Convergence of the fragmentation boundary in self-gravitating discs Collaborators : Phil Armitage - University of Colorado Duncan Forgan- University of St Andrews Sijme-Jan Paardekooper Queen Mary, University of London George Mamatsashvili Tbilisi State University Guiseppe Lodato Universita Degli Studi di Milano Peter Gibbons, Cassandra Hall, Eric Lopez, Beth Biller University of Edinburgh Ken Rice Institute for Astronomy University of Edinburgh Introduction Convergence issue in self-gravitating disc simulations ² Can we constrain the requirements for fragmentation? Does disc fragmentation actually occur? ² Can we find examples of objects formed via GI? 1
Gravitational Instabilities An accretion disc is gravitationally unstable if (Toomre 1964) Q = c s Ω πgσ 1 Q ~ 1 in young protostellar discs The stability of a self-gravitating disc is determined by the balance between heating and cooling (Paczyński 1978; Gammie 2001; Rice et al. 2003) ² important to understand implications of radiative transport! If unstable a disc may either stably transport angular momentum (Lodato & Rice 2004), or it may fragment into bound objects ² protoplanets in protoplanetary discs (Boss 1998, 2000,2002) ² stars in discs around supermassive black holes. Cardiff 22 November 2006 Cooling rates A lot of studies assume we have an adiabatic disk that has a cooling time that is constant when normalised wrt the local orbital period. i.e. t cool ( r) = β Ω 1 where Ω = GM * r 3 du u dt t cool t u exp t cool Implications : Assume pseudo-viscous evolution Dissipation rate D(R) = 9 4 νσω2 1 2 2 Σ 3 = r t r r 1 ν Σ r with ν = αc s H r Thermal equilibrium t cool = 4 9γ γ 1 ( ) 1 αω α = 9 4γ( γ 1)β 2
Long/short cooling times M disk = 0.5 M star, t cool = 7.5 Ω -1 M disk = 0.1 M star, t cool = 5 Ω -1 Fragmentation boundary Disc evolution governed by viscosity ν = αc s H γ = 2 γ = 5/3 γ = 7/5 with 4 1 α = 9γ( γ 1) t cool Ω Fragmentation Fragmentation occurs if the stress (as measured by α) required to balance the cooling, exceeds a well defined maximum value (α max ~ 0.06-0.07). Quasi-steady evolution 3
Perturbation amplitudes Cossins et al. (2009) showed that the rms perturbation amplitudes in a quasi-steady state satisfied δσ Σ 1 β α since α = 4 9γ( γ -1)β where t cool = βω 1 Cossins et al. 2009 Rice et al. 2011 Convergence!!! β crit ~30 Meru & Bate (2011, 2012) suggest that these simulations do not converge! ² As the resolution increases, so does the cooling time for fragmentation ² Maybe fragmentation can occur for any cooling time even extremely long cooling times! 4
Cooling formalism The energy equation in SPH including the cooling term is Ø # P P &! du a 1 u = mb % b2 + a2 (v ab awab a dt 2 b t cool $ ρb ρ a ' Essentially for cooling - assumes that the thermal energy is a delta function at the location of particle a Ø ² ² Problems near shocks and contact discontinuities Produces significant differences in structure for small changes in resolution 500k 1m 2m Smoothed cooling Ø May be more appropriate to smooth the thermal energy to get a more accurate measure of the local thermal energy # P P &! du a 1 1 = mb % b2 + a2 (v ab awab mb ubwab dt 2 b ρ at cool b $ ρb ρ a ' Other suggestions are viscosity (Lodato & Clarke 2010) or fragmentation at boundary between turbulent and laminar flow (Paardekooper, Baruteau & Meru 2011). Ø 500k 1m 2m 5
Convergence with Smoothed Cooling Fragmentation boundary clearly different from that obtained using unsmoothed cooling. Appears to converge to a value of around β crit < 8. Filled symbols unsmoothed cooling. Open symbols smoothed cooling. Rice, Forgan & Armitage 2012 Rice, Paardekooper, Forgan & Armitage 2014 ² Paardekooper (2012) stochastic fragmentation ² Young & Clarke (2015) two-modes of fragmentation. β = 8, N = 10 million Where does fragmentation occur? If convergence issue largely numerical, fragmentation likely occurs at r > 20-30 AU (Rafikov 2005; Clarke 2009; Rice & Armitage 2009) ² Inner disc optically thick, cools slowly. Realistic cooling Fragmentation boundary Fragmentation region 300 AU When considering realistic representations of cooling, the viscous α increases with radius ² Very small inside 10 AU 6
Does fragmentation actually occur? If there is a large population of planetary-mass objects at large radii (r > 50 AU), they should be influenced by even more distant companions. Consider the evolution of outer planets perturbed by more distant stellar companions ² Kozai-Lidov cycles ² Tidal evolution Pure N-body (Parker, Forgan & Rice 2015) illustrates that large eccentricities can be excited. Scattering results Original, largely unperturbed, population 6-7% scattered (and survive) with a < 1 AU. hot Jupiters proto-hoto Jupiters Planets scattered from large initial radii can only be hot Jupiters or proto-hot Jupiters. Only ~ 1% of stars have hot, or proto-hot, Jupiters. 7
Exoplanet properties Most hot and proto-hot Jupiters have masses less than 5 Jupiter masses. Objects that form via fragmentation are expected to typically have masses greater than 5 Jupiter masses (Forgan & Rice 2013). The hot and proto-hot Jupiters have a metallicity distribution consistent with that of the bulk exoplanet population. Planetary-mass bodies scattered from large radii appear to be rare. Consistent with direct-imaging surveys (~ 1% have distant companions). Conclusions The convergence of the fragmentation boundary using standard SPH with basic cooling does not converge with increasing resolution. Difficult to produce a physically plausible explanation for this lack of convergence. Introducing a smoothed cooling implementation (more consistent with the standard SPH formalism) appears to produce convergence ² Fragmentation requiring cooling times β < 10 (α > 0.1). If convergence issue is largely numerical, then fragmentation likely occurs only at large radii ² Possibly even the case if convergence issue not numerical! If disc fragmentation common, where are the resulting planetarymass bodies? ² Sergei s solution: they re all formed via GI! 8