A&A 394, 489-503 (2002)
DOI: 10.1051/0004-6361:20021150
G. M. Beskin^{1,3} - A. V. Tuntsov^{1,2}
1 - Special Astrophysical Observatory, Nizhnij Arkhyz,
Karachaevo-Cherkesia, 369167, Russia
2 - Sternberg Astronomical Institute of the Moscow State University,
Moscow, Russia
3 - Isaac Newton Institute of Chile, SAO Branch programme, Russia
Received 27 November 2001 / Accepted 20 June 2002
Abstract
We consider the gravitational magnification of light for binary
systems containing two compact objects: white dwarfs, a white dwarf and
a neutron star or a white dwarf and a black hole. Light curves of the flares
of the white dwarf caused by this effect were built in analytical
approximations and by means of numerical calculations. We estimate the
probability of the detection of these events in our Galaxy for different
types of binaries and show that gravitational lensing provides a tool for
detecting such systems. We propose to use the facilities of the Sloan
Digital Sky Survey (SDSS) to search for these flares. It is possible to
detect several dozen compact object pairs in such a programme over 5
years. This programme is apparently the best way to detect stellar mass
black holes with open event horizons.
Key words: cosmology: gravitational lensing - black hole physics - stars: binaries: close - stars: neutron - stars: white dwarfs
One of the most important manifestations of gravitational lensing is the visible variability of the astrophysical objects whose emission is affected by the gravitation field of the lens. This effect becomes observable over a reasonable time period, when the velocities of relative motions of the observer, the lens and the source are great enough, i.e. their mutual location changes rapidly. Many cases of this kind of variability have been examined: from light variation of distant quasars to outburst of stars caused by the influence of planets (see, for instance, Zakharov 1997). The brightness variation of the binary system components caused by gravitational lensing was been first considered by Ingel (1972, 1974) and Maeder (1973). They noted that repetition and small characteristic times of the effect make it one of the most accessible to detection and detailed study. It has also been shown (Maeder 1973) that the large amplitudes of brightness variations ( ) can be expected in binary systems consisting of compact objects - white dwarfs, neutron stars, black holes. In fact, only in these cases does the Einstein-Hvol'son radius turn out to be smaller than the lens star radius, and the image of the radiation source is practically not overlapped by it. The latter circumstance makes the gravitational lensing influence in binary systems an effective means for the detection and detailed investigation of compact objects with the aid of photometric methods alone. Their permeability is by higher (as compared to spectroscopy), and, therefore, the number of stars amenable to study is an order of magnitude larger. Here the light curve analysis determines all the parameters of the binary system in full analogy to this task for eclipsing binary systems (with allowance made for its singularity) (see, for example, Goncharsky et al. 1985).
A complete set of data for compact object binaries and their statistical analysis gives a possibility to test binary evolution theories and to investigate the last stages of single star evolution as well. We note that only 10 double white dwarfs were detected during the last 10 years (Maxted et al. 2000) and about 40 pairs of a white dwarf and a neutron star - during 25 years (Thorsett & Chakrabarty 1999). We further note that by means of gravitational lensing it is possible to detect compact object binaries at the same rate (see Sect. 5). Moreover, this may help to discover binaries containing neutron stars with a low magnetic field (not pulsars). Study of these systems in combination with binary star evolution theories gives a possibility to test the detailed cooling models for white dwarfs and neutron stars and the equation of state of the latter (Nelemans et al. 2001; Yakovlev et al. 1999).
Studying compact object pairs is very important to solve a number of astrophysical problems. We present below several such examples.
The "standard candle" of modern cosmology, type Ia supernovae, apparently arises from merging of double CO white dwarfs (Webbink 1984; Iben & Tutukov 1984), which have not been detected at this time.
Close double compact objects have to contribute a significant part of the gravitational wave signal at low frequencies. Thus, white dwarf pairs may be a source of the unresolved noise (Evans et al. 1987; Grischuk et al. 2001). Statistical properties of the closest double compact objects in principle determine parameters of the gravitational wave signal.
Binary systems consisting of a white dwarf and a neutron star (a pulsar) are unique laboratories for high-precision tests of general relativity. To date, post-Keplerian general relativistic parameters have been measured in four such pairs (Thorsett & Chakrabarty 1999). This problem may be solved very easily for binaries with gravitational self-lensing since they are observed nearly edge-on.
It seems that the detection of black holes forming pairs with white dwarfs might be an extraordinarily important result of the search for gravitational lensing in compact object binaries. Despite the opinion of most researchers, in a certain sense black holes have not been discovered so far. Only observational data on the behaviour of matter close to the event horizon showing its presence may testify that a black hole is identified (Damour 2000). There is indirect evidence of the presence of black holes in X-ray binaries and galactic cores based on the "mass-size" relation expected for them (Cherepashchuk 2001). The horizon neighbourhood is seen neither in X-ray binaries nor in the cores of active galaxies because they are screened out by the accreted gas (the accretion rates are very high). This means that only black holes accreting usual interstellar plasma at low rates of /year can be recognized as objects without a surface and with an event horizon (Shvartsman 1971). The black hole companions of white dwarfs in the binaries detected by means of gravitational lensing could be the best objects for horizon study and tests of general relativity in the strong field limit (Damour 2000). It is very easy to measure the mass and size of such a black hole due to binary edge-on orientation and investigate the radiation of gas near the horizon. This matter is accreted from interstellar medium only because its transfer from the white dwarf is absent.
The gravitational lensing effects in binary systems consisting of compact components were studied previously in several papers (Maeder 1973; Gould 1995; Qin et al. 1997). However, the authors did not go further than estimation of the probability of detecting the effect of orientation of the binary system (often underestimating it by a factor of 2-3), which turned out to be very low. Their general conclusion is that the effect cannot be actually recorded. Nevertheless, the data accumulated by now on the evolution of binary systems and their parameters make it possible to define with higher accuracy the probability of detection of the brightness enhancement in binary systems, to find the expected number of such flares and to propose in the final analysis the strategy for their search based on present-day facilities. Our paper is devoted to performing these tasks.
We examine in Sect. 2 the distinguishing features of light magnification in the systems consisting of two white dwarfs, a white dwarf and a neutron star, and also a white dwarf and a black hole. In Sect. 3 we derive probabilities of recording the effect for its different amplitudes in all three cases. In Sect. 4 the numbers of systems of different types which may be detected by means of gravitational lensing are estimated. In Sect. 5 we discuss the possibilities of the quest for brightness magnification in such systems with the aid of the telescope and equipment used in the survey SDSS (York et al. 2000).
We will consider three
types of binary systems consisting of
a) two white dwarfs with masses of
(sometimes
,
b) a white dwarf (
)
and a neutron star
(
),
c) a white dwarf (
)
and a black hole (
).
It is clear that both components of a pair play the part of a gravitational
lens alternately. However, we will mention at once that the source
of radiation will always be the white dwarf to which the approximation
of a uniformly luminous disk can be applied. The amplitude, the shape and
the duration of flares of the lensed object are defined by the relationships
between the parameters of the binary system, the major semiaxis a, the masses
and radii of the components and the orbital plane orientation.
The luminosity magnification of a uniformly radiating disk in gravitational lensing was analysed in detail in the papers by Refsdal (1964), Liebes (1964), Byalko (1969), Ingel (1972, 1974), Maeder (1973) and Agol (2002). On account of the conservation of the surface brightness, the task is reduced in the final analysis to a study of variations of the source image area with allowance made for its occultation by an opaque lens, as the source and lens are moving with respect to the observer (in our case - the orbital motion around the mass centre of the binary system).
Let be the distance from the observer to the gravitational lens, a is the binary system orbit semiaxis (for the sake of simplicity we consider only circular motions), is the lens mass, is its gravitational radius, is the lens radius, is the source radius, is the source mass.
In our case for the light path we can use with high accuracy an approximation of a broken line, the angle between two sections of which is inversely proportional to the impact parameter.
The gravitational lens equation for a point-like source (see, for instance,
Zakharov 1997) is
(1) |
(2) |
(3) |
Figure 1: Schematic view of gravitational lensing. | |
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For the magnification coefficient K, i.e. the ratio of the sum of
brightness of the two images to the point-like source brightness,
we have (Zakharov 1997)
(4) |
For the WD radius we used the relationship (Nauenberg 1972):
(5) |
Pair | |||||
() | |||||
a = 10 | a = 100 | ||||
WD+WD | 0.7 | 10^{-2} | 10^{-2} | ||
WD+NS | 1.4 | 10^{-2} | |||
WD+BH | 10 | 10^{-2} |
It should be noted that when the binary system is viewed
edge-on, at the moment of conjunction, a configuration in which the
observer, the lens and the source are on one line (d = 0) is realized.
It was shown (Liebes 1964) that in this case
,
as a result of averaging of expression (4) over the surface
of the uniformly radiating disk. At
the task gets essentially
more complicated and has no analytical solution. The accurate relationships
for the brightness magnification by a point lens, where elliptical
Legendre integrals of the first and second kinds are used, were derived by
Refsdal (1964), Liebes (1964), Byalko (1969), Ingel (1972, 1974) and
Maeder (1973). There is, however, one more case in which a
simple expression for K exists - the contact between the disk edge
and the lens centre:
(6) |
The light curve of the component-source is the result of variation of
magnification with time variations of d during the system rotation period.
In the approximation of a point source, according to (4)
(7) |
(8) |
Figure 2: Magnification curve for the pair of white dwarfs. The crosshatched circle is the gravitational lensing component. | |
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It is customary to characterize the duration of the flare (brightness
increase) (see Zakharov & Sazhin 1998) by the time
that is taken
by the moving source projection onto the lens plane to cross the Einstein
ring. It can be easily found from the relation
,
that:
(9) |
Note that the brightness magnification of one of the
components by a factor of K corresponds to magnification of the system
brightness (which is the subject of detection in observations) K_{0}times, and
(10) |
Figure 3: Magnification curves of the pair of white dwarfs for different semiaxes a. Dashed lines represent the analytical approximation (7). Solid lines for numerical calculation; fine ones for transparent lens, thick ones for opaque lens. | |
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Present-day telescopes detect rather small light variations during short
exposures (see Sect. 5).
We are guided by these possibilities when determining the minimum
amplitude of detectable flares. Say, if the amplitude of a flare
(K = 1.2), then the trajectory of the source
projection onto the lens plane does not cross the Einstein ring at all,
and such a flare is excluded from the analysis. Therefore, it is necessary
to introduce
at which K = 1.1. Using (4) we derive
(11) |
(12) |
Comparison of the light curves in Fig. 3, plotted from the use of the results of direct calculations and with the use of point approximation (7), shows that they are virtually coincident for . In this case the light curves can be constructed by means of a point approximation. The exception is the region near the maximum at .
Consider the effect of gravitational lensing in a binary system comprising
two equal white dwarfs with
and
.
According to Kepler's third law (in solar units):
(13) |
From (2) and (9), considering that
,
obtain under the same condition (point approximation):
(14) |
Figure 4: Simulated light curves of the pair of white dwarfs. | |
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In Fig. 4 the light curves of this system for different i,
plotted from the data of numerical computations, are displayed.
The light variations are expressed in stellar magnitudes.
The orbit inclination i varies from zero to
with a step of
from the upper to
the lower curve, which is displaced for convenience. The flare shape
variations are due to these changes.
WD+WD | WD+NS | WD+BH | |||||
a_{c} | |||||||
() | () | (s) | () | () | (s) | () | |
4 | 14 | 44 | 0.92 | 2.5 | 12 | 0.13 | |
15.4 | 23 | 170 | 3.15 | 3.2 | 41 | 0.44 | |
62.4 | 49 | 690 | 11 | 10 | 141 | 1.52 |
Figure 5: Simulated light curves of the flares as a result of gravitational lensing in different pairs: a) WD-WD, b) WD-NS, c) WD-BH. Dashed lines show flares with amplitude , dotted lines are used for flares with amplitude and solid lines for flares with amplitude . | |
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One more step brings us closer to the real condition of searching for flares. Having fixed three levels of brightness increase, , and , we determined by two methods the minimum sizes of the systems consisting of two white dwarfs and also a white dwarf and a neutron star whose light rises to these levels (Table 2). From the relation , has been found analytically, while has been derived by direct calculations. The light curves of flares in these edge-on-viewed pairs, which were constructed based on numerical computations, are presented in Figs. 5a,b. It can be seen from Table 2 and this figure that the analytical estimates of the characteristic durations of flares are close to the results of direct calculations. The flares of a white dwarf coupled with a black hole constructed in a similar manner are exhibited in Fig. 5c (Table 2 contains only for this binary system), here the separation of the components , while the amplitudes correspond to the orbital plane inclinations 0.023, 0.015 and 0.01. It is precisely this value of a that was used in the computation since it is close to the minimum size of the pair WD-BH, which "survived" in the process of emission of gravitation waves during the lifetime of the Galaxy. The number of closer pairs drops sharply and the probability of their detection is very low (see Sect. 3).
Figures 5a-c shows that the characteristic durations of flares for pairs of white dwarfs are 50-200 s; for binaries containing NS and BH, the light variations are shorter, 30-130 s and 20-40 s, respectively. Singnificant variations of the flare shape are noticeable in all the cases.
Without dwelling on details, note that the set of parameters of these binary systems obtained in observations - light curves, periods, amplitudes, duration and shape of flares, can make it possible to determine physical characteristics of the components - their masses, sizes, temperatures (e.g. Cherepashchuk & Bogdanov 1995a,b). This problem can be resolved completely for two white dwarfs. At the same time, the determination of, at least, the mass of the lens component and, therefore, its identification as a NS or a BH seems to be self-important for the study of properties of relativistic objects and their extreme gravitation fields.
The number of systems that can be found as a result of brightness increases
in gravitation lensing is determined, first of all, by the probability
F(K_{0},t) of recording a flare with an amplitude larger than the level
K_{0}, during the observing time t of a sample of systems
distributed in some manner over parameters (orbital separation, masses and
component brightness).
This means that
F(K_{0},t) is the probability of a
flare detection from a certain system averaged over such a distribution
(15) |
The first event is consistent with the geometric probability which can be easily found. It follows from (4), (7), (8) and (11) that the values , where , correspond to the flares of the source with amplitudes exceeding K. The locus of the ends of the normals of the orbital planes satisfying this condition is a globe belt of width d_{0} and radius a/2, its area is . The locus of the ends of the normals of any orbit is the surface of a sphere of radius , its area is . Hence . Finally, from (2) .
Note that in determining ,
both analytical expression (11) and the numerical relationship can be applied.
Figure 6 shows the relationships between geometric probabilities and the size
of a system consisting of two white dwarfs which have been derived by
these two methods (analytically in a point approximation - the dashed
line, and numerically - the solid one). The contribution of the dwarf
lens is taken into account here. Minimum sizes of systems capable of
giving rise to a flare of amplitude K correspond to the values
in Table 2. The numerical calculations in systems WD-NS and WD-BH
are virtually the same as the analytical estimates, but
for WD-NS
reduces to zero at
.
Figure 6: Geometrical probability versus semiaxis size for the pair of white dwarfs. Dashed lines for analytical approximation, solid lines for numerical calculation. | |
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The probability
of recording a flare during the time t is
essentially dependent on the relationship of the period of recurrence
of flares T (given by (13)), the duration of the
flare
and the net continuous time of observing t.
Let us lay down certain conditions which must be satisfied to make the
search for flares efficient enough. First of all, it is evident that the
duration of an elementary continuous exposure
must be close to
that of flare .
On the other hand, the interval between the exposures
must be considerably shorter than .
Implying by
the duration of the total exposure when observing a specified sky
region, we obtain for the probability p_{t}:
(16) |
which corresponds to
(16') |
where is the semiaxis of binary with period T=t and we consider only since as .
Hence
(17) |
Due to the serious anisotropy of pair distributions on the sky, the optimum
time of observation of a specified sky field will be shown to be greater than
one night length (at least for WD-WD systems). In this case
a simple expression (16) for p_{t} no longer holds since observations on
individual nights become statisticaly independent.
The probability of detecting at least one flare in n nights is given
by the Bernoulli distribution:
where is the detection probability in (the observational night average length).
Thus, expression for
P(K_{0},n,a) takes the following form
(17') |
where .
For randomization corresponding to (15) we must convolve P(K_{0},t,a)or P(K_{0},n,a) with the pair distribution function in a.
Now we turn to the analysis of this distribution. We must clearly see the difference in distribution for binary systems, progenitors of pairs of compact objects, and the finite distribution of these pairs themselves. In the first case the results of observations yield a distribution uniform in logarithm of semiaxis (Abt 1983): . In the second case the situation is not so clear because of the uncertainty in quantitative characteristics determined in the course of different versions of population synthesis and because of the qualitative distinctions of these versions (Yungelson et al. 1994; Lipunov et al. 1996; Saffer et al. 1998; Portegies Zwart & Yungelson 1998; Fryer et al. 1999; Nelemans et al. 2001). For instance, in different papers the initial mass functions with different indices are used; due to the great uncertainty in the loss of matter, the supposed masses of BH progenitors vary from to . The finite numbers of, say, couples WD-BH contain an uncertainty of 5 orders of magnitude (see Fryer et al. 1999)! Nevertheless, since our analysis is qualitative, one can be content with the main characteristics of samples of objects and rough estimates.
First of all, it will be noted that couples lose energy and angular momentum
radiating gravitation waves (Landau & Livshits 1983; Grishchuk et al. 2001).
Since the rate of these losses depends dramatically on a (a^{-5}),
the stars in close pairs stick together and leave the sample. Using the
expression for gravitational luminosity of a binary system in a circular
orbit
(Landau & Livshits 1983) and taking into account
that the total energy of the system
,
one can
readily derive the value of
of the initial semiaxis of a system
that will merge during 10^{10} years, which is the life-time of the Galaxy:
Pair | |||
() | () | (hours) | |
WD+WD | 0.7+0.7 | 1.82 | 5.77 |
WD+NS | 0.7+1.4 | 2.40 | 7.11 |
WD+BH | 0.7+10 | 5.88 | 12.13 |
The results of population synthesis, which are partially confirmed by observations, suggest that the density destribution of pairs of compact objects rises with decreasing a approximately as , reaches a maximum at , after which it decreases rapidly to (Lipunov et al. 1996; Hernanz et al. 1997; Saffer et al. 1998; Portegies Zwart & Yungelson 1998; Wellstein & Langer 1999).
It follows from the same papers that another important parameter, the maximum size of a binary system , is known to be less than at small eccentricities.
With allowance made for the said above, we have used the function of
distribution in a in a simple form:
(18) |
(19) |
In our further estimations we will use distribution (18), varying the parameter a_{0} in the range , at two values of the parameter and .
Having fixed the distribution f(a), now we have to determine the lower limits of integrating in (15) for different types of binary systems. Referring to Tables 2, 3 and Fig. 6 and allowing for (17) and (18), it can be readily seen that the lower limits for systems consisting of two white dwarfs and a white dwarf in pair with a neutron star coincide with and are 14 and 2.5, respectively. However, for pairs consisting of a white dwarf and a black hole integration is done over the whole range of possible values of a, i.e. from 1 to . The values of binary periods corresponding to are 123, 7.6 and 0.85 hours for WD-WD, WD-NS and WD-BH systems respectively. This leads to the following expressions for flare detection probability obtained from (16), ( ), (17), ( ) and (18).
For WD-WD pairs
when
since
and
,
(20) |
(20') |
For WD-NS pairs
when
,
since
(21) |
when
,
since
(21a) |
(21') |
(22) |
(22a) |
(22b) |
(22') |
Figure 7: Probability F(t) of detecting a flare with as a function of the overall observation time of a specified sky region measured either in hours or in nights. | |
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The values of F(K_{0},t) for
and
are presented in Table 4 for a range of a_{0} and
values.
These rough model estimates may change by a factor of 1.5-2 depending on
the real pattern of the distribution of pairs of compact objects along the
semiaxes.
Parameters | WD + WD | WD + NS | WD + BH | |||
50 | 30 | 50 | 30 | 50 | 30 | |
10 | ||||||
8 | ||||||
7 | ||||||
5.88 | ||||||
4 | ||||||
3 | ||||||
Average | 10^{-5} | |||||
estimate |
Figure 7 represents the behaviour of (for as a function of t and n for the values of a_{0} and which give approximately the average values among those presented in Table 4 for different system types.
We have also estimated the average durations and periods of reccurence of the events considered and their dispersions using (20)-( ). To do this correctly one has to convolve and T not with the f(a)but with the P(K_{0},t,a)f(a)/F(K_{0},t) according to the share each parameter region brings to the probability of detecting a flare. These values, formally, depend on the values of t or n. However, it is clear that their variations cannot influence much the real average characteristic. Actually in these calculations we use the expression for P(K_{0},t,a) and F(K_{0},t) in the region where they are linear in t, which as we will see in Sect. 5 corresponds to the optimal strategy for searching for the flares.
The results are presented in Table 5 for the value of amplitude . The parameters averaged over the range of the used a_{0} (from to ) are in the first and third columns. Variations of flare duration with varying a_{0} do not exceed a factor of 2. In Cols. 2 and 4 the limits of the intervals which contain parameters of 90% of flares are given. The minimum values of the parameters correspond to the closest systems of all types which are capable of giving rise to a flare with the amplitude . In Table 5 we give parameters of events that one can hope to discover in observations.
As we have already noted, the present-day numbers of different pairs of
compact objects are determined in the framework of different versions of
population synthesis (Lipunov et al. 1996; Wellstein & Langer 1999;
Fryer et al. 1999; Nelemans et al. 2001). However, the main objective
of this research is the study of systems which are of interest within the
scope of gravitational-wave astronomy and of the problem of origin of
gamma-ray bursts. These are pairs of neutron stars, neutron stars with black
holes and essentially rarely white dwarfs with black holes. By now
extensive work has been done on the modeling of the origin and evolution of
pairs of white dwarfs. It goes without saying that these binary systems
are best understood; comparison with observations can be made for them
(Hernanz et al. 1997; Nelemans et al. 2001). There are at least two
relatively reliable estimates of the total number of binary white dwarfs
in the Galaxy,
(Lipunov et al. 1996) and
(Nelemans et al. 2001). In principle, these data are sufficient
to determine the number of gravitational-lens flares that can be detected
in this sample. Nevertheless, we have estimated the number of binary
white dwarfs and also white dwarfs in pairs with neutron stars and black
holes in the Galaxy in the framework of assumptions of the progenitors of
binary systems consisting of compact objects. We have followed Bethe &
Brown (1998, 1999) who have determined the number of massive binaries
containing neutron stars and black holes.
Duration (s) | Period of recurrence (h) | ||||
average | interval of | average | interval of | ||
90% | 90% | ||||
WD+BH | 50 | 10-135 | 15 | 1-48 | |
WD+NS | 60 | 20-120 | 42 | 8-120 | |
WD+WD | 135 | 90-180 | 400 | 120-500 |
We will base our discussion on the following assumptions (as previously, solar units are used).
1. The mass distribution of progenitor stars (initial mass function)
obeys Salpeter's (1995) law in the interval
0.1 < M < 120, i.e.
(23) |
At the observed star formation rate and the Galaxy age of 10^{10}years, the number of stars is .
2. About 100% of stars are members of binary systems. This evaluation is used in the population synthesis as well as 50% (see, for instance, Nelemans et al. 2001).
3. The distribution density in the mass ratio, , is constant, i.e. (e.g. Portegies Zwart & Yungelson 1998).
4. The initial masses of progenitor stars must lie within the following
intervals to give:
white dwarf -
1 < M < 10,
neutron star -
10 < M < 25,
black hole - 25 < M.
In the last case the uncertainty is rather great; the values
(van den Heuvel & Habets 1984), and
(Woosly et al. 1995) are also used. As for us, we follow Portegies Zwart
et al. (1997).
5. The neutron star formed in a supernova explosion in a binary system receives a kick. As Cordes & Chernoff (1997) have shown, its distribution is well approximated by the sum of two Gaussians with standard 175 km s^{-1} and 700 km s^{-1}, the share of the former being 80% while that of the latter being 20%. With these parameters 43% of binary systems survive after the explosion (Bethe & Brown 1998). As a black hole is formed, one can believe that the kick will be lower, at least, inversely proportional to its mass (Fryer et al. 1999). For this case, using the relationships between the probability of decay and the additional velocity derived by Bethe & Brown (1998), we have found that 87% of binaries survive after the formation of a black hole.
Now we turn to estimation of the number of pairs of different types.
For the probability
to obtain a binary system with the mass of the
more massive companion (in our case the object is a lens) in the interval
and the mass ratio in the interval
,
considering (23) and the homogeneity of the distribution in q (
is
measured in
), we have
(24) |
Clearly
(25) |
(26) |
For white dwarfs in pair with neutron stars
The same is for pairs containing black holes:
It should be emphasized that the above-given estimates are defined in many respects by the choice of the power index k in the initial function of stellar masses. Clearly they decrease with increasing k. It is likely that the version with k = 2.35 of Salpeter (1995) is most consistent with current observational data (see, for instance, Grishchuk et al. 2001; Raguzova 2001), but, nevertheless, k = 2.5 and also k = 2.7 are used in the population synthesis as well (Bethe & Brown 1998, 1999; Nelemans et al. 2001).
To make the picture complete, we present in Table 6 the estimates of the
numbers of pairs in the Galaxy for the mentioned k.
k | WD-WD | WD-NS | WD-BH |
2.35 | |||
2.5 | |||
2.7 |
On the other hand, in order to determine the accuracy of our estimates, one can compare them (for binary white dwarfs) with the results of population synthesis. In particular, the number of pairs of this type is (Lipunov et al. 1996) and (Nelemans et al. 2001). Note that the last number was obtained for k = 2.7 and it is 2 times as low as our rough value from Table 6. The 2-4-time differences are reasonable, taking into account the uncertainty in the results of modeling of evolution and the qualitative character of our estimates.
To determine the number of objects that may be detected using a certain telescope one has to know not only the overall number but also their distribution in the Galaxy, their luminosity function and the distribution of absorbing matter as well.
We used the star distribution in the Galaxy of the following form:
(27) |
(28) |
(29) |
Note that the part of WD-WD pairs among the complete number of WDs is (7-25)% (Iben et al. 1997; Fryer et al. 1999) and the WD density in the Solar neighbourhood is pc^{-3}(see Nelemans et al. 2001 and references therein). This gives us a range of local density of pc^{-3}. It is very close to our estimate shown in Table 7 and the most optimistic values of density may be 2-2.5 times more than those.
Unfortunately, there is no data to compare with for binaries other
than WD-WD. We suggest only that our estimation range may be 2-3 times
narrower. Therefore, the real number of objects may be a few times higher.
k | WD-WD | WD-NS | WD-BH |
2.35 | |||
2.5 | |||
2.7 |
The insterstellar absorption is proportional to the interstellar medium density which is distributed in the Galaxy by the same (27) law with h=80 pc (Dehnen & Binney 1998). The normalization was done according to the local average value of absorption in the band of the SDSS equipment (Schlegel et al. 1998).
The WD luminosity function is well known and is explained fairly well in the frame of their cooling theory. We used the luminosity function of Oswalt et al. (1996). It is readily seen from this that more than half of the WDs are brighter than and only less than 20 percent are fainter than , where the function reaches its maximum. Thus, the magnitudes of the majority of WDs do not suffer much from the bolometrical corrections which is a few tenth for the stars of these spectral classes (see, e.g. Allen 1973).
In other binary systems, the situation is more complicated. It is only clear that the optical emission from both a NS and a BH can be associated with interstellar gas accretion. In the former case it is likely to be negligibly small because of the high orbital velocity of the NS (>100 km s^{-1}) (Shapiro & Teukolsky 1983). At the same time the orbital velocity of a BH in a pair with a WD is about 50 km s^{-1}, and the luminosity of accretion plasma can reach 10^{30} erg/s by moderate-optimistic estimates (Shvartsman 1971; Beskin & Karpov 2002) and will be by an order of magnitude less by a pessimistic one (Ipser & Price 1982). Apparently, its contribution to the total luminosity of the system is insufficient to correct the depth of the space of detection. However, this level of optical emission makes possible searching for its fast variations with a microsecond time resolution for investigation of accretion processes and strong gravitational fields near the horizon of events (Shvartsman 1971; Beskin et al. 1997; Beskin & Karpov 2002).
Now we have everything we need to calculate the on-sky density of objects
in a given direction (l,b):
(30) |
Figure 8: Magnitudes of detectable flares in the SDSS versus object brightness. | |
Open with DEXTER |
Unfortunately, this integration cannot be done analytically and so we will turn at once to a particular version of the equipment being discussed. In our view, an ideal tool for carrying out the proposed programme is currently the 2.5-metre telescope at APO (New Mexico). It has been used to accomplish one of the most promising projects - the Sloan Digital Sky Survey (SDSS). The telescope has a field of , in which 30 CCD chips of pixels are mounted (York et al. 2000). Sky scanning at a speed of its movement along a strip wide is performed within the framework of the project. During an exposure of 55 s, the limiting stellar magnitude in the filter (it is close to the B filter of the Johnson system) is at a signal-to-noise ratio of 5 (Oke & Gunn 1983; York et al. 2000). We have constructed a relationship between amplitude of flares recorded at a level and stellar magnitude of a quiet object in the band (Fig. 8). In so doing, we used the parameters corresponding to the SDSS project: the seeing -0.8 arcsec, the quantum efficiency of the CCD matrix -80%.
By comparing the data of Fig. 8 with the mean parameters of flares (Table 5) and their light curves for different binary systems (Fig. 5), it is clear that with a particular exposure of time s, the flares with of any pairs that we have examined can be recorded at a significance level of (duration of any flare is a few ). In fact, the level of the flare detection will be better due to registration in the SDSS of the field in several colour bands ( ) and the limiting stellar magnitude is also Moreover, the use and comparison of the data in the and filters give the possibility of easy separation of cosmic ray traces.
On the basis of (30) we have built a map of the expected on-sky densities of
WD-WD pairs up to
which is presented in Fig. 9. The
value of the local density corresponding to the Salpeter mass-spectrum index
of 2.35 is used.
One can see from the map that the maximal density regions have galactic
latitudes of 5-10 degrees and a question might arise if it is possible to
observe in such star-rich regions of the sky. However, even in these latitudes
the average distance between stars of up to
is about 3-4
arcseconds (see e.g., Zombeck 1982, p. 34) and thus, they are resolved with
the SDSS seeing and this crowding therefore cannot have much negative
impact on observations with this equipment.
Figure 9: Simulated on-sky distribution of WD-WD pairs, . Maximum density value is 490 pairs per square degree. | |
Open with DEXTER |
To derive the expected number of events of the discussed type
one
multiplies the probability of detecting it from a randomly chosen pair
F(t) for
(see (15), (20)-(22)
and Fig. 7)
by the number
of such pairs observed during a certain
observational programme that covered the celestial sphere part :
(31) |
(32) |
Figure 10: Expected number of objects in the best (with the highest density of pairs) fields. | |
Open with DEXTER |
Now let the programme last for 5 years. Considering that the matter
in question is detection of rather faint objects, observations have to be
made on dark nights only at moon phases within
from the new moon,
which gives approximately 36% of night time. Assuming the average
observational night of
the 5-year resource of time then will make
nights or
hours. Thus, as every field is
observed for t hours or n nights (and
,
)
the expected number of detected pairs is
(33) |
(33') |
Figure 11: Expected detection numbers in the 5-year programme as a function of total exposure per field. | |
Open with DEXTER |
The expected numbers of detections with optimal strategies along with
the optimal total exposures per field are given at Table 8.
k | WD-WD | WD-NS | WD-BH |
2.35 | 22 | 9 | 16 |
2.5 | 15 | 5 | 8 |
2.7 | 9 | 2 | 3 |
Optimal | |||
exposure time | 6-7 nights | 1-2 nights | 6-9 hours |
per field |
A new project entitled "The Dark Matter Telescope" (http://www.dmtelescope.org/index.htm) being developed involved 8.4 (effectively 6.9) metre telescope with a field of view slightly wider than that of the SDSS. The use of it in a manner similar to that described above increases the number of detectable objects by a factor of 2.2. Other wide field telescopes of 4 metres in diameter being built now, LAMOST (http://www.greenwich-observatory.co.uk/lamost.html) and VISTA (http://www.vista.ac.uk), could detect 1.5 times more binaries than the SDSS.
Thus, for searching for the flares caused by gravitation lensing in binary systems with compact companions, the facilities of the SDSS can be used, having reduced the time of an individual exposure to 10-20 s. It is obvious that after recording a flare in a certain object, another telescope of similar class must be involved in observations. This object should be monitored to prove the effect and for detailed investigation of the detected system.
Thus, within the frame of the proposed programme, it will be possible to find several dozen binary systems comprised of compact objects. It will be recalled that in numerous surveys of microlensing, the number of reliably detected events amounts to about 350 (Alcock 1999), only in two cases the matter in question being of massive lenses (black holes?) (Bennett et al. 2001). When applying the technique suggested, there is a possibility of detecting with absolute assurance more than 10 black holes with open event horizons. An investigation of these objects with high time resolution within the MANIA experiment (Beskin et al. 1997) may permit at last the detection of observational evidence of extreme gravitational fields.
Acknowledgements
This investigation was supported by the Russian Ministry of Science, Russian Foundation of Basic Researches (grant 01-02-17857), Federal Programs "Astronomy" and "Integration" and Science-Education Center "Cosmion". The authors are very grateful to the anonymous referee for very useful observations and E. Agol for important notes. AVT would like to thank D. A. Smirnov and S. V. Karpov for valuable discussions and his especially grateful to I. V. Chilingaryan for his help with various questions. We thank T. I. Tupolova for manuscript preparation.