The r-q Diagram

Orbital Parameters of Binary Systems in r - q Diagram

**Table 4.1:**
Binary	Orbital	q	r	Reference
System	Period (d)
$\beta$ Per	2.87	0.21	0.21	i,j
TX UMa	3.06	0.31	0.16	h
U Sge	3.38	0.33	0.22	a,j
U CrB	3.45	0.29	0.17	a,j
SW Cyg	4.57	0.25	0.15	h
UX Mon	5.90	0.42	0.11	h
TT Hya	6.95	0.23	0.09	k
VW Cyg	8.43	0.28	0.08	h
V356 Sgr	8.87	0.34	0.13	e
RY Gem	9.30	0.16	0.10	d
AD Her	9.77	0.35	0.11	h
AU Mon	11.11	0.25	0.08	h
RX Gem	12.21	0.17	0.08	c
DN Ori	12.97	0.12	0.07	b
RW Per	13.19	0.15	0.08	d
KU Cyg	38.40	0.13	0.04	g
RZ Oph	261.93	0.12	0.02	f

$^{a}$ Albright & Richards (1996)
$^{b}$ Etzel & Olson (1995)
$^{c}$ Hall & Walter (1975)
$^{d}$ Kaitchuck (1989)
$^{e}$ Lane (1996)
$^{f}$ Olson (1987)
$^{g}$ Olson, Etzel & Dewey (1995)
$^{h}$ Peters (1989)
$^{i}$ Richards & Ratliff (1998)
$^{j}$ Richards, Albright & Bowles (1995)
$^{k}$ van Hamme & Wilson (1993)

**Figure 4.4:** *r - q* Diagram. The curve of $\omega_{min}$ represents the minimum distance between the mass-transfer stream and the center of the primary star. The curve of $\omega_d$ represents the calculated outer radius of the accretion disk. Notice that KU Cygni lies well below the curve of $\omega_d$ , indicating that a permanent accretion structure can form.
$\begin{figure}\epsfxsize =5in \epsfysize =7in \begin{center} \leavevmode \epsffile{rq.ps}\end{center}\end{figure}$

Lubow & Shu's calculations of $\omega_{min}$ and $\omega_d$ for various mass ratios can be used to interpret Struve's grouping of Algol-type binary systems, based on H $\alpha$ observations. Figure 4.4, named the r-q diagram, plots the fractional radius, r, of the primary star versus mass ratio, q, for a particular binary system. The mass ratio is defined as the ratio of the secondary mass to the primary mass. Two curves are also drawn, which display Lubow & Shu's calculated values of $\omega_{min}$ and $\omega_d$ , as a function of mass ratio. (Examples of this figure and its application to binary systems can be found in Peters & Polidan 1984, Kaitchuck, Honeycutt & Schlegel 1985, Kaitchuck 1988, Kaitchuck 1989, and Peters 1989). Table 4.1 lists values of r and q for numerous binary systems that are plotted on Fig. 4.4. The largest observational uncertainty is in the mass ratio, q. The masses of the stellar components, as well as the orbital separation, are directly related to the orbital periods of these systems, as stated by Kepler's third law. The short period systems, listed in Table 4.1, are observed to have very transient disk features that change with orbital phase. These short-period systems (P

4.5 days) have small orbital separations and a large primary star radii, so the mass-transfer stream largely impacts their primary stars. In the long-period systems (P

6 days), spectroscopic observations reveal persistent double-lobed H $\alpha$ emission lines at all orbital phases, indicating a permanent disk structure around the primary star. The orbital separation between the components is larger than for the shorter period systems.

The demarcation of these groups is clearly observed in the r-q diagram. The short-period systems are located above the $\omega_d$ curve, implying the radius of the primary star is much larger than maximum radius permitted for the formation of an accretion disk. The position of the systems above this curve reveals that the orbital separation between the two stars is too small for a stable disk to form. The mass-transfer stream hits the surface of the primary star in these systems; therefore, the mass-transfer stream cannot create a stable accretion disk around the primary. On the other hand, systems that are located under the $\omega_{min}$ curve are long-period systems, where the orbital separation is adequately large for a stable disk to form. In these systems, the radius of the primary star is much smaller than the maximum radius of a disk. A permanent structure can be formed because the stream does not hit the primary, but rather loops around the primary to form an accretion disk. For systems that are located between the curves of $\omega_d$ and $\omega_{min}$ , the accretion structures around the gainer star are very unstable and transient in nature. The mass-transfer stream may graze the surface of the primary star and ``splash'' matter into a transient structure. KU Cygni lies well below the $\omega_{min}$ curve, indicating that system should contain a stable disk around the primary star.