Study of superfluidity in magnetars by van Hoven and Levin

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1 Study of superfluidity in magnetars by van Hoven and Levin

2 Overview What is a superfluid and how does it look like in neutron stars? What can happen when both a superfluid and a strong magnetic field are present in a neutron star? Can the presence of a superfluid have some consequences on the beheviour of QPO?

3 Superfluid matter A superfluid is a fluid that flows without viscosity. If the superfluid is formed by charged particles, then it is called superconductor. In neutron stars a superfluid/superconductor can be created just in the core.

4 Superfluid matter This is due to the dispersion relation that links the energy and the momentum of the particles: E=E(p). A superfluid system has usually a dispersion relation that has a bandgap, such has for example: E p=[ p 2 2m 2 2 ] 1 2

5 How does a superfluid move? A system with a band gap, can be describe by a velocity proportional to the gradient of the phase of the wave function: v. This means that the fluid is irrotational: x v=0. The most convenient way to move for the superfluid is hence trought forming quantized vortices.

6 And in a neutron star? In a cool core below the crust of a spinning neutron star superconducting protons coexist with superfluid neutrons: a sea of relativistic degenerate electrons are presents in order to neutralize the system. The density of the neutrons vortices is n v /P cm -2. =10 4 When a magnetic field penetrates the core, the superconductor protons organize themselves in flux tubes (the magnetic field alignes them)

7 The neutron vortices have a different velocity respect with the electrons. This is because a neutron star is usually slowed down by electromagnetic radiation that influences just the electrons: so the vortices are faster than the electrons and they move outward, until they reach the crust-core interface. The short range nuclear force depends on the velocity, so there is a strong interaction between neutron and protons if they come closer each others than about cm. The -11 cm protons flux tubes will be pulled by neutron vortices.

8 This means that the magnetic field will be expelled from the core in a time scale that is proportional to the period of the star. The flux tubes pill up at the crust core interface: they are so dense that can stress the crust and can break it if their magnetic is enough high (usually around B>5* *10 15 G).

9 Could this influence the QPOs? For an uniform magnetic field the lowest QPO is expected at the frequency: ~ B eff 2R 4 c where B eff = BB cr

10 In order to derive a dispersion relation, we need to study a two fluid problems: with D n t v n n =2 v n D p t v p p =2 v p f mf x p f mf = R 1R 2 n n np that is the mutual friction force and R is an adimensional number that measures the drag between protons and neutrons.for R= the vortices get pinned to the plasma only trough the Magnus force. R2 1R 2 f mf ee 2 v p n np

11 Perturbing the dynamical equation and forgetting the plasma viscosisty we get the dispersion relation: = 1 1 x p ± c 2 A k 2 x p The magnetars rotate slowly with 1rad/s so the terms with Coriolis and Magnus forces contribute just for a fraction: x p =0.18 rad / s rad /s x p

12 This means that the superfluidity doesn't affect the propagation of the hydromagnetic waves inside a magnetar and we can use. Do you remember? n x p ~ B eff 2R 4 c For the typical parameter B=10 15 G,R=10Km and x =0.05 we get 24Hz, that is in p agreement with observations.if the superfluidity would have some consequences on frequency, this value would be reduced by

13 Another effect:the Glaberson instability The Glaberson instability arises when, because of turbolence, the normal component develops a meridional circulation current. The component of this current along the superfluid vortices drives the Glaberson instability. The appereance of this instability can explain the timing noise and the glitches in ordinary pulsar.

14 The Glaberson instability sets only if the velocity of the superconductor protons and the superfluid neutrons are misaligned by an angle. This angle must satisfy the following constraint: sin 1 x 2 x 4 p c A 2x 2 p ee ee 1 x 4 p c A x 2 2 ee =0.3 B P G 1 s So for a magnetar this instability is strongly suppressed!