Some Insights into Stellar Structure from Nonlinear Pulsations

Efficient tools of investigation of stellar pulsation are the integral relations which link oscillation frequencies to the static structure of stellar models, as provided by the linear theory of pulsation (for a review, see Saio, this conference).Similarly, oscillation amplitudes and phases, which arise from nonlinear processes, can be related to the stellar structure by means of amplitude equation formalisms (for a review, see Buchler, this conference).For the simple case of a monoperiodic oscillation, involving only one unstable marginal mode, such a formalism shows that the (limit cycle) radius variations, at time t and mass level m, can be approximated, up to second order of approximation, (Buchler and Goupil, 1984; Buchler and Kovàca, 1986) by:where A, R, Ω, ĸ, £r(m) respectively are the amplitude, stellar radius, linear nonadiabatic frequency, growth rate, radius eigenfunction. Second order nonlinearities generated first harmonic oscillations and change in equilibrium radius about which the star oscillates, as represented by the last two terms in (la) respectively. Analogous expressions are obtained for velocity and light variations, that can be compared with observations.

Petitions to the Legislative Assembly of Ontario from Local Governments, 1867–1877: A Case Study in Legislative Participation

In late twentieth-century English-speaking western democracies, the petition is almost exclusively a sporadic, exceptional, and marginal mode of political expression, its legitimacy as an instrument and indicator of public opinion superceded by elected professionals and ubiquitous polls; a tenuous survival from its origin as the universal form of civic supplication. Part and parcel of the democratic revolution that reached its apogee in the nineteenth century, this transition may not have been neatly contemporaneous with the constitutional changes to which it seems collateral. In a recent article in this review, David C. Frederick posited that petitioning effectively disappeared in the United States after the imposition of a “gag-rule” by Congress, imposed in the 1830s as a response to anti-slavery agitation by petition.

Nonlinear Evolution of Internal Ideal MHD Modes Near the Boundary of Marginal Stability

A family of ideal MHD equilibria is considered introducing the concept of a driving parameter λ the increase of which beyond a certain threshold λ0 drives the plasma from a linearly stable to an unstable state. Using reductive perturbation theory, the nonlinear ideal MHD equations of motion are expanded in the neighbourhood of λ0 with respect to a small parameter ε. An appropriate scaling for the expansions is derived from the linear eigenmode problem. Integrability conditions for the reduced nonlinear equations yield nonlinear amplitude equations for the marginal mode. Nonlinearly, the instabilities are either oscillations about bifurcating equilibria, or they are explosive. In the latter case, the stability limit depends on the amplitude of the perturbation and is shifted into the linearly stable regime. Generally bifurcation of dynamically connected equilibria is observed at λ0