Full text of DOI:10.1111/j.1365-8711.2000.03880.x

Mon. Not. R. Astron. Soc. 319, 467±476 (2000)
The outbursts of dwarf novae
M. R. Truss,w J. R. Murray, G. A. Wynn and R. G. Edgar
Department of Physics and Astronomy, University of Leicester, University Road, Leicester LE1 7RH
Accepted 2000 July 10. Received 2000 June 12; in original form 2000 April 26
A B S T R AC T
We present a numerical scheme for the evolution of an accretion disc through a dwarf nova
outburst. We introduce a time-varying artificial viscosity into an existing smoothed particle
hydrodynamics code optimized for two- and three-dimensional simulations of accretion
discs. The technique gives rise to coherent outbursts and can easily be adapted to include a
complete treatment of thermodynamics. We apply a two-dimensional isothermal scheme to
the system SS Cygni, and present a wide range of observationally testable results.
Key words: accretion, accretion discs ± hydrodynamics ± instabilities ± methods:
numerical ± binaries: close ± novae, cataclysmic variables.
example, Rutten et al. 1992). The currently favoured model, and
the model that we apply here, is that of an instability in the
1
I NTRO DUC TION
Dwarf novae are a class of cataclysmic variable that undergo
regular but aperiodic phases lasting several days, during which the
system brightness increases by two to four magnitudes. These are
the well-known normal outbursts of dwarf novae, and they recur
on time-scales of weeks to months. There is a well-known bimodal
distribution of orbital periods of cataclysmic variables, with a
accretion disc itself. Integration of the density profile normal to
the plane of the disc r(z) yields a relationship between surface
density S and temperature T (or mass transfer rate) for an annulus
in the disc. This is the well-known S-curve and is shown
schematically in Fig. 1. It represents the locus of points for which
the annulus remains in thermal equilibrium. The physical basis
that underpins the S-curve is the onset of hydrogen ionization.
Hence the upper and lower branches of the curve correspond to the
high- and low-opacity states of ionized and neutral hydrogen. The
opacity function k(T) is very steep in the range 6000±7000 K;
such an abrupt change leads to the instability. We can define the
local viscous energy generation rate per unit area by
dearth of systems having orbital periods in the range 2:2 < Porb
<
2:8 h: This is known as the period gap. Dwarf novae that lie above
the period gap and only display normal outbursts are classified as
U Geminorum (U Gem) systems, after their template. There is a
roughly linear period±mass ratio relation for interacting binaries
(Frank, King & Raine 1995). The SU Ursae Majoris (SU UMa)
systems, which lie below the period gap and have more extreme
mass ratios q ˆ M2=M1 , 0:25; show longer, slightly brighter
superoutbursts lasting for 10 d or more. These occur in addition
to normal outbursts (typically one superoutburst occurs for every
5±15 normal outbursts) and show a superimposed periodic
variation in brightness at supermaximum (superhumps). In this
paper we introduce a numerical method for studying dwarf nova
outbursts and apply it to a system of the U Gem class. In a future
paper we will present the application to the SU UMa class (Truss,
Murray & Wynn, in preparation).
ꢀ
Q1
ˆ
q1ꢀz† dz;
ꢀ1†
where q1 is the local viscous heat generation rate per unit area,
assumed to be concentrated in the centre of the disc. Heat losses
can be assumed to be from blackbody dissipation from the disc
surface. If the surface temperature is Ts, then the local energy
dissipation rate is
Q2 ˆ 2sT4s ;
ꢀ2†
Historically, two theories have been put forward to explain
Â
the outbursts of dwarf novae. Paczynski, Ziolkowski & Zytkow
where each surface of the disc contributes sT4s : In thermal
equilibrium, Q1ꢀTc† ˆ Q2ꢀTs†; where Tc is the central disc
temperature.
(1969) suggested that a low-mass Roche lobe filling secondary is
potentially unstable, and mass transfer may occur quickly enough
that its convective envelope would become radiative. The
enhanced radiation subsequently stabilizes the mass transfer and
in this way an outburst cycle is initiated. However, observations of
the hotspot (the bright point at which material enters the disc) do
not show an increase in brightness during outburst, as would be
predicted by such a mass transfer instability model (see, for
An annulus residing on the lower (quiescent) or upper (outburst)
branch of the curve subject to perturbations in local surface
density can evolve on a viscous time-scale and remain in thermal
equilibrium. No stable solution is available to an annulus in the
middle branch, however. Here, if the mass transfer rate (and hence
Tc) is increased, Q1 increases faster than Q2 and Tc rises yet
further. Therefore, at the inflection points of the curve, the disc
will evolve on a thermal time-scale (that is, very rapidly) to the
w E-mail: mtr@star.le.ac.uk
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468
M. R. Truss et al.
where the sum is over all the particles. Here, mi and ri are the mass
and position of particle i respectively. W is the interpolation
kernel, which has a characteristic length-scale h, commonly called
the smoothing length. For a general introduction to SPH, the
reader is referred to Monaghan (1992).
In Murray (1996) an SPH code specifically modified for
accretion disc problems was described. The key feature of this
code was the use of an artificial viscosity term in the SPH
equations to represent the shear viscosity n known to be present in
observed discs. The artificial viscosity term introduces, in the
continuum limit, a fixed combination of shear and bulk viscosities
to the fluid. The viscous force per unit mass is
av ˆ kzcL‰72v 1 27ꢀ7 ´ v†Š:
ꢀ6†
Here k is an analytically determined constant that depends upon
the kernel, with k ˆ 1=8 and 1/10 for the cubic spline kernel in
two and three dimensions respectively; c is the sound speed; v is
the fluid velocity; and L is a viscous length-scale. In previous
work L was taken to be equal to the smoothing length h. Here,
however, we relax that constraint. Finally, z is a dimensionless
parameter. Note that in most SPH papers this parameter is denoted
a, but we have followed Murray (1996) and renamed it to avoid a
confusion of subscripts.
Figure 1. The limit-cycle behaviour of an accretion disc in a dwarf nova.
opposite branch. In this way a limit cycle of quiescence and
outburst is established. We interpret the upper and lower branches
of the curve as high- and low-viscosity states. The critical surface
densities at the points of inflection have been calculated by
Cannizzo, Shafter & Wheeler (1988):
In the interior of Keplerian discs, we can neglect the velocity
divergence and we see that the artificial viscosity term generates a
shear viscosity
20:35 20:86
cold
Smax ˆ 11:4R110:05M1
a
g cm22
;
ꢀ3†
ꢀ4†
n ˆ kzcL:
ꢀ7†
Smin ˆ 8:25R110:05M1
a
g cm22
;
hot
20:35 20:8
We can control the shear viscosity throughout the disc by
modifying z and L, and so obtain a functional form very similar to
the Shakura±Sunyaev form used in accretion disc theory.
where R10 is the radius in units of 1010 cm, M1 is the mass of the
primary in solar masses, and ahot and acold are the Shakura±
Sunyaev viscosity parameters in the high and low states (Shakura
& Sunyaev 1973). Note that these are very close to linear radial
dependences. An excellent review of the dwarf nova disc
instability is given by Cannizzo (1993a).
Several tests of this code were presented in Murray (1996). The
code has been used to look at tidally unstable discs (Murray 1998,
2000), tilted discs (Armitage & Murray 1998) and counter-rotating
discs (Murray, deKool & Li 1999) around accreting pulsars.
We begin with a full explanation of our model and the
numerical techniques involved in applying a smoothed particle
hydrodynamics (SPH) approach to the problem. In Section 3 we
demonstrate that our model is physically consistent with a real
dwarf nova, and present a wide range of simulated observables.
Â
Kornet & RozÇycka (2000), using an Eulerian code (quite distinct
algorithmically from the SPH code we use), have reproduced
several of the results of Murray (1998).
2.2 Outbursts
2
MO D E L L I N G DWA R F NOVA OU T B U R S T S
As mentioned in the previous section, quiescence is associated
with a very low value of the Shakura±Sunyaev shear viscosity
parameter (a) but also a low temperature. In outburst, angular
momentum transport is much more rapid because the temperature
is much higher and a is much larger. In previous papers (Murray
1998; Armitage & Murray 1998) preliminary attempts were made
to model outbursts by instantaneously increasing the value of the
shear viscosity throughout the entire disc. Such an approach only
enabled us to model the most basic features of an outburst, and we
could not of course follow the propagation of state changes
through the disc. The principal modification to the code made for
this work was to allow the viscosity to change locally in response
to disc conditions. This was easy to do, simply requiring that each
particle carry a variable z that determined its `viscosity'. To
determine the viscosity of the interaction of any particular pair of
particles we use the harmonic mean of the two z values. Cleary &
Monaghan (1999) found that such a form gave good results for
heat conduction between materials with vastly different properties.
All that remained was to determine how changes in viscosity
were to be triggered. The simplest approach is to let the shear
2.1 The smoothed particle hydrodynamics technique
In this section we discuss substantial developments made to an
existing smoothed particle hydrodynamics accretion disc code that
enable us to model the complete outburst cycle of a dwarf nova.
The principal change is the modification of the dissipation term to
make it a function of local disc conditions. Smoothed particle
hydrodynamics (SPH) is a Lagrangian method for modelling the
dynamics of fluids. A continuous medium is modelled by a
collection of particles that each move with the local fluid velocity.
Fluid properties at any given point are determined by interpolating
from the particle positions. The interpolation takes the form of a
simple summation over the particles with each term weighted
according to distance from the point in question. The weighting
function is known as the interpolation kernel. For example, the
interpolated value for the density at some point r in the fluid is
n
X
rꢀr† ˆ
miWꢀr 2 ri; h†;
ꢀ5†
i
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The outbursts of dwarf novae
469
Figure 2. The simplified surface density trigger conditions used in the
code. Sc and Sh are functions of radius in the disc.
Figure 3. Functional form of the outburst trigger. The viscosity is switched
on a total time-scale tth.
viscosity of each SPH particle be determined by the local surface
density S. If, for some region of the disc in the quiescent state, the
surface density is less than some critical value Sc, that region of
the disc is assumed to be stably quiescent, and the particle's shear
viscosity remains at a value appropriate for a cool disc. Should S
increase to be greater than Sc, then we consider that region of the
disc to have been `triggered' into the hot state. We then let the
particle's shear viscosity increase to a value appropriate for a hot
disc (the details of the transition are described below). The
viscosity continues adjusting to its new value even if S
subsequently drops below Sc. Conversely a hot region of the
disc is stable as long as its surface density remains above a second
critical value Sh , Sc: However, if a portion of the disc has
S , Sh, then that particle's shear viscosity parameter will reduce
to the quiescent value (see Fig. 2). Sh and Sc correspond to Smin
and Smax. We preserve their radial dependence but reduce the
magnitude of their gradients to reduce the run-time of the code. A
full analysis of the scaling is given in Section 3.2.
in which single SPH particles could change state, we implemented
a more coarse-grained approach. The disc was divided into a set of
concentric annuli (typically 100 were used). When the mean
surface density in a given annulus met the triggering conditions,
then all particles in that annulus changed state (again on the
thermal time-scale). This intermediate, azimuthally smoothed,
approach improved the speed of the code by ,50 per cent, and
provides a suitable basis for incorporation of a full thermodynamic
treatment of the thermal instability. The calculations described in
the next section demonstrate that these two approaches to the
viscosity switching are consistent with one another.
As it is written, the triggering routine can simply be replaced by
a more thermodynamically sophisticated routine. Preliminary
calculations have indeed been made. However, as these simula-
tions are proving more computationally demanding, we leave
them to a later paper.
3
R ESULT S
In previous work (Murray 1998; Armitage & Murray 1998) a
was set to change instantaneously between quiescent and outburst
values. In this work, we assume that, when the instability is
triggered, the viscosity changes on the thermal time-scale. As we
have limited knowledge of the mechanism responsible for the high
viscosity of accretion discs, we can only propose an appropriate
functional form for the transition. We use the hyperbolic tangent
function as it allows us to capture both the initial exponential
change in n and a smooth asymptotic approach to its final value.
In terms of the Shakura±Sunyaev viscosity parameter, the
equation for the transition from quiescence ꢀa ˆ acold† to outburst
ꢀa ˆ ahot† is
We show that our fast azimuthally smoothed approximation gives a
good representation of the physical behaviour of the system, and
move on to present a wide range of results and analysis for a
simulation of the dwarf nova SS Cygni. SS Cygni is a very well
observed dwarf nova of the U Gem type. It has an orbital period of
0.275 130d and a mass ratio (defined as the ratio of the mass of the
mass-losing secondary to that of the accreting white dwarf primary)
of 0:59 ^ 0:02; with M1 ˆ 1:19 ^ 0:02 M( (Friend et al. 1990).
All the results presented in this paper are for steady-state discs,
which we define to be those which return to the same quiescent level
between outbursts. We take a constant mass transfer rate of 1029 M(
yr21 throughout the simulations, following Cannizzo (1993b).
ꢁ
ꢂ
ꢀahot 1 acold
†
ꢀahot 2 acold
†
t
aꢀt† ˆ
1
tanh
2 p ;
ꢀ8†
2
2
tth
3.1 Comparison of triggering regimes
and the converse, for the transition from ahot to acold, is
ꢁ
The simulation was performed with both local triggering and
azimuthally smoothed triggering as detailed above. In the case of
local triggering, two regimes are contrasted: (i) ttrigger ˆ 250 s and
(ii) ttrigger ˆ 5000 s: For the azimuthally smoothed case, we
choose a long trigger time-scale of 7500 s. The thermal time-scale
tth and the dynamical (Kepler) time-scale tf are related by
ꢂ
ꢀahot 1 acold
†
ꢀahot 2 acold
†
t
aꢀt† ˆ
2
tanh
2 p :
ꢀ9†
2
2
tth
The functional form of the trigger is shown in Fig. 3. Of course,
tanh does not have compact support, so as soon as a transition has
been triggered we add a small artificial offset to a.
In the following section we describe simulations completed
with the new variable viscosity. As well as using a density trigger
tf , atth:
ꢀ10†
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470
M. R. Truss et al.
SS Cygni has an orbital period of almost 400 min and a Kepler
time-scale of around 250 s at the circularization radius. This is the
radius at which the gas has the same specific angular momentum
as it had on passing through the L1 point (i.e. as it is injected into
the model). It is the radius at which infalling gas would first orbit
the primary before formation of an accretion disc. Hence for
realistic values of a (!1.0), tth will be of the order of a few
thousand seconds, which we use in the azimuthally smoothed code
and case (ii).
that the azimuthally smoothed result is consistent with a local
trigger time-scale somewhere between these two cases. The peak-
to-peak time-scale between outbursts is around 30 d in the
smoothed code, 20 d in the shorter time-scale code, and 40 d in
the longer time-scale code. Our fast azimuthally smoothed code is
consistent with the physically realistic case of a thermal time-scale
greater than the Kepler time-scale.
3.2 Light curves
Fig. 4 contains plots of total viscous dissipation against time for
the different regimes. The outbursts in both cases are coherent, i.e.
a large region of the disc is transformed to the hot state. It is clear
Fig. 5 shows light curves calculated from the SS Cygni simulation
in the U, B, V, J, H and K bands. These are calculated by assuming
that the disc is optically thick, and each annulus of gas in the disc
radiates as a blackbody. We then simply integrate the Planck
function over different wavebands and sum over the annuli to
yield the light curves.
The light curves show regular normal outbursts with a
comparatively rapid rise to maximum and a slightly longer fall
to quiescence. The amplitude of the normal outbursts in the V
band varies around 1.5 mag on our diagram, with a rise time of
,2.5 d and a decay time of ,4.5 d. SS Cygni is observed to
brighten from 12th magnitude in quiescence to 8th magnitude in
outburst. Our result is suppressed by the artificially high value of
the Shakura±Sunyaev `alpha' parameter in the two states, which
we use to reduce the run-time of the code. We can make an
estimate of the compression factor of the amplitude as follows.
The dissipation rate of an accretion disc D(R) is directly
proportional to the torque exerted by one annulus on another G(R):
GꢀR† dV
DꢀR† ˆ
;
ꢀ11†
4pR dR
where V is the angular velocity of the gas in the annulus. The
viscous torque is
dV
GꢀR† ˆ 2pRnSR2
;
ꢀ12†
dR
where n is the viscosity, so
ꢁ
ꢂ
2
1
dV
dR
DꢀR† ˆ nS R
:
ꢀ13†
2
The dissipation rate is therefore proportional to nS. Equation (7)
gives the relationship between n and the `alpha' parameter (z in
equation 7). In an isothermal disc, the ratio of total dissipation in
outburst to that in quiescence is
Doutburst
ꢀaS†
outburst
ˆ
:
ꢀ14†
Dquiescence ꢀaS†
quiescence
We use a simple 1:10 ratio of a irrespective of surface density S.
The density triggers are set at
Smax ˆ 16:67ꢀR=a† g cm22
;
ꢀ15†
ꢀ16†
Smin ˆ 6:250ꢀR=a† g cm22
;
where a is the binary separation. The surface density at the peak of
outburst will be close to Smin, while the surface density in
quiescence just before an outburst will be close to Smax. This is
clearly shown later in Fig. 9. Therefore, in our isothermal disc
simulation,
Figure 4. Top: Light curve (total viscous dissipation) in SS Cygni
simulation with azimuthal smoothing and ttrigger ˆ 7500 s: Middle: Light
curve with local triggering and ttrigger ˆ 250 s: Bottom: Light curve with
local triggering and ttrigger ˆ 5000 s:
Doutburst
ahotSmin
ˆ
. 3:7:
ꢀ17†
Dquiescence acoldSmax
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The outbursts of dwarf novae
471
Figure 5. The U, B, V, J, H and K filter band light curves (azimuthally smoothed model). The light curves show three distinct states ± outburst, quiescence
and mini-outburst.
Real discs, however, are not isothermal. Rewriting equation (7) in
terms of more familiar quantities for a thin Shakura±Sunyaev
disc,
in clusters and are particularly blue, suggesting that only the hot
inner portion of the disc is involved in producing the additional
luminosity. These variations are indeed observed in the V-band
light curve of SS Cygni (see, for example, the AAVSO light
curves).
ꢁ
ꢂ
1=2
R3
n ˆ acsH , ac2s
;
ꢀ18†
GM
It is possible to correlate the number of particles being accreted
on to the white dwarf primary with an expected X-ray luminosity
for the system by simple consideration of gravitational energy
released. For simulation particles of mass Mp, the X-ray lumin-
osity produced is
where cs is the sound speed and H is the scaleheight. In the limit
where we can neglect radiation pressure,
c2s ˆ P=r / Tc;
ꢀ19†
GMp dM
where Tc is the midplane temperature. The critical surface
densities will be given by equations (3) and (4), and in a real
disc at fixed radius,
LX ˆ e
;
ꢀ21†
R
dt
where e is an unknown efficiency factor expected to be !1. An
X-ray light curve can be built up in this way and is presented in
Fig. 6 for our simulation. In quiescence and on the decline from
outburst, the X-ray emission follows the optical emission very
closely; indeed, our X-rays appear to be very sensitive to
fluctuations in the V band in quiescence. However, there is a
noticeable and significant delay in the onset of the outbursts. This
is shown in Fig. 7 for each of the four outbursts. Each plot shows
20 d; referring back to Fig. 5 we have (A) 275±295 d, (B) 305±
325 d, (C) 335±355 d and (D) 360±380 d. The optical rise leads
the X-ray rise by up to 1 d in all our outbursts. This can be
Doutburst
8:25ahotTc;hotah2o0t:8
11:4acoldTc;coldac2o0ld:86
ˆ
:
ꢀ20†
Dquiescence
Using Cannizzo's (1993b) values of Tc;hot . 60 000 K and
Tc;cold . 4000 K; and setting acold ˆ 0:01 and ahot ˆ 0:1; this
ratio is about 13.1. The compression factor introduced in
dissipation amplitude by the code is about 3.5, bringing our
result closer to the equivalent observed amplitude in SS Cygni.
In the quiescent phase, there appears to be an additional
modulation in luminosity. These aperiodic mini-outbursts appear
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472
M. R. Truss et al.
interpreted as the time required for the gas in the part of the disc
that initially goes into outburst to migrate through the inner disc
before it is accreted on to the primary. The measurement of the
delay is in good agreement with observation; Jones & Watson
(1992) found a typical delay of 0.5±1.1 d from EXOSAT
observations. The rise time is also seen to be much faster in
X-rays; we find a rise time of ,12 h in X-ray and ,2.5 d in the
optical. The climb in V-band emission is a gradual process as more
and more gas in the disc is transformed to the high state, whereas
the onset of enhanced accretion on to the white dwarf itself is
instantaneous, triggered when the additional material from the
outbursting disc arrives at its surface.
3.3 Disc analysis
In this section we follow the evolution of an accretion disc through
an outburst cycle. The light curves give much information about
the physical processes taking place in the disc during quiescence
and outburst, but a much clearer representation is obtained by
following the dynamical behaviour of the accretion disc itself.
Fig. 8 is a montage of images of the accretion disc through a
normal outburst. We find that even in quiescence ꢀt ˆ 334:2†;
a small fraction of the inner disc remains permanently locked in
the hot state. The linear relation between the critical surface
density triggers and the disc radius ensures that particles in the
inner disc are continually cycling between the low and high states
(the triggers are very close together here and also very low ± these
conditions are easily satisfied by particles close to the primary).
The rise to outburst is rapid and is initiated in the inner part of the
disc. A density wave (directly analogous to a heating wave in a
Figure 6. X-ray light curve generated from number of simulation particles
accreted on to the primary.
Figure 7. Overlay of V-band (dark line) and X-ray emission for the four outbursts in Fig. 5. Each plot has been normalized to the peak of curve A and shows a
20-d time series.
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The outbursts of dwarf novae
473
Figure 8. Evolution of total viscous dissipation in the disc through an outburst; t is the time in days. The grey-scale is logarithmic, with white at
107 erg21 cm22 through to black at 109 erg21 cm22. The discs are aligned such that the primary±secondary axis is vertical and the hotspot appears at the
bottom of each frame. Spiral shock arms are also clearly evident. Note that the central part of the disc (white) is not modelled in the simulation.
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M. R. Truss et al.
full thermodynamic treatment) moves both outwards and inwards,
transforming a large fraction of the disc to the hot, high-viscosity
state. Subsequently the disc falls (less rapidly) back into
quiescence by another density wave (analogous to a cooling
wave). This pattern is repeated for all the normal outbursts.
Approximately the same fraction of the disc is seen to reach the
high state in each outburst. The maximum extent of the hot portion
is seen to vary to a small degree, but appears ultimately to be
limited to a radius near the position of the spiral shock arms. This
can be readily observed in the light curve ± some outbursts reach a
slightly higher peak luminosity than others, although the
maximum variation is no more than a few per cent.
The outburst is, of course, triggered by the local surface density
in the disc, and as the gas in the disc is accreted inwards on a
viscous time-scale there is a corresponding change in that local
surface density. This response is shown in Fig. 9. The disc clearly
drains during the outburst phase until the local surface density
reaches the lower trigger level, after which the outburst dies away
and the disc is replenished. These density profiles are extremely
similar to those found by Stehle (1999) in a full thermodynamic
one-dimensional treatment (not SPH).
Fig. 10 shows a disc in mini-outburst, and confirms our
observation from the light curve in Section 3.2 that it is the inner
disc that is hot in a mini-outburst. The outburst is initiated very
near the inner edge of the disc and never propagates very far. The
reason for this can be seen in Fig. 9. Just before a normal outburst
(top panel), the surface density profile follows the line of the
upper trigger very closely. Any part of the disc that goes into
outburst will lead to a coherent normal outburst throughout most
of the disc. After a normal outburst (bottom panel), the only part
of the disc near the upper trigger is near the disc centre. All the
remainder of the disc has been drained. Hence, while the centre of
the disc can go into outburst, the majority of the disc cannot. This
is what happens in a mini-outburst.
Disc spectra, temperature profiles and luminosity profiles
clearly show the contrast in the physical properties of the disc
in the different states. The spectra, in Fig. 11, have been computed
for the disc in outburst, quiescence and mini-outburst. The blue,
hot inner disc behaviour in mini-outburst becomes clear in
comparison with the quiescent spectrum.
The luminosity and temperature profiles in Fig. 12 have been
calculated on the rise to a normal outburst, and correspond to the
discs in the first three frames of Fig. 8. All the profiles show an
increased region of viscous dissipation (and temperature) near the
outer edge of the disc, where the surface density peaks as matter
enters the disc from the accretion stream; this is the hotspot. This
normal outburst is apparently triggered at a radius of approxi-
mately 0.15a (near the circularization radius ± 0.11a in SS Cygni),
and the heating wave is seen to propagate in both directions. We
find that in quiescence the disc is typically approximately
isothermal at radii greater than 0.15a at temperatures around
4000 K. In outburst we find temperatures ranging from 6000 to
12 000 K in the inner half of the disc and from 4000 to 6000 K in
the outer half. Bobinger et al. (1997) have performed eclipse
mapping of the dwarf nova IP Peg, which has strikingly similar
parameters to SS Cygni ꢀq ˆ 0:58; P ˆ 0:158 d†; on the decline
from outburst. They found an inner disc at 7000 to 9000 K and an
outer disc with temperatures declining to a quiescent level of 3000
to 4000 K. Our simulated results are in good agreement with these
observations. Fig. 12 also shows that the hotspot is consistently
around 1500 K hotter than the surrounding regions, but the
increase in luminosity from these regions is much more marked ±
the viscous dissipation is 30 times higher here than in the part of
the disc just inside them. The accretion rate from the secondary is
constant in this simulation, so we find a constant hotspot
temperature. It is also interesting to note, however, that the
luminosity generated in the hotspot of the quiescent disc is
comparable to that generated by the very hot inner part of the disc.
Figure 9. Evolution of azimuthally averaged surface density of the disc
through an outburst. The straight lines show the radial dependence of the
upper and lower triggers. The binary separation a ˆ 1:8  1011 cm in SS
Cygni.
4
DI S C U S S I O N
We have performed the first two-dimensional treatment of dwarf
nova outbursts. The method gives light curves in different
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The outbursts of dwarf novae
475
Figure 10. The disc in mini-outburst. The outburst has not propagated far enough to become a normal outburst. The grey-scale is the same as in Fig. 8.
characteristics presented here are just a selection of the possible
areas that can be explored in future work.
We have applied these methods successfully to a study of the
SU Ursae Majoris systems (Truss et al., in preparation) in addition
to the dwarf novae with less extreme mass ratios considered here.
A two-dimensional code is ideal for exploring tidal effects and
instabilities in such systems.
Observations of dwarf novae are by no means comprehensive or
continuous, but this situation should be remedied in the near future
with more telescope time dedicated to them. In consequence, it
will be possible to test several of the results in this paper.
Although our knowledge of the behaviour of dwarf novae in the V
band is good as a result of the efforts of organizations such as the
AAVSO, there are few data available in other bands, so new
observations will be able to probe the inner regions of the disc and
(via X-ray observations) boundary layer. Our findings relating to a
hot inner disc and the occurrence of mini-outbursts should have
Figure 11. Disc spectra in quiescence (bottom curve), mini-outburst
(middle curve) and outburst (top curve). The disc in mini-outburst is bluer
than the quiescent disc.
some testable observational consequences.
The next stage of development of any numerical model such as
this should be two-fold: the incorporation of full thermodynamics
into the simulation, and the extension to three dimensions. This is
not a difficult task with the present scheme, just a lengthy one in
computational terms. The increasing availability of parallel
machines and improvements in CPU performance should make
these refinements a workable proposition.
wavelength bands, disc spectra, density and temperature profiles
that all agree well with observations and full thermodynamic
analyses.
The advantages of using a two-dimensional code to model
dwarf nova outbursts are immediately obvious. Tidal forces are
not approximated as in one-dimensional codes and therefore much
more detail is revealed in the results of these current simulations.
One-dimensional simulations have always predicted extremely
regular outburst behaviour, both in recurrence pattern and in
outburst profile. There is now much more scope for exploring
the behaviour of these systems; the simulated observational
ACKNOW LEDGMENTS
Research in theoretical astrophysics at the University of
Leicester is supported by a PPARC rolling grant. Many of the
calculations were performed using GRAND, a high-performance
computing facility funded by PPARC and based at the University
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M. R. Truss et al.
of Leicester. The authors are grateful to Dr R. Stehle, Professor
J. Pringle, Professor A. King and an anonymous referee for
helpful discussions or comments. MRT acknowledges a PPARC
research studentship and a William Edwards Charitable Trust
bursary for postgraduate study.
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Figure 12. The V-band luminosity and temperature of the disc evolving
through the rise to normal outburst. The inner region of the disc remains in
outburst even during quiescence (solid line) and the normal outburst is
initiated at a radius around 0.15a (points). The dashed line represents the
profile at peak outburst.
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This paper has been typeset from a T X/LT X file prepared by the author.
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