Near-resonant excitation and propagation of eccentric density waves by external forcing

Eve Charis Ostriker, Frank H. Shu, Fred C. Adams

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33 Scopus citations


We review the astronomical evidence that relatively massive, distended, gaseous disks form as a natural byproduct of the process of star formation. We also recall the numerical evidence that SLING-amplified eccentric modes in the outer parts of such disks can drive one-armed spiral density waves in the inner parts by near-resonant excitation and propagation. New numerical calculations demonstrate that SLING-amplification becomes less effective if nebular disks have tapered rather than abrupt edges. We point out that even without modal instability, m = 1 spiral waves can be excited in the inner portions of a nearly Keplerian disk by an external or embedded stellar companion. We then present an ordinary differential equation (ODE) of second order that approximately governs the nonlocalized forcing of waves in a disk satisfying almost everywhere near Lindblad resonance. When transformed and appended with an extra model term, this ODE implies, for free waves, the usual asymptotic results of the WKBJ dispersion relationship and the propagation of wave energy and angular momentum at the group velocity. For forced waves, it yields as well the standard Goldreich-Tremaine formula for the resonant torque exerted at an isolated (localized) Lindblad resonance. Our derivation includes the modification of the effective adiabatic index that results from vertical compression and expansion when one reduces by integration the full three-dimensional problem to two (horizontal) dimensions. We give an analytical solution for the rate of energy and angular momentum transfer by nonlocalized near-resonant forcing in the case when the disk has power-law dependences on the radius of the surface density and temperature (assumed vertically isothermal). Comparison of this solution with the results of the numerical integrations of both the model ODE and the exact set of (linearized) equations shows agreement on the level of about a factor of 2, with the discrepancies dependent on exactly how one treats tunneling through the Q-barrier near the corotation circle and the effect of the corotation resonance itself (neither of which our analytical method of solution handles well). Nevertheless, the simple analytical treatment does provide useful estimates for the overall process for a wide range of physical conditions, as well as insight into the dynamics of the basic mechanism. In particular, we find that the presence of a stellar companion orbiting at a distance of 100 AU or closer can drive waves of sufficient amplitude to cause accretion onto the central star and thereby reduce the mass of an initially heavy disk, over the course of 107 yr or less, to the relatively low fractional levels (a few tenths) that are inferred for the so-called "flat-spectrum" T Tauri stars by submillimeter- and millimeter-wave observations of the thermal radiation from warm nebular dust. The forcing becomes inefficient if the disk mass drops below ̃0.01 M or if the satellite has a mass characteristic of a planet rather than a star. Finally, we comment on the implications that the back reaction of the near-resonant forcing has on the evolution of the orbit characteristics of the satellite body, and we speculate on nonstandard mechanisms for producing forced waves (e.g., irregular infall during the process of single-star formation).

Original languageEnglish (US)
Pages (from-to)192-212
Number of pages21
JournalAstrophysical Journal
Issue number1
StatePublished - 1992

All Science Journal Classification (ASJC) codes

  • Astronomy and Astrophysics
  • Space and Planetary Science


  • Accretion, accretion disks
  • Binaries: general
  • Hydrodynamics
  • Stars: formation


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