X-Ray Surface Brightness of the Warm-Hot Phase: Emission Measures, Count Rates and Exposure Budgets

Emission is the most intuitive way to see the warm-hot intergalactic medium and the hardest to achieve, because surface brightness falls with the square of a density that is already tiny. This paper computes the soft X-ray surface brightness of filament gas on a stated emissivity model — bremsstrahlung with a line-enhancement factor of three for metal-enriched gas — over the phase's density and temperature range, through a two-megaparsec path. At electron density 1e-05 per cubic centimetre the brightness is 1.76e-10 erg per second per square centimetre per steradian at 1e6 kelvin and 5.57e-10 at 1e7; at 1e-04 per cubic centimetre it rises a hundredfold, to 1.76e-08 and 5.57e-08 respectively. For the brightest case, a ten-arcminute patch yields 0.294 counts per second on a thousand-square-centimetre instrument and a formal five-sigma exposure of 99.3 seconds against the stated background; for the faintest case the rate falls to 0.0009312 counts per second and the exposure balloons to 1.47 megaseconds. The exposure budget, spanning four orders of magnitude across the plausible parameter space, is the quantitative reason emission studies of the warm-hot phase target its densest structures and stack.