSOs, but will express most of our results in terms of the following dimensionless parameters: the exponent 2 already defined, the ratio RR = rm/rC (outer envelope radius)/(core radius), the maximum integrated optical depth To in the ra- dial direction, and the values of e and B/Ic' As the line opacity is proportional to the number of ions in the state under consideration, combining equations (I-4), (II-l3), and (II-20) we obtain T a r . (III-1) th In the case 2 = , rr(r) is constant and equal to its maxi- mum, To' In the case 2 = l, rr(r) is linear in r, and hence attains its maximum T0 at r = rm. This points up the tendency of decelerated flows to be more dependent on cut- offs at large r than accelerated ones. Almost all the runs were done with RR = 10, the choice of Kuan and Kuhi, and an economical one computationally. A few were done, however with RR = 3 and with RR = 15, so we can comment on the 39 4O effect of the cutoff. All our envelope models start right at the photosphere, where v = v0. Kuan and Kuhi (1975) applied their model to the case of P Cygni. They took the photospheric radius rc equal to 10 R0 with an effective temperature of 30000 K and an abso- lute bolometric magnitude of -7.4. This value is probably too low, with the correct value closer to -10.4. The visu- al magnitude Mv is about -7.4 (DeGroot 1973), and the bolo- metric correction -3 (Allen 1973). Consequently the radius should be larger than 10 R0. We took this last value for most of our runs, but it is possible to use the results given in the graphs for sets of stellar dimensions and rates that differ from the ones we used in computing them, provided the new sets keep constant the fundamental para- meters like To, which is given by T = nez f. Xion M c(RR)2£-1 o :mc 1j 4n mav v 2 v z r ’ o o c (III-2) where M is the mass loss rate. In all our computations we used N = 1.06 10‘7 solar masses/year and v0 = 3000 km/s. The scaling law for the radiative acceleration is given below. Source Function The source function generally exhibits a sever attenu- ation with radius, as is appropriate for a very extended atmosphere, unless e and B are large. When arc is large, then S 2 B. In the disconnected approximation, it is easy 41 to show that (see Appendix C) for Tr << 1 s g as + W(l - 8) IC , (III-3) while, in the optically thick limit we have for Tr >> 1 and r >> rC 2 r 1.177 1 C B . 1 _—€_— N L = —. - SII—e + Tr I - IT;'tr2 I l-e ’ 1f 2 2 (III 4) r 2 6 0.6095 1 C EB = _ The foregoing approximations support the idea that S falls rapidly with increasing radius, unless 5B is large. These l/rZ effects tend to dominate any of the finer differences that could be caused by the disconnected approximation, and they make graphs of the source function itself rather dull. Therefore we choose to exhibit only a few of the most sig- nificant cases. In Figure 6 we show the source function for the worst example we constructed for failure of the disconnected approximation. The difference in S is at most 0.28 in the logarithm, although it is still growing at rm 0n the basis of Kuan and Kuhi's remark that the approxima- tion is good when the intensity from the core exceeds that from the envelope, one might expect to construct worse cases by making B large. However, we found in that case that if the envelope is thick the local B value dominates S 42 Figure 6. Comparison between the exact solution and the disconnected approximation for a star with RR = 10, rC = 10 R0, To = 5, e = 0, l = 1. In general, "exact solution" refers to including interconnections; the super- sonic approximation is always used. LOG S 0.4 0.0 -0.4 43 (a) exact solution — (b) disconnected approx Figure 6 44 (but not in the force), while if it is thin, interconnec- tions are obviously unimportant. It appears that cases with e = 0 and intermediate optical depths are the worst; also we expect more discrepancy if rm is increased. The reader should not consider that small changes in S where it is small are of no consequence at all, because there is a lot of volume and material out at large radius. In Several runs (not shown) with RR = 3, we found that toward the outer portion of the envelope the source func- tion fell below that in the disconnected approximation. We traced this effect (which ran as high as 15%) to the dis— carding of absorption in inner layers in the disconnected approximation. (Note, however, that Kuan and Kuhi included such absorption in the underlying layer in their formal in- tegration.) The fact that the interconnections become more impor- tant at large RR suggests a bit of caution about the subor- dinate lines being independent of RR as long as it exceeds 10 (Kuan and Kuhi 1975). A change in S for resonance lines far out raises the ground state population for the subordi- nate lines. In Figure 7 we Show the effect of the outer cutoff on the source function in the exaCt solution. The three cases differ only in the value of RR, that is, we have added more material beyond r We see that when we increase rm, the m. value of the source function in the region common with the previous case increases. This is due to the fact that the 45 Figure 7. Effect of the outer radius on the source function. L06 3 (10 -O4 -L2 -L6 -21) 46 I I l I I I I I I I I [=05 €=QOOI B=I 1;=50 ‘0 ‘ +‘RR=3 . . RR=K> ’ - RR=E5 _ 3 I I I I I l I I I I j 50 IOO '50 R/R, Figure 7 47 inner layers receive light from the outer ones, which in- creases the excitation. Remember that in the disconnected case the solution is completely local, i.e. for each layer it does not depend on how much material there is beyond that layer, (see equation II-43). How does the optical depth affect the source function? We compare in Figure 8 three different cases, where all the parameters but the mass loss rate were kept constant, giving maximum optical depths of 0.1, l, and 5. In the region close to the core, the thinner the atmo- sphere the higher the value of the source function. If we refer to Figure l, we see that in the case of point A, the surface able to interact with the atom at A, covers a large portion of the core, decreasing the amount of light re- ceived by the atom, and lowering the excitation. This ef- fect may also be presented in the following fashion. Let us be reminded that the material on curve a, connecting point A in Figure l to the stellar surface has zero velo- city of approach as seen by the material at A. Consider 'now a straight line from A to the stellar surface. If that line crosses the surface a, the optical depth will be large, resulting in a greatly diminished stellar intensity. It is obvious that making the atmosphere thicker will de- crease S even more. When we are far from the core (point C in Figure l) the interacting surface does not cover much of the core, but instead emits (in the case of Figure 8 scat- ters, e = 0) light toward C, increasing the excitation. 48 Figure 8. Effect of the optical depth on the source function. L06 8 0.4 -|.2 49 Figure 8 50 Radiative acceleration 2 de- We found it convenient to take out the strong r- pendence in the radiative acceleration by dividing by the acceleration of gravity GM/rz. For this purpose we chose physical parameters listed in the caption of Figure 9. It is possible to alter the combination in any way that pre- serves To and the product 2 CF = Ic fik Xion rc /M ’ (III-6) where M is the mass of the star and Xion the fraction of all ions in the gas in the form of the chosen ion. (We took mean ionic weight 1.2.) In all cases, we chose para- meters corresponding to the C IV resonance line at 1550 A, fik = 0.286, IC = 5.05 x 10-3 erg cm-2, and Xion in the range 10-3 to 10-7. The value of IC corresponds to a 30000 K black body intensity at the C IV resonance line wavelength. It is easy to convert to any case through equation (III-6). (The mass loss rate also enters the problem, but only in fixing To.) The curious structure in the force law between the photosphere and 25 R0 is present in both the disconnected and regular cases. The dip is approximately where the cone of light from the core is tangent to one of the cones of maximum optical depth given by 51 Figure 9. The ratio of the radiative acceleration to that of gravity for the case shown in Figure 6. We assume a star of 10 solar masses, effective temperature 30000 K, To = 5, corresponding to Xion = 10-6, and other parameters as de- scribed in the text. 44 40 l6 l2 52 l l I I I. ” . \ "I (a) exact SOIUIIOH \ (b) disconnected apprcix ‘w l 1 L l 0 4' 20 40 60 80 IOO r r ° R/R m Figure 9 G) 53 u = i —i—. (III-7) (2 + DI If the Optical depth were independent of u, we should ob— tain a r-2 dependence, that after dividing by the gravita- tional acceleration gives a flat curve. The drop for large r values is due to the increase of Tr (remember that Tr is linear in r for 2 = 1). The force in most of the atmosphere is highly altered in the disconnected approximation. Figure 10 compares the force for atmospheres of var- ious thicknesses. As expected, the force behaves essential- ly as r-2 in the optically thin case, although some of the peculiar structure due to the variation of r with u is pre- sent. Note that even in this case, the radiation force is much larger than the gravity force. Some saturation is present as 1 increases beyond 1. It is worth noting that the mass loss rate or overall density was not changed in these runs, only the abundance of the ion with the reso- nance line. If, instead, the relative abundance were fixed and the total density increased, the average force would remain constant at very small optical depths, and then gradually drop as saturation was reached. In Figures 11 and 12 we compare atmospheres with the same RR, 2, 5, density, and B, with two choices of Xian and also with the interconnections turned off. In the first case, since er is only 0.1, the Planck function is not able 54 Figure 10. Force law for the case RR = 10, rC = 10 R0, 8 - 0, 2 = 0.5, and three values of To, as shown. In runs -6 -6 -7 ion was 5 x 10 , 10 and 10 ’ a, b, and c, X respectively. FORCE I00 80 60 40 20 SS 9....- l l l J 20 40 so 80 IOO R/R, Figure 10 56 Figure 11. Comparison between the exact solution (solid curve) and the disconnected approximation of the force for a star with RR = 10, r = 10 R0, To = 100, e = 0.001, _ - = -4 B - 5 I 2 — 0.5, and X10 10 . FORCE 600 400 200 S7 I l l 0 20 40 60 R/R0 Figure 11 80 |00 58 Figure 12. Comparison between the exact solution (solid curve) and the disconnected approximation of the force for a star with RR = 10, rC = 10 R0, To = 1000, e = 0.001, _ _ = -3 B - 5 I 2 - 0.5, and xion 10 FORCE 2800 2400 2000 l600 I200 800 400 -400 0 59 R/RQ Figure 12 l00 60 to do its maximum at pumping up excitation in the outer parts, while the second case is fairly extreme in that re- gard. In Figure 11, there is already evidence of some in- ward force on the inner layers due to radiation scattered and emitted in the outer layers. This is seen from the fact that near the star the force in the exact case falls below that in the disconnected case (although it is mostly out- ward). Evidently, the force from IC still overcomes the in- ward force in this case. In Figure 12 the inward force has actually dominated the outward near the star, and the disruptive effect of the interconnections is evident. Line Profile We studied a wide selection of cases with a view to determining: (a) how resonance lines behave as compared with the subordinate lines of Kuan and Kuhi, and (b) what the effect is on line profiles of making the disconnected approximation. We also tried to investigate what charac- teristic features of the line profiles could be used when fitting an observed profile with a synthetic one, to deter- mine the parameters defining the velocity law, the condi- tions in the atmosphere and its dimensions. As we said before, the agreement of the disconnected approximation with the exact solution for the line profile was good. Only in a few cases was the error in the equiva- lent width larger than a factor of 2. We show in Figure 13 the line profiles obtained with 61 Figure 13. Comparison of the line profile in the exact solution (solid curve) and the disconnected approximation. The r given is To. The abscissa is (v-vo)/Avm where Avm = (vo/c)vo. 62 PI- S.OOE+O RR L T: 10 1 O O B: 6:: I 0.80 l 0.U0 I 0.00 FREQUENCY Figure 13 I -.Q0 -.80 00.: om.m d ow.m d om.N o:.N xagm m>-¢4mm .20 1.20 00.9 63 the disconnected approximation superimposed on the exact solution. This was one of the cases where we found impor- tant differences between the two line profiles. As Kuan and Kuhi (1975) pointed out, the displacement of the shortward edge (point with frequency +1.0 in Figure 13) is determined by v0, the velocity at the photospheric radius. If v0 is the central frequency, the maximum dis- placement in frequency is Avm = (vO/c)vo. The displacement of the longward edge is given by their equation (14) _ 2/2 .1, Avl - -[2/(I+1)] [1/(2+1)] Avm . (III-8) from which we can determine the exponent L. This is the largest frequency for which the constant velocity surface is completely occulted by the star. We observed, in most of our profiles, an absorption trough on the shortward side with very well defined limit- ing frequencies. When we move from the central frequency toward the blue side, a sudden drop occurs that we inter- preted as reaching the frequency that corresponds to the first velocity surface completely contained within the cut- off radius rm. It can be shown that the frequency of the intensity minimum associated with this sudden drop is given by — 2’ - sz — (re/rm) Avm , (III 9) 64 from which we can determine the ratio rm/rc. When we are at this frequency or beyond, the light from the core will be absorbed by at least one layer of material. The blue edge of this trough appears when we reach the frequency that corresponds to the velocity surface that starts to uncover the star core, that is Av3 = -Avl. In Figure 14 we see how strongly the profile depends on the value of the cutoff radius rm. A run (not shown) with RR = 15 gave a peak of 5.39, and an equivalent width of -0.655. The equivalent widths given here and in the figure captions are defined as v +Av _1 o m FC - Fv EW = (ZAum) I —'—F—— (IV , (III-10) vo-Avm c One might think that rm = 10 rC is a reasonably large value and it is then possible to neglect the existence of materi- al beyond that radius. From our results it appears that the effect of this additional material in the source func- tion up to 10 rC is small, but that it modifies the line profile, raising in some cases the peak value by a factor 1.5, and increasing the equivalent width (see Figure 14). We must note that Kuan and Kuhi (1975) did not find a simi- lar effect for the subordinate lines. Figure 15 shows two profiles calculated with different 2. For both curves, the velocity at the star surface and the opacity at the outer radius were the same. It appears 65 Figure 14. Comparison of two line profiles for atmospheres with To = so, a = 0.001, B(r) = IC(rC/r)I, and I = 0.5. Curve 3 corresponds to an atmosphere with RR = 10, (equiva— lent width (EW) = -O.410); curve b has RR = 3, (EW = -0.0739). Note that the case 2 = l is expected to be more sensitive to the value of RR (see text). RELAHVE FLUX 66 4.0 2.0 ”' |.0 - — (a) RR= IO (b) RR= 3 -- O-a’ -l.2 -0.8 l J l l -0.4 0 0.4 0.8 I.2 FREQUENCY Figure 14 67 Figure 15. Effect of the exponent 2 in the velocity law on the profile. Curve a: 2 = 0.5 (EW = 0.0086), curve b: 2 = 1 (EW = 0.0053). In both cases RR = 10, T = l, e = 0. 68 (a) (b) |.6 - |.2 '- x:...... 8 0. w>_.rLhfijm> OHEJS§§¥. W44 d d «dd a q a 1 1+14 1‘ 1d dd d a d 114 d u 4 a q u «41 q q d a H. q d _ . _ _ a: . _ _ _ _ _ _ ILLL llll I‘I‘N‘N‘I I LJLL ALEWBMM CERHVWHON I I I 1411 T . H t l I 0 VFII O O ITrI LIJI 7 7 LLLL ITTT JLILO ITUT I rrvr '. L14LL11 VIII I bulb—nPDI—b-bb—bLDP—pbpb—yP-Duban l V V I U ...: _ . FCLZCARENK Figure 19 80 Figure 20. The C IV A 1550 resonance line profile observed for the QSO PHL 5200. 81 PHL 5200 Figure 20 82 characteristic frequencies and from them the velocity law and dimensions. But it is obvious that the sharp drop and the absorption trough are quite similar to ours. The Si IV A 1397 line shows a similar behavior. The secondary peak on the red side (left of Figure 20) corresponds to the Fe II A 1608 line. After this work was completed we became aware of the work of Grachev and Grinin (1976), who solved the expanding decelerating case for a somewhat different velocity field and only for e = 0. They arrived at conclusions like ours about the QSO PHL 5200. Image Figures 21 to 24 show the products p It0t(p,-w) and p Ir(p,-m) (curves with R's) versus p/rm, where vO+Avm Itot(p.-«>) = A) m I(v.p.-oo) dv (III-II) v o Ir(p,-m) = I I(v,p,-w) dv . (III-12) vo-Av These are intended to show the brightness variations one could attempt to look for with interferometry, speckle interferometry, or occultation by a companion or by the Moon. We choose wavelength bands intended to capture a reasonable amount of light and yet present as much contrast as possible for observation. We show only the few cases in 83 Figure 21. p Itot and p Ir versus p/rIn for a star with RR = 3, To = 50, e = 0.001, B = Ic’ and 2 = 0.5. 84 90.00 PII 20.00 29.00 28.00 32.00 35.00 16.00 8.00 12.00 4.00 D O ‘ I I cb.00 0.20 aiua 0760 0.80 P/EXTERIOR RnaIus Figure 21 .00 85 Figure 22. p Itot and p II. for a star with RR 100, e = 0.001, B = 5 IC, and I = 0.5. 86 P I 10.00 12.00 19.00 15.00 18.00 20.00 .00 (97 5.00 L1.00 2.00 O O ' I 00.00 0.2 0 0. I40 0. 60 P/EXTERIUR RRDIUS Figure 22 I 0.80 .00 Figure 23. 1000, e = 87 p Itot and p Ir for a star w1th RR 0.001, B = 5 1C, and I = 0.5. 88 .00 I 0.80 I .90 0.60 P/EXTERIOR RRDIUS Figure 23 11 0 0.20 0 00.0m 8 .MA q oo.3m 8 .8 8 .8 1. 90.0: Hum 8 .mm 8.? 8.? oo.m nw 880 89 Figure 24. p Itot and p Ir for a star with RR = 10, r = 1000, e = 0.001, B(r) = S(rC/r)IC, and 2 = 0.5. Note that all the parameters but B coincide with the parameters in Figure 23. 90 20.00 I 19.00 15.00 18.00 12.00 PII 10.00 9.00 6.00 8.00 2.00 ‘1 I I 00.00 0.20 0.00 0.00 0.00 P/EXTERIOR RRDIUS Figure 24 .00 91 which the results were moderately interesting; in a majori- ty of the cases, even with the factor p adding weight to the curves at large impact parameter, most of the light was concentrated centrally (see Figures 21 and 22). The curve p Ir for p < r is a straight line, under c the assumption that IC is constant. Observe that in the case of the infalling matter, we should change the inter- pretation of Ir’ that now becomes the integral over the high frequency side of the line. When 510 << 1, most of the light comes from the core, with only a small contribution from the p > rC region. If arc is increased to approximately 1, the contribution from the core becomes comparatively small, see Figure 23. The appearance of the stars in the two wavelength bands chosen tend to parallel each other in a way that sug- gests there is little observational information to be ob- tained by interferometry in selected bands. On the other hand, we see that the image shown on Figure 23 is very dif- ferent from the one on Figure 24, due to variation of exci- tation with radius, and both contrast strongly with Figures 21 and 22. Thus some hope is offered of utility. While we unfortunately did not construct similar plots for acceler- ating outflows, we may expect that in such cases, when the excitation in the atmosphere is small, one would observe a smaller image in emission near line center and larger ones far from line center. This is due to the concentra- tion of low velocity material near the star core, and to 92 the large extension far from the star core of the high velocity material, in contrast with our decelerating flow case (see Figures 2 and 3). CHAPTER IV CONCLUSIONS AND FUTURE WORK Conclusions In summary, we come to the following conclusions: 1. The value of the source function is not very sensitive to the disconnected approximation, except at large radius. We found that the larger the value of at the better the o, agreement between the approximate and the exact solution. 2. The whole analysis is rather sensitive to the outer cutoff radius. The line profile is the most sensitive part, and can develop an infinitely steep drop at the fre— quency corresponding to the first velocity surface on the near side contained entirely within the cutoff radius. This kind of behavior is hidden, evidently, for subordinate lines (Kuan and Kuhi 1975), and provides a crucial test of the decelerating models. The resonance profiles of N V, observed for several P Cygni stars by Hutchings (1976) do not exhibit the steep slope at the blue edge of emission that is present inall our calculated profiles. We con- clude that insofar as this sample represents P Cygni stars, the model of Kuan and Kuhi does not seem to apply to such stars . 93 94 3. The extremely steep drop observed for large RR is highly suggestive of the C IV profile in the QSO PHL 5200, which Scargle, Caroff and Noerdlinger (1972) could not fit with any of their models. There is still a problem with the large equivalent width of C IV (when C III shows much symmetry), but an envelope somewhat elongated toward and away from the observer could handle that, by diverting C IV radiation to the side. The worst problem has always been the steep dropoff in C IV to the blue of line center. In the present model, this would occur somewhat to the blue of true line center, but we have verified that for a reason- able RR, such as 100, the dropoff can come right at the edge of the QSO rest frequency within the tolerance of the observations. If this model can be further filled in, it could provide important evidence for the masses of QSOs, or the density of the intergalactic medium, whichever caused the deceleration. 4. The force is highly sensitive to any simplifying as- sumption, such as the disconnections. 5. The appearance of the star as could be observed inter- ferometrically is not very exciting. Unless one adds a lot of emission from the envelope, it will be very faint, and the differences among different models are not likely to be decisive observationally. A final point is that our models all apply equally 95 well to the case of accelerating infall. Hopefully, when better observations are available, we can compare profiles with observations of the stars discussed by Walker (1968, 1972). In the case of infall, we may expect more radical changes in ionization conditions at the place identified as r so the cutoff seems more plausible. Furthermore, m’ in the case of infall, the mechanism (gravity) for acceler- ation seems readily comprehensible, while in the case of outflow, we have seen that for most reasonable combinations of parameters, gravity is insufficient to produce required deceleration, due to the much greater luminosity of these stars . Future Work One of the drawbacks of our formulation is the assump- tion that the population of the lower level of the transi- tion is much higher than the population of the upper one (see equation II-49). This is a good approximation for resonance lines or for lines like the C III A 1175, where the lower level is a metastable one. These kinds of lines are usually in the ultraviolet region of the spectrum, which limits tremendously the observing possibilities. Our next step will then be to extend our formulation to the case of subordinate lines. We must find the populations of the upper and lower levels under conditions of Non Local Thermodynamic Equilibrium (NLTE). We have to consider all the processes that tend to 96 populate or depopulate each level and solve simultaneously the rate equations and the transfer equation. Our equa- tions will now be (II-27), (II-28), (II-29), and instead of the two—level atom excitation law (II-30), we will have a system of equations with the populations as unknowns. Following Mihalas, Heasley, and Auer (1975) we consi— der an atmosphere composed of hydrogen and helium. The atomic model for hydrogen assumes 16 levels, allowing de- partures from equilibrium for the first five levels. For helium we assume a 16 level He I atom and a 32 level He 11 ion, with the possibility of having the first two levels of each ion under NLTE conditions. The computation of the rate matrix involves the deter- mination of the collisional excitation and de-excitation rates as well as the radiative transition rates for each pair of levels. Obviously, the fact that we must solve the equations for the whole atmosphere simultaneously, demands a very large storage capability. APPENDICES APPENDIX A Let da be an infinitesimal element of area, and P a point on it (in this Appendix we follow closely Cox and Giuli (1968) and Rybicki (1970)). The unit vector perpen- dicular to the surface is 0', and 0 is another vector that makes an angle 0 with 0'. If dEv is the energy that flows through da in time dt inside the solid angle dm about the direction of 0 with frequency between v and v + dv, we de- fine the specific intensity Iv as dEv = Iv dv da cose dm dt . (A-l) A Iv will be a function of frequency 0, position I, angle n and time t. When the beam of radiation passes through an element of mass dm, the energy in the beam with frequencies be- tween 0 and v + dv contained in the solid angle dw will in- crease due to emission in the element of mass by the amount dE'v during the time dt, and it will be given by dE'V = jv dv dm dw dt , (A-2) jv being the mass emission coefficient. Note that it in- cludes true emission as well as photon scattering into the beam from other directions. Similarly, we have an 97 98 absorption coefficient Kv (true absorption and scattering out of the beam). If no emission occurs, the change in the specific intensity Iv of a beam traveling a distance ds through a medium of density p is given by d1\) = 'Kv p Iv ds , (A‘3) where Kv is the mass absorption coefficient. The corre- sponding change in energy is dE" = -I K 9 ds d0 da cose dm dt . (A—4) v v 0 But p ds da cose = dm, thus dE'v = jv dv p ds da cose dw dt . (A-5) The change in energy in the beam allowing for emission and absorption is then dlvdvda case dwdt = (jV-KvIv)p dvdsda c050 dwdt , (A-6) or equivalently (4-7) this being the general expression of the equation of trans- fer. Let us consider now a frequency V0 corresponding to the transition j + i in a given atom. The mass absorption coefficient KV will be a sharply peaked function with its maximum at the frequency corresponding to the line center 99 v0. The line opacity in the rest frame of the material is k = p K , (A-8) and the integrated line opacity Riff) = IO kvfi?) d0. (A-9) The normalized profile function is defined by , k (1*) dIr,v) = X g . (A-10) k,(r) If the material that is absorbing and emitting is moving at high velocities with respect to the observer, we have to include the effect of the velocity field on the frequency. Different parts of the material will see the same photon at different frequencies. The normalized pro- file will now depend on the direction 0. Considering the Doppler effect to lowest order in v/c ¢(¥,fi,v) = (mi, \7 - 3’59}; - (7(a)) . (A-ll) Due to the sharpness of the absorption profile, this shift in frequency can produce a substantial change in the opti- cal depth along the ray path in a moving atmosphere. We can now write the transfer equation as A n-VI,(?.3) = p m?) - m?) 1,623) , (A-12) 100 \) A A Con-I(¥))[S(?)-IV(?.n)]. (71-13) A +A + + n-VIv(r,n) - k£(r) ¢(r,v- where the source function 8 is defined as ° + N (+ A S(r) = J\)(r) = + 2 r) 21+ , (71-14) Kv(r) N1(r) B12 - N2(r) B21 where N2(;) and Nl(f) are the number density of atoms in the upper and lower level respectively of the transition we are con51der1ng, and A21, B21, and B12 are the E1nste1n co- efficients. If T is a characteristic temperature in the atmo- sphere, we can define a typical Doppler velocity 2kT ‘ Vth = [T]I’ (A'lS) and a Doppler shift 0 V _ 0 th - A0 _ _—77__’ . (A 16) Normalizing with this Doppler shift our frequency, referred to the line center, we obtain the dimensionless variable x = ————° , (A-17) and the line profile 101 76.x) = —__—l— epr-xZ/aZIrII , (A—18) /n S(r) where v (T) Mr) = -t—I‘:—-— - (Ix-19) Vth Using Vth as the unit for velocities, do?) = (I) . (II-20) th <~I <1 the equation of transfer becomes fi-v1x(¥,fi) = kg ¢(f,x-fi-fi(f))[8(f) - Ix] . (A-21) Notice that X' = X-0-0(?) = [v - v - v (A-ZZ) J; 2 o 0 A0 is the frequency seen by the material in the co-moving frame, measured from the line center in units of the ther- mal width. Using x' as independent variable instead of x, the mathematical identity for the gradient becomes 3 31 A A -+ BI “ A A l x' 11° Bu x' n-VI (I n) - n-[ 2 H‘ e - (2 H‘ -—-) I , C O 3 O x J=1 J r1 3 J J r1 3X' + BIX = n-VI , - Q(r,n) , (A-23) 102 with (I A) - I :13- 33 A-24) Q r,n ' h. 3r_ 1 I J J J and _ 3x 2 3y 2 32 2 I . hj _ [(ar. + (31.) + (31.) ] ' (A 25) J J In spherical coordinates, and with 0 = u(r) er 5% = u 66 , 5% = u sine e¢ , 5% = u' er , (A-26) A 2 2 2 u 2 2 Q(r,n) ‘ n U'+ne % + n¢ ; = u U' + (l-u )% . (A-27) being hr = l, h6 = r, h¢ = r sin 9, and nr = u. The gra- dient, keeping x' constant is: A A BI , 2 I 31 . - X - (l-u ) x' n VIx,(r,n) — U 3r r 80 31 , 2 81 _ (l-u ) X' - 0 3r + r 3p ' (A 28) Collecting our results together 103 . = __5_ (1-u ) x' n VIX(r,n) N Br + r Bu aI , -Iu2u'(r)+(1-u2)%15;’f— , (H9) and finally BI 2 81 , 2 2 u 81X, 'I u'(r)+(l-u );I§§T— = C “—r k2 ¢(r, v - 7? n-v(r))[S(r) - Iv(r,0)] . (A-30) APPENDIX B To solve equation (II—40) we approximated it by a sys- tem of linear equations in the n unknowns S(rl), S(rz), ., S(rn), where r1 = rC + (1-l)(rm - rn)/(n-l) . (B-l) For each r = r1 we obtain an integral equation that we dis- cretize, substituting for each integral a sum, obtaining an equation of the form n Z1 aij S(rj) = H(ri) IC + €B(Ti)/(1 - e) . (B-2) To obtain the weights aij we performed the integrals over z using trapezoidal rule, dividing the interval of integra— tion in 250 steps. Given now the point (r,z) there is a constant vZ curve that passes through that point. ~Once this curve is determined we must find its other intersec- tion with the p = constant line, where p = (r2-22)I. This new intersection will be at r2, and r2 is computed from equation (II-22) with p = —(r22 - p2)I/r2. In obtaining r2 we must solve a cubic equation in the L = case, and a Nll—I quadratic equation in the 2 = 1 case. 104 The and the P From the In the 2 u In the 2 ( As we kn we can 1‘ root is 105 line of sight velocity is determined by reg 2 re2 22 z = Vo £+l = vo 2+1 ’ (B-B) r r 2 impact parameter is = (r2 - ZZ)I = (r22 - 2221i . (B-4) se two equations we obtain r 2 _ 22 = r22 _ (_2)2(£+1) Z2 . (B-S) r = 1 case, taking u = (rZ/r)2, (B-5) reduces to z 2 2 z 2 _ _ ;) U ' U + 1 ‘ I?) ‘ 0 9 (B 6) utions u = 1 and = (g)2 - 1 . (B-7) = % case, with w = rZ/r, equation (B-S) becomes 17:.)2 W3 - w2 + 1 - ($2 = 0 . (B-8) ow that w = 1 (known intersection) is a solution, educe (B-8) to a quadratic equation. The positive kn— -——_.- 106 w = I ((g)2 - I + I(§)2-IIiI(§)2+inI . (13-9) Once r2 was found, we determine the two "zones" that comprise this second intersection, i.e. rk < r2 < rk+1. If we are solving the equation corresponding to r = ri, we then have contributions to the terms aik’ and a1 k+l’ that are determined by assigning linear weights depending on how close r2 is to the zones rk and rk+1. In the disconnected case, all terms in equation (II-40) containing a quantity subscripted 2 are discarded, then aij becomes diagonal, and the surviving terms are actually only integrals over 0 = -z/r at fixed r; hence no interpolation procedure is needed. The trapezoidal rule with 100 steps was used. We constructed the aij matrix a row at a time, and at the same time corresponding coefficients for equations (II-51) and (II-52) were computed and stored, later to be folded into the vector of S values, after it is found. A standard IMSL routine, LEQTZF, was used to solve the system of equations. Occasionally, fluctuations were encountered in S(r) and a(r) due to finite zoning, and these decreased slowly but steadily as the zoning was refined. APPENDIX C In the disconnected approximation, equation (II-43) yields 2 EB I I‘ e S(r)[—1—E_:—E + 71; I amaz] = LE + .251,— I G(r)dz , (C-1) -1‘ -I‘ where C(T) is defined in equation (II-38) and r is evalu- ated at r, with direction cosine u, given by u = -z/r. If r << 1, G = l and the integrals are trivial, leading to € _ e Ze + r SIT)I1_€ + 1] - ITE B + Ic(__7F__) , (C'Z) or simply S(r) = eB(r) + IC W(r)(l - a) , (C-3) where W(r) is the dilution factor given by (II~46). If T >> 1, 0(1) 2 T-1 = Tr-l 2-1 |l-uz(£+l)I, (see equation II-22). In this case 107 108 B(r) = 1’ Ir “(I'dz = 1 I1 II-u2(z+1)ldu = .2—1: ‘I‘ ZlTr J_1 1 {I-(MIYIr [ 2(2+1) 11d +I(2+1)—I [1 2(I+1)]d + 2———- u - u 'u u 2Tr J-1 -(g+1)'I 1 2 I ,, [u (z+1)-11du}. (c-4) (2+1) Carrying out the integration gives: B(r) = 1 [§3 (1 - 3L - 1 + 3 (C-5) Err 3 83 S ’ where S = (£+1)I. Defining pc = rC/r, the other integral is - - 2 I 1 I (1 MC ) 22.131. J_1 Il-uz(2+1)Idu . (C-6) If r/rC > [(2+l)/L]I, the integral becomes 2 I 277 I (l-UC ) [02(£+1)-1Idu = r J-1 {7;1 II-(I-uC213/2+(I-ucz)i-II x 75:; . (c-7) For “c small, it is approximately 3%; NC2 = 41r (2912. (c-8) Combining equations (C-l), (C-5), and (C-8), we obtain for 109 T >> 1 and r >> rC 2 1.177 1 r B . 1 SIIFE I r I ”'E?‘ (7 +'18-7: If i = 2 (C'g) I‘ 1‘ I‘ and r 2 8 0.6095 ~ 1 c CB . = _ SIT: + T] — 4Tr r2 + 1_€ , 1f 2, 1. 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