Interpretation of a Doppler Map

Interpretation of a Doppler Map

The velocity coordinates on the Doppler tomogram are $V_x$ and $V_y$ components of a velocity vector in the rotating frame of reference of the binary system. In the Doppler map, the velocity vector of a particle will translate to a point on the map at coordinates ($V_x$, $V_y$). The positive $V_x$-axis defines a line from the primary star to the secondary star. The positive $V_y$-axis defines the direction of orbital motion of the secondary star. The origin in this map is the center of mass of the system (Horne 1991). In a binary system with a circular orbit, the more massive and less massive components orbit around the center of mass with projected velocities $K_1$ and $K_2$, respectively, for a specified orbital inclination.

Figure 5.3 displays the coordinate grid of a Doppler tomogram for KU Cygni. In the tomogram, the orbital velocity of the primary star is located at ($V_x$, $V_y$) = (0, $-K_1$), while the secondary star's orbital velocity is positioned at ($V_x$, $V_y$) = (0, $K_2$). With the assumption of synchronous stellar rotation around the center of mass, the Roche lobes of the stellar components are undistorted when translated to the velocity map. The Roche lobe surface of the secondary star is centered at (0, $K_2$), while the stellar radius of the primary star is centered on (0, $-K_1$). The velocity positions of the primary and secondary stars are marked in Fig. 5.3, as well as the Roche lobe of the secondary star.

The calculated circular Keplerian velocity near the surface of the primary is $\approx$ 437 km/sec. As the angular momentum of disk matter is transferred to the primary star, the rotational velocity of the primary star is expected to increase. Olson et al. (1995) indicated that the primary (gainer) star in KU Cyg rotates supersynchronously ( $\approx 11\times$ synchronous). The synchronous rotational velocity of the primary star is estimated to be $\approx$ 4 km/sec, while the observed velocity is $\approx$ 53 km/sec. However, since this component is so small compared to its Roche lobe, any distortion induced into the accretion disk would be minimal. Supersynchronous rotation of gainers in Algol-type systems are expected.

Figure 5.3: Doppler coordinate map of KU Cygni. The origin in the velocity map is the center of mass. The velocity coordinates of the primary and secondary stars are marked. The concentric circles, centered on the velocity coordinate of the primary star, represent radii of constant Keplerian velocity for fictitous circular orbits around the primary ($\Delta$ R = 5 $R_{\odot }$). The solid-line trajectory, extending left of the secondary star's Roche lobe, is the free-fall velocity path of the stream, while the dashed-line trajectory is the Keplerian image of this path. (Adapted from Horne 1991).

In Fig. 5.3, the solid outer circle represents the circular Keplerian velocity (maximum) at the surface of the primary star ( $v_{surf}\approx437$ km/sec). The dashed innermost circle represents the minimum Keplerian velocity of a particle at the critical Roche lobe radius of the primary star ( $v_{min}\approx128$ km/sec). If emission lines originate in the accretion disk around the primary star, the Doppler coordinates of these particular emission sources should be located between the solid outer circle and the dashed innermost circle. Physically, the inner radius of the accretion disk cannot be smaller than the radius of the primary star. Also, the accretion disk will be contained within the Roche lobe of the gainer star (Lubow & Shu 1975).

In the Doppler map, the velocity components of particles in circular, Keplerian orbits around the primary star will be centered on the velocity coordinates of the primary star. In the tomogram, the velocity radius of these orbits will equal their Keplerian circular velocities around the primary star. In Fig. 5.3 the concentric rings between the outermost and innermost circles are the velocity circles of circular, Keplerian orbits, which are separated in increments of 5 $R_{\odot }$ from the primary star. The Doppler tomogram displays an ``inside-out'' image of the velocity regions within the disk. As dictated by Kepler's third law, the inner disk regions orbit around the primary star with a larger velocity than the outer regions of the disk. Therefore, the inner disk regions will map to a velocity region further from the center of the map than the outer disk regions. Notice that the translation of the secondary star's Roche lobe is found within the innermost dashed circle. The velocity component of the secondary is generally much less than the orbital velocity at the Roche-lobe radius of the primary.

Referring back to Fig. 5.2, the outermost dashed circle is located at a distance of 40 $R_{\odot }$ from the primary. Eight locations (1-8) are identified on this circle. Those same position are numbered in Fig. 5.3, but labeled by their corresponding velocity coordinates. Recall that the velocity vector of a moving particle is tangent to its radius vector. Therefore, positional orientation in the spatial map is rotated 90$^o$ counterclockwise in the Doppler map. For this reason, the position of the Roche-lobe filling secondary star of KU Cygni is centered at (0, $K_2$) in the Doppler map.

In Chapter 4, trajectory calculations of the mass-transfer stream entailed integration of the equations of motion in the restricted three-body problem. Also calculated were the instantaneous velocity components ($V_{x1}$, $V_{y1}$, $V_{z1}$) at position (x, y, z) in the stationary reference frame. Equations 5.6-5.12 were used to project these stationary velocity components of the stream into the rotating reference frame of the system. Then, the coordinates ($V_x$,$V_y$) can be used to plot the stream's velocity path in the Doppler tomogram. Referring back to Fig. 5.2, the stream falls towards the primary star, but is deflected in the direction of orbital motion around the center of mass by the Coriolis effect. Thus, the velocity component of the stream begins in the negative $V_x$ and curves towards the negative $V_y$ direction.

In Fig. 5.3, the solid-line trajectory that extends to the left of the secondary star's Roche lobe is the predicted ballistic (or free-fall) trajectory of the stream in velocity space. The marked positions along the solid-line trajectory identify the same marked positions on the stream trajectory that is displayed in the spatial map (Fig. 5.2), but now, these locations are marked according to their corresponding velocity coordinates. In the spatial map, each position on the stream's trajectory, which is located a certain distance from the primary, can also be identified by a corresponding circular, Keplerian orbital velocity at that distance. Using these corresponding Keplerian velocities, another theoretical velocity trajectory can be mapped on the Doppler tomogram corresponding to the calculated Keplerian velocities along the stream. This trajectory, called the Keplerian image of the stream, is marked by dashed-line to the left of the secondary star's Roche lobe. Recall that the points along the stream marked locations at radius intervals of 5 $R_{\odot }$ from the primary. Notice that these points, marked on the Keplerian trajectory, intersect with the Keplerian velocity rings in the map, as they should. If the mass-transfer stream interacts with the outer disk regions, the stream may be shocked into a near Keplerian flow and appear as an emission region between the two calculated stream trajectories on the Doppler map (Richards et al. 1996).