**Prob. Set 1: Assigned Tues. Sept. 19, Due Thurs. Sept. 28**

Prob. 1.2, 1.3, 2.2, 2.3, 2.4

Prob. 1.2, 1.3, 2.2, 2.3, 2.4

_{ν}(T)/dT. That is, how close are the two peaks?

**Prob. Set 2: Assigned Thurs. Sept. 28, Due Thurs. Oct. 5**

Prob. 2.5, 2.6, 2.9, 2.10b, 2.11

**Prob. Set 3: Assigned Thurs. Oct. 5, Due Tues. Oct. 17 (this assignment is now complete)**

Prob. 2.12, 2.13, 2.14

_{th}/τ

_{nuc}in terms of the stellar radius, R, and mass, M.

(b) From Fig. 2.12, measure (estimate) the power, α, of the relation R ∝ M

^{α}for the ZAMS curve for two regimes: M < 1.3 M

_{Sun}and M > 1.3 M

_{Sun}.

(c) Now re-write τ

_{th}/τ

_{nuc}as a function of mass for the two regimes.

(d) Does this ratio increase or decrease with increasing mass? Comment as to why this might be the case.

Prob. 5. Refer to the figure handed out in class showing the secular behaviour of a number of properties of the center of the Sun with time, namely: X_{H}, κ, T, ε, L, P, and ρ. By `secular', I mean

**Prob. Set 4: Assigned Thurs. Oct. 26, Due Mon. Nov. 6**

Prob. 4.1, 4.2, 4.3, 4.4, and 5.1

**Prob. Set 5: Assigned Tues. Nov. 16, Due Thurs. Nov. 23 (DATES CHANGED)**

**This problem set is finished.**

^{-α}, where C is a constant and α represents a power law slope which varies with mass interval, as specified by the Kroupa handout given in class.

(a) Write expressions for (i) the total number of stars, N, between mass, M1 and M2; (ii) the total luminosity of all stars, L, between M1 and M2; and (iii) the total mass, M, of all stars between M1 and M2.

M should be expressed in units of M

_{Sun}and L in L

_{Sun}.

(b) Evaluate N, L, and M for the range, 0.1 < M/M

_{Sun}< 0.5, and for the range 10 < M/M

_{Sun}< 10.4. (Call them N

_{high}, N

_{low}etc).

(c) Evaluate the ratios: N

_{high}/N

_{low}, L

_{high}/L

_{low}, and M

_{high}/M

_{low}, and comment on your results.

(d) Suppose 25 million years pass. How would you expect these ratios to change, if at all?

_{Sun}star ends at point 2 as given in the handout (Table 13.1). Compare the age of such a star at this time to the age of the universe. Do you ever expect to see a white dwarf that has been formed from such a star?

_{Sun}star that has left the main sequence and is in the subgiant phase in which it has an inert isothermal He core. The core has the following parameters: M

_{ic}= 0.08 M

_{Sun}and T

_{ic}= 19.1 x 10

^{6}K.

(a) Plot P

_{ic}as a function of R

_{ic}. Show the plot over a reasonable range.

(b) What is the value of R

_{ic}for this star? Explain.

_{1}T

_{c}, where C

_{1}is a constant that includes information about the ions. A WD is almost isothermal internally, and will have a central temperature, T

_{c}that is different from the surface temperature. Thus, the luminosity L = C

_{2}T

_{c}

^{7/2}as opposed to a surface temperature dependence of T

_{eff}

^{4}. Here, C

_{2}is another constant.

(a) Derive an expressive for the age, t, of a WD as a function of its initial interior temperature, T

_{c0}and its current interior temperature, T

_{c}. Let t = 0 when T

_{c}= T

_{c0}. Note that dU/dt = -L.

(b) Reasonable values of the constants for a one solar mass star are C

_{1}= 2 x 10

^{40}and C

_{2}= 9.3 x 10

^{5}in cgs units. If a 1 M

_{sun}white dwarf starts with an interior temperature of 10

^{7}K, what would its interior temperature be after 13.7 billion years?