Unveiling Accretion Disks - Physical Parameter Eclipse Mapping of Accretion Disks in Dwarf Novae

Contents

1 Preface

2 Accretion in Cataclysmic Variables
2.1 Cataclysmic Variables
2.2 The observations
2.2.1 The accretion disk
2.2.2 The white dwarf
2.2.3 The boundary layer
2.2.4 The bright spot and the gas stream
2.2.5 The red star
2.2.6 The outburst state
2.3 Theories of accretion phenomena
2.3.1 The viscosity
2.4 The outbursts
2.4.1 The Disk Instability model
2.4.2 The Mass Transfer Burst model
2.4.3 DI model vs. MTB model

3 IP Peg and HT Cas
3.1 The “puzzling” system IP Pegasi
3.1.1 A Dwarf Nova in the spotlight
3.1.2 The spectral appearance
3.1.3 The primary component
3.1.4 The secondary component: The red star
3.1.5 Setting the scene
3.1.6 The eclipses
3.1.7 Summary of the disk parameters
3.2 The “Rosetta Stone” HT Cassiopeiae
3.2.1 An unusual Dwarf Nova
3.2.2 The spectral appearance
3.2.3 The primary component
3.2.4 The secondary component: The red star
3.2.5 Setting the scene
3.2.6 The eclipses
3.2.7 Summary of the disk parameters

4 Eclipse Mapping
4.1 Tomography methods
4.2 Theory
4.2.1 The Entropy
4.2.2 The initial and the default image
4.2.3 Fit to the observations
4.2.4 Illustration of the MEM algorithm
4.3 Images of accretion disks
4.3.1 IP Peg & HT Cas
4.3.2 The general picture

5 Eclipse Mapping in Emission Lines
5.1 Emission Line Mapping of HT Cas
5.2 Discussion

6 Physical Parameter Eclipse Mapping
6.1 The new idea
6.2 Description of the method
6.2.1 Polar grid
6.2.2 Spherical white dwarf
6.2.3 White Dwarf spectra
6.2.4 The uneclipsed component
6.2.5 Use of a grid of model spectra
6.2.6 Use of passband response functions
6.3 Physical Models
6.3.1 Black body
6.3.2 A uniform LTE slab model

7 Application of the method to synthetic data
7.1 Test of the Temperature Mapping
7.2 Comparison with classical Eclipse Mapping
7.3 LTE-slab-version
7.3.1 Study of the model
7.3.2 Test of the LTE slab version
7.3.3 Discussion

8 Application of the method to real data
8.1 IP Peg on decline from outburst
8.1.1 Optically thick accretion disk
8.1.2 Discussion
8.1.3 Comparison to the results from Bobinger et al
8.1.4 Comparison to the superoutburst light curve from HT Cas .
8.2 HT Cas in quiescence
8.2.1 Optically thick disk in quiescence ?
8.2.2 Optically thin solution
8.2.3 Discussion
8.2.4 Fitting the white dwarf simultaneously
8.2.5 Fit with different distances
8.2.6 Comparison to Wood, Horne & Vennes 1992
8.3 Further improvements of the method

9 Discussion

List of astronomical constants

List of Figures

List of Tables

References

So, turning, twisting round and round, for all your life as times pass by, no string holds you and neither ground, the only reason: Dance or Die.

Chapter 1

Preface

When we see the stars flickering above us, so far away that we see no possibility to ever reach them ourselves, we might wonder how the astronomers have found out so much about the universe. The only information we get from the stars and galaxies is the light that we see, either with our naked eye or with telescopes of various kinds. But just this radiation contains a huge amount of information about the physical structure of the universe and physical processes occuring within.

Still, we have the wish to get a picture from the stars as if taken by a photo­grapher from close-by, for the last verification of the truth of our models. This work is aimed at getting such a closer look at a certain kind of astronomical object, the accretion disks1 in close interacting binaries, by producing spatially resolved images.

The following Chapter 2 introduces the reader to the objects under investigation, the cataclysmic variables. The most interesting phenomenon, the accretion of matter through a disk, is described by current theoretical models. Chapter 3 reviews two such systems, the dwarf novae IP Peg and HT Cas, summarizing the information available on them. For the understanding of the following Chapters, this information is not essential and the Sections may be skipped. Chapter 4 explains the Eclipse Mapping method, the algorithm on which my new method is based. Therefore, it is necessary to understand the principles of this classical method. In Chapter 5 I describe the application of this classical method to eclipse light curves in emission lines of the system HT Cas.

The idea of the Physical Parameter Eclipse Mapping method is presented in Chapter 6. In order to interpret real data with confidence, it is very important to understand the behaviour of the method when applied to synthetic data as described in Chapter 7. Finally, in Chapter 8 the application of the method to observations from the dwarf novae IP Pegasi in outburst and HT Cassiopeiae in quiescence is described. The last Chapter 9 summarizes the thesis and discusses the derived results.¹

Chapter 2

Accretion in Cataclysmic Variables

The accretion disks I have investigated are found in cataclysmic variables (CVs). In the following sections I will give a brief overview of the properties of this type of system, describe some of the accretion phenomena and review current theories of the physical processes responsible for them. Comprehensive reviews were given e.g. by la Dous (1989) or Warner (1995).

2.1 Cataclysmic Variables

It is generally accepted that cataclysmic variables are close binaries undergoing mass transfer. They contain a late (close to) main sequence star filling its Roche lobe, the secondary star, which loses mass and an accreting white dwarf. At the inner Lagrangian point where all forces balance, matter can easily transgress from the Roche lobe filling star into the Roche lobe of the white dwarf. Angular momentum conservation, Coriolis forces, and viscosity force the matter into quasi-circular orbits around the central object. By largely unknown processes angular momentum is carried outwards by a small fraction of matter causing the remaining matter to spiral inward towards the white dwarf. Since the matter flow from the secondary is (more or less) permanent, a luminous disk of matter is formed around the central object, the accretion disk.

In this accretion disk the matter loses gravitational energy which is (partly) transformed into radiation leading to a light source which is often brighter than the white dwarf. Only in eclipsing systems, the white dwarf can sometimes be distinguished by pronounced steps in the ingress and egress of the light curve.

The matter transmitted from the red dwarf star into the Roche-lobe of the white dwarf hits the accretion disk at its edge some way from the line combining the two stars in the direction of rotation. Here the kinetic energy of the stream matter in (almost) free fall is partly dissipated and a shock is produced which locally heats up the disk material and leads to a prominent emitting source, the bright spot².

Abbildung in dieser Leseprobe nicht enthalten

Figure 2.1: A cataclysmic variable with HT Cas’s parameters (see Section 3.2) in a sequence through eclipse. The picture is created with K. Horne’s programme cvmovie.

About a handful of systems in each hemisphere of the sky have an inclination large enough to show an eclipse of the white dwarf, the accretion disk, and the bright spot by the red dwarf star. The profile of the eclipse light curve is determined by the geometry and the distribution of light in these components. The white dwarf may cause a symmetric and steep ingress and egress, the accretion disk a symmetric but shallow eclipse, while the bright spot will cause an ingress at slightly retarded time in comparison to the white dwarf and a much more retarded egress. The aim of this work is to extract as much information from these eclipse profiles as possible in order to determine the physical conditions in the eclipsed components. Fig. 2.1 illustrates a sequence through eclipse.

The cataclysmic variables can be categorized into four main subgroups: (1) the dwarf novae (consisting of U Gem and SU UMa stars) which show outburst on time-scales of weeks, months or even years (with the SU UMa stars showing additionally superoutbursts), (2) the nova-like stars which seem to be in a permanent outburst state, (3) the Z Cam stars which alternate between outburst behaviour and standstills and (4) the magnetic dwarf novae where the formation of an accretion disk can be prevented by the strong magnetic field of the white dwarf or at least the inner part disrupted. My concern belongs to all cataclysmic variables which contain an accretion disk, i.e. all except the strongly magnetic systems.

2.2 The observations

This Section summarizes the main features observed in cataclysmic variables: the accretion disk, the white dwarf, the boundary layer, the bright spot and the red dwarf star. Detailed reviews on the dwarf novae IP Peg and HT Cas are given in chapter 3.1 and 3.2, respectively.

2.2.1 The accretion disk

The accretion disk presents itself in the spectrum as a dominating continuum with a maximum in the ultraviolet which falls monotonically towards longer wavelengths. This indicates high temperatures in the disk. The continuum can usually not be fitted with a single black body or power-law spectrum, indicating regions with a variety of temperatures. Superposed onto this continuum are prominent emission lines mainly from Hydrogen (Paschen, Balmer & Lyman series), Helium (He I+II), Calcium (Ca II, ...) and Iron (Fe II, ...) formed in optically thin parts of the accretion disk.

During an eclipse the accretion disk causes a shallow drop and rise in the in­tensity, the widths depending on the wavelength: in the UV, the inner disk with its higher temperatures is visible, producing a narrow eclipse. The longer wave­lengths are indicative of lower temperatures as found in the outer regions of the disk, therefore producing a wider eclipse.

2.2.2 The white dwarf

The white dwarf is often not visible in the optical spectrum, because of the prominent accretion disk, but in the ultraviolet region it may even dominate the spectrum, as seen e.g. in OY Car by Horne et al. (1994). In some cases the emission lines are superposed on broad absorption features which can be attributed to Stark broadened Balmer lines from the white dwarf photosphere (e.g. Lyo in OY Car, Horne et al. or in VW Hyi Sion et al. 1995b, Gänsicke & Beuermann 1996).

In the eclipse light curve the white dwarf often appears as a steep ingress and egress pair symmetric to phase 0³. From these light curves the size and colour of the white dwarf may be derived, leading its mass and temperature (e.g. Wood & Horne 1990, Wood, Horne & Vennes 1992).

2.2.3 The boundary layer

In the absence of a magnetic field, the accretion disk extends down to the surface of the central object. Between the surface of the central object and the disk the accreting matter has to slow down from Keplerian velocities (a few 1000 km/s) to the rotational velocity of the compact star. The latter is believed to be (much) smaller than the corresponding Keplerian velocity for the radius of the star. Usually, it is very difficult or impossible to measure the white dwarf rotational velocity, but Sion et al. (1995a) confirmed this assumption with a nsin¿ = 600 km/s for VW Hyi and Sion et al. (1994) with 150 km/s for U Gem. In case the central object possesses a magnetic field, the formation of an accretion disk around the central star is prevented leading either to an accretion ring or no disk at all. Instead, the matter is accreted onto the central object via an accretion stream following the magnetic field lines and raining onto the magnetic poles of the star. In case the magnetic and the rotational axes are not parallel, the system is observed as a pulsar (this effect is similar to the beaming of a light house with two beams separated by 180°).

Shakura & Sunyaev (1973) predict that the hard radiation from the central re­gions may be re-radiated by the outer parts, because the disk surface has a concave shape. Furthermore, the disk material in the outer parts may be heated and evap­orates leading to an auto-regulation of accretion.

As discussed later, in HT Cas (Wood et al. 1995) and Z Cha (van Teeseling 1997) an eclipse in X-rays has been observed indicating the boundary layer very close to the white dwarf as the X-ray source. In other high inclination systems the X-ray light curve shows no eclipse (UX UMa, Wood, Naylor & Marsh 1995 or OY Car, Naylor et al. 1988) which implies that the boundary layer may be obscured or lacking. The latter occurs, if the white dwarf is rotating rapidly. The X-rays observed then are rather emitted in the disk wind or a corona.

2.2.4 The bright spot and the gas stream

The prominent feature caused by the bright spot is the orbital hump, an increase in brightness just before eclipse. In the region where the gas stream hits the accretion disk, energy is released and radiated an-isotropically away in a direction more or less opposite to the white dwarf. Therefore we usually see the maximum of the bright spot emission at a phase just before the eclipse.

Spatially resolved studies (Rutten et al. 1994) revealed that the bright spot emits a spectrum indicating material that is hotter than in the surroundings, though not as hot as the material in the disk centre.

2.2.5 The red star

The contribution of the red dwarf star is especially visible in the infrared region by the stellar continuum with the typical absorption lines of a late main sequence star. Sometimes absorption lines can even be detected in the optical range, especially if the orbital period is larger than ~ 6h which implies a rather large secondary star.

In the infrared regime the secondary shows up by the ellipsoidal flux variation. This is attributed to its approximately ellipsoidal shape due to the adaptation of its surface to the Roche lobe. The variation in projected surface leads to a phase- dependent contribution of the secondary light with a period of twice the orbital period. Furthermore, the variation of the local gravity leads to flux minima at gravity minima.

In the light curve the secondary appears as the main contributor to the mid­eclipse light, especially in the infrared. In some systems it is partly eclipsed by the accretion disk at its superior conjunction (phase 0.5).

2.2.6 The outburst state

Dwarf novae undergo outbursts on time-scales of days (e.g. VI159 Ori, Patterson et al. 1995) to years (HT Cas, Wenzel 1987), with the majority of dwarf nova showing outbursts cycles of several weeks to a few months. Although the time span between outbursts may vary significantly, the average cycle length stays remarkably stable, even over about 100 years (SS Cyg, Warner 1987). The shape of the outburst light curve does not reproduce from one system to another, yet some characteristic properties may nevertheless be extracted, e.g. the rise time which depends on the quiescent magnitude, or the duration which for a given system shows a bimodality rather than a continuum.

The absolute magnitude in quiescence is correlated with the orbital period and the cycle length. The average absolute magnitudes lie for all dwarf novae (U Gem, SU UMa and Z Cam) below the absolute magnitude for a disk with a critical mass transfer rate .Merit- (Above this value the accretion disk is in steady state and no outbursts occur). Yet, the Z Cam stars always lie close to this limit.

The absolute magnitude at maximum light and the decay time from outburst are well correlated with the orbital period. This reflects a correlation of the size of the accretion disk with the size of the Roche lobe, since the latter is determined by the orbital period and the mass ratio.

The rise into outburst occurs usually first in the optical and up to several hours later in the ultraviolet. This behaviour might also extend to the infrared and the X-ray regimes. Such an UV delay indicates that the outburst starts at large disk radii and moves inwards: Optical or infrared wavelengths are most sensitive to temperatures of a few 1000 К which are found in the outer regions of the disk and as the outburst continues, the temperature increase progresses towards the hot inner disk which is the dominant source at maximum light.

The luminosity of the bright spot stays approximately constant during the erup­tion. This hints that the accretion stream is not connected to the cause of the outburst.

During outburst the radius of the disk increases considerably on a short time scale in early rise to the maximum. After maximum light it decreases slowly until the onset of the next outburst.

As mentioned above, the SU UMa dwarf novae undergo superoutbursts in be­tween the normal outbursts. The superoutbursts last longer, but do not raise the system to a higher luminosity. A necessary characteristic of these superoutbursts is the presence of superhumps, a periodic increase of the system brightness with a superhump period usually slightly larger than the orbital period. These superout­bursts seem to show a more stable cycle length than the normal ones.

2.3 Theories of accretion phenomena

Accretion is at the same time the most interesting part of a dwarf nova and other systems, like AGN or young stellar objects, and the most unknown phenomenon of these systems. Various authors have tried to explain the accretion disk on the basis of observations, as summarized above, and physical models.

Shakura & Sunyaev (1973) were among the first to present a comprehensive model of accretion disks. Up to now there is no good alternative to their so-called a- model. They intended to predict observations of accretion disks around black holes, but extrapolate their results also to systems containing neutron stars. Discounting the inner Sections of the disk, their results apply also to white dwarfs. I will briefly summarize their main ideas and results (as can be found in more detail in Frank, King & Raine 1992).

2.3.1 The viscosity

The main problem in accretion disk physics is the mechanism of angular momentum transport within the disk. Without any outward transport, the matter would not be able to reach the surface of the central object. By an unknown process the viscosity in the disk material is responsible for the dissipation of mechanical energy, leading hereby to the observed flux from the disk. Most probably magnetic fields and/or turbulence in the disk are responsible for the viscosity, i.e. the friction between two adjacent layers which again is necessary for the angular momentum transport.

Shakura & Sunyaev describe the viscosity using the analogy of laminar flows in replacing the molecular viscosity by the turbulent viscosity щ = where p is the density, vt the turbulent velocity, and l the size of the turbulent eddies. The tangential stress in the medium w = z/yc^/R, with vv the azimuthal (Keplerian) velocity and R the radius of the disk annulus, can then be expressed using the thin disk approximation for the disk thickness H = Rcs/vv as i.e. the tangential stress can be parametrisized as w = aP in terms of the perfect gas pressure P = pc2s with a describing the viscosity in the disk material.

Including the contribution of a magnetic field Ti to the angular momentum trans­port mechanism, Shakura & Sunyaev express a as in which the second, magnetic term contains the thermal energy \pc2s = §/°fy- + £r of the matter (er is the energy density of the matter and mp the mass of the particles). It is interesting to note that though Shakura & Sunyaev’s theory predicts a < 1, it does not depend on the exact definition of the viscosity coefficient a.

Usually, the o-Ansatz is now written in the form as follows from equation 2.1 and where the kinematic viscosity z/¿, is replaced by the effective viscosity z/¿, —> v/P- This Ansatz can be understood recalling that the size of the turbulent eddies l cannot be larger then the disk thickness H and with the reasonable assumption that the turbulent velocity vt is subsonic.

The main parameters in the disk are the effective temperature Тед (related to the observable flux via Fq = crT^ which is radiated away from the disk surface), the surface density (i.e. the mass in a column perpendicular to the orbital plane with a cross-section of 1 cm2) and the geometrical half-thickness of the disk H. We can assume that the accretion disks in dwarf novae are geometrically thin. This allows us to integrate the viscosity along the coordinate z perpendicular to the orbital plane: v. As mentioned above, the viscosity determines the emerging flux: in general F0 = ^GJ^3W Ev.

Abbildung in dieser Leseprobe nicht enthalten

For the stationary case with a constant mass accretion rate, conservation laws for the mass, angular momentum and the energy lead to an emerging flux Fq inde­pendent of the viscosity:

Abbildung in dieser Leseprobe nicht enthalten

Here Aiw and Rw are the mass and radius of the white dwarf, Ai the mass accretion rate and G the gravitational constant. If we express the flux Fq through the effective temperature, this yields the steady state radial temperature dependence For large radii, this leads to Te¡¡(R) oc R~ï. Hence, the effective temperature is solely dependent on the mass accretion rate A4 and the radius R. Equation 2.4 and 2.5 do not depend any more on the viscosity. This means on the one hand that we need not specify the viscosity and its origin. On the other hand, steady state disks do not allow us to determine the viscosity through observation. Such disks are believed to be present in nova-like stars and temporarily in dwarf novae during outburst.

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In the time-dependent case it is necessary to consider the addition of material through the gas stream inclusive of angular momentum and the outwards transport of angular momentum by tidal interaction which allows the matter to spiral down onto the white dwarf. For the surface density S this leads to (Smak 1984):

In order to be able to solve the time dependent case, Smak(1984) proposed in analogy to the stellar case with the existence of two boundary conditions another relation between the flux Fo and the surface density: Fo = /(£). This leads now to

Abbildung in dieser Leseprobe nicht enthalten

i.e. among the four parameters ТЬ,Тея, Ez/ and S only one is independent. Smak chooses the surface density.

Numerical integrations (e.g. Meyer & Meyer-Hoffmeister 1981) lead to hysteresis curves in the E-Teff-relations as displayed schematically in Fig. 2.2. Depending on the radius and the assumed value of a the hysteresis curve may shift along the surface density axis log S and slightly along logTefr without a major change in its shape. The hysteresis is caused by the ionization of hydrogen in the temperature range 6000 to 10000 К which in turn causes a very steep dependence of the opacity on the temperature (ос Г10) and to a steep dependence of the effective temperature Teff on the mid-plane temperature T. The location and shape of the bend В can also be affected by convection in the accretion disk which also causes a steep dependence of Teff on T since convection flattens the temperature gradient in the vertical disk structure (Cannizzo & Wheeler 1984).

In the case of deviation from thermal equilibrium in the disk, heating and cooling become important. According to calculations by Faulkner, Lin & Papaloizou (1983) and Smak (1983) the systems undergoing cooling are located left of the Е-Тед rela­tion, while systems with heating are on the right with dT^/dt = /(Е,Тед). This will be important for the disk instability as described in Section 2.4.1. Furthermore, the viscosity Ez/ then also depends on the (Е,Тед) pair and therefore on the radius.

According to equation 2.4 the total energy release is mainly determined by the accretion rate Ai. The mass transfer rate determines whether the disk is thermally stable or not (Smak 1983): A disk with a mass accretion rate above a critical value Alcrit is in a stable state and will not undergo the below described outbursts. This scenario is believed to occur in nova-likes. Systems with a lower mass accretion rate show the characteristic outburst behaviour of dwarf novae. This critical mass accretion rate is given by (Osaki 1996):

Abbildung in dieser Leseprobe nicht enthalten

where R¿ is the disk radius and TeffjCrit the critical effective temperature of an accre­tion disk (Osaki uses logTeffjCrit = 3.9 — 0.1 log Rd,w, where Rd,w is the disk radius in 1010 cm). For values below this no hot state exists.

2.4 The outbursts

At present there are two models which can explain the mechanism of dwarf nova outbursts: The disk instability (DI) model and the mass transfer burst (MTB) model. Although, most accretion disk investigators prefer the DI model, the MTB model cannot be ruled out completely and might be important on certain occasions, like in superoutbursts in SU UMa dwarf novae. Therefore, apart from the description of the DI model, I briefly summarize the mechanism behind the MTB model. Apart from the original literature, comprehensive reviews can be found in e.g. Frank, King & Raine (1992) or Osaki (1996).

2.4.1 The Disk Instability model

Paczynski & Schwarzenberg-Czerny (1980) suggested the disk instability model on the basis of observations of U Gem. Very simplified, it assumes that accretion on the white dwarf is low during quiescence while in outburst the accumulated matter in the disk is falling down on the white dwarf.

As seen in Section 2.3, for the effective temperature Тед and surface density £ a hysteresis curve is realized in the Тед-Е plane with a transition in the region of the ionization temperature of hydrogen. Outside of this region the relation T(£) is strictly monotonous. Qualitatively, the consequences can be described as follows (Osaki 1974): during quiescence, the mass accumulates in the outer regions of the disk, until an unknown instability process allows it to accrete onto the white dwarf. This scenario is supported by the fact that the observed ratio between the luminosity of the disk to the bright spot (~ |) is much larger than the ratio for steady accretion

TjdjQ, Smak 1984). With increasing mass transfer rate the surface density increases as well. This leads to enhanced viscous friction and heating in the disk.

In thermal equilibrium the disk temperature is held constant, because the heat produced in the inner disk (z = 0) is equal to the energy radiated away. A decrease in the viscosity of the material leads to a decrease in the temperature and the mass transport through the disk. However, since matter is streaming in from outside, the surface density increases which leads in turn to an increase in temperature (A —> В in Fig. 2.2).

This thermal equilibrium depends on the slow variation of opacity with temper­ature in this region. Yet, if the temperature increases to the point that the hydrogen starts to ionize, the opacity increases enormously (к oc T^). In this case, the disk is no longer capable of radiating enough energy away to lower the temperature. There­fore, the temperature increases further and the disk becomes thermally instable (B -C).

The instability lasts until the hydrogen is completely ionized. In reaching this stage, the function k(T) becomes much less steep and a new equilibrium state can be assumed (C). The difference between these two equilibrium states is a large difference in temperature which leads to a large difference in the amount of radiation produced.

In this high temperature state the viscosity is much higher than in the low temperature state which leads to a redistribution of matter in the disk and enhanced accretion onto the white dwarf, i.e. the surface density in the disk and with it the temperature decrease (C —> D). If the temperature sinks low enough, the hydrogen can recombine and this instability leads to a steep drop in temperature until the low temperature equilibrium is reached again (D —> A). This is then a new starting point for the next cycle.

The high temperature state can be identified with the outburst while the low temperature state represents quiescence. In quiescence, the mass accretion rate from the secondary is larger than the transport through the disk. This leads to the in­crease in surface density up to the critical value Smax when the steep temperature increase occurs. The system stays in the high state until a second critical surface density, Smin, is reached when the temperature drops again rapidly (see Fig. 2.2). The disk can then alternate between these two states, leading to alternating quies­cence and outburst states observed.

Models like those of Meyer & Meyer-Hoffmeister (1981) can reproduce a length for the outburst state which is about 10 times shorter than quiescence, as is ob­served in dwarf novae. The different shapes of outburst light curves are probably due to varying mass accretion rates and disk radii. However, apart from varying out­burst shapes, irregular cycle lengths and standstills it is not yet possible to exactly reproduce observed outburst light curve profiles.

2.4.1.1 The rise into outburst and the decline

The critical surface density as described in the previous Section varies with the radius from the white dwarf as £crit(-R) oc with £ > 1. This is especially important, if the accretion disk undergoes an outburst, since the critical values Smjn(iž) and Smax(-R) can be reached at different times.

Since the surface density in the disk in quiescence has to obey everywhere £(iž) < Ecrit(Ä), this restricts the surface density distribution within the disk to roughly E(R) oc In quiescence, when the disk is far from steady-state and the viscous time-scale is long, matter will accumulate in the outer regions of the disk. The outburst is initiated at that annulus where the critical surface density Emax is reached first. In this annulus, the temperature and the viscosity increase leading to a steep local gradient in the viscosity. In order to decrease this steep gradient, this annulus expands into the neighbouring disk regions thereby driving heating fronts both in­an d outwards (Cannizzo 1993 and references therein). The speed of this propagation front is roughly acs, i.e. the disk is rapidly transformed into the high state. In this high state, the viscous time-scale is much shorter, leading to an inward movement of the matter and radial dependence of the surface density of £(iž) oc where

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The location of the annulus, where the outburst starts, depends on the mass transfer rate. In systems with low transfer rates, the matter will already redistribute within the quiescence interval and the surface density reaches values close to the critical ones Emax(fž). The outburst can then start at intermediate disk radii, since only a small amount of inward drift will lead to an increase above the local Emax(fž) (inside-out outburst). The outburst observed will have a relatively slow rise, since the progression outwards is relatively slow. As we will see a wee bit later this leads to a symmetric shape of the outburst light curve.

In systems, where the transfer rate is large, the viscous time-scale is too long to lead to a significant redistribution of the disk material in the quiescence state so that the matter is piled up at large radii, where then the critical surface density is reached first (outside-in outburst). This scenario will lead to a fast rise (and asymmetric outburst profile), since the outburst progresses in the same direction (inwards) as the matter in those disk regions which are already in the high state.

On the contrary, the decline always proceeds from large to small radii. This is illustrated in Fig. 2.3: During outburst, the surface density is everywhere £(iž) > Smin(iž) with a distribution £(iž) oc R~^ where ( = | to 1, i.e. the surface density decreases for larger disk radii, while the critical surface density follows Smjn(iž) oc R^ with £ = 1, i.e. it rises with radius. Therefore, the critical value of the surface density Smin(iž) will be reached first at large radii, initiating the end of the outburst through cooling in these regions. A cooling front travels inward. In the outer cooler regions, the viscosity drops radically and the viscous time-scale assumes again large values, preventing much material to accrete onto the white dwarf (only about 10% to 20% of the disk material is actually accreted) and leading to a rather slow decline in the light curve. The decay time is similar to the rise time for the inside-out outburst.

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Figure 2.3: The evolution of the surface density during a decline from an outburst like theory predicts (scanned from Cannizzo 1993) for a bimodal a with acoid = 0.032, = 0.1 and A4 = 1О_9Л40/уг. The numbers denote a sequence in time as follows: 0d0, 2d9, 8d4, 9d9, lld4, 12d6 and 13d4.

2.4.2 The Mass Transfer Burst model

Bath (1973) proposed an alternative to the disk instability model Bath (1973): the mass transfer burst model. This model predicts the instabilities to occur in the atmosphere of the secondary, leading to variable mass transfer. In case of enhanced mass transfer from the secondary, the disk can collapse which leads to the observed outburst. The secondary loses enough matter during this kind of outburst to contract and the only form of mass transfer is maintained via stellar wind. Only after a certain recurrence time is the secondary capable of producing a new outburst.

2.4.3 DI model vs. MTB model

Though no final decision has been made, there are several observational facts sup­porting the DI model. First, it can easily explain the occurances of various types of systems by different mass transfer rates, with Nova-like stars having a mass transfer rate above the critical value, Z Cam stars close to, and dwarf novae below. AM Her stars also fit into this model, since in these systems the white dwarf is preventing the formation of an accretion disk by its magnetic field and consequently, no outbursts are observed. Furthermore, the bright spot has a constant luminosity through the outburst, implying that the accretion stream stays constant, contrary to the predic­tion of the MTB. As seen above, the bimodal outburst behaviour with symmetric and asymmetric light curve profiles can be understood in terms of inside-out and outside-in outbursts. Observations of the radius changes agree with the DI model which predicts a gradual decrease through the quiescence state, while the MTB model proposes a rapid decrease at the end of an outburst (see Section 3.1.3.1).

However, the super outbursts are not understood yet, and it might be that a varying mass transfer rate is at least partly responsible for them.

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Chapter 3

IP Peg and HT Cas

This Chapter gives reviews on the dwarf novae IP Peg and HT Cas. It should be noted that the knowledge of the given information is not essential for the under­standing of the following Chapters. However, for comparison I will later refer to a few of these Sections.

3.1 The “puzzling” system IP Pegasi

3.1.1 A Dwarf Nova in the spotlight

The mostly observed Dwarf Nova is still IP Peg, because it is one of the few eclipsing systems and with a visual magnitude my of 14m in quiescence relatively bright com­pared to other dwarf novae. Other dwarf novae which show as well both an eclipse of the white dwarf and the bright spot, are HT Cas (see next Section 3.2), DV UMa, V2051 Oph and HS 1804+6753 and maybe S 10932 in the northern hemisphere and OY Car, V2051 Oph, Z Cha, VZ Sci in the south.

Though IP Peg is relatively bright, it was discovered only in 1981 by Lipovetskij & Stepanyan (1981) as a possible U Gem variable (registered as SYS 2549) with variation llm to 15m and a spectral type OB. Goranskij et al. (1985) discovered the eclipses and the strong orbital hump caused by the bright spot, later confirmed by many other authors. Since IP Peg is a U Gem dwarf nova it shows only normal outbursts. The orbital period is 3.8h = 3h48m (Goranskij et al. 1985) and lies therefore above the period gap between approximately 2h and 3^.

Since then several optical (Goranskij, Lyutyi & Shugarov 1985, Wood & Craw­ford 1986, Wood et al. 1989b, Wolf et al. 1993, Harlaftis et al. 1994) and infrared (Szkody & Mateo 1986a, Martin, Jones & Smith 1987) photometric as well as spec­troscopic (Martin, Jones & Smith 1987 (LR), Szkody 1987 (UV), Marsh 1988, Martin et al. 1989, Hessman 1989, Piché & Szkody 1989, Marsh & Horne 1990, Harlaftis et al. 1994, Dhillon & Rutten 1995 (spectropolarimetry)) studies of this system have been performed. In the next Sections their findings are reviewed.

In spite of the many observations, IP Peg is still a puzzling system (Piché & Szkody 1989). Some observations are peculiar for this object or at least it shows extreme features which might be connected with the fact that IP Peg was for more than a decade1 the dwarf nova with the longest orbital period known to show eclipses of the white dwarf and the bright spot and the only one above the period gap.

The justification of the title of this Section, though, goes further, because when­ever somebody analyses new observations, some new and unexpected findings are revealed. For example, some of the authors found evidence for confirmation of ei­ther the mass transfer instability (MTB) or the disk instability (DI) to explain the outbursts. Though, the DI model is preferred by many accretion disk investigators, the MTB can not yet be completely banished.

Fig. 3.1 shows a sketch of IP Peg assuming a disk radius of 0.49RL1. Note the difference to HT Cas (Fig. 3.4).

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3.1.2 The spectral appearance

3.1.2.1 The optical spectrum

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The optical spectra show the typical Balmer series in emission with a flat Balmer decrement. In Marsh’s (1988) quiescence spectra (wavelength range 3500 to 5400 A, see Fig. 3.2), up to H13 can be distinguished. Additionally, Marsh sees Ca II H+K, He I AA 4026, 4471, 4921, 5016 and Fe II AA 4924, 5018, 5169 in emission. Unfortunately, most of these non-hydrogen lines are blended, making it more difficult

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Figure 3.2: Spectrum of IP Peg (scanned from Marsh 1988).

to analyse their line profiles. However, all lines seen by Marsh are clearly double- peaked, a usual appearance for disk emission in high inclination systems, reflecting the orbital motion of the disk material.

Dhillon & Rutten (1995) see already in their optical spectra (3800 to 6700 Á) a contribution from the secondary in the form of ТЮ and Na I D absorption lines. Such absorption features are much more prominent in the infrared.

At all wavelengths, the orbital hump is dominating the phase resolved behaviour, as shown for the optical (B and V) by Wood & Crawford (1986), for the UV by Szkody (1987). In the lines, the orbital hump is much less pronounced than in the continuum as seen in Fig. 2 of Marsh (1988).

3.1.2.2 The shape of the emission lines

The shape of the emission lines reflect the physical condition in the accretion disk. The velocity field including possible additional (turbulent) motion, the emission distribution across the disk, inner and outer disk radii, limb darkening, the optical depth of the material for this line and absorption as well as the instrumental profile determine the separation and width of the peaks, the shape and extension of the line wings and the central depletion (see also Stover 1981). In the description of IP Peg’s line profiles, this connection will become clear (see also Marsh & Horne 1990):

The wings of the emission lines of IP Peg are narrow and do not show not especially broad or shallow line wings. Since these wings are produced by high (radial) velocity material usually present in the inner part of the disk, close to the white dwarf. The absence of extended wings indicates therefore an absence of high velocities, due to a real absence e.g. in the form of a hole in the disk or an apparent one e.g. in the form of obscuration of the inner parts. Possibly, the explanation lies in the secrets of the line formation mechanism, since the slope of the wings is also controlled by the distribution of emissivity.

The central cores of the emission lines are not especially deep. The line centres are produced by low (radial) velocity material, like the parts of the disk closest to and farthest from the observer as well as (the centre of) the white dwarf. Since those parts of the disk vary with the orbital motion they are not likely to cause such a permanent feature in the disk, unless it can be explained in general terms of the whole line profile. Therefore, the absorption-line producing white dwarf is to be blamed, and indeed, according to photometric light curves does the white dwarf not contributed more than 10% of the light (Wood & Crawford 1986), while in other dwarf nova it contributes significantly more (60% in Z Cha).

The separation of the peaks is an indicator for the size of the disk: In the parts of the disk in quadrature the radial velocity is equal to the disk velocity v sin i with a slow variation with the sine of the azimuthal angle ip (where ip = 0 for the line from the white dwarf to the observer) making the geometrical area contributing to this radial velocity bin rather large. Assuming Kepler velocities and the mass of the white dwarf, the peaks can therefore be used to calculate the the radius of the disk.

Finally, the width of the peaks is an indicator of the non-Keplerian, i.e. turbulent motion on the disk.

The Ca II lines in IP Peg are possibly narrower than the Balmer lines. According to the above written, this means that the Ca II emission is more concentrated in the cooler outer parts of the disk.

In many systems, a blue/red asymmetry of the emission peaks is visible during the whole or large parts of the orbit (AC Cnc: Schlegel, Kaitchuck & Honeycutt, 1984, Z Cha: Marsh, Horne, Shipman 1987). Though Marsh (1988) and Dhillon & Rutten (1995) see a 10% asymmetry in their averaged spectra, Marsh points out that it is prominent only just before eclipse. This additional emission in the blue peak is attributed to emission from the bright spot. With a 10% asymmetry, the contribution to the line from the bright spot is much less prominent compared to the continuum.

3.1.2.3 The behaviour of the emission lines

In trailed spectra (e.g. Marsh 1988), the orbital motion is clearly visible as a si­nusoidal motion of the whole line with orbital phase. Furthermore, an additional sinusoidal component appears around phase 0.25 called S-wave and attributed to emission from the ’nose’ of the secondary.

For the behavior of the emission lines during eclipse, see Section 3.1.6.

3.1.2.4 The spectrum during outburst

During outburst maximum (Piché & Szkody 1989, Marsh & Horne 1990 three nights later during the same outburst) and on decline (Hessman 1989) IP Peg is unusual for showing an emission spectrum instead of the gradual transition of quiescent emission lines into absorption at the onset of an outburst and the development of slowly growing emission cores on the decline which lead back to the quiescent emission line spectrum (e.g. VY Aqr: Augusteijn 1994, IR Gem: Feinswog, Szkody & Garnavich 1988, SS Cyg: Clarke, Bowyer & Capel 1984). This might be connected with the high inclination angle which causes us to see the emission lines formed in a layer above the accretion disk without the usual background light from the accretion disk itself. Marsh & Horne suggest, foreshortening and limb darkening could suppress the continuum and absorption lines usually seen in the outburst disk. A similar situation is seen in all known eclipsing (i.e. high inclination) dwarf novae which were observed spectroscopically during outburst: V2051 Oph (i = 81°; Warner & O’Donoghue 1987), Z Cha (i = 82°; Honey et al. 1988) and EX Hya (i = 78°; Hellier et al. 1989; inclination angles i from Ritter 1990).

Piché & Szkody note that the Balmer decrement is very much flatter during the outburst than at quiescence and the equivalent widths are much smaller (factor 6 to 9) with a gradual increase during decline (cf. Marsh & Horne and Hessman).

In addition to the unusual Balmer emission lines (and slightly enhanced He I + II lines) a strong blend of high excitation lines, in particular N III A 4640, C III A 4650 blended with He II A 4686 and He I A 4722 is seen. All these emission lines are double peaked. In Piché & Szkody’s and Marsh & Horne’s outburst spectra, the He II emission at 4686 Ais almost twice as strong as H/3, while in Hessman’s decline spectra this line is less strong than H/3, however, due to the blend with nearby lines difficult to distinguish. Piché & Szkody see also faint emission from He I 4471 A, C II 4267 Á, He II 4200 Á and He I 4026 Á.

The outburst spectra all show a narrow chromospheric component in the Balmer lines (though not in the He II 4686 A line) attributed to emission from the secondary which is irradiated by the boundary layer during early outburst (Piché & Szkody, Marsh & Horne) and later by the bright spot (Marsh & Horne, Hessman).

The He II 4686 A emission as seen by Piché & Szkody and Marsh & Horne is broad, but not double-peaked, suggesting an outflowing wind perpendicular to the disk plane. Since the line is significantly eclipsed, the emitting region cannot be extended far above the disk plane. However, Piché & Szkody see an additional narrow component at rest velocity originating in a circumsystem shell and implying a higher mass flow at the beginning of the outburst.

The extension of the line wings of the Balmer and He II 4686 A line show a gradual increase during outburst until quiescence. Piché & Szkody suggest, the inner edge of the disk moves gradually inward during outburst, supporting the disk instability model or just a concentration of the emission farther out which could be caused by an obscuring photosphere, as proposed by Bobinger et al. (1997). The peak-to-peak separation of H/3 is smaller than in later outburst spectra or during quiescence which is caused by a larger disk (see also Section 3.1.3.1).

Both, Piché & Szkody and Marsh & Horne see an asymmetry in the He II line profile implying an asymmetrical distribution of the emitting region in the disk. This could be due to distortions especially in the onset of the outburst. Spiral structures could also explain the higher Id-velocities derived from the emission lines by Piché & Szkody (Steeghs 1997, private communication).

3.1.3 The primary component

3.1.3.1 The accretion disk

Goranskij, Lyutyi & Shugarov (1985) first observed IP Peg photometrically and report a dark, extended ring around the accretion disk in quiescence which let them conclude the confirmation of the MTB model.

Later, Wood & Crawford (1986) observed 6 eclipses in В and V. From the white dwarf eclipse width Аф (which is hampered by the simultaneous ingress of the white dwarf and the bright spot) and a comparison between derived and the observed К-velocity of the white dwarf they conclude that the white dwarf must have been surrounded by an extended boundary layer, at least extending to two white dwarf radii. This boundary layer is relatively massive with a mass between 0.3 and 0.9 Л40 and could be caused by the previous outburst, 5 days before their first observations.

The radius of the accretion disk changes with time after outburst, as Wood et al. (1989b) found by measuring the bright spot egress in spectra taken at various epochs. Wolf et al. (1993) continued these investigation and found a exponential decrease. However, they could not distinguish, whether the decrease is correlated with the time after outburst or the outburst phase, but the decrease reproduces quite well in various outburst cycles (see especially Fig. 6 of Wolf et al.). The changes are larger than observed in Z Cha (Wood et al. 1986, O’Donoghue 1986) or OY Car (Wood et al. 1989a) but similar in behaviour. As Wolf et al. mention this slow, but steady decrease until the next outburst supports the DI rather than the MTB model following comparisons by Ichikawa & Osaki (1992).

In quiescence, the disk radius is ižmin — 0A9RL1 = 0.27a, not much larger than the radius

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needed for a total eclipse (Marsh 1988) which yields for IP Peg Rtot = 0.47RL1 = 0.26a. Just after an outburst the disk reaches a radius i?max > 0.64RL1 = 0.35a, still well below the tidal radius Rt = 0.71RL1 (for IP Peg), a maximal radius at which tidal torques efficiently extract angular momentum from the disk (Paczynski 1977).

From the radius changes Anderson (1988) proposed to derive a lower limit for the mass in the outer accretion disk Aid

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where Rls is the Lubow-Shu radius of a particle in circular orbit around the white dwarf which carries the same specific angular momentum as the material in the gas stream and at the inner Lagrangian point (Lubow & Shu 1975). For IP Peg RLS = 0.19i?!,!, i?max an(i 4?min are maximum and minimum observed radii and At is the time span for the radius decrease. Using Marsh’s (1988) mass transfer rate (see Section 3.1.3.3) of Ài = 2.2 • 10_1ОЛ4@/уг, Wolf et al. found Aid = 0.9 • Ю_1ОЛ40.

Marsh & Horne (1990) and Harlaftis et al. (1994) performed Doppler Mapping of the emission lines in quiescence (Ho, H/3, H7) and outburst (H/3, H7, He II 4686 A). Both groups of authors detected in their quiescent images apart from a symmetric part three regions with enhanced emission, one of them identified with the gas stream. The azimuth of the other regions changed between both observing runs. During outburst, the picture changes completely. Here the main emission in the Balmer lines originates on the surface of the secondary, while He II emission is absent. Harlaftis et al. already observe emission from the secondary shortly (7 days) before an outburst. The accretion disk appears as a lopsided ring in all lines, but especially pronounced in the He II line. Though the exact transformation between velocities of the disk material and spatial coordinates is unknown, there is no doubt that this asymmetric structure transforms into the image in spatial coordinates, indicating some kind of spiral structure as seen more clearly in early outburst (Steeghs 1997, private communication). Marsh & Horne can explain the mechanism to produce the He II emission as photo-ionization by the boundary layer.

From the He II radial intensity profile derived under the assumption of Keplerian orbits, Marsh & Horne (1990) suggest a geometrical disk structure that follows a constant scale height to radius ratio Н/R ~ const in the inner third of the disk and then blows up according to H/R oc R2. This high rim of the accretion disk is able to shield the He Lyman continuum and only part of the hydrogen Lyman continuum from reaching the secondary, therefore explaining the outburst Doppler images.

From the presence of Ha emission on the secondary already several days before the outburst, Harlaftis et al. (1994) propose the secondary as the location of activity and therefore supporting the mass transfer instability model to explain the outburst. However, no emission is seen from the gas stream, so that this conclusion is rather a suggestion.

Bobinger et al. (1997) performed Eclipse Mapping during decline from outburst. Their derived brightness temperature distributions are surprisingly flat with bright­ness temperature values between 7000 and 8500 К in the centre. It is not clear, what causes the flat profile. Bobinger et al. exclude a hole in the disk or a high disk rim which would obscure the inner part. An explanation could be either a flared disk or an optically thick wind sphere, obscuring the central parts. The lack of any white dwarf in- or egress supports this idea, though could be blamed on a low temperature white dwarf. The disk edge appears to move inward with a speed of ~ 800 m/s. This leads them to a rough estimate of the viscosity parameter а = 0.10 — 0.12. The data used by Bobinger et al. are re-analysed in Section 8.1.1.

Ha and He I AA 5876, 6678 were found to be in emission at all disk radii (Haswell, Baptista & Thomas, 1994), not like in UX UMa where a transition from absorption in inner disk to emission in outer disk is observed (Rutten et al. 1994, Baptista et al. 1995). Furthermore, the emission line eclipse maps show asymmetric structures which could be partly interpreted as due to a rotational disturbance in the disk (Baptista 1997, private communication).

3.1.3.2 The bright spot

The prominent orbital hump was already reported from the first observations by Goranskij et al. 1985. Wood & Crawford’s (1986) observations show a variation of about 3 mJy in both В and V or ~ 1™5 (Goranskij et al. 1985) between orbital phases 0.5 and 0.9. In no other dwarf nova such a large hump is visible and it might well be due to the high inclination of IP Peg. The apparition as a hump implies highly anisotropic radiation from the bright spot situated on the rim of the disk, so that in a high inclination system at a certain phase it beams into the direction of the observer. However, the high inclination cannot be alone responsible for the unusually strong orbital hump, because other high inclination systems, like V2051 Oph (Warner & O’Donoghue 1987) and OY Car (Wood et al. 1989a) do not show such a prominent hump, while U Gem with a lower inclination angle (i = 70°, Ritter 1990) does show a strong orbital modulation caused by the bright spot (e.g. Zhang & Robinson 1987).

Szkody (1987) fitted a black body distribution to her IUE spectra and obtained a quite high temperature for the bright spot, or here actually hot spot, of T = 20000K ±1000 К which however underestimates the optical flux (at a different epoch). Marsh (1988) fitted the spectrum of the orbital hump in the optical range with a black body and spectrum of a supergiant. He obtains a much lower temperature of about 12000 K. This latter value is similar to bright spot temperatures of other dwarf novae which lie in the range 12000 to 14000 K (U Gem (Panek & Holm 1984), VW Hyi (Mateo & Szkody 1984), Z Cam (Szkody & Mateo 1986b)). The size of the bright spot calculated by Szkody, though, is comparable with that one of U Gem (Warner & Nather 1971): they calculate a radius of 1.6 • 109 cm.

Marsh concludes from the extraordinary brightness of the bright spot that the quiescence disk is not in a steady state. A crude estimation predicts a much (seven times) higher disk flux density than observed. The higher temperature derived by Szkody would not yield any improvement. Therefore, the mass accretion rate into the bright spot must be much higher than the flow rate through the disk.

Wolf et al. (1993) observed an asymmetry in the bright spot ingress, leading to their assumption that the bright spot consists of a bright head and a tail of cooling gas along the disk edge.

3.1.3.3 The accretion rate

The accretion rate was estimated by Szkody (1987) to be just below the critical accretion rate A4Crit above which no outbursts occur to explain the fact that though PG 1030±590 compared to IP Peg has very similar disk and white dwarf parameters, it is a nova-like system, showing no outbursts. According to Shafter, Wheeler & Cannizzo (1986) this critical mass accretion rate depends only on the orbital period and the white dwarf mass. For IP Peg (with an updated white dwarf mass) their Fig. 2 gives Wicrit — 6 • 10_9A4@/yr. Marsh & Horne (1990) give a range for the mass accretion rate for their outburst observations of 1.5 — 6 • 10_9A4@/yr, consistent with Szkody’s prediction.

Marsh (1988) calculated a mass accretion rate of A4 = 2.2 • 10_loA4@/yr into the bright spot at quiescence from a fit to the spectrum of the bright spot. However, his calculation was inconsistent with the observed flux from the accretion disk itself, suggesting that the mass transfer rate through the disk is lower than into the bright spot and therefore that the disk is far from steady state.

3.1.4 The secondary component: The red star

Szkody & Mateo (1986a) and later Martin, Jones & Smith (1987) detected ellipsoidal variations in their infrared observations, a sign of the secondary with its aspherical geometry. Due to its adaptation to the Roche-lobe, the surface area of the star varies with orbital phase, leading to brightness variation with half the orbital period showing maxima at phases 0.25 and 0.75. These variations have been observed in various systems, like OY Car, Z Cha, CW Mon, X Leo or AF Cam. The amplitude of these variation in IP Peg was found to be 0™2 in К with an average magnitude of 11™9 (Szkody & Mateo), the total range in the Gunn Z-band (0.93 /mi) 2.7 mJy with an average of about 10 mJy (Martin, Jones & Smith)

The К-band observation also revealed an eclipse of the secondary by the accretion disk (which itself contributes much less to the IR than to the optical leading to a much less pronounced primary eclipse depth).

The period-radius relation of Warner (1976) leads to a radius of the secondary of 0.42 Re and absolute К magnitude of the secondary of ~ 12m (Szkody & Mateo). Up to now, no author found evidence that the secondary is not a main sequence star.

The spectral type of the secondary was first estimated by Szkody & Mateo to M4 and later confirmed by Martin, Jones, Smith (1987) to be closer to M4.5 than М3.5. Their spectra taken in the IR show absorption features typical for a M dwarf.

3.1.5 Setting the scene

3.1.5.1 The radial velocity curves

The easiest and most straightforward way to calculate the system geometry param­eters, like the mass ratio and the inclination angle would be by taking advantage of the system being a binary. Usually, masses derived from binary systems are the most reliable ones and other methods often depend on these by the use of prototypes in binary systems.

However, in cataclysmic variables, the radial velocities (RV) of the components are not easily measured. The white dwarf is not visible in the spectrum and RV curves have to be calculated from the lines produced in the accretion disk. Often, the brightness distribution is distorted leading to the typical phase lag of the RV curve of the emission line towards the phase zero or the RV curve of the absorption line component (e.g. Vrielmann 1993). The only way to find a reliable RV curve of the primary component is to measure the line wings produced by material in the disk centre close to the white dwarf.

Systems showing the spectrum of the secondary, especially in the infrared, seem to have at least a reliable indicator for the orbital motion of the late type star. However, the line flux distribution is often not symmetric across the surface of the star due to illumination from the bright spot, the boundary layer and/or the white dwarf. This leads to a depletion or enhancement of lines, depending on the physical process of line formation and illumination (e.g. photo-ionization or heating) and conditions in the stellar atmosphere (e.g. the temperature, abundance). This may lead to a measured radial velocity amplitude which is larger or lower than the true amplitude for the centre of mass of the secondary by a significant amount (Marsh (1988) gives a value of up to 17%).

Various attempts have been made to determine the RV curve of the white dwarf. Marsh (1988) used the double Gaussian method (Schneider & Young 1980) with a set of different separations and determined the (Kx,Ky) pair for each of these separations to interpolate to the white dwarf velocity at (0, — Kw)- This methods yields Kw = 175 ± 15 km/s. Hessman (1989) finds a much lower 17-amplitude of Кw = 118 ± 10 km/s using also the double Gaussian method and a symmet­ric polynomial fit to measure the radial velocity variation of the line wings which are then evaluated via so-called diagnostic diagrams. Since his observations were performed just after an outburst and his (Kx,Ky) diagrams show no clear trend, I prefer Marsh’s (1988) Kw value.

The RV curve of the secondary has been derived by Martin, Jones & Smith (1987) and Martin et ai. (1989) using their IR spectra.

Martin, Jones & Smith (1987) performed cross-correlations of the absorption spectrum with a standard star and found a sinusoidal radial velocity curve (J7a¡,s = 331 ± 7 km) with an significant apparent eccentricity of about 0.075. They blame contamination of the spectra by emission lines from the disk, the gas stream or the bright spot (О I emission around 7770 A). Marsh (1988) modeled the irradiation of the secondary to correct their value J7a¡,s. He excluded heating, because the observations are only compatible with a reduction of the Na I line strength, caused by a more efficient mechanism, possibly by direct ionization. He obtains a correction factor of 1.05 to 1.11 to account for the observed eccentricity, leading to a corrected 17-amplitude Kr of 305 ± 15 km/s.

A further constraint using the eclipse of the emission lines leads to even smaller values of the 17-amplitude of Kr = 293 ± 11 km/s. This agrees well with a mea­surement by Martin et ai. (1989), who measured the RV curve of the chromospheric component of the He I 7065 A and found Kfje = 293±3 km/s. However, He emission is only expected from the secondary surface near the inner Lagrangian point which implies that Marsh over-corrected the secondary 17-amplitude. Assuming only ion­ization of the sodium atoms on the surface of the secondary (and ruling out heating) by irradiation from the primary component, Martin et ai. (1989) found a corrected value of Kr = 298 ± 8 km/s, still in agreement with Marsh’s value.

The analysis of radial velocity curves also leads to the orbital period. However, since they usually show a non-negligible phase shift, the phase zero cannot always be easily derived and in eclipsing systems the ephemeris derived from the eclipse light curve is preferred (see Section 3.1.6).

3.1.5.2 The geometry, masses and radii of the components

The eclipse constraint of Marsh (1988) leads to the 17-amplitude of the white dwarf of Kw = 170 ± 14 km/s. These lead to a mass ratio of q = Mr/Aiw = 0.58 ± 0.06.

[...]

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¹ Though throughout the thesis I stick to the British spelling rather than the American, I make one exception for pure symmetry: The word disk.

² Reading the literature in the held of cataclysmic variables one stumbles over the term used for this spot. Some authors call it hot spot while more recently is is rather called bright spot. Since the temperature of this spot is not necessarily very high I also keep to the expression bright spot. "Usually phase 0 is defined by the white dwarf centre as the superior conjunction of this star.

⁴ until the discovery of HS1804+6753 (Billington, Marsh, Dhillon 1996)

⁵ U Gem has also a large orbital period, but shows only an eclipse of the bright spot (Warner & Nather 1971, Smak 1971).