are neutron stars crushed? gravitomagnetic tidal fields as a...
TRANSCRIPT
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.