Astronomy 20Homework # 4Posted on the class webpage on November 5, 2004Return to TA (!) by 5 pm on Friday, November 12, 20041. A handy unit of energy is 1 Gipper = the amo unt of energy released in a gl obalthermonuclear war (let’s say, 10,000 1-megaton warheads; one of them releases ∼ 4×1022erg if carelessly dropped).(a) What is the solar energy generation rate, in Gippers/sec?(b) What is the fuel consumption rate of the Sun, in tons of hydrogen per sec? Assumethe net efficiency of ∼ 7 MeV per nucleon.(c) How long does it ta ke for the Earth to intercept the amount of sunlight equal to 1Gipper?(d) Consider a neutron star with the mass of 1.4 M⊙and the radius of 30 km. If youdrop a professor (MP rof≈ 80 k g) on it from a safely large distance (i.e., where thebinding energy ≈ 0), what would be the amo unt of energy released upon the impact,assuming that it is all converted into radiation?2. Consider a type Ia supernova, whose progenitor is a carbon-oxygen white dwarf of amass Mwd= 1.4 M⊙, and radius Rwd= 1500 km. In burning of C and O to produceFe-group elements, 7.3 × 1017ergs are produced per gram of fuel.(a) Estimate the binding energy of the white dwarf.(b) What fraction of its mass has to be converted int o iron in order to blow i t apart?(c) Assuming that practical ly all of its mass is ejected with average speeds o f 5000km/s, how much more iron has to be produced in order t o supply the necessarykinetic energy?3. One hypothesis for the orig in of γ-ray bursts (GRBs) is that they represent spiral-downcollisions of two neutron stars, whereby some of the surplus binding energy is releasedin the form of γ-rays. Consider a neutron star binary, wi th each component having amass of Mns= 1.4 M⊙and a radius Rns= 30 km.(a) Estimate the binding energy for each star as if it were isolated. Ignore any relativisticeffects.(b) Assume that after the collision, about 1% of their total initial binding energy isreleased as γ-rays over the burst time of 1 second. Compute this Lγ.(c) Gi ven this Lγ, and assuming the linear scale (i.e., an “effective radius” R ) of theemission region to be about 10 km, what is the effective blackbody temperature Tef f?(d) Assuming (incorrectly, but OK for the homework purposes) that the emission isthermal, what is the energy (in Mev) of individual photons corresponding to thepeak of t he Planck curve for this Tef f? Does this make sense?(more on the other side)Ay 20, HW4, problem 3, continued...(e) Many GRB-detecting satellites have a burst detection limit of Flim= 10−7erg/cm2.How far could they detect such hypothetical merger events, in Mpc?(f) If a mean separation between average galaxies is 5 Mpc, estimate how many galaxiesare thus observable by these satellites.(g) If t he average supernova rate per galaxy is ∼ 1 event per 100 years, how manysupernovæ per day happen among the tot al number of gal axies yo u just computedin (f)? Does this provide an adequate neutron star supply rate, given that we detectabout 1 burst per
View Full Document
Unlocking...