Chapter 2: HISTORY OF KU CYGNI

CHAPTER II


HISTORY OF KU CYGNI

KU Cygni (BD+46$^o$2879) is an Algol-type binary system with a long orbital period of 38.4 days, which exhibits a total eclipse with a duration of 2.98 days (Popper 1964, Olson, Etzel & Dewey 1995, Smak & Plavec 1997). The outside eclipse visual magnitude is 11.0, and the depth of primary eclipse is 1.3 magnitudes. Contact phases of primary eclipse are

$$\phi_{1,4}=\pm0.039 \pm0.001,$$ (2.1)

$$\phi_{2,3}=\pm0.024 \pm0.001,$$ (2.2)

where $\phi_{1,4}$ and $\phi_{2,3}$ refer to the phases of external and internal eclipse contacts, respectively. Popper (1964) published the earliest complete photometric and spectroscopic studies of KU Cyg. He determined the spectral type of the secondary to be K5 III, while the spectrum of the primary resembled a peculiar F4 Ia star.

Popper and others observed variations in the light curve between primary eclipses, which could be attributed to a tidally distorted secondary star. At the time, the ellipticity effect of the secondary could not be adequately calculated because of unknown limb-darkening and gravity-brightening coefficients for a K giant. Popper did note a possible secondary eclipse near $\phi=0.56$, but to acknowledge conclusively its existence, further photometric observations were needed. Near primary eclipse, anomalous brightness variations were observed near 1st contact ($\phi=0.961$) and near 4th contact ($\phi=0.039$), although at mid-eclipse the variations were at a minimum. Popper (1964) notes that the light variations during these contact times suggest that the changes stem from intrinsic variations of the brighter, primary star.

Popper detected a UV excess during mid-eclipse, as determined from the color index values. The value for (B-V) agreed well with the spectral type of the secondary star. But the value of (U-B) was in excess of 0.8 magnitudes. Popper (1964) notes that the ``source of the ultraviolet radiation...is undetermined except that it is not confined to the photosphere of the smaller star.'' His photometric studies suggested that the primary star, which was spectroscopically classified as a supergiant, was very underluminous for its size and color.

Popper (1964) obtained spectroscopic observations of KU Cyg outside of primary eclipse. The spectrum resembled that of an F4 Ia star. Balmer emission lines were also observed. Shortward of 4000 Å, the spectrum during primary eclipse totality had a greater intensity than that expected for the spectral type of the secondary component, which is consistent with the UV excess that was observed photometrically.

Using the spectral features of class A and F supergiants, Popper calculated the Doppler shift of certain absorption features (outside of primary eclipse) as a function of orbital phase to produce a radial velocity curve for the primary (gainer) component of KU Cyg. The velocity variations, even though possibly contaminated by circumstellar material, agreed well with the phases computed from the light curve's ephemeris. Specific orbital elements of a binary system determine the shape of its radial velocity curve. The Heliocentric radial velocity of each stellar component is related to various orbital elements, as follows:

$$V = \gamma + K e cos \omega + K cos(\omega+\nu),$$ (2.3)

where V is the observed Heliocentric radial velocity in km/sec, $\gamma$ is the systemic (center of mass) velocity of the system in km/sec, Kis the semi-amplitude of the velocity curve in km/sec, e is the orbital eccentricity, $\omega$ is the angle of the ascending node to the point of periastron passage, and $\nu$ is the angular position of the observed star around the center of mass, as measured from the periastron point. The periastron point is the point along the orbit where the component is closest to the center of mass. Popper calculated two best-fit shapes of the radial velocity curve (Eqn. 2.3) of the primary star. One solution assumed a circular orbit , and resulted in and . The other solution allowed an eccentric orbit with $^o$, , and . The observed displacements of absorption features from the secondary component yielded a value of $\gamma$ in agreement with the previous calculations obtained from the radial velocity curve of the primary.

The contact phases during primary eclipse constrain the minimum fractional radius of the secondary component in a light curve solution. Assuming that the secondary star fills its Roche lobe, its fractional radius sets the mass ratio of the system. Using the previously calculated value of , and the estimated upper limit for the mass ratio from his series of light curve solutions, Popper (1964) estimated that the value of cannot be much greater than 170 km/sec. Popper obtained very limited spectra of the secondary component in the photographic-blue region during primary eclipse totality. By assuming a circular orbit, he derived a value of $K_2=190$ km/sec. By allowing an eccentric orbit, he obtained values of e=0.25 and $K_2=300$ km/sec. The former circular orbit solution was much more consistent with the value of $K_2$, determined from the photometrically constrained mass ratios. Thus, Popper concluded that the orbital shape is indeed circular, and not eccentric. Note that the Roche-lobe model can only be invoked if the components are orbiting around a center of mass in a circular orbit. Non-circular orbits require a perturbed mathematical model to describe the equipotential surfaces around the components.

Emission lines of H$\beta$ and H$\alpha$ are visible throughout the entire orbital cycle of KU Cyg (Popper 1964, Olson, Etzel & Dewey 1995). Each emission line appears double lobed where the shortward, or violet-shifted (V), emission lobe originates from gas approaching the observer, and the longward, or red-shifted (R), emission lobe originates from gas receding from the observer. At mid-primary eclipse, the intensity of both lobes, as observed on Popper's photographic spectra, was weaker than outside of eclipse. This latter observation suggests that the emitting regions around the primary star are as large as, or slightly smaller than the secondary component as a result of orbital smearing. W. P. Bidelman (Popper 1964) mentioned that ``the spectrum is...probably that of a shell and not that of a star of excessively high luminosity.'' The Balmer emission lines in the spectrum of KU Cyg are normally seen in Algol-type systems with primary stars of spectral type B9 through F5. If the emission lines originated in matter surrounding the primary star, radial velocity information from them should not be utilized in calculations of orbital dimensions.

Popper (1964) determined two sets of orbital elements for KU Cygni, assuming a circular orbit. The first case assumed that the estimated value of $K_{2}=190$ km/sec, was indicative of the true orbital motion of the secondary star. Table 2.1, column 1 (located at the end of this chapter) lists the properties derived by Popper (1964) using this assumption. The values within the parentheses are in the range of acceptable values. The lower limits depend on the minimum value of the secondary radius, as determined from the contact phases of primary eclipse. The upper limits assume that the secondary star slightly overfills its Roche lobe. With this assumption, Popper calculated a large mass for the primary star, which did not agree with its smaller bolometric luminosity, as compared to the secondary star.

Popper's second case determined orbital elements, assuming that the H$\beta$ emission-line ``displacements outside eclipse result from purely gravitational motion of a ring about the smaller star.'' The velocity of the gas ring around the primary star is related to the value of $K_1$ and $K_2$ by the following equation (Popper 1964):

$$K_2(K_1+K_2) = \frac{v^2 r \sin^2i}{(1-e^2)},$$ (2.4)

where v is the velocity in the accretion ring, and r is the radius of the ring, as observed during primary eclipse. Popper constrained the ring radius, r, to equal the secondary radius. Table 2.1, column 2, lists these derived properties of KU Cygni, using the second case assumption. The two calculated values of $K_2$ (84 and 75 km/sec) resulted from assuming a minimum and maximum value for the secondary radius. With either estimated value of $K_2$ in this case, the resulting value for the primary mass is much smaller than the primary mass calculated from the first case. In follow-up study of KU Cygni, Popper (1965) observed the sodium D absorption lines of the cool secondary star near quadrature (outside of eclipse) to determine the value of K$_{2}$ directly, which better constrains the stellar masses. Assuming circular orbits, Popper calculated $K_{2}$ = 92 km/sec and $\gamma$ = -14 km/sec. This independent determination of K$_{2}$ agreed well with the previous value calculated in the second case assumption (Popper 1964).

KU Cygni was largely ignored for 20 years, until Edward C. Olson (1988) observed KU Cyg to describe the emitting and absorbing regions within the accretion disk, by using the uvbyI photometric system. Olson calculated photometric orbital elements using the program WINK (Wood 1972). His derived ephemeris for KU Cyg, starting from mid-primary eclipse, is

$$HJD = 2,433,884.840+38.439484E.$$ (2.5)

The daily and seasonal light variations of KU Cyg, as observed by Popper (1964), were also apparent in Olson's observations. Olson describes the system to exhibit ``low'' and ``high'' states of brightness, which he attributed to variable mass transfer onto the primary star's photosphere. Olson's observations also revealed ``dips'' that appeared before and after primary eclipse and a slight depression in the light curve near phase 0.5. The dips on either side of the primary eclipse appear when the secondary star begins to occult the accretion disk before occulting the primary star (Olson 1988). The broad secondary eclipse begins as the disk around the primary star starts to partially eclipse the secondary star. Olson estimated the mass loss rate ($\dot{M}$ = $\frac{dM}{dt}$) implied from photometric high and low states to be roughly ${10}^{-6}$ $M_{\odot}$ ${yr}^{-1}$. For cool Algols, like KU Cyg, McCluskey (1993) estimates that in active systems, the mass transfer rate ranges from $\dot{M}$ of ${10}^{-5}$ to ${10}^{-7}$ $M_{\odot}$ ${yr}^{-1}$. In less active Algols, $\dot{M}$ ranges from ${10}^{-9}$ to ${10}^{-10}$ $M_{\odot}$ ${yr}^{-1}$. Albright & Richards (1996), Richards & Albright (1995), Richards, Albright, & Bowles (1995) note that the slow mass transfer rates in Algols ranges from ${10}^{-1}$ to ${10}^{-7}$ $M_{\odot}$ ${yr}^{-1}$.

Staszek Zoa (1992) analyzed the uvbyI data, obtained from Olson (1988), to constrain the orbital properties of KU Cyg. Zoa modeled the shape of the light curves using iterative techniques to converge towards elements that would adequately recreate the light curves at different colors. His technique included disk effects on the light-curve shape. The parameters adjusted in this simulation include the mass ratio, temperature of the primary star, inclination angle, the radius and thickness of the disk. Table 2.1, column 3, contains Zoa's calculated orbital parameters. His mass ratio is the most anomalous.

Other spectroscopic studies reveal the existence of the accretion disk in KU Cygni. Etzel, Olson & Senay (1995) discovered that anomalously strong absorption of the O I $\lambda$7774 triplet, or ``oxygen line,'' is a good indicator for the existence of an accretion disk. The formation of absorption lines within the disk can arise if one's line of sight traverses the disk, and ends at the surface of one of the components. These absorptions lines can be utilized to determine the approximate radius and orbital velocity of the disk around the primary star. The excitation of the O I line requires high kinetic temperatures that exist near the massive primary components in Algol binaries. In the O I $\lambda$7774 survey of 20 binary systems, Etzel et al. (1995) describe the equivalent width variations of O I with orbital phase for each of the systems, one of which is KU Cygni. As KU Cyg approaches orbital phase 0.5, the equivalent width of the O I line greatly increases from 1Å to 5Å. Near secondary eclipse, the disk begins to occult the secondary star. In conjunction with the increased equivalent widths, the radial velocity of the O I line changes rapidly from positive to negative values near phase 0.5, implying disk rotation in the same sense as the orbital motion. Using the radial velocity information of O I, Etzel et al. (1995) found evidence for disk infall towards the primary. In longer period Algols, the secondary star is generally larger and more luminous than secondary stars of shorter period systems. Etzel et al. (1995) note that the disk in KU Cyg must eclipse most of the secondary star's equatorial diameter to produce such a deep O I absorption line. The accretion disk must be vertically thick to absorb the radiation emitted by the secondary star and create this deep absorption feature.

Olson & Etzel (1995) studied the fluctuations of the H$\alpha$ emission lines that were observed in the spectra of short, intermediate, and long period Algols. They observed that the equivalent widths of the H$\alpha$ line, outside of primary eclipse, can vary rapidly within one orbital period. Spectroscopic observations obtained outside of eclipse reveal the H$\alpha$ line profile integrated over the entire disk. KU Cyg was found to display variations in the equivalent width of H$\alpha$, although the variations were the smallest, when compared to the variations of the short-period systems. The average equivalent width of the H$\alpha$ line is 5.7Å with a fractional variation of 21% over many orbital periods (Olson & Etzel 1995). This percentage is relatively small as compared to a 46% change in the short-period systems. Outside of eclipse, the O I line does not display similar fluctuations as H$\alpha$, so Olson & Etzel concluded that the outer edges of the disk are more stable than the inner regions of the disk where the H$\alpha$ line forms.

Figure 2.1: Variation in equivalent width of the H$\alpha$ line with orbital phase. The filled symbols represent the V lobe equivalent width values, while the open symbols represent the R lobe width values. Recall that $\phi_{1,4} = \pm 0.039$.

Figure 2.1 displays in three panels their data that reveal the variations in the equivalent width of H$\alpha$ for three observing seasons. In each panel, the filled symbols represent the V emission lobes of H$\alpha$, while the open symbols represent the R emission lobes of H$\alpha$. Notice that the H$\alpha$ equivalent width of the R lobe near primary eclipse increases dramatically as the V lobe decreases. During these phases, the V lobe is eclipsed by the secondary star. Similarly, after primary eclipse, the equivalent width of the V lobe is greater than the R lobe. Now, the R lobe is eclipsed by the secondary star. In Fig. 2.1, notice that the fluctuations do vary from season to season. Olson & Etzel suggest that these variations result from inner-disk instabilities, which may be related to small changes in the mass-transfer rate.

Olson, Etzel & Dewey (1995) presented much improved spectroscopic and photometric solutions to the radial velocity and light curves of KU Cygni. The expanded spectroscopic observations used by Olson et al. (1995) are also the same observations used in this thesis. The program SBOP, a spectroscopic orbit solution based upon Wolfe, Horak & Storer (1967), was used to determine orbital elements from the observed radial velocity curves of the primary (gainer) and secondary (loser) stars. Olson et al. (1995) determined that $K_2=91.2\pm1.9$ km/sec, which agreed well with Popper's (1965, 1964) value. On the other hand, their determination of $K_1 \approx 7.7\pm1.9$ km/sec differed greatly from Popper's value of 17 km/sec. The derived mass ratio, using the above $K_1$ and $K_2$ values, yields a value of 0.085$\pm$0.022. Observation of the Mg II absorption line from the primary (gainer) spectrum yielded an uncertain rotational velocity ( $v_1\sin\emph{i}\approx 53$ km/sec) for the primary star, which suggests supersynchronous rotation ($11 \times$ synchronous).

Olson et al. (1995) re-analyzed the five-color intermediate-band photometric data of Olson (1988) with the Wilson & Devinney (1971) program, which permits semi-detached configurations. Assuming a semi-detached configuration, simultaneous solutions (vbyI filters) for the light curves were obtained for a fixed range of mass ratios ( $0.08\leq\emph{q}\leq0.30$). The solutions for $\emph{q}< 0.10$ diverged noticeably from the observed data. Recall that the spectroscopic solution yielded a mass ratio of 0.085$\pm$0.022. Because of the discrepancy between the photometric and spectroscopic solution for q, Olson et al. (1995) adopted a value of 0.125 for q, which corresponds to $K_1=11$ km/sec. The photometric analysis, presented by Olson et al. (1995), reveal light curve dips in the ultraviolet light curve, which results from a partial eclipse of the disk prior to and possibly after primary eclipse. The disk contributes significantly to the ultraviolet light of the system. With adopted values of $K_1\approx 11$ km/sec and $K_2=91.2\pm1.91$ km/sec, Olson et al. (1995) calculated the masses of the components, the orbital separation of the pair, and the systemic velocity. These values are listed in Table 2.1, column 4. Notice that the secondary (loser) mass is smaller than Popper's original calculations. A summary of results from Olson et al. (1995) can be found in Table 2.1, column 4, which lists their calculated orbital elements. This thesis employs the values determined by Olson et al. (1995) in all further modeling.

Smak & Plavec (1997) investigated the nature of the accretion disk around KU Cyg, based upon a single spectrum taken just before total eclipse ($\phi=-0.044$). By varying model parameters of disk eccentricity, geometric height of the disk, and line-broadening effects, Smak & Plavec fit the emission-line profile of their spectrum of KU Cyg to determine various disk parameters. Their results suggest that the shape of the outer regions is more eccentric than the inner disk regions. The disk eccentricity is estimated to be 0.31$\pm$0.07. Also, as the accretion disk in KU Cyg grows in size, Smak & Plavec suggested that the disk begins to precess in orbit around the primary star. The ``high'' brightness states, noted by Olson (1988), can be explained by a larger accretion disk formed during higher mass-transfer rates, whereas the ``low'' states correspond to a reduction in disk radius (Smak & Plavec 1997). Table 2.1, column 5, lists the results of Smak & Plavec's study.

Summary of Derived Orbital Elements of KU Cygni

Table 2.1:

	1964$^{a}$	1964$^{b}$	1992$^{c}$	1995$^{d}$	1997$^{e}$
i	86$^o$	81$^o$, 90$^o$	82.3$^o$$\pm$0.4	86.5$^o$$\pm$1.5	86.0$^o$$\pm$0.1
	(83$^o$-90$^o$)
q=$M_{2}$/$M_{1}$	0.1 - 0.2	0.1 - 0.2	0.226$\pm$0.013	0.125	0.13$\pm$0.01
$M_{1}$/$M_{\odot}$	33	3.5, 2.5	2.6	3.85	3.83$\pm$0.07
	(19-50)
$M_{2}$/$M_{\odot}$	2.9	0.7, 0.6	0.6	0.48	0.50$\pm$0.04
	(1.7-4.1)
$R_{1}$/$R_{\odot }$	6.8	2.8, 3.0	3.2	3.38	3.20$\pm$0.02
	(4.7-8.3)
$R_{2}$/$R_{\odot }$	33	20, 15	18.0	17.1	17.6$\pm$0.5
	(35-42)
A/$R_{\odot }$	158	77, 70	71	78.1$\pm$0.5	78.02$\pm$0.57
	(130-180)
$K_{1}$(km/sec)	17	17	17	11 $^{f}$	11.8$\pm$0.8
	(14-18)
$K_{2}$(km/sec)	190	84, 75 $^{g}$		91.2$\pm$1.9	90.7
	(160-220)
$T_{1}$(K)	7600		10300	7800
	(6700-8500)
$T_{2}$(K)	3850		3700	3651
	(3400-4150)
$^{a}$ Popper (1964), his Table 12
$^{b}$ Popper (1964), his Table 13
$^{c}$ Zo a (1992)
$^{d}$ Olson et al. (1995)
$^{e}$ Smak & Plavec (1997)
$^{f}$ Adopted value; Measured 7.7$\pm$1.9 km/sec
$^{g}$ Estimated to be 98 km/sec, Popper (1965)