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Tidal Interaction

Everyone is familiar with the effects of weak tidal forces: the high tides and low tides of the oceans and seas are a combined effect of Moon's and Sun's gravitational field. The effect is so weak that tidal forces globally raise a noticeable bulge only on Earth's oceans. An example of stronger tidal effects can be seen in our Solar system: the moon Io orbits Jupiter so close, that planet's tidal forces are strong enough to heat the moon and melt its interior. Consequently, Io is the only moon with active volcanoes. An extreme example of strong tides would be a black hole - normal star binary. Tidal effects on a star encountering a supermassive black hole have been investigated by various authors using different methods with similar results - if the star and the black hole are of the same size, the star is completely tidally disrupted during a close encounter.

Tidal forces not only affect the shape and structure but are also responsible for evolution of orbits and rotation periods of objects involved. It is now known that in the past the Moon used to orbit the Earth much closer than today and that Earth's rotation was much faster. The torques produced by tidal bulges on Earth caused the Moon to slowly drift away from the Earth and slow down Earth's rotation to today's 24 hours. Such torques exist only as long as tidal bulges and the moon are misaligned. The origin of such misalignment is tidal friction.

Evolution of orbits

The first stages of tidal evolution of the orbit can be investigated using Hut's formalism, which is based on the Darwin's weak friction model. In this model the surface of tidally deformed object always takes the equipotential surface it would have had a constant time τ ago, as shown in Figure 1. This model 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 exchanging angular momentum and energy can accurately be calculated by approximating the deformed object by adding two additional bulge masses μ on the surface of the star, as shown in Figure 2.

Tidal bulge
Figure 1: Tidal bulge (blue) on the secondary (m) raised
by the primary (M). Due to viscous friction, the tidal bulge
does not assume the instantaneous equipotential shape.
Instead, at time t it takes the equilibrium shape it would
have had at time t-τ.

Approximation of deformation
Figure 2: The tidal bulge is approximated with two
smaller masses μ. F0, F1 and F2 are the forces
between both bodies. F1 and F2 produce the torque
which transfers angular momentum and energy
between the orbit and the secondary.


It turns out that tidal evolution depends only on one parameter (α), which is the ratio of the orbital angular momentum of the system and spin of both bodies at equilibrium. For a binary system of a black hole and a low-mass satellite the value of α is very small.

Tidal evolution
Figure 3: Tidal evolution
of eccentricity and
periastron.

Tidal evolution
Figure 4: Tidal evolution
for low angular momentum
orbits.

Figure 3 shows tidal evolution toward stable corotating circular orbits at r0 = 450rg. These orbits are initially quite eccentric and have low spin (the upper five curves). The remaining orbits start as circular but with large spin. If the initial spin energy is high enough, it can be transferred to orbit, first elongating it and then circularizing again after joining the circularization track (two upper curves starting to the left). If the initial spin energy is not high enough, the circularization track cannot be reached and the orbit keeps elongating at a slower and slower pace (lower two curves).

Figure 4 shows the evolution of very low orbital angular momentum orbits. Very high initial spin angular momentum and energy can be transferred to orbital momentum and energy, leading to higher periastron orbits (the upper three curves). In the case of small initial spin, the orbital evolution starts with orbital energy dissipation by tidal interaction and little angular mometum transfer, leading to less eccentric orbits with shorter and shorter orbital periods. As a result, the orbital angular momentum eventually transfers to spin as the object is forced into faster and faster corotation.

Tidal evolution
Figure 5: The evolution of radial
turning points in the Kepler (red)
and Schwarzschild effective
potentials (black).

With the solution for rp and e it is possible calculate the evolution of orbital radial turning points. Figure 5 shows this for both the Keplerian and the Schwarzschild effective potentials. The important qualitative difference between the two cases occurs for low values of orbital energy and angular momentum. In the Keplerian case, the effective potential guarantees two turning points for all bound orbits, since it has one minimum and no maxima, while in the Schwarzschild case the effective potential has a minimum and also a maximum. Therefore, the inner turning point disappears in a parametric family of orbits, when the effective potential maximum becomes less than the orbital energy. From the point of view of turning points, Schwarzschild orbits do not necessarily circularize, but the inner turning point disappears when the outer one may still be at approximately 20 rg.

Tidal disruption of stars

A stellar encounter with a massive black hole can be a very energetic event, with energy released and luminosity variations depending primarily on the relative size of the star compared to the black hole. The tidal interaction energy may rise to as high as 0.1 of the total mass-energy of the captured star, which is available when the star is comparable in size to the size of the black hole. This size ratio is also critical to the nature of the disruption.

By simulating the relativistically moving star and its emitted light and taking into account general relativistic effects on particle and light trajectories, our results show that the black hole’s gravity alone induces apparent stellar luminosity variations on typical timescales of a few rg/c to a few 100 rg/c.

Tidal disruption of a star
Figure 6: Tidal disruption of a star - side view.
Tidal disruption of a star
Figure 7: Tidal disruption of a star
-top view.

Figures 6 and 7 show an example of a tidally disrupted star, for two different orientations of the far observer. More results can be found here.

Tidal effects on low-mass satellites

The strength of the tidal interaction depends on the frequencies of quadrupole modes, which are different for gravity and solid state dominated satellites. To get a somewhat qualitative idea of this difference, the effective Roche radius rReff is introduced as the radius at which ωqtf = 1 (tf is the characteristic periastron crossing time, and ωq is the fundamental quadrupole resonant frequency). Both rR and rReff are shown in Figure 8.. This figure shows that gravity dominated satellites, orbiting a supermassive black hole, start rapid tidal evolution when their periastron reaches rp ≈ 10 rg, while solid state dominated bodies may start significant tidal evolution even closer to the black hole. In this case, the effective Roche radius for solid state dominated asteroids is below 2 rg, while the Roche radius for these objects is about 10 rg.

Roche radii
Figure 8: Roche radius and effective Roche radius as a function of mass and radius of the object
for a supermassive black hole with Mbh = 4 × 106 M (left scale) and for a solar-mass black hole with
Mbh = 106 M (right scale). Blue lines show Roche radius for gravity dominated objects and red lines
the effective Roche radius for solid-state dominated objects (MCe is the mass of the asteroid Ceres).
Lines of constant sound velocity cs are plotted in light grey. Some celestial objects are marked by
blue dots: Earth, Jupiter, Sun, Sirius B, Aldebaran, Betelgeuse and Rigel. The green line epresents
zero age main sequence stars and the purple line represents white dwarfs.


As the tidal work is done on the satellite, it eventually melts. Consequently, the objects that were orbiting close to the black hole and well above their solid-state effective Roche radius (e.g. rp ∼ 10 rg for SMBH case), may find themselves below the Roche radius rR, if they are massive enough. The main difference between the SMBH case and a solar-mass black hole case is the size of the objects that are orbiting very closely to the black hole. In case of solar-mass black holes the small meter-sized rocks are strongly affected by tides – note the huge difference between rR and rReff – and may either evaporate or break, while in case of SMBH such objects are completely irrelevant.

The last stages of tidal capture

Consider a solid spherical object moving around a black hole. When tidal forces do enough work, it momentarily melts. After this moment pressure forces become negligible with respect to inertial forces and the particles that constitute the object start to fall freely in the gravitational field of the black hole. A ray-tracing procedure from points on the object’s surface to the distant observer is used to determine its appearance and light curve.

The extent of tidal deformation is estimated from the length and the area of the object’s image as seen by a distant observer above the orbital plane. The results show, that relativistic tidal effects depend predominantly on the ratio ζ = (E/mc2 − Vmin)/(Vmax − Vmin), where Vmax and Vmin are the maximum and the minimum of the relativistic effective potential V. For circular orbits the value of parameter ζ is 0, while ζ ∼ 1 corresponds to semi-stable orbits that extend to Vmax. These are the final orbits resulting from long tidal evolution of bound orbits. Such orbits do not exist in Keplerian case: objects on such orbits partly circle the black hole at radii in the interval 3 rg < r < 6 rg, make short excursions away from the black hole and repeat the circling cycle before either falling inside or flying away.

The ζ ∼ 1 case is represented by a parabolic orbit with E/mc2 = 1 and l/mrgc = 3.999998. This is a ζ = 1.0000066 orbit. The object makes about three to four turns around the black hole at r = 4 rg before falling into it. Here are the results for two different orientations of the observer. The orbits are shown on Figures 9 and 12, the lightcurves on Figures 10 and 13, and the appearance on Figures 11 and 14.

Orbit
Figure 9: The orbit of the object with
E/mc2 = 1 and l/mrgc = 3.999998.

Lightcurve
Figure 10: The light curve of the object of size R = 10−5 rg.
The observer is 5 above the orbital plane. The lightcurve
of melted object is in black, and of rigid sphere in red.


Appearance
Figure 11: The image of a melted object that was initially spherical. The colors correspond to the redshift. The green circle
represents the Schwarzshild radius.


Orbit
Figure 12: The orbit of the object with
E/mc2 = 1 and l/mrgc = 3.999998.

Lightcurve
Figure 13: The light curve of the object of size R = 10−5 rg.
The observer is directly above the orbital plane. The lightcurve
of melted object is in black, and of rigid sphere in red.


Appearance
Figure 14: The image of a melted object that was initially spherical. The colors correspond to the redshift. The green circle
represents the Schwarzshild radius. The second image contains a larger view of the object, to show its deformation and
orientation.


The sharp and narrow peaks in the light curves are due to gravitational lensing and always appear in pairs: the lower one is the contribution of the secondary image and precedes the one from the primary image. The peaks are higher if the observer is closer to the orbital plane. The wider bumps following sharp peaks are the result of Doppler boosting and the aberration beaming of light. The overall increase in luminosity is a consequence of the exponentially increasing surface of the object. To compare results with orbiting blobs of fixed size, the light curves of a freely falling rigid object are also shown.

The corresponding movies, showing the infall of a small object, are here:

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Page last modified on April 16, 2009, at 09:04 PM