Prob. Set 1: Assigned Monday. Sept. 17, Due Wed. Sept. 26:
Do problems: 1.3, 1.5, 2.4b, 2.7, 3.1, and 3.2. For the last question, you may find it useful to read Sect. 3.9 in LeBlanc
Do problems: 1.3, 1.5, 2.4b, 2.7, 3.1, and 3.2. For the last question, you may find it useful to read Sect. 3.9 in LeBlanc
For problem 3.2, it should be to plot log(kappa_R) against log(T) -- not log(R)
Prob. Set 2: Assigned Wed. Sept. 26, Due Thurs. Oct. 4
Do problems: 3.5 (for part c, use a pressure scale height of 0.063 Rsun), 3.8, 3.10, 3.11, and 3.12 (the Sun's thermal timescale is about 32 million years)
Do problems: 3.5 (for part c, use a pressure scale height of 0.063 Rsun), 3.8, 3.10, 3.11, and 3.12 (the Sun's thermal timescale is about 32 million years)
Prob. Set 3: Assigned Thurs. Oct. 4, Due Monday. Oct. 15
Prob. Set 4: Assigned Wed. Oct. 17, Due Wed. Oct. 31
More questions handed out in class.
Prob. Set 5: Assigned Wed. Oct. 31, Due Thurs. Nov. 8 Questions handed out in class.
Prob. Set 6: Assigned Thurs. Nov. 8, Due Mon. Nov. 19
1. Consider a 1 Msun star that has left the main sequence and is in the subgiant phase with an inert isothermal He core. The core has the following parameters: Mic = 0.08 Msun and Tic=19.1 x 106 K.
(a) Plot Pic as a function of Ric. Ensure that you plot over a sufficient range of Ric to show the behaviour of the function.
(b) What is the value of Ric of this star? Explain.
2. The isothermal compressibility of an object is defined as:
κ = -(1/v) (∂v/∂P)|T where v is the specific volume (cm3/g) and P is the pressure.
(a) Rewrite κ in terms of the density, ρ. (b) Find an expression for κ(ρ) for a non-relativistic and relativistic C/O white dwarf with constants evaluated.
(c) Evaluate κ for a typical non-relativistic and relativistic C/O white dwarf. Look up κ for the hardest substance on Earth that you can find and compare. 3. Although the pressure of a white dwarf is produced by electron degeneracy, the internal energy, U, depends on the ion temperature such that U = C1Tint, where Tint is the internal (isothermal) temperature and C1 is a constant. The luminosity of a white dwarf departs somewhat from the Stefan-Boltzmann law when it is related to the internal temperature, i.e. L = C2 Tint7/2, where C2 is another constant.
(a) Derive an expression for the age, t, of a white dwarf as a function of its initial interior temperature, Tint0 and its current interior temperature, Tint. Let t = 0 when Tint = Tint0 and note that dU/dt = -L.
(b) Reasonable values of the constants for a 1 Msun star are C1 = 2 x 1040 and C2 = 9.3 x 105 (cgs). If a 1 Msun white dwarf starts with an interior temperature of 107 K, what would be its interior temperature after 13.7 billion years? Comment on the length of time it would take for a white dwarf to cool to a `cinder'. 4. Estimate the time it takes for the iron core of a 25 Msun star to collapse to a neutron star.
κ = -(1/v) (∂v/∂P)|T where v is the specific volume (cm3/g) and P is the pressure.
(a) Rewrite κ in terms of the density, ρ. (b) Find an expression for κ(ρ) for a non-relativistic and relativistic C/O white dwarf with constants evaluated.
(c) Evaluate κ for a typical non-relativistic and relativistic C/O white dwarf. Look up κ for the hardest substance on Earth that you can find and compare. 3. Although the pressure of a white dwarf is produced by electron degeneracy, the internal energy, U, depends on the ion temperature such that U = C1Tint, where Tint is the internal (isothermal) temperature and C1 is a constant. The luminosity of a white dwarf departs somewhat from the Stefan-Boltzmann law when it is related to the internal temperature, i.e. L = C2 Tint7/2, where C2 is another constant.
(a) Derive an expression for the age, t, of a white dwarf as a function of its initial interior temperature, Tint0 and its current interior temperature, Tint. Let t = 0 when Tint = Tint0 and note that dU/dt = -L.
(b) Reasonable values of the constants for a 1 Msun star are C1 = 2 x 1040 and C2 = 9.3 x 105 (cgs). If a 1 Msun white dwarf starts with an interior temperature of 107 K, what would be its interior temperature after 13.7 billion years? Comment on the length of time it would take for a white dwarf to cool to a `cinder'. 4. Estimate the time it takes for the iron core of a 25 Msun star to collapse to a neutron star.