1. Magnetic Fields Of Neutron Stars
ODIE

2. Outline Introduction (A historical review )
Neutron Star Magnetic Fields (Origin and Evolution)
Conclusion & Further work
Reference
A new form of star (NS)
Pulsar and Pulsar statistics
Millisecond and Binary Pulsars
Origin model
Fields evlouion
Magnetic field decay
G. Chanmugam, Annu. Rev. Astron. Astrophys :143-84
A. G. Lyne & F. Graham-Smith Pulsar Astronomy, Cambridge

3. Introduction A historical review For example:
In 1934, two astronomers, Walter Baade and Fritz Zwicky, proposed the existence of a new form of star, the neutron star, which would be the end point of stellar evolution.
Blackett(1947) proposed that the magnetic moment, μ=BR3, is proportional to its angular momentum. He hence argued that if the angular momentum is conserved, white dwarf with strong magnetic fields may be formed.
Ginzburg(1964) and Woltjer(1964) proposed instead that the magnetic flux(~BR2) of a star is conserved, so that strong magnetic fields would be generated in degenerate stars.
For example:
B~1011~1013G
B~105~107G
B~10~1000G
Magnetic flux conserved ~BR2
B~1016~1018G
B~107~1010G
Angular moment conserved ~BR3
Neutron star
R~106cm
White dwarf
R~109cm
Progenitor star
R~1011cm

4. Shortly, pulsars were discovered serendipitously by Hewish et
al(1968) and identified as rotating neutron stars (Gold 1968).
Their magnetic field were deduced to be B~1011~1013G.

5. Pulsars
If pulsars are assumed to be rotating neutron stars that are spinning down by emitting magnetic dipole radiation, their rotational kinetic energy is E=I Ω2/2, and the energy loss rate of a magnetized neutron star is of order
Therefore, the rate of change of their angular velocity Ω is:
Where I(~1045gcm2) is the moment of inertia of the neutron star and α is the angle between the magnetic dipole and rotation axes. If, for the sake of simplicity, the magnetic field of the neutron star may then be inferred to be B=A(PP)1/2 where A~3.2x1019Gs-1/2. ‧
The field strengths would be different if the spin-down was due to higher multipole radiation (Krolik 1991)

6. Pulsar Statistics
B12/P2～0.2s-2

7. Ostriker & Gunn (1969) proposed that neutron star magnetic field
decay, in order to explain the pulsar statistics at that time.
If the magnetic field decays exponentially on a time scale tD from an
initial value B0, the pulsar period is given by
, as t→∞
If tD ～Myr, this model explains quite naturally the absence of long
period pulsars.

8. Age of pulsars
Gunn & Ostriker (1970) showed that the magnetic field decrease
with the characteristic age τc=P/2P.
The more recent studies show that exponential magnetic field
decay, torque decay, power-law decay are also consistent with
data, while Wakatsuki et al (1992), with somewhat different
assumptions, find that a constant field is consistent with the data.
Pulsar proper motion surveys (Lyne et al 1982, Cordes 1986)
were also thought to provide strong support magnetic field decay.
These surveys enable one to estimate the kinetic age (tk～z/|v|) of
pulsars ‧

9. Assuming pulsars are born in the Galactic plane.

10. Proper motion surveys indicate that a significant number of pulsars are
approaching the Galactic plane possibly because they have undergone
oscillations about it, thereby making such analyses even more complex
(Harrison et al 1992).
Many isolated pulsars may have had their origins in binaries which were
disrupted in asymmetric supernova explosions (Dewey & Cordes 1987,
Bailes 1989). Such pulsars may have different field strengths compared
to isolated pulsars.

11. Millisecond and Binary Pulsars
Several radio pulsars have been discovered to be in binary systems,
with magnetic fields that are weaker than those in canonical neutron
stars. Hulse & Taylor(1975)
In 1982, Backer et al discovered the millisecond pulsars (PSR
, spinning 20 times faster than Crab pulsar (0.033s) with
a period of 1.56ms), whose magnetic fields were found to be only
of order 108~109G. ‧
All of the millisecond pulsars for which P has been measured have
weak magnetic fields (<1000MG).
A large number of millisecond pulsars and binary pulsars have also
been discovered in globular clusters. (Manchester et al 1991,Lyne
1992)

12. Millisecond and Binary Pulsars (cont.)
Two Models
1. Born in original spin (Brecher & Chanmugam 1978)
2. Born in binary (Radhakrishnan & Srinivasan (1982))
Brecher & Chanmugam (1978) pointed out that if the core of the progenitor star had a weak magnetic field, it would be more likely to be spinning fast, because the magnetic field would have enhanced transfer of angular momentum from the core to the envelope during earlier evolutionary phases.
Radhakrishnan & Srinivasan (1982) and Alpar et al (1982) suggested that millisecond pulsars were born in binaries. If such a pulsar is not accreting matter from the companion, it spins down as a radio pulsar. If the companion star transfers matter to the neutron star it adjusts its spin rate. (Ghosh & Lamb 1979):

13. The period will reaches an equilibrium period for disk accretion
According to Ghosh & Lamb’s (1979) calculation, the spin-up time scale
T=-P/P predicted by the present disk accretion model is
where the n is the dimensionless accretion torque, and
The period will reaches an equilibrium period for disk accretion
If the magnetic field decays to a value of about 109G and accretion takes place at the maximum allowed Eddington rate, Peq decrease along what is known as the spin-up line and approaches a few milliseconds. ‧ ‧

14. If the mass transfer stops, the neutron star becomes a millisecond
pulsar.
If the companion can somehow be made to disappear then the isolated
millisecond pulsar would be left behind.
Some theorists (Kluzniak et al 1988, Phinney et al 1988) argue that high
energy radiation and particles emitted by the fast pulsar may succeed in
ablating the companion.
The fact that most millisecond and binary pulsars are to the right of the
spin-up line is consistent with this model.

15. Optical observations of white dwarf companions of some binary pulsars
(e.g. PSR and PSR ) suggest that they are relatively
cold and hence have ages >108yr. Thus the pulsars must be older. Their
magnetic fields ~ G could not therefore have decayed
exponentially with tD~10Myr. ~
Two Models
1. Fields stop decaying (van den Heuvel et al 1986)
2. Accretion-induced collapse (Schatzman et al 1963)
Perhaps the fields stop decaying (or tD lengthens) when they reach the current values of binary pulsars B~ G (van den Heuvel et al 1986, Kulkarni 1986, Bhattacharya & Scrinivasan 1986). But this is rather ad hoc.
An alternative explanation in which the accretion-induced collapse (Schatzman 1963, Canal et al 1990, Nomoto & Kondo 1991) of a white dwarf , whose mass is pushed over Chandrasekhar limit, could produce a rapidly spinning weakly magnetized neutron star if angular momentum and flux was proposed.

16. To sum up the magnetic fields of degenerate stars play a crucial role
in the radiation emission mechanisms of pulsars and accreting
degenerate stars.

17. Origin and Evolution of Neutron Star Magnetic Fields
Almost all models proposed to explain the evolution of neutron star magnetic fields are related to the time scale for the decay of crustal fields.
Two classes
Magnetic field is generated in neutronstar
after it is born.
Neutron star is born with a strong magnetic
field.

18. If magnetic field is generated in neutron star after it is born
Blandford et al (1983) developed a suggestion of Urpin & Yakovlev
(1980) that the strong heat fluxes in the crust of a young rotating
neutron star lead to a possible thermoelectric instability in the solid
crust which causes horizontal magnetic field components to grow
exponentially with time. Once the field was generated in the crust it
was assumed to decay Ohmically in a few million years.
There is evidence that neutron stars become magnetized after
they are formed. Observations of SNR MSH and its
embedded pulsar (Seward & Harnden et al 1982) show a pulsar
with a timing age 1550yr in a SNR estimated to be ~104yr old. A
explanation is if the neutron star is as old as the SNR, but
became a pulsar ~103yr ago when its magnetic field grew to
sufficient strength.

19. If neutron star is born with a strong magnetic field
This proposal also provided an explanation as to why few pulsars
are found in SNRs.
A difficulty with this model is that the Crab pulsar is only 1000yr
old and has a strong magnetic field of ~4x1012G.
If neutron star is born with a strong magnetic field
If neutron star is born with a strong magnetic field because of, for
example, flux conservation (Ginzburg 1964, Wolezjer 1964) and the
field penetrated the regions interior to the crust where the electrical
conductivity is very high, then mechanisms other than simple Ohmic
decay had to be sought to explain the field decay.

20. Field evolution Magnetic field decay
Vandakurov et al (1972) proposed that convective instabilities could
take place in the degenerate interior of a neutron star. The flux
tubes would become buoyant (Parker 1979) and rise rapidly on the
time ~1s until they reach the crust.
Muslimov & Tsygan (1985) assumed that the protons in the interior
of the neutron star form a Type II superconductor. They then argue
that these fluxoids become buoyant and leave the interior on a time
scale < 10Myr, after which they Ohmically diffuse through the crust.
Srinivasan et al (1990) have examined the case when the interior is
superconducting and the magnetic flux tubes are in quantized
fluxoids. They showed the fluxoids cannot easily be buoyant because
they will be pinned to the the quantized neutron vortices which form
in the superfluid interior.

21. Magnetic field decay (cont.)
Ostriker & Gunn (1969) suggested that the field decay takes place
because of Ohmic decay.
It can be shown that the electrical conductivity in side a neutron star is
very roughly
by calculations based on the work of Canuto (1970)
the field decay from Ohmic losses in a spherical star with fixed
conductivity. Consider the lowest decay mode, the solution of the
magnetic field decay time is
If σ is constant, and corresponding to that in the solid crust, R=12km

22. But Baym et al (1969) pointed out that in the region below the crust
the conductivity depends mainly on the proton density and the
temperature and is given by
so that τ0 (τ0>1013yr) should be larger than the Hubble time.
Gunn & Ostriker (1970) suggested that the use of the crustal
conductivity may be more appropriate to determine the decay of
crustal fields.
Detailed numerical calculations which took the variation of σ with
radius r into account showed,however, that τ0>>1012yr
(Chanmugam & Gabriel 1971).

23. Conclusion & Further work

24. Reference G. Chanmugam, Annu. Rev. Astron. Astrophys. 1992. 30:143-84
A. G. Lyne & F. Graham-Smith Pulsar Astronomy, Cambridge

25. Rotation effect
Rotation may stabilize the convective instability if the rotation energy is larger than the magnetic energy. (Chanmugam et al 1979) This means that if the internal field is a factor of about 100 larger than the surface field (~1012G), instabilities may set in when the period of rotation has slowed down to a few seconds.
Therefore, the rotational history of the star is important in determining field evolution.

26. A novel mechanism for field decay was proposed by Flowers &
Ruderman (1977), Ray (1980) and Roberts (1981). They suggested
that if the magnetic flux tubes passed from the star into the
surrounding vacuum and back, internal fluid motions will reduce the
external magnetic field without increasing the corresponding
combined internal and magnetic filed energy. However the formation
of the crust in a few hours or the presence of a crustal toroidal field
would prevent this form happening.

27. Crustal fields decay
For an initial field g(x,0), can be solved directly as an initial value problem.
In the crust
σelectron-phonon：cause from electron and phonon (αT-2)
σimpurity：impurities which form in the crystalline crust in the initially
rapidly cooling star. (Independent of temperature)
The former dominates initially when the star is hot while the latter dominates as the star cools.
For crustal temperature T~109K, Flowers & Ruderman (1977) estimate the impurity fraction in the inner crust χ~10-3.
It has been shown that neutrino losses due to the direct β-decay, hence neutron stars may cool faster than previously thought.

29. X-Ray Binaries X-Ray binaries may be divided into two classes:
1. HMXBs (high-mass X-Ray binaries )
2. LMXBs (low-mass X-Ray binaries )

30. Evolution Field decays exponentially on a time scale tD Ohmic decay
Scientific American, Feb. 2003
Evolution
Field decays exponentially on a time scale tD
Ohmic decay
Torque decay

33. HMXBs Non-degenerate companion star is high mass (O or B type star).
The HMXBs are found to be conceentrated towards the Galatic
disk and many of them display X-ray pulsations, with periods
ranging from 0.069s to 835s, corresponding to the rotation peroid of
the neutron star.
Assuming that the magnetospheric radius rm ~(2-3)R and
LX~ ergs-1 ,
→ B>>100MG

34. LMXBs Non-degenerate companion star is low mass (less than about 2MO).
Few of the LMXBs show X-ray pulsations and the companions
is not seen at optical wavelengths.
They have ages greater than about 5-15 billion years.
Most LMXBs display X-ray bursts which are thought to be caused by
thermonuclear flashes of accreted material on the surface of the
neutron star (Lewin & Joss 1983).
Those that show X-ray pulsations do not display bursts, suggesting
that they have weak fields (B<3x1010G). (Strong magnetic fields
prevent the flow of matter over the surface of the star, and
consequently suppress the bursts)