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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 orbital radius evolves as
The frequency increases
CHIRP
Flusso sistema binario
For binary systems far from coalescence the quadrupole
formalism works.
could these signals be detectable by LISA?
hc  10 23
  1.4 10 4 Hz
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 1913+16
hc  10
23
  1.4  104 Hz
Remember that we are computing the
radiation emitted because of the
Orbital motion ONLY
Flusso sistema binario
could these signals be detectable by LISA?
Cat. Var.
PSR 1913+16
THERE IS HOPE !
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
GW’s 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
Sistemi planetari extrasolari 1
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: P=88 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?
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 formula+point 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:
g   g 0   h
g 0 
    
p  p  p
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 << M*
We perturb Einstein’sequations + Eqs. of Hydrodynamics
We expand
h
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 << 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 r=0, 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
The quasi-normal modes of stars are classified depending on the
restoring force which is prevailing
g - modes
f – mode
..   g n  ..   g1   f   p1  ..   pn  ..
p - modes
w ‘pure spacetime oscillations’
A mode of the star can be excited if the mode frequency
and the orbital frequency
are related by the constraint
k 
G( M *  M p )
R
3
i
(circular orbit)
 k  2i
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
Modi quasi-normali
The quasi-normal modes of stars:
are classified depending on the restoring force which is prevailing
g - modes
..   g n  ..   g1   f   p1  ..   pn  ..
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
LISA
A Brown Dwarf : can stay, for instance, on an orbit resonant with the
mode g4 emitting waves with an amplitude > 2x10-20 for 3 years
Juppiter : g10 mode – with amplitude > 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)
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 Einstein’s 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 << 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)
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  < Mtot < 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-newtonian+perturbative: the signal must be
modeled as a function of (a2 , a2, m1, m2 ), and of the orbital parameters.
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
Perturbative approach:
True star + point mass
We perturb Einstein’s eqs.
+ Hydrodynamical eqs.
We solve them numerically
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 > 0.2
Last 20-30 cycles before
Coalescence!
P(v)=
EGW
/
EORB
picch
i
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
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
EURO
EURO - Third Generation GW Antenna
In May 1999 the funding agencies in Britain, France, Germany and Italy commissioned scientists involved in the
construction and operation of interferometric gravitational wave detectors in Europe (GEO and VIRGO) to prepare a vision
document to envisage the construction of a third generation interferometric gravitational wave detector in Europe on the time
scale of 2010.
In conclusion: gravitational radiation can be studied by using different approaches
-
Quadrupole formalism
Perturbations about exact solutions
Numerical simulations in full GR
1) 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