On the gravitational stability of a disk of stars

“It is well known (Goodman 2003) that the outer parts of disks around SMBHs may be subject to gravitational instability (Safronov 1960; Toomre 1964; Goldreich & Lynden-Bell 1965) when the Toomre Q parameter defined as Q = Ωcs/(πGΣ) drops below unity.”

“The gravitational instability sets in when the Toomre parameter (Toomre 1964) Q = csκ/πGΣ 1, where cs is the sound speed, κ is the local epicyclic frequency and Σ is the mass surface density.”

“However, we expect gravitational instabilities to form at the Toomre critical wavelength, kcr 1⁄4 pGr=X, where G is the gravitational constant, r is the surface density, and X is the orbital frequency (Toomre, 1964).”

“Since the gravitational torque depends on how the disk is gravitationally unstable, αg is often given as a function of Toomre’s Q parameter Q = csκepi/(πGΣ), where κepi = √ (dj2/dr)/r3 is epicyclic frequency (Toomre 1964).”

“The susceptibility to bar formation was convincingly inferred via the combination of two criteria based on the Toomre Q and swing amplification X parameters (for the m = 2 mode; Toomre 1964, 1981).”

“The morphological quenching model is based on the Toomre (1964) theory, which predicts that the development of gravitational instabilities within the gaseous disc is hampered by the gas kinematics and by the presence of dissipative forces induced by the galaxy differential rotation, as shear.”

“As seen in Figure 2d, gravitational instability tends to occur when the disk mass exceeds the protostellar mass (Toomre 1964) and a significant fraction of the disk gas quickly falls onto the protostar.”

“K89 found that a modified Toomre (1964) Q criterion, a simple thin disk gravitational stability condition, could satisfactorily describe the star formation threshold gas surface densities in active star-forming galaxies.”

“The ratio of the CBD mass to the black hole mass is set as MCBD/Mbh = 10 −2, which ensures that the CBD is stable for the self-gravitation because Toomre’s Q value is Q = (MCBD/Mbh) −1(H/r) > 1 (Toomre 1964).”

“1 https://flathub.flatironinstitute.org/jhunt2021 2 https://github.com/treecode/galactics.parallel © 2020 RAS, MNRAS 000, 1–16 Model D1 (MWb in Widrow & Dubinski 2005) is a disc with a high Toomre parameterQ = 2.3 (Toomre 1964) which is stable against bar and spiral formation for several Gyr.”

“Simple estimates of the characteristic fragmentation mass,Mc, in a Q = 1 disk (Toomre 1964) suggest that it should be smaller at high redshift.”

“The Toomre Q stability parameter (Toomre 1964) for the gas component can be expressed as Qgas= σ0κ/πGΣg, where κ is the epicyclic frequency and Σg is the gas surface density.”

“Stars may also form in the outer regions of AGN disks, where the disk is gravitationally unstable and the cooling timescale is short (Toomre 1964; Gammie 2001).”

“The constant factor a= π for a fluid analysis, and a= 3.36 for our distribution function approach (Toomre 1964).”

“We use the Toomre parameter for the self-gravitational stability (Toomre 1964), QT = csκep πG , (1) where κep is the epicycle frequency, the disc surface density, cs = √ kBT /μmH the sound speed, T the temperature at the midplane, μ the mean molecular weight, kB the Boltzmann constant, mH the…”

“On the other hand, in the high-density disk region (r 2 AU), the magnetic field undergoes significant Ohmic dissipation (Nakano et al. 2002) and the angular momentum is mainly transported by the non-axisymmetric structure caused by gravitational instability (Toomre 1964).”

“So as to have Q = 1 (Toomre 1964), the velocity dispersion within the disc is given by q = 6, where the parameter q has been introduced in Eq.”

“The Toomre (1964) Q = c s κ/πGΣ, where c s is the sound speed and κ is the epicyclic frequency, is a parameter that indicates whether a disk is gravitationally stable.”

“The average tilt angle of wa with respect to tangential direction is determined by gradient of the systematic velocity field, corresponding about 20◦ for the Keplerian case; the typical radial spa ing between wakes is close to Toomre’s critical wavelen (Toomre, 1964)”

“To examine the gravitational stability of an accretion flows, we introduce Toomre’s Q parameter (Toomre 1964), Q = csΩ πGΣ ≃ 3.77× 1021αm−1ṁ−1f3/2r̂−3/2 (36) where G is the gravitational constant.”

“…clearly seen when one considers the dispersion relation for tightly wound axi-symmetric disturbances in an infinitesimally thin, rotating disc (cf. Toomre 1964 in the case of a stellar disc, see also Binney & Tremaine 1987 and Bertin 2000 for the fluid case): ω2 = c2sk 2 − 2πGΣ|k| + κ2,…”

“Thus, even if the disc is Toomre unstable (Toomre 1964), proto-fragments which try to condense out undergo an adiabatic bounce and are then sheared apart (Gammie 2001).”

“Figure 7d displays the gravitational stability parameterQ defined by Toomre (1964), Figure 7e displays the QA parameter defined by Elmegreen (1993) for the case on which shear dominates over Coriolis force, and, finally, Figure 7f displays the ratio of the tidal to the gravitation surface…”

“The disc must be gravitationally stable to axisymmetric perturbations (Toomre 1964).”

“The common understanding is that these higher gas fractions and densities lead to gravitationally unstable disks with a Toomre parameter of Q<0.7 (Toomre 1964; Goldreich & LyndenBell 1965; Behrendt et al. 2015) that fragment into a few kiloparsec-sized objects (Bournaud 2016, p. 355 and references…”

“Today, several decades after the pioneering works of Safronov (1960), Toomre (1964) and Goldreich & Lynden-Bell (1965a, b) on local disc instability, and the seminal papers by Lin & Shu (1966) and Jog & Solomon (1984a, b) on the relative contributions of stars and interstellar gas, it is widely…”

“This leads to the global minimum in the Toomre’s parameter Q ∼ 1.6 at R ∼ 2.4h (Toomre 1964).”

“…trend in sV ,0, the combination of our data and these previous measurements agrees fairly well with the expected evolution for marginally stable disks (e.g., Toomre 1964; Förster Schreiber et al. 2006; Genzel et al. 2011; see Figure 8 of Wisnioski et al. 2015, Figure 5 of Turner et al. 2017).”

“The Toomre parameter, Q (Toomre 1964) for a galaxy with a total mass surface density Σ = Σg + Σ?, l.o.s velocity dispersion σ, and epicyclic frequency κ = avc/rd , where a = √ 2 for galaxies with flat rotation curves (Inoue et al. 2016; Leung et al. 2019) is given by Q = σκ πGΣ = √ 2 ( vc σ )−1…”

“In the linear theory, axisymmetric density perturbations grow if Q 1 where Q is defined by σΩ/πGΣ (Toomre 1964) with velocity dispersion σ.”

“Thus, a value of Q < 1 corresponds to instability (Safranov 1960; Toomre 1964).”

“The classic Toomre Q criterion: Q≡ σgκ πGΣgas > 1 (1) where σg is the gas velocity dispersion, κ ∼ Ω the epicyclic frequency, and Σgas the gas surface density (Toomre 1964, 1977; Goldreich & Lynden-Bell 1965) appears to offer some guidance.”

“We parameterize the disk using the Toomre Q parameter (Toomre 1964) ( )Q c G , 10s k p º W S where cs is the isothermal velocity, *GM Rk 3W º is the Keplerian frequency, R is the cylindrical radius, and Σ is the gas surface density.”

“If the SM1 condensation is a disk, its gravitational stability will depend, for example, on whether or not it is in Keplerian rotation, and the mass of the disk relative to the central source (e.g., Toomre 1964).”

“The central radial dispersion was chosen to yield a relatively high Toomre Q parameter (Toomre 1964) across the disc, as can been seen in the bottom panel of Fig.”

“…density for star formation (Kennicutt 1989; Martin & Kennicutt 2001), most likely due to a transition between dynamically unstable and stable regions of the galaxy (e.g., Toomre 1964) or to a phase transition of the gas (e.g., Elmegreen & Parravano 1994; Schaye 2004; Krumholz et al. 2009).”

“Fragmentation happens provided that the Toomre criterion (Toomre 1964) is satisfied, Q ≡ cs(R)κ(R) πGΣ(R) ∼ 1, (1) where cs(R) is the sound speed, κ(R) is the epicyclic frequency, and Σ(R) is the surface density of the disc at a given orbital radius R.”

“…theories to explain their large sizes and separations, such as core/pebble accretion (Johansen & Lambrechts 2017), dynamical scattering (Rasio & Ford 1996; Weidenschilling & Marzari 1996; Chatterjee et al. 2008), disk instability (Kuiper 1951; Cameron 1978), and cloud fragmentation (Toomre 1964).”

“…to simulations without winds likely owes to the higher gas and disk fractions, implying lower Toomre Q parameters for similar mass galaxies (Toomre 1964) and hence increased gravitational fragmentation, and possibly to the injection of energy due to the recycling of the outflowing gas back…”

“In disk galaxies, a possible explanation is that the truncation mass is of order the Toomre mass in the galactic disk (Toomre 1964; Escala & Larson 2008), which is ∼ 106M for the Milky Way, but is significantly larger for present-day starburst/merger galaxies (Larsen 2009) and their high-redshift…”

“The SB approximation, with its emphasis on the effect of shear, is not new and has been introduced in an astrophysical context long ago by Goldreich & Lynden Bell (1965) (see also Toomre 1964) in their study of sheared gravitational instabilities in a galaxy.”

“…fragment: (i) the disc must be Toomre unstable, i.e. it has to be massive enough so that gravity can overcome thermal and local centrifugal support (Toomre 1964) and (ii) the disc must cool efficiently so that protocondensations forming in the disc do not simply undergo an adiabatic bounce and…”

“Eventually, for Ω O > 1.35 × 10−13 s−1 (Runs 5, 6 and 7), the disc becomes unstable against fragmentation, i.e. it satisfies both the Toomre and Gammie conditions (Toomre 1964; Gammie 2001).”

“It is given by Q = csκ 1 πGΣgas , (4) where cs is the sound speed, κ is the epicyclic frequency, and Σgas is the gas surface density (Toomre 1964).”

“The ISM is stable to perturbations with a wavelength longer than the Toomre (1964) length, λT = 4π2GΣ0 κ2 = 2π2GΣ0 Ω2 , (32) and this is therefore the largest scale on which collapse can take place.”

“In contrast, the Toomre stability criterion (Toomre 1964) applies to the linear growth of infinitesimal perturbations in surface density, so it is worthwhile to reconcile the relation between the two criteria here.”

“The standard criterion for the gravothermal stability of a rotating disk, the Toomre criterion (Toomre 1964), can be obtained from the dispersion relation for an axisymmetric density perturbation (i.e., a sound wave) with wavelength l and frequency !”

“The Toomre length (Toomre 1964) is the maximum scale for axisymmetric modes in a thin, cold disk; λT = 4π2GΣ κ2 = 4π2GΣ (4 − 2q)Ω2 , (39) = 1.36 kpc ( Ω 25 km s−1 kpc−1 )−2( Σ 10 M⊙ /pc2 ) .”

“The typical radial scal e of wakes in simulations co responds, at least roughly, to Toomre’s critical wavelen (Toomre, 1964) ,”

“However, Keplerian disks can exhibit transient energy growth despite their asymptotic stability (Goldreich & Lynden-Bell 1965 ; Toomre 1964, 1981), which reveals that the perturbation dynamics are governed by a nonnormal operator (an operator A is nonnormal when it does not commute with its adjoint…”

“GIs can occur in self-gravitating disks whenever the destabilizing effects of gravity become comparable to the stabilizing effects of disk pressure and shear (Toomre 1964).”

“/(Rstr · Lorb)—where the stream is spread over an area7 deter- 7 Rstr · Lorb ends up being comparable to the full ellipse area, ≈ aMbm. mined by the stream radius Rstr and orbital circumference Lorb— we can calculate the Toomre parameter (Toomre 1964; Binney & Tremaine 2008), Q ≡ cs,str Ω/(πG Σ).”

“The question of disk stability has been investigated for more than four decades since the pioneering works by Toomre (1964) for collisionless stars and Goldreich & Lynden-Bell (1965) for gas sheets.”

“An analytic disc model is first created, with a Toomre (1964) Q ∼ 1 for most of the disc.”

“Values of Q s 1 imply the disc is gravitationally unstable (Toomre 1964).”

“For circular orbits this rotation size scale (Toomre 1964; Binney & Tremaine 1987) can be expressed as R G , 1rot 2µ S W ( ) where Ω is the angular rotation velocity of the disk, Σ is the surface density of gas, and G is the gravitational constant.”

“The Toomre parameter in the stars, Qs (Toomre 1964), characterises the stability of the disc to local collapse and is given by Qs = κσR 3.36GΣd (1) where κ is the epicycle frequency, σR the radial velocity dispersion and Σd the disc surface density.”

“We check the stability of each system using the Toomre Q parameter (Toomre 1964), defined by: Q 774.8 ( aQmin 0.1AU )−3/4( M 0.1M ) eaQmin/90 0g , (22) where aQmin is the semimajor axis for which Q has its minimum value and in this case is aQmin = 67.5 AU (Miguel et al. 2011a).”

“Early theoretical studies showed that stellar disks lacking random motions are generally unstable, and rapidly leads to the formation of bars (Toomre 1964; Ostriker & Peebles 1973).”

“To explore the link between disc gravitational instability and the relative mass content of atomic gas, molecular gas and stars in galaxies, we start from the simplest stability diagnostic: the Toomre (1964) parameter, Q = κσ/πGΣ.”

“…property that is not directly observable but is tightly constrained by gravitational instability and other fundamental physical processes: 〈Q⋆〉, the mass-weighted average of the stellar Q stability parameter (Toomre 1964), corrected so as to include disc thickness effects (Romeo & Falstad 2013).”

“Q parameter, where Q = √ 2σvvrot/πGrΣgas (Toomre 1964; Goldreich & Lynden-Bell 1965).”

“We looked at Toomre parameters for MWP14-1, MWP14-2, and MWP14-3 of approximately Q= 1.3, 1.5, and 1.8 respectively (Toomre 1964).”

“The Toomre-Q parameter (Toomre 1964) is a standard metric used to determine whether a disk is gravitationally stable ( Q ≳ 1) or unstable ( Q ≲ 1).”

“Because of its dependence upon the Toomre stability criterion, Q (Toomre 1964), the latter type of morphology-related quenching is sometimes referred to as ‘Q-quenching’.”

“The Toomre (1964) stability criterion measures the susceptibility of the disc to growing perturbations due to self-gravity; whenever Q ≡ 2cs GΣtP < 1, (1) axisymmetric ring distortions will grow in an infinitesimally thin disc.”

“The local stability parameter (Toomre 1964) for the isochrone/5 disc has an near constant value of Q ≃ 1.6, and is Q ≃ 1.2 for the isochrone/8 model.”

“(2) The Toomre Q = κcs(πGΣ) −1 parameter of the disc with cs is the sound speed in the disc (Toomre 1964; Safronov 1960) where a non-self-gravitating disc has Q 1.”

“Observationally, there are hints that the Toomre (1964) Q-parameter of clumpy disks is .”

“The Toomre parameter Q = κσR πGΣ , (3) predicts instability for Q < 1, where κ is the epicycle frequency, σR is the radial velocity dispersion and Σ is the surface density, (Toomre 1964, for a revised criterion for turbulent disks see Romeo et al. 2010 ).”

“We can estimate the conditions at which this occurs by a simple application of the Toomre (1964) criterion, setting Q = σΩ πGΣdust = 1 , (57) where σ is the velocity dispersion in the dust layer and Σdust is the surface density of the dust layer.”

“To assess the stability of the disk as suggested by the estimates of Vrot/σ in both halos, we computed the Toomre stability parameter Q (Toomre 1964).”

“These discs will exist in a state of marginal instability if the local Toomre Parameter (Toomre 1964), Q = csκ πG ∼ 1, (1) where κ is the epicyclic frequency (representing the rotational support, and equal to the angular velocity in a Keplerian disc), cs is the sound speed of the gas (representing…”

“As is well known, interstellar gas in a disk galaxy is stabilized against gravitational collapse by rotation on large scales and pressure on small scales (Safronov 1960; Toomre 1964; Goldreich & Lynden-Bell 1965).”

“(8), the Toomre-Q parameter is (Toomre 1964; Goldreich & Lynden-Bell 1965) Q = Ω2H πGΣ = [ 6561π6 64000 (1 + β)3 G2 Ṁ2 ζ6 α2m(ΩR) 6 ] 1 5 .”

“This is quantified with the Toomre Q parameter, Q = ΩK cs (πG Σ)−1 (Toomre 1964), which simplifies to Q = 21 × ( Σ 10 kg m−2 )−1 × ( r 100 au )−3/2 , (3) following Kimura & Tsuribe (2012).”