Neutrino diffusion in General Relativity Giovanni Camelio with - PowerPoint PPT Presentation

Neutrino diffusion in General Relativity Giovanni Camelio with Stephan Rosswog from Astronomy Department Stockholm University February 6th, 2018 @ Lund Observatory

Motivation There are systems in astrophysics where: matter is opaque to neutrinos + the space-time curvature cannot be neglected For example, hot neutron stars: ◮ core-collapse supernovae → proto neutron stars ◮ binary neutron star mergers → hypermassive neutron stars 1 / 7

Fick’s second law x − 2∆ x x − ∆ x x + ∆ x x + 2∆ x x F ( x − ∆ x / 2) F ( x + ∆ x / 2) N ( t ) � � N ( x , t + d t ) − N ( x , t ) = A d t F ( x − d x / 2) − F ( x + d x / 2) dividing by A · d x · d t ∂ n ∂ t + ∂ F ∂ x = 0 2 / 7

Boltzmann equation Boltzmann equation in Galileian Relativity: ∂ f ∂ t + ∂ f x i − ∂ U g ∂ f � x i = d f � ∂ x i ˙ � ∂ x i ∂ ˙ d t � coll Boltzmann equation in General Relativity: � � ∂ f bc p c ∂ f � = d f e β p b ∂ x β − Γ a � � b ∂ p a d τ � coll e 3 p e 2 ¯ θ e 1 r Lindquist (Annal.Phys., 1966) 3 / 7

Moment scheme ◮ spherical symmetry ◮ multiply the equation by the neutrino mumentum ◮ integrate over the neutrino energy and angle ◮ use baryon number conservation ◮ add electrons ∂ (4 π r 2 α F ν ) ∂ Y L 1 ∂ t + = 0 4 π r 2 n B ∂ r ◮ where Y L = ( n ν + n e ) / n B is the lepton fraction ◮ and α the lapse function (GR term) Thorne (MNRAS, 1981) Pons+ (ApJ, 1999) Shibata+ (PThPh, 2011) 4 / 7

Remarks ◮ there is an analogue equation for the energy ◮ Fluxes can be determined from the Flick first law and with an assumption on the distribution function (closure) ◮ In this way we obtain a parabolic (diffusive) partial differential equation, like ∂ t + D ∂ 2 n ∂ n ∂ x 2 = 0 ◮ Diffusion is tricky for the Courant condition: ◮ explicit integration for short timesteps (easy) ◮ implicit integration for long timesteps (difficult) Thorne (MNRAS, 1981) Pons+ (ApJ, 1999) Shibata+ (PThPh, 2011) 5 / 7

Application 45 00.2 s 40 01.0 s 05.0 s 35 10.0 s 30 20.0 s 30.0 s T [MeV] 25 20 15 10 5 0 0.07 0.06 0.05 0.04 Y ν [#] 0.03 0.02 0.01 0 0 5 10 15 20 25 r [km] Figure: PNS evolution (Camelio+2017) 6 / 7

Outlooks Rotating, hot neutron star evolution (see Miralles+1993): ◮ how good it is an effective inclusion of rotation? (see Villain+2004 & Camelio+2016) ◮ angular momentum redistribution via neutrinos ◮ explore the final phase of a binary neutron star merger 7 / 7

Thanks! Contact: giovanni.camelio@astro.su.se More details in my PhD thesis ( arXiv:1801.01350 )