On the gravitational stability of a disk of stars

“First, the disc must be massive enough so that locally gravity can overcome thermal and centrifugal support (Toomre 1964), i.e.”

“In these GRAPE simulations, a spiral disk with an initial disk mass of 3 × 1010 M⊙ and an initial Q parameter (Toomre 1964) value of 1.0 is composed of 20,000 collisionless particles, and (for self-consistency) these particles respond to the fixed gravitational force of the dark halo and bulge…”

“This only marks the critical case for the m=0 unstable mode in razor-thin disks (Toomre 1964).”

“It is well known since Toomre (1964) proposed this stability criterion, that for an infinitesimally thin disc to be stable to fragmentation, Qc = 1 for axisymmetric disturbances.”

“The local gravitational stability of a rotating disk against axisymmetric perturbations is measured by the Toomre-Q parameter, which is defined by Q = kcs πGΣ , (27) where k is the epicyclic frequency given by k2 = (1/r3)[∂(r4Ωk 2)/∂r] (Toomre, 1964; Goldreich and Lynden-Bell, 1965).”

“Increasing qEOS improves the dynamical stability of the gas and can prevent Toomre (1964) instability even in gas-rich systems (Springel & Hernquist 2003, 2005; Robertson et al. 2004, 2005a).”

“With this temperature profile, the disk initially has a (Toomre 1964) ; marginally unstable disks haveQ ∼ 1.3min -values less than ∼1.5 (Papaloizou & Savonije 1991; Nel-Qmin son et al. 1998; Boss 1998a).”

“The instability model predicts that when > c =3:36G in a one-component stellar disk, spiral arms form easily (Toomre 1964, 1981; Goldreich & Lynden-Bell 1965; Julian & Toomre 1966; Athanassoula 1984).”

“The normalisation constant is set by adopting a certain value of the stability Q-parameter (Toomre 1964) at a particular location in the disk.”

“(3) The dimensionless quantity µ is defined in such a way that µΩr/c ≫ 1 is roughly the largest azimuthal harmonic contributing to the flux (the “torque cutoff”); it is a function of the gravitational stability parameter Q = cΩ/πGΣ (Toomre 1964).”

“The gravitational stability of a disc can be characterized by the Toomre (1964) Q parameter, defined by Q ≈ κσ /πG , where κ is the epicyclic frequency of the galaxy’s rotation, σ is the velocity dispersion, and is the surface mass density.”

“…AU) (Rafikov 2005; Boley 2009; Meru & Bate 2010; Boss 2011) since they are primarily limited by the ability of gas to efficiently cool and by the shear of differential rotation speeds, both of which limit the ability for a protoplanetary clump to exceed the Toomre stability criterion (Toomre 1964).”

“…CFEΓ at the time of GC formation, we use the relation between the truncation mass Mc and the Toomre massMT, i.e. by writing (also see§2.1.1): Mc ≡ Mcl,T = ǫΓMT = ǫΓ(Σ, Q,Ω) π4G2Σ3Q4 4Ω4 , (5) whereΣ is the gas surface density,Q is the Toomre(1964) stability parameter, andΩ is the angular velocity.”

“…to counteract the stabilizing forces of gas pressure and rotational shear, as quantified by Toomre’s Q: Q ≡ csΩ πGΣ < Qo ∼ 1 (1) (Safronov 1960; Toomre 1964), where cs = √ kT/µ is the isothermal sound speed of the gas with mean particle weight µ = 2.3mH appropriate for a molecular gas,…”

“The Toomre criterion Q = csΩ/(πGΣ) for a Keplerian disk is smaller than some critical value Qcr, usually taken to be unity (Toomre 1964).”

“The star formation rates in late types are also lower because late type spirals have lower Σcrit, the critical gas surface densities for star formation (e.g., Toomre 1964; Kennicutt 1989).”

“If the disk accumulates material from the envelope more quickly than mass can accrete onto the star, then it could grow more massive and violate the Toomre stability criterion (Toomre 1964), fragmenting to form a bound companion.”

“Galactic discs at high redshift are expected to develop a gravitational disc instability with a Toomre instability parameter Q smaller than unity (Toomre 1964).”

“…instabilities if its surface mass density exceeds the crit- ical valueΣQ, given by ΣQ(σ) ≈ αQκσ πG , (27) whereκ is the epicyclical frequency κ2 = 2 ( v2 R2 + v R dv dR ) , (28) σ is the characteristic velocity dispersion of the fluid, and the parameterαQ ∼ O(1) (Safronov 1960; Toomre 1964).”

“As discussed by Toomre (1964), a disc is gravitationally unstable as long as the minimum size of shear-stabilizing disturbances, L s G ω−2, is larger than the maximum size of dispersionsupported regions, L d c2s G−1 −1.”

“Classically GI can be parameterized by the Q-value (Toomre 1964) Q = csΩ πGΣ(R) (16) where Ω is the keplerian angular velocity (Ω = √ GM⋆/R3) and cs the sound of speed.”

“In this way, we can reproduce a stiffer equation of state for the dense gas, leading to thicker, more stable discs (Toomre 1964, Robertson et al. 2004).”

“Observational and numerical studies have established that the Toomre (1964) parameterQ ≡ κσ/πGΣgas, whereκ is the epicyclic frequency andσ is the effective sound speed, controls whether or not a given region of a galaxy will actively form stars (Klessen 1998; Li et al. 2005).”

“The stability of rotating discs of gas and collisionless particles to axisymmetric modes has been analysed in classic papers by Goldreich and Lynden-Bell (1965) and by Toomre (1964).”

“G A ] 2 4 by enhancing the Toomre (1964) stability of the disc (Martig et al. 2009), or by harboring a particularly powerful AGN at their centers (Rangel et al. 2014).”

“This is consistent with the 1D axisymmetric thin disc model of Forbes et al. (2014) based on continuous disc instability with Toomre (1964) Q-parameter of the order of unity.”

“This can cause the Toomre stability criterion (Toomre 1964) to exceed unity, implying that the gas is stable against gravitational collapse.”

“MNRAS 000, 1–37 (2019) length to the gas disc scale height and the Toomre (1964) instability length and find that λ matches the gas disc scale height across the full extent of the star-forming disc.”

“…star formation may be related to the global gravitational (in)stability of star-forming discs– if the kinematics of a gas disc are such that it can withstand global collapse, star formation is suppressed (Toomre 1964; Quirk 1972; Fall& Efstathiou 1980; Kennicutt 1989; Martin & Kennicutt 2001).”

“According to the standard Toomre instability analysis (Toomre 1964; Binney & Tremaine 2008, chapter 6), a rotating disc becomes unstable to axi-symmetric modes once the local gravity overcomes both differential rotation and pressure due to turbulence or thermal motions.”

“A Keplerian accretion disc with sound speed cs, surface density Σ, and epicyclic frequency κ will become gravitationally unstable if the Toomre (1964) Q parameter Q = csκ πGΣ (1) is of order unity.”

“Its stellar disk is a Toomre (1964) disk of radial scalelength 5 kpc, truncated at 15 kpc. Gas is distributed in a Toomre disk of scale-length 15 kpc and radius 30 kpc.”

“1 inner disk and the gravitationally stable Q & 1 outer region, where Q is the Toomre (1964) stability parameter Q ≡ κσg πGΣg , (3) where σg is the 1D gas velocity dispersion in the disk plane and κ is the epicyclic frequency: κ = √ 2 vc r ( 1 + r vc dvc dr )1/2 = √ 2 vc r (1 + β)1/2.”

“We estimated the disk stability parameter, Q (Toomre 1964), using the mean velocity dispersion, the azimuthally averaged H i surface brightness, and the angular velocity derived using the rotation velocity.”

“The two canonical modes proposed for gas giant formation are the assembly of a rockice core that can accrete and hold gas (“core accretion”; Pollack et al. 1996) and the direct fragmentation and collapse of material in a Toomre-unstable disk (“disk instability”; Toomre 1964; Boss 2001).”

“Kennicutt (1989) has shown that star formation is strongly suppressed below a critical surface density, which can be modeled by a simple Toomre stability criterion: Σ crit (R) = σ gas κ(R) 3.36 Q G (36) (Toomre 1964).”

“In order for gravitational instabilities (GIs) to form gas giants directly, the Toomre (1964) Q = csκ/πGΣ, where cs is the sound speed, κ is the epicyclic frequency, and Σ is the surface density, must approach unity and the local cooling time must be . the local orbit period (Gammie 2001; Rice et…”

“One significant omission is the disk-instability pathway for bulge formation (e.g., Toomre 1964; Dalcanton et al. 1997; van den Bosch 1998; Parry et al. 2009), which is an internal process where the bulge and disk either form from high- and low-j material, or else exchange j through gravitational…”

“…disks are thought to fragment and become unstable when self-gravity becomes sufficiently strong as measured by a low value of Toomre’s Q parameter: Q = cs,dΩd πGΣd , (1) where Ωd is the disk angular velocity, Σd is the disk surface density, and cs,d is the disk sound speed (Toomre 1964).”

“The accretion flow will become gravitationally unstable if the Toomre parameter (Toomre 1964), QT = csκ πGΣ 1, (3) wherecs is the sound speed andκ the epicyclic frequency.”

“…radial variations in some property of the gas, the length-scale heff at which pressure forces become important is given by heff = 2c2eff/GΣ (e.g., Toomre 1964) where ceff is the effective dispersion of the gas, including both the true thermal sound speed and any non-thermal turbulent motions:…”

“Second, very thin sheets are cold and therefore gravitationally unstable (Toomre 1964).”

“The rotating disc turns unstable and forms clumps when its surface density, Σ, becomes too large, namely when Q ' σΩ/(πGΣ) < 1 [324], with Ω the disc angular velocity, and σ its orthogonal 1-D velocity dispersion.”

“In contrast, fragmentation in the outer, extended disk (r & 50 AU) becomes quite possible if the Toomre (1964) Q can be driven toward unity by, e.g., mass loading (Boley, 2009).”

“Moreover, the critical velocity dispersion for local stability (Toomre 1964) is small, critT1 km s 1.”

“Lin & Shu (1964, 1966) also proposed a global solution for the disc, rather than the local perturbations assumed by Toomre (1964).”

“In particular, we consider theoretical scenarios that assume that the onset of gravitational or axisymmetric instabilities (Safronov 1960; Toomre 1964; Goldreich & Lynden-Bell 1965; Elmegreen 1979) is relevant for the growth of clouds and the onset of SF.”

“A gas disk is will be gravitationally stable if Q > 1 (Toomre 1964; Wang & Silk 1994).”

“The Toomre parameter is calculated from Q = σrκ πGΣ (2) where σr is the one-dimensional random velocity dispersion, κ is the epicyclic frequency and Σ is the surface mass density (Toomre 1964).”

“Toomre 1964).”

“The different stability properties in both cases can be quantified in terms of the eff ctive ToomreQ parameter for the stellar, gaseous, and combined system (e.g., Toomre, 1964; Jog & Solomon, 1984; Wang & Silk, 1994; Elmegreen, 1995).”

“This can be understood in terms of the stability parameter, Q (Toomre 1964), which in the case of gas+stars systems can be extended to include the two components (e.g. Jog & Solomon 1984; Elmegreen 1995; Romeo & Wiegert 2011), as confirmed by observations (Wong & Blitz 2002; Leroy et al. 2008;…”

“Genzel et al. (2011) derived the ToomreQ parameter (Toomre 1964) at locations of observed clumps from the Hα velocity map and Hα surface density.”

“To derive the maximum cloud mass, we adopt the model of Reina-Campos & Kruijssen (2017), which relates MGMC to the largest gravitationally unstable mass in a differentially rotating disc, i.e. the Toomre (1964) mass MT: MGMC = fcollMT, (3) where fcoll is the ‘Toomre mass collapse fraction’.”

“The Toomre (1964) instability quickly causes a fragmentation of the smooth gas disk into clumps, which collapse further under their own self-gravity and form stars on a short timescale.”

“The existence of such a threshold is indeed suggested by dynamical stability analysis of thin, gaseous, differentially rotating disks (e.g. Toomre 1964, Quirk 1972).”

“We expect this to occur via the standard local dynamical instability of self-gravitating disks, i.e. the gaseous version of the Toomre instability (Toomre 1964), although growth of very large masses may perhaps be seeded by other means, for example capture of stars (§4.3).”

“Type 6 Note that the Toomre Q disc instability criterion for star formation (Toomre 1964) is expressed in L-GALAXIES in terms of a critical cold gas mass (Guo et al. 2011, section 3.4).”

“This can be expressed as a condition on the surface density of gas: Σgas > Σcrit = ακσgas πG ≡ QΣgas, (1) (Toomre 1964, Quirk 1972), where σgas is the gas velocity dispersion; α is a dimensionless constant near unity, to account for deviations of real disks from the idealized Toomre thin disk,…”

“For example, ETGs have different gravitational potentials (affecting e.g. the Toomre 1964 stability parameterQ), different elemental abundances (high metallicity and α-element over-abundance), higher prevalence of hot gas (106 K) and active galactic nuclei (AGN).”

“Such disks, characterized by a radially constant accretion rate, are unlikely to extend beyond 10−3–10−2 pc because they would become self-gravitating (Toomre 1964; Kolykhalov & Sunyaev 1980; Goodman 2003).”

“The properties of the disk are chosen so that the Q parameter is initially larger than 1 at all radii and has a minimum value of 1.3 (right panel of Fig 1), implying that the disk is stable to axisymmetric instabilities (Toomre 1964).”

“The instability driving cloud formation may well be gravitational; in this case, the Toomre Q criterion (Toomre 1964) yields a critical gas surface density above which one expects local instability to axisymmetric perturbations.”

“The higher gas mass fraction present in late-type disks makes the disk dynamically cold and lowers the Toomre Q parameter (Toomre 1964), defined as Q ¼ G disk ; where disk is the disk mass surface density, is the gas velocity dispersion, and is the epicyclic frequency.”

“The effective EOS of the gas (for a numerical fit, see Robertson et al. 2004) therefore acts to stabilize dense gas in galactic disks against Toomre (1964) instabilities caused by self-gravitation (Springel & Hernquist 2003; Robertson et al. 2004; Springel et al. 2005b).”

“…to assess the gravitational stability of gaseous discs is the Toomre stability criterion: f g T ≡ κcg πG (7) where g is the gas surface mass density, f is the disc thickness factor, κ is the epicyclic frequency (Toomre 1964), G is the gravitational constant, cg is the gas velocity dispersion.”

“It is also possible to assess the stability of these rotating, turbulent discs through the Toomre Q parameter (Toomre 1964).”

“‖ Which would of course constitute an unstable disk for stars of Solar mass or less (see e.g. Laughlin & Bodenheimer (1994), Toomre (1964)). c© 0000 RAS, MNRAS 000, 000–000 In order to solve the proplyd frequency problem, the initial disk mass required (0.4 M⊙) would imply that 90 per cent of the…”

“We consider the ratio of the observed gas surface density g to the critical gas density c derived from the gravitational instabilitymodels for differentially rotating disks (Safronov 1960; Toomre 1964; Quirk 1972).”

“The Toomre Q-parameter is defined as Q = κcs πΣG (7) with epicyclic frequency κ, sound speed cs, surface density Σ, and Newton’s gravitational constant G. Linear gravitational instability sets in when Q < 1 (Toomre 1964; Goldreich & Lynden-Bell 1965).”

“Formally, the condition Q = 1 marks the stability boundary of discs to axisymmetric instability (Toomre 1964).”

“It has often been suggested that the upper mass limit is related to the maximum unstable length scale in a rotating disc, due to shear and/or the coriolis force (Toomre 1964; Hunter et al. 1998; Dowell et al. 2008).”

“The Toomre (1964) instability is a gravitational instability in a shearing, thin Fig.”