1. Magnetic massive stars as magnetar progenitors
Ren-Yu HU and Yu-Qing LOU
Physics Department and Tsinghua Centre for Astrophysics (THCA), Tsinghua University
(Based on Hu & Lou 2009, MNRAS, in press, astro-ph/ )

2. Compact Stars in the QCD phase diagram II Beijing, China
Acknowledgments
This research was supported in part by Tsinghua Centre for Astrophysics (THCA), by the National Natural Science Foundation of China (NSFC) grants and and by the National Basic Science Talent Training Foundation (NSFC J ) at Tsinghua University, and by the SRFDP and and the Yangtze Endowment from the Ministry of Education and Tsinghua University.
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Compact Stars in the QCD phase diagram II Beijing, China

3. Ultra-Intense Magnetic Fields
Two propositions on the origin of ultra-intense magnetic fields of magnetars (SGRs, AXPs) :
Neutron Star Dynamo
“Fossil-fields”
(Duncan & Thompson 1992; Thompson & Duncan 1993)
Require fast-spining neutron stars at the beginning (Period~1-2 ms)
Dynamo require: large spatial velocity of magnetars and energetic SN remnants.
Based on magnetic flux conservation so far
Supported by population synthesis study (Ferrario & Wickramasinghe 2006, 2008)
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Compact Stars in the QCD phase diagram II Beijing, China

4. Compact Stars in the QCD phase diagram II Beijing, China
Our purpose
Examine quantitatively the “fossil field” scenario
Study the conditions for magnetar formation.
Model magnetized massive stars with a quasi-spherical general polytropic magnetofluid under self-gravity
(Wang & Lou 2008; Lou & Hu 2009)
Our approach
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Compact Stars in the QCD phase diagram II Beijing, China

5. General Polytropic Magnetofluid
A magnetofluid under self-gravity in quasi-spherical symmetry
Mass Conservation
Momentum Equation
Magnetic Induction Equation
General Polytropic EoS
Self-Similar Transformation
k: sound parameter; n: scaling index parameter; \gamma: polytropic index
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Compact Stars in the QCD phase diagram II Beijing, China

6. General Polytropic Magnetofluid
Several integrals
Nonlinear ODEs
nx-v=0 -> m=0 : a central cavity
h: magnetic parameter
q: self-similar parameter
q=2(γ+n-2)/(3n-2)
: singular surface
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Compact Stars in the QCD phase diagram II Beijing, China

7. Compact Stars in the QCD phase diagram II Beijing, China
Progenitor Surface
Finite velocity solutions at large x,
Back to dimensional variables:
x-> ∞ means either t->0+ or r-> ∞
We determine the self-similar parameters from density, temperature, magnetic field and mass loss rate on the surface of progenitor.
We proceed the integration inwards.
We should insert MHD shocks in the solutions to cross the singular surface.
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Compact Stars in the QCD phase diagram II Beijing, China

8. MHD Shock Jump Conditions
We consider different sound speed (temperature) across a shock front.
Entropy increases from upstream to downstream sides.
Quasi-static solutions can cross the singular surface by MHD shocks.
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Compact Stars in the QCD phase diagram II Beijing, China

9. Quasi-Static Solutions
There exists a kind of asymptotic solutions v=LxK, α=A0x-2/n+NxK-1-2/n at small x.
When x->0, t->∞, v->0 (Quasi-Static).
The magnetic Lorentz force and the thermal pressure force are in the same order of magnitude at small x.
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Compact Stars in the QCD phase diagram II Beijing, China

10. Compact Stars in the QCD phase diagram II Beijing, China
One numerical example n=
0.673
q=0 γ=
1.327
h= 1.52×10-4
Rs=
109 cm at 1 s
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Compact Stars in the QCD phase diagram II Beijing, China

11. Compact Stars in the QCD phase diagram II Beijing, China
The numerical example
On the surface of the progenitor (1012 cm):
On the surface of the neutron star (106 cm):
Density = 2.5×10-5 g cm-3, Temperature = 3×104 K,
Mass loss rate = 10-6 solar mass per year, Mass = 5.59 solar mass,
Magnetic field = 103 G ( may arise from dynamo or “fossil fields” from ISM).
Mass = 2.15 solar mass,
Magnetic field = 4.70×1014 G.
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Compact Stars in the QCD phase diagram II Beijing, China

12. Mass Ratio and Magnetic Ratio
The ratio between the neutron star mass and the progenitor mass:
fM = Mi,ult/Mo,ini = (A0/A)λ2/n (ri/ro)3-2n
The ratio between the magnetic field on the neutron star surface and that on the progenitor surface:
fB = <B2i,ult>1/2/<B2o,ini>1/2 = (A0/A)λ2/n(ri/ro)1-2n
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Compact Stars in the QCD phase diagram II Beijing, China

13. Compact Stars in the QCD phase diagram II Beijing, China
Magnetar B-M Relation
Since we have fB/ fM = (ro/ri)2,
Considering the TOV limit of neutron star masses
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Compact Stars in the QCD phase diagram II Beijing, China

14. The Amplification Factor
Magnetic enhancement across a shock
<B2t>1/21/<B2t>1/22 = ρ1/ρ2
= 2/[(γ+1)Mach12]+(γ+1)/(γ-1)
The final magnetic amplification factor depends on n, q, and shock properties.
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Compact Stars in the QCD phase diagram II Beijing, China

15. The Amplification Factor
fB < 1012 =(ro/ri)2
Larger q, smaller fB
Median shock speed corresponds to smallest fB
Here we may resolve the difficulty posed by Petit et al. (2008)
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Compact Stars in the QCD phase diagram II Beijing, China

16. Compact Stars in the QCD phase diagram II Beijing, China
Conclusions
If the progenitor is a magnetic massive star with a surface magnetic field strength of ~ 103 G, it would have a good chance to produce a magnetar at the center of its supernova remnant.
The magnetar magnetic field is proportional to the magnetar mass and the progenitor magnetic field. The TOV limit actually limits the magnetar magnetic field.
If the progenitor is extremely magnetized, the new born magnetar may power the short GRB.
We could naturally expect a continuum from dim isolated neutron stars to magnetars.
In this scenario, magnetars should be slowly rotating neutron stars.
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Compact Stars in the QCD phase diagram II Beijing, China