On the gravitational stability of a disk of stars

“We simulate a selection of 25modelswhich span awide range in Toomre Q (Toomre 1964) and Goldreich-Tremaine X (Goldreich & Tremaine 1978, 1979) parameters and find that a bar develops in virtually all of the cases.”

“(2) The well known stability criterion for axisymmetric disturbances (Toomre 1964) Q = csκ πGΣ > 1 (3) identifies the region of parameter space where the RHS of Eq.”

“There has been debate about whether this critical surface den- c© 0000 RAS sity is best described in terms of a Toomre stability criterion (Toomre 1964) or a constant critical density, and indeed about the physical origin of this critical density (Schaye 2004; Leroy et al. 2008).”

“The disk stability is parameterized by the Q-value (Toomre 1964).”

“We recall that Q ¼ vs= G for a thin disk in Keplerian rotation (Toomre 1964), where is the angular velocity, is the disk surface density, vs is the sound speed (which is proportional to T 1=2, where T is the temperature), and G is the gravitational constant.”

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

“(5) There is strong observational evidence that star-forming discs self-regulate to sustain a Toomre (1964) parameter Q = κcT πGΣg = 2σcT πGΣgr ∼ 1, (6) where κ ≡ √ 4Ω2 + dΩ2/d ln r = 2σ/r is the epicyclic frequency (e.g., Quirk 1972; Kennicutt 1989; Martin & Kennicutt 2001).”

“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).”

“A rotating, symmetric and thin gas disk is unstable to gravitational fragmentation if the Toomre Q-parameter (Toomre 1964) is below a critical value Qcrit.”

“The gravitational stability of discs is normally characterized by the Toomre parameter (Toomre 1964), Q(r) = csκ πGΣ , (2) where cs is the sound speed, κ = √ 2 (Vrot/r) (1 + c© 0000 RAS, MNRAS 000, 000–000 d ln Vrot/d ln r) 1/2 (Binney & Tremaine 1987; Oh & Haiman 2002) is the epicyclic frequency,…”

“We use the Toomre parameter to estimate the stability of the HD 135344B disk against self-gravity which is given by (Toomre 1964) Q(r) = k(r)cs(r) πGΣ(r) , (7) where k(r) is the epicycle frequency, cs(r) the sound speed, G the gravitational constant, Σ(r) the total surface density, and Q(r) .”

“The Toomre stability parameter Q, a parameter for star formation regulation developed by Toomre (1964), is expressed as Q ≡ vsκ πGΣ (25) for a gaseous disk, where vs is the sound speed, κ is the epicyclic frequency, and Σ is the surface density.”

“The physical framework for this equivalence relies on the fact that normal galaxy disks, viewed as dynamically relaxed stable systems, fulfill the Toomre stability criterium (Toomre 1964).”

“In a thin, differentially-rotating disk composed of pure gas (Safronov 1960) or pure stars (Toomre 1964), there is a critical column density Σc above which the disk is unstable to ring-like perturbations in the radial direction.”

“(7) QT = cΩ/(πGΣ) is the standard Toomre (1964) gravitational stability parameter, but unlike standard GI we have no requirement that QT 1.”

“This fast rotation is sufficient to trigger the gravito-rotational instability (Toomre 1964; Bate 1998; Saigo & Tomisaka 2006; Saigo et al. 2008; Tsukamoto & Machida 2011), and the disk becomes nonaxisymmetric in the late phase (Figure 9).”

“The disk is linearly stable for Q > 1 and linearly unstable for Q < 1 (Toomre 1964; Goldreich & Lynden-Bell 1965).”

“The gravitational stability parameterQ (Toomre 1964) is defined as Q = csκΩ πGΣ , (16) whereκΩ is the epicyclic frequency.”

“Following Toomre (1964) we take Σtoomre = ασκ πG (D1) as defining the threshold for stability in a thin rotating gas disk, whereby the ToomreQ=1 when Σtoomre = Σgas.”

“KDM12 also present an alternative prescription for calculating the free-fall time, that assumes that star formation is regulated by the dynamical stability of a continuous disc of gas, which globally should have a Toomre Q parameter (Toomre 1964) of≈1.”

“Our derived value of the surface density exponent α=-1, as opposed to values of 0 or -2, provides disc stability at all radii according to the Toomre stability criterion (Toomre 1964).”

“We took Qg = csκ/πG and Qs = σ Rκ/3.36G where cs is the sound speed, κ is the epicyclic frequency, the surface density and σ R the radial velocity dispersion (Toomre 1964; Goldreich & Lynden-Bell 1965; Binney & Tremaine 1987).”

“The disc self-gravity can be neglected as long as the Toomre parameter (Toomre 1964) is high, Q = hM∗/(πR 2Σ) 1, which holds in our unperturbed discs.”

“…azimuthal averages of the H I column density indicate that the gas density in dwarf irregulars is signiÐcantly lower (by factors of 2È3) than the Toomre (1964) instability criterion throughout the star-forming disk (e.g., & Plummer Zee et al.Hunter 1996 ; van 1997b ; Hunter, Elmegreen, & Baker…”

“A gaseous or stellar disk stable to all axisymmetric perturbations requires the Toomre’s parameters Q > 1 (Safronov 1960; Toomre 1964).”

“This interplay is summarized by the Toomre parameter Q = Lmax/Lmin which is calculated from the velocity dispersion, rotation curve, and mass distribution (Toomre 1964).”

“They could be produced also due to gravitational instabilities in the stars due to interactions (e.g. Toomre 1964).”

“The ratio between tangential and radial velocity dispersion was taken from the epicyclic theory (e.g., Toomre 1964).”

“Gravitational instability to axisymmetric perturbations sets in when the sound speed cs, the rotation frequency , and the surface density satisfy Q cs G < Qcrit ’ 1 ð1Þ (Toomre 1964; Goldreich & Lynden-Bell 1965).”

“…nonaxisymmetric instabilities, akin to the unstable modes that are present in Maclaurin spheriods (see discussion, below), and the Toomre Q parameter (Toomre 1964) is likely to be very small, making the rotationally flattened configuration susceptible to the same types of instabilities observed in…”

“Clump formation and disruption Defining aQ parameter for disk instability (Toomre 1964) of Q ≡ κ √ σ2g + c2s πGΣg (3) whereσg is the local gas velocity dispersion,cs the gas sound speed, and κ = √ 3 √ GMtot(< R) R / R (4) (e.g. Dekel et al. 2009b), we find that the clumps are local minima with Q ≪…”

“Nayakshin (2006) suggested that the luminosity of the young stars which accrete gas from within a star-forming disc is sufficient to heat up the disc, increasing the Toomre (1964) Q-parameter of the disc above unity and hence making the disc stable to further fragmentation.”

“The disks initially have a minimum value of the Toomre (1964)Q gravitational stability parameter of or 1.3, respectively, forQ ∼ 1.5min or 40 K. Note that if K, then the disk massT p 50 T p 280 0 could be lowered below 0.091 and the disk would still haveM, or 1.3.”

“…gas/dust mass ratio for nearby galaxies (Mgas/Mdust = 91+60−36; Sandstrom 53 Usually ascribed to a seminal paper on the stability of stellar disks (Toomre 1964), which contains no mention of Q. et al. 2013), we would determine gas masses for W and T of 19.0 and 11.5×109 M , yielding…”

“(i) The disc must be gravitationally unstable, i.e. massive enough so that gravity can overcome thermal pressure and centrifugal support (Toomre 1964): Q(R) = c(R) κ(R) π G (R) 1, (1) where c is the sound speed, κ the epicyclic frequency, and the surface density.”

“According to Toomre’s criterion (Toomre 1964), the stability of a galactic disk depends on the stellar velocity dispersion σ, the epicyclic frequency k and the local surface density Σ, where the parameter Q = σk/3.36GΣ > 1.”

“…is dependent upon the galaxy rotation curve (Kennicutt 1989; Martin & Kennicutt 2001) and relate c to the surface density, Q, which parameterizes the gravitational instability associated with the Coriolis force (Toomre 1964), Q ¼ c G ; ð2Þ where c is the sound speed and is the epicyclic frequency.”

“(28) Here Qg is a Toomre’s Q parameter for gas, while Qs is different from the Toomre’s Q parameter for the collisionless system, which is given by κσs/(3.36GΣs) (Toomre 1964).”

“DIs are mod- eled using the Toomre (1964) instability criterion.”

“Figure 11 shows Toomre’s Q parameter (Toomre 1964) for the final states of galaxies L1–L3.”

“To confirm this, we have computed the value of the Toomre (1964) Q factor in annular rings in the fiducial model at various times.”

“An additional way to assess the local stability of a circumstellar disk is by evaluating Toomre’s parameter Q (Toomre, 1964).”

“Theory predicts that this mass-scale should correspond to the maximum mass able to overcome the galactic differential rotation and undergo collapse (i.e. the Toomre 1964 mass, see Kruijssen 2014 and references therein).”

“More complex, more realistic treatments of the ISM as a multiphase medium (e.g. Springel & Hernquist 2003a) now make it possible to construct disks with much larger gas fractions that do not violate the Toomre (1964) stability criterion (see, e.g., Fig.6 of Springel et al. 2005b).”

“…spiral galaxies is regulated by the onset of gravitational instabilities in their gas disks, Kennicutt applied the criterion for disk stability from Toomre (1964) to these systems and derived the following threshold density for star formation: Σcrit = α κc 3.36G · (5) κ is the epicyclic frequency,…”

“…Institut Potsdam, An der Sternwarte 16, 14482 Potsdam, Germany, rklessen@aip.de 1 2 (& 1 kpc) from various instabilities (Parker 1966; Toomre 1964; Elmegreen 1991), supernova energy input (e.g., McCray & Snow 1979; MacLow 2002), and stellar winds (Norman & Silk 1980), and continues…”

“6 ˜ G − 1 (Safronov 1960; Toomre 1964).”

“…a rather steep dependence of the star formation rate on the gas surface density, and since surface density and velocity dispersion are linked via Toomre (1964)’s Q, on the velocity dispersion.6 Consequently, velocity dispersions of ∼50 km s−1 can be driven only by star formation rates 100 M…”

“We then find the radius beyond which the Toomre (1964) parameter falls below unity.”

“For a snapshot in time that represents the observation of a high-z galaxy, this equilibrium situation is captured through the Toomre-Q parameter (Toomre 1964), where generally Q Qcrit≈1 indicates gravitational instability.”

“The thickness of the discs is set such that Toomre’s stability criterion (Toomre 1964) is satisfied, which requires, for a stellar disc, Qstar(r ) = σRk 3.36G s > 1, (5) where σ R is the radial velocity dispersion, k is the local epicyclic frequency and s is the stellar surface density.”

“For Keplerian discs, κ approximately equals the angular frequency Ω. Toomre (1964) showed that, for an infinitesimally thin disc to fragment, Q .”

“Requiring tG < 1/x(l) (Toomre, 1964; Goldreich and Ward, 1973) is conservative here because eddies do not truly ‘‘rotate” (many times) with timescale 1/ x(l); rather, 1/x(l) is their existence lifetime before bifurcating.”

“The discs of the two simulations haveQ = 1.4 and 1, respectively, whereQ is the stability parameter introduced by Toomre (1964).”

“In addition to the rotational velocity caused by the gravitational field of the disk, bulge, and dark halo components, the initial radial and azimuthal velocity dispersions are assigned to the disk component according to the epicyclic theory with Toomre’s parameter Q (Toomre 1964).”

“Since such a massive disk is gravitationally unstable, fragmentation tends to occur (Toomre 1964).”

“Using the Toomre (1964) stability model, surface density of the gas in galaxies and a flat exponential disk, Obreschkow et al. (2016) found that the atomic gas fraction in galaxies is∝ (jbar/Mbar)1.12.”

“…crit, is given by crit = κ σg Qcrit πG , (B1) C© 2011 The Authors, MNRAS 416, 1566–1584 Monthly Notices of the Royal Astronomical Society C© 2011 RAS where Qcrit is a dimensionless constant ∼1 (Toomre 1964), σ g is the velocity dispersion of the gas and κ is the epicyclic frequency of the disc.”

“According to the standard Toomre instability analysis (Toomre 1964), a rotating disc becomes unstable to local gravitational collapse if its surface density is sufficiently high for its self-gravity to overcome the forces induced by rotation and velocity dispersion that resist the collapse.”

“Implications for Numerical Simulations of Multiple Star Formation Thermal evolution plays a critical role in determining the onset and outcome of gravitational stability in disks (Toomre 1964; Cassen et al. 1981; Tomley et al. 1991, 1994; Pickett et al. 1998, 1999, 2000, 2003; Gammie 2001).”

“Toomre (1964) showed that a rotating disc becomes unstable to local gravitational collapse once the surface density of gas and “cold” stars (Σ) becomes large enough for self-gravity to overcome the stabilizing effects of pressure (represented here by a velocity dispersion, σ) and centrifugal forces…”

“The runs lasted for 10 planet’s orbital periods, which was long enough to get stationary torques for the largest values of Q, but short enough to avoid a significant growth of nonaxisymmetric perturbations for the lowest values of Q, probably due to swing amplification (Toomre 1964).”

“Briefly, the argument is based on the assumption that clumps form in marginally stable discs described by a Toomre (1964) parameter Q where Q = κσt(R) πGΣ0 ≈ 1, (10) for epicyclic frequency κ ≈ √ 2v/r and velocity v at galactocentric radius r. Hopkins et al. (2012) argued that star-forming galaxies…”

“Kennicutt (1989) used the disk instability relation of Toomre (1964) to show that, for spirals, there is a threshold density for star formation.”

“…mass of the SPH particles scales as u(R) ∝ R−2/3, where the value of the temperature at the outer radius of the disc has been set in order to have the Toomre parameter (Toomre 1964) Q ≥ 3 everywhere, preventing the fragmentation of the disc (the average value of Q over the disc surface is ≈10).”

“…Σgas is the total (atomic plus molecular) gas surface density in units of M⊙ pc −2, fGMC is the fraction of gas in giant molecular clouds, φP̄ is the ratio of the pressure inside a molecular cloud to the pressure at its surface, and Q is the stability parameter as defined by Toomre (1964).”

“Early work on stability and fragmentation focused on smooth media dominated by thermal pressure and/or rotation (e.g. Jeans 1902; Toomre 1964; Goldreich & Lynden-Bell 1965; Lin et al. 1969).”

“10b, Toomre 1964), plasma βp (Fig.”

“…time so that turbulent support dissipates in a crossing time, which is easily satisfied here) will fragment at the Toomre scale, RT ∼ σ2/πGΣ or MT ∼ σ4/πG2 Σ (Toomre 1964); for disks which equilibrate at marginal stability (Toomre Q∼ 1), this is just RT ∼ fgas Rdisk (MT ∼ f 3gas Mdisk).2 In the…”

“Stability of disks against self-gravity requires where (Toomre 1964).”

“The stability of the rings can be crudely estimated with Toomre’s Q parameter (Toomre 1964), where Q = csΩ/πGΣ.”

“Varying T0 has the effect of varying the minimum value of the Toomre (1964) Q stability parameter in each disk (Fig.”

“…the global characteristics of star formation have been suggested: cloud-cloud collisions (e.g., Larson 1988; Tan 2000) and the growth of gravitational perturbations (Toomre 1964), pressure and turbulence regulated media (e.g., Silk 1997, 2001; Elmegreen 2002; Mac Low & Klessen 2004) and others.”

“For example, first consider the Toomre stability criterion, Q, (Toomre 1964).”