Coalescenza quadrupolo

Coalescing binary systems: main target of ground based interferometers

quadrupole approach: point masses on a circular orbit + radiation reactionIf the two stars have different masses

reduced massThe frequency increasesThe orbital radius evolves as

CHIRP

For binary systems far from coalescence the quadrupole formalism works.

could these signals be detectable by LISA?

Hz

hc4

23

104.1

10

Flusso sistema binario

CATACLISMIC VARIABLES semi-detached with small orbital period

Primary star: White dwarf Secondary: star filling its Roche-lobe and accreting matter on the companion

Remember that we are computing theradiation emitted because of the Orbital motion ONLYHz

hc4

23

104.1

10

There exist other sources which may be interesting for LISA

PSR 1913+16

could these signals be detectable by LISA?

Flusso sistema binario

THERE IS HOPE !

PSR 1913+16

Cat. Var.

the quadrupole formalism assumes that

BINARY PULSAR PSR 1913 + 16 OK

PULSATING NEUTRON STARS

WHEN THE SYSTEM IS CLOSE TO COALESCENCE the condition is no longer satisfied: STRONG FIELD EFFECTS

Sistemi planetari extrasolari 1 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 starsEXTRASOLAR PLANETARY SYSTEMS

Discovery: 1992 (Wolsczan & Frail)(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 FEATURESPECULIAR FEATURES::More than 1/3 orbit at a distance smaller than that of Mercury from the Sunsmaller 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?

-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:

Sistemi planetari extrasolari 2

hgg 0

0g

I

Is an exact solution of Einstein + Hydro eqs. (TOV-equations)which describes the central sun-like star

We assume that the star is perturbed by the planet which moves on a circularor eccentric orbit. This is a reasonable assumption because Mp << M*

ppp

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

We perturb Einstein’sequations + Eqs. of Hydrodynamics

We expand in tensor spherical harmonics and separate the equationsh

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

i

3* )(

RMMG p

k

A mode of the star can be excited if the mode frequency

and the orbital frequency (circular orbit)are related by the constraint

ik 2

We find which non-radial mode, i.e. which quasi-normal mode, can be excited

The quasi-normal modes of starsquasi-normal modes of stars are classified depending on the restoring force which is prevailing

g - g - modes

f – f – mode

p - p - modes

w w ‘pure spacetime oscillations’

........11

nn ppfgg

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)

The quasi-normal modes of stars: quasi-normal modes of stars: are classified depending on the restoring force which is prevailing

g - g - modes f – f – modep - p - modes w w ‘pure spacetime oscillations’

........11

nn ppfgg

How much time can a planet stay close to a resonance?

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

The orbital energy is a known function of R0 (geodesic equations)

Modi quasi-normali

The grav. Luminosity is found byNumerical integration

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

LISA

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)

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

Quadrupole formalism + Post-Newtonian corrections Describe with extreme accuracy the coalescence of BLACK HOLES (point masses)

PN -formalism

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

In conclusion: For coalescing, non rotanting BLACK HOLES we know how todescribe the signal up to the ISCO (Innermost Stable Circ. Orbit)

1) What happens after the ISCO is reached?2) What do we know about GW emitted by rotating black holes?

Few events per year detectable by LIGO and VIRGO for systemswith 20 M < Mtot < 40 M

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.

fully non-linear numerical simulations to describe the merging(Grand-Challenge, Potsdam) + perturbative approaches for the quasi-normal mode ringing

The detection of this part of the signal using these templates will allow to determine the total mass of the system

Conclusioni buchi neri

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 theQuadrupole formalism : point masses in circolar orbit + radiation reaction

When they reach distances of the order of 3-4 stellar radiithe orbital part of the emitted energy can be refined by computingthe 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

picchiPerturbative approach:

True star + point mass

We perturb Einstein’s eqs. + Hydrodynamical eqs.

We solve them numerically

We find that differences with respect to black holes due to the

internal structure appearwhen v/c > 0.2

Last 20-30 cycles beforeCoalescence!

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

P(v)= EGW / EORB

discussione

Why are we interested in effects that are so small?

Our knowledge of nuclear interactions at supranuclear densities is verylimited: 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 constraintson 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 consequentlyon 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 AntennaIn 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)

3) The merging phase has to be studied through fully non linear numerical simulations

4) About the excitation of quasi-normal modes, we need tounderstand how the energy is distributed among them in astrophysical situations: known sources need to be studied in much more detail

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