JSTOR

1. AN INTRODUCTION TO CATACLYSMIC VARIABLES

Cataclysmic variable stars (CVs) are very close binary star systems consisting of a compact stellar remnant “primary” star—typically a white dwarf star—that is stripping mass from a less massive lower main‐sequence “secondary” star (Hellier 2001; Warner 1995). They are closely related to low‐mass X‐ray binaries, which instead have neutron star or black hole primaries. CVs display a wide range of behavior, the most dramatic being the “classical nova” explosion, which occurs when the accreted surface hydrogen layer is massive enough to ignite a thermonuclear runaway in the envelope layers of the primary white dwarf star, causing the system to brighten by a factor of ∼10⁴ to 10⁶ over the preeruption brightness. Less dramatic but far more common are the photometric variations caused by the accretion disk itself, which is typically the brightest component of the system. Simple energy arguments (Frank et al. 2002) show that the total accretion disk luminosity is approximately where and are the white dwarf mass and radius, respectively, G is the gravitational constant, and is the accretion rate onto . Roughly speaking, half the luminosity should be radiated in the disk, and half in the boundary layer, where the Keplerian accretion flow settles onto the more slowly rotating white dwarf. Viscosity within the differentially rotating fluid of the disk acts to transport angular momentum outward in radius so that mass can migrate inward. Angular momentum from the outer disk is fed back into the orbital angular momentum of the secondary star by means of tidal torques. The material in the accretion stream is highly supersonic when it impacts the edge of the disk, causing a “bright spot” that, when observed in high‐inclination (i.e., more nearly edge‐on) systems, leads to an observed brightening for that portion of the orbit in which the bright spot is facing the observer. This brightening once per orbit is called an “orbital hump,” and if present, it reveals the orbital period.

Systems with high mass transfer rates , where indicates units of solar mass, are classified as old novae or nova‐like variables, depending on whether a classical nova explosion has been observed or not. The disks in these systems are thought to be in a permanent high‐viscosity state such that mass transfer through the disk and onto is efficient and in a state of quasi‐equilibrium. Most systems with smaller mass transfer rates are so‐called dwarf nova systems and have disks that cycle between high and low states. The mass flow through L1 is not sufficient to keep the disk in a permanently high state, so matter accumulates in the disk until a critical point is reached. Before this critical point is reached, the mass flow rate onto is low, and hence so is the system luminosity. Once that point is reached, however, a heating wave propagates through the disk material, sharply increasing the viscosity and mass flow through the disk and onto the white dwarf, resulting in a rapid brightening of the system by a factor of ∼10² in what is called a “dwarf nova outburst.” The source of accretion disk viscosity is thought to be magnetorotational instability (Balbus & Hawley 1991).

In addition to normal dwarf nova outbursts, many systems with short orbital periods ( hr) are observed to display “superoutbursts,” which are roughly twice as bright and last some 5 times longer (Patterson et al. 2005) than normal dwarf nova outbursts. In all superoutbursting systems, photometric oscillations with periods a few percent longer than the orbital periods are observed to grow to detectability on a timescale of a few tens of orbits. Because they are associated with superoutbursts, these periodic signals are known as “superhumps.” We now understand that superhumps are the result of a driven oscillation of the disk by the rotating tidal field of the secondary star (Whitehurst 1988; Hirose & Osaki 1990; Lubow 1991; Simpson & Wood 1998). For systems with mass ratios , the disk can extend out to the region of the 3:1 corotation radius, and the inner Lindblad resonance can be excited (Lubow 1991).

Cataclysmic variables have long been a favorite target of amateur and professional astronomers, and anyone with a modest‐aperture telescope and a CCD camera can obtain useful data on brighter objects. For example, the Center for Backyard Astrophysics1 is a global network of (mostly amateur) astronomers with small telescopes dedicated to the study of CVs, and the American Association of Variable Star Observers (AAVSO),2 founded in 1911 at Harvard College Observatory to coordinate variable star observations made largely by amateur astronomers, has a large contingent who are specifically interested in CVs.

Our group has been studying the superhump phenomenon numerically using the method of smoothed particle hydrodynamics (SPH). Experiments have confirmed the physical origin of the observed superhump light curves (Simpson & Wood 1998; Wood et al. 2000) and have shown that the light curves should become harmonically more complex for disks observed more nearly edge‐on (Simpson et al. 1998). Our research code computes the hydrodynamics of accretion disks in three dimensions and is highly optimized for serial computers, running as a command‐line FORTRAN program under the Linux operating system. Because of the high level of interest in CVs among observers and the difficulty in visualizing accretion disk dynamics from artists' conceptions and text descriptions in journals, we decided to develop and release a demonstrational version of FITDisk3 that includes a graphical user interface (GUI) and runs under the Windows XP operating system.

In § 2 we briefly introduce the method of SPH as applied to CV accretion disks. In § 3 we discuss the porting of FITDisk to the Windows XP environment and explain how to use the program. We show how our simulation light curves compare with time‐series observations of the helium dwarf nova CR Boo in § 4, and the conclusions are given in § 5.

2. SMOOTHED PARTICLE HYDRODYNAMICS

The SPH method approximates continuum hydrodynamics by using a scattered grid in which the grid points are effectively Lagrangian point markers in the fluid (Lucy 1977; Monaghan 1992, 2005). It is a powerful technique for simulating physical systems that are highly dynamic and bounded by vacuum; thus, for many astrophysical systems, SPH can be a more computationally efficient approach than more conventional Eulerian techniques. SPH replaces the fluid continuum with a finite number of particles that interact pairwise with each other, with the strength of the interaction being a function of the interparticle distance. The function is the kernel W, for which there are many possible choices. The simplest choice would be a Gaussian, but the consequence of this choice is that the interaction force is nonzero for all other particles in the system. It is common to use a polynomial spline function, which approximates a Gaussian but has compact support, being identically zero beyond twice the smoothing length h (Monaghan & Lattanzio 1985). With this choice, interparticle forces need only be calculated for neighbors within of a given particle. In the continuum limit, a field quantity can be estimated by the integral where the integration is over all space and where W is defined such that . To compute a field quantity using the finite number of SPH particles within of , we have where is the mass of the jth particle, is its density, and is its position. A general form for the estimate of the derivative of a field quantity is given by

The general form of the momentum and energy equations that are relevant for accretion disk studies are where u is the specific internal energy, is the viscous force, is the energy generation from viscous dissipation, and are the displacements from stellar masses and , respectively. We assume an ideal gamma‐law equation of state, , where we typically assume .

Our SPH form of the momentum equation for a given particle i is where we use a standard prescription for the artificial viscosity (Lattanzio et al. 1986), where Here is the average of the sound speeds for particles i and j, and we use the notations and . In FITDisk, we fix , but the user can select the viscosity coefficients by using a slider. Note that as is typical in artificial viscosity formalisms, only approaching particles are subject to a viscous force.

The internal energy of each particle is integrated using an action‐reaction principle (Simpson & Wood 1998), which is formally equivalent to the standard SPH internal energy equation but is computationally much more efficient. If this method fails, however, we then use the more standard form Like all hydrodynamics calculations, the time steps in FITDisk are sound‐speed limited. Because physically the orbital period at the surface of a white dwarf is ∼10 s, while the orbital period of the outer disk is ∼1 hr, it is advantageous to let individual particles have time steps as short as necessary. Time steps can be as long as , but a particle can have a shorter time step of , as required by the local environment, where provides sufficient dynamic range for all simulations.

The most common method of observation of superhumping CVs is time‐series photometry; i.e., sampling the brightness hundreds or more times per orbit. In FITDisk, we assume that the sum over all particles of the changes in the internal energies over a time step is directly proportional to the luminosity of the disk over that same time interval. Thus, we can calculate an approximate “light curve” for the simulation:

FITDisk simulates the fluid dynamics of an accretion disk that is subject to the gravitational potential of the two stars in the system. The calculations are made in the inertial frame. We treat the stars as point masses on circular orbits, and the center of mass of the system is the origin of the coordinate system. Particles are injected with thermal velocities of K at random coordinates within a small box at the L1 point. Particles are considered to be lost from the system if they are accreted onto or or are ejected from the system.

Our code specifically models accretion disks in the nova‐like systems. Particles are injected in a burst at a rate of a few thousand per orbit until the desired number of particles is reached. Whenever a particle is lost, a replacement particle is injected at L1. Thus, depending on the parameters of the simulation, the disk will either evolve to a state of quasi‐equilibrium or may eventually be driven into superhump oscillations.

3. USING THE FITDisk GUI

To port our research code from the Linux operating system to the Windows XP operating system, we used Compaq Visual Fortran and added a GUI and visualization using OpenGL. To ensure that the machine requirements (RAM and hard drive space, in particular) are as small as possible while still allowing interesting results, we limit the total number of particles to 25,000, although superhumps can be observed with as few as 1000 particles in a run time of under 5 minutes, allowing direct use in a classroom setting. The program uses about 100 Mbytes of RAM, so we recommend that systems have least 256 Mbytes of RAM for best results. Recording 100 orbits of a 25,000 particle simulation at high resolution will require approximately 7.5 Gbytes of disk space.

We maintain a FITDisk Web page from which the executable code and a reference manual can be downloaded.4 The download file is a compressed (“zip”) archive containing fitdisk.exe and two auxiliary files in a subfolder. These will extract to a directory named FITDisk. The initial release version of FITDisk is version 1.0. We have set up a mechanism at our Web site for users to send feedback, and we will release updates and bug fixes as needed, based on this feedback.

Figure 1 shows the startup screen, including a still image from a 25,000 particle simulation. Figure 2 shows the GUI that appears if “New Disk” is selected from the “File” pull‐down menu. Table 1 lists the adjustable parameters, their ranges, and default values. FITDisk can be used to explore a significant volume of parameter space. When simulating a relatively small number of particles (∼1000), the code runs fast enough to watch the disk build and evolve to superhumps in real time. However, when calculating a simulation with the maximum number of particles being evolved over enough time for superhumps to develop, even users with the latest hardware will probably want to record the results to the hard drive for next‐day playback at a fast frame rate. This is accomplished using standard Windows file requestors.

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Fig. 1.— Opening screen of FITDisk. The image shown is that of a 25,000 particle superhump simulation calculated with FITDisk.

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Fig. 2.— GUI for starting a new disk. The default parameters shown will lead to superhump oscillations within 10 orbits.

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TABLE 1 FITDisk User‐adjustable Parameters

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Simulations are visualized with smooth, double‐buffered graphics. The view can be interactively rotated about a horizontal axis by clicking and dragging the pointer inside the window, using either mouse button. The simulation can be paused (see Fig. 3), providing an opportunity to examine structural details or capture screen shots using the print‐screen key. To provide more insight into the structure and dynamics involved, the fluid particles are assigned colors based on their relative density or temperature, as selected from the “Options” menu. This feature gives an excellent qualitative picture of the radial density and temperature profiles and also helps to reveal features such as spiral shocks and superhumps.

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Fig. 3.— Example of simulations and renderings, which are fully three‐dimensional. At any time during the simulation, the user can pause the run, click‐drag the mouse to change the inclination, and continue the run. Note that the rotation angle is indicated in the title bar.

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Within the program, the frequencies present in the simulation light curves can be plotted and analyzed through the use of a fast Fourier transform (FFT) (see Fig. 4), which has been included. Time‐series plots show both the raw data and a boxcar‐smoothed fit. The smoothed curve can appear markedly similar to the light curves observed in real systems, as we discuss below. The Fourier amplitude spectrum is only shown out to a frequency of five cycles per orbit, to highlight the frequency range of interest. Peaks in the FFT that are 3 or more standard deviations (3 σ) from the mean are labeled with their specific frequencies. In this way, for superhumping systems, the program often identifies the fundamental superhump frequency, as well as the second and third harmonics. Thus, users can explore the relationship between the superhump oscillation period and the orbital period as a function of binary system mass ratio.

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Fig. 4.— Plot the simulation light curve and associated Fourier amplitude spectrum, which can be constructed after a simulation has finished. Peaks in the transform that are more than 5 σ above the mean are labeled.

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When “New Disk” is selected from the “File” menu, the simulations begin and no particles are present. The particles are injected in a burst at the L1 point. The stream initially travels on a roughly ballistic trajectory until the lead particles intersect the stream. Viscosity quickly causes the disk to collapse to the circularization radius, which has the same specific angular momentum as the particles entering at the L1 point. Over the next several orbits, as the mass burst continues, viscosity acts to spread out the mass in both directions from the circularization radius. If the mass ratio and the viscosity parameters α and β are of order unity, then eventually superhump oscillations will be driven into resonance by the rotating tidal field of the secondary.

If the restart option is checked, then at a rate determined by the user, the code will write out a file that can be read in so that the user can continue the simulation from that point. This might be useful if the user did not specify sufficient orbits for superhumps to develop in a high‐resolution simulation run.

4. COMPARISON WITH OBSERVATIONS

I. Silver (2005, private communication) observed the interacting binary white dwarf CR Boo (Wood et al. 1987; Patterson et al. 1997) with the 0.9 m SARA5 Telescope, located at Kitt Peak National Observatory, on the night of 2005 March 10 (UT), using an Apogee AP7p CCD camera with no filter, to maximize photon counts. The exposure time was 20 s, with an 11 s readout time. The data were reduced using IRAF.6 The top panel of Figure 5 shows ∼4.2 hr (∼10 orbits) of time‐series data for CR Boo, which has an orbital period of s and a superhump period of s (Patterson et al. 1997). Note that the observed superhump signal varies considerably from cycle to cycle.

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Fig. 5.— Comparison of the light curve of the helium dwarf nova CR Boo with simulation results. The top panel shows approximately 10 superhump cycles of CR Boo. The middle panel shows the final 10 orbits of a simulation, where a boxcar‐smoothed curve is included to guide the eye. The bottom panel shows the entire simulation light curve.

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To demonstrate that FITDisk generates useful output for comparisons with observations, we simulated a CV with mass ratio for 100 orbits using 25,000 particles, , and viscosity parameters and . The bottom panel of the figure shows the entire simulation light curve, and the middle panel shows the final 10 orbits. Because the simulation is limited to 25,000 particles, the noise band is larger than the observations. To compare with observations, we first summed over four points so that the sample rate per orbit is approximately that of the top panel, and we then included a boxcar‐smoothed line ( points) to guide the eye. Although we have made no special effort to match the observed light curve, the comparison of the simulation output with the observed light curve demonstrates the promise of using FITDisk to constrain system parameters.

5. CONCLUSION

Cataclysmic variables are mass transfer binary star systems that display a rich variety of behavior in their accretion disks. As a class, they have long been a favorite target of professional and amateur astronomers alike. Time‐series photometry of their outbursts and superhump disk oscillations, coupled with detailed numerical simulations, have the potential to reveal not just the structure and evolution of the disks via seismology, but also perhaps the fundamental origin of viscosity in differentially rotating astrophysical plasmas.

We present here the freely available precompiled‐binary code FITDisk, which allows nonspecialist users (e.g., amateur astronomers, students, and classroom instructors) to simulate the dynamics of CV accretion disk dynamics, including superhump oscillations, over a large volume of parameter space. The code is a demonstration version of our fully three‐dimensional smoothed particle hydrodynamics (SPH) code (Simpson & Wood 1998; Wood et al. 2000, 2005), with an intuitive, easy‐to‐use graphical user interface added on and ported to the dominant operating system for PCs. It is our hope that this code will find wide use not only in amateur and professional astronomical circles, but also in educational settings ranging from high schools to universities.