Compact objects from binary self-lensing

Compact object detection in self-lensing binary systems with a main-sequence star

S. Rahvar, A. Mehrabi and M. Dominik
Department of Physics, Sharif University of Technology, P.O. Box 11365–9161, Tehran, Iran
School of Astronomy, IPM (Institute for Studies in Theoretical Physics and Mathematics), P.O. Box 19395-5531, Tehran, Iran
SUPA, University of St Andrews, School of Physics & Astronomy, North Haugh, St Andrews, KY16 9SS, United Kingdom
rahvar@sharif.eduRoyal Society University Research Fellow
Abstract

Detecting compact objects such as black holes, white dwarfs, strange (Quark) stars and neutron stars by means of their gravitational lensing effect on an observed companion in a binary system has already been suggested almost four decades ago. However, these predictions were made even before the first observations of gravitational lensing, whereas nowadays gravitational microlensing surveys towards the Galactic bulge yield almost 1000 events per year where one star magnifies the light of a more distant one. With a specific view on those experiments, we therefore carrry out simulations to assess the prospects for detection of the transient periodic magnification of the companion star, which lasts typically only a few hours binaries involving a main-sequence star. We find that the effect is practically independent of the distance of the binary system from the observer, but a limit to its detectability is given by the achievability of dense monitoring with the required photometric accuracy. In sharp contrast to earlier expectations by other authors, we find that main-sequence stars are not substantially less favourable targets to observe this effect than white dwarfs, not only because of a better achievable photometry on the much brighter targets, but even more due to the fact that there are times as many objects that can be monitored. The requirement of an almost edge-on orbit leads to a probability of the order of for spotting the signature of an existing compact object in a binary system with this technique. Assuming an abundance of such systems about 0.4 per cent, a high-cadence monitoring every 15 min with 5 per cent photometric accuracy would deliver a signal rate per target star of at a recurrence period of about 6 months. With microlensing surveys having demonstrated the capability to monitor about stars, one is therefore provided with the chance to detect roughly semi-annually recurring self-lensing signals from several compact compacts in a binary system. These must not be mistaken for similar signatures that arise from isolated planetary-mass objects that act as gravitational lens on a background star. If the photometric accuracy was pushed down to 0.3 per cent, 10 times as many signals would become detectable.

keywords:
black hole – binary stars – gravitational lensing

1 Introduction

Despite the successful observation of the bending of light by the Sun (Dyson et al., 1920), following the suggestion by Einstein (1911), it required many decades of advance in technology for enabling the detection of this effect for another star, given that ”there is no great chance of observing this phenomenon” (Einstein, 1936). Only following the call by Paczyński (1986) to apply ’gravitational microlensing’ to measure the abundance of potential MACHOs (Massive Compact Halo Astrophysical Objects) in the Galactic halo, the first related experiments were carried out. In fact, a decade of observations of the Large and Small Magellanic Clouds now reveals that there are not enough MACHOs in the Galactic halo to account for the observed flat rotation curve for the Galactic disk (Milsztajn & Lasserre, 2001; Popowski et al., 2005; Moniez, 2009). The gravitational microlensing effect has evolved into an important astrophysical tool for not only studying stellar atmospheres (e.g.  Albrow et al., 1999; Afonso et al., 2001; Gould, 2001; Abe et al., 2003), but also to study populations of extra-solar planets (Mao & Paczyński, 1991; Gould & Loeb, 1992; Dominik, 2010).

In this work, we assess the suggestion to detect Compact Objects (CO), namely black holes (BH), strange (Quark) stars (QS) and neutron stars (NS), by means of their gravitational bending of light received from an observed star that forms a binary system together with the Compact Object in the context of current experiments. The lens action within a binary system of stars or stellar remnants has been discussed in great detail by Maeder (1973). This effect shares many characteristics with the meanwhile common gravitational microlensing events where a foreground star magnifies the light of an unrelated background star, which get aligned on the sky with respect to the observer just by chance. However, the typical duration of the transient brightening is substantially shorter, of the order of a few hours, and the signal repeats periodically (albeit with periods that can be as large as decades). Maeder (1973) moreover found that the smaller the radius of the source star, the larger is the lens effect and its probability of occurence. As a consequence, main-sequence stars (MS) were considered unfavourable candidates as compared to white dwarfs, where however the prospects for MS-BH pairs are substantially better than for MS-NS and MS-WD pairs. As a consequence, Beskin & Tuntsov (2002) have more recently evaluated the detectability of compact objects in a binary system with an observed white dwarf due to gravitational lensing, and in particular looked at the prospects for observing this effect in the Sloan Digital Sky Survey (SDSS), while not considering main-sequence source stars.

However, the chances of success in both cases depend on a number of various factors. First, there is the existing number of respective pairs of binary systems, on which we are currently forced to rely on the best available understanding of stellar evolution. Observations of star forming regions show that 70 to 90 per cent of stars form in the clusters and almost two out of three stars reside in binary systems (Mathieu, 1994). Models of stellar evolution predict that 0.4 per cent of the binary systems will see one of companions turning into a compact object (Hurley et al., 2000; Belczynski et al., 2002), whereas 0.2 per cent of stars end up in a binary system composed of two compact objects. Second, the probability for a signature to be ongoing at any time is given by the product of the probability for the monitored target to show a signal and the ratio between the signal duration and the orbital period. Third, the number of suitable targets that can be monitored plays a crucial rule, and fourth and finally, it cannot be neglected that high-precision photometry on main-sequence stars as far as the Galactic bulge is possible, whereas such an opportunity does not arise for the much fainter white dwarfs.

Gravitational lensing of a star gravitationally bound to a compact object has also been proposed by Campbell & Matzner (1973) as an interpretation of the Weber experiment (Weber, 1970) for the gravitational radiation from the center of Galaxy, where they used the optical approach for calculating the lensing effect in a Schwarzschild metric when the source star is aligned with the massive black hole of the Galaxy and the observer. In the optical approach, the variation of light bundle along the null geodesic describes the intensity of the light. In the extension of this work, Cunningham & Bardeen (1973) obtained the gravitational lensing of a source star rotating around a maximally Kerr metric. The main physical difference between the lensing in the work by Campbell & Matzner (1973) and eclipsing microlensing proposed in this work is that in the former case the source star is orbiting around the black hole with the orbital size in the order of Schwarzschild radius while in later case the source is located in the order of the Astronomical Unit. In this case, the line between the source-lens and the optical axis (line connecting lens to the observer) is small (Bozza & Mancini, 2005).

In contrast to Beskin & Tuntsov (2002), we focus on the self-lensing within binaries that are composed of a compact object and an observed main-sequence star, and on the observability of this effect with current or upcoming microlensing monitoring efforts.

In Sect. 2, we discuss the arising binary self-lensing light curves, and subsequently evaluate the detection probability of such signals using strategies similar to ongoing microlensing efforts in Sect. 3 by means of Monte-Carlo simulations. We briefly discuss the extraction of parameters from the observed data in Sect. 4, before we finally summarize our conclusions in Sect. 5.

2 Self-microlensing within binary systems

As illustrated in Fig. 1, the self-lensing binary system involving the compact object is characterised by its inclination angle with respect to the observer-lens axis (the lens being the compact object), the orbital radius (assuming circular orbits for simplicity), and the Einstein radius

(1)

where

(2)

denotes the Schwarzschild radius of the (lensing) compact object of mass , which evaluates to

(3)

Given that the difference between lens and source distance as compared to their distance from the observer can comfortably be neglected, the Einstein radius becomes a function solely of the lens mass and the orbital radius of the binary system, which means that the observed signature does not depend on its distance from the observer.

Figure 1: Geometrical configuration of lens and source in a binary system. The horizontal line represents the observer-lens line of sight. The binary system with the observer is shown from the side, and denotes the inclination angle of the binary system with respect to observer-lens line of sight. For simplicity, we assume circular orbits with a radius .

With a compact object as lens, we should however be aware of several possible corrections to standard gravitational microlensing light curves: (a) the strong gravitational field of the lensing compact object leads to relativistic images, (b) geometrical corrections due to strong fields, (c) the perturbation effect of the source on the light deflection and (d) the finite-size effect of the source star.

For a black hole, light rays can enter regions with strong gravitational fields near the Schwarzschild radius and reach the observer after a quite complicated track (Chandrasekhar, 1992). Such light rays correspond to relativistic images that exist in addition to the usual weak-field images, and in principle affect the total magnification pattern of the observed source star. For these relativistic images, the relation between the source, image and deflection angle do not satisfy the small-angle approximation, but the lens equation for this configuration is rather given by

(4)

where and are the position angles of image and source, respectively, and is the deflection angle. Integration over the path yields the deflection angle as

(5)

where all distances are in units of the Schwarzschild radius and marks the closest approach of the light ray to the deflector. If observer, lens, and source happen to fall exactly onto a straight line, the condition for the observation of the source essentially becomes , where is the number of turning of the light rays around the black hole (Bozza et al., 2001). For source-lens (line-of-sight projected) separations substantially larger than the Schwarzschild radius, the magnification of the source star due to strong lensing can be neglected as compared to the weak-field images. In this case, the deflection angle is in the order of . With the Schwarzschild radius to be of the order of kilometers and the orbital radius of the order of , the corresponding angles in the lens equation are in the order of , and we find ourselves in the small-angle regime.

The proximity of the source star to the lens may also perturb the gravitational lensing effect. Considering a linear perturbation around the Schwarzschild metric in the weak-field limit, the perturbation on the deflection angle relate to the Newtonian potentials as

(6)

where and are the Newtonian gravitational potentials of the source star and the lens, respectively. For a light ray passing near the Einstein radius , and source and lens object being separated by about an astronomical unit, one finds a relative perturbation on the deflection angle of

(7)

where and are the mass of source star and the lens, respectively. With Eq. (3) one finds a numerical value of , so that the perturbation effect of the companion star does not play a significant role.

Finally we look at the influence of the finite size of the observed source star, which was discussed in detail by Witt & Mao (1994). The relevant parameter is the ratio between the angular radius of the source star and the angular Einstein radius, which simplifies to , given that lens and source distances practically coincide. Eliminating the stellar radius in favour of the stellar mass, using (Demircan & Kahraman, 1991) and using Eq. (3), one finds

(8)

Given that the magnification is limited to

(9)

which is realised for perfect alignment, the signal amplitude is quite substantially suppressed due to the finite size of main-sequence source stars, unless the star is of low mass and/or the compact object is a massive black hole. As pointed out by Maeder (1973), white dwarfs come with a clear advantage of smaller radii, so that larger magnifications occur regularly.

For general separations between lens and source stars, where denotes the angular separation in units of the angular Einstein radius, the magnification for is given by

(10)
(11)
(12)

where , and are the complete elliptic integral of first , second and third kinds respectively and

(13)

whereas for , one finds (Maeder, 1973; Dominik, 1996)

(14)

The centre of the source star is within the angular Einstein radius of the lens star for angles . Therefore, this condition can be used as a reference for the magnification to be substantial. We note that the characteristic inclination angle is independent of the distance of the binary system to the observer. We find an order estimate for the fraction of the binary systems with significant magnification signature in their light curves as . We further find . Using the numerical values for the Schwarzschild radius in the order of a few km and in the order of one tenth of astronomical unit, the fraction of self-lensing binaries with compact objects that provide a signature becomes . Taking 0.4 per cent of binary stars with compact star companions, the probability for the effect to show up amongst all observed stars turns out to be . This number is tiny, but one needs to be aware of the fact that the prospects for observing such an effect crucially depend on the viability of regular monitoring of a huge number of targets, as well as on the frequency of such events to occur.

For a binary system, the angular velocity is given by

(15)

so that the relative transverse velocity of the source with respect to the lens follows as

(16)

and is therefore determined with the choices of the masses and of the components and the orbital radius . This defines an event time-scale

(17)

within which the source moves by . In fact, the motion can be approximated as uniform, where

(18)

with the closest angular approach between lens and source star being

(19)

for a small , which occurs at epoch . Therefore, the signal of eclipsing microlensing resembles an normal extended-source standard microlensing light curve, described by the 4 parameters , , , and .

For reference, the light curve of a binary system with the parameters of , a main sequence star with the mass of , and is shown in Fig. 2. This system has the finite-size parameter and the period of this system is about years. Main-sequence stars are again disfavoured due to their long periods in detectable systems, whereas substantial signals can arise in systems with white dwarfs with much shorter periods.

Figure 2: Gravitational self-microlensing light curve arising from a binary system that involves a black-hole lens of mass and an observed main-sequence star of mass . The orbit is from an edge-on configuration, and the orbital radius is . This yields a finite-size parameter and an orbital period .

3 Detection probability

Let us now investigate the prospects for detecting compact objects by means of binary self-lensing for specific observational strategies. Modelled upon the characteristics of current or upcoming microlensing campaigns, and giving us a hint on the roles of both photometric accuracy and sampling rate, we consider regular monitoring with the following parameters (see also Rahvar & Dominik, 2009): (a) 5 per cent photometric accuracy at 15 min cadence, indicative for high-cadence ground-based surveys (Sumi et al., 2010; Hwang & Han, 2010), (b) 2 per cent accuracy at 2 hr cadence, roughly representative of current follow-up monitoring programmes (Dominik et al., 2002), and (c) 0.3 per cent photometric accuracy at 15-min cadence, reflecting the coming state-of-the-art, including lucky-imaging or spaced-based observations (Jørgensen, 2008; Bennett & Rhie, 2002; Bennett et al., 2003).

For main-sequence stars, we adopt the mass function proposed by Chabrier (2003), namely

(20)

which covers the range of , while we assume a mass-radius relation (Demircan & Kahraman, 1991).

For the compact objects, we adopt the product of the evolution of the zero-age mass function to the final stage of stars (Belczynski et al., 2002) with the mass range of . To estimate the fraction of binary systems with one compact object and one main sequence star, we do a rough calculation for stars in the binaries with the initial masses in the range of for the first star and for the companion star. Star with the larger mass has a relative short life time and will evolve to a compact object while the smaller star stays in the main sequence if we don’t have mass transfer between the two stars. For the binaries located far enough distance from each other (i.e. stellar size should be smaller than the roche lobe), we obtain almost per cent of the stars will end to the binary systems with one compact object and a companion main sequence star.

For the orbital distance within the binary system, we assume a logarithmic distribution in the range of , in accordance with Öpik’s law, while the inclination angle is drawn uniformly from .

Figure 3: Example synthetic light curve as arising from the Monte-Carlo simulation. The adopted parameters are , , and , so that and .

Using these parameter distributions, we generated synthetic light curves by means of Monte-Carlo simulations, where Figure 3 shows an example. With a detection criterion of three consecutive data point being larger than three times of the standard deviation from the base line, we not only obtain the fraction of systems for which the compact object is detectable, but also the distribution of parameters of the expected eclipsing microlensing events.

Figure 4 shows the detection efficiency for the three considered monitoring strategies. One finds that it depends only weakly on the mass of the lens. This is a consequence of the relation between the lens mass and the event time-scale . With and , one finds a weakly-varying . A larger mass of the main-sequence source star implies a larger radius , which diminishes the magnification due to the finite-size effect. Moreover, the event time-scale becomes smaller. On the other hand, a larger source radius enables us to get a signal from a wider range of inclination angles, and the effective signal duration is increased. The gain from a longer signal duration plays a larger role for sparser sampling, while for an inferior photometry the signal drops below the detection threshold earlier.

The effect of the orbital radius of the two companion stars on the observability eclipsing microlensing signal is a function of three factors, namely (a) the dependence of the Einstein radius on the orbital radius as , (b) the relative transverse velocity of the binary system , hence , and (c) . The wider range of suitable inclination angles increases the prospects for a detection in systems with smaller orbital radius. Smaller event time-scales however let signals fall into the gap between subsequent observations. Consequently, we find a rise in the detection efficiency towards smaller orbital radii (and thereby shorter periods) until the signals become to short to be detectable.

Figure 4: The efficiency for revealing the presence of a compact object in a binary system with an observed main-sequence star as a function of (lens mass), (source mass), (orbital radius), and (inclination angle) for three observational setups, characterised by their photometric accuracy and sampling interval (dotted: , ; dashed: , ; solid: , ).

With the detection efficiency and the distribution functions of the adopted parameters, we find the overall probability for detecting binary self-microlensing events. In particular, by multiplying the detection efficiency with the mass function of the lens stars, we obtain the expected distribution of lens masses revealed from observed eclipsing microlensing signals, which is shown in Fig. 5. The mass function of the lens stars were normalized to the overall number of stars. Integrating these histograms results in the total probability of observing eclipsing microlensing events. For our three variants of the adopted observing strategy, we find , , or respectively. With the latter value being close to our earlier thumb estimate, we find a rather good efficiency of the adopted strategy.

We further weigh each detection efficiency with the frequency of the signal, which equals the inverse of the orbital period , i.e. we calculate an average over the realisations arising from the Monte-Carlo simulation, in order to obtain the event rate per observed star as , , or for our three adopted monitoring strategies, which typically find compact objects in binaries with orbital periods of , , or respectively, which equals the period of recurrence of the signals. Naturally, systems with shorter periods dominate the events due to their higher recurrence rate, and the goal of an observational strategy has to be to keep these detectable. The findings of our simulations are summarized in Table 1.

accuracy sampling rate detectability event rate period
Table 1: Fraction of observed systems with a detectable compact companion , event rate per observed system , and ’typical’ period of the signal for the three considered monitoring strategies characterized by the photometric accuracy and the sampling interval , where denotes the detection efficiency for a given configuration, and denotes its orbital period.
Figure 5: Expected distribution of the masses of the detected compact objects that act as gravitational (micro)lenses on the light of observed main-sequence star within a binary system, considering the same observational capabilities as for Fig. 4.

4 Extraction of parameters

The observed light curve allows to extract the 4 standard parameters , , , and , but with not carrying any relevant information about the binary system, we are one parameter short of reconstructing the masses of the components and , the orbital radius , and the inclination angle . Only in the limit , Eq. (17) yields

(21)

In order to go further, one needs to exploit the periodicity of the signal. This again stresses the need for events with shorter periods, not longer than a few years. In fact, any attempt to obtain information by measuring astrometric shifts of the observed source star due to its wobble around the compact object or its radial velocity by means of Doppler-shifts of spectral lines, relates to the orbital period. Withstanding the difficulties in obtaining such measurements for faint stars, the fundamental properties already follow with the orbital period itself.

Kepler’s third law

(22)

would allow to find

(23)

with Eqs. (17) and (19), and one would be able to obtain iteratively

(24)

as well as

(25)

so that with the mass-radius relation for main-sequence stars

(26)

5 Conclusions

Given that the signal amplitude of self-lensing due to a compact object in a binary system is less suppressed by the much smaller finite radius of a white dwarf as compared to a main-sequence star, and moreover the orbital period of detectable systems is smaller (given that the relevance of finite-source effects is quantified by ), and thereby the frequency of signals is larger, Maeder (1973) concluded that white dwarfs are the favourable targets for observing this effect, whereas the prospects for binaries involving main-sequence stars are rather bleak. However, the fortune changes substantially if one looks at the observability of suitable systems. Beskin & Tuntsov (2002) considered the Sloan Digital Sky Survey (SDSS) as most favourable for observing white dwarfs, and in fact, it has dramatically increased the number of known white dwarfs. However, with the sample containing about 15,000 objects (Kleinman et al., 2009), it is times smaller as compared to the stars regularly monitored by current microlensing surveys (Udalski, 2003).

For monitored stars and an event rate per observed star of (for 5 per cent photometric accuracy and 15 min sampling cadence), one finds a total event rate of , where is a coverage factor accounting for the visibility of the Galactic bulge from the respective sites over the year, any losses due to weather or technical downtime, and imperfect cadence or data quality. In contrast to earlier work, we therefore conclude that the detection of compact objects (in fact, predominantly black holes) in binary systems due to self-lensing of an observed main-sequence star companion is possible, provided that a high-cadence sampling substantially below 2 hrs is realised. The upcoming Korea Microlensing Telescope Network (KMTNet) has in fact been designed as a wide-field survey of the Galactic Bulge with 10-minute cadence (Hwang & Han, 2010). Moreover, the MOA (Microlensing Observations in Astrophysics) survey already monitors some of its fields at that cadence (Sumi et al., 2010). Higher photometric accuracies of 0.3 per cent, achievable with space-based observations (Bennett & Rhie, 2002; Bennett et al., 2003) or lucky-imaging cameras (Jørgensen, 2008), could result in 10 times as many observable signals due to self-lensing in binaries with a compact objects, whereas lower accuracies of 20 per cent would lead to about 10 times less objects being detected.

Given that the duration of the expected self-microlensing signals is of the order of a few hours, we issue a note of caution that such is not mistaken for evidence of planetary-mass bodies that pass the line of sight to a background star. In fact, the MOA survey appears to show an excess of short-duration peaks as compared to expectations from stellar populations and the kinematics of the Milky Way (K. Kamiya, private communication).

In practice, one faces a rather hard job to distinguish between usually poorly-covered spikes of different origin. The self-lensing binary signals repeat in principle, but on an initially unknown time-scale of months to years and are rather easy to miss. The discriminating power of the criterion of achromaticity of gravitational microlensing as opposed to stellar variability is also limited due to the lack of detail on the shape of the signal. Only if a period of the binary system can be established, its physical characteristics can be determined.

Acknowledgments

We would like to thank Valerio Bozza for a couple of helpful remarks on this topic.

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