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\begin{document}
%
\title{Tidal effects in the vicinity of a black hole}
%
\classification{ 98.35.Jk, 98.35.Mp, 98.54.-h, 98.54.Cm, 04.70.-s }
%
\keywords{Galaxy: nucleus - galaxies: active - black hole physics}
%
\author{Andrej Cadez}{address={Faculty of Mathematics and Physics,
University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia} }
%
\author{Uros Kostic}{address={Faculty of Mathematics and Physics,
University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia} }
%
\author{Massimo Calvani}{address={INAF - Astronomical Observatory of
Padova, Vicolo Osservatorio 5, 35122 Padova, Italy}}
%
%
\begin{abstract}
The discovery that the Galactic centre emits flares at various
wavelengths represents a puzzle concerning their origin, but at the same
time it is a relevant opportunity to investigate the environment of the
nearest super-massive black hole. In this paper we shall review some of
our recent results concerning the tidal evolution of the orbits of low
mass satellites around black holes, and the tidal effect during their
in-fall. We show that tidal interaction can offer an explanation for
transient phenomena like near infra-red and X-ray flares from Sgr A*.
\end{abstract}
%
\maketitle
%
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%% MAINMATTER
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%
\section{Introduction}
%
It is nowadays widely accepted that most galaxies, if not all, contain a
super-massive black hole at their centre. How these black holes are
formed and how they are related to galaxy formation is still an open
issue, see e.g. \cite{Rees:2006}. The nearest super-massive black hole
(SMBH) is located at the centre of our Galaxy, Sgr A*. Stellar orbits
determinations \cite{Ghez1:2005} show that a central dark mass of
$(3.7\pm 0.2)\times 10^6 [R_0/(8\, {\rm kpc})]^3 M_\odot$ (with an
updated value of $(3.61\pm 0.32)
\times 10^6 M_\odot$ from observations with SINFONI
\cite{Eisenhauer:2005}) is confined within a radius of 45 AU. The
proximity of the Galactic centre ($\approx 8$ kpc) provides us a unique
opportunity to study in detail the environment of massive black holes
(see e.g.\ \cite{Alexander:2005} that contains an exhaustive description
of the Galactic centre), and possibly of their formation. Due to its
proximity, Sgr A* is the only galactic nucleus that can be investigated
using VLBI on sub-AU linear scales, and observations in 2007 have
confirmed structure in Sgr A* on scales of just a few Schwarzschild
radii \cite{Doeleman:2008}.
The recent detection of rapid flare activity coming from Sgr A*
represents additional strong evidence for the presence of a massive
black hole. Their time scales strongly suggest motion only a few
Schwarzschild radii away from the central black hole. If such flares are
uncorrelated single events of accretion, we believe that the most
plausible candidate for their energy source is the tidal part of the
gravitational potential energy. We suggest that Galactic flares are
produced by the final accretion of a single dense object, like an
asteroid or a comet, with a mass of $\approx 10^{20}$g. We explore this
scenario, and we show that the light curve of a flare can be deduced
from dynamical properties of geodesic orbits around black holes and that
it only weakly depends on the physical properties of the source.
%
%
%
\section{Flares from the Galactic centre}
%
Flares from Sgr A* have been detected in X-rays (Chandra and XMM-Newton
satellites) and infra-red (VLT adaptive optics imager NACO, SINFONI
infrared adaptive optics integral spectrometer on the ESO VLT). The
description of this flaring activity can be found in several papers,
e.g. \cite{Baganoff:2001, Genzel:2003, Ghez:2004, Belanger:2006,
EckartB:2006, MeyerA:2006}. These flares are modulated on a short
time-scale with average periods of $\approx 20$ minutes, and
multi-wavelength campaigns found that the time lag between X-ray and NIR
flare emission is very small, strongly suggesting a common physical
origin, see e.g. \cite{Eckart:2004, EckartA:2006, Yusef-Zadeh:2006,
Trippe:2007}.
The main features of the observed flares are: a) duration: $\approx
1/4$h -- 2h; b) frequency: few events per day; c) energy release: of the
order of $10^{35}$ erg/sec; d) linear polarization (NIR, radio): of the
order of few to few ten percent, with stable polarization angle; e)
strong quasi-periodic modulation on time scales of minutes.
Several different models were proposed to explain these phenomena: disk
instabilities (e.g. \cite{Tagger:2006, Yuan:2006, Falanga:2008,
Eckart:2008}), star-disk interaction (e.g. \cite{Nayakshin:2004}),
expanding hot blobs (e.g. \cite{Yusef-Zadeh:2006, Marrone:2008}), hot
spot/ring models (e.g. \cite{Trippe:2007, Eckart:2004, EckartA:2006,
Belanger:2006, MeyerA:2006, MeyerB:2006, Broderick:2006,
Zamaninasab:2008}). Here we do not want to compare and contrast the
above proposed model. Our aim is to propose a scenario in which
Galactic flares are produced by the final accretion of a single dense
object, like an asteroid or a comet. The main motivations of this
approach are: a) the
time-scales of flares strongly suggest motion only a few Schwarzschild
radii away from the black hole; b) the energy release during these
flares corresponds to the source mass of the order of $10^{20}$g, see
e.g.\ \cite{Genzel:2003}; c) it seems reasonable to assume that stars at
the Galactic centre are surrounded by planets and by other small
orbiting bodies, like asteroids and comets (hereafter: LMS, low-mass
satellites); d) tidal evolution of the orbits of LMS will lead to
capturing orbits; e) tidal effects in the vicinity of a black hole can
release a significant amount of energy.
Asteroid-like objects fit both observational facts (time-scales and
energetics): they are of the right mass and they are tidally disrupted
closer to the black hole than gaseous blobs. In active galaxies, such
events would be outshined, but in the inactive centre of our Galaxy, it
is possible to observe them.
In the following Sections we shall expand on the above motivations. Most
of this work is based on our recent papers, e.g. \cite{Gomboc:2005,
Cadez:2008, Kostic:2008, Kostic:2009}, where a thorough discussion and
more details can be found.
%
%
%
\section{LMS at the Galactic centre}
%
According to \cite{Cochran:1995}, the Edgeworth-Kuiper belt of our Solar
System may still (4.5 billion years after its creation) contain as many
as $2\times 10^8$ objects of radii $\lesssim 10$ km. It appears
reasonable to assume that stars at the Galactic centre are surrounded by
planets and by other small orbiting bodies (LMS), like asteroids and
comets, that may be stripped off their parent stars by tidal
interaction, while approaching the black hole. Therefore, we expect that
there must be a considerable number of solid objects that cluster the
Galactic centre.
Little is known about dynamics that determines the fate of such low mass
satellites. However, some hints come from the investigation of core
dynamics at the Galactic centre \cite{Berukoff:2006}; from the
investigation of mechanisms that may cause a star to move inwards into a
massive black hole, in connection with possible detection of
gravitational waves with LISA, see \cite{Hopman:2006} and references
therein; from detailed dynamical studies of the evolution of the Solar
System, see e.g. \cite{Dermott:1988, Faber:2005, Mudryck:2006}.
One might expect to find stellar system satellites all the way down to
the black hole at the Galactic centre, with a fair proportion of them on
low-periastron, highly eccentric orbits. A significant work will be done
by tidal forces near the low periastron. This work partially dissipates
as heat, and as a result it lowers orbital energy and may start
significant tidal evolution of the orbit, see \cite{Cadez:2008}. When
periastron crossing frequency and fundamental quadrupole frequency of
the object are the same, that is at resonance, tidal interaction is
strongest. In this context, one must consider two classes of LMS: those
that are gravity dominated and those that are solid state forces dominated.
For gravity dominated objects the fundamental frequency is $\nu_{\rm g}
\approx 2 \sqrt{G \rho / 3\pi}$, while for solid state dominated objects
it is $\nu_{\rm s}=\frac{1}{4}c_{\rm s}/R$ ($\rho$ is the density of the
body, $R$ its radius and $c_{\rm s}$ the speed of sound). Taking as
typical values $c_{\rm s}\approx 5\ \rm{km/s}$ and $\rho\approx 5\
\rm{g\ cm^{-3}}$, we find that the radius dividing the two classes is
about $1000\ \mathrm{km}$, i.e. the radius of the asteroid Ceres.
Accordingly, all gravity dominated satellites must have about the same
fundamental quadrupole frequency, that corresponds to a period of about
1 hour and all smaller satellites have shorter fundamental periods.
Thus, gravity dominated satellites start a rapid, resonant tidal
evolution when their periastron reaches $r_{\rm p}\approx 10\ r_{\rm g}$
($r_{\rm g} = GM_{\rm bh}/c^2$), while solid state dominated bodies may
undergo significant tidal evolution even closer to the black hole.
The difference between these two classes of objects is better understood
introducing the Roche radius and the effective Roche radius. The
effective Roche radius is the radius at which $\omega_{\rm q} T_{\rm f}
= 1$, where $\omega_{\rm q}$ is the fundamental resonant frequency of
quadrupole modes and $T_{\rm f}=(2 r_{\rm p}^3/ GM_{\rm bh})^{1/2}$ is
the periastron crossing time \cite{Gomboc:2005}. In \reff{Fig:GCobjects}
we show the Roche radius and the effective Roche radius as a function of
object size, mass and density for different bodies expected to populate
the Galactic centre. This graph accentuates the very particular position
of asteroids. With the exception of compact stars such as white dwarfs,
neutron stars or stellar mass black holes, they are the only objects
that can survive the tidal field of the black hole when held together by
solid state forces. All other objects such as planets, main sequence
stars and of course giants find themselves beyond the Roche radius above
ten or a few ten gravitational radii. Note, however, that molten
asteroids become gravity dominated. If melting occurs suddenly below
$\sim 10\ r_{\mathrm{g}}$, the phase change results in a dramatic
rapidly developing tidal disruption. We intend to show that why and how
this may happen.
%
%
\begin{figure}
\includegraphics[width=\columnwidth]{Fig2-new.pdf}
\caption{Astrophysical objects capable of releasing significant tidal
energy before falling down the Galactic centre black hole. The positions
of different celestial objects are also marked in this diagram. Note
that the effective Roche radius for solid state dominated asteroids is
below $2\ r_{\rm g}$, while the Roche radius for these objects is about
$10\ r_{\rm g}$.}
\label{Fig:GCobjects}
\end{figure}
%
%
%
\section{Tidal evolution of LMS orbits}
%
Tidal effects on a star encountering a SMBH have been investigated by
several authors, see e.g. \cite{Gomboc:2005, Brassart:2008} and
references therein. However, tidal forces will not affect only the shape
and the structure of the in-falling body, but are responsible for the
evolution of the orbits and the rotation periods of the involved objects.
The first stages of tidal evolution of LMS orbits can be investigated
using Hut's formalism described in \cite{Hut:1980, Hut:1981, Hut:1982}.
This formalism is suitable for describing slow tides, i.e. tides that
change slowly with respect to the frequency of the fundamental
quadrupole modes of the object that tides are acting upon. In this case
the relevant forces that exchange angular momentum and energy can
accurately be calculated by approximating the deformed object adding two
additional bulge masses $\mu$ on the surface of the star. In this case
tidal evolution of orbits depends only on the parameter $\alpha=l_0
/s_0$, which is the ratio of orbital angular momentum of the system and
spin of the secondary at equilibrium. Therefore, completely different
binary systems with the same value of $\alpha$ behave in the same way.
See \cite{Kostic:2008} for details.
We find that the evolution of orbits strongly depends on the initial
value of $\tilde {\rm r}_{p,0}$, which is the ratio between the
periastron distance ${\rm r}_{p}$ and the radius of the equilibrium
circular orbit ${\rm a}_0$. If an orbit starts with $\tilde {\rm
r}_{p,0}$ below some critical value, it will always end up in
in-spiralling, regardless of the value of $\alpha$. Orbits starting
above this critical value of $\tilde {\rm r}_{p,0}$ will either
circularize, or become unbound.
Tides induce forced oscillations on the object, exciting its eigenmodes.
They are non-resonant, as long as the frequencies $\omega$ of spectral
components of tidal interaction are much lower than the quadrupole
eigenfrequencies $\omega_q$ of the object. If, on the other hand,
spectral components of tides are strong at frequencies close to
eigenfrequencies of the quadrupole oscillations of the object, which
happens when the orbit touches the Roche radius, tides become resonant.
\reff{Fig:FigEvolution} and \reff{Fig:FigResonant} summarize our results.
%
%
\begin{figure}
\includegraphics[width=\columnwidth]{Fig3-new.pdf}
\caption{As tides drive the tidal bulge across the surface of the
solid asteroid, work deforming the body is done at the expense of
orbital energy; little angular momentum is deposited in spinning the
body because of its low moment of inertia (non-resonant phase). When the
body melts and spends more and more time beyond the Roche radius, it
elongates, increases the moment of inertia which allows both orbital
angular momentum as well as orbital energy to be transferred to spin
(resonant phase).}
\label{Fig:FigEvolution}
\end{figure}
%
%
\begin{figure}
\includegraphics[width=\columnwidth]{Fig4-new.pdf}
\caption{Resonant and non-resonant time-scales as a function of
$\beta$ for $\rho = \rho_0$, R $= 10$Km, Q $=$ 200. $\beta$ is the Roche
penetration parameter $\beta = r_R/r_p$ and Q is the resonant damping
factor.}
\label{Fig:FigResonant}
\end{figure}
%
%
As indicated in \reff{Fig:FigEvolution}, tidal evolution of the orbit
has two distinct phases: the non-resonant phase, well describable with
Hut's slow tides formalism and the resonant phase, which was studied
numerically in affine approximation by \cite{1986ApJS...61..219L} and in
Newtonian approximation by \cite{Gomboc:2005}. The energy loss per
periastron passage can in both cases be described by the expression
%
\begin{equation}
\Delta E = - \left({{G M_{\rm bh} mR^2}\over{r_{\rm
p}^3}}\right)\varepsilon^2(\beta_{eff}) \ ,
\label{tidalwork}
\end{equation}
%
where $\varepsilon^2(\beta_{eff})$ is the eccentricity of the ellipsoid
to which the originally spherical object is deformed by tides; it
depends essentially only on the effective Roche penetration parameter
$\beta_{eff}=r_{R_{eff}}/r_p$\footnote{This approximation is valid for
small amplitude resonant tides and is valid as long as most part of the
orbit is outside the Roche radius, i.e. on highly eccentric orbits whose
periastron reaches below Roche radius.}. In the case of gravity
dominated objects $\beta_{eff} = \beta = r_R/r_p$ and in the solid case
$\beta_{eff}$ is smaller; it can be expressed as
$\beta_{eff}=\frac{c_s}{8R}\sqrt{\frac{3\pi}{G\rho}
}\frac{r_R}{r_p}\approx 3300\ \mathrm{km}/R\times \beta$, where the
factor 3300 comes from inserting the approximate value for the velocity
of sound and the density of solid bodies. The time scales for
non-resonant and resonant tidal evolution phase can then be calculated
(cf. \cite{Cadez:2008}). They can be expressed in the form
%
\begin{align}
t_{\rm EN} &= 1.4\times 10^{11}yr\times \frac{Q
f_{EN}(e)}{\beta^8}\left( \frac{10{\rm km}}{R}\right
)^2\left(\frac{\rho_0}{\rho}\right)^{\frac{7}{6}}\\
t_{\rm ER} &= 2.9\times 10^{10}yr\times \frac{
f_{ER}(e)}{\varepsilon^2(\beta)\beta^{\frac{7}{2}}}\left( \frac{10{\rm
km}}{R}\right )^2\left(\frac{\rho_0}{\rho}\right)^{\frac{7}{6}}\ ,
\label{ResCas}
\end{align}
%
where $Q$ is the resonant damping factor, $f_{EN}(e)$ and $f_{ER}(e)$
are factors of order unity depending on the eccentricity of the orbit
($e$), and $\varepsilon^2(\beta)$ is a function that peaks at resonance
i.e. at $\beta = 1$.
These time-scales are plotted in \reff{Fig:FigResonant} for 10 km
objects with $\rho=5g\ cm^{-3}$ and $Q$=200. Also plotted in this figure
is the approximate melting time, i.e. the time it takes to accumulate
enough tidal work inside the object to melt it
($t_{melt}=E_{melt}t_{orb}/\Delta E$; $E_{melt}\sim 10^{-10}m\ c^2$),
and the thermal diffusion time for such an object (assuming the thermal
diffusion coefficient $D\sim 10^{-6}m^2s^{-1}$, characteristic of most
rock). It is clear from this figure that 10 km objects melt for
$\beta>1$, i.e. if their periastra reach below 33 $r_{\mathrm{g}}$ in
less then $3\times 10^5$ years. Asteroids that do not penetrate as deep
or are smaller would not melt, because their thermal diffusion time
scale is too short\footnote{Note that tidal heating is rather slow,
which indicates low tidal stress, so that the break-up of such asteroids
is not expected.}. On the other hand asteroids that are scattered or
released to highly eccentric orbits with pariastra at 10
$r_{\mathrm{g}}$ melt in only a few years. When melted, asteroids become
resonant and their tidal evolution time-scale shortens considerably,
certainly to below a few billion years but possibly even to a much
shorter time scales, depending on their size and most of all on the
periastron of the orbit that they were ejected or scattered to.
Increased tidal heating does not essentially raise their temperature,
because convection sets in so that low temperature ($\sim$1000 K) black
body radiation can dissipate tidal heat. One can conclude that all
asteroids found within $\sim 33\ r_{\mathrm{g}}$ are melted,
non-evaporated and are bound to end up in the black hole.
%
%
%
\section{The free fall phase}
%
At present we do not have a complete theory describing all the steps of
evolution toward plunging into the black hole. Clearly, at $33\
r_{\mathrm{g}}$ asteroids have entered the resonant part of their tidal
evolution in relativistic regime. \reff{Fig:FigEvolution}, based on
Hut's slow tide approximation, indicates that the characteristic of this
part of evolution is the transfer of orbital angular momentum to spin.
In the above resonance regime this transfer can not be effective by
spinning the object faster, but by deforming it into a longer and longer
thread, thus increasing its moment of inertia. Some understanding of
this mechanism can be obtained considering the virial theorem applied on
the self-gravitating mass of an orbiting asteroid. In
\cite{Gomboc:2005}, the authors consider a self gravitating object
moving in the gravitational field of a point mass (Newtonian
approximation). Following the steps similar to those in deriving the
virial theorem, they obtain the equation
%
\begin{equation}
W_{int}+\frac{1}{2}W_G=-G\frac{m_{\rm bh}{\bf R\cdot Q\cdot
R}}{R^5}+\frac{1}{4}\ddot J\ ,
\label{virial}
\end{equation}
%
where $W_{int}$ is the internal energy of the asteroid, $W_G$ is the
self-gravitational potential energy, ${\bf R}$ is the orbital vector
from the black hole to the asteroid, ${\bf Q}$ is the quadrupole tensor
of the deformed asteroid, and $J$ is the momentum scalar ($J=\int \rho
r^2 dV$). If one approximates the deformed asteroid by an axis-symmetric
ellipsoid with the symmetry axis in the direction of $\hat n$, then
${\bf Q}$ can be written in the form ${\bf Q}=q(3\hat n\hat n-{\bf I})$,
where ${\bf I}$ is the identity matrix and
$q=-\frac{2}{5}M^*(r_{pole}^2-r_{eq}^2)$. Furthermore,
$J=\frac{1}{5}(r_{pole}^2+2r_{eq}^2)$, and equation \ref{virial} becomes
%
\begin{align}
&W_{int}+\frac{1}{2}W_G=\nonumber \\
&\frac{M^*}{5}\left(\omega_K^2(r_{pole}^2-r_{eq}^2)(6\cos^2\alpha-2)+\frac{1}{4}({\ddot{r}^2}_{pole}+2\ddot{r}^2_{eq})\right)\
.
\end{align}
%
Neglecting the equatorial radius with respect to the polar one
($r_{pole}\gg r_{eq}$), this becomes
%
\begin{equation}
W_{int}+\frac{1}{2}W_G= \frac{M^*}{5}\left(\omega_K^2
r_{pole}^2(6\cos^2\alpha-2)+\frac{1}{4}{\ddot{r}^2}_{pole}\right)\ ,
\label{virial1}
\end{equation}
%
where $\alpha $ is the angle between ${\bf R}$ and $\hat{n}$ and
$\omega_K=G m_{\rm bh}/R^3$ is the Keplerian angular velocity
corresponding to a circular orbit at radius $R$. In stationary
equilibrium the right hand side of this equation vanishes and the
expression reduces to the well known virial theorem. In the other
extreme, if tides had elongated the object a few times its original
radius, the left hand side of the above equation becomes negligible with
respect to the first term on the right, so that the equation has become
a second order linear differential equation for $r^2_{pole}$. Its
solutions are oscillatory if the average value of $(6\cos^2\alpha-2)$ is
positive, and exponentially expanding if it is negative. For slow tides,
the bulge points almost in the direction of ${\bf R}$ ($\alpha
\rightarrow 0$) and accordingly $r^2_{pole}$ is an oscillatory function.
However, above resonance, the tidal bulge may point more and more in the
direction of motion, which changes the character of the above equation
making the tidal bulge grow exponentially.
The virial theorem is not sufficient to describe this phenomenon in any
detail, because it does not predict the value of the angle $\alpha$.
Therefore, we study the last stages numerically with a fully
relativistic code. This code, described in \cite{Kostic:2009} builds an
image of the tidally evolving object assuming that it starts as a molten
sphere. In calculating the developing shape of the body fully
relativistic dynamics is taken into account, with only one
simplification that the internal pressure can be neglected with respect
to the high kinetic energy density of flow in the tidal bulge. Evolving
images of the object are constructed as they would be seen momentarily
in the frame of a distant observer. A selection of a few slides in
\reff{Fig:Slides} shows such a development and confirms predictions of
exponential growth in the above simple argument. In
\reff{Fig:ExponentialGrowth} we also show the development of the visible
length of an originally spherical object as it is stretched by tides,
obtained in a number of numerical experiments. We find that resonant
tides develop exponentially growing tidal tails which are fastest and
most pronounced in the vicinity of the black hole.
%
%
\begin{figure}
\includegraphics[width=\textwidth]{Slike2-new.pdf}
\caption{A selection of images from simulations of tidal development
of an $0.03\ r_g$ object following an elliptic orbit around a black hole
observed from above the orbital plane (frames on the left) and as seen
from $1^\circ$ above the orbital plane. Captions to left of frames
indicate the size of grid spacings; green grid lines mark coordinates
which are integers of $r_{\mathrm{g}}$. Effects of gravitational lensing
are clearly seen in frames on the right.}
\label{Fig:Slides}
\end{figure}
%
%
\begin{figure}
\includegraphics[width=\columnwidth]{dolzine-new.pdf}
\caption{The length $2r_{pole}$ of the image of a tidally-evolving
object seen above the orbital plane as a function of time for different
orbital parameters: a) elliptic orbit close to innermost stable circular
orbit (ISCO) with $r_{\rm A} = 6.2122\ r_{\rm g}$, $r_{\rm P}=6.2097\
r_{\rm g}$ and $\zeta=2.44\times 10^{-5}$; b) plunging parabolic orbit
with $E/mc^2 = 1$, $l/mr_{\rm g}c= 3.999998$ and $\zeta=1+10^{-6}$; c)
plunging hyperbolic orbit with $r(V_{\mathrm{max}}) = 3.6\ r_{\rm g}$
and $\zeta=1+10^{-6}$; d) elliptic orbit close to ISCO with $r_{\rm A} =
6.1\ r_{\rm g}$, $r_{\rm P} = 5.95\ r_{\rm g}$ and $\zeta=1-3\times
10^{-6}$; e) elliptic orbit with $r_{\rm A} = 20\ r_{\rm g}$ , $r_{\rm
P} = 4.45\ r_{\rm g}$ and $\zeta=1-3\times 10^{-6}$; f) scattering
parabolic orbit with $E/mc^2 = 1$, $l/mr_{\rm g}c= 4.000002$ and
$\zeta=1-3\times 10^{-6}$; g) plunging hyperbolic orbit with
$E/mc^2=3.0$ and $l/mr_{\rm g}c=6$ and $\zeta=6.35$; h) linearly growing
deformation. In cases b,c,e and f, the simulation was halted before the
object evolved beyond a limit of numerical accuracy.}
\label{Fig:ExponentialGrowth}
\end{figure}
At this point it is important to appreciate the difference between
different tidal evolution problems. In the Keplerian case, described so
well by Hut's mechanism, the tidal dissipation of orbital energy and
exchange of angular momentum between orbits and spins brings orbital
parameters closer and closer to equilibrium at the minimum of the
effective potential. This equilibrium may be reached (depending on
initial conditions) either at a distance where the pair is well
separated so that no partner is within each others Roche radius, or on
an orbit where one partner finds itself beyond the Roche radius of the
mutual potential. In the first case, tidal evolution has stopped since
tides can no more dissipate energy (the only dissipation mechanism left
is by gravitational radiation which is much slower in most cases). In
the other case, the member that finds itself beyond its Roche radius
spills over to the companion by building an accretion disk around it.
Unless there is a strong interaction with other bodies in the vicinity,
the accretion phase develops, according to Hut's mechanism, during a
stage when the members of the binary are (almost) co-rotating.
Therefore, according to the above discussion, the angle $\alpha$ remains
close to zero, keeping equation \ref{virial1} in oscillatory regime.
This ensures that the shape of the mass-losing star remains stable. Mass
flow through the inner Lagrangian point is made possible only by the
(slow) transfer of angular momentum enabled by energy dissipation in the
accretion disk. If, on the other hand, the accreting mass is a black
hole and the inner Lagrangian point is close to the maximum of the black
hole effective potential, then most geodesic orbits from the mass-losing
companion lead directly behind the horizon of the black hole. No
stationary accretion disk can form in this case and the mass-losing
companion is sucked to the black hole in one gulp as a characteristic
exponentially elongating string. In our numerical experiments, we found
that the dividing line between non-relativistic and relativistic
accretion is well represented by a relativistic parameter $\zeta$
%
\beq
\zeta=\frac{E/mc^2-V_{min}}{V_{max}-V_{min}}\ , \label{eq:zeta}
\eq
%
where $E$ is the orbital energy of the asteroid and $V_{max}$ and
$V_{min}$ are the maximum and minimum of the relativistic effective
potential. Relativistic accretion occurs only if $\zeta \sim 1$. Even
orbits winding beyond the Roche radius and close to the black hole, but
with orbital energy near to $V_{min}$, do not accrete relativistically.
The elongation of the object does not develop exponentially but
linearly, as shown by the curve 'a' in \reff{Fig:ExponentialGrowth}. The
meaning of $\zeta$ can be visualized in \reff{Fig:pot}. It is clear from
this figure that objects on bound orbits accrete relativistically only
if their orbital angular momentum decreases below $4 m r_g c$. Thus,
relativistic accretion can occur only from orbits with periastra between
4 and 6 $r_g$. Such unstable relativistic orbits wind about the black
hole few times and unwind from there into the black hole. The winding
periods ($T=\frac{1}{c}\sqrt{4\pi^2 r_p^3/r_g}$) on such orbits are
between 15 and 28 minutes (for a $3.6\times 10^6M_\odot$ GC black hole).
%
%
\begin{figure}
\includegraphics[width=\columnwidth]{potencial-new.pdf}
\caption{The Schwarzschild (black) and Kepler (light gray) effective
potentials for different values of dimensionless angular momentum
($\lambda=\frac{l}{m\ r_g c}$)}
\label{Fig:pot}
\end{figure}
%
%
The main conclusion of this section is the realization that tidal
evolution of binary systems with a black hole has three, and not only
two possible outcomes as in Newtonian theory. Namely, accretion of a
smaller object onto a black hole would not occur by gradual Roche lobe
overflow as in a contact binary, but in a single relativistic gulp, if
the Roche lobe overflow has to occur at the inner Lagrangian point
between 4 and 6$r_g$ from the black hole. The energy released during
such an event is probably comparable or more then the energy released
during the whole disk accretion phase and the light-curve is modulated
with the last period of revolution about the black hole. The new
relativistic outcome occurs, if the binary started as detached and the
Roche lobe overflow occurs suddenly (due to melting) near the periastron
which is below $6r_g$.
%
%
%
\section{The role of magnetic fields}
%
NIR flaring emission from $\sgra$ is believed to be produced by
synchrotron radiation of electrons with Lorentz factors $\gamma_e \sim
10^2 - 10^3$ moving in a magnetic field of $10-100\ \mathrm{G}$
\cite{Markoff:2001, Baganoff:2001, Yuan:2004}. In this section we would
like to show that the development of such strong magnetic fields may be
a natural development during the short event of relativistic accretion.
We would also like to show, in a qualitative manner, that if certain
physical conditions are met, then the global magnetic (and electric)
field may play the decisive role in changing the large kinetic energy of
the exponentially extending object into kinetic energy of electrons and
finally into radiation, released just before the object forever
disappears behind the horizon of the black hole.
There is still no general agreement on the strength of the magnetic
field at the Galactic centre. In \cite{LaRosa:2006} it is pointed out
that the field may be strong (of order $1\ \mathrm{mG}$) and globally
organized (see e.g. \cite{Morris:2006}) or globally weak (of order $10\
\mathrm{\mu G}$) with regions where it is stronger (see e.g.
\cite{LaRosa:2005, Boldyrev:2006}). In any case, the observed strength
is orders of magnitude lower than requested by synchrotron models.
Assuming that the object is permeated with such low magnetic field, our
numerical simulations suggest a simple mechanism to increase it, as well
as to provide electrons with high $\gamma_e$.
We have found that objects moving along $\zeta\sim 1$ orbits end up
being exponentially squeezed and elongated. Since the inward squeezing
velocity is low ($v/c\sim 10^{-7}$), it is possible to use the
magneto-hydrodynamic (MHD) approximation to describe the evolution of
the magnetic field
%
\beq \frac{1}{\sigma \mu_0}\nabla\times(\nabla\times\vec{B}) =
-\parc{\vec{B}}{t} + \nabla\times(\vec{v}\times\vec{B})\ ,
\label{eq:induction}
\eq
%
where $\sigma$ is the electrical conductivity and $\vec v$ is the plasma
velocity. Assuming typical values $\sigma=4.14 \times 10^6\
\Omega^{-1}\mathrm{m}^{-1}$ for molten aluminium and $L=100\
\mathrm{km}$ for the size of the object, the magnetic Reynold's number
$R_m$ becomes very large
%
\begin{displaymath}
R_m = \mu_0 \sigma v L \approx 10^7\ .
\end{displaymath}
%
Therefore we may neglect the diffusion term in the above equation
%
\beq
\parc{\vec{B}}{t} = \nabla\times(\vec{v}\times\vec{B})\ ,
\label{eq:induction_frozen}
\eq
%
which, according to Alfv\' en's frozen-flux theorem, leads to magnetic
flux conservation. Consider an object with initial cross section $S_0$
which is permeated with the initial magnetic field $B_0$. If such an
object is exponentially squeezed, magnetic flux conservation increases
the magnetic field density as $B=B_0 S_0/S$. Since the tidal deformation
tensor is traceless, the average cross section area of a squeezed and
elongated object is inversely proportional to its length (in the GR case
this result is only approximate). Therefore, the magnetic field at a
later times can be estimated as
%
\beq
B(t) \sim B_0 \frac{r_{pole}(t)}{R_0} \ ,
\label{eq:B}
\eq
%
where $R_0$ is the initial radius of the object and $r_{pole}(t)$ the
major axis of the ellipsoid, to which the object has been deformed at
time $t$. In \reff{Fig:ExponentialGrowth}, results of dynamical
simulations show length increase to follow the exponential law up to
five orders of magnitude. The limit comes from numerical limitations of
our simulation only - at this factor of expansion the numerical grid
became too coarse to trust further calculations. In fact, in the
approximation taken, where internal pressure was neglected, (almost) no
limit to the expansion factor is expected. Equation \ref{eq:B} obviously
indicates that neglecting he internal pressure has its limits. When the
magnetic pressure becomes comparable to dynamical pressure involved in
driving the expanding flow, the increase of magnetic pressure must be
stopped, impeding further contraction.
In order to estimate dynamical pressure inside the exponentially
expanding object, we consider a simplified problem with a solution that
is similar to the exponentially expanding asteroid solution from
previous section. Consider a spherical self-gravitating incompressible
fluid that suddenly finds itself in a strong field of a cilindrically
symmetric gravitational quadrupole
($\phi_g=\frac{r_gc^2}{r^3}(x^2+y^2-2z^2)$). An exact solution of this
hydrodynamic problem can be found such that the sphere is continually
deformed into and exponentially extending ellipsoid with axes
$r_{pole}(t)=R_0 e^{t/\tau}$ and $r_{eq}(t)=R_0 e^{-t/2\tau}$, with
$\tau=(2 r_g/r^3)^{-1/2}/c $, and the central pressure decreases (for
$t\ \ni$ $r_{pole}\gg r_{eq}$) as
%
\begin{align}
p_c&=\frac{3}{4}\rho\ c^2 \frac{r_g}{r}\left (\frac{r_{eq}(t)}{r}\right
)^2\nonumber \\
&=\ \frac{3}{4}\frac{\rho r^2_{eq}(t)}{\tau^2}
\label{cntralP}
\end{align}
%
Note that the central pressure is 3/4 of what it would be in hydrostatic
equilibrium. If magnetic pressure pushing outward at equator is added
and balances the central pressure, there is no more pressure gradient
toward equator, so radial compression must stop. The negative magnetic
pressure in the axial direction is also working against expansion,
however the expansion velocity along this axis is much larger than the
compression velocity, so expansion can only be halted if the kinetic
energy of expansion is transformed into some other form, as for example
magnetic energy or kinetic energy of electrons. Equating the central
hydrodynamic pressure and the magnetic pressure
$p_m=\frac{1}{2\mu_0}B^2$, one obtains the equation for the equatorial
radius when compression stops ($R_{eq}$)
%
\begin{equation}
\frac{R_{eq}}{R_0}=\left(\frac{2}{3\mu_0
\rho}\right)^{1/6}\left(\frac{\tau}{ R_0}\right )^{1/3}B_0^{1/3}\ .
\label{pressBal}
\end{equation}
%
Inserting the numbers used in previous arguments ($\tau\sim 11r_g\sim
200 {\rm sec}$, $\rho=5 g\ cm^{-3}$) and measure $R_0$ in units of 10 km
and $B_0$ in units of {\rm mG}, then the above becomes
%
\begin{equation}
\frac{R_{eq}}{R_0}=0.0027\times \left(\frac{B_0[{\rm mG}]}{ R_0[10{\rm
km}]}\right )^{1/3}\ ,
\label{presE}
\end{equation}
%
and the ratio of polar ($R_{pole}$) to initial radius follows as
%
\begin{equation}
\frac{R_{pole}}{R_0}=133000\times \left(\frac{B_0[{\rm mG}]}{ R_0[10{\rm
km}]}\right )^{1/3}\ .
\label{presP}
\end{equation}
%
Equations \ref{presE} and \ref{presP} tell that a 10 km asteroid
permeated with an initial magnetic field of 10 {\rm mG} stops
compressing when its equatorial radius is 27 meters, the magnetic field
has increased to 133{\rm G}, the polar radius extended to $1.33\times
10^6{\rm km}$, and the whole event lasted $\sim 40 min$ - less than 3
turns around the black hole. The final velocity of the pole with respect
to the centre of mass is about $6.6\times 10^3{\rm km/sec}$ and the
kinetic energy of expansion of the order $2\times 10^{-4}m\ c^2$.
These 40 minutes are rather eventful. Very few processes in the universe
release so much energy so fast. The work of gravity on the still
extending object can go on to extract in a few more turns up to $ \sim
0.1 mc^2$ in tidal energy (\cite{Gomboc:2005}).
This mechanism works as long as the magnetic dissipation time-scale
$\tau_{\mathrm{D}}=\sigma\mu_0 r^2_{eq}$ is longer than the dynamical
time-scale. Initially $\tau_{\mathrm{D}}/sim 1.2\times 10^8 {\rm sec}$
and this condition is certainly well satisfied. Finally, when the object
is only 27 meters in diameter, the value of $\tau_{\mathrm{D}}$ drops to
$\sim 15{\rm min}$ if the same molten metal conductivity is assumed. It
is quite clear that this assumption can no longer be respected, since
after the release of so much energy, the object can no longer be
considered a molten metal but has obviously turned into hot plasma. The
conductivity probably increases and keeps magnetic flux constant.
As long as the magnetic field is low, it follows the motion of the
plasma. However, when it reaches high enough values, matter starts to
move according to this magnetic field, thus providing a mechanism for
accelerating electrons to relativistic velocities. The gravitational
force which guides the motion of positive charges is negligible for
electrons. They are pushed out by magnetic pressure until charge
separation induces enough electric field to keep them bound to positive
charges beneath. The electrons that are pushed out spiral along magnetic
field lines and experience less scattering by heavy ions. Eventually,
they lose energy mostly by synchrotron radiation. As the electrons
gradually gain speed also in the direction of the magnetic field lines,
the polarization of the emitted radiation also changes.
Of course, only a complete MHD simulation can give precise results,
showing the geometry of currents responsible for such a magnetic field,
and could predict precisely how much energy could be transformed into
radiation, what would be the spectrum and how the polarization would
change with time. The above arguments may only give us some confidence
that such processes actually can occur at the Galactic centre.
%
%
%
\section{A simple scenario}
%
Let us propose the following scenario to explain flares from the
Galactic centre. Let us consider an object (LMS) on a highly eccentric
orbit with periastron below $\sim 10\ \rg$, that got there by
non-resonant relaxation. The tidal force melts the LMS and eventually
brings it on a $\zeta\sim 1$ critical orbit. Just for the sake of
simplicity, let us assume that the critical orbit is a parabolic one,
with the critical radius of $4\ \rg$. Let us assume moreover that the
energy released during this process comes from the gravitational
potential energy of the object and is thus proportional to its mass.
Since potential energy differences on an orbit at $r=4\ r_{\rm g}$ are
of the order of a few percent of $mc^2$, it follows that objects
producing the flares probably have masses of the order of $10^{20}\
\mathrm{g}$. If the sources of flares were gaseous blobs of such a large
mass, they would find themselves below the Roche radius far away from
the black hole, and would therefore be completely disrupted before
producing any modulation of the light curve. We conclude that the source
of the flares is a small and initially solid object that orbits the
black hole above the effective Roche radius, and is then brought close
to the black hole, where it melts.
As the tidal deformations on this orbit grow exponentially with time,
the same happens with the magnetic field and the magnetic pressure,
which resist the tidal squeezing and stretching -- the magnetic field
helps preventing the object from falling apart. The accelerations of the
electrons are the highest at the endpoints of such a thread, because the
magnetic field is highly inhomogeneous there. Consequently, most of the
synchrotron radiation is emitted from that region.
Since the tidal deformation tensor is traceless and the magnetic
pressure tensor is not, the tidal squeezing of the object is stopped by
magnetic pressure, while the tidal stretching along the orbit continues.
Eventually, due to this imbalance, the object breaks into a large number
of smaller pieces, just as in case of Plateau-Rayleigh instability.
Every new piece emits most of the radiation at its endpoints. These
smaller pieces move on the same orbit, because the magnetic field
prevents them from completely spreading around the black hole.
The light-curve of such an event can be constructed in the following
way: the small, radiation emitting pieces are represented by $N$ small
rigid objects, orbiting on a critical orbit with the critical radius of
$4\ \rg$. The orbit is chosen in such a way, to match the time-scales
with the observations, i.e.\ $50\ \rg/c$. Each of the $N$ objects has
the same light-curve with a different phase shift. The objects which are
farther away from the black hole are less deformed, have lower magnetic
field and thus lower luminosity with respect to the ones closer to the
black hole. Due to exponentially increasing tidal deformation, the
light-curves of all the objects are modified by an exponential function,
to reflect this. Finally, all the modified light-curves are integrated
into one.
The appearance and the luminosity of this event is calculated with
respect to the observers at different inclinations and position angles
with respect to the orbit of the in-falling object using ray tracing
techniques. Our simple scenario has five parameters: the mass of the
black hole $\mbh$, the tidal ``squeezing'' time scale $\tau_{\rm h}$,
the length of the tidal tail $l_{\rm tail}$, the inclination of the
orbit $i$ and the longitude of the line of nodes $\Omega$.
%
%
\begin{figure}
\includegraphics[width=\columnwidth]{FitFlara-new.pdf}
\caption{XMM-Newton/EPIC flare of April 4, 2007, fitted with the above
simple model. The inset shows (roughly) the intensity at a fixed
position of the image as a function of time; the rising beginning
corresponds to the ``squeezing'' time-scale and the duration of the
falling edge corresponds to the length of the tidal tail.}
\label{Fig:flare}
\end{figure}
%
%
An example of such a simple model is a fit to an observed Galactic flare
light-curve (flare of April 4,2007, \cite{2008A&A...488..549P}), shown
in \reff{Fig:flare}. This light-curve is consistent with the following
parameters (assuming a Schwarzschild black hole at GC):
$m_{bh}=2.9\times 10^6M_\odot$, $\tau_{\rm h}=4.8{\rm min}$, $l_{\rm
tail}=6r_g$, inclination near $90^{\circ}$ and $\Omega$ is undetermined,
since there are only two peaks in the light curve. The mass following
from the fit does not quite agree with the measured one. The discrepancy
can easily be explained by introducing a Kerr black hole with angular
momentum parameter 0.4. However we prefer not to dwell on such details
before a more detailed understanding is reached.
%
%
%
\section{Conclusions}
%
Given for granted that our Galactic centre harbours a super-massive
black hole, one of its puzzling characteristics are the flares detected
at various wavelengths. Their main characteristics are: a) short
duration; b) quasi-periodic oscillations with the period of 17-22
minutes; c) total energy release of the order $10^{39.5}\ {\rm erg}$; d)
strong indications of the presence of magnetic fields up to 100 gauss.
The exact cause of such flares is still unclear, and several models have
been suggested to explain their origin. We have proposed and explored
the idea that the flares are produced by the final accretion of dense
objects (LMS) with mass of the order of $10^{20}\ {\rm g}$ onto the
massive black hole in SgrA$^*$. To this aim, we have investigated the
effects of black hole tides on small solid objects in the vicinity of a
massive black hole. Such objects are expected to populate the Galactic
centre as a result of being stripped off their parent stars. In
\cite{Cadez:2008}, we have shown that solid objects, in the mass range
$\sim 10^{19}-10^{21}\ {\rm g}$, melt in the vicinity of the black hole.
Their subsequent orbital evolution naturally leads to capture orbits,
most likely via unstable circular orbits. In an exhaustive numerical
simulation we studied the evolution of shapes and light-curves as the
objects progress toward the horizon of the black hole. We find that
tides generally elongate objects into long thin threads, which extend
exponentially on a time scale $\tau_c\approx 11.3\ r_{\rm g}/c$, if the
orbital criticality parameter $\zeta$ is close to 1. Moreover,if the
object is bigger than $\gtrsim 0.3\ r_{\rm g}$, it is completely
disrupted before producing any significant modulation of the light
curve. Therefore the object should be relatively small to produce the
observed modulation.
We studied numerically the tidal evolution of the shapes and the images
of such objects in the gravitational field of a Schwarzschild black
hole. Because of the complexity of the problem, we did not consider the
more general Kerr space-time and limited the analysis to pressure-less
gas. We find, quite generally, that such objects are eventually
elongated into thin threads, which on $\zeta\sim 1$ orbits extend
exponentially.
We point out that during such exponential elongation, conditions for
magnetic flux conservation are met, so that relatively large magnetic
fields can be generated. The exponential growth of magnetic field
density also allows the betatron mechanism which can generate highly
relativistic electrons. We propose this process as a possible source of
galactic flares. We do not have the exact mechanism to extract radiation
from gravitational potential energy, yet we are able to model light
curves of galactic flares with simple assumptions.
Finally, we would like to point out that our scenario differs from other
models in the following respects: 1) Flares originate from marginally
bound and not from marginally stable orbits. This brings them closer to
the black hole and provides a mechanism for fast extraction of energy.
It also agrees with recent measurements of the size of flaring region in
the Galactic centre, showing that it is not larger than $6\ r_{\rm g}$,
see e.g. \cite{Reid:2008}, and relaxes constraints on the angular
momentum of the black hole. 2) Our model does not require a fairly
rapidly rotating black hole as needed e.g. by \cite{Genzel:2003,
Aschenbach:2004, Trippe:2007}. 3) An accretion disk at the Galactic
centre is not needed. In \cite{Yusef-Zadeh:2008} it is highlighted that
``there is no definitive evidence that \mbox{Sgr~$\mathrm{A}^*$}\ has a
disk''. 4) Elongated structures invoked in other models to fit flare
data, e.g. \cite{Falanga:2007, Zamaninasab:2008, Hamaus:2008}, develop
naturally in the proposed model as a result of the tidal evolution of a
melted body. 5) The evolving elongated structures also provide a natural
mechanism for fast changes in magnetic field density inferred from
observations, increasing first as the body is squeezed by tides and
finally decreasing as the body is crossing the horizon of the black hole.
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% BACKMATTER
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\begin{theacknowledgments}
A.\v C. would like to thank the organizers and in particular to prof.
Ruffini for the invitation to take part in the lively meeting honouring
the late prof. Zeldovich.
\end{theacknowledgments}
%
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