INTERFEROMETRIC STELLAR SPECTROMETRY
by
Ian Wynne Wynne-Jones
Astronomy Group Blackett Laboratory, Imperial College, London SW7 2BZ
Submitted to the University of London for the degree of Doctor of Philosophy
London, December 1980 2
"Truth is the child of time, not of authority. dcwHnc^uk Our ignorance is infinite, so let us .diminition by a fraction.
Why try to be so clever now, when at last we can become a little
less stupid?
I have had the unbelievable good fortune to lay my hands on a new instrument by means of which one can see one tiny corner of the universe a little clearer. Not much - but a little.
Make use of it I "
From The Life of Galileo
By Bertolt Brecht. 3
ABSTRACT
The theory of Fourier transform spectrometry is surveyed.
The use of the Michelson interferometer for the Isaac Newton telescope is described. The instrument was used to study stellar spectra at high spectral resolution and high signal-to-noise ratio ^ 100 .
The atmospheric motions of the luminous K stars, Arcturus, Hamal,
Kochab and Pollux were studied. The profiles of their neutral iron absorption lines were measured and synthesized to obtain the microturbulent, macroturbulent and rotational velocities of their atmospheres. A Fourier technique was employed to dissect the line profiles, but it was concluded the method has no advantages compared with more traditional statistical tests. Evidence suggests a more realistic model for spectral line formation in turbulent media is required to synthesize the line profiles consistently.
The hyperfine structures of the interstellar sodium D1 and D2 absorption lines in the directions of *-250 ms ^ ,indicating diffuse interstellar cloud temperatures of 200°K and subsonic internal cloud motions. It was also concluded that the number density of interstellar clouds and variation of cloud column density was greater than previously thought. 4 ACKNOWLEDGEMENTS Firstly, I am indebted to Dr. R.C. Wayte for his guidance and example. I greatly appreciated the opportunity to learn from him and latterly, with him. Secondly, I am most grateful to Professor J. Ring for his supervision and constant encouragement. Thirdly, I would also like to thank Dr. A.P. Thome and Dr H.J. Walker for their constructive criticisms of the draft of this thesis. For their discussion and interest I am grateful to: Professor D. Blackwell, Professor D. Gray, Dr. R.Griffin, Professor B. Pagel, Dr. N.K Reay, Dr. M.A. Smith and Dr. S. Wychoff. To Professor J. Brault, Mr. C. Amos, Dr. K. Hartley, Mr. D. King and Dr. D. Youll I am thankful for advice on computing. I. should also like to thank members of staff and research students in the Astronomy Group at Imperial College and the Royal Greenwich Observatory, Herstmonceux for their help and friendship. Thanks also goes to Mrs. T. Wright for so patiently typing this thesis . X would like to acknowledge the PATT committee for their generous allocation of telescope time on the Isaac Newton telescope and to the Science Research Council for their financial support. Thanks is also due to the R.G.O. and I.C., computing and library sections for use of their resources. Finally, I should like to thank my parents, to whom I dedicate this thesis, for their understanding, and example of diligence and perseverance. 5 CONTENTS Page No CHAPTER 1 FOURIER TRANSFORM SPECTROMETRY 1.1 Introduction ^ 1.2 Theory of the interferogram ^ 1.3 Recovering the spectrum 1.4 The discretely sampled interferogram ^ 1.5 Discrete sampling restrictions ^0 1.6 Flux and acceptance ^4 1.7 Throughput and Luminosity ^9 6 97 1.8 Internal modulation ^ 1.9 Path difference errors 29 1.10 Demodulating the secondary interferogram JO 1.11 Sample interval measurement oJ/ 7 1.12 Interferogram sample signal-to-noise ratio ^7 CHAPTER 2 DATA REDUCTION OF INTERFEROGRAMS 2.1 Introduction 39 ,2.2 Theory 39 2.3 Spike elimination 41 2.4 Normalization of the spectra 48 2.5 The signal-to-noise ratio in the spectrum 50 2.6 Radial velocity measurements 53 CHAPTER 3 SYNTHESIS AND ANALYSIS OF STELLAR LINE PROFILES 3.1 Introduction 57 3.2 Line profile synthesis in local thermodynamic 58 equilibrium 3.2.1 Level populations 60 3.2.2 The absorption coefficient 61 3.2.3 Van der Waa^s broadening 63 6 Contents (Continued) Pa^e No 3.3 Radiative transfer 53 3.3.1 Flux integration gg 3.3.2 Continuous opacities gg 3.4 Model atmospheres gg 3.5 Macro turbulence 7^ 3.6 Rotation 72 3.7 Fourier analysis of line profiles 74 3.8 Macroturbulent and rotation filters 76 3.9 Synthetic line profiles 77 CHAPTER 4 STUDIES OF Fe I ABSORPTION IN LUMINOUS K STARS 4.1 Introduction gp 4.2 Selection of line profiles gg 4.3 Choice of star ^ 4.4 Analysis of line profile data 94 4.5 The Fourier analysis method 4.6 Results and discussion 4.6.1 Accuracy of profile measurements 4.6.2 Abundance and microturbulence ^24 4.6.3 Macroturbulence and rotation ^q 4.7 Conclusions 2.30 CHAPTER 5 INTERSTELLAR Na I D ABSORPTION LINE STUDIES 5.1 Introduction ^32 5.2 Interstellar absorption line profile synthesis ^34 5.3 Instrumental profile broadening ^35 5.4 Saturated line profiles ^3g 5.5 Synthesis of Na I D1 and D2 interstellar lines ^39 5.6 Observational data 7 Contents (Continued) Page No 5.7 Data Analysis 152 5.8 Results of model fitting 5.8.1 The °C Cygni sight line 153 5.8.2 The S Cygni sight line 169 5.8.3 The Tau sight line. 169 5.9 Discussion and comments. 175 8 TABLES 2.1 Number of spikes eliminated versus standard deviation. 47 3.1 Comparison of continuum opacities. 69 401 Atomic transition data of line profiles measured. 92 4.2 Stars observed. 95 4.3 Equivalent width grid for Fe I ^6151 X of Arcturus. 97 4.4 Mean square difference grids for Fe I ^6151 X of Arcturus. 98 4.5 Best fit parameters of line profiles. Ill 4.6 Average values for velocity parameters of each star. 112 4.7 Ratio of collisional to radiative ionization excitation rates 126 in the photosphere of a late type star. 4.8 Equivalent width verses ^29 5ol Atomic parameters used to synthesize Na I D1 and D2 lines. 141 5.2 Details of interstellar Na I D1 and D2 observing runs. 14$ 5.3 Stars observed for their insterstellar spectra. 151 5.4 Properties of well resolved interstellar components. 160 5.5 Seven cloud fit to o( Cygni line of sight. 160 5.6 Nine cloud fit to Cygni line of sight. 170 5.7 Model fits to S Cygni and At Tau. 176 9 FIGURES Page No 1.1 A schematic diagram of a Fourier Transform Spectrometer. 15 1.2 A discretely sampled interferogram instrumental profile. 21 1.3 A continuously sampled interferogram instrumental profile. 22 1.4 The output intensity for an off-axis ray. 25 1.5a Asymmetric instrumental profiles. 31 1.5b Asymmetric absorption profiles. 32 1.6 Internal modulation amplitude error. 35 2.1 An interferogram before spike elimination. 45 2.2 An interferogram after spike elimination. 46 2.3 A spectrum before normalisation 51 2.4 A spectrum after normalisation. 52 3.1 Specific intensity. 64 3.2 Line profile dependence on element abundance. 78 3.3 Line profile dependence on microturbulence. 80 3.4 Line profile dependence on rotation. 84 3.5 Line profile dependence on macroturbulence. 86 3.6 Rotation profile Fourier transform filter. 87 3.7 Macroturbulence profile Fourier transform filter. 88 4.1a The spectrum of Dubhe around ^6065 8. 93 4.1b The spectrum of Aldebaran around "A5379 8. 93 4.2 The Fe I ^6151- 8 line* ofr-Arcturus. 99 4.3a The Fe I ft6151 8 line of Arcturus. 101 4.3b The Fe I ft6151 8 line of Arcturus. 103 10 FIGURES (Continued) Page No. 4.4 The Fe I A6027 X profile of Arcturus. 106 4.5 The Fe I 7\6027 X profile transform of Arcturus. 107 4.6 The Fe I 7\6065 X profile of Arcturus. 108 4.7 The Fe I ft6151 X profile of Arcturus. 110 4.8 The Fe I ^5379 X profile of Hamal. 113 4.9 The Fe I TV 6027 X profile of Hamal. 114 4.10 The Fe I J\6065 X profile of Hamal. 115 4.11 The Fe I 7*6151 X profile of Hamal. 116 4.12 The Fe I ft5379 X profile of Kochab. 117 4.13 The Fe I ft 6027 X profile of Kochab. 118 4.14 The Fe I ft 6065 X profile of Ko chab. 119 4.15 The Fe I ft6151 X profile of Kochab. 120 4.16 The Fe I ft5379 X profile of Pollux. 121 4.17 The Fe I ft6027 X profile of Pollux. 122 4.18 The Fe I A60 65 X profile of Pollux. 123 4.19 Non-LTE Fe I A6151 X profile of Arcturus. 128 5.1 Synthetic interstellar Na I D1 lines for various velocity dispersions. 142 5.2 Synthetic interstellar Na I D1 lines for various column densities. 143 5.3 Instrumental broadening of an unsaturated line. 145 5.4 Instrumental broadening of a saturated line. 146 5.5 Resolving power verses profile accuracy. 147 5.6 The +lkm/s D1 line in Cygni: a) Gaussian velocity distribution, 155 b) exponential velocity distribution. 156 5.7a A typical interferometer passband. 157 11 FIGURES 5.7b Previous high resolution Na D2 spectra of oc Cygni. The ot Cygni sight line. The seven cloud component fit to the interstellar Na D2: 5.8a showing the individual cloud components, 5.8b showing the sum of the components. The seven cloud component fit to the interstellar Na Dl: 5.9a showing the individual components, 5.9b showing the sum of the components. The nine cloud component fit to the interstellar Na D2: 5.10a showing the individual components, 5.10b showing the sum of the components. The nine cloud component fit to the interstellar Na Dl: 5.11a showing the individual components. 5.11b showing the sum of the components. 5.12 The interstellar Na D2 profile in 8 Cygni. 5.13 The interstellar Na Dl profile in 8 Cygni. 5.14 The interstellar Na D2 profile in Taurus 5.15 The interstellar Na Dl profile in Taurus 12 CHAPTER I FOURIER TRANSFORM SPECTROMETRY 1.1 Introduction Michelson (1891, 1892) was first to employ the two beam, inter- ferometer of his invention to analyse spectral lines, by studying their visibility curves. He recognised the superior resolving power of the device over contemporary spectroscopes. He comments that the technique "proved of considerable value especially in cases where the effects to be observed are beyond the power of the spectroscope" (1927). Fourier Transform Spectrometry developed by Fellgett (1956) uses the Michelson interferometer's ability to optically multiplex. Each spectral element is coded as a sinusoidal signal whose modulated frequency is proportional to the optical frequency of the radiation. The signal amplitude is proportional to the flux corresponding to the intensity of the spectral element. The source spectrum is decoded by an harmonic analysis of the output signal which is the sum of many sinusoidal signals. The sinusoidal signals are recorded simultaneously with a single detector. Detector noise is thus distributed amongst all the spectral elements. When limited by detector noise the Fourier Transform Spectrometer (FTS) has a multiplex advantage compared with a sequentially scanning spectrometer. The multiplex gain reduces integration time by a factor equal to the number of spectral elements for a given resolving power and signal-to-noise ratio. * The infra-red spectra of planetary atmospheres obtained by the Connes1 (1969) clearly demonstrate the FTS's attributes of high spectral resolution and multiplex gain. In the visible and ultraviolet spectral regions there is no multiplex gain. The signal-to-noise ratio is limited by photon shot 13 noise and not detector noise. However other properties of Fourier transform spectrometers commend their use. Jaquinot (1954) pointed out that the solid angle of radiation accepted by an optically circularly symmetric interferometer is about two orders of magnitude larger than that of a dispersing spectrometer of the same resolving power. The Jaquinot advantage arises because the interfering beams are separated by intensity modulation rather than by spatial division, Connes (1970) shows it is possible to measure accurately the wavelengths of all the spectral lines in a spectrum by comparison with a single line of standard wavelength. This is especially valuable for determining radial velocities. Ring and Stevens (1972) illustrate the characteristics of the sine function instrumental profile of a Fourier Transform Spectrometer. The sine function instrumental profile causes much less filling in of the cores of sharp absorption lines than the Gaussian Ttype' instrumental profile of a dispersing spectrometer of the same resolution. Brault (1976) mentions that generally the Fourier transform spectrometer instrumental profile degrades partially resolved spectral features considerably less than instrumental profiles of comparable single or double pass grating spectrometers. Spectra obtained with Fourier transform spectrometers are not troubled by scattered light - Brault (1979). The scattered light is not modulated and hence not detected as a signal. The location of zero intensity in the spectrum is well defined, so that equivalent widths and residual intensities are precisely known. With scattered light in grating spectrographs or spectrometers (Griffin, <1969) and parasitic light in poly Fabry-Perot interferometers (Roesler, 1967) this is not the case. The solar spectra obtained by Brault (1979) exemplify these FTSfs properties of high throughput, low instrumental profile distortion of 14 spectral features, accurate wavelength calibration and freedom from scattered light. Having discussed some of the properties of the FTS it is sensible to describe the theory of the instrument^to appreciate its application to the astronomical problems dealt with later in the thesis. 1.2 Theory of the interferogram A schematic diagram of a Fourier transform spectrometer is shown in Fig 1.1 , along with a list of its principal components. Consider a plane wave of amplitude a and wavenumber The two outputs are each composed of two waves. One output, the sum of a wave reflected twice and a wave transmitted twice by the beamsplitters, the second output the sum of waves both suffering one beamsplitter reflection and transmission. There is a phase shift of TT at each reflection so the expressions for the amplitudes of the waves described thus far are:- For the input wave a. = a ei(<° ' " (1) 1 A schematic diagram of a Fourier transform spectrometer 11,2 - Input ports. 0 1,2 - Output ports. B 1,2 - Beamsplitters. Rl,2 - Catseye retroreflectors. P 1,2 - Post monochromator. C 1,2 - Collimating and Camera lenses S - Input and output D 1,2 - Photomultiplier detectors. I,1,2 apertures. 16 For the wave division by the first beamsplitter i(Wt 2 ar = ar1e - "'l'V > (2,i) afc = a t, ' " ™ > (2,ii) For the wave division at the second beamsplitter ^-ViV^"2^'' (3,1) i t 2lT + arr = ae1r1r2e ^ - ^ > (3,ii> i t 27T + art = ae1rlt2e (" - * > (3,iii) a = ae t r ei(Wt 2ir X2 The subscripts r and t denote reflection and transmission and specify the order in which they occur, (0 refers to the angular frequency of the radiation and t time. Combining the two output waves at each of the outputs, the resulting amplitudes are i(U>t-2TTx it it The time average intensity is I = a a where a is the complex conjugate of a. The resulting output intensities are thus I ^A + B cos (2-fT x CT (5,i) I2 = I £c - B cos (2TTx£T)J- (5,ii) where 2 2 2 2 2 2 A = tx r2 e2 + rx t2 e1 (6,i) B = 2rx r2 ^ ex e2 (6,ii) 2 2 2 2 2 2 C = tl t2 e2 + rl r2 el (6,iii) 17 The values A and C may be termed the transmission factors and the constant B the modulation factor of the spectrometer. The output intensities are complementary, beamsplitter absorption is assumed 2 negligible. The reflection coefficient R = r and transmission 2 ... coefficient T = t and the catseye intensity attenuation factor 2 E = e . If e^ =» e2, r^ = r2 and t^ = t^, then A = B (7,i) B = 2ERT (7,ii) C = E(T2 + R2) (7,iii) It is evident the modulation term B is largest for a beamsplitter that reflects and transmits equally. Also the modulation is preserved despite large variations from the optimum, noting that R + T = 1. The modulation is complete for the output of the waves which have undergone one reflection and transmission. Addition of the two output intensities show that the input intensity is conserved apart from losses in the catseyes. 1.3 Recovering the Spectrum Admitting a continuous distribution of waves of source intensity I(tf') the output intensites become I^x) = A.AV(O) + B.M(X) (8,I) I2(x) = C.M(O) - B.M(X) (8,ii) where M (x) = ^ I(0T) cos (2"|\K The expressions for the interferograms I^(x) and ^(x) are of the same form. They are composed of a constant term proportional to the source intensity and a modulation term which is the cosine Fourier transform of the source intensity distribution. The constant term may be subtracted off by measuring the intensity of the interferogram where 18 there is no modulation^enabling /A (x) to be determined knowing B. The source intensity can be divided or it can be separated into odd and even parts, noting I( even P( odd Q( Substituting equation (9,iii) into (8,iii) and simplifying M (X) = J"p(0 cos (2TTXCT) (10) The modulation term is therefore even enabling P (C) to be recovered by evaluating the cosine Fourier transform of M (x), as follows: fur an cwbi-trarift P((T;) = j M CX) cos (2Tx P( P(CT') = P( ^V _ f 1 for = The Dirac Delta (11,iv) L 0 for The intensity I( 2 p( 0 for c $ o In practice the interferogram is neither sampled over an infinite range of path difference or continuously. The effect of the finite number and discrete samples actually used to represent the interferogram on the spectrum must be considered. 19 1.4 The Discretely Sampled Interferogram at Sampling the interferogram a set of K equally spaced intervals A J in path difference starting at zero path difference, the expression for the interferogram modulation, Equation (10) becomes : M(k) = j P«T) e21Pik8 -2lTik8cr' , Multiplying both sides of (13,i) by the factor e where (f^ is the wavenumber of interest, and summing all these equations for each value of k, the left and right hand sides become K-l 2 1 LHS = K^e- " ^^' (U,i) fc = . K-l / RHS = 5Z I P(r)e2 7Tik^(tr-,r )d Changing the order of integration and summation RHS = S (K0 K-l where S (K.jSjflr' ) = £^2TTik$( This is a geometric series which simplifies to c ^ fi - ^TTiKSC^-^) 1 S a, 8> S (KjSjtf') _ cos(TT(K-l) g ( Re ^ S (K j S jV^ is the instrumental profile or scanning function and is shown in Fig. 1.2 for the case when K = 32. Inspection shows that the instrumental profile has transmission maxima at intervals in CT of 1/8 with a maximum centred on . The width of the instrumental profile Act defined as half the separation between a maximum and the nearest zero, then K 9 =1. If X is the maximum path difference then the spectral resolution R = CT /& If we let the number of samples K —^ cO but keeping the maximum path difference fixed so that ^ 0, then Re£S(X, = sinc( fx( CT - ^ )) (17) The discretely sampled interferogram instrumental profile is not a sine function, but approaches it for large numbers of samples. The instrumental profile for the discrete sampling case can be considered as the sum of many sine function instrumental profiles corresponding to the same maximum path difference, placed at intervals of J /S The sine function instrumental profile is shown in Figure 1.3., 1.5 Discrete Sampling Restrictions To measure a spectrum using discrete sampling we are required to place some constraints on the sampling method and/or the spectrum observed. Firstly, to prevent contributions to the spectrum from more than one transmission maximum the bandwidth of the spectrum must be less than the separation of the instrumental profile transmission peaks. Secondly, as the sampling assumes the intensity distribution is an even function to avoid overlapping the authentic spectrum with its reflected duplicate it is necessary to increase the peak separation by a factor of two. If & and are the upper and lower limits of Spectral sample intervals -50 -40 -30 -20 -10 0 10 20 30 40 50 FIGURE 1.2 - Instrumental profile of a discretely sampled interferogram. Number of interferogram samples = 32. Instrumental profile widths 1.0 - u a) 4-1 B FIGURE 1.3 ~ Instrumental profile of a continuously sampled interferogram. 23 the spectrum then the instrumental profile tranmission peaks must be separated by at least 2(0" - (T ) • These constraints are referred U Lri to as the Sampling Theorem. The instrumental profile is scanned across the spectrum by increasing ; however, the values of chosen are not unique and any value such that = Thus if a band limited spectrum ranges from to j rather than measuring the spectrum from 0 to & requiring (2p + 1) = K K samples at intervals of £ l/J^p + 1) 2( C^ - 3 samples at intervals of So = 1/2( 1966) . However, its use requires some qualification in that the instrumental profiles for the two methods differ, since the number of samples are not the same. Turning to the LHS of Equation (14) and evaluating the spectrum at K points such that 1 C = 0~L + j htr j = 0,1, K-l (19,i) where 4 Noting that J0 This summation may be evaluated with the fast Fourier transform FORTRAN routine listed by Gray (1976). The spectrum is evaluated not only for the spectrum itself but also the following mirror alias. 1.6 Flux and Acceptance The previous deductions were for a plane wave travelling parallel to the interferometer optical axis. This corresponds to a point source subtending an infinitesimally small solid angle from the collimating lens. To admit radiation the source is extended so that there is a spread in angles of incidence. For a Michelson interferometer the spread is ,1 normally restricted by a circular aperture of angular radius OC . Also the optics is adjusted so that the two images of the beamsplitter formed by the catseyes are parallel so that the interference fringes are located at infinity. The path difference x ' of a wave incident at an angle Q to the optical axis is x ^ = x cos & . Hence the output intensity for this off axis ray is from Equation (5,i) 1(0) = I + Bcos (2 TTx cos&.cT) J (21) which is illustrated in Figure 1-4 The flux F accepted is found by summing the intensity over all the solid angles that the aperture subtends from the collimating lens. Waves incident at an angle l7" and & + d$ form a beam of solid angle dSL = 2*TT Br d 6. Their path difference assuming & is small is x ' = x(1 - &2/2) = x(1 - JI/2TT ) since the solid angle JL =!T£2. I ' 2 Summing over complete aperture ,JL = "IT ^ . The expression for the flux F becomes 25 0.0 & FIGURE 1.4 - The intensity for an off-axis ray. (a) For 3 cm path difference (b) For 2 cm path difference. (c) For 1 cm path difference. For ^ = 5000 8 and 9 = 10 2 radians. max 26 Jl' F J T (JL) dJL- (22,0 F= J I £ 6 COS (21TX0 - J2-/2T"))J D(22,ii) There are two points to note. Firstly, the modulation is reduced by the factor sinc( Jl? x (T 12) which imposes limits on the spectral resolution. If we consider waves incident at angles & = 0 and & 3 0 of wavenumbers and ^ > then they cannot be distinguished if = cos & • wavenumber resolution is thus = (T^" which for small QC/ gives %(T ~ or for spectral resolution R = 2-JT/Jl/. Multiplying the interferogram modulation term by the sine factor is equivalent to convolving the spectrum by a rectangular function of width ( Alternatively we may consider that the instrumental profile is broadened by the effect of the finite aperture. Secondly, the wavenumber scale is compressed slightly but can easily be corrected by multiplying the wavenumber by the factor (1 + Si! JT ) • 1.7 Throughput and Luminosity The throughput (MertzyL965 ) or Etendue (Bonsquet, 1972) is the product of the solid angle JI subtended by the entrance aperture at the collimating lens and the effective area S of the collimating lens. Provided there is no vignetting it is a geometrical invariant of the optics and determines the quantity or luminosity L of radiation that can be put through a spectrometer. For astronomical observations the telescope and spectrometer throughputs should also be matched. The telescope throughput is the effective area of the primary mirror times the 27 solid angle subtended by the source studied. If D is the transmission of all the non-modulating optics then luminosity output is L = STJ\»D6* The resolution luminosity product is then R,L = 2 "TTSIyR^DGr. There is thus a reciprocity between resolution and luminosity. Although there is theoretically no limit to the resolving power, a practical limit is imposed by the loss of luminosity accompanying an increase in resolving power. 1.8 Internal Modulation To compensate for variations in source intensity, such as stellar scintillation, the path difference X is switched back and forth by i &X Each interferometer output now gives two signals which are separated by phase sensitive detection. The two measurements are made at path differences X + Jx and X - , and at such a switching frequency so as to effectively freeze the star image for consecutive measurements. The output intensities are given by Wayte ('1078). + I1 s <* k I £ A + B cos (2 1T s + I2 r £ k I £ C - B COS (2TT I2" : f k i £ c - Bcos (2lTcr (X-Sx))^ (23, iv) These can be combined to eliminate the atmospheric transmission factor k, and the detector efficiencies and ^ , in the following manner i V'1!" t v-V I = — and I = — i- (24,i +ii) Xx +XX I2 + 1 which yield from Equations 23(i) to'(iv) for a complete spectrum 28 i/oo = /(r8 + Bl<£*>//)) (25,0 # i2 fto = rs (x) / (rB- b rcc*>/c) where isoo - £ w ^T.SX.O (25>ldLi) i Oc) = S IMc*S(2TTxr)M(zir.S*.r)d(r (25>IV) j*l£ These transforms give spectra of the form I( CO = -IS(X)/IB and I2' ft) = IsOO/IB (26,i +ii) For a narrow band alias %X is set equal to l/2( (T + . ( If the samples are measured at intervals of path difference that are integer numbers of the mean wavenumber of the spectrum the complete 29 interferogram modulation is measured. This is analogous to the fringe visibility method used by Michelson (1927). 1.9 Path Difference Errors Path difference errors are of several forms; when the interfero- gram is not sampled at equal intervals, or each individual sample is measured over a region of path difference, optical defects distorting the interfering plane waves, chromatic dispersion and using an incorrect internal modulation amplitude. The phase errors modify the intensity measured in the interferogram and can impose some rather peculiar features on an otherwise normal spectrum. Thus if these phase errors are not accounted for and minimized we may misinterpret effects of instrumental origin to be due to the source. In this application the phase errors could be attributed to some astrophysical phenomenon. If the interferogram is sampled starting at a distance e off zero path difference the intensity modulation from Equation (8) becomes (28) Expanding the cosine term shows that the sine transform is included so that the modulation interferogram now appears asymmetric. For small phase errors where e l/ sine transform is proportional to e from which can be found the condition that (29) to measure the interferogram to a signal-to-noise ratio of SN1?. Applying the same procedure for deriving the instrumental profile for the discretely sampled interferogram, described in Section 1.4, it is found the instrumental profile is modified. The expression for the profile is 30 , Re £ s£K,S, Examples of the asymmetric instrumental profiles and their effect on line profiles are shown in Figures 1.5 and 1.6. To evaluate the instrumental profile the path difference was assumed to be uniform across the area of the collimated beam. As optical surfaces are not perfect the path difference and hence modulation varies across the beam. Adopting the method of Chabbal (1953) the interfero- meter can be considered to be divided up into many elementary interfero- meters of area dS^ each with path difference The intensity of an elementary interferometer of path difference = X + e^ , e is the error in X., is from Equation (5) : e a dS 31 dI(pQi = I £A + B cos (2 7T (X+ i> ">J i < ) The total intensity will be the sum of the intensities from the elementary interferograms. If dS is the total area of the interfero- meter from which 7C lies between e and X+ e + de, the variation of the path difference may be specified by the distribution function H(e) = dS/de. Letting CT = -e /X then d 100 = £a + Bcos (2lTx(cr-cr/))J H( S = ^ H( /—— „ r—— i 1 i IIII 1 i I i i i i i II i FIGURE 1.5(b) - Asymmetry in line profiles caused by asymmetric instrumental profiles, (i) Sampling correctly, (ii) Sampling 7i /50 off the correct positron, (iii) Sampling ?i/25 off the correct positron. 33 The interferogram is thus convolved with the distribution function. The modulation is smeared and becomes drastically reduced if the spread in H(e) is larger than a few wavelengths. If we consider the case where H(e) is a rectangular function, so that |el> h/2 H(e) = J(°> (34) I S/h; | e | ^ h/2 then ; k'l> hr/2*. H «T) = < , , (35) (T/hX ^ |tr'|< The total intensity from Equation (32) then becomes after solving the integral I(X) = SI £a + B sineC]Th The rectangular distribution function corresponds to a spherical 'bowing' of an otherwise plane wave. Its effect on the modulation is similar to that of the finite aperture and is the same when h =J2*X-o This assumes the collimated beam has a circular cross section. Phase errors arise when the path difference is wavenumber dependent. The optical path difference of the waves in the two arms of the inter- ferometer are modified if they transverse different lengths through dispersive media. For a broad band spectrum this results in a 'chirped* interferogram that appears skewed. The location of zero path difference is no longer common for all wave numbers. Dispersive phase errors are generally a weak function of wavelength. The path difference between the two beams becomes T 2 v^f^;-^) (37) where and are the physical path lengths the beams travel in J 34 the medium of refractive index If the path difference is measured with respect to wavenumber T e( If the amplitude of the internal modulation of the path difference is incorrect, it is found that 'ghosts1 appear in the spectra. The amplitude should be set to Sx - l/bCT, however, if there is an error e in the amplitude then applying Equations (25, i-v) for the case of a single spectral element the intensity becomes ! QQ „ -sin(2irx Ix00 = -sin(27TX Harmonics or ghosts of the original interferogram are generated, the amplitude of the first harmonic is proportional to the error in the amplitude of the internal modulation. If the spectrum is aliased the harmonic or ghost spectrum will overlap with the fundamental. The amplitude of is determined by observing an emission line suitably near the wavelength of interest and adjusting Sx until the harmonic spectrum disappears. The condition for the maximum acceptable error in Xx is thus e^ ljC^f (T SNR). Simulations for a broad band aliased spectrum are drawn in Figure 1.6. Sample intervals 0 100 200 300 cd +-u> •H U) Ln < FIGURE 1.6 - The effect of an internal modulation amplitude error on a spectrum. Showing internal modulation amplitudes of 0.3, 0.25 and 0.2 A . 36 1.10 Demodulating the Secondary Interferogram The beamsplitters give two reflections; from the silvered and unsilvered surfaces. The secondary reflection is a few percent of the intensity of the primary reflection and travels an additional optical path in the beamsplitter. Both reflections interfere with the reflection from the other arm of the interferometer. Superimposed on main modulation is a weaker secondary modulation, but with its zero path difference displaced by an amount equal to the optical path of the secondary reflection in the beamsplitter. 4 . This problem is alleviated by utilizing wedged beamsplitters. The fainter secondary modulations, considering the collimated interferometer beam, now appear as a set of straight fringes superimposed on the primary circularly symmetric modulation The average modulation integrating over the circular area of the collimated beam, reduces the secondary modulation to tolerable proportions. The wedge beamsplitter gives a secondary plane wave that is tilted with respect to the primary wavefront by an angle far = n where ^ is the wedge angle of the beamsplitter and n its refractive index. Using Equation (31) and polar co-ordinates to integrate over the area of the collimated beam for an area element dS = rdrd^ gives an error in path difference for the secondary beam of e = J£ + cos(^ (40,i) -C&e. P-iguge 8-u The resulting modulation due to the secondary inter- ference is, for a beam of radius a and wavenumber d" 2TT cu i 100 = A+B COS (2TT 2 .2C . J, (ZTT after some simplification ? cf. Steel (1967). The quantity 1.11 Sample Interval Measurement Use is made of the other interferometer input port to insert a laser reference beam to measure sample intervals. To avoid overlapping laser and signal beams the laser radiation is admitted concentric to the signal beam where there is no signal because of the obscuring secondary mirror of the telescope. The interferogram is sampled at locations that are multiples, m , of the laser wavelength, ^^ This imposes some restrictions on the alias wavenumbers with which a spectrum can be measured. From Equations (18) and (19) if the sample interval is J* then If N samples from zero path difference are made then the spectral resolution R = 0"/is R = 2 p N for Since the wavenumbers of the spectral elements are all coupled to the laser wavenumber, the wavenumber calibration of the spectrum is very accurate. The laser is servo-stabilized by use of the Lamb-dip, so is stable to 1 part in 10^, putting a lower limit on the accuracy of radial velocity measurement at 30 ms 1.12 Interferogram Sample Signal-to-Noise Ratio Individual photon events are counted, so assuming Poisson statistics the result that the standard deviation in counting P photons is JP is used. The sample signal to noise ratio is thus 1/ Jp*. If a fraction f is due to unmodulated scattered light the signal to noise 38 ratio is modified to (1 - f) / J Po It is assumed the phase errors described earlier are sufficiently small to achieve the above signal- to-noise ratio. The Fourier Transform Spectrometer used in these studies is the Michelson Interferometer for the Isaac Newton Telescope, due to Dr. R.C. Wayte. Its use and the data reduction of the interfero- grams are described in the following chapter. ) 39 CHAPTER 2 DATA REDUCTION OF INTERFEROGRAMS 2.1 Introduction In this chapter are described the methods used to obtain spectral line profiles from the interferograms measured with the Michelson Inter- ferometer. There are several stages to the data reduction process; summing interferograms, detecting and eliminating noise spikes (spurious data), transforming to obtain the spectrum, dividing out the post mono- chromator passband and evaluating the signal-to-noise ratio in the final normalized spectrum. For the same spectra observed on different occasions the radial velocity correction must be determined and the spectra shifted correspondingly before co-adding. Consideration is also given to the accuracy to which radial velocities can be measured from spectral line profiles„ 2.2 Theory The amplitude of the interferograms at path difference: ^ are given by -S) -iA A+S) P ,!> anc P are P , 2 3 * 4 t*le samPle<* Photon count at the different phases of the jititey cycle i S • The standard deviations and statistical weights of these measurements are -1/2 (p + (2>i) -1/2 c»54(A) = (P5 + P+) (2,ii) (cr2 = 1/W) I ' 40 W12(>* } = P1 + P2 (3,i) p + p W34(Z>) - 3 4 (3,ii) A single interferogram I(^) may be derived which is the difference of the two complementary interferograms and 134* remembering to add them in proportion to their statistical weights. w X —• w I IU) = ; W (^)=w12+W34 (4,i) W12 + W34 where P -P P -P 1 2 , 3 4 *112 P +P :' 1 34 p +p 2 3 4 which simplifies to: P —P —p +p 1 2 3 4 1(A) = ; w(A) = P + P9 + P, + P, (4,ii) P.+P.+P.+P. 12 3 4 12 3 4 Since the post monochromator does not select the same portion of a stellar spectrum, the interfero grams differ and it is preferable not to superpose them. The interferograms are measured several times from the same path difference and the 'complementary1 interferograms added separately0 As the statistical weights of the interferograms are the sum of the photon counts, this leads to much simplification of the interferogram co-adding. More generally for adding a set of interferograms we have N N N I ' - £ w.I. / £ w. ; cr2 = 1/ X w. (5,i) 1=1 1=1 C =1 41 N Let S. =* \ P.. j A- J1 j =» 1, 2, 3, 4 for each photomultiplier Then I w. (5,ii) 12 12 I w, (5,iii) 34 34 Weighting the interferograms, summing the photon counts separately then calculating the interferograms and transforming to obtain the spectrum requires much less computation than transforming each interfero- gram and superposing the spectra in proportion to their statistical weights. This procedure also allows 1 spikes1 to be eliminated. 2 .3 Spike Elimination Spikes or spurious photon counts can arise from several sources: (i) photomultiplier amplifier oscillation, (ii) pick-up of internal jitter modulation signal, (iii) miscounting of photon integration electronics. These are triggered by pulses in the mains power supply induced by such events as dome rotation, the observatory lift and putting the kettle on. Fortunately, the distribution and behaviour of these noise spikes is non-Gaussian and can be distinguished from the photon shot noise by their improbably large excursions from the mean of the other samples measured. Use of this fact may be made to identify a spurious data sample or 'outliersT and to modify it or not include it in evaluating the final interferogram. The method used here is based on that used by ^oCmolceiv Maokraehen Case Study 7 in his book on Fortran Programming (1962). 42 To eliminate spik.es. from an interferogram we proceed as follows: (a) Sum all the interferograms excluding the one to be checked. (b) Find the difference d of the summed and individual interferograms. (c) Evaluate the standard deviation <5* of the samples. (d) If |d| > 3 (e) If 3C include the sample in the final interferogram. The number of real data points expected to have |d| 3flr is .0027 N where N is the number • of samples. Thus typically summing 10 interferograms with 256 samples, only 7 genuine data points will be rejected. However it is found the number of data points rejected is greater because the noise distribution is not normal through the presence of noise spikes. The above method assumes there are sufficient samples to alleviate the presence of spikes in the sum. A comparison of the complementary interferograms can be used before the above test to remove giant spikes. When spikes occur in both interferograms, because they are complementary they are of opposite amplitude. The noise in ^ A. the spectrum reduced by eliminating the spikes is deemed more advantageous than the signal contributed by the genuine data points sacrificed with them. Dealing with all the data, we sum all the interferogram samples at the same path difference. N 51 = .S*! D12 = VS2 N 5 2 " 2 P2 H W = S +S (M- 53 " £ P3 D34 " VS4 54 = { P4 W34 = S3+S4 JJM 43 The interferogram j to be checked for noise spikes is removed from the sum; D S S S P +P srpij 12j i-"V r 2- li 23 S . W S +S P P 20j S2"p2j 12i l 2- li- 2i <7,i-iv> S.. S.-P.. S.+S.-P-.+P.. 3j 3 3j °34j 3j 4j 3 4 33 43 S. . S.+S.-P-.-P, . • S4"P4i 34j 3j 4j 3 4 33 43 Then the summed interferogram sample and standard deviation and the isolated interferogram sample and standard deviations are evaluated. (S S )/ X34j' " (S3j-S4j)/ (8,i- I 12j = m (W/(W 1 m l/ Now a^^j/ £ °l2i7 where N is the number of interferograms. Thus we compare the difference between the two interferogram samples with the standard deviation, so that if / cU 1 . - 1 3 12j 12j *123 (9,i) I I 3 The procedure is applied to all the interferogram samples and 44 all che interferograms sequentially. Another problem arises when very large spikes or outliers perturb the mean drastically enough so that none of the authentic data points are within 3cr* of the mean. Obviously until the interferogram with the spike is located these genuine data points will be made void. Order sorting the samples would overcome this problem, however computer core limits did not permit this. To reduce this problem the complementary interferograms were used to detect and eliminate such obvious o'utliers. Since the method is automatic it is considerably faster than inspecting the interferograms individually and correcting points. Smaller amplitude spikes are detected and instead of correcting points (usually setting them equal to zero, which may be as erroneous as the original spike) the datum is not included in evaluating the definite interferogram value. A spike in .the interferogram introduces an oscillation in the spectrum whose amplitude is proportional to the spike amplitude and frequency proportional to the path difference at which it occurred. Figures 2.1 and 2.2 illustrate a typical interferogram before and after the elimination of noise spikes. Table 2.1 gives the variation of the number of spikes eliminated with the number of standard deviations tolerated. The method has the advantage that it automatically accommodates for variations in the number of photons counted from interferogram to interferogram and sample to sample. Parallel with the final cleaned interferogram is the data containing the total photons used contributing to each data point. This is by no means the optimum method for outlier detection and spike removal. There are two effects not considered (a) Masking "the tendency for the presence of extreme observations not declared as outliers to mask the discordancy of more extreme Path difference (cms) Figure 2.1 An interferogram before spike removal,adapted, 1.0 X! c 0 •H •P i—i d CT» 1 0.0 I t>0 O J-4 a) mu a) 4-1 a -1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 Path difference (cms) Figure 2.2 An interferogram after .spike elimination 47 TABLE 2.1 The number of genuine data samples eliminated versus standard deviation f _ cr f T E 1.0 0 1.0 0.617 0.5 0.641 0.317 1.0 0.370 0.134 1.5 0.178 0.045 2.0 0.084 0.0124 2.5 0.034 0.0027 3.0 0.013 >3.0 0.006 error distribution from a typical set *E of interferograms. Number of points eliminated r f x Number of samples. 48 observations under investigation as outliers", where there is more than one spike in a set of samples, (b) Slippage "the tendency for spurious data points to perturb the mean value and variance causing other genuine data points to be eliminated". Ideally spikes should be eliminated at source. Otherwise for dis- cordant data the technique of Grubbs(1969) should be used. This improveson Wright's (1884) procedure where observations deviating from the mean by more than three standard devistions or five times the probable error are rejected. Also on Goodwin's method (1913) where an outlying observation in a sample of n is rejected if its deviation from the remaining (n-1) exceeds four times the average deviation of the (n-1) sampleso These procedures fail to distinguish population variance and sample variance. 2.4 Normalization of the spectra For the spectra to be of any value the bandpass of the grating post monochromator must be removed. The bandpass shape is obtained by observ- ing the spectrum of an early type star, or using an artificial light source or fitting a continuum to the observed spectrum by inspection. None of the methods are exact and all give different bandpass shapes, which is disconcerting. Although the reasons why they differ are known, this inexactitude thwarts any objectivity in evaluating the accuracy of the methods. The observed, continuum and artificial spectral bandpass deviate because of differing slit illumination, in terms of the image shape and direction of illumination. The former perturbs the shape of the passband and the latter the position of its centre. The use of any early type star provides similar slit illumination, with assurances that seeing has not changed markedly between successive observations. The direction of illumination may be modified for two reasonso Firstly, because of telescope flexture and misalignment for 49 an object in a different part of the sky0 Secondly, the autoguider guides on a different part of the spectrum of an early type star compared to a late type star since the images are refracted by the atmosphere. The method also assumes there are no spectral features in the early type spectra. The spectra are measured at sufficient resolution to measure the bandpass shape and no higher to reduce observing time. Weak lines may thus escape detection. Inspection of other higher resolution spectra would clarify the issue of weak lines modifying the apparent continuum and hence bandpass shapeo The artificial light source illuminates the slit uniformly, has no spectral features and high signal to noise ratio measurements of the resulting bandpass can be easily obtained. The bandpass usually differs in shape and is wider because the aperture is completely illuminated. The bandpass is displaced if the direction of illumination is off the Goude Polar axis. . There are smooth but systematic differences between the stellar and artificial bandpass, shapes. With the third method a continuum is fitted by inspection to the observed spectrum. However, it requires plenty of authentic continuum in the spectrum and that the continuum is smoothly varying. Ideally the line of interest should be located on a flat portion of continuum where the curvature is minimal. Many trial fits can be attempted to choose the most sensible continuum. Comparison can be made with other late type spectra such as Griffin*s Arcturus Atlas (1968). If S( C(cr ) is the continuum spectrum then I ( I(<*0 = S(cr)/C( cr. = cr + i £ cr 1 o For fitting the continuum spectra to the observed spectrum the 50 spectra are overlaid and the following operations performed on the continuum spectrum to achieve a best fit. Shift or displace the continuum spectrum so its centre coincides with the observed spectrum. Normalise the continuum bandpass height so that the peak heights are the same. Modifying the height and centre of the bandpass enables a best fit to be achieved. Plotting the differences in regions of well defined continuum would assist in obtaining a best fit. Also to accommodate for different bandpass widths a rescaling of the continuum bandpass would be helpful especially when the artificial slit illumination continuum is employed. The procedure for fitting a continuum by inspection is described 09-780 by Wayte (1970) and uses a Fourier Series to generate a suitable continuum bandpass. A spline interpolation would obviate the equal intervals required by the Fourier Series expansion. The errors in the continuum placement are difficult to estimate. Comparison of the methods, even successive attempts by inspection, gives some clues. One particular complication for stellar lines with damping wings is deciding the location of wing and continuum. However, most of the lines studied here are void of wings . See figures 2.4 and 2.5. 2.5. The signal to noise ratio in the spectrum The standard deviation of the normalized spectrum is evaluated by utilizing the portion of the alias where no spectrum is admitted by the post monochromatoro Use is made of the fact that the noise is uniform throughout the spectrum. The spectrum is scaled to unit height and a portion of the alias isolated with no spectrum. The standard deviation and mean are estimated from the noise samples. Ktfj) i = 1, N N N V = — Z ttfri* - 5 * • 5 S ^ <11,i) 1=1 1=1 Data sample 200 500 FIGURE 2.4 The Arcturus OC Bootls Fe I A 6065 8 passband, with the superimposed continuum of an early type star. i FIGURE 2.5 - The Arcturus of Boot Fe I 7i 6065 8 after normalization. The residual intensity of the Ti I ^ 6064.626 8 line is 37%. The residual intensity of the same line in the Arcturus atlas is 42%. This demonstrates that the Fourier transform spectrometer has a greater spectral resolving power than the Mt. Wilson spectrograph employed by Griffin (1968). i 53 The signal to noise ratio is then simply SNR = 1/S This formula assumes there are as many points in the spectrum as there are in the original interferogram. If this is not the case, then the SNR = 1/(S JT) where f is the ratio of the number of points in the spectrum to the interferogram. One disadvantage with this method is that phase errors may give a false under-estimate of the signal to noise ratio due purely to photon shot noise. An alternative means of evaluating the signal to noise ratio in the spectrum makes use of the Rayleigh Theorem for Fourier Transforms, and splitting the measurements into ideal data and noise . I = maximum spectral value (derived from the discrete max cosine Fourier transform of the interferogram modulation) P = integrated photon count per sample SNR = I JT max V Radial Velocity Measurements After identifying lines present in the normalized spectrum the radial velocity of the source may be found. Intuitively, the precision of the radial velocity measurement depends upon the line shape, depth ofJu&CLCCttnTUkj and the signal-to-noise ratio. It is important to obtain an estimate^ tuc e&tiyua^jk to which radial velocities can be measured to compare with the results A obtained in practice, to see if there are other mechanisms shifting the lines. « Taking the example of a Gaussian absorption profile, centred at wavenumber C of line depth £ and half width at 1/e height of c C , the relative flux profile is given by F(O-) = 1 - i exp(-( CT" — cr )/Ac )2) C- c To find the line centre, one method is to fold the profile and 54 minimize the squares of the differences between the two profiles. If (T^ is the centre of the folded profile, then their difference d(tf~) is 2 d( Letting x = oo D(fl-) = t LCT £ [exp(-(x-y)2)-exp(-(x-z)2)J dx —CO which simplifies to 2 (p 00 -j D(cr) = 2£ A <90 2 D( For y-z ^ 1 then letting S „(Sr) . JX (£)*• Remembering that in a real spectrum there is also a contribution to the D(CT) term from the noise, restricting the range of the noise power included to f & Stellar Radial Velocities Star 5379 6027 6065 6151 Allen Average (1973) Arcturus - 5.0 - 4.5 - 4.2 - 5 - 4.6 ± 0.4 Hamal •13.5 •14.2 •13.7 -13.2 -14 -13.7 1 0.4 Ul Ui Kochab 17.9 19.1 18.8 18.1 17 18.5 1 0.6 Pollux 5.2 4.6 6.7 3v* 5.5 ± 1.1 * v - denotes variable i 56 s » . * AT/ (£) To find the precision to which the line centre is found we adopt the criterion that the differences due to noise and profile being off centre are the same, then equating NP and D(&r) we have Sr - ttft If Ar To find the accuracy of a radial velocity measurement Sv replace $r by Sy an to which the radial velocity can be measured is about 200 ms \ _ 1/4 including noise from 4 widths over the profile. The ( II /2) term is dependent on the profile shape, for profiles with sharper transitions from core to wings this term will be smaller. In Table 2.2 the heliocentric velocities for the lines studied in Chapter 4 are listed. The dispersion in the results for the helio- centric velocities between the line profiles, suggests other sources of line shift. If the line profiles are asymmetric they will appear shifted. Alternatively, since profiles are measured on different occasions the shifts may result from the stars motion due to a companion star. The shifts for Arcturus, Hamel and Kochab around 400 ms ^ would appear to indicate the former. However, the shifts for Pollux are considerably larger at 1.1 kms * and indicates the star has a companion*. Allen (1972) notes that Pollux has a variable radial velocity. 57 CHAPTER 3 SYNTHESIS AND ANALYSIS OF STELLAR LINE PROFILES 3.1 Introduction Traditionally non-thermal motions of stellar atmospheres were pictured in terms of micro turbulence and macro turbulence. The concept of microturbulence was introduced by Struve and Elvey (1934) to explain an anomaly in their curve-of-growth studies of supergiant stars. They comment "The Doppler widths inferred from the position of the flat part of the curve-of-growth were far in excess of their thermal value". They attributed this effect to a turbulent motion of small size compared to the scale height of the stellar atmosphere, which enhances the strength of a saturated line. Macroturbulence was proposed by Struve (1946) to account for observations of certain early type stars, where the widths of the line profiles exceeded Doppler-widths derived from curve-of-growth studies. He envisaged that a large scale turbulent motion, that does not modify the strength of a line, would explain this phenomenon. The microturbulence-macroturbulence dichtoray used to describe the turbulence in stellar atmospheres has shortcomings. It is without any foundation on quantitative physical laws and fails to include turbulence of an intermediate scale. Ideally for treating spectral line formation, the equations of hydrodynamics describing turbulence should be derived and solved consistently with the equations of radiative transfer and statistical equilibrium. There appear to be three hindrances to reaching this ideal. The first is to formulate satisfactory models uo describe turbulence; secondly, to measure line profiles with sufficient accuracy to justify such attempts, and finally, provision of the atomic data necessary to solve the radiative transfer and statistical equilibrium equations. Jeffries and White (1967) review the problems of interpreting 58 spectral line profiles. They quote Eddington, whose thoughts on inter- preting line strengths, I believe, should also be adopted for line profiles: "The quantitative calculation of the strength of a line is capable of almost indefinite complication according to the conditions assumed. What is first required is not an elaborate formula providing for all contingencies, but a standard solution for the formation of a line under the most simplified conditions, that do not involve any inconsistency". Alternatives to the micro-macroturbulence model are described by H. and (J. Frisch (1975), Traving et al. (1975) and Mihalas(1980). None of these attempt to solve the hydrodynanical equations, but employ either a stochastic , probabilistic or prespecified description of the stellar velocity field. The former models replace the micro- and macro- turbulence parameters with an overall turbulence velocity and an eddy size. Gail and Sedlmayr (1974) find from their analysis of solar 0 I and Fe I intensity profiles, a turbulence velocity of 22 kms ^ and eddy size of 150 km. Gray (1977) using the micro-macroturbulence model finds - 0.5 kms and jj ^ = 3.1 kms V (RT radial tangential macro- turbulence) by reproducing solar flux profiles. Both models fit the data accurately. Hence, it is difficult to justify on observational grounds, at present for stars other than the sun, which of the models is better. For ease and speed of computing, the simpler of the two models (the rnicro- macrotubrulence model) is described in this chapter and used for analysing the stellar line profile data discussed in the next chapter. 3.2 . Line Profile Synthesis in Local Thermodynamic Equilibrium Local thermodynamic equilibrium (LTE) assumes that the state of a gas and the radiation field are specified by the local temperature, T, and particle density, N. From T and N, the ionization functions and level populations can then be evaluated using Saha and Boltzmann's 59 equations, and the radiation field using the Planck function. With the electron number density N^ the line and continuum opacities may then be evaluated and flux profile calculated for a transition. Although only LTE calculations are described the effects of departures from LTE, as described by Athay and Lites (1972), must be considered when interpreting the data. 60 3.2,1 Level Populations (a) The velocity distribution f(v) of the absorbers is assumed to be Maxwellian m 1/2 ? f(v) dv = ( m ) exp(-mv /2TirkT)dv (1) - - 2~Jr kT - - 1/2 With a most probable speed Vq = (2kT/m) for a particle of mass m or root mean square speed 2.1/ , v1/2 1/2 \Vx2/ = (kT/m) (3) (b) The atoms are distributed over their bound levels according to the Boltzmann excitation equation. If n.. is the number of ij atoms for a given chemical species in excitation state i and ionization state j, ^ the excitation energy relative to the ground state and g^ the statistical weight, then the population for an excited level is n../n . = g../g . exp(— X, .. /kT) (4) U o3 &ij &oj r r ij If N. is the total number of atoms in ionization state i then J J 1 • Znij • f: H hi aW (5) 02 ' the latter sum is termed the partition function u. (T) I /kT) (6) so that the fraction of an atom in a particular excitation state is 61 n /N « g exp(- Y /kT)/U (T) (7) j-j j ij / ij j The Chebyschev partition function approximations evaluated by Traving (197 ) were used. (c) The ionization state is calculated from the Saha ionization equation. If is the ionization potential then (Aller,L953 ) the ratio for number densities of two successive ionization states j and j + 1 is given by N. n ,2 3/2 U. (T) JL. = ( 2 ) _J exp(-I./kT) (8) N. 2 2 TTmkT U.., (T) 3 J+i The fractions of the atoms in.each ionization state follows J from applying the constraint ^ _ £ ^ i.e. the total number of j-1 3 atoms summed over all ionization states is N. j'-t N ' N < N '/N 7J N^/Mj = —J— = 1 ± = (9) N T N. Ni n N3 J Jf-1 j JL + 2 + yy N. v-} /N. 3 EL EL EL 2 1' J 1 J 111 The electron number density calculated from the electron pressure model atmosphere tabulation and using P^3 NMcT. The ionization data was taken from Gray (1976) . 3*2,2 The Absorption Coefficient The integrated absorption coefficient is given by Thorne (1974) 2 e f.. a.. \ a Vdv = ii- U (10) 3 V 4/, 9£ mc o where n^j is the number density of atoms in the lower state of a 62 transition with oscillator strength f... The absorption coefficient ^ J is Doppler broadened because of the line of sight motion of atoms which is assumed to have a Gaussian distribution ^ (v) = exP<-v2/p2> 2 2 2 where = ^T + ^ fc^ s" (12) to include the thermal and non-thermal micro turbulent broadening, m is the atomic mass. If is the frequency of the transition then the Doppler shifted frequency >>' = (1 + v/c) so that dv1 = V^j dv/c, the absorption coefficient is a. . ( V )dV =—g^ftj . — /-WP^^V (13) We now evaluate the mass absorption coefficient ^ required to evaluate the flux profile from a stellar model atmosphere and specify the element abundance as a fraction of the hydrogen abundance. R.(v) = a a (V)/ E (14) J-J ij J-J IK . / £ is the number of absorbers per gram. n.. n.. N. Nfc. N_ = (_il) (_i) (-*) JL (15) e Nj \ nH ? N^/N^ is the element abundance A^ and K K tN • 1 \\ % • Z (16) where yu^ is the atomic mass of element k. The mass absorption 63 coefficient including the effects of stimulated emission is thus 4i)(o)ao= n.j _ , f | - f-Hv^/x.T')) do (l?) 3.2.3 Van der Waals Broadening For including the effects of Van der Waals broadening for the highly excited states of the heavier atoms, we adopt the depth dependence of the damping constant given by Aller (1962) x/. -7/10 f - Cvdw Pg T <18> where P and T are the gas pressure and temperature respectively. This givess a damping parameter such that P T 7/10 a = a (-£-) (-2) (19) op T go where a is the damping parameter for P and T specifying the 0 state of the atmosphere at unit optical depth at = 500 nm. The damping parameter is a free variable in the analysis procedure. The velocity distribution (v) is now replaced by a normalized Voigt function V(V/(J in Equation (13). The values for the Voigt function are calculated from a FORTRAN subroutine given by Drayson (1975). 3.3 Radiative Transfer Fig. 3.1 describes the term specific intensity. If we consider the change in intensity along a distance ds where there are sources and sinks of radiation, then we obtain the radiative transfer equation, for the plane parallel case is given by Woolley and Stibbs (1953). xi = " k I + j (20) ^ ds v v Jy 64 n FIGURE 3.1 - Definition of Specific Intensity. The specific intensity liriutfllJU^g in direction n, with frequency V at time t is the quantity of energy transported by radiation of frequencies V to V across an element of area dS into a solid angle dur in a time interval dt is $E = I(n,y,t) d V dt durdS cos G /v where dS cos & = n.dS , dS^ is the normal to the surface, /4>a> 5 0 65 where jyUVdV dfc/ dt is the energy emitted by the volume element dV and k^I^dVd*\> dw dt is the energy removed by the same. j is the emission coefficient and k^ absorption coefficient at frequency y. Now defining the optical depth dT^ =-k^)ds, so that at the observer =» 0 and increases away from the observer, then the radiation received from a region of total optical thickness (Chandrasekhar, / 1960) C1>r is r -tv C iv -V/" • )e + (21) iv(o,/i> = v\ ^ ^ where I ( T . ) is the intensity on the far side of the region from V vr the region where 'V =» X , the integral is the contribution from ^ r the region. The optical depth is given by V T s 22 v " 1 ¥ < > o For a semi-infinite atmosphere letting S^ = j y/ky we have OQ lv(0,yu)= J Sv(t9)e"V/ dt o S is traditionally called the source function although strictly speaking not used in the mathematical sense of sources and sinks but is rather their ratio. For the LTE approximation the source function is set equal to the Plank function B^(T), (Eisberg, 1974 ), requiring the run of temperature T verses optical depth to be evaluated. B (T) J*h) (23) v 2 it I ehV/kT_1 66 3*3.1 Flux Integration Using the plane parallel approximation, the flux may be computed I3S2 (Kourganoff, 1063), assuming no azimuthal dependence of the flux, from Fv (0) = 2T ^ yo, p) pty (24) 'o or » 1 0) e VJ* dt d V = 2TJ-J Wvve " V / ft * 2 If w = 1/^u, then dw = —dya/yu e then F (0) = 2TT £ Sy(ty) ^ * dw dt (25) V w The inner integral is the second order exponential integral () and is generated using the numerical approximations given by Gray (1975) for and the recursion relation. nE _ (X) = e~X - XE (X) (26) n+i n Hence the surface flux is given by CO Fv(0) = 2T^ S^(ty.) E2(t„.)dtJ, (27) o Considering the large range of optical depths over which the flux originates, logarithmic intervals are used to calculate the flux dty = ty dlog ty ' Fy(0) = 2T1 ^ Sv(tv)E2(ty)ty dlogt^/loge (28,i) Simpson's integration quadrature may then be employed to calculate this N WiS)/ yi^E2 *Vi ^ Vi^0® e + ^ FV (28,ii) 1 Plosive = log - lo^ty. 67 The flux contribution from very opaque optical depths t^ > 20 is negligible. For very translucent optical depths the source function S^ is assumed to be constant, permitting the flux contribution to be evaluated analytically 2TSy(t1) ^ E2(tvi)dtv (28,iii) Af - T Sv(tx)- - 2.E3(tv,)J (28,iv) The optical depth increment i^log t ^ equal to that of the model atmosphere tabulation specifying the run of continuous opacity ^5000^ at ^ - 5000 8, electron pressure (P^) , gas pressure (P^) and temperature (T) as a function of optical depth X ^qqq at ~ 5000 8. Knowing that d-C df V V ds = 11 -2 k k V1 2 then the optical depth scale at an arbitrary wavelength is given from the model atmosphere optical depth scale and the opacities at the standard wavelength and the wavelength of interest by •CV = (29) s Alogts Tv = £ w.i.t • (30) l s v u, 3. i ^ ks1/ ^^ is the optical depth to the first layer of the model atmosphere for which the ratio of the opacities of the two frequencies is assumed constant. Hence 68 is, = iT'1! <31> I S1 Simpson's quadrature weights are used to evaluate T^O^). 3.3.2 Continuous Opacities The opacity is the sum of the continuous bound-free and scattering opacities and bound-bound or line opacity. The continuous opacities included in the line profile calculations are: negative hydrogen H , molecular hydrogen H^ and H^, atomic hydrogen HI and Mg I, Al I, Si I and C I metal opacities. Other opacities included were Thomson scattering by free electrons and Rayleigh scattering by neutral and molecular hydrogen. The H*~ and H I opacities are taken from Gray (1976), the metal opacities Mg I, Al I, Si I and C I from Travis .and Matsushima (1968), the scattering opacities of H. ?H2 and e + from Mihalas (1967) and H2 from Mihalas (1965), and H2 from Sommerville (1964). For late-type stars the scattering opacities dominate in the upper photosphere, the H opacity dominates in the + bulk of the photosphere and H at depth. The metal and H2 opacities are only significant in the ultra-violet. The opacities are compared to those given by Bell (1976) to test for accuracy and examples are listed in Table 3.1 3.4 Model Atmospheres Suitable model atmospheres for the stars observed were taken from the grid of model atmospheres for metal deficient giant stars published by Gustafsson et al. (1975). The model's parameters are effective temperature C^^^) , surface gravity (log g) , metal abundance A/H and microturbulence ( The microturbulence parameter, not usually considered as a model atmosphere parameter, was included since it modifies the degree of line blanketting and backwarming due to atomic and molecular lines. TABLE 3.1 Comparison of Opacities Temperature = 4927 G Gustaffsson et al. (1976). Electron Pressure = 4.25(-l) WJ Author's calculations. Gas Pressure = 3990. A=5000 8 5926 8 = 4162 8 H 9.9(-6) 9.21(-6) 1.61(-5) 1.44(-5) 8.58(-4) 5.60(-4) H~ 1.32(-2) 1.27(-2) 1.54(-2) 1.43(-2) lo03(-2) 1.08(-2) + H 1.24(-4) 2 1.37(-4) 1.04(-4) 1.15(-4) 2.05(-4) 1.60(-4) C I 1.22(-6) 2.87(—6) 1.26(-6) 1.26(-6) 1.57(-6) 7.74(-6) Mg I 6.48(-5) 9.79(-5) 6.26(-5) 1.19(-5) 4.46(-5) 7.12(-6) Al I 1.47(-5) 4.14(-5) 1.57(-5) 6.30(-4) 8.85(-6) 2.92(-5) Si I 1.69(-4) 7.78(-5) 1.72(-4) 6.60(-5) 1.28(-4) 4.59(-5) cr 4.64(-4) 4.53(-4) 2045(-4) 2.31(-4) 9.74(-4) 9„78(-4) 2 1.36(-2) l031(-2) 1.57(-2) 1.47(-2) 1.12(-2) 1.12(-2) G WJ G WJ G WJ i 70 The absorption lines hinder radiative transport through the atmosphere of a star, requiring the radiation to emanate at continuum frequencies with greater flux than if the lines were absent. Since the bandwidth over which the radiation may be emitted is reduced, steeper temperature gradients are required to radiate the same flux. This backwarming effect raises the temperature of the underlying atmosphere. Gustafsson comments that the effects of sphericity perturb the temperature distribution by about -50°K for models with logg =0.75 because of the dilution of the radiation field. This assumes the stars are of solar mass giving & ~2J ft the thickness of the photosphere / stellar radius r^. 0.05 for the worst case. This effect however will be considerably smaller for higher surface gravity giant stars considered herein, where log g 1.5. However, where there is heavy turbulence and chromospheric activity the outer layers of giant and particularly supergiant stars may be considerably more extensive. Also because of the breakdown of the assumptions of LTE and homogeneity Gustafsson states that they should not be used in investigations of spectral features formed in the outer regions of the atmospheres. It is with these model limitations in mind that models were chosen from the Gustafsson grid (1976), since they are the best available to date. Regarding the upper photosphere and chromosphere Kelch (1978) have published semi-empirical models of these regions for late-type stars based on partial redistribution analysis of Ca II K line wings and cores and on the fluxes in the Mg II H and K lines. The presence of a core reversal in the visible Ca II K lines indicates the presence of a chromosphere or * temperature inversion' at shallow optical depths S -4 5000 ^ * temperature inversion for very saturated Fe I results in a core reversal if LTE source functions are employed. However, this will be smeared by macroturbulent motions, as much as 10 kms, reported by Ayres and Linsky (1975) to fit the IR Ca II triplet 71 at ^ 8498, X8662 and X 8542, whose cores are also of chromo- sphere origin. These models derived by Ayres and Linsky could be employed to synthesize other profiles such as Fe I, provided reliable atomic data was available to perform the necessary NLTE calculations required to synthesize saturated Fe I lines. Lites and Cowley (1974) completely ignore chromospheric effects in their NLTE study of Fe I line formation in late type stars. 3.5 Macroturbulence These large scale motions of the atmosphere of a star are considered to be random and anisotropic similar to those observed on the solar surface. One can envisage portions of the atmosphere in upward and downward radial motion such as the two columns of convection cell structure proposed by Dravins (1975). This models the more systematic solar granular motions consisting of 'hot' elements rising and Tcoolf elements sinking; a steady state convective motion with hot gases in the bright ascending granules and cold gases descending in the darker inter-granular lanes that are observed. The model explains observed solar line shift and asymmetries of weak photospheric lines and the dependence of the shift on excitation potential. Such differential displacements can be misinterpreted as an expanding atmosphere. Gray (1976) simplifies the above granulation model in order to model macroturbulence . The direction of motion of the turbulent eddies is considered to be radial or tangential to the stellar surface with upward and downward moving eddies with the same brightness and having a Gaussian velocity distribution. If the approximation that the intensity profile centre-to-limb changes are negligible, then the observed flux profile is the intensity profile convolved with the velocity distribution of the turbulent elements averaged over the disk. Intuitively, considering 72 only radial motion, it is evident the disk average velocity distribution will be a more sharply peaked function because the macroturbulent Doppler broadening decreases approaching the limb. The macroturbulent broadening function (fi)^ ) ^ AX C ®rt(A%) = >i ^ g (32) rfiFt is given by Gray (1976). Smith (1976) recommends using the single parameter exponential distribution. He refers to Lambert and TomkinTs (1974) spectrum synthesis of the K supergiant £ Pegasi. The latter state it was best matched by using an exponential macroturbulent law, as follows Both and a^e similarly peaked functions but, as noted by Smith (1979), their width parameters scale as RT E for the same degree of broadening. The radial-tangential parameter is larger since it specifies the (1/e) half width of the macrovelocity distribution prior to disk averaging. The exponential distribution is already averaged over the disk and it is this macrovelocity distri- bution that is used in this study of luminous K stars. 3.6 Rotation Consider a star to be spherical and rotating as a rigid body. The spectrum from the receding limb will suffer a red Doppler shift and that from the approaching limb a blue Doppler shift. The spectrum is smeared by the effects of rotation, the greater the rotational velocity the greater the extent to which the spectrum appears washed out. If 73 the rotational axis is parallel to the line of sight no rotational broadening is observed,, If SL is the angular rotation vector and R the radius vector, then the velocity at any point on the surface of the star is v =* % R . If SL. lies in the y-z plane and i the angle of inclination to the LOS then _JL = Ji i ± + Jl to* C k (34,i) 9 9 o 1/2 X i + yj. + (R - x - y ) k (34,ii) so that v = i 1 k (35) 0 SLCqS t 2 2 1/2 y (R - x - y ^) The component in the LOS v. = JU suae (36,i) cu The equatorial velocity v = Sim R (36,ii) Hence the Doppler shift = (-) V^VT-C (37) c R Gray (1976) proves that the flux profile of a rotating star is the flux profile of the non-rotating star convolved with the rotation profile G(A) provided the intensity profile shape is the same from centre-to-limb. A linear limb darkening law is used which sour ex. assumes the came- function is of the form S( tp) = a + bTv? = v [a-e + ir cos (16) £ is the limb darkening parameter, a slowly varying function of assumed independent across a line profile, giving (2(1-{ ) J7 + jJTGr ) G( AA) (38,i) IT aMi - e /3) where r (38,iL) (38,iii) 3.7 Fourier Analysis of Line Profiles The technique of Fourier analysis of broadening of spectral line profiles exemplified by Gray and Smith (1976) is employed in this study of luminous K-stars. It is an alternative to fitting profiles in the wavenumber domain. In the Fourier domain it is easier to picture the filtering effects of the macroturbulence, rotation and instrumental profile transforms on the intrinsic flux profile transform than their corresponding convolutions in the wave- number domain of the intrinsic flux profile. The profile is P( F(cr ) = intrinsic flux profile M(cr ) = macroturbulence profile G(cr) = rotation profile I(cr) = instrumental profile denotes convolution which in the Fourier domain simplifies p(s) = f(s).m(s).g(s).i(s) (40) The lower case functions are the Fourier transforms of the functions specified with upper case letters. The Fourier transform is given by 75 ^ 2 TTi C s p(s) » ^ p( If we have N data points at values N-1, then discretizing equation (41) we have N-1 1 s p(s) = T P( If we evaluate N-1 P(k) = £ P(j) e^ijk^As ^ s (43) j=0 th The solution for the k + N term is N-1 2fijk^(rAS 2 TT i j N (44) ZP(j)e e 3=0 For p(k+N) = p(K) for all values of |c we require 2"ir i j N ^ cAS e =1 for all values of j; (45) so that ArAS = 1/N (46) A ^S is the reciprocal wavenumber resolution in the Fourier domain. The method used here differs in detail from Gray's. Gray (1978) uses a deconvolution technique, by dividing out the instrumental profile 76 transform. Having matched the equivalent width, the location of the curve transform at zero cycles and the microturbulence to fit the first natural or thermal zero, the thermal profile is divided out. To the residual profile the rotational and macroturbulence profiles are fitted. Smith (1976) rather than dividing out the thermal and instrumental profiles from the data profile multiplies the theoretical thermal profile by the various filters to fit to the data. One criticism of Gray's method is that it is difficult to estimate the effect of errors in the values derived for the microturbulence and abundance on the errors then used for the rotational velocity and microturbulence. Gray and Smith do not synthesize the macroturbulence and rotation profiles in wavenumber space but use the corresponding analytical expressions for their Fourier transforms. The intrinsic flux profile is computed in wavenumber space, while care is required that they are synthesized at sufficient resolution so that discretization errors, that modify transform shape, are small. Analytical expressions cannot be found for certain filters, in which case we are obliged to use the discrete Fourier Transform to compute the transform filter. The radial analytical model for macroturbulence by Gray is such an example. 3.8 Macroturbulent and Rotation Filters The Fourier transform of an exponential is a Lorentzian (see Bracewell, 1976) so the expression for the macroturbulent filter is m(z) = —j (47, 1 + z where % = 27ft S (47, e The Fourier transform of the rotational profile is given by Bohm (1952) and is of the form 77 J(i) sin (2:) _ cosCHOj g(z) = + (48,i) 1 J?. 2 ' (48,ii) 26 (48,Iii) 0 - e/0 (48,iv) Fig.3.2-7. illustrate the profiles and their corresponding Fourier transforms on modifying the various line profile parameters. 3.9 Line Profiles Gray (1978) has pointed out that there is a range of optimum line strength in order to evaluate the micro-macro turbulent and rotational velocities using the Fourier technique. The optimum line strengths give residuals to the highest Fourier frequencies before merging into the noise. Lines of too great a strength are saturated, resulting in ""zeros in the transform, reducing the residual amplitudes. Lines of too low a strength have low signal—to-noise ratio at all cr ' s and their transforms descend into the noise at a cr" only a little higher than the saturated line case. The optimum line strength is an intermediate case. This criterion depends on the actual extent of the filtering. When the macroturbulent and rotational broadening is large compared to the intrinsic flux profile, a stronger line is best observed, since it will have large Fourier amplitudes for all OT values compared to a weaker line. The lines should not be so strong however that damping wings are present that introduce further filtering at critical frequencies. There are good reasons to observe lines of a range of line w ABUNDANCE -8,-7,-6,-5,-4 DEX 1 1 1 1 l .2 | 1 1 1 1 1 1 1 1 J 1 1 1 1 j 1 1 1 r 1 .0 0.8 I X 0.6 CH 0 .4 h 0.2 h 0 .0 h -0.2 •j—i—i—i—i • < • • J—i—i—i—i—i i -15 -10 -5 0 10 15 KM/S FIGURE 3.2(a) - Line profile dependence on abundance. The abundance scaling factor ranges from -2 to 2 dex in steps of 1 dex. i ABUNDANCE -8,-7,-6,-5,-4 DEX 0 10 NOTES In the line profile transforms the value of the transform at zero s/kra is the equivalent width of the line, given here in Angstroms. The abscissae measured in s/km are proportional to the path difference x in the interferometer. They are related by the expression X. CT S = s/km where c is the velocity of light in km/s and (T the wavenumber, x is measured in cms. S/KM FIGURE 3.2(b) - Line profile transform dependence on abundance. The abundance scaling factor ranges from -2 to 2 dex in steps of 1 dex. i MICROTURBULENCE 0,2,4,6 KM/S l .2 l .0 0.8 0.6 0.2 0.0 °-215 -10 -5 0 5 10 1G Kh/S FIGURE 3.3(a) - Line profile dependence on microturbulence. The microturbulence has values of 0, 2, 4 and 6 km/s. They illustrate the desaturation of the profile with increasing microturbulence. i MICROTURBULENCE 0,2,4,6 KM/S 0 10 T 1 1 I I III T 1 1 1 I I I I o -l =3 10 00 "Y-'v Q_ 21 CE C£ UJ (H -2 ZD 10 n O u_ -3 10 -3 -2 -1 10 10 10 10 S/KM FIGURE 3.3(b) - Line profile transform dependence on microturbulence„ The microturbulence has values of 0, 2, 4 and 6 km/s. i ROTATIONRL VELOCITY 0,5.10,15,20 KM/S 1 .2 1 i 1 1 1 J 1 r T » > » ' J 1 1 1 r 1 .0 0.8 x « t t ZD V' V »" 1- - 0.6 00 ro 0.4 CE _J UJ or 0.2 0.0 -0.2 -i 1 1 L_ 1 " ' ' 1 1 L 1 1 I . • 15 T -J 1 1 L. -10 -5 0 5 10 15 KM/S FIGURE 3.4(a) - Line profile dependence on rotation. The. vsini has values of 0, 5, 10, 15 and 20 km/s. i ROTATION 0,5,10,15,20 KM/ ' I T [IIII ' I I [ I I I l_ 00 CO S/KM FIGURE 3.4(b) - Line profile transform dependence on rotation. The vsini has values O, 5, 10, 15 and 20 km/s. i MRCROTURBULENCE 0.2,4.6,8 KM/S 1 -2 ' 1 T" 1 « « r n 1 r 1 1 r ~> J r~—r 1 • 0 p..WB8 0.8 X ZD 1 0.6 /i UJ 00 > d 0.4 _J LxJ or 0-2 - 0.0 -0.2 -» ' • I —i L -J 1- L -15 -10 "5 0 10 15 KM/S FIGURE 3.5(a) - Line profile dependence on macroturbulence. The macroturbulence has values of 0, 5, 10, 15 and 20 km/s. i MRCROTURBULENCE 0,2,4,6,8 KM/S 0 10 1—i—| i m -i 1—i—» I I I T ~i 1 r—i—i i i i Q^ -1 => 10 00 Ln CL Z= GC or LU • ( (K -2 ID 10 CD Ll_ -3 10 -I 1 I I I I I I _l I .1 I 1 I _i Li—i—1 i i I.I -3 -2 -1 10 10 10 10 S/KM FIGURE 3.5(b) - Line profile transform dependence on macroturbulence. The Macroturbulence has values of 0, 5, 10, 15 and 20 km/s. The equivalent width of the line is not modified. i ROTATION 5 , 10 , 15 ,20 KM/S 10 -frnrr^Ty t _ . —r N N , \ \ \ \ N \ \ \ 4 NOTE \ v V » J i i 4 I I The equivalent width of the line is not 10 h \ A 1 modified. / •;4 • 1 1 ( / 1 tUp -3 i II. IV 10 j L I 11 -3 -2 10 10 10 10 00 S/KM ON FIGURE 3.6 - The Rotation profile Fourier transform filter, vsini has values of 5, 10, 15 and 20 kms""1. i MF1CR0TURBULENCE 2,4,6,8 KM/S 10 \ \ . \ \ * \ \ . \ \ LU . CD , ~ 1 ' \ => 10 \ . \ \ CD CL ' \ -vj 21 CE \ QC 4 \ UJ 1 -2 \ % 10 \ o -3 10 i .j . i -I i i ,i 11 I i i mI -j i—i I i i 11 -3 -2 -1 10 10 10 10 S/KM FIGURE 3.7 - The Macroturbulence profile Fourier transform filter. Macroturbulence has values of 0, 2, 4, 6 and 8 kms i 88 strength. By relating line strength to the depth of line formation in a stellar atmosphere, a test can be made for any depth dependence of the microturbulent and macroturbulent velocities. Also differential line displacements, as predicted by Dravins (1975) can be investigated, and for saturated lines the departures from local thermodynamic equili- brium in the line cores can be studied. In the next chapter, this method of spectral line profile synthesis is used to study the turbulence and rotation of some luminous late type stars. 89 CHAPTER 4 STUDIES OF Fe I ABSORPTION LINES IN LUMINOUS K STARS 4.1 Introduction The Fourier transform spectrometer was applied to measuring neutral iron absorption lines of the spectra of luminous K stars. The aim was not only to measure line strengths but also their profiles or contours. The profiles were then used to investigate the rotational and turbulent motions of the photospheres of the giant K stars. Com- parison of synthetic and observed profiles was used to obtain values for rotation, macroturbulence and microturbulence. Statistical tests were used to compare the profiles directly, and the analysis techniques devised by Smith and Gray (1976) were used to compare the profile Fourier transforms. The synthetic lines were computed assuming local thermodynamic equilibrium to prevail throughout the line formation region of a stellar photosphere. The method for synthesizing the lines is described in the previous chapter. However, consideration was also given to the effects of non-LTE on absorption lines and how they modify the conclusions assuming LTE. 4.2 Selection of Line Profiles The lines were selected firstly by inspecting their appearance in published atlases of stellar spectra. Lines were chosen for their symmetrical appearance and relative freedom from blends. Where possible blends were identified to see whether they were of telluric or stellar origin. Profiles with blends were accepted where suitable corrections could be effected. Blend corrections were made by replacing the blend with the corresponding portion of the profile on the opposite side of the line centre. 90 The lines measured were restricted to those arising from atomic species that would not exhibit any isotope splitting. Lines were also limited to those arising from atomic transitions that were void of hyper- fine structure. Where such effects go unnoticed anomalously large values would be found for the velocity broadening parameters. To measure the microturbulent velocity accurately it is preferable to observe lines due to the heavier elements so that turbulent broadening dominates thermal broadening. The turbulent velocities are common to all atomic species whereas thermal velocities are inversely proportional to the square root of the atomic mass. The rotational and macroturbulent broadening is the same for all the spectral lines of a star. At first sight, it would appear best to observe the narrower spectral lines to measure rotational and macro- turbulence. However the sharper lines are the weaker lines located on the linear portion of the curve of growth. Thus longer integration times are necessary to measure the profile of a weak line compared to that of a saturated line. (The accuracy of a profile measurement is proportional to its equivalent width). At the other extreme, the width of very saturated lines is dominated by microturbulence and Van der Waals broadening. Such lines are located on the upper portion of the plateau and 'square root' region of the curve of growth. Thus there is an optimum line strength for measuring the macroturbulent, microturbulent and rotational velocity parameters. This optimum line strength is located on the shoulder of the curve-of-growth. Inspecting Griffin's curve-of- growth data for Arcturus (1967) the choice of spectral lines measured was thus limited to those with an equivalent width ranging from 60 to 250 m& for the K-stars observed. The published stellar spectra used for finding suitable lines were Delbouille Solar Atlas (1973) and Griffin's Arcturus Atlas (1969). The table of solar spectrum wavelengths of Moore (1966) was invaluable for 91 identifying lines and discerning whether blends were of stellar or telluric origin. The lines observed are listed in Table 1, along with references to the sources of other atomic data. Professor B. Pagel kindly provided a list of elements expected to exhibit isotope structure. These are Li, Ne, CI, K, Cu, Zn, Ga, Se, Br, Kr, Rb, Sr, Ho and Cd. The region of the spectrum from which the lines were selected was confined to that where the instrument was most sensitive and where the star gave the most flux. 4.3 Choice of Star Measurements of line profiles were confined to the spectra of late type stars , to make best use of the resolution that can be achieved with the Fourier transform spectrometer but that is difficult to attain with conventional spectrometers. The broader lines of early type stars are best tackled using instruments of lower spectral resolving power (where multi-element detectors can be used). For the instrument mounted at the Coude focus of the Isaac Newton Telescope, Herstmonceux, a limiting visual magnitude of 2.0 was imposed. The integration times required to achieve a resolution of 2 x 10^ and signal-to-noise rate of 100 were not practical for fainter stars. Binary stars, where the spectrum of the^ secondary star is super- imposed on that of the primary star, were excluded. An example of this is the Ursae Majoris system (Symms, 1969). A portion of its spect- rum is shown in Figure 4.1. The line is evidently assymetric. The results of Gray (1979) for ^ Ursae Majoris should be treated with caution, since no •a-cGounti x?as made for its binary nature. Stars of spectral class later than K were omitted because of the numerous molecular absorption lines that blend with the metallic absorpt- ion lines of interest. Aldebaran K5III was observed and later excluded owing to the numerous blends and attendant difficulty in defining the TABLE 4.1 Atomic transition data of line profiles measured Species Wavelength Oscillator Excitation Multiplet Strength Potential Number X eV Fe I 5379.581 2.95(-2) 3.69 928 Fe I 6027.059 7.8 (-2) 4.07 1018 Fe I 6065.494 9.0 (-2) 2.61 207 rV£o> Fe I 6151.623 4.8 (-4) 2.18 62 1. Bridges, and Kornblitt, 1974. Ap.J. 192, 793. 2. May ,M., Richter,«T., and Winckleman, J*. 1974. Astron.& Astrophys.Supp.Ser.18, 405. 3. Peytremann, E. and Kurucz,R.L. 1975. S.A.0o Special Report, 362, Part II. i 93 - 1.0 0.0 FIGURE 4.1(b) - The Fe I ^ 5379 X line of Of Taurus. The spectrum shovs numerous absorption lines that blend with the line of interest and making it difficult to define the location of the continuum. FIGURE 4.1(a) - The Fe I A 6065 X line of c^ Ursae Majoris. The absorption clearly shows a blueward asymmetry due to the contribution to the spectrum of the secondary star. 94 location of the continuum. Studies were confined to spectral types G 0 through to K4. However because of the limiting magnitude imposed and the telescope declination limit all the stars observed were K giants. Aliens (1976) Astrophysical Quantities, the Astronomical Ephemeris and the Boss General Star Catalogue (1963) were consulted for choosing the stars. The stars observed are listed in Table 4.2 4.4 Analysis of Line Profile Data As described in the previous chapter, theoretical absorption profiles were Fourier transformed to reveal the signatures of the various broadening mechanisms. Similarly, the observed line profiles were trans- formed to identify and quantify the broadening processes. The methods of Smith and Gray were closely followed. However, comparison was also made of the original theoretical and observed profiles before transform- 2 ation. A statistical test, the /C test (Bevington, 1969) was used to test the degree to which theory and observation agree. A comparison of the two methods was then made. Theoretical profiles were fitted to observed profiles in the following manner. Having corrected for blends, the equivalent width of the line was found. A grid of theoretical profiles was then computed to determine the dependence of equivalent width on the element abundance and microturbulence. From the grid suitable combinations of the abund- ance and microturbulence that give the observed equivalent width were selected. For each combination of abundance and microturbulence a second set of grids was computed. These grids map the root-mean-square difference between the theoretical and observed profiles as a function of macroturbu- lence and rotation. Macroturbulence and rotation do not modify the equivalent widths of the lines. By inspection the best combination of abundance, microturbulence, macroturbulence and rotation that gives the least squares fit of the theoretical to observed profiles was then found. TABLE 4.2 Stars Observed Star Right Declination Visual Special Radial Ascension Magnitude Classification Velocity (Epoch 1950) h m s t ii Hamal 2 5 56 23 21 32 2.00 K2 III - I* Pollux 7 43 58 28 4 50 1.15 KO III >? + 3 Arcturus 14 14 39 19 17 47 -0.06 K2p III ** - S Kochab 14 50 45 74 14 43 2.07 K4 III -JK i 96 Further inspection of the root mean square difference maps enabled an objective estimate of the accuracy to which the theoretical profile parameters were measured. The error was defined to be that change in the parameter required to double the root-mean-square difference. However, it was appreciated that these accuracy measurements were underestimates because of the similarity of the broadening mechanisms. Also noted, was that there can be more than one minimum in a root-mean-square difference map. Evidently solutions were not unique, requiring larger error bars to be assigned to the derived profile parameters. When fitting profiles, rather than their Fourier transforms, there is no need to correct for blends. They were still identified, but blended regions were not included when evaluating the root mean square difference. Comparison of various sets of profile parameters resulting after fitting several lines, serves to increase the accuracy., This assumes there is no other systematic difference between various lines. A depth dependence of either macroturbulence or microturbulence could be invoked to explain a dependence of these parameters on line strength. An example of the equivalent width grid and the mean square difference grids are given in Tables 4.3 and 4.4 for the Fe I 6151 X profile of Arcturus. Figures 4.2-3 show the profiles that correspond to the minima in each of the root-mean-square difference maps. The region in parameter space where the percentage mean square difference is less than twice the minimum is marked out, to illustrate the accuracy to which the parameters are determined. 4.5 The Fourier Analysis Method The application of Fourier analysis to astrophysical data, particularly Smith and Qwiy. stellar line profiles, has been actively pursued by Cray and Smith. (1976) . They conclude that the primary benefits of Fourier Analysis are: that astrophysical processes have easily detectable signatures, that the TABLE 4.3 - Equivalent Width Map - Arcturus 6151 & Abundance Microturbulence kms-i lo SlO .0.0 0.33 0.67 1.0 1.33 1.67 2.4 2.33 2.67 -5.00 33.1 33.9 36.1 39.1 42.1 44.9 47.5 49.8 51.8 -4.67 43.4 44.8 480 7 53.9 59.6 65.2 70.6 75.6 80.2 -4.33 52.9 55.1 60.6 67.9 76.0 84.6 93.1 101.4 109.4 -4.00 62.6 65.9 71.8 80.7 91.4 102.7 114.1 125.4 136.6 -3.67 72.0 74.7 81.7 93.0 106.1 119.7 133.9 148.0 162 ol -3.33 79.6 82.4 91.2 105.1 119.7 136.1 152.7 169.6 186.4 -3.00 86.1 89.6 101.1 115.7 132.8 151.9 170.5 190o0 209.5 -2.67 92.5 97.3 110.2 125.1 145.0 165.3 187.0 208.9 231.0 -2.34 99.7 105.3 117.4 134.5 155.6 178.5 201.9 226.2 250.7 Equivalent Width Oscillator Strength 3.0(-4) Ike obs-erwaf is UlouaA6. c-f,TaMj& 4.5. TABLE 4.4 % Mean Square Difference Map - Arcturus Fe I 6151 & 4.4 (i) Abundance =4.6(-3) Microturbulence = 0.8 kms 1 Rotation Macroturbulence -1 -1 kms kms 0.0 0o5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 0 13.4 llo4 8.2 5.0 2.2 1.3 3.2 5.0 6.7 8.1 9.3 1 12.2 10.6 7.7 4.6 1.9 1.4 3.3 5.2 6.8 8.2 9.4 2 9.9 8.8 6.4 8.6 1.3 1.9 3.8 5.6 7.1 8.5 3.6 y£3 CO 3 7.4 6.5 4.6 2.3 1.2* 2.8 4.6 6.2 7.7 8.9 10.0 4 4.7 4ol 2.6 1.4 2.3 4.0 5.6 7.1 8.4 9.5 10.6 5 2.4 2.2 1.9 2.6 3.9 5.4 6.8 8.1 9.2 10.3 11.2 6 2.8 3.0 3.5 4.5 5.6 6.8 8.0 9.1 10.1 11.0 11.9 7 5.0 5.2 5.6 6.4 7.3 8.3 9.3 10.2 11.1 11.9 12.6 8 7.2 7o3 7.7 8.2 8.9 9.7 10.5 11.3 12.0 12.7 13.3 9 9.2 9.2 905 9.9 10.4 11.0 11.6 12.3 12.9 13.5 14.0 10 10.9 10.9 11.1 11.3 11.7 12.2 12.7 13.2 13.7 14.2 14.7 RRCTURUS FE I 6151 1 -2 ~i—i—i—|—i—i—i—i—p ~i 1 1 1 1 1 1 1 r i -0 0.8 x ZD VO 0.6 v£J LU 0.4 R 2-0 KM/S V 1.1 KM/S UJ CK M 2.0 KM/S 0 .2 N 120 MR 0.0 -0.2 -30 -20 -10 0 0 20 30 KM/S FIGURE 4.2 - The Fe I 6151 X profile of Arcturus. In this and following figures the continuous line denotes the data, the short dashed line the synthetic line and the long dashed line the difference curve. The velocity parameters used to synthesize the profiles, namely rotational velocity, microturbulence and macroturbulence are listed in the left hand corner of the graph. This graph shows the best fit synthetic profile. i TABLE 4.4 (continued) -1 4.4(ii) Abundance =1.0(-3) Microturbulence = 1.1 kras Rotation Macroturbulence -1 -1 kras kms 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 0 11.9 10.2 7.2 4.2 1.6 1.8 3.7 5.5 7.1 8.4 9.7 1 10.9 9.5 6.8 3.9 1.4 1.9 3.8 5.6 7.2 8.5 9.7 o 2 8.9 7.9 5.5 2.9 1.1* 2.4 4.3 6.0 7.5 8.8 10.0 3 6.5 5.7 3.8 1.7 1.5 3.3 5.0 6.6 8.0 9.3 10.4 4 4.0 3.4 2.1 1.4 2.7 4.4 6.0 7.4 8.7 9.9 10.9 5 2.0 1.8 1.9 2.9 4.3 5.7 7.1 8.4 9.5 10.5 11.5 6 2.9 3.1 3.8 4.8 5.9 7.2 8.3 9.4 10.4 11.3 12.1 7 5.2 5.4 5.9 6.7 7.6 8.6 9.5 10.4 11.3 12.1 12.8 8 7.4 7.5 7.9 8.5 9.2 9.9 10.7 11.5 12.2 12.9 13.5 9 9.4 9.4 9.7 10.1 10.6 11.2 11.8 12.5 13.1 13.7 14.2 10 11.0 llol 1102 11.5 11.9 12.4 12.9 13.4 13.9 14.4 14.9 ( RRCTURUS FE I 6151 1.2 —•—1—>—'—i—1—•—•—•—\—1—r~—1—1—i—1—1—1—1—i—1—1—1—1—r 1 .0 0.8 X 3 ^ 0.6 LU > £ 0.4 R 2-5 KM/S V 0.8 KM/S LU Ctl M 2-0 KM/S W 123 MR 0.2 0.0 -0.2^ -I I L- J 1 l < I I 1 4 1 1 L -J I I 1 I I . I »'''»' -30 -20 0 0 10 20 30 KM/S FIGURE 4.3(a) - The Fe I ?! 6151 & profile of Arcturus. The abundance of the best fit was reduced and the microturbulence increased to obtain the same equivalent width. The rotational and macroturbulent velocities were adjusted to find a best fit synthetic profile. Table 4.4(ii). TABLE 4.4 (Cont inued) 4.4 (iii) Abundance = 2.1(-4) Microturbulence =1.8 kms -1 Rotation Macroturbulence -1 -1 kms kms 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 0 8.7 7.5 5.3 2.9 1.5 2.6 4.3 6.0 7.4 8.7 9.8 1 8.1 7.0 4.9 2.7 1.5* 2.7 4.5 6.1 7.5 8.8 8.9 2 6.6 508 4.0 2.1 1.7 3.2 4.9 6.4 7.8 9.0 10.1 3 4.8 4.2 2.8 1.7 2.4 4.0 5.6 7.0 8.3 9.5 10.5 roo 4 3.0 2.6 2.0 2.3 3.5 5.0 6.4 7.7 8.4 10.0 11.0 5 2.2 2.3 2.7 3.6 4.9 6.2 7.4 8.6 9.7 10.6 11.5 6 3.5 3.7 4.3 5.2 6.3 7.4 8.5 9.5 10.5 11.3 12.1 7 5.5 5.7 6.2 6.9 7.8 80 7 9.6 10.5 11.3 12.1 12.8 8 7.5 7.6 8.0 8.5 9.2 10.0 10„7 11.5 12.2 12.8 13.5 9 9.3 9.4 9.7 10.1 10.6 11.2 11.8 12.4 13.0 13.6 14.1 10 10.9 lloO 11.2 11.4 11.8 12.3 12.8 13.3 1308 14.3 14.8 i RRCTURUS FE I 6151 1 .0 0.8 xZD 0-6 O UJ U> 0.4 1 .0 KM/S CE 1 .8 KM/S LU en 2-0 KM/S 0.2 125 Mfl 0.0 -0.2 -I 1 L. I L I . I L. -30 -20 -10 0 10 20 30 KM/S FIGURE 4.3(b) - The Fe I "A 6151 X profile of Arcturus. These graphs illustrate the uncertainty of the accuracy to which it was possible to measure the abundance and microturbulence. cf. Table 4.4(iii). i 104 convolution theorem can be applied, and that the behaviour of random noise is simple. They comment further that, "This latter point allows objective error bars to be placed on the velocity fields modelled to the data. It is generally found that these error bars are two or three times smaller than those estimated by visual comparison of model and observed profiles". The important point to note here is the implication that models are fitted by visual comparisons. When statistical tests are employed to test the goodness of a model fit, no such advantage exists. The parameters for the model yielding the best fit profile should be identical to the best fit parameters for the model by fitting the Fourier transforms of the profiles. For blended profiles, where corrections are required before transforming a line profile,there is a distinct disadvant- age of the Fourier analysis method. When using the line profile blends are simply masked off in the analysis presented here, the Fourier method has only been applied to the lines of Arcturus. Since in preference to visual estimates, statistical tests were made, it was not deemed necessary to apply the Fourier method to all the line profiles measured. Thus Fourier transforms were only used to synthesize profiles but not to judge the quality of a model fitted to the data. Gray (1978) describes the method of fitting model profile trans- forms to observed profile transforms. The first step is to note that, the value of the transform at zero cycles in the transform is simply the integral or area of the function. For this inverted profile this is its equivalent width0 Secondly, the frequency at which the transform first merges into the noise, enables the microturbulence to be evaluated. Finally, the rotational and macroturbulent filters are applied to matching the curvature of the model profile transform to that of the data. The method rather hinges on the presence of a sidelobe in the transform. If absent the microturbulence cannot be assigned a unique value. 105 4.6 Results and Discussion Table 4.5 shows the best fit parameters for microturbulence, macro- turbulence and rotation for each of the line profiles observed. Table 4.6 lists the average value for the profile parameters for each star. Fig. 4.4-18 show the best theoretical profile fits to the observed line profiles. In these figures the continuous line is the observed profile, the shorter dashed line is the best fit theoret- ical profile and the longer dashed line the difference curve. Inspect- ion of the difference curve served to confirm the quality of a fit. Fig. 4.4-7 of the Arcturus data also include the Fourier transforms of the line profiles. 4.6.1 Accuracy of profile measurements Where possible these data were compared with the results of Smith (1979) and Gray (1979) . A detailed comparison of the Arcturus Fe I "X 6065 profile has been made with the same profile measured by Gray and by Griffin. See Gray et al. (1979). The line profiles, their Fourier transforms and their relative flux in the line cores were compared. The Fourier transforms were used to compare profiles and the relative flux in the line core used to find difference in position of zero relative flux. Gray used a Digicon detector to record the 9th order spectrum from a 316 1 mm 1 grating blazed at 63° 26f. Order sorting was achieved using interference filters of 280 bandwidth. The average deviation of profiles between Gtay's Digicon and these FTS data is 0.8%. A "scattered light" correction of 0.7% was applied to the FTS data by Gray to match their equivalent widths. The average profile deviation between Gray*s and GriffinTs data was 1.0%, but a scattered light correction of 2.7% was applied to the latter, as might be expected for a traditional grating spectrograph. In view of the difference between the techniques employed and the agreement between these results it was concluded that the flux profiles measured are accurate to about 1%. RRCTURUS FE I 6027 1 .2 i i i | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 r 1 .0 0.8 X O ON ZD 0.6 LU > I 1 h- - CE 0.4 R 2-0 KM/S _J V 1 - 1 KM/S LU DC M 2 . 1 KM/S 0.2 H 100 MA 0.0 0.2 -30 -20 -10 0 10 20 30 KM/S FIGURE 4.4(a) - The Fe I * 6027 % profile of Arcturus. FOURIER AMPLITUDE L 01 RRCTURUS FE I 6065 l .2 i i 1 1 1 1 1 r T i i i i j 1 1 1 1 j r l .0 0.8 X ZD O ^ 0.6 00 LU CE 0.4 R 2.0 KM/S V 1.1 KM/S Od M 2.6 KM/S 0.2 W 182 MR 0.0 -0 .2 i—•—•—.—.—i— -J 1 1 L_ -30 -20 ' I 1 U -10 0 10 20 30 KM/S FIGURE 4-6(a) - The Fe I % 6065 ft profile of Arcturus. FOURIER AMPLITUDE 601 RRCTURUS FE I 6151 S/KM FIGURE 4.7 - The Arcturus Fe I ^ 6151 & profile Fourier transform. TABLE 4.5 - Best fit line profile parameters Star Line W £ VSini mX kms 1 kms 1 kms Hamal Fe I 5379 99 1.5 0.7 3.0 2.8(-3) 6027 93 1.5 1.2 2.0 2.1(-4) 6065 214 1.S 1.2 2.0 1.0(-3) 6151 115 1.5 0.9 2.0 1.0(-3) Pollux Fe I 5379 103 1.7 0.5 1.0 1.0(-2) 6027 90 1.5 1.2 2.0 3.0(-2) 6065 193 1.5 1.3 2.5 1.0(-3) Arcturus Fe I 6027 100 2.3 1.05 2.0 2.0(-4) 6065 182 2.6 1.1 2.0 1.0(-3) 6151 120 2.0 1.1 2.0 4.6(-3) Kochab Fe I 5379 112 2.5 104 1.0 2.0(-4) 6027 92 2.0 1.3 2.0 1.0(-4) 6065 246 2.5 1.5 2.0 3.2(-4) 6151 143 1.5 1.5 4.0 1.0(-3) Error estimates - ±0.5 t0.3 ll.O 112 TABLE 4.6 Average Velocity Parameters for Each Star v smi Star a t -1 -1 kms ^ kms kms Hamal 1.5 1.0 2.3 DG 1.9 1.4 0.0 MS 1.8 2.9 Pollux 1.6 1.0 1.8 DG 1.6 1.2 2.2 MS 2.2 0.8 Arcturus 2.3 1.1 2.0 DG 2.0 1.7 2.8 MS 1.9 2.7 Kochab 2.1 2.1 2.3 DG See Gray (1979) MS See Smith (1979) In order to compare macroturbulent velocities, the results of Smith and Gray using a radial tangential model were divided by 1.5 in order to compare with the exponential model used here. HRMRL FEI 5379 1 -2 i—1—•—r 1 .0 0.8 X ZD 0.6 0.00280 0-5 • u> 1 .00000 cr 0.4 3.0 KM/S 0.6 KM/S LxJ or 1 -5 KM/S 0.2 106 MA 0.0 0 . 2 1—1—1—1—1—1—1—1—•—•—1 « • • i i • • -20 -10 0 10 20 • KM/S Figure 4.8 - The Fe I 5379 X profile of Hamal HRMflL FE I 6027 1 .2 1 1 ' l i i i i | i i - i -• I i i i i - 1 -0 0-8 - X :D ^ 0.6 - ijj > >—• R 2-0 KM/S V/ V 1 .3 KM/S LU - or M 1 -5 KM/S 0.2 N 94 Mfl 0.0 - . i -0.2 • i _j—i— i..... i i i i i i.i i i i i -20 -10 0 10 20 KM/S FIGURE 4.9 - The Fe I y\ 6027 X profile of Hamal HRMflL FEI 6151 1 .2 ' t— 1 1— I | i 1 I 1 | 1 — I 1 • i • 1 ' ' 1 .0 0.8 X ID ^ 0.6 R 0.00250 \ / _ UJ > - . D 0.0 \ / E 1.00000 \ / £ 0-4 R 2.0 KM/S \ / _ _J V 0.9 KM/S \ / UenJ M 1 -5 KM/S V/ - N 112 Mfl 0.2 - s ^ . - - v 0.0 -0.2 L 1. 1 —i L—i 1 1 1 1 i 1 1 1 • ' -20 -10 0 10 20 KM/S Figure 4.11 - The Fe I ^6151 X profile of Hamal. i KOCHAB FE I 5379 1 .2 "i 1 1 1 1—:—i 1 r —i | 1 • . r— n i i r | 1 1 1 r 1 -0 0-8 X ^ 0.6 LU CE 0.4 R 1 .0 KM/S V 1 .4 KM/S Ctl M 2-5 KM/S N 1 12 MR 0.2 0.0 -J —I i L -0.2 j • < 30 -20 -10 0 10 20 30 KM/S FIGURE 4.12 - The Fe I ^ 5379 X profile of Kochab. KOCHRB FEI 6027 1 ~> I " 1 1 1 1 1 1 "i i 1 1 1 1 1 r 1 .0 0.8 X ZD 0.6 R 0.00002 D 0.0 1 .00000 CE 0.4 R 2-0 KM/S V 1 .3 KM/S LU Qd M 2.0 KM/S 0.2 N 92 MR 0.0 -0.2 -I 1 I L_ -J 1 1 I L. -20 .-10 0 10 20 KM/S Figure 4.13 - The Fe I ^ 6027 X profile of Kochab. i KOCHAB FE I 6065 FIGURE 4.14 - The Fe I ?i6065 X Profile of Kochab. KOCHRB FE I 6151 1 -2 t 1 1 1 1 r t 1 r 1 .0 0.8 X ro 0.6 O LU 0.4 R 4 .0 KM/S (X V 1 .5 KM/S cn M 1 -5 KM/S W 143 MA 0.2 0.0 -0.2 -i—i—i—i i i i » « -1 1 1 1 1 I L_ -20 -10 0 10 20 KM/S FIGURE 4.15 - The Fe I 6151 X Profile of Kochab POLLUX FEI 5379 KM/S FlgUre 4"16- " The I ^ 5379 X profile of Pollux. I POLLUX FEI 6027 1 -2 i 1— 1 i • • i i | i • i i 1 | 1 I I 1 1 .0 0.8 - - x ZD 0.6 R 0.00030 \ / LU v > D 0-0 \ / K) i—i E 1.00000 \ / 0.4 - R 2.0 KM/S ^ - cr V 1 .2 KM/S _J LU, M 1 .5 KM/S DC N 89 MR 0.2 - - 0.0 -0.2 I i i , i • X.. .. 1 1 L- -J J 1 1 i.l i i i i -20 -10 0 10 20 KM/S Figure 4.17 - The Fe I 6027 X profile of Pollux, i POLLUX FE I 6065 1 .2 n 1 1 1 I r—T ' ' 1 1 1 1 1 J . r- 1 -0 0-8 X u_ 0.4 h R 2.5 KM/S V . 1.3 KM/S LU CH M 1-5 KM/S 0.2 b N 193 MR 0.0 0.2 —<-—"—i—L—i • • _i_ -20 -10 o 10 20 KM/S FIGURE 4.18 - The Fe I ^6065 X Profile of Pollux. 124 4.6,2 Abundance and Microturbulence For lines on the shoulder and plateau regions of the curve—of— growth, a fractional change in the microturbulence may require an order of magnitude change in the abundance, to restore the line strength. Thus it was expected that abundances would not be determined particularly accurately. Other sources of error arise from uncertainty of the oscillator strength values particularly for the high excitation potential lines, published values vary by a factor of half a dex . Another source of error in estimating the abundance would result if the lines are not formed in local thermodynamic equilibrium, as discussed below. The model used to describe the velocity field has limitations. The micro- macroturbulence model does not account for motions of a stellar photo- sphere of the order of an optical depth scale height. The oscillator strength errors would increase the scatter in the values derived for the abundance. Under-estimating microturbulence and failing to account for departures from LTE would make the abundances appear anomalously large, which is evidently the case. The solar abundance given by Withbroe(1971) is 2.5 (-5). A useful order of magnitude formula for testing the validity of LTE is given by Mihalas (1978) who quotes Bohm (3r9*e) . The formulae are expressions for the ratio of collisional to radiative excitation and ionization rates. For LTE to be satisfied the collisional rates should dominate the radiative rates» The level populations of the atoms should depend on the electron temperature, T , of the surrounding material rather than the intensity of the radiation field passing through the material. For excitation rates; Cu fcj 125 For ionization rates; where A 3 e* m* x he Xij kT„ and W J* BA J^s mean intensity 8/V," Black body intensity YJ = dilutaion factor of the radiation field. Bohm (1970) tabulates the wavelength dependence of the collisional and radiative rates, for the sun and a fixed n and T . Of interest e e here is the depth dependence of these rates through a stellar atmosphere of spectral type KO III, as shown in Table 7, for a fixed wavelength of 6000 X. It is evident that LTE is only valid deep in the photosphere, from where the continuum radiation emanates. In the region of the photosphere where the line radiation escapes, significant departures from LTE occur, both in ionization and excitation. The result of the departures from LTE in excitation is to reduce the line source function toward the boundary of the star. Avrett and Hummer (1965) solved analytically the line source function for a two- level atom where the ratio of collisional to radiative excitation is constant throughout a stellar photosphere. The line source functions S^(^) are given approximately by the expression. (3), T^ is the optical depth at the line centre. TABLE 4.7 The Ratio of Collisional to Radiative Ionization and Excitation Rates in the Photosphere of a Late Type Star. T = 4.500°K, logg=1.5, Z/Zq = -0.5 Optical Depth Temperature Gas Pressure Electron Pressure C../R.. C.,/R., __ U ik lk 1.0 (-4) 3171 6.66 (1) 5.72 (-4) 5.0 (-4) 4.3 (-4) 1.0 (-3) 3486 1.88 (2) 2.75 (-3) 2.1 (-3) 1.8 (-3) 1.0 (-2) 3693 5.78 (2) 9.59 (-3) 608 (-3) 5.7 (-3) 1.0 ("D 3977 1.93 (3) 3.79 (-2) 2.4 (-2) 2.0 (-2) 1„0 (0) 4867 6.42 (3) 2083 (-1) 1.3 (-1) 1.1 (-1) 1.0 (1) 7412 9.60 (3) 8.83 (1) 2.2 (1) 1.8 (1) C. JR.. ratio of collisional to radiative excitation rates. ij iJ Wavelength - 6000 X 127 Slit*)- flf 6a I - bt\ (to i > tt > i //ij" C" where = ^ + ^ Using this formula effects of non-LTE can be simulated by combining the line and continuum source functions as follows : V V "a With the approximate non-LTE source function, the flux profiles of the Fe I \ 6151 line of Arcturus was re-computed for various values of . The profiles are shown in Fig. 4.19 There was a considerable increase in the equivalent width of the line and marked change in the shape of the line, even for optimistic values of the chosen. The change in the equivalent width with is given in Table 4.8* Evidently values derived for the velocity parameters must be regarded with some caution. When considering depth dependent 4jjj and a multilevel model atom depression of the line source function is reduced since 6jj approaches unity at depth. Also higher lying energy levels approach LTE because of their larger collisional cross-sections and their collisional coupling to the numerous adjacent levels. The studies of non-LTE Fe I line formation of Athay and Lites (1972, 1973) and Lites and Cowley (1974) 5 t employ such jL sophisticated multilevel atom. The later comment, "that A A higher excitation lines formed on the linear part of the curve-of-growth RRCTURUS FE I 6151 E=0.01 1 • 2 —•—'—•—•—i—•—'—1—'—i—•—1—•—•—i—'—>—•—r 1 .0 0.8- x ZD ^ 0.6 0.00100 N3 UJ 00 > 0.0 0.01000 0.4- 3.0 KM/S cr 0.7 KM/S UJ Od 2.0 KM/S 119 Mfl 0.2- 0.0 — -0.2 -1 I I 1 I 1 L. -20 -10 0 10 20 KM/S Figure 4.19 - Npn-LTE Fe I 6151 X profile for Arcturus tj Bootis with 6 =0.01. i TABLE 4.8 Equivalent width Versus the Non-LTE parameter For the lines measured in Arcturus Epsilon 6027 6065 6151 WW W n& % mA. mA .1.0 100 181 121 0.1 113 13 184 1.6 128 5.8 0.01 127 27 187 3.3 138 12.3 130 show increases in equivalent widths of characteristically 10-15% over those of LTE computations". Thus departures from LTE cannot be ignored, and merit close study, in parallel with turbulence studies. Velocity fields cannot be studied separately from non—LTE effects and vice versa. 4.6.3 Macroturbulence and Rotation Neglecting effects of non-LTE the mean square difference maps indicate the ability to distinguish and assign values to the macro- turbulent and rotational velocities. There is a valley of minimum mean square difference the minimum of which is described approximately by the expression. <5) Vj^M J VVSlttt,YH is the value at which the minimum mean square difference occurs when VSinC = 0. Likewise, V5iwtm is the. value of the mean square difference when 0. For a unique value to be assigned to the macroturbulence and rotation, there must be a single minimum along the base of the valley. This is the case for the Arcturus Fe I \ 6151 data. However, the error bars may be derived by considering the variation in the parameter required to double the root mean square difference. Refer to Table 4.4 4,7 Conclusions It is concluded from the scatter of velocity measurements from different lines, that the macro-micro turbulence model with the accompany- ing assumption of LTE is not an adequate model describing spectral line formation. Until careful observations were reduced, this fact was not obvious. Profiles should in preference be calculated using a more realistic model for turbulence and determining the effects of departures from LTE. This would then account for the highly individual nature 131 of the profiles. Profiles can be measured more accurately than they can be fitted consistently with the LTE micro-macroturbulence model. Thus for future studies, it is recommended that multilevel model A atomjf be used to synthesize line profiles to account for departures from LTE. Also that a more realistic model for line formation in turbulent media be employed; cf. Gail and Sedlmayr (1975). A useful e starting point would be to study o< Serp>zntis K2 III. This is a super metal rich star where the lines, according to Griffin (private communi- cation) are very deep and have residual intensities approaching zero. This could be explained by a superabundance of metals or departures from LTE, profile measurements would help to resolve the matter. « 132 CHAPTER 5 INTERSTELLAR Na I D ABSORPTION LINE STUDIES 5.1 Introduction Interstellar absorption features are generally very narrow compared to the stellar spectra on which they are superimposed. There- fore to reduce broadening due to the instrumental profile of a spectro- meter, considerably higher spectral resolution is required for measuring interstellar line profiles than stellar line profiles. For stellar profiles a resolution of 10^ suffices but for interstellar lines at least 5 x 10^ is required. At low spectroscopic resolution the equivalent widths of interstellar absorption lines can be measured. At high spectral resolution the line shapes or profiles can be determined. Line profiles give considerably more insight into the physical processes and conditions governing the spectral line formation than equivalent width measures. The Discrete Cloud Model of the interstellar medium proposed by Ambartsumian (1940) is helpful in explaining these insights. The discrete cloud model is composed of interstellar clouds or concentrations of material surrounded by a tenuous intercloud medium. Interstellar absorption features are observed if the line of sight to a star intercepts a cloud. Line profile measurement yields the cloud -2 ... -1 column density, Nm , the velocity dispersion, b kms , of the absorbing atomic species and the radial velocity of the cloud, Tykms \ The velocity distribution of absorbers in the cloud enables the broadening mechanism to be distinguished, be it thermal, turbulent or Shockwave. Individual clouds may be modelled in greater detail with line profiles of the same atomic species but in successive ionization states, enabling the electron pressure and temperature to be estimated. Several clouds are usually observed and are only discerned because their radial velocities differ. The cloud dimensions, 2 pc, and density, ^ cm , 133 can be estimated knowing the distance of the star, d pc, and assuming a 1 dumpiness1 factor, £ , the ratio of the density of the interstellar cloud to the intercloud medium. Then with observations in numerous lines of sight, the effect of galactic rotation of the interstellar medium can be perceived. This is derived from the variation of radial velocities of the interstellar clouds with galactic longitude and latitude. Measurement of several different atomic species provides element abundances and depletion ratios that are valuable for investigating the chemical evolution of the galaxy,. However, deductions concerning element depletions can be no more accurate than our knowledge of the radiation field pervading the interstellar medium. Two extensive surveys of Na I D interstellar absorption lines have been made: Adams (1949) used the Mount Wilson Coude Spectrograph and more recently Hobbs (1969) used the PEPSIOS interferometer of McNutt and Roesler (1966) , The intrinsic width of the interstellar lines was less than the spectral resolving power of the spectrograph employed by Adams, (FHWM $ 6 kms "*") , so he was restricted to equivalent width measurements. With the Doublet - ratio method devised by Stromgren (1948) both cloud column densities and velocity dispersions were estimated for partially saturated D lines. However, the velocity dispersions were overestimated (4 kms because the equivalent widths were of multiple and not individual absorption features0 The variation of the radial velocity of the interstellar clouds with galactic longitude was shown by Munch (1957) to reveal the spiral arm structure of the galaxy. Hobbs (1969 a) showed, with the PEPSIOS observations, that most of the lines listed by Adams as single were in fact multiple. The FWHM of the PEPSIOS spectrometer passband was 0.5 kms"1. Hobbs1 (1969 b) comparison of theoretical and observed interstellar Na I D line profiles gave velocity dispersions ranging from 0.64 to 1.94 kms However, 134 the PEPSIOS spectrometer did not completely resolve these profiles. Results discussed in this chapter, obtained with the Michelson inter- ferometer (Wayte and Ring, 1977), show that there is more valuable information about the interstellar medium to be acquired, at spectral resolutions greater than those achieved with PEPSIOS, i.e. greater than a third of a million. 5.2 Interstellar Absorption Line Profile Synthesis For comparison with observations theoretical line profiles are derived in the following manner. The forward emission of radiation in a line of sight is assumed negligible, so that the solution of the radiative transfer equation, for pure absorption, gives the relative intensity RCX ) » ^ = e" (1) I c L^ refers to the intensity, the subscript c specifying the intensity of the nearby continuum and X denotes the wavelength,. The optical depth C Si-JLNf X>X) (2,ii) 4-^fnc for an isolated interstellar line without any fine structure, where (v) is the absorber velocity distribution and f is the oscillator strength of the transition,, The absorber column density N = £ ng dS where s is the distance along the light of sight and ng the number density of absorbers. The wavelength of the transition is ^^ and e, m, c and are the electron charge, mass, velocity of light 135 and permitivity of free space respectively. These physical constants are tabulated by Woodgate (1970). This method follows the procedure used by Hobbs (1969 b) in his analysis of isolated interstellar lines. Hobbs employed two types of velocity distribution for computing theoretical line profiles. A Gaussian velocity distribution (v) with which thermal and turbulent broadening are associated. The exponential distribution (v) is a simple model of Shockwave broadening, where the root-mean-square absorber velocity is expected to be greater. The normal velocity distributions are V ' a V - Vr v* oo - • -i— Mv'/ fey/r- (3,i) (3,ii) A, 2 The r.m.s. velocity dispersions are VQ = fe/Tz and ^ J2, (3,iii& but the Gaussian t value may be composed of thermal and turbulent components, as described by Hobbs (1969 b) 2kT ^ + (3,v) y m T V is the turbulent r.m.So velocity and k Boltzmann's constant, Tn the atomic mass and T the temperature. For a transition manifesting hyperfine structure that contributes to the overall width of the absorption feature, the absorption due to each hfs component must be evaluated separately. If fj is the oscillator strength of an hyperfine component and X ^ its wave" then if we define as the mean wavelength of all the n hfs transitions, i.e. 136 •HfS j C4,i) setting then the total optical depth is ftnfs 4gomc j-, If there are several clouds in the line of sight then the contributions to the optical depth due to each cloud must be added. If there are fl^ clouds intercepted along the line of sight then the total optical depth is given by ,1 Z H-Zfj^C^j) (4'iii} 5.3 Instrumental Profile Broadening The smearing due to the spectrometer instrumental profile must be allowed for where the absorption features are not completely resolved. The theoretical line profile model R(v) is synthesized at higher spectral resolution than the observed data and is then convolved with the instrumental profile S(v) of the spectrometer. The result of this convolution I(v) is then compared with the observed spectrum. Thus I(v) = SCv-v'Mv' (5,i) where § S(v)dv ® 1 (5,ii) This convolution may be evaluated using two methods, the choice of method dependent on the nature of the spectrometer instrumental profile. The first is by discrete integration and sliding of the spectrometer instrumental profile across the theoretical line profile model, so that I(v) is given by 137 y KVO* £ loj^K-V^S^)^j (6,i) where V. =» V. + j 4 V. J 1 J IAJ j = quadrature weight T and 2>jSfvJ)»l (6,ii) The second method uses the discrete Fourier transform and the convolution theorem which is defined, to quote Bracewell (1978) "If R(v) has the Fourier transform R(s) and S(v) has the Fourier transform S(s); then R(v)*S(v)", (* denotes convolution) "has the Fourier transform R(s)°S(s); that is the convolution of two functions means multiplication of their transforms". The broadened spectrum is recovered by evaluating the inverse Fourier transform of the product R(s)°S(s)0 Thus R(s) = FT £ R(v)| (7,i) S(s) = FT{s(v)} (7,ii) 1 I(v) = FT" £R(S).S(S)} (7,iii) where FT denotes the Fourier transform and FT 1 its inverse. It must be appreciated that both the above methods are approximations, since discrete functions are used in place of continuous. Care needs to be taken that the errors due to this approximation are less than photometric errors in the observations. These errors are made small by synthesizing the theoretical line profile model at a resolution greater than used for the measurements 0 In the analysis present here the theoretical profiles were synthesized at twice the observed resolution. 138 5 .4 Saturated Line Profiles For saturated line profiles where the relative intensity approaches zero at the line centre, a useful expression for the cloud velocity dispersion may be derived. The expression also gives a useful estimate of the resolution necessary to resolve such a profile. Consider an isolated interstellar line with no hyperfine structure and a Gaussian velocity distribution of absorbers, then adapting Equations 1, 2 and 3,i, the relative intensity may be expressed as 2 R(x) = exp(- r6 exp(-x )) (8,i) - N-f where T0 = ^ the line centre optical depth (8,ii) 4£,mc b Jrr and x = Ol-Vr)/fc> (8,iii) For R(x) = 1/2 it follows that x2 = In To- I* (9,i) dR x < > = xIH2. (9,ii) dx Since dR(x) = dv dR(V) _ ^ , dR(v) (9,iii) dx dx dv dv then substituting for x from (8,iii), for dR(x)/dxand for R(x) = 1/2 from (9,ii) and rearranging the terms it is found that the velocity dispersion is given by t = /I jv- 1»V (io,i> where the superscript * denotes the velocity at half the relative intensity. This is an exact expression for the velocity dispersion 139 and is independent of the column density. To evaluate fe for a resolved line then, we only require to measure the half width at the half relative intensity points and the gradient at these points. There is no corresponding expression for an exponential distribution. The column density can be shown to be 2 NM = 4,1*2 feomc. t» • exp( (/h M |J 1) n(10,nn ...) The application of these expressions presume that the relative intensity gradient is not influenced by the spectrometer's instrumental profile. When this is not so the maximum relative intensity gradient is restricted to approximately the spectral resolving power, R, divided by the velocity of light, C„ Placing this result in the expression for the velocity dispersion, it is found that the value derived is an upper limit, bu. . (11,i) and similarly for the column density a lower limit is estimated, N^ . If b is the actual cloud velocity dispersion and N its actual column density, then taking the ratio of N to N£ it is found that if b is overestimated OS !>cl then N can be underestimated by several orders of magnitude. This is evident from the following expression N : b exp 5.5 Synthesis of Na I D1 and D2 Interstellar Lines The Na I D1 and D2 transition lines both show hyperfine structure. They are composed of two hyperfine structure components0 The ratio of 140 the oscillator strengths for both sets of hyperfine structure components is 3:5, the redward component being the stronger. The wave- lengths, oscillator strengths and transition specifications are listed in Table 5.1. Fig 5.1 shows synthetic interstellar D1 absorption 11 -2 profiles with the cloud column density fixed at N = 5 x 10 cm and for velocity dispersions of b = 0.25 to 0.65 kms . They illustrate the desaturation and smearing of' the hyperfine structure for clouds with larger velocity dispersions. Also that the overall width of the line is dominated by the hyperfine structure for all these velocity 1 dispersions. Fig'5.2 - the velocity dispersion is fixed at 110.3 5 -km2 s and the column density has values N = 1.0, 3.0, 10. and 30x10 cm . The profiles saturate with increasing column density and become effectively black at the line centre. It is evident that the gradient at half the relative intensity for these lines increases markedly with column density, and that the characteristic of the hyperfine structure is lost. The profiles were calculated using the prescription described in Section 5.2. A saturated and an unsaturated Na I D1 profile were computed, both with velocity dispersions of 0.35 kms 1 and column densities of 10 x 1011 and 3 x l'O11 cm 2 respectively. They were both then convolved with the Fourier Transform Spectrometer instrumental profile corresponding to resolutions of 0.33, 0.5 and 1.0 kms \ They indicate that the unsaturated profile requires less resolving power than the saturated profile. This is more clearly seen by evaluating the mean square difference between the theoretical and instrumentally broadened instrumental profiles, restricting the range over which the differences are measured to a few Doppler widths or fixed bandwidth. The mean square difference is helpful in deciding the best combination of resolution and signal-to- . noise ratio for the measurement of a profile. The criterion used is that the mean square noise in the observed spectrum is the same as the mean 141 TABLE 5.1 Na I Wavelengths in air and oscillator strengths that were used in model fitting (X) (& D2 5889.939 0.244 5889.958 0.406 D1 5895.911 0.122 5895.932 0.203 Notes: The wavelengths are from McNutt and Mack (1963) . The oscillator strengths that were used in the study differ slightly but not significantly from the values as given by Mashinski (1970). 142 1 .2 1.0- 0.8- 0.6- 0.4- 0.2- 0 .0 -4 -2 0 4 HELIOCENTRIC VELOCITY FIGURE 5al 11 -2 Synthetic profiles of the D1 line with column density N = 5 x 10 cm and velocity dispersions b= 0.25, 0.35, 0.45, 0.55 and 0o65 kms \ The profiles desaturate with increasing velocity dispersion, A Gaussian velocity distribution was used to synthesize the profiles. 143 1 .2 1 .0 >- CO 0.8 LLJ 0.6 LU 0 . 4 az LU 0-2 0-0 -4-2024 HELIOCENTRIC VELOCITY FIGURE 5.2 Synthetic profiles of the D1 line with the velocity dispersion b = 350 ms" 11 -2 and cloud column densities of N = 1.0, 3o0, 10 and 30 x 10 cm . The profiles saturate with increasing column density and exhibit completely opaque cores. A Gaussian velocity distribution was used to synthesize the lines. 144 square difference between the theoretical and broadened profiles. The theoretical and broadened profiles are shown in Figures 5.3/4 . The variation of mean square error with spectral resolving power for the saturated and unsaturated lines are plotted in Figure 5.5., It shows that the resolution required to match a given signal-to-noise ratio is considerably greater for a saturated line. Also that the signal-to- noise ratio increases dramatically with resolving power but that it increases more gradually for the saturated profile. The resolution Of 100 required to match a signal—to-noise for the unsaturated profile is A 0.4 kms 1 and 0.25 kms 1 for the saturated profile. Using expression (11,i) for a resolution of half a million and effective half width at half relative intensity of 1.5 kms 1 (actually full width 4.0 kms 1 including hyperfine structure) then the lower limit to the velocity dispersion is 0.8 kms \ for a very saturated line. If the actual l> value were half this then the derived column density would be over 400 times too small! A resolution of several million would be required to measure the cloud column density and velocity dispersion accurately for such a saturated cloud. Obviously N must be determined using weaker transitions, such as Grutcher (1975) has made of the weak ultraviolet Na I doublet at 71^3302.38 and 3302.99. These column density measurements made from equivalent widths could be combined with Na D1/D2 measurements of high resolving power to obtain the | V*— Vvj parameter and velocity dispersion in the clouds, using Equation (10,ii). This would help to establish if there is any relation- ship between cloud column density and velocity dispersion, 5 .6 Observational Data Observations were made with the Michelson Interferometer for the Isaac Newton telescope during various observing sessions in 1977 and 1978. Details of observing runs are given in Table 5.2. A resolving 145 1 o 2 >- f— CO z LU LU a: LU 0.2h 0-0 HELIOCENTRIC VELOCITY FIGURE 5.3 An unsaturated synthetic D1 profile (solid line), with cloud column density N = 3,0 x 10 11-cm2 and velocit. y dispersion b = 350 ms- 1 , showing the effects of instrumental broadening with resolutions of 0.33(—-—•-), 1 0,5 ( ) and 1.0 (- - -) kms™ 0 146 1 .2 1.0-5? >- t—i CO 0.8- LLJ 0.6- LLJ 0.4- CE _) LLJ CC 0-2- 0.0 -4 -2 0 4 HELIOCENTRIC VELOCITY FIGURE 5.4 12 —2 A saturated synthetic D1 profile with cloud column density N = 1.0 x 10 cm and velocity dispersion b = 350 ms , illustrating the effects of instrumental broadening. Note the negative excursion of relative intensity due to the sine function instrumental profile of the Fourier transform spectrometer. 147 Spectral resolving power /10" 20 10 5 I 10" 10 -1 -2 cn 10 H« OQ 0 10 -4 10 0.2 0.4 0.6 0.8 1.0 Velocity Resolution km/s FIGURE 5.5 The spectral resolving power versus the accuracy of line profile measure- ment, for a saturated and unsaturated D1 line. The velocity dispersion b - 350 ms"*-1- and the column densities are N = 3.0 x ion and 3.0 x cm- 2 respectively. Applying the criterion that instrumental profile distortion and noise be equal, then to measure the profiles to a signal-to-noise of 100 requires a velocity resolution of 0.4 and 0.25 kms~^ for the unsaturated and saturated lines. 148 TABLE 5.2 Details of Observing Runs Star Line Date Universal Radical Velocity Integration Time Correction times h m h tf Cygni D1 8/7/77 24 11.05 2h15m h m 10/7/77 2 15 10.75 3^15m cloudy h 10/7/77 22 10.81 lh15m h J Cygni D1 16/4/78 2 -12.14 2h Om h m 17/4/78 oh -12.24 2 O h 20/9/78 23 8.84 3h45m 22h 21/9/78 8.95 6h15m h m 22/9/78 21 30 9.09 5h30m h m 23/9/78 21 45 9.26 5h Om h m 25/9/78 23 15 9.64 5h30m h 8 Cygni D2 6/10/78 23 11.14 5h Om h m 7/10/78 23 30 11.23 6h30m h m 8/10/78 20 30 11.28 7^15m cloudy h m 11/10/78 22 15 11.69 5h30m h m 12/10/78 22 0 11.79 4h15m h m 17/10/78 21 15 12.22 4h45m h m h 'Vj Taurus D1 19/9/78 5 0 26.24 2 0m h m 21/9/78 4 30 25.81 3h h m 22/9/78 4 30 25.56 ih om h m 23/9/78 4 30 25.30 2h 0m h m 24/9/78 0 30 25.35 "2h30m h m h 25/9/78 4 15 24.81 2 0m h m 26/9/78 3 0 24.62 lh Continued p .2. 149 TABLE 5.2 (Continued) Star Line Date Universal Radical Velocity Integration time' Correction Times h m Taurus D2 6/10/78 4 0 21.34 4h 0m h m 7/10/78 4 0 21.01 4h15m h m 8/10/78 4 15 20.62 4h30m h m 12/10/78 3 45 19.13 5h30m h m 13/10/78 k 30 18.94 lh15m 18/10/78 3h15m 16.72 6h 0m 150 power of 5 x 10^ was used capable of partially resolving the hyperfine structure of the Na I D1 and D2 interstellar absorption features, assuming the velocity dispersion in any clouds present were small enough. A velocity resolution of 0.5 kms 1 was used since a broader instrumental profile would not have revealed any hyperfine structure clearly, neither would the latter give any reliable information about the velocity dispersion since the line width is dominated by the hyper- fine structure itself. Interstellar Na I D1 and D2 spectra were measured towards three stars si. , J Cyg and ^ Tau. Details of these lines of sight are given in Table 5.3 A typical post monochromator plus image slicer bandpass is shown in Fig. 5.7(a). The bandpass width, about 2 X, was kept as small as possible to minimise observing time. It still provided adequate stellar continuum for purposes of normalizing or dividing out the bandpass profile. The bandpass shown is the integration of the Na I D1 line towards Ct Cyg. The D1 profile is off centre. Weak telluric features are identified and hyperfine structure in the weaker D1 components can be discerned. The inte- grations are the sums of several "fast" scans of about 40 minutes duration each. Fast scans were preferred over taking one long integration allowing systematic errors and losses due to weather to be reduced. Also, the change in radial velocity during the night due to the earth's rotation, about 10.2 kms \ is significant when working at very high spectral resolution and leads to spectral degredation for very long integrations. The radial velocity corrections required for co-adding spectra were computed using the program written by David Allen for the WANG desk calculator at the Royal Greenwich Observatory. The accuracy of the program i20 ms 1 is quite sufficient for this application. The radial velocity corrections are included in Table 2. TABLE 5.3 Interstellar Na I D1 and D2 Spectra Observed Star 1(1900) b(1900) Mk type E(B-V) v(helio) vsm 1 r km/s km/s pc Cygni HD197345 84 02 A2 la 1.25 + .09 -5v 15 500 Cygni HD186882 79 10 B9.5 III 2.87 + .01 -21 128 50 Tau HD23630 167 -23 B7 III 2.87 + .03 10 210 120 tn Notes: Mk types are from Kennedy and Buscombe (1974). V magnitudes and B-V data used to compute colour excess are from Johnson (1966), intrinsic B-V data from Johnson (1963). The vsin i values are from Bemacca and Perinotto (1973) for and 8 Cygni, and from Boyartchuck and Kopylov (1964) for ^ Taurus. This latter catalogue quotes vsin i = 185 km/s for & Cygni whilst Uesugi and Fukuda (1970) gives vsin i = 139 km/s. Stellar radial velocities are from Hoffleit (1964) and Lucy (1976) for Cygni; distances are given by Hobbs (1969a). i 152 5 .7 Data Analysis The method used to analyse the interstellar Na I D lines was to synthesize theoretical line profiles, convolve these with the instrumental function and compare the result to the observational data. Multicloud models were, used for the Cyg and A^ Tau spectra. The models have four main parameters, namely, the number of clouds, their radial velocities, column densities and velocity dispersions. The first two parameters were fixed by inspection of the data. Where uncertainty exists in the number of cloud components, more than one model was synthesized. The column densities and velocity dispersions were adjusted to improve the accuracy of the model. The goodness of the model was based upon inspecting the difference between the model and the observed profile and evaluating the mean square difference between the model and data. The difference curve is useful because it helps determine which parameter should be adjusted in order to improve the model. For isolated absorption lines the CURFIT subroutine described by Bevington (1969) was used to improve the model automatically. The code required some modification so that the instrumental profile could be included. The subroutine cannot be used for severely blended profiles because there is no unique solution, in which case the method fails to converge. The Na I D1 and D2 data were treated independently so comparison of the model parameters derived for both spectra gave a measure of the accuracy to which the parameters were measured. Ideally the model parameters should be the same for both spectra. Profiles of isolated interstellar absorption lines were synthesized using an exponential absorber velocity distribution and the results compared with those from a Gaussian distribution. It was concluded for the Na I clouds studied that it is not possible to distinguish between these distributions. Either form fits the data satisfactorily. 153 Even though the theoretical exponential profile reveals the hyperfine splitting more dramatically than does the Gaussian, after convolution with the instrumental profile both give good fits to the observed profile. Higher spectral resolution is required in order to distinguish which velocity dispersion applies. See Figure 5.6. 5,8 Results of Model Fitting (a) The ^ Cygni Sight Line Previous high resolution studies of Na I in this line of sight by Livingston (1969), Hobbs (1969a) and Munch (1968) have shown that there are five major clouds. The Na I profiles obtained from the Michelson interferometric study, over the same portion of spectrum from -22 to +2 kms \ show major differences when compared to the earlier studies which all gave similar results. Firstly, at least two further components are resolved at -5.3 and -1.2 kms 1 in addition to the five strong features which are located at -21.6, -12.6, -8.7, -2.8 and +1.4 kms These new components are comparatively weak but there is little reason to doubt their existence. More components may exist. Secondly, hyperfine structure is seen in the +1.4 and -2.8 kms 1 components and possibly also in the -1.2 kms 1 line, although the latter feature is weak and blended. This study is the first to detect hyperfine splitting of Na I in any inter- stellar cloud. Thirdly, the two strong components at -21.6 and -12.6 kms 1 have completely black cores. In contrast the plots of the inter- stellar spectra in this sight line given by Munch (1968) exhibit core intensities of approximately R ^ Cr* 0.3 and 0.2 for the Kitt Peak and Mount Wilson studies. This suggests that scattered light occurs within the instruments and/or the effective spectral resolution was considerably lower than claimed - possibly because the HWHM is not sufficient to describe the response of a spectrometer. See Figure 5.7. 154 FIGURES 5.6(a) and (b) Notes Comparison of the interstellar +1.4 km/s Na I component in Ot Cygni with a theoretical line profile having (a) a Gaussian velocity distribution and (b) an exponential distribution. In this and subsequent figures the solid line is the observation, the short dashed line is the theoretical line profile and the long dashed line is the theoretical profile after convolution with the instrumental profile. The curve below these profiles shows the difference between observations and theoretical profile after convolution; this difference curve was used to judge the goodness of fit. 155 +1 km/s D1 in a Cyg 1 .0 >- CO 0.8 UJ 0.6 LU 0.4 CE I UJ 0.2 Gaussian J 0.0 0-2 m C_) / • UJ \ 0.0 £ U-. Ll_ -0.25 -2 0 2 4 HELIOCENTRIC VELOCITY FIGURE 5.6(a) The +1 km/s D1 component towards ^ Cygni; Gaussian velocity distribution. 156 + 1km/s D1 in aC LlJ CJ 2: LlJ UJ Ll_ U_ -0.2° HELIOCENTRIC VELOCITY FIGURE 5.6(b) The +1 km/s D1 component towards Cygni: Exponential velocity distribution,. 157 Nal Dl in aCyg FIGURE 5.7(a) A typical interferometer passband showing interstellar Na Dl in Cygni. Weak telluric absorptions are also present. 158 O FIGURE 507(b) Previous high resolution Na I D2 spectra of oC Cygni. Taken from Spectros- copic Astrophysics, pg.298, Ed0 G.H0 Hertig, University of California(1970) . Mt0Wilson: Munch and Vaughan, above reference, Kitt Peak: Livingston and Lynds (1969). Lick Obs0: Hobbs, L.M0, Astrophys.J., (1965) 140, 190. 159 In a preliminary analysis of the +1.4 kms 1 Dl component 11 -2 (Blades, 1978) it was found that N(Na I) = 3.0 x 10 cm and b =0.38 kms Two separate multi-component models have been fitted to the data over the velocity range -25 to +5 kms 1 for both the Dl and D2 spectra. The seven component model is illustrated in Figures 5.8/9 for the Na I D2 and Dl features, individual components are shown in Figures 5.8(a)/9(a) and the overall profile in Figures 5.8(b) and 5.9(b). The column density and velocity dispersion data from this model are listed in Table 5.5. Individual components range in N 1 r\ 0 10 mmO from 3.5 x 10 cm" to 2 x 10 cm" and' ^ / fl from 0.26 to 0.99 kms The total column density in the sight line is N(Na I) 12 -2 6.0 x 10 cm in good agreement with Hobbs1 value (1974) of 12 -2 4.8 x 10 cm . The bulk of the interstellar absorption is contained in only three clouds. The nine component model is shown in Figures 5.10/11 for the D2 and Dl lines, respectively. A better fit to the observations is obtained from this model. The improvement is primarily due to the introduction of the additional component at -10.7 kms \ coupled with the necessary reduction in the b value for the component at -12.6 kms The reason for introducing this component is that it is difficult to see how the -12.6 and -8.6 km/s components could blend to produce the strong absorption with R^(D2) of 0.35 at -11 kms The presence of this component makes a considerable improvement in the fit of the model to the observations, in this region of spectrum (compare the difference curves of the nine and seven component models in the figures) The introduction of the ninth interstellar component at -6.8 km/s makes a minor improvement to the fit. The model parameters for the nine clouds are contained in Table 5.6 . 160 TABLE 5.5 Seven cloud fit to Cygni v(helio) N(Na I)lo"U b/JT" _2 km/s cm km/s -21.6 19.0 0.51 -12.6 20.0 0.71 - 8.7 11.0 0.75 - 5.3 2.85 0.99 - 2.8 3.85 0.30 - 1.2 0.35 0.35 + 1.4 3.0 0.26 TABLE 5.6 Nine cloud fit to Cygni v(helio) N(Na 1)10 "u t/JT _2 km/s cm km/s -21.6 19.0 0.51 -12.9 14.5 0.55 -11.7 2.8 0.51 - 8.4 9.9 0.50 - 6.8 2.3 0.51 - 5.0 1.9 0.51 - 2.8 3.85 0.30 - 1.2 0.35 0.35 + 1.4 3.0 0.26 a Cyg D2 l 0 > CO o 8 UJ 6 UJ 4 o < _J ccill o 2 m 0 0 0-n 29 Q LU CC HO.O LU LL LL -30 -25 -20 -15 -10 -5 0 5 -0 .2 Q HELIOCENTRIC VELOCITY km/s FIGURE 5,8(a) - The Q( Cygni sight line. The seven cloud components fit to the interstellar Na I D2, showing the individual cloud components. i -30 -25 -20 -15 -10 -5 HELIOCENTRIC VELOCITY km/s FIGURE 508(b) - The to . . ' » .M ''' -30 -25 -20 -15 -10 -5 0 HELIOCENTRIC VELOCITY km/s FIGURE 5.9(a) - The t>< Cygni sight line. The seven cloud component fit to the interstellar Na I Dl, showing the individual cloud components. i a Cyg D1 >- 1 .Oh ^Wl A^A H CO Z 0.8 111 h- Z < 0.6 LU > n £ 0.4 in DC 0-2 0.0 0 .2 W o z: / ' . , v - .",., -, UJ 0.0 cc v u 1 » •! < - ' / UJ Li_ U. -0.2Q 30 -25 -20 -15 -10 -5 0 5 10 HELIOCENTRIC VELOCITY km/s FIGURE 5.9(b) - The <* Cygni sight line. The seven cloud component fit to the interstellar Na I Dl, showing the sum of the components. a Cyg D2 0 H>- CO o z: 8 UJ H Z 0 6 LLl > ON Ln LU DC 0 2 h LLi 0 0 Ni \r- 0.2 O LU N 0.0 £ V- ' \ • V » v V I I 1 I , • -J ' 11 > t LL \ LL -0.2 Q -30 25 -20 -15 -10 '-5 0 10 HELIOCENTRIC VELOCITY km/s FIGURE 5.10(a) - The of Cygni sight linea The nine cloud component fit to the interstellar Na I D2, showing the individual cloud components. I H O c RELATIVE INTENSITY On 1 oH- /-s 3* x ft) m r; O VJ O CO tt 3 ff H- o 0 * 05 :m H* H» 3 09 05 3* rt rt 5 ST (I) o 01 C < a rn 0 Hi H r~ 3* n n> O rr o 3 o M. c 3 o n> —I 5 -< T3 O s o o c a a. rt 3 0) o » s T3 O 3 fD 3 r? i-h H. rt rt O 3* n> H« DIFFERENCE 3 rt fD (A 991 aCyg Dl > 1 .0 r 1 H C/.) 2: 0.8 LiJ h- 2 0.6 LLI > H < 0.4 -J 111 DC 0.2 0.0 I V V ' » V «»' 1 ' I / V -0.2Q -30 -25 -20 -15 -10 0 10 HELIOCENTRIC VELOCITY Km/s FIGURE 5.11(a) - The Of Cygni sight line. The nine cloud component fit to the interstellar Na I Dl, showing individual cloud components. a Cyg D1 - ON 00 10-2 8 LU ' / . i v • » l . • V " V '» ' " /• / i , V I I V. I \ n n rc \ > u u .U (Jj LL '-0 -2 Q -30 -25 -20 -15 -10 -5 0 10 HELIOCENTRIC VELOCITY km/s FIGURE 5.11(b) - The DC Cygni sight line, The nine cloud component fit to the interstellar Na I Dl, showing the sum of the components. i 169 The total column densities obtained for both cloud models are similar, however a major change to the individual b values does occur . In the simpler model three clouds have relatively large velocity dispersions 0.71, 0.75 and 0.99 kras Introducing two more clouds reduces all the velocity dispersions to less than 0.55 kms The present data does not permit a definite choice between the two models , A better signal-to-noise is required, particularly in the region from -10 to 0 kms ^ where the presence of the stellar line is troublesome. Removal of the stellar line inevitably increases the noise level in the region at the line centre -5 kms Higher resolution is required to measure the N and b values for the saturated lines at -21.6 kms ^ and -12.6 kms At present only a lower and an upper limit can be respectively assigned to the N and b values derived. (b) The S Cygni Sight Line This sight line only manifests one Na I component at -18.6 km/s. Hyperfine splitting is clearly seen in this component. The data are presented in Figures 5.12 and 5.13 for the D2 and Dl lines, respectively, and the data given in Table 5.7 for the model fit. Hobbs obtained a velocity dispersion of 0.64 kms ^ (1969b) for this cloud which is twice that obtained in this study. This difference in velocity dispersions must be attributed to the broader instrumental profile of the PEPSIOS which did not resolve the hyperfine splitting. The discrepancy in column densities, Hobbs (1974a) gives N(Na I) = 3.6x10"^, could be due to overestimating the amount of scattered and parasitic light of the PEPSIOS spectrometer. (c) The A| Tau Sight Line Here a relatively wide interstellar profile occurs. Two components were used to fit to the data as shown in Figures 5.14 and 5.15 and Table 5.7. This data supports HobbsT(1969b) suggestion that this 170 TABLE 5.7 Model fits to S Cygni and Tau Star v(helio) N(Na I)KfU / 2 _ km/s cm km/s Cygni -18.60 2.55 0.31 Tau +12 o 5 0.5 0.92 +16.2 6.6 0.85 171 5Cyg D2 1 .0 >- CO z 0.8 UJ I- 0 .6 LU > <: 0 .4 LU cc 0 .2 0 .0 LLJ •0.0 £ LUL_ -24 -22 -20 - 18 - 1 6 - 1 -0-4 2 Q HELIOCENTRIC VELOCITY km/s FIGURE 5.12 The fit to the interstellar Na D2 profile in $ Cygni. 172 5Cyg Dl 1 .0 > co 0 .8 z LL! 2 0.6 LLI > 0 . 4 < —i LU cc 0 .2 0 .0 0 .2 LU O / X LU V • \ / 0.0 oc LU Li- LL -24 -22 -2 0 - 1 8 - 1 6 - 1- 04 .2 Q HELIOCENTRIC VELOCITY km/s FIGURE 5.13 The fit to the interstellar Na D2 profile in $ Cygni. 173 t\ TAU D2 > 1 .0 K CD Z 0 .8 LU I- 0.6 LLi > LU CC 0.2 b 0 .0 0.2 LoU •Z. ^ r UJ I . \ 0-0 £ Li. LL. 5 10 1 5 20 2-50 -2 Q HELIOCENTRIC VELOCITY km/s FIGURE. 5_. 14 The fit to the interstellar Na D2 profile in $ Cygni. 174 rj.TAU Dl 1 .0 h > 1 °-8 LLJ 2 0.6 f- LU > h0.4 -j LU c 0 .2 0 .0 10 .2 LU o z i /• U-J 0.0 cr \ i LU LL Li. 5 1 0 1 5 20 2:0.25 5 HELIOCENTRIC VELOCITY km/s FIGURE 5.15 The fit to the interstellar Na D2 profile in $ Cygni. 175 apparently isolated line was actually double. The total column density is N(Na I) 7.1 x 10 cm"2 (Hobbs 1974a, found 9 x 10U cm"2), with the bulk absorption present in one cloud. Both A^ Tau components have large b / J 2 values; the main feature having b / J*2* = 0„85 km/s. 5.9 Discussion and Comments An important result of this study is the discovery that the interstellar Na I D lines are considerably narrower than previously supposed. The widths of the lines arise from thermal (Doppler) motion of the Na I atoms and largescale non-thermal gas motions such as turbulence. Both broadening mechanisms contribute to the observed widths of the Na I lines0 Considering the isolated or well resolved interstellar components selected from the ^ , £ Cygni and Tau spectra listed in Table 5.4 ,four out of the six show hyperfine splitting Despite the limited extent of the survey it is evident that hyperfine splitting of the D lines is commonly revealed in diffuse interstellar clouds. if there were no cloud turbulence then upper limits to the kinetic tempera- ture may be derived using Equation (3,v). This gives for these six components temperatures ranging from 190 to 1990 K, however for those components showing hyperfine splitting the average temperature is less then 260 K0 These values are significantly smaller than those measured by Hobbs (1969b). From a study of eleven components he obtained temperatures from 1100 to 6100 K. Only one line of sight is common to both studies, that of S Cyg, for which the PEPSIOS study gave Tk£ 1100 Ko This study gives Tfc s 270 K0 Spitzer and Cochran (1973) using observations of H^ contained in Spitzer et al0 (1973), derived a mean T^ £ 81 K from thirteen diffuse clouds. Savage et al. (1977) obtained an average value of T, i. 77 K for 61 stars. In general, it can be seen that the typical 176 TABLE 5.7 Properties of well resolved interstellar components Na I b/jr V vt Component rms km/s km/s km/s km/s + 1.4 Cygni 0.26 190 0.20 0.45 - 1.2 OC Cygni 0.35 340 0.31 0.61 - 2.8 Cygni 0.30 250 0.25 0.52 -21,6 <* Cygni £ 0.51 - 720 ^0.48 ^L 0.88 -18.6 S Cygni 0.31 270 0.26 0.54 +16.2 to Tau 0.85 1990 0.83 1.47 1 Notes: Determination of T, in column 3 assumes V = 0 k t o The turbulent velocity in column 4 has been calculated with T. = 80 k. k The three dimensional r.m.s. velocity, V in column 5 is rms given byJ V =» b J 3/2. rms ' 177 kinetic temperature obtained from molecular hydrogen studies is smaller than the corresponding upper limits deduced from the Na I line widths. The diagnostic used for measuring the temperature using molecular hydrogen is the rotational temperature determining the relative populations of the two lowest rotational levels in the ground electronic and vibrational states of So rotational temperature is found from line intensities rather than line widths. Adopting a mean cloud temperature of "Tit — 80 K (Spitzer 1978) the cloud turbulence may be evaluated. For and 8 Cygni the turbulence ranges from 0.20 to 0.48 km$ ; for A^ Tau, however, the turbulence is 0o83 kms . It is reasonable to deduce that both thermal and turbulent motions are important line broadening mechanisms for interstellar Na I. The isothermal sound speed estimated by Spitzer (1978) for a diffuse interstellar cloud is approximately 0.7 kms \ Only the stronger Aa Tau line with 1 a root-mean-square velocity is"larger than this at 1.47 kms Hence, this confirms the conclusions of Blades et al. (1978) that internal cloud motions are subsonic rather than supersonic. 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