Modeling of the cooling of Cassiopeia A neutron star

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Neutron stars as magnetic dipoles

According to the so-called « lighthouse » model schematically plotted in figure 1.3, a pul-sar is a rotating neutron star. As a neutron star spins, charged particles are accelerated along the magnetic field lines and emit electromagnetic radiation. The radiation beam that is formed sweeps the sky like a lighthouse and each time it crosses the observer’s line of sight, a pulse is observed. Therefore the period of the pulses corresponds to the rotational period of the neutron star.

Rotational energy

Assume that a neutron star is a solid ball of mass M and radius R, rigidly rotating with an angular velocity Ω = 2πP , with P the rotational period.

Magnetic dipole radiation

Suppose that a neutron star is a rotating magnetic dipole with α the angle between the rotation and magnetic axis and B the magnetic field strength at the magnetic equator.
Larmor formula for the power of the magnetic dipole radiation is :
with the magnetic dipole moment. For a uniform magnetized sphere, = BR3 sin α.

Surface magnetic field

Assuming that the loss of rotational energy originates from the emission of electromag-netic radiation, Thus, the characteristic pulsar magnetic field BPSR = B sin α is (Haensel et al., 2007) .

Characteristic age

Assuming that the pulsar magnetic field does not change with time, equations (1.15) and (1.17) give
Livingstone et al. (2006) reported that for several pulsars, the magnetic index is well below 3, suggesting that processes other than magnetic dipole radiation are at the origin of the loss of rotational energy and that the model presented here is too simple. It only gives orders of magnitude for the pulsar age and magnetic field.

A variety of neutron stars

Since the discovery of the first pulsar in 1967, neutron stars have been observed in all wavelengths from radio to γ-rays. They show a large diversity in their emission and intrinsic properties. The neutron stars for which the period and period derivative ˙ have been measured are plotted in the − ˙ diagram in figure 1.4. Are also indicated P P the lines of constant BPSR and τP SR.
Based on observations, neutron stars can be classified into different groups, pre-sented in the following (Haensel et al., 2007; Kaspi, 2010).
Rotation-powered pulsars
Rotation-powered pulsars are neutron stars whose emission is powered by the loss of rotational energy due to magnetic braking. They are extremely regular pulsators and emit in all wavelengths. In the − ˙ diagram, one can distinguish two distinct populations (see also part IV) :
Figure 1.4: – ˙ diagram for 1704 objects : 1674 rotation powered pulsars (small black dots), 9 AXPs (blue crosses), 5 SGRs (green crosses), 3 central compact objects (CCO – cyan circles), 6 isolated neutron stars (ISN – magneta squares), and 7 RRATs (red trian-gles) for which these parameters have been measured. Lines of constant BPSR (dashed lines) and τPSR (dot-dashed lines) are plotted. The solid line is a model death-line (see text for details). From Kaspi (2010).
• the normal pulsars, with periods of the order of few seconds and BPSR ∼ 1012 G;
• the millisecond pulsars, in the lower left of the diagram. They have P . 30 mil-liseconds and BPSR ∼ 108 G. They are old neutron stars that have been spun to mil-lisecond periods during an accretion episode. The Fermi Space Telescope showed that some of them are bright γ-ray sources.
They exhibit both steady and pulsed X-ray emission. The former is thought to be the thermal emission from the surface of neutron stars while they cool down (see part II). The latter is non-thermal and pulsed and is due to the pulsar magnetospheric activity.
Since rotation-powered pulsars spin down, their radio emission ultimately turns off when they cross the so-called death line. This is consistent with the lack of observations of pulsars with long periods and small period derivatives. The location of the death-line is model-dependent.

Rotating radio transients

The rotating radio transients (RRATs) do not produce periodic radio emission but exhibit short radio bursts. Whether they are a specific type of rotation-powered pulsars or a distinct population is still unclear.
Magnetars are believed to be young, isolated neutron stars powered by a large mag-netic field BPSR ∼ 1014 − 1015 G. They have long periods 5 . P . 12 s. Two types of magnetars exist :
• the Anomalous X-ray pulsars (AXPs) : they show a pulsed X-ray emission. Bursts were observed from some of them;
• the Soft-Gamma Repeaters (SGRs) : they exhibit highly irregular bursts in soft γ-rays and X-rays.
Observations of bursts from AXPs suggested that AXPs and SGRs belong to the same class of neutron stars.

High-B rotation-powered pulsars

Several radio pulsars have inferred magnetic fields B ∼ 4 × 1013 G, comparable to the lowest values observed for magnetars. They are called high-B rotation-powered pulsars. The observation of a week-long X-ray burst from the young high-B rotation-powered pulsar PSR J1846-0258 (Gavriil et al., 2008) suggests that they could be transient magnetars.

Table of contents :

I Context 
1 Neutron stars : general aspects 
1.1 From theoretical predictions to observations
1.2 Birth of a neutron star
1.2.1 Pre-supernova evolution
1.2.2 Core-collapse supernova explosions
1.3 Neutron stars as magnetic dipoles
1.3.1 Rotational energy
1.3.2 Magnetic dipole radiation
1.3.3 Surface magnetic field
1.3.4 Characteristic age
1.4 A variety of neutron stars
2 A laboratory for physics 
2.1 From microphysics to astrophysics
2.1.1 Structure of a neutron star
2.1.2 Equations for the stellar structure
2.2 A laboratory for microphysics
2.2.1 Mass-radius diagram
2.2.2 Observational constraints
2.3 A laboratory for gravitational physics
2.3.1 Gravitational wave emission
2.3.2 Test of gravitation theories
II Thermal evolution of neutron stars 
3 Cooling of isolated neutron stars 
3.1 A little bit of history
3.2 Thermal evolution modeling
3.2.1 General relativistic heat equations
3.2.2 Modeling
3.2.3 NSCool code
3.3 A toy model
3.3.1 Thermal conductivity
3.3.2 Specific heat
3.3.3 Neutrino emission
3.3.4 Envelope model
3.3.5 Analytical solutions
3.4 Cooling history of a neutron star
3.5 Towards a more realistic model
3.5.1 Superfluidity in neutron stars
3.5.2 Heating processes
3.6 Influence of the microphysics input
3.6.1 Non superfluid stars
3.6.2 Superfluid stars
3.6.3 Influence of the envelope model
3.6.4 Influence of the equation of state
3.6.5 Minimal cooling paradigm
3.7 Observations of the temperature of isolated neutron stars
3.7.1 An observational challenge
3.7.2 Present status
3.7.3 Cassiopeia A neutron star
3.7.4 Future perspectives
3.8 Theoretical modeling versus observations
3.8.1 Modeling of the cooling of Cassiopeia A neutron star
3.8.2 Modeling of all the available data
4 Cooling of young neutron stars 
4.1 Thermal evolution in the early ages
4.2 The specific heat in the crust
4.2.1 The cluster structure of the inner crust
4.2.2 Specific heat in the crust
4.2.3 Specific heat of the superfluid neutrons
4.2.4 Influence of the clusters on the critical temperature
4.2.5 Neutron specific heat in uniform matter
4.2.6 Neutron specific heat in non-uniform matter
4.2.7 Total specific heat in the crust
4.3 Cooling simulations
4.3.1 Neutron star model
4.3.2 Microphysics input
4.3.3 Fast cooling scenario
4.3.4 Slow cooling scenario
4.4 Perspectives
4.4.1 Modeling
4.4.2 Observations
5 Thermal evolution of accreting neutron stars 
5.1 Observations of accreting neutron stars
5.2 Quiescent state of X-ray transients
5.2.1 Nature of the quiescent emission
5.2.2 Deep crustal heating scenario
5.2.3 Atmosphere models
5.3 Heat equation
5.4 Soft X-ray transients
5.4.1 Thermal evolution of a soft X-ray transient
5.4.2 A toy model
5.4.3 Observations & constraints on microphysics
5.5 Quasi-persistent X-ray transients
5.5.1 Observations
5.5.2 Previous modelings of the thermal relaxation
5.5.3 New model for an accreting neutron star
III Rotating elastic neutron stars 
6 Rotating neutron stars 
6.1 3+1 formalism
6.1.1 Spacetime foliation
6.1.2 Induced metric
6.1.3 Eulerian observer
6.1.4 Adapted coordinates
6.1.5 Extrinsic curvature
6.1.6 3+1 decomposition of the stress-energy tensor
6.1.7 3+1 Einstein equations
6.2 Circular, axisymmetric and stationary spacetimes
6.2.1 Stationarity and axisymmetry
6.2.2 Circular spacetime
6.2.3 Metric
6.2.4 Maximal slicing
6.3 Einstein equations for rotating stars
6.4 Perfect fluid
6.4.1 Circularity
6.4.2 Decomposition of the fluid velocity
6.4.3 Energy-momentum tensor
6.4.4 Fluid equilibrium
6.4.5 Global properties
6.5 (2+1)+1 formalism
6.5.1 Foliation of the t hypersurfaces
6.5.2 Induced metric
6.5.3 Adapted coordinates
6.5.4 Extrinsic curvature
6.6 Numerical resolution with LORENE
6.6.1 Spectral methods
6.6.2 LORENE library
6.6.3 Block diagram of the Nrotstar code
6.6.4 An example
6.7 Constraints on the equation of state for dense matter
6.7.1 Observations of millisecond pulsars
6.7.2 Maximum rotational frequency
6.7.3 Influence of rotation of the M − R diagram
7 Newtonian and relativistic elasticity
7.1 Solid phases in neutron stars
7.1.1 Glitches
7.1.2 Solid crust
7.1.3 Liquid or solid core ?
7.1.4 Observational consequences
7.2 Newtonian models of elastic neutron star
7.2.1 Newtonian elasticity in a nutshell
7.2.2 Models of neutron stars with a (partially) solid interior
7.3 Elasticity in General Relativity
7.3.1 Previous formulations
7.3.2 Carter & Quintana formalism
7.3.3 Karlovini & Samuelsson formalism
7.3.4 Relativistic formulation of starquakes
8 Rotating neutron stars with a solid interior 
8.1 Elastic deformation of rotating stars
8.1.1 Small deformations
8.1.2 Eulerian variation
8.1.3 Lagrangian variation
8.1.4 Semi-Lagrangian variation
8.2 Rotating Elastic neutron stars
8.2.1 Metrics
8.2.2 Quasi-isotropic coordinates
8.2.3 Strain tensors
8.2.4 Relative strain tensor
8.2.5 Shear tensor
8.2.6 Energy momentum tensor of an elastic fluid
8.2.7 Circularity condition
8.2.8 Einstein equations
8.2.9 Equation for equilibrium
8.2.10 Boundary conditions
8.3 Newtonian limit
8.3.1 Equation for equilibrium
8.3.2 Boundary conditions
8.4 Numerical resolution
8.4.1 Block diagram of the Elastar code in LORENE
8.4.2 KADATH
8.5 Perspectives
IV Spin-up of accreting neutron stars 
9 Formation of millisecond pulsars 
9.1 Accretion in binary systems
9.1.1 Roche-lobe overflow
9.1.2 Mass transfer dynamics
9.1.3 Disk formation
9.1.4 Neutron star recycling
9.2 Evolution of neutron star binaries
9.2.1 The population of millisecond pulsars
9.2.2 The different cases of Roche lobe overflows
9.2.3 Neutron star X-ray binaries & millisecond pulsars
10 Model of accreting magnetized neutron stars 
10.1 Spin-up modeling
10.1.1 Mass increase and accretion rate
10.1.2 Angular momentum evolution
10.2 Accretion disk model
10.2.1 Magneto-hydrodynamic equation
10.2.2 Inner radius of the accretion disk
10.2.3 Relativistic specific angular momentum
10.2.4 Magnetic torque
10.2.5 Total angular momentum equation
10.2.6 Degeneracy parameter
10.3 Magnetic field evolution of accreting neutron stars
10.3.1 Accretion-induced magnetic field decay
10.3.2 Model of magnetic field decay
10.4 Models of neutron stars
10.4.1 Equations of state
10.4.2 Rotating neutron star configurations
10.5 Block diagram of the Evol code
11 Application to the spin-up of neutron stars 
11.1 PSR J1903+0327
11.1.1 An eccentric millisecond pulsar
11.1.2 Formation scenarios
11.1.3 Results
11.1.4 Conclusions
11.2 The extreme-mass millisecond pulsars
11.2.1 The less massive millisecond pulsar : PSR J0751+1807
11.2.2 The most massive pulsar : PSR J1614-2230
11.2.3 Modeling
11.3 Perspective : sub-millisecond pulsars
Conclusion and perspectives

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