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
Giovanni Camelio with Stephan Rosswog from Astronomy Department Stockholm University February 6th, 2018 @ Lund Observatory
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
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x − 2∆x x − ∆x x N(t) x + ∆x x + 2∆x F(x − ∆x/2) F(x + ∆x/2) N(x, t + dt) − N(x, t) = Adt
∂n ∂t + ∂F ∂x = 0
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Boltzmann equation in Galileian Relativity: ∂f ∂t + ∂f ∂xi ˙ xi − ∂Ug ∂xi ∂f ∂ ˙ xi = df dt
Boltzmann equation in General Relativity: pb
b
∂f ∂xβ − Γa
bcpc ∂f
∂pa
dτ
r e1 e2 e3 p ¯ θ Lindquist (Annal.Phys., 1966)
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◮ spherical symmetry ◮ multiply the equation by the neutrino mumentum ◮ integrate over the neutrino energy and angle ◮ use baryon number conservation ◮ add electrons
∂YL ∂t + 1 4πr2nB ∂(4πr2αFν) ∂r = 0
◮ where YL = (nν + ne)/nB is the lepton fraction ◮ and α the lapse function (GR term)
Thorne (MNRAS, 1981) Pons+ (ApJ, 1999) Shibata+ (PThPh, 2011)
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◮ 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 ∂n ∂t + D ∂2n ∂x2 = 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)
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5 10 15 20 25 30 35 40 45 T [MeV] 00.2 s 01.0 s 05.0 s 10.0 s 20.0 s 30.0 s 0.01 0.02 0.03 0.04 0.05 0.06 0.07 5 10 15 20 25 Yν [#] r [km]
Figure: PNS evolution (Camelio+2017)
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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
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Contact: giovanni.camelio@astro.su.se More details in my PhD thesis (arXiv:1801.01350)