REGLAS RELATIVAS A LA ADMINISTRACIÓN DEL RIESGO CIRCULAR EXTERNA 049 DE 2006.

In order to compute synthetic light curves for binary systems we must have some knowledge of the radiative properties of the components. In PHOEBE these cal- culations are performed in general, the radiative properties of the objects are computed in an aspect (observer) independent fashion, and from this the flux in a given direction, at a given time, is computed.

In order to compute the flux, PHOEBE makes use of the model atmospheres developed by Kurucz (1970). These atmospheric models are commonly used, and provide a good estimate of the emergent intensity from a stellar atmosphere (see Fig. 2.9). In close binary systems the assumptions upon which these models rest may be broken, however for a detached system like ζ Aurigae the models should provide reliable results (Sivieroet al., 2004).

In order to obtain accurate light curves we must take account of the distribution of flux across the stellar disk. This distribution is not uniform, and is altered by an effect known as limb-darkening. Limb-darkening occurs as a result of the fact that we see to the same optical depth at all points on the disk of a star (τ = 2/3), but this does not correspond to the same physical depth, and hence

temperature/brightness. As such there is a fall off in intensity as we move from disk centre to limb.

Another important effect which must be accounted for is gravity darkening. Gravity darkening arises from the fact that the intensity distribution depends on the energy transfer mechanism in a stellar envelope. The work of von Zeipel (1924) showed that the flux distribution over the surface is proportional to the effective gravity, Fλ =− 16σT3 3¯κρ dT dΨg β (2.148) where σ is the Stefan-Boltzmann constant, T is the local temperature, ¯κ is the Rosseland mean opacity,ρis the density of the gas, Ψ is the gravitational potential, and β is the gravity darkening coefficient. von Zeipel (1924) demonstrated that for a radiative envelopeβ = 1, and later work by Lucy (1967) derivedβ = 0.32 for convective envelopes. This expression can be rewritten for local temperature as

T4_{(θ, φ) =T}4 pole g(θ, φ) gpole β (2.149) where θ and φ are polar coordinates on the stellar surface.

These radiative properties are phase dependent only as a result of the tidal deformation of the objects, in a circular orbit the face of one star would always appear the same to the other. In PHOEBE these radiative effects are accounted for by the following expression

Lnorm = Z 2π I(θ) I(0)dθ Z 2π Z 2π T4(θ, φ)dθdφ (2.150) where Lnorm is normalised luminosity. This normalised luminosity is then used to scale the Kurucz luminosities. These radiative properties are discussed further in the specific instances where they are used in the work.

Finally, in systems where the separation of the objects is.5 times greater than the radii of the components (i.e. ζ Aurigae) we must take account of the mutual irradiation of the components by one another. This irradiation serves to increase the temperature, and hence the luminosity of the objects, and is usually referred to (somewhat misleadingly) as reflection. This effect was first derived explicitly by Wilson (1990). In order to take account of this effect we must compute the

irradiation of each part of one object due to the second (integrating over the surfaces visible one from another). We define the total reflective excess for each object as R1 = 1 +A1 F2→1 F1 R 2 = 1 +A2 F1→2 F2 (2.151) where A is the albedo of the star, F is its flux, and Fi→j is the flux from i falling

onj,

Fi→j =Ri

Z

S

Fjcosφdσj (2.152)

where we are integrating over the visible surface of the star, S, Fj is the flux

emerging from a surface element dσj, and φ is the angle between the surface

element and i. Since both stars irradiate one another these equations are solved iteratively until the values for R have converged. A lengthy discussion of the reflection can be found in Chapter 5.

Now that we have described much of the underlying theory and mathematics which will be used in this thesis we will proceed with discussing the instrumentation we have used.

3

Instrumentation

In this chapter we describe the instruments used to make the observations upon which this thesis is based. We briefly discuss the theory underlying the instru- ments, their technical construction, and the reduction of their raw output to produce science data. We will begin by discussing discussing ground-based spec- troscopy, produced at the Dominion Astrophysical Observatory and other sites, which have been integral in our study of ζ Aurigae. We also describe the Hubble Space Telescope, focusing on the particular instrument we have used, and outlining the reduction pipelines. This work also relies on observations at radio wavelengths, made using CARMA and APEX, and we will describe these telescopes in detail, again providing a description of how the calibrated radio fluxes are produced. In this discussion we also describe the theory and methods underlying the interfer- ometric observations used in this thesis, including observations made at optical, infrared, and radio wavelengths.

In this work we have made use of a number of observational methods in order to study the structure of late-type chromospheres, observations at many wavelengths, encompassing spectrometers — both analogue and digital — radio bolometers, and interferometers. In this chapter we will describe these methods, beginning by discussing spectra, before moving on to radio observations, and finally providing a discussion of optical spectro-interferometers.

3.1

Spectrometers

The results of this work rely heavily on optical spectroscopic observations. We have made use of archival ground-based observations, using the Dominion Astrophysical Observatory (DAO) and other sites, as well as the Hubble Space Telescope (HST). We will discuss each in turn.