On the Gravitational Stability of a Disk of Stars

“For a thin, rotating disk, rotation stabilizes self-gravitational contraction for wavelengths greater than the Toomre length λT ≡ 4π2GΣ/κ2, where κ is the epicyclic frequency (Toomre, 1964).”

“For example, the dynamical stability of a pure gas disc to axisymmetric perturbations is determined by the condition (Toomre 1964) Q ≡ csκ πG > 1, (26) where cs is the sound speed, κ is the epicyclic frequency and is the gas surface density.”

“Following Toomre (1964), the condition for instability in a thin gas disk is Q gas = σ g κ π G Σ gas 1 .”

“This disk may become gravitationally unstable if the surface density exceeds a critical value given by the epicyclic frequency and the sound speed (Safranov, 1960; Toomre 1964) and fragment into multiple objects (as summarized by Bodenheimer et al. 2000).”

“Toomre (1964) showed that violent instability sets in when Q ≡ σr/σcrit → 1, where σcrit = 3.36Gµ/κ. Gas inflow increases both the density and the epicyclic frequency, but density wins and σcrit increases as the evolution proceeds.”

“The clump formation can be understood analytically in the context of the Toomre (1964) Q ≡ csΩ/πGΣ, where cs is the sound speed, Ω is the angular speed, and Σ is the local surface density.”

“&(Safronov 1960 ; Toomre 1964 ; Goldreich Lynden-Bell for a recent review of the criteria applied1965 ; to gravitationally coupled stars and gas in a disk, see Jog If Q\ 1, the structure is unstable, and large, massive1996). star clusters may form.”

“…high during the initial stages of formation (§3), the Toomre Q parameter, Q(R) = cΩ/πGΣ where c is the sound speed and Ω the orbital angular velocity (Toomre 1964), is generally much greater than unity for these Class II YSO disks implying that they are gravitationally stable at all radii…”

“Stars form efficiently only in regions where the surface mass density exceeds a critical value which is plausibly related to the Toomre (1964) threshold for local instability of a rotationally supported disk.”

“The Toomre parameter Q is defined as (Toomre 1964) Q g G g ¼ ffiffiffi 2 p g G g ; ð35Þ where ffiffiffi 2 p is the epicyclic frequency, ¼ vrot/r is the angular velocity, and r is the galactocentric radius.”

“Instability Criterion According to the standard Toomre instability analysis (Toomre 1964; Binney & Tremaine 2008, Chapter 6), a thin rotating gaseous disk becomes unstable to axisymmetric modes once the local gravity overcomes both differential rotation and pressure due to turbulence or thermal…”

“Instability in a Keplerian disk sets in where the sound speed c s , the rotation frequency Ω, and the surface density Σ satisfy Q ≡ c s Ω πGΣ < Q crit ≃ 1 (1) (Toomre 1964; Goldreich & Lynden-Bell 1965).”

“The gravitational stability parameter (Toomre 1964) for our disk models is Q = csΩK πGΣg ≃ 40f−1disk ( a 1AU )−1/4 .”

“In a marginally stable galactic disc, the Toomre (1964) Q-parameter, Q = κσ/(πG gas), is near unity.”

“Moreover, the progenitor disks were constructed to have reasonable Toomre (1964) stability parametersQ and realistic star formation rates; this required careful tuning of the prescriptions for star formation and eergy feedback.”

“They were all constructed with a Toomre (1964) “stability” parameter Q = 1.5 and run in isolation for 2 Gyr before being set into orbit in the cluster.”

“Gravitational instability is a necessary condition for efficient star formation in a galactic disk (e.g., Elmegreen 2002; Li et al. 2005; Kawata et al. 2007) The instability of a single-component, rotating disk at radius r is characterized by the Toomre Q parameter (Toomre 1964): Q = κσ ǫGΣ .”

“We assume that star formation in the disk is governed by local gravitational instability as described by Toomre’s Qparameter (Toomre 1964).”

“As proposed by Quirk (1972), such a threshold may be related to the parameter Qg for gravitational instability in the gas layer (Safranov 1960; Toomre 1964): Qg ≡ Σcrit Σgas , where Σcrit = κcg πG .”

“The stability ofthe disk can be described in terms of the Toomre parameter (Toomre 1964) Q = csΩk πG Σg .”

“The giant star-forming clumps in the observed z ∼ 2 SFGs reside in regions where the Toomre Q-parameter is at or below unity.”

“The existence of a surface density threshold for SF has long been explained in terms of the Toomre (1964) criterion for instability (e.g. Quirk 1972; Kennicutt 1989; Martin & Kennicutt 2001), usually under the assumption of a constant velocity dispersion.”

“Disk gravitational stability Other interesting physical property that can be calculated from a detailed disk model is the gravitational stability parameter introduced by Toomre (1964), which quantifies the fluid reaction against axisymmetric gravitational perturbations.”

“The velocity dispersion of solids determines the Toomre (1964) gravitational stability, or Q, parameter for thin disks.”

“…nor pressure can stabilize a thin gaseous differentially rotating disk if the Toomre Q parameter, Q(r) cs G g ; ð1Þ is less than unity (Safronov 1960; Toomre 1964; Goldreich & Lynden-Bell 1965; Binney & Tremaine 1987), where the effective sound speed cs, the epicyclic frequency , and the gas…”

“…only disk galaxies a prediction for the evolution of ionized gas velocity dispersions can be derived by rewriting the Toomre stability criterion (Toomre 1964) as vrot σ0 = a fgas(z)Qcrit , (7) where a = √ 2 for a disk with constant rotational velocity and Qcrit = 1.0 for a quasi-stable thin gas…”

“…threshold effects we calculated for each region the expected critical density using the relation of Kennicutt (1989), which is based on applying the Toomre (1964) gas stability criterion for an isothermal disk of gas clouds: Σc = α κc πG (9) where Σc is the critical (total) gas surface density for…”

“For example, an isothermal gas disc is expected to become unstable when Σg > csκ πG , (3) which is equivalent to Q < 1 in terms of the Toomre parameter (e.g., Toomre, 1964; Thompson et al., 2005; McKee & Ostriker, 2007).”

“Motivated by this, we adopt a functional form proposed by Gammie (1996), and allow α to spatially vary with the Toomre Q-parameter evaluated at each cell on the equator, α(Q(r)).”

“Here crit is the star formation threshold density, modelled as the critical surface density given by Toomre’s stability criterion (Toomre 1964; Kennicutt 1989).”

“Critical column densities are derived to parameterize instabilities or magnetic fields with respect to self-gravity (Toomre 1964, Mouschovias & Spitzer 1976).”

“(10) 8 Kratter & Lodato From this requirement alone we thus recover the usual stability parameter for GI (Toomre 1964).”

“The gaseous progenitor disks in model GAn are not stable against Toomre (1964) instability as the sound speed of the weakly pressurized gas cannot support the massive gas disk against self-gravity.”

“Support for this idea comes from a stability analysis of our model disks using the Toomre criteria for the stability of a rotating stellar disk (Q4 for stability where isp Vr i/3.36G&[ 1 [Toomre 1964], p Vrthe radial velocity dispersion of the stars, and i is the epicyclic frequency).”

“The linear stability of an initially axisymmetric thin disk, rotating with angular velocity Ω(r), is quantified by the Toomre Q parameter (Toomre 1964), Q ≡ csκ πGΣ , (10) where κ, the epicyclic frequency, is the frequency at which a radially displaced particle oscillates.”

“Theoretically, such a threshold could be imposed by the Toomre (1964) stability criterion, which predicts the conditions under which disks are stable to collapse and star formation (Schaye 2004).”

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

“All new galactic fuel needs to pass through a galaxy’s halo, where the gravitational potential will draw the gas to the disk and help maintain spiral structure by keeping the Toomre Q parameter low (Toomre, 1990, 1964).”

“Regions in the disk are unstable to axisymmetric gravitational perturbations if the Toomre parameter Q = csΩ πGΣ (20) (Toomre 1964) is less than unity.”

“…followed the method described by Hernquist (1993), with the following small differences: (i) For the radial velocity dispersion Hernquist (1993) adopts σ2R(R) = Cexp(−R/h), where the constant C is normalized so that the Toomre Q parameter (Toomre 1964) has a prescribed value at a given radius.”

“The idea that gravitational instability might determine the critical gas density for star formation was introduced by Spitzer (1968) and Quirk (1972) soon after local stability criteria for differentially rotating disks were developed (Toomre 1964; Goldreich & Lynden-Bell 1965).”

“The condition of the gravitational instability is 1 > Q = csΩK/πGΣg ∝ M 1/2 ∗ M 1/2 ∗ /Σg = M 1−β ∗ , and its radial wavelength λ = 2π2GΣg/Ω 2 K ∝ M β−1 ∗ (Toomre 1964) where Σg ∝ M β ∗ .”

“If the disk reaches marginal stability in terms of Toomre’s Q parameter (Toomre 1964), the relationship between /vc and the gas fraction fgas can be written as =vc fgasQ =a; ð6Þ where a ¼ 2 1 is a dimensionless parameter depending on the assumed distribution of gas and gravitational potential.”

“Our finding that the Milky Way’s disk has a short scale length raises the question of whether such a massive disk is stable to axisymmetric perturbations, i.e., whether Toomre’s Q > 1 (Toomre 1964).”

“Fragmentation of marginally Toomre-unstable disks (Toomre 1964; Goldreich & Lynden-Bell 1965) is thought to produce a mass distribution weighted toward larger and not smaller masses.”

“The fiducial run 0 has σ = 9 km s−1, fG = 0.5 and H/D = 0.5, the resulting Toomre X-parameter (Toomre 1964) is shown as a function of radius on Figure 2; this parameter is defined by: X(r) = rκ(r)2 2πGmµ(r) (1) where κ and µ are the epicyclic frequency and surface density, respectively.”

“The setup of the stellar disk is complete once the Toomre parameter, Q(R), is assigned (Toomre 1964).”

“We have verified that our large disc models do not become gravitationally unstable anywhere in the disc (Toomre 1964), see for example Fig.”

“…convective instability (Cameron & Pine 1973; Lin & Papaloizou 1980; Ruden, Papaloizou & Lin 1988; Ryu & Goodman 1992), gravitational instability (Toomre 1964; Lin & Pringle 1987; Papaloizou & Savonije 1991), the global and subcritical baroclinic instabilities (SBI; Klahr & Bodenheimer 2003;…”

“Relatively massive protostellar disks around young stars (>0:1M ) also have a proper mechanism for transporting angular momentum via self-gravity (Toomre 1964; Laughlin & Bodenheimer 1994).”

“Equation (8) encapsulates the physics of our model, but to compare with observed planets, we re-express this result in terms more easily related to the expected population of protoplanetary disks from which planets form—Toomre’s Q parameter (Safronov 1960; Toomre 1964) and the disk aspect ratio H/r.”

“…dispersion relation for axisymmetric waves: ω2 = c2k2r − 2πGΣ|kr| + κ2 (21) which informs us that pressure stabilizes short wavelengths, rotation stabilizes long ones, and self-gravity de-stabilizes intermediate wavelengths when Q ≡ cκ πGΣ 1 (22) (Toomre 1964, Goldreich & Lynden-Bell 1965).”

“(1) (Safronov 1960; Toomre 1964).”

“In the order of appearance, these are the star formation law, the Toomre (1964) Q parameter, the time-evolution of the CFE, and some miscellaneous parameters that have a weak influence on the CFE.”

“Assuming the dust traces 1% of the gas mass, the disk structures presented here are stable against gravitational fragmentation, with the minimum local Toomre Q values ranging from ∼5-50 in most cases (Q = csΩ/πGΣ, where Ω is the Keplerian orbital velocity; Toomre 1964).”

“…the expectation based on theory and numerical simulations (Goldreich & Lynden-Bell 1965; Kim & Ostriker 2001) in thin, single-phase gaseous disks is that gravitational instabilities leading to star formation would grow only if the Toomre parameter (Toomre 1964) Q ≡ vthκ/(πGΣ) is sufficiently small.”

“Q = σR κ/(3.36GΣ) (Toomre 1964)—for a flat rotation curve is Q = 1.72 ( σR 32 km s−1 ) ( Vc 220 km s−1 ) ( R 8 kpc )−1 ( Σ 50M⊙ pc−2 )−1 .”

“Toomre’s Q in Outer Disks Toomre’sQ parameter (Toomre 1964), which measures the stability of a thin axisymmetric disk (or ring, if the galactocentric radius is kept as a free parameter), has been closely linked to the decline of star formation in outer disks (e.g., Martin & Kennicutt 2001).”

“As explained shortly, the gas disk is assumed to stay at the marginal stability deÐned by Q\ 1, where Q denotes the stability parameter introduced by Toomre (1964).”

“The code does not suppress the Jeans instability on scales λJ > ǫ; on such scales, the gas is usually stabilized by the combined effects of pressure and rotation (Toomre 1964).”

“From the rotation curve and from the H i+He surface density, we can derive the classical disk stability parameter Q (Toomre 1964, Kennicutt 1989).”

“It has long been known that local vorticity is a factor in gravitational instability ( Toomre 1964, Goldreich and Ward 1973).”

“The axisymmetric stability of a thin disc with respect to its own self-gravity is determined by the parameter Q (Toomre 1964), defined as Q = csκ πG , (1) where cs is the effective thermal speed of the disc, is the surface density and κ is the epicyclic frequency (which, for a Keplerian disc, is…”

“ForToomre (1964) an axisymmetric, thin, collisionless disk, Toomre (1964) showed that a nonaxisymmetric perturbation will be unstable if Q\ 1.”

“…has a rotational velocity V at a&gasradius R, its gas-to-dynamical mass ratio within R is The stability parameter Q of theMgas/Mdyn\ nGR&gas/V 2.same disk (Safronov 1960 ; Toomre 1964) is Q\ where i and p are epicycle fre-ip/nG&gas B 2V p/nGR&gas,quency and gas velocity dispersion, respectively.”

“This parameter is a measure of susceptibility to local gravitational instability (Toomre 1964) : Q4 c s i nG& , (7) where i, the epicyclic frequency, equals ) in nonÈself- gravitating disks.”

“Its stellar disk is a Toomre (1964) disk of radial scalelength 5 kpc, truncated at 15 kpc. Gas represents 8% of the disk mass, and is distributed in a disk of 30 kpc radius.”

“…as described by (1), the code computes the mean circular velocity in each cell, the rotational frequencyΩ, the epicyclic frequency κ, etc. Random motions are given to all particles to prevent axisymmetric instabilities, their magnitude being at least equal to the critical speed (Toomre 1964).”

“There are several equivalent ways of expressing this condition, the classical Toomre (1964) Q criterion being the best known (Binney & Tremaine 1987).”

“is the so-called Toomre parameter (Toomre 1964),”

“The parameter W is the analog of the Toomre (1964) parameter Q ≡ κσR/(πGΣ) for vertical rather than horizontal oscillation frequency and velocity dispersion; i.e.”

“If the disc is massive enough, gravitational instability leads to additional transport (Toomre 1964; Laughlin & Bodenheimer 1994; Nelson et al. 1998; Pickett et al. 2000).”