Trajectory Computations

Trajectory Computations

Trajectories of the mass-transfer stream through the L1 point for KU Cygni are computed by numerically integrating the equations of motion for an infinitesimal particle in the gravitational field between two massive objects. The equations of motion in the restricted three-body problem were stated in the previous section of this chapter. Ronald Polidan (1985) coded trajectory solutions of Eqns. (4.1-4.2) and Eqn. 4.9. Equations 4.1 and 4.2 describe the acceleration of the particle at a given location, while Eqn. 4.9 describes the velocity components of this particle. Stable, non-intersecting particle trajectories are calculated for user specified initial position coordinates and velocity components. The Cartesian coordinates are the same as defined in the computation of zero-velocity curves around two point masses (refer to Fig. 4.1). Motion that is towards the more massive object will have a negative $V_x$ component, while motion in the direction of orbital revolution will have a positive $V_y$ component. The trajectory of the mass-transfer stream is confined to the orbital plane of the binary system. To perform these calcuations, the primary and secondary parameters of mass and radius, as well as orbital period of the system are required. The initial values of (x,y,z) and ($V_x$, $V_y$ $V_z$) are also required for the infinitesimal particle. The ballistic trajectory of this fictituous particle represents the average motion of the center of the mass-transfer stream (Lubow & Shu 1975). In order to calculate the stream's trajectory, these three equations must be numerically integrated for a given or calculated time step. The equations are solved using the fourth-order Runge-Kutta numerical integration technique.

Lubow & Shu (1975) discuss the necessary assumptions required to perform these trajectory calculations. These authors admit that their results present a simplistic view of mass transfer in semi-detached binaries. Nevertheless, their results yield adequate information pertaining to the gas dynamics of mass transfer. Their conclusions were applied to the study of mass transfer in KU Cygni. As stated previously, Lubow & Shu (1975) surmised that matter-transfer stream travels across the L1 point at nearly sonic to super-sonic velocities. Therefore, in these calculations, the miniumum initial velocity of the stream, $v_s$, is given as the speed of sound, $v_c$ ($10$ km/sec). For a given mass ratio, Lubow & Shu (1975) calculated the angle of ejection of the matter stream with respect to the x-axis. Lubow & Shu define their mass ratio as the mass ratio of the detached component to the contact component. Thus, for KU Cygni, $M_D/M_C$ is 8.021 from Olson et al. (1995). For this mass ratio, the mass-transfer stream makes an exit angle, $\theta_s$, of $-21.486$$^o$ to the x-axis.

The transfer stream exits at the inner Lagrangian (L1) point, located at coordinates $(x,y)=(0.588709, 0)$. The initial velocity vector of this stream equals the speed of sound, exiting at an angle, $\theta_s$. The $V_x$ and $V_y$ components of the velocity vector are calculated for this given angle. Initially, the stream falls towards the primary star, that is, in the negative $V_x$ direction. Because of orbital revolution, the Coriolis effect diverts the stream in the direction of motion, or in the positive $V_y$ direction. The initial velocity components inputted into Polidan's code are ($V_x$, $V_y$, $V_z$)=(-9.3051, 0.0356, 0). (Note that the program requires that velocities are inputted as unit velocity values). The program outputs the timestep, in fractions of an orbital period, the spatial coordinates of the stream, as well as the velocity components. A check for constant C and constant angular momentum at every timestep is determined throughout the calculations.

Lubow & Shu calculated the relationship between the mass ratio and dimensions of the accretion disk. They calculated a value $\omega_{min}$, in fractional units of the orbital separation between the components, which describes the minimum distance between the mass-transfer stream and the gainer star. If the surface of the detached gainer star lies in the path of the stream, then the stream will impact the star at high velocity. If the surface of the detached gainer is smaller than $\omega_{min}$, then stream will eventually loop around the gainer and eventually intersect with itself. This merger between the returning stream and incoming stream will travel in a periodic orbit around the gainer star. This periodic orbit can be described by an equivalent circular orbit of radius, $\omega_{d}$ (in fractional units), that defines the outer disk radius. (Note that circumstellar matter could still exist beyond $\omega_d$ out to the Roche limit of the gainer). Using Lubow & Shu's tabulated values of $\omega_{min}$ and $\omega_d$ for various mass ratios, the interpolated values for KU Cyg are as follows:

$\omega_{min} = 0.1278 \times A = 9.98$ $R_{\odot }$,
$\omega_d = 0.2093 \times A = 16.346$ $R_{\odot }$,

where A is the orbital separation between the components in solar radii. The trajectory of the mass-transfer stream in KU Cygni is displayed in Figure 4.3. In this figure, the radius of the primary star and its critical Roche lobe are shown. Circles of radius $\omega_{min}$ and $\omega_d$ about the primary star are also displayed. Notice that the stream does not collide with the photosphere of the primary star, but rather loops around the gainer star and eventually collides with itself. As displayed in Fig. 4.3, the minimum distance between the stream trajectory and the primary star, as calculated using the restricted three-body problem, agrees well with Lubow & Shu's (1975) estimated value of $\omega_{min}$ for KU Cygni's mass ratio.

Figure 4.3: Trajectory of the mass-transfer stream in KU Cygni. The stream exits from the L1 point at the speed of sound. Two circles, centered on the primary star, with radius $\omega_{min}$ and $\omega_d$ are labeled. Notice that the stream does not impact the primary star surface, but rather, loops around the star. The stream is confined within the Roche lobe of the gainer star. Orbital revolution is counterclockwise around the center of mass.