On the gravitational stability of a disk of stars

“In that scenario, such large clumps arise from gravitational instabilities in gas-rich disks (Toomre 1964; Noguchi 1999; Genzel et al. 2011), Electronic address: tljohn@umich.edu * Based on observations made with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science…”

“…of gravitational forces promoting star formation and energy and momentum feedback by recently formed stars counteracting gravitational collapse (Toomre 1964; Ostriker et al. 2010; Kim et al. 2011; Hopkins et al. 2011; Romeo & Wiegert 2011; FaucherGiguère et al. 2013; Agertz et al. 2015;…”

“The disc can become gravitationally unstable to axisymmetric instabilities if (Toomre 1964) Q = csκ πG 1, (1) where cs is the local sound speed, κ is the epicyclic frequency and is the surface density of the disc.”

“The local linear stability of selfgravitating discs is simple (Toomre 1964): it depends upon the parameter, Q ≡ csκ πGΣ , (1) wherecs andΣ are the sound speed and surface density of a lowmass disc (Mdisc ≪ M∗), andκ is the epicyclic frequency which, in a Keplerian disc, is the same as the angular…”

“Gravitational instability occurs when the destabilizing self-gravity of a disk dominates over the restoring forces of gas pressure and differential rotation (Toomre 1964; Goldreich & Lynden-Bell 1965), a requirement that roughly translates into Mdisk/Må0.”

“We now compute the Toomre parameter for local gravitational stability, Qg = vsκ/(πGµg) (Toomre 1964), in the same manner as usually done for galactic centers, i.e. assuming a gas density µg equal to the average, and the epicyclic angular velocity κ = 2V/R in the linear part of the rotation curve.”

“Toomre (1964) showed that for an infinitesimally thin disc to fragment, the stability parameter must be less than a critical value, Qcrit ≈ 1.”

“In a thin, di†erentially rotating disk composed of pure gas (Safronov 1960) or stars only (Toomre 1964), there is a critical surface density above which the disk is& cunstable to ringlike perturbations in the radial direction.”

“To see this I note that the condition for stability against self–gravity for a gaseous disc can be expressed as csΩ πGΣ > 1 (1) (Toomre, 1964), where cs,Ω and Σ are the local sound speed, orbital frequency and surface density respectively, and G is the gravitational constant.”

“This is measured by the Toomre Q parameter (Toomre, 1964) Q = Ωσr 3.36GΣ , (6) where σr is the particle radial velocity dispersion.”

“The conditions for gravitational instability of a Keplerian rotating disk are given by (Toomre 1964) &0Z c S )K nG , (7) where and are the surface density of the nebular&0, cS, )Kgas, the sound velocity, and the Keplerian orbital frequency, respectively.”

“This value is calculated using the formula of Toomre (1964), Q = σvsκep πΣsG , (9) where σvs is the velocity dispersion of the solid bodies (their standard deviation), Σs is the solids surface mass density, G is the gravitational constant, and κep is the epicyclic frequency, which in a Keplerian…”

“The local stability parameter, Q = σR/σR,crit (Toomre 1964), is postulated to be Q & 1 in the outer disc and to rise steeply to Q > 2 near the centre.”

“Martin & Kennicutt (2001) suggest that the threshold density is determined by the Toomre criterion (Toomre 1964) for gravitational instability.”

“The emergence of the gravitational instability within a disc is governed by the parameterQ (Toomre, 1964), which for a gaseous Keplerian disc is given by Q = csΩ πGΣ .”

“It is has long been suggested that there is a threshold column density below which star formation is quenched (Toomre 1964; Spitzer 1968; Quirk 1972).”

“…Q parameter, defined as Q(r) = Ω(r) cs(r)/(π GΣ(r)) (where Ω(r) is the angular velocity, cs(r) the sound speed, G the gravitational constant and Σ(r) is the local surface density, Toomre 1964), is Q(r) >∼ 1 at radii r ∼ 0.15, 0.1, 0.2 and 0.1 pc for run A, B, D and E, respectively (see Fig.”

“We fix the Toomre (1964) stability parameter Qstars = 1.5 (constant in space), and then obtain velocities of particles by finding the time-independent axisymmetric solution of the Jeans equations; the ratio of azimuthal-to-radial velocity dispersions is computed using the epicyclic approximation…”

“For example, one class of theoretical models assumes that the velocity dispersion and/or Toomre (1964) Q parameter in galactic disks is regulated by star formation feedback, and use this assumption to deduce a star formation law (e.g., Thompson et al. 2005; Ostriker et al. 2010; Ostriker & Shetty…”

“According to standard analyses of the problem, beginning with those of Toomre (1964) and of Goldreich & Lynden-Bell (1965), the disc should be stable wherever its surface density lies below a critical value Σc = α κcs 3.36G (4) where cs is the velocity dispersion of the gas, κ the epicyclic…”

“(34) Thus, is equal to the standard Q parameter \Qlop & Solomon for theic/nGkun (Toomre 1964 ; Jog 1984a)axisymmetric case ; divided by fk.From the variation in with / it can be seen thatfk (° 2.7),the region /\ 0¡È90¡ and /\ 270¡È360¡ would show lower values of than the axisymmetric case,…”

“Mergers and slow tidal interactions between galaxies are a well studied phenomenon that can produce dramatic tidal tails of stellar debris (c.f. Toomre 1964, Barnes & Hernquist 1992 and references within).”

“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.”

“In discs undergoing Keplerian rotation, local regions may be unstable to axisymmetric perturbations if the Toomre parameter, Q (Toomre 1964), is Q = csκ πGΣ 1, (1) where cs is the sound speed, κ is the epicyclic frequency (the orbital velocity Ω for a Keplerian disc), G is the gravitational…”

“…this is set by the motion of the gas under the influence of the total potential and is therefore proportional to Σ(Σ + Σ )], and we have omitted an additional dependence on the Toomre (1964) Q stability parameter (also see e.g. Ostriker and Shetty 2011; Faucher-Giguère et al. 2013; Kim and…”

“Gravitational instability arises when Q = Ωcs/πGΣ < 1 (Toomre 1964) and is known to lead to fragmentation in the outer regions of active galactic nuclei (see e.g. Shlosman et al. 1990) and of young stellar objects (see e.g. Adams & Lin 1993).”

“The unstable length is derived from the Toomre (1964) analysis for a centrifugally supported disk, kJ ¼ 2c2s G (1þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Q2 p ) ; ð11Þ where cs, , and Q denote the sound speed at the central plateau, the surface density of the adiabatic disk, and the…”

“GIs, MRI, FU Orionis Outbursts, and Chondrules Gravitational instabilities ca activate in a disk when the Toomre (1964) parameterQ¼ cs / G P1:7 for a thick disk (see Durisen et al. 2007), where cs is the sound speed, is the surface density, and is the epicyclic frequency.”

“To understand the stability of the disk, Toomre’s parameter (Toomre 1964) defined by Q = csΩ πGΣ , (2) is often useful, where Ω is the orbital frequency and Σ is the surface density of the disk.”

“A self-gravitating disc becomes gravitationally unstable if (Toomre 1964): Q = csκ πG ∼ 1, (1) where cs is the local sound speed, is the surface density and κ is the epicyclic frequency (for Keplerian discs, this is equal to the angular frequency ).”

“The Toomre Q parameters defined by equation (6) give the surface densities relative to the critical values at Qg = Qs = 1 for axisymmetric gravitational instability in a razor-thin, gas-only or star-only disk (Toomre 1964; Binney & Tremaine 1987).”

“…left panel of Figure 5 shows the Toomre parameter, averaged in annular bins, of the disk gas at the final simulation redshift, Q = c s κ/ ( πG Σ), where c s is the adiabatic sound speed, κ = (4Ω 2 + rd Ω 2 /dr ) 1 / 2 the epicyclic frequency, and Ω the angular velocity of the disk gas (Toomre 1964).”

“6 is calculated as Toomre (1964) Q(r) = csκ πG , (4) where cs is the sound speed, κ = √ 2 (Vrot/r) (1 + d ln Vrot/ d ln r)1/2 (Oh & Haiman 2002; Binney & Tremaine 2008) is the epicyclic frequency, Vrot is the rotational velocity, is the surface mass density and r is the radius.”

“…can scale the Jeans equations by the constant factor φP from Equation (1), to obtain the scale height hg as hg = σg√ 2πGφP ρg , (3) which leads to the definition of ρg (Equation 36 in Krumholz & McKee 2005), ρg = φPκ 2 2πQ2G , (4) whereQ = κσg/πGΣg is the Toomre stability parameter (Toomre 1964).”

“One reason for this is that higher densities would violate the Toomre (1964) gravitational stability criterion (e.g., Kuchner 2004; Rafikov 2005), while the vast majority of PPDs observed in young stellar associations were found to be gravitationally stable at all radii (e.g., Isella et al. 2009).”

“This is necessary in order to ensure that the disc is Toomre stable (Toomre 1964).”

“On the other hand, the wavelength λcr is given as (Toomre 1964) λcr = 4π2GΣ κ2 , (21) where epicyclic frequency κ is equal to the angular velocity Ω0 in Saturn’s ring.”

“Rotating disks are stable to gravitational fragmentation as far as the Toomre (1964) parameter s pº W SQ G2 exceeds the critical values - -0.7 1 2 (for thick, thin, and composite disks, respectively), where Ω≡v/R;j/R2 is the angular rotation velocity, σ is the intrinsic velocity dispersion of the…”

“The stability of a disk against fragmentation can be quantified via the Toomre Q parameter (Toomre 1964): k p = S Q c G ,s where cs is the sound speed of the gas, κ is the epicyclic frequency (equal to the angular velocity Ω in a Keplerian disk), and Σ is the surface density of the gas.”

“frommicrolensingsurveys. DiskInstability Thediskinstabilityscenario[64,65,66,317,316]invokesalocalizedregionof thediskthatbecomesgravitationallyunstable.Thisisusuallyquantifiedwiththe ToomreQ criterion[475]: c s ˇG &lt;Q crit ˇ1; (44) wherec s isthesoundspeed, isthelocalorbitalfrequency,G isthegravitational constant,an…”

“The Toomre stability criterion can be used to estimate the stability of discs against self-gravity (Toomre 1964).”

“For thin disks, stability is determined by the Toomre parameter QT = κcs/πGΣ [21, 22], where cs is the sound speed of the disk, κ is the epicyclic frequency, Σ is surface density, and G is the gravitational constant.”

“From the mass model of the galaxy and from profile (12) we can calculate the radial profile of the Toomre parameter Q (Toomre 1964), shown in figure 5.”

“An accretion disc can become gravitational unstable if (Toomre 1964) Q = csκ πGΣ ∼ 1, (6) where κ is the epicyclic frequency and can be replaced by the angular frequency, Ω, in a Keplerian disc.”

“…taking Σs(a) = fdgΣQ=1(a) in Equation (14), where fdg is the dust-to-gas ratio, which we take to be 10−2, and ΣQ=1(a) is the gas surface density at which the disk is marginally gravitationally unstable, which happens when the Toomre Q ≡ Ωcs,0/(πGΣQ=1) is of order unity (Safronov 1960; Toomre 1964).”

“Furthermore, the truncation of the disk at inner radii introduces an artiÐcial o†-axis minimum in the systemÏs gravitational potential, inducing stellar motion and the artiÐcial generation of SLINGampliÐed instabilities et al. et al.(Heemskerk 1992 ; Nelson While the combination of these assumptions enables1998). exte…”

“The Toomre Q parameter indicates the susceptibility of axisymmetric perturbations to grow in uniform thin disks: for Q 1, the surface density is sufficiently large for gas self-gravity to overwhelm the restoring effects of Coriolis forces and pressure (Toomre 1964).”

“The Toomre (1964) Q-parameter must satisfy Q = cs πG ≈ H R M∗ Md 1.5 , (4) where cs and are the disc sound speed and surface density, respectively.”

“The disks’ stability in the context of self-gravity was checked by use of the Toomre criterion Q=κRCs/(πGΣ) 1 (Toomre 1964), where = ¶ ¶S˜C Ps represents a 2D speed of sound (Pringle & King 2007).”

“There is a threshold surface density (depending on the local epicyclic frequency and the dispersion of the random velocity component, Toomre 1964; Kennicutt 1989), above which gas is unstable.”

“Using linear perturbation theory applied to an axisymmetric, razor-thin disk, Toomre (1964) showed that the exponential growth of local density perturbations is controlled by the parameter Q ≡ csκ πGΣ (1) where cs is the sound speed in the disk, κ is the epicyclic frequency, which equals the…”

“The susceptibility of an infinitesimally thin disc to gravitational instability can be measured using the Toomre Q parameter (Toomre 1964): Q = csκ πGΣ , (1) wherecs is the local sound speed,Σ is the disc surface density and κ is the epicyclic frequency (equal to the angular frequencyΩ in Keplerian…”

“The first step, linear gravitational instability (GI), requires that the disk be Toomre unstable: Q = csΩ/πGΣ ≤ Q0 ≈ 1 (Toomre 1964; Safronov 1960), where cs = √ kT/µ is the isothermal sound speed of the gas with mean particle weight µ and temperature T , G is the gravitational constant, Σ is the…”

“Larger dispersions, expressed as a larger Toomre (1964) Q value, lead to a more stable disc (Sellwood & Carlberg 1984).”

“The study of instabilities in disks has a long history, following the seminal work of Toomre (1964), with a considerable literature on many aspects of it.”

“The conditions which determine the stability of clouds for global star formation in galaxy discs have been shown by Kennicutt (1989), following Toomre (1964), to depend on the relationship between the gas surface density and the velocity dispersion in the plane.”

“Furthermore, the models presented in Krumholz & Burkhart (2016) rely on the Toomre stability parameter for gas (Qg), stars (Q∗) and the galaxy disc (Q), where Q−1 = Q−1∗ + Q−1g (see Toomre 1964; Wang & Silk 1994), assuming Q and Qg to be equal to unity for the gravity and feedback models, respectively.”

“Another factor that may be affecting the intrinsic scatter is Toomre’s Stability Criterion (Safronov 1960; Toomre 1964), whose parameter Q is related to gas fraction (since it depends on the gas surface density).”

“…with Galactic orbits similar to that of the Sun (low inclination, low eccentricity), dynamical encounters with giant molecular clouds, spiral structure and other gravitational potential gradients scatter stars stochastically (Spitzer & Schwarzschild 1953; Toomre 1964; Junqueira et al. 2013).”

“The Toomre Q parameter (Toomre 1964) with values smaller than 1.5 is a good indicator for gravitationally unstable regions.”

“Writing down the Toomre equation (Toomre 1964), this becomes, 𝑓𝑔,𝑄 𝜔𝑐𝜎𝑔 𝜋𝐺Σ𝑔 = 𝑄min .”

“For such instabilities leading to the formation of bound objects in a rotating disk to arise, the key requirement is a Toomre (1964) parameterQ below unity3.”

“(5) Here λcrit is given by λcrit = 4π2GΣ κ2 , (6) which local theory predicts is the longest unstable wavelength of axisymmetric Jeans instabilities in a purely rotationally supported disk (Toomre 1964).”

“For a putative rotating disk, we estimate the Toomre Qcriterion for disk stability (Toomre 1964),Q = vcvT/πGrΣ, where the circular velocityvc ∼ 200 km s−1 , the turbulent velocity vT ∼ 50− 100 km s−1 , the disk radiusr ∼ 3 kpc and the disk surface densityΣ∼ 200 M⊙ pc−2 .”

“Hydrodynamical simulations show that the gas becomes dense enough for star formation to occur in the spiral arms, providing the Toomre criterion (Toomre 1964) is satisfied (Q .”

“For many years most of the star-forming disk of M 33 has been known to be stable according to the simple Toomre gravitational stability criterion (Toomre 1964) if only the gas surface density is considered (Elmegreen 1993a).”

“They applied two standard criteria for gravitational collapse: the Roche (or critical) density and the Toomre (1964) Q-parameter.”

“The factor 3.36 is for a stellar fluid (Toomre 1964); it should be π for a gaseous fluid (Safronov 1960).”

“This outer dense band of gas is from the Toomre instability (Toomre 1964) during the disc’s initial fragmentation and therefore not as realistic an environment for cloud formation.”

“Discs E-mail: dittmann@astro.umd.edu become unstable when the Toomre criterion (Toomre 1964) is satisfied, Q ≡ csκe/(πG ) 1 for a gas disc, where cs is the sound speed, κe is the radial epicyclic frequency, is the surface density, and G is Newton’s constant.”

“The gravitational instability is well characterized with the Toomre’s Q parameter (Toomre 1964): Q = κcs πG , (25) where κ is the epicyclic frequency (κ = if the rotation profile is Keplerian), cs the sound speed of the disk, and is the surface density, respectively.”

“While the instability against self-gravitation can be measured by Toomre’s parameter Q (Toomre 1964), the optical thickness at the upper boundary z∗ is fixed by the boundary condition τ∗ = 2/3.”

“To sustain a global spiral wave, the Toomre (1964) length, 2πGΣ/κ2 (for epicyclic frequency κ and total disk surface density Σ) has to be comparable to the disk scale length.”

“Axisymmetric gravitational stability in a razor-thin disk requires that Toomre’sQ parameter exceeds unity, where Q 0:30 e M ðaÞa2 ð22Þ (Toomre 1964; Binney & Tremaine 1987); here ðaÞ is the surface density at radius a and e is the rms value of one component of the eccentricity vector or 2 1=2 of…”