Reviews in Modern Astronomy Vol. 16

Hot Stars:
Old-Fashioned or Trendy?

A. W. A. Pauldrach
Institut für Astronomie und Astrophysik der Universität München
Scheinerstraße 1, 81679 München, Germany
UH10107@usm.uni-muenchen.de, http://www.usm.uni-muenchen.de/people/adi/adi.html

Abstract

Spectroscopic analyses with the intention of the interpretation of the UV-spectra of the brightest stars as individuals - supernovae - or as components of star-forming regions - massive O stars - provide a powerful tool with great astrophysical potential for the determination of extragalactic distances and of the chemical composition of star-forming galaxies even at high redshifts.

The perspectives of already initiated work with the new generation of tools for quantitative UV-spectroscopy of Hot Stars that have been developed during the last two decades are presented and the status of the continuing effort to construct corresponding models for Hot Star atmospheres is reviewed.

Because the physics of the atmospheres of Hot Stars are strongly affected by velocity expansion dominating the spectra at all wavelength ranges, hydrodynamic model atmospheres for O-type stars and explosion models for Supernovae of Type Ia are necessary as basis for the synthesis and analysis of the spectra. It is shown that stellar parameters, abundances and stellar wind properties can be determined by the methods of spectral diagnostics already developed. Additionally, it will be demonstrated that models and synthetic spectra of Type Ia Supernovae of required quality are already available that make it possible to tackle the question of whether Supernovae Ia are standard candles in a cosmological sense and the SN-luminosity distances thus indicate accelerated expansion of the universe.

In detail we discuss applications of the diagnostic techniques on two of the most luminous O supergiants in the Galaxy and a standard Supernova of Type Ia. Furthermore, it is demonstrated that the spectral energy distributions provided by state-of-the-art models of massive O stars lead to considerably better agreement with observations if used for the analysis of H II regions. Thus, an excellent way of determining extragalactic abundances and population histories is offered.

Moreover, the important role that Hot Stars play in a broad astrophysical context will be discussed. As they dominate the physical conditions of their local environments and the life cycle of gas and dust of their host galaxies, special emphasis will be given to the corresponding diagnostic perspectives. Beyond that, the relevance to cosmological issues of Hot Stars will be considered.

1  Introduction

As is well known, Hot Stars are not a single group of objects. Instead, the term covers sub-groups of objects in different parts of the HR diagram and at different evolutionary stages. The most important sub-groups are massive O/B stars, Central Stars of Planetary Nebulae, and Supernovae of Type Ia and II. All these sub-groups have in common that they are characterized by high radiation energy densities and expanding atmospheres. Thus, the state of the outermost parts of these objects is due to these properties characterized by non-equilibrium thermodynamics. In order to cover the best-known fundamental stages of Hot Star stellar evolution in sufficient depth this review will be restricted to the discussion of O stars and Supernovae of Type Ia; we will not discuss objects like Wolf-Rayet stars, Luminous Blue Variables, Be-stars, Supernovae of Type II, and others. Beyond that, Hot Stars play an important role in a broad astrophysical context. This implies that a complete review covering all aspects in theory and observation is not only impossible, but also beyond the scope of this review. Thus, this review will focus on a special part of the overall topic with the intent of concentrating on just one subject; the subject chosen is UV spectral diagnostics. It will be shown, however, that this subject has important implications on astrophysical topics which are presently regarded as being ''trendy''.

But first of all we have to clarify what is meant by UV spectral diagnostics. This is best illustrated by the really old-fashioned (1977, Morton & Underhill) UV spectrum of one of the brightest massive O stars, the O4 I(f) supergiant z Puppis. As is shown in Figure 1, expanding atmospheres have a pronounced effect on the emergent spectra of hot stars - especially in the UV-part. The signatures of outflow are clearly recognized by the blue-shifted absorption and red-shifted emission in the form of the well-known P Cygni profiles. It is quite obvious that these kind of spectra contain information not only about stellar and wind parameters, but also about abundances. Thus, in principle, all fundamental parameters of a hot star can be deduced from a comparison of observed and synthetic spectra.

The key for this purpose are realistic synthetic UV spectra for Hot Stars. Such powerful tools, which, however, are still in development and not yet in widespread use, offer the opportunity to determine the stellar parameters, the abundances of the light elements - He, C, N, O, Si - and of the heavy elements like Fe and Ni, quantitatively, as is indicated not only by IUE, but also more recent HST, ORFEUS, and FUSE observations of hot stars in the Galaxy and Local Group galaxies. All these observations show that the spectra in the UV spectral range are dominated by a dense forest of slightly wind-affected pseudo-photospheric metal absorption lines superimposed by broad P Cygni line profiles of strong, mostly resonance lines formed in different parts of the expanding atmosphere. Thus, the obvious objective is to investigate the importance of these lines with respect to the structure of the expanding atmospheres that are characterized by the strength and the velocity of the outflow, which in turn mainly determines the shape of the spectral lines. It is actually not surprising that it turned out within the last 30 years that this objective, which will also be the primary objective of this paper to review, represents not an easy task.

Before we discuss in detail the status quo of the diagnostic tool required (Section 6), we will first examine in the following parts of this review whether such a tool has been obsoleted by the general development in astrophysics or whether it is still relevant for current astronomical research.

For this purpose we first discuss the diagnostic perspectives of galaxies with pronounced current star formation. Due to the impact of massive stars on their environment the underlying physics for the spectral appearance of starburst galaxies are rooted in the atmospheric expansion of massive O stars which dominate the UV wavelength range in star-forming galaxies, and that therefore the UV-spectral features of massive O stars can be used as tracers of age and chemical composition of starburst galaxies even at high redshift (Section 2).

With respect to the present cosmological question of the reionization of the universe - which appeared to have happened at a redshift of about z ~ 6 - the ionization efficiency of a top-heavy Initial Mass Function for the first generations of stars is discussed (Section 3).

Starting from the impact of massive stars on their environment it is demonstrated that the spectral energy distributions provided by state-of-the-art models of massive O stars lead to considerable improvements if used for the analysis of H  ii regions. Thus, the corresponding methods for determining extragalactic abundances and population histories are promising (Section 4).

Regarding diagnostic issues, the role of Supernovae of Type Ia as distance indicators is discussed. The context of this discussion concerns the current surprising result that distant SNe Ia appear fainter than standard candles in an empty Friedmann model of the universe (Section 5).

Finally, we discuss applications of the diagnostic techniques on two of the most luminous O supergiants in the Galaxy; additionally, basic steps towards realistic synthetic spectra for Supernovae of Type Ia are presented (Section 7).

2  The impact of massive stars on their Environment -
UV Spectral Analysis of Starburst Galaxies

The impact of massive stars on their environment in the present phase of the universe is of major importance for the evolution of most galaxies. Although rare by number, massive stars dominate the life cycle of gas and dust in star forming regions and are responsible for the chemical enrichment of the ISM, which in turn has a significant impact on the chemical evolution of the host galaxy. This is mainly due to the short lifetimes of massive stars, which favours the recycling of heavy elements in an extremely efficient way. Furthermore, the large amount of momentum and energy input of these objects into the ISM controls the dynamical evolution of the ISM. This takes place in an extreme way, because massive stars mostly group in young clusters, producing void regions around themselves and wind- and supernova-blown superbubbles around the clusters. These superbubbles are ideal places for further star formation, as numerous Hubble Space Telescope images show. Investigation of these superbubbles will finally yield the required information to understand the various processes leading to continuous star formation regions (cf. Oey and Massey 1995). The creation of superbubbles is also responsible for the phenomenon of galactic energetic outflows observed in starbursts (Kunth et al. 1998) and starburst galaxies even at high redshift (Pettini et al. 1998). It is thus not surprising that spectroscopic studies of galaxies with pronounced current star formation reveal the specific spectral signatures of massive stars, demonstrating in this way that the underlying physics for the spectral appearance of starburst galaxies is not only rooted in the atmospheric expansion of massive O stars, but also dominated by these objects (cf. Figure 2 from Steidel et al. 1996, for star-forming galaxies at high redshift see also Pettini et al. 2000, and for UV line spectra of local star-forming galaxies see Conti et al. 1996; note that the similarity of the spectra at none/low and high redshifts suggests a stellar content which is similar or not too different).

What can be learned from an analysis of these spectra can nevertheless be investigated in a first step by using observed UV spectra of nearby O-type stars as input for the corresponding population synthesis calculations instead. In the frame of this method stars are simulated to form according to a specified star-formation history and initial mass function and follow predefined tracks in the HR-diagram. The integrated spectra are then built up from a library of observed UV spectra of hot stars in the Galaxy and the Magellanic Clouds. The output of this procedure is semi-empirical UV spectra between 1000 and 1800 Å at 0.1 to 0.7 Å resolution for populations of arbitrary age, star-formation histories and initial mass function. The computational technique of this method is described in the literature comprehensively and we refer the reader to one of the latest papers of a series (Leitherer et al. 2001).

The conclusion from these examples is that this kind of analysis is very promising, but relies on observed UV spectra, which are just available for a small number of metallicity values, namely those of the Galaxy and the Magellanic Clouds. In order to make progress in the direction outlined before, realistic synthetic UV Spectra of O-type stars are needed. This is in particular the case for high-redshift galaxies observable spectroscopically when the flux is amplified by gravitational lensing through foreground galaxy clusters, since in this cases the expected metallicities of starbursting galaxies in the early universe (cf. Pettini et al. 2000) are most probably different from local ones.

3  First generations of Stars - Ionization Efficiency of a Top-Heavy Initial Mass Function

Apart from the short evolutionary timescale of massive stars it is obviously the metallicity, and connected to that the steepness of the Initial Mass Function, which is responsible for the rarity of these objects in the present phase of the universe. It has to be the metallicity, because very recently strong evidence has been found that the primordial IMF has favored massive stars with masses > 10² M_sun (cf. Bromm et al. 1999). Thus, in the early universe, when just primordial elements left over from the Big Bang, nature obviously preferred to form massive stars. This prediction is based on the missing metallicity which leads to a characteristic scale for the density and temperature of the primordial gas, which in turn leads to a characteristic Jeans mass of M_J ~ 10³ M_sun (Larson 1998).

Finally, due to these physical conditions the Initial Mass Function becomes top-heavy and therefore deviates significantly from the standard Salpeter power-law (see, for instance, Bromm et al. 1999, 2001). Such an early population of very massive stars at very low metallicities (i.e., Population III stars) which have already been theoretically investigated by El Eid et al. (1983), recently turned out to be also relevant for cosmological issues. The most important of these issues concerns the cosmological question when and how the cosmic ``dark ages'' ended (Loeb 1998). We know they ended because the absence of a Gunn-Peterson trough (Gunn & Peterson 1965) in the spectra of high-redshift quasars implies that the universe was reionized again at a redshift of z > 5.8 (Fan et al. 2000). Carr et al. (1984) suspected since a long time that the first generations of stars have been relevant to control this process. Thus, Population III stars could contribute significantly to the ionization history of the intergalactic medium (IGM), but the contribution of the first generation of stars to the ionization history of the IGM depends crucially on their initial mass function. With regard to the first generations of stars the ionization efficiency of a top-heavy Initial Mass Function therefore will have to be investigated.

It is the enormous amount of UV and EUV radiation of these very massive stars which could easily change the status of the cold and dark universe at that time to become reionized again. This is indicated in Figure 5 which also shows that the total spectral luminosity depends solely on the total amount of mass, if the mass of the most massive stars exceeds 300 M_sun (cf. Bromm et al. 2001). In this case of a top-heavy Initial Mass Function the total spectral energy distribution deviates significantly from that obtained with the standard Salpeter power-law, as is shown in Figure 6, and the flux obtained can contribute the decisive part to the unexplained deficit of ionizing photons required for the reionization of the universe (cf. Bromm et al. 2001).

Figure 6: Predicted flux from a Population III star cluster at z=10. A flat universe with W_L = 0.7 is assumed. The cutoff below l_obs = 1216 Å is due to complete Gunn-Peterson absorption, and the observable flux is larger by an order of magnitude for the case of the top-heavy IMF when compared to the case of the standard Salpeter power-law. Figure from Bromm et al. 2001.

However, in order to be able to make quantitative predictions about the influence of this extremely metal poor population of very massive stars on their galactic and intergalactic environment one needs primarily observations which can be compared with the predicted flux spectrum.

Future observations with the Next Generation Space Telescope of distant stellar populations at high redshifts will principally give us the opportunity to deduce the primordial IMF from these comparisons. That this is generally possible has been shown very recently by Kudritzki (2002), who calculated state-of-the-art UV spectra for massive O stars in a metallicity range of 1-10^-4 Z_sun. From an inspection of his spectra he concluded that significant line features are still detectable even at very low metallicities; thus, there will be diagnostic information available to deduce physical properties from starbursting regions at high redshifts that will eventually be observed with the Next Generation Space Telescope in the infrared spectral region.

As a second requirement we need to determine the physical properties of Population III stars during their evolution. A key issue in this regard are realistic spectral energy distributions calculated for metallicities different from zero for the most massive objects, since the assumption of a metallicity of Z = 0 is certainly only correct for the very first generation of Population III stars.

4  Theoretical Ionizing Fluxes of O Stars

Although less spectacular, we will now investigate the impact of massive stars on their environment in a more direct manner. Apart from the chemical enrichment of the ISM, the large amount of momentum and energy input into the ambient interstellar medium of these objects is primarily of importance. Especially the radiative energy input shortward of the Lyman edge, which ionizes and heats the Gaseous Nebulae surrounding massive Hot Stars, offers the possibility to analyze the influence of the EUV radiation of the photoionizing stars on the ionization structure of these excited H  ii regions.

The primary objective of such investigations are studies of theoretical models of starburst regions, which for instance are used to determine the energy source in ultra-luminous infrared galaxies - ULIRGs - (cf. Lutz et al. 1996; Genzel et al. 1998). The interpretation of the corresponding extra-galactic observations obviously requires understanding the properties of the spectral energy distributions (SEDs) of massive stars and stellar clusters. Thus, the quality of the SEDs has to be probed in a first step by means of investigations of Galactic H  ii regions.

4.1  The Excitation of Galactic H II Regions

Giveon et al. (2002) recently presented a comparison of observed [Ne  iii] 15.6 mm / [Ne  ii] 12.8 mm and [Ar  iii] 8.99 mm / [Ar II] 6.99 mm excitation ratios, obtained for a sample of 112 Galactic H  ii regions and 37 nearby extragalactic H  ii regions in the LMC, SMC, and M33 observed with the ISO-SWS observatory, with the corresponding results of theoretical nebular models.

The authors have chosen infrared fine-structure emission lines for their investigation because these lines do not suffer much from dust extinction and the low energies of the associated levels make these lines quite insensitive to the nebular electron temperature. Moreover, the relative strengths of the fine-structure emission lines chosen are ideal for constraining the shape of the theoretical ionizing fluxes, since the [Ne  iii] 15.6 mm / [Ne  ii] 12.8 mm line ratio depends on photons emitted at ³ 3 Rydbergs, while [Ar  iii] 8.99 mm / [Ar  ii] 6.99 mm is sensitive to the region ³ 2 Rydbergs. Thus, these line ratios are extremely useful probes of the physical properties of H  ii regions and their associated ionizing sources especially with regard to the ionizing spectral energy distributions.

The diagnostic diagram clearly shows that the predicted nebular excitation increases with increasing effective temperature of the photoionizing star, and it also shows that the high excitation [Ne  iii] emission observed in H  ii regions is by far not reproduced by nebular calculations which make use of the ionizing fluxes of LTE models - the line ratios are under-predicted by factors larger than 10. This result is clearly an example of the well-known Ne  iii problem (cf. Baldwin et al. 1991; Rubin et al. 1991; Simpson et al. 1995).

The reason for this improvement are the spectral shapes of the NLTE fluxes shortward of the Ar  ii and Ne  ii ionization thresholds which are obviously somewhat more realistic in the NLTE case. This is illustrated in Figure 8 where the spectral energy distributions of an LTE and an NLTE model are compared for a dwarf with an effective temperature of 35 000 K. As is shown the SEDs harden in the NLTE case, meaning that the LTE model produces much less flux above the Ne⁺ 40.96 eV threshold than the NLTE model - this is the behavior that is essentially represented in the excitation diagram.

4.2  Spectral Energy Distributions of Time-Evolving
Stellar Clusters

The improvement obtained for the diagnostic diagram of the analysis of Galactic H  ii regions in Section 4.1 also has important implications for determining extragalactic abundances and population histories of starburst galaxies.

With regard to this, Thornley et al. (2000) carried out detailed starburst modelling of the [Ne  iii] 15.6 mm / [Ne  ii] 12.8 mm ratio of H  ii regions ionized by clusters of stars. As was shown above, the hottest stars in such models are responsible for producing large nebular [Ne  iii] 15.6 mm / [Ne  ii] 12.8 mm ratios. Hence, the low ratios actually observed led to the conclusion that the relative number of hot stars is small due to aging of the starburst systems. Thus, the solution of the Ne  iii problem has important consequences for the interpretation of these extragalactic fine-structure line observations, since the conclusion that due to the low [Ne  iii] 15.6 mm / [Ne  ii] 12.8 mm ratios obtained, the hottest stars as dominant contributors to the ionization of the starburst galaxies have to be removed, has obviously to be proven.

Figure 9: Calculated spectral energy distribution of a typical O star. The spectrum represents the result of the WM-basic model (Pauldrach et al. 2001), and the thick line represents the result of a corresponding COSTAR model (Schaerer & de Koter 1997).

In order to tackle this challenge, Sternberg et al. (2002) computed spectral energy distributions of time-evolving stellar clusters, on basis of a large grid of calculated NLTE spectral energy distributions of O and early B-type stars in the hot, luminous part (T_{ef f} > 25 000 K) of the HR diagram, from which models of stars following evolutionary tracks are suitably interpolated, and which are the major contributors to the Lyman and He  i fluxes. The SEDs used rely on recent improvements of modelling expanding NLTE atmospheres of Hot Stars (Pauldrach et al. 2001). As an example of the individual models used, Figure 9 shows the calculated spectral energy distribution of a typical O star compared to the corresponding result of an NLTE model of Schaerer & de Koter (1997).

Concerning the continuous mode, the star-formation rate is assumed to be constant with time - this regards time scales which are longer that the lifetimes of massive stars - and it is further assumed that the cluster forms stars at a rate of 1 M_sun per year. In the impulsive mode, all stars are formed ''instantaneously'', i.e., on time scales which are much shorter than the lifetimes of massive stars; the total mass of stars formed is 10⁵ M_sun, and the cluster is assumed to evolve passively thereafter. Furthermore, the computations are based on a Salpeter initial mass function, where for each mode two cases, one with an upper mass limit of M_up = 120 M_sun and the other one with an upper mass limit of M_up = 30 M_sun, are assumed.

Two striking effects are seen in Figure 10:

1. In the continuous mode, the cluster ionizing spectrum becomes softer - i.e., steeper - between 1 and 10 Myr. This is due to the increase of the relative number of late versus early-type OB stars during the evolution, since the less massive late-type stars have longer lifetimes. This effect is particularly noticeable for the cluster model with M_up = 120 M_sun.

2. In the impulsive mode, the magnitude of the Lyman break increases drastically, because the massive stars disappear with time.

An answer to this question requires an ultimate test !

This ultimate test is only provided by a comparison of the observed and synthetic UV spectra of the individual massive stars, due to two reasons:

1. This test involves hundreds of spectral signatures of various ionization stages with different ionization thresholds which cover a large frequency range.

2. Almost all of the ionization thresholds lie within the spectral range shortward of the Lyman ionization threshold (cf. Figure 12); thus, the ionization balances of all elements depend sensitively on the ionizing radiation throughout the entire wind.

The ionization balance can be traced reliably through the strength and structure of the wind lines formed throughout the atmosphere. Hence, it is a natural and the only reliable step to test the quality of the ionizing fluxes by virtue of their direct product: the UV spectra of O stars.

But before we turn to this test we will continue our discussion with another astrophysically important stellar object.

5  Supernovae of Type Ia as Distance Indicators

In order to complete the discussion of Hot Stars in the context of modern astronomy, we will concentrate now, as a last point, on the subject of Supernovae of Type Ia. With respect to diagnostic issues we will investigate the role of Supernovae of Type Ia as distance indicators. The context of this discussion regards, as a starting point, the current surprising result that distant SNe Ia at intermediate redshift appear fainter than standard candles in an empty Friedmann model of the universe.

Type Ia supernovae, which are the result of the thermonuclear explosion of a compact low mass star, are currently the best known distance indicators which due to their large luminosities (L_max ~ 10⁴³ erg/s) reach far beyond the local supercluster (cf. Saha et al. 2001 and references therein). It is thus not surprising that Type Ia supernovae have become the most important cosmological distance indicator over the last years, and this is not only due to their extreme brightness - it is primarily due to their maximum luminosity which can be normalized by their light curve shape, so that these objects can be regarded as standard candles. That this actually is the case has been shown by observations of local SNe Ia which define the linear expansion of the local universe extremely well (Riess et al. 1999), and this is convincing proof of the accuracy of the measured distances. On the other hand, observations of distant SNe Ia (up to redshifts of about 1) have yielded strong evidence that the expansion rate has been accelerated 6 Gyr ago (cf. Riess et al. 1998 and Perlmutter et al. 1999).

This surprising result becomes evident if we look at the Hubble diagram in the form of distance modulus versus redshift (cf. Figure 13). As is shown in the diagrams of the lower panels, which are normalized to a cosmological model for an empty universe, most SNe Ia at intermediate redshift are positioned at positive values of the normalized distance modulus; thus, the distant supernovae appear fainter than what would be expected in a empty universe. This means that deceleration from gravitational action of the matter content does not take place. Moreover, the SNe Ia appear even more distant indicating an accelerated expansion over the last 6 Gyr.

The implications of this are immediately recognized in the cosmological diagram on the right-hand side of Figure 13, where W_L versus W_Mass is plotted. The SN Ia results obviously exclude any world model with matter without a contribution by a cosmological constant; hence, the need for an energy contribution by the vacuum is shown. Thus, the SN luminosity distances indicate accelerated expansion of the universe!

However, this is not the only interpretation of the result obtained. There are other astrophysical explanations, such as obscuration by intergalactic dust or evolution of supernovae of Type Ia (cf. Leibundgut 2001). The latter point regards primarily the absolute peak luminosity of SNe Ia which might have changed between the local epoch and a redshift of 1.0, especially in case of a decrease - the fainter supernovae would not be a signature of larger distances, but rather of an evolution of these objects. Although present SN Ia models do not favor such an evolution, it cannot be excluded, because neither the explosions nor the radiation emerging from the atmosphere are, up to know, understood in detail. Consequently, serious caveats for the cosmological interpretation of distant supernovae exist.

Thus, we are faced with the question, are SNe Ia standard candles independently of age, or is there some evolution of the SN luminosity with age? Among other things spectroscopy is certainly a powerful tool to obtain an answer to this question by searching for spectral differences between local and distant SNe Ia.

Unfortunately, there is at present not a clear picture about the exact physical processes which take place in a SN Ia explosion and how the radiation released from it should be treated in a proper manner (for a recent review see Hillebrandt and Niemeyer 2000). In order to get an impression about the basic physics involved that affect the atmospheric structure of a SN Ia, a quick glance at an observed spectrum will be helpful to point out the necessary further steps for their analysis.

To answer the question of whether SN Ia are standard candles in a cosmological sense, realistic models and synthetic spectra of Type Ia Supernovae are obviously required for the analysis of the observed spectra. As just shown, however, these models will necessarily need to be based on a similar serious approach for expanding atmospheres, characterized by non-equilibrium thermodynamics, as is the case for Hot Stars in general.

It will be shown in Section 7.1 that such spectra are in principle already available.

6  Concept for Consistent Models of Hot Star
Atmospheres

In order to determine stellar abundances, parameters, and physical properties (and from these, obtain realistic spectral energy distributions) of Hot Stars via quantitative UV spectroscopy, a principal difficulty needs to be overcome: the diagnostic tools and techniques must be provided. This requires the construction of detailed atmospheric models.

As has been demonstrated in the previous sections, such a tool has not been obsoleted by general development in astrophysics; on the contrary, it is becoming of ever greater relevance for current astronomical research as spectral analysis of hot luminous stars is of growing astrophysical interest. Thus, continuing effort is expended to develop a standard code for providing the necessary diagnostic tool. In the following part of this section we describe the status of our continuing work to construct realistic models for expanding atmospheres.

Before we focus on the theory in its present stage, we should not forget previous fundamental work which turned out to be essential in elaborating the theory. In this context I want to emphasize publications which regard key aspects of theoretical activity. The starting point of the development of the radiation-driven wind theory is rooted in a paper by Milne (1926) more than 80 years ago. Milne was the first to realize that radiation could be coupled to ions and that this process subsequently may eject the ions from the stellar surface. The next fundamental step goes back to Sobolev (1957), who developed the basic ideas of radiative transfer in expanding atmospheres.

Radiation pressure as a driving mechanism for stellar outflow was rediscovered by Lucy and Solomon (1970) who developed the basis of the theory and the first attempt at its solution. Beyond that, the pioneering step in the formulation of the theory in a quasi self-consistent manner was performed by Castor, Abbott, and Klein (1975). Although these approaches where only qualitative due to many simplifications, the theory was only developed further owing to the promising results obtained by these authors. Concerning key aspects of the solution of the radiative transfer the work of Rybicki (1971) and Hummer and Rybicki (1985) has to be emphasized. With regard to the scenario for considering the soft X-ray emission of O stars (detected by Seward et al. 1979 and Harnden et al. 1979), Cassinelli and Olson (1979) have investigated the possible influence of X-rays on the ionization structure. And Lucy and White (1980) and Owocki, Castor, and Rybicki (1988) have have distinguished themselves with basic theoretical investigations of time-dependent radiation hydrodynamics which describe the creation and development of shocks (for more details of the role of X-rays in the atmospheres of Hot Stars see the reviews of Pauldrach et al. 1994b and Kudritzki and Puls 2000).

6.1  The general method

The basis of our approach in constructing detailed atmospheric models for Hot Stars is the concept of homogeneous, stationary, and spherically symmetric radiation driven winds, where the expansion of the atmosphere is due to scattering and absorption of Doppler-shifted metal lines. Although these approximations seem to be quite restrictive, it has already been shown that the time-averaged mean of the observed UV spectral features can be described correctly by such a method (Pauldrach et al. 1994, 1994b).

However, developing such a method is not straightforward, since modelling hot star atmospheres involves replicating a tightly interwoven mesh of physical processes: the equations of radiation hydrodynamics including the energy equation, the statistical equilibrium for all important ions including the atomic physics, and the radiative transfer equation at all transition frequencies have to be solved simultaneously. Figure 15 gives an overview of the physics to be treated.

The principal features are:

The stellar parameters T_{ef f} (effective temperature), logg (logarithm of photospheric gravitational acceleration), R_* (photospheric radius defined at a Rosseland optical depth of 2/3) and Z (abundances) have to be pre-specified.

The hydrodynamic equations are solved (r is the radial coordinate, r the mass density, v the velocity, p the gas pressure and · M the mass loss rate). The crucial term is the radiative acceleration g_rad, which has minor contributions from continuous absorption, and major contributions from scattering and line absorption. For each line the oscillator strengths f_lu, the statistical weights g_l, g_u, and the occupation numbers n_l, n_u of the lower and upper level enter together with the frequency and angle integral over the specific intensity I_n and the line broadening function j_n accounting for the Doppler effect.

The occupation numbers are determined by the rate equations containing collisional (C_ij) and radiative (R_ij) transition rates. Low-temperature dielectronic recombination is included and Auger ionization due to K-shell absorption (essential for C, N, O, Ne, Mg, Si, and S) of soft X-ray radiation arising from shock-heated matter is taken into account using detailed atomic models for all important ions. Note that the hydrodynamical equations are coupled directly with the rate equations. The velocity field enters into the radiative rates while the density is important for the collisional rates and the equation of particle conservation. On the other hand, the occupation numbers are crucial for the hydrodynamics since the radiative line acceleration dominates the equation of motion.

The spherical transfer equation which yields the radiation field at every depth point, including the thermalized layers where the diffusion approximation is applied, is solved for the total opacities (k_n) and source functions (S_n) of all important ions. Hence, the influence of the spectral lines - the strong EUV line blocking - which affects the ionizing flux that determines the ionization and excitation of levels, is taken into account. Note that the radiation field is coupled with the hydrodynamics (g_rad) and the rate equations (R_ij).

Moreover, the shock source functions (S_n^S) produced by radiative cooling zones which originate from a simulation of shock heated matter, together with K-shell absorption (k_n^K), are also included in the radiative transfer. The shock source function is incorporated on the basis of an approximate calculation of the volume emission coefficient (L_n) of the X-ray plasma in dependence of the velocity-dependent post-shock temperatures and the filling factor (f).

The temperature structure is determined by the microscopic energy equation which, in principle, states that the luminosity must be conserved. Line blanketing effects which reflect the influence of line blocking on the temperature structure are taken into account.

The iterative solution of the total system of equations yields the hydrodynamic structure of the wind (i.e., the mass-loss rate · M and the velocity structure v(r)) together with synthetic spectra and ionizing fluxes.

Essential steps which form the basis of the theoretical framework developed are described in Pauldrach et al. 1986, Pauldrach 1987, Pauldrach and Herrero 1988, Puls and Pauldrach 1990, Pauldrach et al. 1994, Feldmeier et al. 1997, and Pauldrach et al. 1998.

The most complicating effect in the system is the overlap of the spectral lines. This effect is induced by the velocity field of the expanding atmosphere which shifts at different radii up to 1000 spectral lines of different ions into the line of sight at each observer's frequency. Since the behavior of most of the UV spectral lines additionally depends critically on a detailed and consistent description of the corresponding effects of line blocking and line blanketing (cf. Pauldrach 1987), special attention has to be given to the correct treatment of the Doppler-shifted line radiation transport of all metal lines in the entire sub- and supersonically expanding atmosphere, the corresponding coupling with the radiative rates in the rate equations, and the energy conservation. Another important point to emphasize concerns the atomic data, since it is quite obvious that the quality of the calculated spectra, the multilevel NLTE treatment of the metal ions (from C to Zn), and the adequate representation of line blocking and the radiative line acceleration depends crucially on the quality of the atomic models. Thus, the data have to be improved continuously whenever significant progress of atomic data modelling is made. Together with a revised inclusion of EUV and X-ray radiation produced by cooling zones which originate from the simulation of shock heated matter, these improvements have been implemented and described in a recent paper (Pauldrach et al. 2001). Very recently we have focused in addition on a remaining aspect regarding hydrodynamical calculations which are consistent with the radiation field obtained from the line-overlap computations and incorporated this improvement into our procedure (see Pauldrach and Hoffmann 2003).

This solution method is in its present stage already regarded as a standard procedure towards a realistic description of stationary wind models. Thus, together with an easy-to-use interface and an installation wizard, the program package WM-basic has been made available to the community; download is possible from the URL given on the first page.

In the next section it now has to be investigated thoroughly whether these kind of models already lead to realistic results.

7  Synthetic Spectra and Models of Hot Star
Atmospheres

The first five sections of this review inevitably lead to the realization that line features of expanding atmospheres are one of the most important astronomical manifestations, since they can easily be identified even in spectra of medium resolution in individual objects like supernovae or in integrated spectra of starburst regions even at significant redshift. As a consequence, they can be used to provide important information about the chemical composition, the extragalactic distance scale, and several other properties of miscellaneous stellar populations. All that is required are hydrodynamic NLTE model atmospheres that incorporate the effects of spherical extension and expansion as realistic and consistent as possible. After almost two decades of work these model atmospheres now seem to be available, and in the following we will investigate thoroughly whether basic steps towards realistic synthetic spectra for Supernovae of Type Ia can already be presented, and whether UV diagnostic techniques applied to two of the most luminous O supergiants in the Galaxy lead to synthetic spectra which can be regarded as a measure of excellent quality.

7.1  Synthetic and Observed Spectra of Supernovae of Type Ia

Prominent features in the spectra of Type Ia supernovae in early phases are characteristic absorption lines resulting from low ionized intermediate-mass elements (such as Si  ii, O  i, Ca  ii, Mg  ii, cf. Figure 17). These absorption lines show characteristic line shapes due to Doppler-broadening resulting from the large velocity gradients and the formation of these lines results in a pseudo-continuum that is set up by the enormous number of these lines. Supernovae appear in this `photospheric' epoch for about one month after explosion and the spectra at this epoch contain useful information on the energetics of the explosion (luminosity, velocity- and density-structure) and on the nucleosynthesis in the intermediate and outer part of the progenitor star. The primary objective, in order to analyze the spectra, therefore is to construct consistent models which link the results of the nucleosynthesis and hydrodynamics obtained from state-of-the-art explosion models (cf. Reinecke et al. 1999) with the calculations of light curves and synthetic spectra of SN Ia. In a fist step, however, this is done by using the standard hydrodynamical model W7 by Nomoto et al. 1984, and Figure 16 exemplifies the status quo of the synthetic Spectra of Type Ia Supernovae at early phases obtained from this first step, compared to observations.

Figure 16: Synthetic NLTE spectra of a Supernova of Type Ia. Top : The dotted lines show the spectrum from a model without line blocking (left panel) and from one where line blocking is treated consistently (right panel), both compared to the observed spectrum of SN 1992A (solid line). (Figure from Pauldrach et al. 1996.) Left : The two curves in the lower part show a synthetic spectrum for a W7 model at 23 days compared to the observed spectrum of SN 1992A. (Figure from Nugent et al. 1997.) The observed spectrum is from Kirshner et al. 1993 (5 days after maximum light).

The first comparison shown (top left panel of Figure 16) is obviously not at all the outcome of a realistic atmospheric model. The synthetic spectrum shown has been obtained for a model without line blocking, i.e., only continuum opacity has been considered in the NLTE calculation. Thus, the drastic effects of line blocking on the ionization and excitation and the emergent flux can be verified by comparing the calculated flux with the observed flux also shown - the huge difference observed in the UV is due to the missing contribution of the lines to the opacity. It is an essential by-product of this result that the observed line features in the optical are also not reproduced: the two strongest features in the synthetic spectrum are due to Fe  iii lines, and this indicates that the ionization is too high because of the excessive UV flux. This behavior nicely illustrates that the ionization balance depends almost entirely on the ionizing UV flux and that this influence can be traced by the spectral lines in the observable part of the spectrum, as has been stated in Section 4.2.

That the difference in the UV observed for this model is indeed due to line blocking becomes obvious if this important effect is properly taken into account, as is the case for model shown in the upper right panel of Figure 16. The agreement of the resulting spectrum of this model with the observations is quite good, demonstrating that line blocking alone can diminish the discrepancy between the synthetic and the observed spectra. Both the flux level and most line features are now reasonably well reproduced throughout the spectrum (cf. Pauldrach et al. 1996). Additionally, Figure 16 shows in the lower left panel a spectral fit to the same supernova observation by Nugent et al. (1997) which is based on a completely independent approach to the theory. As is shown, the resulting spectrum is also quite reasonable.

Figure 17: Comparison of the most recent synthetic NLTE spectrum at early phases and the observed spectrum of SN 1992A, illustrating that the method used is already on a quantitative level. The rest wavelengths of various metal lines are indicated by vertical bars. Figure from Sauer and Pauldrach 2002.

Finally, Figure 17 shows a recently calculated synthetic spectrum that is based on the important improvements to the theory described in the previous section, compared to the observed ``standard'' supernova spectrum (SN 1992A). As is shown, the synthetic spectrum reproduces the observed spectrum quite well, and the overall impression of this comparison is that the method used is already on a quantitative level, indicating that the basic physics are treated properly (cf. Sauer and Pauldrach 2002).

Although there is still need for improvements regarding, for instance, a proper treatment of the g energy deposition in the outermost layers and the use of current hydrodynamic explosion models, the basis for the primary objective of the subject, to search for spectral differences between local and distant SNe Ia, is formed. Concerning the facilities of spectral analysis we are thus close to the point where we can tackle the question of whether SNe Ia are standard candles in a cosmological sense.

7.2  Detailed Analyses of Massive O Stars

The objectives of a detailed comparison of synthetic and observed UV spectra of massive O stars are manifold. Primarily it has to be verified that the higher level of consistent description of the theoretical concept of the effects of line blocking and blanketing and the involved modifications to the models leads to changes in the line spectra with much better agreement to the observed spectra than the previous, less elaborated and less consistent models. Secondly it has to be shown that the stellar parameters, the wind parameters, and the abundances can be determined diagnostically via a comparison of observed and calculated high-resolution spectra covering the observable UV region. And finally, the quality of the spectral energy distributions has to be verified. As was illustrated in Section 4.2 the latter point is, however, a direct by-product of the other objectives in case the complete observed high-resolution UV spectra are accurately reproduced by the synthetic ones.

In the following, the potential of the improved method will be demonstrated by an application to the O4 I(f) star z Puppis and the O9.5 Ia star a Cam.

7.2.1  UV Analysis of the Hot O Supergiant z Puppis

At this point we return to the beginning, as we now start to examine carefully the old-fashioned UV spectrum of z Puppis. But before we do that, we will first briefly review the improvements of UV line fits which followed from stepwise improvements of the methods used.

The first serious attempt to analyze the UV spectrum of this standard object goes back to Lamers and Morton in 1976. From a present point of view their model can not be regarded as a sophisticated one, because they assumed the complete model structure - the dynamical structure, the occupation numbers, and the temperature structure - and the line radiative transfer was treated in an approximative way. What we learned nevertheless from this kind of spectrum synthesis was that a solution to the problem is feasible, and that at least the dynamical parameters - the mass-loss rate (· M) and the terminal velocity (v_¥) - can in principle be determined from UV lines quite accurately. The UV line fits of Hamann in 1980 have been based on the same assumptions apart from a very detailed treatment of the the line radiative transfer, and as a result of this the calculated resonance lines of some ions were already quite well in agreement with the observed ones. But the obvious conclusion from these comparisons was that a more consistent treatment of expanding atmospheres is required for a detailed analysis.

The situation improved considerably with the first attempts at a consistent solution, meaning that the equations of hydrodynamics, non-equilibrium thermodynamics, and radiative transfer have been solved in a consistent way. Pauldrach (1987) introduced for the first time a full NLTE treatment of the metal lines driving the wind into his consistent method, and Puls (1987) investigated the important effect of multiple photon momentum transfer through line overlaps caused by the velocity-induced Doppler-shifts for applications in stellar wind dynamics. The outcome of this procedure have already been dynamical parameters, constraints on the stellar parameters, and as has been verified by a comparison of a sample of calculated spectral lines with the observed spectrum, an ionization equilibrium and occupation numbers which were close to a correct description. But, among other approximations the models still suffered from the neglect of radiation emitted from shock cooling zones and a very approximate treatment of line blocking.

After this step it turned out to be a long road to reproduce most of the important observed individual line features in the UV. This required a lot of effort in atomic physics, in improved NLTE multilevel radiative transfer, and in spectrum synthesis techniques. An important step towards this objective was the paper by Pauldrach et al. (1994), where for the first time stellar parameters and abundances have been determined from individual UV line features. Figure 18 shows the corresponding synthetic spectrum that was already in overall agreement with the observations. Nevertheless, the treatment was still affected by a number of severe approximations regarding especially the neglect of the line blanketing effect, an approximate treatment of the line radiative transfer, and a simple treatment of emitted radiation from shock cooling zones. Thus, the conclusion remained that a more consistent treatment of expanding atmospheres was required for a detailed analysis.

Consequently, we regard these kind of quantitative spectral UV analyses as the ultimate test for the accuracy and the quality of theoretical ionizing fluxes (cf. Figure 20), which thus can be used as spectral energy distributions for the analysis of H  ii regions.

7.2.2  UV Analysis of the Cool O Supergiant a Cam

That the method also works for cool O supergiants like a Cam is shown in Figure 21. We recognize again that the synthetic spectrum is quite well in agreement with the IUE and the Copernicus observed spectra (for a discussion see Pauldrach et al. 2001). In the uppermost panel of the spectrum simulations of interstellar lines have additionally been merged with the synthetic spectrum in order to disentangle wind lines from interstellar lines (for a discussion of this point see Hoffmann 2002). Moreover, our new realistic models applied to this object along with UV spectrum synthesis techniques allowed also in this case to determine stellar parameters, wind parameters, and abundances from the UV spectra alone. (The operational procedure of the method begins with realistic estimates of R, T_{ef f}, M, and a set of abundances, with which the model atmosphere is solved and the velocity field, the mass loss rate · M, and the synthetic spectrum is calculated. The parameters are adjusted and the process is repeated until a good fit to all features in the observed UV spectrum is obtained. For a compact description of this procedure see Pauldrach et al. 2002). It turned out that the effective temperature can be determined to within a range of ±1000 K and the abundances to at least within a factor of 2.

7.3  Wind properties of massive O stars

As a last point we want to illustrate the significance of the dynamical parameters of radiation-driven winds. The intrinsic significance is quite obvious, because it is the consistent hydrodynamics which provides the link between the stellar parameters (T_{ef f}, M, R) and the wind parameters (v_¥, · M), and the appearance of the UV spectrum is thus determined by the interplay of the NLTE model and the hydrodynamics (as was discussed in Section 6.1, the hydrodynamics is controlled by the line force which is primarily determined by the occupation numbers and the radiative transfer of the NLTE model, but the hydrodynamics affects in turn the NLTE model via the density and velocity structure).

In a first step, the tool for illustrating the significance of the dynamical parameters is offered by the so called wind-momentum-luminosity relation.

Moreover, it is important to note that as a by-product of this relation it can be used as an independent tool for measuring extragalactic distances up to Virgo and Fornax, since it is independent of the observationally unknown masses (for a recent and comprehensive review on this subject see Kudritzki and Puls 2000).

For this investigation the same stellar parameters as have been used to obtain the observed wind parameters must be used as input for our models. On basis of this requirement Figure 24 shows in the upper panel that the observed and predicted values of the terminal velocities (v_¥) are in agreement within 10%. Since v_¥ is proportional to the escape velocity (v_esc)

v¥ µ vesc = æ ç è 2GM R ( 1 - G) ö ÷ ø 1/2

which strongly depends on the mass of the objects, the mass is thus determined very accurately by the predicted values (G is the gravitational constant and G is the ratio of radiative Thomson to gravitational acceleration). Concerning the mass-loss rates we found agreement within a factor of two as is shown in Figure 24 lower panel. Due to the strong relation between the mass-loss rates and the luminosities (· M µ L) the luminosities thus can be determined precisely.

Computing the wind dynamics consistently with the NLTE model therefore permits not only the determination of the wind parameters from given stellar parameters, but conversely makes it possible to obtain the stellar parameters just from the observed UV spectrum alone. As stated above, this means that in principle the stellar parameters can immediately be read off simply by comparing an observed UV spectrum to a proper synthetic spectrum. This, however, is not always really simple, since in most cases a comprehensive grid of models and some experience will be required to do so. Although this idea is not new (cf. Pauldrach et al. 1988 and Kudritzki et al. 1992), only this new generation of models has reached the degree of sophistication that makes such a procedure practicable instead of purely academic. The corresponding conclusion is that realistic models are characterized by at least a quantitative spectral UV analysis calculated along with a consistent dynamics.

8  Conclusions and Outlook

The need for detailed atmospheric models of Hot Stars has been motivated in depth and a diagnostic tool with great astrophysical potential has been presented.

It has been shown that the models of the new generation are realistic. They are realistic with regard to a quantitative spectral UV analysis calculated, for the case of O stars, along with consistent dynamics, which, in principle, allows to read off the stellar parameters simply by comparing an observed UV spectrum to a suitable synthetic spectrum. The new generation of models has reached a degree of sophistication that makes such a procedure practicable.

The astronomical perspectives are enormous. Of course, there is still a large amount of hard and careful astronomical work that needs to be done. The diagnostic tool for expanding model atmospheres is in our hands, but this is (as usual) just the starting point; further elaboration, refinements, and modifications are required before the results are quantitatively completely reliable.

However, this is not the handicap for the future. The present handicap are future observations. What is most urgently needed is the Next Generation Space Telescope.

With respect to the topics discussed in this review it is needed for a variety of reasons:

To tackle the question whether Supernovae Ia are standard candles in a cosmological sense realistic models and synthetic spectra of Type Ia Supernovae require spectral observations of objects at mid-redshift and sufficient resolution. The context of this question regards the current surprising result that distant SNe Ia at intermediate redshift appear fainter than standard candles in an empty Friedmann model. Consequently, the SN-luminosity distances indicate accelerated expansion of the universe.

In order to be able to make quantitative predictions about the influence of very massive, extremely metal-poor Population III stars on their galactic and intergalactic environment one primarily needs observations which can be compared with the predicted flux spectrum already available for zero metallicity and principally producible for metallicities different from zero. Observations with the Next Generation Space Telescope (NGST) of distant stellar populations at high redshifts will principally give us the opportunity to deduce the primordial IMF from these comparisons, and thus allow to investigate quantitatively the ionization efficiency of a Top-heavy IMF via realistic spectral energy distributions of these very massive stars.

NGST observations are further important for determining extragalactic abundances and population histories of starburst galaxies from an analysis of H  ii regions. This task also requires energy distributions of time-evolving stellar clusters, which are calculated on basis of a grid of spectral energy distributions of O and early B-type stars. A crucial point with respect to this is whether the spectral energy distributions of massive stars are already realistic enough to be used for diagnostic issues of H  ii regions. It has been shown in this review that the ultimate test is provided by a comparison of observed and synthetic UV spectra of individual massive stars, since the ionization balance can be traced reliably through the strength and structure of the wind lines formed throughout the atmosphere.

NGST is also important for exploiting the diagnostic perspectives of galaxies with pronounced current star formation directly. It has been demonstrated that massive stars dominate the UV wavelength range in star-forming galaxies, and that therefore the UV-spectral features of massive O stars can be used as tracers of age and chemical composition of starburst galaxies even at high redshift. This is in particular the case when the flux from these galaxies is amplified by gravitational lensing through foreground galaxy clusters; corresponding observations render the possibility of determining metallicities of starbursting galaxies in the early universe via realistic UV spectra (in the rest frame) of massive O stars.

As a final remark it is noted that the solution method of stationary models for expanding atmospheres is in its present stage already regarded as a standard procedure towards a realistic description. Thus, together with an easy-to-use interface and an installation wizard, the program package WM-basic has been made available to the community; download is possible from the URL given on the first page.

Acknowledgments

I wish to thank my colleague Tadziu Hoffmann for carefully reading the manuscript, and I am grateful to Amiel Sternberg and Claus Leitherer for providing me with figures from their publications. This research was supported by the Sonderforschungsbereich 375 of the Deutsche Forschungsgemeinschaft, and by the German-Israeli Foundation under grant I-551-186.07/97.

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