Why do IMSP’s have weaker magnetic fields compared to those of ordinary pulsars? B ≈ 2.59 x 108 G
Why do IMSP’s spin so rapidly?
P ≈ .005 s
Why do IMSP’s have transverse velocities smaller compared to those of ordinary pulsars? Vt ≈ 60.11 km/s
(Average values of 9 IMP’s with reported transverse velocities)
During its birth process a neutron star experienced: 1. An increase of its period from the initial value P0 to the current value Ps (a change of rotational energy)
2. An exponential decay of its magnetic field from the initial value B0 to the current surface value Bs (a change of radiative energy)
3. An increase of its space velocity from the initial value V0 to the current value V (a change of kinetic energy)
4. These birth energy changes are connected by
where and are the radius and mass of the neutron star; the speed of light and the characteristic time of the exponential field decay and the initial velocity is taken to be zero. According to the green formula, the radiation loss and increase of kinetic energy are both at the expense of rotational energy. [A similar equation but with a different radiative term is the basis of the “Rocket Model” proposed by Harrison ad Tademaru, ApJ , 201, 447 (1975), See Eq. (12)]
For the Crab pulsar the equation yields if and
For the magnetar J18091943 the equation gives if and
For the IMSP B1257+12 the equation yields if and
The characteristic time is consistent with the idea that all neutron stars are born with magnetic fields in the range of and initial periods in range
The time is the shortest theoretical time for a physical kick
From the exponential law and It follows that is the time decay from to . For field decays from one to eight orders of magnitude one has and therefore indicating an ultrafast magnetic field decay!
With the energy conversion takes the form
This red formula is the fundamental equation in the birthultrafastmagneticfielddecay model of neutron stars
Using the canonical values
into the reed formula it implies
where is the Lambert function, defined as the inverse of the function satisfying
and
To apply the yellow formula for one needs to consider neutron stars whose are known.
IMSP Ps Bs Vt
Averages Bs ≈ 2.59 x 108 G Ps ≈ .005 s Vt ≈ 60.66 km/s
The yellow formula and the average values yield the initial magnetic fields for the IMPs
B0 ≈ 1.3 x 1016 G if P0 ≈ .004 s
B0 ≈ 3.64 x 1015 G if P0 ≈ .0049 s
At the end of its formation, a neutron star may increase its rotational period from P0 ≈ .0049 s to Ps ≈ .005 s during ~ 104 s and then the rotational energy released can be transformed into kinetic and radiative energies in such a way that the IMPS acquires its transverse velocity Vt ≈ 60.66 km/s provided it has the initial magnetic B0 ≈ 3.64 x 1015 G, which is in the range of magnetars.
The generally accepted scenario of millisecond pulsar creation involves a long period of accretion in a low mass binary system. But Miller and Hamilton (2001) have proposed that the PSR 1257+12 was born with approximately its current period and magnetic field
“some and perhaps all isolated millisecond pulsars may have been born with high spin rates and low magnetic fields instead of having been recycled by accretion.”
email: ricardoherasosorno@gmail.com email: ricardoherasosorno@gmail.com
