Are neutron stars crushed?Gravitomagnetic tidal fields as a

mechanism for binary-induced-collapse

Marc FavataCornell University

APS Meeting, April 2005, Tampa FL

Motivations

• Wilson-Mathews revised simulations still show small compression.

• Over 15 papers refuted their original claims• A “loophole” remains:

– all analytic papers neglected gravitomagnetic-velocity couplings– numerical papers assumed corotation or irrotation or used

ellipsoidal stellar models– [some evidence for binary-induced-collapse in particle clusters;

(Shapiro, Duez et al., Alvi & Liu)]

• Question of principle: is there any way to compress a NS?

How to crush a neutron star?Gravitomagnetic tidal forces act on the star:

aexti = ¡ri©ext ¡ _³ext

i + (v £ B)i

³exti = ¡2

3²ipqBplxqxl

©ext =12Eabxaxb

B = r£ ³ext

induced velocity field

tidal acceleration

Sij =Zx(i²j)abxa½vb d3x

How to crush a neutron star?

Sij =Zx(i²j)abxa½vb d3x

Gravitomagnetic tidal forces act on the star:

aexti = ¡ri©ext ¡ _³ext

i + (v £ B)i

³exti = ¡2

3²ipqBplxqxl

©ext =12Eabxaxb

B = r£ ³ext

resulting change in central density:

±½c½c

= c1Eij(t)E ij(t) + c2Bij(t)Bij(t)

O

"µRd

¶6#

O

"µRd

¶7#

Details I: Eulerian Perturbation expansion

start with fluid equations:

expand them:

@½@t

+ri(½vi) = 0

@vi@t

+ (vkrk)vi = ¡riP½

¡ri© + aexti

r2© = 4¼½

aexti = "

23²ijk _Bjl xkxl ¡ 2²ijkBklvjxl

½(t; x) = ½(0) + "½(1) + "2½(2) + ¢ ¢ ¢P (t; x) = P (0) + "P (1) + "2P (2) + ¢ ¢ ¢©(t; x) = ©(0) + "©(1) + "2©(2) + ¢ ¢ ¢v(t; x) = v(0) + "v(1) + "2v(2) + ¢ ¢ ¢

Details I: Eulerian Perturbation expansion

This leads to an induced current quadrupole moment:

where

is the gravitomagnetic Love number.

Analogous to the Newtonian Love number k2 :

at orderO("0) :

at orderO("1) :

riP (0) = ¡½(0)ri©(0) ; _½(0) = 0

½(1) = P (1) = ©(1) = 0

v(1)i = ¡³ext

i =23²ijkBj lxkxl

Sij = °2MR4Bij °2 =215

R »10 »6µn d»»61 jµ0(»1)j

Iij = ¡13k2R5Eij

See poster by Hinderer & Flanagan: Monday

Details I: Eulerian Perturbation expansion

at orderO("2) :@½(2)

@t+ri[½(0)v(2)

i ] = 0

@v(2)i

@t+riP (2)

½(0) +ri©(2) +½(2)

½(0)ri©(0) = atoti

atoti = (v(1) £B)i ¡ (v(1) ¢ r)v(1)

i

Can convert this to an equation for first-order Lagrangianperturbations of a spherical star [ch. 6 of Shapiro & Teukolsky]

Ä»i + Lij [»j ] = atoti

Solve for the fundamental radial mode to determine the change in central density.

Results I: Change in central density

±½c½c

= ¡2:280µM 0

M

¶2µRd

¶6

+ 1:157µMR

¶2µ1 +

M 0

M

¶µM 0

M

¶2 µRd

¶7

Tidal stabilization term

[Lai; Taniguchi & Nakamura]

Gravitomagnetic compression term

Although compression increases with mass ratio, tidal disruptionor ISCO is always reached before compression dominates.

Exception: contrived configurations in which Newtonian tidal field vanishes.

v0 =2X

m=¡2

B2mv (t; r)Y B;2m

Details II: Pre-existing velocity fieldHowever, it is possible to get a net compression if there is a pre-existing velocity field.

v0(t; x) =X

lm

Elmv (t; r)Y E;lm +Blm

v (t; r)Y B;lm +Rlmv (t; r)Y R;lm

Consider a general expansion of the stellar velocity in vector spherical harmonics:

Suppose that in isolation, this velocity satisfites: r ¢ (½v0) = 0 ; v20 ¿M=R

The only angle-averaged radial force comes from the Lorentz-like term in the external accelaration: v0 £B

O

"µRd

¶6#

O

"µRd

¶7=2#

±½c½c

= c1Eij(t)E ij(t) + c02Sij(t)Bij(t)

Results II: Change in central density

Tidal stabilization term

[Lai; Taniguchi & Nakamura]Gravitomagnetic compression term

±½c½c

= ¡2:280µM 0

M

¶2µRd

¶6

+5:493VSµMR

¶1=2µM 0

M

¶µ1 +

M 0

M

¶1=2 µRd

¶7=2

where we defined a characteristic velocity of the current quadrupolar motions:

jSij j ¼MR2VS

Results II: Change in central density

VS = 0:1µMR

¶1=2

andµM 0

M

¶= 1; 3; 5; 10

±½c½c

= ¡2:280µM 0

M

¶2µRd

¶6

+5:493VSµMR

¶1=2µM 0

M

¶µ1 +

M 0

M

¶1=2 µRd

¶7=2

Conclusions

• For non-rotating stars that are unperturbed in isolation, a compressive force does exist, but it is always smaller than the Newtonian tidal stabilization. (Except for contrived situations.)

• If the star has a pre-existing current-quadrupolar velocity component, net compression is possible. The change in central density is small, but could, in principle, cause collapse in a star very, very close to its maximum mass.

• What does this say about the Wilson-Mathews compression: probably nothing, but binary-induced compression is possible at least in principle.

• It would be worthwhile to perform binary simulations without constraints on the hydrodynamics. These effects (and r-modes too) vanish for irrotational configurations. Probably unimportant for NS/NS mergers, but might be important for NS/BH mergers.