Physics 521, Stars Stanimir Metchev, Nov 3, 2011

Midterm Due at 10am on Friday, Nov 4, 2011

You can attempt any combination of problems up to 50 points. You are not allowed to collaborate with other students. Completed midterms should be handed to, or slipped under the office door of, Prof. Jin Koda in ESS 455.

Brown dwarfs. As a protostar radiates its thermal energy it contracts and its central density and temperature rise. If these become high enough, nuclear reactions begin to occur rapidly, and the resulting nuclear ht source prevents further contraction on the thermal timescale. However, another possibility is that electrons become degenerate at the center of the protostar before the temperature becomes high enough for nuclear reactions to occur at a significant enough rate. Then further contraction would be prevented by the degeneracy pressure of the electrons, and the protostar would never ignite nuclear fusion at its center. Such failed stars are called brown dwarfs. You will calculate some approximate properties of brown dwarfs. A fact about brown dwarfs that will be very important for you below is that they are fully and efficiently convective from their centers to their photospheres.

1. (20 points) In ionized non-degenerate gas of cosmic abundance (i.e., near solar X, Y, Z, µe = 1.15 for fully ionized gas) the electrons’ contribution to pressure is comparable to that of the ions. When the electrons are degenerate, their degeneracy pressure is much larger than the thermal corrections, but the ions continue to have ordinary gas pressure. Show that the following expression gives the correct approximation for the total pressure in the non-degenerate and degenerate limits:

5 3 2 13$ # ' $ * ' –2 (1) P "10 & ) &1 +* + ) dyn cm , % µe ( % 1+* (

–6 2/3 –2/3 where ψ ≈ 8×10 Tµe ρ is the degeneracy parameter (ψ << 1 for degenerate matter, ψ >> 1 for non-degenerate ideal gas electrons), and where the first term comes from! electron degeneracy pressure, the second from ion gas pressure, and the third (at most a factor of 2 correction to the others) is the thermal contribution of the electron pressure. We have neglected powers of µe in the thermal contributions.

2. (10 points) Brown dwarf interiors are in a state of degenerate coulomb plasma, with the entropy per nucleon given by

(2) s/k = 2.2ln(T /" 0.63 ) #11.6.

Using the first law of thermodynamics, show that for non-degenerate ionized gas, s/k = A ln (T/ρ0.66) – B (you need not compute A or B). Thus, in both cases, show that ψ is nearly !constant if s is. Use fact above to argue that s is nearly constant throughout the brown dwarf, and thus show that they are n = 3/2 polytropes.

3. (10 points) Use the relations for polytropes to show that the radius of a brown dwarf is # " 2 & (3) R = R0%1 +" + ( , $ 1+" ' 9 1/3 –5/3 where R0 = 2.7×10 (MSun/M) µe cm.

4. (10 points) Also, use the ratio of central to mean density for an n = 3/2 polytrope to ! derive a relation between ψ, the central temperature Tc, and the mass and radius:

$9 4 / 3 2 $8 / 3 (4) " = 2 #10 Tc (MSun / M) (R /R0 ) µe .

5. (10 points) Use equations (2)–(4) to plot Tc versus R with M held fixed and R varied from very large to very small values (as would be the case for a contracting protostar; ! large and small being relative to R0). Make plots for M = 0.04, 0.08, 0.16, and 0.32 MSun, assuming cosmic abundance of H and He. Show that Tc initially increases, but then decreases. Show that Tc is maximized when R/R0 = 1.75 (ψ ≈ 0.55), and that

7 4 / 3 (5) Tc,max = 8 "10 (M / MSun ) K .

6. (30 points) Use your knowledge of the temperature dependence of the p-p reaction rates to argue that this sets a lower limit to the mass of hydrogen burning main sequence! stars, and that protostars with masses below this critical mass will become brown dwarfs. What is the critical mass? Hint: a star is destined to become a brown dwarf when its fusion luminosity, at its peak (when T = Tc,max), can not supply the energy released by gravitational contraction to R(Tc,max) during the contraction phase. Assume that the contraction phase lasts up to the current age of the universe. Note that this implies that the definition of a brown dwarf changes as the universe ages. Finally, make an appropriate assumption for the size of the nuclear burning region to get the fusion luminosity.