On the gravitational stability of a disk of stars

“An alternative suggestion (Kuiper 1951; Boss 1998) is that gas giant planets may form via direct gravitational collapse in discs that aresufficiently massive so as to sustain a gravitational instability (Toomre 1964).”

“Fourth, we control the strength of the particle self-gravity by using the dimensionless parameter Here, Q is the Toomre parameter (Toomre 1964) for the gas disk.”

“The stability of the disc is described in terms of the Toomre parameterQ (Toomre 1964).”

“• Instabilities: A two-component Toomre Q value (Toomre 1964; Romeo & Wiegert 2011) for each annulus is routinely calculated.”

“This idea was originally introduced by Safronov (1960) and further quantified by Toomre (1964) for a disk of stars, and has since been used in various applications ranging from planet formation to galaxy dynamics.”

“Studies of Toomre's instability parameter Q (Toomre 1964) show that the Galaxy is (marginally) stable at all radii (Lewis & Freeman 1989; Fuchs & von Linden 1998).”

“1). c© 2013 RAS, MNRAS 000, 1–6 The outer parts of the dead zone become self gravitating when the Toomre (1964) parameter, Q = csΩ/πGΣ, is below its critical value of Qcrit = 2.”

“The Toomre parameter Q (Toomre 1964) can be used as a measure of the level of gravitational instability within a disc: Q = csκ πGΣ (1) where cs is the sound speed, κ is the epicyclic frequency, G is the gravitational constant, and Σ is the disc surface density.”

“Thus, the optical thickness τT of the torus plays an important role as an optically thin self-gravitating torus would likely collapse into a thin disk with subsequent star formation (Toomre 1964).”

“Yet another effect to consider is the scale height of the disk, which tends to stabilize the disk (Toomre 1964; Jog & Solomon 1984).”

“This cutoff was interpreted in terms of the Toomre Q parameter (Toomre 1964), wherein galaxy disks are unable to form stars below a critical density for instability (see also Quirk 1972).”

“…= λ/∆v = Ω −1(λ/H)2(∆Σ/Σ)−1 (5) Since in a gravitationally unstable disc the scale length for density inhomogeneities is of order the most unstable wavelength, which, in marginally unstable discs is of orderH (Toomre 1964, Lodato 2007), we can setλ ∼ H and hence tmin = Ω −1 ( ∆Σ Σ )−1 (6)”

“To confirm the disk stability, we estimate the mass-weighted Toomre Q parameter (Toomre 1964), which is averaged over the entire disk R Rdisk as ò ò k p = S S S Q c G dS dS , 4disk disk s disk ( ) ( ) ( ) where cs is the sound speed, G is the gravitational constant, Σ is the surface density, and k…”

“Self-gravitational effects will not affect the disk structure as the surface density in our models is always orders of magnitude smaller than any critical value (see Toomre 1964).”

“…* Fellow of the International Max Planck Research School for Astronomy and Cosmic Physics at the University of Heidelberg (IMPRS-HD) by Toomre (1964) as: Q = csΩ πGΣ , (1) where Ω = √ GM/R3 is the orbital Keplerian frequency of the disk, stabilizing the disk to density perturbations…”

“To configure the final parameter, σR0, we aim on getting an initial Toomre Q value (Toomre 1964) at the reference radius (2.5Rd), Q0, of ∼ 1.2.”

“…∼ 10 km s−1 (e.g. Tamburro et al. 2009; Ianjamasimanana et al. 2012; Caldú-Primo et al. 2013), and the velocity dispersions are such that these galaxies’ Toomre Q parameters (Toomre 1964) are about unity, indicating that the galactic discs are marginally gravitationally stable (Leroy et al. 2008).”

“There is a long history of theoretical studies of the dynamical effects of disc non-axisymmetries in two dimensions in the Galactic plane, dating back to the seminal works of, e.g., Lin & Shu (1964) and Toomre (1964) on spiral arms.”

“To examine whether gravitational instability may be important in driving accretion or fragmentation, we examine the value of the Toomre (1964) instability parameter Q = κcs/πGΣ, (8) where G is Newton’s gravitation constant, Σ is the disk surface density, κ is the epicyclic frequency, and cs is the…”

“The Toomre (1964) characteristic wavelength is defined as λ∗ = 4πGΣ/Ω 2 r = 4h/Q.”

“(3) This quantity was first derived by Toomre (1964) for a thin disc of stars (where σ → σR, i.e. the stellar radial velocity dispersion, and the denominator has π replaced by 3.36), and we hence refer to it as Toomre’s Q.”

“Disks exhibit gravitational instability (GI) when their Toomre’s Q value (Toomre 1964) fulfills the following condition Q ≡ csκep πGΣgas .”

“We checked the Toomre (1964) stability parameter Q = cs πG (7) for each of the models that we will discuss, finding that they are all gravitationally stable (Q > 1) at all radii in the disc.”

“This instability occurs when the Toomre parameter Q = √ 2σΩ/πGΣdisk < 1 [79, 80], where Ω = vc/Rdisk is the angular frequency.”

“This dead zone may become self gravitating if sufficient mass builds up such that te Toomre (1964) parameter,Q, is less than a critical value that we take to be 2.”

“Gaseous discs in Keplerian rotation become unstable to self-gravity when the Toomre (1964) Q parameter is less than some critical value of order unity.”

“This can be interpreted as spiral structure preferentially forming in Toomre unstable disks (Toomre Q stability criterion; Toomre 1964).”

“In this context it is also interesting to look at Toomre’s stability parameter for axisymmetric stability of stellar disks (Toomre 1964), QToomre ≡ σR κ 3.36GΣ > 1, (2) whereσR is the radial velocity dispersion,Σ the surface density, and κ2 = 3 R ∂Φ ∂R + ∂2Φ ∂R2 (3) is the epicycle frequency.”

“…dynamical disk, yield a column density∼ 1021 atoms cm−2 – sufficiently above the “star-formation threshold” of∼ 1020 cm−2 (Martin & Kennicutt 2001), or above the Toomre criterion for gravitational instability in a rotating disk (Toomre 1964), to form stars (Jimenez et al. 1997; Verde et al. 2002).”

“…perturbations if cs < cT = πGσ0 κ , (30) or Toomre’s Q ≡ cs/cT-value, Q = csκ πGσ0 < 1, (31) respectively, where cT is the Safronov–Toomre (Safronov 1960, 1980; Toomre 1964; Lin et al. 1969; Rohlfs 1977; Binney & Tremaine 2008) critical sound speed to suppress the instability of only m = 0…”

“If we assume that the molecular gas in the HCG galaxies are mainly rotationally supported, we can use the Toomre criterion (Q> 1; Toomre 1964), which describes the balance between rotation and turbulence, and gravitational binding energy, to determine the required energy budget necessary to…”

“…broadly argued for to explain velocity dispersions in Milky Way-like galaxies: (i) Marginal Gas Stability (Q̃gas = 1): Gas drives itself to marginal stability against gravitational fragmentation and collapse (per the earlier discussion in § 1), as measured by the Toomre-Q parameter (Toomre 1964).”

“…Q and Ω is almost linear, because the kpc-scale velocity dispersion and surface density of the gas disc are roughly constant with galactocentric radius (see Section 3.3), leaving the gravitational stability to vary with the degree of centrifugal support (Toomre 1964), which is proportional to Ω.”

“…impose that a gaseous disc annulus becomes eligible for star formation only once its surface densities surpass a critical threshold crit given by the Toomre (1964) criterion: crit = σgas κ(R) 3.36 G Q , (26) where κ(R) is the epicycle frequency (see vdB01 for details), Q ∼ 1.5 is a constant and σ…”

“Once a certain threshold density is reached, massive star formation is initiated since the velocity dispersion no longer can neutralize the self-gravity of the gaseous disk (Toomre 1964; Kennicutt 1989).”

“But a self-gravitating disc is expected to be gravitationally unstable as soon as some criterion is fulfilled (Goldreich & Lyndell-Bell 1965; see also Toomre 1964), usually this is Q = Ω 2 πGρ ≤ 1, where Ω is the Keplerian angular velocity and ρ is the local mass density.”

“Conversely, in all our simulations with a multiphase ISM model, gravitational disc instabilities (Toomre 1964) are promoted as a consequence of a better description of density and temperature gradients (Scannapieco et al. 2006).”

“This is quantified by the parameter Q0 of Toomre (1964).”

“In contrast, Galactic rotation provides a stabilizing effect (Toomre 1964).”

“To check this, we made use of the Toomre criterion (Toomre 1964) that describes when a disk becomes unstable through gravitational collapse.”

“The velocity dispersion of the gas is also an important parameter for star formation laws based on the Toomre Q criterion (Toomre 1964; Kennicutt 1989).”

“To check if the Toomre criterion for disk fragmentation is fulfilled in our model, we calculate the Toomre Q-parameter and search for the disk regions where Q 1.0 (Toomre 1964).”

“The “classic” single-component Q parameters for purely stellar or purely gaseous disks (Toomre 1964; Goldreich & LyndenBell 1965) are defined such that local axisymmetric perturbations in infinitely thin, homogeneous disks are gravitationally unstable when Q .”

“The Toomre Q-parameter describes the stability of the disk against linear perturbations and is given by Q ¼ turb / G , where is the epicyclic frequency and is the surface density of the gas, both determined locally (Toomre 1964; Goldreich & Lynden-Bell 1965).”

“…supermassive black hole (Taniguchi & Murayama 1998), gas in the torus rotates in a Keplerian manner and the gravitational stability parameter Q (Toomre 1964) is described as Q ¼ = G gas, where , , and gas are the gas velocity dispersion, angular velocity, and gas surface density, respectively.”

“The horizontal wavelengths a comparable to the critical wavelength for axisymmetric stabilities(Toomre, 1964) 2 ( )3 rings are significantly enhanced by multiple scattering relative to that of the C ring.”

“To find a more physical maximum disk mass, we calculate the Toomre Q parameter Toomre (1964) using the surface density and midplane temperature profiles from our models.”

“The Toomre parameter in the stars, Qs (Toomre 1964), characterises the stability of the disc to local collapse and is given by Qs = κσR 3.36GΣ0 (5) where κ is the epicycle frequency, σR the radial velocity dispersion and Σ0 the disc surface density.”

“If the progenitors of BCDs are compact Irrs, the high circular-velocity gradient would imply high values of the critical surface-density threshold for gravitational instabilities Σcrit (Toomre 1964; Kennicutt 1989) and make the gaseous disk relatively stable against large-scale perturbations.”

“The criterion that dictates disk stability is given by Q = csκ πGΣ , (21) where cs is the thermal sound speed, κ is the epicylic frequency and Σ is the surface density (Toomre, 1964).”

“The disc was initially set as stable to self-gravitating spirals, such that the Toomre parameter (Toomre 1964),Q is Q = csκ πGΣ & 2 (1) data from /Users/ch427/work/kinematics_gi/production_data/d30/13CO/data_13C16O Bottom from…”

“These in turn could be explained by the presence of a large-scale gravitational stability criterion (Safronov 1960; Toomre 1964), though other models have also been proposed to account for the thresholds (Elmegreen & Parravano 1994; Schaye 2004).”

“Axisymmetric Instability Consider Ðrst the noncontroversial case of axisymmetric instability (Toomre 1964 ; see also Schmitz 1986, 1988, 1990 ; Lemos, Kalnajs, & Lynden-Bell 1991).”

“The regions active to MHD turbulence are described using the α-prescription described in Section 3.2.2, but in addition the model also takes into account the effects of self-gravity using a prescription dependent on the Toomre Q parameter (Toomre 1964): Q = cs πG .”

“The Toomre (1964) model of a differentially rotating thin disk appears to explain star formation in the inner disks of spirals (Kennicutt 1989).”

“…cluster mass follows from the maximum giant molecular cloud (GMC) mass as Mc = ǫsfΓMGMC, with ǫsf the star formation efficiency (SFE) and MGMC the maximum GMC mass, which is proposed to be set by the Toomre (1964) mass, i.e. the largest self-gravitating scale in a differentially rotating disc.”

“They further argue that for the thin stellar disc to be stable against axisymmetric perturbations, it should satisfy the Toomre (1964) criterion: σR > 3.36GΣ/k, where σR is the radial component of the dispersion, G is the gravitational constant, Σ is the surface mass density of the disc, and k is…”

“We also show the Toomre parameter QT=κ 〈σ〉r (3.36 G Σ (R))−1 (Toomre 1964) calculated with the azimuthally averaged surface mass density profile Σ (R) and the radial velocity dispersion 〈σ〉r from the disc particles and the epicycle frequency κ from the total circular velocity.”

“…understanding of binary formation invokes two formation channels; (i) fragmentation of the proto-stellar core, systems ≥100 au (Bodenheimer & Burkert 2001) (ii) via gravitational instability and fragmentation of the proto-stellar accretion disc of the primary star, systems ≤100 au (Toomre 1964).”

“The use of a such a threshold is motivated both theoretically (e.g. Toomre 1964; Kennicutt 1989; Schaye 2004) and observationally (e.g. Martin & Kennicutt 2001).”

“A gaseous disc with sound speed cs, surface density Σ, and epicyclic frequency κ is described as self-gravitating if the Toomre (1964) Q parameter, Q = csκ πGΣ , (1) is of order unity.”

“Since the maximum value of the mass of the discs considered is relatively high and since this can lead to the occurrence of gravitational instabilities, our model checks, ac- cording to the Toomre criterion (Toomre 1964) given by Q = csΩk πGΣg , (40) the stability of our discs.”

“At later stages, hot stars increase the velocity dispersion of the disks (and hence the Toomre Q-parameter, Toomre (1964) and Genzel et al. (2011)) to stabilize them.”