Coalescenza quadrupolo

1
Coalescenza quadrupolo
? Coalescing binary systems main target of
ground based interferometers
quadrupole approach point masses on a circular
orbit
radiation reaction If the two stars have
different masses
reduced mass
The frequency increases
The orbital radius evolves as
CHIRP
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Flusso sistema binario
For binary systems far from coalescence the
quadrupole formalism works.
could these signals be detectable by LISA?
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There exist other sources which may be
interesting for LISA
CATACLISMIC VARIABLES semi-detached with
small orbital period Primary
star White dwarf Secondary
star filling its Roche-lobe and accreting matter
on the companion
PSR 191316
Remember that we are computing the radiation
emitted because of the Orbital motion ONLY
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Flusso sistema binario
could these signals be detectable by LISA?
Cat. Var.
PSR 191316
THERE IS HOPE !
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the quadrupole formalism assumes that
? BINARY PULSAR PSR 1913 16 OK
? WHEN THE SYSTEM IS CLOSE TO COALESCENCE the
condition is no longer satisfied STRONG FIELD
EFFECTS
? PULSATING NEUTRON STARS
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Sistemi planetari extrasolari 1
GWs emitted by a binary systems carry
information not only on the features of the
orbital motion, but also on processes that may
occur inside the stars

• EXTRASOLAR PLANETARY SYSTEMS
• Discovery 1992 (Wolsczan Frail)
• Since then 60 have been discovered in our
  neighbourhood
• Solar type star one or more planets
• 46 with mass 0.16-11 Juppiter mass
• 12 with bigger masses (brown dwarfs)

PECULIAR FEATURES More than 1/3 orbit at a
distance smaller than that of Mercury from the
Sun Some of them have an orbital period of the
order of hours (Mercury P88 days) Mass and e
radius of the central star mass and orbital
parameters of the planets can be inferred from
observations THEY ARE VERY CLOSE TO US!!! D ?
10 pc
Could a planet be so close to the central star as
to excite Its proper modes of oscillation?
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Sistemi planetari extrasolari 2

• Could a planet be so close to the central star as
  to excite its proper
• modes of oscillation?
• How much energy would be emitted in GWs by a
  system in this
• resonant condition with respect to the energy
  due to orbital motion
• (quadrupole formulapoint particle
  approximation)?
• - for how long can a planet stay in this resonant
  situation?

A more appropriate formalism to describe these
phenomena is based on a
PERTURBATIVE APPROACH
Is an exact solution of Einstein Hydro eqs.
(TOV-equations) which describes the central
sun-like star
I
We assume that the star is perturbed by the
planet which moves on a circular or eccentric
orbit. This is a reasonable assumption because Mp
ltlt M
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We perturb Einsteinsequations Eqs. of
Hydrodynamics
We expand in tensor
spherical harmonics and separate the equations
We obtain a set of linear equations, in r and t,
which couple the perturbations of the metric with
the perturbations of the thermodynamical variables
On the right hand side of the equations there is
a forcing term the stress-energy tensor of the
planet moving on a circular or elliptic
orbit the planet is assumed to be a point mass
Mp ltlt M.
The perturbed equations are solved numerically
to find the GW signal
As a first thing we find the frequencies of the
quasi-normal modes they are solutions of the
perturbed equations, which satisfy the condition
of being regular at r0, and that behave like a
pure outgoing wave at radial infinity. They
belong to complex eigenfrequencies the real part
is the pulsation frequency, the imaginary part is
the damping time, due to the emission of
Gravitational Waves
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The quasi-normal modes of stars are classified
depending on the restoring force which is
prevailing g - modes f mode p - modes
w pure spacetime oscillations
A mode of the star can be excited if the mode
frequency and the orbital frequency

(circular orbit) are related by the
constraint
We put the planet on a circular orbit at a given
radius and check, by a Roche-lobe analysis, if
it can stay on that orbit without being disrupted
by the tidal interaction, i.e. without accreting
matter from the star (and viceversa)
We find which non-radial mode, i.e. which
quasi-normal mode, can be excited
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Modi quasi-normali
The quasi-normal modes of stars are classified
depending on the restoring force which is
prevailing g - modes f mode p - modes
w pure spacetime oscillations
A planet like the Earth can stay on an orbit such
as to excite a mode g4 or higher, whithout
melting or being disrupted by tidal forces A
Juppiter like planet can excite the mode g10 or
higher
How much time can a planet stay close to a
resonance?
The orbital energy is a known function of R0
(geodesic equations)
The grav. Luminosity is found by Numerical
integration
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LISA
A Brown Dwarf can stay, for instance, on an
orbit resonant with the mode g4 emitting waves
with an amplitude gt 2x10-20 for 3 years Juppiter
g10 mode with amplitude gt 3x10-22 for 2
years
V. Ferrari, M. D'Andrea, E. Berti Gravitational
waves emitted by extrasolar planetary
systems Int. J. Mod. Phys. D9 n.5, 495-509
(2000) E. Berti,V. Ferrari Excitation of g-modes
of solar type stars by an orbiting
companion Phys. Rev. D63, 064031 (2001)
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PN -formalism
Can we obtain better estimates of the radiated GW
for binary systems close to coalescence?
Post-Newtonian formalism The equations of
motion and Einsteins eqs are expandend in
powers of V/c to compute energy flux and
waveforms. In this manner the treatment of the
radiation due to the orbital motion is refined

• NON-ROTATING BODIES
• - test-particle (m1 ltlt m2) everything is
  known up to (V/c)11
• equal masses
• orbital motion up to (V/c)6 (3PN) beyong
  Newtonian acceleration
• GW- emission up to (V/c)7 (3.5PN) beyond the
  quadrupole formula

Quadrupole formalism Post-Newtonian corrections
Describe with extreme accuracy the coalescence
of BLACK HOLES
(point masses)
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Conclusioni buchi neri
In conclusion For coalescing, non rotanting
BLACK HOLES we know how to describe the signal up
to the ISCO (Innermost Stable Circ. Orbit)
The detection of this part of the signal using
these templates will allow to
determine the total mass
of the system
Few events per year detectable by LIGO and VIRGO
for systems with 20 M ? lt Mtot
lt 40 M ?
1) What happens after the ISCO is reached? 2)
What do we know about GW emitted by rotating
black holes?
fully non-linear numerical simulations to
describe the merging (Grand-Challenge, Potsdam)
perturbative approaches for
the quasi-normal mode ringing
Much work to do post-newtonianperturbative
the signal must be modeled as a function of (a2
, a2, m1, m2 ), and of the orbital parameters.
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Pert. Stelle di neutroni1
WHAT DO WE KNOW ABOUT THE COALESCENCE OF
NEUTRON STARS?
When they are far apart, the signal is correctly
reproduced by the Quadrupole formalism point
masses in circolar orbit radiation reaction
When they reach distances of the order of 3-4
stellar radii the orbital part of the emitted
energy can be refined by computing the
post-newtonian corrections (same as for BH)
At these distances, the tidal interaction may
excite the quasi-normal modes of oscillation
of one, or both stars
This process can be studied by a
perturbative approach
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Perturbative approach True star point
mass We perturb Einsteins eqs.
Hydrodynamical eqs. We solve them numerically
P(v) EGW / EORB
picchi
We compute the orbital evolution, the waveform
and the emitted energy for different
EOS Gualtieri, Pons, Berti, Miniutti, V.F. Phys.
Rev D, 2001, 2002
We find that differences with respect to black
holes due to the internal structure appear when
v/c gt 0.2 Last 20-30 cycles
before Coalescence!
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discussione
Why are we interested in effects that are so
small?
Our knowledge of nuclear interactions at
supranuclear densities is very limited we do not
know what is the internal structure of a NS
Observations allow to estimate the mass of NS
(in some cases) but not the RADIUS we are
unable to set stringent constraints on the EOS of
nuclear matter at such high densities.
If we could detect a clean GW signal coming
from a NS oscillating in a quasi-normal mode, we
could have direct information on its internal
structure and
consequently on the EOS of matter in extreme
conditions of density and pressure
unaccessible from experiments in a laboratory
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Phase transitions from ordinary nuclear matter
to quark matter, or to Kaon-Pion condensation,
occurring in the inner core of NS at
supranuclear densities, would produce a density
discontinuity. A g-mode of oscillation would
appear as a consequence
Miniutti, Gualtieri, Pons, Berti, V.F. Non radial
oscillations as a probe of density discontinuity
in NS
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EURO
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• In conclusion gravitational radiation can be
  studied by using different approaches
• Quadrupole formalism
• Perturbations about exact solutions
• Numerical simulations in full GR

• To study the coalescence of BH-BH binaries
  post-newtonian calculations have to be extended
  to the rotating case (already started)

2) To study the coalescence of NS-NS or NS-BH
binaries, the perturbative approach has to
be generalised to the case of equal masses and
to rotating stars
3) The merging phase has to be studied through
fully non linear numerical simulations
4) About the excitation of quasi-normal modes, we
need to understand how the energy is distributed
among them in astrophysical situations known
sources need to be studied in much more detail