Ay101 Set 1 solutions

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1 Ay11 Set 1 solutions Ge Chen Jan The scale of the Sun a 3 points Venus has an orbital period of 5 days. Using Kepler s laws, what is its semi-major axis in units of AU Start with Kepler s third law: Ω GM + M a 3 1 First, consider the Earth-Sun system, assuming that M E M : Ω E GM a 3 E Now, Ω E can be calculated from known values Earth s period is 1 year and a E is known it s 1 AU, so can solve for M : M Ω E a3 E G Then consider the Venus-Sun system, again assuming that M V M : 3 Ω V GM a 3 V 4 Then solve for a V, substituting equation 4 for M and converting angular velocities to periods P π Ω : a V 1/3 GM 5 Ω V Ω E a 3 1/3 E 6 PV Ω V P E /3 a E 7 Plug in known values P E days, a E 1 AU and given values P V 5 days, and find that a V.74 AU. b points At conjunction with the Sun, astronomers on Earth emit a radio pulse directed toward Venus. It takes 76 seconds to detected the radio waves that reflect off Venus. Assuming circular orbits for the Earth and Venus, use this information to compute the distance in 1 AU. Note: conjunction Venus is directly between Earth and the Sun. Call the distance between Earth and Venus d V. The light takes time t to travel distance d V : d V ct cm 8 1

2 From the previous problem, we can compute d V in terms of AU: d V a E a V 1.74 AU.76 AU 9 Then combine equations 8 and 9 to convert AU to cm: 1 AU cm. c 3 points Use your results above to compute the absolute mass of the Sun. Using its measured angular radius of 16 arcmin, compute the radius of the Sun. Kepler s third law: M Ω E a3 E G Plug in known values in cgs; find that M g. Definition of angular size: R a E tan θ Plug in known values in cgs; find that R cm. d points The surface temperature of the Sun is T 577K. Using the blackbody radiation law, compute the luminosity of the Sun in cgs units. From the Stefan-Boltzmann equation: L 4πR σ B T 4 eff 1 Plug in values in cgs; find that L erg s 1. Stars are gases a 6 points Provide a quantitative relation between the temperature and density of a star which indicates when we can treat it as a gas as opposed to liquid or solid throughout its interior, in spite of the very high densities. Is our assumption valid at the center of the Sun, where the density is about 1 times the average density? Note that your answer should only involve classical physics, no quantum mechanics. To check if the center of the Sun can be treated as a gas, we can compare Coulomb energy to thermal energy. The ideal gas law is reasonable when the thermal energy T is larger than the Coulomb energy Ze /r. This occurs when kt Ze /r 11 r Ze /kt 1 where r is the interparticle distance and Z is the charge of the ion Z 1 for a gas composed only of ionized hydrogen. By thinking of the number density n as r 3, we can write r in terms of mass density: r n 1/3 ρ/ 1/3. Then we can rewrite equation 1: ρ k/e 3 T 3 13 For atomic hydrogen, µ.5. Plugging this in, our condition for treating a star as an ideal gas is in cgs units: ρ T 3 14 The sun s central temperature is T c 1 7 K. By equation 14, we require ρ c g cm 3. Since the sun s central density ρ c is only 15 g cm 3, we may treat the sun as an ideal gas throughout, and need not consider Coulomb interactions between particles. b 4 points If all stars have roughly the same central temperature that of the Sun, use a scaling argument to determine the stellar mass at the which the simple noninteracting ideal gas assumption breaks down. Now we want to know how ρ scales with stellar mass M. In the following derivation we only care about approximate scaling arguments, so don t worry about prefactors. The internal energy can be approximated as U T M at the central temperature T. Now recall that by the virial

3 theorem, the internal energy is approximately the gravitational energy Ω GM /R. Solve for T and find that T M/R. We can then assume that all stars have roughly the same central temperature which is actually a good approximation for main-sequence stars, so the central density M R. Then ρ M/R 3 M, so M ρ 1/. 1/ Plugging in values for the Sun, we find M lim ρlim M. This yields a limiting mass of M.16M, which is about the mass of brown dwarfs or giant planets not stars!. Therefore, we will never have to consider Coulomb interactions for main sequence stars. 3. A toy star Assume that a star obeys the density model ρr ρ c 1 r. 15 R a 3 points Find an expression for the central density in terms of R and M. Solve for total mass M by integrating over the density profile: M π 4π dφ R π 4πρ c R 3 ρ sin θdθ ρ c r r3 R 3 R3 4 R r ρrdr 16 dr π 3 ρ cr 3 19 Then solve for central density: ρ c 3M πr 3. b 6 points Use the equation of hydrostatic equilibrium and zero boundary conditions to find the pressure as a function of radius. Your answer will be in the form P r P c fr/r, where fx is a function that you will determine. What is the dependence of the central pressure P c in terms of M and R? Doing the same integral as in the previous problem, we know that mr 4π 3 ρ cr 3 1 3r 4R Now use the equation of hydrostatic equilibrium: dp dr Gm ρr 1 r G 4π 3 ρ cr 1 3r ρ c 1 r 4R R 4π 3 Gρ cr 1 7r 4R + 3r 4R 3 Integrate equation 3 to get the total pressure: P r 4π 3 Gρ c r 7r 4π [ r 3 Gρ c 4R + 3r3 4R 7r3 1R + 3r4 16R + C dr 4 ] 5 3

4 Use the zero boundary condition P R to solve for the integration constant C: P R 4π [ 1 3 Gρ cr C ] R 6 C R R 8 Now let s solve for the central pressure P c P r : P c 4π 3 Gρ cc 9 5π 36 Gρ cr 3 Okay, finally we can substitute stuff into equation 5 to write the full equation for pressure: P r 4π [ r 3 Gρ c 7r3 1R + 3r4 16R 5 ] 48 R 31 [ P c 1 4 r 8 r 3 9 r ] R 5 R 5 R So we find that P r P c f r R as expected. Now plug in the answer for part a to get P c as a function of M and R: We can simplify this to P c 5 GM 4π R. 4 P c 5π 3M 36 G πr 3 R 33 c 3 points What is the central temperature T c, assuming an ideal gas equation of state? How does it scale with mean particle mass? Ideal gas: P ρt Solve this for central temperature, plugging in answer from b for P c and answer from a for ρ c : This simplifies to T c 5GM 1R T c P c ρ c 34 5π 36 Gρ cr 35 5π 36 G 3M πr R µm 3 p 36 which scales as T c. is the mean particle mass. d 3 points Find the ratio of radiation pressure to gas pressure at the center of the star as a function of total stellar mass in M. At what mass does radiation pressure become comparable to the ideal gas pressure? Radiation pressure is given by 1 3 a ot 4. The ratio at the center of the star is therefore T 4 c 1 P gas 3 a o 37 5 GM 4π R 4 4

5 Then plug in T c from part c: P gas 1 3 a o 4 5GM 1R 5 GM 4π 15π 1555 a og 3 M 38 R 4 4 µmp 39 Assuming solar composition µ.6, we can rewrite this in terms of solar masses as: P gas M M When P gas 1, the mass of the star is M 37M. Note that this is not an exact result, since our formula for T c assumes that radiation pressure is negligible. 4. Stellar coronae a 3 points Consider a spherically symmetric stellar corona and assume that the temperature in the corona is constant and that the mass of the corona is negligible compared to the mass of the star M. Solve hydrostatic equilibrium for the density profile as a function of radius given a density ρ b at the base radius r b R. Start with hydrostatic equilibrium and assume an ideal gas equation of state with constant temperature T. Also assume that the mass of the corona is negligible compared to the mass of the star M, so mr M: dp dr T dρ dr dρ dr Gmr r ρr 4 GM ρr 41 r GM r ρr 4 T Equation 56 is a separable differential equation: dρ GM dr ρ T r 43 GMµmp ρr Aexp 44 r T Now use the given boundary condition to solve for integration constant A: GMµmp ρ b Aexp r b T A ρ b exp GMµm p r b T So the final equation is ρr ρ b exp GMµm p GMµmp exp r b T r T 47 b points What is the pressure in the corona as r? Note that it is finite! Assume further that the star is surrounded by a low density and low pressure interstellar medium ISM, such that the ISM pressure is much less than asymptotic coronal pressure inferred here. Comment on the implications for the dynamics of the stellar corona. 5

6 As r, we find the limit ρr ρ b exp P ρt, this yields a pressure P ρ b T exp GMµmp r b T GMµmp r b T. Since we ve assumed an ideal gas. This is finite! Since the stellar corona is surrounded by an ISM with much lower pressure, we expect the corona to be able to produce a stellar wind. 5. Using the MESA stellar evolution code a 4 points Run the default stellar model located in mesa/star/work/. At time step 1, what is the core temperature and surface temperature of the model? The core temperature is log T c 5.48 [K], or T c K. The surface temperature is T eff 345 K. b 6 points Evolve a 1M model up the red giant branch. What is the surface temperature and radius of the star when its luminosity reaches 5L? Copy the directory work and make your own model in the new directory. Change the starting mass to 1M in inlist project. Disable the create pre main sequence model option in inlist project. Change the stopping condition so that the model will evolve beyond core hydrogen depletion.! stop when the star nears red gaint L nuc /L > 1 Lnuc div L upper limit 1 When L 5L, T eff 4393 K and log R 1.87, or R 1.R. 6