VALORAR CON TRATAMIENTOS

In this thesis we have made use of interferometric observations at both radio and optical wavelengths. In this section we will provide a background to the underlying theory of interferometry, and describe the CHARA optical interferometric array, ahead of a more detailed description of our radio interferometric observations in the next section.

3.2.1

Principles of an Interferometer

An astronomical interferometer is an array of two or more telescopes whose light is combined to provide angular resolution equivalent to a single, larger telescope. An N-element interferometer can be treated as N(N ₋1)/2 two-element interfer- ometers, and as such we will begin by examining this simple case.

Figure 3.4: Diagram of Young’s double slit experiment. On the left-hand side of each image we see the source, and on the right-hand side we see the resultant interference pattern. Note that as the source size and slit separation is varied the interference pattern is modulated. Image Credit: From Jackson (2008), adapted by E. O’Gorman.

The two element interferometer is precisely analogous to the Young’s double slit experiment. The double slit experiment can be seen in Fig. 3.4. This figure demonstrates a number of key ideas in interferometry. We begin with a point source the light from which passes through the two slits, and we obtain an interference pattern on our detector. This pattern emerges as a result of the different path lengths travelled by the light from each slit in reaching the detector. This causes a phase shift in the wave trains, and hence constructive or destructive interference. This interference is referred to as the fringe pattern in interferometry. We also note that as the source becomes more extended (which we treat as a series of point sources), that the fringe pattern becomes less clear, to the point that we do not see any interference. In the final case we see that reducing the separation of the slits allows us to see the fringe pattern from the more extended source. From

Figure 3.5: Diagram of a two element interferometer. Light enters the two tele- scopes, labelled 1 & 2, and is combined, resulting in the interference pattern shown.

Image Credit: National Radio Astronomy Observatory.

this we see the spatial resolution to which the fringe pattern is sensitive is inversely related to the separation of the slits,

θR = λ

s [radians] (3.2)

where θR is angular resolution, the minimum angular separation which can be

resolved, ands is the separation of the slits.

a baseline, b. In this configuration the two telescopes are analogous to the slits, and the light from each is combined. In the case that the source is not directly above the telescopes path length of the light to each will be different (by a factor of bcosθ), causing a fringe pattern (shown in the diagram), as in the double slit experiment. As such, the output of an interferometric instrument is not a direct measure of the magnitude of the incident brightness, but rather a series of fringes of light and dark, corresponding to the interference of the wave trains. This is the principal observable in interferometry, the ratio of maximum to minimum intensity of the fringes, known as the interferometric visibility:

V = Imax−Imin

Imax+Imin

(3.3) At this point it is convenient to establish a coordinate system for our interfer- ometer. Traditionally the ground coordinates, the positions of the telescopes, are denoted by the coordinates (u, v, w), whereuis oriented East, andv North, andw

is the relative height. These coordinates are in units of the observing wavelengths, such that

B = bcosθ

λ =

√

u2_+v2_+w2 _(3.4)

In the plane-of-sky the coordinates (l, m) are used to describe the brightness distribution, I(l, m), where these are angular coordinates (as we will see they are the Fourier counterparts of (u, w)). In this coordinate system the visibility can be written as V(u, v, w) = Z ∞ −∞ Z ∞ −∞ I(l, m)e−2πi(ul+vm+w(√1−l2_−m2₎₎ dldm √ 1₋l2_−m2 (3.5) In the case that the field-of-view, i.e. _|l_| and _|m_|, is small, and the telescopes are co-planar, i.e. w= 0, this equation becomes

V(u, v) = Z ∞ −∞ Z ∞ −∞ I(l, m)e−2πi(ul+vm)_{dldm (3.6)} This is known as the van Cittert–Zernike theorem (van Cittert, 1934; Zernike, 1938), and it states that the visibility measured by an interferometer is the Fourier transform of the brightness distribution on the sky. In our work we will mostly be interested in circularly symmetric objects, and it is useful to recast this equation

in polar coordinates, where θ is the angle coordinate on the sky, φ is the angle coordinate on the ground (both with respect to a common reference direction) and,

r=√l2_+m2 _(3.7)

q =√u2 _+v2 _(3.8)

After some manipulation this gives us (remembering that the source is symmetric inφ, and using µ= cosθ as in the previous chapter)

V(q) = 1 A Z 1 0 I(µ)J0(2πqR∗ p 1−µ2_{)µdµ (3.9)} where J0 is a zeroth order Bessel function. Equations of this type are known as Hankel transforms. Here A is simply a normalising factor, it is the integral evaluated where theJ0(x) = 1.

We note that we measure a single visibility for each pair of telescopes, deter- mined by their (u, v) coordinates (other thanu =v = 0, which returns the flux). In order to reconstruct the brightness distribution by the inverse Fourier trans- form, we must measure a large number of visibilities. This can be achieved by having a large array of telescopes (recall N telescopes gives N(N ₋1)/2 unique baselines/pairs), by having an array which can be reconfigured, or by using the rotation of the Earth to alter the projected baseline. This sampling of the (u, v) plane is known as aperture synthesis.

3.2.2

Interferometry with CHARA

The Center for High Angular Resolution Astronomy (CHARA, McAlister et al. (2005)) is an optical/near-infrared interferometeric array located on Mt. Wilson, California. The array consists of six 1 m telescopes, arranged in Y-shaped configu- ration (quite common for interferometers, as it provides good (u, v)-plane coverage under rotation while avoiding redundancy), with (6×5/2 =) 15 possible baselines. The baselines range from 34 m to 331 m giving a maximum resolution of 1.4 mas at K-Band, and 0.3 mas at V-Band.

CHARA is equipped with three near-infrared instruments (CLASSIC, FLOUR, and MIRC) and two visible, PAVO and the instrument we will be most concerned with in this work, the Visible Spectrograph and Polarimeter (VEGA (Mourard et al., 2009)). VEGA provides spectral resolution up to R = 30,000 in the wave- length band 450 nm–850 nm, corresponding to 60 km/s–10 km/s. With this spec- trograph it becomes possible to measure the interferometeric visibility as a function of a wavelength across a spectral line, hence providing a direct measure of the ex- tent of the line-forming region. We will make use of this diagnostic later in this work.