Chapter 5: DOPPLER TOMOGRAPHY OF KU CYGNI

CHAPTER V


DOPPLER TOMOGRAPHY OF KU CYGNI

Eclipsing binary systems naturally provide astronomers with a unique opportunity to probe the system from different observing angles within the orbital plane. As the stellar components orbit around the center of mass, full orbital coverage through the orbital plane is possible because the orbital inclination of an eclipsing binary system is approximately 90$^o$ to the sky plane. Figure 5.1 displays the different observing angles through the orbital plane of a binary system during one orbit around the center of mass. Recall that the orbital phase, $\phi$, is a fraction of the orbital period with $\phi=0$ at mid-primary eclipse. Spectroscopically, observations obtained at different orbital phases will record total radiative flux at a given wavelength. For a given spectral feature, the wavelength scale of a spectrum can be translated into a corresponding radial velocity scale centered at the rest wavelength of that spectral feature. In Fig. 3.9, a trailed spectrogram for KU Cygni displays the radial velocity variations of H$\alpha$ with orbital phase. The apparent sinusoidal variation indicates periodic motion of emission regions towards and away from the observer, as the system orbits around the center of mass.

Figure 5.1: Perspective of the binary system with orbital phase $\phi$.

The H$\alpha$ emission-line profile of KU Cygni depends on the overall distribution of line flux over the entire disk (Marsh & Horne 1988). The spectra that were obtained of KU Cygni can be utilized as a whole to construct a velocity map of the H$\alpha$ emission regions of the accretion disk. This 2-dimensional map displays the intensity of emission sources with a particular velocity ($V_x$,$V_y$). The intensity of any point on the velocity map depends on the total amount of flux emitted by parcels of gas with those velocity components. In order to utilize the H$\alpha$ emission lines to create a velocity map, the accretion disk is assumed to be essentially optically thin at H$\alpha$, so that emission by each parcel of gas is observable at any given orbital phase (Horne 1991). Each spectroscopic observation displays a one-dimensional projection of the velocity map along a given observation angle through the system. The projection angle through the disk depends on the orbital phase of the observation. This two-dimensional velocity map, or Doppler tomogram, can be reconstructed from the radial velocity information of the H$\alpha$ line. Marsh & Horne (1988) introduced the methods required to analyze spectroscopic data and create a Doppler tomogram from emission-line profiles. This form of accretion disk analysis was initially applied to cataclysmic variable (CV) systems, such as U Gem (Marsh et al. 1990), DQ Her (Kaitchuck et al. 1994), and IP Peg (Kaitchuck et al. 1994). Kaitchuck et al. (1994) created an atlas of Doppler tomograms for numerous CV systems, using emission lines of H$\beta$ and He II. Application of Doppler tomography to Algol-type systems has been recently accomplished in H$\alpha$ for $\beta$ Per, or Algol (Richards, Jones & Swain 1996, Richards & Ratliff 1998), TT Hya (Albright & Richards 1996), U CrB (Richards, Albright & Bowles 1995, Albright & Richards 1996), and other Algol-type systems. In the studies presented above, TT Hya is the only long-period system analyzed. Therefore, KU Cygni will be only the second long-period Algol-type binary system, analyzed using Doppler tomography. The following sections will introduce the methodology required to produce the Doppler tomogram of the disk in KU Cygni.

In the field of medicine, the required mathematical techniques of Doppler tomography have been applied in the creation of computerized axial tomography (CAT) scans and magnetic resonance imaging (MRI) scans. In these scans, an object of interest is located between an emission source, usually X-ray radiation, and a detector. The object is exposed to the emission source. As the radiation penetrates the object, interior structures absorb some radiation as the radiation travels through the object. A detector records any emission that emerges on the other side of the object. A traditional 2-dimensional X-ray image is created for the object at one given projection angle. Then the object is rotated (or the detector and emission source are rotated) and the process of exposure and detection are repeated for a different projection angle. These projections are two-dimensional slices of the object along the given angle. These slices are obtained for many projection angles, which are then used to recreate the three-dimensional image of the absorbers within the object. In the case of binary systems, the velocity map of H$\alpha$ emission regions only contain two-dimensional information. Therefore, a projection of this velocity map along one axis will create a one-dimensional velocity profile. Spectra represents this one-dimensional collapse of the velocity map, or Doppler tomogram, along one axis through the map.

Doppler tomograms reveal the location of emission regions in velocity space, not in Cartesian space. The spectroscopic observations reveal the total observed motion of emission regions at any one given time, via radial velocity information. The tomogram cannot be easily transformed from velocity coordinates into spatial ones without knowledge of the velocity field within the disk. The velocity field of the accretion disk is not necessarily Keplerian in nature. But, knowledge of the theoretical circular Keplerian velocities will assist in the interpretation of emission regions in the Doppler tomogram.

The Keplerian velocity of a disk particle in circular orbit around the primary star of mass, $M_1$, at a distance, $R_1$, is given by

$$V_{Kep} = \sqrt \frac{GM_1}{R_1}.$$ (5.1)

The magnitude of the radius vector and the position angle, $\theta_1$, for a given Cartesian coordinate, centered on $M_1$, are determined by the following equations:

$$R_1= \sqrt{x^2_1 + y^2_1},$$ (5.2)

$$\theta_1 = \tan^{-1}\left(\frac{y_1}{x_1}\right).$$ (5.3)

Figure 5.2 displays the circular, Keplerian orbits of fictitious accretion rings around the primary star of KU Cygni, as well as the trajectory of the mass-transfer stream. The critical Roche lobe of the primary is located approximately 45 $R_{\odot }$ from its center, as estimated from its mass ratio, q (Kopal 1959, p.135). In Fig. 5.2, the radius of the closest circle to the surface of the primary is 10 $R_{\odot }$ from the its center. The radii of the other circles are separated by 5 $R_{\odot }$, with the outermost dashed circle, located 40 $R_{\odot }$ from primary center. The intersection of the mass-transfer stream with each circular orbit is identified by a dot. These positions along the stream will be utilized later in interpreting the Doppler tomogram. Also, eight locations (1-8) on the outermost circle are numbered to later demonstrate the translation of each position into the Doppler map.

Figure 5.2: Spatial coordinate map of KU Cygni. The origin of this map is the center of mass. Seven circular orbits, representing fictitious accretion rings of radii 10 - 40 $R_{\odot }$, are displayed around the primary star. Interval between circles is 5 $R_{\odot }$. The mass-transfer stream is shown with six marked points that intersect the orbits. Orbital revolution is counterclockwise in this figure. (Adapted from Horne 1991).

For a given position on a circular orbit of radius, $R_1$, from the primary star, the Keplerian velocity components ($V_{x1},V_{y1}$) of the particle are related to its polar coordinates. Each position on this orbit will have a velocity vector, $V_{Kep}$, tangent to its radius vector, $R_1$. The velocity components in this stationary reference frame are determined as follows:

$$V_{x1} = (-V_{Kep}) \sin{\theta}$$ (5.4)

$$V_{y1} = (V_{Kep}) \cos{\theta}.$$ (5.5)

Each position on the circular orbit has an additional orbital velocity component arising from synchronous rotation of the system around the center of mass. Thus, velocity projections into the rotating frame of reference must be computed for every parcel of gas. Every position in the binary system has orbital velocity vector, $V_{orb}$, tangent to its radius vector, $R_{CM}$, which is measured from the center of mass. Assuming circular, orbital motion around the center of mass, the additional orbital velocity components are determined by the following equations:

$$R_{CM} = \sqrt{x^2 + y^2} ,$$ (5.6)

$$\theta_{CM} = \tan{^-1}\left(\frac{y}{x}\right),$$ (5.7)

$$V_{orb} = \frac{2\pi R_{CM}}{P},$$ (5.8)

$$V_{x(orb)} = (-V_{orb}) \sin{(\theta_{CM})},$$ (5.9)

$$V_{y(orb)} = (V_{orb}) \cos{(\theta_{CM})}.$$ (5.10)

Thus, in the rotating reference frame of the system, the observed velocity component at a certain position on the circular orbit around the primary star is given by the total of its Keplerian velocity and orbital velocity, or

$$V_x = V_{x1} + V_{x (orb)}$$ (5.11)

and

$$V_y = V_{y1} + V_{y (orb)}.$$ (5.12)

Although the previous paragraphs referred to locations on circular, Keplerian orbits around the primary, the value of ($V_x$, $V_y$) for any other parcel of gas is determined in a similar manner. In that case, the values of ($V_{x1}$, $V_{y1}$) are the velocity components of that gas parcel in the stationary reference frame. Referring to Eqns. 5.11-5.12, two or more distinct parcels of gas can have the same velocity coordinates in the rotating reference frame. Therefore, multiple values of ($V_{x1}$, $V_{y1}$) and ($V_{x (orb)}$, $V_{x (orb)}$) can yield the same velocity coordinates ($V_x, V_y$).