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Astron. Astrophys. 327, 428-431 (1997)

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2. Stochastic GR background produced by rotating single NS

Let us assume a stationary situation, i.e. that the number of NS in the Galaxy is determined by a constant formation rate [FORMULA]. Assuming the present star formation rate in the Galaxy 1 [FORMULA] /yr, Salpeter mass function [FORMULA] (Salpeter 1955), and the minimal mass of the star [FORMULA] [FORMULA] (such a choice yields the total stellar mass in the Galaxy [FORMULA] during [FORMULA] years), we find that the mean formation rate of massive stars ([FORMULA] 10 [FORMULA] to produce NS) are about 1 per 30 years. Below we shall normalize all calcualtions to this rate, [FORMULA] yr [FORMULA] s-1.

A rotating non-axisymmetric NS with the ellipticity [FORMULA] loses energy in the form of GR at a rate (Shapiro & Teukolsky 1983)

[EQUATION]

where G and c are the Newtonian gravity constant and speed of light, [FORMULA] is the NS rotation frequency, I is the moment of inertia.

If the NS were emitted GR at strictly twice rotational frequency and this frequency were not changed, in principle each star may be distinguished by an ideal detector provided its frequency band is sufficiently narrow. However, the rotating NS radiates both at [FORMULA] and [FORMULA] (and possibly at other higher harmonics if its form is more complex) and its rotational frequency are constantly changing by the energy conservation law

[EQUATION]

where we have explicitly written down possible rotational energy losses - electromagnetic ([FORMULA]) and others. In the ideal case of GR being the only source of energy loss we would retain only first term in the expression above. We should note that the spin evolution of a magnetized NS becomes much more complicated when the NS is in a binary system (e.g. Lipunov 1992); however, their fraction among the total number of NS is hardly higher than [FORMULA], and we will not consider them here.

As laser interferometers are broad-band detectors ([FORMULA]), a large number of sources within the sensitivity band would produce a stochastic background. Long-term continuous observations, however, allow to make the sensitivity band efficiently narrower provided that the faweform of the signal is known (in fact, as [FORMULA], where T is the integration time; this permits to increase signal-to-noise ratio for continous source observations as [FORMULA] using match filtering thechnique). For old NS, however, match filtering technique of data analysis would be excessively demanding for modern computers (Schutz 1997) (a priori we do not know the form of the signal, its frequency, source location on the sky, etc.), so for old NS the interferometer always works as a broad-band detector.

Clearly, the condition that a stochastic signal appears within the detector band reads

[EQUATION]

where [FORMULA] is the time for a typical source to pass through the detector band [FORMULA]. This time is determined by a particular mechanism of energy losses, and we examine it separately for GR and electromagnetic losses.

1. Elecromagnetic losses. They are described by the law

[EQUATION]

Here [FORMULA] is the dipole magnetic moment of the NS. The solution to the equaiton (4) reads

[EQUATION]

Assuming [FORMULA] we find the upper frequency of the stochastic background

[EQUATION]

with [FORMULA].

2. GR losses. These are

[EQUATION]

Then

[EQUATION]

and under the same assumption about [FORMULA] we find

[EQUATION]

with [FORMULA].

Therefore, for plausible values of the NS magnetic fields ([FORMULA] - [FORMULA]) and ellipticities ([FORMULA] - [FORMULA]), at any frequency [FORMULA] [FORMULA] Hz we deal with stochastic backgrounds from galactic NS. Physically, this is due to the inability of old NS to leave frequency interval [FORMULA] during the typical time between consecutive supernova explosions. This is not so for young NS (see, e.g., Lai & Shapiro 1995).

Now we ask the question: how many sources with changing frequency are to be simultaneously observed within a frequency interval [FORMULA]? The answer is immediate: Under stationary conditions the continuity equation implies

[EQUATION]

Here we assumed that all sources come into the interval through its upper boundary. This assumption is correct if the upper boundary of the interval lies sufficiently far from the initial frequency of NS (i.e. less than about 100 Hz). Now, if the number of sources within a frequency interval [FORMULA] interval is more than unity, the resulting GR signal at the frequency [FORMULA] would read

[EQUATION]

where dimensionless strain amplitude from one source relates to the energy flux [FORMULA] at the frequency [FORMULA] as

[EQUATION]

where r is the distance to the source.

If we perform a long-term search for the background ([FORMULA] day) and thus not interested in the possible modulation of the signal by Earth rotation or if old NS populate an extended halo, Eq. (11) may be rewritten in the form

[EQUATION]

where [FORMULA] and [FORMULA] is the inverse-square average distance to the typical source. Using Eq. (2) and (10), we obtain to within an unimportant trigonometrical coefficient of order unity (see Giazotto et al. (1997) for more detailed calculations of this factor)

[EQUATION]

where we omitted all but electromagnetic and GR loss terms. For purely GR-driven NS spin-down the resulting spectrum is independent of the unknown value of [FORMULA] in the NS population. (The independency of the resulting signal on the ellipticity when the pulsar spin-down is governed by GR losses only was noted by Thorne (1987) with the reference to private communication from R. Blandford in 1984). Note that any additional braking mechanism always lowers the resulting signal. For example, taking typical values [FORMULA] g cm2, [FORMULA] yr-1 we obtain

[EQUATION]

(here we assumed the characteristic distance to NS population of order 10 kpc). The GR background of such strength could be detected by the advanced LIGO/VIRGO interferometers in one year integration (Thorne 1987; Giazotto 1997).

What kind of losses governs the NS spin evolution for realistic NS parameters? The ratio of electromagnetic losses to GR losses [FORMULA] is

[EQUATION]

with A and B determined as above, and for typical parameters [FORMULA] and [FORMULA] we find

[EQUATION]

that is electormagnetic losses becomes insignificant only at frequencies

[EQUATION]

i.e. they dominate practically always. If we would take [FORMULA] and [FORMULA] as in millisecond pulsars, we would obtain [FORMULA] Hz, however millisecond pulsars are spun up by accretion in binary systems and are not considered here.

Therefore, for realistic NS we must consider the case [FORMULA]. Using Eqs. (14) and (17) we derive that the stochastic background from old NS is

[EQUATION]

Note the frequency dependence appeared in this expression. If we take the estimate of magnetic field from observations of pulsar periods P and their change rate [FORMULA]: [FORMULA] (here [FORMULA]) and assume maximum possible ellipticity allowed by P - [FORMULA] observations: [FORMULA], Eq. (19) immediately yields the same Eq. (15) for [FORMULA] as above.

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© European Southern Observatory (ESO) 1997

Online publication: April 8, 1998
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