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Astron. Nachr. (2000) 3, 181–192
Nonlocal density wave theory for gravitational instability of protoplanetary diskswithout sharp boundaries
G. Rudiger, Potsdam, Germany
Astrophysikalisches Institut Potsdam
L.L. Kitchatinov, Irkutsk, Russia
Institute for Solar-Terrestrial Physics
Received 2000 June 30; accepted 2000 July 17
The stability of a self-gravitating infinitesimally thin gaseous disk rotating around a central mass is studied. Our globallinear analysis concerns marginal stability, i.e. it yields the critical temperature for the onset of instability for any given ratioof the disk mass to the central mass. Both axisymmetric and low-m nonaxisymmetric excitations are analysed. When thefractional disk mass increases, the symmetry character of the instability changes from rings (m = 0) to one-armed trailingspirals (m = 1). The distribution of the surface density along the spiral arms is not uniform, but describes a sequence ofmaxima that might be identified with forming planets. The number of the mass concentrations decreases with increasingfractional disk mass. We also obtain solutions in the form of global nonaxisymmetric vortices, which are, however, neverexcited.
Key words: accretion disks – instabilities – planetary systems
1. Introduction
Studies of the general structure of planetary systems are strongly stimulated by detections of exosolar planets likethat around 51 Peg. The number, size and orbital geometry of the planets in various types of planetary systemsmay now be expected to form a database that can be used to test theoretical concepts. The first question is, ofcourse, whether the properties of our solar system are the rule or rather an exception.
The idea to consider the solar system as the relic of a global instability is not new in the context of the densitywave theory. Such a concept has its roots in the fascinating order of the Titius-Bode law, suggesting a globalformation process rather than a local one. The density wave theory in its short-wave approximation, however, ishere faced with difficulties. Conditions on both the mass and temperature of the protoplanetary disk prove to berather restrictive if the theory is applied to a thin Keplerian disk. Modes with wavenumber
kcrit =1π
M∗Σ0r3
(1)
become unstable if the Toomre parameter,
Q =Ω cac
π G Σ0, (2)
falls below unity (Toomre 1964, Safronov 1969, Goldreich & Ward 1973). With the standard notation, G is thegravity constant, M∗ is the central mass, cac is the speed of sound, Ω is the angular velocity and Σ0 is the surfacedensity of the disk. Eq. (1) can easily be rewritten as
Mdisk M∗/kr, (3)
so that the ‘observed’ pattern structure with
∆r/r 0.5 (4)
(Nieto 1972, Lissauer & Cuzzi 1985), i.e. kr ≈ 10, forces the disk to be rather massive in contrast to the normallyassumed low mass disk (Mdisk M). Moreover, too massive disks do not exhibit a Keplerian rotation law. Thereare thus difficulties applying eq. (1) to the observed scales (Polyachenko & Fridman 1972, Cameron 1978).
182 Astron. Nachr. 321 (2000) 3
On the other hand, writing the sound speed as cac =√RT , the instability condition reads
T <∼π2Σ2
0Gr3
RM≈ 10−2Σ2
3r3AU (5)
with Σ3 = Σ0/(103g cm−2) and rAU = r/1AU. In the Earths neighbourhood (Σ3 ≈ 1) the disk must thus be muchcooler than it is reasonable to assume (Goldreich & Ward 1973). As numerical simulations with a large number ofmass points (Cassen et al. 1981) generally confirm the stability criterion, Q > 1, there is little hope for a gaseousdisk to be sufficiently cold except its outer regions (Bell et al. 1997).
With the most optimistic estimates, however, a very thin dust layer, which forms slowly in the disk’s midplane,could be cold enough. It may have the wanted low temperature according to cac HΩ with H as the disk’svertical half-thickness of the order of a few meters (Nakagawa et al. 1981). Once the dust disk is sufficiently thinand the gas is laminar, the parameter Q becomes subcritical and the disk begins to fragment into a number ofrings (Goldreich & Ward 1973, Lissauer 1993). The ring distances, however, are too small as the surface densityof the dust disk is low. With the often quoted value of 100 g cm−2 for the surface mass density, Weidenschilling(1977) obtained kr ∼ 104. The resulting mass concentrations are maximally 1018 g in the Earth neighbourhoodand must be defined as ‘planetesimals’ (Safronov 1969, Safronov & Ruzmaikina 1985, Hayashi et al. 1985). Thelarge value of kr also ensures here the applicability of the short wavelength approximation on which all the aboveconsiderations are based.
In the context of the density wave theory for planetary systems the question remains open as to whether or notthe above mentioned problems also apply to a nonlocal formalism. A single wavenumber has a restricted meaningin such a formulation. Only a spectrum characterises the resulting global pattern. The critical temperature belowwhich a global structure is excited is the main output of such a formulation.
Using a global approach a significant progress has been made in explaining the spiral structure of galaxies inthe context of the density wave theory. In addition to giving a detailed discussion of the dispersion relation fortightly wound spirals, the instability of global modes has been analysed with considerable success (Lin & Shu 1964,Aoki & Iye 1978, Aoki et al. 1979, Haass 1983, Meinel 1983, see also Binney & Tremaine 1987).
For protoplanetary disks Rudiger & Tschape (1987) tried to overcome the shortcomings of the short-waveapproximation by employing a relation between potential disturbances and density disturbances given by Bertinand Mark (1979). In contrast to the earlier treatments (cf. Shu 1985), it incorporates terms up to the secondorder in kr. The resulting dispersion relation has been solved yielding indications for an instability with non-shortring-like structures. One of the two basic difficulties of the density wave theory for protoplanetary disks seems tobe reduced in this way. It might thus to be worthwhile to develop a truly nonlocal formulation of the theory ofgravitational instabilities.
The main idea of the present work is to find out whether axisymmetric or nonaxisymmetric modes are moreunstable. Are rings or spirals excited once the cooling of a protoplanetary disk brings it to an unstable state?From this point of view there are close relations to the phenomena of symmetry breaking by hydrodynamic andmagnetohydrodynamic instabilities, to the planetary and stellar mean-field dynamo theory, to the rotating proto-stars (Yang et al. 1991) and to spiral wave excitation in galaxies (Hohl 1971, Bertin 1983, Monaghan & Lattanzio1991). Also the stability or fragmentation problem for uniformly rotating self-gravitating stellar or gaseous disks(Vauterin & Dejonghe 1995, Kohler 1995, Fuchs 1996) generally belongs to the theory developed in the presentpaper. For simplicity we are neglecting here the thickness of the disk so that the influence of a second constituentsuch as a gas must be excluded.
We also mention the suggestion by Morfill et al. (1993) that the existence of ring-like structures may help toovercome the main difficulty of the standard-accretion disk theory, i.e. the disagreement between the theoreticallypredicted and observed radial temperature profiles (Horne 1993, Beckwith 1994).
2. The basic equations
Small disturbances in an infinitesimally thin inviscid disk in nonuniform rotation are considered. We start withthe linear relation between the pressure and density perturbations,
p′ = c2acρ′, (6)
valid for isothermal as well as for isentropic disks. Then the linearised equations for the radial (u′) and azimuthal(v′) velocity disturbances and the mass conservation equation read
∂u′
∂t+ Ω
∂u′
∂φ− 2 Ω v′ +
∂ψ′
∂r+
∂
∂r
c2acΣ′
Σ0= 0,
G. Rudiger and L.L. Kitchatinov: Nonlocal density wave theory for protoplanetary disks 183
∂v′
∂t+ Ω
∂v′
∂φ+
κ2
2Ωu′ +
1r
∂ψ′
∂φ+
c2ac
Σ0r
∂Σ′
∂φ= 0,
∂Σ′
∂t+ Ω
∂Σ′
∂φ+
1r
∂(u′rΣ0)∂r
+Σ0
r
∂v′
∂φ= 0, (7)
with κ being the epicyclic frequency,
κ2 =2Ωr
d(r2Ω
)dr
. (8)
Σ denotes the surface density with Σ0 and Σ′ being its equilibrium value and disturbance, respectively, and ψ isthe gravity potential. The potential and the density are related by the Poisson equation,
∆ψ = 4πGΣδ(z) (9)
(cf. Mohlmann 1985, Schmit & Tscharnuter 1995). The modes with different azimuthal wave numbers, m, can beconsidered independently. Then the Toomre (1964) solution for the Poisson equation applies
ψ′ = −2πG
∞∫0
S(k)Jm(kr)eimφ−|kz|dk, (10)
where S(k) is the density spectrum in the Fourier-Bessel transform, and
Σ′ =
∞∫0
Jm(kr)kS(k) dk eimφ . (11)
The existence of the general solution (10) of the Poisson equation (9) demonstrates the convenience of the Fourier-Bessel transform. However, the velocity components, u′ and v′, do not match transformations like (11). It isappropriate to describe the flow in terms of scalar potentials for the momentum density disturbances, i.e.
Σ0u′ =
∂Φ∂r
+1r
∂V
∂φ, Σ0v
′ =1r
∂Φ∂φ
− ∂V
∂r. (12)
The potentials V and Φ define the vorticity and divergence of the flow, respectively,
div (Σ0v′) = ∆2Φ, rotz (Σ0v
′) = −∆2V, (13)
where v′ is the 2D velocity vector and
∆2 =1r
∂
∂rr∂
∂r+
1r2
∂2
∂φ2(14)
is the 2D Laplacian operator.With eqs. (12) to (14) the system (7) can be reformulated in terms of the flow potentials. Then it reads
∂ (∆2Φ)∂t
+ Ω∂ (∆2Φ)
∂φ+(κ2
Ω− 2Ω
)(∆2V ) −
(κ2
Ω− 4Ω
)∂
∂r
(∂V
∂r− 1
r
∂Φ∂φ
)+
+dΣ0
dr∂ψ′
∂r+ Σ0 (∆2ψ
′) + ∆2
(c2acΣ
′) +1r
∂
∂r
(nc2acΣ
′) = 0,
∂ (∆2V )∂t
+ Ω∂ (∆2V )
∂φ− κ2
2Ω(∆2Φ)−
(ddr
κ2
2Ω
)(∂Φ∂r
+1r
∂V
∂φ
)− 1
r
dΣ0
dr∂ψ′
∂φ+
nc2ac
r2
∂Σ′
∂φ= 0,
∂Σ′
∂t+ Ω
∂Σ′
∂φ+ ∆2Φ = 0, (15)
where
n(x) = − r
Σ0
dΣ0
dr(16)
is the radial slope of the surface density profile.
184 Astron. Nachr. 321 (2000) 3
The linear stability problem reduces to a mathematical eigenvalue problem by adopting an exponential time-dependence. With the Fourier mode ansatz in the azimuthal coordinate φ all the perturbations become proportionalto exp(ωt + imφ). In dimensionless units our equation system turns to
(ω + imΩ(x)
)(∆2Φ) +
(κ2
Ω− 2Ω
)(∆2V )−
(κ2
Ω− 4Ω
)∂
∂x
(∂V
∂x− im
xΦ)
+
+ 2f
(dΣ0(x)
dx∂ψ′
∂x+ Σ0(x) (∆2ψ
′)
)+
1M2
(∆2
(TΣ′
)+
1x
∂
∂x
(nTΣ′
))= 0,
(ω + imΩ(x)
)(∆2V ) − κ2
2Ω(∆2Φ)−
(ddx
κ2
2Ω
)(∂Φ∂x
+imx
V
)− 2f
imx
dΣ0
dxψ′ +
imnT
x2M2Σ′ = 0,
(ω + imΩ(x)
)Σ′ + ∆2Φ = 0. (17)
Here the characteristic scale, R, of the background density profile is used to measure distances, r = Rx, and Σ0(R)is applied to normalise densities, Σ0(x) = Σ0(R)Σ0(x). Ω0 = Ω(R) scales the angular velocity, Ω = Ω0Ω(x), Ω−1
0
scales the time, Ω0R the velocity, and 2πGΣ0(R)R the gravity potential. T (x) is the normalized temperatureprofile defined by the equation, c2ac = C2
acT , where C2ac is the (constant) sound velocity at r = R, so that T (1) = 1.
We also keep in eqs. (17) the same notations for the normalized flow potentials and normalized density and gravitydisturbances as used before for the physical variables. The basic dimensionless parameters of eqs. (17),
M =RΩ0
Cac, f =
πΣ0G
RΩ20
, (18)
are the Mach number and the f-parameter, which for Kepler disks with Ω20 = GM∗/R3 becomes the fractional disk
mass, f = πR2Σ0/M∗. For the early solar nebula f in any case exceeds the value 0.03 (Dubrulle 1993). M and fcan be combined into the global Toomre parameter (2),
Q = 1/(fM). (19)
3. Spectral formulation
There are two clear advantages with the spectral formulation by applying the Fourier-Bessel transforms. First, ityields the simple solution (10) of the Poisson equation (9) for the gravity potential. Second, typical singularitiesof the problem like that of the Keplerian angular velocity, Ω ∼ x−1.5, at x = 0 are integrable with the Besselfunctions. Hence, we do not need any internal boundary condition. Moreover, the internal boundary representedby a central star is also not very important because its radius is very small compared with the disk radius.
Both the temporal and azimuth dependencies in the Fourier-Bessel expansions are dropped, so that
∆2Φ(x) =
∞∫0
Jm(kx)k Φ(k) dk,
∆2V (x) =
∞∫0
Jm(kx)k V (k) dk,
Σ′(x) =
∞∫0
Jm(kx)k S(k) dk,
ψ′(x) = −∞∫0
Jm(kx) S(k) dk (20)
results, where the last equation solves the Poisson equation (9). Note that Φ is the spectrum for ∆2Φ, while thespectrum for Φ is −k−2Φ, and the same about V .
The expansions (20) lead to the final spectral formulation of our eigenvalue problem:
ωΦ(k) = −im
∞∫0
R(k, k′) Φ(k′) dk′ − 2
∞∫0
R1(k, k′) V (k′) dk′ +
G. Rudiger and L.L. Kitchatinov: Nonlocal density wave theory for protoplanetary disks 185
+ 2f
∞∫0
D(k, k′) S(k′) dk′ +1
M2
∞∫0
T (k, k′) S(k′) dk′,
ωV (k) = −im
∞∫0
R2(k, k′) V (k′) dk′ +
∞∫0
R3(k, k′) Φ(k′) dk′ +
+ i2mf
∞∫0
D1(k, k′) S(k′) dk′ +imM2
∞∫0
T1(k, k′) S(k′) dk′,
ωS(k) = −im
∞∫0
R4(k, k′) S(k′) dk′ − Φ(k), (21)
where the rotational kernels (R, R1, ..., R4), the density kernels (D, D1), and the temperature kernels (T, T1) aregiven in the Appendix.
All the kernels in eqs. (21) are real functions. Therefore, if S(k), Φ, V is a solution with the eigenvalue ω =γ + imΩp, then S∗,−Φ∗, V ∗ is also a solution with the eigenvalue −ω∗ = −γ + imΩp.1 In other words, for eachtrailing global spiral pattern with growth (decay) rate, γ, and the pattern speed, Ωp, there exists a correspondentleading pattern with exactly the same pattern speed and decay (growth) rate −γ. This statement known asthe anti-spiral theorem for tightly-wound spiral waves (Lynden-Bell & Ostriker 1967) applies also to the globaldisturbances. The theorem is fully symmetric about leading and trailing patterns. The computations in Section5.2., however, provide a remarkable difference between the two types of excitations. Only trailing structures proveas unstable. The leading spirals either decay or have zero growth rates. Also our global simulations had to theknown observational fact that in general spiral arms are trailing.
For uniform mean quantities such as angular velocity, surface density and temperature (i.e. when the typical% ∼ k−1 is small compared to the scales of the background parameters) the system (21) reduces to an algebraicequation and reproduces the well-known dispersion relation of the Lin theory for tightly-wound spiral density waves.
3.1. Axisymmetry
For known profiles of the angular velocity the background density and temperature the kernels of the integral eqs.(21) can be defined and the solution of the eigenvalue problem must be found numerically.
Drastic simplifications appear for axisymmetric ring-like disturbances. For m = 0 the first two equations of thesystem (21) become
ωΦ(k) = −2
∞∫0
R1(k, k′) V (k′) dk′ + 2f
∞∫0
D(k, k′) S(k′) dk′ +1
M2
∞∫0
T (k, k′) S(k′) dk′,
ωV (k) =
∞∫0
R3(k, k′) Φ(k′) dk′. (22)
The remaining equation relates the divergence and density spectra, i.e.
ωS(k) = −Φ(k). (23)
Substitution into (22)2 gives the integral relation between vorticity and density spectrum,
V (k) = −∞∫0
R3(k, k′) S(k′) dk′. (24)
With (22)1 we find the equation for the density spectrum, i.e.
ω2S(k) = −k
∞∫0
R(k, k′) S(k) dk′ + 2f
∞∫0
D(k, k′) S(k′) dk′ − 1M2
∞∫0
T (k, k′) S(k′) dk′. (25)
1∗ means complex conjugate
186 Astron. Nachr. 321 (2000) 3
In the axisymmetric case the kernels read
R(k, k′) =
∞∫0
κ2(x)xJ1(kx)J1(k′x) dx,
D(k, k′) = kk′∞∫0
Σ0(x)xJ1(kx)J1(k′x) dx,
T (k, k′) = kk′∞∫0
T (x)(kxJ0(kx)− n(x)J1(kx)
)J0(k′x) dx. (26)
For Kepler rotation, κ = x−1.5, we find
R(k, k′) =kk′
2 (k + k′)F
(12,32; 3;
4kk′
(k + k′)2
), (27)
with the hypergeometric function F . The same approximation applied to an isothermal disk provides
T (k, k′) = k2δ(k − k′). (28)
All the kernels in (25) are here symmetric. The symmetry means that ω2 is real so that overstability cannot appear.Only stable oscillations (negative ω2) or continuously growing or decaying eigenmodes (positive ω2) are possible.
4. The model
The rotation law of the disk is defined by the radial equilibrium condition
rΩ2(r) =GM∗r2
+dψdr
+c2ac
Σ0
dΣ0
dr. (29)
It depends on the basic parameters (18) of the problem. This involves similar dependence in rotational kernels ofthe governing integral eqs. (21). Self-gravity and pressure contributions are providing deviations from the Keplerrotation. Deviations from the Kepler law are computed and discussed by Heemskerk et al. (1992) – they are sosmall that their influence to the desired eigenvalues can be expected as unimportant. For our density model, (30),the rotation profiles as a solution of (29) are given in Fig. 1 for various disk masses. As already Adams et al.(1989) stressed the rotation law nearly remains a power law, it nearly remains Keplerian. Though the presentedglobal formulation also works with non-Keplerian rotation laws, we have always applied the Kepler law in orderto simplify the numerics considerably. At least the rotation of disks with up to 50% of the central mass can beconsidered as Keplerian in a high degree of accuracy (cf. Fig. 1). In particular, the Kepler rotation forms a verygood approximation for the disk mass f 0.16 for which we shall show as our main result that the ring instabilityswitches to a spiral instability.
The surface density distribution is parameterised as
Σ0 =x−p
(1 + xlq)1/l. (30)
It describes the power law x−p for the inner region x < 1, changing to x−(p+q) for x > 1. The parameter l definesthe transition between the slopes. We adopt l = q = 10 to model rather sharp transitions. Hence, x = 1 definesthe outer disk edge in our model. So the formulation of outer boundary conditions can be avoided by formallyconsidering an infinite disk.
For the power index of the density profile inside the diskWeidenschilling & Cuzzi (1993) favor p 1.5 consideringthe planet distribution in the solar system. Cassen & Moosman (1981) suggest values of 0.5 . . .2 as possible forprotostellar disks (see Boss, 1998, for a detailed discussion). We assume p = 1 in the present model. Thetemperature is assumed as uniform through the entire disk. At least in the axisymmetric case the local diskstability is controlled by the Toomre parameter. The global parameter is defined by eq. (19). The local values canbe found by multiplying the global Q by the profile of Fig. 2. The region around the outer edge of the disk shouldobviously be ‘most unstable’. The location of the minimal Q at the outer edge x = 1 seems to be natural, butmore calculations should reveal the role of this model attribute. On the other hand, for p > 1.5 our model cannotwork as then the Toomre parameter would vanish for x = 0.
G. Rudiger and L.L. Kitchatinov: Nonlocal density wave theory for protoplanetary disks 187
Fig. 1: Rotation profiles for disks withthe density law (30) for ratio of diskmass to central mass of f = 0 (solid,Kepler law), f = 0.5 (dashed) and f =1 (dotted). Mach number M is 10.
Fig. 2: Radial profile of the ‘local’Toomre parameter. There is alwaysa minimum as long as the density fi-nally sinks faster than the Kepler law.For our model this minimum appearsat x = 1
188 Astron. Nachr. 321 (2000) 3
Fig. 3: The marginal stability lines forrings (m = 0) and low-m spirals forvarious disk masses. The axisymmetryswitches at f 0.16. For the massivedisks on the right, however, one has toinclude non-Kepler rotation and finitedisk thickness.
Kepler rotation and thin-disk approximation imply that the fractional mass (18) is small and Mach number islarge. We shall, however, consider fractional masses up to f = 1 in the hope that the model remains qualitativelyvalid. Any short wave contribution in the integral eqs. (21) and (25) is neglected by replacing infinity in the upperlimits of the integrals by some large but finite value k0. Typically, k0 ∼ 50 . . .100 for the axisymmetric modes andk0 ∼ 100 . . .200 for nonaxisymmetric excitations provide satisfactory accuracy.2
5. Results and discussion
Fig. 3 gives the stability diagram of the Kepler model. The disk is stable for low Mach numbers, i.e. if the tem-perature is high enough. During the cooling process the location of the disk moves upwards with time. Eventuallythe instability region is reached. We expect that the type of the pattern excited by the instability is defined bythe first marginal stability line. In Brandenburg et al. (1989) is shown with global nonlinear dynamo models thatmain features of the solution are fixed by the bifurcation diagram rather than the kinematic growth rates. Thereal mixture of the modes can only be found with nonlinear calculations.
We expect, therefore, that the axial symmetry of the instability depends on the fractional mass. Massive disksprefer one-armed spirals (m = 1) while the rings (m = 0) are excited in preference by the non-massive disks.According to our computations the change of the axisymmetry-character occurs at f 0.16.
5.1. Rings
The local stability criterion, Q > 1, is reproduced only for very small fractional masses. The critical value of theglobal Toomre parameter (19) decreases slowly with f to become Q = 0.60 for f = 1. Probably the decrease isrelated to the increase of the scales of the linear excitation pattern as can be seen in Fig. 4. After Fig. 2 the localToomre parameter attains its minimum close to the edge of the disk. Only the excitations strongly concentratedto this point can be unstable with Q below unity. When the fractional mass is not too small, the scales of thering pattern become comparable to the disk size. Hence, the excitation of the global pattern requires the globalQ-parameter to sink considerably below unity (cf. Kikuchi et al. 1997).
In Fig. 4 one finds one or two rings of maximal density.3 When a disk becomes cool enough for instability, oneor two rings are produced at its outer edge. The rest of the disk has a reduced mass. Hence, it requires furthercooling for the gravitational instability to produce more and more rings.
2The integral equations are solved with the Nystrom method using Gauss-Legendre quadrature rules.3Our linear analysis can not define the sign of the eigenmodes so that profiles with inverse sign also represent eigensolutions.
G. Rudiger and L.L. Kitchatinov: Nonlocal density wave theory for protoplanetary disks 189
Fig. 4: Marginally stable density ring-like patterns for f = 0.5 (left) and f = 0.1 (right). The radial scales of the eigenmodesgrow with the disk masses.
5.2. Spirals
The spectrum of spiral-type excitations proved to be discrete. The matrix equation in the numerical code has, ofcourse. always a discrete spectrum. However, if the resolution is increased for a given range of wavenumbers thenumber of spirals remains roughly constant indicating the existence of a discrete spectrum. In accordance withthe anti-spiral theorem mentioned above the spirals are always found in pairs with opposite sense of spiralling.The problem is, however, that the simulation with finite numerical resolution produces small but finite false realparts of the eigenvalues, γ ∼ 0.01 . . .0.1. By this reason, even for low Mach numbers with definite stability, thenumerically-defined eigenvalues of the spiral and related anti-spiral are not identical but possess small real partswith opposite signs. As the Mach number is increased, the pairs of eigenvalues are proceeding along the imaginaryaxis, (ω) = 0, until two pairs meet each other. After that, one pair remains on the imaginary axis while two othereigenvalues start moving perpendicular to that axis in opposite directions. Their real parts increase rapidly. Weidentify the bifurcation with the onset of instability.
We never found growth of a leading spiral. They have zero growth rate or decay when the corresponding trailingspiral becomes growing. The examples of growing modes for the zonal wavenumbers m = 1, 2 are shown in theright panels of Fig. 4. This result is an interesting confirmation of similar former results (Lau & Bertin 1978, Aokiet al.1979, Binney & Tremaine 1987), and may serve as a positive test of our mathematical concept.
The gravitational instability represents a possible mechanism for planet formation. Though the alternativeconcept of planetesimals coagulation seems to be now in favor, the instability can not be finally rejected. The one-armed spirals is the most promising mode in this context. On the right-top part of Fig. 4 we see that the unstablemode has several density maxima which could be identified with future planets. The number of maxima dependson the fractional mass. Fig. 6 illustrates the tendency: the number of the density concentrations decreases withf . The massive disks, therefore, can be suspected to produce binary companions or giant planets while multiplesmaller planets can be formed in the more light disks.
Our model is too simple in order to reproduce the Titius-Bode law (cf. Dubrulle & Graner 1994). It maybe noted, however, that Figs. 4 and 6 display the correct tendency. The radial separation of subsequent densitymaxima increases with axial distance.
The preference of the one-armed nonaxisymmetric disturbances is, probably, a special feature of the disksaround the massive central bodies. If the central mass were absent or small, the conservation of the mass centerposition would preclude the m = 1 instability in favor of the two-armed modes. In our case the central mass playsthe role of the second arm. Also indirect potential neglected in our model reinforces the instability of the m = 1modes (Adams et al. 1989, Laughlin & Bodenheimer 1994). This effect included the transition from rings to spiralsin Fig. 3 should be shifted to a smaller fractional mass.
5.3. Vortices
There is still another type of nonaxisymmetric eigensolutions. Because of their strong correlation between densityand vorticity disturbances we call them vortices. In contrast to spirals, the eigenvalues of vortices are not degen-erated and their real parts are strictly zero within the numerical precision. If resolution of the numerical modelis increased, say, two times, the number of vortex-type eigenvalues is also roughly doubled indicating continuous
190 Astron. Nachr. 321 (2000) 3
Fig. 5: Decaying (left) and growing (right) one-armed (top) and two-armed (bottom) spirals for f = 0.3. Only the positivedensity disturbances are shown. The dashed circle is the disk edge. The background rotation is anti-clockwise.
Fig. 6: Growing one-armed modes for f = 0.2 (left) and f = 0.4 (right). More surface density maxima are produced withsmaller fractional masses.
G. Rudiger and L.L. Kitchatinov: Nonlocal density wave theory for protoplanetary disks 191
spectra. The circulation around the density maxima has the same vorticity sign as the background rotation. Asthese circulation directions are opposite to that required for the geostrophic balance (Adams & Watkins 1995) ourvortices are not geostrophic ones. Accordingly, the divergence of the flow excitation is not small but has the sameorder of magnitude as vorticity. The vortices were never found to be excited, hence external driving is needed fortheir generation and maintenance.
Acknowledgements. The authors acknowledge support by the Deutsche Forschungsgemeinschaft and by the Russian Foun-dation for Basic Research (Project 96-02-00010G). M. Schultz and H.-E. Frohlich are acknowledged for technical support.
A The kernel functions
The rotational kernels of the integral equation system (21) read
R(k, k′) =
∞∫0
Ω(x)
[2kxJm+1(kx)Jm+1(k
′x)− 2(m − 1)k′
(kJm+1(kx)Jm(k
′x) + k′Jm(kx)Jm+1(k′x))+
+
(4m(m − 1)
k′x− k′x
)Jm(kx)Jm(k
′x)
]dx, (31)
R1(k, k′) =
∞∫0
Ω(x)
[kxJm+1(kx)Jm+1(k
′x) +m(m − 1)
k′(kJm+1(kx)Jm(k
′x) + k′Jm(kx)Jm+1(k′x))−
− 2m2(m − 1)
k′xJm(kx)Jm(k
′x)
]dx, (32)
R2(k, k′) =
∞∫0
Ω(x)
[2kxJm+1(kx)Jm+1(k
′x)− 2(m − 1)k′
(kJm+1(kx)Jm(k
′x) + k′Jm(kx)Jm+1(k′x))+
+
(4m(m − 1)
k′x− kx
k
k′
)Jm(kx)Jm(k
′x)
]dx, (33)
R3(k, k′) =
∞∫0
mΩ(x)
[− 2kxJm+1(kx)Jm+1(k
′x) +2(m − 1)
k′(kJm+1(kx)Jm(k
′x) + k′Jm(kx)Jm+1(k′x))
−(4m(m − 1)
k′x− x
(k2 + k′2)k′
)Jm(kx)Jm(k
′x)
]+ kx
κ2
2ΩJm+1(kx)Jm+1(k
′x)
dx, (34)
R4(k, k′) =
∞∫0
Ω(x)k′xJm(kx)Jm(k′x) dx. (35)
For Kepler rotation they can be expressed in terms of the hypergeometric functions similar to eq. (27). Reliable numericalroutines exist to compute the functions. An examination of the bulky expressions for the rotational kernels reveals that theyare always linear combinations of simple common blocks. The same is true for the density and temperature kernels:
D(k, k′) =
∞∫0
Σ0(x)
[− xkk′Jm+1(kx)Jm+1(k
′x) +mkJm+1(kx)Jm(k′x) +mk′Jm(kx)Jm+1(k
′x) −
− 2m2
xJm(kx)Jm(k
′x)
]dx, (36)
D1(k, k′) =
∞∫0
Σ0(x)
[2m
xJm(kx)Jm(k
′x) − kJm+1(kx)Jm(k′x)− k′Jm(kx)Jm+1(k
′x)
]dx, (37)
T (k, k′) = kk′∞∫0
T (x)
[(kx+
mn(x)
kx
)Jm(kx)Jm(k
′x)− n(x)Jm+1(kx)Jm(k′x)
]dx, (38)
T1(k, k′) = −k′∞∫
0
T (x)n(x)
xJm(kx)Jm(k
′x) dx. (39)
192 Astron. Nachr. 321 (2000) 3
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Addresses of the authors:
Gunther Rudiger, Astrophysikalisches Institut Potsdam, An der Sternwarte 16, D-14482 Potsdam, Germanye-mail: [email protected]
Leonid L. Kitchatinov, Institute for Solar-Terrestrial Physics, P.O. Box 4026, Irkutsk, 664033, Russiae-mail: [email protected]
