Here, we select Fig. In contrast to Fig. The first orbit is just very weakly chaotic in comparison with the other two. See Section 3 for more discussion. It is worth to note the difference between the second and the third rows: in Fig. This confirms that orbit E involves more correlation on short time-scales, but on larger scales the correlation washes away; this is consistent with the orbit's time evolution: we checked that in Fig.
Note that in Fig. It is also interesting to subject to such a comparison the quantifiers provided by the recurrence analysis. We can, for instance, directly compare top row of fig. Note that three particular trajectories occurring in these plots are shown in Fig. Abstracting from the somewhat different colour tones chosen, all the four plots apparently carry the same information.
One possible message which stems from this comparison is a further confirmation of efficiency of the simplest recurrence quantifier DIV. Such an adjusting is possible since the MEGNO indicator approaches a distinct universal value namely 2 for regular orbits, whereas for chaotic motion it converges towards larger values typically of the order of hundreds to thousands. This is its main advantage over the FLI, while otherwise both columns are seen to provide the same information.
The colours going from blue to red in the visible-spectrum order correspond to FLI increasing in the range 0— left and to MEGNO increasing in the range 0— after adding to every value above 4, which makes the blue of regular islands more contrasting. Zoom of the most complex regions is added in the second row. The current models of galactic nuclei suggest that there typically occurs an inner hot accretion disc, having radius of light days at most, not very heavy but perhaps reaching down to the vicinity of the black hole horizon, and a much more massive molecular torus at much larger radius 50 light years, say.
If the environment is sufficiently rarefied so that the physical interaction with gas and radiation is negligible, the motion of an individual star can be approximated by a geodesic in the gravitational field dominated by the black hole plus possibly the outer torus and perturbed by the accretion disc. Yet the nucleus of our galaxy is a complex region with a number of components.
Section (B): Normal rings—The Stacks project
In comparison with the cold tori, the hot inner accretion structure around the black hole is very small it reaches up to few hundreds or thousands Schwarzschild radii of the central hole and bears just small fraction of the black hole mass. We will examine which of the above structures could have some effect on particle dynamics.
Approximating the centre by a static originally Schwarzschild black hole, the tori by the Bach—Weyl ring solution and the inner accretion disc by the inverted first Morgan—Morgan disc solution, we will ask whether some of these sources surrounding the black hole can perturb the motion of an orbiting star treated as a test particle characterized just by mass so as to become chaotic. The star is supposed to orbit above the inner accretion zone but below the CNR. Note that we do not take the other stars into account, describing the test-star orbit as a time-like geodesic in the space—time generated by the black hole plus the surrounding ring s or disc alone.
This is one of our future plans. This is also our aim here, but rather than perturbations coming from fine details of the very centre black hole plus the inner cluster , we are interested in the effect of larger circumnuclear structures cf.
In other words, the above parameters do not correspond to actual relations in the Galactic nucleus, but they are also not orders of magnitude different. Colours going from blue to red across the visible spectrum correspond to average MEGNO ranging from 0 to , but the values above 4 are increased by in order to better distinguish between regular and chaotic regions thus the range is expanded to The figures show that the effect of the ring on the motion of particles is quite pronounced and strongly depending on the ring location with respect to the accessible region.
Generically, the ring affects mainly those orbits which pass in its vicinity.
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For given fixed constants of geodesic motion energy and angular momentum at infinity , the system thus appears the most chaotic when the ring passes just through the centre of the accessible region. Then the primary island with period 1 shrinks and then disappears completely. When the ring position approaches the outer boundary of the lobe, a regular region inside grows in the form of period-two islands. Finally, a secondary lobe enclosing just the ring detaches, leaving the primary lobe almost regular except for a small territory near the separatrix between the two regions ; the secondary lobe is heavily chaotic.
Shifting the ring still farther, the primary lobe stabilizes further, while the chaotic lobe tied to the ring gradually shrinks and becomes more regular as well.
Namely, while varying the mass by a factor of 10, one proceeds from a completely regular to a rather chaotic system. The phase-space structures evolve in a similar way as in the preceding case. The phase spaces of the 0. Actually, the chaotic region around the origin is connected, by these orbits and via the narrow peripheral layer, with the chaotic vicinity of the ring.
This is why we also added a plot the last one where the motion is confined to a smaller region not reaching to the very ring. For such a motion everything is regular. More precisely, there do exist a chaotic region tied to the ring, but this is not connected with the main accessible region and hence the latter remains regular. The hot accretion disc in the black hole vicinity does not appear to have a significant gravitational effect on the dynamics of stars orbiting the centre in the distance range of the well-known S-stars or somewhat farther. As the mass 0. We again do not attach the resulting figures and indicator values.
We have subjected to such a study the time-like geodesic dynamics in the static and axisymmetric field of a Schwarzschild black hole perturbed by the presence of a concentric thin disc or ring. Of these, only the smaller CNR have been found to be able to partially destabilize the motion of stars treated as test particles , but only if these can approach it closely. Since the parameters corresponding to our Galactic nucleus are not that far from the range where chaos begins to occur, at least when speaking about the smaller CNR, it is likely that in some galaxies a similar analysis of orbital dynamics would yield interesting results.
The last remark concerns the validity of exact models exact solutions of Einstein's equations we are using to describe the gravitational field. This would be rather easy if the cluster were only approximated by a kind of spheroidal potential. Secondly, it would certainly be more realistic to consider thick toroids instead of infinitely thin rings to model the circumnuclear galactic structures, at least if stars may get to their close vicinity.
This is also a feasible task, though the metric is then much more demanding for practical computations see e. Thirdly, a physical interaction of stars with circumnuclear environment should also be accounted for somehow, but it is clearly beyond our present scope to discuss it here. Let us also mention several related results which have appeared in the literature recently.
They found, among others, that the black hole rotation rather attenuates the instability, consistently with previous experience acquired in this respect. We used the code for geodesic motion in Weyl space—times originally written by M. The plots were produced with the help of the gnuplot utility and D. Krause's bmeps program. We thank D. Movie 2 Sukova-semerak-video2. Please note: Oxford University Press are not responsible for the content or functionality of any supporting materials supplied by the authors.
Any queries other than missing material should be directed to the corresponding author for the paper. Oxford University Press is a department of the University of Oxford. It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide.
New Ring of Dust Discovered in the Inner Solar System
Sign In or Create an Account. Sign In. Advanced Search. Article Navigation. Close mobile search navigation Article Navigation. Volume Article Contents. Free motion around black holes with discs or rings: between integrability and chaos — III P. Oxford Academic. Google Scholar. Cite Citation.
Permissions Icon Permissions. Abstract We continue the study of time-like geodesic dynamics in exact static, axially and reflection symmetric space—times describing the fields of a Schwarzschild black hole surrounded by thin discs or rings. The formulas describing the exact superposition of a vacuum static and axially symmetric originally Schwarzschild black hole with a concentric thin disc or ring were given in previous papers of this series together with the original literature , so we will only list them very shortly for reference.
We consider superpositions with the inverted first member of the Morgan—Morgan counter-rotating thin-disc family and with the Bach—Weyl thin ring. The LCEs describe the rate of orbital divergence in the neighbourhood of a given trajectory. In relativity, the main issue is the choice of time t. In stationary space—times, there exists a privileged, Killing time which offers a natural option for a study of test-particle dynamics. They also claim i that it is sufficient to follow this evolution in the configuration space i. The main disadvantage of the original LCEs is their very slow convergence to the final value, which means that for weakly chaotic systems a very long integration time is often necessary to prove the nature of their orbits.
Discovering new things in space is a regular occurrence. Astronomers keep finding more distant objects in the outer reaches of the Solar System. Two recent scientific papers are filling in some gaps in our understanding of the inner Solar System. Both papers are centered around the dust that populates the Solar System.
And that dust is ancient. The Solar System used to be a much more chaotic place that it is now. In fact, after the Sun flared to life, but before any of the planets formed, there were vast amounts of dust, and of course gas, swirling around in a huge disk.
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Much of that dust coalesced into the rocky planets. Some of it comes from asteroid collisions, some from comets, and some from other events like meteor impacts. And most of it is ancient. On its long journey toward the Sun, the dust gets trapped by the gravity of the planets. Astronomers have known this for a while. Twenty-five years ago, scientists discovered that Earth orbits the Sun along with a giant ring of dust. More recently, one was discovered near Venus.
The Venus ring was only confirmed in Mercury is so close to the Sun that astronomers thought no dust ring could exist there.
But a new paper presents evidence that the closest planet to the Sun does indeed have a dust ring. In it they present their evidence for a dust ring new Mercury. Ironically, they were looking for a dust-free area. Finding the dust-free region would back up the popular theory that the force of the Sun should create a dust-free region close to the Sun itself.
The idea was that the size of the area, and how far it was from the Sun, would tell us something about the nature of the dust itself, and how it was shaped by the force of the Sun. And since the dust is ancient, it would tell us something about the evolution of our Solar System. Stenberg and Howard wanted to develop a way to remove unwanted dust from images. The observatories follow highly elliptical geocentric orbits. Over time, one of them pulls farther ahead of Earth while the other falls further behind.
This means that together they provide stereo images of the Sun. The problem is, the light from the dust is about times brighter than the coronal light.