The birth-ultrafast-The birth-ultrafast-magnetic-field-decay magnetic-field-decay

model applied to model applied to isolated millisecond isolated millisecond

pulsarspulsars

Ricardo HerasRicardo HerasPreparatoria Abierta SEIEM Edo. MexicoPreparatoria Abierta SEIEM Edo. Mexico

Pulsars with a rotational period in the Pulsars with a rotational period in the range of about 1-20 milliseconds and range of about 1-20 milliseconds and magnetic fields in the range of 10magnetic fields in the range of 1088- 10- 1099 G. G.

They are taught to begin life as longer They are taught to begin life as longer period pulsars but are spun up through period pulsars but are spun up through accretion in a binary relationship.accretion in a binary relationship.

This talk deals with three This talk deals with three fundamental questions on isolated fundamental questions on isolated

millisecond pulsars millisecond pulsars (IMSP’s) (IMSP’s)

Why do IMSP’s have weaker magnetic fields Why do IMSP’s have weaker magnetic fields compared to those of ordinary pulsars? compared to those of ordinary pulsars?

B ≈ 2.59 x 10B ≈ 2.59 x 108 8 G G

Why do IMSP’s spin so rapidly?Why do IMSP’s spin so rapidly?

P ≈ .005 s P ≈ .005 s

Why do IMSP’s have transverse velocities smaller Why do IMSP’s have transverse velocities smaller compared to those of ordinary pulsars? compared to those of ordinary pulsars?

VVtt ≈ 60.11 km/s ≈ 60.11 km/s

(Average values of 9 IMP’s with reported transverse (Average values of 9 IMP’s with reported transverse velocities) velocities)

Basic assumptions in the Basic assumptions in the Birth-ultrafast-magnetic-field-decay Birth-ultrafast-magnetic-field-decay

model:model: See R. Heras, “Pulsars are born as magnetars” in ASP conference See R. Heras, “Pulsars are born as magnetars” in ASP conference

20122012

During its birth process a neutron star During its birth process a neutron star experienced:experienced:

1. An increase of its period from the initial value 1. An increase of its period from the initial value PP00 to the current value to the current value PPss (a change of rotational energy)(a change of rotational energy)

2. An exponential decay of its magnetic field from 2. An exponential decay of its magnetic field from the initial value the initial value BB00 to the current surface value to the current surface value BBs s

(a change of radiative energy)(a change of radiative energy)

3. An increase of its space velocity from the initial 3. An increase of its space velocity from the initial value value VV00 to the current value to the current value VV (a change of kinetic energy)(a change of kinetic energy)

Basic assumptions……Basic assumptions……

4. These birth energy changes are connected by4. These birth energy changes are connected by

where and are the radius and mass of the where and are the radius and mass of the neutron star; the speed of light and the neutron star; the speed of light and the characteristic time of the exponential field decay and characteristic time of the exponential field decay and the initial velocity is taken to be zero. the initial velocity is taken to be zero. According to According to the green formula, the radiation loss and increase of the green formula, the radiation loss and increase of kinetic energy are both at the expense of rotational kinetic energy are both at the expense of rotational energy.energy.

[A similar equation but with a different radiative term is the basis of the “Rocket [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)]Model” proposed by Harrison ad Tademaru, ApJ , 201, 447 (1975), See Eq. (12)]

Implications of the model:Implications of the model:

For the For the CrabCrab pulsar the equation yields pulsar the equation yields if and if and

For the magnetar For the magnetar J1809-1943J1809-1943 the equation gives the equation gives if and if and

For the IMSP For the IMSP B1257+12B1257+12 the equation yields the equation yields if and if and

The characteristic time is consistent with the The characteristic time is consistent with the idea that all neutron stars are born with magnetic fields idea that all neutron stars are born with magnetic fields in the range of and initial periods in range in the range of and initial periods in range

The time is the shortest theoretical time for a The time is the shortest theoretical time for a physical kickphysical kick

Implications of the model……..Implications of the model……..

From the exponential law and From the exponential law and It follows that is the time decay from It follows that is the time decay from to . For field decays from one to eight orders of to . For field decays from one to eight orders of magnitude one has and therefore magnitude one has and therefore indicating an ultrafast magnetic field decay! indicating an ultrafast magnetic field decay!

With the energy conversion takes the form With the energy conversion takes the form

This red formula is the fundamental equation in the This red formula is the fundamental equation in the birth-ultrafast-magnetic-field-decay model of neutron birth-ultrafast-magnetic-field-decay model of neutron stars stars

Initial magnetic field of neutron starsInitial magnetic field of neutron stars

Using the canonical valuesUsing the canonical values

into the reed formula it impliesinto the reed formula it implies

where is the Lambert function, defined as where is the Lambert function, defined as the the inverse of the function satisfying inverse of the function satisfying

and and

To apply the yellow formula for one needs to To apply the yellow formula for one needs to consider neutron stars whose are consider neutron stars whose are known. known.

Application to IMSP’sApplication to IMSP’s

IMSP PIMSP Pss B Bss V Vtt

AveragesAverages

BBss ≈ 2.59 x 10 ≈ 2.59 x 108 8 GG

PPss ≈ .005 s ≈ .005 s

VVtt ≈ 60.66 km/s ≈ 60.66 km/s

Application to IMSP’sApplication to IMSP’s

The yellow formula and the average values yield the The yellow formula and the average values yield the initial magnetic fields for the IMPsinitial magnetic fields for the IMPs

BB00 ≈ 1.3 x 10 ≈ 1.3 x 1016 16 G if G if PP00 ≈ .004 s ≈ .004 s

BB00 ≈ 3.64 x 10 ≈ 3.64 x 1015 15 G if G if PP00 ≈ .0049 s ≈ .0049 s

At the end of its formation, a neutron star may At the end of its formation, a neutron star may increase its rotational period from increase its rotational period from PP00 ≈ .0049 s ≈ .0049 s to to PPss ≈ .005 s ≈ .005 s during during ~ 10~ 10-4-4 s s and then the rotational and then the rotational energy released can be transformed into kinetic and energy released can be transformed into kinetic and radiative energies in such a way that the IMPS radiative energies in such a way that the IMPS acquires its transverse velocity acquires its transverse velocity VVtt ≈ 60.66 km/s ≈ 60.66 km/s provided it has the initial magnetic provided it has the initial magnetic BB00 ≈ 3.64 x 10 ≈ 3.64 x 1015 15

G,G, which is in the range of magnetars. which is in the range of magnetars.

Changing paradigms!Changing paradigms!

The generally accepted scenario of millisecond pulsar creation involves a long period 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 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 proposed that the PSR 1257+12 was born with approximately its current period and magnetic field magnetic field

““some and perhaps all isolated millisecond pulsars may have been born with high spin 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.” rates and low magnetic fields instead of having been recycled by accretion.”

The model proposed here predicts that very tiny fractions of second after their formation, isolated millisecond pulsars display their observed small periods and low magnetic fields.

Birth magnetorotational instabilities!Birth magnetorotational instabilities!

Spruit (2008) has suggested that a differential rotation in the final stages of the core collapse process can produce magnetic fields typical of magnetars. Some form of magnetorotational instability may be the cause of an exponential growth of the magnetic field. “Once formed in core collapse, the magnetic field is in danger of decaying again by magnetic instabilities.”Geppert & Rheinhardt (2006) have discussed a magnetohydro dynamical process (MHD) that significantly reduces the initial magnetic field of a newly-born neutron star in fraccions of a second. Such a field reduction is due to MHD-instabilities, which are inevitable if neutron stars are born as magnetars.

Birth loss of rotational energy!Birth loss of rotational energy!

The idea of a loss of rotational energy during the birth process of millisecond pulsars was already considered by Usov (1992). “Once formed such rapidly rotating and strongly magnetized neutron stars [~1015 G] would lose their rotational energy catastrophically, on a timescale of seconds or less”

Final comment: The model proposed here also applies to other families of NS such as magnetars or radio pulsars. The idea that all NS are born with magnetic fields typical of magnetars and periods typical of millisecond pulsars accounts for a grand unification of NS

Thank you for your time!Thank you for your time!

e-mail: e-mail: ricardoherasosorno@gmail.com

website: website: http://www.ricardoheras.com