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Section 1
Setting the scene
1.1
Introduction
The broad purposes of this course are twofold:
• first, we want to be able to understand the principles of how to model stellar spectra
(such as that in Fig. 1.1) in order to infer photospheric properties;
• secondly, we aim to understand these stellar properties in the context of our theory of
stellar structure and evolution. In effect, this means assembling a set of tools that allow
us to interpret the distribution of stars in the Hertzsprung-Russell diagram (Fig. 1.2).
The content is therefore divided into two parts, delivered before and after Reading Week:
• we will first study how we can model stellar atmospheres and stellar spectra.
• then we will study the equations for the structure of the stellar’s interior and its evolution.
1.1.1
Stellar Atmospheres
In the broadest terms, a stellar atmosphere can be considered as the transition region from the
stellar interior to the interstellar medium. The behaviour of the atmosphere is controlled by the
density of the gases in it and the energy escaping through it. These quantities depend
proximately on the effective temperature, the surface gravity, and the atmospheric abundances
(and potentially other factors, such as mass loss, or magnetic fields), which in turn are
ultimately determined by the mass and age of the star (and potentially other factors, such as
chemical composition and rotation rate).
1
Figure 1.1: A short section of the solar flux spectrum around the strong Hα line (adapted from Wallace
et al. 2011).
Figure 1.2: The Hertzsprung-Russell diagram for stars within 100 pc of the Sun (distances from Hipparcos).
2
A model atmosphere is a numerical simulation of a real stellar atmosphere, typically presented
as the run of physical parameters (such as temperature) as a function of depth; here ‘depth’
generally refers to optical depth (§3.4), measured inwards.
Observationally, the most easily accessible part of the atmosphere is the photosphere, because,
by definition, the visible spectrum originates there. (We will generally use ‘atmosphere’ and
‘photosphere’ more or less interchangeably.) The photosphere is strongly affected by its
characteristic temperature. Typically, the temperature drops by a factor ∼ 2 from the bottom
to the top of the photosphere, so the local temperature is not of itself a useful parameter to
characterize the star. Instead, we can define (and measure) the effective temperature, Teff , as
the total power, per unit area, leaving the star:
Z ∞
4
(1.1)
Fν dν ≡ σTeff
0
where Fν is the flux emitted by the star, at frequency ν. The luminosity is then given by1
4 . It is easy to recognize that T
L = 4πR2 σTeff
eff corresponds to the temperature T of a black
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body having the same power output as the star.
In the Sun, the photosphere is around 500 km thick3 – that is, about half the distance from
Land’s End to John O’Groats, and less than about ∼0.1% of the solar radius
(R⊙ ≃ 695,800 km). Because of this, the atmosphere is well approximated as a plane parallel
medium. For other stars, the thickness scales with the effective temperature and surface gravity
(∼ M/R2 ), and with the opacity (Section 3.3), as a consequence of hydrostatic equilibrium
(Section 4.1). In principle, therefore, by studying the photosphere we can infer information on
the gravity (stellar mass and radius) and on the opacity (temperature and abundances).
1.1.2
Stellar structure and evolution
The information we can infer from the spectrum is, literally, superficial: it tells us something
about the surface conditions of the star. To connect these to global stellar parameters, such as
mass, radius, luminosity, we need to know something about how these fundamental parameters
relate to the photospheric conditions. This is what underpins the investigation of stellar
structure; but the structure (and hence the photospheric properties) changes with time, as the
nuclear fuel powering the star is consumed. Hence the studies of stellar structure and of stellar
evolution are inextricably linked.
1
This simple formulation is true only for spherical stars. This is usually an excellent approximation, but can
break down for very rapidly rotating stars, or stars in close binary systems, where Teff can vary from point to
point, and an explicit integration over area is required.
2
Appendix C discusses black-body radiation.
3
As determined from an analysis of limb darkening; cf. Section 8.3
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1.1.3
Linking to observations
Although the fundamental physical properties are parameters such as M , R, and L, these
translate into observational parameters such as absolute magnitude and colour index
(cf. Appendix E), or spectral type. The ultimate goal is to relate these observationally
accessible quantities to the physical parameters, and thence to make inferences about stellar
structure and evolution.
The spectrum is a particularly powerful tool in this endeavour, because, as already noted, the
continuum and line spectra depend principally on the gravity, temperature, and composition
(as well as rotation, magnetic field, mass loss, maculation, and other properties). The line
spectra are used to classify stars, according to their temperature and photospheric pressure.
With this criterion, we can assign a spectral type to a star. Originally, such classifications used
photographic records (spectrograms) that centred in the blue region of the spectrum. Spectra
toward the cool end of the sequence are called ‘late-type’ spectra, those toward the hot end are
‘early-type’ (for historical reasons; they bear no relationship to actual ages or evoluitionary
stages). There are ∼60 categories of stars, from O2 to M8. From the hottest to coolest, the
standard temperature sequence is: OBAFGKM. (Additionally, types C [and R, N] and S are
used for spectroscopically distinct objects at M-star temperatures; types L, T, and Y are used
for brown dwarfs.)
In the short introductory sections 1–4, we’ll briefly recap some basic material (which should be
familiar from PHAS 2112) in order to ensure we’re fully equipped for subsequent topics.
Section 5 then examines opacity sources in some detail, introducing the concept of the
Rosseland Mean Opacity; while Section 6 reviews LTE. Sectionch:RadTrans introduces some
basic concepts in radiative transfer.
We then look at ‘real world’ model atmospheres.
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