Unless you are among the crew of space-enterprise enthusiasts, that is a peculiar species of kids from the late 1950s to which I myself belong, very few remember how the Salyut7 Soviet low-Earth orbiting space station famously “died” in 1985, when a loss of power from a short-circuit shut down all of its systems, and the spacecraft begun to drift out of its orbit. Later that year, two cosmonauts risked their lives to revive the silent space station. Vladimir Dzhanibekov and Viktor Savinykh were sent to the difficult mission which, according to science historian and writer David Portree, was “one of the most impressive feats in space travel history.” And following their story beats any of your recent, special-effects-laden Hollywood movies about space adventures. (You may also watch a Russian movie about the mission, there exists an online pirate version with French subtitles that I cannot make public here, for obvious Hadopi reasons!)

Docking a crewed spacecraft to the Salyut7 was a supremely dangerous maneuver. A failed docking could cripple the ship, strand the crew in orbit, and likely kill them outright. Soviet spacecraft usually depended on an automated docking system, but that relied on computers aboard both vessels being in constant communication with each other. Fortunately, Dzhanibekov had previously performed a manual docking exercise with the functioning Salyut7. Then, the two cosmonauts in charge of the rescue mission trained extensively on new protocols developed for the planned docking with the lifeless Salyut7, and on June 6, 1985, the pair launched aboard Soyuz T-13. Just by using a handheld optical rangefinder (talk about space heroes!) Dzhanibekov manually nestled his Soyuz onto Salyut7, linking the two crafts at the forward docking port. By methodically moving through a series of hatches, by paying careful attention to equalize the pressures at each step, Dzhanibekov and Savinykh finally reached the work area of Salyut7.

Due to the sub-zero temperature, the two cosmonauts wore wool hats and heavy winter coats, which made them look more like country peasants rather than intrepid space explorers. The station was dark. All of its water supplies had frozen. The instruments and walls were covered with a fine layer of frozen moisture — a picturesque scene that indeed belied the severe risk of an electrical short. The cosmonauts checked the air quality, confirming it was breathable, and opened the shades to allow sunlight to help warm the station. Although Salyut7 had no power left, they found some operable batteries onboard, connected them to the solar panels, and by using the Soyuz 13 thrusters, they moved the entire station to properly align with the Sun. Once the batteries charged up, they began to bring Salyut7 vital systems back online. One at a time, they revived the lights, communications, water storage and delivery apparatus, and so on. Working tirelessly, and under the harshest of harsh conditions, Dzhanibekov and Savinykh astonishingly resuscitated all of Salyut7 with ten days of painstaking work.

After such an impressive feat, they remained onboard respectively for 110 and 168 further days. While spending months yawning in space, Dzhanibekov noticed a peculiar fact, which led to a scientific “discovery” that was later named after him. Many supplies from Earth were secured with wingnuts, and when for a little extra fun he unscrewed the wingnut by giving it a fast spin, thus making the thing to skip out of the screw, it kept spinning about the screw axis forever, because of the zero gravity. However, he also noticed that while fast-spinning, the wingnut also experienced a periodic slow tumble about another axis, perpendicular to the rotation, which made it tumble back and forth at nearly regular intervals. A weird idea also spread around, that the Soviets would have withheld such a “discovery” from public as a topmost military secret, something which frankly speaking is hard to believe. (You can see a zero-gravity experiment with a fast-spinning handle in space, showing the same “Dzhanibekov effect” here https://www.youtube.com/watch?v=1n-HMSCDYtM .)

The Dzhanibekov effect has become known also as the “twisting tennis racket” problem, after the solution to this apparent mystery was published in a 1991 paper bearing the same title [Ashbaugh, Chicone & Cushman, J. Dyn. Diff. Equat. 3, 67 (1991)]. Imagine a racket with a black and a white side, and draw a {xyz} cartesian frame such that, for example, the x is parallel to the handle and the z is perpendicular to the strings plane. You can spin the racket about the x or z axis as many turns as you want, you’ll get no trouble; however, if you flip the racket just once about the other y direction, invariably you will get it back with the white side up if you started from the black one, or viceversa. In other words, while turning about y, the racket also makes a tumble about x, no matter how carefully you try to avoid it. The Dzhanibekov effect is also of great concern to skateboarders, who have trouble with their “impossible monster flip”. (Have a look at this other video https://www.youtube.com/watch?v=vkMmbWYnBHg in which the skater always fails in getting the board to flip about its short length, unless it makes two consecutive flips and the thing falls back on the wheels; you can check this by playing the video in slow-motion at minute 1:40.)

However, this “problem” was not really a problem, since its solution has indeed been known for about 150 years. Louis Poinsot published his Théorie nouvelle de la rotation des corps already in 1834, in which he beautifully showed how to obtain the equation of motion for the angular velocity vector, for a rotating body of arbitrary shape. Even without solving the Euler equations of rigid body dynamics (see below), Poinsot made a smart graphical construction, called the Poinsot ellipsoid, based only on the conservation of kinetic energy and angular momentum as constraints for the angular velocity vector. As it turns out, the angular velocity vector for a body of arbitrary shape cannot be constant in the absence of external forces, but describes a circle in space, and the motion is mirror-symmetric about the plane of the circle. Although Poinsot did not notice it explicitly in his book, it is this symmetry that is enforced when the spinning wingnut periodically tumbles about a perpendicular axis. And, contrary to what stated in the 1991 paper by Ashbaugh et al. [“The twisting phenomenon seems to be new. It is not mentioned in a recent article on the Eulerian wobble (Colley, 1987), in general texts on classical mechanics (Arnol’d, 1978; Goldstein, 1950; Landau and Lifschitz, 1976)”], this is instead reported in any decent book of solid mechanics, for example Landau-Lifschitz chapter 37 page 118 of the very 1976 edition.

This funny Dzhanibekov, or Poinsot, or whatever-you-want-to-call-it effect has to do with the spinning rotation of rigid bodies as ruled by their moments of inertia. In general, according to the three cartesian directions of space we can define three axes about which a body of whatever shape can rotate. For a body of arbitrary shape there are of course an infinite number of moments of inertia (MOI), according to any arbitrarily chosen axis of rotation. However, if you imagine breaking down this generic shape into tiny masses, each located at a point {xi,yi,zi} in space, it can be proved that any distribution of matter sums up to a 3×3 matrix of MOIs, which can be further reduced to diagonal form, thus identifying the three principal MOIs, conventionally labelled I1I2 and Ifor the three cartesian directions. If the three numbers are equal, we have a spherically symmetric object; if two out of three coincide we have the so-called symmetrical top, such as a diabolo; if all the three are different, that is the most generic case famously represented by the tennis racket and the skateboard, we have an asymmetrical top.

A moment of inertia is the product of mass times the square of length, and especially in the case of the asymmetrical top, the masses can be distributed very differently along the three axes. For the largest and the smallest MOI, the mass distribution is respectively the farthest away from, or the closest to, the center of mass. The corresponding rotation velocity achieved for a given kinetic energy (torque) imparted, will be the lowest or the largest, for the two cases. However, things are different for the “intermediate” MOI. In fact, the Euler equations Iiwi‘+(IkIj)wjwk=Ti (where wi stands for the xyz components of angular rotation, and each index i,j,k run cyclically on 1,2,3), which equate the combination of MOI and rotation velocities to the applied torques Ti (that is, ultimately the kinetic energy) can be turned with a little bit of manipulation into harmonic equations for the three rotation accelerations, wi“= ­–ciwi (i.e., angular acceleration equals rotation angle times the negative of a constant). The constant ci is crucially proportional to the difference (Ik–Ij)(Ik–Ii). Now, for Ik equal to either the largest or the smallest MOI, ci is always positive and the harmonic equation is therefore stable; however, for the “intermediate” MOI with a value between the largest and the smallest, ci is negative and the angular acceleration has the same sign as the rotating motion. In this condition, any infinitesimal perturbation gets amplified: the fast-rotating asymmetric body is walking on the edge of a knife, and will tumble about this unstable “median-MOI” axis with regular periodicity.

Now that the mystery is no longer a mystery, the only mystery left to solve is why somebody would think that the Soviet government could have considered such a thing a military top-secret. As the legend goes, one reason would have been the fear that the Earth, a rather asymmetrical top itself, might suddenly flip north over south… What a difference would make keeping such a thing secret, it’s difficult to have a clue about. We know that randomly, with average about every 450,000 years, the Earth’s magnetic poles experience a sudden (on geological scale) inversion; as far as we know, this is due to disruptions of the magnetic flow, possibly associated with perturbations in the inner fluid core. However, as much as we know, the Earth mass axis never flipped over, to make Australians the top-abovers. Some scientists suggested that the Earth’s axis could move in response to changes in the distribution of weight on its surface, such as the shifting continents, and in the 1990s it was found geological evidence that a tilt as large as 50 degrees could have occurred some 800 million years ago. Anyway, we can be sure that the Earth would not care whether the Dzhanibekov effect was kept secret or public, and would flip or not, only according to physics.

Earth doesn’t flip, you wingnut!

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