Using Temporal Variations of the Nonthermal Redshifted Lyα Emission to Deduce Properties of Proton Beams Injected into a Stellar Atmosphere

Abstract

We present theoretical calculations of temporal variations in the nonthermal redshifted Lyα emission due to time-invariant proton beams injected into a stellar atmosphere during the impulsive phase of a flare. The computations are performed for a power-law spectrum of nonthermal proton energies injected into a model stellar atmosphere consisting of pure hydrogen in local thermodynamic equilibrium; beam-induced variations in temperature and particle densities at all depths and for all times are calculated with the Saha equation. We characterized the injected model proton beams with the total energy flux and the power-law index δ and computed time-dependent nonthermal redshifted Lyα emission profiles for five different values of and three different values of δ. Based upon trends evident in the resulting emission, proton beam properties can be deduced from sufficiently high-quality observations of the nonthermal redshifted Lyα profile. The beam penetration depth initially decreases with time, but in most cases it increases again after reaching some minimum value. This behavior is due to changes in the ionization and temperature of the atmosphere. The Lyα intensity also initially decreases with time, but in most cases it reaches a relative minimum, increases again to a secondary relative maximum, and then slowly but steadily decreases thereafter. Observable properties of this time-dependent emission, such as the ratio of the profile's peak spectral intensity at relative minimum to that at beam onset (Irelmin/I0), the difference between the profile's width at beam onset and that at the secondary relative maximum (FWHM0-FWHMrelmax), and the difference between the profile's centroid wavelength shift at beam onset and that at relative minimum (Δλ0-Δλrelmin) can be used to deduce δ. Once δ is known, can be deduced from other observable properties such as I0 and the times since beam onset at which the Lyα intensity reaches its relative minimum and secondary relative maximum values (trelmin and trelmax).