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An orbit can be understood without first drawing the orbit itself. For a fixed angular momentum, its possible radial motion is encoded in the effective potential: minima support stable circular orbits, maxima support unstable ones, while the energy determines which radial regions can be reached. This turns orbital dynamics...

10,600 次观看 • 2 天前 •via X (Twitter)

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Dundadah Bartholomew 的头像
Dundadah Bartholomew2 天前

If you apply the concept of effective potential to life it turns personal dynamics into an intuitive map of energy, boundaries and habits. Just as a planet navigates a gravitational well, we navigate a psychological & behavioral landscape shaped by our drives and external forces.

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Physics feels stable, predictable, and well-behaved largely because we learned it in 3D. That comfort hides a trap. In 1907, Paul Ehrenfest pointed out something unsettling. If you take the laws we treat as fundamental and transplant them into a different number of spatial dimensions, they often stop working the way we expect. Not just numerically different but qualitatively different. The issue isn’t the force law by itself. It’s geometry. Gauss’s law ties inverse-square forces to the surface area of spheres, and sphere geometry depends on dimension. Change the dimension, and the same-looking force produces a different potential, a different balance of attraction and inertia, and a different fate for motion. You can see this cleanly with a single problem of central force motion. In d spatial dimensions, flux conservation gives F(r) ∝ 1 / rᵈ⁻¹ so the potential scales as V(r) ∝ −1 / rᵈ⁻² (for d ≠ 2) Now add angular momentum. The effective radial potential becomes V_eff(r) = L² / (2 m r²) − C / rᵈ⁻² In 3D, those two terms balance in just the right way to allow stable bound orbits. Small perturbations stay small. Atoms don’t collapse. Planets don’t spiral away. In other dimensions, that balance breaks. In 2D, the force becomes 1/r, the potential becomes logarithmic, and bound motion sits on a knife edge. In 4D and higher, the attractive term becomes too steep. The centrifugal barrier loses the fight. Orbits plunge or escape. Same equations. Same initial conditions. Different dimension. Different physics. This isn’t science fiction. It’s a warning label. So it's clear that a lot of what we call physical intuition is really three-dimensional intuition wearing a lab coat. #Physics #MathematicalPhysics #ClassicalMechanics #DynamicalSystems #Geometry #Ehrenfest

Mathelirium

32,305 次观看 • 8 个月前

Astronomers have discovered an extraordinary star orbiting Sagittarius A*, the supermassive black hole at the centre of the Milky Way, on the most extreme stellar orbit observed there so far. The star, designated S301, was identified using the GRAVITY instrument and its upgraded GRAVITY+ system on ESO’s Very Large Telescope Interferometer in Chile. What makes S301 particularly important is not simply its enormous speed, but how deeply its orbit carries it into the strongly curved spacetime surrounding the black hole. S301 completes one orbit in only about 8.7 years, the shortest known period for a star around Sagittarius A*, and during its closest approach it passes roughly 1.78 billion kilometres from the black hole, only about 12 times the Earth–Sun distance and comparable to the distance between Saturn and the Sun. At that point it reaches around 25,000 km/s, more than 8% of the speed of light, making it the fastest known star in the Milky Way. Sagittarius A* contains approximately 4.3 million times the mass of the Sun, compressed into a region small enough to behave observationally as a black hole. Astronomers have been studying stars around it for decades because their trajectories provide exceptionally clean tests of gravity. The most famous example is S2, whose 16-year orbit has already allowed researchers to detect gravitational redshift and relativistic orbital precession exactly where general relativity predicts them. S301 takes this experiment much further. Its orbit is extremely elongated, with an eccentricity of about 0.98, and at pericentre it approaches Sagittarius A* roughly ten times more closely than S2 in terms of Schwarzschild radii. The resulting relativistic effects should therefore be considerably stronger. The most interesting consequence is that S301 may allow astronomers to directly measure the spin of Sagittarius A*. According to general relativity, a rotating black hole does not simply curve spacetime through its mass; its rotation also drags the surrounding spacetime with it. This phenomenon, known as frame dragging or the Lense–Thirring effect, produces an additional precession in the orbit of an object moving close to the black hole. The effect becomes rapidly weaker with distance, which is why it has been extremely difficult to detect using previously known stars around Sagittarius A*. S301 travels close enough that the change in its orbit caused by the black hole’s rotation may become measurable within roughly the next decade. Importantly, the researchers have not yet measured the spin of Sagittarius A* from S301. Rather, they have discovered a star whose orbit is sensitive enough to that spin that such a direct measurement may now be realistically achievable. The discovery was technically difficult because S301 is extraordinarily faint. In the infrared K band it has a magnitude of about 19.3, and ESO notes that it appears roughly two billion times fainter than Betelgeuse in the sky. GRAVITY achieves the necessary angular resolution by combining the light from four 8.2-metre Unit Telescopes of the VLT through interferometry, effectively producing a virtual telescope with far greater resolving power than any individual telescope. The team first clearly identified S301 in observations from 2023 and subsequently followed it during 2024 and 2025. Once its preliminary orbit was established, astronomers were able to trace it retrospectively in earlier data from 2021 and even find evidence for it in observations obtained in 2017. Altogether, 19 astrometric measurements were used to constrain its orbit. There is still an important limitation: S301 is currently too faint for researchers to obtain a reliable spectrum and radial velocity. Without that information, two possible three-dimensional orientations of its orbit remain compatible with the observations. Future instruments should resolve this problem. In particular, MICADO on ESO’s Extremely Large Telescope should be sensitive enough to obtain spectroscopy of S301 and determine its radial velocity, while continued observations with GRAVITY+ will refine its astrometry. Its next pericentre passage is expected in 2031, and observing at least two complete orbits should give researchers the precision needed to search for the subtle additional precession produced by the spin of Sagittarius A*. S301 may also provide clues about how stars end up so close to a supermassive black hole. Stars are unlikely to form normally at such small distances because the black hole’s tidal forces make the collapse of ordinary star-forming clouds extremely difficult. The researchers instead favour a scenario involving the Hills mechanism. S301 may originally have belonged to a tight binary system that approached Sagittarius A*. The black hole’s tidal gravity could have torn the binary apart, capturing S301 onto its present highly eccentric orbit while ejecting its companion at enormous velocity, potentially fast enough for that star to escape the Milky Way entirely. The observed orbit of S301 is consistent with this interpretation. The importance of the discovery therefore goes beyond setting a speed record. S301 effectively acts as a natural test particle moving through one of the strongest gravitational fields that astronomers can study using individual stars. Tracking its motion could provide the first direct stellar-dynamical measurement of the rotation of Sagittarius A*, improve tests of general relativity in the strong-field regime and, over longer timescales, potentially probe more subtle properties predicted for rotating Kerr black holes. Instead of observing the black hole itself, we can use the trajectory of S301 to map how Sagittarius A* deforms and twists the spacetime around it. 👉

Erika 

305,127 次观看 • 1 个月前