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«DAVID J. JEFFERY AND DAVID BRANCH Department of Physics and Astronomy University of Oklahoma Norman, Oklahoma 73019, U.S.A. Reference Jeffery, D. ...»

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Department of Physics and Astronomy

University of Oklahoma

Norman, Oklahoma 73019, U.S.A.


Jeffery, D. J., & Branch, D. 1990, in Jerusalem Winter School for Theoretical Physics, Vol. 6, Supernovae, ed. J.C. Wheeler, T. Piran, and S. Weinberg (Singapore: World Scientific), p. 149.



Department of Physics and Astronomy University of Oklahoma, Norman, Oklahoma 73019, U.S.A.


A simple and fast analysis procedure for supernova spectra is presented. Included in the presentation is an introduction to the Sobolev method of radiative transfer, a description of a simple model of a supernova, a catalogue of line profiles illustrating line formation in the simple supernova model, and an example analysis of the SN 1987A photometry and optical spectra. The SN 1987A analysis is intended as a benchmark to which more advanced and detailed analyses can be compared. Synthetic spectra calculated in the SN 1987A analysis using solar-composition thermodynamic-equilibrium predicted line optical depths are in rough qualitative agreement with the observed spectra; the agreement becomes poorer after day 100 after the explosion. It is probable that for the period from day 16 to about day 100 after the explosion that the SAAO photometric radius is close to being the true photospheric radius: earlier than day 16 the photometric radius is too small.


This article presents a simple and fast analysis procedure for supernova spectra. The procedure uses the Sobolev method of radiative transfer and a simple supernova model in order to calculate synthetic spectra. Comparisons and fits of the synthetic spectra to observed spectra allow determinations of supernova parameters. Because of the many simplifying assumptions made in the analysis procedure, the determinations the procedure makes will be mainly qualitative. More exact, but more difficult, determinations require a more advanced procedure (see, e.g., H¨flich 1988; Eastman and Kirshner 1989).

o In § II of this article, the Sobolev method is presented. The supernova model, here called the ES model, that is used in the analysis procedure is described in § III. In § IV, a catalogue of line profiles that illustrates line formation in the ES model is presented and discussed. As an example of the analysis procedure, an analysis of the photometry and optical spectra of SN 1987A is given in § V. This analysis is also intended to serve as benchmark to which the more advanced and detailed analyses of SN 1987A can be compared. The book Stellar Atmospheres (hereafter referred to as SA) by Mihalas (1978) is used throughout this article as a reference for the concepts and equations of radiative transfer and for well-known results of radiative transfer analysis.


The Sobolev method is an approximate technique that is used to calculate line radiative transfer in atmospheres in which there are macroscopic velocity fields with large velocity gradients. It originated with Sobolev (1947) and has been extended by others (e.g., Castor 1970; Rybicki 1970; Lucy 1971; Rybicki and 1 2 Hummer 1978; Olson 1982; Hummer and Rybicki 1985; Puls and Hummer 1988; Jeffery 1988, 1989, 1990;

Mazzali 1988, 1989). It has been used to calculate synthetic line spectra for stars with strong winds (e.g., Castor and Nussbaumer 1972) and has also been used in radiation-driven wind calculations (e.g., Pauldrach, Puls, and Kudritzki 1986). For supernova spectrum analyses, the Sobolev method has been used by Branch and collaborators (Branch 1980; Branch et al. 1981, 1982, 1983, 1985). Radiative transfer techniques that are more accurate for moving atmospheres than the Sobolev method are the comoving-frame formalism (SA, p. 490) and Monte-Carlo simulations (see, e.g., Natta and Beckwith 1986). In this section, the basic Sobolev method expressions are derived and the procedure for Sobolev method calculations is outlined. The derivations and notation of this section largely follow those of Rybicki and Hummer (1978).

Consider an atmosphere with a macroscopic velocity field with large velocity gradients. In addition to the macroscopic velocity field, there will be the random thermal motions of atoms and there may be a microturbulent velocity field: i.e., a random motion of fluid elements. It is assumed that the characteristic velocities of the macroscopic velocity field are much larger than the thermal velocities or any microturbulent velocities. For the moment, the only source of opacity that will be considered is the opacity of a single atomic line transition with a line center wavelength of λ0. The line opacity is described by a line absorption profile φ(λ) that is normalized to 1, that is a maximum at the line center wavelength, and that decays rapidly as wavelength decreases or increases from the line center wavelength. Now consider a specific intensity beam I(λ, n, s) where n points in the propagation direction and s is the beam path coordinate. The component of ˆ ˆ the macroscopic velocity in the n direction on the beam path is v(s) = n · v(r ). The Doppler shift that the ˆ ˆ photons experience in a reference frame in which except for random motions the matter is at rest (hereafter called the comoving frame) causes a wavelength shift of the line center wavelength and the line absorption profile in an observer’s reference frame. If it is now assumed that the velocity gradient is never zero, then the observer frame line center wavelength will equal the specific intensity beam wavelength only at isolated points. Provided that only the first order Doppler formula is required, these points, called resonance points, can be determined from the equation λ0 = λ [1 + v(s)/c]. (1) It is at the resonance points that the beam photons will interact most strongly with the line. However, since the line opacity is distributed over a range of wavelength according to the line absorption profile φ, the varying Doppler shift due to the varying macroscopic velocity will bring each part of the profile into resonance with the specific intensity beam at a different spatial point. Thus, there will be a spatial region where the specific intensity beam will interact significantly with the line. Such spatial regions are called resonance regions.

The confinement to resonance regions of the interaction of line transitions and specific intensity beams is a great simplification for a radiative transfer technique. However, even with this simplification, the general problem of radiative transfer in an atmosphere with macroscopic velocity fields is still difficult. The Sobolev method is a technique that exploits the confinement to resonance regions, but in its usual formulation obviates much of the difficulty of the general problem with the following assumptions. (1) The first order Doppler formula is sufficiently accurate. This means that the velocities in the atmosphere must be much less than the speed of light. In most astrophysical atmospheres of interest this condition should be met.

(2) The integrated line opacity K, directional velocity derivative (dv/ds) = i,j ni nj (∂vi /∂xj ), and line source function S can be approximated for a whole resonance region by their resonance point values. Since K and, in general, S depend on the occupations numbers (i.e., the populations of atomic levels and ionization stages), the condition on K and S implies that the occupation numbers be approximately constant across the resonance regions. (3) Complete redistribution in wavelength and angle in the comoving frame, hereafter simply referred to as complete redistribution, is a valid approximation for line scattering. This assumption is valid for many but not all atmosphere calculations (see SA, p. 411, and references therein).

Before deriving the Sobolev method, it is useful to give some further information about the line absorption profile φ and the integrated line opacity K. The absorption profile that applies in most astrophysical cases is the Voigt profile which has Lorentzian wings due to natural and collisional damping and a Gaussian core due to thermal and, perhaps, microturbulent Doppler line broadening (SA, p. 279). The √ characteristic width associated with the line profile, called the linewidth, is usually 2 times the standard deviation of the Gaussian core. The Gaussian linewidth of convolved thermal and microturbulent Gaussian 3

–  –  –

where A is the atomic mass and T is in Kelvins. If the integrated line opacity is very large, the Lorentzian wings of the line may have significant opacity. In such cases, the Gaussian core linewidth cannot be a completely adequate characteristic linewidth. The integrated line opacity is given by

–  –  –

In order to derive the basic Sobolev method results, consider again the system consisting of a specific intensity beam propagating along a beam path in the n direction through an atmosphere that has a ˆ macroscopic velocity field and large velocity gradients. Let s∗ = s(r ) be a resonance point for wavelength λ∗ ; from an alternative point of view, λ∗ is thought of as the resonance wavelength for point s∗. Using the first order Doppler formula and the assumption that (dv/ds) is constant in the resonance region that centers on s∗, the line absorption profile in the observer frame in the neighborhood of s∗ is φ {λ + λ0 [v∗ + (dv/ds) (s − s∗ )] /c}, (6) where v∗ = v (s∗ ). Note that in writing the argument of equation (6), λ0 has been used instead of λ as the coefficient of the second term. The consequence of this substitution is that one or more factors of (1 + v∗ /c) are neglected in the definitions of τ, β, and Jβ introduced below. Given the assumption that the velocities in the atmosphere are much less than the speed of light, the neglect of the (1 + v∗ /c) factor is only a small error.

Both the λ and s parameters occur linearly in the argument of equation (6), and thus both the line absorption profile and what can be called the spatial line profile will have the same shape. Moreover, with the assumption that the integrated line opacity is constant across the resonance region, the monochromatic line opacity profile in both wavelength and spatial coordinate spaces will have the same shape. This wavelength-spatial symmetry implies that there is a resonance region width

–  –  –

where ∆v is the velocity width of the convolved thermal and microturbulent Gaussian velocity distributions.

Now ∆s is not the actual size of the resonance region. If the optical depth in the spatial line profile Lorentzian wings can be considered negligible, then about 55 % and about 85 % of the total resonance region optical depth (see definition below) are contained in regions centering on the resonance point of sizes ∆s and 2∆s, respectively. It is shown below that if the Lorentzian wings are negligible, then it is reasonable to say that 4

–  –  –

With the Sobolev method assumptions, the argument of the exponential in equation (18) is just τ H± [x(λ, s − s∗ )]. Thus, the formal solution to equation (17) for a resonance region centered on s∗ is

–  –  –

Equation (20) is the formal Sobolev method general expression for radiative transfer through a resonance region centered on s∗. Given the Sobolev method assumptions, equation (20) will in general be most accurate for λ equal to the resonance wavelength λ∗ and will in general decrease in accuracy as λ departs from λ∗ since this causes the resonance point for λ to move away from s∗ where S and τ have been evaluated. However, again given the Sobolev method assumptions, equation (20) will in general have some reasonable accuracy for all wavelengths λ that have resonance points that are inside the resonance region centered on s∗.

In most calculations, the spatial width of the resonance region is considered to be negligible. Therefore, what is needed is the expression for the specific intensity of wavelength λ∗ at points well past the resonance point for λ∗ : i.e., at points where s is effectively positive infinity. From equation (20) with s set to positive infinity, one obtains

–  –  –

where the subscripts i and j, counting in the −ˆ direction, number the resonance regions the specific intensity n beam has traversed, where each si is a resonance point for λ∗, and where the source functions are subscripted since they will in general be due to different line transitions.

Equation (22) can be used to determine the emergent specific intensity of a model atmosphere. However, usually one wants to know the net emergent flux directed toward a distant observer. This is obtained by integrating the observer-directed emergent specific intensity over the projected surface of the atmosphere.

It is for this integration that the concept of CD (common-direction) velocity surfaces becomes useful. In an atmosphere with a macroscopic velocity field where the velocity gradient is never zero, the matter with a given velocity component in the observer’s direction will lie on geometrical surfaces. It is these surfaces that are called CD velocity surfaces. It is clear that atoms lying on a CD velocity surface will have a common observer frame line center wavelength for a given line transition. Thus, the CD velocity surface is a surface 6

–  –  –

where t∗ is the optical depth to the resonance point. Equation (37) is exactly the Sobolev method result for a weak line (see eq. [21]). Clearly, for the weak line case, the Sobolev method approximation of using the resonance point value of S will to be tend to be valid even if S and K vary linearly with s over distances of order 2∆s.

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