Modeling W44 as a Supernova Remnant in a Density Gradient with a Partially Formed Dense Shell and Thermal Conduction in the Hot Interior. I. The Analytical Model

We show that many observations of W44, a supernova remnant in the Galactic plane at a distance of about 2500 pc, are remarkably consistent with the simplest realistic model. The model remnant is evolving in a smooth ambient medium of fairly high density, about 6 cm^-3 on average, with a substantial density gradient. At the observed time it has an age of about 20,000 yr, consistent with the age of the associated pulsar, and a radius of 11-13 pc. Over most of the outer surface, radiative cooling has become important in the postshock gas; on the denser end there has been sufficient compression of the cooled gas to develop a very thin dense half-shell of about 450 M_☉, supported against further compression by nonthermal pressure. The half-shell has an expansion velocity of about 150 km s^-1 and is bounded on the outer surface by a radiative shock with that speed. The deep interior of the remnant has a substantial and fairly uniform pressure, as expected from even highly idealized adiabatic models; our model, however, is never adiabatic. Thermal conduction, while the remnant is young and hot, reduces the need for expansion cooling and prevents formation of the intensely vacuous cavity characteristic of adiabatic evolution. It radically alters the interior structure from what one might expect from familiarity with the Sedov solution. At the time of observation, the temperature in the center is about 6 × 10⁶ K, the density about 1 cm^-3. The temperature decreases gradually away from the center, while the density rises. Farther out, where cooling is becoming important, the pressure drops precipitously, and the temperature in the denser gas there is quite low. We provide several analytic tools for the assembly of models of this type. We review the early evolution and shell formation analyses and their generalizations to evolution in a density gradient. We also calculate the density and temperature that should be present in the hot interior of a remnant with thermal conduction. We supply the van der Laan mechanism in a particularly useful form for the calculation of radio continuum from radiative remnants. Finally, we estimate the optical emission that should be present from fluorescence of UV light, emitted by the forming shell and the radiative shock and absorbed in the cold shell and the ambient medium, and the associated 63 μm [O I] emission. Both are in agreement with the intensity and spatial structures found in recent observations. Neither requires interaction with a dense molecular cloud for its generation. We calculate the gamma rays that should be emitted by cosmic-ray electrons and ions in the shell, interacting with the cold material, and find each capable of generating about 25% of the flux reported by EGRET for the vicinity.