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During a time when it was fashionable to `explain' the main-
tenance of spiral structure by magnetic fields, Lindblad per-
sisted in the belief that gravitation was the dominant factor,
and now we have come full circle back to this view.
E.M. Burbidge 1971, p.266

2.1 Regenerative spirals by Lynden-Bell

We deduce that our galaxy is likely to have had spiral arms
for most of its lifetime and that as old arms coil up so new
uncoiled arms must start to form from their corpses. The
problem of describing such a mechanisms we call the
regeneration problem.
Lynden-Bell 1960b

In 1960 Lynden-Bell presented at the University of Cambridge his PhD thesis "Stellar and Galactic Dynamics" (Lynden-Bell 1960b) 37 considering some general aspects of stellar-dynamical and ergodic theories. Its separate part "Cosmogonical gas dynamics" was on the spiral problem. It stated, echoing the stress of the day, that "the arms are primarily the seat of gas and dust" (so that the lenticular galaxies, deprived of them, "can no longer give birth to a spiral structure"). It found the cosmogonical approach the most convenient - in case of full denial from Jeans' classic scheme as inoperable in the presence of differential rotation.

"It seems impossible that the protogalactic gas was uniformly rotating when the stars formed. It seems more likely that as the primordial gas broke up into condensations [protogalaxies] each fluid element tended to preserve its angular momentum about the centre of the local condensation. The equilibrium reached is then one in which centrifugal force nearly balances gravity and the pressure is mainly important in preventing the system from becoming very flat."

Lynden-Bell analyzed realistic equilibrium configurations of a frictionless gas system and derived "an energy principle which should provide a powerful means of determining the equilibria on a computer". Any such configuration, when achieved by the system, is exposed to a slow secular evolution that "will not be determined by shrinkage due to the radiation of energy as in Jeans case, but by the transfer of angular momentum due to friction" neglected in the equilibrium derivations. The system "must: i) concentrate its angular momentum into a very small fraction of its total mass, and ii) leave the remainder a more concentrated uniformly rotating or pressure supported body. This is borne out by observation on both the scale of the solar system and that of the galaxy. [...] We should thus expect a uniformly rotating central condensation surrounded by a differentially rotating disc" (Lynden-Bell 1960b).

It is with such an evolved disk of gas that Lynden-Bell linked his spiral considerations. In shearing deformation - a point-blank menace to `any structural irregularity' - he, unlike many workers of the day, saw not an antagonist to the persistence of spiral arms, but a factor of their cyclic regeneration created through gravitational instability of the gaseous subsystem in a combined star-gas galactic disk (the stellar component being liable for gas equilibrium rather than for any collective dynamics). In such a setting, the problem needed a global stability analysis of a system in differential rotation, which technically was not feasible. That is why for want of the better Lynden-Bell employed the methods that had served Fricke (1954) with his Omega = const model; this led to a necessary and sufficient condition of Jeans' stability, Omega2 / pi G rho0 > 2/3 (cf. Sect. 1.3), and instructed the growth rate for unstable stages to be gamma leq 2Omega. An m = 2 mode at k cong 1/3 kpc-1 was found the most important, it fell down towards the disk edge and center, being long-wave and therefore fast-growing. This was in substance Lindblad's bar mode, one specified by a pair of condensations placed oppositely at r cong 9 kpc from the center. Before density had grown by a factor e, rotation turned the system through 180° (at gamma = 2Omega). But as this passed, effects of shear (excluded from the strict stability analysis) just wound the "azimuthally independent structure" round the galaxy, at least once. This meant a grave radial-wavelength reduction, which was expected to be a cause for slowing down the growth rate as effectively as to turn off instability altogether. In this event, the spiral arms would expand back "to form the sheet from which we started", and the whole process might then recur. However, a more careful analysis confirmed the dependence of gamma on k only "for systems very close to stability". This would be "far too sensitive to give the great variety of spirals" and could not apply "for any part of the observed spiral arms". The regeneration theory proposed, Lynden-Bell (1960b) concluded, was "therefore untenable".

But as it turned out later, this pessimism was rather excessive, since it became clear eventually that there was a good deal of wisdom even in such regenerative thoughts. This, however, is not how things developed immediately, because, as we will see in the forthcoming section, the old idea of steady spiral modes was about to gain a new and important burst of enthusiasm.

37 Leon Mestel was his advisor. Back.

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