Question 4 [12 marks]: Consider a sphere of ionized hydrogen plasma that is undergoing spherical gravitational collapse. The sphere is held at constant isothermal temperature 7o, uniform density and constant mass Mo, during the collapse, and has decreasing radius R(t). The sphere cools by emission of bremsstrahlung in its interior. At to the sphere is optically thin. (a) [4 marks] What is the total luminosity of the sphere as a function of Mo, R(t), and 7o while the sphere is optically thin? Assume that the Gaunt factor =1.2. (b) [3 marks] What is the luminosity of the sphere as a function of time after it becomes optically thick? (c) [2 marks] Give an implicit relation, in terms of R(1),for the time, when the sphere becomes optically thick. (d) [3 marks] Draw a qualitative curve of the luminosity as a function of time. Make sure to label the graph. Stefan-Boltzmann law: j=σT Magnitude scale: mx=-2.5 log10 Sx Sx.0 Measurable properties of Thermal Bremsstrahlung: • Luminosity of an isothermal hydrogen plasma cloud L(T) =C29 (T) T¹² dV where C2=1.44×10 40W m³ K-1/2 1 32 2л3 (4лE₁)³ 3 3 km³) • Volume emissivity j,(v) dv=g(v,T,Z). 3 1/2 z²e 2 6 -hv/kT e 3 nen T1/2 -dvi -hv/kT T-1/2 e =C₁g(v,T,Z)Zn¸n, -dv where C₁ = 6.8×1051J m³ K1/2 Constants G=6.674 × 10" N m² kg²² = Lsun Lo 3.828 × 1026 W (or J/s) Msun Mo 1.989 × 1030 kg = c 2.998 x 108 m/s h=6.626 × 1034 m² kg s²¹ k = 1.381 x 1023 JK-1 σ 5.670x108 W m²K-4 = me 9.109 x 1031 kg = 511 keV/c² = mp 1.673 x 1027 kg -3 E = 8.854 × 10-12 A² s4 kg"¹ m²³ Ho 1.257 x 106 TmA¹ Units 1 eV 1.602 x 10-19 J 1 light-year 9.461 × 1015 m 1 Jy 1.0 x 1026 W m² Hz