Our Plan We want to study rigorously (more than just using the global virial relations) the stability/instability of dense cores. Conditions for instability correspond to finding conditions for the collapse of the core into a star. Since for cores T <~ U ~ W we can consider only the thermal content U of the cloud to simplify the problem. Later in the course we will introduce also the contribution of rotational energy and magnetic energy…

Isothermal Sphere I “at” = sound speed Spherical case: r=r(r) Treat cores as SPHERICAL, NON-TURBULENT gas clouds with CONSTANT TEMPERATURE (no bad approximation for dense cores, would be bad for GMCs instead). We want to find the density structure and the potential of such a cloud in equilibrium “at” = sound speed Spherical case: r=r(r) Boundary condition: Fg=0 at r=0 and r= rc at r=0

Isothermal Sphere II Boundary conditions: f(0) = 0 f’(0) = 0 Note different notation. compared to book for adimensional potential Boundary conditions: f(0) = 0 f’(0) = 0 Gravitational potential and force are 0 at the center. --> L-E equation must be integrated numerically with the above boundary cs.

Isothermal Sphere III Recall r/rc= exp (-f) Figure describes an infinite sequence models parametrized by r/rc (density contrast) i.e. for any r/rc I can build infinite models with arbitrary values of rc and the density at the boundary, r0 Recall r/rc= exp (-f)
Density and pressure (P=ra2) increase monotonically towards the center. --> important to offset inward pull from gravity for grav. collapse.
After numerical integration of the Lane-Emden equation, one finds that the density r/rc approaches asymptotically 2/x2 (x >> 1) Hence the dimensional density profile of the singular isothermal sphere is: r(r) = a2/(2pGr2) (called singular because density diverges for r=0) Useful formula to describe density profile (except at very small r).

Isothermal Sphere IV With: r = √(at2/(4πGrc))*x r = rc exp(-f) We now want to isolate the dependency of m on r/rc, i.e. the function m(r/rc)

Isothermal Sphere V Read the above function in the following way; mass function of density which in turn is function of radius (all expressed via the adim. variables) Left boundary value of function is for a radius x0=0, hence rc/r0=1 and m=0. For increasing rc/r0 m then increases until rc/r0=14.1, corresponding to the dimensionless radius x0=6.5. The function has an oscillatory behaviour to the right with a sequence of maxima and minima and tends asymptotically to the (nondimensional) mass of a singular isothermal sphere (m=0.798)

Jeans Instability -- Standard (rigorous) derivation considers an infinite, homogeneous gaseous static self-gravitating medium and studies first order perturbative solutions of the physical equations valid for a self-gravitating fluid. -- Heuristic (approximate) derivation; consider a gas sphere and perturb the sphere, then compare either: Gravitational (Fg1) and pressure (Fp1) force for the perturbed system and find condition for Fg1 > Fp1 - instability regime=collapse (2) Sound crossing time and free fall time in perturbed system and find condition for which tdyn < tsound - instability regime=collapse Using either (1) or (2) one finds that the system is unstable (i.e. collapses) on a scale r given by; r2 > at2/Gro

Infinite homogeneous fluids: can they be static? (a) Consider a charged fluid (plasma) with no gravitational field; if the fluid is neutral there is no net electromagnetic force - electrostatic infinite homegeneous plasma is a sensible physical system (b) Consider now a charge-free self-gravitating fluid; gravity cannot be neutral since it is always attractive so a net force will arise  the fluid has to move  static infinite homogenous self-gravitating fluids cannot exist However mathematically static homogeneous equilibria are very useful to study evolution of perturbations. Need to use Jeans swindle to carry out the analysis = assume that unperturbed gravitational field is zero, Fo = 0 + Poisson eq. only valid for perturbed quantities There is no general justification for this but the Jeans swindle is reasonable in many situations. In fact: (1) In real systems the unperturbed gravitational field is balanced by other forces (e.g. pressure, centrifugal rotation) so it is like the net force field is zero (e.g. pressure/density gradients of Earth’s atmosphere balance Earth’s gravitational field) and (2) if the scale of these other forces vary is much larger than the scale of interest (e.g. wavelength of sound waves in Earth’s atmosphere) we can treat the medium as infinite and homogeneous.

Jeans instability Jeans swindle: in homogeneous medium not generally true, but usually justified Linearize equations of motion for this medium G G

Gravitational stability Gravity dominated Pressure dominated
Low-density contrast cloud: Increasing outer pressure P0 causes a rise of m and rc/r0. Since P=rat2, the internal pressure P rises more strongly than P0 and the cloud remains stable (no collapse) However the physical radius r0 (radius of the cloud) is related to x0 and rc like r0 = sqrt(at2/(4pGrc)) * x0 Since rc increases faster than x0 (e.g. fig. 9.1), the core actually shrinks with increasing outer pressure P0. The same as Boyle-Mariotte law for ideal gas: PV=const. --> P * 4/3pr3 = const.
High-density contrast clouds: gravity more important and if rc/r0> 14.1 (x0> 6.5) clouds are gravitationally unstable. The associated critical mass is the Bonnor-Ebert mass (Ebert 1955, Bonnor 1956) MBE = (m1at4)/(P01/2G3/2)

To see that MBE is the maximum mass for a gravitationally stable cloud one has to perturb the Euler (no magnetic field), Poisson and continuity equations for a spherical cloud and solve the linearized perturbed equations to determine the time evolution of perturbations. Similar to Jeans instability but for finite, pressure bounded system (i.e. with boundary conditions P0 and r0) Can start from an ansatz for the perturbation, e.g. normal mode (= sinusoidal mode in which each physical variables oscillate with same frequency in the indipendent on position but different variables (e.g. r, P) can have different phases of the oscillation): r(r,T) = req(r) + dr(r) exp(iwt) req(r) = unperturbed function Solving the linearized equations one determines dr(r) (eigenfunction) and w2 (eigenvalue). Different modes are identified by different values of w2 (w2 > 0 stability, w2 < 0 instability). Unstable modes have “nodes”, i.e radii where the amplitude of the perturbation is zero. The first order or fundamental mode (smallest value of w2) has zero nodes which means entire cloud breathes (like a hearthbeat) in and out. Cores with M > MBE are unstable to this “breathing mode” so the entire cloud contracts as a response to the global oscillation. Higher order modes (i.e. modes with increasing number of nodes) are “activated” as a new peak of the function m is encountered going right on the rc/ro axis. Therefore a cloud with M < MBE can become unstable as long as it has a strong enough density contrast (instability to higher order modes)

Gravitational stability: The case of B68 Optical Near-Infrared Starless Bok Globule x0=6.9 is only marginally about the critical value 6.5 gravitational stable or at the verge of collapse