A&A 476, 121-135 (2007)
DOI: 10.1051/0004-6361:20077105
P. B. Ivanov^{1,2} - J. C. B. Papaloizou^{1}
1 - Department of Applied Mathematics and Theoretical Physics, CMS, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK
2 - Astro Space Center, P. N. Lebedev Physical Institute, Profsouyznaya St., 84/32 Moscow, Russia
Received 15 January 2007 / Accepted 31 August 2007
Abstract
Aims. We consider how tight binaries consisting of a super-massive black hole of mass M=10^{3}-
and a white dwarf in quasi-circular orbit can be formed in a globular cluster. We point out that a major fraction of white dwarfs tidally captured by the black hole may be destroyed by tidal inflation during ongoing tidal circularisation, and therefore the formation of tight binaries is inhibited. However some fraction may survive tidal circularisation through being spun up to high rotation rates. Then the rates of energy loss through gravitational wave emission induced by tidally excited pulsation modes and dissipation through non linear effects may compete with the rate of increase of pulsation energy due to dynamic tides. The semi-major axes of these white dwarfs are decreased by tidal interaction below a "critical'' value where dynamic tides decrease in effectiveness because pulsation modes retain phase coherence between successive pericentre passages.
Methods. We estimate the rate of formation of such circularising white dwarfs within a simple framework, modelling them as n=1.5 polytropes and assuming that results obtained from the tidal theory for slow rotators can be extrapolated to fast rotators.
Results. We estimate the total capture rate as
yr^{-1}, where
and r_{0.1} is the radius of influence of the black hole expressed in units 0.1 pc. We find that the formation rate of tight pairs is approximately 10 times smaller than the total capture rate, for typical parameters of the problem. This result is used to estimate the probability of detection of gravitational waves coming from such tight binaries by LISA.
Conclusions. We conclude that LISA may detect such binaries provided that the fraction of globular clusters containing black holes in the mass range of interest is substantial and that the dispersion velocity of the cluster stars near the radius of influence of the black hole exceeds 20 km s^{-1}.
Key words: black hole physics - gravitational waves - stellar dynamics - white dwarfs - galaxies: star clusters - stars: oscillations
There are some observational indications and theoretical suggestions that favour the presence of black holes in the mass range 10^{2}- in the centres of globular clusters. The observational arguments supporting this hypothesis relate to kinematical phenomena observed in the centres of some globular clusters (e.g. Gebhardt et al. 2000; Gebhardt et al. 2002) and the presence of X-ray sources not associated with the central nuclei in certain galaxies (e.g. Fabbiano 1989; Matsumoto et al. 2001; Ghosh et al. 2006, and references therein). There are also some theoretical models of the formation of such systems (e.g. Miller Hamilton 2002). A review of the observational and theoretical aspects of this problem has been recently given by van der Marel (2004).
In this Paper we assume that there is a black hole of mass - in a star cluster and estimate the rate of capture of white dwarfs of mass m by the black hole. The capture rate is always determined by the interplay of two processes. These are the effect of distant two body gravitational encounters changing the orbital angular momenta of the stars and an interaction associated with the presence of the black hole which removes the orbital energy of a star and is effective only when the orbital angular momentum is sufficiently small. This type of interaction may result either through tidal interactions or by the emission of gravitational waves induced by the stellar orbital motion.
The efficiency of tidal interactions is determined by the ratio of orbital pericentre distance to the tidal radius - the latter being the distance from the black hole below which significant disruption of the star through mass loss induced by tides occurs. On the other hand, the efficiency of orbital energy loss due to gravitational wave emission is determined by the ratio of the orbital pericentre distance to the gravitational radius of the black hole.
Since the tidal radii corresponding to white dwarfs for the range of black hole masses considered here are larger than their gravitational radii, orbital energy is changed mainly through the action of dynamical tides. This is in contrast to the case of the more massive black holes residing in galactic centres where emission of gravitational waves is more effective for changing the orbital energy of white dwarfs (e.g. Ivanov 2002; Freitag 2003, and references therein).
Since the relative contribution of two body gravitational encounters to the orbital evolution decreases very sharply for small angular momenta, a star with sufficiently small angular momentum loses orbital energy through tidal interaction while the orbital angular momentum remains approximately unchanged. The latter occurs because the star cannot store a significant amount of angular momentum in comparison to the orbit. Accordingly, the orbital eccentricity decreases during this process which will be referred to as orbital circularisation.
As a result of this process a tight quasi-circular orbit around a black hole may be formed. A white dwarf on such an orbit can emit gravitational radiation in the frequency band of order of 10^{-2} Hz which is the most favoured for the planned LISA space borne gravitational wave antenna. In principal this is able to detect gravitational waves with dimensionless amplitude as small as 10^{-24}, for an observational time of one year. That means that the presence of such a white dwarf can be detected from distances of order of 10^{3} Mpc. Taking into account the fact that globular clusters form a very abundant population of cosmic objects such systems may contribute significantly to the budget of sources of gravitational radiation available for LISA.
There is one principal obstacle inhibiting formation of a binary pair consisting of a black hole of mass M and a white dwarf of mass m in a tight orbit around it. Because it is produced through tidal interaction, its semi-major axis will be close to the tidal radius. There the ratio of orbital binding energy to gravitational energy of the white dwarf is of order of . Thus, for such an orbit to be produced, an amount of energy far exceeding the internal binding energy of the white dwarf must be removed from it. When tides are effective, orbital energy is transferred to pulsation modes and thence to the internal energy of the star. Therefore, without an effective energy loss mechanism, the white dwarf could be easily unbound and thus destroyed by the tidal input of energy or tidal inflation.
Here we propose and discuss such a mechanism, which may, in principle, allow unbinding due to tidal energy input to be circumvented for a range of orbital parameters of the star. The operation of this mechanism depends on the interplay of several factors influencing the orbital evolution of star which we introduce below.
The character of orbital evolution under the influence of tides is mainly determined by three factors: 1) the time scale for decay of the stellar pulsations excited by tidal interaction, 2) the orbital parameters of the star, and 3) the rotation of the star.
Let us first consider the decay of pulsations in a non-rotating star. In this paper we assume that decay of stellar pulsations is caused both by the emission of gravitational waves that occurs because of the time-dependent density perturbations associated with them, and also by the dissipation of pulsation energy leading to its conversion into internal energy of the star. The latter decay channel is assumed to result through non-linear effects. Since its properties are very poorly understood at the present time, we shall consider the corresponding dissipation time scale to be a free parameter. But we shall assume that it is larger than the orbital period of the star. However, this is not an essential assumption for orbital evolution at sufficiently large semi-major axes, see below.
Initially the stellar orbit will be highly eccentric with a tidal interaction that excites stellar pulsations occurring impulsively at pericentre passage (e.g. Lai 1997; Ivanov Papaloizou 2004, hereafter IP). For long decay time scales, pulsations will always be present in the star, and as a result of every periastron passage, a new perturbation excited by tides is added.
When orbital semi-major axis is sufficiently large or the orbital period sufficiently long, it has been established that tidally induced changes to it cause the phase correlation between preexisting pulsations and freshly excited ones to be lost. Then, both the energy content of excited pulsation modes and the orbital energy of the star evolve in a stochastic manner (e.g. Kochanek 1992; Kosovichev Novikov 1992; Mardling 1995). Under this evolution, the mode energy and the orbital binding energy of the star grow on average, with a part of the mode energy being transferred to the internal energy of the star. Therefore the stochastic exchange of energy between the orbit and stellar pulsations leads to a decrease of the orbital semi-major axis and period as well as an increase of the internal energy of the star.
However, once the orbital period is sufficiently short, or the semi-major axis is below a critical value, pulsation modes can maintain phase coherence between successive pericentre passages, stochastic evolution ceases and dynamic tides are expected to become less efficient. At this point, the tidal evolution rate is determined by the natural decay timescales of the pulsation modes. A steady pulsation energy typical of that induced through one pericentre passage may be maintained, rather than the growth that occurs through the cumulative effect of mode energy inputs proceeding over many pericentre passages when the semi-major axis is large.
An important issue is whether the internal energy added to the star during the phase of stochastic evolution is enough to cause its destruction. For a non-rotating star the internal energy obtained by the star when the critical semi-major axis is reached is larger than the stellar gravitational energy. Therefore, such a star could be disrupted by tidal heating before the critical semi-major axis is reached.
However, during orbital circularisation orbital angular momentum is also transferred to the star until an equilibrium rotation rate is attained. This requires a rotation rate corresponding to corotation at periastron or faster. Thus the star can be spun up to high rotation rates. When measured in terms of the amount of energy input per periastron passage, the efficiency of tidal interaction is minimised for such a rotating star. See for example Fig. 8 of Lai (1997) and also IP. Accordingly the time scale of circularisation is increased.
In this situation, when dissipation of the mode energy as a result of non linear effects is not effective, the time scale for transmitting orbital energy to the pulsation modes may be larger than the time scale for removal of the pulsation energy through gravitational waves emitted because of the time-dependent perturbation of the star. In this situation the white dwarf may reach the critical semi-major axis without internal dissipation causing tidal inflation because of cooling by emission of gravitational waves. In the opposite limit of effective mode dissipation and internal heating, the rapidly rotating white dwarf may attain the critical semi-major axis through the emission of gravitational waves induced by orbital motion rather than through tidal interaction.
Thus taking into account the reduction in effectiveness of the tidal energy transfer brought about by the effect of stellar rotation, we find that for sufficiently large orbital angular momenta the semi-major axis can be decreased below the critical one without significant heating of the star.
After the critical semi-major axis is reached, again, because of the reduced efficiency of the tidal interaction, orbital evolution is governed by the emission of gravitational waves determined by the orbital motion of the white dwarf. This process can further reduce the orbital semi-major axis and lead to formation of a tight quasi-circular orbit.
We formulate the criterion for white dwarf "survival'' treating dynamic tides within the framework of the simplest possible model of tidal interactions. We assume that the internal structure of a white dwarf is the same as that of a n=1.5 polytrope. We also assume that the results obtained from the theory of dynamic tides in slowly rotating stars can be extrapolated to high rotation rates for the purpose of making approximate estimates. Based on these assumptions we estimate the formation rate of tight pairs which turns out to be an order of magnitude smaller than the total rate of tidal capture, for typical parameters of the problem. This allows us to obtain an estimate of probability of detection of such sources of gravitational waves by LISA. We conclude that LISA could, in principal, detect such a source provided that there is a significant fraction of globular clusters containing black holes with masses 10^{3}- , and with stellar velocity dispersions in their innermost regions exceeding 20 km s^{-1}.
There are a number of different possibilities and branchings associated with the orbital evolution subsequent to tidal capture. Each of these requires consideration of several physical processes and is described by a number of algebraic expressions. In order to clarify the situation, we provide a more transparent summary of the proposed paths to a circularised orbit with disruption of the star avoided. In Fig. 1 we give a diagrammatic illustration of the possible orbital evolutionary paths that can be taken by a white dwarf subsequent to tidal capture by an intermediate mass black hole. Initially, impulsive energy and angular momentum exchanges between the orbit and star that occur every pericentre passage spin it up and excite modes of oscillation. As a result the rate of tidal evolution of the orbit decreases such that gravitational radiation can become more important. This may also be important for damping the oscillation modes. If the tidal capture occurs at sufficiently large pericentre distance, the importance of gravitational radiation during the orbital and pulsation mode evolution may allow the star to survive the regime of impulsive energy input at pericentre passage and become circularised without disruption and be a potential LISA source.
Figure 1: A schematic illustration of the possible orbital evolutionary paths of a white dwarf that is tidally captured by an intermediate mass black hole. The star is initially scattered into a highly elongated trajectory that evolves due to impulsive energy and angular momentum exchanges between the star and orbit that occur every pericentre passage, the pericentre distance remaining very nearly fixed. As a result of these, the internal energy and angular momentum increase. However, this process is slowed as the star spins up and under favourable conditions gravitational radiation can become important so that it controls the orbital evolution and/or damps the excited pulsation modes. The latter case a) leading to cases a1) and a2) occurs where other non linear processes are ineffective at dissipating the pulsations. Case b) leading to cases b1) and b2) occurs when such processes are more effective. For both cases a) and b) the star may survive the orbital evolution if the tidal capture occurs at sufficiently large pericentre distance. | |
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The plan of the paper, in which the above phenomena are considered in more detail, is as follows. In Sect. 2 we describe the model of the stellar cluster we use and the effect of distant two body encounters. In Sect. 3 we go on to discuss the tidal interaction of a white dwarf with the central black hole and pulsation mode excitation through dynamic tides. Then in Sect. 4 we consider the way in which gravitational wave emission/non linear effects may lead to pulsation mode amplitude limitation. In Sect. 5 we go on to explore the conditions for safe circularisation or tidal disruption of the white dwarf. In Sect. 6 we discuss the rate of tidal capture of white dwarfs by the black hole and estimate the fraction that can circularise safely, with technical details being relegated to an Appendix. In Sect. 7 we estimate the probability of detection of this source of gravitational radiation. Finally in Sect. 8 we discuss our results.
In order to discuss the rate of orbital circularisation for white dwarfs it is first necessary to consider the properties of the star cluster in which they are situated. This is because dynamical interaction with the stars in the system is responsible for the production of orbits for which tidal interaction with the central black hole becomes significant.
The radius of influence of the central
black hole, r_{a}, is defined as the radius
within which the total mass of the stars is
the same as the mass of the central black hole:
.
As has been noted by many authors, when
r < r_{a}, a "cusp''
in the stellar distribution of stars is formed with a density distribution
such that
In this paper we make only approximate estimates of the tidal circularisation rate. Thus for simplicity, we have adopted p=0 for all components of the stellar system. We note that if a steeper cusp profile were to be adopted, the number of tidally captured stars at any particular time would increase by a numerical factor of order unity.
Making the usual assumption that the phase space distribution function for
"typical'' stars depends on the binding energy per unit mass only
and the distribution of white dwarfs is the same as
that of typical stars, it can be easily shown from (1) with p=0that the distribution function for the white dwarfs
over binding energy per unit mass and specific angular momentum,
,
is
The assumption of isotropy of the phase space distribution function
is applicable only for estimates of quantities
determined by the bulk of the stars, such as e.g. quantities
characterising distant gravitational interactions, see Eq. (5) below.
It becomes invalid
for stars with sufficiently low orbital angular momentum which
can either be tidally disrupted or
directly captured by the black hole.
The presence of the
black hole thus leads to the formation of a "loss cone''
such that Eq. (2)
over-estimates the number of stars having low angular momenta.
Assuming that stars with sufficiently
small specific angular momenta
are absent,
the presence of the loss cone may be easily
accounted for by a correction factor (e.g. Lightman
Shapiro 1977)
such that the distribution function of the white dwarfs is modified to become
In order to make order of magnitude estimates we use the simplest possible model for the white dwarf. We assume zero temperature so that the pressure is due to completely degenerate electrons. In this case the equation of state is baratropic and gravity or g modes are absent. Furthermore we approximate the structure of the white dwarf as that of a n=1.5 polytrope. In addition to the mass m and radius we introduce the two parameters and cm). The effectiveness of tidal interactions depends significantly on the mean density of the white dwarf. To characterise low and high density cases we consider two sets of values for and The "low density'' case will be characterised by parameters appropriate for a "typical'' white dwarf for which and , and for the "high density'' case we use parameters corresponding to Sirius B for which and .
For a baratropic star only fundamental (f), pressure (p) and inertial modes can be excited by tidal interaction. Since p-modes have large eigen-frequencies and inertial modes are significant only for rather large orbital angular momenta (Papaloizou Ivanov 2005), their respective contributions to the tidal exchange of energy and angular momentum are small compared to the contribution of the f-modes for the range of the orbital parameters of interest, so they are not considered further.
To make our estimates, we use the linear theory of tidal perturbations induced in a slowly rotating star and consider only the leading order quadrupole response. The assumption of slow rotation is certainly valid in the early stages of orbital evolution due to tides. In this regime orbital changes induced by distant gravitational interactions and tidal effects compete with each other. Then we may assume that the star is non-rotating when calculating the rate of tidal circularisation.
However, the star can be spun up because of the effect of tides during orbital circularisation, achieving a considerable rotation rate during the later stages. As we will see below this may have important implications for the outcome of the orbital evolution of the star. In particular, it may play a role in determining whether is it possible or not for the star to survive in a tight quasi-circular orbit. Therefore, it is directly related to the problem of formation of sources of gravitational radiation. Unfortunately, a theory of tidal perturbations of a fast rotating star is practically undeveloped in the present time. Accordingly, we shall assume that the results obtained from the theory of slowly rotating stars can be extrapolated to the case of fast rotation for the purposes of making order of magnitude estimates.
During the early stages of tidal circularisation, the orbit of the star is highly eccentric. In this case tidal interactions are only important near pericentre. Because of this, in order to calculate the energy and angular momentum exchange between the orbit and the modes of pulsation of the star it is possible to consider it as undergoing a sequence of successive flybys of the black hole each of which produces an impulsive change. This formulation of the problem allows us to use the theory of tidal excitation and dissipation of the fundamental modes in a baratropic star moving on a highly eccentric or parabolic orbit developed elsewhere (see e.g. Press & Teukolsky 1977, hereafter PT; Lai 1997, IP). This theory is based on consideration of a single pericentre passage where an amount of energy and an amount of angular momentum is transferred from the orbit to the star. These quantities depend on the amplitudes and phases of the pulsation modes before the passage and on the radius of pericentre, .
Let us temporarily assume that the amplitude of the pulsation mode
before pericentre passage is negligible and the stellar model
and the angular velocity of the star are fixed. In this case, it can be shown that the quantities
and
are determined by the value of dimensionless parameter (PT)
The quasi-static part of the energy transfer , can be written in the form: , where is some "reference'' energy associated with the star and the dimensionless coefficient is determined by the dissipative processes operating in the star. An explicit expression for can be found in IP. During the flyby energy loss occurs as a result of viscous dissipation in the bulk of the white dwarf and in its convective envelope and also due to emission of gravitational waves. Since the coefficient of dynamic viscosity in the bulk of the white dwarf appears to be very small (10^{4}-10^{6} in cgs units, e.g. Chugunov Yakovlev 2005, and references therein) and convective envelopes of white dwarfs also have a rather small estimated turbulent viscosity and a very small relative mass, their respective contributions to the coefficient are quite small. The main contribution appears to come from the emission of gravitational waves generated by time dependent perturbation of the white dwarf (see Osaki Hansen 1973, hereafter OH). However, a simple estimate shows that the value of determined by this process leads to a value of much smaller than that associated with dynamic tides, for relevant values of . Therefore, the overall contribution of quasi-static tides appears to be negligible and will not be considered further.
Now let us consider dynamic tides associated with
the fundamental quadrupole mode of pulsation. As was shown by IP
for the case of a slowly rotating star,
the quadrupole mode propagating in the direction of orbital motion
(the so-called prograde mode) determines the exchange
of energy between the orbit and the star provided that
the angular frequency of the star,
,
is
smaller that its "equilibrium'' value
,
where
is a "reference''
frequency associated with the star. As it
will be shown later, in our problem a typical value
of
,
and the equilibrium angular frequency
is formally larger than
the angular frequency at rotational break-up of the star,
.
Although the theory
leading to this conclusion is not valid at such high rotation
rates, it seems reasonable to assume that the energy exchange
is mainly determined by the prograde mode unless a rate of
rotation very close to
is reached.
Considering excitation of only this
mode and assuming that the star is not
perturbed before pericentre passage, it easy to see
from results given in IP that the energy gained by the
star per unit of mass,
can be expressed as
The characteristic pericentre distance below which tidal disruption is expected
is determined as the distance where the tidal force acting on an element of the star near its surface is comparable to the gravitational force due to the star itself. Thus it is given by
The black hole mainly tidally disrupts the white dwarfs
provided that its mass is sufficiently small.
To see this we note that the specific orbital angular
momentum of stars directly captured by the black hole must be smaller
than
.
Define the "critical'' mass
by the condition
that
when
One obtains
As indicated above, the rotation rate of the star has an
important effect in determining the strength of the tidal interaction.
From the results given in IP it follows that the corresponding
correction factor
entering in Eq. (7) has the form
Let us consider a sequence of many pericentre flybys assuming that the energy per unit mass contained in modes of oscillation, E_{m}, does not decay significantly between successive pericentre passages. In this case there is interference between the preexisting wave perturbation in the star and the wave excited in the vicinity of pericentre as a result of tidal perturbation. In this case, simple addition of (7) to E_{m} after the pericentre passage to obtain the new E_{m} needs further justification. In this context it is important that the orbital period is changing with time as a result of tidal interaction and loss of orbital energy due to emission of gravitational waves^{}, and therefore the the phase change of the oscillating mode generated between two successive pericentre passages, , is also changing with time. When the change of during one orbital period, is larger than , the mode phase at the time of the next pericentre passage is essentially a random quantity, and accordingly, the mode energy per unit mass evolves as a stochastic variable. In this case we can add expression (7) to at each pericentre passage on average, assuming that all resulting expressions are valid in some statistical sense being averaged over many realisations of the stochastic process. The process of stochastic evolution of the mode energy will be refereed to as "stochastic instability''.
A condition for the stochastic instability to operate can be
readily found from the condition
and the expression (3) for the orbital period in the form^{}
It is interesting to note that the condition (13) can be also derived from quite a different approach based on the treatment of the tidal interaction as occurring through resonances between the mode frequency and the changing orbital frequency of different order n, such that . The condition of overlap of these resonances given that is changing is This is equivalent to the condition (13).
As we have mentioned above the most important linear decay channel of
the fundamental mode appears to be through emission of
gravitational waves. The corresponding decay time of the fundamental
mode has been calculated by OH for two models of iron
white dwarfs. The results can be extended to the range of
masses and radii appropriate for "standard'' CO white
dwarfs with help of the expression for the gravitational
wave luminosity, L_{GW}, produced by the oscillating mode and the resulting decay time
can be expressed as
In this situation the mode energy can attain a large value as a result of the cumulative effect of multiple tidal interactions occurring at pericentre passage. At high amplitudes non linear effects may lead to additional dissipation and mode decay. We characterise such effects by adopting a mode decay time scale t_{nl} which is a function of the mode energy. The non linear theory for these pulsation modes is not yet adequately developed to provide a form for t_{nl}. Therefore, we regard this as a free parameter and consider two limiting cases: a) the time t_{nl} is assumed to be large compared to , being taken to be comparable to , the actual time scale of orbital evolution which is the smaller of the times required to change the orbit significantly through tidal interaction or to make it evolve as a result of the emission of gravitational waves induced by that orbital motion, see below Eq. (22). Assuming that all quantities characterising the mode of pulsation and the star evolve on the time scale of order of , we estimate below that the condition is equivalent to the condition that the rotational angular momentum of the star is the same order of magnitude as the angular momentum corresponding to the mode, see Eq. (18) below. Thus, in this situation the degree of non-linearity adjusts in a way that does not allow the mode angular momentum to significantly exceed that in the stellar rotation. b) The time is assumed to be short compared to both and the orbital evolution time scale However, in both cases we shall assume that t_{nl} is much larger than the orbital period of the tidally interacting star. But note that this is not strictly necessary during the first phase of evolution when the semi-major axis is large (see Sect. 4.1 below)
Here we estimate the amount of mode energy
that is stored in the star when there is a balance between
the build-up of mode energy due to
the stochastic instability and decay due to the emission of gravitational
waves or non linear effects.
Provided this balance can occur with a stellar
mode angular momentum content that is small enough to avoid break up
of the star,
it is possible for it to
avoid disruption by having to absorb a large amount
of released orbital binding energy.
In a stationary state
the mode energy per unit of mass can be estimated from the balance equation
The rate of transfer of angular momentum to the star is
related to the rate of dissipation of mode energy
(e.g. Goldreich & Nicholson 1989) and thus determined by
t_{nl}. The rate of transfer of specific angular momentum is
and accordingly the
total specific angular momentum transferred to the star in a time
interval
is estimated to be
Estimating the moment of
inertia of the star per unit of mass as
and
using the standard relation
,
from Eqs. (16)-(18) we obtain
the associated dimensionless angular velocity of the star
to be given by
In general the time scale of orbital evolution
is given by
Recalling that the break up angular velocity
,
and assuming that
the criterion
provides the condition for the
white dwarf to survive the circularisation process,
we obtain a limitation on the orbital binding energy per unit mass
of the white dwarf, during the phase of stochastic instability,
to be given by
In the opposite limit corresponding to case b) we have
Then Eq. (23) gives
Note too that we have neglected the change of the white dwarf radius due to tidal heating. Assuming this change is modest, we can take it into account by considering the white dwarf to have a slightly smaller mean density when discussing conditions for safe circularisation, see the next section.
The change of orbital energy per unit mass
due to emission of gravitational waves generated by orbital motion
of the star can be obtained from results given by
Peters (1964). In the case of a highly eccentric orbit,
the change per orbital period, which is mainly induced at pericentre
may be written in the form
On the other hand, the star is spun up by tides during ongoing orbital circularisation and t_{T} is increased as a result. Therefore, at a later stage, the orbital evolution may be governed by emission of gravitational waves when the orbital binding energy is sufficiently large, depending on parameters of the white dwarf and of the star cluster. Moreover, when the binding energy exceeds the value corresponding to the onset of stochastic instability, the tidal response becomes quasi periodic and ineffective so that the orbital evolution is solely determined by the emission of gravitational waves.
Before going on to estimate of the rate of tidal capture and circularisation we use the results of the previous sections to examine more closely the conditions required for mode amplitude limitation due to gravitational wave emission to enable safe circularisation with the possibility of tidal heating, inflation and disruption avoided.
During the early evolutionary phases, subsequent to tidal capture, the star's orbit is highly eccentric, and the orbital angular momentum is approximately conserved during the orbital evolution while the semi-major axis/binding energy as well as the angular velocity of the star change with time (see e.g. IP). When tides dominate over the emission of gravitational waves induced by orbital motion, the evolution time scale of the semi-major axis is given by Eq. (20) and we have . When gravitational radiation controls the orbital evolution, which happens at large enough orbital binding energy, we have where t_{GW} is given by Eq. (29). For a given value of the orbital angular momentum, the orbital evolution time depends on the orbital energy E, and thus, on the star's semi-major axis a. This dependence can be found from Eqs. (19)-(25) and (29). For illustrative purposes we plot this dependence as well as that of other characteristic time-scales in Figs. 1 and 2 for a "low density'' white dwarf orbiting around a black hole. The cusp size is taken to be 1 pc. All dependencies are calculated in assumption that the stochastic instability operates for the shown range of binding energies.
Figure 2: Characteristic time scales in years as functions of the dimensionless orbital energy Plotted here are , the decay time of the pulsation mode due to emission of gravitational waves associated with the stellar pulsation, , the time scale for evolution of the orbit, t_{T}, the time scale of build-up of the mode energy and , the time scale of orbital evolution due to emission of gravitational waves induced by orbital motion. The horizontal dotted line 1 represents . The two solid curves 2a and 2b represent calculated assuming that the stellar rotation frequency is given by Eq. (19). The curve 2a taking on larger values at small values of represents the case , and the other curve represents the case . The inclined dotted line 3 represents t_{GW} which becomes equal to at the larger binding energies where gravitational radiation controls the orbital evolution. The dashed and dot dashed lines 4 and 5 represent t_{T} calculated for and , respectively. The dot dot dashed curves 6a and 6b represent t_{T} calculated for the star rotating with angular frequency given by Eq. (19) for values of binding energies larger than that corresponding to the intersection of and . For smaller values of energies we have . Note the slower evolution due to tides for the faster rotation rate. This results in gravitational radiation ultimately controlling the orbital evolution. All curves are calculated for , , , r_{1}=1 and M_{4}=1. | |
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In Fig. 1 the characteristic time-scales corresponding to a star with are shown. The horizontal line indicates the decay time scale of the stellar pulsations due to the emission of gravitational waves, , given by Eq. (15). The inclined dotted line indicates the dependence of the characteristic time t_{GW} on and the inclined dashed and dot-dashed lines show the dependence of t_{T} on for (dashed line) and (dot-dashed line). We recall that the timescale t_{T} applies to the evolutionary phase just after tidal capture which can exhibit stochastic instability. It may not be applicable to the later stages, when is large, and the conditions for additive impulsive energy inputs at pericentre passage are not satisfied, see discussion below.
For a given value of , the dot-dashed line corresponding to a non rotating star, gives the smallest possible value of . For fixed , stars with values t_{T} exceeding that given by the dashed line, would have angular velocities exceeding , and so would be disrupted. Therefore the dashed curve gives the largest allowed value of .
The solid curves indicate the evolution of with which is given by Eq. (19). The two different curves correspond to the different assumptions about the value of the mode decay time due to non-linear effects, t_{nl}. The curve which is uppermost at small values of corresponds to the case a) where we assume that while the other solid curve corresponds to the case b) for which t_{nl} is much less than both and . One can see from Figs. 2 and 3 that the behaviour of the characteristic time-scales is rather similar for these two cases. When is very small, rotation of the star is small, and both curves are close to the dot-dashed line. Also, for small values of we have .
Figure 3: Same as Fig. 1 but for . Note that in this case we have for the whole range of binding energies shown. | |
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When increases, the star is spun up by tides and t_{T} gets larger. When attains a value the solid curves, which for smaller values, correspond to tidally driven evolution cross the dotted line representing evolution controlled by the gravitational radiation time scale t_{GW}. For larger values of the orbital evolution is mainly determined by emission of gravitational waves, and we have . In this regime the star continues to spin up by tides, and the tidal time scales shown by dot dot dashed curves get larger with increasing of . When the dot dot dashed curves cross the inclined dashed line, the binding energies are equal to given by Eqs. (24) and (26), and, according to our criterion, the star is disrupted at energies .
In Fig. 2 we show the same quantities calculated for a star with a smaller value of orbital angular momentum corresponding to . In this case the evolution is always controlled by tidal effects and for the whole range of energies shown and, accordingly, the the dot dot dashed curves coincide with the solid curves.
As follows from this discussion and Figs. 2 and 3, the white dwarf has the possibility of surviving the tidal evolution when the energy scale corresponding to the onset of the stochastic instability (see Eq. (13)) is smaller than - the energy corresponding to break-up rotation. When , disruption of the star is avoided during the phase of stochastic instability. In this case, when the orbital evolution proceeds mainly through emission of gravitational waves. Therefore, for our purposes it is very important to establish under what conditions the inequality holds.
As can be seen from Figs. 2 and 3, there are four different possibilities: case a1) where and tides determine the orbital evolution when the orbital energy of the star is close to ; case a2) where and the gravitational radiation determines the orbital evolution; case b1) when and tides determine the evolution, and the case b2) where and gravitational radiation controls the evolution. Note that it is straightforward to see that case b1) is always associated with disruption of the star. Indeed, in this case the factor entering Eq. (26) is of order of unity. That means that the energy is always small when compared to , for typical parameters of the system. Therefore, the condition cannot be satisfied in this case and the case b1) is not considered further. On the other hand, in the opposite case b2) the factor can be large when estimated at energies of the order of , see Fig. 1, and the condition can be fulfilled, see Appendix for details. In Appendix it is also shown that circularisation is possible for the cases a1) and a2) provided that the orbital angular momentum of the star and, accordingly, , is sufficiently large. Thus, circularisation can, in principal, be achieved for the cases a1), a2) and b2).
The condition
can also be reformulated in terms of
semi-major axes. Using the expression (23) we can easily find the characteristic
semi-major axis
corresponding to the binding energy per unit mass
When the non linear dissipation time scale t_{nl} is small (the cases b1 and b2),
does not depend on t_{nl} and we have
In order to estimate the rate of tidal capture by the black hole we assume that the stars in the cusp are slowly rotating. Therefore, at the beginning of orbital circularisation the effects of rotation are not important so we neglect it when estimating the rate of capture into circularising orbits, setting in the expression (21) for the time scale of orbital evolution. Then from the discussion in Sect. 5 and data plotted in Figs. 2 and 3 it follows that during this phase orbital energy loss resulting from tides is more important than that due to gravitational radiation.
The rate of capture into circularising orbits, as a function of initial
binding energy per unit mass of the star, depends on whether the tidal time scale (21) is larger than the orbital period
.
Equating (21) to (3) we find the binding energy per unit mass
where these two times are equal to be given by
At first we assume that
for a range of angular momenta of interest.
A star gets tidally captured by the black hole
when the characteristic time scale of evolution of its angular
momentum due to distant gravitational encounters with other stars
in the cluster
When the tidal evolution time scale is approximately equal to the orbital period and therefore the size of the tidal circularisation loss cone is determined by Eq. (33) which is, in this case, considered as an implicit equation for for a given value of . Stars with these energies which have calculated in this way have relatively large energy changes induced within a single orbit.
When
the number of stars tidally captured by the
black hole per unit of time, with typical energies of order of
some E,
,
can be estimated to be
For the case
it can be easily seen that
for any value of J.
This situation is analogous to the full loss cone regime
in the theory of tidally disrupted stars (e.g. Frank
Rees
1976; Lightman
Shapiro 1977). In this case the rate of tidally captured stars can be estimated as
Thus, the rate of tidal circularisation is mainly determined by
energies
(Novikov et al. 1992),
and therefore it important to obtain
an explicit expression for the value of
.
As we
discussed above this value is determined by the condition that
the timescale for orbital evolution due to tides be equal to the orbital period
while being on the boundary of the tidal loss cone.
It is determined by Eqs. (33) and (37) which can be rewritten in the form
This is different from the situation where only the effect of emission of gravitational waves is taken into account as a mechanism for changing the orbital energy. In that case the energy scale analogous may also be obtained from the requirement that the orbital evolution time be equal to the orbital period on the boundary of the corresponding circularisation loss cone. Obviously, our estimate of the circularisation rate is valid only when the size of the circularisation loss cone is larger than the size of "true'' loss cone, corresponding to direct capture or disruption of stars by the black hole. However, for typical parameters expected for gravitational waves to be the dominant process of orbital evolution ( and ) the energy scale is smaller than another energy scale defined by the condition that the sizes of the circularisation and true loss cones coincide at . When the stars are mainly destroyed by the black hole and the process of circularisation is strongly suppressed. In such a situation the capture rate is mainly determined by energies (Hopman Alexander 2005). Because of this difference, the tidal circularisation rate in our case, (see Eq. (51) below) differs from the estimate of the capture rate due to emission of gravitational waves provided by Hopman Alexander (2005), see also Novikov et al. (1992).
Equation (41) gives
The maximal rate of capture ,
can be estimated
a little more accurately as
According to the discussion in Sect. 5 and the Appendix,
such white dwarfs have a considerable survival probability,
for cases a1) and a2) corresponding to large t_{nl,} when
When the above conditions are not satisfied,
the typical binding energy of white dwarfs having
a considerable probability of survival, ,
has to be larger than
the critical value obtained from Eq. (49) and so their capture rate
is reduced by a factor
when compared to that given by Eq. (53) (see Eq. (39)). Accordingly their
capture rate is given by
We go on to obtain explicit expressions for the rate of capture of white dwarfs that may survive the circularisation process in that case. This rate will be referred to as "the circularisation rate''.
First let us consider the case a) for which
.
As follows from Eq. (A.8)
when
.
Since the systems containing sufficiently
massive black holes are likely to provide the bulk of the sources of
gravitational radiation, we assume hereafter that
and set, accordingly,
.
In this case the condition (54) leads to
We use Eqs. (44), (47), (53) and (57) to obtain the
the suppressed capture or circularisation rate in the form
A comparison of Eq. (53) with Eqs. (59)-(61), (63)-(65) shows that our criterion of survival of the white dwarfs during the process of circularisation typically results in an order of magnitude decrease of the circularisation rate. However note that for case b2) when t_{nl} is small, the suppression is not significant for the ``high density'' white dwarfs..
The probability of formation of observable sources of gravitational waves is determined by several important factors such as the rate of formation of white dwarfs on circularising orbits calculated above, the subsequent orbital evolution of the white dwarfs, the abundance of globular clusters having sufficiently massive black holes and the properties of a gravitational wave antenna receiving the signal. In what follows we make a rough estimate of the probability assuming that the gravitational wave antenna has characteristics close to what has been indicated for the future LISA space borne gravitational wave antenna.
LISA has its maximal sensitivity in the frequency range 10^{-2} Hz for which the dimensionless amplitude of the gravitational
waves, h, can be as small as
Now, substituting (70) in (66) we obtain
minimal values of the amplitude h, ,
which can
be detected by LISA
In order to find the total volume of space, available
for detection of the signals by LISA, we should relate the quantities
to the distance from a source,
.
For that
the amplitude h of radiation emitted by the source should be known
as a function of distance and of parameters of the source.
The appropriate relation has been obtained, e.g., by Nelemans et al. (2001). In our units it has the form
In order to find the total number of potential sources,
,
we use an estimate obtained by Portegies Zwart
McMillan (2000) for
the average number of globular clusters in the Universe,
for the value of the Hubble constant
H=70 km s^{-1} Mpc^{-1}^{}. We obtain from Eq. (73)
For case a) with
,
using Eqs. (60), (61) and (74) we have
For case b) with
we proceed in a similar manner
but use Eqs. (64) and (65) instead of Eqs. (60) and (61), respectively. We find
As for case a) "typical'' white dwarfs have a much larger probability of detection. However, for case b) this probability depends explicitly on the black hole mass when the cusp size is expresses in terms of the typical stellar velocity dispersion . As follows from Eqs. (79)-(81) when and km s^{-1} this probability is about two times larger than that corresponding to case a). But the probability ratio decreases with the black hole mass if is kept fixed.
In this paper we have given a qualitative analysis of the problem of tidal circularisation of white dwarfs in globular clusters containing black holes with masses in the range 10^{3}- and estimated the rate of production of white dwarfs that begin to circularise their orbits for specified parameters for the cluster, white dwarf and black hole, see Eq. (53). We also proposed a simple criterion for "survival'' of a white dwarf during the stage of orbital circularisation, and found production rates for white dwarfs, which can, in principal, settle down in a quasi-circular orbit thus forming sources of gravitational radiation, see Eqs. (59-61) and (63-65). These rates are order of magnitude smaller than that given by Eq. (53). We made a simple estimate of the probability of detection of systems containing white dwarfs on quasi-circular orbits and found that these systems can, in principle, be detected by LISA provided that the globular clusters containing black holes are sufficiently abundant and their stellar velocity dispersion near the radius of influence of the black hole is sufficiently large, see Eqs. (78) and (81).
Our results should be treated with caution. Many processes occurring during the formation of circularising stars and during ongoing tidal circularisation depend strongly on the parameters of the problem and this leads to uncertainties in the results. Let us consider some of them.
Finally we comment that in a recent paper Baumgardt et al. (2006) have calculated the tidal capture rates for stars of different types with help of N-body simulations, during first 12Myrs of evolution of the system. They have considered black holes with masses in the range 10^{3}- . Giant stars and low mass main sequence stars with mass <0.4 have been modelled as n=1.5 polytropes. The low mass stars seem to be absent in the cusp due to the effect of mass segregation operating during formation of the cusp. However, their capture rate of red giants can be compared with what is given by Eq. (53). They have obtained a capture rate for red giants of the order of a few events per run. Assuming that a typical red giant has a radius 10 , mass , taking the black hole mass and the cusp size to be equal to and to 0.1 pc, respectively, and substituting these values into Eq. (53), we obtain yr^{-1}, where is the number fraction of the red giants in the cusp. Our results appears to be in full agreement with the results of N-body simulations provided that .
Acknowledgements
We are grateful to A. G. Polnarev for fruitful discussions to A. G. Doroshkevich for useful remarks and to the referee for valuable comments. P.B.I. has been supported in part by RFBR grants 04-02-17444 and 07-02-00886.
As indicated in Sect. 3.7, there is a phase of evolution at large semi-major axis subsequent to tidal capture where the internal pulsation mode energy may increase in a stochastic manner. This requires the semi-major axis to exceed above which the pulsation mode does not maintain phase coherence between successive pericentre passages.
Here we obtain an explicit expression for the semi-major axis
defined through Eq. (13). At first let us assume that
during the orbital
evolution. This situation is illustrated in Figure 2. In this case we substitute Eq. (7) in
Eq. (13) and obtain
As indicated in Sects. 4 and 5, when a white dwarf has a possibility of surviving the process of tidal circularisation only if it is cooled efficiently by emission of gravitational waves for values of its semi-major axes > . Therefore, in order to reach a quasi-circular orbit, the white dwarf must have orbital parameters such that where is given by Eq. (32). This inequality leads to an inequality of the form (or equivalently the fixed pericentre distance must exceed a certain value), where is defined by the condition . In general, is a monotonic function of the stellar rotation rate with smaller values of corresponding to larger values of . This is because tides weaken with increasing enabling safe circularisation starting with smaller pericentre distances.
Thus in order to take into account all possible evolutionary
tracks leading to formation of circular orbits, we calculate the lower
boundary of
assuming that
.
In this case the
factor
entering (A.1) takes the form
In the subsequent discussion we
use the quantity
defined in Eq. (8)
rather than .
This quantity can be approximately represented
in the form
Now let us assume that gravitational waves determine the orbital evolution
when a rapidly rotating white dwarf has its semi-major axis near to
and
,
see Fig. 1. Proceeding as in the preceding section, we
obtain an equation for the critical value of that is analogous to Eq. (A.4) in the form
The inequality (A.8) implies that for a sufficiently large values of the black hole mass and the rotation of the white dwarf the orbital evolution is mainly determined by emission of gravitational waves when the semi-major of the orbit is close to the characteristic semi-major axis corresponding to the onset of stochastic instability. For the parameters corresponding to a "typical'' white dwarf with and , we have , and for a dense white dwarf of Sirius B type with and , we have M_{*}=0.13. Both characteristic values of M_{*} are smaller than the black hole masses of systems expected to produce a large amount of gravitational radiation during the last stages of orbital evolution of the star. Therefore, the condition is more important for our purposes.
Now let us consider the case when . The discussion proceeds in a similar way to the previous case but now is given by Eq. (31). As we have mentioned above it can then be shown that gravitational waves dominate the orbital evolution when the semi-major axis is provided that the black hole mass exceeds , and accordingly, .
Therefore in order to obtain an equation for the parameter
separating orbits leading to safe circularisation from those
leading to disruption of the star, we equate expressions (31) and (A.2),
and use Eqs. (7), (28), (29) and (A.3),
to obtain