Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
1. Sirius B has a temperature of 27000K, and a luminosity of only 3% solar. How big is it? From the Stefan Boltzmann law: L 4R 2T 4 1/ 2 L R RSun LSun 0.03 1/ 2 T T Sun 2 4.682 0.0079 The average density is 3M 3M Sun 2.9 109 kg / m3 . 3 3 4R 4 0.0079RSun =16 (for oxygen), the number density is therefore: n mH 2.9 109 kg / m3 1.0 1035 m 3 . 16mH Using 2. Calculate the conditions for degeneracy pressure to dominate: The Fermi energy is 2 2 Z F 3 2me A mH 2/3 The average thermal energy of an electron is 3/2 kT. Thus a gas is degenerate when 3 kT F 2 T 2/3 For Z/A~0.5, T 2/3 2 3 2 3kme mH Z A 2/3 1.3 103 Km2 kg 2 / 3 E.g. in the centre of the Sun, T=1.6x107K and =1.62x105 kg/m3. T 2/3 5384 Whereas in a white dwarf, T=7.6x107K and =3x109 kg/m3 so T 2/3 36.5 3. Evaluate the central pressure of a degenerate star with constant density. dP GM r G 4r 3 4G 2 r dr r2 r 2 3 3 5/3 If the pressure is given by degeneracy pressure: P dP 2 / 3 d so dr dr Setting the two pressure gradients equal to one another gives: d 2/3 2r dr 4 / 3 or d rdr so R 2 1 / 3 M 3 R 1 / 3 M R 3 MV constant 4/3 If you use the relativistic expression, P you get M R2 2 / 3 3 R 2 2 2 / 3 R R M M constant 1 / 3 4. Calculate the cooling time of a white dwarf The thermal energy is primarily in the nuclei, since the electrons are degenerate. 3 U N nuclei kT 2 M AmH 3 kT 2 Equating the change in thermal energy to the luminosity: dU L dt d M 3 M T kT 0.03LSun 7 dt AmH 2 M Sun 2.8 10 dT KT 7 / 2 dt 7/2 LSun 2 AmH k 1 where K 3 0.03 M Sun 2.8 107 7/2 4.0 10 36 A T 2.5 5K t C 2 5 T T0 1 KT02.5t 2 2.5 L t A T0 1 1.2 7 9 L0 12 10 K 10 yr 2 / 5 2.5 t A T T0 1 1.2 70 9 12 10 K 10 yr 2 / 5 7 / 5 5. a. If pulsars were binary stars, what would be the size of the semimajor axis? a3 P M 2 so for a solar-mass system with P=2s, we would need 2s a 1AU 1 year 2/3 0.003RSun . Since the stars would have to be smaller than this, the density would be >5.0x1010 kg/m3, and it would have to be a neutron star. b. How fast can a star rotate before it breaks up? Equate centripetal and gravitational accelerations: 2 max RG M R2 Pmin 2 R3 GM For white dwarfs, Pmin ~ 7s, while for neutron stars, Pmin ~ 0.0005 s. 6. How much energy is output by the Crab pulsar, with dP/dt=4.21x10-13? The kinetic energy of a rotating sphere is K 1 I 2 2 2 I2 2 with I 2 MR 5 2 So the rate of energy loss is: M 1.4M Sun For the Crab pulsar: R 10 4 m P 0.0333s P 4.2110 13 dK 4 2 IP 8 2 MR 2 P dt P3 5P 3 , so dK 5.0 1031W dt P