Gravitational radiation from triaxial neutron stars: Implications for data analysis

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Gravitational radiation from triaxial neutron
stars Implications for data analysis S. Frasca
C. Palomba Universita La Sapienza and INFN
Roma1, Roma, Italy
Abstract The rigid crust of neutron stars can
sustain triaxial shapes. The compact object
detected in SN 1987 A could be an example of such
kind of objects. The gravitational signal emitted
by a triaxial neutron star is is characterized by
a frequency modulation which produces a splitting
of spectral lines. We investigate the
implications of frequency modulation for data
analysis discussing, in particular, how it
impacts on the hierarchical procedure for
detection of periodic sources. Here we assume the
reader knows how the hierarchical method for
detection of continuous waves works.
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• Motion of a triaxial body
• The free motion of a triaxial body has three
  components
• rotation around a principal axis of inertia (
  )
• rotation around the fixed direction of the
  angular momentum ( )
• nutation ( )
• We define two asimmetry parameters
• ellipticity
• oblateness

For oblate bodies ( ) it must be We
assume being the
rotational energy of the star.
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The analytic expressions for the Euler angles are
the following
Jacobi functions
Period of Jacobi functions
Complete elliptic integral of the 1st kind
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• the time derivatives of the Euler angles are
  periodic functions of time (with period
  )
• the global motion of the body is not periodic.
• these properties disappear for an axisymmetric
  body
• ( ) or for zero wobble
  angle.

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For realistic values of ellipticity and
oblateness As a consequence, the gravitational
signal is emitted, basically, at frequencies
and . The frequency modulation
will produce a splitting of the spectral lines.
We indicate with the modulation frequency
and with the amplitude of the
modulation. We find where
are functions of the initial wobble angle. The
number of side frequency lines and their power
content are connected to the modulation index
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For two side bands appear at
frequencies For four side
bands appear at frequencies
. The fraction of power contained at
frequencies is,
respectively, . Regarding the
dependency of on the initial wobble angle,
it must be noted that the maximum wobble angle
that the crust of a neutron star can sustain
without cracking can be estimated (Jones and
Andersson) as Where the maximum strain
is largely unknown but probably in the range
.
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As a consequence, depending on the initial
frequency a given range of initial wobble
angles is permitted. In particular, for higher
rotation rates the range of possible initial
angles is smaller. This implies that the
modulation index takes larger values for smaller
rotation frequencies. In the following graphs
the shift frequency and the modulation
index are plotted as a function of the
ratio between the oblateness and the
eccentricity, for different values of the initial
gravitational wave frequency and of the initial
wobble angle. We assume
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Curves of different style/color correspond to
different initial values of the gravitational
wave frequency. For each frequency, three curves,
corresponding to different values of the initial
wobble angle, are plotted. The three values of
wobble angle span the range of allowed initial
values. Note that in the plot of b the curves
for frequencies below 125Hz are no more dependent
on the frequency.
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• Impact on data analysis
• We describe now how a frequency modulation could
  impact on the hierarchical procedure for
  detecting periodic gravitational signals.
• First step
  
  In the first step of the hierarchical procedure
  the FFT length is chosen, for each frequency
  band, in such a way that the power of a signal
  would be all confined into a single frequency
  bin. This condition gives an upper limit to the
  duration of FFTs
• The power spread due to the side bands puts a
  stronger limit on the maximum length of FFTs

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• From the previous plots we see that
• the modulation index takes high enough values
  (gt0.1) for some ranges of the ratio h/e at
  frequencies below 500Hz
• the frequency shift is almost everywhere larger
  than the no modulation bin width.
• We conclude that frequency modulation could
  really affect the choice of FFTs length (at least
  if braking strains near the upper limit are
  considered).
• The length (or, equivalently, duration) of FFTs
  at the first step of the hierarchical procedure
  (remember it is taken as the nearest power of 2
  to the maximum length) determines the sensitivity
  of the whole procedure
• The minimum detectable amplitude increases, with
  respect to the
• case of optimal search, by a factor
  where is the total observation time.

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• If, due to modulation, we have to decrease the
  duration of FFTs built in the first step, a
  further loss of sensitivity will follow.
• Second step
• Candidates selected at the end of the first step
  are followed in the second step. Correcting the
  data according to the parameters of candidate
  sources allows us to build longer FFTs and
  obtain, at the end, more refined candidates.
• However, if side peaks of a gravitational signal
  are present they will be no more confined in a
  frequency bin. This could imply a loss in
  signal-to-noise ratio proportional to the power
  content of side peaks.

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• Third step
• In the third step the signal-to-noise ratio
  increases again because no further dispersion of
  signal power takes place. The false alarm
  probability at the end of the third step
  decreases to negligible values.
• If enough power is contained in side peaks they
  could produce new source candidates. If one
  realize that different candidates are, in fact,
  the same source emitting at more than one
  frequency, a lot of information on the source
  parameter can be obtained.
• Concluding remark we have argued that
  gravitational emission from triaxial neutron
  stars can affect the data analysis procedure. We
  have shown what we expect to happen in the
  various steps of the hierarchical method. Details
  on what to do in practice must be still worked
  out.

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• Conclusions
• the gravitational radiation emission from
  triaxial rotating neutron stars poses interesting
  problems for data analysis
• the sensitivity could be reduced with respect to
  the case of an axisymmetric star, due to the
  spread of signal power into side peaks
• a lot of information on the star parameters can
  be obtained if a frequency modulated signal is
  detected
• the compact remnant detected in SN 1987 A could
  be an example of a triaxial neutron star
  (spinning down mainly due to gravitational
  radiation emission)
• work in progress