Doppler Tomogram Analysis

Doppler Tomogram Analysis

Figure 6.1: Doppler tomogram of KU Cygni in H$\alpha$ with coordinate grid. This image displays the intensity distribution of H$\alpha$ emitting regions in KU Cygni, as a function of velocity coordinates, in the rotating reference frame of the system. Notice the spot of enhanced H$\alpha$ emission located near ($V_x$, $V_y$) = (-240, 33) km/sec.

Figure 6.1 displays the same Doppler tomogram of KU Cyg in H$\alpha$, as Fig. 5.7, but with the addition of the Doppler coordinate grid. The positions of the stellar components, mass-transfer stream trajectory, and circular, Keplerian orbits on this grid will assist in the interpretation of the Doppler map of KU Cygni. Recall that the outermost solid circle on this grid represents the maximum Keplerian velocity at the surface of the primary star. The innermost circle represents the minimum circular, Keplerian velocity at the critical Roche lobe of the primary star. The velocity components of emission regions within the disk are not expected to be greater than the Keplerian velocity at the surface of the primary, nor less than that at the Roche lobe of the primary. Therefore, the intensity distribution of these regions in velocity space should lie between these two velocity circles. As evident in Fig. 5.7, the H$\alpha$ emission regions map into a ``donut-like'' or toroidal shape in the tomogram, which is located between the assumed minimum and maximum Keplerian reference circles. The majority of the H$\alpha$ intensity contributions are located between the second outermost and the innermost reference circles, as seen in Fig. 5.7. The second outermost circle corresponds to the circular, Keplerian velocity of an accretion ring located 5 $R_{\odot }$ from the primary star. The appearance of emission-line intensity near this circle indicates the presence of gaseous regions, moving in near-Keplerian orbits, close to the primary star surface ($R_{prim}=3.38$ $R_{\odot }$).

The apparent toroidal shape of the emission regions in the Doppler tomogram indicates the existence of a somewhat symmetric accretion disk around the primary star. Looking at Fig. 5.7, the emitting regions extend down to the innermost circle, which corresponds to minimum Keplerian velocities at the Roche lobe radius of the primary. Therefore, the accretion disk probably nearly fills the entire Roche lobe of the primary. In other long-period Algol-type systems, such as TT Hya ($P=6.95d$) and RZ Oph ($P=261.93d$), the accretion disks are estimated to fill 90-95% of their primary star's Roche lobe (Peters 1989).

In short-period Algol-type systems (eg. $\beta$ Per, U CrB, U Sge), emission regions originating along the mass-transfer stream trajectory, stream-star impact region, localized regions, and chromospheric activity of the secondary star have been imaged in Doppler tomograms (see Richards, Albright & Bowles 1995, Richards, Jones & Swain 1996, Albright & Richards 1996). Because KU Cygni is a long-period Algol-type binary system, a stable accretion disk is expected to be observed in this system. Spectroscopically, the V and R emission lobes reveal the existence of an accretion disk, which orbits around the primary (gainer) star in a prograde fashion. With such a large orbital separation, the mass-transfer stream should not impact the primary star, but rather interact with the outer regions of the accretion disk. In their study of TT Hya, Albright & Richards (1996) were unable to identify conclusively any region of enhanced emission in the Doppler tomogram that would correspond to the stream interaction with the outer edge of the accretion disk.

In the Doppler tomogram of KU Cygni (Fig. 5.7), one area on the map appears to contribute more to the intensity of H$\alpha$ emission than other areas. One small area of enhanced emission near ($V_x$, $V_y$) = (-240, 33) km/sec lies close to the intersection point of the predicted ballistic and Keplerian velocity trajectories of the mass-transfer stream. At this location the velocity components of the stream and the Keplerian disk would be identical. This coincidence suggests that the stream's velocity, although non-Keplerian as it exits the inner Langrangian (L1) point, will eventually merge into the accretion disk and move at near Keplerian velocities. The region where the stream encounters the disk may interact with the surrounding areas; thus, the enhanced emission may result from this gaseous turbulence. There is no compelling evidence for emission along the stream's trajectory as it travels away from the L1 point. The stream's total volume is small compared to that of the accretion disk.

Refering back to Figs. 5.6 and 5.7, this enhanced emission region appears in both Doppler tomograms. This region of enhanced emission may result from unequal phase sampling of the spectroscopic data; therefore, the region could be an artifact and not be realistic. In Fig. 5.6 this brighter ``spot'' is also accompanied by three (less-apparent) ``spots''. The symmetrical nature of the velocity positions of these four ``spots'' could be attributed to the inadequate sampling between phase intervals of 0.30-0.39 and 0.60-0.69. With (somewhat) equal phase sampling the Doppler tomogram in Fig. 5.7 still displays the brighter ``spot'' of emission, even though the symmetrical nature of the disk is more apparent. Therefore, this brighter ``spot'' is not an artifact, but probably an enhanced region of emission. To conclusively identify this enhanced emission region, supplemental spectroscopic observations within the aforementioned phase intervals are required to reduce imaging artifacts.

The enhanced H$\alpha$ emission source appears to be related to the mass-transfer stream impacting the outer regions of the accretion disk. Since two parcels of gas can have the same Doppler coordinates on the velocity map, determination of the spatial location for this emission source can be ill-defined. Using the Keplerian-velocity reference circles on the Doppler map, a simple inspection reveals that the source is located between 10-15 $R_{\odot }$ from the primary star. For an initial analysis, the velocity field within the accretion disk is assumed to be nearly Keplerian. Therefore, the velocity coordinate of this enhanced emission region can be identified with a certain spatial location within the disk. The central velocity coordinates of this enhanced emission source are estimated to be ($V_x$, $V_y$) = (-240, 33) in units of km/sec. Assuming that the above velocity coordinates corresponds to a particle in motion around the primary star, the velocity radius of its corresponding circular, Keplerian orbit is calculated as the follows:

$$V_{Kep} = \sqrt{V^{2}_{x} + (V_{y}+K_1)^2},$$ (6.1)

and

$$R_1 = \frac{G M}{V^{2}_{Kep}}.$$ (6.2)

The Doppler tomogram in centered on the center of mass for the system, and not the primary star's orbital velocity. For this reason, the semi-amplitude velocity of the primary star was added to $V_y$, in order to determine the approximate spatial location around the primary. The calculated radius of an accretion ring that corresponds to the circular, Keplerian velocity ($V_x$, $V_y$) in the rotating frame of reference is $\approx$ 12.5 $R_{\odot }$. The enhanced emission region would be spatially located $\approx$ 80$^o$ counterclockwise from the x-axis and centered on the primary star. Note that this spatial location is found on the trailing side of the disk (R lobe during primary eclipse). The Keplerian location of this enhanced emission region coincides roughly with the value of $\omega_d$=16.3 $R_{\odot }$. From disk-eclipse timings, the estimated outer radius of the accretion disk is $R_{disk}$=22 $R_{\odot }$. Keep in mind that $\omega_d$ defines an effective (outer) radius of the accretion disk, as determined from the assumed mass ratio of KU Cyg (Lubow & Shu 1975). A majority of the disk material contributing to the relative intensity of H$\alpha$ emission can be located near this effective (outer) radius, as noted by the Keplerian location of the enhanced emission region. As revealed in the Doppler tomogram, contributions to the relative intensity from the outer edges of the Roche lobe (innermost circle on Doppler tomogram) indicate that the disk is larger than $\omega_d$. Also note that stable, Keplerian orbits with radii greater than $\omega_d$ exist, as determined from restricted three-body orbital calculations.

The enhanced emission region is located on the trailing side (R lobe near primary eclipse or V lobe near secondary eclipse) of the accretion disk, as determined from its coordinate on the Doppler map. The mass-transfer stream is predicted to impact the edge of an accretion disk on this side of the disk. Evidence for this impact region is supported by the spectroscopic observations of KU Cyg, analyzed in Chapter III, as well as photometric observations. As illustrated in Fig. 2.1, the equivalent width of the V emission lobe decreases dramatically near primary eclipse, as compared to the persistent strength of the R lobe. Prior to primary eclipse, the leading side of the disk (V lobe, in this instance) will be occulted by the secondary star, while the trailing side (R lobe, in this instance) will be unocculted and visible. The mass-transfer stream will be receding from the observer prior to 2nd contact of primary eclipse; thus, the stream can also contribute to the R lobe profile. Also, referring to Fig. 3.2, the departure velocity of the R lobe increases dramatically prior to primary eclipse. In effect, the observer is ``looking downstream'' into the receding mass-transfer stream. Similarly, near phase 0.50, the departure velocity of the V lobe is greater than that of the R lobe, which suggests that the observer is ``looking upstream'' into the approaching mass-transfer stream. In Fig. 3.8, the intensity ratio, V/R, of the emission lobes decreases progressively from phase 0.8 to the start of primary eclipse totality. After phase 0.91, the V lobe will be occulted, thus increases the apparent intensity of the R lobe, as indicated by the decrease in V/R. The gradual increase can be attributed to an enhanced emission source, as it comes into view along our line of sight. The ultraviolet light curve of KU Cyg reveals the existence of an enhanced emission source. Referring to the u-band light curve analyzed by Olson, Etzel & Dewey (1995, Fig. 9), the light curve during primary eclipse minimum displays a slight, downward slope in magnitude from beginning to end of eclipse totality. This downward slope implies that, after mid-primary eclipse, the secondary star eclipses a region on the receding side (R lobe) of the disk that is slightly more luminous than in the approaching side (V lobe). The more luminous source could be the stream-disk impact region on the trailing side of the disk, imaged in the Doppler tomogram.