On the gravitational stability of a disk of stars

“The classical Toomre gravitational instability model (Toomre 1964; Quirk 1972), which seems to explain the SF threshold successfully in spiral galaxies (Kennicutt 1989; Martin & Kennicutt 2001) and elliptical galaxies (e.g. Vader & Vigroux 1991), predicts that the gas surface density of dIrr…”

“To evaluate the gravitational stability, the Toomre’sQ parameter (Toomre 1964) was computed under the assumption of a g/d of 100.”

“Kennicutt (1989) derived coherent results when applying the Toomre (1964) “Q” criterion Q = vgasκπGΣgas – where κ is the epicyclic frequency, vgas the velocity dispersion of the gas, and Σgas the gas surface density – to a sample of spirals.”

“The dead zone can become self gravitating when the Toomre parameter (Toomre 1964) is sufficiently small, Q 2.”

“In this case, SFR ∝ [ gas − crit], (2) where the critical density crit is evaluated using the disc stability criterion given in Toomre (1964).”

“The Toomre instability criterion for a gas disk is QGas ≡ csΩ πGΣgas < 1 (5) (Toomre 1964; Goldreich & Lynden-Bell 1965).”

“…that this suppression is due to the gas density falling below the critical density for global disc stability as given by the Toomre criterion (Toomre 1964) crit = σgasκ(R)/3.36GQ, (21) where σ gas is the velocity dispersion of the gas, κ is the epicyclic frequency and Q is the Toomre Q…”

“Star formation in a galaxy disc takes place when the Toomre (1964) Q parameter drops below ≃ 1.”

“Most disks, however, are well below the Toomre criterion for global gravitational instability (Q < 1, Toomre 1964; Boss 1997), with a typical Mdisk/M∗ of only ∼ 5±3%.”

“In summary, galactic disks should settle close to the gravitational stability line that is determined by the Toomre criterion (Toomre 1964; Wang & Silk 1994) Q ≡ κ πG ( Σg σg + Σ∗ σ∗ )−1 ≤ Qc. (20) κ is the epicyclic frequency that is related to the local angular circular velocity Ω at radius r…”

“The first is the stability parameter (Toomre 1964), Q = csκep πΣG , (1) where cs is the sound speed in the disc, κep is the epicyclic frequency, which for Keplerian discs is approximately equal to the angular frequency, Ω, Σ is the surface mass density and G is the gravitational constant.”

“The high gas fraction together with the high density in the early Universe, leads to a violent gravitational disc instability (Toomre 1964), which involves giant clumps and operates on short, orbital timescales (Dekel, Sari & Ceverino 2009; Ceverino, Dekel & Bournaud 2010), as opposed to the slow,…”

“We therefore appeal to theory to guide us. c© 0000 RAS, MNRAS 000, 000–000 In general, galactic discs appear to be well-described by a Toomre Q parameter near unity (Toomre 1964; Goldreich & Lynden-Bell 1965), corresponding to a state of marginal gravitational stability.”

“The self-gravity parameterG̃ is connected to the more commonly usedQ (whereQ > 1 is the axisymmetric stability criterion for a flat disc in Keplerian rotation, see Safronov, 1960; Toomre, 1964) throughG̃ ≈ 1.6Q−1.”

“However, the basic ““ Toomre criterion ÏÏ (Toomre 1964) for a rotating disk with rotation velocity andvrote†ective ““ sound speed ÏÏ requires that for a disk of surfacev sdensity & to be gravitationally unstable, must be smallerv sthan v s ¹ f(G&R/vrot)D 11&23 R0A.5/v250 (km s~1) , (8) where is the…”

“…Kennicutt (1989) showed that star formation is abruptly suppressed below a critical surface density, which is associated with the onset of large-scale gravitational instabilities, and which can be modeled by a simple Toomre disc stability criterion: Σcrit(R) = σgas κ(R) 3.36GQ (12) (Toomre 1964).”

“For a razor-thin disk, the instability threshold is atQc ¼ 1 (Toomre 1964); for a purely self-gravitating (s0 ¼ 1), vertically stratified disk, the threshold is at Qc 0:676 (Goldreich & Lynden-Bell 1965a; Gammie 2001); and for a disk subject to both self-gravity and external gravity with s0 ¼ 1,…”

“The formation of the dense clump and excitation of the spiral arms are examined here based on the linear stability of Toomre (1964).”

“The Toomre (1964)Q parameter ranges from 130 at 0.1 AU to a minimum of 1.4 at 28 AU, so the disk remains gravitationally stable by a small margin.”

“Locally, the stability of a flat disk against Jeans instabilities is quantified by the parameter Q = σ κ (3GΣ)−1 (Toomre 1964), where σ is the local velocity dispersion, κ is the orbital frequency, and Σ is the local surface density.”

“These are canted relative to the azimuthal direction, with a typical length scale of about Lw ∼ 10(2) m, one Toomre critical wavelength [63].”

“The system becomes unstable when the Toomre parameter (Toomre 1964) Q gas (R) ≡ Σ crit (R) Σ gas (R) = κ(R)σ gas (R) πGΣ gas (R) , (5) describing the competition of pressure and rotation on the one side and gravitation on the other side, at a given radius , is less than 1, i.e. when the gas surface…”

“Disc stability is described by the Toomre parameter (Toomre 1964) defined as Q = κcs πΣG (11) with the epicyclic frequency κ, sound speed cs, surface density Σ, and gravitational constant G. Gravitational instability sets in when Q 1.”

“…Published by Oxford University Press on behalf of the Royal Astronomical Society at T he A ustralian N ational U niversity on M arch 24, 2016 http://m nras.oxfordjournals.org/ nloaded from which in turn leads to galaxies with low values of the Toomre Q parameter (Toomre 1964), QToomre = κσd πG d .”

“The yardstick for dynamical instabilities, λcrit = 4π2GΣ κ2 (8) (Toomre 1964), decreases with the disk mass (for a fixed rotation curve).”

“Gravitational instability (GI) in astrophysical disks has been a subject of investigation for more than fifty years since the seminal work by Safronov (1960) and Toomre (1964).”

“= 0.1Σg/tdyn , (A1) (more commonly used in cosmological simulations), which is appropriate for a marginally Toomre stable disk (Toomre 1964), i.e. the vertical dynamical time is close to the orbital time scale.”

“Figure 4 shows the Toomre Q parameter Q = Ωσ/(πGΣ) (Toomre 1964), where Ω is the angular velocity, σ is the velocity dispersion, and Σ is the surface density.”

“The instability is controlled by the two parameters Q and J, where Q = κσ πG (10) is the usual Toomre (1964) parameter, and J = √ T1 kcrit , (11) with T1 = (1 − β)(2m /κr)2 and kcrit = κ2/2πG , is a dimensionless measure of the strength of shear in the disc.”

“The large star-formation rate and low value of the Toomre (1964) Q parameter ( 1) implied a gas rich disc forming stars in-situ rapidly and suggested continuous fuelling by cosmological accretion.”

“For example, there has been some evidence for positive metallicity gradients in some samples (Cresci et al. 2010; Queyrel et al. 2012; Toomre 1964) interpreted as arising from cold flows.”

“One traditional way of quantifying such stability is via the Toomre ‘Q’ parameter (Toomre 1964) which effectively assesses the mass of the disk with respect to the total gravitating mass of the system: Q = V 2 GMdisk/rdisk = Mdyn Mdisk (5) Typically, values of Q 1 represent dynamically unstable…”

“We compute the expected separation and mass of large scale structures resulting from gravitational instabilities, as well as the stability of the disc, as measured by the Toomre parameter Q (Toomre 1964).”

“(45) For a vertically isothermal disk this can be integrated (Spitzer 1942) to ρsg(r, ϕ, z) = ρ0(r, ϕ) cosh−2 ( z Hsg ) , (46) where the vertical scale heightHsg is defined by Hsg = c2s πGΣ = csΩK πGΣ H QH , (47) with the Toomre parameterQ (Toomre 1964).”

“Furthermore, given the disc mass and temperature considered, we neglect disc self-gravity (Toomre 1964).”

“…20 AU to 60 AU), the disk mass to stellar mass ratios Md/Ms , the initial disk temperatures Ti, and the initial minimum and maximum values of the Toomre (1964) Q gravitational stability criterion, increasing monotonically outward from unstable Q = 1.1 values at 20 AU to marginally stable Q ∼…”

“Progressing from the third row, showing the case with highest surface density, to the top row, showing the case of lowest surface density we note an increasing tendency of the disk around the primary star to fragment and generate spiral arms characteristic of a Toomre-unstable disk (Toomre 1964).”

“In Figure 7 we show the initial radial profile of the Toomre stability parameter Q (Toomre 1964) for the gas, modified for a composite disc made out of gas and stars (Wang & Silk 1994; Naab & Ostriker 2006): Qg = κσg πGΣg ( 1 + Σ∗σg Σgσ∗ ) .”

“This structure is considered to be developed by gravitational instability (Toomre 1964) because the disk mass is comparable to the protostellar mass c© 0000 RAS, MNRAS 000, 000–000 during the (early) Class 0 stage (Inutsuka et al. 2010; Inutsuka 2012).”

“The gravitational coupling between stars and gas does not alter the form of the local axisymmetric stability criterion, Q ≥ 1 (Toomre 1964), but makes the Q stability parameter dependent on the radial velocity dispersions and surface mass densities of the two components (Bertin & Romeo 1988;…”

“The latter depends on the maximum size of a region that will overcome the shear of the disk and the kinetic pressure of the gas and start collapsing (Toomre 1964).”

“The standard theory of gravitational instability for a thin disk (Toomre 1964; Paczyński 1978) predicts that the minimum mass of the gas in the disk that would make it gravitationally unstable for the parameters appropriate to our Galactic Center is ∼104 M (Sect.”

“This observation is traditionally interpreted as the result of local instabilities developing in differentially rotating discs within a threshold radius where the gas density is larger than a critical density (Toomre 1964).”

“In particular, the Toomre (1964) Q-parameter must be less than unity: Q = csΩ πGΣ 6 1 , (2) where cs is the gas isothermal sound speed; Ω = (GM∗/R 3 p) 1/2 is Kepler’s angular frequency; M∗ is the mass of the star, assumed to be larger than that of the disc, and Σ = 2Hρ is the surface density of…”

“Requiring Ωcs/πGΣ < 1 [316, 123] is a necessary, but not sufficient condition [239]; see Zink et al.”

“In their approach, they consider that two criteria need to be satisfied simultaneously to form a planet by GI: the Toomre parameter (Toomre 1964) needs to be small enough for an instability to occur, which translates into a minimum mass of the resulting planet, and the cooling timescale of the disk…”

“Thus we arrange that the Toomre Q parameter (Toomre 1964) exceeds a minimum value (> 1) throughout the disc.”

“Gravitationally unstable discs must satisfy the Toomre criterion (Toomre 1964): Q = csκe πGΣ 1.5 − 1.7, (1) whereQ is the Toomre parameter,cs is the local sound speed of the gas,κe is the local epicyclic frequency (equal to the angular velocity Ω in Keplerian discs) andΣ is the disc surface…”

“According to the Toomre stability criterion (Q ∝ Vrot · σR; Toomre 1964) the thick disk should be slightly more stable than the thin disk (Qthickdisk ≈ 1.6 ·Qthin disk) and less disturbed by gravitational influences from the Galactic bar.”

“The local stability criterion, first established by Toomre (1964), ensures stability against axisymmetric perturbations in a stellar/gaseous galaxy disk when the column density of stars/gas (N∗,g) is smaller than a critical value (N crit∗,g =κσv/πG), i.e., when the so-called Toomre parameter Q∗,g=N…”

“To understand the relevance of gravitational instabilities in the simulated multi-phase disks, we use the Toomre parameter (Toomre 1964) defined, for gas (Goldreich & Lynden-Bell 1965a), as Qg = κcs πGΣg , (8) where cs is the sound speed of the gas.”

“…the amount of gas in the galaxy, in broad agreement with the observed relationship between the size of gas discs and the amount of gas in those discs (e.g. Wang et al. 2016), and potentially in agreement with arguments based on the Toomre (1964) stability parameter (Schmitz et al., in preparation).”

“For example, Toomre (1964) formalism allows us to estimate the critical density above which a gas disc is gravitationally unstable and the shear is low enough to have SF (e.g. Kennicutt 1989; Romeo 1990, 1992; Hunter et al. 1998; Martin & Kennicutt 2001).”

“However, contrary to their local counterparts, they are highly turbulent with high average velocity dispersions resulting in marginally stable discs (e.g. Wisnioski et al., 2015) in terms of the Toomre (1964) Q parameter.”

“Typically, ~t tg orb in normal galaxies, i.e., the Toomre parameter is of order unity (Toomre 1964).”

“…column densities at the largest radii M33 is forming stars to the expected critical surface density derived by Kennicutt (1998) based on the Toomre (1964) criterion and confirmed previous findings that the observed HI column (3 - 5 M⊙ pc −2) is three times lower than this expected critical…”

“The initial surface density profile is Σgas ∝ R −1, the initial temperature profile is T ∝ R−0.5, and the temperature is normalised such that the disc is initially gravitationally stable at all radii (Toomre 1964).”

“In normal galaxy discs, the parameter used to describ this balance is the ToomreQ parameter (Toomre 1964): Q = σκ 3GΣ whereσ is the gas velocity linewidth,κ is the epicyclic frequency,G is the gravitational constant, andΣ is the gas surface density.”

“A radially varying velocity dispersion is chosen that ensures the stellar distribution is Toomre stable (Toomre 1964) at all radii.”

“For this reason we consider the Toomre parameter (Toomre 1964) describing the stability of the disc: Q = κcs πΣG (4) with the epicyclic frequency κ, sound speed cs, surface density Σ, and gravitational constant G. Gravitational instability sets in when Q 1.”

“The model assumes a thin ‘α-disc’ that becomes self-gravitating beyond the radius where the Toomre Q parameter (Toomre 1964) equals unity.”

“At these satellite sites, gas-rich proto-galactic disks could form that do not fragment or cool but rather go dynamically unstable (Q-unstable, see Toomre 1964), leading to rapid runaway accretion of gas to the center and to the formation of a massive DCBH seed.”

“(6) Relating σ T back to the disc surface density with a modified Toomre-Q parameter (Toomre 1964), Q = κ √ c2s + σ 2T πG disc , (7) where κ is the epicyclic frequency ∼√2 for galactic potentials with flat rotation curves, cs is the sound speed, and disc ≈ gas + is the disc surface density.”

“Boss (2002b) presented results for models with initial To ¼ 40 and 50 K (as in models eb and ab), leading to initial Toomre (1964) Q stability values of Qmin ¼ 1:3 and 1.5, respectively.”

“However, these observations also show strong vertical velocity dispersions (e.g. Barro et al. 2014), verging on or exceeding the Toomre stability criterion (Toomre 1964), i.e. if these are discs they are highly unstable and unlike any type of disc seen locally.”

“Some models derive the KSR and its normalization by assuming that stellar feedback regulates ISM turbulence so as to maintain vertical and/or Toomre (1964) equilibrium within gaseous disks (Ostriker & Shetty 2011; Faucher-Giguère et al. 2013; Hayward & Hopkins 2017).”

“It has been known for a long time (Safronov 1960; Toomre 1964) that massive rotating disks are prone to gravitational instability (GI), which sets in when the destabilizing effect of the selfgravity in the disk exceeds the combined restoring action of the pressure and Coriolis forces.”

“Disc stability can be parametrized in terms of the Toomre Q parameter (Toomre 1964) given by Q = σκ πGΣ , (1) where σ is the 1D velocity dispersion, κ is the epicyclic frequency (which is related to the circular frequency κ ∝ Ω), and Σ is the mass surface density.”

“The two dominant processes for angular momentum transport in disks, viscous torques due to turbulence (Balbus & Hawley, 1994) and gravitational instability (GI) (Toomre, 1964; Laughlin & Bodenheimer, 1994), both produce variability in the accretion flow.”