Astroparticle Physics

212 9 The Early Universeϱ = π 230 g ∗T 4 . (9.55)Work out the expansion rate H as a function of the temperatureand Planck mass using the Friedmann equation!4. In Sect. 9.1 the Planck length has been derived using theargument that quantum and gravitational effects becomeequally important. One can also try to combine the relevantconstants of nature (G, ¯h, c) insuchawaythatacharacteristic length results. What would be the answer?5. Estimate a value for the Schwarzschild radius assumingweak gravitational fields (which do not really apply tothe problem). What is the Schwarzschild radius of theSun and the Earth?Hint: Consider the non-relativistic expression for the escapevelocity and replace formally v by c.6. A gas cloud becomes unstable and collapses under itsown gravity if the gravitational energy exceeds the thermalenergy of its molecules. Estimate the critical densityof a hydrogen cloud at T = 1000 K which would havecollapsed into our Sun! Refer to Problem 8.2 for the stabilitycondition (Jeans criterion).7. Occasionally it has appeared that stars move at superluminalvelocities. In most cases this is due to a motion ofthe star into the direction of Earth during the observationtime. Figure out an example which would give rise to a‘superluminal’ speed!