E. Canuto, Former Faculty, Politecnico di Torino
Explorer I spin anomaly
After the Sputnik shock (October 4, 1957), well expressed by a small poem of the Michigan governor, G. M. Williams:
“Oh little Sputnik, flying high
With made-in-Moscow beep,
You tell the world it’s a Commie sky
and Uncle Sam’s asleep.
You say on fairway and on rough
The Kremlin knows it all,
We hope our golfer knows enough
To get us on the ball.”
the US Defense Department approved funding for a satellite project Explorer, in alternative to the Vanguard project, which failed by rocket explosion on December 6, 1957. In less than three months, the Explorer Consortium made by the Jet Propulsion Laboratory (spacecraft), the Army Ballistic Missile Agency (rocket) and the University of Iowa (scientific instruments), succeeded in launching from Cape Canaveral on January 31, 1958, 10:48 PM EST (Eastern Standard Time) the 2 m long, 13.5 kg, pencil-shape Explorer I, into a 0.58 rad inclined orbit with 0.14 eccentricity and 7830 km semi-major axis. Orbit inclination was limited below by the launch site latitude (about 0.50 rad).
The spacecraft mounted four flexible whips operating on 108.00 MHz, two of which in Figure 1 are bending down. Once in orbit, the four whips should stay orthogonal to the spinning spacecraft axis and radially oriented due to the centrifugal force of the 78.5 rad/s spin rotation, thus forming a turnstile antenna. Actually, being flexible, they underwent bending and heating leading to kinetic energy dissipation, increase of the nutation angle between the inertial angular momentum and the current spin axis, and spin rate decay to about 0.95 rad/s. The unexpected phenomenon was discovered on ground  by noticing a slow periodic fluctuation of the received signal intensity in addition to the flutter caused by the 78.5 rad/s spin motion, which however damped out just during the second orbit. Also the satellite structure was not perfectly rigid and thus subjected to deformation, but the main source of energy dissipation became soon clear to be the four flexible whips. They where eliminated on-board of Explorer III (launched on March 26, 1958), which started to tumble only at the end of the third orbit day .
Figure 1. Explorer I spacecraft already mounted on the launch rocket (from Wikipedia).
Let us assume the moment of inertia ratio and the initial rotational kinetic energy . Since the residual energy amounted to about , in practice all the kinetic energy was lost during the first orbitperiod of about 6870 s, at a mean rate of about 26 mW.
Explorer I revealed that a rotation (spin) about the principal axis with the minimum moment of inertia (the minor axis) of a torque-free rigid body cannot be conserved if the kinetic energy is transformed into heat and dissipated. in other terms, under kinetic energy dissipation, the spin motion about the rigid-body minor axis is Lyapunov unstable, and the residual kinetic energy transfers to the major axis (the principal axis with the maximum moment of inertia)  whose spin is Lyapunov stable. It is a common opinion that this theoretical result remained unknown until Explorer I unexpected demonstration.
In summary, only the major axis of a rigid body is gyroscopically Lyapunov stable, i.e. conserves the spin direction, since any real body is not perfectly rigid and thus subjected to elastic deformations converting kinetic energy into heat. The term gyroscope was coined by the French physicist L. Foucault in 1856, borrowing from the ancient Greek gyros (circle) and scopein (to observe). In 1852, after the visible and measurable proof of the Earth’s rotation given by a long swinging pendulum suspended to the Pantheon dome in Paris (1851), L. Foucault devised another conceptually simple but hard-to-observe proof of the Earth’s rotation by reinventing and naming the gyroscope. Until the advent of electric motors in 1860, the gyroscope spinning mass could remain in motion less than 10 minutes, corresponding to less than 45 mrad of the Earth’s rotation, and thus required a microscope for observing (scopein) the circular motion (gyros) of the gimbal around the spinning mass.
Figure 2 – A late copy (1867) of the Foucault gyroscope and microscope conserved in Paris (from Wikipedia)
Rigid body rotation stability under energy conservation
At the Explorer I epoch, the known gyroscopic stability theorem only concerned torque and dissipation-free rigid bodies. It affirmed that only spin motion about the intermediate axis (the principal axis with the intermediate moment of inertia) could not be conserved (unstable), unlike the rotation about minor and major axes that was stable and conserved. A formal and modern proof can be found in , Section 7.4.2. The theorem should have been known at least since the publication in 1851 of the Poinsot’s geometric constructions  .
Figure 3 (from , page 338) shows the sphere of the normalized angular momentum in body coordinates and four energy-constant trajectories of (*) . The principal body axes are denoted by (minor axis), (intermediate axis) and (major axis). Trajectories 1 (blue) and 4 (cyan) about major and minor axes are Lyapunov stable: the angular momentum, though misaligned from a principal axis, moves in the neighborhood of the same axis. Viewed from an inertial frame where is fixed, minor and major body axes slide on a cone around (the precession cone): the variable cone semi-aperture is the nutation angle. The cone shrinks to under pure spin motion. Trajectories 2 and 3 (red) are Lyapunov unstable, i.e. the equilibrium condition (here, the spin motion about a principal axis) cannot be conserved. The trajectories were obtained by slightly perturbing a pure spin motion about . As a result, the angular momentum does not remain in a neighborhood of , but migrates to encircle one of the stable axes (which axis depends on the perturbation). During each half of the encirclement the spin motion of the intermediate axis damps out and then increases again but in the opposite direction, since halfway the rotation axis becomes orthogonal to . Moreover, the rotation speed about this orthogonal axis, which is composition of minor and major axis, reaches its maximum just halfway, which makes the spin direction to switch rapidly.
Figure 3. Angular momentum sphere and energy-constant trajectories about the body principal axes (from ).
The ‘intermediate axis theorem’ is sometimes known as the ‘tennis racket theorem’ , since the principal moments of inertia of tennis racket are each other different. The polar (or twistweight) moment about the axis aligned with the handle is the minimum, the lateral (or spinweight) moment about the axis orthogonal to the racket head is the maximum, and the transversal (or swingweight) moment about the axis lying in the head and orthogonal to the handle is the intermediate, the moment being slightly smaller than the lateral. Thus, a given spin evolves regular about polar and lateral axes, whereas it flips in direction about the transversal axis. That space stations are ideal laboratories for such demonstrations, was discovered the first time in 1985 by the Soviet cosmonaut V.A. Dzhanibekov; hence the alternative name of ‘Dzhanibekov’s effect’.
A simulation of the Explorer I spin anomaly
A lecture with animations of Explorer I spin anomaly can be found in . Here, we provide a simple animation of a spinning spacecraft moving along the assumed orbit of Explorer I and subjected to energy dissipation. The spacecraft shape is similar to Explorer I, but the four antenna wires are kept rigid. Moments of inertia and spin rate have been tuned to the simulation frequency of 10 Hz (), which cannot accommodate the initial Explorer spin frequency of about 12.5 Hz. The inertia ratio has been chosen to be , and the initial spin rate of about 0.3 Hz. Following the literature (see ), energy dissipation has been generated by placing in the spacecraft center of mass a spherical slug of inertia matrix surrounded by a fluid with viscous matrix , where . The initial angular rate vector is misaligned from the body axis of about 7 mrad. In view of the animation, the orbital period was reduced to about and the orbit semi-major axis was enlarged to about 1.33 times the Earth equatorial radius. The magnification scale of the spacecraft dimensions is about .
Figure 4 shows one minute animation of the earlier two and half orbits of Explorer I. The animation was built by the Simulink 3D animation tool of an academic version of Matlab 2019b. The red trace indicates the inclined orbit, the white trace describes an equatorial orbit for better enlightening the red orbit inclination. The spacecraft orbit starts with the spacecraft spinning about the minor axis and ends with the spacecraft spinning about the major axis. The Earth rotation rate is coherent with the simulated orbital period.
Figure 4. Animation of the earlier Explorer I orbits .
Figure 5 shows the time profile of the spacecraft angular rate in body coordinates: the minor axis is , the major axis is and the intermediate axis is . Intermediate and minor axis rates tend to zero together with the angular momentum components in Figure 6. Figure 6 plots the normalized angular momentum body coordinates , and the normalized magnitude . Figure 7 shows the profile of the normalized kinetic energy .
Figure 5 – Angular rate body components.
Figure 6 – Normalized angular momentum.
Figure 7 – Normalized kinetic energy.
Thanks to F. Maddaleno, Politecnico di Torino, for his corrections.
 W.K. Victor, H.L. Richter and J.P. Eyraud, Explorer satellite electronic, IRE Trans. on Military Electronics, April-July 1960, pp. 78-85.
 E. Canuto, C. Novara, L. Massotti, D. Carlucci and C. Perez Montenegro, Spacecraft Dynamics and Control: the Embedded Model Control approach, Butterworth-Heinemann, 2018.
 M. Poinsot, Théorie nouvelle de la rotation des corps (Première et deuxième Parties), Journal de Mathématiques pures et appliquées. Première série, Vol.16, 1851, pp.9-129.
 M.S. Ashbaugh, C.C. Chicone and R.H. Cushman, The twisting tennis racket, Journal of Dynamics and differential equations, Vol. 3, No. 1, 1991, pp. 67-85.
 D. Levinson, Lecture on the Explorer I anomaly, November 2012, Video posted by J. Moore, https://www.youtube.com/watch?v=RdDJtUxLwqQ.
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