Maximum elastic deformations of relativistic stars
Nathan K. Johnson-McDaniel
Theoretisch-Physikalisches-Institut
Die Friedrich-Schiller-Universität Jena
SFB/TR7 video seminar
October 24th, 2011
In collaboration with Benjamin J. Owen
History and motivation
I
Deformed rotating neutron stars have been considered as
potential sources of gravitational radiation ever since pulsars
were realized to be spinning neutron stars.
[Shklovskii, Ostriker and Gunn, Ferrari and Ruffini, and Melosh, all 1969.]
I
...and deformed neutron stars in general are very convenient
sources of gravitational radiation: In particular, with known
pulsars, we have precise sky locations, and know how fast
they’re spinning (and spinning down, or glitching), which
greatly aids the analysis, and allows it to take full advantage
of the persistent nature of the signals.
I
While we won’t discuss it much in this talk, one could also
obtain bursts of GWs from the violent relaxation of a
deformation, as in magnetar flares (SGRs).
Motivation (cont.)
I
I
The Crab pulsar, in particular, is an iconic potential GW
source, having been the subject of directed searches as far
back as 1972 (Levine and Stebbins, with a 30 m
interferometer).
And there are many ongoing searches for continuous GWs
from pulsars and neutron stars both known (e.g., the Crab,
Vela, Cas A) and unknown (e.g., with Einstein@Home) in
LIGO/Virgo data. (There are also ongoing searches for GWs
from magnetar flares.)
Cas A NS and nebula, Chandra
Crab pulsar and nebula, Chandra
Motivation (cont.)
I
But can we reasonably expect a detectable continuous GW
signal from the Crab, or any other known pulsar?
I
Well... Often the electromagnetically determined spindown
constrains any GW emission to be well below what
LIGO/Virgo could detect.
I
And this spindown limit is likely a rather generous upper
limit—one expects the pulsar’s electromagnetic and particle
emission to contribute significantly to the spindown.
I
Indeed, for the pulsars for which one can measure the braking
index (n := ν ν̈/ν̇ 2 ), one finds that it is less than 3, the value
for dipole radiation (n = 5 for quadrupole radiation). (For
instance, n ' 2.5 for the Crab.)
Motivation (cont.)
I
But the spindown limit gives the experimenters a number to
aim for when searching for these waves.
(Recall that h ∼ Q22 ω 2 /r . We know ω and generally r for known pulsars, and get the spindown Q22 up
to some slight uncertainty about the pulsar’s moment of inertia.)
Figures from Pitkin MNRAS 415, 1849 (2011)
Motivation (cont.)
I
Thanks to LIGO, we now know that GWs contribute less than
2% of the Crab’s spindown power. Or, alternatively, that the
Crab has a nonaxisymmetric (Newtonian) ellipticity (i.e.,
Q22 /Izz ) of . 10−4 .
I
However, is there any reason to expect the Crab (or any
isolated NS) to have such a large deformation? For
comparison, the Crab’s internal field would generate an
ellipticity of ∼ 10−11 (assuming that the internal field is
comparable to the ∼ 1012 G external field). So is it at all
possible to obtain such a large ellipticity?
I
As we shall show, the elastic properties of exotic cores can
support deformations of this size, with plausible theoretical
input.
Caveats!
I
But note that there is no reason to assume that a given pulsar
will be significantly deformed, even if its constituents could
support a large deformation.
I
Thus, despite claims to the contrary in the literature, upper
limits on the GW emission from known pulsars can only
constrain “mountain building” mechanisms, at best—they do
not constrain the stars’ constituents!
I
Contrariwise, the observation of a large deformation would
allow one to conclude that the star contained some sort of
solid exotica—or a very large internal magnetic field
(∼ 1016 G for an ellipticity of ∼ 10−4 ).
Calculational generalities
I
Leaving aside the problem of how to generate large
deformations (and how long they last, before subsiding due to
viscoelastic creep, etc.), the problem of knowing the
maximum deformation possible divides cleanly into two parts.
I
The first part involves the nuclear and condensed matter
physics of neutron stars—i.e., what sorts of matter are inside
neutron stars, and how resistant are they to shears?
I
In particular, for a given neutron star model that contains
some solid portion, one wants to know the solid’s shear
modulus (as a function of density)—i.e., how strongly it
resists being sheared—and its breaking strain—i.e., how much
one can shear it before it yields.
(Really, one wants the full elastic modulus tensor, since these solids are likely not isotropic, but stellar
perturbation calculations have not yet progressed to that level of sophistication.)
Calculational generalities (cont.)
I
With these quantities in hand, one then turns away from
messy condensed matter physics to the far, far cleaner
problem of stellar perturbation theory, and asks how large a
quadrupole deformation the given shear modulus and breaking
strain could support on a star of a given mass.
Dany Page’s neutron star schematic
Solid portions:
I
Lattice of nuclei in the crust
I
Possible lattice of exotica in
the core
Shear moduli of solids in NSs
Let us first review the possible theoretical scenarios for solid phases
in neutron stars:
The crust
Theoretical descriptions of the neutron star crust are quite
well-developed, and they predict a solid lattice of (heavy!) nuclei in
the crust (with possible pasta phases at higher densities). [Shear modulus
calculated by Ogata and Ichimaru PRA 42, 4867 (1990), with charge screening included by Horowitz and Hughto
arXiv:0812.2650, and quantum effects by Baiko MNRAS 416, 22 (2011); both of these are small effects]
But the shear modulus of the crust is (relatively!) small; to obtain
a large shear modulus, one has to move inward, to far less
well-understood densities...
Shear moduli of solids in NSs (cont.)
The core
In the neutron star core, one can obtain significantly larger charge
separations (and thus larger shear moduli) from exotic phases, but
with much attendant theoretical uncertainty.
I
We will consider the hadron–quark mixed phase.
[Shear modulus calculated by NKJ-M and Owen]
I
Other possibilities include various meson condensates (pions
or kaons). [Estimates of the shear modulus are given in Haensel, Potekhin, and Yakovlev Neutron
Stars 1, Chap. 7]
CSC
Finally, one can have crystalline superconducting quark matter,
either throughout a strange quark star (very speculative!) or in the
core of a hadron–quark hybrid star. [Shear modulus calculated by Mannarelli, Rajagopal,
and Sharma, PRD 76, 074026 (2007)]
Large shear moduli, from the lab to the heavens
35
10
Diamond
Steel
12
Crystallized C WD
NS crust
CSC SQM
Hybrid core (Hy1)
30
-3
µeff (ergs cm )
10
25
10
20
10
15
10
10
10
0
10
2
10
4
10
6
10
8
10
-3
ρ (g cm )
10
10
12
10
14
10
16
10
A very brief overview of the hadron–quark mixed phase
I
Due to asymptotic freedom, one expects to obtain deconfined
quarks in cold nuclear matter at sufficiently high densities
I
...and these densities may be present in compact (“neutron”)
stars. [Collins and Perry PRL 34, 1353 (1975)]
I
In particular, as first proposed by Glendenning [PRD 46, 1274 (1992)],
the nucleation of a large region of quark matter is favored as a
way to reduce isospin asymmetry.
I
This region of hadron–quark mixed phase has the same
“pasta” structure proposed for the nuclei at the bottom of the
crust.
figure from Sonoda et al. PRC 77, 035806 (2008)
...and of our shear modulus calculation
I
Given the lattice structure (obtained from energy arguments
and using standard models to describe the hadronic and quark
phases), one can calculate the mixed phase’s angle-averaged
shear modulus by generalizing the standard Fuchs calculation
[PRSA 153, 662 (1936)] to this more involved lattice structure.
I
We also include charge screening, which decreases the shear
modulus, and contributions from changing the cell size for the
lower dimensions, which increase it.
I
Of course, the input parameters to the models are rather
uncertain (as are the models themselves), but one can still
obtain large regions of mixed phase with large shear moduli
with reasonable input parameters. (And the greatest effect of
the parameters on the maximum quadrupole is generally on
the extent of the mixed phase; their effect on the magnitude
of the shear modulus is usually secondary.)
Breaking strain
There has only been one calculation of the breaking strain, this for
the crust, by Horowitz and Kadau PRL 102, 191102 (2009).
They find that the crust is very strong (breaking strain of ∼ 10−1 ,
compared to terrestrial materials, with a maximum breaking strain
of ∼ 10−2 ).
This strength is attributed to the great pressure to which the
system is subjected, which prevents most of the standard failure
modes (e.g., dislocations), so while the Horowitz-Kadau calculation
is only directly applicable to the crust, such high breaking strains
seem likely to be generically applicable to the core, as well. (And
we shall take them to be so.)
Maximum quadrupoles
The first calculation of the maximum elastic quadrupole was
carried out by Ushomirsky, Cutler, and Bildsten (UCB) [MNRAS 319, 902
(2000)], in the context of accreting neutron stars. Here they used
Newtonian stellar perturbation theory in the Cowling
approximation (i.e., neglecting the self-gravity of the perturbation).
The UCB formalism has been used in all subsequent calculations of
the maximum quadrupoles (Owen, Lin, Knippel and Sedrakian,
Horowitz, 2005-2010), except for two by Haskell et al. that drop
the Cowling approximation (but remain Newtonian).
We give a direct, fully relativistic generalization of the UCB
formalism with no Cowling approximation. (This also gives a
simpler formalism for the Newtonian no Cowling computation than
the Haskell et al. result.)
Calculating the maximum quadrupole (without the
Cowling approx.)
I
Use first order stellar perturbation theory (treating the shear
modulus as a perturbation), and express the quadrupole in
terms of the surface value of the gravitational perturbation.
In the relativistic case, we use the standard Thorne and Campolattaro Regge-Wheeler gauge formalism
[ApJ 149, 591 (1967)]. As first shown by Ipser [ApJ 166, 175 (1971)], we can use a nonrotating
background since all the stars we are interested in are rotating much slower than their breakup speed.
I
Decompose the elastic portions of the stress-energy tensor
into tensor spherical harmonics, as
rr
r⊥
Λ
+ tr ⊥ Yab
+ tΛ Yab
.
δTab = δρt̂a t̂b + δp(gab + t̂a t̂b ) + trr Yab
[Since we only consider the maximum possible quadrupole, one thus actually doesn’t need a full relativistic
theory of elasticity here, though this result can be obtained from, e.g., the Carter-Quintana version.]
I
Express δρ and δp in terms of the t• using stress-energy
conservation.
Calculating the maximum quadrupole (cont.)
I
Use the Green function for the gravitational perturbation to
write the quadrupole as an integral over the t• and their
derivatives.
I
Integrate by parts to put the integral in the form
Z ∞
Q22 =
[()trr + ()tr ⊥ + ()tΛ ]dr
0
(This also takes care of any distributional contributions.)
I
Note that all the coefficients have the same sign, so maximum
uniform strain ⇒ maximum quadrupole.
2
(We have t• = 2µσ• , and the von Mises breaking strain is given by σab σ ab = 2σ̄max
.)
I
We can thus write the maximum quadrupole as
Z R
max
Q22
= σ̄max
µ(r )G(r )dr .
0
Quadrupole integrands for the SLy EOS:
0.124M star; compactness of 0.013 (= 2GM/Rc 2 )
14
Newt. Cowling
Newt. no Cowling
GR δρ + δp
GR stresses
10
8
3 2
Q22 integrand / µσmax (10 m s )
12
8
6
4
2
0
0
5
10
15
r (km)
20
25
30
Quadrupole integrands for the SLy EOS:
0.500M star; compactness of 0.12
Newt. Cowling
Newt. no Cowling
GR δρ + δp
GR stresses
4
6
3 2
Q22 integrand / µσmax (10 m s )
5
3
2
1
0
0
2
4
6
r (km)
8
10
12
Quadrupole integrands for the SLy EOS:
1.40M star; compactness of 0.35
Newt. Cowling
Newt. no Cowling
GR δρ + δp
GR stresses
1.25
6
3 2
Q22 integrand / µσmax (10 m s )
1.5
1
0.75
0.5
0.25
0
0
2
4
6
r (km)
8
10
12
Quadrupole integrands for the SLy EOS:
2.05M star; compactness of 0.60
7
Newt. Cowling
Newt. no Cowling
GR δρ + δp
GR stresses
5
5
3 2
Q22 integrand / µσmax (10 m s )
6
4
3
2
1
0
0
1.5
3
4.5
6
r (km)
7.5
9
39
10
-6
10
Newt. Cowling
Newt. no Cowling
GR (total)
εLIGO
2
Q22 (g cm )
Maximum crustal quadrupoles (SLy EOS)
38
10
-7
1.2
1.3
1.4
1.5
1.7
1.6
M (solar masses)
1.8
1.9
2
10
2.1
These were computed using the detailed composition results for
this EOS due to Douchin and Haensel [A&A 380, 151 (2001)] along with
the Ogata and Ichimaru result for the shear modulus.
Here LIGO =
p
8π/15Q22 /Izz denotes the “LIGO ellipticity,” which is computed using the fiducial moment of
inertia of Izz = 1038 kg m2 = 1045 g cm2 . It does not reflect the star’s actual shape in the relativistic case!
Maximum hybrid star quadrupoles
43
10
-2
10
Newt. Cowling
Newt. no Cowling
GR (total)
42
-3
10
41
10
-4
10
40
10
-5
10
39
10
-6
10
38
10
1.8
1.85
1.9
1.95
M (solar masses)
Computed for the Hy1 EOS
2
2.05
εLIGO
2
Q22 (g cm )
10
Maximum hybrid star quadrupoles
generic
generic´
Hy1
Hy1´
LKR1
42
10
-3
10
41
-4
10
40
10
-5
10
39
10
-6
10
38
10
-7
10
1.2
1.3
1.4
1.5
1.7
1.6
M (solar masses)
1.8
1.9
2
εLIGO
2
Q22 (g cm )
10
Conclusions
I
Deformed neutron stars are an observationally convenient
potential source of gravitational waves.
I
We have made the first relativistic calculation of the
maximum elastic quadrupoles of neutron stars, as well as the
first careful calculation of the shear modulus of hadron–quark
hybrid matter, and found that:
I
Hybrid stars can sustain large (i.e., detectable in current
LIGO/Virgo searches) quadrupoles for reasonable parameter
values...
...even with the relativistic suppression of the quadrupole for
large masses.
I
I
This suppression is due to “no-hair”-type considerations, in
addition to the increased gravity of relativistic stars.
Conclusions (cont.)
I
The relativistic suppression is even more severe for crustal
quadrupoles, due to the sharp changes in density at the
crust-core transition.
I
Additionally, hybrid star deformations can also store a large
amount of elastic energy. While we have not yet made a
careful calculation of this (due to technical complications),
naı̈ve estimates suggest that the maximum will be around
1049 ergs (applying the Horowitz-Kadau breaking strain). If a
significant fraction of this energy was released in GWs, it
could have been detected in certain LIGO/Virgo searches for
GWs coincident with magnetar flares.