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An optical study of the flickering, flaring, and oscillations in the unusual cataclysmic variable AE Aquarii
Welsh, William F., Jr., Ph.D.
The Ohio State University, 1993
UMI 300 N. Zeeb Rd. Ann Arbor, MI 48106
An O ptical Study of the Flickering, Flaring, and O scillations in the Unusual Cataclysmic Variable AE AQUARII
dissertation
Presented in Partial Fulfillment of the Requirements for
the Degree Doctor of Philosophy in the Graduate
School of The Ohio State University
By
William F. Welsh, Jr.,
The Ohio State University
1993
Dissertation Committee: Approved by
Dr. Bradley M. Peterson
Dr. Gerald Newsom A d viser Dr. Richard Pogge Department of Astronomy To
My Parents
and
Natasha
ii A cknowledgements
Unquestionably, the person I am most indebted to is Keith Horne. Many years ago (or so it seems) Keith’s encouragement and enthusiasm prompted me to leave my research assistant position at Space Telescope and to head off to graduate school.
That in itself would have earned much gratitude. But as fate would have it, three years later I returned to Keith, and this time he was my thesis advisor. For his moral
(and monetary!) support, expert guidance, attention to detail, torrents of ideas, and infectious enthusiasm, I am greatly in his debt.
I gratefully acknowledge the support of the Space Telescope Science Institute, and specifically the Graduate Student Program, without which my apprenticeship under
Keith would not have been possible. I must also thank the OSU Astronomy Depart ment for their support and for allowing me leave to study under Keith. In particular,
I wish to bestow many, many thanks to Brad Peterson, whose moral support saved me during my darkest hours, and whose insight, guidance and knowledge were invalu able during my tours of the AGN universe. I also want to acknowledge the guidance provided to me by Bob Wing and Ron Kaitchuck.
I want to acknowledge the enormous efforts of Richard Gomer, whose technical expertise was crucial to obtaining the finest quality data available on AE Aquarii.
Richard is the “hero behind the scenes” for the Cable Experiment. I wish to thank Bev Oke for donating time on the 200” and the MCSP. Janet Mattei must also be thanked for providing the AAVSO observations of AE Aquarii.
There are dozens of people who have played important roles over the years and should be mentioned. These include (but are not limited to) the grad students at
OSU (especially Mark, Mike F., Andy, Mike 0., and of course, Pedro), my many friends at Space Telescope (including Christina, Jen, George, and the rest of the
“Tribal Intergalactic Psycho Climbers”), “Team CV” (Keith, Phil, Carole, Fu-Hua,
Kirk, Mike, and Raymundo) and the folks at the Mansion Theater (John, Jane, Skizz,
Joe, Todd, Kyle, April, Jeanine, Poli, Marlin, Laura, Ed...).
To my brother Jim for showing me that it is possible to be excellent and yet crazy, to my sister Anna for showing me that is is possible to be crazy and yet excellent, and to my parents for their love and support (and patience during my junior scientist days) - I thank you, and cannot overemphasize how important you’ve been to me.
To Natasha, I thank you so much just for being Natasha.
And lastly, a special thanks to Lee, whose influence shall never be forgotten. V it a
04-March-1964 ...... born in New York; U.S. citizen, married
1982-1986 ...... B.S. Physics and Astronomy State University of New York at Stony Brook Stony Brook, New York USA.
1986-1987 ...... Research Assistant, Space Telescope Science Institute Baltimore, MD USA
1987-1990 ...... Teaching and Research Assistant, Department of Astron omy Ohio State University, Columbus, Ohio USA
1990 ...... Complex Systems Summer School, Sante Fe Institute, Santa Fe, NM, USA
1990-present ...... Graduate Student Fellow Space Telescope Science Institute, Baltimore, MD USA
v P ublications
“On the Location of the Oscillations in AE Aquarii” Welsh, W.F., Horne, K., and Gomer, R. 1993 ApJ Letters 410, L39.
“On the Mass of Nova DQ Her (1934)” Horne, K., Welsh, W .F., and Wade, R.A. 1993 ApJ 410, 357.
“Optical Spectrophotometry of Oscillations and Flickering in AE Aquarii Welsh, W.F., Horne, K., and Oke, J.B. 1993 ApJ 406, 229.
“Anisotropic Line Emission and the Geometry of the Broad-Line Region Active Galactic Nuclei”, Ferland, G.J., Peterson, B.M., Horne, K., Welsh, W .F., and Nahar, S.N. 1992 ApJ 387, 95.
“Echo Images of Broad-Line Regions in Active Galactic Nuclei” Welsh, W .F., and Horne, K. 1991 ApJ 379, 586.
“Echo Mapping of Broad H/? Emission in NGC 5548” Horne, K., Welsh, W.F., and Peterson, B.M. 1991 ApJ Letters 367, L5.
F ie l d s o f S t u d y
Major Field: Astronomy
Studies in: Cataclysmic Variables - Dr. Keith Horne
Active Galactic Nuclei - Dr. Bradley M. Peterson T a b l e o f C o n t e n t s
DEDICATION ...... ii
ACKNOWLEDGEMENTS ...... iii
VITA ...... v
LIST OF FIGURES ...... x
LIST OF T A B L E S ...... xiii
CHAPTER PAGE
I INTRODUCTION ...... 1
1.1 Overview ...... 1 1.1.1 Cataclysmic Variables and the Big P ic tu re ...... 7 1.1.2 Flickering and Flaring ...... 9 1.1.3 O scillations ...... 12 1.2 AE A q u a rii...... 15
II OSCILLATIONS AND FLICKERING IN AE AQUARII...... 24
2.1 Introduction ...... 24 2.2 Optical Spectrophotometry of Oscillations and Flickering in AE Aquarii 25 2.2.1 O b se rv atio n s ...... 27 2.2.2 Spectra ...... 31 2.2.3 Light Curves ...... 31 2.2.4 Power Density Spectra ...... 37 2.2.5 The Secondary Star Spectrum ...... 39 2.2.6 The Disk Spectrum ...... 45 2.2.7 The Flare Spectrum ...... 49 2.2.8 The Oscillation Spectrum ...... 54
vii 2.2.9 Conclusions ...... 61 2.3 The Experimental Control ...... 63 2.4 Additional Results on Flickering . . 67 2.5 Using the Mg I b-complex as a Diagnostic of the Secondary Star . 75 2.6 Brief Comments on the Inclination and Secondary Star Size .... 76
III ON THE LOCATION OF THE OSCILLATIONS IN AE AQUARII . . 79
3.1 On the Location of the Oscillations in AE Aquarii ...... 80 3.1.1 Introduction ...... 80 3.1.2 Observations ...... 82 3.1.3 The Orbital Motion of the Secondary Star ...... 83 3.1.4 Discussion ...... 87 3.2 Ephemeris D etails ...... 93 3.3 Note on the Mass Determination ...... 108
IV DETERMINING THE SYSTEM PARAMETERS OF AE AQUARII . 112
4.1 Introduction ...... 112 4.1.1 Motivation and Overview ...... 112 4.2 Observations ...... 115 4.3 Absorption Line Modelling ...... 118 4.4 The Spherical Approximation ...... 122 4.4.1 Spherical Modelling D etails ...... 122 4.4.2 Spherical Modelling Results ...... 127 4.5 The Roche Lobe Approximation ...... 136 4.5.1 Motivation and General Considerations ...... 136 4.5.2 The Models ...... 140 4.5.3 Roche Lobe Modelling Details ...... 142 4.5.4 Roche Lobe Modelling Results ...... 144
V C O N C L U S IO N S ...... 149
5.1 S u m m a ry ...... 149 5.2 What next ? Unanswered Questions and Ideas ...... 152
APPENDICES
viii A The “Cable Experiment” ...... 157
A.l Introduction ...... 157 A.2 Cable Experiment High-Speed Data Acquisition ...... 157 A.3 Photometry ...... 158 A.3.1 Power Density S p e c tra ...... 163
B Ha Flares ...... 165
B.l Introduction ...... 165 B.2 Ha Line Profile Variability ...... 165 B.3 Time Evolution of the Ha and Continuum F lares ...... 167
C Ephemeris Data Base and Details ...... 174
C.l The Data B ase ...... 174 C.2 Confusion ...... 176 C.3 Discrepencies ...... 176
BIBLIOGRAPHY ...... 187 L i s t o f F i g u r e s
FIGURE PAGE
1 The Standard Model of a Non-magnetic Cataclysmic Variable. . . . 5
2 AAVSO Light Curve of AE Aquarii ...... 17
3 A Multiwavelength View of AE Aquarii ...... 21
4 MCSP Spectrum of AE Aquarii ...... 32
5 Continuum Light Curves ...... 35
6 Line Light Curves ...... 36
7 MCSP Power Density Spectra ...... 38
8 MCSP PDS on log-log scale ...... 40
9 Reduced \ 2 °f LTE Disk -f Template Star M odels ...... 43
10 Low Temperature LTE Disk + Secondary Star M odel ...... 46
11 High Temperature LTE Disk + Secondary Star Model ...... 47
12 Flare Spectrum ...... 52
13 Oscillation Pulse Profiles ...... 55
14 Oscillation Spectrum ...... 56
15 Comparison of Power Density Spectra ...... 64
16 AE Aquarii Log Power Density Spectra ...... 65
x 17 IID 19445 Log Power Density Spectra ...... 66
18 RMS Flickering Spectra ...... 70
19 Ha Power Spectrum ...... 74
20 The Mg I b-complex as a Diagnostic ...... 77
21 Mean Spectrum of AE Aquarii from Cable Data ...... 83
22 Radial Velocity Curves ...... 87
23 Phase Offsets ...... 92
24 Periodogram of Cable Data ...... 94
25 Periodogram of Joy’s Absorption Line Data ...... 95
26 Periodogram of Chincarini & Walker’s Absorption Line Data, 96
27 Periodogram of Combined Absorption Line Data ...... 97
28 Detail of Periodogram of Combined Absorption Line Data. . 98
29 Periodogram of Joy and RSB’s Emission Line Data ...... 100
30 Radial Velocity Curve — Joy’s Data ...... 102
31 Radial Velocity Curve — Chincarini h Walker’s Data. . . . 103
32 Radial Velocity Curve — Cable Data ...... 104
33 Radial Velocity Curve — Combined Data Sets ...... 105
34 Stellar Mass Constraints for Inclination i = 70° ...... 110
35 Stellar Mass Constraints for Inclination i = 56.7° ...... 111
36 Trailed Spectrogram Emphasizing the Absorption Lines. . . 119
37 The VTOt sin i — q Relation ...... 121
38 Spherical Modelling Fits ...... 128
xi 39 Spectral Type vs. x2 ...... 131
40 Spherical Modelling x2 Surface ...... 133
41 Ellipsoidal Variations — Integrated Line Flux. . 138
42 Ellipsoidal Variations - Rescale Factor ...... 139
43 Line Broadening Functions ...... 141
44 Predicted Data from Roche Lobe Model ...... 143
45 Observed and Computed Ellipsoidal Variations. 147
46 Radial Velocity — Continuum Level Correlation, 148
47 High-Speed Photometry Night 1 ...... 159
48 High-Speed Photometry Night 2 ...... 160
49 High-Speed Photometry Night 3 ...... 161
50 High-Speed Photometry Night 4 ...... 162
51 Cable Photometry Power Density Spectra. . . . 164
52 Ha Trailed Spectrograms ...... 166
53 Example of Ha Profile Variations ...... 168
54 Ha and Continuum Light Curves ...... 169
55 Cross-correlation Nights 1 &: 2 ...... 170
56 Cross-correlation Nights 3 h 4...... 171
xii L i s t o f T a b l e s
TABLE PAGE
1 Multi-Channel Spectrophotometry of AE Aquarii ...... 30
2 Confidence Intervals for Blackbody Fits to Oscillation Spectrum. . . . 60
3 Orbital Ephemeris of AE Aquarii ...... 86
4 Orbital Elements of AE Aquarii ...... 106
5 Testing the New Ephemeris ...... 107
6 Cable Experiment Instrumentation ...... 116
7 The Template Stars ...... 126
8 Spherical Model Results ...... 129
9 Comparison of AE Aquarii Data Sets ...... 179
10 Cable Experiment Radial Velocities ...... 180 C H A PT E R I
INTRODUCTION
1.1 Overview
Estimates of the fraction of nearby main sequence stars in binary or multiple systems are on the order of 50% (Mihalas & Binney 1981) and the value could be much higher depending upon selection effects. Of these multiple star systems, a small subset are known as the “interacting binaries”, the adjective “interacting” indicating that because of their proximity to each other, the stars profoundly influence one another in appearance and evolution. A particularly spectacular subset of the interacting binaries are the “cataclysmic variables”, hereafter referred to as “CVs”. About 1 in every 10s main sequence stars belongs to a CV system (Wade & Ward 1985), so there are on the order of 1 million CVs in our Galactic disk. Downes & and of these approximately 168 have a measured orbital period (Ritter 1990).
Historically, the name “cataclysmic variable” was given to this class of stars be cause they exhibited large and sudden changes of brightness. Yet after one of these outbursts, the system would return to its previous quiescent state. This differentiated them from the class of objects known as “catastrophic variables”, referred to today as supernovae.
1 Although both the novae and dwarf novae (defined below) were observed a hundred years prior, it wasn’t until the mid 1950’s that a viable model for the CV phenomenon was proposed. Preceeded by photometric observations which revealed the ubiquity of rapid variability and the binary nature of some novae and dwarf novae (via eclipses),
Crawford & Kraft (1956) used spectroscopic evidence to deduce that one of the stars fills its Roche lobe and transfers mass to the other. [A Roche surface is a surface of equipotential in a rotating, two point-mass system, including the effects of both gravity and centripetal force, (but not the Coriolis force). The Roche lobe denotes the critical surface which contains the inner Lagrangian point L\, where the oppos ing gravitational forces between the two masses and the centripetal force are just balanced. In the Roche approximation, isobaric and equipotential surfaces coincide.
Because the shape of a star is given by the surface of zero pressure, the shapes of the stars in a close binary system are given by the Roche equipotential surfaces. For a detailed discussion see chapter 1 in Frank, King, & Raine (1992) and/or Pringle
(1985).]
Figure 1 shows the components of the standard model for a non-magnetic CV.
A late-type main sequence star, typically a K or M dwarf, and a white dwarf orbit each other. The orbits are circular, because tidal dissipation would quickly circularize them otherwise. The white dwarf is the more massive of the two stars, and is called the primary star; likewise, the red dwarf is referred to as the secondary star. The secondary star fills its Roche lobe and therefore has a teardrop-like shape. Gas escapes from the secondary star at the Li point and flows toward the primary star. Because of angular momentum, the stream cannot flow directly from one star to the other, but instead orbits around the white dwarf, eventually crashing into itself, forming a ring. Viscosity acts to dissipate and redistribute the angular momentum, allowing the gas to spread out into a flat disk. As energy is dissipated and angular momentum redistributed, the matter slowly spirals down the white dwarf’s potential well, radiating profusely along the way. As more material is transferred from the secondary star, a quasi-steady “accretion disk” forms. The location where the mass transfer stream collides with the disk is a site of enhanced emission, and is called the “bright spot” or “hot spot”. In some systems, the bright spot can be a major contributor to the total optical flux of the system (e.g. U Gem). The interface between the white dwarf and the inner edge of the accretion disk is known as the boundary layer. Theoretically, up to 50% of the bolometric luminosity can be emitted from the boundary layer. However observations seem to indicate a value much less than that [suggesting that either the white dwarf is very rapidly rotating or the energy dissipation is non-radiative, e.g. winds. For a discussion, see Cordova (1993) and also Hoare & Drew (1991)]. For the magnetic systems, the inner part of the disk is disrupted, and the accreting gas is forced to flow along the magnetic field lines, eventually crashing down near the magnetic poles. If the magnetic field strength of the white dwarf is sufficient, the entire disk is disrupted and the accreted material is confined along field lines from near the L\ point. Thus in the weakly magnetic systems the disk has a hole in the center, and the strongly magnetic systems have no disk at all. A detailed discussion about accretion and accretion disks is given in Frank, King, & Raine (1992).
CVs are classified into several types based on observational properties. These include the novae, recurrent novae, dwarf novae, nova-likes, AM Herculis stars (also known as “polars”), DQ Herculis stars (or “intermediate polars”), and other types and subclasses. The fine details of the differences between these classes are unimportant for this dissertation, and only the major classes are briefly defined below.
Novae display the largest amplitude brightness changes, 9-20 magnitudes (Wade
& Ward 1985) and are caused by runaway thermonuclear fusion on the surface of the primary star. Although no nova outburst has been observed to occur twice, the hibernation hypothesis predicts a recurrence timescale of a few thousand years
(Shara 1989). Dwarf novae have much smaller amplitude outbursts, typically 2 - 5 magnitudes. The exact mechanism responsible for the outburst is unknown, but it is generally believed to be due to temporary enhanced mass accretion onto the surface of the primary star. Dwarf novae have a recurrence timescale on the order of weeks to months. The AM Her stars are systems in which the white dwarf possesses a strong magnetic field (greater than ~ 107 Gauss) that dominates the accretion flow and thus creates a morphology quite different from the non-magnetic CVs. The DQ Her stars have weaker but non-negligible magnetic fields, and it is believed that they contain a disk whose inner region is disrupted by the magnetic field. Nova-likes are systems that appear similar to nova systems after the nova explosion has subsided. As their name implies, they may be pre- or post- novae. This class also contains systems which do not naturally fall into any other category. A class of objects closely related 5
q = 0 .6 5 i = 5 5 °
Prifv\ary S'fa.r (wKfTe dwaff )
i
bf^ki" Spot
Secondary 5"fa«
WELSH 6-JUL-19M 22:16
Figure 1: The standard model of a non-magnetic cataclysmic variable. A K or M main sequence secondary star fills its Roche lobe and loses mass through the L\ point. The mass forms an accretion disk around the white dwarf primary star. A “bright spot” is formed where gas stream crashes into the accretion disk. The sense of rotation in this figure is counterclockwise as viewed from above, and the mass ratio q, inclination i, and orbital phase <{> are as indicated. to the CVs are the low mass x-ray binaries. The essential difference between CVs and “LMXRBs” is that the primary star in an LMXRB is a neutron star (or black hole). They will not be discussed further in this dissertation. For further information on CVs the reader is referred to the following sources: Livio (1993); Cordova (1993);
Bode Sz Evans (1989); Wade & Ward (1985); and references therein.
It is important to understand the scales involved with CVs. Orbital periods range from ~ 1 - 10 hours, with most under 4 hours. The white dwarf masses are less than the Chandrasekhar limit (~ 1.4M0 ), and the secondary star masses are typically less than ~ 1M®. The binary separation is on the order of 1R0. Because they typically have solar luminosities, most known CVs are relatively nearby (within a few hundred parsecs).
The orbital velocities are on the order of tens to hundreds of km s-1. Because the secondary star’s rotation is tidally locked to the orbital period, it is forced to rotate at a much larger velocity than are isolated field stars of similar spectral type.
A typical rotation velocity is ~ 120 km s-1, compared to a few km s-1 for late-type field stars. (For example Dempsey et al. (1993) give rotation velocities for five K V stars, all of which have rotation velocities less than 10 km s -1.)
The accretion disk itself contains relatively little mass (~ 1O_1OM0) and is fed with a mass transfer rate of ~ 10-8M0?/r-1 for the high M systems and about three orders of magnitude smaller for the low M systems (Patterson 1984). The orbital period at the outer edge of the disk is roughly an hour, while at the inner edge it is a few seconds. The disk temperature varies as a function of radius, ranging from a few tens of thousand Kelvin in the inner disk to a few thousand at the outer edge.
1.1.1 Cataclysmic Variables and the Big Picture
Aside from their intrinsic appeal, CVs are interesting because of their relationship to other fields of study in astronomy and astrophysics. Some of these important relationships are enumerated below.
(1) Although there are some serious systematic effects, CVs can be used to place constraints on the masses and radii of white dwarfs and late-type main sequence stars.
(2) CVs challenge theories of stellar evolution and binary star formation simply by their existence: How did the stars come so close to one another? Why is one star a white dwarf while the other is still a main sequence star? Could merging binaries become “blue stragglers”? (For a comprehensive review of CV evolution, see
Patterson 1984.)
(3) The secondary stars are forced to be non-axisymmetric, are spun-up to large rotation velocities, and are irradiated — it is probable that these extreme environmen tal conditions strongly affect the star and pose quite a challenge to stellar atmosphere theory.
(4) Novae are important objects because they enrich the interstellar medium with metals (M artin 1989) and are sites of dust formation (Bode h Evans 1989). The metals in the nova shell ejecta have very different isotopic abundances than solar
(Livio 1993). It may be possible to use nova outbursts as cosmological candles (Livio
1993, Shara 1989). (5) Supernovae type I may result from the white dwarf accreting enough mass to push it over the Chandrasekhar limit (see Livio 1993 for a discussion).
(6) And finally, CVs offer an excellent opportunity to study the accretion process in detail. CV accretion disks offer the advantages of being in a regime where (special and general) relativity, self gravity, and radiation pressure can be ignored. There are also the observational advantages of having timescales and luminosities such that one can observe the accretion process with high spectral and high temporal resolution.
It is even possible to obtain spatial information via eclipses and tomography (see
Horne 1985, and Marsh & Horne 1988). Thus it is in the CVs that the cleanest measurements of accretion disks can be made.
Accretion disks are believed to play important roles in star and planetary forma tion, and because of the very high efficiencies possible for converting gravitational potential energy into electromagnetic energy, accretion disks are invoked to power ac tive galactic nuclei. Perhaps the most important aspect of CV accretion disks however is deceptively the most obvious — CV accretion disks exist. While in some CVs the evidence for a disk is weak or non-existent (see Williams 1989 for example), there are a few systems in which the accretion disk hypothesis seems irrefutable. For example, in Young, Schneider, h Shectman’s (1981) study of HT Cas, first the blue then the red portion of double-peaked emission lines disappear during eclipse ingress. During egress, the blue portion reappears followed soon afterward by the red portion of the line. The disk hypothesis naturally explains this as occultation of the approaching half of the rotating disk followed by occultation of the receding half of the disk by 9 the secondary star.
1.1.2 Flickering and Flaring
“Flickering” is the term used to describe the small amplitude (less than ~ 1 mag), seemingly random brightness variations seen in CVs. Flickering occurs over a large range in timescale (less than a second to hours), but because the power spectrum of the flickering is red (i.e., power density falls off quickly with increasing temporal frequency) the long-period trends dominate the appearance of light curves. Bruch
(1992) gives a qualitative definition of flickering as “continuous variations which ap pear as a statistical superposition of individual flares of quite different amplitudes and durations.” To quantify the flickering, Bruch uses a procedure based on identifying the local extrema of the light curve. The main virtue of his “sensible” definition is that it is not rigid, i.e., it contains a free parameter which defines a minimum ampli tude necessary to identify a fluctuation as a real. (This free parameter is defined in part by the the amount of noise in the data.) This flexible definition of flickering is intrinsically ill-defined and this is a consequence of our lack of understanding of both flickering and flaring.
W ith the advent of photoelectric photometry in the late 1940s and 1950s, it be came clear that flickering was a phenomenon common to all CVs. [For an excellent description of the historical development of observations of flickering in CVs, see
Warner (1988).] Because of its ubiquity, understanding flickering will be vital to un derstanding accretion itself. Unfortunately the field has degenerated into the following state: a lack of theoretical interpretation has dissuaded observers from pursuing the 10 topic, while lack of observations have left theorists with no data to interpret (e.g. direct quote from the literature: “The origin of flickering is certainly an important unsolved problem. But it is someone else’s problem...” Patterson 1981). Part of the difficulty is that because of its statistical nature, it is hard to extract information from the flickering.
Attempts have been made to determine the location of the flickering in a few CVs.
In U Gem the flickering appears to be associated with the bright spot (Warner 1988), while in HT Cas (Patterson 1981) the flickering seems to be associated with the white dwarf. In RW Tri (Horne & Stiening 1985) and V2051 Oph (Warner & Cropper 1983) the flickering appears to originate in the inner disk, and not too surprisingly on the white dwarf in AM Her (Priedhorsky &; Krzeminiski 1978).
Only a few attempts have been made to determine the colors of the flickering.
Using ubvr photometry, Szkody (1976) found that the flickering amplitude increases from the r to u filter and interpreted this as due to a temperature change, assuming the flickering is caused by free-free emission. Middleditch and Cordova (1982), using
U, Cyan and R filters, found that in SY Cnc the flickering is blue with a Balmer jump in emission and could not be reproduced with any single-component thermal or free-free spectrum, modulated in either intensity or temperature. Bruch (1992) carried out an extensive photometric observational program to study the flickering in numerous systems and found that for every system the flickering was blue, and as an ensemble, could not be fit with either a blackbody or power-law model. Flickering has been observed in the Balmer lines by Chincarini and Walker (1981) and Walker (1981).
Just as in the case of flickering, little is known about flares in CVs. [The term
“flare” is defined as a rapid increase then decrease in flux, where rapid means short compared to the orbital timescale, and the change in flux must be much larger than the typical size of statistical fluctuations due to noise. The definition used here is intentionally crude, so as not to imply more than is actually known. The term “flare” is used in the context of CVs in a fashion analogous to its use in solar and stellar flares, though this may be a misnomer. Crudely speaking, changes in brightness due to flares are larger than those due to flickering, and flares are always considered as positive increases in brightness while flickering encompasses both increases and decreases. The topic of CV flickering and flaring is further discussed in this dissertation.] Flares and flickering may be different manifestations of the same phenomenon in that flickering may be caused by many superimposed small flares, and/or flares might be large, isolated “flicker” events. If the latter is true, then flares are a very clean way to study the physical conditions involved in flickering (one can study a single event, rather than time-averaged data).
It is not known where the flares in CVs are coming from. It is possible that the flares arise on the secondary stars (in analogy with the dMe flare stars), and not in the disk. An argument against the secondary star hypothesis is that typically flare stars are M-stars and have a mass less than 0.6 M® (Elsworth Sz James 1982), while many CV secondary stars are known to be K-stars. However this argument can be easily countered by the fact that the secondary stars are forced to rapidly rotate, 12 which may induce enhanced magnetic activity.
As mentioned earlier, the power density spectra of the flickering drops off quickly with increasing temporal frequency, and can be well represented with a power-law:
P(f) oc / “, where a is typically between -1 and -2. (Note that it is important to distinguish between Fourier power and Fourier amplitude, both of which are quoted in the literature. The amplitude density is essentially the square root of the power density, so power-law exponents quoted for amplitude spectra need to be doubled to compare with exponents of power spectra.) Bruch (1992) gives a mean exponent
-1.0 ±0.2 for the amplitude spectrum of the CVs he surveyed, but cautions that there are considerable deviations between individual systems as well as somewhat smaller night-to-night variations in the same system. Elsworth & James (1982) determined a value of -1 for the exponent of the amplitude spectrum in AE Aquarii. Note: the astronomical use of the term “flickering” has a more general meaning and should not be confused with the more common term “flicker noise”. Flicker noise, also called
“one-over-f noise” or “pink noise”, has by definition P(f) oc f~l. As stated above, flickering in CVs seems to be better represented by P(f) oc / ~ 2, which is sometimes referred to as “Brownian noise”, a term illustrative of its random walk nature. “White noise” has P{f) independent of /, and is sometimes called “Johnson”, “thermal”, or
“shot” noise, depending upon its source.
1.1.3 Oscillations
Oscillations are sinusoid-like brightness variations seen in CVs, most conspicuously in the DQ Her class. Some oscillations maintain fixed periods year after year, while others are short-lived and last only a few cycles (the “quasi-periodic oscillations” or
QPOs). Both exhibit amplitude and waveform fluctuations on a short timescale, and this lack of stability distinguishes them from pulsating stars. In addition, there are oscillations associated with dwarf nova outburst, the “DNOs”. The rapid QPOs and
DNOs are very poorly understood, and may be caused by short lived inhomogeneities in the inner accretion disk, boundary layer instabilities, non-radial oscillations in the disk, or a number of other possibilities (see Warner 1988 for a discussion of QPOs and DNOs, and references therein). The DNOs and QPOs will not be discussed here; instead the focus lies on the more stable oscillations.
The fixed-period oscillations are associated with the magnetic systems, where we are seeing the effects of accretion along field lines anchored to the rotating white dwarf.
Because the accreted gas is funneled onto the white dwarf near its magnetic poles, the emitted radiation pattern is not azimuthally symmetric (assuming the magnetic dipole axis is not exactly perpendicular to the plane of the disk). As the white dwarf rotates, these sites of localized emission rotate with it, and the radiation sweeps across the disk and/or our line of sight. Theoretical calculations indicate that the infalling material will shock above the surface of the white dwarf, reaching temperatures of
108/lT and therefore the bulk of radiation is in the hard X-ray wavelength range
(Frank, King, &; Raine 1992). These hard X-rays will illuminate the surface of the white dwarf and be reprocessed into softer radiation. The disk will also be a site of reprocessing. Evidence for the above scenario is seen in DQ Her where white-light photometric observations reveal that the oscillations undergo a phase shift through disk eclipse, shifting from zero to +90° during eclipse ingress and from —90° back to zero during egress (this effect is known as the “eclipse-associated phase shift” and is discussed in Warner et al. 1972 and Patterson, Robinson, k Nather 1978). This immediately tells us that the entire disk is involved, and supports the rotating searchlight model.
Additional evidence comes from observations of the He II 4686 A emission line in DQ
Her by Channan, Nelson, k Margon (1978) which show a “velocity-associated phase shift” across the line in the sense that the red portion of the line (receding) pulses earlier than the blue portion (approaching). An irradiating beam originating at the surface of the prograde-rotating white dwarf can explain these effects, provided that the disk is flared and tilted (inclination ^ 90°) so that the back half of the disk is more visible than the front.
The spectrum of the oscillations has never been measured, though broadband op tical photometry indicates that they are blue (Martell 1993, private communication).
Coherent oscillations have been detected in the X-ray band for the polars and the intermediate polars (Norton k Watson 1989; Eracleous, Patterson, k Halpern 1991;
King k Shaviv 1984). Norton k Watson find that for the intermediate polars the amplitude of the modulation decreases with increasing energy in the 0.05 to 10 keV range, and cannot be due solely to self-occultation. The fractional amplitudes of the oscillations in the X-rays are very large, 25-100%; they are typically much smaller in the optical (less than ~ 1%). The sinusoidal shape of the oscillations suggests that the emission region is extended, and occupies a large fraction of the white dwarf’s surface (King & Shaviv 1984, Norton & Watson 1989), though Rosen, Mason, &
Cordova (1988) argue that the accretion stream scenario should be replaced with an accretion curtain model, and the fractional area is much smaller (~ 1%).
The period of the oscillations should give the white dwarf spin period, if one is careful to disentangle the confusion due to the spin—orbit sidebands (Warner 1986).
[In fact, Hellier (1992) uses the absence of certain sidebands to demonstrate that disks exist in the DQ Her stars, and are not completely disrupted by the white dwarf’s magnetic field, as in the case of AM Her stars.] A further confusion of a factor of two in the spin period is possible if the emission originating near each magnetic pole is equal in amplitude. The spin period of the white dwarfs in the intermediate polars range from 33 seconds to 1255 seconds (King & Shaviv 1984).
I.2 AE Aquarii
AE Aquarii is one of the brightest CVs, with a quiescent V magnitude of about
II.6 (R itter 1990) and peak brightness of about 9.8 during flares. Figure 2 shows a light curve of AE Aquarii determined by the American Association of Variable Star
Observers (AAVSO). Nearly 22 years of data are shown, and the estimated error per point is ~ 0.3 mag. It is immediately apparent that although large and frequent brightness changes are visible, no simple recurrence pattern is seen and the light curve does not resemble the light curve of dwarf novae. The approximate times of the observations used in this dissertation are shown (“MCSP” denotes the multichannel spectrophotometry discussed in Chapter 2 and “Cable” denotes the high-resolution Ha spectrophotometry discussed in Chapters 3 and 4).
AE Aquarii is classified as a nova-like object, but in many ways this object is unique and defies this classification. For example, the nova-like objects have high mass-transfer rates in general, but the mass-transfer rate in AE Aquarii is rather low: 1.6 x 10-10 to 6 x lO-11M0r/r-1 (Lamb & Patterson 1983; Jameson, King, &
Sherrington 1980).
Extensive photometry has been obtained on AE Aquarii (e.g. Patterson 1979;
Van Paradijs, Kraakman, & Van Amerongen 1989) that shows large amplitude flares
(factor of 2 or more increase in flux) that occur on a timescale as short as ten minutes, and occur at all orbital phases. This flaring amplitude is large compared to other flares seen in CVs (Bruch 1992). In addition, AE Aquarii seems to be the only
CV that has extended periods of time when no flaring or flickering is present at all
(Bruch 1991). When folded on the orbital period, the photometry shows a well- defined lower envelope, a sinusoid-like curve which is interpreted to be caused by the varying contribution from the non-spherical secondary star (see Van Paradijs et al.
1989 for an example of these “ellipsoidal variations”).
The optical spectrum of AE Aquarii shows strong Balmer lines in emission and strong absorption lines from the secondary star (Joy 1954; Crawford & Kraft 1956;
Chincarini & Walker 1981; Robinson, Shafter, & Balachandran 1991). He I and the
Ca II H and K lines are also in emission. None of the lines are double peaked. The
IUE UV spectrum shows He II, strong N V and Si IV lines, and a (surprisingly weak)
C IV line in emission on top of a weak continuum (Eracleous et al. 1993; Jameson, 17
oo MCSP
Q> "tiE o ‘c o»o E
CM
0 2000 4000 6000 <3000 TO -2^39 2 Figure 2: AAVSO Light Curve of AE Aquarii. The light curve of AE Aquarii spanning nearly 22 years is plotted as visual magnitude versus Julian date. The approximate times of the observations used in this dissertation are shown. Time zero corresponds to Julian D ate 2439295. 18 King, & Sherrington 1980). Mg II and Ca II are also observed to be in emission, and their presence suggests that a two-phase model is needed to produce both these low and high ionization lines (Jameson, King, & Sherrington 1980). Semi-forbidden lines are also detected. The spectral type of the secondary star can be roughly determined from the ab sorption lines (Crawford &; Kraft 1956; Chincarini & Walker 1981; Bruch 1991), from infrared photometry where the disk contribution is presumed small (Tanzi, Chincar ini, L Tarenghi 1981), or from flux ratio diagrams (Wade 1982) and is found to be in the range K1 to K5. The luminosity class is IV or V, as the star needs to be slightly evolved to fill its Roche lobe (post ZAMS, but not yet terminal-age main sequence; see Patterson 1979, 1984). Chincarini & Walker (1981) suggested an al ternative explanation, that the secondary star does not fill its Roche lobe, and that mass transfer occurs via mass ejections due to prominence activity over a large area of the secondary star. AE Aquarii exhibits coherent oscillations at 16.5 and 33.0 seconds (Patterson 1979; Robinson, Shafter, &; Balachandran 1991) and these have been detected in the X-ray band (Patterson 1980; Eracleous, Patterson, & Halpern 1991; De Jager 1991). The fractional amplitude is typically 0.1 - 0.3% in the optical, and up to 28% in the X-ray (Einstein 0.16 - 3.5 keV IPC band). This oscillation period is the shortest of any CV, making the white dwarf in AE Aquarii the fastest spinning white dwarf known (Abada-Simon et al. 1993). Detection of pulsed gamma ray emission has also been reported (Meintjes et al. 1992) and, if confirmed, may be a very important clue 19 as to the mechanism ultimately responsible for the oscillations. The authors claim that the TeV gamma rays are correlated with the optical flares. No other CV has been detected in the gamma ray band. AE Aquarii is unique in another respect —- it has very large, rapid radio flares (Bookbinder & Lamb 1987; Bastian, Dulk, & Chanmugam 1988; Abada-Simon et al. 1993). Flux density changes of up to a factor of 12 were observed and have been interpreted as synchrotron emission from discrete, expanding blobs of electrons that have been explosively accelerated to relativistic velocities. Flare-like disruption of the inner disk, such as those inferred in Cygnus X-3, are believed to be responsible for the particle accelerations. The gamma-ray observations support this model, as do the very rapid bursts seen in the optical, whose rise and decay times are ~ 2 seconds with durations of minutes or less (De Jager & Meintjes 1993). The only other CV that has detectable radio emission is the highly magnetic system AM Herculis. The multicolor photometry of Van Paradijs et al. (1989) and Bruch (1991) show that the amplitude of the flaring and flickering is greatest in the blue and monoton- ically decreases to the red. Chincarini &; Walker (1981) show that the flares occur chiefly in the Balmer continuum and lines. They claim that the flaring events are more apparent at orbital phases 0.3 and 0.7, a statement contradicted by Van Paradijs et al. (1989). [In this dissertation the photometric convention is followed, defining phase zero as the phase of primary eclipse (if the system were eclipsing, which it is not), i.e., when the white dwarf is farthest away from us.] The color of the oscillations has never been measured in AE Aquarii prior to the results of this thesis. 20 Figure 3 shows a multiwavelength view of AE Aquarii. Assembled from the pub lished sources mentioned above plus the infrared photometry of Szkody (1977), HST ultraviolet spectra of Eracleous et al. (1993), the Einstein X-ray bremsstrahlung model of Eracleous (1993), plus work described in this dissertation, the figure reveals the remarkable energy distribution of AE Aquarii. No other CV has been detected across such a wide range of energies. The TeV gamma ray detection (Meintjes et al. 1992) has been omitted for clarity — it would lie approximately at a frequency of 1026'8 H z and vf„ = 10-1° erg cm 2 s -1 Robinson, Shafter, h Balachandran (1991) carried out an intensive study of the oscillations in AE Aquarii with the hope of using the pulse arrival time delays as a way to accurately measure the apparent orbital motion of the white dwarf. [The pulse arrival times vary in a sinusoidal fashion with the orbital period, due to the light-crossing time of the white dwarf’s orbit. This effect was noted in AE Aquarii by Patterson (1979).] The pulse-timing orbit has the potential of giving a very accurate measure of the motion of the white dwarf. They also studied the He* emission line and from it derived a new orbital ephemeris and apparent orbital velocity. Robinson, Shafter, & Balachandran (1991) discovered the following very remark able result: the pulses did not coincide with the expected position of the white dwarf. The disagreement was large, ~ 60° in phase (as measured from the center of mass of the system towards the white dwarf). In addition, the amplitude of the pulse-timing delay curve (2.3 light-seconds) gave an apparent orbital velocity significantly smaller than the Ha emission-line velocity (at the 2cr level). Taken at face value, these two 21 AE Aquarii op tic al O IR photometry UV (IUE) m e a n O optical flare UV flare O optical flickering CN ray" CD o oscillations S 'o secondary star O in CD co radio flares CD 1 0 9 1 0 10 1 0 11 1 0 12 1 0 13 1 0 14 1 0 15 1 0 16 1 0 17 1 0 18 x ' WELSH 8—JUL—1993 10:47 Figure 3: A multiwavelength view of AE Aquarii. The observed spectral energy distribution of AE Aquarii is shown as v fv versus i> on a log-log plot. References for the data are given in the text. The data were not obtained simultaneously. The curves labelled “optical”, “optical flare”, “optical flickering”, “secondary star” and the optical portion of the “oscillations” are derived in the following chapters. 22 results imply that the region producing the optical oscillations is located just outside the white dwarf’s Roche lobe, and leading the white dwarf by ~ 60°. However this result seemed implausible to Robinson et al., who interpret the result as implying that reprocessing of hard photons is taking place in the disk, so that we are observ ing a combination of both direct and reprocessed pulsed light. The authors state the they know of no reason why this particular region in the disk should reprocess hard photons more than any other, but note that the ballistic trajectory of the gas stream, if not impeded by the disk, would swing past the white dwarf and loop around roughly in the same location as the inferred reprocessing site. Another possible in terpretation is that a strong wind is emitted by the secondary star that collides with the disk. The relative velocity differences between the wind and disk are greatest on the leading edge of the disk, again approximately where the inferred reprocessing site is located, perhaps inducing a localized density enhancement. Robinson, Shafter, &; Balachandran conclude “Whether or not the [reprocessing] model is correct, however, our basic conclusion is unchanged: the pulse-timing orbit represents the true orbit of the white dwarf poorly”. In Chapter 3 of this dissertation, the above statement is shown to be false. But before proceeding to that, it is first shown in Chapter 2 that the reprocessing model is incorrect (in the sense that reprocessing does not occur over a large area of the disk and is most likely associated with the white dwarf’s surface). Also in Chapter 2, spectra of flickering and flaring are presented and analysed. In Chapter 4, questions concerning the values of the system parameters (mass ratio, inclination, stellar masses, etc.) are addressed. Chapter 5 concludes this dissertation and gives a summary along with some unanswered questions. Several Appendices are included to give further details concerning the observations, and to discuss more on the flaring. C H A PT E R II OSCILLATIONS AND FLICKERING IN AE AQ UARII 2.1 Introduction This Chapter focuses on the nature of the oscillations and flickering in AE Aquarii, and also on the contribution of the disk and secondary star to the optical spectrum. Section 2 of this Chapter contains previously published material, reproduced with permission from the Astrophysical Journal, American Astronomical Society. The name of the paper and its reference is: “Optical Spectrophotometry of Oscillations and Flickering in AE Aquarii”, Welsh, W. F., Horne, K. & Oke, J. B. 1991, ApJ 406, 229. The version given here is nearly identical to the published version. The remainder of the Chapter contains results not included in the published paper. Section 3 includes a comparison with a non-variable star observed during the same night and Section 4 presents additional results on flickering. Section 5 contains a discussion on the distance and spectral type of the secondary star. Section 6 consists of a brief comment on the inclination and the size of the secondary star. 24 25 2.2 Optical Spectrophotometry of Oscillations and Flicker ing in AE Aquarii Studies of AE Aquarii led Joy (1954) and Crawford & Kraft (1956) to develop the standard model of cataclysmic variables (CV’s). Yet AE Aquarii remains a most puzzling CV. We can be fairly confident that it has an orbital period of 9.88 hours, a K star secondary (Chincarini &; Walker 1981), and an inclination i ~ 60° ± 10° (Patterson 1979). The mass ratio (secondary star/white dwarf) lies in the range q ~ 0.88 ± 0.06 based on the observed radial velocities of the emission lines from the disk and absorption lines from the secondary star (Robinson, Shafter, & Bal achandran 1991, hereafter RSB). But AE Aquarii exhibits a host of extraordinary phenomena which are not understood. These include large amplitude flares and flick ering (e.g. see Patterson 1979; Van Paradijs, Kraakman, & Van Amerongen 1989; Bruch 1991); emission lines that are not double peaked; variable flare-like radio emis sion (Bookbinder & Lamb 1987; Bastian, Dulk, h Chanmugam 1988); variable linear polarization (Szkody, Michalsky, & Stokes 1982) and perhaps circular polarization (Cropper 1986). AE Aquarii also exhibits rapid, coherent oscillations with periods of 16.5 and 33.0 seconds in the optical, X-ray, and possibly TeV gamma-ray bands (Patterson 1979; Patterson et al. 1980; RSB; Meintjes et al. 1992; De Jager 1991). Transient quasi-periodic oscillations near these periods are also seen. The 16.5 and 33 second oscillations have been attributed to accretion onto the magnetic poles of the white dwarf, which produces localized emission of EUV/X-rays. As the white dwarf rotates, 26 emission from these regions acts as “searchlight” beams sweeping across the disk, and the reprocessing of these hard photons in the disk produces optical oscillations. Strong evidence for this “oblique rotator” model is found in DQ Herculis, where an eclipse- associated phase shift of the 71-second optical oscillations (Patterson, Robinson, & Nather 1978) and a velocity-dependent phase shift in pulsed He II A4686 emission (Chanan, Nelson, & Margon 1978) are observed. No X-ray oscillations from DQ Her have been detected, but this may be due to the high inclination of the system. If the optical pulses arise from a region concentric with the white dwarf, one expects to see cyclical variations in the arrival times of the pulses, reflecting the orbital motion of the white dwarf. In AE Aquarii a ±2.3 second variation in the pulse arrival time is detected, but surprisingly, the inferred orbital motion is shifted in phase by ~ 60° with respect to the orbit of the white dwarf (RSB). The pulse-timing orbit seems to have an amplitude ~ 13%(2cr) smaller than that of the emission- line orbit and appears to be distorted from a circular orbit. RSB’s straightforward interpretation of this is that the optical pulses arise from a combination of direct emission from near the white dwarf and reprocessed emission from a target at the outer edge of the disk, comoving in the binary frame. This reprocessing model predicts that the optical pulses should have a slightly longer period than the X-ray pulses. However the observed X-ray period agrees with the optical period (Eracleous, Patterson, & Halpern 1991; De Jager 1991), suggesting that both arise from the same region. This result leads to a perplexing dilemma: if we are seeing reprocessing (from a target fixed in the binary frame, as the pulse-timing orbit suggests) there should be a period 27 difference between the X-ray and optical pulses; if we are not seeing reprocessing, then the pulses originate near the white dwarf’s surface and should be in phase with the emission-line orbit. Neither alternative is supported by the data (but see Marsh 1992). In this Chapter we present the optical spectrum of the oscillations in AE Aquarii and attempt to gain insight into the oscillation-producing mechanism. In addition, we estimate the spectrum of the secondary star, the disk, and a flare. First we present high-speed 30-channel spectrophotometric data and power density spectra which clearly show the oscillations. We then describe the methods used to decompose the spectra into various components. We analyse these components and in Section 2.2.9 we summarize our findings. 2.2.1 Observations We observed AE Aquarii with the MCSP multichannel spectrophotometer (Oke 1969) on 26 October 1979 UT 2.83-4.50 with the Hale 5m telescope at Mt. Palomar. Light curves were obtained simultaneously in 30 wavelength channels: channels 1- 16 (3267-5667A) at 80A resolution and channels 17-30 (5934-10414A) at 160A resolution, with gaps between the channels. The sky was clear with 2” seeing and the aperture (diaphragm) size was 10.13”. Sky subtractions were accomplished by continuously chopping at 30 Hz between star and sky apertures. Integration times were 3.0 seconds followed by 1.3 seconds of dead/readout time. The 1201 observations obtained cover orbital phase 0.438 to 0.607. We use here the ephemeris of RSB: IIJ D = 2,439,030.830(±3) + 0.4116580(±2)i?, in which phase Certain limitations of the data uncovered in the course of this analysis must be borne in mind. To shorten the dead time, parts of the normal MCSP data acquisition software were disabled during the high-speed observations. This caused occasional halts during the cycle of repeat integrations. As a result, our light curves have numerous gaps of variable duration between sections of evenly spaced data. Table 1 lists the 30 channels, their central wavelengths and resolution. Channel 22 appears to suffer from a flux calibration error of unknown origin (possibly due to poor A-band extinction correction) and so the normalization is in error, but the shape of the light curve may be correct. The shape of the light curve from channel 23 is peculiar (perhaps the phototube was unstable) and so data in this channel should be used with caution. Some channels longward of about 8500A may be less photometrically reliable due to contamination by telluric water vapor. Higher resolution spectra were constructed from a series of 60 second observations taken before and after the high-speed run. The central wavelengths of the 30 channels in each of these observations were shifted to fill in the gaps, so that when combined, a spectral resolution of 40A/80A shortwards/longwards of 5767A was obtained. Four sets of observations were made in this fashion. Because of the longer integrations and the time needed to shift the channels, it was possible for AE Aquarii to vary sig nificantly between observations, producing a spectrum with a sawtooth appearance, as noted by Oke & Wade (1982). Both sets taken prior to the high-speed run were found to have this problem, and are not used in the following analysis. The two sets taken after the high-speed run have been combined into one spectrum. Photometric reliability of some of the reddest channels has been compromised by contamination due to variable telluric water vapor. 30 Table 1: Spectrophotometry of AE Aquarii channel A AA mean flare oscillation (A) (A) (mJy) (■mJy) (mJy) 1 3267 80 13.43±0.04 7.98±0.28 0.12±0.04 2 3427 80 14.89±0.02 9.54±0.16 0.10±0.02 3 3587 80 17.65±0.02 11.35±0.15 0.12±0.02 Hoo 4 3747 80 19.77±0.02 10.49±0.14 0.12±0.02 5 3907 80 25.05±0.02 11.18±0.13 0.12±0.02 6 4067 80 37.42±0.02 9.74±0.14 0.09±0.02 H<5 7 4227 80 35.93±0.02 1.98±0.14 0.07±0.02 8 4387 80 50.93±0.02 2.15±0.12 0.12±0.02 9 4547 80 64.99±0.03 2.36±0.19 0.08±0.03 10 4707 80 72.30±0.03 2.00±0.19 0.05±0.03 11 4867 80 96.28±0.03 17.54±0.22 0.06±0.03 E/3 12 5027 80 79.49±0.03 5.35±0.20 0.09±0.03 13 5187 80 76.64±0.04 3.79±0.23 0.11±0.04 Mg I b 14 5347 80 102.34±0.04 3.02±0.25 0.11±0.04 15 5507 80 111.19±0.05 2.94±0.32 0.02±0.05 16 5667 80 120.64±0.05 2.66±0.33 0.11±0.05 17 5934 160 126.58±0.03 6.99±0.23 0.08±0.03 Na I, He I 18 6254 160 140.84±0.04 3.34±0.26 0.04±0.04 19 6574 160 208.83±0.05 16.33±0.30 0.12±0.05 H a 20 6894 160 173.71i0.06 4.82±0.37 0.05±0.05 21 7214 160 192.09±0.04 6.07±0.27 0.08±0.04 22 7534 160 243.36±0.07 7.81±0.43 0.01±0.06 bad calibration? 23 8014 160 213.08±0.07 -2.11±0.44 0.05±0.07 unstable pmt? 24 8334 160 233.05±0.29 9.20±1.90 0.24±0.29 25 8654 160 244.76±0.23 9.99±1.50 -0.18±0.22 26 8974 160 266.23±0.48 17.24±3.18 -0.28±0.47 27 9454 160 279.92±0.41 13.82±2.73 -0.09±0.40 28 9774 160 265.12±0.28 7.24±1.79 0.41±0.27 29 10094 160 300.06±0.33 13.91±2.11 -0.55±0.32 P 6 30 10414 160 338.59±0.47 6.67±3.00 0.23±0.46 2.2.2 Spectra Figure 4 shows the “high-resolution” spectrum of AE Aquarii, obtained just after the high-speed run, revealing a red continuum, strong Balmer emission lines and weak Balmer jump in emission. The horizontal bars (connected with the dotted lines) show the mean level and width of each of the 30 channels from the high-speed observa tions. These mean fluxes and lcr error bars are listed in column 4 of Table 1. Note that channels 11 and 19 cleanly isolate the H/3 and Ha emission lines. Notice that the Mg I b absorption line complex at 5200A is very prominent. Also shown is the spectrum of HD 154712 (I<4 V) from the Gunn-Stryker Atlas (Gunn & Stryker 1983. The spectrum has been scaled to approximate the spectrum of the secondary star in AE Aquarii, as described in Section 2.5 of this Chapter. The Na I D and the Ca II H and I< absorption lines are strong in the spectrum of the K4 V dwarf but are not seen in the AE Aquarii spectrum due to superimposed He I A5876 and Ca II H and K emission from the disk. As mentioned above, contamination due to water vapor is a problem, and the shape of the AE Aquarii spectrum in these affected regions therefore does not represent the true energy distribution (for exam ple, the features redward of about 7500A are not due to TiO from the secondary star). 2.2.3 Light Curves Continuum and emission-line light curves are presented in Figure 5 and Figure 6, respectively, as a function of orbital phase. Continuum light curves redward of about 32 o o AE Aquarii Ha to o o “3 CM £ HD 154712 O O jHnJ He + Co“ I f Y Nal D T Mgl b O 4000 6000 8000 ,4 wavelength (A) WELSH 10-SEP-1992 11:24 Figure 4: The mean spectrum of the high speed run, shown as horizontal bars con nected by dotted lines. Channel 22 has an “X” superimposed to stress that the flux level is not reliable. Superimposed is the “high resolution” spectrum. Also shown is HD 154712, scaled to match the Mg I feature in AE Aqr. 33 4200A are all similar, rising by ~ 16% over the duration of the observations, at first rapidly and then more slowly. The phase coverage is insufficient to positively identify the source of this rising red component. Since the light curves strictly increase, and do not show a minimum at phase 0.5, this increase is not due primarily to ellipsoidal variations of the secondary star, such as seen in the light curves in Van Paradijs et al. (1989). This assumes the assigned orbital phasing is correct, an assumption that is addressed in detail in Chapter 3. The Balmer continuum light curve AA3227-3627 exhibits rapid variations (i.e. flickering) and appears quite different from the Paschen continuum light curves as represented by the other light curves. Most notable of these variations are a sharp, brief flare at (f> ~ 0.465, a flare starting at 4> ~ 0.505, a shallow flare centered at (Patterson 1979; Van Paradijs et al. 1989; Bruch 1991). This large flare is also visible at a low level in all but the reddest continuum bands. Unfortunately, our high-speed observations ended before this flare subsided. The H/? and Her emission lines show variations similar to those seen in the Balmer continuum. To extract the line fluxes, we subtracted the secondary star spectrum and then measured the excess flux in channel 11 or 19 above a linear continuum fitted to neighboring channels (the Mg 1 b-line strength was used to determine the secondary star’s contribution — see the discussion in Section 2.5 in this Chapter and note that the scale value used here corresponds to an intermediate temperature disk). The relatively large amplitude fluctuations seen in the Balmer emission lines and continuum indicate that flickering arises from an optically thin region with T ~ 104K. The flare amplitude of 50% in Ha compared with 70% in the Balmer continuum suggests that Ha is saturated. The Ha/H/? ratio varies by ±35% around a mean value of ~ 3, and is inversely correlated with the line fluxes, consistent with there being a higher optical depth in Ha than in H/?. The observed Ha/H/3 ratio is much closer to the case B recombination value, 2.85 for T=104K (Osterbrock 1989), than in most CV’s (Williams 1980). The ratio drops noticeably during the large flare at the end of the run. Wade (1982) found Ha/Il/3 = 2.19 in AE Aquarii, lower than our mean value, yet more importantly, still higher than the typical value of ~ 1 for dwarf novae in quiescence. The main spectral feature attributable to the secondary star in our data is the Mg I b absorption complex near 5200A. The bottom panel of Figure 6 shows the light curve of the Mg I b flux deficit. The light curve was obtained by computing a linear fit to the continuum regions (channels 9-10 and 15-16), then taking the difference between the interpolated continuum and the actual flux in channel 13. The Mg I feature grows stronger between phases ~ 0.44 and ~ 0.56, then decreases. The shape of the Mg I light curve is vaguely similar to that of the Ha/H/? line ratio light curve, and the inverse of the Ha and H/5 light curves. Like the red continuum light curves, the behavior of the Mg I flux deficit is not consistent with ellipsoidal variations of the secondary star. 35 “i 1----- 1-----1----- 1----- 1-----r~ I I i pM7932-9532A V* lO CM CM HS& r *■"■ '" i i i i I i ■-»------1— t I i t i i ■ « » - *X6814-7294A U2LftK _ : _____i_____L_ JL _L XX5467-5707A I i i i t _i I I l_ _ J I I r **4187-4747A AX3227—3627A 0 . 4 5 0.5 0.55 0.6 orbital phase 0 WELSH 10—SEP—1992 11:26 Figure 5: Continuum light curves plotted against orbital phase. The points at phase 0.63 are derived from the high resolution spectrum. For scale, 0.01 of an orbital phase equals 6 minutes. 36 oo CO o ET CD ro 0 . 4 5 0 . 5 0 . 5 5 0.6 orbital phase 96 WELSH 10 —SEP—1992 11:33 Figure 6: Upper Panel: Ha and H/? emission-line light curves. Points at phase 0.63 are derived from the high resolution spectrum. The cross denotes the lcr uncertainty in the absolute orbital phase and the typical Ha error bar. The brackets above the H/3 light curve define the regions used to compute the flare spectrum. Middle Panel: Ratio of Ha to H/? line fluxes. The dashed line at 2.85 shows the ratio expected for case B recombination of hydrogen at T=10 x 103K. Lower Panel: Flux deficit (mJy) in the Mg I b absorption line (interpolated continuum minus observed flux in channel 13). 37 2.2.4 Power Density Spectra Power density spectra (PDS) of the continuum light curves of Figure 5 are shown in Figure 7 where the dotted vertical lines mark the expected positions of the 16.53837 and 33.076737 second oscillation periods (Patterson 1979). The oscillations are clearly strongest in the blue. The power is defined here in the RMS sense — a sinusoid of (semi)amplitude A mJy will have an integrated power f P(f)df = A2/2 mJy2 (where / is the temporal frequency). We have chosen two different vertical scales for the PDS because the white noise background level is not the same in all the continuum bands. An 8th degree polynomial fit was used to remove trends in the light curves prior to computing the PDS. Note that despite this low-frequency filtering, the power sharply increases at the lower temporal frequencies due to flickering, particularly in the Balmer continuum band. Power spectra of the Ha and H/? light curves are similar to the Paschen continuum PDS shown in the upper two panels, lacking any clear evidence of oscillations. To better investigate the flickering, we constructed a light curve from the entire data set by averaging the different wavelength channels using weights inversely pro portional to the Poisson noise variance. The PDS is shown on a log-log scale in Figure 8. The horizontal dashed line denotes the background white noise level ex pected from Poisson noise, showing that the flickering is considerably above the white noise background. Power-law models P(f) oc 1// and P(f) oc l / / 2 were added to this expected noise background and normalized by eye to match the observed PDS. Elsworth k James (1982) found P(f) oc l / / 2 in their white-light observations of AE 38 o “l------1------r- ro XX6814-7294A o CN w i l k ^ XX5467-5707A M nz CNJ 1 fO oo I M41 87-4747A O to co 0 0 . 0 5 0 .1 frequency f (Hz) WELSH 1 0 —SE P—19 9 2 1 4 :0 0 Figure 7: Power density spectra at various continuum wavelengths. The dotted verti cal lines show the expected frequencies of the 16.5 and 33.0 second oscillations based on Patterson 1979. 39 Aquarii. Our results seem to favor a PDS that falls off more like 1//, but we note that the slope of the observed PDS is sensitive to how much power the white noise background level contains. In the limit of no noise, P(f) oc 1 / / yields a reasonable fit, while in the limit that the noise equals the power of the observed PDS near the Nyquist frequency, P(f) oc l / / 2 is more suitable. Because we do not clearly observe the white noise level in our data, we cannot be absolutely sure of its placement, hence the ambiguity in our determination of the slope of the PDS. We conclude that the power of the flickering in AE Aquarii falls off with frequency between 1 / / and l / / 2. 2.2.5 The Secondary Star Spectrum To isolate the disk spectrum and estimate the distance to AE Aquarii, we need to determine the secondary star’s contribution. We model AE Aquarii’s spectrum as a sum of a scaled template star from the Gunn-Stryker Atlas (Gunn & Stryker 1983) plus a model disk spectrum (discussed in Section 2.2.6 below). The Gunn-Stryker Atlas is ideally suited for this purpose because the observations therein were made with the same MCSP equipment as our observations. A downhill simplex “amoeba” algorithm (Press et al. 1986) was used to minimize the %2 of this model with respect to the observed high-resolution spectrum. Spectral fitting was accomplished using the XCAL synthetic photometry package (Horne 1990). The Balmer and Na I D lines were masked out so that the emission lines do not affect the fitting procedure. This is important because the physics of emission line formation in accretion disks is complex (Horne h Marsh 1986) and detailed modelling requires many effects to be taken into account (e.g. metallicity, turbulent 40 o CM o o IO Q. o 0.01 0.1 frequency / (Hz) WELSH 10-SEP-1992 14:05 Figure 8: Power density spectrum of the (inverse variance) weighted mean of the 30 light curves plotted on a log-log scale. The horizontal dotted line shows the expected white noise background level based on the uncertainties in the data. The two solid curves show the power density spectrum of 1 / / and l / / 2 power-laws added to this white noise background. The vertical dotted lines show the expected frequencies of the 16.5 and 33.0 second oscillations. and Stark broadening, shear effects, inclination, etc.). Because the reductions were unable to remove the variable telluric contamination in sections of the high-resolution spectrum, these sections were replaced with values from the mean of the high-speed observations. These changes are as follows : high-resolution data between AA8614- 8934 were replaced with channel 25, AA8934-9254 by channel 26, and AA9334-9574 by channel 27. In addition, the region AA7494-7814 was omitted altogether. To place a further constraint on the models, we included the JHK infrared data of Tanzi et al. (1981) and Szkody (1977). Although these data are not contemporaneous with our data, most of the light in these bands originates from the secondary star and should not vary greatly. The results are summarized in the upper panel of Figure 9 where the reduced X 2 for the best fit models is plotted against spectral type of the Gunn-Stryker (GS) dwarf star templates. The spectral type of the best fit template (GS 58) is not known, but from the reduced x2 curve one can deduce that the spectral type of the secondary star is in the range K4-K5, in accord with various previous estimates (e.g. Crawford & Kraft 1956, Chincarini & Walker 1981, Tanzi et al. 1981, Bruch 1991), but we caution that the spectral types are not well determined. Giants from the GS atlas were also considered because there is evidence that the secondary star in AE Aquarii is somewhat evolved (Crawford & Kraft 1956; Patterson 1979, 1984; Echevarria 1983; and our own calculations). A spectral type of K4 III was implied, but in general the giants did not fit as well as the dwarfs and so will be ignored hereafter. Focusing our attention to GS star 58 (HYAD 185), in the lower panel of Figure 9 we show the reduced x 2 (257 degrees of freedom) of the fits computed holding the disk temperature fixed. This figure reveals a characteristic of all of the models we tried — an asymmetric, broad x2 minimum with a strong lower temperature cutoff and a weak upper temperature limit. Because of this lack of tight constraint on the models, the fits do not yield an unambiguous division of the light between the disk and the secondary star. Reasonable fits are obtained with the secondary star contributing ~ 85% of the light at 5500A if the disk temperature is ~ 10 x 103K. However, for disk temperatures ~ 6 x 103I< the secondary star contribution declines to ~ 65%. This ambiguity is illustrated in Figure 10 and Figure 11, where we have plotted the AE Aquarii spectrum, the model, the secondary star, the disk spectrum and the residuals. The uncertainty in the visual magnitude of the secondary star results in a corre sponding uncertainty in the distance. To find the distance, we cannot use a simple main-sequence color—absolute luminosity relation, since the secondary star in AE Aquarii is somewhat evolved. Instead we employ the Barnes-Evans relation (Barnes, Evans, & Moffett 1978), which calibrates optical color indices versus surface bright ness, and use the fact that the secondary star fills its Roche lobe. This method is similar to that used for the infrared distance determinations of Bailey (1981), ex cept that we use optical rather than infrared colors. Using the Barnes-Evans R-I relation, colors from the GS Atlas, and Eggleton’s formula for the radius of the Roche lobe (Eggleton 1983), we find the distance D=(95 ± 10) x (0.86/F) parsecs, 43 o o 40,53 50 52 63,64 59 61,65 CM GS 51 CM.'x in -o CD CJ G5V GB G9 K0 K2 K3 K4 ? K5 K7 K8 MO spectral sequence o CO O GS 5 8 LTE DISK CM "O *— oa> “O=3 O a> ■«— o oo o CD 5 0 0 0 10^ 1 .5x10 disk temperature WELSH 1 0—SEP—1992 14:11 Figure 9: Reduced x2 °f various models fit to the “high resolution” spectrum. The models consist of a scaled star from the Gunn-Stryker (GS) Atlas + hydrogen slab in LTE. upper panel: minimum x 2 plotted against spectral sequence using the GS templates indicated, lower panel: minimum x2 versus disk temperature using the best tem plate, GS 58. 44 where F is the fraction of light at 5500A from the companion, constrained by the fits to be between 0.64 and 0.86. If we use the Barnes-Evans B-V relation, we find D=(110 ± 10) x (0.86/F). In both cases we adopted These determinations are not very sensitive to q, but are mildly sensitive to errors in 1 / 3 M2 (note distance oc M2 roughly); the estimated uncertainties quoted are based on letting M 2 vary from 0.6 to 0.8 M q. Because the Mg I b-complex is very sensitive to spectral type, we attempted to determine the contribution of the secondary star based solely on this feature, thus independent of any disk models. (The Ca II II and K and the Na I D lines are un suitable for this purpose because of contamination by strong disk emission.) Each GS template was scaled to match the Mg I absorption in our high-resolution spectrum of AE Aquarii in the region between 5067A and 5267A. GS star 57, HD 154712 (K4 V), gave the best approximation for the secondary star, and this scaled template is shown in Figure 4. This method agrees reasonably with the more global template + disk model method. But we caution the reader that a single observation of the Mg I b-lines may yield an unreliable measurement of the contribution from the secondary star. The line may be partially filled in by chromospheric emission from the secondary star or disk, which would result in underestimating the secondary star’s contribution and deducing an incorrect spectral type. For example, Echevarria et al. (1989) have published a spectrum of AE Aquarii observed during orbital phase 0.46 - 0.57 in which the Mg I b-lines are nearly absent. And in AH Her, the Mg I b-lines appear to be abnormally weak relative to the Fe I lines (Horne, Wade, & Szkody 1986). As 45 noted earlier, the behavior of the Mg I light curve is not consistent with that expected from ellipsoidal variations, and there seems to be an anti-correlation with the Ha and H/3 line flux. As these lines grow stronger, the Mg I flux deficit grows weaker, lending support to the idea that the Mg I b-complex may be contaminated with emission lines. 2.2.6 The Disk Spectrum As discussed above, we have constructed a model consisting of the sum of a template spectrum from the Gunn-Stryker Atlas and a disk spectrum. Fits were made to the high-resolution spectrum, and it should be kept in mind that effects from the large flare prior to this observation may still be present. For the disk spectrum we use a face-on uniform slab of pure hydrogen gas in LTE. (“LTE” is used in the sense that the Saha equation is used to compute the ionization states and the Boltzmann equation is used to compute the level populations.) The free parameters of this model are the template star, its scale factor, and the temperature T, baryon column density Nb, and radius of the disk Rd (the thickness L was held fixed at 10 9 cm). This simple parameterization is not expected to provide a detailed description of the disk, but rather, a global characterization. As mentioned earlier, we found no unique best-fit template star + LTE disk model. This ambiguity can be understood in the following way. The data have a Balmer jump in emission, which cannot be produced by any scaling of the secondary star’s spectrum. While matching this Balmer jump, disk models with T ~ 6 x 103K 46 T = 6 0 0 0 K H-& SECONDARY STAR O O DISK oO V) « - ~oD Oin *w P u o n 4000 5000 6000 7000 8000 9000 wavelength (A) WELSH 10-SEP-1992 14:07 Figure 10: Figures 10 and 11 show two different disk -f secondary star models, il lustrating the wide range of possible solutions. Shown in the upper panels of these figures are the AE Aqr spectrum (histogram, with error bars), the fit (solid curve), the scaled spectrum of HYAD 185 (dotted) and the disk spectrum (dashed). The lower panels show the residuals of the fit. This figure displays a low temperature disk model (T = 6 x 103K), showing the maximum disk contribution to the continuum. 47 o o n T= 10,000 K s ; S T CM E SECONDARY STAR O o DISK o tn t- -g§ in o 'in *- o 50004000 6000 7000 0000 9000 wavelength (A) WELSH 10-SEP-1992 H:07 Figure 11: High temperature disk model (T'=10 x 10 3K), showing the minimum disk contribution to the continuum. (See caption to Figure 10.) 48 produce considerable Paschen continuum emission, while models at T ~ 10 x 103I< do not. To compensate for these varying amounts of Paschen continuum emission, the contribution from the secondary star is altered. Hence the scale factor is correlated with the temperature of the LTE slab. The nature of the solution is such that the X2 surface for the determination of the temperature has a very broad minimum (see bottom panel of Figure 9), allowing a wide range of temperatures within our simple parameterization. Because the models are not at the optically thin limit, the degeneracy between area, density and thickness is lifted, and their values can (in principle) be determined independently. This permits an additional constraint, the requirement that the radius of the disk be less than the radius of the Roche lobe of the white dwarf. Using RSB’s determination of the system parameters and assuming the disk radius to be 0.8Rli, the radius falls between ~ 6-7 x lO 10 cm (using Table Al from Pringle & Wade 1985). This rules out all models using a K4 V or earlier template, but places no constraint on models using a later template. Permitted solutions give characteristic temperatures of the disk roughly in the range of 6—10 x 103 K. Using template GS 58 (HYAD 185), the cool disk (7 = 6 x 103 K) solution gives R d=3.7 x lO 10 cm and iVf,=2.0 x 10 24 nucleons cm~2, while the hot disk (7=10 x 10 3 K) solution gives Rd =5.5 x 109 cm and W(,=1.4x 1023 nucleons cm~2. The overall best fit model had 7=6360 K, Rd=2.2 x lO10 cm, and iV&=2.1 x 1024 nucleons cm~2. In each case there is a strong Balmer jump in emission. Because the models considered only continuum emission of a pure hydrogen gas, the 49 residuals of the fits show the Balmer lines and the Ca II H and K lines in emission (see lower panels of Figure 10 and Figure 11). The Balmer decrement determined from the residuals is fairly steep — Ha :: H {3 :: H7 :: H£ is 1.93 :: 1.00 :: 0.78 :: 0.46. The conspicuous absence of the Na I D line in the secondary star spectrum results from a gap in the spectral coverage in GS 58, and is not intrinsic to the star itself. Thus He I 5876A does not reveal itself in the residuals, though it must be present. If the characteristic temperature of the disk is ~ 9 x 10 3 K, then the disk con tributes only a small amount of Paschen continuum (ratio disk to total flux in the V band ~ 16%), in agreement with Wade’s (1982) estimate of 7 — 16%. However, at temperatures closer to 6 x 103 K we find a surprisingly red disk spectrum. For these cool disks, the ratio of disk flux to total flux in the V band is ~ 35%. Because the secondary star is somewhat evolved, the main sequence color - absolute magnitude relation given by Wade’s equation (2) may not be valid for AE Aquarii, explaining his lower estimate of the disk contribution. If the disk is in fact this red, then the near-infrared contribution from the disk is not negligible - the disk may contribute as much as one third of the flux in the R and I bands. At the H band (Ae// = 16500A) the contribution has dropped to less than 5% for both the hot and cool disks. 2.2.7 The Flare Spectrum Under the assumption that the light produced by a flare is simply added to the quiescent disk spectrum, the difference between the mean spectrum at the end of the run and the mean spectrum near phase 0.54 yields an estimate of the spectrum of a flare. The actual portions of the light curves used are marked in Figure 6 above the H/? light curve, and it is more correct to call this the spectrum of the rising portion of a flare. This flare spectrum, shown in Figure 12, has a large Balmer jump in emission, strong Balmer emission lines, He I A5015 and A5876 emission and a rising Paschen continuum. The resolution is too poor and the error bars are too large to be certain about the Paschen jump at 8200A, though P£ (A10049A) appears to be present in emission. Channel 23 (at A8014A) is negative, and because of its anomalous behavior as noted earlier, it is discarded in the following analysis. Fluxes and uncertainties are listed in column 5 of Table 1. We attempted to fit the flare spectrum using the same LTE slab model we previ ously used to model the disk. We fit only to the continuum regions to estimate the temperature, radius, baryon column density, and thickness of the slab. We found the reduced x 2 surface has a broad minimum (reduced x 2 ~ 8 with 18 degrees of freedom), suggesting flare temperatures between 7 x 10 3I< and 15 x 103K. These limits arise from the size of the Balmer jump and the slope of the Balmer and Paschen continuua (the higher temperatures have flatter continuua). The requirement that the flare region be smaller than the area of the disk adds an additional but weak constraint, ruling out temperatures less than 8 X 103K for the thin (L less than 10 8 cm) models. Because of the correlations between parameters (e.g. a high temperature and low column density model will conspire to give the same x 2 as a low temperature and high column density model), tighter constraints could not be placed on the parameters without using the Ha and H/? emission lines. With the parameters set by the continuum fits, we fit the lines by varying only the line broadening for the T = 8 , 10 and 15 xl03K models. The line broadening is parameterized as the Mach number of micro-turbulence, which allows the fluxes of saturated lines to increase. The 8 x 103I< model gave an accept able fit to the lines (reduced x 2=0.94 with 1 degree of freedom) with a turbulence of Mach 46. This Mach velocity suggests supersonic expansion or non-thermal broad ening in the flare region. The 10 x 103K model was unable to reproduce the observed line strengths (minimum reduced x2=15.8, Mach = 8 turbulence) and likewise for the 15 x 103K model (minimum reduced x2=604, zero turbulence). Because the LTE models do not include Stark broadening, the line broadening may be underestimated. While this may make the 8 and 10 xl03K constraints less stringent, it does help to rule out the 15 x 10 3I< model - the inclusion of Stark broadening could only make the fit become worse by allowing more flux in the line, and already with zero turbulent broadening the model predicts lines that are much stronger than observed. We find that for temperatures less than ~ 15 x 103K, the values of the emission measure times the flare area are not constant, implying that the gas is not at the optically thin limit, thus allowing radius, thickness and density to be approximately determined. The best fit model gives T —8 x 103K, i?=1010 cm, L=108 cm, Nb= 4 x 1022 nucleons cm ~2 and Mach=46 turbulence, but we strongly caution that these results should not be over-interpreted, especially since the reduced x 2 formally rejects these models, and at these temperatures one does not expect to produce the observed He I emission lines. 52 n FLARE SPECTRUM O Csl O O 4000 6000 8000 wavelength (A) WELSH 10-SEP-1992 14:02 Figure 12: The spectrum of the large flare at the end of the observing run (horizontal bars connected by dotted lines). The solid curve shows a fit using a hydrogen slab in LTE at T = 8 x 103 K. The squares represent the fit rebinned into channels identical to the data. The observed luminosities of the Ha and H/? lines are 1.63 and 1.85x lO 30 (d/100pc )2 erg s-1, respectively (assuming isotropic emission into 47 r steradians). Although the time between the spectra defining the pre-flare state and the flaring state was 38 minutes, the flare itself had a rise time of about 10 minutes. If we assume the de cline timescale is similar to the observed rise time, as seems plausible given the flare profiles seen in extensive white light photometry (Patterson 1979, Van Paradijs et al. 1989), then we can estimate the total energy emitted in Ha to be 10 33 erg radiated in a timescale of roughly 20 minutes. Note that because the duration of the flare is most likely longer than 20 minutes, this approximation gives a crude lower limit to the flare energy. This amount of energy is large compared to the typical Ha energy released in solar or stellar flares (e.g. the “great” flare on AD Leo (Hawley & Pet- terson 1991) resulted in a total of 7.5 x 10 32 erg in Ha using the crude conversion L(H7 ) = 0.33L(Ha) given by Butler, Rodono, & Foing 1988. For a review of solar and stellar flares see Haisch, Strong, h Rodono 1991). The U band energy is roughly 8 x 1033 erg , again large compared to typical stellar flare values (6.3 x 10 33 erg for the aforementioned flare on AD Leo). The Ha/H/? flux ratio is ~ 0.88, quite different from the mean disk spectrum, but not unusual for stellar flares (e.g. Mochnacki & Zirin 1980). It would be very interesting to see how closely stellar flares and flares seen in CV’s are related. For example it is known that the soft x-ray luminosity in stellar and solar flares is linearly correlated with the H 7 luminosity over four orders of magnitude in luminosity (Butler, Ronodo, & Foing 1988). It is also known that in stellar flares the 54 Balmer lines and the Ca II H and K lines rise to, and decline from, the flare peak more slowly than the U band continuum (see figure 8 of Hawley & Petterson 1991 for example). Because of the large and numerous flares and the fact that the system is a known x-ray emitter, AE Aquarii seems an ideal candidate for simultaneous x- ray and optical spectroscopic observations of the evolution of a flare to investigate a possible CV—stellar flare connection. 2.2.8 The Oscillation Spectrum The oscillation pulse profile at each wavelength was computed by removing trends in the light curves and folding onto the known 33.076737 second period using the ephemeris of Patterson (1979). Figure 13 shows the pulse profile for various continuum regions. As expected from the appearance of the PDS, the oscillations are present above the noise only in the shorter-wavelength light curves. The phase-folded light curves from different wavelength channels indicate that the data are consistent with a wavelength-independent pulse profile. A weighted mean pulse profile was therefore constructed and is shown in the bottom panel. The mean pulse profile shows a large peak at - - - 0 0.2 0 0.2 0.2 0 0.2 0.2 0 0.2 0.5 u r silto phase oscillation .1 Wi6814-7294 1.5 . n XX41 87-4747. Wv5467-5707_ XX3227—3627 moan 2 CM o CM 55 56 OSCILLATION SPECTRUM in o \/ o in o o o 4000 6000 8000 wavelength (A) WELSH 14-SEP-1992 17:12 Figure 14: The spectrum of the oscillations (horizontal bars connected with dotted lines) and aT = 19xl03K blackbody fit (dark solid curve). Also shown are T=10,800K and T =oo blackbody curves. By scaling this mean pulse profile to fit the pulse profile at individual wavelengths, the oscillation spectrum was estimated, and is shown in Figure 14. The scaling factor at each wavelength multiplied by the RMS of the mean pulse profile gives the RMS amplitude of the oscillations at that wavelength. Because the oscillations are near to or below the noise level in the red portion of the data, the error bars become very large. Fluxes and uncertainties are listed in column 6 of Table 1. The oscillation spectrum shows a blue featureless continuum with no obvious emis sion lines or Balmer discontinuity. A blackbody fit to the oscillation spectrum yields a temperature of 19 x 10 3K, and is shown in Figure 14 (the reduced x 2 is 0.74 with 28 degrees of freedom). Using this temperature and the observed flux, the projected area of the emitting surface is 1.1 x 1018 (d/100pc )2 cm2 (this is an RMS area; peak- to-peak area will be larger by a factor of ~ 2.8). Because the parameter space is highly nonlinear in temperature, in Table 2 we quote confidence intervals rather than simply giving ±lcr error estimates. Under the assumption that the oscillations are due to modulations in area (rather than temperature), the area inferred for the region producing the oscillations is small compared to the projected area of the accretion disk (~ 2 x 1021 cm2). This relatively small area argues against the hypothesis that the optical oscillations result from reprocessing of harder photons over a large area of the disk. However a power-law (/„ oc i/+“) with exponent a = O^Oi8;8^ also fits the data. If the optical oscillations are due to direct observation of entrained gas accreting onto the white dwarf, then the fact that the two pulses are not 180 degrees out of phase (see Figure 13) suggests that the accretion sites are not precisely on opposite sides of the star. This could arise if the magnetic dipole axis does not pass through the geometric center of the white dwarf. An alternative explanation is that the pulsed radiation is emitted anisotropically. The difference in the strength of the two pulses is most likely due to geometric viewing angle effects. White dwarfs with masses in the range allowed by RSB (~ 0.72 — 0.83714®) have projected surface areas of 1.7-1.4 x 10 18 cm2, hence the pulse emitting area is similar to the projected area of the white dwarf. This result is reminiscent of the conclusion of King &: Shaviv (1984) who argue that the hard X-ray emitting regions in interme diate polars occupy a large fraction of the white dwarf surface. We stress that our result for the emitting area is based on the observed flux and distance to the system (under assumption that the emission region radiates like a blackbody), and does not make use of any information from the shape of the pulse profiles. Because King and Shaviv’s polecap and occultation model has some difficulty with the observed energy dependence of the amplitude of the X-ray oscillations, an alternative model has been proposed (Rosen, Mason, & Cordova 1988) in which an arc-shaped accretion curtain replaces the cylindrical accretion column (for a discussion of the merits of these mod els see Norton & Watson 1989). The accretion curtain model predicts that the X-ray emission comes from a very small fractional surface area (~ 1%). So it would seem that the accretion curtain model favors the reprocessing of X-rays either in the disk or on the surface of the white dwarf as the source of the optical oscillations, rather than the direct observation of matter being accreted onto the white dwarf’s surface. Although our oscillation spectrum is noisy, the blackbody fits constrain the emit ting area to be a few times the white dwarf’s projected surface area. While this does not necessarily rule out the reprocessing model, it does not allow reprocessing to take place over a large area of the disk. However one must keep in mind the limitations of the data — a power law model fits equally as well as the blackbody model. Because the spectrum of the oscillations indicates that the amplitude of the os cillations increases toward the ultraviolet, observations in the UV and EUV should 59 be able to place far tighter constraints on the physical mechanisms producing the oscillations. Simultaneous high-energy observations will also be particularly useful for solving the oscillation problem, especially since we know the oscillations in this energy range have quite a different character — the X-ray (and gamma ray) oscilla tions show only one peak per 33-second cycle (Patterson et al. 1980; Eracleous et al. 1991; De Jager 1991). 60 Table 2: Confidence Intervals for Blackbody Fits to Oscillation Spectrum confidence level temperature (K) area (xlO18 (d/100pc )2 cm2) 68.3% (±1ct) 14,900 < T < 27,300 2.0 > A > 0.54 95.4% (±2or) 12,400 < T < 57,000 3.3 > A > 0.17 99.7% (±3 We have investigated rapid spectral variations in AE Aquarii using a 30-channel spectrophotometer on the Hale 5m telescope. 1.7 hours of observations were obtained at 4.3 second time resolution, covering the wavelength range 3227-10494A and orbital phases 0.438 to 0.607. We found the following: (1) The 16.5 and 33 second optical oscillations in AE Aquarii are very blue. We estimated the spectrum of the oscillations and found that a blackbody with a tem perature between 12-57 x 103K and area 0.2-3 x 10 18 cm2 fits the observed spectrum, as does a power-law f v oc i/+0-9. If the blackbody interpretation is correct and the modulation is total, the emitting surface area is a small fraction of the surface area of the accretion disk and is comparable to the projected area of the white dwarf. This suggests that the optical oscillations are produced on or near the surface of the white dwarf as a result of direct accretion, rather than by the reprocessing of harder photons over a large area in the disk. We also find that the two peaks in the 33 second oscillation pulse profile are not exactly 180° out of phase, and that the data are consistent with a wavelength-independent pulse profile. (2) The Balmer continuum light curves exhibit far more rapid variability than the Paschen continuum light curves. The Balmer continuum variations are smaller than those in the Balmer emission lines, both exhibiting flaring and flickering. The mean Ha/II/? ratio is ~ 3 but drops down to ~ 2.2 during a large flare. The wavelength- dependence of the variability suggests that flickering arises from an optically thin region with a temperature on the order of 10 4 K. 62 (3) The power density spectrum P(f) appears to be well represented by a power- law that falls off between 1/ / and l / / 2, the ambiguity due to the uncertainty in the determination of the white noise background level. (4) Using a uniform slab LTE model, we find a range of permissible temperatures characterizing the disk spectrum. All feasible disk models possess strong Balmer lines and Balmer jump in emission. The lowest temperature solutions (T ~ 6 x 103 K) have considerable and quite red Paschen continuum emission, contributing as much as 35% of the total flux in the V band. The hottest disk solutions (T ~ 10 x 103 K) have much less Paschen continuum emission, and contribute less than 16% of the total flux in the V band. (5) Modelling the AE Aquarii spectrum as the sum of a K star and a hydrogen slab in LTE, we find that the spectral type of the secondary star is most likely in the range K4-K5 V, and contributes between 64—86% of the light at 5500A from the system. Using the Barnes-Evans relation and the equivalent volume Roche lobe radius of the secondary star, we estimate the distance to AE Aquarii to be ~ 95 parsecs if the disk is hot or ~ 135 parsecs if the disk is cool. (6) A large flare with a ~ 10 minute rise time shows a Balmer jump in emission, strong Balmer emission lines, He I in emission, and a rising Paschen continuum. The Ha/H/3 ratio of the flare is ~ 0.9, quite different from the mean disk spectrum. LTE fits to the continuum yield a temperature between ~ 7-15 x 10 3 K. The Ha and Kj3 emission-line strengths confine the temperature to be closer to ~ 8 x 103 K, and their strengths suggest supersonic expansion velocities. 63 2.3 The Experimental Control As a control in the experiment, the star HD 19445 was observed at high speed with the same set-up for ~ 18.4 minutes, which was about 18% as long as the run on AE Aquarii. To check for instrumental effects, the power density spectra of AE Aquarii and HD 19445 were compared. Channels 1 - 3 (wavelengths 3227 - 3627 A) were combined for this test, as this wavelength range contains the strongest flickering and oscillation signal. The mean was subtracted from the data and a 10% split-cosine bell taper was applied prior to computing the power spectrum. A comparison of the power density spectra shown in Figure 15 reveals the following: (1) neither of the oscillations in AE Aquarii are instrumental in origin; (2) the spike at ~ 0.01 Hz is probably instrumental in origin, although the corresponding period of 97 seconds has no apparent origin; (3) there is considerably more power at low frequencies in the control star than expected, indicating that the light curve had long period drifts, perhaps due to atmospheric transmission changes. In Figure 16 and Figure 17 these power spectra are plotted on a log-log scale to help reveal the noise characteristics. Here we see that the control star is much flatter than AE Aquarii indicating a much “whiter” light curve. The horizontal line is the expected noise level based on Poisson statistics, and the high frequency variations seen in the control star are consistent with this noise. This suggests that the variations are not due to atmospheric scintillation. The AE Aquarii PDS lies above this level, indicating that flickering power is present up to the Nyquist frequency of the data. 64 comparison of AE Aquarii and HD 19445 PDS 1 g £ o 0k_) O5 a . H i 11*- I - I * - - . j - ■ , V) c ■oa> o iWVinrnjw^rwi^ 0 0.02 0 .0 4 0 .0 6 0 .0 8 0.1 frequency (Hz) WELSH 30-JUL-1993 14:20 Figure 15: Comparison of power density spectra of AE Aquarii (upper panel) and a comparison star HD 19445 (lower panel). The 16.5 and 33.0 s oscillations lie at frequencies 0.06 and 0.03 Hz and are easily seen in the AE Aquarii PDS. The vertical scale is in units of mJy2/Hz~1. 65 AE Aqr Nyqu st freq H 1------1------1------1------H H 1 i 1------1------1------H fL •oQ) L. CN s 1 o a -3 .5 - 3 -2 .5 - 2 -1 .5 - 1 log frequency WELSH 18-JUL-1MS 01:30 Figure 16: Power density spectra of AE Aquarii is shown on a log-log scale.. The dotted vertical lines indicate the 16.5 and 33.0 second oscillations, and the Nyquist frequency. The horizontal line indicates the noise level expected for pure Poisson noise. The horizontal axis is in units of (log) Hz and the vertical scale is in units of (log) mJy2/Hz~1. Compare with Figure 17. ih iue 16. Figure with Figure 17: Power density spectra of HD 19445 is shown on a log-log scale. Compare Compare scale. log-log a on shown is 19445 HD of spectra density Power 17: Figure log power density CN O 3. 3 - .5 -3 I I E N D 94 CHANNELS 1+2+3 MEAN 19445 HD 2.5 -2 o frequency log -2 1.5 -1 ES 1-U-93 01:18 WELSH 18-JUL-1993 1 - yus freq st Nyqu 66 67 2.4 Additional Results on Flickering In Section 2 of this chapter a flare spectrum was presented and analysed. As men tioned in the introduction, the difference between flaring and flickering may be only semantics. Here it is suggested that indeed the flaring and flickering spectra are the same. The published paper omitted an analysis of the flickering spectrum because of difficulty determining the correct error bars. The flare spectrum, with its much higher signal-to-noise ratio, was very similar in appearance. Because the flare spectrum was so easy (and intuitive) to determine, and had well-defined error bars, it was studied in place of the flickering spectrum. The spectrum of the flickering can be determined in the following way. Each of the 30 wavelength channels produce 30 independent light curves. The root-mean-square (rms) of each light curve gives a measure of the amplitude of the fluctuations in that channel. These amplitudes, plotted as a function of wavelength, give the spectrum of the fluctuations, i.e, the rms spectrum is the spectrum of the flickering. A few comments are necessary. The rms is a biased statistic in the sense that taken at face value, it overestimates the degree of variability. This can easily be seen in the case where the measured data consist of noise but no signal. The rms statistic will always yield a positive value, even though there is no signal. However if the characteristics of the noise are known, than this bias can be corrected for, and a “noise-corrected rms” can be used. 68 It is possible to weight the rms (whether or not noise-corrected), simply by mul tiplying each term in the sum by a certain weight and normalizing the results by the sum of the weights. An optimal choice for the weights is usually the inverse-variance of the data. When computing the rms, the question of “rms with respect to what?” should come up. If it is computed with respect to zero, the rms measures the total power. It may also be computed with respect to the mean, giving the rms amplitude of the fluctuations about some constant level. But there is no requirement that either of these must be used. By computing the rms with respect to higher order fits (e.g. parabolic, cubic, etc.) the rms becomes less and less sensitive to slow trends. The data are effectively high-pass filtered, and the rms then measures only the faster variations. We now give a precise definition of the “noise-corrected, weighted rms”: IZlridi-ffy/af] N rms - [S £i(*-fl)2/«*l N p - \ | [. EZhiM £ i iM . J LX e i ^ l i i 1 M.M. where ff is a fit of degree p to the data d,, are the uncertainties in the data and N is the number of data points. The first term measures the fluctuations in the data and the second term corrects for the presence of noise. If the residuals (dt- — ff)2 are on average equal to the of, then both the fluctuation and noise terms are equal and the rms is zero. For the rms spectra discussed below, a parabolic fit was used to remove trends from the data (p = 3). The flickering spectrum is shown in the upper panel of Figure 18. The Balmer jump and lines are clearly seen in emission. The cause of the large emission feature at 9000A is unknown, but is is suspected to be atmospheric, as discussed below. The flickering spectrum is similar to the flare spectrum shown in Figure 12. The slope of the Balmer continuum is steeper in the flare spectrum, and the H/? line is stronger with respect to the Ha line, so it is not expected that the temperatures and densities derived from the two spectra should exactly agree. But qualitatively they are the same, in the sense that both result from optically thin gas around T = 10,000 K. Because the large flare at the end of the run may dominate the rms, the rms was recomputed using data which omit the last 150 out of the 1201 points (the light curves were truncated at roughly phase 0.59). This flickering spectrum is denoted with the open circles in the upper panel of Figure 18. It is also apparent that there is signal in the control star. This is unexpected. If the fluctuations seen in HD 19445 are consistent with photon counting noise, the noise- corrected rms spectra should be consistent with zero. Two possible explanations for this anamolous behavior are: ( 1) the formula given above for the noise-corrected rms fails to remove enough of the noise contribution and ( 2) there are variations in the data which are not properly modelled by assuming photon counting noise. Explanation (1) can easily be tested by creating fake data sets with known noise properties. A fake light curve was produced that had an rms = 1.0, a 1/ / PDS spectrum, and was sampled in the same way as the MCSP data. Gaussian white noise ((7=0.8) was added to the data to simulate observational data, so now the uncorrected rms = 1.6306. The noise-corrected rms was then computed and gave rms = 0.9953 ±0.0296. Even though this was only one test, it shows that the method 70 AE AQR and HD 19445 RMS Flickering Spectra oo ro o 4 0 0 0 0 0 8 0 0 060 10' wavelength (A) Figure 18: RMS Flickering Spectra. Noise-corrected, weighted RMS flickering spectra for AE Aquarii {upper panel) and the comparison star HD 19445 {lower panel). The Balmer lines and jump are clearly seen in emission in AE Aquarii. An unexplained red component is seen as well. The open symbols in the upper panel denotes the AE Aquarii flickering spectrum derived from data that exclude the large flare at the end of the run. 71 is not grossly incorrect. The remaining explanation, that there are additional fluctuations in the data, is probably correct. Assuming that the variations are not intrinsic to the star, the ad ditional noise must be instrumental and/or atmospheric. As the PDS in the previous section showed, the noise is probably not scintillation or very high frequency noise, but rather low frequency drifts. These may be due to changing atmospheric condi tions. It is interesting to note that the spectrum of these variations is red. Although a cause for this noise is not known, it does suggest that the red rise seen in the AE Aquarii flickering spectrum is not real. This has the effect of flattening the spectrum, making it easier to be fit with an LTE recombination spectrum. But not all of the redness can be ascribed to instrumental/atmospheric effects because there is a red component present in the flare spectrum, as well as both AE Aquarii rms spectra. As stated earlier, a quadratic fit was used to remove trends from the data. The results are qualitatively the same if up to about a 6th order polynomial is used. Far above that the spectrum becomes featureless within the noise except for the rising red component. In fact this red component remains in the rms spectra even for high- order polynomial fits (p > 12) which suggests a third origin for its existence: (3) The error bars on the red portion of the data are underestimated. This could result from a calibration error in the reduction process. Before making a digression on rms calculations, the flickering spectra results can be summarized as follows. The spectrum of the flickering in a 1.7-hour segment of AE Aquarii’s light curve has been extracted via a noise-corrected, weighted rms. 72 The spectrum shows strong Balmer lines in emission and a Balmer jump in emission. A red component is seen in the spectrum, and its origin is not understood, either intrinsic or extrinsic. The flickering spectrum is similar to the spectrum of a flare seen in AE Aquarii, suggesting that a common mechanism is responsible for both. Now for a short digression on rms calculations. In addition to high-pass filtering, low-pass filtering is possible. This is done by replacing the data with a high-order fit to the data. These “new data” are now insensitive to any variations faster than the order of the fitting function — in other words, it has been smoothed. The rms of this smoothed data measures only the slow variations. By combining the low- pass and high-pass filtering, the rms can be tuned to any specific frequency interval and is therefore “notch filtered” or “bandpass filtered”. (There are of course certain limitations. For very high order polynomial fitting, i.e. the degree of the polynomial is larger than ~ 2 y/N where N equals the number of data points, the polynomials tend to develop numerical oscillations, thus introducing spurious high frequency noise.) While sinusoids could be used as the fitting functions (thus giving the Fourier components and making filtering very simple), they have the big disadvantage of not representing the data well until a large number of terms are used. Slow trends are particularly poorly represented, thus rendering sinusoids ill-suited as general purpose filtering functions. When filtered versions of the data are used, the data points are no longer uncorrelated and using inverse-variance weights is no longer optimal. For filtered rms calculations, equal weights may be used. Computing the error bars of a noise-corrected, filtered rms is tricky. The filtering leaves power only in a certain 73 frequency window. If the noise is white, then it is possible to simply scale the noise power by the ratio of the width of the bandpass window to the full window out to the Nyquist frequency. Because polynomials and splines do not have sharp frequency cut-offs, simulations with white noise models should be used to numerically determine the bandpass of the filtering when these functions are used. As discussed earlier, Elsworth & James (1982) found P{f) oc l / / 2 in their white- light observations of AE Aquarii, while our data seem to favor a PDS that falls off more like 1//. This is a tricky measurement to make, however, and is very sensitive to the placement of the white noise background level. The differences between these two results are perhaps a result of this background placement. However, it may be possible that both results are correct, in which case two explanations come to mind: (1) It may be that the PDS changes from night to night. This does not seem unlikely for a random-walk-like process, and exactly this effect was noted by Bruch (1991). (2) The discrepancy may be due to differences in the temporal frequency windows in which AE Aquarii was observed — our data extend to significantly lower temporal fre quencies than Elsworth and James’, and they have sensitivity to significantly higher frequencies. There is considerable overlap, however, and this explanation would re quire that the faster flickering have a different power distribution than the somewhat slower flickering, i.e., the power in the faster flickering rolls off more rapidly with temporal frequency than the slower flickering. If true, this would be very interesting. 74 Ha Power Density Spectrum (O 0 0.02 0.04 0.06 0.08 frequency (Hz) WELSH 20-JUL-1993 03:07 Figure 19: The Ha (channel 19) power density spectrum is shown. The dotted vertical lines show the expected frequencies of the 16.5 and 33.0 second oscillations. No evidence of oscillations is seen. Earlier in this Chapter it was stated that no pulsations were seen in either the H a or H /3 light curves above those in the nearby continuum. In Figure 19 we show the Ha (channel 19) power spectrum, computed after removing trends the light curve with a 6th degree polynomial. As claimed, no oscillations are apparent above the noise background. 75 2.5 Using the Mg I b-complex as a Diagnostic of the Sec ondary Star In the published section of this Chapter, the derivation of the secondary star’s spec trum was based on a global stellar template + LTE disk model. A brief mention of using the Mg I b-lines to determine the properties of the secondary star was made, and the agreement between the two methods was noted. However, to be conservative, the method relying only on the Mg b-complex was discarded as unsafe because of possible contamination by disk emission. This argument was supported by the un explainable shape of the Mg I light curve. However, with the new orbital phasing derived with the new ephemeris discussed in Chapter 3, the Mg I light curve behaves as expected for ellipsoidal variations, and the worry that the line is contaminated is now substantially weaker. Thus it is perhaps not too unsafe to use the line as a diagnostic. Spectra of numerous G, K and M stars from the Gunn-Stryker atlas (Gunn & Stryker 1983) were scaled so that their Mg I b-complex best matched the Mg I b- complex in our AE Aquarii spectrum. (A linear fit to the continuum was removed before the scaling.) These rescaled spectra were then used as models for the secondary star in AE Aquarii. Of course not all models were acceptable. Most were easily rejected because in order to match the Mg I absorption in AE Aquarii they required a rescaling that resulted in a spectrum that exceeded the observed flux from AE Aquarii. The acceptable models were found to lie in the range K4 V to MO V. Other criteria were applied to the models (such as how well the model Mg I b-complex matched that 76 of AE Aquarii) but it was not possible to claim one model was significantly better than another on the basis of the MCSP data alone. The results of this modelling is illustrated in Figure 20. The circles represent dwarfs and the squares represent giants and subgiants from the Gunn-Stryker atlas. The vertical placements of the stars was determined by rescaling the V magnitudes of the stars by the same same scale factor needed to match the Mg I absorption complex. In essence this gives the V magnitude the star would have if it were at the distance of AE Aquarii. The R - I colors were computed for each star using synthetic photometry (via the software package XCAL, Horne 1990). The dotted lines are lines of equal distance, computed using the Barnes-Evans relation (Barnes, Evans, h Moffett 1978). The filled-in symbols represent the acceptable models. As stated earlier, we could not rule out any of these models, though the models at smaller values of R - I are slightly favored (earlier spectral type and more distant). From this figure, we estimated the distance to AE Aquarii to lie between ~ 80 and ~ 150 pc. Note that the lower the star lies below the dashed line, i.e. the larger the V magnitude, the larger is the accretion disk contribution to the V magnitude. For comparison with Figure 9, acceptable models were produced using the stars GS 57 through GS 65. 2.6 Brief Comments on the Inclination and Secondary Star Size RSB give upper and lower limits on the mass ratio and inclination. The system must be rather close to the lower mass ratio and lower inclination or else an eclipse of the disk would occur. It must remembered that the work of Chanan, Middleditch, & 77 05 .100 o '1.40 !*•< >... '1B0 CM •HD'154712 0.2 0.4 0.6 0.8 1 R—I WELSH 30-JU L-1M 3 12:54 Figure 20: Stars from the Gunn-Stryker atlas are plotted in a color — magnitude diagram (R - I color index versus rescaled V magnitude). The rescaling is done by matching the strength of the Mg I b absorption-line complex with that seen in AE Aquarii. The filled symbols give acceptable solutions. The squares represent giants and subgiants; the circles represent dwarf stars. Lines of equal distance are shown as dotted lines. The horizontal dashed line shows the total V magnitude of the system and represents a hard upper limit to the contribution from the secondary star. 78 Nelson (1976), which constrains q and i from the eclipse width, is valid only for a point source at the location of the primary star. An extended disk requires a lower inclination or mass ratio in order to be consistent with the lack of eclipses seen in AE Aquarii. It is claimed in the literature that the secondary star in AE Aquarii must be somewhat evolved in order for it to fill its Roche lobe (Patterson 1984; Chincarini &; Walker 1981). This statement was checked using a wide range of secondary star masses (0.6 - 0.8 M q), mass ratios (<7=0.65 - 0.94), and assumed mass-radius relationships (the empirical ZAMS of Patterson 1984; the empirical ZAMS of Lacy 1977; the model ZAMS, model TAMS, and empirical ZAMS of Demircan & Kahraman 1991; and Echevarria 1983). In all cases, the inferred radius of the secondary star based on its mass is smaller than its Roche lobe. If the secondary star does not fill its Roche lobe, then it probably maintains a near-spherical shape and ellipsoidal variations should not be seen. We will return to this point again in Chapter 4. On the other hand, if the secondary star does fill its Roche lobe, then it must be slightly evolved, and therefore it is dangerous to use a main-sequence mass-radius relationship to determine any of the system parameters. CHAPTER III ON THE LOCATION OF THE OSCILLATIONS IN AE AQUARII This Chapter focuses on determining the location of region responsible for producing the optical oscillations in AE Aquarii. Research on this topic was inspired by the very puzzling findings of Robinson, Shafter, &; Balachandran (1991): the oscillation time-delay curve is phase shifted by ~ 60° with respect to the emission-line radial velocity curve, although both should track the orbital motion of the white dwarf. In this Chapter we present simultaneous absorption and emission-line radial velocities, and from the absorption lines we derive an improved orbital ephemeris. We find that our emission-line velocities are phase shifted by ~ 75°, and hence are unreliable tracers of the orbital motion of the white dwarf. However the oscillation orbit is shifted by only 5° ± 3°, confirming that the oscillations arise from a region concentric with the white dwarf. Thus the arrival times of the oscillations yield a measure of the white dwarf’s apparent orbital velocity, independent of emission-line radial velocity measurements. This Chapter contains previously published material, reproduced with permission from the Astrophysical Journal Letters, American Astronomical Society. The name of the paper and the reference is: “On The Location of the Oscillations in AE Aquarii”, 79 80 Welsh, W. F., Horne, K. & Gomer, R. 1993 ApJ 410, L39. The version given here is nearly identical to the published version. Section 2 contains fine details concerning the ephemeris determination and in Section 3 a brief note on the mass determinations is presented. 3.1 On the Location of the Oscillations in AE Aquarii 3.1.1 Introduction The nova-like variable AE Aquarii is an extraordinary member of the class of in teracting binary stars known as the cataclysmic variables. The system has a rela tively long orbital period (9.88 hr) and is non-eclipsing. Among the many exotic properties of this system are the 33.08 s oscillations detected in the X-ray bands (Patterson 1980) and accompanied by a first harmonic at a period of 16.54 s in the optical and UV (Patterson 1979; Eracleous et al. 1993). Detection of these oscil lations has also been reported in the 7 -ray band (Meintjes et al. 1992). Patterson (1979) discussed much about the phenomenological nature of the optical oscillations and found that (1) the period of the oscillations is extremely stable (Q = | 1/ P | ~ 8 x 1011) and ( 2) the relative amplitudes and waveforms of the oscillations can vary by large amounts in a short time. These facts led Patterson to conclude that the mechanism driving the oscillations is rotation of the white dwarf. He examined the pulse arrival times and found them to agree with the expected orbital motion of the white dwarf, and used the pulse-timing orbit to define the motion of the white dwarf (K\ = K puise = 127 ± 5 km s -1 ). 81 However Robinson, Shatter, & Balachandran (1991, hereafter RSB) derived a new orbital ephemeris and found the oscillations were not coincident with the white dwarf. This discrepancy was very large — the oscillations led the white, dwarf by ~ 60° in orbital phase. Using the amplitude and phase of the pulse-timing delays, the data suggested that the oscillations originated from a region outside the white dwarf’s Roche lobe (see RSB’s Fig. 6). Because this seemed implausible, RSB concluded that the observed optical oscillations could be a mixture of direct and reprocessed light. Marsh (1992) developed a model in which reprocessing from a region near the mass transfer stream impact spot could produce the observed pulse timings. As RSB point out, if the optical oscillations are reprocessed EUV/X-ray photons, there may be a period difference between the optical and X-ray oscillations. A slightly longer optical period is predicted if reprocessing is taking place on a target fixed in the binary frame (the first lower sideband of the orbital and white dwarf spin periods). Eracleous et al. (1991) and De Jager (1991) address this issue and find no difference in the X-ray and optical periods. However the question of what is the true spin period of the white dwarf and what is the sideband period has not been definitively answered, and so the issue is not settled. Welsh, Horne, & Oke (1993, hereafter WHO) determined the optical spectrum of the oscillations and found them to be quite blue: /„ oc voa. A small, hot blackbody (T 12,400 and Area ~ 3.3 x 1018 cm2) could also produce the observed optical spectrum. If the blackbody interpretation is correct, their result argues against the reprocessing model and in favor of a compact region close to the white dwarf. 82 In this Chapter we reexamine the question of where the oscillations arise. In Section 3.1.2 we describe briefly the details of the observations and in Section 3.1.3 we describe the methods used to obtain an improved orbital ephemeris. In Section 3.1.4 we show that the oscillations arise on or near the white dwarf, and discuss the implications, including deriving new system parameters for AE Aqr. 3.1.2 Observations For brevity, details of the “Cable Experiment” data acquisition will be discussed elsewhere (Welsh, Horne, &; Gomer, 1993b) [see Chapter 4 and Appendix A]. High speed spectroscopy was obtained with the Mt. Wilson 2.5 m Coude Shectograph on 27-30 July 1981, covering wavelengths ~ 6360 — 6820 A at a resolution of ~ 1.5 A. Simultaneous photometry (bandpass matching the spectrograph) was obtained on the Mount Wilson 1.5 m using the data acquisition system described in Horne &; Gomer (1980). A signal was sent from the Shectograph to the photometer via a shielded cable indicating exactly when simultaneous data were being recorded. This allowed us to synchronize the spectroscopy and the high speed photometry to roughly 0.01 ms. The spectroscopy was flux calibrated in the standard manner and then rescaled to agree with the photometry, thus negating any possible slit losses or guiding errors, resulting in spectroscopy that is truly photometric. For the purposes of this Chapter, each night’s high speed data were rebinned to 10 km s~l pixel-1 and into 50 orbital phase bins, resulting in a total of 200 spectra. Figure 21 shows an orbit-averaged spectrum folded on the ephemeris derived in Section 3.1.3. Note by comparison with the spectrum of the K5 V star 61 Cyg A 83 o o CO o o o o OJ o 66006500 6700 wavelength (A) WELSH 3-FEB-198J 14:48 Figure 21: Mean spectrum of AE Aqr, after removing the orbital motion of the secondary star. Also shown is 61 Cyg A (K5 V) (arbitrarily rescaled) for comparison. the numerous absorption lines arising from the secondary star. 3.1.3 The Orbital Motion of the Secondary Star To determine the secondary star’s apparent orbital velocity K 2, we computed the cross-correlation between the AE Aqr spectra and 61 Cyg A, whose radial veloc ity is —64.5 ± 2.2 km s-1 (Beavers & Eitter 1986; Abt 1973). A K5 V star was chosen because there is good evidence that the secondary star is approximately of 84 this spectral type (WHO, Bruch 1992, Chincarini & Walker 1981, hereafter abbre viated as C&sW). A linear fit to the continuum was subtracted from each spectrum, and using the spectral regions 6400-6530 A, the cross-correlation was computed for shifts up to ±1500 km s-1. Once the cross-correlation peaks were found and obvious misidentifications corrected or deleted (a total of 13 points needed correction, and of these, 3 were deleted), the data were folded onto RSB’s ephemeris (To=2439030.830, P=0.4116580) and a sinusoid of fixed period was fitted to the radial velocity curve. We noticed a large offset between the expected and actual phasing of our data (A(/>o = 0.1886 ± 0.0010), much larger than the formal uncertainty (~ 0.0103) at the time of our first observation (HJD 2445177.72113). It was not possible to revise the ephemeris because too few orbital cycles were spanned by the Cable data alone. To extend the baseline of radial velocity measurements, we combined our data with the absorption-line velocities measured by Chincarini & Walker (1981) and Joy (1954). [Note that C&W’s data are not included in their published paper (Chincarini & Walker 1981), but are given in their ESO preprint (Chincarini & Walker 1981b). We shall refer to both of these publications by the same symbol “C&W”, with the explicit assumption that the reader should refer to Chincarini &: Walker 1981b for access to the data.] These measurements were of (mostly Fe I) absorption lines in the 4045-4415 A range and span HJD 2430980 to 2438979. Along with the Cable data, a total of 14201 days (~ 39 years) are spanned by 552 measurements. The revised ephemeris is given in Table 3. Both the period and the epoch have been improved (To is defined as the superior conjunction of the white dwarf with respect to the secondary star, i.e., phase zero defined as the time of eclipse of the white dwarf if AE Aqr were an eclipsing system). We folded the combined data on this improved period and determined improved values of To, K 21 and 7 (see Table 3). The 1 1983). The resulting radial velocity curves for the Cable data and for the combined data sets are shown in the lower panel of Figure 22. To verify that our ephemeris was an improvement, we folded the combined data set using the periods given by RSB, C&W, Feldt Chincarini (1980, hereafter F&C), and Van Paradijs, Kraakman, & Van Amerongen (1989) and computed a best fit sinusoid. The residuals indicate the best fit is indeed generated by our new ephemeris [see Table 5 and Section 3.2 in this Chapter]. In addition, we recomputed the phases of WHO’s multichannel photometry and found that the unexplained rising red component seen in their data can readily be explained by ellipsoidal variations expected from the secondary star. Other details of this ephemeris determination will be discussed in Section 3.2, Appendix C, and in Welsh, Horne, &; Gomer 1993b. 86 Table 3: Orbital Ephemeris of AE Aquarii To Period K 2 7 2 (HJD) (days) (km s -1) (km s " 1) 2439030.7879 0.411655601 157.9 -66.7 ± 0.0011 ±0.000000056 ± 0.8 ±0.7 87 o t ■ i — 1_ t _- > » i i ■ i i i— » < i ■ . . i ...i -1 -0.5 0 0.5 1 Binary Phase weush 7- f e b - i w j 21.24 Figure 22: Radial velocity curves, using the ephemeris given in Table 3. (a) Si multaneous absorption-line (filled circles) and emission-line (open circles ) velocity measurements obtained from the Cable data, (b) The combined absorption-line mea surements of Joy, C&W, and the Cable data. 3.1.4 Discussion Using our new orbital ephemeris, we computed the phase difference between the sec ondary star orbit and the pulse-timing orbit as given by RSB: To=JD©2445172.284(3). Here To is defined to be the superior conjunction of the pulse-timing orbit, i.e., when the region producing the oscillations is farthest away from us. We found a phase 88 difference A We believe that our results differ from RSB’s because RSB used the Ha emission line to compute their ephemeris, while we used absorption lines. Absorption lines originate on the secondary star, and are quite reliable for determining the ephemeris. However the emission lines originate in the disk, and because the disk can have non-axisymmetric features (e.g. the bright spot, resonances and tidal disturbances, magnetic distortions, etc.) the emission lines may not reliably track the radial velocity of the white dwarf (for a discussion see Shafter 1991). RSB were of course aware of this and relied on very careful measurements of the emission line using the double- Gaussian technique and a “diagnostic diagram” (Schneider & Young 1980; Stover et al. 1980; Shafter 1983). The radial velocity curve they obtained is sinusoidal and free of distortions. Even more convincingly, the phasing derived from the emission lines agreed with the phasing derived from the absorption lines given by F&C. Thus it appeared that the emission-line velocities were not biased, and a reliable white dwarf radial velocity curve could be determined from emission-line velocities. However folding Joy’s (1954), C&W’s, and the Cable data onto F&C’s ephemeris revealed significant absorption-line phase offsets. This is not true when the data 89 sets are folded onto the new ephemeris. This suggests that F&C’s ephemeris lacks sufficient accuracy, and if true, then RSB’s conclusion that the emission-line radial velocity measurements track the motion of the white dwarf becomes less robust. It is then possible that the emission lines are in fact biased and cannot be used to derive an accurate orbital ephemeris. To test this hypothesis, we computed Ha radial velocity measurements from our Cable data via the double Gaussian method using Gaussian widths of 90 km s _1 and separations of 800 and 1200 km s -1 (similar results were obtained when Gaussians of 270 km s -1 were used). The data were then folded on our ephemeris and the result is shown in Figure 22a. The Ha radial velocity curve shows a large phase shift and is rather distorted (error bars are ~ the size of the symbol). The phase shift seen in our simultaneous data clearly demonstrates that the Ha line does not reliably track the white dwarf and cannot be used to derive an accurate ephemeris for AE Aqr. In Figure 23 we show the phase shift associated with each of the data sets. The RSB and Cable Ha measurements show the same phase shift, lagging the absorption-line orbit by ~ 70—80°. Disagreement between the phasing of the emission lines and the absorption lines (or eclipses) is not uncommon — for example the radial velocity curve derived from the Ha emission line in the nova-like variable PG0027-(-260 (a “SW Sex” star) appears undistorted but nonetheless is 76° out of phase with the secondary star (see Fig. 5 of Thorstensen et al. 1991). Currently there is no widely accepted explanation of the cause of these phase shifts. Note that the emission-line data of Joy (1954), based on 90 numerous low-resolution lines shortward of ~ 5000 A (mostly Ca II and Balmer lines), is not phase shifted. This suggests that the Ha line may be particularly sensitive to emission that is not centered on the white dwarf. The phase lag of ~ 75° suggests that the Ha line is strongly contaminated by emission from the bright spot, where the accretion stream from the secondary star collides with the edge of the disk. We have found that even though averaged Ha line profiles can look rather symmetric (e.g., Figure 21 and RSB’s Fig. 1), individual spectra can show enormous variability (see Appendix B and also Welsh, Horne, & Gomer 1993b). Combining these new results with those of WHO’s, there is now very strong evi dence that the origin of the optical oscillations is localized to either the white dwarf’s surface or the innermost regions of the accretion flow. The equality of periods of the X-ray and optical pulses is naturally explained, both equal to the spin period of the white dwarf. Because the optical oscillations arise very close to the white dwarf, the pulse- timing orbit should give an unbiased measure of the white dwarf’s orbital motion. Yet the pulse-timing orbit is surprisingly “noisy” and distorted (RSB). It is not clear whether these are intrinsic fluctuations, possibly due to individual accretion events, or simply statistical noise. In consideration of the problems that arise when using emission lines to determine orbital velocity, we are not surprised by RSB’s result that the orbital velocity derived for the emission lines Remission — 141 ± 8 km s -1 is considerably larger than K puise — 122 ± 4 km s-1. Note that from examining Fig ure 21 it appears that the secondary star should contribute a significant amount of 91 narrow Ha absorption, anti-phased with the Ha emission. The effect would be to bias radial velocity measurements high, thus one should expect K emissi0n > Kpulse■ Defin ing K\ to be RSB’s determination of Kpuue we obtain q = M2/M i = 0.773 ± 0.026 and M \sinzi = 0.528 ± 0.017Mq. Using the constraints that the inclination i must be less than 70° and M2 ~ 0.7 Mq (see RSB), we find 0.64 < M i /M q ~ 0.91, 0.49 < M2/M q ~ 0.70, and 56° ~ i < 70°. 92 + .cco 3x104 3.5x104 4x104 4.5x104 HJD - 2 4 0 0 0 0 0 WELSH 7-FEB-199J 21:30 Figure 23: Phase offsets between various data sets and the ephemeris given in Table 3. The filled symbols represent absorption-line data and the open symbols represent emission-line data (Joy: squares, C&W: circles, Cable: stars, RSB: diamond. The emission-line phases have been corrected to account for the expected 180° phase shift. 93 3.2 Ephemeris Details In this Section further information on the ephemeris derivation is presented. Details concerning the observation will be presented in Chapter 4, and the very fine details of the data are left for Appendix C. It should be first noted that the radial veloci ties were measured by cross-correlating the data with the K5 V star 61 Cyg A. A Gaussian fitting routine (ANTARES) was used to find the principle peaks of each cross-correlation function, using a Gaussian of width 20 km s-1. This narrow Gaus sian insures that very little of the wings of the cross-correlation function contaminates the determination of the peak. It has been empirically determined that only the peak of the cross-correlation function should be used (see Wade k Horne 1988). Using the wings of the cross-correlation function degrades the determination of the peak, as should be expected if one includes data whose correlation is by definition low. The radial velocities obtained from the “Cable Experiment” data were not suffi cient to revise the orbital ephemeris because they span only 4 days. This is demon strated in Figure 24, where the broad minimum extends over many hundredths of a day (tens of minutes). The dotted vertical line is the best period solution obtained by combining all the data, as described in Welsh, Horne, k Gomer (1993a), hereafter referred to as WHG. In Figure 25, the periodogram for Joy’s (1954) absorption-line data is shown, and in Figure 26, the periodogram for Chincarini k Walker’s (1981) absorption-line data is shown, on the same horizontal scale. Figure 27 shows the periodogram for the combined data sets, covering a very wide range in period, 0.1 to 1.5 days. 94 Cable data o oo ta a t o 0.4 0.405 0.41 0.415 0.42 Trial Period (days) WELSH 30-JUL-1993 14:46 Figure 24: Periodogram of Cable Data. Periodogram of the absorption line radial velocities in the Cable Experiment data. The broad minimum is tens of minutes wide, far too big to accurately constrain the orbital period. The dotted vertical line is the period determined by WHG. than in the Cable data periodogram, allowing a far better determination of the orbital orbital the of determination better far a allowing periodogram, period. data Cable the in than Figure 25: Periodogram of Joy’s Absorption line data. Much finer structure is present present is structure finer Much data. line Absorption Joy’s of Periodogram 25: Figure x2 / 101 0.405 o' asrto—ie data absorption—line Joy's ra Pro (days) Period Trial 0.41 0.415 ES 3-U—93 1*:38 WELSH 30-JUL—1993 95 with Figure 24 and Figure 25. Figure and 24 Figure with Figure 26: Periodogram of Chincarini Chincarini of Periodogram 26: Figure x2 / 246 m o o 0.4 .0 0.41 0.405 &' asrto-ie data absorption-line C&W's ra Pro (days) Period Trial h Walker’s Absorption Line Data. Compare Compare Data. Line Absorption Walker’s .1 0.42 0.415 E S 30J 19 14:41 -1993 L WELSH 0-JU 3 96 97 combined data sets ooO i A o CD 00 iD \ •Vi § o 0.6 0.8 1 Trial Period (days) ...... WELSH 30-JUL 30-JU L -1993 15:02 Figure 27: Periodogram of Combined Absorption Line Data. The data sets of Joy, Chincarini & Walker and the Cable experiment were combined, thus the data span nearly 40 years. The horizontal scale on this figure is much wider than in the previous three periodograms, to show the more global characteristics of the periodogram. 98 combined data sets oco ID 00 m 2.4286 2.4288 2.429 2.4292 2.4294 2.4296 2.4298 2.43 Trial Frequency (cycles/day) WE1J5H 30-JUL-1993 15:14 Figure 28: Detail near x 2 minimum of the periodogram of the combined absorp tion-line data. Note the horizontal scale on this figure is in frequency units. The dotted lines correspond to RSB’s period (left) and WHG’s period (right). In Figure 28 we zoom in on the minimum of the periodogram. The abscissa is now in frequency units rather than period units to facilitate in searching for cycle-count errors. Plotted against frequency, the periodogram has equally separated minima, with (A frequency )-1 being equal to the gap in time between data sets. The dotted lines show the solutions obtained by RSB and WHG. 99 The incorrect ephemeris derived by RSB seemed somewhat surprising. Because RSB published a table of their radial velocity measurements, we were able to repeat all of their analysis (with and without the emission-line data of Joy 1954). We obtained identical results. We also folded RSB’s data onto both RSB’s and F&C’s ephemeris and verified that the phase shift is very small. We wish to make it explicitly clear that RSB’s analysis was entirely correct, given the data available to them. This is graphically illustrated in Figure 29 where we show the periodogram based only on the emission line data of Joy and RSB. We decided to check the ephemeris given by Feldt & Chincarini (1980). We were unable to reproduce their results. We therefore suspect that the F&C ephemeris is incorrect. In fact, folding C&W’s absorption-line data on F&C ephemeris yields a large phase offset (~ 33°), rather surprising considering F&C used these data to derive their ephemeris. We can now see how this insidious pulse-timing phase shift discrepancy arose. RSB’s work was based on the Ila emission line, which was phase shifted. They did not suspect a phase shift however, because their phasing agreed with F&C’s, which probably is in error. This agreement was accidental and unfortunate. Because we repeated RSB’s analysis we could check to see if the period was con fused with a nearby alias or sidelobe. This was not the case. We did notice that the beating pattern in the x 2 curve of the best-fit sinusoids was not symmetric about RSB’s best-fit period (see Figure 29). Rather, the lower envelope of this beating pattern was centered on a slightly higher frequency, closer to our revised estimate. lines correspond to RSB’s period period RSB’s to correspond lines Figure 29: Periodogram of the combined Joy and RSB emission line data. The dotted dotted The data. line emission RSB and Joy combined the of Periodogram 29: Figure x2 / 123 to oo 2.4286 .28249249 .242.4296 2.4294 2.4292 2.429 2.4288 S ad o' eiso—ie data emission—line Joy's and RSB ra Feuny (cycles/day) Frequency Trial (left) and W HG’s period period HG’s W and (right). .282.43 2.4298 ES 3 J 19 17:09 -1993 L WELSH -JU 30 100 Figure 30 through Figure 32 show the individual radial velocity measurements of Joy (1954), C&W and the Cable experiment folded on the WHG ephemeris. The sine curve shown is the adopted best-fit solution to the combined data sets. The bottom panel in each figure shows the residuals (observed - computed) and also the rms scatter in each tenth of a phase bin. Note that the zero level of the rms is the bottom of the figure (-100 km s -1 according to the labelled vertical scale) and the rms has been multiplied by a factor of 2 to make it more visible. For example, the first bin (phase -1.0 to -0.9) in Figure 30 (Joy’s data) has an rms scatter of ~ 42 km s-1. These are not noise corrected rms values. From these figures we see that the if 2 amplitude and phasing of the ephemeris is good for each data set, not just the ensemble. There is no obvious trend in the residuals with orbital phase, and distortions resulting from a non-circular orbit are not present. If examined carefully, the figures show that Joy’s and the Cable data lie systematically above the best fit solution, while C&W’s lie below. This is due to a systemic velocity (7 ) difference between the different data sets, and is discussed in Appendix C. For completeness, the entire combined data set is shown in Figure 33. In Table 4 we list the orbital elements found in the literature, plus our own de terminations. In Table 5 we compare the various ephemerides found in the literature using the combined absorption-line data we constructed. As claimed earlier, the WHG ephemeris is indeed an improvement. It should be noted that the phasing of the pulse-timing orbit derived by RSB depends explicitly upon the orbital period they derived from their emission-line data. Fortunately the difference between using their 102 JOY.FOLD1 o CNo o o o o o “D O in o>*W m o I -1 -0.5 0 0.5 1 binary phase Figure 30: Radial Velocity Curve — Joy’s Data. Joy’s absorption-line radial velocity measurements have been folded on the WHG orbital ephemeris. The sine curve is the WHG solution. The bottom panel shows the velocity residuals. The histogram in the bottom panel shows the rms scatter at that phase, multiplied by two for clarity. orbital period and ours over the ~ 417 days of pulse-timing measurements amounts to a maximum of only 0.006 orbital phases, which is negligible in determining the pulse-timing orbital phasing. pare with Figure 30 and Figure 32. Figure and 30 Figure with pare inln rda vlct esrmnsfle n h WG ria ehmrs Com ephemeris. absorp orbital C&W’s WHG the Data. on folded Walker’s & measurements Chincarini velocity — radial Curve Velocity tion-line Radial 31: Figure residuals radial velocity (k m /s) 0.5 0 5 . 0 - 1 - -50 50 -200 -100 0 100 2 00 binary phase phase binary CW.F0LD1 residuals radial velocity (k m /s) 0. 1 .5 0 0 5 . 0 - 1 - -50 50 -200 -100 0 100 200 l I I l l i l l l i l l l I l l i i l l l l I l I l l E , i H r J : A ^ ' ' r i i ■ i■ i ■ i i ■■ ■ i i i i i ■ d \ ' ' '\\ o .$ rot . . . . binary phase phase binary j ' CABLE.F0LD1 : '' r j$r. . r $ :j ■ ■ i■ i i i | i ;■■■' j i i ■ i ■ "i i i i...... i I . • j> < ■ ■ ■ ■ ' e f ' w i i twti ...... ES 3-UL-93 17:39 -1993 L WELSH 30-JU , ...... _____ i 104 H obtl peei. opr ih iue 0 Fgr 1 n Fgr 32. Figure and 31 Figure 30, Figure with Compare ephemeris. orbital WHG radial velocity measurements of Joy, C&W, and the Cable experiment folded on the the on folded experiment Cable the and absorption-line C&W, Combined Joy, of Sets. ata D measurements Combined — velocity Curve radial Velocity Radial 33: Figure residuals radial velocity (k m /s) 5 0 5 1 .5 0 0 .5 0 - 1 - -50 50 -200 -100 0 100 ■ 1 « ■ I ■ ■ ■ I ■ I ■ ■ I I - ■ ■ « -1 1 ■ ■ I"— “ I " I III! E q ALLDATA.FOLD1 Aqr AE ------binary phase phase binary | ~ I ‘ I ■ 11 I'— t T - t '"— 'I 1,1 I ■ I I ‘ l‘" I ~ I | 1 j> < ES 30J 19 17:1* -1993 L WELSH 0-JU 3 2 x rms 105 106 Table 4: Orbital Elements of AE Aquarii authors To Period k 2 72 Remission 7i (HJD) (days) (km s-1) (km s-1) (km S_1) (km s-1) Joy 2431328.7014 0.701024 151 -52.6 146 -43 c & w t 2439030.984 0.4116537 159.4 -69.3 135 -32 ±0.003 ±0.0000042 ±0.9 ± 0.6 ±5 ±3 (2439030.934) F&Ct 2439030.621 0.41165794 159.7 -64.0 —— ±0.0000042 — ±9.6 — — VPKA 0.4116560 --- ______±0.0000005 ------— --- RSB 2439030.830 0.4116580 --- __ 141 -39 ±0.003 ± 0.0000002 ------± 8 ±5 WHG 2439030.7879 0.411655601 157.9 -66.7 138* -7* ± 0.0011 ±0.000000056 ± 0.8 ±0.7 ± 6 ±4 f To defined such that superior conjunction of the white dwarf with respect to the secondary star occurs at phase 0.5 * K\ and 71 are included only for comparison. The sinusoid fit to these data is so poor that these values are almost meaningless. We do not believe that the emission- line radial velocity curve accurately reflects the orbital motion of the white dwarf. Table 5: Testing the New Ephemeris authors Period reduced x 2 RMS residuals phase offset (days) (549 d.o.f.) (km s -1) (cycle) C&W 0.4116537 4.237 39.579 -0.351 F&C 0.41165794 5.774 45.327 0.397 VPKA 0.4116560 1.422 20.508 -0.356 RSB 0.4116580 6.010 46.301 - 0.111 Cable 0.41159 55.691 128.922 0.45 WHG 0.411655601 1.315 19.684 0.000 108 3.3 Note on the Mass Determination In Section 3.1 a range of possible masses for the two stars is given. Although the determination of the masses is dealt with in more detail in Chapter 4, it is useful to see how these estimates were made. The various factors needed to determine the masses are the period, apparent orbital velocities, and inclination. If the mass ratio is known, then only one apparent orbital velocity is needed. The period is assumed to be known very well and is taken to be WHG’s value. The mass ratio is determined by the ratio of apparent orbital velocities q = M2/M 1 = K \fK 2. The apparent orbital velocities are measured from the amplitude of the radial velocity curves. Note that the inclination cancels out in this expression. The inclination is very difficult to determine and is allowed to be a free parameter. The inclination is discussed in more detail in Chapter 4. In Figure 34 and Figure 35 we show these simultaneous constraints. The curve labelled UK2" is the constraint using K 2, and likewise for K\. The curve labelled “K\ + K 2n comes from the constraint derived from the sum of the orbital velocities and the curve labelled “ q” comes from the q constraint (neither are independent constraints). The thick vertical line at 0.7M 0 is (approximately) an upper limit to the mass of the secondary star based on its spectral type (~K5 V). The intersection of these constraints gives allowable solutions for the masses. Figure 34 shows these constraints for an inclination of i = 70°, which is an upper limit based on the no eclipse constraint. Figure 35 is for i = 56.7°, which is a lower limit based on the constraint that M2 must be less than 0.7 M q. This is a weak constraint, but probably not too bad. To be consistent with RSB, the results quoted earlier (0.64 < Mi/M® ~ 0.91, 0.49 < M2/M 0 ~ 0.70) are for the intersection of the best estimates of the parameters, and does not include uncertainties in these parameters. Thus there is some freedom to extend the acceptable range for the masses by about 0.05 M®. The reader may notice that a mass range is quoted, rather than a mean and an uncertainty. This was not accidental, but rather reflects the ignorance in these determinations. The probability distribution between these limits is not known and quoting a mean with an uncertainty implies that the distribution is Gaussian (or at least peaked at the mean). We have no evidence that any location within these limits is a better estimate than another. 110 M^-Mj plane co d - \ o CO o o 0.4 0.6 0.8 1 M,z /' o WELSH_ lfl-JUl-1993 13:42 Figure 34: Stellar Mass Constraints for Inclination i = 70°. Constraints on the stellar masses in AE Aquarii are drawn with their la uncertainties. The curves are labelled according to which constraint is being applied. The orbital period is assumed to be known exactly. The thick vertical line at 0.7M® is an approximate upper limit for the secondary star mass based on its spectral type. The intersection of the curves gives the allowed solution for the masses, which can be read off of the axes. I ll Mi“M2 plone GO o o 2 2 co o o 0.4 0.6 0.8 1 ^2 / WELSH 19-JUL-1993 13:43 Figure 35: Stellar Mass Constraints for Inclination i = 56.7°. The stellar masses for this inclination are considerably larger than the i = 70° case. Compare with Figure 34. C H A PT ER IV DETERMINING THE SYSTEM PARAMETERS OF AE AQUARII 4.1 Introduction In this chapter we attempt to accurately determine the system parameters in AE Aquarii, with emphasis on determining the stellar masses. We attempt to do this without using emission-line measurements which, as discussed in Chapter 3, give unreliable estimates of the white dwarf’s orbital motion. In Section 4.2 we describe the acquisition and reduction of the “Cable Experiment” data. In Section 4.3 we begin our discussion of the modelling of the absorption lines and in Section 4.4 we describe and present our results of spherical modelling. In Section 4.5 we introduce a more sophisticated Roche lobe model, and discuss the results. 4.1.1 Motivation and Overview Fundamental to our understanding of any interacting binary star system is a de termination of the system parameters (period, mass ratio, stellar masses, distance, etc.), and this is especially true when the binary system is peculiar. Because the cataclysmic variable AE Aquarii is an unusual system and has gained much recent attention, in this Chapter we attempt to make accurate measurements of the system 112 113 parameters. The optical spectrum of AE Aquarii contains both emission lines from the accre tion disk and absorption lines from the secondary star, and so one might expect it to be relatively easy to measure the system parameters. However this is not true for a number of reasons. The emission lines are phase shifted with respect to the absorption lines, indicating that they do not accurately track the motion of the white dwarf (Welsh, Horne, k Gomer 1993a, hereafter referred to as WHG). Because the cause of this phase shift remains unknown, any estimate of the system parameters derived using the emission lines are suspect. Thus even (spectroscopic) estimates of the mass ratio are poorly determined. The emission lines are not double-peaked, and hence constraints based on the outer disk radius do not exist. The system is non-eclipsing, and so one can only place an upper limit of ~ 70° on the inclination, depending upon the value of the mass ratio (Chanan, Middleditch, k Nelson 1976). It is possible that the amplitude of the ellipsoidal variations, such as those seen in the photometry of Van Paradijs et al. 1989, can be used to constrain the inclination. However they may be contaminated or even dominated by bright spot emission (an elongated bright spot with an anisotropic emission pattern could produce the observed variations), so one must use caution. Fortunately AE Aquarii gives us another way to constrain the system parameters — the 16 and 33s oscillations. Interpreted as being the rotation period of the white dwarf (see Chapter 2 and 3) , the 33s oscillations allow a lower limit on the mass of the white dwarf to be determined: 0.6M© (Robinson, Shafter, k Balachandran 1991). Also, the time delay of the pulse arrival time (due to the orbit light-crossing time) can be used to determine the apparent orbital velocity of the white dwarf. Various estimates for the amplitude of the pulse-timing orbit K puise are: 127 ± 5 km s -1 Patterson (1979); 122±4 RSB (1991); 108±7 De Jager (1993); and 102±2 Eracleous et al. (1993). These estimates should be reliable, as the pulses do seem to originate close to or on the white dwarf’s surface (WHG). We will not discuss the determination of KpUise, but rather simply adopt the results of Eracleous et al.. We choose their K Puise estimate because they used UV observations of the oscillations that are of a much higher quality than the optical data used by the other authors. Henceforth we shall use K\ = 102.0 ± 2.0 km s-1 . In this Chapter we attempt to determine the system parameters via a detailed study of the absorption lines. We use two methods, both capitalizing on the fact that the rotation of the secondary star is tidally locked to the orbital period, and thus the absorption lines are rotationally broadened. First, assuming a spherical secondary star we derive an estimate of VTOt sin i which we use to constrain the mass ratio q. Second, the spherical assumption is dropped and we model the phase-dependent mod ulations of the absorption lines. Though far more computationally costly, the second method allows us in principle to estimate the inclination i. In both cases the data are compared to model spectra which are produced by convolving a slowly rotating reference star spectrum with a rotationally-induced line-broadening function. 115 4.2 Observations Simultaneous high-speed photometry and spectroscopy of AE Aquarii were obtained at the Mt. Wilson 1.5 and 2.5m telescopes, respectively on the nights of 27 — 30 July 1982. The photometric observations were recorded with the Cassegrain S-20 photometer and the high-speed data acquisition system described in Horne & Gomer (1980). An interference filter with a FWHM of ~ 600A centered at 6600A was used to approximately match the wavelength coverage of the spectroscopy (6288 - 6900A on night 1 and 6357 - 6826A on nights 2 , 3 Sz 4). The spectroscopy was obtained using the Coude Shectograph. Details of the spectrograph set up are given in Table 6. The Shectograph was programmed to take exposures of duration 1.9 seconds on nights 1 & 2 and 3.3 seconds on nights 3 & 4. During the ~ 3.2 seconds in which data were being written to tape, no data could be taken, thus giving us a time resolution of 5.1 and 6.5 seconds. Argon arc lamp spectra were taken roughly every half hour while the tape was being rewound. Guiding corrections for the photometry were made at this time to minimize data loss (the spectra were guided continuously). A ( 7th order) polynomial fit to the arc emission lines (18 lines on Night 1, 17 lines on Nights 2, 3 & 4) determined the pixel to wavelength calibration. Linear interpolation in time was made between pairs of wavelength calibrations defining the wavelength scales applied to the spectra. The flux standard stars BD +26° 2606 and +17° 4708 (Oke h Gunn 1983) were used to flux calibrate the spectra. Tungsten lamp spectra were used to determine the flat—field correction. 116 Table 6: CABLE EXPERIMENT INSTRUMENTATION 27 June 1982 28 June 1982 29 June 1982 30 June 1982 Night 1 Night 2 Night 3 Night 4 grating “46B” 600 1/mm “4IB” 400 1/mm “4 IB ” “4 IB ” blaze Ca ) 7000 12,500+ 12,500+ 12,500+ slit dimensions arcsec x mm) 1.3x28.2 1.3 x 25.5 1.3 x 25.0 1.3 x 25.0 seeing (arcsec) 1 .5 - 2 7 1 .5 - 4 0.8 - 1.0 exposure (») 1.90 1.90 3.3 3.3 UT start - stop 5:07 - 10:55 5:33-11:15 9:46- 11:56 5:47 - 12:02 wavelength (A) 6288 - 6900 6357 - 6826 6357 - 6826 6357 - 6826 dispersion pix 1 0.085 0.064 0.064 0.064 (A/pix) pix 1872 0.219 0.165 0.165 0.165 pix 3744 0.09 3 0.093 0.093 0.093 resolution pix 1 31.6 33.7 33.7 33.7 (FWHM) pix 1872 65.3 52.5 52.5 52.5 (km s-1) pix 3744 56.6 43.3 43.3 43.3 1 used in 2nd order with GG495 in place to block 3rd order 117 The Shectograph has two arrays, recording the object and sky spectra simultane ously. Normal procedure is to alternate between the arrays, so that any differences between sensitivity cancel out. However to maximize the time resolution, we did not alternate between the arrays during our high speed runs. Instead we intercalibrated the arrays by observing bright twilight spectra in both, then applied a balance fac tor to the sky channel prior to subtraction. Because the signal in each high-speed spectrum was so weak, we did not attempt to remove any telluric lines. The exactly simultaneous photometry and spectroscopy covered a similar wave length range which permitted us to renormalize the spectroscopy to match the pho tometry, thus compensating for any guiding errors or other slit loss effects. Because the photometric bandpass was not identical to the spectroscopic bandpass, we made a small extrapolation of each spectrum to match the photometry. We did this by assuming the continuum shape to be that of a I<5 V star. Though this is clearly an approximation, the error should be small because the wavelength range over which we are extrapolating is rather small, the secondary star contributes ~ 65 - 85% of the continuum light at these wavelengths (WHO), and a K5 V star is an acceptable estimate for the secondary star (WHO, Bruch 1991, Section 4.42 of this Chapter). We then calculated synthetic photometry using the extrapolated spectra and a trace of the filter passband used for the photometry. The ratio of the actual photometry to the synthetic photometry derived from the spectroscopy gave the correction factors by which we multiplied each spectrum. This careful procedure was required because we are interested in cleanly separating the line and continuum variations. 118 For the purposes of the absorption line modelling in this chapter, the data were combined into bins l/50t/l of an orbital cycle long (approximately 6 minutes), giving a total of 200 spectra. In Figure 36 we show the data displayed as a trailed spectrogram, with a greyscale stretch than emphasizes the appearance of absorption features. The absorption lines from the secondary star are clearly visible, showing orbital radial velocity shifts of ~ 160 km s~l . Numerous stationary telluric absorption features are also present, most notably those at velocities ~ -2000, -2500, -3100, -3750, and +2500 km s-1. He I 6678A emission is also present, with an absorption line superposed. The He I emission shows a similar variability pattern as Ha. 4.3 Absorption Line Modelling The secondary star absorption lines are much broader than those of single stars with a similar spectral type (see Figure 21 in Chapter 3 for example). This difference is attributed to the rotational broadening of the secondary star. Because of tidal locking, the rotational period of the secondary star is equal to the orbital period, which is in general much shorter than the rotation period of single stars of similar spectral type. We can model the secondary star’s line profile as the convolution of a line from a slowly rotating star with a line-broadening function. Because the rotation is rapid, the line broadening is dominated by rotational broadening, though limb darkening, Roche geometry and gravity darkening all play important roles, as does the inclination. Note that unlike the stellar rotation analogue, one cannot normalize the continuum level to unity because the continuum does not arise from the secondary star alone; the disk+primary star usually contribute a majority of the continuum (though this 119 *i > J $ j l H i ,1s '-f O S- ; ' 1 J } 4 i M hV •«. '* ;i U ‘ I !U’J >{iJ'l'i-i: i KH* \ o i > 3iLI f f t b k w - I w., \ % ... Q) .$ '3 E ( P i m ,1 i! <"* >lf / \ • - 5 0 0 0 5 0 0 0 velocity (km s-1) WELSH 2 0 -JUL-1993 23:54 Figure 36: Trailed spectrogram showing all 4 nights of data. The intensity levels have been set to emphasize the absorption lines (dark bands). The sinusoidal motion of the absorption lines is due to orbital motion. The core of the Ha emission line is extremely saturated on this scale, but the wings are easily seen. The bottom panel shows the mean spectrum. Because the orbital motion was not removed before averaging, the stellar lines are smeared. However the telluric absorption features remain sharp. 120 is not the case in AE Aquarii; nevertheless the disk contribution is not negligible). For a given period, the larger the size of the Roche lobe the broader the line profile will be. Because the size of the Roche lobe relative to the binary separation is a function of the mass ratio q alone, the line profile width can be combined with the orbital velocity to determine q via the relation Vrot sin i/ if 2 = (1 + <7)(Reg/o) > where Vrot sin i is rotational broadening velocity, Req is the radius of a sphere with the same volume as the Roche lobe and a is the orbital separation of the two stars. Because Vrot sin i and K 2 are observables, the ratio can be used to determine q independent of I This method was employed by Horne, Welsh, & Wade (1993), where they rota- tionally broadened the IR Na I absorption lines of an isolated star (of similar spectral type as the secondary star) until they matched the observed line profiles in their data (DQ Her). The value of q they obtained was consistent with direct spectroscopic measurement and also with the eclipse width. In Figure 37 we plot the quantity Vrot sin i/ K 2 against q. The two curves corre spond to using the formula of Eggleton and the formula of Paczynski to compute R eq (see Eggleton 1983). The horizontal dashed line represents the observationally deter mined value K 0tsin ifK 2 (discussed in Section 4), and the vertical dashed line shows the estimated values of q. The lcr uncertainties are shown as dotted lines. Implicit in the calculation is the assumption that the spherical approximation for the Roche lobe is acceptable. 121 CO d c* id o o CM O 0 0.2 0.4 0.6 0.8 1 Cl ( M 2/M 1) WELSH 19-JUL-1993 15:33 Figure 37: The Vrotsm i — q Relation. The observable quantity VTOtS\ni/K 2 is plotted against the mass ratio q. Our best estimate for Vrot sin ijK i is drawn in as a horizontal line (with ± 1 a uncertainties) and the corresponding value of q is denoted by vertical lines. 122 4.4 The Spherical Approximation In this Section we describe the methods necessary to compute models using a spherical approximation for the secondary star and compare them with a mean AE Aquarii spectrum. 4.4.1 Spherical Modelling Details A mean AE Aqr spectrum was constructed by first rebinning the data to a uniform velocity scale (20 km s -1 pix ~l) then removing the orbital motion of the secondary star and averaging the data. A quadratic continuum fit with only the Ha and He I 6678 lines masked at ±2000 km s -1 (and no sigma rejects) was subtracted from the mean. The resulting weighted mean spectrum defines the data against which the models are judged. Aside from the orbital motion correction, the comparison star spectra were processed in an identical manner, defining the template spectra. Note that the fit will fall below the actual continuum level because the absorption lines have not been masked. This is true in both the template spectrum and AE Aqr. If we attempted to mask out the absorption lines prior to the continuum fit, we would undoubtedly miss the very weak ones hidden in the noise, and the fit would then be biased low in spectra with higher signal-to-noise ratio. However the above method of dealing with the continuum levels is not sufficient. Because of spectral type mismatch between the template stars and the secondary star (because the spectral types are intrinsically different, or the secondary star is peculiar due to being in an interacting binary system, or elemental abundance effects), the relative strengths of the absorption lines do not agree. As a result, the continuum fits will reflect this difference. For example, say a particular line is much stronger in AE Aqr secondary star’s spectrum than in the template star’s spectrum. Because the line is not masked, the continuum fit to AE Aqr will be relatively lower than in the template. This continuum mismatch will not allow any rotational broadening of the template to give a good fit to the secondary star’s spectrum. We deal with this rather insidious problem in the following way. We convolve the template with the rotational broadening function and scale it to best match the data. The residuals show the effects of the continuum error. A least-squares polynomial fit is made to the residuals, and this fit is then scaled and added to the template, producing a corrected template. The procedure is repeated thrice to insure a correction term of sufficient quality. (The fitting process is non-linear, so the coefficients of the polynomial fit to the second round of residuals (after the initial correction has already been applied) are not necessarily zero. However, we find that the process converges rapidly, and two or three iterations are sufficient.) The corrected template is then rotationally broadened and scaled to give the final model. Each line is treated independently, because each one has a different continuum level correction. Earlier attempts to fit the entire residuals spectrum with a single high-order polynomial or spline fit induced undesirable curvature and oscillations in the continuum correction, while low order polynomials could not fit the residuals properly. We conclude that the best approach was to treat each line separately, using either a constant or linear continuum correction. Higher-order polynomials, of course improve the fit, but begin to overfit and remove flux from the line (hence an F-test cannot be used to determine what order polynomial is optimal). Though the linear case has the potential to bias the position of the line center and induce a 7 velocity shift, the effect is generally small compared to the uncertainty in determining 7 . [The worst case gave a gamma shift of 5 km s-1. For the K5 template, the shift is less than 0.1 km s-1.] This rather messy technique for dealing with the continuum was not anticipated — we were forced to adopt such a procedure because the results of our model fitting were dominated by systematic errors. One cannot escape the conclusion that the continuum level is part of the problem and must be solved for as a nuisance parameter. A serious problem which can bias the models is the presence of telluric absorption features (due mostly to atmospheric water vapor). These lines are clearly visible as straight lines in the trailed spectrogram (see Figure 36, especially near velocities -2100 and -3800 km s-1). The strength of these lines vary depending upon the observing conditions, and cannot readily be removed from the data. To avoid the serious biases induced by these contaminants, we were forced to mask out any regions where telluric contamination was seen or even suspected, using the atlas of the solar spectrum as a guide (Kurucz et al. 1984, Beckers et. al 1976, Moore et. al 1966). (Particularly troublesome are the telluric absorption lines which blend with the secondary star’s absorption lines.) Unfortunately, this removes most of the data, leaving only a few lines to work with. Our analysis will employ four lines which were chosen because they are a priori free of any telluric contaminants and a postiori they reveal no systematic trends in the residuals. These are the Fe I blend at ~6421 A, the Ca I line at 6439.1 A, the Ca I line at 6449.8 A and the Ca I line at 6717.7 A. While other lines may work, we know that at least these four are free of obvious biases. We fit all four lines simultaneously, using an “amoeba” simplex (Press et al. 1986) to search through parameter space, and note that results for each individual line are consistent with the combination of all four. The template stars span the spectral range from K2 V to K7 V and are listed in column 1 of Table 7. We have found radial velocity measurements in the literature for all of these stars (GCRV; Abt 1973; Beavers & Eitter 1986), except for BD +26° 3003. Radial velocity of AE Aquarii were then computed. It is worth noting that aside from HD 165341B which gives a surprisingly poor fit, all the 7 velocities agree. This is evidence that 7 is orthogonal to the spectral type in our fitting procedure. We also note that the K5 star 61 Cyg A is known to be chromospherically inactive and have a very slow rotation velocity (Dempsey et al. 1993). 126 Table 7: The Template Stars /<2 — 158 km a-1 ; U = 0.5; npoly=2 spectral type star 7 (km s *) x 2/93 I<2 V HD 166620 - 66.1 4.73 K3 V 190470 -63.2 2.30 K4 V 65341B -90.83 3.39 I<5 V 61 CYG A -65.4 1.34 K6 V +26 3003 (?—63.2) 2.62 K7 V 61 CYG B -64.6 3.02 127 In summary, the parameters in our spherical model are the amount of rotational broadening Vrot sin i (determined by the line width), the spectral type of the template star (determined by the line strengths and ratios), the systemic velocity 7 (determined by the shift of line centers), and the linear limb darkening coefficient U (determined by the line shape and width). The apparent orbital velocity of the secondary star K 2 is also a parameter, but is fixed at a particular value before the parameter search (consistent with the radial velocity measurements of Ki). The rescale factor is op timized for every model, and is essentially a measure of the distance to the system. The continuum correction levels are the remaining (nuisance) parameters, and are determined by the template mismatch. 4.4.2 Spherical Modelling Results Figure 38 shows the data and the best-fit model for the four lines which were used to determine the system parameters. The parameters themselves are listed in Table 8 . As will be discussed below, we cannot independently determine U, so we let U take on the values of 0.0, 0.5 and 1.0. Allowing U to vary between 0.0 and 1.0 produces a change in the value of Vrot sin i of about 12 km s-1 or ~ 14%. For a given value of U however, the uncertainty in VTOt sin i is considerably smaller. 128 DATA and MODEL FITS Vrot sin i = 89.6 o in o mo o o T) o I 3 * 2 to CM o o 11 ro CMI CM 01 ■M- I T -x 6800 -6600 -6400 -5800-5600-5400-5200 6600 6800 7000 velocity (km /s) wrt Ha WELSH 0-JU L-1993 19:12 Figure 38: Spherical modelling fits to the absorption lines are shown (data — heavy lines, fit = thin lines). The parameters used are K 2 = 157.9 km s -1, V r o t sin i — 89.5 km s -1 , U = 0.5, and a linear continuum correction term. From left to right these are the Fe I blend at ~6421 A, the Ca I line at 6439.1 A, the Ca I line at 6449.8 A and the Ca I line at 6717.7 A. The lower panel shows the residuals. 129 Table 8 : Spherical Model Results I<2 = 158 km s -1 Udark V r o t sin i 7 npoly Xred n.d.o.f. (km 5—1) (km s -1) 0.0 85.1 - 0.88 1 1.50 96 0.0 85.1 -0.87 2 1.34. 93 0.5 89.5 -0.93 1 1.50 96 0.5 89.5 -0.93 2 1.34 93 1.0 96.7 -1.06 1 1.51 96 1.0 96.7 -1.07 2 1.35 93 Figure 39 shows a plot of the reduced x 2 versus the spectral class of the template stars. Six curves are shown, corresponding to the degree of the continuum correc tion polynomial and two assumed values of 772 . As expected, the results favor using the linear continuum correction method, though the final results are insensitive to whether a constant or linear correction is applied. We therefore adopt the linear con tinuum correction method. The models used values of K 2 = 157.9 and 163.5 km s -1, and as will be discussed below, the lower value is favored. The results of the mod elling strongly suggest that the secondary star in AE Aquarii is approximately K5. (Because all of our templates are main sequence dwarfs, we cannot address the is sue of luminosity class.) Because this spectral type produces the most appropriate template, we henceforth restrict ourselves to models using the K5 V star 61 Cyg A. Using the I<5 template, we ran models with 772 = 152.0, 157.9 and 163.5 km s -1, letting U take values 0.0, 0.5 and 1.0. As expected, U and K 2 are correlated — a larger value of U creates a wider line, and the data can be fit with a smaller 772 . However the correlation is weak and K 2 can be accurately determined. For the linear continuum correction method, parabolic fits to the reduced x 2 points give 772 = 158 .3, 158.1 and 158.0 km s _1 for U = 0.0, 0.5 and 1.0, respectively. (The constant correction method gave a slightly lower value of K 2 ~ 156.4 km s~l .) The results show that the assumption 772 = 157.9 is valid, and suggests that perhaps the cross-correlation method used to determine K 2 overestimates its amplitude. For the remainder of the spherical model discussion we will work under the assumption that K 2 — 158 km s -1. 131 oo Spherical model . npoly«0 co npoly«*1 npoly=2 . K2V K3V K4V K5V K6V K7V spectral sequence WELSH 1O-JUN-1093 16:06 Figure 39: Spectral type of the template star versus reduced \ 2 f°r the spherical models. The three curves correspond to different continuum correction methods, with polynomial coefficients indicated. Each of these methods employed two values of K%, 157.9 and 163.5 km s-1, denoted by open and filled symbols respectively. A limb darkening coefficient U = 0.5 was assumed in each case. The horizontal dashed lines show the expected range in reduced x 2- 132 There is insufficient information in the data to separate the effects of rotational broadening and limb darkening, i.e., we are unable to distinguish between a line that is rotationally broadened with no limb darkening and a line that is less rotationally broadened but is highly limb darkened. The two parameters are not completely correlated, because they alter the line profile in slightly different ways. The correlation is on the order of ~ 10 - 15%, as demonstrated in Figure 40, where we show the x 2 contours as a function of Vrot sin i and U. In this figure, 7 , the scale factor, and the continuum correction terms are optimized for each grid point. From Figure 40 we can estim ate the value of VTOt sin i: Vrot sin i = 85.1±8.6 km s-1 for U—0.0, and 96.8 ± 9.9 km s _1 for (7=1.0. (Note that the la error estimates are not actually symmetric, but the difference is not important at this level of precision.) Because we cannot a priori know what U is, the acceptable 1 Vrot sin i < 106.8 km s-1. Combining this estimate with K 2 we are able to determine the mass ratio: 0.51 < q < 0.85. The uncertainty is largely affected by the ambiguity in the limb darkening — for example, for a limb darkening of U=0.5, the value of the mass ratio is q = 0.65 ± 0.10. This absorption line determination of q agrees with the q derived from the pulse arrival time delays, q = 0.646 ± 0.013. Though our error bars are larger, this is an independent measurement of q, and the technique is not limited to CVs in which rapid oscillations from the white dwarf can be used to determine K j. If we include the additional information provided by the pulse timing delays, we are able to independently determine the mass ratio q, and thus fix a value for Vrot sin i. Vrot sin i (km/s) WELSH 3 0-JU L -1W J 10:01 Figure 40: x2 surface of the spherical models showing the acceptable values of Vrot sin i and the correlation with U. The value of q determined from K puhe/K 2 is shown along with the corresponding value of the limb darkening U. 134 This in turn allows us to determine the linear limb darkening coefficient U. We find that for q = 0.646 db 0.013, Vrot sin i = 88.8 ± 0.5 km s _1, and thus U = 0.40 ± 0.05. A number of tables of theoretical limb darkening coefficients have been published (e.g. Wade & Rucinski 1985; Rubashevskii 1990) and from these one can check to see if the above value agrees with the theoretical values. [Note that the theoretical values themselves are uncertain at the few percent level due to systematic differences in the methods of computation (see Claret & Gimenez 1990), but these differences are unimportant for our comparison purposes]. However, Claret h Gimenez (1990) con sider the case where irradiation from the primary is included (using a plane-parallel approximation) and find that the limb darkening coefficient is very sensitive to irra diation. They also find that the linear coefficients decrease as a result of irradiation, i.e., the brightness distribution becomes more uniform (the amount depends of course on the distance and effective temperature of the irradiating star, and the cosine of the angle of the incident flux). Because of this sensitivity, a comparison with theoretical limb darkening coefficients is not very fruitful. Using P = 0.4116550601 ± 5.6 x 10~8days, q = 0.646 ± 0.013, and I<2 = 157.9 ± 0.8 km s-1, constraints on the stellar masses can be made. We 4ind that M\ sin3 i = 0.455 ± 0.01A/©. Assuming the secondary star fills its Roche lobe, the lack of white dwarf eclipse implies that the inclination must be less than about 72° (Chanan, Middleditch, h Nelson 1976). The inclination must in fact be a bit smaller, else eclipses of the extended disk would occur. RSB have computed lower mass limits for the white dwarf based upon its 33 second rotation period: M\ > 0.6M© assuming 135 a helium white dwarf or > 0.4M@ if an iron white dwarf. If a helium white dwarf, the mass limit can also be used to constrain the inclination : i < 65.8°. To keep the mass of the white dwarf below the 1.44M0 Chandrasekhar limit, the inclination must be greater than 42.9°. Although these constraints are firm, they are not tight, and to do better one must make assumptions about the secondary star’s mass (only weakly constrained now to be 0.39 < < 0.93). If we assume that the upper limit for the secondary star’s mass is 0.7Mq, then the inclination must be greater than ~ 48°. Applying all of these constraints yields 0.60 < M\/M q < 1.08 and 0.39 < M2/M© < 0.70 for inclinations in the range 48.5° < i < 65.8°. Obviously the Holy Grail in this game is to get the inclination. Even with these weak constraints we can still make the interesting observation that unless the the secondary star is at its maximum permissible mass limit (O.93M0 ), the secondary star will not fill its Roche lobe, as previously noted by Patterson (1979) and Chincarini k Walker (1981) and mentioned at then end of Chapter 2. This remains true, even though the mass ratio q is considerably smaller than previously thought, and requires that either mass transfer occurs via a wind/flare ejection or that the secondary star is slightly evolved. This is an important fact concerning our modelling: if the secondary star does not fill its Roche lobe, then the spherical approximation may in fact be superior to the Roche approximation. A final note concerning the spherical models: the careful reader will notice that although the best-fit reduced x2 — 1-35 seems quite reasonable, it is 2.4cr too large (<7 = i/2 /ridof), and implies that the la uncertainties we quote on the parameters may 136 be slightly underestimated. There are a number of possible explanations of why the reduced x 2 is too large — the presence of hidden telluric features, slight abundance differences between template and AE Aqr, small spectral type or luminosity class mismatch, etc.. And of course it is possible that we are seeing the limitations of the model — the secondary star may not be perfectly spherical! 4.5 The Roche Lobe Approximation 4.5.1 Motivation and General Considerations As Figure 37 shows, the error in q depends strongly on the error in the ratio of the rotational to orbital velocities. This problem is further compounded by the fact that Vrot sin i is difficult to determine accurately, and even if it were known exactly, the vol ume equivalent sphere is only an approximation to the Roche-lobe shaped secondary star. If the secondary star is spherical there would be no ellipsoidal variations. As mentioned earlier, continuum measurements have shown what appear to be ellipsoidal variations (Van Paradijs et al. 1989; Bruch 1991). But it is not known whether these are due to the disk or secondary star. The fact that the two peaks are not equal in height suggest that there is disk contamination (Van Paradijs et al. 1989). Measurements of the absorption-line flux deficit should settle the issue, as the absorption lines are known to arise on the secondary star. Because of our care in ensuring that the Cable spectroscopy is photometric, we can compare our absolute absorption-line flux deficits at different times. In Figure 41 and Figure 42, we show the variations detected in the absorption line flux. Figure 41 is a direct integration of the line flux, which is negative because these are absorption lines. In Figure 42 we 137 compute the variations by rescaling the mean spectrum to best match each individual spectrum. This has the advantage of improving the signal-to-noise ratio. Note that if there were no variations, the points would scatter about a level of 1.0. Both these figures suggest that we have detected absorption-line ellipsoidal variations in our data. Because secondary stars in CVs are believed to be non-spherical the observed properties vary as a function of viewing phase. As the star rotates, we view different amounts of surface area (causing ellipsoidal variations). Consequently the rotational line-broadening function must be dependent upon the orbital phase We attempted to solve for the line broadening function directly using the maxi mum entropy method as formulated in the software MEMECHO (see Horne, Welsh, h Peterson 1991). The virtue of this approach is that it lets the data themselves define the line-broadening function. At first we placed little constraint on the default image, allowing it to be a smoothed version of the current solution. The calcu lated line broadening functions, shown in Figure 43, were very broad and Gaussian in appearance. We then tried forcing the solution to be as close as possible to the spherical case (i.e., classical rotational broadening) by forcing the default image for each iteration to be the spherical rotational broadening function that best matched the current solution. [This was done via the Levenberg-Marquardt non-linear least squares method as implemented in Press et al. 1986.] The solutions we obtained were consistent with simple spherical rotational broadening, though we note that MEME CHO had difficulty converging. Thus we concluded that the signal-to-noise ratio in 138 o CM O O h-* O fO - 0.6 - 0 .4 - 0.2 0 0.2 0 .4 0.6 Binary Phase Figure 41: Ellipsoidal variations seen in the direct integration of flux across the Ca+Fe absorption-line blend at 6486 - 6500 A. Note that traditionally the line flux deficit is plotted, so this figure must be inverted to compare with continuum ellipsoidal variations. 139 Absorption line ellipsoidal variations 10 o -+o -» **—o 0) o CO to o o 0 0.2 0.4 0.6 0.8 1 orbital phase r WELSH 21-JUL-1993 02:23 Figure 42: Ellipsoidal variations seen in the spectral region 6300 - 6500A. The vertical axis is a measure of the rescaling required to match the mean spectrum with the individual spectra. The histogram is the binned version of the data. 140 the Cable data was not sufficient to force the solutions to deviate significantly from the bias we introduced toward spherical rotational broadening. We then turned to modelling the line profile variations. 4.5.2 The Models The approach we adopted is similar to that of the spherical model: we construct a rotational line-broadening function and convolve it with a template spectrum to compute a predicted spectrum, which is rescaled to best match the data. Comparing the predicted line-profile variations with the observed data enabled us to find the parameters defining the line broadening function via x 2 minimization. Unlike the spherical case, the line broadening is no longer a simple function of VTOt and U. Rather, it is constructed by summing all the visible surface elements of a Roche- lobe shaped, limb- and gravity-darkened model star. Defining the line broadening function are 7 , K%, q, i, U, the gravity darkening index /?, and the rescale factor (the period P is assumed to be known). A few comments about these parameters are in order. The systemic velocity is & essentially a nuisance parameter, defining zero velocity, and is well constrained by the data. The mass ratio defines the shapes and relative sizes of the Roche surfaces, while /<2 (and P) define its scale. The line width (or equivalently, Vrot) is the major observable that constrains q. The inclination is constrained primarily by the ampli tude of the ellipsoidal variations. Both gravity darkening and limb darkening tend to decrease the rotational broadening of the absorption lines, and hence compete with q. While the effects of the limb and gravity darkening are most pronounced near the 141 . 1 I -i • , • , j r . . y • 1 ' ll ' ' ' 1 |J ■ ' ' | : ' at 1 ' 1 : yiuM Mdmmtk y i - t L ^ . 7 5 | 1 -; . ir . . . 1 . i r . . . . i . . ^ . . i ? . \ .ifl. i . 1 . . PI . i - | | r - \ - i / V i i-----1 i - H * I ' 1 - A.;• 0=0.00 1 : - 2 0 0 0 200 -7000 -6000 -5000 -4000 -3 0 0 0 -2000 V (km s') Figure 43: The four left-hand side panels show the line broadening functions cal culated using the maximum entropy technique. The right-hand panels show the template star spectrum (bottom panel) and four AE Aquarii spectra at the orbital quadrature phases indicated. The solid line is the MEMECHO fit. For the spectra, the vertical scales are in units of mJy\ the horizontal scale is in units of velocity (km s-1) with respect to Ha. For the broadening functions, the horizontal scale in in units of (km s~1) and the vertical scale units are arbitrary (though the same in each panel). 142 L\ point, this is also the region with the least emission due to geometrical projection effects. Note that VTOt sin i is not a parameter: for a non-axisymmetric star, Vrot has no well-defined meaning. VTOt is not computed since the model depends on q explic itly. Lastly, the rescale factor is not used to define the line broadening function but is needed to account for the distance to AE Aqr. It is optimized for every permutation of the parameters in our calculations. 4.5.3 Roche Lobe Modelling Details The Roche lobe surface is divided up into a grid of 160 longitudinal and 80 latitudinal segments for a total of ~ 12800 surface elements. The “poles” of the grid occur at the L\ point and the back of the star (side facing away from the white dwarf). The grid is equally spaced in latitude but the longitudinal grid varies in size in an elliptical fashion. This is required so that the “front” and “back” of the star are properly resolved in this coordinate system. Each surface element is assumed to produce a Gaussian line profile with cr=l km s-1. Each surface element is limb darkened, gravity darkened and foreshortened along our line of sight according to the orbital phase and inclination. In Figure 44 we show a predicted trailed spectrogram computed by convolving the continuum-subtracted template star with the line broadening function and rescaling to best match AE Aquarii. One can easily see the Doppler shift due to orbital motion. Although somewhat difficult to see, the change in line shape and strength as a function of phase is visible. Compare these predicted data to the actual data, shown in Figure 36. A predicted trailed spectrogram such as this is created for every 143 JZ o> ro •C g> z JOo> “3>N E -5000 0 5000 velocity (km s-1) WELSH 21-JUL-1W3 00:23 Figure 44: Predicted trailed spectrogram constructed by convolving the Roche lobe line broadening functions with 61 Cyg A. For this simulation, we let q = 0.65, i = 60° and U = 0.5. Compare with Figure 36. set of input parameters and is multiplied by a scale factor to best match the data. The result is then compared against the data, resulting in a x2 statistic with 9844 degrees of freedom (55 spectra x 179 pixels per spectra minus 1 for the scale factor). Because noise in the predicted line profiles (stemming from noise in the template star) are correlated we cannot easily use them in our rescaling procedure. We assumed that the template star was noise-free, resulting in a y 2 statistic which is biased, but 144 as the signal-to-noise ratio is much greater in the template than in the data, the bias is small. 4.5.4 Roche Lobe Modelling Results Because of the large computing time, it was not possible for us to fully explore this multidimensional parameter space. To reduce the number of free parameters, we fixed the spectral type of the template star to that deemed best by the spherical method, the K5 V star 61 Cyg A. The gravity darkening index is held fixed at /? = 0.08, appropriate for stars with a convective envelope (see Hill & Hutchings 1973). We allowed the limb darkening exponent to take on values of 0.0, 0.5, and 1.0. We incorporated a downhill simplex “amoeba” to search the remaining parameter space for the x2 minimum. Initial results were highly encouraging, but upon closer scrutiny it became appar ent that the solutions were highly dependent upon the spectral region of the data we used. In retrospect this is not surprising, considering this is the same difficulty the spherical models ran into. [In fact the Roche lobe models were constructed first, and through frustration we were driven to construct the spherical models. We should note that the spherical models themselves contain 4 free parameter plus 2 nuisance parameters and are not all that much less sophisticated. The real difference is that in the spherical case we model a single orbit-averaged line profile, while in the Roche models we model each phase separately. This additional freedom allows the Roche models to be sensitive to the inclination.] As in the case for the spherical modelling, we suspect that the residuals to the fits may be dominated by telluric contamination 145 and absorption-line strength mismatches between the template star and AE Aquarii. In Figure 45, we plot the observed ellipsoidal variations and two extreme cases of predicted ellipsoidal variations. The predicted ellipsoidal variations were computed by summing the visible surface elements of the Roche lobe model (essentially integrating the line profile) and rescaling to best fit the observed data. We show the cases where q = 0.70, U = 0.0, and i — 45° which gave the smallest amplitude variations (the triangles), and q = 0.55, U = 1.0, and i = 65° which gave the largest amplitude variations (the stars). The reduced x 2 values were nearly identical, 1.973 and 1.977 respectively, indicating that we are rather insensitive to the constraints provided by the data. The fact that the reduced x 2 lies so many sigma away from 1.0 indicates that the models are not good representations of the data. The data in Figure 45 follow an ellipsoidal light curve with unequal maxima, the absorption lines being stronger at phase 0.75 than at phase 0.25. This suggests that the leading face of the secondary star produces deeper absorption lines. (“Leading face” denotes the side of the star facing the direction of orbital motion.) This effect was noticed by Van Paradijs et al. (1989) in the continuum, and they concluded that it was due to bright spot contamination. Now that it is observed in the absorption lines, it seems unlikely that the bright spot is responsible. We also notice that the deeper minimum is phase shifted, occurring about a tenth of a cycle too early. While these effects suggest non-axisymmetric features on the secondary star, we caution that the data are sparse, and the inferred features may be an artifact of night-to- night variations in the strength of the absorption lines. For example, all the data 146 from phase ~ 0.25 to ~ 0.55 come from night 1 only. Since this region defines the global minimum, solutions are particularly sensitive to these data. While geometric, gravity, and limb darkening effects are accounted for in our Roche lobe modelling, there can be additional effects that the models cannot reproduce. The Roche model has bilateral symmetry, so any deviations from this cannot be accounted for. Examples of this symmetry breaking are starspots and flares, which have the additional nasty complication that they can be transient. Irradiation from the disk and white dwarf can also induce an asymmetry by heating the inward face of the star. Davey & Smith (1992) have found that in five of eleven dwarf novae they observed, strong heating efFects were present (as measured by the eccentricity of the IR Na I doublet radial velocity curve). They also noticed that the Na I distribution is far more asymmetric than expected for irradiation by the white dwarf and bright spot. The region where the Na I absorption is suppressed is not symmetric about the Li point, but rather more along a line 45° away from vector pointing towards the primary star (and in the direction of the orbital motion). The authors speculate that this is perhaps due to circulation currents on the leading side of the star. We investigated the possibility that irradiation may affect the absorption-line strengths. One might expect this, for example, if a disk flare irradiates the secondary star, causing a change in the photospheric structure. The absorption lines will then either be enhanced or suppressed on the side facing the disk, shifting the center of light away from the center of mass, and producing a radial velocity offset (the “K correction”, a la Wade & Horne 1988). Figure 46 shows the (0 - C) deviations 147 Data and Model Ellipsoidal Variations "o Oo o o M—o Oo o o ctf 0) 00 00 o (0 Oo 00 oo 00 o Oo CO 0 0.2 0.4 0.6 0.8 1 orbital phase WELSH 21-JU L -1993 02:31 Figure 45: Observed ellipsoidal variations, shown as circles and rebinned into the histogram. Two extreme cases of ellipsoidal variations computed from Roche lobe models are shown as the stars and triangles. (To save computing time, the models were evaluated only at the same phases as the data, thus the model curves have breaks.) 148 VELOCITY-FLARE CORRELATION o o “I 1--- 1----1----1----1----T" T- O i (0 in 5 o o ° o w m > I o - o _J______I______I______I______I______I______1______I______I______l_ 100 200 300 400 continuum f (mJy) WELSH 6-APR-19B3 17:21 Figure 46: Radial velocity residuals plotted versus continuum level. No correlation is apparent, implying that irradiation from the primary star and disk do not affect the absorption lines. from the best sinusoidal fit plotted against the continuum flux (AA 6400 - 6800A with Ha and He I 6676A masked). The fit is based on the Cable data only. Along with evidence from Figures 30 through 33 which show no systematic radial velocity residuals as a function of orbital phase, we conclude that no flare-radial velocity correlation is present and that irradiation by the flares is unimportant. C H A PTER V CONCLUSIONS 5.1 Summary Although AE Aquarii is a rather unusual CV, it is unlikely that it contains any truly unique physics. Most of the phenomena it exhibits are known to occur in other CV systems, though usually not as extreme. In this dissertation we have intensely studied AE Aquarii and have learned much about this particular CV. However much of the value of this work lies in the insight we have gained on CV systems in general. In particular, we have learned that the well-accepted model for the origin of the optical oscillations (i.e., the reprocessing of harder photons by the accretion disk) is not always correct. We have also found evidence that suggests that the flares seen in CVs are caused by the same mechanism responsible for stellar flares. This intriguing possibility may open a new realm for flare physics because the physical conditions in accretion disks are very different from those of stellar atmospheres. We have also further developed a general method by which the mass ratios in CVs can be determined without using measurements of the emission lines. And finally, while our attempt to determine the inclination of AE Aquarii has failed, we have developed a technique which should allow us to measure this highly valuable parameter in other 149 150 CVs. In the remainder of this Section, we summarize our most important results. We give quantitative results where relevant so that these values can be readily found without having to search through this dissertation. In Chapter 2 we determined the optical spectrum of the oscillations in AE Aquarii, and it is the only spectrum of oscillations determined to date for any CV. A blackbody model gives a good fit to the spectrum, yielding a temperature of ~ 12000 K and an area of ~ 3.3 x 1018 cm2. This area is very small compared to the surface area of the accretion disk, which leads us to believe that the optical photons are not produced as a result of the reprocessing of harder photons by the disk (contrary to what is commonly believed). We also extracted a spectrum of a flare, and found that it shows a Balmer jump in emission, strong Balmer emission lines, He I in emission, and a rising Paschen continuum. LTE modelling indicates that the flare consists of supersonically expanding gas at a temperature of ~ 8000 K. The energetics, timescale, and Ha/H/? line ratio of the flare are similar to those of the largest stellar flares, and suggest that a similar mechanism may be responsible for both phenomena. This CV — stellar flare analogy is further supported by the time evolution of the Ha line during a flare (see Appendix B). As in stellar flares, the Ila line in AE Aquarii rises after, and decays slower, than the continuum intensity. The problem of locating the origin of the optical oscillations in AE Aquarii is discussed in detail in Chapter 3. Using a straightforward technique, Robinson, Shafter & Balachandran (1991) determined that the optical oscillations originated outside the Roche lobe of the white dwarf. This very puzzling result was difficult to understand and all interpretations required substantial reprocessing of radiation in the accretion disk. We combined new data from the “Cable Experiment” (see Chapter 4 and Appendix A) with data found in the literature and computed an improved orbital ephemeris based on absorption-line radial velocities. In addition to revealing that the Ha emission-line orbit is phase shifted by ~ 75° with respect to its expected position, this new ephemeris solves the puzzle of the location of the oscillations: the oscillations are found to arise from a region within 5° of the white dwarf as seen from the center of mass of the system. Along with the results presented in Chapter 2, this inferred location of the oscillation-producing region strongly implies that the optical oscillations are not due to reprocessing in the disk, but rather originate on or very close to the white dwarf. The methods by which system parameters of CVs (mass ratio, inclination, masses, etc.) are estimated are discussed, and in Chapter 4 we employed these techniques to estimate the system parameters of AE Aquarii. We used methods that are inde pendent of emission-line measurements because the emission lines are poor tracers of the orbital motion of the white dwarf (see Chapter 3). Models were constructed that attempted to reproduce the observed characteristics of the secondary star’s ab sorption lines. This was done by convolving the spectrum of an isolated star (of similar spectral type as the secondary star AE Aquarii) with a line broadening func tion. Note that the line broadening function is dominated by Doppler broadening because of the rapid rotation of the secondary star. We used a spherical approxima tion and found that the projected rotation velocity of the secondary star is between 77 and 107 km s-1, and constrained the mass ratio, q, to be 0.51 < q < 0.85. We then used the additional information supplied by the pulse-arrival-time orbit to find q = 0.646 ±0.013, and Vrot sin i = 88.8 ±0.5 km s-1. We also enabled us to measured the linear limb darkening coefficient: U = 0.40 ± 0.05. An improved estimate for the mass of the white dwarf gives Mi sin 3 i = 0.455 ± 0.01 A/©. We dropped the spherical assumption with the intent of determining the inclination from the phase-dependent modulations of the absorption lines (i.e., ellipsoidal variations). We found that the observed variations could not be reproduced by a limb- and gravity-darkened Roche lobe model. This implies that there are deviations from a uniform distribution of absorption-line strength across the secondary star’s surface. 5.2 What next ? Unanswered Questions and Ideas As is often the case, as some questions are answered, others arise. Concerning the oscillations, a number of questions immediately come to mind. Does AE Aquarii really emit pulsed TeV gamma rays? If so, this is an important clue as to what is ultimately responsible for the oscillations. Also, at the claimed flux level, a large fraction of the total energy budget is emitted by the gamma rays, and so they cannot be treated as an interesting, but insignificant, phenomenon. The oscillations detected in the X-ray band show only the 33s pulse, and not the 16.5s harmonic. Why? Is one of the oscillation-producing sites occulted? Does this tell us something about the geometry of the inner part of the disk? Do the oscillations show any dependence on orbital phase? This would also tell us about the geometry of the inner part of the 153 disk. Do the oscillations show any dependence on the flaring state of the system? Can oscillation “echoes” be seen coming from the secondary star? De Jager (1993) claims to have detected a large spin-down (period increase) of the oscillations. If true, this P term can provide much information on the coupling between the accretion disk and the white dwarf. But one must be wary that a P term can also be produced by an acceleration of the binary system caused by a third body in the system; in other words, if the binary system AE Aquarii was actually part of a triple star system, orbital motion could induce a spurious P measurement. Because the oscillation amplitude is much larger in the UV than in the optical (~ 40% versus ~ 0.1%; Eracleous et al. 1993), the UV is probably the best place to study the oscillations. Can we extend the stellar flare analogy to CVs? If the same flare-producing mechanism is operating in CVs (i.e., a sudden release of magnetic energy rapidly heats the gas), then two simple observational predictions can be made: (1) the Ca II K line in CVs flares should to rise and decay even more gradually than the Balmer lines, with little increase in their widths, and (2) the integrated soft X-ray flare energy should be correlated with the energy released in H 7 or the Ca II K line (see Hawley & Pettersen 1991 and Butler, Rodono, & Foing 1988). Although not discussed earlier in this dissertation, we briefly addressed the ques tion of whether the flares are coming from the disk or from the secondary star. Doppler tomograms (see Marsh & Horne 1988) of the Ila emission line were constructed for each night in the hope that the location of the flares in velocity space could be dis cerned, but the flares were not localized and the source of the flares could not be identified. The lack of localization does suggests that the flares do not originate on the secondary star. Caution must be used in interpreting this result, as an assumption built into the Doppler tomography method is that line profile changes are a result of viewing angle changes caused only by orbital motion. Since the timescale of line pro file changes due to flares can be short compared to the orbital period, this assumption is violated. The standard Doppler tomography technique is not, in retrospect, likely to shed light on the origin of the flares. And so the big questions remains unanswered: “Where are the flares coming from?” As shown in Chapter 3, the Ha line shows a orbital phase shift of ~ 75° from the position of the white dwarf. This effect is also seen in other CVs, particularly the SW Sex stars. What causes this phase shift? Because AE Aquarii is one of the brightest CVs, it may be the best place to look for the answer to this question. Also, what causes the line to be so broad and variable? Because the lines show no signs of being double-peaked, and the inner part of the disk is probably disrupted by the magnetic field of the white dwarf, the disk in AE Aquarii may be rather feeble. This claim is supported by the fact that the secondary star is the major contributor to the optical spectrum, not the disk. Could a large part of the Ha line be produced by a stellar wind from the secondary star or a wind from the disk, and could this emission fill in the cores of double-peaked emission lines making them appear single-peaked? The C IV A1549 doublet is a good diagnostic of winds from CV disks (see for example Drew 1991). In AE Aquarii the C IV doublet shows no signs of a P-Cygni profile (Jameson et al. 1980; Eracleous et al. 1993), nor do any other optical or ultraviolet 155 emission lines. This implies that it is unlikely that a strong disk wind is present, but it does not completely exclude the possibility of a wind. Is the Ha line different from the other lines in the Balmer series, i.e., is it more contaminated by secondary star chromospheric emission or emission from the bright spot? And finally, is magnetic Zeeman-effect broadening important? The secondary star apparently does not fill its Roche lobe unless it is somewhat evolved. Is the secondary star really evolving off the main sequence? Does it really fill its Roche lobe? How is mass transferred? Could it be via a stellar wind, as in the (high-mass) X-ray binaries? In Chapter 4 evidence was presented that showed that the secondary star was not featureless. An obvious thing to do is to construct an image of the secondary star from the absorption line variations via a method akin to the maximum entropy version of the Doppler imaging technique (see Vogt, Penrod, & Hatzes 1987 for a discussion). Are flickering and flaring related? The evidence presented in Chapter 2 hinted that this might be the case, but the results are far from conclusive. If they are related, is this a general feature in CVs or is it only true in AE Aquarii? A recent phenomenological model for solar flares has been proposed which incorporates the ideas of self-organized criticality (the “avalanche model” — see Lu & Hamilton 1991, Crosby et al. 1993, and Lu et al. 1993). Does the flickering in CVs obey a similar relationship, and can these ideas be applied to CV disks? What is the nature of the rapid flares/spikes seen in the optical and UV? How are they related to the flares and flickering? Are they correlated with radio emission and gamma-ray bursts? What is 156 their spectrum? And finally, it should be noted that a “World Astronomy Days” campaign to simultaneously observe AE Aquarii at many wavelengths (ROSAT, IUE, HST, vari ous radio, optical and IR bandpasses, air shower gamma rays and the Whole Earth Telescope network) is scheduled for October 1993. The uninterrupted “Whole Earth Telescope” monitoring should prove to be particularly useful for studying the oscilla tions, and the multiwavelength coverage should be extremely helpful is studying the nature of the flares. Hopefully the answers to many of the above questions will be available in the near future. A ppendix A The “Cable Experiment” A.l Introduction In Chapters 3 and 4 we presented results based on the “Cable Experiment” data. For both of these Chapters, the data were combined in time into bins appropriate for resolving phenomena on the orbital timescale (the orbital period is 9.88 hours). The full time resolution of the data were not used. In this Appendix we describe the high-speed data acquisition method and briefly look at the data with high time resolution. A.2 Cable Experiment High-Speed Data Acquisition As discussed in Chapter 4, the Cable data consist of exactly simultaneous photometry and spectroscopy. Here we describe how this was possible. Absolute timing was essential for our goal to measure possible signatures of the rapid variations, but the Shectograph data system was not controlled by a stable clock. We therefore made a special effort to record in the photometer datastream the exact times when the Shectograph was taking data. This was done by linking the two data systems with a shielded twisted-pair cable and sending a signal from the Shectograph (at the 2.5m telescope) to the photometer (on the 1.5m telescope). 157 158 The strong microwave background from TV broadcast antennae on Mt. Wilson made it necessary to use a battery powered optoisolator as a differential signal to a sec ond optoisolator at the photometer. The Shectograph shutter signal received at the photometer was used to modulate the 100kHz signal derived from the 1MHz master clock of the photometer system and this was recorded as a second photometer chan nel. Ultimately this allowed us to synchronize the spectroscopy and the high speed photometry to roughly 0.01ms, and obtain high-speed photometric spectroscopy. A.3 Photometry In Figures 47 through 50 we show the high-speed photometric observations of AE Aquarii obtained with the Mt. Wilson 1.5m telescope. The data have been rebinned in such a way as to match the simultaneous high-speed spectroscopy (i.e., rebinned to 1.9s on nights 1 and 2, and rebinned to 3.3s on nights 3 and 4). Notice the large amplitude flares and the periods of quiescence. The curvature during the quiescent periods is probably due to (ellipsoidal-like) variations of the secondary star. The flares can be of short duration (tens of minutes) or considerably longer (few hours). The flaring and quiescence seen in these data are typical of AE Aquarii (see Patter son 1979 or Van Paradijs et al. 1989). Note that 175 mJy corresponds to about 2600 counts/2s, giving an (assumed Poissonian) error bar of ~ 3.4 mJy. The vertical spread (i.e. the band-like appearance) of the data away from the flares appears ~4 times greater that this, but this is in fact consistent with a Poissonian distribution. This implies that the power in the flickering is less than the photon counting noise power for integrations shorter than a few seconds. aeeghbn fr ih 1 Ntc tedsic pros ffaig n quiescence. and flaring of periods distinct the Notice 1. night for band wavelength Figure 47: The Cable experiment high-speed photom etry in the the in etry photom high-speed experiment Cable The 47: Figure Photometric mJy o to o 100 150 200 250 6 5 7 T fr 07-27 2 - 7 -0 2 8 9 1 for UTC E Aqr AE 8 9 r\j 6000 - 7000A 7000A - 6000 11 159 wavelength band, obtained on night night on obtained band, wavelength Figure 48: The Cable experiment high-speed photometry in the 6000 - 7000A 7000A - 6000 the in photometry high-speed experiment Cable The 48: Figure Photometric mJy o in O 100 150 200 250 5 7 6 T fr 07-28 2 - 7 -0 2 8 9 1 for UTC 2 . AE Aqr 8 9 i ' S •i- 10 160 161 V603 Aql and AE Aqr UTC for 1982-07-29 Figure 49: The Cable experiment high-speed photometry, obtained on night 3. Ob servations of AE Aquarii occupy the latter half of the night; the earlier light curve is of V603 Aquilae. Notice that it also flickers. iue5: h al xeiethg-pe htmtyotie nngt4 Com 4. night 49.on Figure obtained and 48, photometry Figure 47, high-speed Figure with experiment pare Cable The 50: Figure Photometric mJy o 100 200 .300 7 6 T fr 0 3 - 7 0 - 2 8 9 1 for UTC 8 E Aqr AE 9 10 11 162 163 A.3.1 Power Density Spectra Power density spectra created from the high speed photometry are shown in Figure 51. In each case a rapid rise towards low temporal frequency is clearly visible. This is due to flickering, and is common in CVs. These power density spectra show no evidence for coherent oscillations or QPO’s near the 33 or 16.5s periods in any data segment or for any night as a whole. This null result is not surprising if one considers the spectrum of the oscillations. WHO show that the spectrum of the oscillations is quite blue (/„ oc ^+1), and so near Ha the amplitude of the oscillations is small, ~ 0.085 mJy. [Note that this is a peak- to-peak amplitude, not rms value. In WHO, the rms value from their blackbody fit is 0.06 m Jy, and the data actually give 0.12 ± 0.05 mJy. A power-law fit to WHO’s data give an Ha oscillation peak amplitude flux of 0.09 m Jy.] The sensitivity of the photometry allows us to detect a Fourier amplitude of approximately 0.2%, and for the photometric flux variations in the range ~ 130 - 300 mJy, the detection threshold is ~ 0.26 - 0.6 mJy, comfortably above the expected oscillation amplitude. However it is still possible that the Ha line is modulated, because the equivalent width of the line is much smaller than the bandpass of the photometry. 164 AE Aqr Night 1 JlliAiilLlLL* • I i ■ in f p ’IIV ‘i » '|n ' »|\'| J n J i i iJu i i T ik4i.liLt.ib .V J I Mfrjr'iji' Ii .j, i.iil - JllilyLlliiilil .lli’it j J ,'iiLj, i J.ill.■!>l; . . t . u l i l Night 2 UliJUiliiiiLtiiiiWLjik.t .ilill.t Jkh^UuiuU J 1 liiiu.. jLI ill J J.LL Lilli. 4 ..-I fcA ^ L.i ^iui jj.li'1 Jlu a . .iL* i h l Night 3 Night 4 ll t i i l i ktlLlllilJN I .1*. Jil k ilUbij., U ilJ i^ L jILl..illl i I. LiJiij.i 40 60 80 100 Frequency (mHz) Figure 51: Power density spectra of the high-speed photometry for each night. No detection of the oscillations was seen (the expected frequencies occur at 30.2 m H z and 60.5 mHz). A ppendix B H a Flares B.l Introduction In this Appendix, we show examples of the large amplitude Ha flares in AE Aquarii, as seen in the Cable data. We find that the Ha flares peak after and are of longer duration than the continuum flares, further strengthening the CV flare and stellar flare analogy as discussed in Chapter 2. B.2 Her Line Profile Variability- Trailed spectrograms of the Ha emission line for each of the four nights are dis played in Figure 52 versus orbital phase. The data have been rebinned in velocity to 20 km s -1 pix -1 and in time to 1 /50th of an orbit (approximately 6 minutes). To show the line variability, the continuum has been subtracted off each spectrum. The grayscale intensities are the same for each night (0 - 800 m Jy, linear stretch). The Ha line shows large variations in both amplitude and profile. Individual flare events can be seen. The variations are not limited to the line core, but can be seen to extend to ±1500 km s-1, and are not necessarily symmetric. Upon visual inspection, it is not apparent that the flares follow any pattern in velocity. 165 -1000 0 1000 -1000 0 1000 -1000 0 1000 -1000 0 1000 velocity (km s-1) WEISH 20-JUL-1B93 02:0S Figure 52: Trailed spectrograms of the Ha line plotted against orbital phase (time increases towards the top). The time resolution is ~ 6 minutes. Large amplitude flares extending to velocities over 1000 km s-1 are present. 167 As an example of the extreme line profile variability, we show two Ha continuum- subtracted line profiles in Figure 53. These two spectra correspond to the maximum flare and minimum quiescent levels on night 4. The spectra were taken 2 hours apart. We looked at several “quiescent” spectra over the four nights of observations and found that even in quiescence the line profile can have very different shapes. Note that in Figure 53, the orbital motion has not been removed; thus some of the variation is simply due to Doppler motion. However this can account for a Doppler shift of up to about 100 km s-1, which is only half a tick mark on the figure and therefore is not im portant. B.3 Time Evolution of the Her and Continuum Flares In Figure 54 we show the continuum and Ha light curves plotted against orbital phase (using the WHG ephemeris). The curve with the large dots represents the continuum measurements. The scale is the same in each panel, with the continuum flux scale on the right and the integrated line flux scale on the left. The fractional variation in the Ha line is considerably larger than in the continuum. We notice a remarkable phenomenon: the continuum flares are sharper and slightly precede the Ha flares. This is best seen in night 4. In Figure 55 and Figure 56 auto- and cross-correlations are plotted for each of the four nights. The data were linearly interpolated to uniform time sampling. The dots are the continuum auto-correlations, the triangles are the line auto-correlations and the crosses are the cross-correlations. In all cases the cross-correlation shows a strong asymmetry, with the Ha line lagging the continuum. spectrum preceeded the quiescent spectrum by two hours. two by spectrum quiescent the preceeded spectrum Figure 53: Continuum -subtracted H a line profile variations on night 4. The flare flare The 4. night on variations profile line a H -subtracted Continuum 53: Figure (mJy) -2000 O CM o o o o (O o o co o o 1000 -1 a ie rfl Variations Profile Line Ha eoiy k s 1) s (km velocity 0 1000 2000 168 169 Continuum and Ha flares O O o continuum . O Ha O to CN O O N I oE ■ night 2 cn Q> Z3E oO ‘ night 3 nr o o o o o LO CN I night 4 O o 0 .5 1 Binary Phase WELSH 20—JUL—1993 02:21 Figure 54: Integrated Ho: flux and mean continuum level plotted against orbital phase. The integrated line flux scale is on the left and the continuum scale is on the right. Notice that the continuum flares are more sharp and precede the Ha flares. 170 Nights 1&2 Ha and continuum cross correlations in o o in o o - 5 0 0 50 lag (minutes) WELSH J-AUG-1993 17:0B Figure 55: The X’s show the cross-correlation of the integrated Ha line flux against continuum flux for night 1 (upper panel) and night 2 (lower panel). The cross-correlation is asymmetric and the peak lags behind the continuum. Also shown are the continuum auto-correlation (dots) and the line auto-correlation (triangles). 171 Nights 3&4 H« and continuum cross correlations o o - 5 0 0 5 0 lag (minutes) WELSH 3-AUG-1993 17:09 Figure 56: The X’s show the cross-correlation of the integrated Ha line flux against continuum flux for night 3 (upper panel) and night 4 (lower panel). The scale is the same as in Figure 55. Also shown are the continuum auto-correlation (dots) and the line auto-correlation (triangles). This time-delay effect is also seen in stellar flares, where the Balmer lines tend to lag and decay more gradually than the continuum (see for example Figs. 3 and 4 in Mochnacki & Zirin 1980). Thus the time evolution of the Ha flares adds support to the hypothesis that a similar mechanism is responsible for both kinds of flares (recall in Chapter 2 that it was found that the optical spectrum of a flare in AE Aquarii resembles that of a stellar flare.) Other support comes from the UV spectrum, which shows very strong Si lines and Fe II features (Jameson, King, & Sherrington 1980, Eracleous et al. 1993) which are strongly enhanced in flare stars during flares (e.g. see Hawley &; Pettersen 1991). From the time delay it is possible to make a crude estimate of the lower limit of the density of the flaring gas using the approximate relationship that the recombination time is ~ NeaB, where Ne is the electron number density and ag is the case B recombination coefficient for Ha (see Table 4.2 in Osterbrock 1989). Time delays in the range 1 to 10 minutes give electron number densities between 7.5 x 109 and 2.7 x 1011 cm-3 for temperatures between 5,000 and 20,000° A. Case A recombination yields densities about a factor of ~ 1.5 times larger. It should be cautioned that these lower limits are based on the assumptions that the time delay is a result of the recombination time required for the gas to react to ionization and that the observed continuum varies simultaneously with this ionizing radiation. Because the physical processes producing the flares are not known, this assumption may or may not be valid. Nevertheless these values can be compared with those typical of stellar flares. By examination of stellar flare spectra, Kunkel (1970) estimates electron densities in the range 1013 to 1015 cm-3, and Haisch, Strong, &; Rodono (1991) quote a value of ~ 1012 cm-3 based on an analysis of various time-resolved lines. It is not known whether the smaller values for N e determined for the Ha flares in AE Aquarii are due to intrinsic differences in the flare mechanism, or to the different methods used to determine Ne. A ppendix C Ephemeris Data Base and Details C.l The Data Base HJD’s and absorption-line radial velocity measurements used by Chincarini & Walker (1981) are listed in Table 5 of the ESO preprint by Chincarini & Walker (1981b). These also include values obtained by Joy (1954). These 370 measurements span nearly 22 years, and incorporating them into our ephemeris calculations expands our time coverage from four days to nearly 39 years. But in order to use the data from C&W’s Table 5, a few corrections had to be made, which we enumerate below so that others may more easily use this wealth of observational data. (1) In the calculation of phases, a P term was used, although its significance is dubious, and it seems that all the phases have been truncated rather than rounded off. For HJD (243000.+)8587.827 the phase is incorrect. For these reasons, we recommend that the user re-compute the phases using the given HJD’s. (2) For HJD’s prior to 2438160, the values listed in C&W are taken from Joy (1954). Comparing with Joy’s Table 1, we find that the HJD listed as (2430000.+) 1011.6880 in C&W should be 1011.685, and that the velocity listed for HJD 1328.6530 should probably be -150 rather than -159 km s~1. 174 175 (3) The fifteen measurements made between HJD 8160 and 8179 were found to have inconsistent phases listed. After recomputing the phases, we found that the velocities were also inconsistent with expectations. This leads us to believe that it is the HJD’s that are incorrect. These values have been omitted from our analysis. (4) The HJD 8577.6965 should be changed to 8677.9656 (5) HJD’s 8607.8702 and 8607.8689 are in reverse order. The associated phases are correct, so we assume that the velocities are also correct. We believe that the measurements are valid, but have simply been printed out of order. To use these data, they must be properly weighted. Unfortunately neither Joy (1954) or C&W list the uncertainties in their measurements, so we are forced to estimate them. By making the assumption that the absorption-line velocities reflect a circular orbit, we fit the data with a sinusoid, and use the square root of the reduced X 2 to scale the error bars until the reduced x2=1.0. This was done independently for Joy’s data and C&W’s data, yielding estimates of the la uncertainties of ±22.71 and ±10.55 km s-1 respectively. The error estimates for Joy’s data reflect the fact that long exposures were used (1 or 2 hours) which allowed considerable smearing of the lines. Because we are combining data sets, an identical method for estimating the errors was carried out on the Cable experiment data as well, but in this case the error bars are not all identical. The Cable experiment radial velocities are listed in Table 10. 176 C.2 Confusion Unfortunately,' a number of typographical errors have been made in the literature. In C&W, the HJD of To is given as 2439030.934 in the text, but as 2439030.984 in their Table 6. Van Paradijs et al. (1989) use C&W’s epoch, but quote a value of 2439030.983, and in an erratum (Van Paradijs et al. 1991) they state To=2439030.934. Here 7o is defined such that if an eclipse of the white dwarf could occur, it would happen at phase 0.5. C&W used both absorption-line and emission-line data in their ephemeris deter mination and found a period of 0.4115637 days. Using only the absorption-line data from C&W and Joy (1954), F&C give To = 2439030.621 and period 0.41165794 days, which, as discussed in Chapter 4, may be in error. Combining their photometric data with Joy’s and C&W’s, Van Paradijs et al. (1991) found a period of 0.4116560 days. RSB quote F&C as using To=2439030.827, which is F&C’s value advanced a half cycle (using P=0.411658) so that white dwarf eclipse would occur at phase 0.0. In a “note added in proof”, Bruch (1991) states that the discrepancy in his data between the expected and observed phases of the maxima of the secondary star’s ellipsoidal light curve is due to a slight error of the orbital period as given by VPKA, but no new period is given. C.3 Discrepencies Some puzzling discrepancies were noticed between the radial velocity measurements of C&W, Joy (1954) and our own. Folding each data set separately onto the adopted ephemeris, we notice small, but formally statistically significant, differences in the gam ma and K 2 velocities (the phase offsets agreed, indicating that our ephemeris is correct). These values are compared in Table 9. Note that the values in this table for Joy’s and C&W’s data are not identical to those published by said authors because we are folding their data on an improved period. (Also, C&W combined their data with Joy’s and also with emission line measurements.) The gamma velocity of C&W’s data (-70 km s-1) is much larger in amplitude than that of Joy’s (-52 km s-1) or our data (-54 km s_1). The disagreement between Joy and C&W was noted by C&W, and neither we nor C&W can offer an explanation. It is highly unlikely that the systemic velocity of our reference star (61 Cyg A) could be in such large error that the discrepancy between the C&W’s and the Cable data could be explained. Even if this were the case, the discrepancy with Joy’s data would still have to be explained. If we assume for the moment that C&W’s data have a systematic velocity error, which we then correct so that C&W’s gamma velocity equals that of the Cable data, a new orbital solution can be computed and compared with our adopted values. The results of this test are consistent with our adopted values (except for gamma, of course, which was -54 km s-1). It may be possible that because the absorption lines lie on a red continuum, all velocity estimates that do not remove the continuum beforehand will systematically measure line centers that are too blue, hence introduce a bias towards negative gamma velocities. However a shift of 0.14A is needed to produce a 10 km s -1 bias at A 4250A, and it would affect not only C&W’s measurements but Joy’s as well. It is interesting to note that the gamma velocity determined from our spherical star 178 modelling (Chapter 4) is -65.4 km s-1, which does not agree with the gamma velocity determined from the absorption-line radial velocity measurements. Because gamma does not play a role in determining the dynamics of the binary system, we make no further effort to improve its estimate. We note in passing that the the observed value of ~ 55 km s~l is rather large for a disk population star. The K 2 velocity on the other hand is a crucial parameter and any discrepancy warrants attention. Although we forced the reduced y2= l for each of the fits, the RMS residuals are not identical. C&W’s data show far less scatter, and therefore should best represent the (assumed) circular orbit. The adopted K 2 velocity reflects this, and in fact is identical to C&W’s estimate. We are somewhat concerned that the I<2 velocity based only on the Cable data lies nearly 3.5 This may reflect the technique used to measure the radial velocities; indeed, various methods of locating the peak of the cross-correlation function can induce differences in I<2 for the Cable data of up to 2 km s-1. The spherical modelling (Chapter 4) gave K2 ~ 158 km s-1, again suggesting that the cross-correlation method somehow biases K 2 too large. 179 Table 9: Comparison of AE Aquarii Data Sets. Data Set number of K 2 72 phase offset RMS residual points (km s -1) (km s -1) (cycle) (km s -1 ) Joy 105 153.5 -51.9 -0.0114 23.45 3.2 2.4 0.0070 C&W 250 157.9 -70.8 -0.0003 10.88 0.9 0.7 0.0021 Cable 197 163.2 -54.3 0.0005 14.02 1.5 1.1 0.0031 Combined 552 157.9 -66.7 0.0004 19.68 0.8 0.7 0.0021 Table 10: Cable Experiment Absorption Line Radial Velocities (with respect to 61 Cyg A) HJD orbital velocity ±<7 phase (km s -1) (km s -1 )t 2445177.721130 0.2231 191.1 10.7 2445177.722311 0.2259 180.7 10.3 2445177.729837 0.2442 59.0 -11.0 2445177.731975 0.2494 190.8 7.3 2445177.736008 0.2592 167.6 7.0 2445177.740141 0.2692 160.6 5.3 2445177.744244 0.2792 150.0 -9.1 2445177.746695 0.2852 140.1 7.8 2445177.756741 0.3096 175.5 7.8 2445177.760651 0.3191 165.4 7.3 2445177.764814 0.3292 132.8 8.7 2445177.768920 0.3392 134.9 6.5 2445177.771592 0.3456 47.3 -12.4 2445177.778758 0.3631 224.1 -10.5 2445177.781297 0.3692 163.1 7.1 2445177.785401 0.3792 129.8 7.3 2445177.789584 0.3893 113.7 7.8 2445177.793667 0.3993 109.6 6.2 2445177.806123 0.4295 82.6 6.6 2445177.810093 0.4392 83.4 5.4 2445177.814311 0.4494 38.9 7.3 2445177.818125 0.4587 49.2 5.5 2445177.827395 0.4812 2.8 5.1 2445177.830672 0.4892 32.0 7.2 2445177.834818 0.4992 -2.5 6.1 2445177.838940 0.5092 6.9 8.1 2445177.842218 0.5172 -37.4 6.9 2445177.852384 0.5419 10.1 8.0 2445177.855376 0.5492 -28.2 5.4 2445177.859499 0.5592 -40.7 3.8 Table 10 (continued) HJD orbital velocity ±< 7 phase (km s -1) (km s_1)t 2445177.863568 0.5691 -55.2 6.0 2445177.866791 0.5769 -49.6 6.5 2445177.876526 0.6006 -74.2 6.2 2445177.880298 0.6097 -89.8 7.0 2445177.884202 0.6192 -113.4 8.5 2445177.888335 0.6292 -97.4 6.3 2445177.891435 0.6368 -132.2 8.0 2445177.897793 0.6522 -115.6 7.5 2445177.900639 0.6591 -125.2 8.1 2445177.904789 0.6692 -115.3 6.9 2445177.908893 0.6792 -131.7 6.2 2445177.912672 0.6884 -107.5 4.4 2445177.921617 0.7101 -152.4 5.9 2445177.925366 0.7192 -128.0 8.2 2445177.929499 0.7292 -147.1 4.8 2445177.933617 0.7392 -137.0 6.6 2445177.936497 0.7462 -164.1 5.8 2445177.943565 0.7634 -108.4 6.3 2445177.945927 0.7691 -125.7 8.7 2445177.950060 0.7792 -148.2 6.4 2445177.954193 0.7892 -138.3 7.0 2445177.958296 0.7992 -153.0 5.9 2445177.960510 0.8046 -100.0 -11.3 2445178.717365 0.6431 -140.2 7.0 2445178.719785 0.6490 -143.0 6.8 2445178.723918 0.6591 -126.4 7.4 2445178.728050 0.6691 -140.7 6.0 2445178.731238 0.6768 -133.0 6.0 2445178.737816 0.6928 -140.7 8.5 2445178.740384 0.6990 -143.3 6.9 Table 10 (continued) HJD orbital velocity ±(7 phase (km s -1) (km s_1)t 2445178.744487 0.7090 -167.6 6.8 2445178.748620 0.7191 -172.8 7.1 2445178.752693 0.7289 -166.1 6.4 2445178.761147 0.7495 -149.4 7.6 2445178.765073 0.7590 -174.8 7.5 2445178.769176 0.7690 -147.0 6.0 2445178.773308 0.7790 -140.2 6.6 2445178.776024 0.7856 -186.9 6.3 2445178.782264 0.8008 -144.6 7.8 2445178.785659 0.8090 -137.8 6.2 2445178.789792 0.8191 -169.7 7.0 2445178.793895 0.8291 -138.7 6.8 2445178.797139 0.8369 -141.9 7.2 2445178.803524 0.8524 -108.5 9.8 2445178.806240 0.8590 -130.1 6.9 2445178.810343 0.8690 -113.4 6.0 2445178.814476 0.8790 -99.3 7.2 2445178.818402 0.8886 -105.3 6.4 2445178.826990 0.9094 -74.9 7.1 2445178.830945 0.9191 -83.0 6.1 2445178.835048 0.9290 -69.7 5.8 2445178.839151 0.9390 -58.5 6.0 2445178.841867 0.9456 -44.8 7.5 2445178.849096 0.9631 -46.8 5.0 2445178.851517 0.9690 -24.2 4.6 2445178.855620 0.9790 -13.5 4.9 2445178.859753 0.9890 -2.3 4.7 2445178.863886 0.9991 23.3 6.1 2445178.866042 0.0043 21.5 9.3 2445178.872565 0.0201 41.2 6.3 Table 10 (continued) HJD orbital velocity ± 2445178.917378 0.1290 139.7 6.0 2445178.921501 0.1390 157.7 7.0 2445178.925616 0.1490 127.6 6.4 2445178.929720 0.1590 146.2 7.0 2445178.931993 0.1645 188.9 8.5 2445178.938019 0.1791 140.0 -9.8 2445178.942063 0.1890 157.8 6.5 2445178.946196 0.1990 168.2 6.1 2445178.950330 0.2091 179.2 7.8 2445178.952927 0.2154 173.8 7.3 2445178.959073 0.2303 175.2 8.2 2445178.962626 0.2389 164.0 6.8 2445178.966798 0.2491 147.9 5.4 2445178.970911 0.2590 166.7 7.2 2445178.973981 0.2665 172.4 7.4 2445179.914285 0.5507 -70.1 6.9 2445179.917634 0.5588 -57.7 8.0 2445179.921744 0.5688 -52.2 5.2 2445179.925854 0.5788 -57.7 7.4 2445179.929963 0.5888 -63.4 6.0 Table 10 (continued) HJD orbital velocity ± 2445179.979379 0.7088 -141.4 6.7 2445179.983489 0.7188 -155.0 7.3 2445179.987598 0.7288 -139.1 7.5 2445179.989974 0.7346 -143.6 7.5 2445179.996821 0.7512 -137.7 8.0 2445179.999951 0.7588 -146.6 6.9 2445180.002703 0.7655 -150.7 7.3 2445180.746886 0.5733 -33.5 11.2 2445180.749111 0.5787 -74.5 7.1 2445180.753221 0.5887 -63.0 7.1 2445180.756841 0.5975 -80.1 7.3 2445180.763149 0.6128 -113.3 9.3 2445180.765562 0.6186 -114.5 7.2 2445180.769673 0.6286 -130.0 6.4 2445180.772727 0.6361 -112.3 7.9 2445180.779063 0.6514 -126.7 9.6 2445180.782042 0.6587 -137.8 6.9 2445180.786152 0.6687 -129.3 6.1 2445180.788828 0.6752 -137.3 7.7 2445180.795059 0.6903 -131.7 8.7 Table 10 (continued) HJD orbital velocity dbcr phase (km s -1) (km s -1)t 2445180.798490 0.6986 -148.4 7.5 2445180.802600 0.7086 -150.9 6.9 2445180.808571 0.7231 -143.4 5.9 2445180.810871 0.7287 -144.4 7.9 2445180.814981 0.7387 -150.0 8.0 2445180.817998 0.7460 -143.5 9.0 2445180.824370 0.7615 -136.3 8.0 2445180.827311 0.7686 -166.9 7.0 2445180.831422 0.7786 -148.5 7.1 2445180.835569 0.7887 -179.1 7.8 2445180.839679 0.7987 -141.6 6.4 2445180.843714 0.8085 -142.2 6.7 2445180.853860 0.8331 -103.2 10.3 2445180.856123 0.8386 -135.2 6.6 2445180.860233 0.8486 -130.5 6.7 2445180.864343 0.8586 -101.7 6.5 2445180.868491 0.8687 -117.6 4.9 2445180.872602 0.8787 -110.7 5.4 2445180.875279 0.8852 -117.0 4.9 2445180.882458 0.9026 -72.8 6.5 2445180.884947 0.9087 -76.9 6.7 2445180.889057 0.9186 -56.0 6.0 2445180.893167 0.9286 -72.3 6.5 2445180.897278 0.9386 -55.1 5.7 2445180.901426 0.9487 -37.6 6.0 2445180.909868 0.9692 -29.5 6.9 2445180.913754 0.9786 -8.7 8.2 2445180.917868 0.9886 -9.5 6.3 2445180.921978 0.9986 2.8 5.7 2445180.926126 0.0087 18.6 5.4 Table 10 (continued) HJD orbital velocity ±cr phase (km 6-1) (km s-1)t 2445180.929180 0.0161 29.3 6.4 2445180.935773 0.0321 36.4 7.7 2445180.938451 0.0386 61.2 6.3 2445180.942562 0.0486 57.5 4.8 2445180.946673 0.0586 79.0 6.5 2445180.950823 0.0687 76.5 6.6 2445180.954910 0.0786 84.2 4.9 2445180.957208 0.0842 98.5 8.4 2445180.963611 0.0997 112.6 6.8 2445180.967271 0.1086 121.1 7.0 2445180.971385 0.1186 118.3 7.0 2445180.975507 0.1287 133.2 5.9 2445180.979620 0.1386 134.9 7.0 2445180.982977 0.1468 146.5 7.1 2445180.989465 0.1626 152.5 8.9 2445180.991954 0.1686 136.9 7.9 2445180.996068 0.1786 135.1 7.0 2445181.000186 0.1886 157.0 8.7 2445181.004334 0.1987 155.0 1.7 2445181.006861 0.2048 166.1 8.3 t 1 a uncertainty based on determining the peak of cross-correlation function. 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