EDP Sciences
Free Access
Volume 406, Number 2, August I 2003
Page(s) 735 - 740
Section Diffuse matter in space
DOI https://doi.org/10.1051/0004-6361:20030822

A&A 406, 735-740 (2003)
DOI: 10.1051/0004-6361:20030822

On the generation of Alfvén waves by solar energetic particles

R. Vainio

Department of Physical Sciences, PO Box 64, 00014 University of Helsinki, Finland
(Received 18 March 2003 / Accepted 16 May 2003 )

A simple analytical theory of Alfvén waves amplified by streaming solar energetic particles (SEPs) is studied. It is pointed out that a finite time-integrated net flux of energetic protons has to pass each point in space before we can expect Alfvén waves to be significantly modified by the streaming instability. The time-integrated net proton flux needed for the time-integrated wave growth rate (or wave growth, for short) to exceed unity is evaluated. Assuming that protons stream much faster than the waves, we evaluate the wave growth as a function of position and wavenumber for a specified proton injection energy spectrum, dN/dE. The wave growth is found to be proportional to $vp\, \mathrm{d}N/\mathrm{d}E$, where v and p are the particle speed and momentum, and to the local Alfvén speed VA. Thus, maximum wave growth is achieved at the location of maximum VA (at a few solar radii), and the minimum value of dN/dE required for the wave growth to exceed unity there is a few times 10 32/vp protons per unit solid angle (in coordinate space) at the solar surface. If dN/dE is below this value, test-particle theory is a valid description of particle transport and acceleration. The value is not exceeded (above 1 MeV energies) in small gradual SEP events having peak 1-MeV proton intensities below $\sim $10 protons (cm $^{2}\, $sr s MeV) -1 at 1 AU. The spatial and momentum dependence of the wave growth can also be used to estimate the maximum emission strength of a moving proton source in the interplanetary medium. For a strong source moving through the solar wind at constant super-Alfvénic speed, the number of escaping particles per unit time and flux-tube cross section is approximately constant in time, predicting a plateau-type time-intensity profile observed ahead of the source. The model reproduces observations of streaming-limited intensities at energies around 10 MeV and explains the double peaked injection profiles observed in large SEP events.

Key words: instabilities -- Sun: particle emission -- turbulence

© ESO 2003